Advances in Structural Vibration: Select Proceedings of ICOVP 2017 9789811558610, 9789811558627

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Lecture Notes in Mechanical Engineering

Subashisa Dutta Esin Inan Santosha Kumar Dwivedy   Editors

Advances in Structural Vibration Select Proceedings of ICOVP 2017

Lecture Notes in Mechanical Engineering Series Editors Francisco Cavas-Martínez, Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini, Dipartimento di Ingegneria, Università di Modena e Reggio Emilia, Modena, Italy Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Vitalii Ivanov, Department of Manufacturing Engineering Machine and Tools, Sumy State University, Sumy, Ukraine Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland

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Subashisa Dutta Esin Inan Santosha Kumar Dwivedy •

•

Editors

Advances in Structural Vibration Select Proceedings of ICOVP 2017

123

Editors Subashisa Dutta Department of Civil Engineering Indian Institute of Technology Guwahati Guwahati, India

Esin Inan Department of Civil Engineering Işık University Istanbul, Turkey

Santosha Kumar Dwivedy Department of Mechanical Engineering Indian Institute of Technology Guwahati Guwahati, India

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-15-5861-0 ISBN 978-981-15-5862-7 (eBook) https://doi.org/10.1007/978-981-15-5862-7 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

On the auspicious year of silver jubilee celebration of Indian Institute of Technology, Guwahati, we proudly introduce these two books on vibrational problems, containing selected and peer-reviewed papers presented in 13th International Conference on Vibrational Problems (ICoVP) 2017 conference held in the institute, and technical collaboration with IŞIK University, Istanbul, Turkey. The two books document the technical contributions of scientists, research scholars, academia, and industry participants from India and abroad in the field of vibrational problems. This book entitled Advances in Structural Vibration consists of 42 contributed chapters and includes different structural vibration problems such as dynamics and stability of structures under seismic loading and wave propagation. The book also highlights the use of new materials such as composite, piezoelectric, and functionally graded materials for improving the stiffness and damping properties of micro to macro scale structures in Industries and earth-quake resistance buildings. These chapters represent recent analytical, experimental, and engineering practices in these fields with a number of illustrations, case studies, and data analysis techniques. These books will particularly help young researchers and engineers in obtaining technical know-how in these fields. The success of organizing this conference and bringing out these two books on vibrational problems goes to many. First, a heart-filled gratitude to the founding members of the ICoVP conference: Prof. M. M. Banerjee and Prof. B. Biswas, who initiated the conference way back in 1990 in Jalpaiguri (West Bengal), India. A large number of members in the steering committee, international advisory committee, local organizing committee, and manuscript review committee are highly acknowledged for their time-to-time help and support for these technical events. In addition, participants from India and abroad who made four days of international scientific gathering successful by delivering a series of lectures on these topics are acknowledged for their scientific endeavor. Last but not the least, the editors sincerely appreciate the significant contribution of Dr. Arunasis Chakraborty and Dr. Sandip Das, faculty in the Department of Civil Engineering, IIT, Guwahati, for conducting the conference and bringing out these books. Mr. A. Anjaneyulu, a research scholar in the department, has contributed v

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significantly in managing all these chapters and communicating with the authors for shaping out these books, and his effort is sincerely appreciated. Before concluding this Preface, we sincerely thank Ms. Sushmitha Shanmuga Sundaram, Ms. Rini Christy Xavier Rajasekaran, Ms. Muskan Jaiswal, and Dr. Akash Chakraborty from Springer Nature for their contributions in making these two books an archived technical reference document for the next-generation readers/researchers. Guwahati, India Istanbul, Turkey Guwahati, India

Subashisa Dutta Esin Inan Santosha Kumar Dwivedy

Contents

Building Structures Propagation of Viscoelastic Waves in a Single Layered Media with a Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pankaj Kumar, Anirvan DasGupta, and Ranjan Bhattacharyya

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Axial Resistance of Short Built-up Cold-Formed Steel Columns: Effect of Lacing Slenderness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Adil Dar, Dipti Ranjan Sahoo, Arvind K. Jain, and Sunil Pulikkal

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Analysis of Stepped Beam Using Reduced Order Models . . . . . . . . . . . . Rahul Kumar, Sayan Gupta, and Shaikh Faruque Ali

21

Numerical Analysis of Geosynthetic Strengthened Brick Masonry Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hasim Ali Khan, Radhikesh Prasad Nanda, and Diptesh Das

35

Studies on Dynamic Characteristics of Sandwich Functionally Graded Plate Subjected to Uniform Temperature Field . . . . . . . . . . . . . . . . . . . E. Amarnath and M. C. Lenin Babu

43

Numerical Model and Dynamic Characteristics Analysis of Cable-Stayed Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antara Banerjee and Atanu Kumar Dutta

55

Breathing Crack Detection Using Linear Components and Their Physical Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Prawin and A. Ramamohan Rao

73

Optimal Design of Structure with Specified Fundamental Natural Frequency Using Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . Kandula Eswara Sai Kumar and Sourav Rakshit

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Non-linear Dynamic Analysis of Structures on Opencast Backfilled Mine Due to Blast Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Kumar, S. C. Dutta, S. D. Adhikary, and M. A. Hussain

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Seismic Response of Shear Wall–Floor Slab Assemblage . . . . . . . . . . . . 105 Snehal Kaushik and Kaustubh Dasgupta Comparison Between Two Modeling Aspects to Investigate Seismic Soil–Structure Interaction in a Jointless Bridge . . . . . . . . . . . . . . . . . . . 117 S. Dhar and K. Dasgupta Probabilistic Flutter Analysis of a Cantilever Wing . . . . . . . . . . . . . . . . 133 Sandeep Kumar, Amit K. Onkar, and M. Manjuprasad Optimum Support Layout Design for Periodically Loaded Structures Using Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A. T. Korade and S. Rakshit Seismic Behaviour of RC Building Frame Considering Soil–Structure Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Nishant Sharma, Kaustubh Dasgupta, and Arindam Dey Inelastic Time History Analysis of Mass Irregular Moment Resisting Steel Frame Using Force Analogy Method . . . . . . . . . . . . . . . . . . . . . . . 171 S. S. Ningthoukhongjam and K. D. Singh Modification and Modeling of Experiments with Bi-directional Loading on Reinforced Concrete Columns . . . . . . . . . . . . . . . . . . . . . . . 185 Subhadip Naskar, Sandip Das, and Hemant B. Kaushik Fatigue Life Estimation of an Integral RC Bridge Subjected to Transient Loading Using Ansys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 M. Verma and S. S. Mishra Free Vibration of Composite Sandwich Beams with Microstretch Viscoelastic Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 S. Aydinlik, A. Kiris, and E. İnan Seismic Response Mitigation of Structure by Negative Stiffness Devices via Mid-Story Weakening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Arijit Saha and Sudib Kumar Mishra Development of a Novel Viscoelastic Nanocomposite and Investigation of Its Damping Capacity for Large Frequency Band . . . . . . . . . . . . . . . 233 Nitesh Shah, Bishakh Bhattacharya, and Husain Kanchwala Effect of Pulse-Type Ground Motions on the Rocking Response of Rigid Blocks on Deformable Media . . . . . . . . . . . . . . . . . . . . . . . . . . 253 V. H. Sheikh and S. Mukhopadhyay

Contents

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Vibration Control of Seismically Excited Adjacent Buildings Prone to Pounding by Use of Friction Dampers . . . . . . . . . . . . . . . . . . . . . . . . 261 N. K. Dutta and A. D. Ghosh Influence of BNWF Soil Modelling on Dynamic Behaviour of Pile Foundation for RC Frame with Structural Wall . . . . . . . . . . . . . . . . . . 277 A. Sinha, N. Sharma, K. Dasgupta, and A. Dey Aerospace Structures Control and Limiting Strategies for Random Vibration Tests on Spacecraft Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 A. R. Prashant, K. Sreeramulu, B. R. Nagendra, S. Ramakrishnan, M. Madheswaran, V. Rameshnaidu, P. Govindan, and P. Aravindakshan Dynamic Characterization of Large Structures and Its Application to Solid Rocket Motor Testing and Launch . . . . . . . . . . . . . . . . . . . . . . 305 Venkata Ramakrishna Vankadari, Dileep Pasala, Jopaul Ignatius, and Sankaran Sathiyavageeswaran A New Multi-shaker System Development For Testing Launch Vehicle Subassemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 G. Meghanath, V. Venkata Ramakrishna, A. Veerraju, and Jopaul K. Ignatius Dynamic Characterisation as a Tool for Avoiding Vibration Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Ajay Kumar Panda, Asir Nesa Dass N, R. Balaji Srinivas, Arunkumar R, and M. Vasudevan Unni A Simplified Impact Damping Model for Honeycomb Sandwich Using Discrete Element Method and Experimental Data . . . . . . . . . . . . 337 Nazeer Ahmad, R. Ranganath, and Ashitava Ghosal Aeroservoelastic Analysis of RLV-TD HEX01 Mission . . . . . . . . . . . . . 355 Mahind Jayan, P. Ashok Gandhi, Sajan Daniel, and R. Neetha Time Domain Aero Control Structure Interaction Studies of Indian Reusable Launch Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 P. Ashok Gandhi, Mahind Jayan, Sajan Daniel, and R. Neetha Vehicle Dynamics Polynomial Neural Network Based Stochastic Natural Frequency Analysis of Functionally Graded Plates . . . . . . . . . . . . . . . . . . . . . . . . . 379 Pradeep Kumar Karsh, Abhijeet Kumar, and Sudip Dey

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Comparative Study Among Different Vehicle Models in Case of High-Speed Railways and Its Experimental Validation . . . . . . . . . . . 387 B. Pal and A. Dutta Evaluation of Ride Comfort in Railway Vehicle Due to Vibration Exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 S. Pradhan and A. K. Samantaray Non-linear State Space Formulation Simulating Single Station Ride Dynamics of Military Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Saayan Banerjee, V. Balamurugan, and R. Krishna Kumar Delamination Growth Behaviour in Carbon/Epoxy Composite Road Wheel of an Armoured Fighting Vehicle Under Dynamic Load . . . . . . . 429 Sarath Shankar, Subodh Kumar Nirala, Saayan Banerjee, Dhanalakshmi Sathishkumar, and P. Sivakumar Torsional Vibration Analysis of Crank Train and Design of Damper for High Power Diesel Engines Used in AFV . . . . . . . . . . . . . . . . . . . . . 443 N. Venkateswaran, K. Balasubramaniyan, R. Murugesan, and S. Ramesh Fluid Structure Interaction Dynamic Behavior of Swaged Plates in Water-Immersed Condition . . . 457 G. Verma, S. Sengupta, S. Mammen, and S. Bhattacharya Love Wave Propagation in an Anisotropic Viscoelastic Layer Over an Initially Stressed Inhomogeneous Half-Space . . . . . . . . . . . . . . . . . . 469 Bishwanath Prasad, Prakash Chandra Pal, and Santimoy Kundu Propagation of Edge Wave in Homogeneous Viscoelastic Sandy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Pulkit Kumar, Amares Chattopadhyay, and Abhishek Kumar Singh Love-Type Wave Propagation in Functionally Graded Piezomagnetic Material Resting on Piezoelectric Half-Space . . . . . . . . . . . . . . . . . . . . . 495 J. Baroi and S. A. Sahu Fluid–Body Interactions in Fish-Like Swimming . . . . . . . . . . . . . . . . . . 509 Dipanjan Majumdar, Chandan Bose, Prerna Dhareshwar, and Sunetra Sarkar Comparison of Stochastic Responses of Circular Cylinder Undergoing Vortex-Induced Vibrations with One and Two Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 M. S. Aswathy and Sunetra Sarkar

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An Exact Solution for Magnetogasdynamic Shock Wave Generated by a Moving Piston Under the Influence of Gravitational Field with Radiation Flux: Roche Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 G. Nath and Sumeeta Singh Isogeometric Collocation for Time-Harmonic Waves in Acoustic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 M. Dinachandra, S. S. Durga Rao, and R. Sethuraman Effect of Seismic Excitation on Bubble Behavior in Liquid Between Fuel Rods of Boiling Water Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 S. P. Chauhan, M. Eswaran, and G. R. Reddy Experimental Study on Shallow Water Sloshing . . . . . . . . . . . . . . . . . . 567 Saravanan Gurusamy and Deepak Kumar

About the Editors

Dr. Subashisa Dutta is currently a professor at the department of civil engineering, Indian Institute of Technology Guwahati. He completed his B.E (civil engineering) from Sambalpur University (Odisha) and was awarded PhD on computational hydraulics from IIT Kharagpur. He worked as a scientist in Space Application Centre Ahmedabad, ISRO, on satellite remote sensing and geospatial technology for water resources management. Later in 2003 he joined as an assistant professor in Department of Civil Engineering, IIT Guwahati. He is working in the field of morphodynamic oscillations of large braided river systems like Brahmaputra, fluvial hydraulics, hillslope hydrology, remote sensing applications in hydrology and river morphology. He has guided 6 PhD students, 25 Master’s students and currently 10 PhD research scholars are working under his guidance. He has published more than 100 papers in international journals and conferences, and several technical reports and book chapters. He is a reviewer of more than 10 international journals. He received the S.N. Gupta memorial award in 2018 and R.J. Garde award in 2012. He was the editor of “Advances in Water Resources Engineering and Management” published by Springer Nature. Besides he has also been the associate editor of Journal of Hydrology (Elsevier) since 2016. Dr. Esin Inan is currently a professor at the Department of Mathematics, Faculty of Art and Science, IŞIK University. She completed her B.S. Engg and M.E. from Technical University of Istanbul (ITU). She obtained her PhD from Technical University of Istanbul (ITU) then joined as assistant professor in Technical University of Istanbul (ITU) in the year 1970 and she is professor since 1983. She is working in the field of continuum mechanics: classical and non-classical models. She has published more than 90 papers in international journals, conferences and book chapters. Currently, she is a board member of the Science Center Foundation of Turkey and Foundation of the Istanbul Technical University.

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About the Editors

Dr. Santosha Kumar Dwivedy is currently a professor at the Department of Mechanical Engineering, Indian Institute of Technology Guwahati. He completed his B. E (mechanical engg.) from University College of Engineering Burla (Odisha) and M.E (mechanical engg. with specialization in machine design and analysis) from Regional Engineering College Rourkela (now NIT Rourkela). He obtained his Ph.D. from the mechanical engineering department, IIT Kharagpur in the year 2000 working in the field of Nonlinear Dynamics of Parametrically excited systems. He joined as a faculty member in the Department of Mechanical Engineering, IIT Guwahati in the year 1999. He became professor in the same department in the year 2008. He is working in the field of nonlinear vibration, robotics and biomechanics. He guided 6 PhD students, 45 M. Tech students and 40 B. Tech projects and currently 12 PhD students are working under his guidance. He has published more than 150 papers in international journals, conferences and book chapters. He was one of the editors for two volumes of the books “Recent Advances in Computational Mechanics” published by I.K. International in 2006. He was also the guest editor for the Procedia Engineering, (Proceedings of the International Conference on Vibration Problems 2015), Elsevier Publication, 2016. He is reviewer of more than 30 international journals.

Building Structures

Propagation of Viscoelastic Waves in a Single Layered Media with a Free Surface Pankaj Kumar, Anirvan DasGupta, and Ranjan Bhattacharyya

Abstract In this work, a wave propagation formulation in a layered half space is presented. A viscoelastic layer is considered to be fixed to an elastic half space. The inhomogeneity in various reflected and refracted waves is caused due to the viscoelastic layer. In the analysis, the low-loss approximation is considered for the viscoelastic layer. Using the matrix formulation of Thomson [1] and Haskell [2], the free surface displacement functions are derived by considering boundary conditions. The effect of medium parameters and wave types on the wave propagation behavior is studied numerically and conclusions are drawn. Keywords Bulk waves · Layered media · Viscoelasticity

Nomenculture ψi jkl   λ,μ Q pL aL c L ,cT

Relaxation tensor Scalar potential Vector potential Lam´e parameters Quality factor Propagation vector Attenuation vector Wave speeds

P. Kumar (B) · A. DasGupta · R. Bhattacharyya Department of Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India e-mail: [email protected] A. DasGupta Center for Theoretical Studies, IIT Kharagpur, Kharagpur 721302, India

© Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_1

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P. Kumar et al.

1 Introduction The study of bulk waves in viscoelastic media is important in many aspects. In materials characterization and non-destructive testing, the understanding of the wavemedium interaction plays a significant role. Also, due to propagating bulk waves, the free surface displacement can develop a driving force along the propagation, which can be used in applications like positioning, segregation, transportation, etc. This motivates the present analysis. Anelasticity introduces inhomogeneity in various waves associated with viscoelastic layer. The inhomogeneity can be described by the general theory of viscoelasticity. In addition, the theory gives the basis to explain the attenuation in bulk waves. In viscoelastic medium, the bulk wave behavior has been evaluated by many researchers. Some of the earlier studies on bulk waves in viscoelastic media dealt with its energy relation and physical characteristics [3–6]. Buchen [3] and Borcherdt [4] considered harmonic P and SV wave, and details are provided on associated energy with these waves. Cooper [5] and Borcherdt [6] investigated the propagation of bulk waves at a planner interface composed of two viscoelastic half spaces. In some works [7, 8], reflection and refraction of waves and their frequency relation in a composite medium are considered. A composite consists of alternating layers of both elastic and viscoelastic materials, which is simplified to a homogeneous composite. Recently, problems on wave propagation in layered media have been studied in the context of welded interface in Kaur et al. [9] and of imperfect interface in Liu et al. [10]. In this paper, we are interested in the study of free surface displacement. The excitation is caused by an incident harmonic wave in the half space. The extended matrix formulation of Thomson [1] and Haskell [2] is presented and discussed to analyze attenuation in viscoelastic waves. To keep the problem simple, we consider a viscoelastic layer on an elastic half space. The analysis is carried out considering an incident P and SV waves.

2 Problem Formulation In a linear viscoelastic medium, the time dependent stress σi j and εi j can be related by the following constitutive equation σi j = ψi jkl ∗ ε˙ kl ,

(1)

where the most general relaxation tensor ψi jkl can be expressed in terms of the bulk relaxation ψ1 , and the shear relaxation ψ2 as ψi jkl = (ψ1 − 23 ψ2 )δi j δkl + ψ2 (δik δ jl + δil δ jk ).

(2)

Propagation of Viscoelastic Waves in a Single Layered Media …

5

Substituting (2) in (1), and using the relation εi j = 21 (u i, j + u j,i ) for an infinitesimal displacement u i , the stress-divergence is then obtained as     σi j, j = ψ1 − 23 ψ2 ,t ∗ (∇ · u),i + (ψ2 ),t ∗ ∇ 2 u i + (∇ · u),i .

(3)

In absence of body forces, using (3) in equation ρ u¨ ,i − σi j, j = 0, one obtains the equation of motion as   ρ u¨ − ψ1 − 13 ψ2 ,t ∗ (∇ · u) − (ψ2 ),t ∗ ∇ 2 u = 0,

(4)

using Helmholtz theorem, the displacement field u in (4) can be rewritten as u = ∇ + ∇ × ,

(5)

where  and  are, respectively, the scalar and vector potentials with ∇ ·  = 0. Now, substituting displacement (5) into equation of motion (4) yields on simplification ¨ − (ψ2 ),t ∗ ∇ 2 ] = 0, ¨ − (ψ1 − 4 ψ2 ),t ∗ ∇ 2 ] + ∇ × [ρ  ∇ · [ρ  3

(6)

Hence, the equation of motion (4) is satisfied by the displacement field (5) only if the particular vector quantities in (6) vanish. With these, Eq. (6) may be rewritten in the standard form of wave equation as ¨ − cT2 ∇ 2  = 0, ¨ − c2L ∇ 2  = 0, and  

(7)

here the complex speeds c2L = F(ψ˙ L )/ρ = (λ + 2μ)/ρ and cT2 = F(ψ˙ T )/ρ = (μ)/ρ are in general frequency dependent. The constants λ and μ are known as Lam´e parameters. The operator F(·) denotes the Fourier transform, and the quantities ψ L = ψ1 − 43 ψ2 and ψT = ψ2 . Hence, one can also define the viscoelastic moduli for P and SV waves using the velocity relations. Now, consider the general harmonic viscoelastic P wave as  = 0 (ω) exp(−ikL · e),

(8)

where kL = pL − iaL , with pL and aL as the propagation and attenuation vectors, respectively. In the expression of complex wave number (k L − ω/c L ), the complex velocity c L is defined earlier in (7). Substituting (8) in (7)1 and using velocity relation yields on simplification     [k L2 ] = [k L2 ] 1 − i Q −1 , k L2 = kL · kL = [k L2 ] 1 + i L 2 [k L ]

(9)

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where [·] and [·] denote real and imaginary parts, respectively, and the quality factor is Q L = −([k L2 ]/[k L2 ]). In a viscoelastic medium, the quality factor also depends upon the angle γ L between the vectors pL and aL . Next, extracting the real and imaginary parts from (9), and after solving and simplification for p L and a L , leads to    2γ , sec 2 p 2L = [k L2 ] 1 + 1 + Q −2 L L   (10)  −2 2 2 2 2a L = [k L ] −1 + 1 + Q L sec γ L . Similarly, using SV wave solution, the expressions for SV wave can be derived. The present study considers the reflection and refraction of bulk waves in a layered viscoelastic medium, and hence the direction of vectors p and a for each propagating waves must be defined. The Snell’s law along with the boundary conditions will give these vectors separately. Consider next the case of elastic half space (Q ≈ ∞), attenuation vector will be zero, and all other fields are modified accordingly. Figure 1 shows the geometry of a viscoelastic layer of thickness h, with lower boundary fixed with an elastic half space. We assumed that a harmonic wave is incident on the interface x3 = h at any arbitrary angle θ . Let ρ, c L , and cT denote density, and wave speeds for the viscoelastic layer, and ω is incidence frequency. The corresponding primed variables stand for the elastic half space. In the layer, one can write the total displacement potential as  = [1 exp(iαx3 ) + 2 exp(−iαx3 )] exp(−ikx1 ),

= x2 = [ 1 exp(iβx2 ) + 2 exp(−iβx2 )] exp(−ikx1 ).

(11)

  where α = k L2 − k 2 and β = k T2 − k 2 , and the amplitudes i and i (i = i, 2) are in general complex, and k is horizontal wave number. The propagator term in (11) is dropped since it is common in all fields. Since the wave numbers are complex in the viscoelastic medium, the quantities α and β are given by the corresponding principal values.

Fig. 1 Wave incident on the interface of layered half space

Free surface

x1 Welded interface

Incident Bulk wave

Viscoelastic Layer

θ

x3

Elastic half space

h

Propagation of Viscoelastic Waves in a Single Layered Media …

7

Let us write the various fields in the layer using (3), (5), and (11) in matrix form as ⎡

⎤ ⎡ ⎤⎡ ⎤ β sin βx3 k sin αx3 −iβ cos βx3 u1 −ik cos αx3  1 + 2 ⎢ u 3 ⎥ ⎢ −α sin αx3 −ik cos βx3 iα cos αx3 ⎢ ⎥ k sin βx3 ⎥ ⎢ ⎥=⎢ ⎥ ⎢ 1 + 2 ⎥ ⎣σ33 ⎦ ⎣ − cos αx3 2iβkμ sin βx3 −i sin αx3 2iβkμ cos βx3 ⎦ ⎣1 − 2 ⎦ σ13 2ikαμ sin αx3 μ cos βx3 2kαμ sin αx3 iμ sin βx3

1 − 2

(12) where  = (k 2 λ + α 2 (λ + 2μ)), and  = (β 2 − k 2 ). For simplicity, (12) in a compact form is represented as X(x3 ) = C(x3 )A.

(13)

Now, one can relate easily the fields at x3 = 0 and x3 = h, by using (13). Thus, eliminating the amplitude vector yields a linear relationship between the fields at x3 = 0 and x3 = h as (14) X(h) = C(h)C(0)−1 X(0). Next, using the fixed boundary conditions at the interface x3 = h, we obtain the amplitude relation for the half space as A = C (0)−1 C(h)C(0)−1 X(0),

(15)

where A = [1 + 2 1 + 2 1 − 2 1 − 2 ] is the amplitude vector in the half space. In elastic half space, the matrix C is calculated similarly as calculated in (12) for viscoelastic layer. In (15), the field vector X(0) is [u 1 u 3 0 0]T (since the stresses are zero at free surface, x3 = 0). 2.1 Incident P wave: Let us consider an incident longitudinal wave in the half space. This obtained by setting 1 = 0 in (15). Solving for free surface displacements u 10 and u 20 in terms of incident amplitude from (15) yields u 10 = −(D12 D31 − D11 D32 )−1 D32 1 , u 20 = (D12 D31 − D11 D32 )−1 D31 1 .

(16)

2.2 Incident SV wave: The expression for this case is obtained by setting 1 = 0 in (15). The resulting displacements are calculated in terms of 1 as u 10 = (D12 D31 − D11 D32 )−1 D12 1 , u 20 = −(D12 D31 − D11 D32 )−1 D11 1 .

(17)

where matrix D = C (0)−1 C(h)C(0)−1 depends upon both medium and incident wave parameters.

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Table 1 Elastic and viscoelastic material parameters Material ρ(kg/m3 ) c L (m/s) cT (m/s) Layer (E&C Epoxy) Half space (Steel)

1600 7800

2960 6020

1450 3218

QT

36 –

50 –

5

1 80

QL

10

3

10 7

5 9 7

10

5 9

40

L (Degree)

L (Degree)

2

9 7

60

3

2 5

10 7

7 9

3

7

40

10

10

3

7

2 9 7 3

3

60

3

9

20

20

2

9

9

1

2

f

5

5

10

7

0 0

5 7

80 3

1 3

2

0 4

0

1

f0

(a) Vertical displacement

9

2

f

3

4

f0

(b) horizontal displacement

Fig. 2 Normalized absolute displacement variation for incident P wave

3 Results and Discussions We have analyzed the free surface displacement characteristics for a viscoelastic layered half space. Numerical results are calculated from (16) and (17) and presented to show the behavior of surface displacement. The material parameters for the layered medium are listed in Table 1. In Fig. 2, the displacement variations are shown as a contour map for an incident P wave. The displacement components are plotted as separate map against incidence angle and normalized incident frequency f / f 0 , where f is incidence frequency in Hz, and f 0 = c L /4h and cT /4h, for an incident P and SV wave, respectively. It is observed that the vertical displacements curves are nearly zero valued at grazing incidence. For an incident SV wave, the variation of the vertical and horizontal parts of displacement is shown in Fig. 3. For grazing incidence, the horizontal displacement components are zero valued. There are some interesting observations to note in this case. Due to the higher quality factor and travel time, the effect of anelasticity becomes more pronounced here. The displacements fluctuate notably when the incident angle reaches its critical value. Here, the critical angle in this case is calculated as sin−1 (cT /cL ) = 32.3◦ . The results are provided for this incident angle in Fig. 4. Also, Rayleigh wave mode will appear thereafter this critical angle. In the layered

Propagation of Viscoelastic Waves in a Single Layered Media …

8 1

7

3

4

40 1 5

3 8

20

4

1

12 6 1 4 8

6 2

3

7

7 1

2

f

5 7

6 82 57 5

3

4 40 1

3

4

6

7

3

5

1

1 6

2

20

8

2

46 5 8 12

4 5 9 8 5

6

4

6

2

0 0

5 6 2

60

5

Degree

Degree T

3 4

6

3

1

80

6

4

7 60

8

T

80

5

5

2

9

1 1

0 0

4

f0

9 2

f

3

2 4

f0

(b) horizontal displacement

(a) Vertical displacement

horizontal displacement

vertical displacement

Fig. 3 Normalized absolute displacement variation for incident SV wave

10 8 6 4 2

12 10 8 6 4 2 0

1

2

f

3

f0

4

1

2

f

3

4

f0

Fig. 4 Variation of normalized displacement for incident SV wave at θT = 32.3◦

medium, the motions of the material points on the free surface are typically elliptical; this is also due to the effect of anelasticity. In a similar way, this elliptical motion can be analyzed by plotting the phase difference between horizontal and vertical components of displacement.

4 Conclusions In the present work, the matrix method is expanded to study the bulk wave characteristics in a viscoelastic layer that is fixed from below by an elastic half space. The free surface displacement distributions are derived analytically in closed form. The results for the normalized horizontal and vertical displacements are plotted and analyzed. The effective anelastic constants along with wave constants in the viscoelastic layer are derived. When the incidence angle reaches close to 90◦ , the vertical displacement

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components are zero for an incident P wave, and the horizontal displacement components are zero for an incident SV wave, respectively. For a critical incident SV wave, the displacement curves fluctuate suddenly near that angle, for which the results are also provided. These analyses have potential applications in non-destructive testing, material evaluation, ultrasonic assessment, and others. Also, wave assisted motion and handling can be improved based on our understanding of the free surface motion characteristics. The analysis of the interfacial weakness, interfacial bonding, and friction is an important aspect for future applications.

References 1. Thomson WT (1950) Transmission of elastic waves through a stratified medium. J Appl Phys 21:89–93 2. Haskell NA (1962) Crustal reflection of plane P and SV waves. J Geophys Res 67:4751–4767 3. Buchen PW (1971) Plane waves in linear viscoelastic media. Geophys J Int 23:531–542 4. Borcherdt RD (1973) Energy and plane waves in linear viscoelastic media. J Geophys Res 78:2442–2453 5. Cooper HF Jr (1967) Reflection and transmission of oblique plane waves at a plane interface between viscoelastic media. J Acoust Soc Am 42:1064–1069 6. Borcherdt RD (1982) Reflection-refraction of general P-and type-I S-waves in elastic and anelastic solids. Geophys J Int 70:621–638 7. Shumilova VV (2015) Reflection of plane a plane sound wave from the boundary of a heterogeneous medium consisting of elastic and viscoelastic layers. Comput Math Math Phys 55:1188–1199 8. Shamaev AS, Shumilova VV (2017) Plane acoustic wave propagation through a composite of elatic and Kelvin-Voigt viscoelastic material layers. Mech Solids 52:25–34 9. Kaur T, Sharma SK, Singh AK (2017) Shear wave propagation in vertically heterogeneous viscoelastic layer over a mcropolar elastic half-space. Mech Adv Mater Struc 24:149–156 10. Liu J, Wang Y, Wang B (2010) Propagation of shear horizontal surface waves in a layered piezoelectric half-space with an imperfect interface. IEEE T Ultrason Ferr 57:1875–1879

Axial Resistance of Short Built-up Cold-Formed Steel Columns: Effect of Lacing Slenderness M. Adil Dar, Dipti Ranjan Sahoo, Arvind K. Jain, and Sunil Pulikkal

Abstract Cold-formed steel (CFS) construction has become popular in the recent decades mainly due to their favorable features like higher yield strength, easy fabrication/handling, faster cost-effective construction, etc. Built-up members are often used as columns in structures with demands to resist large axial forces. Past studies have mainly stressed on evaluating the performance of battened CFS columns. Therefore, there is a need to examine the behavior of built-up laced CFS columns under the axial loading conditions. This study stresses on the numerical investigation of the influence of lacing element slenderness on the compression capacity of built-up CFS columns. Cold-formed steel columns consisting of four angle sections were connected by single lacing configuration. Test results from literature were used to calibrate the nonlinear finite element model developed. The design strengths predicted by North American Standards (AISI S-100) are calculated and compared with the test results. Keywords Built-up columns · Cold-formed section · Lacings · Numerical modeling · Strength

1 Introduction The adoption of cold-formed steel (CFS) sections in low-rise residential buildings has been extensive due to their favorable features like higher yield strength, easy fabrication/handling, faster cost-effective construction, etc. Premature instability due to thin-walled nature of CFS members is their primary limitation. Past research has identified efficient shaping and judicious stiffening as the measure to overcome these buckling issues. Suitably modified design and detailing of thin-walled CFS members helps in delaying/eliminating their buckling instability, thus significantly enhances their strength and stiffness characteristics [1]. Proper designed CFS sections can develop resistance up to their plastic moment strength [2]. Adequately designed CFS timber composite sections perform better with an advantage of easy fabrication, and M. A. Dar (B) · D. R. Sahoo · A. K. Jain · S. Pulikkal Department of Civil Engineering, IIT Delhi, New Delhi 110016, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_2

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thus have the potential to replace hot rolled steel for better economy and utility [3]. Significant enhancement in load-carrying capacity can be achieved through partial stiffening and strengthening of CFS sections with a higher strength to weight ratio [4, 5]. The use of built-up members as columns is recurrent when demands for resisting axial force in the structures are high. Past studies have mostly stressed on the evaluation of the performance of CFS columns comprising of two channels connected by batten plates. Vijayanand and Anbarasu [6] investigated CFS built-up columns with variation in the number of spacers. Numerous other parameters were varied too. It was observed that an increase in column slenderness ratio led to a drop in the axial resistance. EI-Aghoury et al. [7, 8] studied the behavior of pin ended CFS battened columns under concentric and eccentric loading conditions. Specimens varied in chord width to thickness ratio for both short and medium member slenderness. Outstanding width to thickness ratio, the overall slenderness ratio of column, and global slenderness ratio of angle were the main parameters governing the ultimate strength and behavior of battened columns. The column strength dropped due to the interactive buckling comprising of torsion and flexure. Dabaon et al. [9] evaluated the axial strength of battened CFS pin ended built-up columns fabricated with two channels back to back at different spacing. The batten plates were welded to the channels. The variation in the spacing between the channels was responsible for transforming the failure modes from one mode to the other. Anbarasu et al. [10] examined the behavior of CFS web stiffened built-up battened columns with varying slenderness ratio. When the effect of batten plate depth and the number of batten plates used was neglected, the DSM design strengths obtained were found to be conservative. This trend increased with an increase in the slenderness ratio. When elastic buckling analysis and modified slenderness approach of the NAS [11] Specification was used to calculate the critical buckling load for the columns, un-conservative design strengths were obtained. By increasing the overall slenderness of the columns, the ultimate resistance dropped. Intermediate stiffening of the web helped in eliminating its local buckling completely. Dar et al. [12–14] conducted a detailed numerical study to explore the performance of closed battened CFS columns comprising of two plain channels with pin ended support conditions. On increasing the toe to toe channel spacing, the load-carrying potential of the built-up columns enhanced. Furthermore, very limited research on CFS laced columns has been reported [15–21]. Keeping in view the focus of past research on built-up battened CFS columns, there is a need to evaluate the behavior of CFS built-up laced columns comprising of four angles under the axial loading conditions. Although, there are various parameters that affect the behavior of a laced built-up column, the slenderness ratio of the lacing element does influence substantially. The upper limit on the slenderness ratio of single lacing system specified by ANSI/AISC 360 [22] and IS-800 [23] is 140 and 145, respectively. However, it is applicable to hot rolled steel sections only. Accordingly, in this study, the influence of lacing slenderness on the compression capacity of CFS built-up laced columns with single lacing arrangement is investigated. These members comprise of four angle sections connected by lacing bars. In the current study, a nonlinear numerical model was developed to investigate the effect of lacing slenderness on the behavior of built-up columns. The models were

Axial Resistance of Short Built-up Cold-Formed Steel Columns …

13

(a) SL-60-20-805(a)

(b) SL-60-20-1455 Fig. 1 Failure mode comparison-experimental versus FEM

calibrated against the relevant experimental results available in the literature. In the parametric study, the lacing slenderness ratio variation was adopted from 79 to 395. In addition to this, the numerical results were compared with strength predictions made by North American Specification [11], American National Standard [22], and Indian Standards [23] (Fig. 1).

2 Finite Element Modeling Finite element method is used to numerically investigate the behavior of built-up laced columns. ABAQUS [24] a commercial general-purpose program was adopted for carrying out the finite element analysis. The details of the finite element model are given in Table 1. The test results of Dar et al. [16] and EI-Aghoury et al. [7] were used for calibrating the finite element model developed for conducting the parametric study in order to investigate the role of lacing slenderness in influencing the axial resistance of built-up CFS columns. The comparison between the ultimate loads obtained from the test results of Dar et al., 2018 [16] and the results obtained from finite element analysis is shown in Table 2. Figure 1 shows the failure mode comparison of specimens SL-60-20-805(a) and SL-60-20-1455. The comparison between the ultimate load obtained from the test result of the battened column (1000B80L30B6) conducted by EI-Aghoury et al. [7] and the result acquired from the finite element analysis is presented in Table 3. Figure 2 shows the comparison of the load vs. displacement between the test result of EI-Aghoury et al. [7] and FEM.

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Table 1 Details of the finite element model Detail type

Feature adopted

Detail type

Feature adopted

Type of element

Shell element (S4R5)

Support condition at the other end

Pined end

Mesh size & Aspect ratio

5 × 5 mm2 & 1.0

Application of axial load

By applying an axial displacement at loaded end

Material model

Elastic perfectly plastic

Connection between the lacing and sections

Mesh independent fastener option available in ABAQUS

Modulus of Elasticity 2 × 105 MPa (E)

Local imperfection

0.006 × w × t [17]

Yield stress (fy )

450 MPa

Global imperfection

1/500 of column height [25]

Support condition at loaded end

Pined end with displacement in the longitudinal direction permitted

Types of analysis

Linear buckling analysis, followed by a nonlinear analysis

Table 2 Comparison of test results with numerical results for laced column

Specimen

PuExp (kN)

PuFEM (kN)

PuExp /PuFEM

SL-60-20-805(a)

68.9

65.83

0.98

SL-60-20-1455

44.6

48.78

0.94

PuExp: Ultimate load obtained from experiment PuFEM: Ultimate load obtained from FEM Table 3 Comparison of test result with numerical result for battened column

Researcher

PuExp (kN)

PuFEM (kN)

PuExp /PuFEM

EI-Aghoury

52

53

0.98

Dar

52

51.5

1.00

PuExp: Ultimate load obtained from experiment PuFEM: Ultimate load obtained from FEM 60

Load (kN)

Fig. 2 Comparison of experimental and FEM result for battened column 1000B80L30B6

Test FEM

40 20 0

0

0.5 1 1.5 Axial Deformation (mm)

2

Axial Resistance of Short Built-up Cold-Formed Steel Columns …

15

Fig. 3 Numerical model of the laced built-up column

The results shown in Tables 2, 3 and Fig. 2 indicate that the agreement between the results of FEM and the experiments is good. Hence, this model can be used for parametric study.

3 Parametric Study In the parametric study, numerical models were developed to carry out analyses on the cross-section of the specimen SL-60-20-805(a) used by Dar et al. [16] (See Fig. 3). The slenderness ratio of the lacing elements was varied from 395 to 79. The results of this variation in lacing slenderness on the behavior of laced built-up columns including the comparison of the FEA predicted strengths with the design strengths predicted by AISI S-100, ANSI/AISC 360, and IS-800 are given in Table 4. The load versus axial deformation curves for Model-I with single lacing arrangement are shown in Fig. 4. The models were labeled as Model type, type of lacing arrangement, and slenderness of the lacing element. For example, the label “Model-I-S-395” defines the following: Model-I indicates the type of model to choose from the tested specimens (all the dimensions of Model-I to be used), S indicates the type of lacing arrangement (S stands for single lacing) used, and 395 indicates the slenderness ratio of the lacing element.

4 Discussion There was a slight drop in the initial stiffness of the columns between 45 to 55 kN until their ultimate capacities. As the lacing slenderness increased from 79 to 395, the ultimate capacities of the columns dropped from 65.8 kN to 57.7 kN (around 12.3%). However, it was observed that post the peak load, with an increase in the lacing slenderness, the stiffness of the curve dropped. The variation of strength with

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Table 4 Results of parametric study Model-I Lacing PuFEM (kN). PuFEM /PuNAS PuFEM /PuAISC PuFEM /PuIS-800 Mode of slenderness failure S-395

395

57.74

0.86

0.91

0.84

L

S-263

263

58.91

0.87

0.93

0.86

L

S-197

197

58.99

0.88

0.93

0.86

L

S-158

158

60.72

0.90

0.96

0.88

L

S-132

132

62.52

0.93

0.99

0.91

L

S-113

113

62.55

0.93

0.99

0.91

L

S-99

99

64.46

0.96

1.02

0.94

L

S-88

88

64.61

0.96

1.02

0.94

L

S-79

79

65.83

0.98

1.03

0.96

L

PuNAS Design strength predicted by NAS S-100 PuAISC Design strength predicted by ANSI/AISC 360 PuIS-800 Design strength predicted by IS-800 PuFEM Ultimate load obtained by finite element analysis L Local buckling

Fig. 4 Effect of lacing slenderness

80

395 263

Load (N)

60

197 158 132

40

113 99

20

88

0

79

0

1

2

3

4

Axial Deformation (mm)

respect to lacing slenderness in a single lacing arrangement is shown in Fig. 5. The failure mode in all the models was governed by local buckling as shown in Fig. 6a for Model-I-S-395. For Model-I-S-395, the buckling of lacing during failure was prominent as shown in Fig. 6b.

5 Conclusions This research has stressed on the influence of lacing slenderness on the compression resistance of cold-formed steel built-up columns using FEA. The cold-formed steel columns consisted of four angle sections connected by a single lacing configuration. The following were the main conclusions drawn out of the present study.

Axial Resistance of Short Built-up Cold-Formed Steel Columns …

17

67

Load (kN)

63

59

55

0

100

200

300

400

Lacing Slenderness Fig. 5 Variation of strength with respect to lacing slenderness

(a) Failure mode in Model-I-S-395

(b) Lacing buckling in Model-I-S-395

Fig. 6 Failure in model-I-S-395

• The parametric study confirmed that slenderness of the lacing element affects the behavior of built-up columns, particularly their strength. • With the increase in lacing slenderness from 79 to 395, the ultimate capacities of the columns dropped from 65.8 kN to 57.7 kN (around 12.3%) • Post the peak load, with an increase in the lacing slenderness, the stiffness of the curve dropped. • With reference to the upper limit of 140 and 145 specified on the slenderness ratio of a single lacing system by ANSI/AISC 360 and IS-800, respectively, a slenderness ratio σ 3 and so on). Since through experiments, CFD simulations and full order FE simulations, etc., the snapshot matrix is known in advance (through FE analysis in this study). After SVD, the energy carried by each mode is calculated, then the transformation matrix is formed based on the dominating modes. The transformation matrix (T pca ) will consist of the first e columns of the Q, where e ≤ k. The transformation matrix is given as       Tpca  = Q1 Q2 Q3 . . . . . . Qe

(24)

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R. Kumar et al.

where Qe is the eth column of the left orthogonal singular matrix. Assuming, X r = T pca up and substituting if back in system Eq. (1) with multiplication with T Tpca  , the final reduced equation of motion is given as  pca ¨ p + C pca ˙ p + K pca  p = fpca M  u  u  u 

(25)

where T   T   T M   = Tpca  T Kr Tpca   = Tpca  r Tpca  pca M  = T pca  , C pca  , = T pca  , f pca  Cr T pca  f r

(26)

Equation (26) is used for obtaining reduced mass, stiffness, and damping matrices, and Eq. (25) is the dynamic equation for the reduced system.

3 Numerical Results The above mentioned ROMs are applied to a test problem of simply supported stepped beam which is shown in Fig. 1. Firstly, this system is analyzed using full order FE analysis. The results of this full model obtained through FE analysis are used as the benchmark for comparison with that obtained through ROMs. The beam shown in Fig. 1 has two different cross-sectional areas A1 and A2 . The uniformly distributed sinusoidal load is placed on the mid-span of the middle section (A2 ). For full order FE analysis, the system is discretized into two-dimensional elements. Each node has two DOFs. The parameter value used in the analysis is given in Table 1. The system has 5% proportional damping. Figure 2 shows the amplitude of FRF of the displacement at the midpoint of the middle span for the convergence of the number of elements for the analysis. The FRF of response for a different number of elements is almost coinciding to one over the other. So, for FE analysis, 24 elements (50 DOFs) are taken. The normalized singular values are plotted in Fig. 3. The singular value contains the information of the energy carried by each mode. From Fig. 3, it is clear that the first mode is carrying

Fig. 1 Schematic of simply supported Euler–Bernoulli stepped beam

Analysis of Stepped Beam Using Reduced Order Models Table 1 Parameters of the system

Parameters

29 Value

Damping ratio

0.05

Amplitude of load

1000 N

Youngs’ modulus (E)

25e9 Pa

Density of material (ρ)

2400 kg/m3

Cross-sectional area (A1 )

4e-2 m2

Cross-sectional area (A2 )

8e-2 m2

Length of each section (L)

2m

Area moment of inertia (I1 )

1.33e-4 m4

Area moment of inertia (I2 )

10.67e-4 m4

Fig. 2 FRF of the system for convergence with various number of elements

maximum energy (99.7%). The other modes are having very less contribution and are of the order of 10−15 . Figure 4(a) shows the original snapshot matrix with all the principal components. Top right and the other two bottom plots contain the one, two, and three principal components, respectively. There is no significant difference in the responses of other plots as compared to the original response. Figure 4b–d show responses with one, two, and three principal components, respectively. The first five natural frequencies obtained through SEREP (7 DOFs used) and CMS (13 DOFs used) are compared with that obtained through full order FE analysis. The comparison is shown in Fig. 5. As discussed in PCA that only one mode is dominating, so in further analysis, only one equation is solved. The natural frequency obtained in PCA is 60.58 rad/s with an error of 0.8%. SEREP shows a very good

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Fig. 3 Energy contained in each mode shown on a logarithmic plot

Fig. 4 a Original snapshot response, b response with one principal component, c response including two principal components, and d response using three principal components

Analysis of Stepped Beam Using Reduced Order Models

31

Fig. 5 Comparison of frequencies obtained through different ROMs

match of value of natural frequencies with that of the benchmark value as compared to CMS (Fig. 6). It is evident from Fig. 5, and it can be seen that the first two natural frequencies obtained through CMS are almost close to the benchmark value, but for higher mode, CMS shows considerably large deviation. The accuracy here depends on the number of modes considered for the analysis. So if a larger number of DOFs are considered in CMS, a better match is expected. For the beam shown in Fig. 1, the maximum deflection is expected at the mid-span. The displacement at the corresponding point is obtained through FE model first and then the same at that point using ROMs. The ROMs are showing a very good match qualitatively with that of the FEM results.

4 Conclusions In this paper, the principle and mathematical formulations for ROMs like CMS, SEREP, and PCA are discussed. To study the efficiency of these ROMs, an Euler— Bernoulli beam with a stepped section has been considered. The beam has sinusoidal loading conditions. In CMS, the DOFs corresponding to the boundary of substructure always remain in the final reduced subspace, and it cannot be eliminated, which is the disadvantage. The accuracy of results in CMS depends on the number of DOFs assumed for the analysis. SEREP shows a very good match with results obtained through FE model. In PCA, only one equation has solved as only one mode was dominating. Therefore, it reduces the computational effort sufficiently. Further, these methods can also be used to solve the complex engineering problems. The first two

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Fig. 6 Displacement at midpoint of the beam obtained through a FEA, b CMS, c SEREP, and d PCA

methods, i.e., CMS and SEREP are useful for linear systems, whereas the last one (PCA) is also used for nonlinear systems.

References 1. Guyan RJ (1965) Reduction to stiffness and mass matrices. AIAA J 3(2):380 2. Hurty WC (1965) Dynamic analysis of structural systems using component modes. AIAA J 3:678–685 3. Mauricio GS, Bladh R, Hans M, Paul PR, Torsten F, Damian MV (2017) Forced response analysis of a mistuned, compressor blisk comparing three different reduced order model approaches, Journal of engineering for gas turbines and power, 139 4. Yang MT, Griffin JH (1999) A reduced order model of mistuning using a subset of nominal system modes. In: Proceedings of international gas turbine and Aeroengine congress and exhibition 5. Catanier MP, Pierre C (2006) Modeling and analysis of mistuned bladed disk vibration: status and emerging directions. J Propul Power 22(2):384–392 6. Friswell MI, Garvey SD, Penny JET (1995) Model reduction using dynamics and iterated irs techniques. J Sound Vib 186(2):311–323

Analysis of Stepped Beam Using Reduced Order Models

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7. Friswell MI, Garvey SD, Penny JET (1998) The convergence of the iterated irs method. J Sound Vib 211(1):123–132 8. O’Callahan JC, Avitabile P, Madden R (1989) System equivalent reduction expansion process. In: Seventh international modal analysis conference, Las Vegas, Nevada 9. Yi dong L, Adam M, Lorenz TB, Sorin M, Jens IM, Stephen EZ (2009) Reduced order model based on principal component analysis for process simulation and optimization. Energy Fuels 23:1695–1706 10. Holmes P, Lumley JL, Berkooz G (1996) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge 11. Griffith DT (2009) Analytical Sensitivities for principal components analysis of dynamical systems. Proceedings of the IMAC—XXVII, Florida USA 12. Ma X, Vakakis A (1999) Karhunen—Loeve decomposition of the transient dynamics of a multibay truss. AIAA J 37(8):939–946 13. Stone JV, Independent component analysis: a tutorial introduction. The MIT Press, Cambridge 14. Strang G (1988) Linear algebra and its application. Thomas Learning INC 15. Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analysis. AIAA J 6:1313–1319 16. Craig RR, Kurdila A (2006) Fundamentals of Structural Dynamics. Wiley, New Jersey

Numerical Analysis of Geosynthetic Strengthened Brick Masonry Panels Hasim Ali Khan, Radhikesh Prasad Nanda, and Diptesh Das

Abstract Un-reinforced masonry (URM) structure is extremely vulnerable to seismic actions. Their susceptibility to collapse has provided the concussion to develop strengthening techniques to strengthen URM buildings. The numerical analysis of the in-plane behaviour of un-reinforced and geotextile strengthened brick masonry specimen, using a 3D macro nonlinear model, is presented in this paper. All specimens are subjected to diagonal compression tests. Two different patterns viz. parallel and diagonal are strengthened. Numerical analyses are carried out to verify the efficiency of the reinforcement with geosynthetic. From the investigation, it is noticed that geosynthetic strengthening enhanced the load-bearing capability, diagonal shear strength and stiffness remarkably. It is estimated that the diagonal shear strength enhanced from 36% to 39%. Hence, masonry strengthened with geosynthetic will perform better in the seismic prone area. Keywords Strengthening · Geosynthetic · Geotextile · Masonry panels · Diagonal compression test

1 Introduction URM buildings are weak to failure in an earthquake. Un-reinforced masonry (URM) is widely used in the world. Mortar is a weak part of masonry. Two types of collapse are commonly noticed during seismic prone areas. In-plane and out-ofplane collapses are noticed in URM structures [1]. The in-plane collapse mode is vitally significant in URM walls under seismic action. Past investigator shows that

H. A. Khan (B) · R. P. Nanda · D. Das Department of Civil Engineering, National Institute of Technology Durgapur, Durgapur, India e-mail: [email protected] R. P. Nanda e-mail: [email protected] D. Das e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_4

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Fig. 1 In-plane failure techniques of un-reinforced brick walls a Shear failure, b Sliding failure, c Rocking failure, d Toe-crushing failure

throughout an earthquake, the principal collapse mode is shear [2–4]. The vital inplane collapse strategies of URM walls exposed to seismic actions are presented in Fig. 1 [5, 6]. Application of geosynthetic products has been utilized widely in numerous civil engineering construction viz. retaining walls, embankments, soil backfill [7]. Geosynthetic enhances the execution of roads inside the base course [8, 9]. Geosynthetic, as a form of base isolation, has been investigated by Yegian and Kadakal [10]. There is an expansion to develop new applications to resolve different civil engineering problems [11,12]. In-plane strength is significantly enhanced by using Geosynthetic products [13]. This aim of this study to calculate the in-plane shear behaviour of un-strengthened and strengthened brick walls by application of geotextile with various patterns numerically using ANSYS [14]. To get a diagonal shear collapse mode of a masonry wall, a force can be applied along diagonal of masonry panel as per ASTM E519 [15].

2 Numerical Model 2.1 Introduction Masonry is an anisotropic component found by the assemblage of bricks and mortar. Numerical models of un-strengthened and strengthened masonry have been formed by finite element analysis in ANSYS. Therefore, numerical models of masonry habitually display a reasonable level of complexity. Generally, three different methods are implemented for the modelling of masonry. The modelling approaches are complete micro-modelling, simplified micro-modelling and macro-modelling [16– 18]. In this research, a macro nonlinear 3D model has been formed to determine the in-plane performance of un-strengthened and geotextile strengthened brick masonry specimen.

Numerical Analysis of Geosynthetic Strengthened Brick … Table 1 Composition of the masonry constituents [13]

Table 2 Composition of nonwoven polypropylene geotextile [13]

37

Properties Density, ρ

Brick

Mortar

(kg/m3 )

1750

2150

Elasticity modulus, E (MPa)

2020

4050

Poisson’s ratio, ν

0.15

0.22

Ultimate tensile strength, ft (MPa)

1.66

0.86

Ultimate compressive strength, fc (MPa)

8.93

3.49

Property

Unit

Value

Tensile strength

MPa

0.18

Young’s modulus

MPa

15850

Poisson’s Ratio

–

0.35

Thickness (at 2 kPa)

mm

1.5

Mass per unit area

g/m2

262

Elongation

%

85

2.2 Parameters Utilized in Masonry The parameters utilized in masonry are determined experimentally. The material utilized in the research is illustrated in Table 1. The parameters of the nonwoven geotextiles utilized in the present study are presented in Table 2.

2.3 Model Description To determine the in-plane performance of masonry panel under the diagonal compression test, a 3D macro model is analysed. The masonry is assumed as a homogenous considered. The mechanical parameters of the whole structure being homogeneous elements. The dimensions of the masonry panel are 600 mm × 600 mm × 125 mm. Figure 2 illustrates the detail description of the setup and boundary limitations. The compressive loads are applied simultaneously along one diagonal. Therefore, one diagonal gets contracted, and other gets extended. In this model, masonry specimen is modelled with a higher-order 3D, 10 nodes of SOLID187 tetrahedron elements Fig. 3a. Geotextile is utilized with SHELL 63 element due to its bending and membrane capacities in the ANSYS, and the detail description is depicted in Fig. 3b. The interface between masonry and geotextile is modelled with CONTA174 element Fig. 3c. The Drucker–Prager formulation is considered for the masonry specimen [13]. Nonlinear behaviour is investigated.

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H. A. Khan et al.

Fig. 2 Masonry specimen loaded diagonally

Fig. 3 Elements taken of model a SOLID 187 b CONTA174 c SHELL63

Figure 4a–c depicts finite element mesh utilized for the masonry panel before and after strengthening correspondingly. Nonlinear static analyses are adopted by using the Newton–Raphson iteration method.

(a) UR

(b) RHSS

Fig. 4 Mesh details of a UR, b RHSS, c RXSS

(c ) R X S S

Numerical Analysis of Geosynthetic Strengthened Brick …

39

3 Results and Discussion The strengthening of masonry specimen was evaluated. Figure 5 depicts the distribution of shear stress. Shear capability is enhanced from UR to RHSS and RHSS to RXSS correspondingly. The load-deformation graphs evaluated along the compressed diagonal is investigated. The load-deformation capacities increase with strengthening giving maximum in case of the diagonal pattern. Figure 6 depicts the graphs of comparative load-deformation performances for masonry specimen. The comparison indicates almost equal before and after strengthening in collapse load and the stiffness correspondingly.

(a) UR

(b) RHSS

Fig. 5 Shear stress distribution for a UR, b RHSS, c RXSS at collapse

Fig. 6 Diagonal compression of masonry specimen

(c) RXSS

40

H. A. Khan et al.

Table 3 Evaluated diagonal strength Strengthening pattern Diagonal shear strength (MPa) Increase in Diagonal shear strength over UR (%) UR

0.23

–

RHSS

0.3128

36

RXSS

0.3197

39

Fig. 7 Comparison of the Diagonal shear strength of one-side strengthened specimen

As per ASTM E 519, the distribution of shear stress for masonry specimen is evaluated and shown in Table 3 and Fig. 7. The Diagonal shear strength enhanced from 36 to 39%. Furthermore, it was also illustrated that the panel with diagonal strengthening gives better performance. Figure 7 illustrates the comparisons of the diagonal shear strength of the oneside strengthened specimen. RXSS indicates the highest stiffness and deformation capability.

4 Conclusions The numerical observation was investigated under diagonal compression tests to develop the diagonal shear strength of masonry specimen. Based on the analysis, the following findings are pointed out: • The strengthened specimen enhanced the failure load and deformation from UR to RHSS and RHSS to RXSS correspondingly. • The diagonal shear strength enhanced from 0.3128 to 0.3197 MPa. • Furthermore, diagonal strengthening has more stiffness than others.

Numerical Analysis of Geosynthetic Strengthened Brick …

41

• Brittle failure noticed for un-strengthened panel while strengthening enhanced its deformation capability. • It is also pointed out that load-carrying capability, deformation capability, diagonal shear strength and stiffness are significantly increased from parallel to diagonal, respectively.

References 1. Khan HA, Nanda RP (2020) Out-of-plane bending of masonry wallette strengthened with geosynthetic. Constr Build Mater 231:117198 2. Khan S, Khan AN, Elnashai AS, Ashraf M, Javed M, Naseer A, Alam B (2012) Experimental seismic performance evaluation of unreinforced brick masonry buildings. Earthquake Spectra 28(3):1269–1290 3. Khan HA, Nanda RP, Roy P (2016) Retrofitting of Brick Masonry Panels with Glass Fibre Reinforced Polymers: IOSR Journal of Mechanical and Civil. Engineering 1:11–18 4. Nanda RP, Khan HA, Pal A (2017) Seismic-retrofitting-of-unreinforced-brick-masonry-panelswith-glass-fibre-reinforced-polymers. Int J Geo Earth Eng 8(1):28–37 5. ElGawady MA, Lestuzzi P, Badoux M (2007) Static cyclic response of masonry walls retrofitted with fibre-reinforced polymer. J Compos Constr 11(1):50–61 6. Kalali A, Kabir MZ (2012) Cyclic behaviour of perforated masonry walls strengthened with glass fibre reinforced polymers. Scientia Iranica A 19(2):151–165 7. Kalali A, Kabir MZ (2012) Experimental response of double-wythe masonry panels strengthened with glass fibre reinforced polymers subjected to diagonal compression tests: Eng. Struct. 39:24–37 8. Koerner RM (2000) Emerging and future developments of selected geosynthetic applications. J Geotech Geoenviron 126(4):293–306 9. Giroud JP, Han J (2004) Design method for geogrid-reinforced unpaved roads. I. Development of design method. J Geotech Geoenviron 130(8):775–786 10. Yegian MK, Kadakal U (1998) Geosynthetic interface behaviour under dynamic loading. Geosynth Int 5(1):1–16 11. Majumder S, Saha S (2020) Behaviour of reinforced concrete beam strengthened in shear with geosynthetic. Adv Struct Eng 23(9):1851–1864 12. Palmeira EM, Tatsuoka F, Bathurst RJ, Stevenson PE, Zornberg JG (2008) Advances in Geosynthetics materials and applications for soil reinforcement and environmental protection works. Electron J Geotech, Eng 13. Khan HA, Nanda RP, Das D (2017) In-plane strength of masonry panel strengthened with geosynthetic: Const. Build Mat 156:351–361 14. ANSYS. 2017. Release 17.0. ANSYS Inc 15. American Society for Testing and Materials, ASTM E 519: Standard test method for diagonal tension (shear) in masonry assemblages (2001) 16. Zucchini A, Lourenḉo PB (2004) A coupled homogenisation-damage model for masonry cracking. Struct 82:917–929 17. Khan HA, Nanda RP, Das D (2019) Numerical Analysis of Capacity Interaction of Brick Masonry Wallettes Strengthened with Geosynthetic. In: Proceedings of the 13th North American masonry conference, Salt Lake City, Utah, Paper No-178,1554–1564 18. Khan HA, Nanda RP, Das D (2018) Numerical Analysis of Geosynthetic Strengthened Brick Masonry Wallettes Subject to In-Plane and Out-of-Plane Loading. In: Proceedings of the 16th symposium on earthquake engineering, Indian Institute of Technology Roorkee, paper Id-398

Studies on Dynamic Characteristics of Sandwich Functionally Graded Plate Subjected to Uniform Temperature Field E. Amarnath and M. C. Lenin Babu

Abstract The effect of uniform temperature on dynamic characteristics of a sandwich plate with functionally graded (FG) core is analyzed by finite element methods. The top and bottom of the sandwich plate are considered as metal and ceramic, and the core is considered as functionally graded. Aluminum and alumina materials are used as top and bottom face sheets, and the same materials are graded through thickness for FGM core. The material properties of FG core vary throughout the thickness direction according to the power law index. The effective material properties are calculated using the Voigt technique. The Buckling temperature of sandwich plate is analyzed for different boundary conditions and used as criteria to study the effect of temperature on natural frequencies and mode shapes. Keywords Free vibration · Sandwich functionally graded plate · Thermal environment · First-order shear deformation theory

1 Introduction Functionally graded materials (FGM) concept was introduced in 1984 by Japan for aircraft designs. FGM are the class of composites which continuously change properties throughout its thickness direction. FGM are widely preferred in automobile and aerospace industries because of their strength, corrosion resistance, and thermal barrier capabilities. Gasik [1] reviewed different micro mechanical modeling approaches and discussed about the problems involving in the design of functionally graded materials. Victor and Larry [2] reviewed FGM structures with the viewpoint of dynamic characteristics and various manufacturing methods. Udupa et al. [3] overviewed the various concepts involved in designing of functionally graded composites and their applications. Watanabe [4] studied about the fabrication of functionally gradient materials by powder metallurgical processing technique and

E. Amarnath · M. C. L. Babu (B) School of Mechanical and Building Sciences, VIT, Chennai 600127, Tamil Nadu, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_5

43

44

E. Amarnath and M. C. L. Babu

showed it provides a wide range of compositional and microstructural control alongwith shape forming capability. Thai and Kim [5] critically reviewed FG plates by considering modeling and loading methods as criteria. Zhao et al. [6] investigated the thermal buckling behavior of functionally graded plates numerically, and the plate was modeled using first-order shear deformation theory (FSDT). Influence of volume fraction and hole geometry on buckling load was also analyzed by them. Kiani et al. [7] studied the thermal and mechanical instability of the sandwich functionally graded plate when resting on the elastic base. Zenkour and Mashat [8] investigated the thermal buckling behaviur of functionally graded plate (FGP) by considering the effects of shear deformation and thickness stretching. Bouhadra et al. [9] calculated the thermal buckling temperature of FGP with uniform temperature. In their FGP model, the material distribution was considered by power law index and Voigt mode, and varied throughout the thickness direction of the plate. Numerical analysis was carried to study the effect of plate aspect ratio and power law index on critical buckling temperature. Reza [10] did a detailed investigation about a thermal buckling of beams, plates, and shells. Bhagat et al. [11] determined the thermal buckling and natural frequency of uniformly heated isotropic cylindrical shell. Mayandi and Jeyaraj [12] investigated about the buckling and natural frequency of composite beam under non-uniform thermal load. Jeyaraj et al. [13] discussed about the vibration and acoustic characteristics of the viscoelastic sandwich plate under the thermal environment. Li et al. [14] considered two different types of sandwich functionally graded plates to study the free vibration characteristics by three dimensional theory of elasticity. Simply supported and clamped boundary conditions were used to find the free vibration characteristics under the influence of volume fraction, thickness– length ratios, aspect ratio, and layer thickness ratios of sandwich FGP. Chandra et al. [15] determined the analytical solution for vibration and acoustic response of FGP by using FSDT. They showed that the increase in power law index leads to a decrease in natural frequency for simply supported boundary condition. Jeyaraj et al. [16] found that the vibration and acoustic properties of steel plate are significantly affected by the temperature. For this study, they have used critical buckling temperature as a parameter. In this work, a sandwich functionally graded plate is considered to study the effect of uniform temperature influences on free vibration characteristics. By using power law index, material properties varied throughout the thickness direction on functionally graded core. The effective properties are determined by using the Voigt technique. Influence of power law, different boundary conditions (CCCC, CCFC, FCFC, and CFFC), and temperature rise on mode shapes and natural frequencies is also studied in detail.

2 Problem Description A sandwich plate with functionally graded core is considered for this study with top and bottom face sheets considered to be metal and ceramic as shown in Fig. 1.

Studies on Dynamic Characteristics of Sandwich Functionally …

45

Fig. 1 Sandwich plate

Volume fraction of alumina (Vc ) across the thickness direction is given in Eq. 1, w here p denotes power law index, z is the height of each layer from the mid plane, and he total thickness of the plate along z coordinates is h.  z p Vc = 0.5 + h

(1)

Vm = (1 − Vc )

(2)

P(z) = (Pc − Pm )Vm + Pm

(3)

Effective Young’s modulus (E), density(ρ), Poisson’s ratio (v), and coefficient of thermal expansion (α) were derived using the Voigt model as given in Eq. 3, where c and m denote ceramic and metal, respectively. In this work, four different sandwich configurations such as 4-2- 4, 3-4-3, 2-6-2, and 1-8-1 are considered to get critical buckling temperature. In each configuration, the numbers represent the metal face sheet thickness (h1 ), FG core thickness (h2 ), and ceramic face sheet thickness (h3 ) of the sandwich FGP.

3 Methodology As given in the equation critical thermal buckling temperature is calculated with considering the pre-stress effects [12]   [Kss ]+λi [Kgs ] {Ψi }= 0

(4)

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E. Amarnath and M. C. L. Babu

where [Kss ] denotes structural stiffness matrix, [Kgs ] is geometric stiffness, λi is the eigenvalue, and Ψi is the eigenvector. Then the natural frequencies and mode shapes were analyzed by assuming the critical buckling temperature as a parameter. The effect of temperature is included in the form of the geometric stiffness matrix as shown in Eq. 5. 

  [K ss ] + [K gs ] − ωi2 [M] {Φi } = 0

(5)

where ωi denotes natural frequencies and [M] is structural mass matrix. Finally, the influence of power law index on critical buckling temperature and free vibration characteristics are analyzed for different boundary conditions like CCCC, CCFC, FCFC, and CFFC. The entire analysis carriedout by using ANSYS finite element software.

4 Validation Studies Bouhadra et al. [9] analyzed critical buckling temperature for Al/Al2 O3 FGP and Li et al. [14] examined non-dimensional natural frequencies of sandwich FGP. These two cases are used for validating the above finite element formulation. The physical properties of Al and Al2 O3 are given in Table 1. Validation of critical buckling temperature on FGP is presented in Table 2, and non-dimensional natural frequencies of sandwich FGP are shown in Table 3. The present study shows good agreement with the results considered for validation. Table 1 Physical properties of Alumina (Al2 O3 ) and Aluminum (Al) Properties

Ceramic (Alumina)

Metal (Aluminum)

Young’s modulus, GPa

380

70

Poisson’s ratio

0.3

0.3

3800

2707

7.4 × 10−6

23 × 10−6

Density,

kg/m3

Coefficient of thermal expansion, 1/°C

Table 2 Validation of critical buckling temperature of all side clamped square FGP a/h

Theory

p=0

p = 0.5

p=1

p=2

p=5

50

Bouhadra et al. [9]

181.3

102.7

84.3

74.7

76.9

Present study

180.7

102.7

84.1

74.4

76.4

100

Bouhadra et al. [9]

45.5

25.8

21.156

18.7

19.3

Present study

45.4

25.8

21.121

18.6

19.2

Studies on Dynamic Characteristics of Sandwich Functionally …

47

Table 3 Validation of non-dimensional natural frequency of clamped square sandwich FGP h/a

Theory

p = 0.5

p=1

p=2

p=5

p = 10

0.01

Li et al. [14]

2.45

2.54

2.64

2.80

2.91

Present study

2.42

2.5

2.61

2.76

2.87

0.1

Li et al. [14]

2.24

2.34

2.45

2.60

2.70

Present study

2.21

2.32

2.43

2.58

2.67

0.2

Li et al. [14]

1.86

1.97

2.09

2.22

2.28

Present study

1.83

1.96

2.07

2.20

2.27

5 Results and Discussion A rectangular sandwich FGP having dimensions of (0.5 × 0.4 × 0.01)m corresponding to length, width, and thickness directions is considered for the detailed study. After a careful examination, a converged 45 × 45 finite element mesh size is used.

5.1 Critical Buckling Temperature Critical buckling temperature is analyzed for different power law indexes, boundary conditions, and core thicknesses as shown in Figs. 2b–d. It is clear that an increase in power law index tends to an increase in critical buckling temperature because of the increase in stiffness due to the ceramic content. From Fig. 2a, one can see that at a lower power law index, the critical buckling temperature is high for 4-2-4 configuration with CCCC boundary condition. In contrast to the above at higher power law index, the critical buckling temperature is found to be high for 1-8-1 configuration. This is due to the variation of ceramic content in different configurations considered for the analysis. Figures 2b–d shows the variation of critical buckling temperature of sandwich FGP for CCFC, FCFC, and CFFC boundary conditions. It is clear that the variation of thermal buckling strength in other boundary conditions is following the trend of CCCC and indicates higher strength for CFFC condition. This may be due to the free expansion of adjacent free edges.

5.2 Free Vibration Characteristics In order to show the influence of temperature on free vibration characteristics, it is decided to use 1-8-1 sandwich FGP configuration since it shows higher thermal buckling strength than others. Then the obtained critical buckling temperature is used as a parameter and applied over all the nodes of the plate uniformly. For this

48

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Fig. 2 Critical buckling temperature, Tcr (°C) for different boundary conditions

purpose, 0*Tcr, 0.25*Tcr, 0.5*Tcr, 0.75*Tcr and 0.99*Tcr uniform temperature rise are considered. Then the natural frequencies are obtained by performing the modal analysis by considering pre-stress effects due to the applied temperature. Effect of temperature on natural frequencies of sandwich FGP for different boundary conditions, power law indexes, and uniform temperature rises is shown in Tables 4, 6, 8 and 10. From Table 4, one can see that the natural frequencies are increasing with an increase in power law index. It is due to the increase in ceramic content than metal and leads to an increase in stiffness of sandwich FGP. At the same time, Table 4 shows that the natural frequencies are decreased when the applied temperature is increased. This may be due to the influence of pre-stress effect on bending stiffness. A similar kind of variation is observed in other boundary conditions also and showed in Tables 6, 8, and 10. Next, the influences of uniform temperature rise on mode shapes is studied for different boundary conditions. The analysis has been done for different power law indexes; however, for power law index 1, the material properties are varying linearly with respect to thickness. So in this study, only the power law 1 is chosen to study the effect of temperature on free vibration characteristics. Tables 5, 7, 9, and 11 show the results of the effect of temperature on the first five mode shapes for different

Studies on Dynamic Characteristics of Sandwich Functionally …

49

Table 4 Comparison of natural frequencies for (1-8-1) sandwich FGP with CCCC boundary condition Power law

Tcr (°C)

Uniform temp rise

First mode

Second mode

Third mode

Fourth mode

Fifth mode

0.5

101.7

0*Tcr

635.4

1115.1

1452.8

1889.2

1894.7

0.25*Tcr

552.5

1020

1359.2

1790.2

1794.5

0.5*Tcr

453.2

914.3

1258

1685.1

1688

0.75*Tcr

322.2

793.6

1147.3

1572.4

1574.1

1

10

107.8

162.3

0.99*Tcr

64.8

655.9

1029.2

1455.6

1456.2

0*Tcr

658.4

1155.9

1506.4

1959.5

1965.1

0.25*Tcr

572.5

1057.3

1409.3

1857

1861.2

0.5*Tcr

469.6

947.8

1304.6

1748.1

1751

0.75*Tcr

333.8

822.8

1189.9

1631.5

1633

0.99*Tcr

67.2

680.2

1067.6

1510.6

1510.9

0*Tcr

755.6

1326.8

1729.3

2249.8

2256.2

0.25*Tcr

657

1213.6

1617.9

2132.1

2136.9

0.5*Tcr

538.9

1087.9

1497.6

2007

2010.3

0.75*Tcr

383

944.4

1365.9

1873.1

1874.9

0.99*Tcr

76.9

780.7

1225.5

1734.2

1734.6

Table 5 Mode shapes variation of sandwich FGP with uniform temperature rise by CCCC (1-8-1) Uniform temp

First mode

Second mode

Third mode

Fourth mode

Fifth mode

0*Tcr

658.4

1155.9

1506.4

1959.5

1965.1

947.83

1304.6

1748.1

1751

680.2

1067.6

1510.6

1510.9

0.5*Tcr

0.99*Tcr

67.2

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E. Amarnath and M. C. L. Babu

Table 6 Comparison of natural frequencies for (1-8-1) sandwich FGP with CCFC boundary condition Power law

Tcr (°C)

Uniform temp rise

First mode

Second mode

Third mode

Fourth mode

Fifth mode

0.5

113

0*Tcr

494.7

700.1

1177

1327.2

1552.9

0.25*Tcr

435

636.7

1114.4

1248.3

1471.2

0.5*Tcr

361.7

565.8

1047.4

1161.7

1384.7

0.75*Tcr

261.4

484.5

975.2

1065

1292.6

1

10

119.8

180.3

0.99*Tcr

53.4

391.4

899.8

959.5

1197.8

0*Tcr

512.5

725.5

1220.1

1376

1610.4

0.25*Tcr

450.7

659.8

1155.3

1294.3

1525.7

0.5*Tcr

374.8

586.4

1085.9

1204.6

1436.2

0.75*Tcr

270.8

502.2

1011.1

1104.5

1340.8

0.99*Tcr

55.4

405.8

933.1

995.2

1242.6

0*Tcr

588.2

832.7

1400.7

1579.6

1848.8

0.25*Tcr

517.3

757.3

1326.2

1485.8

1751.6

0.5*Tcr

430.1

673

1246.6

1382.8

1648.8

0.75*Tcr

310.7

576.4

1160.7

1267.8

1539.3

0.99*Tcr

63.5

465.8

1071.1

1142.5

1426.5

Table 7 Mode shapes variation of sandwich FGP with uniform temperature rise by CCFC (1-8-1) Uniform temp First mode

Second mode

Third mode

Fourth mode

Fifth mode

0*Tcr

512.5

725.57

1220.1

1376

1610.4

0.5*Tcr

374.8

586.4

1086

1204.6

1436.2

0.99*Tcr

55.9

405.9

933.1

995.3

1242.7

boundary conditions. Results show the movement of nodes and antinodes is not much significant for CCCC boundary conditions under the influence of temperature. This may be due to its symmetry boundary conditions about x-direction and y-direction. However, this is not true for CCFC boundary condition, first and fourth

Studies on Dynamic Characteristics of Sandwich Functionally …

51

Table 8 Comparison of natural frequencies for (1-8-1) sandwich FGP with FCFC boundary condition Power law

Tcr (°C)

Uniform temp rise

First mode

Second mode

Third mode

Fourth mode

Fifth mode

0.5

111.9

0*Tcr

472.5

530.8

758.9

1234.5

1300.8

0.25*Tcr

412.3

480.5

715.3

1198.9

1222.5

0.5*Tcr

339.5

423.3

668.9

1137.6

1162

0.75*Tcr

242.3

355.8

618.9

1044.5

1123.8

1

10

118.6

178.6

0.99*Tcr

48.8

274.1

566.9

944.9

1077.3

0*Tcr

489.5

550

786.6

1279.8

1348.6

0.25*Tcr

427.2

497.9

741.5

1242.9

1267.5

0.5*Tcr

351.7

438.7

693.3

1179.6

1204.7

0.75*Tcr

251

368.7

641.6

1083.1

1165.1

0.99*Tcr

50.6

284.1

587.7

980.03

1117.4

0*Tcr

561.8

631.3

902.9

1469.3

1548.2

0.25*Tcr

490.3

571.5

851.1

1426.9

1455

0.5*Tcr

403.7

503.5

795.9

1354.1

1383.1

0.75*Tcr

288.1

423.2

736.5

1243.3

1337.6

0.99*Tcr

58.2

326.1

674.7

1125

1282.8

modes, antinodes are moving toward the clamped edge from the free edge, and for second, third, and fifth modes, no significant effects of shifting occur as seen from Table 7. Similarly, for FCFC boundary condition, it has been observed that second and third modes are not affected much than other modes. Antinodes are moving toward the center from the free boundaries for the first mode, and the modes are swapped between fourth and fifth modes when the temperature is increased which can be seen from Table 9. Nodes and antinodes of CFFC boundary condition are moving toward the fixed edge corner from the free edge corner for the first mode. This may be due to the free expansion of adjacent free edges. However, one cannot observe a similar variation for other modes except minor shifting.

6 Conclusion The effect of uniform temperature field on dynamic characteristics of rectangular sandwich plate with functionally graded core material is analyzed using finite element methods. Critical buckling temperature and free vibration characteristics of sandwich FGP for temperature rise, material distribution, and different boundary conditions (CCCC, CCFC, FCFC, and CFFC) are studied in detail. It is clear from the results when the power law increases, natural frequencies are increased irrespective of applied temperature. The natural frequencies are decreased when the applied

First mode

489.58

351.7

50.6

Uniform temp

0*Tcr

0.5*Tcr

0.99*Tcr

284.1

438.7

550

Second mode

587.7

693.3

786.6

Third mode

Table 9 Mode shapes variation of sandwich FGP with uniform temperature rise by FCFC (1-8-1)

980

1179.6

1279.8

Fourth mode

1117.4

1204.7

1348.6

Fifth mode

52 E. Amarnath and M. C. L. Babu

Studies on Dynamic Characteristics of Sandwich Functionally …

53

Table 10 Comparison of natural frequencies for (1-8-1) sandwich FGP with CFFC boundary condition Power law

Tcr (°C)

Uniform temp rise

First mode

Second mode

Third mode

Fourth mode

Fifth mode

0.5

256.2

0*Tcr

120.4

368.5

512.6

820.3

900.6

0.25*Tcr

113.4

349.2

496.7

786.3

870

0.5*Tcr

103.3

322.8

476.3

748.2

832.6

0.75*Tcr

85.4

284.5

449.3

705.8

785.7

1

10

271.6

408.8

0.99*Tcr

21.8

227.4

413.4

660.6

729

0*Tcr

124.7

381.8

531.1

850.2

933.2

0.25*Tcr

117.5

361.8

514.5

815

901.5

0.5*Tcr

107

334.5

493.5

775.6

862.8

0.75*Tcr

88.4

294.8

465.5

731.7

814.3

0.99*Tcr

22.6

235.7

428.4

685

755.6

0*Tcr

143.2

438.2

609.5

976

1071.3

0.25*Tcr

134.8

415.2

590.5

935.6

1034.9

0.5*Tcr

122.7

383.9

566.4

890.4

990.3

0.75*Tcr

101.4

338.3

534.3

839.9

934.7

0.99*Tcr

25.9

270.5

491.8

786.3

867.4

Table 11 Mode shapes variation of sandwich FGP with uniform temperature rise by CFFC (1-8-1) Uniform temp

First mode

Second mode

Third mode

Fourth mode

Fifth mode

0*Tcr

124.7

381.8

531.1

850.2

933.2

0.5*Tcr

107

334.5

493.5

775.6

862.8

0.99*Tcr 22.6

235.7

428.4

685

755.6

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uniform temperature approaches critical buckling temperature. Also, the influences of temperature rise on mode shapes are studied for different boundary conditions. Results showed that the temperature rise significantly altered the mode shape patterns. Among all boundary conditions considered, FCFC showed much variation in the shifting of antinodes and nodes under the influence of temperature.

References 1. Gasik MM (1998) Micromechanical modelling of functionally graded materials. Computat Mater Sci 13:42–55 2. Birman V, Byrd LW (2007) Modeling and analysis of functionally graded materials and structures. Appl Mech Rev 60:195–216 3. Udupa G, Shrikantha Rao S, Gangadharan KV (2014) Functionally graded composite materials: an overview. Procedia Mater Sci 5:1291–1299 4. Watanabe R (1995) Powder processing of functionally gradient materials. MRS Bulletin, 32–34 5. Thai H-T, Kim S-E (2015) A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct 128:70–86 6. Zhao X, Lee Y, Liew KM (2009) Mechanical and thermal buckling analysis of functionally graded plates. Compos Struct 90:161–171 7. Kiani Y, Bagherizadeh E, Eslami MR (2011) Thermal and mechanical buckling of sandwich plates with FGM face sheets resting on the Pasternak elastic foundation. J Mech Eng 226:32–41 8. Zenkour AM, Mashat DS (2010) Thermal buckling analysis of ceramic-metal functionally graded plates. Natural Sci 2(9):968–978 9. Bouhadra A, Benyoucef S, Tounsi A, Bernard F, Bouiadjra RB, Ahmed Houari MS (2015) Thermal buckling response of functionally graded plates with clamped boundary conditions. J Thermal Stresses 38:630–650 10. Reza Eslami M (2018) Buckling and postbuckling of beams, plates, and shells. Springer Nature 11. Bhagat V, Jeyaraj P, Murigendrappa SM (2015) buckling and free vibration characteristics of a uniformly heated isotropic cylindrical panel. Procedia Eng 144:474–481. 12th International conference on vibration problems, ICOVP 2015 12. Mayandi K, Jeyaraj P (2013) Bending, buckling and free vibration characteristics of FG-CNTreinforced polymer composite beam under non-uniform thermal load. J Mater: Design Appl 229:13–28 13. Jeyaraj P, Padmanabhan C, Ganesan N (2011) Vibro-acoustic behavior of a multilayered viscoelastic sandwich plate under a thermal environment. J Sandwich Struct Mater 13:509–537 14. Li Q, Iu VP, Kou KP (2008) Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vibr 311:498–515 15. Chandra N, Raja S, Nagendra Gopal KV (2014) Vibro-acoustic response and sound transmission loss analysis of functionally graded plates. J Sound Vib 333: 5786–5802 16. Jeyaraj P, Padmanabhan C, Ganesan N (2008) Vibration and Acoustic response of an isotropic plate in a thermal environmental. J Vibr Acoust 130:1–6

Numerical Model and Dynamic Characteristics Analysis of Cable-Stayed Bridge Antara Banerjee and Atanu Kumar Dutta

Abstract The aim of this work is to numerically model a cable-stayed bridge, study its dynamic characteristics using two engineering tools viz. MATLAB® and SAP2000® and compare the results with those from the experimental analysis. The bridge considered for the study is a two-span cable-stayed bridge over the Garigliano River along the coast road between Rome and Naples. The structural details including material or sectional properties, boundary conditions and experimental responses are adopted [2]. It is observed that the present numerical model captures the experimental data well with minor discrepancies, which can be attributed to lack of proper data in numerical modelling. The finite element model can be further improved with the access of more structural data and can be effectively used for the determination of dynamic response due to earthquake and wind. Keywords Cable-stayed bridge · Finite element model · Dynamic characteristics · MATLAB® · SAP2000® · Dynamic response

1 Introduction 1.1 General The first modern cable-stayed bridge construction dates back to the year 1955 at Stromsund in Sweden. Thereafter, cable-stayed bridges due to its beauty and highly efficient use of materials are gaining popularity in the practical field of bridge engineering all over the world. This type of bridge for relatively long spans was often found to be more structurally efficient and economical in comparison to other types of bridges. It mainly comprises of tower, deck, girder and cables. The tower spread out A. Banerjee (B) Structure Engineering, Jorhat Engineering College, Jorhat, India e-mail: [email protected] A. K. Dutta Jorhat Engineering College, Jorhat, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_6

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straight cables which are tightly anchored on the deck. Cable-stayed bridges are prone to the exposure of dynamic actions like earthquake, wind and traffic induced vibration which are considered equivalent static loads in the design of bridge for simplicity [1]. The complex nature of dynamic response of these bridges under dynamic loads induces the necessity to look into the actual dynamic behaviour which is again defined mainly by the dynamic characteristics, i.e. Mode shapes, natural frequencies and damping ratios of the bridge based on the bridge data and the local environmental hazards.

1.2 Objective of the Study The main objective of this study can be enumerated as stated below: i.

To model a 3-D finite element model of a cable-stayed bridge and determine its dynamic properties using the software SAP 2000® . ii. To model a 3-D finite element model of the same cable-stayed bridge and determine its dynamic properties using the software MATLAB® iii. To compare both the above obtained analytical results with the experimental results given in [2]

2 Description of the Structure 2.1 Introduction The bridge considered here for the study is a two-span cable-stayed bridge over the Garigliano River, along the coast road between Rome and Naples. The structural details including material or sectional properties and experimental responses are provided here, adopted from the detailed description given in [2]. The view of the Garigliano cable-stayed bridge is shown in Fig. 1.

2.2 Layout Details The length of the bridge is 180 m, with each span being 90 m. The width of the deck is 26.1 m. The pre-stressed pre-fabricated concrete box girder is simply supported at both ends and fully constrained with the tower. The girder is suspended by 18 couples of cables with 9 couples of cables for each span, starting at a different section of the pylon but not evenly spaced along the deck. The cables are spaced 0.85 m from the centreline of the cross-section of the bridge. The height of the pylon is 10.85 m from the foundation to the deck extrados and 30 m from the deck. The lower part from the

Numerical Model and Dynamic Characteristics Analysis …

57

Fig. 1 Garigliano Cable-Stayed Bridge, Italy, Rome

foundation to 5 m above the deck is made of concrete, and the remaining part is a steel box beam. The view and plan of the bridge are shown in Fig. 2, whereas the cross-section of girder is shown in Fig. 3.

Fig. 2 View and plan of the bridge

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Fig. 3 Cross-section of the girder

Table 1 Cross-sectional properties of the deck or girder Cross-sectional area, A (m2 )

Ix (m4 )

Iy (m4 )

Torsion moment of inertia, J (m4 )

16.52

13.3

650.7

30.0

Table 2 Cross-sectional properties of the tower

Table 3 Cross-sectional areas of the cable

A (m2 )

Iz (m4 )

Ix (m4 )

0 < y < 10.85

23.66

178.37

11.68

10.85 < y < 15.85

10.80

16.70

5.60

15.85 < y < 25.85

0.75

1.00

0.80

y > 25.85

0.81

0.833

0.677

Y

Cable

A (cm2 )

A and B

67.5

C to H

82.5

I

70.5

2.3 Geometrical Properties Geometrical properties of the structure are adopted from [2]. The cross-sectional properties of the deck, tower and cables are listed in Tables 1, 2 and 3.

2.4 Material Properties Material properties of the structure are adopted from [2]. The material properties of the deck, tower and cables, i.e. elastic longitudinal and tangential moduli of materials are listed in Table 4.

Numerical Model and Dynamic Characteristics Analysis … Table 4 Elastic moduli of the materials

59

Structural element

E (MPa)

G (MPa)

Girder

42,200

17400

Cables

200,000

77000

Concrete

36000

15600

Steel

200000

77000

Tower

2.5 Boundary Conditions Boundary conditions of the structure are adopted from [2]. The girder is fully fixed to the tower. The girder is simply supported at two other ends where torsional rotation isn’t allowed. The tower is fixed at the foundation.

3 Modal Analysis The dynamic response characteristics depend upon the vibration characteristics of a structure. For a proper study of the dynamic behaviour of a cable-stayed bridge, a modal analysis of the bridge is essential. The cable-stayed bridge structure consists of a system with a continuous distribution of stiffness and mass. In the structural analysis of any cable-stayed bridge, modelling is important as the model, if can simulate or reflect the stiffness and mass of the actual structure accurately, might seriously affect the accuracy of the calculation. Mass, structural stiffness and boundary conditions consideration are essential in the modal analysis as because these are related to the structural characteristics of the cable-stayed bridge. Thus, the structure has to be divided into finite elements for its better and comprehensive study. The results are not only reliable but also easier to determine than by using traditional empirical formulas. The dynamic equilibrium equation for the model is as follows: ◦   ¨ [M] U + [C] U + [K]{U} = {P(t)} where [M] = mass matrix of the structure [C] = damping matrix of the structure [K] = stiffness matrix of the structure {U} = displacement vector of each node ˚ = velocity vector of each node { U} {Ü} = acceleration vector of each node Ignoring the resistance, we get the equation as below:

(1)

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A. Banerjee and A. K. Dutta

  [M] U¨ + [K]{U} = 0

(2)

Taking [U(t)] = {} sinωt, then solving the differential equation, we get   [K] − ω2 [M] {Φ} = 0

(3)

Thus, we get natural frequencies of the system ωi (i= 1, 2, 3…N) and the vibration modals {i }(i= 1, 2, 3…N) of the structure from the above equations.

4 Modelling and Studying the Dynamic Characteristics of Garigliano Cable-Stayed Bridge The modelling and analysis of the cable-stayed bridge is performed using SAP2000® and MATLAB®. Since Finite Element Method package is not provided by MATLAB® so to model and analyse the bridge, the method adopted in this regard is from a formulation, i.e. the toolbox given by Dr. Juan Martin Caicedo Washington University in Saint Louis Structural Control and Earthquake Engineering Laboratory. It was created in January 2002. A 3-D finite element model of Garigliano cable-stayed bridge is developed using SAP2000® and MATLAB® and the discretised model is shown in Figs. 4 and 5. The bridge is modelled based on the information obtained regarding section properties, material data and support conditions from Clemente et.al [2]. The deck is modelled as single spine employing beam elements for girder where cross-sectional area A = 16.52 m2 , flexural moments of inertia Iy = 13.3 m4 ,

Fig. 4 Finite Element Model in SAP 2000®

Numerical Model and Dynamic Characteristics Analysis …

61

Fig. 5 Finite Element Model using MATLAB®

Ix = 650.7 m4 and torsion moment of inertia J = 30.0 m4 . The nodes are taken at certain intervals along the spine as well as at the cable ends that are connected to the spine through rigid links where cable ends are at a distance of ± 0.85 m from the centre line. The spine nodes are considered as the master nodes and the nodes connecting the cables to the spine via rigid links are considered as the slave nodes for the evaluation of stiffness. The basic unknowns are determined at the nodes or at any point inside the element in terms of the nodal values of the element. In MATLAB® , the distribution of lumped masses both translational and rotational mass is done along the spine. The translational mass as concentrated mass lumped along the spine takes into consideration the cross-sectional area of the deck and the density of the concrete surface for the defined spine nodes as mentioned above. The rotational mass moment of inertia is calculated using the equation Imξ =

n 

(Imi + m i ri2 )

(4)

i=1

where I mξ = actual mass moment of inertia of deck between the cable anchor points about ξ axis (x, y or z axis) Imi = mass moment of inertia of ith element about its own centroidal axis mi is elemental mass and r i is radial distance of ith mass from the centre of rotation. The cables are modelled using linear elastic truss elements where cable sag is considered negligible. Each cable is modelled using a single truss element. The cable

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Fig. 6 Line diagram of the bridge with cable type

element is restrained so that no compression deformation occurs thus simulating its practical condition of the real bridge. The cable at one end is attached to the pylon and on the other end to the deck, both the points being away from the neutral axes of their supporting pylon and deck elements. Therefore, rigid links were used to connect the cables to the neutral axis of the pylon and the deck. Cables are not considered to be pre-tensioned. Since the mass density of cables and cable tension data is not available so in MATLAB® , the mass density of cable is assumed as that of steel, i.e. 7850 kg/m3 . The tower is modelled using three-dimensional beam elements with nodes introduced at locations of cable attachments and sectional or material changes. The torsional moment of inertia of tower is not available and thus the modelling in MATLAB® is done by assuming a value which is actually the summation of Iy and Ix . The boundary conditions are taken as mentioned in Clemente et.al [2]. The girder is fixed fully to the tower and the two ends of the girder are simply supported where torsional rotation is not allowed. The tower at its base is fixed at the foundation w.r.t all displacement components. The lower part of tower from foundation to 5 m above deck is assigned concrete, whereas the upper part of tower is assigned steel. The girder is assigned concrete (Figs. 6 and 7).

5 Observation and Results The natural frequencies of the first six significant vibration modes and the shapes of six modes obtained from modelling the structure in SAP2000® and MATLAB® are presented in Figs. 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19.

Numerical Model and Dynamic Characteristics Analysis …

63

Fig. 7 Nodes defined for modelling the bridge

6 Comparison of Results A comparison of the results obtained using the two engineering tools, i.e. SAP2000® and MATLAB® with that of the experimental analysis is presented below in the following tables which shows the frequencies and mode shapes of Garigliano cablestayed bridge in its six vertical modes (Tables 5 and 6).

64

Fig. 8 Mode 1—Frequency = 1.02 Hz

Fig. 9 Mode 2—Frequency = 1.07 Hz

A. Banerjee and A. K. Dutta

Numerical Model and Dynamic Characteristics Analysis …

65

Fig. 10 Mode 3—Frequency = 1.71 Hz

7 Conclusion In this study, Garigliano cable-stayed bridge is modelled and its dynamic characteristics are studied using MATLAB® and SAP2000® with the available geometrical and material properties from the standard literature. An insight is obtained in finite element formulation of this complicated structure and a comparison also has been drawn from the results obtained using the two engineering tools with those from the experimental analysis of established literature. It is observed that the present numerical model captures the experimental data well with minor discrepancies in MATLAB® , which can be attributed to lack of proper data in numerical modelling. The model can be further improved with the access of more structural data and can be effectively used for the determination of dynamic response due to earthquake and wind.

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Fig. 11 Mode 4—Frequency = 1.99 Hz

Based on the results of this study, the following conclusions are drawn: • Calculation of rotational mass moment of inertia is important as this leads to the torsional mode of vibration of the structure • SAP2000® is unable to reflect the torsional behaviour of the structure as the rotational mass moment of inertia cannot be imparted to the spine model in SAP2000® • It has been observed that from the point of capturing modal characteristics as well as being a less expensive finite element model with less number of elements and degrees of freedom, the numerical modelling in MATLAB® excels over that of in SAP2000®

Numerical Model and Dynamic Characteristics Analysis …

Fig. 12 Mode 5—Frequency = 2.12 Hz

Fig. 13 Mode 6—Frequency = 2.34 Hz

67

68

Fig. 14 Mode 1 (Frequency = 0.94 Hz)

Fig. 15 Mode 2 (1.35 Hz)

A. Banerjee and A. K. Dutta

Numerical Model and Dynamic Characteristics Analysis …

Fig. 16 Mode 3 (2.28 Hz)

Fig. 17 Mode 4—Frequency = 2.89 Hz

69

70

Fig. 18 Mode 5—Frequency = 3.02 Hz

Fig. 19 Mode 6—Frequency = 3.16 Hz

A. Banerjee and A. K. Dutta

Numerical Model and Dynamic Characteristics Analysis …

71

Table 5 Comparison of experimental and analytical frequency and mode shape results Mode

Results of Modal frequency (Hz) and mode shapes Experimental

SAP 2000®

MATLAB®

1

0.90 Hz, flexural

0.94 Hz, flexural

1.02 Hz, flexural

2

1.30 Hz, flexural

1.35 Hz, flexural

1.07 Hz, flexural

3

2.50 Hz, Torsional mode but tower moves in transverse direction

2.28 Hz, tower moves in transverse direction

1.71 Hz, Torsional mode but tower moves in transverse direction

4

–

2.89 Hz, flexural

1.99 Hz, flexural

5

2.69 Hz, torsional

3.02 Hz, flexural

2.12 Hz, torsional

6

2.78 Hz, torsional

3.16 Hz, flexural

2.34 Hz, flexural

Table 6 Comparison of Finite Element Model characteristics

Finite Element Model characteristics

SAP 2000®

MATLAB®

Number of nodes

790

93

Number of degrees of freedom

2880

558

Number of elements

1583 beam elements 585 cable elements 240 rigid links

52 beam elements 36 cable elements 40 rigid links

The model in MATLAB® can be used for the determination of dynamic response due to wind and earthquake. The model can further be used to find a control strategy against such excitations and also can help in long-term health monitoring of the structure.

References 1. Malla S (1988) Safety and Reliability of the cable system of a cable stayed bridge under stochastic earthquake loading. Asian Institute of Technology, Bangkok, Thailand 2. Clemente P, Marulo F, Lecce L, Bifulco A (1998) Experimental modal analysis of the Garigliano cable-stayed bridge. Elsevier Soil Dyn Earthq Eng 17(1998):485–493 3. Wilson JC, Gravelle W (1991) Modelling of Cable stayed bridge for dynamic analysis. Earthq Eng Struct Dyn 20:707–721 4. Hua C-H, Wang Y-C (1995) Three dimensional modelling of a cable stayed bridge for dynamic analysis 5. Clemente P, Çelebi M, Bongiovanni G, Rinaldis D (2004) Seismic analysis of Indiano cable stayed bridge. In: 13th world conference on earthquake engineering, Vancouver Canada

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6. Macdonald JHG, Daniell WE (2005) Variation of modal parameters of a cable-stayed bridge identified from ambient vibration measurements and finite element modelling. Eng Struct Elsevier 27:1916–1930 7. Chen G-D, Wang W-J (2006) Three dimensional finite element model of the Bill Emerson cable stayed bridge and its calibration with field measured data. In: 4th International conference on earthquake engineering, Taiwan 8. Wei L, Cheng H, Li J (2012) Modal analysis of a cable stayed bridge. In: International conference on advances in computational modeling and simulation Elseiver 9. Krishnamurthy CS, Tata Mc-Graw-Hill Publishing Company Ltd (2001) Finite element analysis: theory and programming, 2nd Edn, New Delhi 10. Chandrupatla TR, Belegundu AD (2002) Introduction to finite elements in engineering, 3rd edn. Prentice Hall Inc., New Jersey

Breathing Crack Detection Using Linear Components and Their Physical Insight J. Prawin and A. Ramamohan Rao

Abstract Identification of fatigue-breathing cracks at the time of initiation that develop in structures under repetitive loading is desirable for successful implementation of any health monitoring system. The presence of higher order harmonics and sidebands apart from the fundamental excitation harmonic in the Fourier power spectrum subjected to harmonic excitation are widely used as breathing crack damage indicators. The majority of the existing breathing crack detection and localization techniques use the amplitudes of the super harmonics/modulation components obtained spatially across the structure from the current and healthy states. In the present work, instead of using nonlinear harmonic components for breathing crack detection, we exploit the decrease in the Fourier power spectrum amplitude of the linear components due to the transfer of energy from linear components to nonlinear harmonic components in the presence of breathing crack. Two new indices quantifying the ratio of energy variations in linear and nonlinear components have been proposed to highlight the effectiveness of the proposed concept. Numerical simulation and experimental investigations established the fact that the energy variation in linear components between varied crack depths shows significantly higher sensitivity than the energy variation due to nonlinear harmonic components. Keywords Breathing crack · Nonlinear · Intermodulation · Super-harmonics · Bitone

J. Prawin (B) · A. Ramamohan Rao Structural Health Monitoring Group, CSIR Structural Engineering Research Centre, Chennai, India e-mail: [email protected] A. Ramamohan Rao e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_7

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1 Introduction Detection of incipient damage at the earliest possible stage is desirable for successful implementation of any health monitoring system. The fatigue-breathing crack is one of the most common damages that occur in civil structures due to dynamic loading. These fatigue cracks exhibit repetitive crack open-close breathing like phenomenon and induce bilinear rather nonlinear behaviour. The bilinear dynamic behaviour of the structure with breathing cracks makes them difficult to detect when compared to their counterpart open cracks, using the traditional vibration-based damage detection techniques. Therefore, the detection, localization and severity quantification of fatigue cracks using nonlinear sensitive features poses a greater challenge and new damage diagnostic techniques are being developed. [1–4]. When the structure is healthy (i.e. intact and linear), and subjected to harmonic loading with excitation frequency ωexc , it vibrates at same excitation frequency. In contrast, the cracked response contains not only input excitation frequency ωexc , but also their super harmonics (nωexc , where n is an integer). Similarly, when the system is excited simultaneously with two different frequencies ωprob and ωpump (bitone: ωprob > ωpump ), the cracked response contains not only input frequencies but also their harmonics and modulation. These modulations lead to the presence of side bands (ωprob ± nωpump ,). The presence of super harmonics (i.e. higher order harmonics) and sidebands apart from fundamental excitation harmonics confirms the bilinear behaviour of the structure due to breathing crack. The majority of the existing breathing crack detection and localization techniques are based on the comparison of the amplitudes of the first one or two super harmonics and sidebands components obtained spatially across the structure from the current and healthy states [1–4]. Further, these nonlinear harmonics (i.e. higher order or super harmonics) rather depends heavily on the spatial location and the depth of the crack, excitation frequency and as well the amount of damping. However, extracting these super harmonics and sidebands may mislead the damage diagnostic process due to highly corrupted noisy measurements and environmental variability. It may be noted here that the Fourier power spectrum amplitude of the linear components is of the higher order of magnitude than that of the higher order harmonics and sideband components. Hence, alternatively in this paper, instead of using nonlinear harmonic components for breathing crack detection, we exploit the decrease in Fourier power spectrum amplitude of the linear components due to the transfer of energy from linear components to nonlinear harmonic components in the presence of breathing crack.

Breathing Crack Detection Using Linear Components …

75

2 Breathing Crack Detection Using Linear Harmonic Components The uniqueness of the present study is based on the identification of breathing crack using the nonlinearity reflected on linear responses rather than the nonlinear harmonic/modulation components being used popularly. The presence of breathing crack causes reduction in Fourier power spectrum amplitude of the linear responses and increase in Fourier power spectrum amplitude of higher order harmonics and sidebands. Therefore, there occurs the transfer of energy from the fundamental linear excitation harmonic component to higher order superharmonic and sideband harmonics components due to nonlinear behaviour of the structure in the presence of breathing crack. Therefore, the energy decrease in the linear component response is definitely larger than the energy increase in the super harmonic and sideband components generated due to nonlinearity. First, a damage index proposed based only on the linear components of the healthy and cracked structure response is defined as D Ilinear

  n  cracked   Ai,linear   = 1 − healthy /n;  Ai,linear  i=1

(1)

cracked where Ai,linear is the response amplitude of linear harmonics of ith degree of freedom of the structure with breathing crack and n indicates the total number of degree healthy of freedom. Similarly Ai,linear corresponds to the healthy structure. The value of the damage index lies in the range 0 ≤ D Ilinear ≤≈ 1. The value of D Ilinear corresponding to 0 and close to 1 indicate the healthy and the completely damage state of the structure. The value of D Ilinear increases with increase in crack depth. It can be easily explained by the fact that due to the generation of nonlinear harmonics or modulations, there will be reduction in the amplitude/energy of linear components in the case of cracked structure when compared with the healthy structure. The energy reduction occurs due to transfer of energy from linear component to nonlinear harmonic/modulation components in the case of structure with breathing crack. It may be noted here that the amplitude of linear harmonics includes both amplitudes of probing and pumping frequencies in the case of structure subject to simultaneous excitation of two different frequencies. In order to demonstrate that the linear components are more sensitive to breathing crack than nonlinear harmonics or intermodulation components, a damage index based on the ratio of the energy of amplitude of nonlinear harmonics to linear harmonics, widely used by the researchers is used [1–4]. It is defined as

D Iharmonics/modulations =

n cracked  Ai,nonlinear i=1

cracked Ai,linear

/n.

(2)

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3 Numerical Investigations The steel cantilever beam, considered by popular researchers earlier [1, 4] for identification of breathing crack in engineering structures is used in the present work. We consider the same dimensions as the length of the beam as 0.3 m and the area of the beam as 2.5e−4 m2 . The finite element discretization of the cantilever beam along with the node numbers and degrees of freedom per node is shown in Fig. 1. Standard 1D Euler-Bernoulli beam elements are employed to model the beam and shear deformation is neglected. There are two nodes per element and each node has three degrees of freedom: longitudinal displacement, translational displacement and bending rotation as shown in Fig. 1. The bilinear behaviour of the breathing crack is shown in Fig. 2. Rayleigh damping is considered. The beam is idealized with 10 elements. The fundamental frequencies of the uncracked beam are found to be 92.172, 577.650, 1617.8 and 3172.4 Hz. The damaged element stiffness matrix of a plane beam element, with the breathing crack simulated by the Heavy side step function is given below   H θi − θ j  = 1, θi > θ j H θi − θ j = 0, θi < θ j ⎤ 0 0 6Iu ⎥ u − 12I ⎥ l2 l ⎥ 6Iu (Iu − Id ) − l 2Iu ⎥ ⎥, μ = 0 0 ⎥ Iu 12Iu 6Iu ⎥ ⎦ − 2 l l 4Iu

  K d = K u − H θi − θ j K c : ⎡

0 0 0 0 ⎢ 12Iu 6Iu 0 ⎢ l2 l μE ⎢ 4Iu 0 ⎢ Kc = ⎢ 0 l ⎢ ⎢ ⎣ sym



(3)

(4)

where E, L, K, I, H indicates the Young’s Modulus, span, element stiffness, moment of inertia and Heaviside step function. The subscripts ‘u’ and ‘d’ indicate the

Fig. 1 Finite element model of the cracked Cantilever beam

Breathing Crack Detection Using Linear Components …

77

Fig. 2 Bilinear behaviour of breathing crack

undamaged and damaged states of the structure. The index ‘μ’ represents the nondimensional flexural damage [3, 5]. The acceleration time history responses before post-processing are polluted with 15% noise level (i.e. SNR = 25) to test the applicability of the proposed approach in the presence of noise. The system is excited at free end with 90 Hz excitation and as well as bitone with frequencies of 90 and 1710 Hz. The breathing crack is induced in element no.5 and three different breathing crack depths considered for investigation are about 7%, 20% and 41% of the overall depth of the beam (i.e. with non-dimensional flexural damage μ = 0.2, 0.5and 0.8). Three different crack depths have been considered to demonstrate the sensitivity of the above proposed two damage indices. The free end power spectrum response of the healthy and damaged beam subjected to harmonic excitation of 90 Hz is shown in Fig. 3. Similarly, the Fourier Power spectrum responses of the structure measured at free end when excited simultaneously with two different frequencies of 90 and 1710 Hz is shown in Fig. 4. While Subplots (a) of Figs. 3 and 4 indicate the complete Fourier power spectrum and (b) indicate the zoomed power spectrum. The following are the observations 1. The Fourier power spectrum shows peaks with high amplitude only at excitation frequency (i.e. 90 Hz in case of harmonic excitation with single frequency and 90 Hz and 1710 with respect to simultaneous excitation of structure with two frequencies) for the healthy beam. 2. The Fourier power spectrum of the cracked structure, shown in Fig. 3 shows peaks at higher order harmonics of 90 Hz (i.e. 180, 270, 360 Hz and so on apart from excitation frequency. 3. The Fourier power spectrum of the response shows more super harmonics with the increase in crack depth

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J. Prawin and A. Ramamohan Rao 4

5

1x10 healthy

cracked (mu=0.5)

healthy

cracked (mu=0.5)

cracked (mu=0.2)

cracked (mu=0.8)

cracked (mu=0.2)

cracked (mu=0.8)

Amplitude

Amplitude

1x10

1

1x10

0

-3

1x10

2

1x10

0

100

200

300

400

1x10

500

80

120

Frequency (Hz)

160

200

240

280

320

Frequency (Hz)

(a) Complete spectrum

(b) zoomed spectrum

Fig. 3 Power Spectrum— single tone—numerical cantilever beam 5

5

1x10 healthy

cracked (mu=0.6)

healthy

cracked (mu=0.5)

cracked (mu=0.3)

cracked (mu=0.9)

cracked (mu=0.2)

cracked (mu=0.8)

Amplitude

Amplitude

1x10

1

1x10

-3

-3

1x10

1

1x10

0

600

1200

1800

2400

3000

1x10 1400 1500 1600 1700 1800 1900 2000

Frequency (Hz)

(a) Complete spectrum

Frequency (Hz)

(b) zoomed spectrum

Fig. 4 PowerSpectrum—bitone— numerical cantilever beam

4. The Fourier power amplitude of super harmonics increases and amplitude of linear harmonics decreases with increase in crack depth. 5. Overall, irrespective of any crack depth, the amplitude of higher order harmonics is of less order in magnitude than that of the linear harmonic component. Further, the Fourier power spectrum amplitude of higher order harmonics is of very low

Breathing Crack Detection Using Linear Components …

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Table 1 Energy Variation Indices Harmonic excitation

Single tone

Crack depth (μ)

0.2

0.5

0.8

Bitone 0.2

0.5

0.8

..

0.0371

0.0664

0.275

0.0528

0.0802

0.314

D Iharmonics/modulations

0.0128

0.0255

0.0576

0.0026

0.0082

0.0175

magnitude with respect to the fundamental harmonics even for higher crack depth, which is evident from Fig. 3. 6. The Fourier power spectrum of the cracked structure shown in Fig. 4. shows high amplitudes at two excitation frequencies (i.e. 90 and 1710 Hz) an low amplitude at its corresponding sidebands frequencies (both left and right sidebands) of probing frequency (i.e. 1800, 1890, 1980, 1620, 1530 Hz and so on). 7. The Fourier power spectrum amplitude of both left and sidebands are also of lower magnitude when compared to the Fourier power spectrum amplitude of probing frequency. 8. The presence of super harmonics (i.e. higher order harmonics) and sidebands apart from fundamental excitation harmonics confirms the bilinear behaviour of the structure due to breathing crack. The frequency domain responses shown in Figs. 3 and 4 indicate that there is amplitude reduction in linear components and amplitude increase in nonlinear harmonic/modulation components with increase in crack depth for both single as well as bitone excitation. The results of the damage indices are presented in Table 1. For the crack depth of 41%, the energy variation in the case of harmonic excitation with single frequency is found to be 27.5% for damage index based on linear components alone, while it is around 5.76% for damage index based on nonlinear harmonics. Similarly, the energy variations corresponding to bitone harmonic excitation are found to be 31.4 and 1.75% for damage index based on linear harmonics and nonlinear harmonics. This clearly demonstrates that the damage index based on reduction in energy of the linear components due to breathing crack is more sensitive than the increase in energy of the nonlinear harmonics/modulations. It can be concluded that the bitone harmonic excitation can be preferred due to their higher sensitivity in contrast to the harmonic excitation with single frequency.

4 Experimental Validation A cantilever aluminium beam with and without breathing crack have been experimentally validated to demonstrate the proposed concept of breathing crack identification using linear harmonic components apart from the numerical investigation. Two aluminium alloy beams (i.e. two, three or many pieces) are bonded to form a single beam in contrast to the single unified beam. Bonding is carried out using Araldite epoxy adhesive. There may be two or three plates at the top depending upon

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the number of crack, where in the faces of the plates are in contact but not bonded to induce breathing behaviour. The faces of the plate in contact helps in simulating opening and closing behaviour of the beam under dynamic loading. Similarly, the healthy beam is formed by only bonding two plates (i.e. bottom and top) instead of a single plate to have equivalent uncracked beam as same as that of cracked beam. The experimental set up in the present work has been followed as per the guidelines by Prime et al. [6] and Douka et al. [7] for validation of their damage diagnostic techniques. The schematic illustration of the instrumentation set up of the PZT accelerometers (PCB393B04) placed longitudinally across the test cantilever beam and experimental specimen is presented in Fig. 5a. The beam is tightly clamped using C- clamp with the steel test bench for cantilever action as shown in Fig. 5b. The Data acquisition and spectrum analyzer used during experimentation is shown in Fig. 5c. The spacing of the accelerometers is given in Fig. 6. The cross section of the beam is 25.4 mm × 9.505 mm and the span is 1000 mm. The distance of the various aluminium plate pieces on the top dictates the spatial location and number of the crack. In the present test setup, the crack is located at 10 mm from the fixed end of the cantilever beam. The thickness of the top plate is 3.175 mm, which results in

(a) Test cantilever Beam setup

(b) Clamped end and crack Fig. 5 Experimental cantilever beam

(c) Data acquisition system

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Fig. 6 Instrumentation Setup

crack depth equal to 33% of total depth of the beam in the case of the first cracked specimen. The second cracked specimen is formed similarly with both top and bottom plates having thickness 4.7525 mm each. This results in crack depth of 50% of total depth of the beam. The other properties of the beam and crack location are same as that of the previous cracked specimen. The natural frequencies (i.e. first four) of the healthy experimental beam are 7.782, 48.77, 136.61, and 267.85 Hz. The beams with and without breathing crack are subjected to harmonic excitation of 115 Hz and as well simultaneously excited with two frequencies of 8 Hz and 145 Hz. The free end power spectrum response of the healthy and the cracked beam subjected to harmonic excitation of 100 Hz is shown in Fig. 7. The results 10000 1000 Healthy

Healthy

33% cracked

33% cracked

50% cracked

100

50% cracked

100

PSD amplitude

PSD amplitude

1000

10 1 0.1 0.01

10

1

1E-3 1E-4

0

200 400 600 800 1000 1200 1400 1600

Frequency (Hz)

(a) Complete spectrum

0.1 90

180

270

Frequency (Hz)

(b) zoomed spectrum

Fig. 7 Power Spectrum—single tone—experimental cantilever beam

360

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10 Healthy 33% cracked 50% cracked

0.1

0.01

1E-3

1E-4

Healthy 33% cracked 50% cracked

1

PSD amplitude

PSD amplitude

1

0.1

0.01

1E-3

0

1E-4 130

45 90 135 180 225 270 315 360 405 450

140

Frequency (Hz)

150

160

170

180

Frequency (Hz)

(a) Complete spectrum

(b) zoomed spectrum

Fig. 8 Power Spectrum—bitone—experimental cantilever beam

corresponding to responses measured under harmonic excitation of two frequencies of 8 and 145 Hz is shown in Fig. 8. The response of the healthy beam shows a single maximum peak at 115 Hz only. While the plot related to damaged beam, (i.e. both 33% and 50% cracked beam) shows peaks at 115, 230, 345 Hz and so on. The cracked beam exhibits more number of higher order harmonics for higher crack depth beam. Further, the amplitude of superharmonics increases and amplitude of linear harmonics decreases with increase in crack depth. The energy of higher order harmonics is very low with respect to the linear harmonic even in the case of beam with higher crack depth. The Fourier power spectrum reveals two peaks at 8 and 145 Hz for the healthy beam which is evident from Fig. 8. While the plot related to the cracked beam shows peaks at sidebands (ωprob ±nωpump ) apart from the peaks at each individual excitation frequencies. Similar to single tone harmonic excitation case, the Fourier power spectrum amplitude of both left and right sidebands is of very low magnitude when compared to the Fourier power spectrum amplitude of probing frequency excitation component. The results of the damage indices estimated for the cracked experimental specimens are given in Table 2. Table 2 establishes the fact that the energy variation in linear components (i.e.D Ilinear ) between varied crack depths shows significantly higher sensitivity than energy variation due to nonlinear components Table 2 Energy Variation Indices Harmonic excitation

Single tone

Crack depth (μ)

0.2

Bitone 0.5

0.2

0.5

D Ilinear

0.03688

0.122

0.062

0.148

D Iharmonics/modulations

0.00824

0.0218

0.00104

0.00785

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(i.e.D Iharmonics/modulations ). This clearly emphasis the fact that the damage index based on reduction in energy of linear components due to breathing crack is more sensitive than increase in energy of nonlinear harmonics/modulations.

5 Conclusions A unique breathing crack damage diagnostic technique exploiting the possibility of utilizing linear components instead of conventional nonlinear components is developed. Numerical investigations on a cantilever beam with breathing crack near the centre of the beam with varied crack depth are carried out to understand the transfer of energy from linear components to super harmonics and sideband components in the presence of nonlinearity due to breathing crack. Two damage indices; one based on linear components and the other considering nonlinear harmonics and modulations/sidebands, have been presented and compared to investigate the sensitivity of the response components for fatigue-breathing crack identification. This is further validated through experimental investigations. Both the numerical and experimental investigations conclude that there occurs transfer of energy from the fundamental linear excitation harmonic components to nonlinear harmonic components induced by the breathing crack. The decrease in Fourier power spectrum amplitude in the linear components due to energy transfer is highly sensitive than the Fourier power spectrum amplitude increase in the case of higher order harmonics and sideband nonlinear harmonic components. Therefore, the linear components can be effectively employed for damage diagnosis of breathing crack. Further, this eliminates the complex process of reliable extraction of nonlinear harmonics under harsh noisy environment. Acknowledgments The authors would like to thank and acknowledge the help received from their colleagues in the ASTAR and SHML laboratories, CSIR-SERC

References 1. Bovsunovsky A, Surace C (2015) Non-linearities in the vibrations of elastic structures with a closing crack: a state of the art review. Mech Syst Signal Process 62–63:129–148 2. Kim GW (2011) Localization of breathing cracks using combination tone nonlinear response. Smart Mater Struct 20:055014 3. Giannini O, Casini P, Vestroni F (2013) Nonlinear harmonic identification of breathing cracks in beams. Comput Struct 129:166–177 4. Broda D, Pieczonka L, Hiarkar V, Staszewski WJ, Silberschmidt VV (2016) Generation of higher harmonics in longitudinal vibration of beams with breathing cracks. J Sound Vib 381:206–219 5. Prawin J, Rama Mohan Rao A (2015) Nonlinear identification of structures using ambient vibration data. Comput Struct 154:116–134

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6. Prime MB, Shevitz BW (1996) Linear and nonlinear methods for detecting cracks in beams. In: Proceedings of 14th International modal analysis conference, pp 1437–1443 7. Douka E, Hadjileontiadis LJ (2005) Time-frequency analysis of the free vibration response of a beam with a breathing crack. Nondestructuctive Testing Eval 38:3–10

Optimal Design of Structure with Specified Fundamental Natural Frequency Using Topology Optimization Kandula Eswara Sai Kumar and Sourav Rakshit

Abstract Resonance occurs when the natural frequency of the system matches with the vibrating frequency. It may cause structural instabilities. To avoid this, engineers maximize the first natural frequency of the system. In many applications, the natural frequency is pre-designed. Structural engineers aim to reduce the weight of structures subject to functional and safety constraints. This motivates us to modify the frequency optimization problem to weight minimization problem, for a specified fundamental natural frequency. In this paper, we solve for weight minimization using topology optimization subject to lower bound constraint on fundamental frequency. Keywords Topology optimization · Eigen frequency · Resonance · Optimum volume fraction · Method of moving asymptotes

1 Introduction The phenomenon of resonance causes structural instabilities and it has to be avoided while design of structures. To avoid resonance, the Eigen frequency is taken as the objective function to maximize, using structural topology optimization [1]. Frequency optimization is importance in designing the structures under dynamics loads. In many engineering applications, the fundamental natural frequency of the structure is pre-designed [2]. In such cases, structural engineers aim to minimize the weight of structures. The structure of minimum weight saves the cost of the material and improves the efficiency when they used in machine components [3]. In this paper, we solve the weight optimization problem for a clamped beam using topology optimization subject to a lower bound constraint on natural frequency, i.e. the fundamental natural frequency of the structure is greater than the specified frequency. K. E. S. Kumar (B) · S. Rakshit Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India e-mail: [email protected] S. Rakshit e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_8

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2 Topology Optimization Formulation 2.1 Problem Definition In this paper, we pose an alternative approach to optimization of structures to avoid the resonance. Minimization of the volume fraction: min V =

N 

ϑe ρe

e=1

  subject to K − ω12 M ∅1 = 0 ωg ≤ ω1

(1)

where K and M are global stiffness and mass matrices, respectively, ω1 is the fundamental natural frequency of the structure, ωg is the specified frequency value, ∅1 is the Eigen vector corresponding to fundamental natural frequency, ϑe is the volume of each element, N is number of elements, V is the volume fraction and ρe is the density (design variable).

2.2 Topology Optimization Topology optimization is an iterative optimization method, and it finds the optimal material distribution in the design domain subjected to the constraints [4]. We supply the dimensions of the design domain, material properties and a specified frequency as an input to the topology optimization. After that, we discretize the design domain into finite elements and assign design variable i.e. density variable to each element. It calculates the fundamental Eigen frequency of the structure by the finite element analysis [5]. This paper uses the MMA as an optimization solver to update the design variable [6]. If the density value is 1, it represents the material point i.e. solid (in black color) and if it is 0, it represents the no material, i.e. void (in white color). Topology optimization suffers with two kinds of mathematical instabilities, named checkerboard pattern and mesh dependency problem [7]. To avoid these, we apply filtering techniques to the sensitivities [8]. Figure 1 shows the flow chart of the topology optimization process [4].

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Initialization

Finite Element Analysis

Objective function and its sensitivities

Sensitivity filtering

Update design variables

Convergence? No Yes End

Fig. 1 Flow chart of topology optimization

2.3 Sensitivity Analysis 2.3.1

Sensitivity of Objective Function

Objective function V =

N 

ϑe ρe

(2)

e=1

∂V = ve ∂ρe

(3)

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2.3.2

Sensitivity of Constraint

Equality Constraint is given as (K − ω12 M)φ1 = 0

(4)

Assume λ1 = ω12 then, Eq. (4) becomes (K − λ1 M)φ1 = 0 ∂ (K − λ1 M)φ1 = 0 ∂ρe   ∂K ∂M ∂λ1 ∂φ1 − λ1 − M φ1 + (K − λ1 M) =0 ∂ρe ∂ρe ∂ρe ∂ρe

(5)

In the above equation, the second term will become zero. By pre-multiplying the Eq. (5) with φ1T becomes,   ∂K ∂M ∂λ1 φ1T − λ1 − M φ1 = 0 ∂ρe ∂ρe ∂ρe     ∂λ1 ∂K ∂M φ1 M φ1 = φ1T − λ1 φ1T ∂ρe ∂ρe ∂ρe 

∂λ1 T φ Mφ1 ∂ρe 1



 = φ1T

 ∂K ∂M φ1 − λ1 ∂ρe ∂ρe

(6)

but, φ1T Mφ1 = 1, then   ∂K ∂λ1 ∂M φ1 = φ1T − λ1 ∂ρe ∂ρe ∂ρe

(7)

The sensitivities of objective function and equality constraint is given by Eqs. (3) and (7) respectively [9], [10].

2.4 Design Domain and Properties Figure 2 shows a clamped beam, for which we minimize the weight using topology optimization of a mesh size 280 × 40. The material properties are, Young’s Modulus, E = 25e7 Pa, Poisson’s ratio, υ = 0.3, Mass density, ρ = 250 kg, Beam thickness, t = 0.1 m, Beam length, l = 280 m, Beam width, b = 40 m.

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Fig. 2 Design domain for clamped beam of mesh size 280 × 40

Fig. 3 Design domain for cantilever beam of mesh size 150 × 75

Figure 3 shows a cantilever beam, for which we minimize the weight using topology optimization of a mesh size 150 × 75. The material properties are, Young’s Modulus, E = 25e7 Pa, Poisson’s ratio, υ = 0.3, Mass density, ρ = 250 kg, Beam thickness, t = 0.1 m, Beam length, l = 150 m, Beam width, b = 75 m.

3 Results 3.1 Example 1: Clamped Beam We solve Eq. (1) for different specified frequency values, starts from 10 to 270 rad/s with a range of 20 rad/s. Table 1 lists the optimal volume fractions and the corresponding converged fundamental natural frequency of the optimal topologies for the above-mentioned cases. From Table 1, it is observed that in some cases, the converged natural frequency is more than the specified frequency, and in others, it is less than the specified frequency. Later case violates the frequency constraint of optimization problem. For the specified frequencies (in rad/s) of 10, 30, 50, 70, 90, 110, 130, 150, 170, 190, 210 the optimization converges, whereas for the specified frequencies (in rad/s) of 230, 250 and 270 the optimization diverges because it violates the lower bound constraint on Eigen frequency. From the above all solutions, we choose the solution obtained for a specified natural frequency of 170 rad/s as design because the objective is to minimize the volume and it has the minimum volume fraction value.

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Table 1 Optimal volume fraction and converged fundamental natural frequencies for different specified frequencies starts from 10 to 270 rad/s with a range of 20 rad/s for a clamped beam Optimal volume Fraction ‘V’ Converged fundamental natural Specified fundamental natural frequency ‘ω1 ’ (in rad/s) frequency ‘ωg ’ (in rad/s) 0.6593

214.5122

10

0.6495

214.1080

30

0.6684

219.9565

50

0.6198

214.8531

70

0.6581

215.3216

90

0.6286

215.8456

110

0.6531

215.1164

130

0.6159

219.4630

150

0.6152

219.9838

170

0.6384

217.7039

190

0.6441

216.1638

210

0.6446

215.9406

230

0.6421

216.3036

250

0.6374

217.9464

270

Fig. 4 Optimal topology of clamped beam with an optimized volume fraction of 0.6152 with natural frequency of 219.9838 rad/s for a specified frequency of 170 rad/s

Figure 4 shows the optimal topology obtained for a specified fundamental natural frequency of 170 rad/s and the convergence history of both volume fraction and converged fundamental natural frequency is shown in Fig. 5.

3.2 Example 2: Cantilever Beam We solve Eq. (1) for different specified frequency values, starts from 40 to 520 rad/s with a range of 40 rad/s for the cantilever domain. Table 2 lists the optimal volume fractions and the corresponding converged fundamental natural frequency of the optimal topologies for the above-mentioned cases. From the above all solutions, we choose the solution obtained for a specified natural frequency of 360 rad/s as design because the objective is to minimize the volume and it has the minimum

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Fig. 5 Convergence history of Eigen frequency and volume fraction for 200 iterations for clamped beam

Table 2 Optimal volume fraction and converged fundamental natural frequencies for different specified frequencies starts from 40 to 520 rad/s with a range of 40 rad/s for a cantilever beam Optimal volume Fraction ‘V’ Converged fundamental natural Specified fundamental natural frequency ‘ω1 ’ (in rad/s) frequency ‘ωg ’ (in rad/s) 0.2947

525.1562

40

0.2845

532.9132

80

0.2797

502.6193

120

0.2884

520.9923

160

0.291

512.3198

200

0.2944

535.881

240

0.2811

540.9427

280

0.2804

536.5597

320

0.2784

542.1122

360

0.2919

502.2156

400

0.2791

519.8919

440

0.2832

531.7349

480

0.2849

509.5215

520

volume fraction value. Figure 6 shows the optimal topology obtained for a specified fundamental natural frequency of 360 rad/s and the convergence history of both volume fraction and converged fundamental natural frequency is shown in Fig. 7.

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Fig. 6 Optimal topology of cantilever beam with an optimized volume fraction of 0.2784 with natural frequency of 542.1122 rad/s for a specified frequency of 360 rad/s

Fig. 7 Convergence history of Eigen frequency and volume fraction for 200 iterations for cantilever beam

4 Conclusions This paper presents a new method to avoid resonance while designing the structures using topology optimization of weight minimization problem. A clamped beam of mesh size 280 × 40 and a cantilever beam of mesh size 150 × 75 solved by using this proposed method. The optimal volume fraction is 0.6152 and the converged fundamental natural frequency is 219.9838 rad/s found for a specified natural frequency of 170 rad/s for the clamped beam. The optimal volume fraction for cantilever beam

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is 0.2784. The converged fundamental natural frequency is 542.1122 rad/s found for a specified natural frequency of 360 rad/s.

References 1. Zargham S, Ward TA (2016) Topology optimization: a review for structural designs under vibration problems. Struct Multidiscipl Optim 53(6):1157–1177 2. Thomson WT (1972) Theory of vibration with application. Prentice-Hall, Englewood Cliffs 3. Sui YK, Yi GL (2013) A discussion about choosing an objective function and constraints in structural topology optimization, 10th World Congress on Structural and Multidisciplinary Optimization. Orlando, Florida, USA 4. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications 5. Du J, Olhoff N (2007) Topological design of freely vibrating continuum structures for maximum values of simple and multiple Eigen frequencies and frequency gaps. Struct Multidiscipl Optim 34(2):91–110 6. Svanberg K (1987) The method of moving asymptotes: a new method for structural optimization. Int J Numer Method Eng 24:359–373 7. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Multidiscipl Optim 16(1):68–75 8. Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscipl Optim 21:120–127 9. Mada Y, Nishiwaki S, Izui K, Yoshimura M, Matsui K, Terada K (2006) Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmodes shapes. Int J Numer Method Eng 67:597–628 10. Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscipl Optim 20(1):2–11

Non-linear Dynamic Analysis of Structures on Opencast Backfilled Mine Due to Blast Vibration S. Kumar, S. C. Dutta, S. D. Adhikary, and M. A. Hussain

Abstract The structures nearby mining area are generally subjected to blast-induced ground vibration. Thus, this is imperative to understand the behaviour of such structures through numerical analysis as an experimental study would be expensive and time consuming. Hence, this paper presents a three-dimensional non-linear finite element analysis of a two-storey reinforced concrete (RC) frame structures. Global responses in terms of storey displacements, drifts as well as local responses in terms of stress and strain of concrete at each floor level are extracted and discussed. It is observed from storey drift that there is an excessive deformation of structures and it exceeds the permissible limit. Moreover, through the analysis of strain data, it is observed that the structures already reach its plastic limit zone. Therefore, the structures will collapse due to normal gravity-based design of structures. Blast-resistant design and detailing criteria should be kept in mind when designing such structures in mine regions. In future, efforts will be driven to find out the safe permissible distance of residential structures in mining areas.

1 Introduction The activity of coal extraction is inevitable due to the rapid increase in population and its energy demand. Blasting is a very common activity in the coal industry for the removal of overburden. This, in particular, helps in efficient extraction of coal.

S. Kumar (B) · S. C. Dutta · S. D. Adhikary · M. A. Hussain Indian Institute of Technology (ISM), Dhanbad, India e-mail: [email protected] S. C. Dutta e-mail: [email protected] S. D. Adhikary e-mail: [email protected] M. A. Hussain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_9

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Explosives are loaded into the drilled holes and are further detonated. The detonation leads to the release of energy and a portion of the energy is converted into wave energy with compression waves, shear waves and Rayleigh waves transmitted in all the direction from blast source. Rayleigh wave receives the maximum percentage of energy and causes the most destruction as they travel along the surface. Vibration energy that travels beyond the zone of rock breakage causes damage to surface structure and annoyance to the resident in the vicinity of mine areas. Completion of excavation work without endangering the safety of the surrounding structures is of great concern to all. Ground vibration is mainly influenced by parameters like distance between the blast and observation points, geological characteristics properties of the rock mass and nature of explosives [1, 2]. Two very important parameters that characterize the blast-induced ground vibration is peak particle velocity (PPV) and frequency. The damage potential due to blast vibration is generally qualified in terms of only PPV [3–7]. Thus, forecasting the level of ground vibration at different distances from blasting location for the safety of structures becomes an important issue. The allowable vibration levels in terms of PPV for a residential structure near an underground explosion have been suggested based on many field observations to be 11 cm/s for those on rock and 6 cm/s for those on soil [4–11]. However, these values obtained were mainly based on field observation of low-rise residential building. Although, they are excellent principal standard for structural safety but the application is limited in practice. This is fundamentally due to the fact that it overlooks the effect of conditions like structural type, structural condition, site conditions, ground vibration frequency content and ground vibration duration on structural response. It is very much familiar that stress waves generated from an underground explosion are different from that of the seismic waves. Vibration due to explosion generally contains high-frequency energy and its energy is distributed over a wider frequency band. Comparatively, the distances concerned are of an order of hundreds of metres, its amplitudes are much higher and the duration is much shorter than seismic waves. Further, as far as acoustic wave generated due to BIGM is considered it can be overlooked as the underground blasting has negligible influence on structures located at far of distance from the site. The opencast mines after extraction of coal are generally left open and filled with mine spoil. The filled up opencast mines due to rapid increase in population are seeking attention in terms of rehabilitation for best possible use. The issue for raising structure on such filled up mine area is its vicinity with the ongoing mining activity in parallel. One such site, where planning for construction is going on, is a site at Talcher, Orissa, India. The construction site is about a distance of 100– 300 m where planning for construction is ongoing. The non-linear analysis of a 3-D framed structure has been considered using finite element package ABAQUS. The acceleration time history for a blast loading has been considered from a well-accepted literature [12]. Further, the computational study has been carried out about stresses and strain in concrete, displacements and drift in the structure at a different storey. Thus, this paper as a whole can give us a clear idea about the behaviour of structure in the vicinity of the mining area.

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2 Underground Blast Explosion and the Considered Problem Figure 1 shows the general site configuration of the underground explosion that is similar to mine explosion. The setup for such explosion consists of a soil layer above the rock mass, underground chamber in rock mass and the layer of rock mass. Explosives are placed in the chamber floor for detonation. DSD indicates the surface distance measured from the point directly above the explosive, and DS indicates the depth of the soil layer. The present paper considers the explosion data obtained from well-accepted literature [12]. The acceleration time history data of the literature is obtained by field experimental results. Further, the results so obtained are also well verified by the literature through AUTODYN [12]. The other acceleration time history data after verification were generated for different soil layer thickness using AUTODYN. Figure 2 shows the plot of acceleration time history for a different case of soil layer thickness DS and safe surface distance from explosion DSD . The following data obtained from the literature has been considered for time history non-linear analysis of 2-storey frame structure. The frame structure considered for non-linear analysis is 1-bay and 2-storey building as shown in Fig. 3. The height of each storey is 3 m, while the length and the breadth along each bay is 3 m. The various loading considered is based on as per Indian Standard Code [13, 14]. The grade of concrete considered is M20 and the corresponding elastic modulus of concrete (E) in the frame section is 22361 N/mm2 . The steel grade considered for reinforcement in concrete is Fe415. The structure is

Safe surface distance DSD

Ds

Soil layer

Dc

Rock

Explosive Chamber Fig. 1 Site configuration of underground explosion [12]

S. Kumar et al.

Ground acceleration (m/s 2 )

Ground acceleration (m/s 2 )

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Time(s)

Time(s)

Fig. 2 Surface ground motions induced by underground explosion [12]

300 mm 3m 300 mm

3m

3m 3m

20 mm reinforcement

3m

Fig. 3 Frame section of a two-storey building and the chosen rectangular frame section

modelled in the commercial Finite Element code ABAQUS with the assigned materials (steel and non-linear reinforced concrete), geometries (multi-storey frames) and loading scenarios (static loads, gravity wall load, and blast load). The governing equation following the non-linear analysis of frame structure is given by Eq. (1). Non-linear dynamic structural responses are solved by a step-by-step integration procedure using the Newmark-β method. At time t, the incremental equation of dynamic equilibrium for a discrete structural system can be written as M x(t) ¨ + C x(t)+K ˙ x(t) = −F(t)

(1)

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x(t) ¨ nodal acceleration vector, x(t) ¨ nodal velocity vector, x(t) ˙ nodal displacement vector. where, in this study, [M] is a constant lumped mass matrix; [C] is a viscous damping matrix; [K] is a current tangent stiffness matrix; F(t) is the effective incremental load calculated from ground motions.

3 Results and Discussion 3.1 Drift at Each Storey Drift resistance is one of the major factors in assessing whether a proposed structure is suitable for a high-rise building. Figure 4 shows the drift of a framed structure measured at each storey with respect to its neighbouring down floor. Drift at each storey as shown in Fig. 4 is defined as the difference in the spatial displacement at up storey (U up ) and down storey (U down ) divided by storey height. The storey drift so calculated is shown in Fig. 5, where this can be observed that storey drift is maximun in the top floor of the building. The other interesting observation is that the drift

Uup

Udown

Fig. 4 Drift mechanism in a framed structure

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0.35 0.3 Storey drift

0.25 0.2 D11

0.15

D22

0.1

D33

0.05 0 0.0327771 0.0608017 0.0828464 0.121023 0.147505 0.178364 0.205814 0.232223 0.264763 0.293001 0.318241 0.338678 0.352436 0.363506 0.375997 0.383133 0.388747 0.394372 0.399232

0

time(sec) Fig. 5 Drift at each storey due to blast-induced ground motion

in the frame continues to have its impact even after the ground motion seizes. The drift at different storey increases linearly with the time and becomes constant at the end of duration. The drift value so obtained for such high impact underground blast loading exceeds the allowable limit as per the codal provision of the earthquake. The structure at such a limit of drift crosses the elastic limit and reaches the plastic range. The spatial displacement time history at different nodes obtained from the base of the framed structure is shown in Fig. 6. U1, U2, U3 shows the nodal displacement

0 0.0449071 0.0731832 0.119886 0.154557 0.192741 0.230632 0.274036 0.305901 0.337137 0.355187 0.369699 0.382412 0.389582 0.396624

spa al displacewment at nodes (mm)

me(s)

0 -200 -400 -600 -800

SpaƟal sisplacement U1 SpaƟal displacement U2 spaƟal displacement U3

-1000 -1200 -1400 -1600

Fig. 6 Spatial nodal displacement measured with the base of frame structure

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at first, second, and third floor of the framed structure with respect to the base of the structure. Figure 6 clearly shows that spatial displacement is maximum at the topmost floor. While the least spatial displacement can be observed on the first floor. The maximum spatial displacement at the topmost floor is due to the summation of all the spatial displacements at each floor measured with respect to ground. Thus, leading to maximum floor displacement at top, this exceeds the permissible limit of spatial displacement.

3.2 Stresses and Strain in Frame The RC frame is subjected to blast-induced ground motion simulated at a distance of 150 m (DSD = 150 m) from the blasting source. The depth of the soil layer is considered for the two cases, namely, at Ds = 0 m and Ds = 20 m. Time history dynamic analysis is carried out for such a combination of explosion distance and thickness of soil layer. The stress and strain response histories at different points in the frame are extracted from the output. Figure 4 shows the stress history at different elemental points at each storey aligned along one direction of the frame at the junction of column and beam at each storey. The stresses at different elemental point including the base of the frame is represented by S0, S1, S2 and S3 which, are points at the base of frame, first floor, second floor and third floor. Figure 4 shows a large variation in the stresses at each storey and no strict pattern is followed in its behaviour. However, the maximum stress can be observed to be at the second storey of the frame (S3). The stresses can significantly found to increase after the blast-induced ground motion seizes and becomes constant at the end of its duration with minor fluctuation. However, in a situation when the depth of soil layer considered is 20 m, i.e., Ds = 20 m. The variation in stresses is quite unpredictable at all the points S0, S1, S2 and S3. The stresses at the base of the frame is observed to be maximum, i.e., S0. The behaviour of the distribution of stresses with time, however, remains the same with the maximum occurring after the ground motion seizes, rises to its peak, and then remains constant. Strain histories of the extreme fibres at the most critical location in the beam and the column are also included in the figure. As can be seen, strain induced after the major shock period is larger than that induced during the major shock period. The maximum strains at some locations are larger than yielding strain of the reinforcing bars. Hence, the formation of plastic hinges at these locations cannot be ruled out. As strains in all locations exceed cracking strain of concrete, cracks are expected to appear throughout the frame (Figs. 7, 8, 9 and 10).

S. Kumar et al.

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

S0 S1 0 0.021 0.042 0.063 0.084 0.105 0.126 0.147 0.168 0.189 0.21 0.231 0.252 0.273 0.294 0.315 0.336 0.357 0.378 0.399

Stress at different storey in frame (N/mm2)

102

S2 S3

me (S)

Fig. 7 Stress history at each storey for integration point at each floor for Ds = 0 m

2 1

S0

0 -1 -2

0 0.0299646 0.0560325 0.0766024 0.114966 0.13634 0.164957 0.188773 0.21546 0.246693 0.272164 0.296964 0.319368 0.338127 0.350785 0.361407 0.372324 0.381261 0.386415 0.391594 0.396399

Stresses at different storey in frame(N/mm2)

3

S1 S2 S3

-3 -4

me(s)

Fig. 8 Stress history at each storey for integration point at each floor for Ds = 20 m

4 Conclusion The above study is a finite element analysis of three-dimensional concrete frame structures for the impact of short duration blast-induced ground motion in the mining area. The study is mainly focussed on parameters like storey drift, spatial displacement at each storey and stress and strain behaviour at each storey. The storey drift so observed for short duration blast vibration is very high and exceeds the allowable limit. The spatial displacement as observed on the first floor is not normal and almost is a sign of the collapse of the structure. The study gives us an idea that the structure reaches its plastic limit zone as can be observed from the strain value obtained. The plastic zone in the entire frame is reached and the structure will collapse as for a

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5.00E-04 4.00E-04 E0

3.00E-04

E1

2.00E-04

E2

1.00E-04

E3

0.00E+00 -1.00E-04

0 0.023 0.046 0.069 0.092 0.115 0.138 0.161 0.184 0.207 0.23 0.253 0.276 0.299 0.322 0.345 0.368 0.391

Strain at different storey

6.00E-04

Time(s) Fig. 9 Strain history at each storey for integration point at each floor for Ds = 0 m

0.025 0.02 0.015

E0

0.01

E1

0.005

E2

0

E3

-0.005

0 0.0347303 0.0653455 0.0892151 0.127773 0.152338 0.182705 0.21434 0.247368 0.276268 0.303668 0.330638 0.345282 0.35959 0.370224 0.381105 0.386766 0.392817 0.397725

Strain at different storey

0.03

Time (sec) Fig. 10 Strain history at each storey for integration point at each floor for Ds = 20 m

normal design for a 2-storey building. Thus, special frame sections with a high grade of concrete are proposed to be investigated from the viewpoint of structural safety.

References 1. Elevli B, Arpaz E (2010) Evaluation of parameters affected on the blast induced ground Vibration by using relation diagram method. Acta Montanistica Slovaca 15(4):261e8 2. Liang Q, An Y, Zhao L, Li D, Yan L (2011) Comparative study on calculation methods of blasting vibration velocity. Rock Mech Rock Eng 44(1):93e101

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3. Edwards AT, Northwood TD (1960) Experimental studies of the effects of blasting on structures. Engineer 210:538e46 4. Duvall WI, Fogelson DE (1962) Review of criteria for estimating damage to residences from blasting vibrations. US Bureau of Mines 5. Chae YS, Design of excavation blasts to prevent damage. Civil Engineering, ASCE 48:77e9 6. Esteves JM (1978) Control of vibrations caused by blasting. Memoria 409. Lisbon, Portugal: Laboratorio de Engenharia Civil, Ministerio de Habitacao e Obras Publicas 7. Langefors U, Kihlstrom B (1978) The modern technique of rock blasting. Wiley, p 405 8. Dowding CH (1985) Blast vibration monitoring and control. Englewood Cliffs, NJ: Prentice– Hall 9. Nicholls HR, Johnson CF, Duvall WI (1971) Blasting vibrations and their effects on structures Bureau of Mines, Bulletin 656, Washington DC 10. Odello RJ, Origins and implications of underground explosives storage regulations Technical memorandum, No. 51-80-14, Naval facilities engineering command, USA 11. Gustafsson R (1973) Swedish blasting technique. SPI, Gothenburg, Sweden 12. Ma G, Hao H, Zho Y (2000) Assessment of structure damage to blasting induced ground motions. Eng Struct 22:1378–1389 13. IS: 875-1987 (Part II). Code of Practice for design loads (other than earthquake) for building and structures. Bureau of Indian Standards, New Delhi, India 14. IS: 456-2000 Plain and Reinforced Concrete - Code of Practice. Bureau of Indian Standards, New Delhi, India

Seismic Response of Shear Wall–Floor Slab Assemblage Snehal Kaushik and Kaustubh Dasgupta

Abstract Reinforced Concrete (RC) structural walls are commonly used in tall RC frame–wall buildings in severe seismic zones for enhancing lateral strength and stiffness of the buildings. Conventionally, the structural walls in multi-storeyed buildings are designed in the same way as isolated shear walls. However, due to the presence of floor slabs at different levels, there is an increase of stiffness locally at each slab–wall junction, which may lead to a significantly different response of the assemblage as compared to the isolated shear wall. Also, the floor slabs tend to partition the slender wall into a number of smaller panels between successive floor slabs. The present study aims to investigate the seismic behaviour of such multi-storeyed slab–wall assemblage using non-linear time history analyses of three different models under ground motions recorded during a past earthquake. Keywords Time history analysis · Shear walls · Slab–wall junction · RC buildings

1 Introduction Reinforced Concrete (RC) structural wall is widely used in the lateral force-resisting system for multi-storeyed buildings located in the earthquake-prone regions. In such buildings, the wall is connected to the RC floor slab at every floor level. The junction region of shear wall and floor slab constitutes an important link in the load path from slab to the wall during earthquake shaking, thereby influencing the pattern of lateral load distribution in the various structural members of the system. Consequently, the behaviour of the lateral load resisting element affects the seismic performance of the overall building. During the Chile earthquake of 3 March 1985, many moderate-rise RC buildings got severely damaged. Most of them were designed S. Kaushik (B) · K. Dasgupta Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India e-mail: [email protected]; [email protected] K. Dasgupta e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_10

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with structural walls to resist both gravity and seismic loads. Walls and slab–wall connections sustained extensive cracking during that earthquake [1]. Also during the 27 February 2010 Chile earthquake, more than a hundred highrise RC shear wall buildings got severely damaged. Cracking of concrete walls and cracking of floor slabs occurred throughout the damaged corners of many buildings, causing significant building distortions. Damage caused to the buildings was mostly due to the compression failure of thin shear walls at the lower levels of the buildings [2]. From the past experimental research on a single-storey wall–slab assemblage, it was observed that the shear wall–slab junction experienced large stress concentration under combined axial and cyclic lateral loading [3]. The shear wall considered for analysis was squat in nature, and no study has been carried out on slender shear walls with connected floor slabs. Although various studies have been carried out on coupling action of the beam and floor slab in building with shear walls, the behaviour of shear wall–floor slab junction has not been studied extensively. Using finite element modelling and experimental studies, the bending stiffness of floor slab and its effects on the distribution of bending moments and stresses in slab have been investigated [4–7]. None of the past studies has focused on the detailed investigation of the behaviour of floor slab and shear wall junction under earthquake shaking. To investigate the behaviour of the shear wall–slab junction for rectangular walls in multi-storeyed buildings, non-linear time history analyses are carried out for three different models under ground motions recorded during a past earthquake.

2 Modelling Details and Parameters A hypothetical five-storeyed RC frame–wall building is assumed to be located in Seismic Zone V as per the Indian Earthquake Code [8]. Three different models, namely, (a) five-storeyed building, (b) an exterior wall–slab assemblage (EWSC), and (c) coupled wall slab (CWSC) assemblage where two shear walls are coupled with the slab in between (Fig. 1), are considered for carrying out time history analysis. Beams and columns are modelled using 2-node linear beam elements (B31) while floor slab and shear wall are modelled using 4-node doubly curved thin or thick shell element with reduced integration (S4R) in the ABAQUS/Standard [9] finite element program. The mentioned models are analysed under seven different ground motions, recorded during the 2011 Sikkim earthquake in India, using the dynamic implicit method. In the dynamic analysis, acceleration time history is applied at the base of each specimen and it is increased with a smooth amplitude curve varying over time (in seconds). Each ground motion is scaled to the arithmetic mean linear-elastic 5%damped spectral acceleration of the ground motion ensemble at the fundamental period of the structure being analysed. The elastoplastic material properties are assigned to the beams and columns of the full building, while the Concrete Damaged Plasticity (CDP) [10] model for concrete is assigned to the shear wall and the floor slab of the sub-assemblage model. The various ground motions are first scaled with

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Roller Support Ux =0

Base Acceleration along Z direction Fully restrained support Full Building

(a)

EWSC

(b)

Fully restrained support CWSC

(c)

Fig. 1 Geometry and boundary conditions: a Full Building, b EWSC and c CWSC

the S¯a (T0 ) method (corresponding to the fundamental natural period) [11]. Then, the entire ground motion ensemble is scaled for the second time by different factors so the Peak Ground Acceleration (PGA) values become 1 g for all the records. Pseudo-acceleration spectra for the entire ground motion ensemble are developed using SeismoSignal [12] program. The elastic modulus of concrete is considered as 25,000 MPa. Steel reinforcement is modelled with the material property assigned using the plasticity model in FE program. The yield stress, ultimate stress of steel are considered as 415 MPa and 527 MPa respectively. The modulus of elasticity for steel is considered as 2 × 105 MPa. The gravity loads (both dead and live loads) on the slab are assigned as pressure loads on the surface of solid elements. The total intensity of loading on slab including live load and floor finish is considered as 4 kN/m2 . The translational and rotational degrees of freedom are restrained at the bottom nodes of the wall. As the present study intends to investigate the non-linear behaviour for in-plane analysis of wall, the outer edges of slabs are supported on rollers, and the out-of-plane bending of the shear wall is prevented. The assemblage analysed in the current study has a characteristic cube compressive strength of concrete as 25 MPa. The tensile strength of concrete is assigned as 3.5 MPa. In the current study, the dilation angle is assumed as 55°, eccentricity  as 0.1, viscosity parameter as 0.01, shape factor (K c ) as 0.667 and stress ratio σb0 σc0 as 1.16 [13].

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2.1 Dynamic Analysis The dynamic analysis procedure in the Abaqus program uses the implicit direct integration operator of Hilbert, Hughes and Taylor method [14]. The operator is an extension of the trapezoidal rule, with an additional parameter that can be varied to introduce different levels of numerical dissipation. In an implicit dynamic analysis, the integration operator matrix needs to be inverted and a set of non-linear equilibrium equations are solved at each time increment. The implicit operator options available in Abaqus/Standard are unconditionally stable and, thus, there is no limit on the size of the time increment for most of the analyses (accuracy of the results depends on the time increment in Abaqus/Standard).

2.2 Scaling of Ground Motion Records Selection and scaling of strong ground motion time histories are critical and important to the time history analyses of structures. The scaling procedure used should be simple and should be implemented in such a way that the frequency content of the records is not changed. Real earthquake records are selected to match specific features of the ground motion, generally based on either response spectrum or an earthquake scenario with the minimum parameter being the magnitude, distance and site classification [15]. For the analysis and design, actual time histories are recommended to be used. The records should not be manipulated in the frequency domain but should be adjusted arithmetically in the time domain to match the desired spectral characteristics at the periods of most interest, or within a range around the period of interest [16]. To specify and predict the desired level of performance (degree of damage) of a structure for a specific level of ground motion intensity, non-linear time history analyses conducted using ground motion records that are scaled to adequately define the damage potential for the given site conditions and structural characteristics. Previous research describes many scaling methods of ground motion records. There are several methods that can be adopted to scale the ground motions in ensembles to produce a mean spectrum that satisfies the requirement of the studies. In the current study, the various ground motions are first scaled to the arithmetic mean linear-elastic 5% damped spectral acceleration of the ground motion ensemble at the fundamental period of the structure being analysed, ( S¯a (T0 )). The ( S¯a (T0 )) method depends on the structural properties (i.e. (T0 )) as well as the ground motion characteristics. The ( S¯a (T0 )) parameter is also referred to as the structure-specific ground motion spectral intensity. Thus, the entire ground motion ensemble is scaled for the second time by different factors so that the PGA value becomes 1 g for all the records. Pseudo-acceleration spectra for entire ground motion ensemble are developed using SeismoSignal program. The pseudo-acceleration spectra from the original ground motion record are shown in Fig. 2.

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Fig. 2 Pseudo-spectral acceleration for original ground motion ensemble

The fundamental time periods for the full building, EWSC and CSWC models are obtained as 0.456 s, 0.041 s and 0.057 s, respectively. Each ground motion is scaled using ( S¯a (T0 )) method of scaling, such that the spectral acceleration at the fundamental time period for all the specimens is equal to the mean spectral acceleration. The scaled pseudo-acceleration spectra for three above mentioned specimens are shown in Fig. 3. After scaling the original ground motions using ( S¯a (T0 )), WAVGEN [17] program is used to generate seven (7) acceleration time histories to fit the scaled spectral acceleration. WAVGEN modifies a recorded accelerogram to make it compatible with a given Pseudo Spectral Acceleration (PSA) spectrum. Figure 4 shows the original ground motion records, the target pseudo spectra and the WAVGEN generated ground motion records developed from the target pseudoacceleration spectra for 2011 Sikkim earthquake at Chungthang station. Similarly, for other stations, the ground motions are generated using WAVGEN. The specimens EWSC and CWSC are having very less fundamental time period, to achieve the expected damage in the specimens, these targeted ground motions require to scaling up for the second time. The entire targeted ground motion ensemble was scaled for the second time by different factors so the PGA value becomes 1 g for all the records. The factors of scaling and various characteristics of the ground motion records are given in Table 1 for different recording stations. These scaled ground accelerations are employed to the two specimens, namely, (a) EWSC and (b) CWSC, to perform nonlinear analyses in the time domain by using as input at the base of the specimens. The behaviours of the two models (EWSC and CWSC) are compared with the behaviour of the five-storied frame–wall building analysed under a single ground motion record. Figure 5 represents the recorded and scaled Fourier amplitude spectrum of the recorded ground motions at Chungthang station of the 2011 Sikkim earthquake. It

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Fig. 3 Scaling of ground motion based on spectral acceleration: for a five-storeyed building b EWSC model and (c) CWSC model

Fig. 4 Spectral response scaling at the fundamental natural period of the model for Chungthang station recorded during the 2011 Sikkim Earthquake; a EWSC model and b CWSC model

is seen that the energy content of the accelerogram gets changed after applying the fundamental period scaling procedure. After the application of the second scaling factor, the energy level increases significantly. It is also observed that the band for the frequency level where the energy is maximum is the same for the recorded and the scaled ground motions. The value of the Fourier amplitude changes with the scaling techniques.

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Table 1 Characteristics of selected ground motion records for the 2011 Sikkim Earthquake Station

PGA (g)

Scaling factor using ( S¯a (T0 ))

Second time scaling factor

EWSC

CWSC

EWSC

CWSC

Chungthang

0.351

1.06

0.64

30.00

18.39

Gangtok

0.235

1.10

1.31

47.56

56.48

Geizing

0.422

1.33

1.13

30.92

26.31

Mangan

0.277

1.54

1.87

53.90

63.94

Melli

0.228

1.04

1.18

46.44

52.57

Silliguri

0.039

0.20

0.14

53.28

37.65

Singtam

0.200

0.66

0.69

34.20

35.85

Fig. 5 Comparison of Fourier amplitudes for ground motions at Chungthang station

3 Finite Element Analyses Results 3.1 Five-Storey Shear Wall Building Model The non-linear time history analysis of the five-storey building is carried out using a scaled acceleration ground motion of PGA value 1.12 g. The selected ground motion was recorded at Jellapur station during the 1997 Indo-Burma earthquake and the PGA was observed as 0.14 g. During the analysis, no damage was observed in the structure using the 0.14 g ground acceleration. To study the behaviour of the shear–wall slab junction, the PGA value is arithmetically scaled in the time domain up to 1.12 g. The input acceleration time history used for the analysis and the corresponding tensile damage pattern at the time instance of PGA are shown in Fig. 6a, b, respectively. From the analysis, the damage is observed to initiate at the base of the shear wall first and then moves to the floor levels, mainly at the wall–slab junction. The damage starts at 0.2 s of the time interval at the base of the wall and reaches the wall–slab junction at the time instance of 1.46 s. The cumulative damage is determined and is observed to increase at the wall–slab junctions from the lower level. The cumulative damage, represented in Fig. 7a, shows that the damage starts earlier at the base of the shear wall as compared to the beginning of damage in the upper storeys. The maximum damage level is observed around the time instance of the PGA. It is observed that average tensile damage in the slab on the first floor level is more

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Fig. 6 a 1997 Indo-Burma (Jellapur) ground acceleration time history and b tensile damage pattern at the time instance of PGA

Fig. 7 a Comparison of average tensile damage parameter for five-storey building and b displacement at the top node of the five-storey building

severe than the other floor levels, mainly due to the stress concentration in the slab. Figure 7b represents the relative displacement at the top node of the building with respect to time. It is observed that the maximum displacement of 270 mm occurs in the negative X-direction, while 67 mm in the positive X-direction, around the time instance of PGA.

3.2 Shear Wall–Slab Junctions for ESWC and CSWC Models Seven ground motion records from 18 September 2011 Sikkim earthquake of magnitude Mw 6.9 are selected to carry out the non-linear time history analysis of two different shear wall–slab assemblages (ESWC and CSWC). Figure 8 compares the tensile damage pattern of EWSC and CSWC specimens at the time instance of PGA.

Fig. 8 Comparison of tensile damage pattern a ESWC model, b CSWC model at time instance of PGA for different earthquake stations

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For both the models, it is observed that the damage at the base of the shear wall started well before the time instance of PGA. While the development of maximum tensile stress and tensile damage in the slab region started after reaching the PGA value, it can be concluded that the specimen undergoes maximum damage after the time instance of PGA. A similar damage pattern is observed at the wall–slab junction region considering different stations. The predicted tensile damage pattern at the shear wall–slab junction for both the specimens shows the effect of higher stress concentration at the junction region. The damage increases until the peak acceleration value is reached, thereafter it remains constant. It implies the achievement of maximum damage state in the structure. For the CSWC model, with an elapsed time of shaking, the tensile damage moves to the upper floor levels. The damage reaches the fourth floor for the ground motion records at Geizing and Melli stations. For other ground motion records, the damage does not proceed beyond the second or third floor level.

4 Conclusions Based on the present study, the following salient conclusions are drawn: • For the scaled-up recorded ground motion of the 1997 Indo-Burma earthquake at station Jellapur (PGA 0.14 g), the cracking is observed to begin at the base of the wall and then get initiated at the slab–wall junctions. The tensile damage also gets propagated in the floor slabs due to the combination of slab displacement and flexural displacement of the shear wall. • For the ESWC and CWSC specimens, analysed using ground motion ensembles of the 2011 Sikkim Earthquake, the maximum stresses and the tensile damage get developed at the base of the shear wall first and then get developed at the wall– slab junction region. Maximum stresses are developed at the first floor level. The damage increases till the attainment of the PGA value of the ground motion thereafter, it remains constant.

References 1. Riddell R (1992) Performance of R/C buildings in the 1985 chile earthquake, proceedings of tenth world conference on earthquake engineering, Madrid, Spain, 19–24 July 1992 2. Sherstobitoff J, Cajiao P, Adebar P (2012) Repair of an 18-story shear wall building damaged in the 2010 chile earthquake. Earthq Spectra 28(S1):S335–S348 3. Pantazopoulou S, Imran I (1992) Slab-wall connections under lateral forces. ACI Struct J 89(5):515–527 4. Coull A, Wong YC (1985) Effect of local elastic wall deformations on the interaction between floor slabs and flanged shear walls. J Build Environ 20:169–179 5. Qadeer A, Smith BS (1969) The bending stiffness of slabs connecting shear walls. ACI Structural Journal 66(6):464–473

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6. Schwaighofer J, Collins MP (1977) Experimental study of the behavior of reinforced concrete coupling slabs. ACI Struct J 74(3):123–127 7. Paulay T, Taylor RG (1981) Slab coupling of earthquake-resisting shear walls. ACI Struct J 78(2):130–140 8. IS: 1893 (Part 1)-2016, Indian Standard Criteria for Earthquake Resistant Design of Structures. Part 1: General Provisions for All Structures and Specific Provisions For Buildings, Bureau of Indian Standards (BIS), New Delhi, India, 2016 9. ABAQUS, ABAQUS Analysis User’s Manual, Hibbitt, Karlsson, and Sorenson, Pawtucket, R.I. (2011) 10. Lubliner J, Oliver J, Oller S, Onate E (1989) A plastic-damage model for concrete. Int J Solids Struct 25(3):299–326 11. Kurama YC, Farrow KT (2003) Ground motion scaling methods for different site conditions and structure characteristics. Earthq Eng Struct Dynam 32:2425–2450 12. Seismosoft, Seismosignalv.5.1.0, www.seismosoft.com (2013) 13. Gulec CK, Whittaker AS (2009) Performance-based assessment and design of squat reinforced concrete shear Walls, MCEER Technical Report-09-0010. MCEER, Buffalo 14. Hilber HM, Hughes TJP, Taylor RL (1978) Collocation, dissipation and overshoot for time integration schemes in structural dynamics. Earthq Eng Struct Dynam 16:99–117 15. Fahjan YM, Ozdemir Z, Keypour H (2007) Procedure for real earthquake time histories scaling and application to fit iranian design spectra. In: Fifth International Conference on Seismology and Earthquake Engineering, Tehran, Iran, 13–16 May 2007 16. Lew M, Naeim F (1996) Use of design spectrum-compatible time histories in analysis of structures. In: Eleventh world conference on earthquake engineering, Acapulco, Mexico, 23–28 June 1996 17. Mukherjee S, Gupta VK (2002) Wavelet-based generation of spectrum-compatible timehistories. Soil Dynam Earthq Eng 22(9):799–804

Comparison Between Two Modeling Aspects to Investigate Seismic Soil–Structure Interaction in a Jointless Bridge S. Dhar and K. Dasgupta

Abstract Seismic waves propagate through a series of rock and soil layers before they interact with the foundation and superstructure. Besides the original characteristics of the earthquake motion at the instant of fault rupture, it is also essential how the soil site responds in terms of amplification or de-amplification for different frequency contents. A coupled soil–structure model is required to capture the dynamic behavior of the entire system efficiently, considering the possible nonlinear response of soil and structure. This paper focuses on the comparison of two modeling strategies for Soil–Structure Interaction (SSI) aiming to define the behavior of a jointless bridge, namely, (a) one with an explicit full-scale soil domain with bridge model and (b) another with Beam on Dynamic Winkler’s Foundation (BDWF)/nonlinear soil springs. Finally, the structural components that affect the overall behavior of superstructure are compared between these two models, and the variation of seismic response from the performance-based study is discussed. Keywords Beam on dynamic Winkler foundation (BDWF) · Soil continuum · Jointless bridge · Abutment–backfill interaction

1 Introduction Creation of full-scale soil domain as continuum requires significant attention and expertise, and the analyses are numerically costly for SSI investigation. In particular, two models of a specific bridge are investigated by using a structure that resembles the well-known Humboldt Bay Middle Channel (HBMC) Bridge in California. Though, modeling the target bridge through the proper definition of all details to describe its S. Dhar (B) · K. Dasgupta Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India e-mail: [email protected] K. Dasgupta e-mail: [email protected] S. Dhar University of California Los Angeles, Los Angeles, CA 90095, USA © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_11

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seismic response was already achieved in the past studies [1, 2] and is not the goal of the present work. In the first approach, a full-scale soil-foundation-bridge with or without abutment–backfill interaction (i.e., full SSI with/no BA models) is modeled in OpenSees [3, 4] and the second approach, nonlinear springs are introduced to represent soil stiffness, replicating Soil–Pile Interaction (SPI) and Abutment–Backfill Interactions (ABI) (i.e., FB_SD no/with BA models). Hence, in the first and the second modeling approaches, different structural parameters are compared to investigate the overall response of the bridge structure. Modeling of full SSI no/with BA models is computationally expensive and time-consuming. Thus, modeling of continuous soil domain to consider SSI is not a very common practice in design firms, currently. So, to incorporate simplistic SSI in seismic analysis of the bridge, continuous soil domain has been replaced with nonlinear spring-dashpots to take care of SSI in FB_SD no/with BA models. Further, comparisons are made with and without ABI between these two modeling approaches.

2 Selection of Ground Motions A Uniform Hazard Response Spectrum (UHRS) for bedrock-level ground motions is used as the target spectrum to select and scale the input ground motions for the analyses as discussed in Dhar et al. [5]. The UHRS is developed from the 2008 United States Geological Survey [6] national seismic hazard maps for the Humboldt Bay area for rock outcrop assuming VS, 30m = 800 m/s (according to NEHRP [7], site class B). The corresponding 5% damped elastic displacement response spectrum has been given as target to REXEL-Disp [8] to select the ground motions for dynamic analysis from strong ground motion database SIMBAD [9]. The input parameters in REXEL-Disp to find the ground motions are: magnitude = 5.5−7.5; fault to site distance = 0−30 km; spectrum matching tolerance = ±20%; spectrum matching period = 0.2−5 s; site specification = EC8 site class A; probability of exceedance = 10% in 50 years, representing the return period of 475 years. Seven real record ground motions are chosen for horizontal input motion by scaling their respective displacement spectra within the period of interest, such that the average displacement spectrum lies within the tolerance limits. Different parameters of selected motions are given in Table 1. The corresponding 5% damped elastic displacement spectra with the average of the ground motions are shown in Fig. 1. Table 1 shows the set of selected and scaled rock outcrop ground motions [5]. The ground motions which are highlighted in gray are chosen to discuss in detail the soil and structural response. The Acceleration Time Histories (ATHs) of the chosen ground motions and their Fourier transform are shown in Fig. 2. The two decided ground motions (GM#1 and GM#2) correspond to the same station but with different Peak Ground Acceleration (PGA) amplitudes. Thus, the focus of this paper is to identify similarities and differences between the two different modeling techniques in the light of the observed response.

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Table 1 Different parameters of selected ground motions [5] Epicentral PGA, m/s2 distance, km

Scaled PGA, m/s2

23.77

0.54

3.46

6.5

5.25

3.39

3.06

May 29, 2008

6.3

8.25

3.28

5.47

Olfus

May 29, 2008

6.3

7.97

5.00

7.06

BSC

Irpinia

November 6.9 23, 1980

28.29

0.95

0.68

ST_106

South Iceland

June 21, 2000

6.4

21.96

0.51

0.73

LPCC

Christchurch February 21, 2011

6.2

1.48

9.16

12.64

6.5

13.85

3.26

4.73

Station ID

Earthquake name

Date

Mw

ALT

Irpinia

November 6.9 23, 1980

ST_106

South Iceland

June 17 2000

ST_112

Olfus

ST_101

Mean values:

0.4

Spectral Displacement (m)

Fig. 1 Displacement spectra of all the ground motions used in the present study; the period of interest is shown within a blue shadow. Chosen ground motions for discussing the results are also marked

0.3

0.2

Target Spectrum Upper Tolerance Lower Tolerance Average Spectrum Period Range of Interest GM#1

0.1 GM#2

0.0 0.0

2.5

5.0

Period (s)

3 Modeling 3.1 Structural Modeling As illustrated in detail in Zhang et al. [1], the Humboldt Bay Middle Channel Bridge, located near Eureka in California (USA), is 330 m long, 10 m wide, and 12 m high (average height over mean water level). The bridge superstructure consists of nine spans with four precast prestressed concrete I-girders and cast-in-place concrete

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1.5

Fourier Amplitude

1.0 0.5 0.0

(b)

1.5

Fundamental frequencies FB_SD no BA 1.0 full SSI no BA 0.5 0.0 0.1

1

(d)

10

Frequency, Hz Fig. 2 Rock outcrop earthquake (scaled) records used in this study: a strike-slip fault of the June 17, 2000 South Iceland Earthquake; Ground Motion (GM) #1, b Fourier amplitude spectrum of GM#1, c strike-slip fault of the June 21, 2000 South Iceland Earthquake; Ground Motion (GM) #2, and d Fourier amplitude spectrum of GM#2 with the fundamental frequency (marked) for different considered models

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slabs. The bridge deck is resting on two integral abutments monolithically connected with continuous deck, and the shear keys are exempted at the superstructural level, to serve the purpose of our present study which is to investigate the behavior of an integral bridge [10]. Pile caps of 1 m thickness are supported by deep foundations consisting of driven precast prestressed concrete pile groups. For the sake of simplicity, only the longitudinal dynamic response is analyzed in this study. Two finite element models of the bridge are considered: a linear model with elastic beamcolumn elements for both the superstructure and substructure (piers) and a nonlinear model where piers are modeled using force-based fiber elements. The elastic properties of structural elements are adapted from Zhang et al. [1] as A (area, m2 ) = 12, 4.56, 3.4 and I (moment of inertia, m4 ) = 1.44, 3.212, 0.8188 for abutment, deck and pier sections, respectively. All the concrete elements have the same elastic modulus of 28 GPa. The details of the pile group modeling have been discussed in Dhar et al. [11]. In the nonlinear simulations, force-based fiber elements [12] with five integration points are used in piers. Pier cross section is discretized as shown in Fig. 3a. KentPark-Scott [13] concrete model is used to model nonlinear concrete material with degraded linear unloading/reloading stiffness, and no tensile strength is considered. Giuffre-Menegotto-Pinto [14] steel material is specified with 0.8% isotropic strain hardening for reinforcement bars with 200 GPa elastic modulus and 276 MPa yield strength. The properties of confined and unconfined concrete used in the study are the same as those adopted by Zhang et al. [1]. Compressive strengths of confined and unconfined concrete are 34.5 MPa and 27.6 MPa, respectively. The simulations have been performed using the finite element program OpenSees [4]. Rayleigh damping scheme is introduced as viscous material damping to calibrate the Rayleigh damping parameters. The damping ratio is prescribed as 5% at 0.5 Hz and 5.0 Hz for full SSI no/with BA models in OpenSees shown in Fig. 3b.

Rayleigh Damping (%)

10.0

7.5

5.0

2.5

Fundamental frequencies full SSI no BA FB_SD no BA

0.0 0.0

(a)

2.5

5.0

frequency, Hz

7.5

10.0

(b)

Fig. 3 a Fiber-based discretization of the pier cross section with reinforcement and b Rayleigh damping considered for the SSI analyses

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3.2 Geotechnical Modeling Two-dimensional soil modeling has been carried out in OpenSees. Soil domain is 1500 m wide (evaluated through iterations to reach free-field motion at the boundary) and 220 m in depth. The entire soil domain consists of 4 different layers (Fig. 4a) having the static and dynamic properties (Table 2), in which the geotechnical constitutive parameters are adapted from Zhang et al. [1]. Pressure independent multi-yield material has been used to describe the soil behavior through a formulation based on the multi-surface plasticity concept [15] with associative flow rule, inbuilt in OpenSees. The yield surfaces are of the Von Mises type. Since total stress analyses

(a)

1.0 20

Damping ratio (%)

G/Gmax

0.8 0.6 0.4 0.2 0.0

1E-4

(b)

OL/SM SP/SM CL SP

1E-3

0.01

0.1

Shear Strain (%)

1

15 10 5 0 1E-4

1E-3

0.01

0.1

Shear Strain (%)

1

(c)

Fig. 4 a Full SSI no BA model in OpenSees (all dimensions in m), b Normalized shear modulus degradation versus shear strain and c Damping ratio versus shear–strain curves for different soil layers (Darendeli 2001). Explicit details of the numerical modeling are provided in [11]

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Table 2 Properties of different soil layers used in the present study (modified after [1]) Soil layer

Elastic properties Maximum shear modulus (MPa)

Poisson’s ratio

Nonlinear properties Total unit weight (t/m3 )

Undrained shear strength, (kPa)

Shear modulus ratio (Fig. 4b, c)

Plasticity Index

10

OL/SM

76

0.45

1.9

30

Cyan Line

SP/SM

171

0.45

1.9

11.9

Yellow Line

0

CL

288

0.45

1.8

100

Orange Line

30

SP

525

0.45

2.1

52.5

Brown Line

0

are carried out, any direct consequence of significant excess pore water pressure generation is naturally neglected in the present study. To represent the nonlinear nature of the soil domain, variation of shear modulus degradation and damping ratio with shear strain are adapted as per the proposition of Darendeli [16], shown in Fig. 4b, c, respectively. The soil domain lateral boundary conditions are implemented by Tied Degrees of Freedom (TDOF) [2, 17] at the lateral two ends of the soil domain. Thus, the soil domain follows the pattern of a 2D shear beam constraints, in which generally the horizontal response dominates over the vertical response. At the base level, classical Lysmer and Kuhlemeyer [18] type absorbing boundary conditions are applied in the horizontal direction by adequately calibrating the dashpot coefficients together with the classical vertical displacement restraints. The dynamic base input motion is given in horizontal direction to study the horizontal response of the soil–structure system. In the case of full SSI no/with BA models, the precast prestressed concrete pile groups, considered linear elastic in all the analysis, are analyzed per [19−21] with an equivalent pile group of stiffness 1.1 GPa. The precast driven piles are of 5.2 m in length and floating type. In the case of full SSI with BA model (Fig. 5), abutment–backfill dynamic properties are kept similar to the soft clayey soil (topmost layer) of the soil domain. The effects of gaps or interfaces are excluded at the abutment–backfill interface.

Fig. 5 Schematic diagram of a soil–pile interaction and b abutment–backfill interaction modeled in the present study (adapted from [11])

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In the simplified models, on the other hand, FB_SD no BA model represents the Soil–Pile Interaction (SPI), and FB_SD with BA serves SPI and ABI, together. In the models above, SPI and ABI are represented through classical nonlinear twonoded zero-length links and dashpots to represent idealized SSI at far and near fields. In the near field spring-dashpot system, hysteretic damping is considered through nonlinearity of multi-linear plastic uniaxial material, inbuilt in OpenSees. Lateral and vertical springs are modeled as per API-rp2a [22] in parallel to represent lateral load-bearing capacity and skin friction of pile shaft or pile tip end bearing, respectively. Far-field soil stiffness and radiation damping are modeled using the coefficients provided by Gazetas and Dobry [23]. Far-field spring-dashpots are linear elastic and modeled in series with near-field hysteretic springs. Far-field spring and dashpots are modeled in a parallel configuration. At the fixed end of far-field springdashpots, free-field motions are applied based on the exact depth of soil column at the corresponding depth of springs. Detailing of springs and dashpots with the schematic diagrams of SPI (modeled explicitly with near-field and far-field) and ABI are discussed in Fig. 5a, b, respectively; where K s is the linear stiffness, C s is the radiation dashpot coefficient of far-field spring and SF for skin friction of pile. It was proved in past researches [24−26] that force–displacement relationships from API overestimate the soil stiffness to address SPI. Therefore, the implementation of API curves in FB_SD no/with BA models gives a conservative response. Moreover, far-field spring-dashpots are linear elastic. Thus, to reduce this error, 10% Rayleigh damping is introduced in FB_SD no/with BA models. In FB_SD with BA model, nonlinear springs are added to model abutment–backfill soil. The abutment is considered as frame-type abutment, and nonlinear springs are positioned at 1 m distance along with the height of abutment (Fig. 5b). The force– deformation curves for the springs are calculated as per the Highway Agency [27].

4 Analysis During the first phase of the study, two models are compared, namely, (a) full SSI no BA model and (b) FB_SD no BA model. Initially, a single-step static analysis is carried out. Then, input motion is applied as force–time history at the base of the soil domain in full SSI with BA model. For FB_SD with BA model, after static gravity analysis, free-field input motions are applied at different corresponding depths as Displacement Time Histories (DTHs) extracted from 2D ground response analysis of similar soil column performed in OpenSees. For full SSI with BA and FB_SD with BA models, salient parameters are compared in the following sections, and differences in structural behavior from nonlinear THA are discussed.

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5 Results 5.1 Comparison of the Response of Full SSI no BA and FB_SD no BA Models From Fig. 6a, b under GM#1, eighth pier top and bottom ATHs are compared and in (c) and (d) their respective Fourier Transforms (FT) are shown. The observed ATHs at the bottom of the pier have similar values (Fig. 6a, c), but at the deck level (Fig. 6b, d) FB_SD no BA model shows a more amplified response as compared to the full SSI no BA model. However, the peaks of Fourier amplitude occur at 1.78 Hz and 2.29 Hz for FB_SD no BA and full SSI no BA models, respectively. Foundation of FB_SD no BA model is found to be stiffer and nonlinearity still not developed in the underlying soil because the API curves overestimate soil stiffness at different depths of piles. Moreover, a resonance effect is also noted near 2 Hz, which is not present in the fully coupled SSI counterpart. FB_SD no BA full SSI no BA

Acceleration, m/s2

2 1 0 -1 -2 -3

0

10

20

30

40

0

10

20

(a)

Time, s

(b)

1

10 0.1

1

30

40

Fourier Transform, m/s

4 3 2 1 0 0.1

(c)

Frequency, Hz

10

(d)

Fig. 6 Acceleration time history (ATH) a at the bottom, b ATH at the top of eighth pier, c FT of the ATH shown at (a) and (d) FT of the ATH in (b) under GM#1

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Shear Force, kN

1000 500 0 -500

Peak 1300.4 kN 1485.3 kN

-1000 -1500 0

10

20

30

40

1.0 normalized fourier amplitude

1500

0.8 0.6 0.4 0.2 0.0

(a)

time, s

0.1

1

10

Frequency, Hz

(b)

Moment, kNm

10000 5000 0 -5000

-10000 -0.004

residual FB_SD no BA residual full SSI no BA -0.002

0.000

(c)

0.002

0.004 -0.004

-0.002 -1

Curvature, m

0.000

0.002

0.004

(d)

Fig. 7 a Shear force time history at the top of eighth pier and b Fourier transform of the SFTH. Moment curvature response of eighth pier at c top and d bottom under GM#1

A similar response is observed for Shear Force Time History (SFTH) plots in Fig. 7a. Due to seismic waves amplification at the deck level, SFTH is higher in FB_SD no BA model. From the SFTH, it can be stated that FB_SD no BA model shows significantly less nonlinearity in the bridge substructure component as compared to the full SSI no BA model. From normalized FT of SFTHs in Fig. 7b, it is observed that the peak of the Fourier amplitude of FB_SD no BA model is at 1.78 Hz, thus the magnitude of Fourier amplitude is significantly higher. This implies that once the seismic waves propagate from the foundation to the deck level, FB_SD no BA model shows a more amplified response as compared to the full SSI no BA model. This is also evident from the moment–curvature response at the top and bottom sections of the eighth pier (Fig. 7c, d). Moment–curvature response at the top of the eigth pier is found to be underdamped in FB_SD no BA model and mainly forms the negative moment–curvature loops in the third quadrant. At the base of the eighth pier, moment–curvature response (Fig. 7d) shows a similar pattern on the opposite quadrants. After investigating several parameters at the top and base of the pier, it can be stated that FB_SD no BA model is getting more amplified from the foundation to the deck level as compared to the full SSI no BA model; thus, mismatches in response arise for different response parameters at the deck level.

Shear Force, kN

1000 500 0 -500 Peak 1266 kN 966 kN

-1000 -1500

0

10

20

time, s

30

40

(a)

127

Normalised Fourier Amplitude

Comparison Between Two Modeling Aspects …

1.0 0.8

FB_SD no BA full SSI no BA

0.6 0.4 0.2 0.0 0.1

1

Frequency, Hz

10

(b)

Fig. 8 a SFTH at the top of eighth pier and b FT of the SFTHs in (a)

Under GM#2, SFTHs at the top of the eighth pier (deck level) are shown for the two models in Fig. 8a. The difference in the energy content of the response is observed through the normalized FT of SFTHs in Fig. 8b.

5.2 Comparison of the Response of Full SSI with BA and FB_SD with BA Models For comparison, the ATHs are shown at the top and the bottom of the eighth pier in Fig. 9a, b, and the corresponding Fourier amplitudes of ATHs are shown in Fig. 9c, d under GM#1. ATH at the base of the pier in FB_SD with BA model is marginally higher than full SSI with BA model. Thus, in Fig. 9c, the peaks of the amplitudes are higher from 0.6 Hz onwards. However, at the top of the eighth pier (Fig. 9b) ATHs are quite comparable for both the models. In Fig. 9d, peak Fourier amplitudes for FB_SD with BA and full SSI with BA models are at 2.83 Hz and 2.12 Hz, respectively, which indicate that FB_SD with BA model is marginally stiffer as compared to the full SSI with BA model. Under GM#2, similar responses are observed, at the top and bottom of first pier ATHs in Fig. 10a, b and their corresponding FTs in Fig. 10c, d, respectively. Due to low-intensity input motion, the ATHs at the base of the pier are quite similar for both the models along with their FTs. At the top of the pier, ATHs have two different peaks at different time instants. In Fig. 10d, peak amplitudes occur at the frequencies of 1.48 Hz and 3.07 Hz for FB_SD with BA and full SSI with BA model, respectively. At the low intensity of PGA input, foundation soil is expected to behave in the linear elastic range. As the nonlinearity of the foundation soil does not influence the behavior of the superstructure, the bridge is observed to be more flexible for FB_SD with BA model as compared to the full SSI with BA model. Thus, the peak of FT at

Acceleration, m/s2

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1 0 -1 -2

Fourier Transform, m/s

0

10

FB_SD with BA full SSI with BA 20 30 40 0 Time, s (a)

10

20

30

40

(b)

2.0 1.5 1.0 0.5 0.0 0.1

1 (c)

10 0.1 Frequency, Hz

1 (d)

10

Fig. 9 ATH a at the top and b at the bottom of eighth pier, c Fourier transform of the ATH shown in (a) and (d) Fourier transform of the ATH in b under GM#1

deck level in FB_SD with BA model is at lower frequency instant than full SSI with BA model. Moreover, amplitudes are quite the same both spectrally and temporally. At the top of the eighth pier of FB_SD with BA model in Fig. 11a under GM#1, both the peak and the residual shear forces are higher as compared to the full SSI with BA model; this shows that more nonlinearity develops in the FB_SD with BA model under GM#1. The Fourier amplitude plot of SFTHs in Fig. 11b shows that in FB_SD with BA model, the seismic forces are higher as compared to the full SSI with BA model with a frequency of 0.8 Hz, beyond which the SSI model shows higher seismic force. The peaks of Fourier amplitudes occur at 0.42 Hz and 1.06 Hz for FB_SD with BA and full SSI with BA models, respectively. Thus, it signifies that in the former model, the seismic response is being more amplified at deck level and more nonlinearity develops at pier sections than the full SSI with BA model. Under GM#2, the variation of SFTHs at the top of the eighth pier shows that the peak shear force in FB_SD with BA model is higher than the full SSI with BA model (Fig. 11c); also the FB_SD with BA model shows a higher residual response. Thus, in FB_SD with BA model piers have developed significant nonlinearity as compared to the full SSI with BA model. From the comparison of Fourier amplitudes of SFTHs in Fig. 11d, the seismic force content is observed to be higher in FB_SD with BA model up to a frequency of 1 Hz, beyond which the full SSI with BA model carried higher forces

Comparison Between Two Modeling Aspects …

Acceleration, m/s2

1.5

FB_SD with BA full SSI with BA

1.0 0.5 0.0 -0.5 -1.0 -1.5

0

10

20

(a) Fourier Transform, m/s

129

30

40

0

Time, s

10

20

30

40

(b)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.1

1

(c)

10 0.1

Frequency, Hz

1

10

(d)

Fig. 10 ATH a at the top, b ATH at the bottom of first pier, c Fourier transform of the ATH shown at (a) and (d) Fourier transform of the ATH in (b) under GM#2

in the high-frequency range. Thus, the FB_SD with BA model is weaker and more flexible than full SSI with BA model.

6 Conclusion Based on the nonlinear time history analysis of the bridge-soil system under the two selected ground motions, the two mentioned modeling approaches yield reasonably close response. The salient conclusions are stated as follows: • As compared to the full SSI no/with BA models, shear forces and bending moments at piers are higher for FB_SD with/no BA models because foundation soil stiffness is more elevated in API force–displacement curves for clayey soil. • The full SSI with BA model is observed to show the lowest nonlinear dynamic response for the bridge structure. As the backfill soil provides longitudinal restraints to the bridge, the seismic force from the superstructure dissipates into the backfill soil through passive resistance.

S. Dhar and K. Dasgupta 1500

500 0 -500 -1000

FB_SD with BA full SSI with BA

-1500 0

10

20

30

Time, s

40

Shear Force, kN

350 0 -350 -700 -1050 0

10

20

time, s

1

0.1

0.01

1E-3 0.1

(a)

30

40

(c)

Normalised Fourier Amplitude

Shear Force, kN

1000

Normalised Fourier Amplitude

130

1

10

Frequency, Hz

(b)

1

0.1

0.01

1E-3 0.1

1

Frequency, Hz

10

(d)

Fig. 11 a SFTHs at the top of eighth pier and b Fourier transform of the SFTHs in (a) under GM#1, (c) SFTHs at the top of eighth pier and d Fourier transform of the SFTHs in c under GM#2

• FB_SD no BA model shows a higher response as compared to the full SSI no BA model as the prescribed API guidelines overestimate the soil stiffness for the foundation soil. The structural response is amplified significantly at the deck level for FB_SD no BA model and results in higher forces and moments at pier–deck junctions. • For FB_SD with BA and full SSI with BA models, the former model is stiffer at superstructure level and deforms in higher curvature after dynamic analysis; thus, the mobilized nonlinearity and the residual response are observed to be higher for this modeling approach. Due to insufficient soil nonlinearity at the foundation and backfill components, the FB_SD with/no BA models exhibit overall higher stiffness. • In full SSI with/no BA models, due to significant soil nonlinearity, bridge response does not amplify at deck level, thus in this type of modeling bridge response is lower than the simplified SSI modeling approach. A more complete and detailed discussion of the problem has been provided in [11] for more transparent representation.

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Acknowledgments Part of this research was carried out while the first author (SD) was visiting Politecnico di Milano, Italy under INTERWEAVE Project, Erasmus Mundus Program during 2014−2017. The first author would like to thank Dr. Ali Gunay Ozcebe, Prof. Roberto Paolucci, and Prof. Lorenza Petrini from PoliMi for their valuable comments during the work and Ministry of Human Resource and Development (MHRD) for the scholarship during her Ph.D.

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18. Lysmer J, Kuhlemeyer RL (1969) Finite dynamic model for infinite media. J Eng Mech Div 95(EM4):859–877 19. Ariyarathne P, Liyanapathirana DS, Leo CJ (2013) Comparison of different two-dimensional idealizations for a geosynthetic-reinforced pile-supported embankment. Int J Geomech 13(6):754–768 20. Mokwa RL (1999) Investigation of the resistance of pile caps to lateral spreading. Dissertation, Dept. of Civil Engineering, Virginia Polytechnic Institute and State University, VA 21. AASHTO LRFD (2012) Bridge Design Specifications. American Association of State Highway and Transportation Officials; Washington DC 22. API RP2A-WSD (2003) Recommended practice for planning, designing and constructing fixed offshore platforms-working stress design. Am. Petrol. Ins. Washington DC 23. Gazetas G, Dobry R (1984) Horizontal response of piles in layered soil. J Geotech Eng Div 110(1):20–40 24. Granas JL (2016) Undrained lateral soil response of offshore monopile in layered soil. M.Tech Thesis Dept. of Civ. & Transport. Eng. NTNU 25. Brødbæk KT, Møller M, Sørensen SPH, Augustesen AH (2009) Review of p-y relationships in cohesionless soil. DCE Technical Reports No. 57, Dept. of Civ. Eng., Aalborg University 26. Monkul MM (2008) Validation of practice oriented models and influence of soil stiffness on lateral pile response due to kinematic loading. Marine Geores Geotech 26(3):145–159 27. Highway Agency BA 42/96 (2003) Amendment No. 1, highway structures: Approved procedures and general design. Sec. 3.5. The Stationary Office, London

Probabilistic Flutter Analysis of a Cantilever Wing Sandeep Kumar, Amit K. Onkar, and M. Manjuprasad

Abstract A probabilistic flutter analysis of geometrically coupled cantilever wing is carried out using first-order perturbation approach by considering bending and torsional rigidities as Gaussian random variables. The unsteadiness in the aerodynamic flow is modeled using Theodorsen’s thin airfoil theory. The probabilistic response of the wing is obtained in terms of mean, standard deviation, and coefficient of variation (COV) of real and imaginary parts of the eigenvalues at various free stream velocities. The perturbation results are also compared with Monte Carlo simulations. It is observed that the probabilistic response obtained from the perturbation approach is very accurate up to 7% COV in bending rigidity but in the case of torsional rigidity, it starts losing accuracy after 3%. Keywords Probabilistic flutter · Perturbation · MCS · Goland wing

1 Introduction In the design of aircraft, one of the critical failure phenomena, which happens in the aeroelastic system due to the fact that power pumped by aerodynamic flow is not completely able to be dissipated by the dissipative mechanism of the aeroelastic system, and the structure fails catastrophically due to diverging large amplitude vibrations. The flow velocity at which there is power balance, is called flutter velocity. In general, the material properties of the structure is not unique, and it depends on various factors such as method of manufacturing, method of testing and S. Kumar · A. K. Onkar (B) · M. Manjuprasad Academy of Scientific and Innovative Research (AcSIR), Bengaluru 560017, Karnataka, India e-mail: [email protected] S. Kumar e-mail: [email protected] M. Manjuprasad e-mail: [email protected] CSIR-National Aerospace Laboratories, Bengaluru 560017, Karnataka, India © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_12

133

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test conditions, human errors in calculation, etc. So the material properties must be modeled as stochastic parameters in order to get more realistic results. Pettit [1] showed the importance and challenges of uncertainty quantification in aeroelasticity and potential future application of uncertainty-based aircraft design over the conventional factor of safety-based design. Kurdi et al. [2] considered the box type of Goland wing with spars and ribs, and their thickness and area were considered as Gaussian random variables. For probabilistic response analysis, a Monte Carlo Simulation (MCS) approach was used where free vibration and flutter analyses were conducted using MSC-Nastran and ZONA 6 module of ZAERO, respectively. Khodaparast et al. [3] used Nastran-based doublet lattice method for aerodynamic modeling to carry out both probabilistic and non-probabilistic flutter analysis of aircraft wing. The wing with spar, ribs, upper and lower skins, and their thickness and area were considered as random variables. For probabilistic analysis, perturbation method and for non-probabilistic analysis, interval and fuzzy logic were used to determine the bounds on the flutter mode. Borello et al. [4] considered both isotropic and composite wing with structural uncertainties for flutter analysis, and the reliability analysis was carried out using various approaches such as First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM), Response Surface Method (RSM), and compared with MCS. Cheng and Xiao [5] proposed a hybrid method based on RSM, Finite Element Method (FEM), and MCS to carry the probabilistic free vibration and flutter analysis of suspension bridge. Castravete and Ibrahim [6] investigated the effect of stiffness uncertainties on flutter of a cantilever wing using MCS and first-order perturbation technique in time domain. From the literature, it is observed that most of the work reported on probabilistic flutter analysis has been done using commercial software. In the present work, probabilistic flutter analysis of a cantilever wing is carried out using physics-based approach in the frequency domain by considering both bending and torsional rigidity as independent Gaussian random variables. The impetus of the present work is on modeling two-dimensional Theodorson’s unsteady aerodynamics [7] in a strip theory approximation containing frequency-dependent terms. In the present first-order perturbation approach, the frequency-dependent aerodynamic terms are also modeled as random variables and their effect on the probabilistic flutter characteristics of the wing is studied.

2 Mathematical Modeling The Goland wing is a metallic wing [8] and its schematic diagram is shown in Fig. 1. In the figure, points F, G, and H are the aerodynamic center, center of mass, and location of elastic axis at the section, respectively. The dimensionless parameters a and e (−1 ≤ a ≤ 1 and −1 ≤ e ≤ 1) determine the location of elastic axis and inertia axis, respectively, at the section. The kinetic energy (T ) of the wing is defined as

Probabilistic Flutter Analysis of a Cantilever Wing

135

Fig. 1 Schematic representation of cantilever wing (Goland wing)

T =

1 2



l



l

I p α˙ 2 dy +

0

mxα bh˙ αdy ˙ +

0

1 2



l

m h˙ 2 dy

(1)

0

where I p , m, b, and xα (= e − a) are the mass moment of inertia per unit length about elastic axis, mass per unit length of the wing, semi chord of the wing, and dimensionless static unbalance respectively. The (˙) over h and α denotes the time derivatives of heave h(y, t) and pitch displacement α(y, t), respectively. The potential energy (V ) of the wing is expressed as V =

1 2





l

EI 0

∂2h ∂y 2

2 dy +

1 2





l

GJ 0

∂α ∂y

2 dy

(2)

The virtual external work done (δWext ) by the system is given as 

l

δWext = −



l

Ldyδh +

0

Mdyδα

(3)

0

where L and M = (M1/4 + L(0.5 + a)b) are the lift and moment per unit length, respectively. Using Hamilton’s principle, the governing equations of motion of aeroelastic system can be obtained as   ∂2h ∂2 ¨ (4) m h + mxα bα¨ + 2 E I 2 + L = 0 ∂y ∂y ∂ I p α¨ + mxα bh¨ − ∂y



∂α GJ ∂y

 −M =0

(5)

According to thin airfoil theory [7], the lift and moment per unit span at the aerodynamic center can be represented as

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      1 − a α˙ + πρ∞ b2 h¨ + U α˙ − ba α¨ (6) L = 2πρ∞ bU C(k) U α + h˙ + b 2  M1/4 = −πρ∞ b

3

   1¨ 1 a h + U α˙ + b − α¨ 2 8 2

(7)

where U is the free stream velocity and C(k) appeared in Eq. 6 is the complex function called Theodorsen’s function in which k(= bω/U , ω is the frequency of aeroelastic system) is the reduced frequency. Theodorsen’s function is represented for numerical computation as [12]: 

0.6724 0.1757 − + C(k) = 1 + 1.099 1 + ( k ) 1 + ( 0.243 ) k

 2.41 2.41 − i 1 + ( 0.214 ) 1 + ( 0.156 ) k k

(8)

The weak form of Eqs. 4 and 5 can be obtained after substituting the expressions of L and M from Eqs. 6 and 7. The elemental Finite Element (FE) equations can be written as Msb1 {w¨e } + E Mab1 {w¨e } + Msc1 {α¨e } + C Mac1 {α¨e } + (B1 Cac1 + B2 Cac1 ) {α˙e } (9) + DCab1 {w˙e } + AK ac1 {αe } + K sb1 {we } = {Fi } Mst2 {α¨e } + R Mat2 {α¨e } + Msc2 {w¨e } + T Mac2 {w¨e } + (Q 1 Cat2 + Q 2 Cat2 ) {α˙e } + SCac2 {w˙e } + P K ac2 {αe } + K st2 {αe } = {τi }

(10)

In Eqs. 9 and 10, {we } and {αe } are elemental bending and torsional degrees-offreedom, respectively, and {Fi } and {τi } are the internal load vectors, respectively. The terms appearing in Eq. 9 can be written as  Msb1 = m

yi+1 yi

 E Mab1 = πρ∞ b

2

 Msc1 = mxα b

{Nw }Nw dy yi+1

yi yi+1 yi

 C Mac1 = −πρ∞ b3 a

{Nw }Nw dy

{Nw }Nα dy yi+1

yi

{Nw }Nα dy 

B1 Cac1 = U C(k)πρ∞ b (1 − 2a) 2

yi+1

yi

{Nw }Nα dy

(11)

Probabilistic Flutter Analysis of a Cantilever Wing

137

 B2 Cac1 = U πρ∞ b2

yi+1 yi

 DCab1 = U C(k)2πρ∞ b

yi+1 yi

 AK ac1 = U C(k)2πρ∞ b 

yi+1

{Nw }Nw dy

yi+1

2

K sb1 = E I

{Nw }Nα dy

yi 

{Nw }Nα dy 

{Nw }Nw dy

yi

and the terms in Eq. 10 can be written as  Mst2 = I p

yi+1

yi

{Nα }Nα dy

   yi+1 1 R Mat2 = πρ∞ b4 a 2 + {Nα }Nα dy 8 yi  Msc2 = mxα b

yi+1 yi

 T Mac2 = −πρ∞ b a 3

 Q 1 Cat2 = −U C(k)πρ∞ b3

Q 2 Cat2

{Nα }Nw dy yi+1

yi

{Nα }Nw dy

  yi+1 1 + a (1 − 2a) {Nα }Nα dy 2 yi

   yi+1 1 = −U πρ∞ b − + a {Nα }Nα dy 2 yi 3

 SCac2 = −U C(k)2πρ∞ b

2

1 +a 2

 P K ac2 = −U 2 C(k)2πρ∞ b2  K st2 = G J

yi+1 yi



1 +a 2 

yi+1

yi



{Nα }Nw dy

yi+1

yi 

{Nα }Nα dy

{Nα }Nα dy

(12)

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where Nw and Nα are the bending and torsional shape functions, and ( ) represents the derivative with respect to y. Now assembling the elemental form of Eqs. 9 and 10, we get the assembled form of FE equation as ¨ + (U [Ca ] +U C(k) [Caω ]) {q} ˙ ([Ms ] + [Ma ]) {q}   + [K b ] + [K t ] + U 2 C(k) [K aω ] {q} = {0} (13) In Eq. 13, the displacement vector {q} contains all bending and torsional degrees-offreedom. [Ms ], [Ma ], U [Ca ], [K b ], and [K t ] are the structural inertia, aerodynamic inertia, aerodynamic damping, bending stiffness, and torsional stiffness matrices, respectively, and U C(k)[Caω ] and U 2 C(k)[K aω ] are frequency-dependent damping and stiffness matrices respectively. Let {q} be represented by harmonic function {q} = {q}e ¯ λt , where λ = −ζω + iω and in other form λ = λ R + iλ I , and substituting it in Eq. 13 gives the following expression as λ2 ([Ms ] + [Ma ]) +λ (U [Ca ] + U C(k) [Caω ])  

+ [K b ] + [K t ] + U 2 C(k) [K aω ] {q} ¯ = {0}

(14)

The above equation can be solved using state-space approach as an eigenvalue problem.

3 Stochastic Modeling The stochastic modeling of the geometrically coupled cantilever wing is based on first-order perturbation approach. In this approach, the random variables are expanded using Taylor’s series as β = βo +

∂β |r =ro δr ∂r

(15)

where βo is the mean of random variables and r denotes independent random parameters. The derivatives of the random variables are evaluated at the mean value of the random variables. In the present problem, both bending rigidity and torsional rigidity are considered as Gaussian random variables. The random bending stiffness, torsional stiffness, eigenvalue, eigenvector, and frequency-dependent Theodorsen’s function expanded via Taylor’s series truncated to first order term can be written as  ∂[K b ] |r =ro δr [K b ] = K bo + ∂r  ∂[K t ] |r =ro δr [K t ] = K to + ∂r

Probabilistic Flutter Analysis of a Cantilever Wing

λ = λo +

∂λ |r =ro δr ∂r

139

(16)

∂{q} ¯ |r =ro δr ∂r   b ∂C(k) ∂λ I C(k) = C(k o ) + |r =ro δr U ∂k ∂r {q} ¯ = {q¯ o } +

Now substituting terms from Eq. 16 to Eq. 14 and separating zeroth-order and firstorder terms: Zeroth order:   (λo )2 ([Ms ] + [Ma ]) + λo U [Ca ] + U C(k o ) [Caω ]   

+ K bo + K to + U 2 C(k o ) [K aω ] {q¯ o } = {0} (17) First order:   (λo )2 ([Ms ] + [Ma ]) + λo U [Ca ] + U C(k o ) [Caω ] +



  ∂{q} ¯ K bo + K to + U 2 C(k o ) [K aω ] ∂r

∂λ o 2λ ([Ms ] + [Ma ]) + U [Ca ] + U C(k o ) [Caω ] {q¯ o } ∂r

b ∂C(k) 2 ∂λ I b ∂C(k) o λ U [Caω ] + U [K aω ] {q¯ o } + ∂r U ∂k U ∂k   ∂[K b ] ∂[K t ] + {q¯ o } =− ∂r ∂r

+

(18)

Now multiplying Eq. 18 by adjoint eigenvector or left eigenvector transpose { X¯ ad j }T [9–11], we get the following equation as:   { X¯ ad j }T (λo )2 ([Ms ] + [Ma ]) + λo U [Ca ] + U C(k o ) [Caω ]    ∂{q} ¯ + K bo + K to + U 2 C(k o ) [K aω ] ∂r

∂λ ¯ T o { X ad j } 2λ ([Ms ] + [Ma ]) + U [Ca ] + U C(k o ) [Caω ] {q¯ o } + ∂r b ∂C(k)

b ∂C(k) 2 ∂λ I ¯ { X ad j }T λo U [Caω ] + U [K aω ] {q¯ o } + ∂r U ∂k U ∂k  ∂[K b ] ∂[K t ] T ¯ + {q¯ o } (19) = −{ X ad j } ∂r ∂r

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The eigenvector derivative coefficient matrix in Eq. 19 becomes zero. Equation 19 can be rewritten as

∂λ ¯ { X ad j }T 2λo ([Ms ] + [Ma ]) + U [Ca ] + U C(k o ) [Caω ] {q¯ o } ∂r b ∂C(k)

b ∂C(k) 2 ∂λ I ¯ { X ad j }T λo U [Caω ] + U [K aω ] {q¯ o } + ∂r U ∂k U ∂k  ∂[K b ] ∂[K t ] T ¯ {q¯ o } = −{ X ad j } + ∂r ∂r

(20)

Substituting complex form of eigenvalue (λ = λ R + iλ I ) in Eq. 20 and solving for real and imaginary part of eigenvalue derivatives as φ(γ − ϕ) − η(ψ + ξ) ∂λ R = ∂r ϕ(γ − ϕ) − ξ(ψ + ξ)

(21)

∂λ I ηϕ − φξ = ∂r ϕ(γ − ϕ) − ξ(ψ + ξ)

(22)

where terms ξ, ϕ, γ, ψ, η and φ can be obtained from the expression given below:

ξ + iϕ = { X¯ ad j }T 2λo ([Ms ] + [Ma ]) + U [Ca ] + U C(k o ) [Caω ] {q¯ o } b ∂C(k)

b ∂C(k) 2 λo U [Caω ] + U [K aω ] {q¯ o } U ∂k U ∂k   ∂[K b ] ∂[K t ] T ¯ + {q¯ o } η + iφ = −{ X ad j } ∂r ∂r

γ + iψ = { X¯ ad j }T

(23)

The variance of the real part and imaginary part can be obtained as  V ar (λ R ) =  V ar (λ I ) =

∂λ R ∂r ∂λ I ∂r

2 V ar (r )

(24)

V ar (r )

(25)

2

where variance of λ I also represents variance of frequency (ω).

Probabilistic Flutter Analysis of a Cantilever Wing

141

4 Results and Discussion First, the mean flutter analysis of the wing is performed using zeroth-order equation based on pk method using the mean data given in Table 1. Figure 2 shows the variation of mean damping (λoR ) and frequency (λoI ) of the wing at various free stream velocity U . From the figure, the mean flutter velocity is found to be 137.38 m/s, which matches well with those given in [12].

Table 1 Properties of Goland wing [12] Parameters Description EI GJ m xα

Values

Span-wise bending stiffness Span-wise torsion stiffness Mass per unit span Dimensionless static unbalance Elastic axis location parameter Semi-reference chord Span Mass moment of inertia per unit span Free stream density

a b l Ip ρ∞

9.77 × 106 Nm2 0.988 × 106 Nm2 35.719 Kg/m 0.33 −0.2 0.9144 m 6.09 m 6.5704 Kgm2 /m 1.225 Kg/m3

10

o

λR

0 −10 −20 −30 −40

0

50

100

150

100

150

Velocity (U) 140

Mode 1

100

Mode 2

λ

o I

120

80 60 40

0

50

Velocity (U)

Fig. 2 Variation of real and imaginary part of mean eigenvalue with free stream velocity (U )

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Table 2 Variation of mean and SD of eigenvalues with free stream velocity for COV of E I = 0.05 Velocity U (m/s)

Perturbation approach Mean (λoR )

Mean (λoI )

100

−8.9547

125 135

Monte Carlo Simulation (with 10,000 Samples) SD (λ R )

SD (λ R )

SD (λ I )

SD (λoR )

Mean (λ I )

Mean (λoI )

50.5290 0.0276

0.9734

−8.9568

50.5050 0.0275

0.9741

−17.0811 57.6563 0.0424

0.9228

−17.0839 57.6268 0.0415

0.9242

−26.4397 60.3614 0.0346

0.8351

−26.4374 60.3338 0.0364

0.8369

100

−5.9584

113.8010 0.0928

0.4779

−5.9591

113.8010 0.0927

0.4779

125

−5.9187

96.2131 0.1907

0.5428

−5.9215

96.2191 0.1904

0.5427

135

−1.9540

84.9500 0.2509

0.7361

−1.9678

84.9576 0.2516

0.7346

Mode 1

Mode 2

The probabilistic flutter analysis of cantilever wing is performed by treating E I and G J as stochastically independent Gaussian random variables with 5% COV in E I and G J . The mean and Standard Deviation (SD) of the first two eigenvalues obtained using first-order perturbation approach and MCS are shown in Tables 2 and 3, respectively. From the tables, it is observed that the mean and SD of eigenvalues obtained using the perturbation approach matches well with MCS at various free stream velocities which validate the present first-order perturbation approach for probabilistic analysis. From the tables, it is also observed that there is an increasing trend of SD of damping (real part) for second eigenmode (flutter mode) with increasing velocity, which indicates that eigenmodes are sensitive to probabilistic variations which can have an influence on the design of aircraft with proper flutter margin. Comparing the SD of the real part for both mode 1 and mode 2, the standard deviation of mode 2 is greater than mode 1 corresponding to a particular velocity

Table 3 Variation of mean and SD of eigenvalues with free stream velocity for COV of G J = 0.05 Velocity U (m/s)

Perturbation approach

Monte Carlo Simulation (with 10,000 Samples) SD (λ R )

SD (λ I ) Mean (λoR )

Mean (λoI )

SD (λ R )

SD (λ I )

Mean (λoR )

Mean (λoI )

100

−8.9547

50.5290 0.2654

0.1501

−8.9787

50.5406 0.2729

0.1528

125

−17.0811 57.6563 1.5942

0.7578

−17.3649 57.6105 1.7244

0.6612

135

−26.4397 60.3614 3.2967

0.6939

−26.5630 59.9716 3.1417

0.8463

100

−5.9584

113.8010 0.1154

3.6420

−5.9461

113.7181 0.1198

3.6478

125

−5.9187

96.2131 0.7781

4.9591

−5.7294

96.0161 0.9259

4.9599

135

−1.9540

84.9500 2.8628

5.1211

−1.6325

85.2785 2.6318

4.8720

Mode 1

Mode 2

Probabilistic Flutter Analysis of a Cantilever Wing

143

−3

5

x 10

0.035

COV of λI of Mode 1

COV of λR of Mode 1

0.03 4

3

2

0.025 0.02 0.015 0.01

1 0.005

Perturbation MCS (10000 Samples) 0

0.02

0.04

0.06

COV of EI

0.08

0

0.1

Perturbation MCS (10000 Samples) 0.02

0.04

0.07

0.012

0.06

0.01

0.05 0.04 0.03 0.02 0.01 0

Perturbation MCS (10000 Samples) 0.02

0.04

0.06

COV of EI

(c)

0.08

0.1

(b)

COV of λI of Mode 2

COV of λR of Mode 2

(a)

0.06

COV of EI

0.08

0.1

0.008 0.006 0.004 0.002 0

Perturbation MCS (10000 Samples) 0.02

0.04

0.06

COV of EI

0.08

0.1

(d)

Fig. 3 COV of real and imaginary part of eigenvalues of various modes for different COV of bending rigidity at U = 125 m/s a λ R of mode 1, b λ I of mode 1, c λ R of mode 2, and d λ I of mode 2

for the variation in bending rigidity and mode 1 in the case of variation in torsional rigidity. This means that uncertainty in bending rigidity affects the real part of mode 2 whereas torsional uncertainty affects the real part of mode 1. Figure 3 shows the variations in the real and imaginary part of eigenvalues for different COV of E I at velocity 125 m/s and a very good agreement with MCS is observed. The COV follows a linear relationship for most of the cases except for λ R of the first mode, which starts significantly deviating after 7% COV of E I . Figure 4 shows the variation in the real and imaginary part of eigenvalues at a free stream velocity of 125 m/s for different COV of G J . From the figure, it can be observed that the COV of λ R of mode 1, mode 2, and λ I of mode 1 are accurate up to 3% of COV of G J , and λ I of mode 2 is accurate for all value of COV of G J . A linear relationship between COV of eigenvalues up to 3% COV of G J can be also observed. Figure 5 shows the COV of real and imaginary part of eigenvalues at various free stream velocities due to variation in E I and G J . From the figure, it can be observed

144

S. Kumar et al. −3

0.1

14

x 10

0.09

COV of λ of Mode 1

0.07 0.06

10

I

COV of λR of Mode 1

12 0.08

0.05 0.04 0.03

0.01 0.01

0.02

0.03

COV of GJ

0.04

6 4

Perturbation MCS (10000 Samples)

0.02

8

2 0.01

0.05

Perturbation MCS (10000 Samples) 0.02

(a)

0.05

0.055 0.05

0.16 0.14

COV of λ of Mode 2

0.12 0.1

0.045 0.04 0.035

I

COV of λR of Mode 2

0.04

(b)

0.18

0.08 0.06 0.04 0.02 0.01

0.03

COV of GJ

Perturbation MCS (10000 Samples) 0.02

0.03

COV of GJ

(c)

0.04

0.05

0.03 0.025 0.02 Perturbation MCS (10000 Simulations)

0.015 0.01 0.01

0.02

0.03

COV of GJ

0.04

0.05

(d)

Fig. 4 COV of real and imaginary part of eigenvalues of various modes for different COV of torsional rigidity at U = 125 m/s a λ R of mode 1, b λ I of mode 1, c λ R of mode 2, and d λ I of mode 2

that the COV of λ R for mode 2 has very high value near to flutter velocity due to variation in E I and G J . It is also observed that the COV of λ I due to bending rigidity uncertainty is low for mode 2 in comparison with torsional rigidity uncertainty. This may be due to the fact that eigenvalues are more sensitive due to variation in torsional rigidity near to flutter velocity. Since mode 2 is the flutter mode, we further investigate the probability density function (pdf ) of second mode at various velocities. Figure 6 shows the pdf of real and imaginary part of eigenvalues for COV of E I (= 0.05) at 125 m/s and 137 m/s obtained from MCS with 10,000 simulations. From the figure, it is observed that the pdf of λ R of mode 2 at velocity 137 m/s is slightly skewed than at velocity 125 m/s. It is also observed that λ I of mode 2 at velocity 137 m/s shows a wider band than at velocity 125 m/s. Figure 7 shows the pdf of real and imaginary part of eigenvalues for COV of GJ (= 0.05) at 125 m/s and 137 m/s obtained from MCS with 10,000 simulations. From the figure, it is observed that the pdf of λ R of mode 2 at velocity 137 m/s shows a wider band than at velocity 125 m/s. The pdf of λ R at velocity 137 m/s indicates

Probabilistic Flutter Analysis of a Cantilever Wing

145 200

COV of λR

COV of λR

15 10 5 0 0

50

Velocity (U)

100

0 0

150

0.08

COV of λ

I

I

0.03

COV of λ

100

0.02 0.01

Mode 1 Mode 2

0 0

50

Velocity (U)

100

0.06

50

Velocity (U)

100

150

100

150

Mode 1 Mode 2

0.04 0.02 0 0

150

50

Velocity (U)

(b)

(a)

Fig. 5 COV of real and imaginary part of eigenvalues at various free stream velocities due to a COV of E I = 0.05, b COV of G J = 0.05 2.5

0.8 0.7

2

0.6 0.5

pdf

pdf

1.5

1

0.4 0.3 0.2

0.5

0.1 0 −7

−6.5

−6

λR

−5.5

0 94

−5

95

96

λI

97

98

99

(b) U = 125 m/s

(a) U = 125 m/s 1.8

0.5

1.6 0.4

1.4

0.3

1

pdf

pdf

1.2

0.8

0.2

0.6 0.4

0.1

0.2 0 −1.5

−1

−0.5

λR

0

(c) U = 137 m/s

0.5

0 80

81

82

83

λI

84

85

86

87

(d) U = 137 m/s

Fig. 6 pdfs of real and imaginary part of eigenvalue of mode 2 for COV of E I = 0.05 a λ R at U = 125 m/s, b λ I at U = 125 m/s, c λ R at U = 137 m/s, and d λ I at U = 137 m/s

146

S. Kumar et al. 0.7

0.08

0.6

0.07 0.06 0.05

0.4

pdf

pdf

0.5

0.3

0.03

0.2

0.02

0.1 0 −8

0.04

0.01 −4

−6

λR

0

−2

0 70

2

90

80

λI

110

100

120

(b) U = 125 m/s

(a) U = 125 m/s 0.09

0.14

0.08

0.12

0.07 0.06

0.08

pdf

pdf

0.1

0.06

0.05 0.04 0.03

0.04

0.02 0.02 0 −10

0.01 −5

0

λR

5

(c) U = 137 m/s

10

0 70

75

80

85

λI

90

95

100

105

(d) U = 137 m/s

Fig. 7 pdfs of real and imaginary part of eigenvalue of mode 2 for COV of G J = 0.05 a λ R at U = 125 m/s, b λ I at U = 125 m/s, c λ R at U = 137 m/s, and d λ I at U = 137 m/s

that there is a chance of occurrence of flutter due to uncertainties in the bending and torsional rigidities as seen in Figs. 6c and 7c (+ ve region under pdf of λ R ). The figures also indicate that chances of occurrence of flutter is high in the case of variation in G J as compared to E I due to large positive area under pdf of λ R of mode 2.

5 Conclusions The probabilistic flutter analysis of geometrically coupled cantilever wing has been carried out using first order perturbation approach considering bending rigidity and torsional rigidity as Gaussian random variables. The probabilistic response of the wing has been obtained in terms of mean and SD of the real and imaginary part

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of eigenvalues for different COV of random variables at a particular velocity using the perturbation approach. The results are compared with MCS and found to be in very good agreement for variation in E I for all COV considered. In the case of the variation in G J , the results show accuracy up to 3% COV in G J beyond which it loses its accuracy, which limits the applicability of perturbation approach. From the results obtained, we can conclude that the COV of λ R of the flutter mode near the flutter velocity starts increasing, and theoretically becomes infinite at the flutter velocity. The pdf of λ R for flutter mode shows that there are chances of occurrence of flutter at lower velocities in the presence of uncertainties. Hence, consideration of uncertainties is very important in the design of the aircraft for proper flutter margin.

References 1. Pettit CL (2004) Uncertainty quantification in aeroelasticity: recent results and research challenges. J Aircraft 41(5):1217–1229. https://doi.org/10.2514/1.3961 2. Kurdi M, Lindsley N, Beran P (2007) Uncertainty quantification of the Goland+ wing’s flutter boundary. In: AIAA atmospheric flight mechanics and exhibit, hilton head, South Carolina, 20–23 August 2007. https://doi.org/10.2514/6.2007-6309 3. Khodaparast H, Mottershead J, Badcock K (2010) Propagation of structural uncertainty to linear aeroelastic stability. Comput Struct 88(3-4):223–236. https://doi.org/10.1016/j.compstruc. 2009.10.005 4. Borello F, Cestino E, Frulla G (2010) Structural uncertainty effect on classical wing flutter characteristics. J Aerospace Eng 23(4):327–338. https://doi.org/10.1061/(ASCE)AS.19435525.0000049 5. Cheng J, Xiao RC (2005) Probabilistic free vibration and flutter analyses of suspension bridges. Eng Struct 27(10):1509–1518. https://doi.org/10.1016/j.engstruct.2005.03.016 6. Castravete SC, Ibrahim RA (2008) Effect of stiffness uncertainties on the flutter of a cantilever wing. AIAA J 46(4):925–935. https://doi.org/10.2514/1.31692 7. Theodorsen T (1935) General theory of aerodynamic instability and the mechanism of flutter. Tech. Rep, NACA, p 496 8. Goland M, Buffalo N (1945) The flutter of a uniform cantilever wing. J Appl Mech-T ASME 12(4):A197–A208 9. Nelson RB (1976) Simplified calculation of eigenvector derivatives. AIAA J 14(9):1201–1205. https://doi.org/10.2514/3.7211 10. Ji-ming L, Wei W (1987) First-order perturbation solution to the complex eigenvalues. Appl Math Mech 8(6):509–514. https://doi.org/10.1007/BF02017399 11. Adhikari S (1999) Rates of change of eigenvalues and eigenvectors in damped dynamic system. AIAA J 37(11):1452–1458. https://doi.org/10.2514/2.622 12. Irani S, Sazesh S (2013) A new flutter speed analysis method using stochastic approach. J Fluid Struct 40:105–114. https://doi.org/10.1016/j.jfluidstructs.2013.03.018

Optimum Support Layout Design for Periodically Loaded Structures Using Topology Optimization A. T. Korade and S. Rakshit

Abstract Optimal support layout design of structures under forced vibration is determined using topology optimization method. The vibrations of a structure can be reduced by adding external supports to the structure. Linear elastic springs are used as the external supports and the objective function used is dynamic compliance of structure. Solid Isometric Material with Penalty (SIMP) method is employed to solve the topology optimization problem. The optimal support locations are determined for cantilever beam and plate structures subjected to harmonic loads. Keywords Topology optimization · Dynamic compliance · Solid Isometric Material with Penalty (SIMP) · Modal analysis

1 Introduction Optimal support layout design is an important area of research for structures subjected to both static and dynamic loading. In this work, the formulation is given for optimal support layout of structures under forced vibration using topology optimization method. Topology optimization is classified based on four methods used, namely, Homogenization method, SIMP (Solid Isotropic Material with Penalization) method, ESO (Evolutionary Structural Optimization), method and Level set method [1]. In this work, the SIMP method is used to formulate the objective function. Optimal support location of structures is determined using various objectives, e.g., maximization of the Eigen frequencies [1, 2], minimization of the maximum deflection of a structure [3], and minimizing the dynamic compliance of a structure [4]. Homogenization method can be used to determine the optimum topology of structures under vibration for different objectives, for example, to maximize the stiffness, Eigen frequency maximization [5]. In this paper, the objective function A. T. Korade (B) · S. Rakshit Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600036, India e-mail: [email protected] S. Rakshit e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_13

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used is the dynamic compliance of the structure. Dynamic compliance can be stated as external force times the displacement [6] or the average input power over a cycle [4]. The former definition is used as the objective function to determine the optimal support location for cantilever beam and plate as shown later in the paper.

2 Problem Formulation 2.1 SIMP Model and Finite Element Discretization “Topology optimization of solid structures involves the determination of features such as the number and location and shape of holes and the connectivity of the domain with predefined boundary conditions” [7]. In this work, topology optimization is used to determine the optimal support location to minimize the dynamic compliance of the structure. In SIMP model, density variable with penalization is used as a design variable. Design domain is discretized using finite elements and the nodes of the FE discretization are considered as a possible location of supports, which are modeled using linear springs. Following the SIMP method, spring stiffness can be modeled as ki = xi∝ ko

(1)

where xi is the design variable corresponding to node i, ko is stiffness of a spring, α is penalization factor. Since xi is a real number varying continuously between 0 and 1, and the presence or absence of a support at a particular node is a binary problem, xi , may be taken as the probability of positioning a support at node i. Governing equation of vibrating structure (without damping) is ¨ + [K ]{u} = { f e } [M]{u}

(2)

where [M] = Global mass matrix, [K ] = Global stiffness matrix, { f e } = External excitation force = { f }cos(wt), f = Force amplitude, w = Excitation frequency. Equation (2) can be decoupled using modal analysis, which converts the physical coordinate system “u” to modal coordinate system “y”. Hence, governing equation in the modal coordinate system is given as [I ]{ y¨ } + [K  ]{y} = { f mod }

(3)

Optimum Support Layout Design for Periodically Loaded …

151

where [I ] = Identity matrix,   2 K = Diagonal matrix with diagonal entries as wni , wni = Natural frequency corresponding to each mode, i = 1, 2, . . . . . . , (NMOD) { f mod } = Modal force = {F mod }cos(wt) NMOD = Number of modal masses of system depends upon the finite element discretization. Solution to the system of Eq. (3) is given by yi (t) = (yio − Ai )cos(wni t) +

y˙io sin(wni t) + Ai cos(wt) wni

(4)

where yio = Initial displacement, i = 1, 2, . . . . . . , (NMOD) y˙io = Initial velocity, i = 1, 2, . . . . . . , (NMOD) F

i Ai = w2 mod , F mod i = Magnitude of modal force corresponding to node “i”. ( ni −w2 ) In this paper, the initial displacement and velocity in the physical coordinate system “u” are taken as zero. Hence in the modal coordinate system “y”, initial conditions will also be zero. The optimal support location is determined for minimizing the dynamic compliance of the structure. The dynamic compliance (DC) is given by

DC =

NMOD 

 2π

2

w

∫ f ei (t)u i (t)dt

(5)

0

i=1

The above Eq. (5) is in physical coordinate system. In modal coordinate system, the dynamic compliance can be stated as DC =

NMOD 

2

 2π w

∫ f 0

i=1

mod

i

(t)yi (t)dt

(6)

Solution to Eq. (6) after applying initial conditions yio = 0 and y˙io = 0 is DC =

NMOD  i=1



 F mod i Ai

 2

π wni 2π wni sin + 2 2 w w w − wni

(7)

Next, it is proved that dynamic compliance in physical coordinate system and modal coordinate system is the same, i.e., Eq. (5) = Eq. (6).

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2.2 Proof Dynamic compliance is given by DC = (u T f e )2

(8)

By using modal analysis, we can write f mod = ∅T f e ,

(9)

u = ∅y

(10)

∅ = Normalized Eigenvector matrix ∅ = {∅1 , ∅2 , . . . . . . . . . ., ∅NMOD } ϕi ∅i = √ mi ϕi = Eigenvector corresponding to ith mode. m i = Modal mass corresponding to ith mode, ϕiT Mϕi Hence, from Eqs. (8), (9), and (10), DC can be written as   2 NMOD DC = y T f mod = (yi f modi )2

(11)

i=1

Integrating over a time period

 2π w

⎧

DC =

will give energy over a time period, i.e.,

⎪ w NMOD  ⎨ i=1

⎪ ⎩

2π

0

f modi (t)yi (t) dt

⎫2 ⎪ ⎬ ⎪ ⎭

(12)

2.3 Problem Statement The optimization is initialized with spring at all nodes, e.g., Fig. 1b represents the ground structure of beam with springs supports. In this work, a number of external springs used are constrained. As the sum of all probabilities of “n” design variables should be less than or equal to the maximum number of springs N, the constraint on the number of supports is given as

Optimum Support Layout Design for Periodically Loaded …

(a)

153

(b)

Fig. 1 a Finite element discretization of beam. b Ground structure of beam with springs supports n 

xi ≤ N

(13)

i=1

Thus, by using Eqs. (12) and (13) the optimal support layout design problem can be stated as ⎧

Minimize: min x

NMOD ⎨ w  ⎪ i=1

Subject to:

⎪ ⎩

2π

f modi (t)yi (t) dt

0

n 

⎫2 ⎪ ⎬ ⎪ ⎭

(14)

xi ≤ N

i=1

3 Examples Here results are presented for the cantilever beam and plate to determine the optimal support location using optimization problem stated in Eq. (14). The constrained optimization problem is solved using MATLAB’s fmincon optimization solver.

3.1 Beam Consider a cantilever beam of length “L” = 0.45 m, width “b” = 0.02 m, thickness “h” = 0.003 m, material density “d” = 7850 kg/m3 and Young’s modulus E = 2.1 × 1011 Pa. Stiffness of external spring support used is k o = 2.8 × 106 N/m. As shown in Fig. 1b, optimization starts with the springs at all nodes, i.e., xi = 1, except where external forces or moments or lumped mass are applied and at fixed nodes. Figure 2, shows optimal location of two support springs shown as “*”, obtained for the external excitation frequency close to second natural frequency of beam, i.e., w = 487.39 Rad/s. The beam is subjected to point force and moment of magnitude

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Fig. 2 Optimal location of two support springs (shown by “*”) for a cantilever beam loaded with a harmonic force of 100 N at the middle (shown by “×”) and harmonic moment of 100 N-m (shown by “∇”) at the end, both with an excitation frequency of w = 487.39 Rad/s. Second mode shape of the cantilever beam is shown by a solid line and the undeformed beam is shown by a dashed line

100 N and 100 Nm, respectively, as shown in Figs. 2 and 3. Similarly Fig. 3 shows the optimal location of two support springs (shown as “*”), when a lumped mass of 2 kg is placed at the center of the beam and external excitation frequency is close to second natural frequency of system, i.e., for w = 233.3004 Rad/s.

3.2 Plate Consider a cantilever plate of length “L” = 1 m, width “W ” = 1 m, thickness “h” = 0.003 m, material density “d” = 7850 kg/m3 and Young’s modulus E = 2.1 × 1011 Pa, Poisson ratio, “u” = 0.3. Stiffness of external spring support used is ko = 2.8 × 106 N/m. Figure 4 shows the cantilever plate with the finite element mesh. Optimal location of two support springs are shown as blue dots. The plate is subjected to a harmonic point force of 1500 N and a harmonic moment of 600 N-m both with excitation frequency of w = 1755.1562 Rad/s. The external excitation frequency is close to the first natural frequency of plate. Solid gray line represents the clamped side of plate. Here also, the external force and moment locations are excluded from the design domain.

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155

Fig. 3 Optimal location of two support springs (shown by “*”) for a cantilever beam with lumped mass of 2 kg at center loaded with harmonic force of 100 N at middle (shown by “×”) and harmonic moment of 100 N-m (shown by “∇”) at the end, both with an excitation frequency of w = 233.3004 Rad/s. Second mode shape of the cantilever beam is shown by a solid line and the undeformed beam is shown by a dashed line

Fig. 4 Optimal locations of spring supports (shown by blue dots) for a cantilever plate subjected to external harmonic force of 1500 N (location shown by F and down arrow) and harmonic moment of 600 N-m (location shown by M and curved arrow) with an excitation frequency of w = 1755.1562 Rad/s. The fixed end is shown by a solid gray line. The grid represents finite element mesh

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4 Conclusions In this paper, optimal location of spring support is determined for cantilever beam and plate structures subjected to harmonic loads. Dynamic compliance of the structure is minimized with bounds on the number of external spring supports to determine the optimal locations. From the examples, it is observed that optimal locations are near to the maximum deflection points or near to the point of external excitation force.

References 1. Zargham S, Ward TA, Ramli R, Badruddin IA (2016) Topology Optimization: a review for structural designs under vibration problems. Struct Multidisc Optim 53:1157–1177 2. Jihong Z, Weihong Z (2006) Maximization of structural natural frequency with optimal support layout. Struct Multidisc Optim 31:462–469 3. Won KM, Park YS (1998) Optimal support position for a structure to maximize its fundamental natural frequency. J Sound Vib 213:801–812 4. Ma ZD, Kikuchi N (1995) Topological design for vibrating structure. Comput Methods Appl Mech Eng 121:259–280 5. Wang D (2004) Optimization of support positions to minimize maximal deflection of a structure. Int J Solids Struct 41:7445–7458 6. Jog CS (2002) Topology design of structure subjected periodic loading. J Sound Vibr Eng 253:687–709 7. Bendsoe MP, Sigmund O (2002) Topology optimization theory, methods and application. Springer, Denmark

Seismic Behaviour of RC Building Frame Considering Soil–Structure Interaction Effects Nishant Sharma, Kaustubh Dasgupta, and Arindam Dey

Abstract Reinforced Concrete (RC) frame buildings constitute a large fraction of the urban building stock in India. During past earthquakes, a number of these buildings have been observed to suffer extensive damages. Although conventional codeprescribed seismic design methodology does not account for consideration of soil– structure interaction, the presence of soil can cause a significant change in the seismic behaviour of the buildings. The present article investigates the seismic behaviour of an RC building frame under the influence of nonlinear Soil–Structure Interaction (SSI). Finite element analysis of a five-storeyed building frame is carried out under applied ground motions to simulate the possible effects of earthquake shaking. Analysis of various response entities reveals the mechanisms by which the influence of SSI affects the structural behaviour. Moreover, the analysis demonstrates crucial aspects of the nonlinear behaviour and energy dissipation characteristics of the building frame under the influence of SSI. The study shows that seismic soil–structure interaction cannot be ignored, contrary to the present state of practice and guidelines of the design codes of various countries. Keywords Soil–structure interaction · Frame building · Ground motion · Viscous boundaries · Nonlinear behaviour · Time history analysis

N. Sharma · K. Dasgupta (B) · A. Dey Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India e-mail: [email protected] N. Sharma e-mail: [email protected] A. Dey e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_14

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1 Introduction Effect of seismic soil–structure interaction on the behaviour of the structure has been in debate for decades and the present state of practice is to ignore the effect of SSI for the seismic analysis and design of building structures. However, for a flexible foundation soil system, it is imperative that the interaction mechanism would play a role in the behaviour of the structure in the event of an earthquake. Also, very few past studies have simultaneously considered coupling of structural and soil nonlinearities in SSI problems. Therefore, behaviour of the structure considering nonlinear soil– structure interaction has not been investigated in detail. The present article attempts to investigate and understand the non-linear behaviour of the structure considering SSI effects by modelling the soil as a continuum along with the structure in the finite element based software framework, OpenSEES [1]. The structure chosen is modelled with various soil conditions and subjected to earthquake excitation using a ground motion record. The present article aims to understand how the non-linear behaviour of an RC building frame is modified with the inclusion of SSI effects.

2 Modelling Two-dimensional modelling of the structure, foundation-soil system has been carried out in OpenSEES. The modelled SSI system along with the adopted mesh is shown in Fig. 1, and the modelling aspects are discussed in the following subsections.

Fig. 1 SSI system and meshing adopted

Seismic Behaviour of RC Building Frame Considering … Table 1 Details of reinforced concrete frame sections

159

Member

Size (mm2 )

Main r/f

Beam

250 × 400

4@20 mm ϕ 2 legged 8 mm@ (+) 100a

Shear r/f

4@20 mm ϕ (−) Column

400 × 400

8@16 mm ϕ 3 legged 8 mm@ 75*; @ 200** Uniformly distributed

a Uniform

spacing of stirrup in millimeter (mm); * Spacing of stirrup near ends of the member in mm; ** Spacing of stirrup elsewhere in member in millimeter (mm); (+) Tension reinforcement; (−) Compression reinforcement; ϕ = dia

2.1 Structural System The structural system considered in the present study is a five-storied RC building frame with five bays. The uniform storey height and bay width are considered as 3 m, respectively. The structure is located on a soft soil site in Seismic Zone V as per the Indian seismic design code IS 1893: Part I [2]. For the purpose of design and analysis of the structure, relevant Indian standards have been referred [2–5]. The sectional details of the beams and the columns are shown in Table 1. Grade of concrete and reinforcing steel used are considered as M25 and Fe415, respectively. Various column locations have been marked as C1-C6 (Fig. 1). C1 and C6 are the exterior most columns, C2 and C5 are intermediate columns, and C3 and C4 are innermost columns.

2.2 Foundation Soil System Rectangular sandy soil domain of length 10 times the structural base width (10 × 15 m = 150 m) is considered (Fig. 1). Bedrock is assumed to be at a depth of 30 m from the surface of the ground. Four-node quadrilateral elements, with bilinear isoparametric formulation, are used to model the soil as a continuum. A non-uniform meshing is adopted to appropriately capture the soil behaviour in the region of interest. In total, 3822 nodes and 3650 elements are used for representing the soil domain. The size of the smallest element used is 0.375 m. Pile foundation is used for supporting the structure on the soil medium. The lateral force estimation and design of pile group have been done using IS 2911: Part I/Sec I [6] and other relevant Indian standards. Since significant nonlinearity is not expected in the pile groups, the pile elements are assigned linear elastic sectional properties. The piles are connected to the soil elements using zero-length rigid link member and interface nonlinearity has not been considered in the analysis. The pile groups are

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Table 2 Details of soil properties and pile foundation Soil type

ρ (t/m3 )

ϕ

ν

e

Gr (kPa)

Dia (m)

Length (m)

n

Soft Soil (SS)

1.7

29

0.33

0.85

5.5 × 104

0.5

8.0

3

104

Med Soil (MS)

1.9

33

0.33

0.70

7.5 ×

0.5

7.0

3

Med Dense Soil (MDS)

2.0

37

0.35

0.55

1.0 × 105

0.5

6.0

3

Dense Soil (DS)

2.1

40

0.35

0.45

1.3 × 105

0.5

5.0

3

ρ = Density; ϕ = Friction angle; ν = Poisson’s ratio; Gr is reference low strain shear modulus measured at 80 kPa reference pressure, n = number of piles in a group

connected to each other using grade beams of size 0.4 m × 0.4 m. In the present study, four different types of soil have been considered and for each soil condition, the pile groups have been designed. In practice, it is common to keep the diameter of the piles as the same for various locations and to adjust the length of the piles for obtaining the appropriate design capacity of the pile foundation. Therefore, in the present study, the pile groups for different soil conditions have been designed keeping the diameter as 0.5 m and appropriate lengths. Table 2 shows the basic soil properties considered and the details of pile groups designed.

2.3 Material Properties

30

500

25

400

Stress (MPa)

Stress (MPa)

PressureDependMultiYield material has been used to simulate the nonlinear behaviour of the soil. The plastic behaviour in this material model follows the Drucker–Prager yield surfaces (nested yield surface) criteria. Stress–strain data for confined and unconfined concrete are obtained using the relationships prescribed by Chang and Mander [7] and they are shown in Fig. 2a. The stress–strain relationship used to model the reinforcing steel [8] is shown in Fig. 2b. The stress–strain values are assigned to the fiber section for modelling the beam and the column sections.

20 15 10 Unconfined Concrete

5 0

300 200 100

Fe 415

Confined Concrete

0.0

0.2

0.4

0.6

0.8

0

0.0

Strain (%)

(a) Fig. 2 Stress–strain relationship for a concrete b rebar steel

0.2

0.4

0.6

0.8

Strain (%)

(b)

1.0

1.2

Seismic Behaviour of RC Building Frame Considering …

161

2.4 Plastic Hinge The nonlinearity in the building frame is considered in the form of lumped plastic hinges that develop over a length at the ends of the member. The length of the plastic hinge for the members is obtained using the relationship shown in Eq. (1) proposed by Paulay and Priestley [9]. l p = 0.08l + 0.022db

(1)

where l p is the length of plastic hinge, l is the distance between the points of contraflexure and db is the diameter of the longitudinal bar used. The assignment of plastic hinge length ensures that the nonlinearity developed in the building frame members is localized at the end regions.

2.5 Soil Domain Boundaries For SSI studies, modelling of the boundaries is very important to simulate the effect of radiation damping and the application of excitation input. Also, proper modelling of the boundaries allow the truncation of the soil domain to a finite extent. In the present study, the vertical and horizontal boundaries have been modelled using Lysmer– Kuhlemeyer viscous dashpots [10] to arrest the waves at the boundary in the transverse and longitudinal directions and to prevent the same from reflecting back into the soil medium after being incident at the far-off boundaries. The ground motion input, for to the SSI cases, is applied in the form of equivalent nodal forces using the procedure outlined in [11]. For the structure supported on rock (R), it is appropriate to restrain the translational and the rotational degrees of freedom at the column bases to simulate the characteristics of rocky medium (R).

3 Rayleigh Damping The presence of nonlinearity in the soil produces high-frequency spurious oscillations, due to underdamped modes, in the numerical solution of the SSI system. To overcome the issue, the HHT-α method for time step integration [12] may be used. For cases wherein the HHT-α method is ineffective for removal of the spurious oscillations, incorporation of a small amount of Rayleigh damping is useful. Therefore, in the present study, Rayleigh damping has been considered. It is assumed that all the contributing modes are having approximately the same damping ratio of 5%. For the fixed base analysis, the frequencies of the various modes of the structure can be estimated using the conventional eigenvalue analysis. However, for the SSI system,

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Table 3 Rayleigh damping coefficients Soil type

Avg. vs (m/s)

ω1 (rad/s)

ω2 (rad/s)

a0

a1

Soft Soil (SS)

190

29.85

49.74

1.86

0.001

Med Soil (MS)

220

34.56

57.60

2.16

0.001

Med Dense Soil(MDS)

250

39.27

65.45

2.45

0.001

Dense Soil (DS)

280

48.93

73.30

2.75

0.001

the conventional eigenvalue analysis cannot be applied. Hence, the theoretical relationship mentioned in [13] is used. The frequencies corresponding to the first and the second mode are chosen for the estimation of Rayleigh damping coefficients using the relationships mentioned in [14]. Based on the damping ratios and the frequency of the modes, the coefficients are estimated to form the damping matrix. Table 3 shows the details of the frequencies and Rayleigh damping coefficients, corresponding to the structure and the soil used for the formation of the damping matrix.

4 Gravity and Time History Analyses To conduct a dynamic analysis of the structure–soil system, it is a prerequisite to carry out static gravity analysis in a staged manner [15]. Moreover, before conducting a full-fledged analysis of the soil–structure system, it is necessary to ensure accurate incorporation of boundary conditions. For this, a linear elastic soil model (without structure) with sine wavelet as input has been analyzed and the model is validated for the response in the centre of the soil domain as shown in Fig. 3a. Figure 3b shows the ground motion selected for performing time history analysis to study the soil–structure interaction effects after performing gravity analysis. To reduce the computational time, only the significant duration of the ground motion has been 1.00

Present Study

Acceleration (g)

Total acceleration

Zhang et al. [15]

0.50 0.00 -0.50 PGA= -0.86g

-1.00 0

0.2

0.4

0.6

0.8

Time (s)

(a)

1

1.2

1.4

1.6

0

5

10

15

20

Time (s)

(b)

Fig. 3 a Validation of the SSI model b ground motion used for time history analysis

25

30

Seismic Behaviour of RC Building Frame Considering …

163

used in conducting the time history analysis. The significant duration is the time duration of the ground motion during which the Arias Intensity is above 5% but not more than 95% of the total Arias Intensity developed over the duration of the entire ground motion.

5 Results and Discussion 5.1 Time History Response

Roof Displacemnt (m)

Figure 4a and 4b shows the comparison of roof displacement response and shear developed at the base of first storey columns, respectively. It can be observed that the response of the structure supported on Soft Soil (SS) lags as compared to the structural response with other types of supporting soil (MS, MDS and DS) or rocky condition (R). This is due to the fact that in soft soil the propagation of the shear waves is slower due to its density being less than other stiffer soils or rocky medium. The peak roof displacement is highest for Medium Soil (MS) and for other soil conditions (SS, MDS and DS) the value is slightly lower, and it is lowest for structure supported on rock (R). For the structure supported on rock, a larger number of well-defined peaks and crests are visible in the response as compared to those for the structure supported on soils (SS, MS, MDS and DS). Except for the absolute maximum value of the response, most peaks are greater in magnitude for the structure on rock (R). It can be observed that many small peaks, observed for the structure on rock (R), are subdued for the structure supported on soil (SS, MS, MDS and DS). Moreover, the peaks forming in the duration of 4–8 s (as seen for rocky site) result in the build up of a larger peak for the structure supported on soil. This leads to the peak response 0.30

SS MS MDS DS R

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(b) Fig. 4 Comparison of time histories for a roof displacement b base shear

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in the structure supported on soil (SS, MS, MDS and DS) to be greater than that of the structure supported on rock (R). The observed phenomenon can be explained by relating to the nonlinear behaviour exhibited by the soil. When a wave propagates towards the structure, it forces the structure to get displaced in one direction. A wave having sufficient energy remaining, after undergoing radiation damping and hysteretic damping, tends to displace the structure in a particular direction. If the wave does not have sufficient energy remaining after the dissipation, then, the structure is unable to undergo significant displacement and a potential peak is unable to develop. As observed in the case of rocky strata, for example, in Fig. 4a, it can be seen that the wave after 4.5 s pushes the structure supported on rock as well as soil in the direction of positive displacement. Once the peak is attained, the wave tries to push the structure back in the negative displacement direction. The mentioned phenomenon is observed for structure supported on rock (R). However, for structure supported on soil (SS, MS, MDS and DS), sufficient energy is not available after hysteretic damping and radiation damping so as to displace the structure in the negative direction (as observed in the case of the structure supported on rock). This inhibits the formation of a peak in the negative direction. At the same instant, another wave strikes and tries to displace the structure in the positive direction and leads to the development of a low frequency and high amplitude wave, as can be seen from Fig. 4a, for the structure supported on soil. The inability of the seismic waves to displace the structure in the negative direction causes the structure to displace further in the positive direction. A similar process is repeated for the next wave as well resulting in an overall buildup of the displacement in the positive direction for the structure supported on soil. This causes the displacement of the structure supported on soil to be higher than that of the structure supported on rock even though the latter is subjected to higher energy from the ground motion. Similar observations are made for base shear (Fig. 4b), and can be explained likewise. From Fig. 4a and 4b, it can also be seen that as the stiffness of the soil is reducing from DS to MS, the peak displacement and base shear tend to increase. However, on further reducing the stiffness of the soil from MS to SS, a slight drop in the peak values can be observed. This can be due to the fact that the reduction in the stiffness of the soil from MS to SS allows for higher nonlinear hysteretic behaviour in soil, leading to higher energy dissipation and reduction in the energy content of the waves being transmitted to the structure. Hence, reducing the peak displacements/base shear for the structure supported on SS compared to that of the structure supported by MS. Figure 5a and 5b presents the maximum floor level accelerations (amax ) and root mean square acceleration (arms ), respectively, in the structure for different soil conditions. It can be seen that amax for the structure on rock is highest for all storey (floor) levels. For a structure supported on soft soil (SS), the value is the least for most of the storey levels. As the stiffness of the soil increases from SS to DS, the profile of amax for various storey levels approaches that of the structure supported on rock (R). For structures supported on SS and MS, amax is highest at the topmost storey level. However, for structure supported on MDS, DS and R, amax increases till the second storey level thereafter it reduces till the fourth storey and again increases at

3

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0 0.00

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Fig. 5 Comparison of storey level peak and root mean square responses. a acce1eration amax , b acceleration arms , c interstorey drift ratio IDRmaz , d interstorey drift ratio IDRrms

the fifth storey level. To get an idea of the magnitude of the floor level accelerations developed over the entire duration of the time history, root mean square acceleration (arms ) is obtained and plotted at various storey levels in Fig. 5b. It can be seen that for SS and MS, higher storey levels develop greater accelerations. As the stiffness of the soil is increased from MS to R, there is a reduction in arms at higher storey levels but the same increases at the lower storey levels. The average arms are obtained by averaging the arms values for all the storey levels. It has been observed that the structures supported on MS, MDS and DS experience 11%, 10% and 5% more acceleration whereas for the structure supported on SS the average arms experienced is 6% lesser than that compared to the structure supported on rock. The difference in the trend for softer and stiffer soils may be due to the fact that the structure supported on SS and MS deform primarily according to the fundamental mode shape as the high-frequency oscillations are filtered out in the presence of soft or loose soil. On being excited by the fundamental mode, it is imperative that the higher storey levels develop larger accelerations. For structure supported on MDS, DS and R, apart from the fundamental mode shape, higher modes may also get excited during the deformations. The excitation of the higher modes allows for the development of larger accelerations in the intermediate storey levels, leading to the trends as observed for amax and arms . The two entities (amax and arms ) provide a qualitative estimate of the forces being experienced by the structural frame for different soil conditions as inertial forces are directly proportional to the acceleration. Figure 5c and 5d shows the maximum and root mean square Interstorey Drift Ratios (IDRmax and IDRrms ), respectively. For lower storey levels, the structure supported on soil is subjected to greater IDRmax . However, for higher storey levels storey levels, it is the structure supported on rock, which is subjected to higher IDRmax . It can also be seen that as the stiffness of the soil gets reduced from type R to type MS, IDRmax tends to increase for the lower storey levels and get reduced for the higher storey levels. On further reduction of the soil stiffness, a reduction in IDRmax for lower storey levels is observed. Figure 5d shows the comparison of IDRrms for various storey levels. It

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provides an estimate of the average interstorey drift ratio experienced by the structure over the entire duration of the time history. A drastic change in the IDR at a storey level indicates the occurrence of large deformations at that level. For the structure supported on soil, a sharp drop in the IDRmax and IDRrms values can be seen at the level of the third storey. However, for the structure supported on rock, a drop is seen at the third storey level and a sharper drop is seen at the fourth storey level. Large storey deformations can occur if the column members at that storey level yield significantly. Further investigation of nonlinear structural behaviour is discussed in the following subsection.

5.2 Structural Nonlinearity

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The nonlinearity in the structure is defined in terms of the plastic hinge locations and to study the nonlinear behaviour of the structure, the mobilized moment–curvature (M-ϕ) relationships are obtained. Since it is not possible to discuss the M-ϕ response at all the column sections, only some noteworthy results are discussed herein. Figure 6a and 6b shows the M-ϕ response at the column locations C1 and C6 at the base of the first storey, respectively. Similarly, Fig. 6c and 6d shows the M-ϕ response at the column locations C1 and C6 at the top of the third storey, respectively. It can be observed that the columns of the building frame supported on soil show unsymmetrical behaviour as compared to that of the structure supported on rock (R),

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Fig. 6 Comparison of moment-curvature response at base of first storey columns. a C1, b C6 third storey columns c C1, d C6 e third storey column C4 f fourth storey column C2

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i.e. large curvature in one particular direction and small curvature along the other direction. This can be attributed to the modification of the structural response due to interaction effects as observed in Fig. 4. The development of a low-frequency, high-amplitude wave pushes the structure in one particular direction and causes the development of moments and curvatures in that direction. This leads to such unsymmetrical M-ϕ behaviour, which is not observed in the columns of the building frame supported on rock (R). It can also be observed that among all the soil conditions, it is the columns of the structure supported on medium soil (MS) that develop the largest curvature and hence its sections yield more as compared to the columns of the building frame supported on other rock or soil conditions. This is due to the high acceleration values developed in the structure supported on the medium soil as discussed for Fig. 5a and 5b. Higher accelerations induce greater inertial forces within the structure and consequently develop larger displacements and curvatures. It is to be noted that the curvatures and moments developed for the third storey columns are in the opposite direction to those developed in the first storey columns, indicating a change in the curvature of the deformed shape. Such behaviour is expected in a frame wherein the redundancy provided by various columns and beams does not allow the frame to deform like a cantilever. It can be seen that although both the columns are under similar gravity loading condition, the mobilized moment capacity of columns at C6 is higher than that of the columns at C1. This is because, on being subjected to the ground motion, the structure displaces more in the positive direction over the entire duration due to which the exterior columns at C6 are subjected to additional axial compressive forces while the column at C1 is subjected to a reduction in compressive force at the same time. The additional compressive forces prevent the fibres of the section to undergo failure and hence are responsible for the increased moment capacity of the columns at C6. Figure 6e and 6f represents the nonlinear behaviour at the top of the third storey and the fourth storey columns, respectively. It can be seen that for the structure supported on soil (SS, MS, MDS and DS), the third storey columns show significant yielding. For the structure supported on rock (R) besides the third storey columns, the fourth storey columns also yield significantly. To confirm the observed trend, the energy dissipated by the frame members is estimated from the hysteresis loops exhibited by the M-ϕ relationships of the various members. Figure 7a shows the comparison of the storey-wise energy dissipated by the structure. Figure 7b and 7c shows the contribution of the energy dissipated by the beams and columns, respectively. Figure 7d shows the gross total energy dissipated by the structure for the different soil conditions. From the figures, it can be observed that the energy dissipated by structure on MS is the maximum followed by the structures supported on R, MDS, SS and DS types of soil. Storey wise, it is the first storey which undergoes the highest nonlinearity followed by the third storey and subsequently the other storey levels. It can be seen that the first and the third storey columns undergo significant nonlinearity for the structure supported on soil. For the structure supported on rock, the second, third and the fourth storey columns exhibit a similar extent of nonlinearity. This is in agreement with the observations in Fig. 5e and 5f. In addition, for the structure supported on rock, significant yielding of columns occurs for

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Fig. 7 Comparison of a storey-wise total energy dissipation b storey-wise energy dissipation in beams c storey-wise energy dissipation in columns d gross total energy dissipated e maximum storey shear f RMS storey shear

the second, third and fourth storey levels. However, for the structure supported on soil, the yielding of columns of the second and the fourth storey levels are relatively less than the yielding of columns in the third storey. For the structure supported on rock, the fourth storey is susceptible to greater damage due to the reduced moment capacity of the columns, as the gravity load is reduced at higher storey levels. Thus, for the rocky condition, although the second, third and fourth storey levels dissipate similar amounts of energy, still, the columns at the fourth storey undergo larger rotations/curvatures as seen from Fig. 5f. Also, due to the large rotations developed at the fourth storey level, a sharp drop in the interstorey drift is observed for the structure supported by rock. For the structure supported on soil, a sharp drop in the interstorey drift ratio is observed at the third storey level due to the columns undergoing larger nonlinear deformations at that storey (Fig. 5c and d). Hence, the failure of the columns get shifted to the lower storey levels when the structure is supported on soil or vice versa. This can be understood with the help of Fig. 7d and e, which shows the storey-wise developed maximum and root mean square shear forces, respectively. From the figures, it can be seen that for the structure supported by rock, shear developed in the fourth storey is comparable to the shear developed in the third storey of the structure supported by soil. This leads to the development of higher nonlinearity

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at the higher storey levels for the structure supported on rock. The observation can be related to the sharp increase in the maximum acceleration developed at the fifth storey for the structure supported on rock (R). The corresponding induced inertial forces developed at the fifth storey may have been enough to cause the failure at the fourth storey level in the structure supported by rock (R). For the structure supported on soil, the inertial forces from the fifth storey may not have been sufficient to develop failure at the fourth storey columns. However, the combined inertial forces from the fourth and the fifth storey levels may have caused significant yielding in the columns at the third storey level, leading to the development of greater interstorey drift and curvature in the columns at that level. For beam members, it can be seen that all the storey levels develop comparable levels of nonlinearity in their sections.

6 Conclusions From the present study, it can be concluded that soil–structure interaction can significantly modify the structural response and failure patterns. The study shows that the belief of soil flexibility not being detrimental rather beneficial has been found to be contradicted. It is possible that particular soil conditions could produce situations that may cause greater damage to the structure supported on soil than that supported on rock. Nonlinear soil–structure interaction has been found to modify the structural response, giving rise to low-frequency, high-amplitude excitations, which could develop larger forces at particular instants of time. This may be sufficient to push the structure towards failure especially at specific storey levels of the structure. The concentrated failure of the columns at a particular storey level may lead to the collapse of the structure. Hence, it is inferred that ignoring SSI may prove to be detrimental in certain cases and it would be wise to assess the problem on a case-by-case basis without generalizing the problem of soil–structure interaction as a whole.

References 1. Mazzoni S, McKenna F, Scott MH, Fenves GL (2006) OpenSEES command language manual. Pacific Earthquake Engineering Research (PEER) Center, USA 2. IS 1893: Part I (2016) Indian standard, criteria for earthquake resistant design of structures. Bureau of Indian Standards, New Delhi, India 3. IS 875: Part 2 (1987) Indian standard, code of practice for design loads (other than earthquake) for building and structures: Imposed loads. Bureau of Indian Standards, New Delhi, India 4. IS 456 (2000) Indian standard, plain and reinforced concrete-code of practice. Bureau of Indian Standards, New Delhi, India 5. IS 13920 (2016) Indian standard, Ductile detailing of reinforced concrete structures subjected to seismic forces-code of practice. Bureau of Indian Standards, New Delhi, India 6. IS 2911: Part I/Sec 1 (1979) Indian standard, code of practice for design and construction of pile foundations. Bureau of Indian Standards, New Delhi, India

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7. Chang G, Mander J (1994) Seismic energy based fatigue damage analysis of bridge columns: Part I—Evaluation of seismic capacity. NCEER Technical Report 94-0006 8. Filippou FC, Popov EP, Bertero VV (1983) Effects of bond deterioration on hysteretic behavior of reinforced concrete joints. Report EERC 83-19. Earthquake Engineering Research Center, University of California, Berkeley, USA 9. Paulay T, Priestley MJN (1992) Seismic design of reinforced concrete and masonry buildings. Wiley, USA 10. Lysmer J, Kuhlemeyer RL (1969) Finite dynamic model for infinite media. J Eng Mech Div 95(EM4):859–877 11. Joyner WB (1975) Method for calculating nonlinear seismic response in 2-dimensions. B Seismol Soc Am 65(5):1337–1357 12. Hilber HM, Hughes TJ, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq Eng Struct D 5(3):283–292 13. Kramer SL (1996) Geotechnical earthquake engineering. Prentice Hall, USA 14. Chopra AK (1995) Dynamics of structures-theory and applications to earthquake engineering. Pearson Education Inc., India 15. Zhang Y, Yang Z, Bielak J, Conte JP, Elgamal A (2003) Treatment of seismic input and boundary conditions in nonlinear seismic analysis of a bridge ground system. In: 16th ASCE engineering mechanics conference, University of Washington, Seattle, USA, 16–18 July 2003

Inelastic Time History Analysis of Mass Irregular Moment Resisting Steel Frame Using Force Analogy Method S. S. Ningthoukhongjam and K. D. Singh

Abstract In this paper, inelastic time history analysis of a six-storey moment resisting steel frame adopted from a hospital building located in Woodland Hills California (Wong and Yang, J Eng Mech 125:1190–1199 [1]) has been investigated considering both material and geometric nonlinearities, via an implemented Force Analogy Method (FAM) Matlab code. The reference steel frame has been analyzed by considering Kobe earthquake time history as the input ground motion. Storey mass has been taken as the parameter to study the effect on the structural response (e.g., displacement, etc.). Profiles of elastic, inelastic, and total displacement components of floors have been examined when irregular mass is located at different floor levels. Keywords Force analogy method · Vertically irregular frame · Geometric nonlinearity · Elastic and inelastic displacement components · Time history analysis

1 Introduction Vertically mass irregular building frames occur when a single building floor or multiple floors are used for specific purposes like car parking, shopping mall, equipment area, etc. As per the Uniform Building Code (UBC) [2], vertical irregularity in mass of a building frame shall be considered to exist when the effective mass of any storey is more than 150% of the effective mass of an adjacent storey. However, it may be noted that the current seismic codes [2, 3] do not address adequately the inelastic behaviors of mass irregular building frames during seismic excitation. Therefore, it is imperative to explore other methods, which can give accurate results and are

S. S. Ningthoukhongjam (B) Department of Civil Engineering, Manipur Institute of Technology, Manipur, India e-mail: [email protected] K. D. Singh Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_15

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more computationally efficient in estimating the inelastic seismic behavior of mass irregular building frame structures. Investigations on seismic response of vertically mass irregular building frames have been reported out by several researchers. Based on Equivalent Lateral Force (ELF) method, Valmundsson and Nau [4] observed that the response of elastic deformation is more affected when mass irregularity is located at higher floor levels in a building, while the response of inelastic deformation is mainly influenced by mass irregularity located at lower floors of the building. Al-Ali and Krawinkler [5] reported that the effect of mass irregularity on both elastic and inelastic deformations of building is generally small, although a relatively increased roof and storey drifts were observed when mass irregularity is located at top floors as compared to mid-height or base of the building. Magliulo et al. [6] conducted seismic analysis with a very large variation of mass distribution in elevation, and observed negligible changes in seismic responses with large variation (200%) of mass distribution in the building. Choi [7] concluded that mass irregularity is an important aspect in affecting seismic responses of building frame structures. With the help of DRAIN2D+ program, Choi [7] found the structural responses in terms of drift ratio, plastic rotation, energy distribution, and stress in element levels to be most critical if vertical mass irregularities are located at upper and lower floors rather than at building mid-height. Karavasilis et al. [8] performed an extensive analytical parametric study on plane steel Moment Resisting Frames (MRF) with vertical mass irregularities using DRAIN-2DX to examine and evaluate the seismic inelastic deformation demands. Based on the study, it has been concluded that, in general, height-wise distributions of the elastic component of deformations are uniform, while the inelastic component of deformation seemed to concentrate at the lower storey levels only. The accuracy in estimation of inelastic responses of structures during seismic excitation depends upon how nonlinearity behavior is integrated in the method of analysis. The traditional method of incorporating material nonlinearity is performed by changing stiffness to capture force reduction when yielding occurs in the structural members. Such a method invites a stability question on the appropriate use of stiffness matrix to represent coupled material and geometric nonlinearities, such as in the case of near-collapse situation. In the literature (e.g., [9, 10]), P- and geometric stiffness approaches are the two commonly used methods for analyzing geometric nonlinearity effects. In P- approach, only large P- effect is considered by assuming first-order strain approximations. On the other hand, in geometric stiffness approach, both large P- and small P-δ effects are considered, via second-order strain approximations (with all higher order terms truncated); however, the stiffness has a simple linear relationship with axial force. Among software packages, [9, 10] Perform-3D and Open Sees (Regents of UC 2000) use P- stiffness matrix while SAP2000 uses the geometric stiffness matrix to incorporate geometric nonlinearity effects resulting in lack of consistency in their outputs. It is worth mentioning that addressing adequately the effects of geometric nonlinearity is required for structures with significant lateral deformation, and simply linear or second-order approximations of the geometric stiffness may not be able to predict the near-collapse behavior

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of the structure accurately. Hence, Wong and Speicher [11] developed a new formulation based on Force Analogy Method combined with stability functions to capture large displacement responses more precisely as compared to other small displacement methods.

2 Force Analogy Method The original Force Analogy Method (FAM) for material nonlinearity was developed by Wong and Yang [1]. FAM is a nonlinear analytical method, which is very efficient in computational time because in this algorithm only initial stiffness is used throughout the entire nonlinear analysis. For a particular problem, Li et al. [12] reported that solution based on FAM substantially cut down (~70%) solution computational time as compared to that of SAP2000. A brief discussion on FAM [9] is presented here. Consider a moment resisting frame with n Degrees Of Freedom (DOFs) and m Plastic Hinge Locations (PHLs). The total displacement at each DOF is given by the summation of elastic displacement x and inelastic displacement x x = x  + x 

(1)

Similarly, the total moment, M at each PHL is represented as the summation of elastic moment M and inelastic moment M M = M + M

(2)

The displacements in Eq. (1) and the moments in Eq. (2) are related by the relation as M  = K T x  ,

M  = −[K  − K T K −1 K  ] , x  = K −1 K  

(3)

where  is the plastic rotation at each PHL, K is n × n global stiffness matrix, K is n × m stiffness matrix formed by relating plastic rotations at the PHLs with the restoring forces at the DOFs, and K is the m × m stiffness matrix formed by relating plastic rotations with corresponding residual moments at the PHLs. By matrix simplification, the governing equations of FAM with respect to an external applied force Fa can be represented as 

K K K T K 



x −



 =

Fa M

 (4)

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2.1 Consideration of Geometric Nonlinearity in FAM Wong and Speicher [11] incorporated geometric nonlinearity on Force Analogy Method (FAM) to capture large displacement responses more precisely than other small displacement methods. In the equation of motion, the stiffness force is calculated by multiplying the stiffness matrix K(t) with the elastic displacement x (t) in FAM. For an n-DOF system, the equation of motion subjected to earthquake ground motions can be represented by M x(t) ¨ + C x(t) ˙ + K (t)x  (t) = −Mg(t) − Fa (t)

(5)

where M is n × n invertible mass matrix, C is n × n damping matrix, x(t) ˙ is n × 1 velocity vector, x(t) ¨ is n × 1 acceleration vector, g(t) is n × 1 earthquake ground acceleration vector, and F a (t) is n × 1 additional force vector imposed on the frame due to geometric nonlinearity (mainly due to P- effect). The relationship between this additional force vector F a (t) and the lateral displacement vector x(t) can be represented as Fa (t) = K a x(t)

(6)

where K a is n × n stiffness matrix which is a function of the gravity loads and the corresponding storey height, but K a is not a function of time. The stiffness matrix K(t) in Eq. (5) considers both large P- and small P- effects and it can be represented in the form: K (t) = K L + K G (t)

(7)

where K L represents the elastic stiffness matrix due to the gravity loads only, and K G (t) represents the stiffness matrix which changes as the axial load on members changes during the earthquake loading. Here the elastic stiffness matrix K L is constant which is computed once at the beginning and can be used throughout the entire nonlinear time history analysis of the structure, which is one of the important advantages of Force Analogy Method.

3 Problem Description A six-storey steel structure hospital building located in Woodland Hills, California has been taken as the reference frame for the problem in this paper (see Fig. 1) [1, 9]. The building properties are shown in Table 1. The relevant material and geometric data are obtained from Li and Wong [9] and are briefly mentioned here. Considering all the members as axially rigid, the total number of DOFs and PHLs are 30 (i.e.,

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Fig. 1 A six-storey hospital building located in Woodland Hills, California Table 1 Properties of building frame

Mode

Periods

Damping (%)

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1.00

2

2

0.36

2

3

0.20

2

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0.14

2

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0.11

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0.09

2

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n = 30) and 84 (i.e., q = 84), respectively. Further, an elasto-plastic steel behavior with yield stress of 250 MPa has been assumed. Additionally, updation of geometric nonlinearity effects due to changes in axial forces in columns (i.e., the effect of follower load) has been ignored. An approximated uniformly distributed load of 21.89 KN/m is applied on all the beams to give axial compressive forces in the columns. Mass moment of inertia at the rotational joints is neglected and hence by static condensation 24 rotational DOFs (i.e., r = 24) are eliminated resulting only in 6 translational DOFs for the frame (i.e., x 1 , x 2 … x 6 ). The details of the section members used, damping ratio, and time periods are also shown in Fig. 1. The storey mass for each floor has been considered as 200,000 kg which gives a total mass of 1,200,000 kg for the entire building frame. Hence, the mass matrix, M dd of the entire building frame becomes ⎡

Mdd

200 ⎢ 0 ⎢ ⎢ ⎢ 0 =⎢ ⎢ 0 ⎢ ⎣ 0 0

0 200 0 0 0 0

0 0 200 0 0 0

0 0 0 200 0 0

0 0 0 0 200 0

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ × 1000 kg 0 ⎥ ⎥ 0 ⎦ 200

Kobe earthquake (1995) (shown in Fig. 2) has been taken as the input ground motion for the problem in this paper. Inelastic time history analysis has been carried out via an implemented Force Analogy Method (FAM) Matlab code. The Matlab code has been validated with the results of Example 7.8 of [9]. 1

Ground Acceleration (g)

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0

2

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6

8

10

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Fig. 2 Kobe earthquake (1995) ground motion

12

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3.1 Vertically Mass Irregular Building Frame For the parametric study, the six-storey hospital building frame has been converted into six vertically mass irregular building frames by increasing the storey mass up to 200% at each floor level. The new six frames are labeled as 2.00M1, 2.00M2, 2.00M3, 2.00M4, 2.00M5, and 2.00M6, where 2.00M indicates 200% increase in storey mass, and 1, 2, …6 indicate floor levels. Hence, a total of seven frames (six irregular and one regular frames) have been analyzed for full inelastic time history. Each of the seven frames has been subjected to input ground motion of 0.5 times the amplitudes of Kobe Earthquake ground acceleration and the responses (both elastic and inelastic components of floor deflections) are analyzed. The scaling with a factor of 0.5 is essentially to limit the instability associated with very large plastic displacements (to discuss later).

4 Results and Discussion Time history responses of both regular and six irregular frames have been obtained via implemented FAM and are presented below.

4.1 Regular Frame Time history plots of roof floor deflection (x 6 ) for regular frame have been plotted in Fig. 3. This roof floor deflection (x 6 ) is composed of an elastic component (x6 ) and an inelastic component (x6 ), i.e., x6 = x6 + x6 . Similarly, time history plots of elastic, inelastic, and total deflections of all six floors for the regular frame have been plotted together in Fig. 4, Fig. 5, and Fig. 6 respectively. It can be observed from Fig. 4 that in case of regular frame, elastic components of all six floors deflections increase proportionately as the floor level moves from first floor to sixth floor. It can be seen from Fig. 5, that regular frame behaves in an elastic manner until ~3.5 s of input ground motion. Inelastic deformations start after ~3.5 s and an abrupt increase of inelastic deformations can be seen at ~4 s for all the floor levels. Beyond 4 s, the inelastic behavior remains more or less stable (or plateauing effect), indicative of the plastic hinge formation at critical locations. The storey drift component of inelastic deformations decreases as the floor level moves from first to sixth floor. From Fig. 6, it can be observed that the total floor deflections of all six floors of

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regular frame increases as the floor level moves from first to sixth floor but the amplitude of total floor deflections decreases as the floor level moves upward. The maximum total floor deflection is observed at roof level (x 6 ). The maximum roof deflection is recorded as ~260 mm at 3.9 s of the input ground motion.

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4.2 Time History Plots of Elastic, Inelastic, and Total Deflections of All Six Irregular Frames Figure 7 shows time history plots of elastic components of floor deflections of all six irregular frames. It can be seen from Fig. 7a and f that the elastic component of all floors deflections for frames 2.00M1 and 2.00M6 are found to vibrate near about the mean position. However, for other frames (2.00M2, 2.00M3, 2.00M4, and 2.00M5), an initial shift in the center of vibration away from the mean position, followed a return toward the mean, with the shift being relatively more prominent for higher floors, can be observed. In general, beyond the initial elastic deformation (i.e., after

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~3.5 s), the amplitude of vibration of all the floors increases with increasing floor height. Time history plots of inelastic components of floor deformations of all six irregular frames are shown in Fig. 8. As in the case of the regular frame (see Fig. 7), inelastic deformations are observed to occur from ~3.5 s onward. In general, an increase in inelastic deformation can be seen to occur during ~3.5–10 s, after which a stable value of deformation is achieved. In all the frames considered, the direction of inelastic deformation for Frame 2.00M1 is in the opposite sense (shown by +ve value in Fig. 8a) to those of the rest. It is also seen that Frames 2.00M1 and 2.00M5 showed the minimum (~50 mm) and maximum (~220 mm) inelastic deformations, respectively. This may be related to the repeated occurrence of plastic hinges corresponding to the subsequent peaks in amplitudes of ground acceleration during this period of input ground motion (see Fig. 2). Although a gradual increase in inelastic deformation component is observed from Frames 2.00M1 and 2.00M5 (i.e., when the mass irregularity moves upwards), a decrease has been seen for Frames 2.00M6 or when the mass irregularity is located at the top floor. A sudden large increase of inelastic deformation is observed between first and second floors in all irregular

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frames. The maximum inelastic floor deflection is observed at roof level in irregular frame 2.00M5. The patterns of total deflection (see Fig. 9) of all floors increase as the mass irregularity moves upward from Frame 2.00M1 until 2.00M6. However, a slight decrease of deflection in all floors is observed in Frame 2.00M6. A large storey drift at lower floors can be observed in frames 2.00M2 and 2.00M3. This is due to relatively larger inelastic component of floor deflections at lower storey levels. The maximum total floor deflection is observed at the top floor of irregular frame 2.00M5 because inelastic component of floor deflection is highest at this location. The maximum total floor deflection is found to be ~370 mm.

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5 Conclusions Time history analysis of a six-storey moment resisting steel frame has been investigated considering both material and geometric nonlinearities, via an implemented Force Analogy Method (FAM) Matlab code. Storey mass has been taken as the parameter to study the effect on the floor deflections of the moment resisting steel frame. Based on the FAM analyses, the following conclusions are made • Lateral floor deflection increases as the mass irregularity moves upward from bottom to top of the building frame. However, a slight decrease in floor deflections is observed when mass irregularity is at the top of the building. • The profile of all floors deflection in a frame depends on the location of mass irregularity along the height of the building and the pattern of input ground motion. • The center of vibration of floor deflections moves away from the mean position as the mass irregularity moves upward from bottom to top of the building. • A large storey drift at lower floors is observed when mass irregular is located at lower floor of the building as observed in frames 2.00M2 and 2.00M3. Thus

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suggesting that increase of mass irregularity at lower floors is relatively more critical. • The amplitude of vibration of all floors first decreases as the mass irregularity moves upward from bottom to mid-height of the building and then increases on further movement of mass irregularity from mid-height toward the top of the building. • The maximum total floor deflection of ~370 mm is observed at the top floor when mass irregular is located at fifth floor of the building as observed in frame 2.00M5. Further work is underway for increasing the number of storey in the building.

References 1. Wong KKF, Yang R (1999) Inelastic dynamic response of structures using force analogy method. J Eng Mech 125:1190–1199 2. UBC (1997) Uniform Building Code. In: International conference of building official. Whittier, California 3. Eurocode 8—Part 1 (2004) Design of structures for earthquake resistance: General rules, seismic actions and rules for buildings 4. Valmundsson EV, Nau JM (1997) Seismic response of building frames with vertical structural irregularities. J Struct Eng 123:30–41 5. Al-Ali A, Krawinkler H (1998) Effects of vertical irregularities on seismic behavior of building structures. Report No. 130, John A. Blume Earthquake Engineering Center, Standard University, Stanford CA, p 396 6. Magliulo G, Ramasco R, Realfonzo R (2002) Seismic behaviour of irregular in elevation plane frames. In: 12th European conference on earthquake engineering, Paper reference 219 7. Choi BJ (2004) Hysteretic energy response of steel moment-resisting frames with vertical mass irregularities. Struct Des Tall Spec Build 144:123–144 8. Karavasilis TL, Bazeos N, Beskos DE (2008) Estimation of seismic inelastic deformation demands in plane steel MRF with vertical mass irregularities. Eng Struct 30:3265–3275 9. Li G, Wong KKF (2014) Theory of nonlinear structural analysis: the force analogy method for earthquake engineering. Wiley, Singapore Pte. Ltd 10. Wong KKF, Speicher MS (2015) Dynamic effects of geometric nonlinearity on inelastic frame behavior for seismic applications. In: Proceedings of the annual stability conference structural stability research council, Nashville, TN, USA, 24–27 March 2015 11. Wong KKF, Speicher MS (2015) Dynamic effects of geometric nonlinearity on inelastic frame behavior for seismic applications. In: Proceedings of the annual stability conference structural stability research council, Nashville, TN, 24–27 March 2015, pp 1–20

Modification and Modeling of Experiments with Bi-directional Loading on Reinforced Concrete Columns Subhadip Naskar, Sandip Das, and Hemant B. Kaushik

Abstract Capacity evaluation of bi-directionally loaded column is important not only for the performance-based seismic design of structures but also for the estimation of structural damage. In this paper, an experimental study has been carried out on full-scale columns with different axial stress ratios, followed by the development of an analytical model to predict the lateral load response of the column under bi-directional loading after taking care of the effect of the functional interactions between the two loading actuators and the column specimen. These interactions, if not taken into account, result in apparent underestimation of ultimate strength and overestimation of maximum displacement capacity of the test specimen, thereby demanding unnecessary changes in model parameters for the purpose of calibration. It is also found that the analytical model accounting for the aforementioned functional interactions leads to a more realistic and different dynamic structural response from that obtained using the analytical model ignoring the interactions. Keywords Bi-directional loading on RC columns · Capacity evaluation · Pseudo-static cyclic load · Cyclic degradation of strength · Seismic damage index

1 Introduction Performance-based seismic design of any civil structure requires capacity parameters, such as lateral displacement capacity and lateral strength capacity. Estimation of structural damage (or any other response-dependent performance parameter) due to seismic ground motions also needs the capacity evaluation of the structures. Therefore, since the last few decades, many researchers have done experiments on S. Naskar · S. Das (B) · H. B. Kaushik Civil Engineering Department, IIT Guwahati, Guwahati 781039, Assam, India e-mail: [email protected] S. Naskar e-mail: [email protected] H. B. Kaushik e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_16

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cantilever column subjected to uniaxial or biaxial lateral force under the action of constant or varying axial load to study the cyclic behavior, ultimate strength, and ultimate displacement of RC members. Vertical cantilever specimen fixed at a concrete base [1, 2], strong beam-weak column sub-assemblages [3–5], was used to study the effect of uniaxial flexural load in presence of constant axial load. But the inelastic response of column is greatly influenced by the variation of axial forces during the cyclic response, depending on the relative magnitude of lateral and gravity loads and the proportioning of the force members of the frame. Therefore, experiments were carried out under different axial load conditions varying with respect to transverse displacement or transverse load [6–10]. Later to get a more realistic prediction of response of RC structural members during two-dimensional seismic excitations, many researchers have studied the behavior of cantilever column [11–21] and beamcolumn sub-assemblages [22–25], subjected to biaxial flexure and constant axial load. Very few researchers carried out experimental studies of structural response due to biaxial lateral loading condition with varying axial force [26–29]. These experimental studies have been used further for validation of many different types of numerical modeling such as material level modeling at the point-by-point basis, member-bymember type of modeling considering one-to-one correspondence between elements of the model and members of the structure, relatively simple few degrees-of-freedom models. A detailed report on such validation of different modeling can be found in the state-of-the-art report on RC frames under earthquake loading [30]. It is understood that under large deformation or high axial load ratio the subtle change in loading directions (of the actuators) might affect the load-deformation behavior and warrant some interaction among the actuators and the specimen. It is important to decouple such interaction effect by means of some essential kinematic corrections before developing analytical models from the experimental data. However, no such modifications of the experimental results have been reported for bi-directionally loaded cantilever column subjected to large lateral deformation or high axial load ratio. In the present study, different nonlinear monotonic and pseudo-static cyclic experiments on cantilever column specimens under the action of different constant axial stress ratios are conducted. A methodology of kinematic correction is proposed to account for the aforementioned interactions before developing an analytical model. Further, the effects of such interactions on the analytical model and the structural response thereof, including structural damage, under the action of seismic motion are also studied.

2 Experimental Details 2.1 Experimental Setup Monotonic and cyclic experiments on a full-scale cantilever column have been carried out, by simulating the lateral load under the action of different levels of axial load

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Fig. 1 Detailed view of the typical experimental setup

on the specimen. A displacement controlled MTS actuator of 250 kN capacity and another displacement controlled MTS actuator of 1000 kN capacity has been used to simulate the horizontal force and vertical axial force condition, respectively (Fig. 1).

2.2 Loading Characteristics and Material Details Full-scale cantilever column has been used to study the behavior of column of moment resisting frame to reduce both cost and time of carrying out the experiments, as the deflected shape of a cantilever column, subjected to a transverse load, resembles the same of both ends fixed column considering the fixed end to the point of contraflexure. A vertical axial load is given to the free end of the cantilever specimen to simulate the effect of gravity load. Table 1 shows the different loading characteristics of monotonic and cyclic experiments. Mix design of concrete for the test specimens has been carried out as per IS:10262-2009. The yield stress, ultimate stress, modulus of elasticity of longitudinal reinforcement of the columns have been found as 535.28 MPa, 641.40 MPa, and 215840.32 MPa, respectively, from test results.

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Table 1 Loading characteristics and characteristic strength of test specimens Test no.

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3 Modification of Experimental Results During the experiment, both horizontal and vertical actuators make an angle with the horizontal or vertical axes (see Fig. 2), due to which actual horizontal and vertical forces become the sum of horizontal and vertical components of both actuators, respectively. Similarly, the horizontal component of horizontal actuator should be used as the horizontal displacement of the cantilever column. Also, an extra moment, generated due to the inclined position of the actuators, is responsible for additional deformation of the cantilever column. Therefore, some rigorous correction, depending on the geometry of the experimental setup, kinematic constraint, and basic equilibrium conditions at each time

Fig. 2 Schematic diagram of undeformed and deformed test specimen

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step, is required to estimate the exact value of horizontal force, vertical force, horizontal displacement, and nullify the effect of the additional moment before using these experimental values further.

3.1 Correction Due to Vertical Actuator In Fig. 2, A represents the column tip, B represents the hinge joint between the vertical actuator and rigid prefabricated connection of the column before deformation and D represents the hinge joint between the vertical actuator and the strong wall. A and B represent the location of A and B after deformation. From the geometry, moment due to inclined vertical actuator (MV ), force components of vertical actuator (PH , PV ), and the actual horizontal displacement () can be expressed as MV = P(h 1 + h v ) sin(θ + φ);

PH = P · sin φ; PV = P · cos φ;  = δh · cos α (1)    + (h 1 + h v ) sin θ (2) φ = sin−1 L v + δv

where h 1 = length of the vertical actuator head, h v = distance between the center of column stub and bottom end of the vertical actuator, P = force reading of the vertical actuator, δh , and δv = displacement reading of the horizontal and the vertical actuator, respectively; L v = initial length of the vertical actuator before test, θ = tip rotation of the column, and φ = angle between the vertical actuator and the vertical axis.

3.2 Correction Due to Horizontal Actuator In Fig. 2, C represents the hinge joint between the horizontal actuator and rigid prefabricated connection of column before deformation and E represent the hinge joint between the horizontal actuator and the strong wall. C represents the location of C after deformation. Similarly, the moment due to the inclined horizontal actuator (M H ) and force components of the horizontal actuator (FH , FV ) can be expressed as given below: M H = F(h 2 + b/2) sin(θ + α); α = sin−1



FH = F · cos α;

  h 2 + b/2 sin θ L h + δh

FV = F · sin α

(3) (4)

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where h 2 = length of the horizontal actuator head, b = width of the column stub, F = force reading of the horizontal actuator, L h = initial length of the horizontal actuator before test, and α = angle between the horizontal actuator and the horizontal axis. Angle parameters α and φ should be known to estimate actual horizontal force and displacement components. But they are a function of unknown tip rotation θ , which can be expressed as the sum of elastic tip rotation (θe ) and plastic tip rotation (θ p ). Further, θe is the resultant effect of both horizontal force (H = PH + FH ) and moment (M = MV + M H ) acting at the tip of the column. Therefore,   H L 2c M Lc ; θeM = θ = θe + θ p = θeH + θeM + θ p ; θeH = 2E c Ie E c Ie

(5)

where L c = length of the column, E c = elastic modulus of concrete and Ie = effective moment of inertia of the column. Now, Δ can be also be expressed in terms of the elastic portion due to horizontal force (eH ), moment (eM ), and the plastic portion ( p ) as   H L 3c M L 2c ; eM =  = e +  p = eH + eM +  p ; eH = 3E c Ie 2E c Ie

(6)

Considering a new factor c = θe L c /e , it can be obtained from Eqs. (5) and (6): c=

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(7)

Since no additional tip moment is encountered just at the start of the experiment, the initial value of factor c can be taken as c0 = 1.5.

3.3 System Kinematics From the geometry of the experimental setup, the following expressions can be obtained: p

e = δhe · cos α; cos α = p

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(8)

where δhe and δh = elastic and plastic portion of δh , respectively. But, initially only total displacement of the horizontal actuator (δh ) is known, so an initial ratio r = δhe /δh is assumed. The assumed value of r has been cross-checked with the estimated value by established elastic theory and iterated accordingly until the exact value of r has been found. Therefore,

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p (1 − r )θe L c (1 − r )θe L c ; θp = = cr L0 cr (L c − L p )

(9)

where L 0 = length of the test specimen participating in plastic deformation and L p = the plastic hinge length [31], as given below: L p = 0.08z + 0.022db f y

(10)

where z = the distance from the critical section to the point of contraflexure, db = the largest dia. of the longitudinal reinforcement and f y = the yield strength of steel. Since δh comprises only elastic portion just at the start of the experiment, the initial value of r can be taken as r0 = 1.0. Now, from Eqs. (4) and (8), it can be found that a1 sin2 (a2 θe ) + a3 θe2 = 1  a1 =

h 2 + b/2 L h + δh

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Lc cδhe

2 (12)

Further, Eq. (11) can be simplified as follows by assuming a2 θe = x: a1 a22 sin2 x + a3 x 2 − a22 = 0

(13)

Since the exact solution of the above transcendental equation is not possible, Maclaurin series of sin x [32] has been used to get the following expression: ∞

2 (−1)n x 2n+1 + a3 x 2 − a22 = 0 a1 a22 + 1)! (2n n=0

(14)

Newton–Raphson method has been used in this paper to get θe by solving the above equation, from which the complete orientation of the test specimen during a particular time step can be found and therefore, the actual horizontal force (H), actual vertical force (V = PV − FV ), and additional tip moment (M) can be evaluated.

3.4 Effect of Large Displacement of Horizontal Actuator During cyclic loading with small displacement amplitude, the cantilever tip movement can be analyzed by following a straight path. But the tip movement starts to follow a curvilinear path with subsequently increasing the amount of lateral drift, especially during monotonic tests.

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Fig. 3 Vertical lowering of column tip due to large lateral displacement

In Fig. 3 the column tip, A moves to the point A p after a plastic rotation θ p about the plastic hinge, P and further, it moves to the point A e+p after an elastic rotation θ e . Thus, the modified vertical length of the cantilever column (L c ) can be expressed as L c = L c −  L c  L c = L 0 (1 − cos θ p ) + e sin θ p ; e =

(15)

(H cos θ p + V sin θ p )L 3c M L 2c + 3E c Ie 2E c Ie (16)

where Δ L c = total vertical shortening due to both plastic and elastic rotation, e = elastic displacement due to the loads acting normally to the direction of PAp and the moment acting at Ap . Further L c can be used to evaluate the displacement removing the effect of the additional tip moment ( ) as given below:  =  −

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(17)

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Fig. 4 Experimental and modified monotonic and cyclic pushover curves

4 Results and Discussions 4.1 Modification of Experimental Data Figure 4 clearly shows that direct incorporation of experimental results will underestimate the lateral strength and overestimate the lateral displacement capacity of the system. Neglecting the horizontal force component of the vertical actuator and displacement due to additional tip moment generated by the inclined position of both actuators are the main reason for underestimation of lateral strength and overestimation of the lateral displacement capacity of the system, respectively. Clearly, the modification scheme makes the original system stiffer. Further, this modification has been found to be more significant in the presence of high axial load acting at the column (see Table 2). During cyclic experiments, a lesser amount of pinching and considerably higher amount of strength degradation has occurred in the presence of higher axial load.

4.2 Analytical Model Using OpenSees Software Two different analytical models have been developed in OpenSees platform by calibrating with experimental data as well as modified data. The analytical model with exact material properties has been found to produce similar results as the modified

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Table 2 Comparison of underestimation of lateral strength and overestimation of lateral displacement capacity for different tests M1 Experimental results Modified results

M2

C1

C2

Fmax (kN)

37.88

31.52

−34.00

35.88

−35.36

35.84

δmax (mm)

144.80

58.32

−30.41

29.80

−30.08

30.08

Hmax (kN)

39.86

37.30

−35.75

37.81

−39.04

39.95

max

135.1

46.14

−28.66

27.54

−26.17

25.65

% increase in Fmax

5.23

18.34

5.15

5.38

10.41

11.47

% decrease in δmax

5.75

20.88

5.75

7.58

13.00

14.73

(mm)

Table 3 Details of different ground motions [33] Sl. no.

Ground motion name

Location

1.

Bhuj, India (2001)

23.02 N, 72.38 E

2.

Loma Prieta, USA (1989)

32.05 N, 181.80 W

◦

◦

◦

◦

Magnitude

PGA (cm/s2 )

Nearest dist. to fault (km)

7.0 Mb

−103.82

239.0

7.0 Mw

469.38

2.8

data rather than the experimental data, which definitely signifies the importance of the described modification scheme. Further, a column, with one end fixed and another end rotationally constrained (to achieve the deformed shape with double curvature), of length equal to 3 m has been considered and the analytical models have been used to find the behavior of the column subjected to two different ground motions (see Table 3). Though the responses of two different analytical models are found to be similar when subjected to a far-field ground motion with lesser PGA value, two models behave differently being subjected to a near-field ground motion with higher PGA value, as shown in the Fig. 5. Experimentally calibrated analytical model overestimates the displacement response, but underestimates the lateral force demand and the residual inelastic displacement, due to the less inherent stiffness of the model compared to the analytical model calibrated with the modified data.

4.3 Estimation of Seismic Damage Index Most widely used modified Park and Ang damage index [34], as given below, has been estimated in this paper: DI =

δm − δ y β + δu − δ y Q y δu

 dE

(18)

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Fig. 5 Different responses of two analytical models subjected to different ground motions. a Bhuj, India (2001). b Loma Prieta, USA (1989)

where δm = the max. deformation under seismic loading condition, δ y and δu = the yield and the ultimate deformation under monotonic loading condition, β = non-negative strength degrading parameter (taken as 0.15 in this paper), d E total absorbed hysteresis energy, and Q y = yield strength under monotonic loading. Table 4 shows the estimation of considered damage index for two different analytical models. It has been found that the experimentally calibrated analytical model estimates incorrect seismic damage index by a significant amount, which is not desirable. Table 4 Comparison of damage indices for both experimental and modified model subjected to two different ground motions Parameters from static pushover

Bhuj, India (2001)

Qy (kN)

δy (mm)

δu (mm)

δm (mm)

Exp model

31.76

15.49

144.79

Mod model

33.43

14.74

135.77

% change



Loma Prieta, USA (1989)

dE (kN, mm)

DI

δm (mm)

5.58

450.58

0.0147

88.47

9759.75

0.8828

5.40

427.23

0.0141

82.69

11010.30

0.9253

−3.23

−5.18

−4.08

−6.53

+12.81

dE (kN, mm)

DI

+4.81

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5 Conclusions Nonlinear monotonic experiments and pseudo-static cyclic experiments have been carried out in the present study for various axial load ratios. A methodology of kinematic correction has been proposed to account for the kinematic interaction between inclined horizontal and vertical actuators, to refine the response of reinforced concrete cantilever columns under the action of bi-directional loading. It has been found that such interactions, if not taken into account, adversely affect the loaddeformation behavior and the analytical model calibrated by using it and subsequent response prediction. The material model developed based on the proposed refined experimental data has been found to be more realistic as it can address the issue with underestimation of strength and overestimation of deformability associated with kinematic interactions.

References 1. Otani S, Cheung V (1981) Behavior of reinforced concrete columns under biaxial lateral load reversals test without axial load. Department of Civil Engineering, University of Toronto, Canada 2. Saatcioglu M, Ozcebe G (1989) Response of reinforced concrete columns to simulated seismic loading. Struct J 86:3–12 3. Zagajeski SW, Bertero VV, Bouwkamp JG (1978) Hysteretic behavior of reinforced concrete columns subjected to high axial and cyclic shear forces. University of California, Berkley, p 78 4. Park R, Zahn F, Falconer T (1984) Strength and ductility of reinforced and prestressed concrete columns and piles under seismic loading. In: Proceedings of 8th world conference on earthquake engineering, San Francisco, USA 5. Rabbat B, Daniel J, Weinmann T, Hanson N (1986) Seismic behavior of lightweight and normal weight concrete columns. J Proc 83:69–79 6. Gilbertsen ND, Moehle JP (1980) Experimental study of small scale R/C columns subjected to axial and shear force reversals. University of Illinois Engineering Experiment Station, College of Engineering, University of Illinois at Urbana-Champaign 7. Kreger M, Linbeck L (1986) Behavior of reinforced concrete columns subjected to lateral and axial load reversal. In: Proceedings of 3rd US national conference on earthquake engineering, vol 11. Charleston, South Carolina, USA 8. Ristic D et al (1986) Effects of variation of axial forces to hysteretic earthquake response of reinforced concrete structures. In: Proceedings of 8th European conference on earthquake engineering, vol 4. Lisbon, Portugal 9. Abrams DP (1987) Influence of axial force variation on flexural behavior of reinforced concrete columns. Struct J 84:246–254 10. Saadeghvaziri MA, Foutch DA (1990) Behavior of RC columns under nonproportionally varying axial load. J Struct Eng 116:1835–1856 11. Takizawa H, Aoyama H (1976) Biaxial effects in modelling earthquake response of R/C structures. Earthq Eng Struct Dyn 4:523–552 12. Otani S, Cheung V, Lai S (1979) Behavior and analytical models of reinforced concrete columns under bi-axial earthquake loads. In: 3rd Canadian conference on earthquake engineering, Montreal 13. Zahn F, Park R, Priestly M (1989) Strength and ductility of square reinforced concrete column sections subjected to biaxial bending. Struct J 86:123–131

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14. Bousias S, Verzelleti G, Fardis M, Magonette G (1992) RC columns in cyclic biaxial bending and axial load. In: 10th world conference on earthquake engineering, Madrid 15. Kim JK, Lee SS (2000) The behavior of reinforced concrete columns subjected to axial force and biaxial bending. Eng Struct 22:1508–1528 16. Qiu F, Li W, Pan P, Qian J (2002) Experimental tests on reinforced concrete columns under biaxial quasi-static loading. Eng Struct 24:419–428 17. Tsuno K, Park R (2004) Experimental study of reinforced concrete bridge piers subjected to bi-directional quasi-static loading. Struct Eng/Earthq Eng 21:11S–26S 18. Kawashima K, Ogimoto H, Hayakawa R, Watanabe G (2006) Effect of bilateral excitation on the seismic performance of reinforced concrete bridge columns. In: 8th US national conference on earthquake engineering, San Francisco, CA, USA 19. Chang SY (2009) Experimental studies of reinforced concrete bridge columns under axial load plus biaxial bending. J Struct Eng 136:12–25 20. Rodrigues H, Arede A, Varun H, Costa AG (2013) Experimental evaluation of reinforced concrete column behavior under biaxial cyclic loading. Earthq Eng Struct Dyn 42:239–259 21. Campione G, Cavaleri L, Di Trapani F, Macaluso G, Scaduto G (2016) Biaxial deformation and ductility domains for engineered rectangular rc cross-sections: a parametric study highlighting the positive roles of axial load, geometry and materials. Eng Struct 107:116–134 22. Suzuki N, Otani S, Kobayashi Y (1984) Three-dimensional beam-column sub-assemblages under bi-directional earthquake loading. In: 8th world conference on earthquake engineering, San Francisco, USA 23. Monti G, Nuti C (1992) Nonlinear cyclic behavior of reinforcing bars including buckling. J Struct Eng 118:3268–3284 24. Li L, Mander JB, Dhakal RP (2008) Bidirectional cyclic loading experiment on a 3D beamcolumn joint designed for damage avoidance. J Struct Eng 134:1733–1742 25. Akguzel U, Pampanin S (1987) Effects of variation of axial load and bidirectional loading on seismic performance of retrofitted reinforced concrete exterior beam-column joints. J Compos Constr 14:94–104 26. Low SS, Moehle JP (2004) Experimental study of reinforced concrete columns subjected to multi-axial cyclic loading. University of California, Berkeley 27. Li KN, Aoyama H, Otani S (1988) Reinforced concrete columns under varying axial load and bi-directional lateral load reversals. In: 9th world conference on earthquake engineering, Tokyo–Kyoto, Japan 28. Bechtoula H, Kono S, Watanabe F (2005) Experimental and analytical investigations of seismic performance of cantilever reinforced concrete columns under varying transverse and axial loads. J Asian Archit Build Eng 4:467–474 29. Rodrigues H, Furtado O, Arede A (2015) Behavior of rectangular reinforced-concrete columns under biaxial cyclic loading and variable axial loads. J Struct Eng 142 30. Comite Euro-International Du Beton (CEB) (1996) RC frames under earthquake loading: vol. 2. ASCE, Publication Sales Department, Thomas Telford, London 31. Paulay T, Priestley MN (1992) Seismic design of reinforced concrete and masonry buildings. Wiley New York, USA 32. Zwilinger D (2011) CRC standard mathematical tables and formulae. CRC Press, USA 33. http://www.strongmotioncenter.org/vdc/scripts/earthquakes.plx. Accessed 27 Oct 2017 34. Kunnath SK, Reinhorn AM, Lobo R (1992) IDARC version 3.0: a program for the inelastic damage analysis of reinforced concrete structures. Technical Report NCEER, US National Center for Earthquake Engineering Research, 92

Fatigue Life Estimation of an Integral RC Bridge Subjected to Transient Loading Using Ansys M. Verma and S. S. Mishra

Abstract Integral bridges are becoming popular day by day, as they are easy to construct and require lesser maintenance efforts due to absence of bearings. There is an increasing tendency to construct long-span bridges. However, due to movement restraints, fatigue stresses build-up that leads to a reduction in the useful life. In this study, an effort has been made to estimate the fatigue life of an integral bridge subjected to transient loads. In this paper, the results of transient analysis of an integral bridge of total length 156 m having five continuous spans with a maximum span of 40 m has been done using ANSYS. The roles of deformation and von-Misses stress that occur in the bridge have been found to influence fatigue life. Further, midpoint deflection in the longest span, its variation with loading history, and its influence on fatigue life have been analyzed and found to match satisfactorily with standard results, and the same process is applied on various length of a longer span. Keywords Integral bridge · Fatigue life · Transient loading · ANSYS

1 Introduction Integral bridges in the simplest term can be classified as bridges that are constructed without joints between pier and deck. In Integral bridges, the monolithic connection is established between the superstructure (deck and girder) and the substructure (piers and abutments). Starting from one abutment these bridges are constructed without joints to another abutment. They do not have any joints between other intermediate piers. In Integral bridges, the bearings are removed to eliminate the problems associated with the installation, maintenance, and replacement of bearings, sometimes they are very costlier and become uneconomical to repair further, the jointless bridges that M. Verma (B) · S. S. Mishra Department of Civil Engineering, NIT Patna, Patna 800005, Bihar, India e-mail: [email protected] S. S. Mishra e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_17

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are used to avoid and eliminate the characteristic problems related with installing, maintaining, and repairing deck joints and bearings [11, 12]. Transient loading can be defined as loads and forces that occur and vary over a short time interval. A transient load can be to any load that will not remain on the bridge forever. Mostly, these loads include vehicular live loads and their tributary effects including dynamic load allowance, braking force, centrifugal force (caused in the curved section only), and live load surcharge. Additionally, there also exist pedestrian live loads, force effects due to uniform temperature, and gradient of temperature, force effects because of settlement of piers, river water force and pressure due to water stream on piers, wind loads on structure, wind on live load, friction forces that are generated between vehicle and deck pavement, ice loads in some areas, vehicular collision forces occur during accidents, vessel collision forces, and earthquake loads. For most ordinary bridge case, there are a few transient loads which are likely to be considered and that are live loads of vehicle and their subsequent effects including braking force, centrifugal force, and dynamic load allowance. These subsequent effects shall always be collectively considered with the gravity effects of live loads. For this study, only the vehicular load is provided according to Indian-standard IRC-6:2014 and the permanent load that includes material load of bridge structural components and nonstructural components, load of material used in wearing coat of deck surfaces that is laid over deck are considered, for simplicity down drag forces, horizontal earth pressure loads, vertical pressure due to a dead load of earth fill [3], earth surcharge load, force effects due to creep, shrinkage, secondary forces that are generated from post-tensioning of members, and other forces that are introduced due to construction process are not taken into consideration. Due to the above applied transient loading, the bridge deck deflects and expands and other parts of the bridge also show such tendency of deflection and expansion. This deflection and expansion from deck and girder are transferred to the piers because of fixity. The bridge girder in this transient case shows a behavior similar to a fixed beam [2]. As in the fixed beam, a large amount of deformation occurs at the mid-span and this deformation results in the plastic deformation of the beam [5]. A similar case will occur in the integral bridge, but under a different type of geometrical consideration like bridge length, span, or various other things are considered for later result verification. Depending upon these variations in expansion and deflection, the bridges may show different types of mechanical responses [1]. There are three methods of analyzing the fatigue life and these are strain life, stress life, and fracture mechanics [10]; In ANSYS 17.1 Fatigue Module, only strain life and stress life methods are available. At present, the approach to strain life is commonly used. Strain life mainly deals with the occurrence of less number of fatigue cycle so-termed as low-cycle fatigue, by using this approach the fatigue life can be predicted to an acceptable limit. The strain life-based approach is based on crack initiation. Whereas the stress life is depending upon the total life and has nothing to do with the crack initiation so it deals with the high number of cyclic loads before failure so it results in high cycle fatigue. Low-cycle fatigue refers to 16 ton) and an overturning moment capacity shaker system. India’s advanced launch vehicle is termed as GSLV MK III. To meet such extreme test requirements, single shaker testing is not ideal, as it needs large expanding fixtures, high capacity bearings and corresponding auxiliary systems. A similar multi-shaker testing philosophy is adapted by European Space Agency (ESA) and NASA for large articles [2]. Hence, a multi-shaker system with two shakers is used for qualification and acceptance vibration testing of GSLV Mk-III launch vehicle subassemblies. Multi-shaker operating system is crucial to maintain the amplitude and phase variations within 2% and 5° , respectively, for the dual shakers, as the two shakers are connected rigidly to a common platform. Vibration testing of several subassemblies of a GSLV Mk-III launch vehicle is performed using the analog multi-shaker control system meeting the diverse requirements of testing subassemblies with large CG offset and sensitive avionics packages. The current feedback signals from the dual shakers in uncoupled mode are fed to the analog multi-shaker control system, and tuning procedure is carried out to reduce the phase and amplitude differences between the signals. The tuning procedure is conducted in the uncoupled mode using a potentiometer phase and amplitude controls. The deviations in phase and amplitude of the signals are nullified for a bandwidth of lower and higher frequencies. The process of tuning between the feedback signals from the dual shakers is repetitive in action and needs expertise. In this context, a new digital multi-shaker control system is realized with an objective to have improved features compared to the existing analog multi-shaker system. The new digital multi-shaker system automatically corrects the deviations in phase and amplitude of the dual shakers, allows to specify the frequency bands for correction, defines threshold limits for phase and amplitude deviations, ensures simultaneous switch ON/OFF of both the power amplifiers and shaker systems, interlocks cross-coupling, ensures shut down of both systems in case of failure in any of the auxiliary systems, monitors the amplifier output signals to the dual shakers and also has a provision to limit the peak currents which helps to safeguard the power amplifiers and shaker systems. Finally, the safety parameters are given utmost importance for the smooth shut down of the entire system in case of any exigency.

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2 Analog Multi-shaker Control System The analog multi-shaker control system controls two 16-ton electrodynamic shakers as shown in the block diagram (Fig. 1). In order to vibrate the dual shakers synchronously, the drive signals are passed through a set of band pass filters of low and high-frequency ranges covering the operating range of the system. Then the amplitude and phase of these drive signals are adjusted by the gain and phase controlling potentiometers in the controller to ensure the current feedback signals are synchronous with respect to phase and amplitude. The tuning procedure allows to neutralize the phase and amplitude deviations in an iterative method. Various steps involved in the procedure are shown in Fig. 2. The tuned signals from the analog multi-shaker control system are fed to power amplifiers and drives the dual shakers synchronously. The amplitudes of the output signals driven to the shakers are continuously monitored against the defined threshold limit to prevent damage to the test equipment. The accelerometer feedback in Fig. 1 from the transducer is acquired, conditioned and processed in a separate vibration control system in real time to qualify the test article for the recommended test levels. Figure 3 shows the current feedback signals and accelerometer feedback signals which are tuned with an existing analog system and monitored in an oscilloscope at 400 Hz. Applying the tuned settings of an analog multi-shaker system, large launch vehicle subassemblies of 4 m diameter, 10 m tall and mass up to 2.5 ton have been successfully tested. Figure 4 shows the setup for vibration testing of LVM3 equipment bay in the lateral axis.

Fig. 1 Analog multi-shaker control system

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Fig. 2 Tuning procedure of analog multi-shaker control system

Accelerometer feedback Current feedback

Fig. 3 Phase and amplitude deviations of the two shakers at 400 Hz

2.1 Limitations of Analog Multi-shaker Control System In the analog multi-shaker control system, tuning of the dual shakers is done in manual mode by adjusting the amplitude and phase potentiometers which is a tedious job and is more susceptible to errors. Secondly, the existing system is a custom-built, imported hardware and is very costly to maintain. Since the operating range is covered only in two bands, it is not possible to achieve closer matching of both the signals, leading to more phase and amplitude difference.

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Fig. 4 Vibration testing of equipment bay in lateral axis

3 Design of New Digital Multi-shaker Control System Considering the merits and limitations of the analog multi-shaker system, a new digital multi-shaker control system is proposed and developed. The principle of operation of the new control system is shown in Fig. 5. It employs a repetitive compensation control strategy in which one of the feedback signals is considered as reference and the other as measured. The operation of the system involves two processes. In the first process, the phase and amplitude deviations between the two current feedback signals for discrete exciting frequency in the digital domain are calculated using a sample and gain estimator block as shown in Fig. 5. Based on the estimated deviations, the adjustments required in phase and amplitude at discrete exciting frequencies are derived automatically in the digital domain and updated in the correction table. In the second process, while operating the shakers in coupled mode, based on the frequency of COLA, using the correction table, the real-time controller corrects the output drive signals to the dual shakers. The phase difference is corrected by delaying the samples in a selected buffer and the amplitudes are scaled proportionately. The synchronous phase and amplitude of the dual shakers are attained in an automatic routine in the digital domain. The steps involved in the new tuning procedure are shown in Fig. 6.

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User Gain Control

Phase Correction

Frequency Generator (DDSO) Write Buffer (Addr. Generation)

Buffer Read (Reference Channel)

X

Addressing: FIFO order (No shift)

Multiplier

(Data to write, Address)

Decimator

Sample Delta

Buffer Read (Correction Channel) Addressing: FIFO order +/- Sample Delta

Ratio

X

X

Ratio Multiplier

Multiplier

(Ref/Corr.)

Frequency(COLA)

Scaling

AO 1

Power Amp. Drive

Gain

Sample & Gain Delta estimator All Analog Inputs

Power Amp. Drive

Gain Correction

Lag/Lead Analog Input Acquisition

AO 0

User selected Circular Buffer

Master Ctrl Signal

Control Signal Selector

AO Generation

Phase (PA O/Ps) Measurement • Frequency Trip Current (PA O/Ps) (Difference) • Phase Monitoring • Current • G (Accel.) Acceleration

Fig. 5 Principle of digital multi-shaker control system

Fig. 6 Tuning procedure of digital multi-shaker control system

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3.1 Control Technique of Digital Multi-shaker Control System Considering the current feedback signals from the dual shakers are x1 [n], x2 [n] master and slave, respectively [3] x1 [n] = k1 sin(n + φ1 )

(1)

x2 [n] = k2 sin(n + φ2 )

(2)

where k1, k2 are amplitudes and φ1 , φ2 are phases of the respective sine data sequences. The sine sequences are sampled at frequency s and the exciting frequency is . Length of the sine sequence block N0 is given by N0 = s / 

(3)

Considering x2 [n] lags/leads x1 [n], let the phase difference observed be φ = φ1 ± φ2

(4)

φ is computed by extracting a single tone using FFT-based function; once the φ is given, the phase of x2 [n] will be φ2 = φ1 ± φ

(5)

Phase of x2 [n] is adjusted by shifting the calculated samples given by N f = s ∗ (φ1 ± φ)/  ∗ 2π

(6)

For the assumed buffer size N, N f < N /2.

(7)

3.2 Implementation of the Digital Multi-shaker Control System The digital multi-shaker control system is implemented on National instruments hardware and programmed using the LabVIEW platform. It consists of a real-time controller with an embedded operating system along with high-speed DAQ modules. The software consists of two modules; first one is a host module, which provides the

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Fig. 7 Overview of digital multi-shaker control system

user interface. It sends commands to the RT controller, receives and displays data from RT and performs data logging. In the second one, the RT module is executed in the PXI controller. It receives commands from the host and performs acquisition, generation and control tasks. The communication between the host GUI and RT module will be performed over TCP and UDP links. The following block diagram in Fig. 7 gives an overview of the system.

3.3 Testing and Qualification of New Multi-shaker Control System The qualification of a new multi-shaker control system is done as per the setup shown in Fig. 8. The output drive signals of the multi-shaker control system are fed to the input of the power amplifier. The deviations in phase and amplitude of current feedback signals observed at various discrete frequency bands are corrected by the new multi-shaker control system.

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Fig. 8 New multi-shaker control system

In order to test the developed method, a constant voltage signal of 1 V rms, 400 Hz is generated by an internal oscillator and fed to the input of the power amplifier. The phase and deviation between the current feedback signals at 400 Hz are estimated as 3° in which the second current feedback signal leads the first, and the amplitude ratio of the first current feedback signal to the second is estimated as 0.8. To nullify the 3° phase difference caused by the second current feedback signal, the samples to be shifted are calculated by the phase estimator block, and the waveform is generated accordingly. Now the generated waveforms in which the second feedback signal is delayed by 3° (approximately 2.97° ) and similarly the amplitude ratio is also adjusted to 1.2 (approximately 1.18) to compensate the amplitude deviation. The results are shown in Fig. 9. The verification test results show that the new multishaker control system can compensate phase and amplitude deviations at discrete frequencies quantitatively, meeting our requirement.

4 Conclusions For vibration testing of large subassemblies of a launch vehicle, a multi-shaker control system is required to meet the test requirements. In the present paper, the development of a new multi-shaker control system has been discussed. The new system adjusts the phase and amplitude differences in a quantitative manner compared to the existing system. It helps to achieve better synchronal control between current feedback signals at various bands of frequencies over the operating range. Simultaneous switching ON/OFF of the power amplifiers, interlock cross-coupling which shuts down the shaker system in any exigency and software options to limit the peak currents for safeguarding the systems are additional features.

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Shaker – 1 current COLA

Shaker – 2 current

Fig. 9 Corrected output waveforms at 400 Hz

References 1. Nasa handbook–7005, Dynamic environmental criteria, March 2001 2. Popvitch A (2003) Multi shaker upgrade at ESTEC, ESA 3. YaoqiFeng, Hanping (2014) Research on dual shaker sine vibration control. In: International congress on sound and vibration, July, 2014

Dynamic Characterisation as a Tool for Avoiding Vibration Related Problems Ajay Kumar Panda, Asir Nesa Dass N, R. Balaji Srinivas, Arunkumar R, and M. Vasudevan Unni

Abstract In aerospace applications, mathematical models used for the dynamic analysis of launch vehicles are validated through dynamic characterization tests for better confidence in the models. The models are being used for flight critical studies; hence, the accuracy of the models in predicting the behavior of the system correctly is essential. This paper describes the dynamic characterization tests used for validating the mathematical model of the system from a dynamics point of view. In the paper, tests on three different test specimens with different requirements and different modes of testing are presented. The objective of the tests is to mainly obtain the frequency, mode shapes of the specimen, and get an estimate of the damping of the system, if required. The test methodologies are dependent on the test article configuration, which includes its material, overall layout, attachment points, etc. In the three test cases presented, the first test was an impact hammer excitation test carried out with the test specimen suspended with slings. In the second test case, the test was carried out by mounting it over the slip table of a high capacity shaker, and provided sine & random excitation through the shaker. For the third case, the specimen was fixed rigidly at the base, and excitation was carried out with small capacity shakers to estimate frequency response functions (FRFs) at various points. Keywords Dynamic characterization · Mode shape · Modal analysis

A. K. Panda (B) · A. N. D. N · R. B. Srinivas · A. R · M. V. Unni ISDTF/SDMG/STR, VSSC/ISRO, Trivandrum 695022, India e-mail: [email protected] A. N. D. N e-mail: [email protected] R. B. Srinivas e-mail: [email protected] A. R e-mail: [email protected] M. V. Unni e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_27

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1 Introduction In aerospace and other critical applications, any new design or major modifications in the existing design of parts or assemblies has to be qualified through tests. The qualification tests include static, dynamic, acoustic tests, etc. In aerospace applications, dynamic test plays a major role in determining the flight worthiness of the test assembly. For airborne structures, the estimation of dynamic characteristics is necessary for updating the mathematical model, which can be used for response studies and digital autopilot design with full confidence. The present paper describes three test cases with different test objectives, different specimen configuration and material, and hence different test setup and procedures were adopted for extracting the dynamic characteristics of the specimens.

2 Test Case I This section describes the dynamic characterization testing of a composite thin cylindrical structure that has undergone a modification in design. The objectives of the tests were (1) to obtain the dynamic characteristics like frequency, mode shape, and damping of the structure to be used for finite element model updating and (2) to verify the health of the structure after the static test to confirm structural integrity. Hence, tests were carried out prior to the structural test to obtain the pre-static test response and the same set of tests were repeated after the structural test for structural health monitoring.

2.1 Test Setup The test article was a thin cylindrical structure made of composite material. The overall diameter of the structure is about 3000 mm and the height is about 1200 mm. Total mass of the test article was about 175 kg. On the aft end of the test article, a metallic ring was provided for additional stiffening. Tests were carried out with free-free boundary conditions by suspending the article from crane through elastic chords. The free-free condition was preferred to eliminate the uncertainty of support stiffness. Figure 1 shows the test article in a free-free condition, with 4 elastic chords. Impact excitation was used wherein the test article was excited using an instrumented hammer. Tests were carried out with the excitation at four locations, two on Fore End side and two on Aft End ring. The transfer function for all the response locations was determined with each of these impact locations as reference. The objective of multiple reference points is to have redundant measurements to enhance the confidence in the test results. After the tests, the Frequency Response Function

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Elastic Chord

Modal testing and Data Acquisition system

Fig. 1 Test article with suspension

(FRF) data obtained from all the reference locations were compared and the best FRFs were selected using the Modal Assurance Criterion (MAC). These FRFs were used for estimating the resonant frequencies and corresponding mode shapes. The bold dots in Fig. 2, show the four excitation locations. The arrows show the 4 planes along the height of the test article, where measurements were taken.

Fig. 2 Response measurement Locations and modal model [Case 1]

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To effectively capture shell modes of such a large structure, response measurements were carried out using high sensitivity ICP type accelerometers having TEDS support. Figure 2 shows the response locations as grid points. Radial measurements were taken at four planes, with 72 measurements in each plane at 5° interval (Total 288 measurements). From the measured input (force) and output (response), FRFs were computed. With 4 reference locations (impact excitation points) and 288 response measurements, 4 × 288 = 1152 FRFs were obtained. The FRFs corresponding to each reference location were separately analyzed using Modal Analysis Software for determination of the dynamic parameters, i.e., frequencies, associated mode shapes, and damping ratios.

2.2 Test Procedure The tests were carried out in 2 phases to verify the structural integrity of the article. (i) Pre-static test dynamic characterization and (ii) Post-static test dynamic characterization. The test setup is shown in Fig. 1. The excitation was provided with an instrumented hammer with rubber tip and the transfer function for 288 channels with respect to each excitation location was recorded from 0 to 100 Hz, in each test. (these data were generated from 8 sets of tests with a 32 channel acquisition system). The same procedure was repeated for the post-static test dynamic characterization and transfer functions with respect to the same 4 reference locations were obtained.

2.3 Data Analysis and Results In data analysis, the first task was to select the best set of transfer functions to interpret the dynamic characteristics of the test article correctly. With more detailed analysis, the transfer functions with respect to location 9 were selected to be the best ones from satisfying MAC [1]. PolyMAX method of Frequency Domain Parameter Identification method was used for the analysis and frequency band of 1–100 Hz with a model size of 120 [1]. The stabilized poles were selected from the stabilization diagram which corresponds to the resonant frequency of the structure. The mode shapes and the damping were estimated for all the selected frequencies. The same method of analysis was followed for dynamic characterization after the static structural test, for FRFs with all 4 reference locations. The dominant and global modes were selected for comparison with the post-structural test results. The pretest and posttest FRFs at location 40 were compared in Fig. 3. It shows a close match indicating no deviation in dynamic characteristics of the structure due to static load test.

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Fig. 3 Comparison of FRFs for pre-static and post-static test case (Location 40)

3 Test Case II For multiple satellite missions, different configurations of adaptors were designed to accommodate and ensure the safe deployment of each satellite in their respective orbits without any collision. The mathematical model of the new adaptor was validated through dynamic characterization tests. The adaptor was made of the honeycomb structure. The main objective was to estimate the frequencies and mode shapes of the satellite simulator and masses mounted on the adaptor. Due to the honeycomb structure, impact excitation was not suitable for this specimen; hence the test was carried out by mounting it on a shaker and provide base excitation.

3.1 Test Setup The test article was mounted on the slip table of an electro-dynamic shaker using a conical fixture (Fig. 4). Three triaxial accelerometers were mounted on each of the heavy masses, as well as satellite simulators. One accelerometer each is mounted at the base of these balancing masses and satellite simulators. Three accelerometers were mounted on the cylindrical portion in two perpendicular directions to identify both bending and torsion modes of the cylinder.

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Satellite Simulators, 4 Nos Heavy masses, 2 Nos

Honeycomb Cylinder Interface Plate

Conical Fixture

Fig. 4 Test setup (lateral axis)

Table 1 Test matrix for case II

Sine

5–200 Hz

0.1 g, 0.25 g

Random

10–2000 Hz

0.5 grms

3.2 Test Procedure Low-level Sine and Random Tests were carried out in two lateral axes. The details of the tests were provided in the Table 1. The data were acquired in a data acquisition system. Time and frequency domain data were acquired.

3.3 Test Analysis and Results The results were analyzed and the modal properties were extracted from the test data and validated using modal validation methods. Control channel (input) was used as the reference for generating Frequency Response Functions, which were used to identify the modes. Typical FRFs were shown in Figs. 5 and 6, where a number of closely spaced modes can be seen. To distinctly identify each mode, mode shapes were generated using coordinates of the location and the FRF data. All the mode shapes were compared with finite element analysis results (a typical comparison of Mode shape as shown in Fig. 7). The difference clearly shows the importance of tests to validate and update the model if necessary.

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Fig. 5 FRF at a typical location (Tangential measurements)

Fig. 6 FRF at a typical location (Radial measurements)

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Test Mode: 16.2 Hz

Fig. 7 Typical mode shape from finite element analysis and test

4 Test Case III In this test case, the test article was a large structure that was designed as a gridded structure of honeycomb to keep the weight minimum. It was attached at the bottom to an adaptor which attaches this structure to the vehicle during flight. On top of it, the main satellite was attached through an adaptor. On the gridded specimen, three small satellites were mounted. Hence, the structure was critical for the flight point of view, and hence it’s FE model has to be validated through dynamic characterization tests. The different options for excitation included either mounting on a shaker or using impact excitation through a hammer or exciting with small shaker while fixing it rigidly at the base. Impact excitation was not a good choice as the size of the subassembly was too large. Due to the lack of interfaces in large slip tables at smaller diameters, it cannot be mounted on large shaker systems. Also, due to heavier mass, small shakers cannot be used, and thus this option of using a shaker has been ruled out. Hence, it was decided to fix the subassembly at the base and excite it using small portable shakers for exciting all the required modes.

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4.1 Test Setup The specimen was mounted rigidly on a fixture to ensure minimum flexibility at the base. The base flexibility adds an uncertainty while matching the analysis results with the test results. On the top of the specimen, a satellite simulator with the same mass and moment of inertia as the satellite was attached to get a flight equivalent configuration. This was important to ensure that both the test and analysis represent the same configuration. The instrumentation locations were numbered as shown in Fig. 7. The measurements were carried out on two perpendicular directions to find lateral, as well as torsion modes. Triaxial measurements were also placed near the satellite simulators for each small satellite. ICP type accelerometers were used for the measurements. Three Small Shakers (200 N force rating) were used for exciting the specimen in axial, lateral, and torsion modes. The input force was measured using force gage connected between the rod connecting the shaker to the specimen (Fig. 8).

4.2 Test Procedure Band -imited random excitation was used for the specimen from 5 to 100 Hz for the lateral modes and from 5 to 300 Hz for torsion modes estimation (Fig. 9).

4.3 Test Analysis and Results The test results were analyzed and modal parameters were extracted using the good data. The frequency and mode shapes were compared with Finite element analysis and a good match was observed (Figs. 10, 11).

5 Conclusions This paper provides details of the dynamic characterization of three different test specimens. In the first test case, a composite thin cylindrical structure was tested in a free-free condition. The pre- and post-static test results were compared and a good match in frequency and mode shapes was observed. This confirmed the structural integrity of the test article, which was later confirmed by NDT techniques also. In the second test case, a satellite adaptor was tested by mounting on a vibration table. By taking FRFs with respect to the shaker input (control channel) the frequency and mode shapes were obtained and compared with the Finite element results. The FE model was then updated to match the test results. In the third case, a gridded assembly of honeycomb structure was subjected to dynamic characterization tests using small

334

Fig. 8 Instrumentation for test case III

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Fig. 9 Excitation for test case III 0.10

g/N Log

FRF Pitch:53:+Y/Pitch:52:+Y

10.0e-6

11.33

33.59

11.33

33.59

∞ Phase

180.00

-180.00 5.00

Fig. 10 Typical FRF for lateral excitation

Hz

100.00

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Fig. 11 Comparison of first and second bending modes (Finite Element and Test Mode shapes)

shakers suspended with cranes. The global frequencies, as well as frequencies of each small satellite branch mounting, were also estimated accurately. The test data was matching with the FE analysis after modifications. As we have seen from the above three test cases, dynamic characterization tests can be effectively used to validate mathematical models, and these models can be used for all critical studies with confidence to avoid problems to the systems and assemblies by proper management before the actual flight/ground use.

Reference 1. LMS user manual, LMS international, 2010

A Simplified Impact Damping Model for Honeycomb Sandwich Using Discrete Element Method and Experimental Data Nazeer Ahmad, R. Ranganath, and Ashitava Ghosal

Abstract Honeycomb sandwich laminates with aluminum and carbon fiber reinforce polymer (CFRP) face—sheets are widely used in spacecraft structures and aerospace industries. The damping behavior of such structures is reported to improve when the granular particles, called damping particles, are inserted in the honeycomb cells. The discrete element method (DEM) has been successfully used and found to give a reasonably accurate estimate of the impact damping. In DEM formulation, Newton’s laws of motion are used to obtain the equations of motions of each damping particle considering the contact forces from immediate neighboring particles and other sources, if any. The use of DEM for the real structure where the number of particles is of order 108 or more is inefficient and impractical to perform optimization. In this paper, a damping model dissipating equivalent energy is presented for a system consisting of a small honeycomb sandwich coupon filled with damping particles and has resonance frequencies beyond the bandwidth of the model. The coupon is subjected to a range of harmonic excitations (varying frequency and amplitude). The energy dissipated by the damping particles is estimated by DEM. The normal and tangential components of contact forces are modeled using Hertz’s nonlinear dissipative and Coulomb’s laws of friction, respectively. Then the parameters of the equivalent damper are obtained which dissipates the same energy. The damping model presented incorporates the effect of fill fraction, particle size, and material, as well as the amplitude and frequency of excitation. The comparisons of the DEM model for some of the load cases are done with the experimental data showing reasonably good agreement. The model presented could be readily incorporated in the FEM model like zero-stiffness proof-mass actuator, and the effect of impact damping can be studied without actually solving the DEM governing the motions of the particles. Keywords Impact damping · Discrete element method · Honeycomb sandwich · Granular damping particles · Passive vibration isolation · Spacecraft structure N. Ahmad (B) · R. Ranganath ISRO Satellite Centre, Old Airport Road, Bangalore 560017, India e-mail: [email protected] A. Ghosal Indian Institute of Science, Bangalore 560012, India © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_28

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1 Introduction Composite laminates with aluminum/CFRP face sheets and honeycomb core are widely used in aerospace industries owing to its lightweight and excellent mechanical properties. However, in general, a honeycomb sandwich composite possesses very small, less than 2%, inherent structural damping, which results in excessive resonance responses leading to failure of a structure or the mounted subsystems. The damping characteristic of honeycomb is reported to improve when granular particles are inserted in the core [1, 2]. This technique of using granular particles to enhance the damping characteristics of structures is called particle impact damping (PID). The PID is simple, low cost, and effective in extreme environmental conditions. The damping particles dissipate energy in the form of heat and sound after acquiring it from the vibrating structures by momentum exchange. The dissipation is highly coupled and nonlinear depending mainly on parameters: level and frequency of excitation, density of damping particles, fill fraction, mass ratio, and location of filling. The large number of parameters affecting the damping performance of the particles makes it difficult to develop a model, which could capture the complex interactions taking place. The different modeling techniques available in the literature can be found in [6, 8, 11, 12]. One of the methods that is widely used in the particle assemblage simulation is the discrete element method (DEM) [3]. The DEM alone takes into account the particle-toparticle level interaction, enabling to study the dependence of energy dissipation on a large number of parameters. In DEM, equations of motions of damping particles are obtained using Newton’s laws assuming that only the particles in immediate neighborhood affect the motion. The contacts of the particle-to-particle and particleto-cell walls are modeled using the Hertz’s theory and Coulomb’s law. The energy dissipation is evaluated for each contact occurring during the vibration. As these computations, require to solve the coupled dynamics of the particle and structures, for large structures the number of damping particles runs into millions, and thus the DEM becomes inefficient. Generally, honeycomb sandwich panels that are used in spacecraft are large, and thus the number of damping particles required to effect the damping characteristic is huge. Thus, the use of DEM is computationally very expensive. Therefore, the energy dissipation by damping particles filled in a small honeycomb coupon of size 100 mm x 100 mm under sinusoidal excitation is estimated experimentally, for some load cases, and using DEM. Furthermore, the dependence of energy dissipation on the various parameters is studied. Finally, the parameters of an equivalent viscous damper are estimated, which could be readily integrated like a proof-mass actuator enabling prediction of structural responses without solving the actual DEM problem.

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2 Mathematical Formulation A small square-shaped coupon of the honeycomb sandwich, shown in Fig. 1, is considered for assessing the dissipation of energy by damping particles. The coupon is 100 x 100 × 25.4 mm dimension. The coupon is very stiff; a normal mode analysis with a free-free boundary condition shows that the first natural frequency is at 6235.2 Hz and the corresponding mode shape is shown in Fig. 2. In this study, we intend to study the damping behavior of the coupon up to 1000 Hz. The coupon is assumed to be rigid, and therefore, cells of the honeycomb do not rotate and undergo deformation. The equations of the cell walls with respect to a local coordinate system, which is at the geometric center of the cell with an axis parallel to the global axis, is shown in Fig. 1. The cell walls are defined by Eqs. (1). The x-axis of the global coordinate system is along the L-direction of the core and the y-axis is along the W-direction of the core. h z¯ ± √ 2 3 y¯ ± R 2 y¯ x¯ + √ ± R 3 y¯ x¯ − √ ± R 3

=0 =0 =0 =0

(1)

The damping particles are constrained to move inside the cell as shown in Fig. 3, when the coupon is vibrated. The damping particles in the cells collide and rub with the walls of the cells and face sheets, as well as between themselves. The rubbing and collision result in momentum transfer and energy dissipation. An impact results

Fig. 1 Honeycomb coupon and axis definition

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Fig. 2 First mode of the coupon

Fig. 3 Motion of damping particles in a cell

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in normal and tangential forces, the normal force is modeled by Hertz’s nonlinear dissipative contact model defined in [9], as      1/4  3/2 finj = − kn δinj + α m i∗j kn δinj δ˙i j ni j

(2)

where δ˙inj is the local indentation velocity and α is the damping constant related to normal restitution coefficient en [9], defined as.  α = − ln(en )

5 ln(en )2 + π 2

(3)

The Hertz’s constant kn can be found in [4], and the equivalent mass m i∗j in Eq. (2), is defined as m i∗j =

mi m j mi + m j

(4)

The tangential contact force is modeled the coulomb’s law of sliding friction [4], given as   Vit j fitj = −μfinj   t  V 

(5)

ij

where μ is the coefficient of friction and Vit j is the relative tangential velocity of contact points. The change in the velocity and evolution of the forces/moments during an oblique contact process is given in Figs. 4 and 5. Figure 4a–d present the velocity and forces/moments when a damping particle collides with a velocity of [0 0.5–0.1] m/s to the plane, z = −h/2. Figure 4c and d show the effect of nonlinear dissipative terms present in the expression for normal force, due to this dissipative term, the relative velocity reaches to zero well before the end of the contact process that can be seen as a small loop at the end of the contact process. The pre and post-collision velocity and √ force distributions of the same damping particle colliding again with the plan: y¯ − 3/2R = 0 is given in Fig. 5a–d. As it is known that the Coulomb’s model of friction force, for smaller incidence angles, typically less than 30o [4], does not predict correctly the post collision velocities. However, due to its simplicity and speed, it is extensively used by researcher in vibration problems. One of the consequence of Coulomb’s force is the oscillation of tangential force; this phenomenon is clearly visible in time-tangential force plot in Fig. 5d. The motion of the DPs in the cell can be described by the force and moment balance equations as [2].

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Fig. 4 Change in velocities of a particle colliding walls of the cells

m pi p¨ i = −m pi g+

n1 j=1

¨i = Ii 

fi j +

n2

fiw

w=1

 n1

n2 δi j ri − ni j × fi j + (ri − δiw )niw × fiw 2 j=1 j=1

(6)

where the mass of the damping particle is represented by m pi , radius ri , and moment of inertia Ii . The position vector and angular acceleration of the damping particle ¨ i , respectively. ni j and niw are the unit vectors. g represents is given by pi and  the acceleration due to gravity and the contact forces due to neighboring particle and walls are represented by fi j and fiw , respectively. The local normal indentations against damping particle and wall are represented by, δi j and δiw , respectively. The DEM formulation requires a very small time step in the integration of equations typically an order less than that of the contact period. In this work, a time step

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Fig. 5 Change in velocities of a particle colliding walls of the cells (2nd collisions)

of 2 × 10−6 s has been used. The selection of time step is crucial for the success of the DEM, as the new contacts are formed and old contacts are broken leading to change in differential equations being integrated. The energy dissipated per unit area by the contact forces during the impact as a result of vibration can be written as  Nc tc   α m i∗j kn δ 1/4 δ˙i j δ˙i j + f itj δ˙i j dt dt

Ed =

k=1 0

A

(7)

where tc is the contact duration, Nc is the number of contacts, A is the area of the coupon and E d is the energy dissipated. The energy dissipated by Hertz’s and Coulomb’s forces for a single particle collision described above in two events are given in Fig. 6.

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1.4

10

-6

1.2

1 Particle KE + Losses

Energy (J)

0.8

Particel KE Hertzian dissipation

0.6

Coulomb's dissipation 0.4

0.2

0 4.5

5

5.5

Time (s) 10

1.4

10

-4

-6

1.2 Particle KE + Losses 1

Particel KE

Energy (J)

Hertzian dissipation 0.8

Coulomb's dissipation

0.6

0.4

0.2

0 2.8

2.81

2.82

2.83

Time (s)

2.84

2.85 10

-3

Fig. 6 Energy dissipation prediction by Eq. 7 (1st and 2nd collision)

3 Criterion for PID Performance The PID dissipates energy by Collison and friction which results in a damping effect on the structure as it takes energy from the structures. As the process is highly nonlinear, the criterion for performance assessment should hold for harmonic, as well as transient vibrations. Specific damping capacity (SDC) is one such parameter

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which is used for assessment of the performance of a PID [5]. It is defined as η=

E E

(8)

where the kinetic energy dissipated per cycle is represented by E, and E is the maximum kinetic energy during the cycle. If the structure is subjected to harmonic excitation of constant acceleration amplitude, then E = E d

(9)

and E is given by E=

1 mca2 2ω2

(10)

The specific damping capacity is related to the loss factor as η/2π and to the linear damping as − ln(1 − η)/4π .

4 Specific Damping Computation and Experimental Validation The specific damping capacity computation is performed for the acrylic damping particles. The properties of the damping particles are given in Table 1, and the properties of the coupon are given in Table 2. The specific damping capacity is studied with respect to excitation acceleration level, frequency of excitation, and fill fraction as these are the parameters on which SDC is strongly dependent. It is reported in the literature that the density of the DP affects the performance but in the context of the honeycomb structures where it cannot be loaded with metallic particles as it Table 1 Properties of damping particles Properties

Units

Aluminum

Acrylic

Radius

mm

1

1.25

Density

kg/m3

2850

1180

Young’s modulus

N/m2

70 × 109

2.84 × 109

Poisson’s ratio

–

0.33

0.402

Coefficient of sliding friction

Normal restitution coefficient

0.50

0.85

Material pairs Aluminum—aluminum

–

Acrylic—acrylic

–

0.096

0.70

Acrylic—aluminum

–

0.14

0.70

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Table 2 Properties of honeycomb coupon Properties

Units

Face-sheet (AA 2024 T3)

Honeycomb core (CR 3/16-5056-0.0007-P-32)

Thickness

mm

0.25

25.4

Density

kg/m3

2800

32.1

Young’s modulus

N/m2

72 × 109

E x x = E yy = E zz = 10000

0.33

νx y = ν yz = νx z = 0.3

N/m2

–

Poisson’s ratio Shear modulus

G x y = 10000 G yz = 0.89 × 108 G x z = 1.85 × 108

Diameter of inscribing circle of hexagonal cell

mm

–

4.76

will drastically increase the weight of the structure nullifying the advantage it offers due to its lightweight. Therefore, in this study, light particle like acrylic is used and study with respect to the density of DP is ignored.

5 Experimental Setup The honeycomb coupon was mounted on a modal shaker (make: M B Dynamics, model: 2050A, Force rating: 100 N) fixed at the center of the coupon. An impedance head (make: PCB, model: 288D01) for measuring the input acceleration and force was fixed between stinger and honeycomb coupon. For measurement of velocity, a PDV-100 Portable Digital Vibrometer was used. The LMS system was used for all data acquisitions. The setup is shown in Fig. 7.

5.1 Computing the Loss Factor Using Experimental Data The loss factor can be computed from the direct measurement of velocity by laser vibrometer and the input force sensor fixed between the stinger and coupon. Let the f(t) and v(t) represents the instantaneous signals from the force sensor and laser vibrometer, respectively, then the complex power Pc can be expressed as [7, 10] 1 Pc = T

T  ∞ 0

n=0

f n e j (nωt−φ f )

 ∞ m

 vm e j(mωt−φv )

(11)

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Fig. 7 Experimental setup

The loss factor can be obtained from the complex power Pc as it is the ratio of the real and imaginary parts of the complex power given by Eq. (11). The loss factor can be related to SDC as discussed in Sect. 3. The SDC obtained for some of the load cases is given in Table 3. Three levels of harmonic input acceleration of constant amplitudes of [1 5 10]g at frequency points[50 100 500 1000] Hz for varying fill fractions are computed using the DEM and results are given in Table 3, along with the measured values. The coupon contains 441 cells and each cell can accommodate a maximum of 36 damping particles (100% fill fraction). As the DEM takes 12–16 h of computational time for each load case, and SDC depends on a range of parameters predominantly on fill fraction frequency of excitation and input acceleration amplitude, a multivariate interpolation function is proposed. The interpolating function is obtained using the data given in the Table 3, and the values of SDC at intermediate data points are generated using the interpolation function.

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Table 3 Specific damping capacity Frequency (Hz)

Acceleration (g)

Packing ratio

Specific damping capacity (DEM)

50

1

5

10

100

1

5

10

500

1

5

10

25

1.5702e−4

50

0.0018

75

0.0051

90

0.0089

25

0.2210

50

0.1856

75

0.2904

90

0.3490

25

0.1150

50

0.2783

75

0.3914

90

0.4832

25

2.9705e−4

50

0.0032

75

0.0109

90

0.0195

25

0.1836

50

0.1891

75

0.3158

90

0.3190

25

0.1211

50

0.2648

75

0.3849

90

0.4321

25

0.0034

50

0.1443

75

0.4963

90

0.6503

25

0.1965

50

0.5417

75

0.8044

90

0.8359

25

0.0975

50

0.2770

75

0.4145

Experimental 0.10

0.23

0.31

0.01

0.21

0.32

0.22

0.65

0.35 (continued)

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Table 3 (continued) Frequency (Hz)

Acceleration (g)

Packing ratio

Specific damping capacity (DEM)

1000

1

5

10

90

0.4756

25

0.0505

50

0.6138

75

0.8078

90

0.8010

25

0.1968

50

0.5612

75

0.8720

90

0.8843

25

0.0911

50

0.2886

75

0.4444

90

0.5305

Experimental

0.71

0.63

0.30

6 Variation of SDC with Input Acceleration Amplitudes Figures 8, 9, 10, 11, 12 show the variation of SDC with respect to input acceleration levels at fill fractions varied from 25%, 50%, 75%, and 90%, respectively. For all the fill fractions SDC increases as acceleration level increased up to 5 g, and thereafter, it is seen decreasing till 10 g. The levels computed using DEM and interpolated are shown in the legend. For low fill fractions, a lower value of SDC can be attributed

Specific damping capacity

0.25

0.2

1g DEM 5g DEM 10 DEM

0.15

2g Interpolation 3g Interpolation 4g Interpolation

0.1

6g Interpolation 7g Interpolation 8g Interpolation

0.05

9g Interpolation

0 0

200

400

600

Frequency (Hz)

Fig. 8 SDC at 25% fill fraction

800

1000

350

N. Ahmad et al. 0.7

0.6

specific damping capacity

1g DEM 5g DEM

0.5

10g DEM 0.4

2g Interpolation

0.3

4g Interpolation

3g Interpolation

6g Interpolation 7g Interpolation

0.2

8g Interpolation 9g Interpolation

0.1

0 0

200

400

600

800

1000

Frequency (Hz)

Fig. 9 SDC at 50% fill fraction 1

0.8

1g DEM

Specific damping capacity

5g DEM 10g DEM 0.6 2g Interpolation 3g Interpolation 4g Interpolation

0.4

6g Interpolation 7g Interpolation 8g Interpolation

0.2

9g Interpolation

0 0

200

400

600

Frequency (Hz)

Fig. 10 SDC at 75% fill fraction

800

1000

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1

1g DEM

Specific damping capacity

0.8

5g DEM 10g DEM 0.6

2g Interpolation 3g Interpolation 4g Interpolation

0.4

6g Interpolation 7g Interpolation 8g Interpolation

0.2

9g Interpolation

0 0

200

400

600

800

1000

Frequency (Hz)

Fig. 11 SDC at 90% fill fraction

Specific damping capacity

1

0.8 25% Packing ratio DEM 50% Packing ratio DEM

0.6

75% Packing ratio DEM 90% Packing ratio DEM 40% Packing ratio Interpolation

0.4

60% Packing ratio Interpolation 70% Packing ratio Interpolation

0.2

0 0

200

400

600

800

1000

Frequency (Hz)

Fig. 12 SDC at 1 g acceleration level

to a lesser number of particles in the cell, and thus less number of collisions, and therefore, smaller values of SDC. The value of SDC appears almost constant in the frequency range of study. However, for the fill fractions 50–90%, SDC increased up to 500 Hz and thereafter remains nearly constant.

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7 Variation of SDC with Varying Fill Fraction Figures 12, 13, 14 present the variation of SDC with respect to frequency when the amplitude of harmonic input acceleration is kept constant and the packing ratio is varied. The SDC is seen increasing with respect to frequency at lower acceleration levels for all fill fractions. However, the rate of increase with respect to acceleration level decreases as the acceleration increase. At an acceleration level of 10 g, SDC appears to be independent of frequency. The likely reason for such behavior could be the fact that particles remain most of the time in the cavity space and colliding less frequently with the structure.

Specific damping capacity

1

0.8 25% Packing ratio DEM 50% Packing ratio DEM

0.6

75% Packing ratio DEM 90% Packing ratio DEM 40% Packing ratio interpolation

0.4

60% Packing ratio interpolation 70% Packing ratio interpolation 80% Packing ratio interpolation

0.2

0 0

200

400

600

800

1000

Frequency (Hz)

Fig. 13 SDC at 5 g acceleration level

Specific damping capacity

0.6 0.5 25% Packing ratio DEM

0.4

50% Packing ratio DEM 75% Packing ratio DEM

0.3

90% Packing ratio DEM 40% Packing ratio interpolation 60 Packing ratio interpolation

0.2

70 Packing ratio interpolation 80 Packing ratio interpolation

0.1 0 0

200

400

600

Frequency (Hz)

Fig. 14 SDC at 10 g acceleration level

800

1000

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8 Conclusions The dissipation of energy by the damping particles filled in a small coupon of honeycomb is studied with discrete element method and experimentally. The coupon is vibrated with different levels of constant amplitude harmonic acceleration in a frequency band of 50–1000 Hz with varying amounts of damping particles in the cavity. The energy dissipation is estimated in terms of specific damping capacity and it is found to be dependent on predominantly three parameters: fill fraction, amplitude, and frequency of the input acceleration. A multivariate interpolation model of SDC is worked out using “pchip” interpolant. Using the interpolation, SDC is predicted and presented for various combinations of the variables. The interpolation function developed herewith for SDC can be used for the prediction of the structural response of any honeycomb structure treated with damping particles under harmonic, transient excitation loads. Acknowledgments This research work is partially funded by ISRO-IISc Space Technology Cell (ISTC/MME/AG/394). The authors would like to thank ISRO Satellite Center, Bangalore, for providing laboratory facilities.

References 1. Ahmad N, Ranganath R, Ghosal A (2016) Assessment of particle damping device for large laminated structures under acoustic excitations. Presented at the proceedings of 14th ISAMPE national conference on composites (INCCOM-14), Hyderabad 2. Ahmad N, Ranganath R, Ghosal A (2017) Modeling and experimental study of a honeycomb beam filled with damping particles. J Sound Vib 391: 20–34. 2017/03/17 3. Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29:47–65 4. Johnson KL (1985) Contact Mechanics. Cambridge University Press 5. Mao K, Wang MY, Xu Z, Chen T (2004) Simulation and characterization of particle damping in transient vibrations. J Vib Acoust 126:202 6. Olson SE (2003) An analytical particle damping model. J Sound Vib 264:1155–1166 7. Romdhane MB, Bouhaddi N, Trigui M, Foltête E, Haddar M. (2013) The loss factor experimental characterisation of the non-obstructive particles damping approach. Mech Syst Signal Process 38: 585–600 8. Saeki M (2005) Analytical study of multi-particle damping. J Sound Vib 281:1133–1144 9. Tsuji Y, Tanaka T, Ishida T (1992) Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol 71:239–250 10. Wong CX, Daniel M, Rongong J (2009) Energy dissipation prediction of particle dampers. J Sound Vib 319:91–118 11. Wu CJ, Liao WH, Wang MY (2004) Modeling of granular particle damping using multiphase flow theory of gas-particle. J Vib Acoust 126:196 12. Zhang C, Chen T, Wang X, Li Y (2014) Discrete element method model and damping performance of bean bag dampers. J Sound Vib 333:6024–6037

Aeroservoelastic Analysis of RLV-TD HEX01 Mission Mahind Jayan, P. Ashok Gandhi, Sajan Daniel, and R. Neetha

Abstract Study of Aeroservoelastic (ASE) interactions is of prime importance in modern aircrafts employing autonomous flight control systems. The complex nature of unsteady aerodynamic forces can induce adverse ASE coupling effects leading to mission failure. This study discusses the ASE analysis of Reusable Launch Vehicle Technology Demonstrator Hypersonic Experiment (RLV-TD HEX01) of the Indian Space Research Organisation (ISRO). Pertinent modeling philosophy adopted for various subsystems, analysis methodology, validations, and simulations carried out to establish closed loop stability of RLV-TD system is discussed in detail. The results of the study clearly indicate the absence of adverse modal coupling in the presence of unsteady aerodynamic and control forces. The existence of adequate closed loop damping for critical structural modes is established through simulations to ensure adverse interaction free environment in the experimental flight.

1 Introduction RLV-TD HEX01 of ISRO employs autonomous navigation, guidance, and control systems in the descent phase of flight. These systems have to perform with a high degree of reliability under demanding re-entry environment. The flight control system uses motion sensors (accelerometers, pitch, roll, and yaw rate gyros) to measure aircraft rigid body responses which are then processed by the control law embedded in the digital flight control computer to provide appropriate feedback signals to the primary control surface actuators to stabilize and control the aircraft. As the total vehicle response is the sum of a rigid body and elastic body responses, the motion sensors also pick-up the airframe dynamic responses due to elastic vibration modes of the structure at the sensor locations. These signals when processed by the digital flight control computer and fed back to the actuators may generate an undesirable M. Jayan · P. Ashok Gandhi · S. Daniel · R. Neetha (B) Structural Dynamics and Analysis Group, Structural Engineering Entity, Vikram Sarabhai Space Centre, Thiruvananthapuram, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_29

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effect on the aircraft responses if adequate attention is not paid on the closed loop stability of the system. The adverse effects include degradation in the handling qualities, adverse modal coupling, and increased elastic vibrations or, in extreme cases, dynamic instability of the aircraft [1]. RLV-TD descent phase configuration is a system with strong aerodynamic and structural coupling. These characteristics were established and validated from the series of wind tunnel and ground vibration tests conducted. The unsteady aerodynamic effects present in the descent phase can induce modal coupling which can prove detrimental to the mission [2]. Aeroservoelasticity represents this mutual interaction between the flight control system and the airframe dynamic response due to inertial and unsteady aerodynamic forces.

2 Mathematical Models The primary prerequisite for a linear ASE stability analysis is the availability of a mathematical model representing the rigid body and elastic structural vibration modes of the vehicle. The unsteady aerodynamic loads produced by airframe oscillations and dynamic characteristics of the closed loop control subsystem including control actuator dynamics are to be incorporated for generating an integrated ASE computational model. MSC.NASTRAN™ 2014, has been used as the analysis environment due to the proven capability of the software in capturing the integrated effects of structural dynamics system, unsteady aerodynamic system, and servo dynamic system [3]. Figure 1 illustrates the different component models that are assembled together to create the integrated aeroservoelastic model.

2.1 Integrated Finite Element Model A detailed Integrated Finite Element Model (IFEM) was created to capture the rigid body and flexible body dynamics of RLV-TD descent phase configuration. Eigenvalue analysis was performed on the model, and the relevant elastic mode shapes and corresponding frequencies were established. The results of the Eigenvalue analysis were validated against those obtained from Ground Resonance Tests (GRT). Modal parameters of critical structural modes were matching closely with the experimentally estimated values. Figure 2 illustrates the IFEM highlighting major structural components. Rudder and elevon are the control surfaces employed during the descent phase of flight.

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Structural Model Ground Resonance Test (GRT)

Mass, Centre of Gravity, Inertia and Geometric Data

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Linear Spline Coupling Aerodynamics Model Wind Tunnel Flutter Tests

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Actuator Dynamics Command to Actuator Control System Model Control Law

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Fig. 1 ASE model synthesis

Fig. 2 RLV-TD integrated finite element model

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Fig. 3 RLV-TD aerodynamic mesh

2.2 Unsteady Aerodynamics Model Unsteady aerodynamic forces are produced by flow disturbances due to elastic vibrations and due to turbulence where the flow itself is unsteady [4–6]. Unsteady aerodynamic forces have a characteristic phase lag with respect to the aerodynamic excitation which depends mainly on the flow Mach Number and reduced frequency [7]. Steady and quasi-steady aerodynamic models are not adequate to capture the effects of phase lag, and therefore, proper simulation of unsteady aerodynamic effects is a prerequisite for ASE study of winged body vehicles. Aerelasticity module of MSC.NASTRAN™ 2014, has been used to generate the complex unsteady aerodynamic influence coefficient matrices for various regimes of flow. The unsteady aerodynamic mesh used for RLV-TD is shown in Fig. 3. Linear spline theory has been used for coupling the structural degree of freedom displacements of RLV-TD IFEM to the aerodynamic degree of freedom displacements and thereby synthesize the aeroelastic model. The present analysis has used ZONA51 method for the supersonic regime and Piston Theory for the hypersonic regime of flight, respectively, for generating aerodynamic influence coefficient matrices [8].

2.3 Control System Model Schematic block diagram of RLV-TD descent phase integrated digital flight control system is shown in Fig. 4. The case under study uses acceleration and rate feedback from respective sensors for generating the control surface deflection commands. Numerical models of various control loop elements viz. sensors, filters, actuators are represented in the Laplace domain as second order transfer functions. Control system elements with higher order transfer functions are cascaded and idealized as a number of second order systems. The Transfer Function (TF) option in MSC.NASTRAN™ 2014, is used to represent all the elements of the control system [9]. Control system degrees of freedom are modeled as Extra Points (EP) in the input driver deck. The output variable of a transfer function is represented in the form of linear combination of input variables as in Eq. (1).

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Fig. 4 Integrated ASE model



n     Md s 2 + C d s + K d u d − Mk s 2 + Ck s + K d u ik = 0

(1)

k=1

In Eq. (1) u d is the output/dependent variable and u ik is the kth input/independent variable. Terms {M, C, K} correspond to the equivalent mass, damping, and stiffness terms, respectively, of the associated transfer function. These terms are assembled and added appropriately to the global aero elastic matrices to generate the aeroservoelastic matrices. Actuator modeling is an important aspect of the ASE analysis [10]. In line with other control system elements, actuators have been modeled as shown in Eq. (1). Actuator transfer function is modified so that the dependent variable is a moment acting on the control surface shaft. This allows the combined dynamics of actuator, control surface, and support structure stiffness to be captured in the analysis.

3 Mathematical Formulation of ASE Problem The synthesis of ASE problem lends itself amenable to frequency domain solution schemes. The integrated ASE matrices have the following form as in Eq. (2).   [Ms + ρ Ma + Mc ]q¨ + [Cs + ρvCa (χ ) − Cc ]q˙ + K s + ρv 2 K a (χ ) − K c q = 0 (2) In Eq. (2) {M, C, K} correspond to the equivalent mass, damping, and stiffness matrices of the structural, aerodynamic, and control subsystems with the subscript {s, a, c} representing structure, aero, and control, respectively. ρ and v represent the free stream density and flow velocity, respectively. The nondimensional parameter χ is the reduced frequency which is a measure of time taken by the flow to travel a chord length of the lifting surface to the time period of corresponding modal oscillation.

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The magnitude of reduced frequency determines the level of unsteadiness in the flow. Aerodynamic matrices generated depend on χ and the free stream Mach Number (M). For a given operating point Eq. (2), can be reduced to Eq. (3) 

 Mλ2 + Cλ + K {∅} = 0

(3)

where λ is the complex Eigenvalue of the system and is of the form given by Eq. (4). λ = α ± iω

(4)

Stability of the system defined by Eq. (2), can be evaluated by estimating the complex Eigenvalue, λ of the system. The real part of,λ i.e., α of a structural mode is a measure of closed loop damping of the corresponding mode. For a closed loop ASE system to be stable, the condition for stability is given by Eq. (5). α≤0

(5)

ASE analysis aims at finding the effective closed loop damping of all critical structural modes under the influence of unsteady aerodynamics and control system feedback. The stability of the system is ensured if all structural modes have adequate positive damping ratio which is equivalent to the condition in Eq. (5) being satisfied.

4 Simulation Studies Complex Eigenvalue Solution sequence, SOL 145 of MSC.NASTRAN™ 2014, has been used for estimating the system Eigenvalues employing the PK method as a solution scheme. Initially, complex Eigenvalue analysis was performed on the closed loop servo elastic system to estimate the changes in modal characteristics due to combined servo elastic system dynamics. It was observed that the structural mode shapes and frequencies of the servo elastic system are very close to values observed from GRT and real Eigenvalue analysis of the IFEM. Critical mode shapes are shown in Fig. 5. Analysis of the open loop structural frequencies highlights adequate spacing between the structural modes and control modes. Hence, the frequencies of close loop servo elastic system are expected to be sufficiently close to frequencies of the structural system, and this was established through Eigenvalue analysis of servo elastic system. Figure 5 shows the existence of lifting surface modes and control surface modes with significant elastic coupling. As unsteady aerodynamic effects can induce modal coupling due to the induced phase lag comprehensive ASE analysis, simulation studies were carried out at critical time instants to ensure the absence of adverse coupling.

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Fig. 5 Critical lifting surface mode shapes

Minimum margin cases with respect to control system stability were selected for assessing the ASE stability. These cases correspond to supersonic and hypersonic regimes of descent phase flight trajectory. ASE analysis was carried out at trajectory conditions corresponding to the minimum margin cases by using the corresponding flight dynamic pressure, Mach Number, and control system gains. Simulations were carried out with a conservative modal damping ratio of 0.5%. For the minimum margin case, all the critical structural modes have positive damping indicative of system stability. Further, perturbation studies were carried out by variation of flight velocity around the neighborhood of the minimum margin point. Modal frequency versus Flight velocity (v–f) and Modal Damping versus Flight velocity (v–g) curves was extracted to understand the associated variation in closed loop frequency and damping with variation in flight velocity. Figure 6 depicts v–f and v–g curve for a minimum margin case. A closer study of the damping curves in Fig. 6, reveals that effective close loop damping of the ASE system is more than the assumed modal damping ratio. Hence, it can be inferred that for the ASE system under study aerodynamic damping from unsteady aerodynamics and control system damping are augmenting the structural damping. All the critical structural modes studied showed damping to increase with flight velocity which implies higher aerodynamic damping due to an increase in dynamic pressure. Most of the critical modes exhibit a linear increase in damping with flight velocity as expected for a case with minimal aeroelastic coupling. However, the elevon symmetric mode shown by magenta dash dot lines and elevon asymmetric mode shown by blue dash lines displays a nonlinear variation in damping against flight velocity.

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Fig. 6 Minimum margin case v–f and v–g curves of critical modes

Coupled dynamics of the actuator, control surface, and support structure stiffness combined with flow unsteadiness will have a pronounced effect on the control surface modes leading to the nonlinear variation seen. Nevertheless, the values of closed loop damping observed are within acceptable limits. High frequency modes of vertical tail and rudder show a tendency of modal frequency coalescence at higher flight velocities, however, higher positive damping for these modes with increasing damping trend assures modal stability.

5 Conclusions This paper presents an approach to evaluate closed loop stability of an aeroservoelastic system. The problem formulation, ASE model synthesis and implementation, simulation studies, and discussion on results have been presented comprehensively.

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Closed loop stability of critical structural modes of RLV-TD HEX01 mission for descent phase has been estimated and found to be within acceptable limits. The robustness of servo elastic design by allowing for significant bandwidth between control modes and structural modes has helped in achieving adequate margins from the vehicle stability point of view. Signatures of aeroservoelastic interactions clearly seen for control surface dominated modes are explained, and stability has been ascertained for various perturbations. Acknowledgements The authors would like to acknowledge the contributions of Shri. M. V. Dhekane (Former Director, IISU, ISRO) for comprehensive reviews of the RLV-TD Hex Mission ASE analysis. Authors express their gratitude to RLV-TD digital auto pilot design team of CLD division CGSE entity for the various inputs received for ASE analysis. We also acknowledge the contributions of Shri. S. Balakrishnan, GSLV, VSSC during the early developmental phase of ASE analysis.

References 1. Upadhya AR, Madhusudan AP (2003) Analysis of aeroservoelastic interactions in a modern combat aircraft. IE (I) J AS 84 2. Pak C-G (2008) Aeroservoelastic stability analysis of the X-43A stack. NASA/TM-2008214635 3. Patil MJ, Hodgesy DH (2000) On the importance of aerodynamic and structural geometrical nonlinearities in aeroelastic behavior of high-aspect-ratio wings. In: 41st AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Atlanta, April 2000 4. Bisplinghoff RL, Ashley H, Halfman RL (1996) Aeroelasticity. Dover Publications, New York 5. Gulcat U (2010) Fundamentals of modern unsteady aerodynamics. Springer 6. Botez R, Biskri D, Doin A (22 May, 2012) Closed-loop aeroservoelastic analysis validation method. J Aircraft 41(4). Engineering Notes 7. Wright JR, Cooper JE (2007) Introduction to aircraft aeroelasticity and loads. Wiley, New York 8. Rodden WP (1994) MSC/NASTRAN Aeroelastic Analyisis: Users Guide, Version 68, MacNeal-Schwendler Corporation USA 9. Reymond M, Miller M (eds) (1996) MSC/NASTRAN quick reference guide version 69. The MacNeal-Schwendler Corporation, USA 10. Britt RT, Volk JA, Dreim DR, Applewhite KA (2015) Aeroservoelastic characteristics of the B-2 bomber and implications for future large aircraft, RTO AVT specialists’ meeting on “Structural Aspects of Flexible Aircraft Control”, Ottawa, Canada, October 2015

Time Domain Aero Control Structure Interaction Studies of Indian Reusable Launch Vehicle P. Ashok Gandhi, Mahind Jayan, Sajan Daniel, and R. Neetha

Abstract Reusable Launch Vehicle Technology Demonstrator Hypersonic Experiment (RLV-TD HEX01) mission of the Indian Space Research Organisation (ISRO), was the maiden flight which employed a winged body configuration. The vehicle faces multiple excitations during its atmospheric phase of flight. In this perspective, structural vibrations on the vehicle arising out of external excitations have to be adequately stabilized to prevent adverse vibrations. A flexible vehicle response analysis package: FLXTRJ-RLV was developed to assess the aero control structure interaction characteristics of the vehicle. Results of simulation studies ensures the increase in closed-loop damping and improvement in disturbance rejection characteristics of the vehicle, thereby reducing the in-flight loads acting on the vehicle. Comprehensive post-flight analysis studies were carried out on the flight-experienced responses. The flight observed rates highlighted a benign interaction-free environment during the ascent phase of flight. Keywords Reusable launch vehicle · Control structure interaction · Time-domain response · Closed-loop damping

1 Introduction RLV-TD HEX01 was ISRO’s maiden flight, which employed a winged body configuration (Fig. 1). In this mission, the Technology Demonstrator Vehicle (TDV), was boosted to hypersonic Mach number using a solid booster stage. TDV being a winged P. Ashok Gandhi · M. Jayan · S. Daniel · R. Neetha (B) SDSD/SDAG/STR Entity, VSSC, Thiruvananthapuram, Kerala, India e-mail: [email protected] P. Ashok Gandhi e-mail: [email protected] M. Jayan e-mail: [email protected] S. Daniel e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_30

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body configuration is significantly different from conventional launch vehicles with respect to aerodynamic characteristics and control strategy. TDV is connected to the solid booster through an inter-stage structure. Dynamic characterization of the vehicle using analytical FE models and exhaustive GRT tests helped in understanding the structural dynamic behavior of this configuration. It has been observed that significant bending is seen in proximity to the inter-stage structure for all the significant global modes of the vehicle. Hence, the proximity of the structural dynamic and control dynamic response ranges are seen. TDV has a double delta wing and twin vertical tails. Four fins, which comprise of a fixed portion and a movable portion, attached to the base shroud will provide the pitch, yaw, and roll control till solid booster separation. This configuration of TDV heightens the importance of aerodynamics in vehicle stability owing to the significantly higher value of aerodynamic moments about the vehicle center of gravity. Aerodynamic characterization of the vehicle through simulation and wind tunnel studies have clearly established the existence of coupled aerodynamic excitations arising out of pitch plane angle of attack (α) and yaw plane angle of attack (β). Aerodynamic damping and stiffness matrices also cause modal coupling which has to be carefully studied form the modal stability point of view (Fig. 1). The TDV follows a wind biased trajectory during the ascent phase by employing an autonomous close-loop flight control system. Navigation guidance and control unit (NGC), uses motion sensors (normal and lateral accelerometers, pitch, roll, and yaw rate gyros) to measure TDV responses which are processed by onboard digital autopilot software (DAP) to provide appropriate command signals to the primary control system actuators. Since the sensors pick up the elastic response also, adequate phase stabilization and gain stabilization schemes have to be built in for achieving a controllable vehicle. This calls for rigorous validation of the control system during the atmospheric phase, to make sure the design of filters is taken care of and proper gain and phase stabilization strategy has been employed [1]. Though the control system design is carried out in frequency domain and validated through short-period analysis in the time domain, it is essential to validate the control design along the trajectory including vehicle flexibility effects in a long period simulation Base Shroud

Fuselage Wing Solid Booster Vertical Tail

Rudder Elevon

Fig. 1 Vehicle configuration

Fins

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where the time-varying plant dynamics are captured. In this aspect, aero control structure interaction studies assume significance for ensuring the robust controllability of the vehicle. This time-domain long period simulation including vehicle flexibility is carried out through flexible vehicle response analysis package FLXTRJ-RLV. FLXTRJ for RLV-TD has been developed around the conventional launch vehicle FLXTRJ package existing for PSLV and GSLV. Here the time domain aero control structure interaction studies carried out for RLV-TD in the ascent phase using in-house developed software, FLXTRJ-RLV is presented. The major challenges with respect to aerodynamic modeling of RLV-TD updates required to simulate the new facets of aerostructure interaction problem, the strategies adopted to address those challenges, and results of closed-loop interaction studies are discussed here. The results go a long way in establishing the adequacy of the automatic flight control system for achieving the mission objectives.

2 Analysis Framework The atmospheric phase of flight is perhaps the most critical regime of flight for any launch vehicle. Owing to typical launch vehicle configurations effect of vehicle flexibility is most pronounced during the atmospheric regime of flight [2] Higher altitude regimes of flight are usually dominated by rigid body responses when seen from the mission planning point of view. Hence, an integrated analysis environment is required to estimate the vehicle responses along the trajectory, which may be dominated, by rigid body responses or elastic responses as the case may be. In line with legacy trajectory simulation packages used in ISRO, the package developed for RLV-TD follows an approach wherein the results of flexible vehicle response package are superposed to the rigid body responses of an integrated 6 degree of freedom trajectory simulation package (6D package). The 6D package essentially estimates the rigid body dynamic response of the vehicle by incorporating time-varying lumped inertial and excitation characteristics as per the chosen trajectory, whereas the flexible vehicle response package (FLXTRJ-RLV) simulates the elastic structural response using distributed flexibility and distributed aerodynamic data. The formulation of flexible vehicle problem, numerical solution scheme employed, aerodynamic, and structural dynamic models and modeling of control structure interaction in FLXTRJ-RLV are further discussed. The governing equations of motion in FLXTRJ are derived in the Lagrangian dynamics framework by evaluating the integrals for total kinetic energy, potential energy, and dissipation energy of the structure considering both rigid body and elastic body contributions [3]. Classical mode superposition theory is used to model the elastic body response. Application of the classical Lagrangian formulation to the global set of equations yields a second order ordinary differential equation of the form (1) which is solved numerically using Runge–Kutta method. In Eq. (1), M is general-

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ized mass matrix, C is damping matrix, K is stiffness matrix, q is generalized coordinate, and Fg is total generalized force. The generalized force includes the contribution from aerodynamic excitation, control force, and booster thrust excitations [M]q¨ + [C]q˙ + [K ]q = Fg

(1)

An important objective of flexible vehicle simulation is the incorporation of models to capture changes in structural loading due to flexibility. For a controlled vehicle the angle of attack along the vehicle can be represented by Eq. (2) [4]. In Eq. (2), x denotes the spatial coordinate and t denotes time and α(x, t)flex denotes the local flexible angle of attack profile along the length of the structure. Figure 2 depicts the angle of attack profile developed along the length of the vehicle due to its flexibility when the elastic response is dominated by the fundamental mode. The total aerodynamic force (F aero ) arising out of the induced angle of attack and the control force (F control ) are the major components in generalized force that determine the lateral elastic response of the vehicle. α(x, t) = α(t)error + α(t)wind + α(x, t)flex [4]

(2)

Through the flexible vehicle response analysis using appropriate distributed aerodynamic data, vehicle structural mode shapes and relevant excitations, Equation (1) is solved along the vehicle trajectory to estimate α(x, t)flex . The estimated α(x, t)flex is then further used to determine the additional loading arising out of vehicle flexibility. The computed additional loading is then appropriately added to the global

VW V

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Vertical Tail and Rudder Fins Wing Fuselage and Booster

Fig. 3 First global bending mode

forcing functions for estimating vehicle response. The various sub-system models in FLXTRJ-RLV are discussed.

2.1 Modeling of Structural Dynamic System During Structural dynamic modeling adequate care needs be taken so that it represent both global vehicle modes and the lifting surface modes accurately. To ascertain the modal characteristics of RLV-TD HEX01vehicle, a detailed FE model was developed, and the frequency and mode shapes are validated through Ground Resonance Test (GRT). For the purpose of FLXTRJ-RLV simulations, the centerline mode shapes along the core of vehicle and mode shapes along the different chord of the lifting surface are provided. Figure 3 shows the first global bending mode of the vehicle. Being a winged body configuration, it is important to include along with global modes, lifting surface modes, such as wing and vertical tail symmetric and asymmetric mode, control surface modes, such as elevon and rudder symmetric and asymmetric mode, Fin related modes both bending and torsion. Apart from this, all the coupled modes such as wing asymmetric along with vertical tail bending modes are also included. The mode shape data in the structural points are interpolated to the aero points using linear spline functions. Along the trajectory, the frequency and generalized mass is made available at each time instant by linear interpolation.

2.2 Modeling of Aerodynamic System For winged body configuration, the idealization of aero points to capture the overall aerodynamics properly becomes vital, and at the same time introducing more points

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Wing and Elevon

Fuselage

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Fig. 4 Aerodynamic model idealization

leads to an increase in simulation run time. To have a balance between the two, the aero points are optimally chosen, so that they can represent the aerodynamics properly. The lifting surfaces are divided into various aero panels (Fig. 4). The adequacy of aerodynamic representation is validated by comparing the integrated aerodynamic coefficient (C N and X cp ) with the wind tunnel test, and it was found to be very close. Sufficient Mach numbers are considered particularly in the transonic regime to aptly capture the aerodynamic variation along the trajectory. The generalized aerodynamic force acting on the vehicle is as explained in Eq. (3). The generalized aerodynamic force is evaluated as an integral and made available at each time instant by linear interpolation   dC Nα dC Sβ αtotal (x)d x φz + S βtotal (x)d x φ y FA = Q ∫ S dx dx

(3)

dC S

where, Q is Dynamic pressure, dCd xNα and d x β are normal aerodynamic distribution along the length of the vehicle, S is the reference area, φ z and φ y are the modal displacements of the launch vehicle αtotal and βtotal are the total angle of attack (as explained in Eq. 2). The flexible angle of attack at any point along the vehicle is, αflex

    {∅ z }T {q} ˙ ∂∅ z T {q} + = − ∂x Vm,w

(4)

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 z are the modal displacements and slope, respectively, q and q˙ where, ∅ z and ∂∅ ∂x are generalized coordinate and its first derivative with respect to time, respectively, and Vm,w is the relative velocity of the vehicle. The flexible angle of attack is used to evaluate the additional force and moment due to flexibility. Using the aerodynamic and structural dynamic models the software has the capability to compute elastic load at any point on the vehicle. The elastic load at any station is computed by Eq. (5). I n 5, {m i , xi } denote the lumped mass and total lateral acceleration, respectively, at the aero points for the corresponding station. 

{Elastic Normal Shear Force}station = Q

S

  p dC Nα αtotal (x)d x − m i x¨i dx i=1

(5)

2.3 Interaction with Control System The interaction of structural dynamic system with the control system completes the closed-loop behavior of the coupled aero control structure model. Solution of the modal equations of motion using the aerodynamic and structural dynamic models described earlier yield accurate estimates of the vehicle flexible rates at the sensor locations. Equation (6) is evaluated with mode slopes at the respective sensor locations and summed over all the modes (n) considered to get the total flexible rate of the vehicle.   n   ∂φ {q} ˙ Flexible attitude rates = ∂x at sensor locations i=1

(6)

The computed flexible body rates are added to the rigid body rates computed by a 6D package and passed on to the navigation module as shown in Fig. 5. This process couples the rigid body and flexible body responses, and the flight control system will feel the rate as experienced in the actual flight condition. In addition to the flexible body rates, the vehicle also experiences incremental forces and moments at the global level due to vehicle flexibility. These incremental effects are modeled by Eqs. (7) and (8). ∂C N ∝ {αflex }d x ∂x 

{Incremental Moment} = {Incremental Force} ∗ xcg − x {Incremental Force} = ∫ Q S

(7) (8)

As depicted in Fig. 5, the computed incremental effects are passed on to the 6D package and added as external disturbances acting on the vehicle. This completes the interaction model where in the control system will feel the total elastic forces, as

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Navigation Flexible rate FLXTRJ

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Fig. 5 Interaction diagram between flexible vehicle program, 6D package and digital flight control system

well as external disturbance arising out of vehicle flexibility. Hence, in the simulation testbed, the inputs felt by the flight control system will have the effects arising out of vehicle flexibility and so will the generated control force; which is then passed on to FLXTRJ, thereby completing the closed-loop feedback path as in the actual vehicle.

3 Simulation Results Flexible vehicle response simulation is carried out for RLV-TD HEX01 vehicle and the two main outcomes of the simulations are confirming the increase in closed-loop damping and improved disturbance rejection characteristics of the vehicle, thereby reducing the in-flight loads acting on the vehicle.

3.1 Control Structure Interaction Studies Initial simulation of RLV-TD HEX01 vehicle revealed a reduced closed-loop damping in the yaw plane toward the end of the atmospheric phase (Fig. 6). At this time, there is not much disturbance from aerodynamics, only structure and control systems are involved. To quantify the effect of reduced damping, a disturbance in the form of a standard gust of 9 m/s is introduced in the wind and the response at sensor locations (Fig. 7), is observed. The structural damping considered during the simulation is 1% and it is observed that the closed-loop damping is found to be 0.4% in yaw. Subsequent to this, update in Digital Auto Pilot (DAP) design was called to get improved closed-loop damping.

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Sensor yaw rate

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Resp 0.4

0.7% Close Loop Damping

0.4% Close Loop Damping

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Time a) Before DAP update

b) After DAP update

Fig. 6 Comparison of system damping due to update in DAP design

3.2 Mission Simulation Studies Apart from aero Control Structure interaction studies, the load computation subroutine in FLXTRJ-RLV is used to estimate elastic loads acting at various critical stations. These computed loads are checked against the design limit as a part of the launch clearance activity. Prior to that, during the preflight phase, several simulations are carried out with various winds perturbed in Monte Carlo sense to find the maximum load expected in the vehicle due to wind variation. In addition to this, disturbance in the form of standard gust (Fig. 7), is introduced to the worst-case measured winds to find the robustness of the Mission design. During one such simulation using measured wind with gust, the angle of attack increased up to 62% and fin deflection increased more than 100% in transonic regime, with respect to the case where gust was not considered. The increase in the angle of attack and fin deflection resulted in normal fin loads exceeding the design limits. This increase in the angle of attack is on the higher side and several options are studied to reduce this. Three major design updates gave the maximum benefit in terms of reducing the angle of attack. First, the disturbance rejection in transonic regime was improved in DAP by increasing the aero margin at the expense of gain margin, so that error buildup is reduced. The second update was in the philosophy of steering generation scheme. Initially, the philosophy of steering generation scheme was to Fig. 7 Standard gust model

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Angle of attack reduces by 40% Time

Fig. 8 Angle of attack profile comparison between the initial and updated design

Normal Force on fin

fly the vehicle at zero normal force, which required the vehicle to fly at a specified angle of attack profile (α0 ). Due to intentional α0 steering at the transonic regime to make the normal force zero, the error was building up. This steering generation scheme is later updated from the normal force zero profile to the angle of attack zero profile (α = 0). The third update was on the wind smoothening scheme. The wind smoothening scheme was updated from the existing five-point average scheme to Savitzky–Golay filter scheme [5]. With all the above three updates the angle of attack reduces by 40% (Fig. 8), the fin deflection reduces by 50%, and the fin loads reduce by 44% (Fig. 9). The pitch rate corresponding to the updated design is shown in Fig. 10. It is observed that the first mode is getting excited due to the application of gust, and the peak overshoot due to elastic response is within acceptable limits. Further, in the capture phase of rates, it can be observed that the active damping provided by the control system dissipates the energy of elastic modes within a few cycles of operation. The high temporal convergence seen for sensor elastic rate pattern supports this fact.

Design limit Fin load reduces by 44% Time

Fig. 9 Fin normal force profile comparison between the initial and updated design

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Time Fig. 10 Sensor pitch rate for the updated design

4 Conclusions A flexible vehicle response analysis package: FLXTRJ-RLV was developed to assess the aero control structure interaction characteristics of the vehicle. This software was extensively used for simulation studies during the mission design and launch clearance phase. The initial simulations brought out the existence of reduced closedloop damping, which was later improved and further verified through simulation. The structural load estimation capability of the software along the trajectory was useful in assessing the performance of the steering program during the ascent phase. The valuable outcome of the software leads to improvement in DAP design, steering generation, and wind smoothening scheme The flight observed rates from the postflight analysis highlighted a benign interaction-free environment during the ascent phase of flight in conformance with preflight studies. Acknowledgments The authors would like to acknowledge the valuable suggestions and guidance of Shri. K. L. Handoo, Former Deputy Director, VSSC (STR), during the detailed reviews. The authors express their gratitude to Shri. M. V. Dhekane, Former Director (IISU/ISRO), for the comprehensive reviews conducted. The authors also acknowledge the contribution of Shri. S. Balakrishnan, SAT/OLV/VSSC during the initial phase of FLXTRJ development for RLV-TD. Contributions of FLXTRJ Task team lead by Shri. K. L. Handoo toward the conceptualization, development, and realization of FLXTRJ software in use for ISRO’s legacy launch vehicle projects are also acknowledged.

References 1. NASA Space Vehicle Design Criteria (1970) Effects of structural flexibility on launch vehicle control systems. NASA SP8036, February 1970 2. Greensite AL (1 August 1967) Analysis and design of space vehicle flight control systems— elastic body equations. NASA CR-834, vol XV 3. Handoo KL et al (1989) 6-D trajectory analysis of launch vehicles incorporating vehicle flexibility effects. In: Details of mathematical formulation and the software package, vol 3

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4. Suresh BN, Sivan K (2015) Integrated design for space transportation system. Springer Publishers, India 5. Savitzky A, Golay MJE (1964) Smoothing and differentiation of data by simplified least squares procedure. Anal Chem 36(8):1627–1639

Vehicle Dynamics

Polynomial Neural Network Based Stochastic Natural Frequency Analysis of Functionally Graded Plates Pradeep Kumar Karsh, Abhijeet Kumar, and Sudip Dey

Abstract The present article deals with the stochastic approach for natural frequency (NF) analysis of functionally graded (FG) plates by employing polynomial neural network (PNN) surrogate model combined with finite element (FE) method. The surrogate model for NF analysis of FG plates is validated with the original FE method. Both individual and mixed variation of material properties are taken into account. The present PNN model significantly rises the computational efficiency, and the computational cost decreased in comparison to Monte Carlo Simulation (MCS).

1 Introduction In the recent era, plenty of applications of advanced materials such as functionally graded material (FGM) structures are gaining popularity in the field of aerospace, automobile, medical optoelectronics, and many other engineering applications due to high stiffness [1]. FGM is a major type of composite material in which microstructure is varying continuously throughout the section [2]. In FGM, there is no internal boundary so stress concentration is negligible [3, 4]. FGM is the new composite materials composed of two different materials to obtain the functional requirements. The FGM is inhomogeneous composite materials with property gradient depending upon the chemical composition, atomic order, and microstructure. There are two materials namely metal and ceramic, which are smoothly and continuously distributed throughout the volume of the plate. The mechanical properties of FGM are better than the laminated composite materials due to absence of interlaminate joint, internal stresses, delamination, improper bonding. The FGM have good thermal resistance properties provided by ceramic material, while high mechanical strength is given by metal constitute. Some researchers worked on NF analysis of different materials

P. K. Karsh (B) · A. Kumar · S. Dey National Institute of Technology, Silchar, India e-mail: [email protected] Parul Institute of Engineering & Technology, Parul University, Vadodara, India © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_31

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using different models. Escobedo-Trujillo et al. [5], presented a hybrid model integrates to increase the coefficient of performance forecasting for solar refrigeration system. Zjavka [6] have used polynomial neural networks for forecasting wind speed in order to improve forecasting. Han et al. [7] developed a PNN model for sequential processes of silicon solar cell fabrication. Fazel Zarandi et al. [8] proposed a fuzzy PNN to forecasting the strength of concrete. Zhang et al. [9] used single-output Chebyshev PNN for pattern classification. Haiyan et al. [10] introduced orthogonal polynomial neural networks for modeling of polymer molecular weight distribution. Roha et al. [11] introduces a new method for designing of fuzzy radial basis function approach. Dorn et al. [12] used the PNN for the forecasting of approximate threedimensional structures of polypeptides. Xu and Meng [13] determined properties of FGM beams by employing the different distribution laws and performed the dynamic analysis by employing the analytical method. Moita et al. [14] carried out structural and sensitivity analysis along with the material modeling and optimization of FG structures. The power-law index and thickness are taken as design parameters, while mass, displacement, and NF are taken as response parameters. Attia and Rahman [15] carried out NF analysis of FG nano-beams with considering the influence of microstructure rotation and surface energy by using the Bernoulli–Euler beam theory and illustrates the influence of the power-law exponent, damping, surface elasticity, thickness, and Poisson’s effects. Wali et al. [16] carried out NF analysis of FG shell by employing the 3D-shell model, while Kim [17] applied analytical model for the free vibration dynamic analysis of FG cylindrical shell. In the past, many researchers applied stochastic approach for dynamic analysis of structures [18–27]. In this article, stochastic first three NF of FG plates are determined computationally by employing the PNN surrogate model. Steps required to do stochastics NF analysis by employing the PNN approach are shown in Fig. 1.

2 Mathematical Formulation The displacement field can be written as U (x, y, z) = U0 (x, y) − zβx (x, y)

(1)

V (x, y, z) = V0 (x, y) − zβ y (x, y)

(2)

W (x, y, z) = W0 (x, y) = W (x, y)

(3)

where U, V, and W are the displacements in the three directions. U0 , V0 , and W0 are the displacements of the mid plane and βx , β y are the rotations in x and y direction, respectively. The stress–strain is expressed a

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Fig. 1 Steps for stochastic NF analysis by employing PNN

εx =

∂ 2 W0 ∂ 2 W0 ∂ V0 ∂ 2 W0 ∂U0 ∂ V0 ∂U0 −z − z + − 2 , ε = , γ = y x y ∂x ∂x2 ∂y ∂ y2 ∂y ∂x ∂ x∂ y

(4)

Equation of dynamic equilibrium given by [28] ¨ + [K ]{u} = {F} [M]{u}

(5)

where {u} is displacement, {F} is external load, [M] represents the mass matrix, and [K] represents the stiffness matrix. For free vibration, Eq. (6), can be written as ¨ + [K ]{u} = 0 [M]{u}

(6)

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Table 1 Properties of the FG components at 300 K [29]

Material Metal Ceramic

υ

ρ (kg/m3 )

70

0.25

2707

151

0.3

3000

E (GPa)

3 Stochastic Approach by Employing PNN Model In this article, the properties of FG plate are taken as stochastic input in which both individual and mixed variations are taken as: I. Individual variation of properties   only E 1 ( ) = E 1(1) , E 1(2) , . . . .E 1(i) II. Mixed variation of all properties g{ E 1 ( ), G 12 ( ), μ( ), ρ ( )}       1  E 1(1) , E 1(2) , . . . .E1(i) , 2 G 12 (1) , G 12 (2),  . . . . . G 12 (i) , = 3 μ(1) , μ(2) . . . . μ(i) , 4 ρ(1) , ρ(2) . . . . ρ(i) where E, G, μ, and ρ are the elastic modulus, shear modulus, Poisson’s ratio, and material density of the FG plate (Table 1).

4 Results and Discussion In this article, the computational cost is reduced by using the PNN model. Figure 2 illustrates the comparative results between the surrogate PNN model and MCS results by employing the probability density function (PDF) plot of fundamental NF by considering the mixed variation of properties with various sample sizes (N). Further, the scatter plot of fundamental natural frequency between the PNN model and the original FE model is shown in Fig. 3 with considering N = 1024 for combined variation of properties. Individual variation of properties (longitudinal elastic modulus) with taking different values of Stochasticity as 10, 20, and 30% also carried out. Figure 4 illustrates the PDF for first three NF due to individual variation of longitudinal elastic modulus (E1 ) for FG plate. The PDF shows that sample size (N) 2048 have close results with the original MCS.

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Fig. 2 PDF of fundamental NF (rad/s) due to the effect of mixed variation of properties for the PNN model and MCS model

Fig. 3 Scatter plot between the PNN model and the original FE model of fundamental NF considering N = 1024

5 Conclusions The novelty of this article includes the application of PNN based surrogate model in conjunction with the FE method for the stochastic NF analysis of FG plates. The computational efficiency is significantly increased by employing the PNN based model since computational time is decreased as compared to the MCS method. Combined and individual variation of input properties is considered. This PNN based surrogate approach can be applied for more complex structures and systems in the future.

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Fig. 4 PDF plots of first three NF (rad/s) due to the effect of variation of longitudinal elastic modulus (E1 ) only with considering 10, 20, and 30% of stochasticity

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References 1. Xu Y, Qian Y, Song G (2016) Stochastic finite element method for free vibration characteristics of random FGM beams. Appl Math Model 40:10238–10253 2. Ziane N, Meftah SA, Belhadj HA, Tounsi A, Bedia EAA (2013) Free vibration analysis of thin and thick-walled FGM box beams. Mech Sci 66:273–282 3. Hien TD, Noh HC (2017) Stochastic isogeometric analysis of free vibration of functionally graded plates considering material randomness. Comput Methods Appl Mech Eng 318:845–863 4. Dey S, Naskar S, Mukhopadhyay T, Gohs U, Spickenheuer A, Bittrich L, Adhikari S, Heinrich G (2016) Uncertain natural frequency analysis of composite plates including effect of noise—a polynomial neural network approach. Compos Struct 143:130–142 5. Escobedo-Trujillo BA, Colorado D, Rivera W, Alaffita-Hernandez FA (2016) Neural network and polynomial model to improve the coefficient of performance prediction for solar intermittent refrigeration system. Solar Energy Mod. 129:28–37 6. Zjavka L (2015) Wind speed forecast correction models using polynomial neural networks. Renew Energy Mod 83:998–1006 7. Han S-S, Kim I, You C, Joung J (2012) Polynomial neural network modeling for sequential processes of silicon solar cell fabrication. Front Comput Educ 133:651–658 8. Fazel Zarandi MH, Turksen IB, Sobhani J, Ramezanianpour AA (2008) Fuzzy polynomial neural networks for approximation of the compressive strength of concrete. Appl Soft Comput 8:488–498 9. Zhang Y, Yin Y, Guo D, Yu X, Xiao L (2014) Cross-validation based weights and structure determination of Chebyshev-polynomial neural networks for pattern classification. Pattern Recogn Mod 47:3414–3428 10. Haiyan W, Liulin C, Jing W (2012) Gray-box modeling and control of polymer molecular weight distribution using orthogonal polynomial neural networks. Process Control Mod 22:1624–1636 11. Roha S-B, Sung-Kwun O, Pedrycz W (2011) Design of fuzzy radial basis function-based polynomial neural networks. Fuzzy Sets Syst Mod 185:15–37 12. Dorn M, Braga Andre LS, Llanos CH, Coelho LS (2012) A GMDH polynomial neural networkbased method to predict approximate three-dimensional structures of polypeptides. Expert Syst Appl 39:12268–12279 13. Xu XJ, Meng JM (2018) A model for functionally graded materials. Compos B Eng 145:70–80 14. Moita JS, Araujo AL, Correia VF, Soares CMM, Herskovits J (2018) Material distribution and sizing optimization of functionally graded plate-shell structures. Compos B Eng 142:263–272 15. Attia MA, Rahman AAA (2018) On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int J Eng Sci 127:1–32 16. Wali M, Hentati T, Jarraya A, Dammak F (2015) Free vibration analysis of FGM shell structures with a discrete double directors shell element. Compos Struct 125:295–303 17. Kim YW (2015) Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge. Compos B Eng 70:263–276 18. Dey S, Mukhopadhyay T, Sahu SK, Li G, Rabitz H, Adhikari S (2015) Thermal uncertainty quantification in frequency responses of laminated composite plates. Compos B Eng 80:186– 197 19. Dey S, Mukhopadhyay T, Sahu SK, Adhikari S (2016) Effect of cutout on stochastic natural frequency of composite curved panels. Compos B Eng 105:188–202 20. Dey S, Mukhopadhyay T, Adhikari S (2015) Stochastic free vibration analyses of composite shallow doubly curved shells—a Kriging model approach. Compos B Eng 70:99–112 21. Karsh PK, Mukhopadhyay T, Dey S (2018) Spatial vulnerability analysis for the first ply failure strength of composite laminates including effect of delamination. Compos Struct 184:554–567 22. Dey S, Mukhopadhyay T, Sahu SK, Adhikari S (2018) Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading. Eur J Mech/Solids 67:108–122

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23. Karsh PK, Mukhopadhyay T, Dey S (2018) Stochastic dynamic analysis of twisted functionally graded plates. Compos B Eng 147:259–278 24. Karsh PK, Mukhopadhyay T, Dey S (2019) Stochastic low-velocity impact on functionally graded plates: probabilistic and non-probabilistic uncertainty quantification. Compos B Eng 159:461–480 25. Mukhopadhyay T, Naskar S, Karsh PK, Dey S, You Z (2018) Effect of delamination on the stochastic natural frequencies of composite laminates. Compos B Eng 154:242–256 26. Kumar RR, Karsh PK, Pandey KM, Dey S (2019) Stochastic natural frequency analysis of skewed sandwich plates. Eng Computations 36(7):2179–2199 27. Karsh PK, Kumar RR, Dey S (2019) Radial basis function-based stochastic natural frequencies analysis of functionally graded plates. Int J Computational Methods. p. 1950061. https://doi. org/10.1142/S0219876219500610 28. Meirovitch L (1992) Dynamics and control of structures. Wiley, New York 29. Singh H, Hazarika BC, Dey S (2017) Low velocity impact responses of functionally graded plates. Procedia Eng 173:270–364

Comparative Study Among Different Vehicle Models in Case of High-Speed Railways and Its Experimental Validation B. Pal and A. Dutta

Abstract A comparative study of different existing vehicle models like moving load, moving mass, discrete sprung mass and moving system are presented for addressing problems related to high-speed railways. Finite element framework is used, where the bridge deck is modelled using Bernoulli–Euler beam elements. MATLAB code has been developed for different vehicle models to obtain the bridge and vehicle responses. Such responses of bridge and vehicle, obtained for different vehicle models, show that the prediction of bridge responses can be reliably done using moving load model. On the other hand, there is a significant underestimation in the calculation of vehicle body acceleration, if one goes with the sprung mass model instead of 10 DOFs interaction model. Vehicle’s pitching effect might be the reason behind such underestimation. In addition to that, for the purpose of validation of such 10 DOFs vehicle-bridge interaction model, the response of bridge obtained through such interaction model are compared with measured response data on an existing steel girder bridge. A good matching between experimentally and theoretical evaluated data is observed, which indicates the suitability of the adopted Finite Element model in the practical field. Keywords Vehicle model · Vehicle-bridge interaction · Vehicle acceleration · Bridge responses

1 Introduction Constantly growing requirements on improving infrastructures related to railways are observed in different parts of the world, which is primarily based on increasing demand on the speed of railway locomotives. Development of high-speed railway B. Pal (B) · A. Dutta Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India e-mail: [email protected] A. Dutta e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_32

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lines are getting a major thrust. The train line, which permits to run above 250 km/h throughout the trip or a substantial part of the trip is termed as high-speed line as per UIC [1]. In this regard, it is required to mention that the acceleration of vehicle body governs the design for high-speed railways, because such vehicle body acceleration determines passengers comfort level. Some of the most important aims of current research on railway bridges are to enhance the understanding of the interaction among vehicle, track and bridge, to increase the loading capacities and to reduce the deterioration of such structures. Two clearly distinguishable phases can be seen in the research related to the evaluation of dynamic responses of bridges under the action of vehicles in motion. The advent of digital computers actually led to the development of these two phases. The initial approaches were primarily analytical based on some simplified or approximate assumptions and were confined to the solution of most simple and fundamental problems. Such approaches were thus not applicable in addressing the more complex treatment problems like vehicle-bridge interaction (VBI). The introduction of digital computers enabled researchers to address realistic problems with the more complex vehicle and bridge parameters. Researchers [2–4] developed many models over the years. To begin with, it was focused on a single part or element of the structure. However, subsequently, with the increase in computational power, numerical models with more and more interactions between different parts evolved. Experimental studies were also carried out for validation of those numerical models. The requirement of high-speed railways led to an increased motivation to model and analyze VBI in a more realistic manner. The aim of this study is to examine the acceleration of the vehicle, which is an important indicator of passenger’s comfort. Thus, acceleration of the vehicle along with the response of the bridge become design criteria. In general, two approaches namely, analytical and numerical methods were used in these investigations. Analytical methods [3] are simple, but may not be suitable for evaluating the complex behaviours of bridges under moving trainloads. However, the finite element method (FEM) has found significant application in this domain as a powerful numerical technique by many researchers [5, 6]. Based on FEM and structural dynamics, Yang and Yau [5] introduced a bridgevehicle element to model the bridge-vehicle dynamic interaction. In this study, the train was modelled as a series of sprung masses lumped at the bogie positions and the bridge by beam elements. Thereafter, Yang and Wu [6] improved the bridge-vehicle element by considering each vehicle of the train comprising of a car body supported by two wheelsets through one stage suspension system (4 DOFs vehicle model). The authors also showed the effect of different vehicle models (Moving Load, sprung mass and Moving system model with 4DOFs) on the bridge and vehicle responses. Zheng et al. [7] conducted the instability analysis of an axially compressed rail subjected to moving trainload. Analytical approach was adopted, wherein the rail was modelled as an infinite beam resting on a viscoelastic foundation. Wu and Yang [8] studied 2D steady-state response and riding comfort of a train considering each car as 10 DOFs vehicle model which consists of one car body supported by two

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bogies and ultimately by four sets of the wheel, i.e. two stages of suspension system was considered for the vehicle model. The rail track system was idealized as an infinite beam supported on spring-dashpot units, which are uniformly-distributed. On the other hand, using the concept of stationarity of total potential energy of the system subjected to dynamic load, Lou and Zeng [9] evaluated the equations of motion with time-dependent coefficients for two different vehicle models—sprung mass model and 10 DOFs moving system model. In consequence of that, the authors separately analyzed the effect of these two vehicle models. Lou [10] examined the responses of the dynamic interaction system comprising of a train, rail and the supporting bridge by FE method. Lou et al. [11] developed a special rail-bridge coupling element, which could be regarded as an extension of the theory presented in [9, 10]. In addition, a double-layer track model with sleeper was considered. The research work [10, 11] led to the development of an interactive model, ‘train–track– bridge’ system with a 10 DOFs vehicle model. A study, therefore, is lacking which could compare the effect of sprung mass model and 10 DOFs vehicle model on the dynamic behaviour of ‘train–track–bridge’ interaction system and its experimental validation. The primary focus of this paper is to thoroughly examine the different methodologies that were used for modelling of the train along with the bridge. Then, it is intended to show that how the complex 10 DOFs vehicle model differs in terms of the vehicle-bridge responses from the conventional Moving load model and also from that of simplest interaction vehicle model—the Sprung massass model. Finally, using this complex vehicle-bridge interaction model, responses of a realistic train–bridge system is simulated and compared with that of the measured in field responses.

2 Different Train Modelling Methodologies 2.1 Moving Load Model Moving load (or moving force) model is considered as the simplest model for evaluation of dynamic behaviour of vehicles and bridges. Axle forces are considered to move in series over the bridge at a constant speed (Figs. 1 and 2). Thus, no interaction between vehicle and bridge is considered and the inertia effect of the vehicle is also

Fig. 1 Moving load model [4]

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Fig. 2 Bridge–Train model: a a general model, b simplified sprung mass model [6]

ignored. Such approach does not lead to any major anomaly for the smaller ratio of the mass of the vehicle to that of the bridge, and also for the specific objective of evaluation of bridge response alone [5, 12, 13]. Being simple, the moving load method finds a large scale application, though the approach is approximate. In order to find out the bridge responses, well-known Finite Element (FE) method is used, where the bridge is modelled as a 2-D elastic Bernoulli–Euler beam element and the effect of each of the load at any instant is transferred to the elemental degreeof-freedom (DOFs) through proper shape functions. Details can be found in [14].

2.2 Moving Mass Model If the vehicle mass to the bridge mass is not small and inertia effect of the vehicle cannot be ignored, moving mass model is found to be quite useful. However, vehicle inertia cannot be accommodated, if the response of the vehicle is to be determined. This is due to the assumption of the no-jump condition of the moving mass at the contact point of the bridge. For finding the responses of the bridge, similar approaches as that of Moving Load model can be taken. The only thing which differs is the force due to inertia effect of the vehicle (i.e., due to the mass lumped at each wheel position) and this has to be considered separately apart from the gravitational weight of the vehicle. Details can be found in [15].

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2.3 Sprung Mass Model A sprung mass model is a type of vehicle model that considers the effect of the suspension system. A train travelling over the bridge with constant speed v is modelled by a sequence of lumped sprung mass units of regular intervals. Each sprung mass unit is used to either represent the front or rear half of a train car, which consists of two concentrated masses. The two masses are connected by a set of spring and dashpot that serves to represent the vehicle’s suspension and energy dissipation mechanism. This is a very general type of a model that considers the interaction between the train and bridge. For the purpose of finding the responses of both train (better to say car body) and bridge, the concept of contact force (which acts through the wheel and the corresponding bridge element in contact) is used [6].

2.4 Moving System Model A moving system model having many DOFs is a more involved model as compared to the sprung mass model. Discrete masses connected by suspension systems are considered to model various parts of the vehicle. Appropriate stiffness and damping properties are assigned to simulate the suspension system. A model capable of including the pitching effect of the vehicle body is shown in Fig. 3, which is a 4-DOF model. It has three vertical (one for car body and one for each wheelset) and one rotational (or pitching for car body) DOF. Two suspension systems (linear spring and dashpot) are used to support the body of the train, which is modelled as a rigid beam. Exhaustive train models are considered only for the analysis of the bridge–vehicle interaction of high-speed trains. A typical one car of a train composed of one vehicle body, two bogies and four wheelsets with 10-DOFs [9, 10], connected by linear springs and dashpots is shown in Fig. 4. Vertical and rotational (or pitching) DOFs Fig. 3 A car model with 4-DOF [6]

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Fig. 4 Ten DOFs model of one car of a train [9]

Fig. 5 A typical train–track–bridge interaction system with double-layer track model [11]

are considered for the vehicle body and bogies, while only vertical DOFs are adopted for wheelsets. Figure 5 shows a typical train–track–bridge interaction system, where the train consists of a series of identical four-wheelset vehicles rest on a series of multi-span continuous beams (representing railway bridges) with two approach embankments, i.e. the part of the rail track outside the bridge range. A mass-spring-damper system is used to model a vehicle of a train comprising of a car body, two bogies, four wheelsets and two stage suspensions. Linear elastic Bernoulli–Euler beam having finite length is used to model the rail, while Bernoulli– Euler beams are used to model the multi-span continuous bridge deck. The separation between the wheelset and the rails are not considered in the analysis. The elasticity and damping properties of the rail bed are modelled as discrete springs and dampers at the sleeper position as shown in Fig. 5. Here, only the vertical motion of sleeper is considered. Using the principles of the stationary value of total potential energy of the dynamic systems, the equation of motion for the train–track–bridge interaction system can be obtained as [11]

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Mvv 0 0 0 ⎢ 0 ⎢ 0 Mrr 0 ⎢ ⎣ 0 0 Mss 0 0 0 0 Mbb

⎤⎧ X¨ v ⎪ ⎪ ⎪ ⎥⎨ ⎥ X¨ r ⎥ ⎦⎪ ⎪ X¨ s ⎪ ⎩ X¨ b

⎫ ⎪ ⎪ ⎪ ⎬

⎡

Cvv Cvr 0 0 ⎢ 0 ⎢C C C + ⎢ r v rr r s ⎪ ⎣ 0 Csr Css Csb ⎪ ⎪ ⎭ 0 0 Cbs Cbb

⎤⎧ X˙ v ⎪ ⎪ ⎪ ⎥⎨ ⎥ X˙ r ⎥ ⎦⎪ ⎪ X˙ s ⎪ ⎩ X˙ b

⎫ ⎪ ⎪ ⎪ ⎬

393 ⎡

K vv K vr 0 0 ⎢ K K 0 ⎢K + ⎢ r v rr r s ⎪ ⎣ 0 K sr K ss K sb ⎪ ⎪ ⎭ 0 0 K bs K bb

⎤⎧ Xv ⎪ ⎪ ⎪ ⎥⎨ ⎥ Xr ⎥ ⎦⎪ Xs ⎪ ⎪ ⎩ Xb

⎧ Fv ⎪ ⎪ ⎪ ⎨ Fr = ⎪ ⎪ Fs ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ Fb ⎫ ⎪ ⎪ ⎪ ⎬

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(1) where K, M, and C represent stiffness, mass and damping matrices, respectively, F and X indicate force and displacement vectors, respectively, and subscripts ‘b’, ‘s’, ‘r’ and ‘v’ represent bridge, sleeper, rail and vehicle, respectively. Detailed formulation and expressions of all the matrices can be found in [9–11].

3 Numerical Study In this section, several numerical examples are presented based on the theory presented for different vehicle models, and the MATLAB codes are developed for each of these vehicle models. In this regard, both simply supported and continuous bridges are taken. Along with such comparative studies, the bridge responses obtained from the numerical study is validated with real data. It is considered that the bridge and the rail resting on it, are at rest when the train just enters. The parameters of vehicle, track and bridge are taken from [10], if otherwise not mentioned. The dynamic equilibrium equation of the bridge-rail-train system is solved by Wilson-θ method, where θ = 1.4. In the finite element analysis of the rail-bridge model, length of each element of the rail and bridge is taken as 1 m. • Comparison of moving load versus Moving mass versus Discrete Sprung mass model First of all, a simply supported bridge of length of 30 m is taken to examine the influence on the bridge responses of different models like Moving Load, Moving Mass and Discrete Sprung Mass. The train is assumed to be consist of five identical vehicles. Displacement time history at the mid-span of the bridge is shown in Fig. 6, for the three types of vehicle models, for the train moving at a speed of v = 100 km/h. In case of moving load and moving mass models, total lumped load or mass at each wheelset position is taken as the total load or mass of the upper part (M v ) and lower part (M w ) of the discrete sprung mass model (Fig. 2). Further, the variation in maximum bridge displacement at different speeds is shown in Fig. 7. It can be seen from Figs. 6 and 7, that the moving load model can be confidently adopted to estimate the response of bridge, as hardly any significant differences exist among the solutions obtained from these three types of vehicle modelling approaches. The sprung mass model brought in the inertial and interaction effect of the moving vehicles, resulting in a minor reduction of the peak response of the bridge.

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Fig. 6 Comparison of bridge displacement for a speed of 100 km/h

Fig. 7 Comparison of maximum bridge displacement at various speeds for different vehicle models

• Comparison of Sprung mass model versus Moving system model In this example, sprung mass model and moving system model with 10 DOFs are considered and the responses of both train and bridge for these two different vehicle models are presented. Here, two different types of bridges are taken—one is three single span simply supported bridge and the other one is 3-span continuous bridge having a span of 30 m each. The train here is also assumed to consist of five identical vehicles. In case of Sprung mass model (Fig. 2), mass of the upper part (M v ) is taken as half of the mass of the car body (i.e. 0.5M c ) of the 10 DOFs Moving system model (Fig. 4), and for lower part (M w ), it is taken as twice the mass of each wheelset, i.e. 2mw . However, in order to maintain the total mass of each 10 DOFs vehicle, the half of the mass of each bogie (i.e. 0.5mt ) is assumed to be lumped with the upper part of the sprung mass model (i.e. total mass of upper part becomes 0.5M c + 0.5mt ) and half with the lower part of the sprung mass model (i.e. total mass of lower part becomes 2mw + 0.5mt ). The suspension stiffness k v of discrete sprung mass can be

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obtained by considering k ct and k tw in series in the two stage suspensions of 10 DOFs model, i.e. k v = k ct · k tw /(k ct + k tw ). Similarly, the damping coefficient cv can be obtained as cv = cct · ctw /(cct + ctw ). To study the effect of sprung mass model and moving system model along with the two different bridges (i.e. 3-span continuous and three single span simply supported), both the bridge responses, as well as vehicle responses are plotted in Figs. 8, 9 and 10 at different speed for the two types of bridges. It may be seen from Figs. 8 and 9, that the effects of sprung mass model and moving system model on bridge response in terms of deflection, as well as acceleration are insignificant. However, the significant effect of vehicle model types can be observed on the acceleration (vertical) of the body of the car as presented in Fig. 10. The noninclusion of the pitching effect of the car body along with the interaction between the front and rear bogies of the four-wheelset vehicle, as represented by the sprung mass model, may result in substantial underestimation of the response of the vehicle, which may not be accepted from the viewpoint of design. Further, by comparing Figs. 8, 9 and 10 for the two different types of bridges (3-span continuous and three single span simply supported), it is observed that all the responses of the bridge,

Fig. 8 Vertical deflection at the mid-point of central span of the bridge. a Continuous, b Simply supported

Fig. 9 Vertical acceleration at the mid-point of the central span of the bridge. a Continuous, b Simply supported

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Fig. 10 Vertical acceleration of the last car body at the position of the rear bogie. a Continuous, b Simply supported

as well as vehicle in continuous span bridge, are smaller than those for the simple beam model. Therefore, a continuous type of bridge may be more preferred than simply supported one to reduce the dynamic response of the bridge, as well as the acceleration of the car body. • Comparison of numerical results versus Measured responses In this section, a realistic bridge is chosen for which in situ data for bridge responses are available. The bridge is a steel girder type bridge with three simply supported span of 12 m (Fig. 11), for which 2D spline model is good enough in order to model the bridge. The location of different sensors is shown in Fig. 12.

Fig. 11 The test bridge under a train (Report No. 527/02, NF Railway, India)

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Fig. 12 Locations showing different sensor placement (Report No. 527/02, NF Railway, India)

The responses of this train–bridge interaction system are calculated considering each car of the train as a 10 DOFs vehicle model and compared with the measured data. In order to address train–bridge interaction involving complex bridges, one can take the advantage of commercial finite element software like ANSYS for bridge modelling. Therefore, apart from developing MATLAB code for the whole train– bridge system, one can first make a bridge model in ‘ANSYS’, and import the mass, damping and stiffness matrices of the bridge to MATLAB based code along with the nodal coordinates and element connectivity matrices. Using these imported matrices and position of train at an instant, it would then be possible to develop a MATLAB code through which the system force vector, and finally the responses of the system can be easily found out [16]. Here also the bridge is modelled in ‘ANSYS’ using 2D Euler-Bernoulli beam element and the required matrices are taken in ‘MATLAB’ code for finding the system responses. The bending strain and vertical displacement of the bridge, evaluated at the bottom flange of side-span centre (sensor 4 and 5, Fig. 11), for the train moves over the bridge at a speed of 50 km/h, are shown in Figs. 13a and 14a, with those measured at the site are in Figs. 13b and 14b. By looking at Figs. 13 and 14, one could easily say that the simulated results, obtained using 10 DOFs vehicle model along with the bridge, are almost comparable with the experimental data. However, the small discrepancies might be due to the combined effect of unavailability of exact test train data, running speed, instrument sensitivity, etc.

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Fig. 13 Bending strain at the bottom flange of first-span mid-section a calculated, b measured

Fig. 14 Vertical deflection at the bottom flange of first-span mid-section a calculated, b measured

4 Conclusions In this study, various existing train–bridge interaction models have been implemented in the finite element framework. In this regard, various comparative studies have been performed in terms of both bridge and vehicle responses for different types of bridges (e.g. simply supported, continuous). Further, an existing bridge has been taken and a study has been performed in order to compare the numerical value of bridge responses obtained using 10 DOF vehicle model with the measured responses. The responses obtained for a three span simply supported bridge have been validated with the measured data. By comparing such different results, the following conclusions can be drawn: • If the vehicle responses are not of major concerns (a case not related to highspeed railways), analysis of the train–track–bridge interaction system could be done using the moving load or moving mass models. It is, therefore, not required to conduct such analysis considering a complex vehicle model. • In order to determine vehicle responses, though, discrete sprung mass model could be used. However, it is suggested to conduct such analysis by considering the

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vehicle as a moving system. Since the analysis performed with the discrete sprung mass model could lead to a significant underestimation of vehicles responses. • Continuous bridge instead of simply supported bridge, could be used to reduce the vehicle responses (which serves as passenger’s comfort level) along with bridge responses. • In the 2D context, 10 DOFs vehicle model can suitably be used in the practical field for realistic determination of train responses.

References 1. UIC (2011) High-speed lines in the world. UIC High-Speed Department 2. Frýba L (1996) Dynamics of railway bridges. Thomas Telford House, London 3. Frýba L (1999) Vibration of solids and structures under moving loads. Czech Republic, Thomas Telford House 4. Yang YB, Yau JD, Wu YS (2004) Vehicle-bridge interaction dynamics: with applications to high-speed railways. World Scientific Publishing, Singapore 5. Yang YB, Yau JD (1997) Vehicle-bridge interaction element for dynamic analysis. J Struct Eng 123(11):1512–1518 6. Yang YB, Wu YS (2001) A versatile element for analyzing vehicle–bridge interaction response. Eng Struct 23(5):452–469 7. Zheng DY, Au FTK, Cheung YK (2000) Vibration of vehicle on compressed rail on viscoelastic foundation. J Eng Mech (ASCE) 126(11):1141–1147 8. Wu YS, Yang YB (2003) Steady-state response and riding comfort of trains moving over a series of simply supported bridges. Eng Struct 25:251–265 9. Lou P, Zeng QY (2005) Formulation of equations of motion of finite element form for vehicletrack-bridge interaction system with two types of vehicle model. Int J Numer Meth Eng 62(3):435–474 10. Lou P (2007) Finite element analysis for train–track–bridge interaction system. Arch Appl Mech 77(10):707–728 11. Lou P, Yu ZW, Au FTK (2012) Rail-bridge coupling element of unequal lengths for analyzing train–track bridge interaction systems. Appl Math Model 36(4):1395–1414 12. Wu JS, Dai CW (1987) Dynamic responses of multi-span non-uniform beam due to moving loads. J Struct Eng ASCE 113(3):458–474 13. Wang RT (1997) Vibration of multi-span Timoshenko beams to a moving force. J Sound Vib 207(5):731–742 14. Lin YH, Trethewey MW (1990) Finite element analysis of elastic beams subjected to moving dynamic loads. J Sound Vib 136(2):323–342 15. Esen I (2011) Dynamic response of a beam due to an accelerating moving mass using moving finite element approximation. Math Comput Appl 16(1):171–182 16. Pal B, Dutta A (2016) Seismic response analysis of train–track–bridge interaction system to evaluate running safety of train in case of high-speed railways. Bridge Struct Eng J (INGIABSE) 46(2):12–22

Evaluation of Ride Comfort in Railway Vehicle Due to Vibration Exposure S. Pradhan and A. K. Samantaray

Abstract Air spring is one of the important components in the modern railway vehicle, which affects passenger comfort. It comprises both stiffness and damping and placed between the bogie and car body. The main function of air spring is to isolate vibration, which transmits from bogie frame to car body. Deep knowledge is essential in order to study the influence of the parameters on the comfort. The vertical stiffness of air spring is calculated from force–displacement hysteresis loop, and the influences of stiffness at different frequencies are estimated for different preloada. Finally, comfort felt by the passengers is calculated by Sperling’s ride comfort for both straight and curved tracks. For better performance, optimization of preload along with the effective area of air spring is necessary. Keywords Ride comfort · Air spring · Dynamic performance · Railway vehicles

1 Introduction Nowadays air spring is used as a secondary suspension system instead of coil spring in high-speed trains, inter-cities, and commuters to improve passenger comfort. Passengers feel discomfort when constantly exposed to vibration and different individuals feel differently as it varies with seat design; the operating condition of the vehicle; track irregularities; gender, mass, and physique of the human body and environmental conditions; etc. As rigid modes of car body/coach are not appropriate to estimate the passenger comfort level, the flexibility of the car body is taken into account to calculate the same. Carlbom [1] has investigated the consequences of the flexibility of the car body on the performance of the railway vehicle by both simulation and measurement approaches. Diana et al. [2] have also emphasized on the requirement of considering the car body flexibility to evaluate the ride comfort of railway vehicles. S. Pradhan (B) · A. K. Samantaray Indian Institute of Technology, Kharagpur 721302, India e-mail: [email protected] A. K. Samantaray e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_33

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The activities such as health, comfort, and performance of the passenger are greatly affected under the frequency range of 0.5–100 Hz. Many researchers have focused on low frequency range (0.5–20 Hz) vibration, as the human body is most sensitive under this situation. Generally, the vibration is mainly transmitted to passengers through seat, backrest, and floor of the car body through feet. The main functions of the air spring are to manage ride height, increase the preload capacity, and isolate the airborne noises and vibration. The vibration transmission from bogie to car body has been reduced by using air spring as it eliminates the high frequency vibrations. Currently, it is more essential for the higher deadweight (due to weight of the car body and passengers’ weight), higher speed, and higher cant deficiency (higher curving speed) as well as for improving the ride comfort. Several air spring models such as the Nishimura, the vampire, and the Berg models are used worldwide. The aim of this paper is to find out the effect of air spring on the dynamic performances of railway vehicles. In addition, it is used to determine the effect of parameters such as the effective area of the air spring and the weight of the car body including passengers’ weight. Air spring has also frequency-dependent behavior, which needs to be accurately modeled. The estimation of ride comforts both in lateral and vertical directions in the presence of irregularities has been studied for both straight and curved tracks.

2 Modeling In this section, we have described the multi-body railway vehicle system, which is coupled with both railway vehicle and track. The details are discussed below.

2.1 Vehicle Model In this chapter, the modified European Rail Research Institute (ERRI) vehicle is chosen as a base model for the dynamic analysis. The full vehicle consists of two bogies (front and rear) and a flexible car body. Each bogie comprises primary and secondary suspensions, wheel-set (two wheels along with axle), axle boxes, etc. The primary and secondary suspension systems lie between wheel-set and bogie frame and bogie frame to car body, respectively. The pitch-plane vehicle model is shown in Fig. 1. In this analysis, the influence of air spring on the performance of the railway vehicle has been studied by varying different parameters through commercial multibody simulation software VI-Rail (ADAMS). Bogie and track design parameters and track irregularities, operating speed, standard wheel and rail profiles s1002 and UIC 60, respectively, are the input parameters to the full vehicle model. The full vehicle model along with the bogie model is depicted in Fig. 2, and details of the bogie design parameters are given in Table 1.

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Fig. 1 Model of the full vehicle model

Fig. 2 a Significant components of bogie and b MBS Full vehicle model built in VI-Rail Table 1 Parameter values of important components

Sl no

Parameters

Values (in SI units)

1

Car body mass (M c )

32,000

2

Car body inertia (I c )

I xx = 5.68 × 104 I yy = 1.97 × 106 I zz = 1.97 × 106

3

Bogie mass (M b )

4

Bogie moment of Inertia (I b )

5

Wheel mass (M w )

2615 I xx = 1722 I yy = 1476 I zz = 3067 1500

Moment of Inertia of wheel (I w )

I xx = 810 I yy = 810 I zz = 112

6

Primary spring stiffness (K p )

K x = 6.8 × 106 K y = 3.92 × 106 K z = 5.756 × 105 K  = K α = 63.5

7

Length of the car body

20.72

Note The stiffness, K x, K y , and K z are in x, y, and z directions, respectively, the torsional stiffness K θ and K α are in x and y directions, respectively

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2.2 Air Spring Model Air spring is the main part of the secondary suspension in the railway vehicle. The important components of air spring are surge reservoir, surge pipe, leveling valve, orifice and air bags, etc. The presence of surge reservoir and surge pipe increases the total air volume, which considerably softens the air suspension and improves the vibration isolation. The presence of orifice may increase the damping properties at lower frequencies (≈1 Hz) [3]. The load-leveling device is connected between the bogie frame and base plate of air spring (as shown in Fig. 3) that controls the height of the air spring by controlling the air pressure in the bellows. The behavior of air spring is dependent on the fluid dynamic and thermodynamics mechanism. Nishimura air spring is commonly used that is implemented in MBS software VI-Rail (ADAMS), and corresponding data are given in Table 2. The stiffness and damping constants are given by Eq. (1) [4].   K 1 = n A2e VPr0   K 2 = n A2e VPb0

d A2

⎫ ⎪ ⎪ ⎪ ⎬

⎪ K 3 = (P0 − Pa ) dze ⎪ ⎪ ⎭ 0.126 2 2 C = R f Ae ρ0 g = d 3 Ae ρ0 g

,

(1)

s

where Ae is the effective area (m2 ), P0 and Pa are the initial absolute pressure and atmospheric pressure (Pa), respectively, n=1.32, polytropic coefficient of the air in the spring, d s is the diameter of the surge pipe (m), g is the gravitational acceleration (m2 /s), Rf is the flow resistance coefficient, ρ0 is the mean density of the air (kg/m2 ), Fig. 3 Air spring system

Table 2 Significant parameters of air spring [data from VI-Rail software]

Parameters

Values (SI)

Orifice diameter (d)

0.015

Reservoir volume (V r )

0.025

Ambient air pressure (Pa )

1.0532 × 105

Ambient air temperature (T a )

298

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Fig. 4 Equivalent mass-damper representation of Nishimura air spring with linear damper

Vb = Vb0 − Z Ae + Z s As , is the air bag volume (m3 ), and Vr = Vr 0 − Z s As, volume of the reservoir (Z and Z s are the deflections as shown in Fig. 4). The frequency range of our interest is 0–20 Hz, which is crucial from the vehicle dynamic and human comfort point of view. The vertical preload of air spring is about 50–150 kN per air spring which may be calculated as F=Ae (P0 − Pa ) [5].

2.3 Track Model

Vertical irregularities

Track is an integral part of the simulation, which is built separately in VI-Rail. The track consists of straight, transition, and curved portions. The design parameters are cant/ super-elevation, curve radius, rail inclination, gage width of the track, etc. Different irregularities such as sinusoidal, PSD, and stochastic (measured) types are given in the vertical or lateral directions of both sides of the track for the accurate dynamic analysis. In this article, we have focused on the sinusoidal and stochastic type of irregularities for different purposes. The sinusoidal type of irregularities (as shown in Fig. 5a) is used for calculating stiffness for different preloads at different

0.002

0 -0.002

-0.004

0

1000

2000 3000 Track abscissa

(a) Fig. 5 Types of irregularities a Sinusoidal b Measured-type irregularities

(b)

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frequencies, and stochastic type of irregularities (as shown in Fig. 5b) is used for estimating ride comfort for both straight and curved tracks for different preloads.

3 Simulation and Results Preload of each air spring is due to the weight of the car body as well as passengers’ per air spring. The initial absolute pressure/static pressure is calculated as P0 = ((mcp ×g)/Ae ) + Pa , where mcp stands for the mass of car body and passengers’. The variations of pressure with effective area for different preloads are given in Fig. 6. Static pressure increases with an increase in load and decreases with effective area. However, it is noticed from the simulation that pressure is increased suddenly when the effective area is about 0.25 m2 . The main aim of this chapter is to analyze the influence of air spring on the dynamic behaviors such as critical speed, derailment speed, and ride comfort with varying effective areas. For critical speed analysis, a ramp type irregularity of width 5 mm and height 5 mm is given to the flexible straight track in the vertical direction on the right side of the track. The variation of the critical speed with effective area for different preloads is given in Fig. 7a. Critical speed increases with increases in preload and independent of effective area, i.e., the critical speed is constant with an increase in effective area. Derailment speed is estimated in the curved track with a sharp curve radius of 320 m. Figure 7b shows that derailment speed decreases with preload and is independent of effective area of the air spring. The key behavior of air spring is explained in terms of stiffness in the vertical direction (from force–displacement loop, hysteresis loop) that depends on excitation amplitude, frequency, orifice diameter, and preload. The axial air spring stiffness is almost proportional to pressure, and damping increases with an increase in preload.

Fig. 6 Variation of air spring pressure with effective area for different loads (weight of car body and passengers)

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42 Derailment speed (m/s)

Critical speed (m/s)

130

407

120 110 100 90

39

107.9 KN

78.48 KN 127.53 KN

157 KN

36

0.2

0.22

0.24 0.26 0.28 Effective area (m2 )

0.3

0.2

0.22

0.24 0.26 Effective area (m 2 )

(a)

0.28

0.3

(b)

Fig. 7 Variation of a critical and b derailment speed with effective area for different preloads

The damping behavior of air spring is not discussed in the article. The stiffness is dependent on air spring height (displacement in z-direction) and pressure, which are the outputs of the dynamic analysis of the MBS software. Preload and displacement dependent forces have been represented in the air spring model. The dynamic stiffness (K) is calculated as K = (F max − F min )/(Z max − Z min ), where F and Z represent force and displacement, respectively, in a cycle and subscripts min and max denote maximum and minimum values [6]. It is also noted that the viscous behavior is periodic but non-linear due to the interaction between air bag and reservoir volume. The inertial effect is noticeable for low amplitude excitation and no orifice damping. The force–displacement, hysteresis loop is given in Fig. 8a. Stiffness increases for higher frequencies up to 10 Hz and then decreases with frequencies. At higher preload, the stiffness increases due to increase in pressure in the air spring as shown in Fig. 8b. The curve (Fig. 8b) consists of both, low frequency relates to the excitation of bellow and tank, and the high frequency corresponds to the excitation of bellows alone [7].

84100 84000 83900 0.00

d=15 mm

10

84200

Stiffness, K z (N/m)

Force in Z direction (mm)

×105

0.20

0.40

Displacement z (mm)

(a)

0.60

8 6 80 kN

100 kN

120 kN

4 0

5 10 15 Frequency (Hz)

20

(b)

Fig. 8 a Vertical force–displacement loop for preload = 78.5 kN and orifice diameter 15 mm subjected to harmonic excitations and b Variation of stiffness with frequency for different preloads

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Different countries use different standards of ride comfort such as International Union of Railways (UIC) 513, International Organization for Standardization (ISO) 2631, European Committee for Standardization (CEN) ENV 12299, and Sperling ride index. However, we have focused on the Sperling ride index to measure comfort in this analysis. The weighting function (B) in both vertical and horizontal directions is different. The B in horizontal (lateral) direction is given by Eq. (2) [8]. 

1/2 2  1.911 f 2 + 0.25 f 2 , Bh = 0.737  2  2 1 − 0.277 f 2 + 1.563 f − 0.0368 f 3

(2)

where f is the frequency in Hz. Similarly, B in vertical direction is calculated as Bv = Bh /1.25. Ride comfort for each individual frequency in the corresponding direction is expressed as W z = (a2 B2 )1/6.67 . The ride comfort for n frequencies is estimated as [8] 1 6.67  Wz_total = W z 16.67 + W z 26.67 + W z 36.67 + W z 46.67 + . . . + W z n6.67 /

(3)

Although the frequency range 0.5–100 Hz affects whole body vibration, low and medium frequency range such as 0.5–20 Hz affects severely on the ride comfort and other activities during traveling. For estimating ride comfort both in straight and curved tracks, stochastic type of irregularities are given to both vertical and horizontal directions of the track. In straight track, the vehicle runs at 100 m/s (360 kmph), and corresponding ride comfort is shown in Fig. 9. From Fig. 9a, it is shown that the ride comfort is independent of effective area and increases with increase in preload in only lateral direction. Conversely, it is very difficult to predict the passenger comfort in the vertical direction by varying the effective area. It is noticed from the simulation that ride comfort is better at low effective areas for larger preload. The ride comfort has been estimated in the curved track also whose curve radius is 320 m as shown in Fig. 10. In the curved track, it is not easy to predict the trend of the

Wz in vertical direction

Wz in lateral direction

2.5

2

1.5 78.48 KN 127.53 KN

1 0.2

107.9 KN

157 KN

2.3 2.1 1.9

78.48 KN 127.53 KN

107.9 KN

157 KN

1.7

0.22

0.24 0.26 Effective area (m 2 )

(a)

0.28

0.3

0.2

0.22

0.24 0.26 Effective area (m 2 )

0.28

0.3

(b)

Fig. 9 Variation of ride comfort for straight flexible track with effective area for different preloads a in lateral direction and b in vertical direction

2.5

Wz in vertical direction

Wz in lateral direction

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2.3 2.1 1.9 1.7 1.5 0.2

107.9 KN

78.48 KN 127.53 KN

0.22

157 KN

0.24

0.26

0.28

0.3

409

2.5 2.3 2.1 1.9 78.48 KN 127.53 KN

1.7

107.9 KN

157 KN

1.5 0.2

0.22

0.24

0.26

Effective area (m 2 )

Effective area (m2 )

(a)

(b)

0.28

0.3

Fig. 10 Variation of ride comfort for curved flexible track with curve radius 320 m with effective area for different preloads a in lateral direction and b in vertical direction

ride comfort both in vertical and lateral directions. In lateral direction, it is almost independent of effective area except large preload (157 kN) and it does not give any trend like lateral direction on the straight track. In vertical dirction, at low (78.48 kN) and high preload (157 kN), ride comfort felt by passenger is better at the medium range effective area (0.24–0.28 m2 ) which is not possible in other cases. Hence, for better dynamic performances, the parameters (preload and effective area) should be optimized.

4 Conclusion In this article, we have focused on the influence of change in effective area with different preloads in the dynamic performance of the vehicle such as estimation of critical speed, derailment speed, and ride comfort. Critical and derailment speeds are independent of effective area. Critical speed increases with preload, and derailment speed decreases with preload. While estimating ride comfort in both straight and curved tracks, the influences of effective area and preload have been noticed. Stiffness in the vertical direction is calculated from force–displacement loop (hysteresis loop) for a particular preload and frequency. It is also found from the simulation that the stiffness in the vertical direction is frequency dependent. In a particular preload, at a certain effective area, the vehicle gives better ride comfort. Hence, optimization is required for effective area with respect to preload, which may be considered as future work.

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References 1. Carlbom P (2001) Combining MBS with FEM for rail vehicle dynamics analysis. Multibody Syst Dyn 6:291–300 2. Diana G, Cheli F, Andrea C, Corradi R et al (2002) The development of a numerical model for railway vehicles comfort assessment through comparison with experimental measurements. Veh Syst Dyn 38(3):165–183 3. Berg M (1999) An air spring model for dynamic analysis of rail vehicle. KTH, Stockholm 4. Presthus M (2002) Derivation of air spring model parameters for train simulation, Master’s Thesis, Luleal University of Technology, Sweden 5. Quaglia G, Sorli M (2001) Air suspension dimensionless and design procedure. Veh Syst Dyn 35(6):443–475 6. Mazzola L, Berg M (2012) Secondary suspension of railway vehicles-air spring modeling: performance and critical issues. J Rail Rapid Transit 1–17 7. Docquier N, Fisette P, Jeanmart H (2007) Multi-physical modeling of railway vehicles equipped with pneumatic suspensions. Veh Syst Dyn 45(6):505–524 8. Garg VK, Dukkipati RV (1984) Dynamics of railway vehicle systems. Academic Press, Canada

Non-linear State Space Formulation Simulating Single Station Ride Dynamics of Military Vehicle Saayan Banerjee, V. Balamurugan, and R. Krishna Kumar

Abstract Military vehicles are generally equipped with hydro-gas suspension systems that exhibit better shock-absorbing capability over drastic dynamic environments compared to linear suspension. In order to implement suspension semiactive/active control in the future, it is required to develop the mathematical model of the vehicle using non-linear state space approach by incorporating the hydro-gas suspension trailing arm dynamics in the governing equations of motion. The present study formulates the non-linear state space approach which simulates single station ride dynamics of military vehicles. Incorporating the developed trailing arm kinematics and non-linear suspension characteristics, non-linear state space approach has been used to formulate the sprung and unsprung mass governing equations of motion. The multi-body dynamics model for the single station is established in MSC.ADAMS in order to validate the non-linear state space model. The mathematical model is solved using MATLAB and compares well with the multi-body model simulations. The entire military vehicle non-linear state space model can also be developed which would be suitable for carrying out vehicle dynamics control studies with active or semi-active suspension systems. Keywords Military vehicle · Hydro-gas suspension · Non-linear state space · Multi-body dynamics · Ride dynamics

S. Banerjee (B) Centre for Engineering Analysis and Design, Combat Vehicles R&D Estt., DRDO, Chennai 600054, India e-mail: [email protected] V. Balamurugan Aircraft Projects Division, Combat Vehicles R&D Estt., DRDO, Chennai 600054, India e-mail: [email protected] R. Krishna Kumar Department of Engineering Design, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_34

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1 Introduction The military vehicles are generally equipped with hydro-gas suspension systems in order to have a better shock-absorbing capability over the drastic dynamic environment compared to that of linear suspension. Therefore, for future implementation of semi-active or active control system, it is an important pre-requisite to determine the transmitted vibration levels to the vehicle with passive suspension. Gillespie [1] provided a detailed overview of the fundamental theory of vehicle dynamics. Solomon and Padmanaban [2] have proposed a polytropic gas compression model to describe the spring characteristics and hydraulic conductance model to represent the damper characteristics of hydro-gas suspension system. The above spring and damper models have been incorporated in a vehicle dynamic in-plane math model and carried out simulations for sinusoidal and Axle Proving Ground (APG) terrains as well as validated with experimental measurements [2]. Dhir and Shankar [3] have derived the tracked vehicle math model by using the Lagrangian method over a hard terrain and constant vehicle speed. The track–terrain interaction has been considered and ride dynamic analysis of an Armoured Personal Carrier vehicle has been carried out [3]. Rakheja et al. [4] have carried out a comparative ride dynamic studies of a non-linear vehicle dynamic in-plane model with active and passive suspensions. However, in [2–4], the trailing arm dynamics for each of the suspension stations have not been considered. The equivalent vertical stiffness and damping parameters have been derived from the trailing arm kinematics. The effect of trailing arm dynamic behaviour will be quite different from the vertical suspension system which requires to be mathematically formulated. Moreover, the above studies do not include the roll mode of the vehicle. Sujatha et al. [5] have carried out an experimental ride dynamic evaluation of a 6 station military vehicle with torsion bar suspension. Accelerations have been obtained at specified locations over various dynamic environments and analysed [5]. Balamurugan [6] has developed a finite element model of a highly mobile military-tracked vehicle over a hard road for estimating the ride characteristics. Hada [7] has derived the dynamic behaviour of a 12 station tracked vehicle with torsion bar suspension. It is observed that [5–7] does not take into account non-linear effects of the hydro-gas suspension. Subburaj et al. [8] have described the solution methodology for structural dynamics problems through implicit procedures with a detailed description on the solution stability. Paduart et al. [9] have proposed a state space methodology that deals with Multiple Input and Multiple Output (MIMO) systems. Reddy [10] provided a detailed overview on the finite element methods and practice. Banerjee et al. [11] have described the trailing arm suspension dynamics for a single station of a tracked vehicle. In [2–7], the vehicle dynamics have not been modelled using non-linear state space approach. It is noteworthy that extensive research has undergone in the area of vehicle dynamics. The present study describes the non-linear state space mathematical method for formulating the military vehicle single station ride dynamics by incorporating the trailing arm kinematic and dynamic behaviour. Subsequent validation studies have been carried out with numerical experiments which are based on the developed

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MSC.ADAMS multi-body dynamics model. Reference [11] provided a systematic approach to bring out the significance of the physical dynamic behaviour of the trailing arm suspension and effects of inertia coupling between the sprung and upsprung masses on the ride dynamics. However, in the present paper, the developed state space model of single station with trailing arm dynamic effects can be used for implementing semi-active and active control techniques for ride vibration control unlike the mathematical model described in [11]. The above mathematical state space ride model when extended for a full military vehicle model would also serve as a useful input to a driving simulator.

2 State Space Approach for Single Station Representation Initially, the hydro-gas suspension kinematics which was derived in Sect. 2.1 [11] has further been simplified for implementation in the state space matrix. The stiffness non-linearities which pertain to various charging pressures have been expressed as binomial series in terms of axle arm angular displacement. The angular motion behaviour of trailing arm unsprung mass has been expressed with the Taylor series expansion in terms of axle arm angular displacement for suitable implementation in state space matrix. Following the above mathematical representations, non-linear state space domain matrices are formulated and solved in MATLAB. The sprung mass dynamics responses have been compared and validated with the MBD model which is subjected to standard terrain excitations.

3 Implementation of Suspension Kinematics in State Space Matrix Figures 1 and 2 represent the hydro-gas suspension assembly and kinematic description of the same. Referring to Sect. 2.1 [11], the trailing arm suspension kinematic relations can be expressed as   2  x = sqrt L 2p − L c {cos(β + γ ) − cos(β − ϕ + γ )} + L p sin(δ) +L c {sin(β + γ ) − sin(β − ϕ + γ )} − L p cos(δ).

(1)

For the convenience of incorporation of suspension kinematics in the non-linear state space matrix, a factor f is determined from Eq. (1) such that f = x/ϕ. Differentiating x in Eq. (2) with respect to time t,

(2)

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x˙ =

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d x dϕ dx = . dt dϕ dt

∴ x˙ = f 1 · ϕ, ˙

(3)

where   L c {cos(β + γ ) − cos(β − ϕ + γ )} + L p sin(δ) {L c sin(β − ϕ + γ )} dx  f1 = =  2  dϕ sqrt L 2p − L c {cos(β + γ ) − cos(β − ϕ + γ )} + L p sin(δ) +L c cos(β − ϕ + γ ). It is observed from Eq. (3) that the piston velocity can be related to the axle arm angular displacement and velocity. Suitable kinematic simplifications are carried out through Eqs. (2) and (3) for incorporation in the state space matrix. It may be noted that hydro-gas suspension kinematics have a mild non-linearity. The following sections describe the method of incorporating non-linearity due to suspension stiffness in state space matrix domain.

4 Determination of Non-linear Stiffness Characteristics of Suspension Referring to Sect. 4.2 [11], the suspension kinematics have been used to determine the non-linear stiffness characteristics which are highlighted through Eqs. (4)–(6) as Pr eb Von = P(Vo − V1 )n ,

(4)

F p = Pπ d 2 /4,

(5)

T = Fp L o .

(6)

Due to static angular rotation of the axle arm by ϕst , the actuator piston displacement xst takes place in line with the actuator cylinder axis. The reaction moment about pivot point ‘O’ is Tst at a statically settled position. Subsequent to the static settlement, the axle arm rotation is described by ϕl due to terrain excitation. This in turn causes the reaction moment Tl due to displacement of the actuator piston by x in line with the actuator cylinder axis. From Eqs. (4)–(6), Tst = 

π  n d 2 L 0 , 4 st 4

pr eb V0n V0 −

π d2x

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Fig. 3 Gas restoring force variation with piston displacement for rebound pressures of 11.4 and 13 MPa

Tl = 

π pr eb V0n  n d 2 L 0 .  V0 − π d 2 xst 4 − π d 2 x 4 4

(8)

Using Eq. (2), Eq. (8) may be written as Tl = 

π pr eb V0n   n d 2 L 0 . V0 − π d 2 xst 4 − π d 2 f ϕl 4 4

(9)

The non-linear force–displacement characteristics can be derived from the above equations which are highlighted in Fig. 3 for different rebound gas pressures.

4.1 Taylor Series Approximation of Hydro-Gas Suspension Spring Restoring Moment Subsequent to the achievement of the static equilibrium configuration (described in [11]), the hydro-gas suspension spring restoring moment about the static position (Tl − Tst ) is required to be expressed in terms of ϕl in order to facilitate the transformation of the coupled equations of motion into matrix domain. From Eqs. (8) and (9), if Vo π 2 d 4 Then

pr eb V n π − xst = l2 and  π on d 2 L o = K , d2 4 4

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(Tl − Tst ) =



−n K K  K x 1 − − = − 1 . l2 (l2 − x)n (l2 )n (l2 )n

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(10)

  −n Also, the expression 1 − x l2 in Eq. (10) may be expanded in binomial series which is truncated after the fifth term as  −n nx n(n + 1) 2 n(n + 1)(n + 2) 3 1 − x l2 =1+ + x + x l2 2l22 6l23 n(n + 1)(n + 2)(n + 3) 4 n(n + 1)(n + 2)(n + 3)(n + 4) 5 + x + x . 24l24 120l25 

The truncation was based on a trial and error method. Simulations have also been carried out with higher order terms in the state space matrix; but, an insignificant difference in results has been obtained. Therefore, the binomial series have been truncated accordingly. This is a reasonable approximation by considering the dynamic range of angular operation of the axle arm which is normally limited within 30° from static equilibrium position. Using the above binomial series and Eq. (2) in Eq. (10), (Tl − Tst ) = K (ϕl )ϕl ,

(11)

where n(n + 1) f 2 n(n + 1)(n + 2) f 3 2 K nf + ϕ + ϕl K (ϕl ) = n l l2 l2 2l22 6l23 +

n(n + 1)(n + 2)(n + 3) f 4 3 ϕl 24l24

+

n(n + 1)(n + 2)(n + 3)(n + 4) f 5 4 . ϕ l 120l25

Equation (11) relates the spring dynamics restoring moment and rotational angle of the axle arm. Equation (11) is of non-linear nature in terms of the angular rotation of axle arm about the static position. This relation may be directly implemented while transforming the coupled equations of motion into non-linear state space matrix domain.

5 Formulation of State Space Mathematical Single Station Ride Model The model description is similar to Sect. 5 [11]. It may be noted that in practice, the unsprung mass is distributed partly on the axle arm. However, the axle arm mass is less compared to that of the road-wheel and track pad. Therefore, the unsprung

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mass has been lumped at the road-wheel centre. However, the present mathematical formulation may similarly be extended for a distributed unsprung mass.

5.1 Single Station Mathematical Model Using Non-linear State Space Approach The governing differential equations of motion which are described in Sect. 5.2 [11] have been transformed into non-linear state space domain for both the sprung and unsprung masses. The state space approach has been followed subsequent to the static settlement of the two degree of freedom single station model. The static equilibrium equations have been derived in Sect. 5.1 [11]. With reference to Sect. 5.3 [11], the dynamic behaviour of the single station model as well as free body diagrams for the sprung and unsprung masses are shown in Fig. 4a, b and c, respectively. The state space mathematical model comprises similar nomenclature as described in Table 1 [11].

5.2 Sprung Mass Bounce Motion in the State Space Domain With reference to Sect. 5.3 [11], the sprung mass bounce is described by Eq. (12) as   (M + m l ) X¨ + m l X¨ ϕ + kt X + X ϕ − Y = 0 ,

(12)

where X ϕ = Lcos(ρl ) − Lcos(ρl + ϕl ); ρl = αl + ϕst ,

X¨ ϕ = L ϕ˙l2 cos(ρl + ϕl ) + L ϕ¨l sin(ρl + ϕl ) = L ϕ˙l2 cos(ρl + ϕl ) + L ϕ ϕ¨l , L ϕ = Lsin(ρl + ϕl ). Now, cos(ρl + ϕl ) = cos(ρl ) cos(ϕl ) − sin(ρl )sin(ϕl ). Also, cos(ϕl ) and sin(ϕl ) may be expanded into the Taylor series as shown below (neglecting higher order terms), cos(ϕl ) = 1 −

ϕl2 ϕ4 + l, 2 24

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Fig. 4 a Application of ground excitation to the single station model, b Sprung mass force representation, c Unsprung mass moment representation

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sin(ϕl ) = ϕl −

ϕ5 ϕl3 + l . 6 120

The truncation of the above trigonometric expressions for ϕl is based on trial and error method which is explained in Sect. 4.1. Using the above conversions and expressing trigonometric expressions of variable ϕl as the Taylor series expansion,  

ϕl2 ϕl4 ϕl5 ϕl3 − + k2 ϕl − + . kt X ϕ = k1 − k1 1 − 2 24 6 120

(13)

∴ kt X ϕ = kt (ϕl )ϕl ,

(14)

where k1 = kt Lcos(ρl ) and k2 = kt Lsin(ρl ),  

ϕl2 ϕl3 ϕl4 ϕl kt (ϕl ) = k1 − + k2 1 − + . 2 24 6 120 Therefore, Eq. (14) may be written as (M + m l ) X¨ + m l X¨ ϕ + kt (ϕl )ϕl + kt X = kt Y. ∴ (M + m l ) X¨ + m l ϕ¨l L ϕ + m l L ϕ˙l2 cos(αl + ϕst + ϕl ) + kt (ϕl )ϕl + kt X = kt Y. (15)

5.3 Unsprung Mass Rotational Dynamics in the State Space Domain With reference to Sect. 5.4 [11], unsprung mass rotation is described in Eq. (16) as m l L 2 ϕ¨l + m l X¨ L ϕ + (Tl − Tst ) + c x˙ L 0 + (Tt − TY ) = 0, where L ϕ = Lsin(ρl + ϕl ),   (Tt − TY ) = kt X + X ϕ − Y L ϕ X ϕ = Lcos(ρl ) − Lcos(ρl + ϕl ) Using the previous expressions and Eq. (3), Eq. (16) may be written as

(16)

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m l L 2 ϕ¨l + m l X¨ L ϕ + K (ϕl )ϕl + c f 1ϕ˙l L 0 + kt1 (ϕl )ϕl + kt L ϕ X = kt L ϕ Y. ∴ m l L 2 ϕ¨l + m l X¨ L ϕ + {K (ϕl ) + kt1 (ϕl )}ϕl + c f 1ϕ˙l L 0 + kt L ϕ X = kt L ϕ Y, (17) where kt1 (ϕl ) = kt (ϕl )L ϕ .

5.4 Transformation into Non-linear State Space Domain Equations (15)–(17) can be transformed into state space domain which is highlighted in Eqs. (18)–(19), respectively. (M + m l ) X¨ + m l ϕ¨l L ϕ = −m l L ϕ˙l2 cos(ρl + ϕl ) − kt (ϕl )ϕl − kt X + kt Y.

(18)

m l L 2 ϕ¨l + m l X¨ L ϕ = −{K (ϕl ) + kt1 (ϕl )}ϕl − c f 1ϕ˙l L 0 − kt L ϕ X + kt L ϕ Y. (19) Define ϕ˙l = u 1 and X˙ = u 2 Substituting for ϕ˙l and X˙ in Eqs. (18) and (19), the following state space matrix is obtained: 1 {A(ϕl )Z + B(ϕl )U}, Z˙ = M1 where ⎡

⎤ u1 ⎢ u2 ⎥ ⎥ Z=⎢ ⎣ ϕl ⎦and U = [Y], X ⎡ ⎤ m l L ϕ (M + m l ) 0 0 ⎢ ml L 2 ml L ϕ 0 0 ⎥ ⎥, M1 = ⎢ ⎣ 0 0 1 0⎦ 0 0 01 ⎤ −kt (ϕl ) −kt −m l L ϕ˙l cos(ρl + ϕl ) 0 ⎢ −c f 1L o 0 −{K (ϕl ) + kt1 (ϕl )} −kt L ϕ ⎥ ⎥ A(ϕl ) = ⎢ ⎣ 1 0 0 0 ⎦ 0 1 0 0 ⎡

(20)

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⎤ kt ⎢ kt L ϕ ⎥ ⎥ B(ϕl ) = ⎢ ⎣ 0 ⎦. 0 ⎡

The non-linear state space form of the equations which comprises the non-linear suspension characteristics is described by Eq. 20. The matrices A(ϕl ) and B(ϕl ) vary with ϕl and get updated with every time increment.

5.5 Solution for the Non-linear State Space Model The solution technique for the single station non-linear state space model is similar to that described in Sect. 5.5 [11]. Since the matrices are of non-linear nature, therefore, at every computation time, the matrices get altered as per the change of axle arm angular rotations about static position which results from base excitation.

6 Single Station Multi-body Dynamics Model Referring to Sect. 7 [11], the multi-body dynamics model which is developed in MSC.ADAMS for the non-linear state space model validation has been shown in Fig. 5. The multi-body model has been solved using a similar approach as described in [11]. The MBD model can be considered to be a numerical experiment that is solved through an implicit time integration scheme.

7 Comparative Responses Between State Space and Multi-body Dynamic Models The multi-body dynamics model for a single station has been used to validate the non-linear state space model through simulations over APG profile at vehicle speeds of 20 and 30 kmph. Generally, the vehicle negotiates APG terrain at average speeds of 20 and 30 kmph. Therefore, simulations have been carried out at the above speeds. Similar parameters which are described in Table 2 [11] have been used for the state space model.

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Fig. 5 MSC. ADAMS single station MBD model

7.1 Dynamics Analysis at 20 Kmph Over APG The APG (Axle Proving Ground) excitation profile (shown in Fig. 6) is applied to the single station model at a vehicle speed of 20 kmph. The simulation is carried out for 20 s. In the present case, suspension charging pressure is 11.4 MPa. The input has been applied as a vertical base displacement with respect to time. The time-domain variation is obtained by dividing the spatial distance between terrain elevation profiles by the corresponding vehicle speed. For the present model, vertical base displacement input with respect to time has been applied by considering a

Fig. 6 Representation of the APG terrain

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spatial variation on the left side of the APG terrain. The vertical displacement input due to APG at 20 kmph is shown in Fig. 7. The comparative sprung mass bounce displacement and acceleration responses are shown in Figs. 8 and 9, respectively. The maximum vertical accelerations over APG at 20 kmph which are obtained from the MBD and mathematical state space models are about 26.7 m/s2 and 28.8 m/s2 , respectively. The RMS of vertical accelerations over APG at 20 kmph which are obtained from MBD and mathematical state space models are about 4.7 m/s2 and 5.2 m/s2 , respectively. The dynamic responses from state space model compare well with the MBD model. Figure 10 shows the frequency-domain comparison of the sprung mass bounce acceleration response.

Fig. 7 Vertical displacement input due to APG at 20 kmph

Fig. 8 Sprung mass bounce displacement response at CG for 20 kmph

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Fig. 9 Sprung mass bounce acceleration response at CG for 20 kmph

Fig. 10 Frequency spectrum for sprung mass bounce acceleration response at CG for 20 kmph vehicle speed

7.2 Dynamics Analysis at 30 Kmph Vehicle Speed Over APG Comparative dynamic simulations have been carried out over APG at 30 kmph vehicle speed. The simulations have been performed for 14 s. The suspension characteristics and the displacement input procedures are similar to those described in Sect. 7.1. The vertical displacement input which pertains to 30 kmph vehicle speed over APG has been shown in Fig. 11. Figures 12 and 13 highlights the comparative sprung mass bounce displacement and acceleration responses. The maximum vertical accelerations over APG at 30 kmph which are obtained from MBD and mathematical state space models are about 32.1 m/s2 and 29.1 m/s2 , respectively. The RMS of vertical accelerations over APG at 30 kmph which are obtained from MBD and mathematical state space models are 6.3 m/s2 and 5.8 m/s2 , respectively. The sprung mass dynamics responses which are obtained over APG at 30 kmph show good comparison with the

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Fig. 11 Vertical displacement input due to APG at 30 kmph

Fig. 12 Sprung mass bounce displacement response at CG for 30 kmph

Fig. 13 Sprung mass bounce acceleration response at CG for 30 kmph

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Fig. 14 Frequency-domain variation for sprung mass CG bounce acceleration response for 30 kmph speed

MBD model. Also, sprung mass responses obtained from the models over APG at 30 kmph are comparatively higher than that at 20 kmph. The frequency spectrum which pertains to the sprung mass comparative bounce acceleration response at CG is shown in Fig. 14. The above simulations have also been carried out by using higher order terms in the non-linear state space model. However, insignificant difference is observed in the results with higher order terms when compared to those highlighted above. One of the reasons for possible deviations in response is due to differences in solvers (numerical solution and interpolation techniques) which are used in math and Adams models. Very marginal response difference may also be attributed to the dropping of higher order terms during the Taylor series and binomial expansions. Irrespective of the above deviations, there is a close agreement in response between the state space model and MBD model.

8 Conclusions The non-linear state space mathematical model for the single station incorporates the kinematics and dynamics effects of the trailing arm suspension. The non-linear equations for the single station in state space matrix domain are reformulated so that it is directly feasible for future control related studies without compromising much on accuracy. The maximum vertical accelerations over APG at 20 kmph which are obtained from the MBD and mathematical state space models are about 26.7 m/s2 and 28.8 m/s2 , respectively. The RMS of vertical accelerations over APG at 20 kmph which are obtained from MBD and mathematical state space models are about 4.7 m/s2 and 5.2 m/s2 , respectively. The maximum vertical accelerations over APG at 30 kmph which are obtained from MBD and mathematical state space models are about 32.1 m/s2 and 29.1 m/s2 , respectively. The RMS of vertical accelerations

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over APG at 30 kmph which are obtained from MBD and mathematical state space models are 6.3 m/s2 and 5.8 m/s2 , respectively. The sprung mass peak and RMS vertical acceleration responses which are obtained at 30 kmph speed are more when compared to that at 20 kmph. From the above results, it is observed that the non-linear mathematical state space model shows a close agreement with Adams MBD model. The entire military vehicle non-linear state space model which involves many degrees of freedom can also be derived from the present model. The present non-linear state space model provides the design and development motivation for semi-active and active suspension system in the military vehicles.

References 1. Gillespie TD (1992) Fundamentals of vehicle dynamics. Society of Automotive Engineers, USA 2. Solomon U, Padmanabhan C (2011) Hydro-gas suspension system for a tracked vehicle: Modeling and analysis. J Terramechanics 48(2):125–137 3. Dhir A, Sankar S (1995) Assessment of tracked vehicle suspension system using a validated computer simulation model. J Terramechanics 32(3):127–149 4. Rakheja S, Afonso MFR, Sankar S (1992) Dynamic analysis of tracked vehicles with trailing arm suspension and assessment of ride vibrations. Int J Veh Des 13(1):56–77 5. Sujatha C, Goswami AK, Roopchand J (2002) Vibration and ride comfort studies on a tracked vehicle; Heavy vehicle systems. Int J Heavy Veh Syst 9(3):241–252 6. Balamurugan V (2000) Dynamic analysis of a military tracked vehicle. Def Sci J 50(2):155–165 7. Hada MK (1996) Tracked vehicle motion dynamics. Institute of Armament Technology, Pune, M.Tech Thesis 8. Subburaj K, Dokainish MA (1989) A survey of direct time integration methods in computational structural dynamics II Implicit methods. Comput Struct 32(6):1387–1401 9. Paduart J, Schoukens, Pintelon R (2005) Non-linear state space modeling of multivariable systems. In: International symposium on nonlinear theory & its applications (NOLTA 2005), pp 18–21 Bruges, Belgium 10. Reddy JN (2005) An introduction to nonlinear finite element analysis. Department of Mechanical Engineering, Texas A&M University 11. Banerjee S, Balamurugan V, Krishnakumar R (2014) Ride dynamics mathematical model for a single station representation of tracked vehicle. J Terramechanics 53:47–58

Delamination Growth Behaviour in Carbon/Epoxy Composite Road Wheel of an Armoured Fighting Vehicle Under Dynamic Load Sarath Shankar, Subodh Kumar Nirala, Saayan Banerjee, Dhanalakshmi Sathishkumar, and P. Sivakumar Abstract The primary objective of this paper is to find the threshold size of delamination as well as critical delamination location for composite road wheel of the Armoured Fighting Vehicle (AFV) under severe loading condition to which an AFV is subjected to, during the operation. The composite material considered in the study is carbon/epoxy composite (AS4/3501-6). Finite element analysis was performed using Ansys Workbench 17.2 to predict the delamination location and to study the delamination growth behaviour. The delamination onset spot was predicted using the stress-based failure criteria developed by Puck. Various delamination sizes were modelled at the critical location and the Strain Energy Release Rate (SERR) with respect to the fracture modes, viz., mode I, mode II and mode III were found using Virtual Crack Closure Technique (VCCT). The 3D failure criterion which was developed based on Benzeggagh–Kenane (B-K) mixed-mode I and II failure criterion was employed to predict the delamination growth based on which the threshold delamination size was found. Keywords Delamination · Composite · Road wheel · Dynamic load · Fracture · Strain energy release rate

1 Introduction The mobility and ride performance besides firepower and protection of an Armoured Fighting Vehicle (AFV) are of great importance for its survivability in the battlefield. The running gear system of the vehicle plays a major role in providing better mobility and ride performance to the vehicle. The road wheel is a part of the running gear system and a dynamic load bearing member which transmits ground reaction to the suspension unit. Since the quantity of road wheel is always more in any AFV, it is developed using carbon/epoxy composite material for achieving overall weight reduction. Carbon/epoxy composite is chosen owing to its excellent specific strength S. Shankar (B) · S. K. Nirala · S. Banerjee · D. Sathishkumar · P. Sivakumar Combat Vehicles Research & Development Establishment, DRDO, Ministry of Defence, Avadi, Chennai 600054, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_35

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and specific modulus. Also, weight reduction of the road wheel improves the ride performance and reduces the acoustic signature of the vehicle as the road wheel is an unsprung mass. However, the most common threat to the use of composite materials in dynamic applications is the presence of internal defects, mainly delamination, as they affect the performance of the system. In a laminated composite, delamination is an adhesive failure at the interface between two adjacent layers. It can be considered as a crack of definite length. Composite components may contain delaminations due to bad lay-up and defects during manufacturing and assembling processes. Delaminations may also develop in a composite structure due to in-service impacts, static overloads and fatigue [7]. Dynamic load can cause the delamination to grow in size, leading to a rapid reduction in the mechanical properties and in turn catastrophic failure of the composites. Several researchers have studied the propagation behaviour of delamination in composite materials under various load conditions. Leif et al. [14] have carried out an experimental study on the propagation of mode I, mode II and mixed-mode delamination in HTA/6376C composite material under fatigue load case based on which the threshold values of the SERRs for delamination propagation were determined. The main finding was that the initial delamination mechanism under static and cyclic loading is identical which was demonstrated by comparing the fracture faces formed due to static load and fatigue load. Under cyclic loading, the specimen was shown to be sensitive to edge effects and, consequently, would not be representative of loading conditions within a composite structure. Balcioglu et al. [2] have studied lateral buckling of laminated composites having square and circular delamination experimentally as well as numerically. The authors used four aspect ratios to investigate the shape effect of fixed delamination area on the critical lateral load. It was concluded that the critical lateral load increases with an increase in aspect ratio and decreases with an increase in delamination size. Also, square-shaped delamination has more effect on the critical lateral load than circular-shaped delamination. Wimmer et al. [12] have examined delamination onset and growth in fibre-reinforced laminated composite by analysing a curved laminate. Puck failure criterion is used for predicting the starting delamination, and VCCT is used for modelling the growth of delamination. The relation between size of an initial delamination and stability of crack growth is studied and a limit initial delamination size is found. For the verification of the proposed simulation procedure, the authors used Cohesive Zone Method (CZM) and the same results were obtained. Marcello et al. [6] have used the configuration of double cantilever beam and assessed in detail through finite element method the mode I fracture of composite laminates made using carbon/epoxy composite material with fibre-reinforced in z-direction. The authors found that propagation of mode I fracture can be resisted to a great extend by reinforcing fibre in z-direction. However, the effect of z-direction reinforcement on the onset of delamination propagation from an initial crack was found to be insignificant. Omid et al. [15] have studied the dynamic behaviour of crack, debond and delamination in different type of materials (from normal homogeneous material to complex composite laminates) through nonlinear finite element dynamic analysis using Ansys software. It was concluded that dynamic analysis can be applied to a wide variety of materials, including composite materials

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for finding material properties and fractures, mainly delamination without destructing the material. From the review of previous studies on the delamination growth in composite materials, it is apparent that majority of the studies were restricted to mode I and mode II fracture regimes as most of the toughness data were limited to this 2D regime. The present study discusses the investigation of critical delamination location and the threshold delamination size by performing finite element analysis for the composite road wheel of the AFV under severe loading conditions and subsequently using a 3D failure criterion considering the effect of fracture modes I, II and III.

2 Background The running gear is an essential sub-system of the AFV, which ensures smooth movement of the vehicle by efficient usage of torque delivered by the power train to the sprocket of the vehicle. Road wheel is a part of running gear sub-system, which supports the vehicle and transmits the ground reaction to the suspension unit. This ensures a smooth ride with less noise. Road wheel is connected to the suspension unit through axle arm as shown in Fig. 1. Combat Vehicles Research and Development Establishment (CVRDE) is one of the premier establishments of Defence Research and Development Organization (DRDO) which is involved in the development of primarily AFVs, and Arjun Main Battle Tank (MBT) is one of its flagship products. Towards the effort of achieving overall weight reduction of Arjun MBT, the road wheel using carbon/epoxy composite material is being developed by CVRDE. Subodh et al. [10] have studied in detail, the strength and performance of carbon/epoxy composite road wheel of AFV for various load cases. The authors have performed a preliminary analysis assuming that there are no internal defects like delamination in the composite wheel and concluded that road wheel can be suitably made using the composite material Fig. 1 Suspension unit of the AFV

Suspension Unit Axle Arm

Road Wheel

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for achieving overall weight reduction and improving the ride performance of the vehicle. Unlike in homogenous metals where crack propagation is rapid under inservice load, the growth of delamination in a laminated composite material is gradual, which means that the composite material with delamination can still withstand load for an appreciable period of time. Hence, it is pertinent to understand the delamination growth behaviour of composite components under external loading for damage tolerant design and reliability assessment.

3 Analysis Methodology Ansys Workbench 17.2 with Ansys Composite Prepost (ACP) module was the tool used in this paper for carrying out finite element analysis. The location of the delamination onset was predicted by employing the lamina failure criterion developed by Puck. It is a local strength criterion for tri-axial stress states at the lamina level. This criterion is built on a physically based phenomenological model and is suitable for long fibre-reinforced polymers. The delamination growth was simulated by VCCT for both static as well as dynamic load conditions. VCCT is a numerical simulation technique which is based on the Griffith criterion for crack propagation. In recent years, the VCCT has been widely used for understanding fracture problems, such as the behaviour of composite materials in the presence of delamination and the behaviour of dissimilar materials in the event of adhesive failure. In these cases, the VCCT is used to compute total SERR of the three fracture modes, viz., mode I (opening), mode II (sliding) and mode III (tearing) [8]. VCCT is based on the assumption that for an extremely small crack opening, the strain energy released is equal to the work required to be done to close the crack [5]. Various delamination sizes are introduced in the critical location, and VCCT is employed to find SERR corresponding to the three fracture modes, viz., mode I, mode II and mode III. The representative fracture modes are shown in Fig. 2.

Fig. 2 a Mode I fracture b Mode II fracture c Mode III fracture

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B-K mixed-mode I and II criterion is a 2D failure criterion which has been used by numerous researchers for studying the growth behaviour of delamination in composite materials. James [5] has proposed a new 3D failure criteria by using B-K criterion, which assumes that the relationship between mode I and mode III toughness is similar to that between mode I and mode II toughness and the relation between mode II and mode III toughness can be obtained by using a linear interpolation. This criterion considers the joint effect of mode I, mode II and mode III loading. The new 3D failure criterion, as given in Eq. (1) [5], is employed for delamination growth prediction for various sizes. Accordingly, the threshold delamination size is obtained. 

GT

GIc + (GIIc - GIc ) GGIIT + (GIIIc - GIc ) GGIIIT



GII + GIII GT

η−1 ≥ 1

(1)

GI , GII and GIII are the strain energy release rates corresponding to mode I, mode II and mode III loadings, respectively. The total strain energy release rate (GT ) is obtained by the addition of GI , GII and GIII . GIc , GIIc and GIIIc are the critical strain energy release rates with respect to mode I, mode II and mode III loadings, respectively. η is the curve fitting parameter, which is found by plotting the fracture test data of the material in mixed-mode diagrams. These data are obtained through mixedmode testing of the material [5]. The mathematical expression on the left-hand side of Eq. (1) is the failure index, which when exceeds 1, the delamination growth takes place leading to the failure of laminated composite structure.

4 Finite Element Analysis 4.1 Material Model In this study, carbon/epoxy (AS4/3501-6) unidirectional lamina was used as the composite material owing to its high specific properties and its suitability in dynamic structural application. The material properties are listed in Table 1. The curve fitting parameter (η) in Eq. (1) for the above material is 1.75 [11]. The co-ordinate system used in this analysis for defining the material properties was 1-2-3, where axis-1 represents the direction in which fibre is laid, axis-2 represents in-plane transverse direction and axis-3 represents the direction of stack-up. The composite road wheel was constructed layer-wise using a reference surface as per the lay-up sequence. The outer most surface of the wheel was used as the reference surface for stacking the lamina. The lay-up sequence used in this study is [0°/+45°/90°/−45°/90°/+45°]2s . 0° means that the fibre is laid along the axis-1 and 90° means that the fibre is laid along the axis-2. Figure 3 shows 0°, +45°, 90° and −45° lamina.

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Table 1 Material Properties [3, 4, 11] Elastic constants E1 (MPa)

E2, E3 (MPa)

G12, G13 (MPa)

G23 (MPa)

ν12, ν13

ν23

147000

10300

700

3100

0.27

0.4

Strength data (MPa)

Critical strain energy release rate (kJ/m2 )

σ1t

σ2t = σ3t

σ1c

σ2c = σ3c

τ12 = τ13

τ23

GIc

GIIc

GIIIc

2280

46

1725

228

76

32

0.0816

0.554

0.886

Fig. 3 Fibre orientation in a 0° b +45° c 90° and d −45° lamina

Fig. 4 3D model showing a the reference surface b meshing of the reference surface

The meshing of reference surface was done using 4 node shell 181 elements. The reference surface and the meshing are shown in Fig. 4. The co-ordinate system used for defining the geometry, load and boundary condition was X-Y-Z co-ordinate system.

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4.2 Delamination Location The road wheel of an AFV is subjected to three severe load conditions, viz., bump, steering and pivot turning during the operation. It experiences maximum stress and deflection during bump condition as compared to the other two conditions. This is the case where the vehicle encounters with a vertical obstacle which causes the maximum vertical displacement of the wheel. The load conditions are shown in Table 2 [10]. In this study, only the load corresponding to the bump condition was considered to predict the location of delamination. These loading conditions are equally shared by two wheels as each suspension station of the vehicle contains two road wheels. In this analysis, only one road wheel was considered. Hence, half of the load value corresponding to bump condition as mentioned in Table 2, i.e. 12.5 t was considered for the analysis. The load was applied on a contact patch and fixed boundary condition was applied on the inner web surface of the wheel as shown in Fig. 5. In reality, the wheel contains mounting holes on the inner web area which are used for mounting it on to the vehicle. In this study, mounting holes were not considered for the ease of analysis as the stresses acting on this area are less [10]. Using Puck failure criterion, it was found that the critical location with respect to delamination onset was the outer rim region of the wheel near the contact patch between 11th and 12th lamina, as shown in Fig. 6. Table 2 Load per station for different load conditions

Load condition

Vertical load (t)

Bump

25

Steering

10

6

6

12

Pivot turning

Fig. 5 Load and boundary condition applied on the reference surface

Side load (t) 0

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Fig. 6 Critical location with respect to delamination onset

4.3 Static Analysis A static analysis employing VCCT numerical simulation technique was done to find out the impact of external load on the delamination growth behaviour. VCCT requires a delamination area, as shown in Fig. 7, to be introduced in the geometry to simulate the delamination growth. On the critical location predicted using Puck criterion, delamination of various sizes, viz., 10 mm width and length of 5, 8, 10, 15 and 20 mm were modelled. Load corresponding to the bump condition was applied on the contact patch along with the boundary condition. The SERR values for the fracture modes I, II and III were obtained for the above delamination sizes, which are plotted in Fig. 8. Fig. 7 Delamination modelled on the critical location

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Fig. 8 Variation of SERR corresponding to a Mode I and Mode II fracture b Mode III fracture with respect to delamination length

Fig. 9 Variation of failure index with respect to delamination length

Using the above SERR values and Eq. (1), the failure index for the wheel with the above delamination sizes was calculated which is as mentioned in Fig. 9. The value of failure index was observed to exceed 1 beyond 8 mm delamination length. The failure index value for 8 mm delamination length was found to be 0.975.

4.4 Dynamic Analysis To evaluate the dynamic load coming on the road wheel, a preliminary multi-body dynamic analysis technique was done using MSC Adams ATV software. The terrain considered for the analysis was Aberdeen Proving Ground (APG) which is used for the qualification of the suspension system. APG consists of staggered bumps with

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Fig. 10 Transient vertical displacement acting on the composite road wheel

each bump having a height of 150 mm and a span of 1800 mm. It is a terrain with the variable frequency spectrum of excitation, which is used for military vehicle dynamics studies as well as for the development of suitable control techniques pertaining to gun stabilization. It can excite the major suspension modes of vibration of the vehicle, viz., bounce, roll and pitch. A time-domain dynamic simulation of the vehicle integrated with composite road wheel running on APG terrain at a speed of 30 km/h was carried out to obtain the dynamic load acting on the wheel. The dynamic load in the form of transient vertical displacement is shown in Fig. 10. A transient structural analysis was carried out by considering road wheel with pre-modelled delamination, axle arm and the suspension unit to examine the impact of dynamic load on delamination growth. The above dynamic load input was applied on the contact patch of the wheel. Road wheel was connected to axle arm through revolute joint. The other end of the axle arm was constrained using revolute joint with respect to the ground to which a torsional stiffness was applied to take into account the effect of suspension unit. The dynamic simulation was done for the delamination sizes, viz., 10 mm width and length of 5, 8, 10, 15 and 20 mm for which the SERRs corresponding to the three fracture modes, namely mode I, mode II and mode III were obtained. Using these SERR values and the critical SERR values as given in Table 1, the failure index for the wheel for the above delamination sizes was calculated using Eq. (1) which is mentioned in Fig. 11. It was clearly evident that the failure index exceeded 1 beyond 5 mm delamination length. The failure index value for 5 mm delamination length was found to be 0.952.

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Fig. 11 Variation of failure index with respect to delamination length

5 Discussion The finite element analysis using the Puck failure criterion shows that the critical location with respect to delamination onset is between 11th and 12th lamina which are the middle layers in the wheel laminate. This is due to the presence of high interlaminar shear stresses in these layers. This shows good agreement with literature [12, 13]. In this study, the Puck delamination index was observed to be 0.59, which was the maximum when compared to the other regions in the wheel. Between 11th and 12th lamina, delamination onset was observed on the outer rim area near the contact patch, where the load was applied. This is because of the stress concentration in this area, which is due to the sharp edge of the wheel geometry. This could be avoided by providing proper fillet, which results in a variable cross-section wheel geometry. Generally, variation in the cross-section is accomplished by stopping the ply laying at appropriate areas during manufacturing. This method is called ply drop-off technique [9]. From the graphs shown in Fig. 8, in case of static analysis, the value of SERR corresponding to fracture modes I, II and III, increases with the delamination length. It shows that the SERR is directly proportional to delamination length which has good agreement with literature [1]. This is because, when delamination is formed, the free surface is created which causes some portion of the material to become unloaded. As a result, a portion of stored strain energy is released. An increase in delamination causes an increase in the SERR because of the increase in the area of the free surface created. The increase in GIIIc was observed to be predominant when compared to that of GIc and GIIc . Similarly, Fig. 7 shows an increase in the failure index value in relation to the delamination length and it exceeded 1 beyond a delamination size of 8 mm × 10 mm which is the threshold size. From the result of dynamic analysis as shown in Fig. 11, it was observed that the failure index increases with respect to the delamination length and the threshold delamination size noticed was 5 mm × 10 mm. This size is less when compared to

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the threshold size obtained from static analysis. This shows that the effect of dynamic load on delamination propagation is more severe as compared to that of static load.

6 Conclusion In this study, the critical delamination location and its growth behaviour in composite road wheel of AFV under severe loading conditions are investigated through finite element analyses. Following conclusions can be drawn from the above analyses: • The onset of delamination was observed on the outer rim region of the wheel near the force patch between 11th and 12th lamina. • SERRs for the three fracture modes, namely modes I, II and III and the failure index were observed to be directly proportional to delamination length. • In comparison with a static load, the dynamic load has a severe effect on delamination propagation in laminated composites. • The threshold delamination size, as a result of dynamic analysis, was found to be 5 mm ×10 mm. The results of this study will be helpful in enhancing the design and manufacturing process of a composite road wheel. In addition, the value of threshold delamination size forms an input to formulate the acceptance criteria for the non-destructive tests like radiography test and ultrasonic test of composite road wheel during the production process. Acknowledgments We express our appreciation to Shri. Sakthivel, M/s. CoreEl Technologies, Bangalore for his valuable guidance during the course of work. We are immensely grateful to Shri. V. Kavivalluvan, Sc ‘E’, Running Gear Division, CVRDE for all kind of support being extended since the inception of the project. Finally, the composite team feels extremely thankful to Director, CVRDE for his suggestions, support and dynamic guidance.

References 1. Atodaria DR, Putatunda SK, Mallick PK (1999) Delamination growth behavior of a fabric reinforced laminated composite under mode i fatigue. J Eng Mater Technol 121(3):381–385 2. Balcioglu HE, Aktas M (2013) An investigation on lateral buckling of laminated composites with delamination. Indian J Eng Mater Sci 20: 367–375 3. Becht GJ, Gillespie JW Jr (1989) Numerical and experimental evaluation of the Mode III interlaminar fracture toughness of composite materials. Polym Compos 10(5):293–304 4. Daniel IM, Ishai O (2013) Engineering mechanics of composite materials. Oxford University Press, India 5. Elisa P (2011) Virtual crack closure technique and finite element method for predicting the delamination growth initiation in composite structures. In: Dr. Pavla Tesinova (Ed) Advances in composite materials-analysis of natural and man-made materials. ISBN: 978-953-307-449-8, pp. 463-480

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6. Grassi Marcello, Zhang Xiang (2003) Finite element analyses of mode I interlaminar delamination in z-fibre reinforced composite laminates. Compos Sci Technol 63(12):1815–1832 7. Khan R, Rans CD, Benedictus R (2009) Effect of stress ratio on delamination growth behavior in unidirectional carbon/epoxy under mode i fatigue loading. In: International conference on composite materials. Edinburh, UK 8. Krueger Ronald (2004) Virtual crack closure technique: History, approach, and applications. Appl Mech Rev 57(2):109–143 9. Mukherjee A, Varughese B (2001) Design guidelines for ply drop-off in laminated composite structures. Compos Part B 32(2001): 153–164 10. Nirala SK, Shankar S, Sathishkumar D, Kavivalluvan K, Sivakumar P (2017) Carbon fiber composites: a solution for light weight dynamic components of AFVs. Def Sci J 67(4): 420–427 11. Reeder JR (2006) 3D Mixed-mode delamination fracture criteria–an experimentalist’s perspective. In: 21st Technical conference of the American society for composites 2006, vol 1 of 3, Dearborn, Michigan, USA, September 17–20 12. Wimmer C, Schuecker HE (2006) Pettermann numerical simulation of delamination onset and growth in laminated composites. The e-J Nondestruct Test 11:1–10 13. Wisnom MR (2012) The role of delamination in failure of fibre-reinforced composites. Phil Trans R Soc A 370:1850–1870 14. Asp LE, Sjögren A, Greenhalgh ES, Delamination growth and thresholds in a carbon/epoxy composite under fatigue loading. J Compos Technol Res 23(2): 55–68 15. Zargar OA (2014) Finite element dynamic analysis of composite structure cracks. Int J Mech Aerosp Ind Mech Manufact Eng 8(5): 1005–1017

Torsional Vibration Analysis of Crank Train and Design of Damper for High Power Diesel Engines Used in AFV N. Venkateswaran, K. Balasubramaniyan, R. Murugesan, and S. Ramesh

Abstract This paper analysed the torsional vibration of a 12-cylinder V-engine used for tracked vehicles and design of a suitable damper to reduce the amplitude of vibration. The entire engine crank train was modelled analytically, and mass properties are calculated using CREO software. The crank train was analysed using ABAQUS software for calculating the natural frequency of the engine. The results of the analysis were compared and validated using Holzer’s method. Suitable damper characteristics were designed using ABAQUS and MATLAB software to reduce the amplitude of vibration. Physical dimensions such as damper inertia, damper size, damping ratio and damping coefficient of the damping medium were determined to achieve the designed damper characteristics. Keywords Torsional vibration · Damper · Mass properties · ABAQUS · Holzer’s method

1 Introduction The torsional natural frequencies (torsional critical speeds) of any rotating system depend on the torsional dynamics of all other rotating systems coupled with it, i.e., it depends on the torsional stiffness and mass moment of inertias of all the rotating components coupled with the rotating system under consideration, which is the crankshaft. It is very hard to detect the torsional vibration without special N. Venkateswaran (B) · R. Murugesan · S. Ramesh Centre for Engineering Analysis and Design, Combat Vehicles R&D Estt., DRDO, Chennai 600054, India e-mail: [email protected] R. Murugesan e-mail: [email protected] S. Ramesh e-mail: [email protected] K. Balasubramaniyan Anna University CEG, Chennai 600025, India © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_36

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equipment. However, the higher amplitude of vibration can be destructive without adequate damping. Therefore, the prediction of natural frequency is very important in order to reduce or minimize the effect of resonance. Since at the resonance, minimum energy is sufficient to cause the large amplitude of the vibration. Torsional vibration analysis is very important in the design of crankshaft of internal combustion engines because it can cause the failure of crankshaft when the frequency of the vibration matches the torsional resonant frequency of the crank. In this article, the necessary calculations were performed using ABAQUS to determine the natural frequency of the system Initially, the equivalent mass moment of inertia and equivalent torsional stiffness were calculated in order to conduct vibration analysis.

2 Literature Survey Wu and Chen [1] have presented the natural frequency calculation using Holzer’s method for undamped and damped systems. Guangming and Zhengfeng [2] have presented the empirical formula for calculating the torsional stiffness of the crankshaft. Buczek [3] has presented the empirical modelling of multi-cylinder engine. Nestorides [4] has explained about the phase vector summation. Ker Wilson [5] has presented the methodology for carrying out the harmonic analysis. Cagri Cevik et al. [6] has presented the approach to find the stiffness calculation of a crank with static finite element methodology. Xingyu et al. [7] has presented the literature on torsional vibration issue published in recent years, which summarizes on the modelling of torsional vibration, corresponding analysis methods, appropriate measures and torsional vibration control pointing out the problems to be solved in the study and some new research directions. ABAQUS Tutorial guide—Release 6.142, August 2014 [8] presented the methodology for carrying out the modal and torsional vibrational analysis.

3 Analytical Model of Crank Train The crankshaft to be analysed for studying the requirement of a damper is shown in Fig. 1. Analytical model is preferred for the vibration analysis of crank train since it involves the variation in natural frequency during the revolution of crank train. Therefore, it is computationally very expensive to conduct the quasi-static modal analysis for such a huge crank train mechanism using the FEA method. Hence, it is desirable to simplify the system into an equivalent torsional rotor system with associated stiffness. Each piston assembly and flywheel was considered as an equivalent rotor. Therefore, the crank train was modelled as thirteen degrees of freedom equivalent spring rotor system. The equivalent piston and connecting rod mass are assumed concentrated at the crank pin centres. Figure 2 shows the division of crank train for equivalent inertia and

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Fig. 1 Solid model of crank train assembly

Fig. 2 Division of crank train for analytical modelling

equivalent stiffness calculation. Each piston assembly has half journal shaft, a crank web, half crank pin, connecting rod and piston assembly. The crankshaft portion between straight lines considered for equivalent inertia calculation and the length of each crankshaft portion between cylinder centers is considered as an equivalent length for the calculation of equivalent stiffness of the spring, which is connecting the equivalent rotors.

4 Natural Frequency and Critical Speeds The equivalent inertia of each rotor was lumped at 13-points corresponding to their position. The points were connected by torsional springs, and the stiffness of each spring was assigned according to the calculated value of equivalent stiffness. Then the modal analysis was performed, and the natural frequency and corresponding mode shapes were extracted. The equation of motion was solved using Holzer’s method in

446 Table 1 Natural frequency

N. Venkateswaran et al. S. No

Natural frequency using ABAQUS (Hz)

Natural frequency by Holzer method (Hz)

1

0

0

2

201.48

201

3

502.42

502

order to verify the ABAQUS results. Table 1 shows the first three natural frequencies. The first natural frequency of crank train (201.48 Hz) in rpm is 12060 rpm and the maximum speed of the engine is 3200 rpm. In any torsional vibration, analysis of an engine system, order number and multiples of order number of the engine is an important factor to be considered. The Order number is defined as the number of disturbing pulses per revolution of the crankshaft. The order number is also defined as a harmonic number divided by two in case of four-stroke engine and order number equal to the harmonic number in case of twostroke engine. A four-stroke engine has power stroke once in every two revolutions, which means the order numbers are ½ and multiples of ½. Critical speed is the speed at which the excitation frequency coincides with the natural frequency of the system. Order number and the critical speed is related by Critical speed = fn/order numbers where fn = natural frequency of nth mode. In order to conduct the forced vibration analysis, the determination of excitation torque is very important and it is explained subsequently.

5 Generation of Torque Versus Theta Curve Simulation of crank train arrangement with a single cylinder is required to generate the torque. It is done using ADAMS/VIEW software. The solid model of engine parts was assembled in order to study the dynamics of crankshaft due to inertia and gas pressure. Single cylinder arrangement is sufficient for analysis since the behaviour of all cylinders is the same. Figure 3 shows the solid model assembly of single cylinder and piston arrangement. After importing the solid model assembly to ADAMS, the pressure vs crank angle data was given as input to find out the torque as output. The output torque was extended for 12 cylinders and it is shown in Fig. 4. Forced vibration analysis was carried out using the Finite Element Analysis (FEA) method. The amplitude of vibration was found to be 0.5°, which is higher than the allowable limit of 0.3°. In order to reduce the amplitude of vibration, a suitable damper has to be designed. The vibratory work at the resonance solely depends on a specific harmonic component which is exciting the natural frequency. Not all harmonic components are critical

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Torque (N-mm)

Fig. 3 Single cylinder and piston assembly

6.E+06 5.E+06 4.E+06 3.E+06 2.E+06 1.E+06 0.E+00 -1.E+06 0 -2.E+06 -3.E+06

100

200

300

400

500

600

700

Crank Angle (Deg) C1

C2

C3

C4

C5

C6

C7

C8

C9

C 10

C 11

C 12

Fig. 4 Excitation torque for 12 cylinder

to the engine. Depending on the firing order and phase between the harmonics of individual cylinder, the torque is getting multiplied or cancelled with each other. Hence, it is required to carry out the signal processing to find out the harmonics of excitation torque.

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6 Fast Fourier Transformation In order to carry out the forced vibration analysis, the harmonic components of excitation torque corresponding to various orders are required. A harmonic analysis converts a non-harmonic tangential pressure curve of an engine cylinder into harmonic components of tangential pressure by Fourier analysis from which the harmonic components of excitation torque corresponding to various orders can be determined. Fast Fourier Transformation is carried out using ADAMS/POST PROCESSING software. Table 2 shows the harmonic component of torque, and Fig. 5 shows the graphical representation of harmonic torque with respect to the various orders. Table 2 Harmonic of torque Harmonic number

Order number

Harmonic excitation torque (N-m)

Harmonic number

Order number

Harmonic excitation torque (N-m)

1

0.5

1841

11

5.5

288

2

1

2119

12

6

247

3

1.5

1290

13

6.5

213

4

2

1056

14

7

185

5

2.5

325

15

7.5

160

6

3

705

16

8

138

7

3.5

500

17

8.5

123

8

4

465

18

9

105

9

4.5

400

19

9.5

94

10

5

337

20

10

82

Harmonic torque 2500

Torque(N-m)

2000 1500 1000 500 0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Order No Fig. 5 Graphical representation of harmonic torque with respect to the various orders

7.5

8

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Fig. 6 Excitation torque and harmonics of excitation torque vs crank angle

Figure 6 shows the graphs of excitation torque and some of its harmonic components (up to 3rd harmonic) of excitation torque vs crank angle.

7 Critical Order Fourier analysis finds out the harmonic components of excitation torque corresponding to various orders. Not all the orders are critical to engine, since some of the torque vectors will be cancelled and some of the torque vectors will be added. These orders in which the vectorial torque addition is critical to the engine, the excitation source for torsional vibration is mainly due to harmonics of gas pressure and inertia

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torque. Hence, the phasing between the harmonic torque components of individual cylinder and firing order is very important. The phase angle of the torque vectors depends on the phase of the harmonics relative to the firing positions of their respective cranks. Phase vector sum depends on two factors, namely firing sequence and angles between the various cranks. Figure 7 shows the phasing diagrams for various orders. It is found that both 4.5 and 12th order are very critical to the engine. Then the resultant torque to be calculated. Figure 8 shows the graphical representation of resultant torque with respect to various order. It is understood that the resultant torque obtained for the 12th order is negligible for this engine configuration.

Fig. 7 Phasing diagrams for various orders

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Resultant Torque

3500 3000

Torque(N-m)

2500 2000 1500 1000 500 0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Order No

Fig. 8 Graphical representation of resultant torque with respect to various order

8 Design of Damper It is essential to control the amplitude of vibration in the crankshaft in order to ensure the safe operation of the engine near the critical speeds of the engine. Theoretically, the vibration amplitude reaches infinity and practically to a very high value in resonance condition. This may lead to the failure of the crankshaft. Dampers are the devices used to control the effects of the torsional vibration. Dampers of this selected type consist of an annular seismic mass enclosed in a casing. The peripheral and lateral gaps between these two members are filled with a viscous fluid. The damper is untuned since there is no elastic member between seismic mass and the casing; only the viscous torque transmitted by the fluid acts upon the seismic mass. Figure 9 shows the 3-rotor system consisting of Flywheel, engine and damper. The damper consists of free mass, which is keyed to the shaft. Normally, the disc rotates at the shaft speed owning to the viscous drag of oil between the disc and the case. However, if the shaft vibrates torsionally, the viscous action of the oil between the disc and casing gives a damping action. In order to calculate the optimum-damping ratio, the crank train was modelled as a 3 DOF spring rotor system. The damper is to be designed for the more critical order, which is nothing but 4.5 order. Fig. 9 Three rotor system consisting of Flywheel, Engine and Damper

452

N. Venkateswaran et al. 0.01

Torsional Amplitude (rad)

c = zero 0.008

c = infinite ζ =0.2

0.006

ζ =0.3 ζ =0.4

0.004

ζ =0.5 ζ =0.6

0.002

0 160

170

180

190

200

210

220

230

Frequency (Hz) Fig. 10 Optimum damper characteristics

Table 3 Technical parameters of damper

Damper type

Torsional untuned viscous damper

Damper stub shaft stiffness

4.177 + 9 N-mm/rad

Inertia of damper

Jd = 552 N mm s2

Optimum-damping coefficient

430503.75 N-mm-s/rad

To design a suitable damper, the forced vibration analysis was carried out for different damping ratios. Figure 10 shows the optimum damper characteristics. In this, the intersection point of zero and infinite damping curve is called as optimum point. For all value of the damping coefficient, the curve will pass through this point. The damping coefficient will be called as optimum when maxima of the Frequency response curve lies on the optimum point. In order to find optimum-damping coefficient for viscous damper, the system was modelled as a 4 DOF spring rotor system. Table 3 shows the optimum damper characteristics viz., Inertia, stiffness and the damping coefficient.

9 Conclusion In the present study, the torsional vibration analysis was carried out for a 12-cylinder V-engine of an armoured fighting vehicle (AFV). The critical speeds were estimated by modal analysis using ABAQUS software. From the analysis, it has been concluded that there are many critical speeds within the operating speed range of the engine. The harmonic analysis of excitation torque was carried out, and harmonic components of torque were found. From the forced vibration analysis, it is found out that the

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amplitude of vibrations in the crank train exceeds the permissible limits within the operating speed range. In order to control the effects of critical orders and to minimize the amplitude of vibration, a suitable damper has been selected and introduced in the crank train. Subsequently damper characteristics were designed. Acknowledgments The authors are grateful to V. Balamurugan, Outstanding Scientist, Director, CVRDE for his guidance, support and encouragement. The authors are also grateful to the contributions made by divisions of ENGINE and CEAD of CVRDE.

References 1. Wu JS, Chen WH (1982) Computer method for torsional vibration of propulsive shafting system of marin engine with or without damper. J Ship Res 26:176–189 2. Guangming Z, Zhengfeng J (2009) Study on torsional stiffness of engine crankshaft. International forum on computer science-technology and applications, Chongqing, China, pp 431–435 (25th– 27th December 2009) 3. Buczek K (2008) Torsional vibration analysis of crankshaft in heavy duty six cylinder inline engine. Czasopasimo Techniczne 8:193–207 4. Nestorides (1958) A handbook on torsional vibration. B.I.C.E.R.A Cambridge University Press, 663 pp. ISBN-13: 978-0521203524 5. Ker Wilson W (1963) Practical solutions of torsional vibration problems, vol 2. Chapman & Hall Ltd., pp 194–291 6. Cagri Cevik M, Pischinger S, Rebbert M, Maassen F (1918) Borderline design of crankshafts based on hybrid simulation technology. In: SAE international 7. Xingyu L, Gequn S, Lihui D, Bin W, Kang Y (2011) Progress and recent trends in the torsional vibration of internal combustion engine. Adv Vib Anal Res 04:245–272 8. ABAQUS SYS Inc (2014) ABAQUS Tutorial guide—Release 6.142

Fluid Structure Interaction

Dynamic Behavior of Swaged Plates in Water-Immersed Condition G. Verma, S. Sengupta, S. Mammen, and S. Bhattacharya

Abstract Fuel assemblies with swaged thin fuel plates are often used in high flux research reactors. In swaging operation, the fuel plates are inserted into grooves created all along the length of the plates and then the swage is rolled such that the movement of the plates is restricted. This results in a CFCF-type condition for each fuel plate. It is important to note that the boundary condition developed using swage joints is different in comparison to that of a welding joint where all the six degrees of freedom get fixed. The present work deals with the investigation of vibration characteristics of swaged plates in air as well as in water-immersed conditions for different water immersion ratios using a commercial code. This involves a realistic representation of a swage joint using a finite element model further employing coupling techniques to achieve fluid–plate interaction.

1 Introduction A swage joint is a technique to join thin plates when methods such as welding cannot be applied either due to space limitation or because of functional requirements. This type of joining technique is widely utilized in plate-type fuel assemblies keeping the core compact resulting in high neutron flux. In fuel assemblies, the thin fuel plates (~1.2–2 mm) are swaged to the comparably thick side plates (~4 mm) by inserting the fuel plates into the corresponding slots of the side plates and roll swaging it. A swage joint almost results in a perfectly clamped condition (depending upon the quality of the swaging operation). A CFCF-type condition is established for the fuel plates where all the translational movements are prevented and all the rotational movements (except one along the axis parallel to the swage joint) are prevented. Caresta and Wassink [1–3] suggested that for small rotations, a perfect swage joint could be modeled using a torsional spring whose stiffness is proportional to the quality of swaging. Accordingly, if the quality of the swage joint were high, the joint G. Verma (B) · S. Sengupta · S. Mammen · S. Bhattacharya Research Reactor Design and Projects Division, Bhabha Atomic Research Centre, Mumbai 400085, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_37

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stiffness would be high tending toward a perfect clamp condition and vice versa for the lower quality swage joint. The present work aims to investigate the vibration characteristics of swaged plates in air and in water-immersed condition for different water immersion ratios (ratio of elevation of water to the elevation of the plate when measured from the bottom) using a commercial code. This work involves a realistic representation of the swage joint using finite element techniques and further employing coupling techniques to achieve fluid–plate interaction. In order to validate the finite element model of the swaged plate, benchmarking of its free vibration behavior is done with the works of Caresta and Wassink [2]. Further, fluid–plate interaction coupled dynamic characteristic is benchmarked with the works of Jeong et al. [4]. Many researchers had developed analytical and experimental methods to determine the dynamic vibration characteristics for coupled fluid–plate interaction problems [5–7]. Nevertheless, all these works involved clamping boundary condition with all the degrees of freedom fixed. Present work deals with coupled fluid–plate interaction problems where the plate possesses swaged boundary condition along the edges.

2 Benchmarking 2.1 Free Vibration Characteristics of CFCF-Type Swaged Plate In an effort to validate the code for the modeling procedure and verify the results, the variation of the frequency parameter with the spring stiffness for a swaged plate having an aspect ratio of 9.3:1 is analyzed as shown in Fig. 1. The results are compared Fig. 1 Variation of the frequency parameter with the spring stiffness

1.0

0.9

MODE

1st

4th

Present work Caresta et al.

Normalised n

0.8

0.7

0.6

0.5

0.4 -2 10

10-1

100

101

102

103

104

Distributed Spring Stiffness (N/rad)

105

106

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459

with that of Caresta and Wassink [2] for the same geometry of the plate. It is observed from the figure that for higher spring stiffness, the frequency parameter λn approaches unity, which is the fundamental frequency of the fuel plate for the perfectly clamped case. λn = ωn d 2 (ρh/J )1/2

(1)

where ωn is the circular natural frequency, d is the width of the plate of thickness h, and ρ is the density of the material. J is the flexural rigidity given by Eq. (2) J = Eh 3 /12(1 − υ 2 )

(2)

where E and υ are Young’s modulus and Poisson’s ratio. The parameter “normalized λn ” is the ratio of λn for a given stiffness to the perfectly clamped condition.

2.2 Fluid–Plate-Coupled Free Vibration Characteristics of Clamped Plates In order to check the validity of the code for the coupling technique, the following problem of Jeong et al. [4] is selected. Two identical rectangular Aluminum plates with dimensions 320 mm × 240 mm × 2 mm are kept at a distance of 40 mm. The gap between the two plates is completely filled with water up to the elevation of the plates and is bounded. The fluid movement along the rigid walls is allowed only in the direction normal to the plates. All the degrees of freedom of the plate edges were constrained realizing a completely clamped condition. The results obtained are shown in Table 1. It is observed that the In-Phase natural frequencies of the present work are in agreement with that of Jeong et al. within 2% variation. However, the error range for the Out-of-Phase frequencies is of the order of 20%. Jeong et al. reasoned that the approximation of the Out-of-Phase modes with the dry beam functions is Table 1 Comparison of the natural frequencies for fluid–plate-coupled vibration Mode

In-phase

Out-of-phase

n

m

Present work (Hz)

Jeong et al. (Hz)

Variation (%)

Present work (Hz)

Jeong et al. (Hz)

Variation (%)

0

0

113.44

114.1

0.58

–

–

–

1

192.67

194.5

0.94

72.90

61.2

2

327.47

330.9

1.03

176.73

194.6

9.18

0

273.50

275.4

0.69

109.29

108.4

0.82

1

348.90

353.9

1.41

180.19

176.6

2.03

2

479.89

488.5

1.97

304.50

278.5

9.33

1

19.11

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Fig. 2 a In-phase modes (i) 113.44 Hz, n = 0, m = 0; (ii) 192.67 Hz, n = 0, m = 1, b Out-of-phase modes (i) 109.29 Hz, n = 1, m = 0; (ii) 180.19 Hz, n = 1, m = 1

inappropriate, as the approximation method was not able to fulfill the fluid-volume conservation for numerous modes. Figure 2a, b shows some of the modes of the fluid–plate-coupled system.

3 Finite Element Modeling of the Swaged Plate Coupled with Fluid The finite element model of the swaged fuel plate (Fig. 3) is designed as per the geometry with four-noded shell elements. The edges along the width of the plate are kept free, whereas the ones along the length of the plate are provided with the specified boundary condition. The three translational and two rotational degrees of freedom of the edge nodes are fixed, and a torsional spring with variable torsional stiffness is attached to account for the rotation along the axis of the swaged edge. In order to simulate either a simple support or a perfect clamp boundary condition, the value of the spring stiffness is varied accordingly (Caresta and Wassink [2]). In case of immersed plates, the fluid in the system is confined, coupled, and modeled using eight-noded fluid elements having three degrees of freedom at each node. The fluid movement is restricted normal to the plates and to the rigid walls but allowed along the walls and plates (Jeong et al. [4]). Eigenvalues and Eigenvectors for the coupled fluid–plate system are obtained through the Block Lanczos mode extraction method.

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461

y

z

x

Fig. 3 Finite element model of the swaged single fuel plate

4 Results and Discussion In order to investigate the vibration characteristics of swaged plates in air and in water-immersed conditions under variable stiffness, the following case studies are performed for the Swaged Single Plate System with different immersion (liquid depth) ratios.

4.1 Swaged Single Plate System in Air In this numerical simulation, a modal analysis of a single swaged plate is performed and its results are documented. Figure 4 describes the variation of the normalized frequency parameter of the first and the fourth mode with respect to the increasing distributed torsional spring stiffness. It can be seen from Fig. 4 that at lower spring stiffness, the edge conditions tend toward a simply supported condition. As the stiffness increases, the conditions tend towards a clamped condition. It is also important to note that with an increase in the mode of vibration, the value of the normalized frequency parameter for smaller stiffness values increases. Figure 5a–d shows the four mode shapes of the system in air. Here, it is observed that because of the edge condition (CFCF-type), nodal lines only in one direction (i.e., in x-direction) are observed. Hence, for all the mode shapes, n = 0 (i.e., no nodal line in y-direction is observed).

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0.9

Normalised n

0.8

0.7

0.6 1st Mode

0.5

0.4 100

4th Mode

101

102

103

104

105

106

Distributed Spring Stiffness (N/rad)

Fig. 4 Variation of normalized frequency parameter with distributed spring stiffness of swaged single plate system in air

Fig. 5 Mode shapes of swaged single plate system in air a first; n = 0, m = 0, b second; n = 0, m = 1 c third; n = 0, m = 2 d fourth; n = 0, m = 3

4.2 Swaged Single Plate System in 25% Immersion Ratio In this case, modal analysis is performed for the swaged plate which is immersed in water having an immersion ratio of 25%. Figure 6 describes the variation of the normalized frequency parameter with respect to the spring stiffness for the first and fourth mode. From the simulation, it is observed that the first three modes are affected by the hydrodynamic mass of the coupled water resulting in the lower modal frequency compared to the dry modal analysis of case 4.1. This phenomenon

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1.0

Normalised n

0.9

0.8

0.7

0.6 1st Mode

0.5

0.4

4th Mode

100

101

102

103

104

105

106

Distributed Spring Stiffness (N/rad)

Fig. 6 Variation of normalized frequency parameter with distributed spring stiffness of swaged single plate system in 25% immersion ratio

is justified by Fig. 7a through c where it is seen that only 25% of the plate region (wetted with water) is excited. From fourth mode onwards, the effect on the remaining 75% of the plate is observed. Figure 7d displays that effect. Also, it can be seen from Fig. 6 that there is an increase in the normalized frequency for the fourth mode when compared to the case where the swaged plate was oscillating in air.

Fig. 7 Mode shapes of swaged single plate system in 25% immersion ratio a first; n = 0, m = 0 b second; n = 0, m = 1 c third; n = 0, m = 2 d fourth; n = 0, m = 0

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4.3 Swaged Single Plate System in 50% Immersion Ratio Now, in this case, the water immersion ratio is increased from 25 to 50%, and the variation of the normalized frequency is plotted for the first and seventh modes in Fig. 8. Similar to the case of 4.2, in this case, the first six mode excitations of the plate are localized to 50% of the wetted plate length. However, the first modal frequency value observed for the present case is found to be lower than case 4.2. The seventh mode shows complete plate excitation, which is shown in Fig. 9a through d. Additionally, it is also observed that the value of the λn is higher compared to the 1.0

Normalised n

0.9

0.8

0.7

0.6 1st Mode

0.5

7th Mode

0.4 100

101

102

103

104

105

106

Distributed Spring Stiffness (N/rad)

Fig. 8 Variation of normalized frequency parameter with distributed spring stiffness of swaged single plate system in 50% immersion ratio

Fig. 9 Mode shapes of swaged single plate system in 50% immersion ratio a n = 0, m = 0 b n = 0, m = 1 c n = 0, m = 5 d n = 0, m = 6

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1.0

Normalised n

0.9

0.8

0.7

0.6 1st Mode

0.5

10th Mode

0.4 100

101

102

103

104

105

106

Distributed Spring Stiffness (N/rad)

Fig. 10 Variation of normalized frequency parameter with distributed spring stiffness of swaged single plate system in 75% immersion ratio

fourth mode value in case 4.2 suggesting that with an increase in the mode number, the value of the λn also increases for increasing water immersion ratio.

4.4 Swaged Single Plate System in 75% Immersion Ratio For the increase in the water immersion ratio to 75%, the normalized frequency variation with respect to the spring stiffness is plotted in Fig. 10 for the first and tenth modes. As per the trend followed in the previous cases, in this case also, the first nine mode shapes of the plate are present within the 75% of the wetted plate region and participated in the excitation. Also, the first natural frequency of the plate for the present case was observed to be lower than the previous case of 4.3. From tenth mode onwards, complete plate excitation is visible and is shown in Fig. 11a through d.

4.5 Swaged Single Plate System in 100% Immersion Ratio In this case, the complete plate is immersed in water with an immersion ratio of 100%. The variation of the normalized frequency is plotted in Fig. 12 for the first and fourth modes. In this case, partial participation of the plate in the modal analysis is not observed as the whole plate is immersed in water (full plate length coupled with the hydrodynamic mass). Also, the first modal frequency value observed was lower than the previous case of 4.4. From Fig. 13, it can be noted that the normalized frequency parameter for the fourth mode is lower than the value observed in Fig. 6

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Fig. 11 Mode shapes of swaged single plate system in 75% immersion ratio a first; n = 0, m = 0 b second; n = 0, m = 1 c ninth; n = 0, m = 8 d tenth; n = 0, m = 9 1.0

Normalised n

0.9

0.8

0.7

0.6 1st Mode

0.5

0.4 0 10

4th Mode

101

102

103

104

105

106

Distributed Spring Stiffness (N/rad)

Fig. 12 Variation of normalized frequency parameter with distributed spring stiffness of swaged single plate system in 100% immersion ratio

Fig. 13 Mode shapes of swaged single plate system in 100% immersion ratio a first; n = 0, m = 0 b second; n = 0, m = 1 c third; n = 0, m = 2 d fourth; n = 0, m = 3

Dynamic Behavior of Swaged Plates in Water-Immersed Condition

467

for lower stiffness values. The first four mode shapes of the swaged plate system are shown in Fig. 13a through d. From Fig. 13a, it can be compared that the first three mode shapes of the swaged plate oscillating in water are completely distinguishable from that observed in Fig. 5a–c.

4.6 Comparison of Normalized Frequency Parameter for All Swaged Single Plate Systems Immersion Conditions Figure 14 compares the normalized frequency parameter with distributed spring stiffness of Swaged Single Plate System for all immersion ratios. The first mode of the Swaged Single Plate System in air has a higher normalized frequency parameter value in comparison to the first mode of the Swaged Single Plate with a 100% immersion ratio for a given spring stiffness. In Fig. 14, it is further observed that as the immersion ratio increases, i.e., for immersion ratios 25, 50, and 75%, the first mode of vibration in which the maximum mass participation occurs (full plate vibrates) also increases as 4th mode, 7th mode, and 10th mode, respectively. In addition, the normalized frequency parameter values for each of these modes increase with spring stiffness in order of the increasing mode values. The normalized frequency parameter values for water excited modes (i.e., the first mode for each of the immersion ratios) decrease with increasing immersion ratio for a given spring stiffness. 1.0

0.9

Normalised n

0.8

0.7 1st Mode-SSPS in Air 1st Mode-SSPS in 25% immersion 4th Mode-SSPS in 25% immersion

0.6

1st Mode-SSPS in 50% immersion 7th Mode-SSPS in 50% immersion 1st Mode-SSPS in 75% immersion

0.5

10th Mode-SSPS in 75% immersion 1st Mode-SSPS in 100% immersion

0.4 0 10

SSPS: Swaged Single Plate System 1

10

2

10

3

10

4

10

5

10

6

10

Distributed Spring Stiffness (N/rad)

Fig. 14 Comparison of normalized frequency parameter with distributed spring stiffness of swaged single plate system for all immersion ratios

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5 Conclusion In the present work, a numerical method to model a swaged plate with variable stiffness is established, and subsequently, a technique of coupling the swaged plate with liquid immersed condition was developed. Various case studies investigating the dynamic characteristics of the swaged plate under different liquid immersion ratios in terms of a normalized frequency parameter are performed. It is observed that with an increase in the immersion ratio, the fundamental natural frequency of the plate decreases. Also, increase in water immersion ratio, increases the number of plate modes directly affected with coupled hydrodynamic mass. Further, it is observed that with an increase in the mode number, the value of the normalized frequency parameter also increases for increasing water immersion ratio. A distinguishable change in the first three mode shapes of the Swaged Single Plate System is also observed in the case of dry and wet modal analyses.

References 1. Caresta M, Wassink D (2010) Vibrational characteristics of roll swage jointed plates. Acoustics Australia 38(2):82–86 2. Caresta M, Wassink D (2012) Dynamic characterization and longitudinal strength of swaged joints. Appl Acoust 73(5):484–490. https://doi.org/10.1016/j.apacoust.2011.12.002 3. Caresta M, Wassink D (2013) Structural analysis of plate-type fuel assemblies and development of a non-destructive method to assess their integrity. Nucl Eng Des 262:209–218. https://doi. org/10.1016/j.nucengdes.2013.05.003 4. Jeong KH, Yoo GH, Lee SC (2004) Hydro-elastic vibration of two identical rectangular plates. J Sound Vib 272:539–555. https://doi.org/10.1016/S0022-460X(03)00383-3 5. Jeong KH, Kim JW (2009) Hydro-elastic vibration analysis of two flexible rectangular plates partially coupled with fluid. Nucl Eng Technol 41(3):335–346. https://doi.org/10.5516/NET. 2009.41.3.335 6. Jeong KH, Kang HS (2013) Free vibration of multiple rectangular plates coupled with a liquid. Int J Mech Sci 74:161–172. https://doi.org/10.1016/j.ijmecsci.2013.05.011 7. Verma G, Eswaran M, Sengupta S, Reddy GR, Mammen S (2017) Dynamic characteristics of immersed plate-type fuel assemblies under seismic excitation. Nucl Eng Des 314:11–28. https:// doi.org/10.1016/j.nucengdes.2017.01.005

Love Wave Propagation in an Anisotropic Viscoelastic Layer Over an Initially Stressed Inhomogeneous Half-Space Bishwanath Prasad, Prakash Chandra Pal, and Santimoy Kundu

Abstract Love-type surface wave in a model comprising of an orthotropic viscoelastic layer supported by a half-space is investigated. The layer and half-space both are heterogeneous, and the half-space is in the state of initial stress. Employing relevant boundary conditions, frequency equation is derived, based on which numerical computations are carried out to analyze the impact of different parameters on Love wave speed. It is discovered that dissipation function and heterogeneity of the layer and initial stress and heterogeneity of the half-space has a substantial effect on phase velocity. Keywords Love wave · Inhomogeneity · Orthotropic · Viscoelastic · Initial stress

1 Introduction The study of seismic waves gives us the most accurate results and information regarding Earth’s interior. It is a well-known fact that the Earth’s crust is not uniform; different sorts of heterogeneity and anisotropy exist within it. The presence of inhomogeneity and anisotropy drastically affects the seismic wave propagation. Detailed information related to seismic wave propagation in different types of medium is available in Ewing et al. [1], Love [2], Bullen [3], Gubbins [4], etc. The intrinsic viscosity of the Earth layer along with different parameters, such as heterogeneity and anisotropy, has a crucial effect on earthquake waves. Therefore, while investigating seismic waves, one should proceed in such a way that the studies interpret these factors simultaneously. Investigations are made by several authors (Cooper [5]; Shaw and Bugl [6]; Buchen [7]; Schoenberg [8]; Borcherdt and B. Prasad (B) · P. C. Pal · S. Kundu Department of Applied Mathematics, IIT (ISM) Dhanbad, 826004 Dhanbad, Jharkhand, India e-mail: [email protected] P. C. Pal e-mail: [email protected] S. Kundu e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_38

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ˇ Wennerberg [9]; Carcione [10]; Cervený [11]) to analyze the effect of viscoelasticity of the Earth on the propagation of earthquake-generated waves. Chattopadhyay et al. [12] investigated shear waves in a model comprising of a layer over a half-space where both the mediums were viscoelastic and the boundary separating them was irregular. Kakar [13] discussed Love waves in a layer of Voigt-viscoelastic type supported by a gravitating half-space. Pandit et al. [14] investigated Love waves in an anisotropic viscoelastic layer over a half-space. The layer considered by him was prestressed having the orthotropic type of anisotropy, whereas the half-space was porous in nature saturated by fluids. Prasad et al. [15] discussed the scattering of SH waves through viscoelastic and reinforced media where the boundary separating the two mediums was sinusoidal in nature. Stresses, which are present in a body even when there are no external forces, are known by the name of initial stress. The presence of initial stresses in solid bodies can have a significant effect on their subsequent response to applied loads. Therefore, while dealing with many problems of science and engineering, it is imperative to consider their effects. The fact that our Earth is highly initially stressed was first predicted by Love [2]. Atmospheric pressure, creep, gravity variations, temperaturedifference are different factors which cause initial stress to be present within the Earth. These stresses profoundly affect the characteristic of earthquake-generated waves. Biot [16] showed that elastic wave propagation in the presence of initial stress was different and the classical linear-elasticity theory was unable to explain that. Biot [17] described the theory of incremental deformation and later many researchers applied this theory to deal with surface waves in prestressed bodies. Du et al. [18] researched Love waves in the prestressed magneto-electro-elastic medium. Abd-Alla et al. [19] illustrated shear waves in an inhomogeneous anisotropic incompressible medium under the influence of gravity. Chattraj et al. [20] conducted investigations regarding Love waves in an irregular shaped anisotropic layer interposed between two uniform half-spaces. The interposed layer considered by him was a prestressed porous medium saturated with fluid. Gupta et al. [21] examined Love wave in a heterogeneous slab supported by a prestressed heterogeneous half-space without considering the viscoelasticity. Kakar [22] explored the propagation of Love wave in a uniform slab separating two different half-spaces. The half-space over the uniform slab was taken as orthotropic, whereas the half-space below the slab was initially stressed and heterogeneous. In a recent study, Kundu et al. [23] examined shear waves in an inhomogeneous viscoelastic layer beneath a porous layer and overlying a prestressed uniform half-space. Keeping the above facts and the inhomogeneous nature of Earth in mind, in the present problem, an attempt is undertaken to investigate Love-type surface wave in an inhomogeneous viscoelastic orthotropic layer supported by a prestressed inhomogeneous half-space. Employing relevant boundary conditions, frequency equation is derived, and detailed numerical simulations are presented to interpret the effect of different parameters, e.g., inhomogeneity, initial stress on Love wave speed. The analytical derivations in this work may be useful for seismologists, structural engineers, and those working on geophysical prospecting.

Love Wave Propagation in an Anisotropic Viscoelastic Layer …

471

Fig. 1 Problem model

2 Problem Formulation An inhomogeneous orthotropic viscoelastic layer of uniform thickness d resting over a prestressed heterogeneous half-space is considered (Fig. 1). Let (u 1 , v1 , w1 ) and (u 2 , v2 , w2 ) be the displacement vectors in the half-space and layer, respectively. In the two-dimensional x − z plane and for Love waves, we have u i = 0, vi ≡ vi (x, z, t), wi = 0 (i = 1, 2)

(1)

3 Dynamics of the Half-Space The variable rigidity (μ) and density (ρ1 ) of the lower medium are defined as μ = μ (1 + lz), ρ1 = ρ1 (1 + mz),

(2)

where l and m are constants having a dimension that is inverse of length. Let P be the initial compressive stress acting along x-axis. Neglecting body forces, the field equation for the half-space is (Biot [17]).   ∂σ yz ∂σ yx P ∂ 2 v1 ∂ 2 v1 = ρ1 2 , + − 2 ∂x ∂z 2 ∂x ∂t

(3)

where σi j are components of incremental stress. Using the relations σ yx = 2μe yx

    ∂u 1 ∂w1 ∂v1 ∂v1 + , σ yz = 2μe yz = μ + , =μ ∂x ∂y ∂z ∂y

and Eqs. (1) and (2), Eq. (3) becomes

(4)

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 2 P ∂ v1 ρ1 (1 + mz) ∂ 2 v1 ∂ 2 v1 l ∂v1 1− = + + . 2μ (1 + lz) ∂ x 2 ∂z 2 1 + lz ∂z μ (1 + lz) ∂t 2

(5)

For solving Eq. (5), we put v1 = V1 (z) exp{ik(x − ct)},

(6)

where V1 , k and c, respectively, are amplitude, phase velocity, and wave number. Using Eq. (6) in Eq. (5) gives     l d V1 P d 2 V1 ρ1 (1 + mz) 2 k 2 V1 = 0 + − 1 − + c dz 2 1 + lz dz μ (1 + lz) 2μ (1 + lz)

(7)

 1 2 Introducing V1 = ψ(z) (1 + lz) / into Eq. (7) gives     P c2 (1 + mz) l2 d 2 ψ(z) 2 1− − 2 ψ(z) = 0, + −k dz 2 2μ (1 + lz) c1 (1 + lz) 4(1 + lz)2 (8)  where c12 = μ ρ1 .

Substituting S = 1 − we get

P 2μ (1+lz)

−

c2 m c12 l

1/ 2

,η=

2Sk(1+lz) , l

and ω = kc in Eq. (8),

  1 R 1 d 2ψ + ψ = 0, + − dη2 2η 4η2 4

(9)

where R = ωc2(l−m) . 2 1l k S Equation (9) is the well-known Whittaker’s equation. Thus 2

ψ(η) = A1 W R / 2,0 (η) + B1 W−R / 2,0 (η)

(10)

where A1 , B1 are arbitrary constants and W R / 2,0 , W−R / 2,0 are Whittaker’s functions. Keeping in mind that V1 (z) → 0 as z → ∞, i.e., ψ(η) → 0 as η → ∞, the appropriate solution is ψ(η) = A1 W R/2,0 (η). Hence v1 (x, z, t) = V1 (z) exp{ik(x − ct)} =

A1 W R / 2,0 (η) exp{ik(x − ct)}. (1 + lz)1/ 2

Expansion of Whittaker’s function to the second term gives

(11)

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 v1 (x, z, t) = A1 exp

−Sk(1 + lz) l



473

   2Sk 1/ 2 Sk(1 + lz) exp{ik(x − ct)}. 1 + (1 − R) l l

(12)

4 Dynamics of the Surficial Layer Field equation of upper layer is ∂τ yz ∂τx y ∂ 2 v2 + = ρ2 2 , ∂x ∂z ∂t

(13)

where τx y = (C66 + iωM66 )

∂v2 ∂v2 , τ yz = (C44 + iωM44 ) . ∂x ∂z

(14)

C44 and C66 are elastic coefficients, M44 and M66 are viscoelastic coefficients, v2 is the y component of displacement, ρ2 is the density and ω is the angular frequency. The heterogeneity in the layer is defined as Ci j = Ci j eaz , Mi j = Mi j eaz , ρ2 = ρ2 eaz .

(15)

Using Eqs. (14) and (15) in Eq. (13), we get   2 ∂v2     ∂ 2 v2 ∂ 2 v2  ∂ v2  = ρ2 2 . + a + C + iωM C66 + iωM66 44 44 2 2 ∂x ∂z ∂z ∂t

(16)

Considering the harmonic solution of Eq. (16) as v2 = V2 (z) exp{i(kx − ωt)}, we get    2  ∂ 2 V2 ∂ V2 2 ρ2 c − C 66 + iωM66 +k +a = 0.   ∂z 2 ∂z C44 + iωM44

(17)

Solution of Eq. (17) is given by V2 = e−az / 2 (A2 cos(ksz) + B2 sin(ksz)),

(18)

where A2 and B2 are arbitrary constants, and

s=

 21    + iωM66 ρ2 c2 − C66 a2 − .   4k 2 C44 + iωM44

(19)

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Hence v2 = e−az / 2 (A2 cos(ksz) + B2 sin(ksz)) exp{i(kx − ωt)}

(20)

5 Boundary Conditions and Dispersion Relation To determine the dispersion relation, the following boundary conditions will be used (i) Top surface of the surficial layer is stress less, i.e., τ yz = 0 at z = −d (ii) Displacement and stress continuousness at the common boundary, i.e., v1 = v2 and σ yz = τ yz at z = 0 Using Eqs. (12) and (20) in these conditions gives three simultaneous equations as follows: a   a cos(kds) − ks sin(kds) A2 − sin(kds) + ks cos(kds) B2 = 0, 2 2 μ k



2k l



 21

2S0 k l

    a   S0 k  A1 − C44 − A2 + ks B2 = 0, exp − + iωM44 l 2  21     S0 k S0 k 1 + (1 − R0 ) exp − A1 − A2 = 0, l l

where

  3 −1/ 2 1 l −3 2 S / − 1) − ζ S0 + ζ R0 S0 − 4 4 k 0    k 3/ 2 1 −1 1 2 ζ S0 + S0 S0 (1 − R0 ) + S0 / , + 2 l

3 2 =S0 / (R0



c2 m S0 = 1 − ζ − 2 c1 l

1/ 2

, R0 =

ω2 (l − m) P and ζ = . 2μ c12 l 2 k S0

Elimination of arbitrary constants from the above equations gives the dispersion relation for Love waves as follows: μ s φ(k, c) = tan kds −    1/ 2 a 2    S0 2 1 + (1 − R ) S0 k + C44 + iωM44 + s 0 l 2 4k

a  μ 2k

= 0. (21)

The above dispersion equation is complex due to dissipation.

Love Wave Propagation in an Anisotropic Viscoelastic Layer …

475

k being a complex quantity is expressible as k = k1 + ik2 = k1 (1 + iδ),  where k1 and k2 are real quantities and δ = k2 k1 (1) is the attenuation coefficient. Thus, the phase velocity can be expressed as c = ω k1 . Splitting the obtained dispersion relation into real and imaginary part gives [φ(k, c)] = 0, [φ(k, c)] = 0.

(22)

The first relation in Eq. (22) is the dispersion equation, which gives the dispersion curves, whereas the second absorption  one  is the  relation yielding the damping curves.   C66 = Q −1 say Q −1  1 is the dissipation function, The quantity ωM66 which qualifies the an-elastic behavior due to internal friction in the viscoelastic layer.

6 Computational Results and Discussions In this section, numerical calculations are performed to illustrate the influence of different factors on the phase velocity of Love waves. For numerical computations, the following values are considered. For the half-space (Gubbins [4]) μ = 6.34 × 1010 N/m2 , ρ1 = 3364 kg/m3 . For the upper layer (Yu et al. [24])    = 6.12 × 109 N/m2 , C66 = 3.32 × 109 N/m2 , M44 = 0.02 × 109 Ns/m2 , C44  M66 = 0.009 × 109 Ns/m2 , ρ2 = 1500 kg/m3 .

Moreover, unless otherwise stated the other relevant parameters and values are taken as a  = 0.2, l  = 0.1, m  = 0.1, ζ = 0.1, δ = 0.1, Q −1 = 0.02, where a  , l  and m  are dimensionless inhomogeneity parameters defined as    a  = a k1 , l  = l k1 and m  = m k1

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      Using Mathematica software, deviation of c β β = C66 ρ2 with real wave number k1 d for different values of inhomogeneity parameters, initial stress and dissipation function is sketched out in Figs. 2, 3, 4, 5 and 6. All the figures reflect the tendency that as the wave number rises, the non-dimensional phase velocity of the Love wave goes down. Fig. 2 Impact of inhomogeneity of the layer on phase velocity

1.70

1. a ' 2. a ' 3. a ' 4. a ' 5. a '

Dimensionless phase velocity c

1.68

1.66

0.200 0.202 0.204 0.206 0.208

1.64

1 2 3

4

5

1.62

1.60 1.0

1.5

2.0

2.5

3.0

Dimensionless wave number k1 d

Fig. 3 Impact of inhomogeneity in rigidity of the half-space on phase velocity

1.70

1. l ' 2. l ' 3. l ' 4. l ' 5. l '

Dimensionless phase velocity c

1.68

1.66

0.100 0.102 0.104 0.106 0.108

1.64

1

2

3

4

1.62

5

1.60 1.0

1.5

2.0

2.5

Dimensionless wave number k1 d

3.0

Love Wave Propagation in an Anisotropic Viscoelastic Layer … Fig. 4 Impact of inhomogeneity in density of the half-space on phase velocity

477

1.70

Dimensionless phase velocity c

1.68

1 2 3 4

1.66

1. m ' 2. m ' 3. m ' 4. m ' 5. m '

0.100 0.102 0.104 0.106 0.108

5 1.64

1.62

1.60 1.0

1.5

2.0

2.5

3.0

Dimensionless wave number k1 d

Fig. 5 Impact of the dissipation factor of the layer on phase velocity

1.70

1 2 3 4 5

Dimensionless phase velocity c

1.68

1.66

1. Q

1

0.016

2. Q

1

0.018

3. Q

1

0.020

4. Q

1

0.022

5. Q

1

0.024

1.64

1.62

1.60 1.0

1.5

2.0

2.5

3.0

Dimensionless wave number k1 d

Figure 2 demonstrates the impact of the non-dimensional heterogeneity parameter a  associated with the surficial layer. It is noticed that the rise in heterogeneity parameter a  escalates the phase velocity. Figures 3 and 4 show the impact of the non-dimensional heterogeneity parameters l  and m  pertaining to the rigidity and density of the half-space, respectively. It is noted that the rise in inhomogeneity parameter corresponding to the rigidity raises

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Fig. 6 Impact of the presence of initial stress in the half-space on phase velocity

1.70

Dimensionless phase velocity c

1.68

1.66

1.

0.02

2. 3.

0.06 0.10

4.

0.14

1, 2, 3, 4

1.64

1.62

1.60 1.0

1.5

2.0

2.5

3.0

Dimensionless wave number k1 d

the phase velocity, whereas the rise in inhomogeneity parameter corresponding to the density reduces the phase velocity. Figure 5 depicts the response of the dissipation function associated with the layer. It is clear that phase velocity goes down with a rise in dissipation function. Figure 6 demonstrates the impact of initial stress present in the half-space. It is noted that as the magnitude of the initial stress parameter ζ increases, phase velocity decreases.

7 Conclusions In this paper, Love wave in an inhomogeneous orthotropic viscoelastic layer lying over a prestressed inhomogeneous half-space is studied. Using appropriate boundary conditions, a general frequency relation for Love wave is established. Using this frequency relation and Mathematica software, numerical simulations are carried out to identify the impact of different parameters associated with the layer and the halfspace. The present study reveals that the inhomogeneity parameters of the layer and the half-space have a significant impact on the phase velocity of Love wave. Phase velocity rises with an increase in the inhomogeneity parameter linked with the rigidity of the half-space, whereas the inhomogeneity parameter related to the density of the half-space has a reverse effect. Also, phase velocity rises with increasing values of the inhomogeneity parameter of the upper layer. Moreover, the dissipation factor of the surficial layer and initial stress of the half-space significantly reduce the phase velocity.

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References 1. Ewing WM, Jardetzky WS, Press F (1957) Elastic waves in layered media. McGraw-Hill, New York 2. Love AEH (1944) A treatise on mathematical theory of elasticity, 4th edn. Dover Publication, New York 3. Bullen KE (1963) An introduction to the theory of seismology. Cambridge University Press, London 4. Gubbins D (1990) Seismology and plate tectonics. Cambridge University Press, Cambridge 5. Cooper HF (1967) Reflection and transmission of oblique plane waves at a plane interface between viscoelastic media. J Acoust Soc Am 42(5):1064–1069 6. Shaw RP, Bugl P (1969) Transmission of plane waves through layered linear viscoelastic media. J Acoust Soc Am 46(3B):649–654 7. Buchen PW (1971) Plane waves in linear viscoelastic media. Geophys J Roy Astron Soc 23(5):531–542 8. Schoenberg M (1971) Transmission and reflection of plane waves at an elastic-viscoelastic interface. Geophys J Roy Astron Soc 25(1–3):35–47 9. Borcherdt RD, Wennerberg L (1985) General P, type-I S, and type-II S waves in anelastic solids; inhomogeneous wave fields in low-loss solids. Bull Seismol Soc Am 75(6):1729–1763 10. Carcione JM (1990) Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields. Geophys J Int 101(3):739–750 ˇ 11. Cervený V (2004) Inhomogeneous harmonic plane waves in viscoelastic anisotropic media. Stud Geophys Geod 48(1):167–186 12. Chattopadhyay A, Gupta S, Sharma VK, Kumari P (2010) Propagation of shear waves in viscoelastic medium at irregular boundaries. Acta Geophys 58(2):195–214 13. Kakar R (2016) Love waves in Voigt-type viscoelastic inhomogeneous layer overlying a gravitational half-space. Int J Geomech 16(3):4015068–1–9 14. Pandit DK, Kundu S, Gupta S (2017) Propagation of Love waves in a prestressed Voigttype viscoelastic orthotropic functionally graded layer over a porous half-space. Acta Mech 228(3):871–880 15. Prasad B, Pal PC, Kundu S (2017) Propagation of SH-waves through non planer interface between visco-elastic and fibre-reinforced solid half-spaces. J Mech 33(4):545–557 16. Biot MA (1940) The influence of initial stress on elastic waves. J Appl Phys 11(8):522–530 17. Biot MA (1965) Mechanics of incremental deformations. Wiley, New York 18. Du J, Jin X, Wang J (2007) Love wave propagation in layered magneto-electro-elastic structures with initial stress. Acta Mech 192(1–4):169–189 19. Abd-Alla AM, Mahmoud SR, Abo-Dahab SM, Helmy MI (2011) Propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field. Appl Math Comput 217(9):4321–4332 20. Chattaraj R, Samal SK, Mahanti NC (2013) Dispersion of Love wave propagating in irregular anisotropic porous stratum under initial stress. Int J Geomech 13(4):402–408 21. Gupta S, Majhi DK, Kundu S, Vishwakarma SK (2013) Propagation of Love waves in nonhomogeneous substratum over initially stressed heterogeneous half-space. Appl Math Mech 34(2):249–258 22. Kakar R (2015) Dispersion of Love wave in an isotropic layer sandwiched between orthotropic and prestressed inhomogeneous half-spaces. Lat Am J Solids Struct 12(10):1934–1949 23. Kundu S, Alam P, Gupta S, Pandit DK (2017) Impacts on the propagation of SH-waves in a heterogeneous viscoelastic layer sandwiched between an anisotropic porous layer and an initially stressed isotropic half space. J Mech 33(1):13–22 24. Yu JG, Ratolojanahary FE, Lefebvre JE (2011) Guided waves in functionally graded viscoelastic plates. Compos Struct 93(11):2671–2677

Propagation of Edge Wave in Homogeneous Viscoelastic Sandy Media Pulkit Kumar , Amares Chattopadhyay, and Abhishek Kumar Singh

Abstract The objective of this work is to investigate the propagation characteristics of edge wave in a composite structure comprised of two uniformly homogeneous viscoelastic incompressible sandy plates of finite width and infinite extent. An analytical approach is used to deduce the closed form of frequency equation concerning to phase as well as damped velocities and analyzed the various affecting parameters. The effects of influencing parameters such as viscoelastic parameter, sandy parameter of both plates and wave number on the phase as well as damped velocities of edge wave are depicted graphically with the help of numerical simulation. It has been found that all the affecting parameters have significant effects on the edge wave propagation in the composite structure. Keywords Edge wave · Viscoelasticity · Sandy parameter · Frequency equation

1 Introduction The propagation of surface waves in elastic media with boundaries is more complicated than general cases. It is because of numerous reflections of waves from the neighboring boundaries and causes differences of velocity at the edge and toward the core/boundary. Waves confined to the edges of plates are significant for understanding the problems of the edge impact of plate-like structures. These types of waves are known as edge waves. Investigations into the edge wave propagation in viscoelastic and sandy media are quite important for the development of fundamental work on the mechanics of a solid deformable body and the applications in diverse areas of science and engineering like geotectonic, geophysics, seismology, and defectoscopy. Numerous researchers [1–5] have been discussed the propagation of edge wave in distinct material plates such as poroelastic plate, incompressible anisotropic plate, fiber-reinforced plate under the effect of initial stresses. P. Kumar (B) · A. Chattopadhyay · A. K. Singh Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, Jharkhand, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_39

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Earth material behaves nearly elastically under the application of small transient forces but viscously under the application of long duration of forces. Materials such as coal tar, salt, and sediments, which are buried beneath the Earth surface can be modeled as viscoelastic materials. The property of materials of being viscoelastic represents attenuation in the propagation of elastic waves through them. The general theory of viscoelasticity describes the linear behavior of both elastic and inelastic materials and provides the basis for describing the attenuation of seismic waves due to inelasticity. As seismic wave propagates, the attenuations happen due to the energy dissipation [6, 7]. It is also established that this attenuation is mainly due to the imperfect elasticity of the crust and upper mantle. In addition to this, the lithosphere–asthenosphere is represented by a viscoelastic half-space or an elastic layer in welded contact with viscoelastic half-space below it or a viscoelastic layer free to slide over the material or else a layered viscoelastic half-space. Chattopadhyay et al. [8, 9] examined the shear wave propagation in a viscoelastic structure due to the point source having a layer half-space model. All the materials of the Earth are not always elastic, isotropic, for which dispersion of the waves may be affected by the proximity of initial friction to the dry sand. It is imperative to suppose the layer of the soil in Earth to be more sandy than elastic. A dry sandy mantle may consist of sand particles or sandy parameter and has been defined by a parameter η. Weiskopf  [10] discussed the mechanics of dry sandy soil and proposed the relationship E μ = 2η(1 + σ ) which is appropriate for the dry sandy soil, where η > 1, E is Young’s modulus of elasticity, μ is the rigidity, and σ is the Poisson’s ratio. Localized waves near the stress-free surface or the free edge of a solid with a thin nematic coating are investigated by Zakharov [11]. Fu [12] discussed the existence and uniqueness of flexural edge wave propagation along the edge of an anisotropic elastic plate. Pal and Ghorai [13] expound the propagation of Love wave in a sandy layer lying over an anisotropic porous half-space under gravity. The propagation of edge wave in two uniform isotropic, homogeneous, viscoelastic, incompressible, sandy plates of finite width and infinite length has been investigated. An analytical approach is used to find the closed form of frequency equation relating to phase and damped velocities of Edge wave with the affecting viscosity parameter and sandy parameter. The effects of viscosity parameters and sandy parameters of both plates on the phase and damped velocities of edge wave have been depicted graphically through numerically computation. The propagation of edge wave in viscoelastic sandy medium is yet unexplored in two plates, which adds the novelty to the present problem.

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2 Formulation and Solution of the Problem Let us consider the propagation of edge wave in a composite structure comprised of two uniform homogeneous viscoelastic incompressible sandy plates M1 and M2   having finite width H1 2 and H2 2 with infinite extent. Introducing Cartesian coordinate system in such a way that the x-axis is taken at the common interface line of both plates, y-axis is vertically upwards, and z-axis is considered mutually perpendicular as shown in Fig. 1. The propagation of edge wave is confined along the direction of x-axis. Let u j , v j , and w j be the displacement components along the direction of x, y, and z axes while the subscript j is formed to distinguish the components in both the plates as j = 1 (for upper plate) and j = 2 (for lower plate). By the characteristics of edge wave propagation, we have u j = u j (x, y, t), v j = v j (x, y, t), w j = 0 and

∂ ≡ 0. ∂z

(1)

The stress-displacement relations for viscoelastic sandy material are ⎫   ∂u ∂v j τx x = η j λ¯ j + 2μ¯ j ∂ xj + η j λ¯ j ∂ yj , ⎪ ⎪ ⎬   ∂u ∂v j τx y = η j μ¯ j ∂ yj + ∂ xj ,   ∂v ⎪ ⎪ ∂u j τ yy = η j λ¯ j j + η j λ¯ j + 2μ¯ j j ⎭ ∂x

(2)

∂y

  λ where τx x , τx y , τ yy are the stress components; λ¯ j = λ j 1 + λ jj ∂t∂ , μ¯ j =     μ μ j 1 + μ jj ∂t∂ with λ j and μ j are Lamé constants, μj μ j and λj λ j are shear and volume viscosity parameters, and η j are sandy parameters for upper and lower plates with j = 1, 2 respectively. The equation of motion for the propagation of edge waves are defined as j

j

∂ 2u j ∂τx y ∂τx x + = ρj 2 ∂x ∂y ∂t

Fig. 1 Geometry of the problem

(3)

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and j

j

∂ 2v j ∂τx y ∂τ yy + = ρj 2 ∂x ∂y ∂t

(4)

where ρ j are the densities of the plates.

2.1 Solution for Upper Homogeneous Incompressible Viscoelastic Sandy Plate M1 For the viscoelastic sandy incompressible homogeneous plate M1 , choosing j = 1 and using the stress-displacement relations, the non-vanishing equation of motion with the help of Eqs. (3) and (4) result in  ∂ 2u1  ∂ 2 v1  ρ1 ∂ 2 u 1 ∂ 2u1  ¯ 1 + μ¯ 1 = + μ ¯ + λ λ¯ 1 + 2μ¯ 1 1 ∂x2 ∂ y2 ∂ x∂ y η1 ∂t 2

(5)

  ∂ 2 v1  ∂ 2u1 ρ1 ∂ 2 v1 ∂ 2 v1  ¯ 1 + μ¯ 1 = λ¯ 1 + 2μ¯ 1 + μ ¯ + λ . 1 ∂ y2 ∂x2 ∂ x∂ y η1 ∂t 2

(6)

and

The condition of incompressibility for the propagation of edge wave is defined as ex(1)x + e(1) yy = 0

(7)

∂v1 1 where ex(1)x = ∂u and e(1) yy = ∂ y . ∂x In view of Eq. (7), Eqs. (5) and (6) yield

1 ∂ 3 φ1 ∂ 3 φ1 ∂ 3 φ1 = + ∂ y3 ∂ x 2∂ y η1 β¯12 ∂t 2 ∂ y

(8)

∂ 3 φ1 1 ∂ 3 φ1 ∂ 3 φ1 = , + ∂x3 ∂ y2∂ x η1 β¯12 ∂t 2 ∂ x

(9)

and

1 1 where u 1 (x, y, t) = − ∂φ , v1 (x, y, t) = ∂φ , β¯1 = μρ¯ 11 , and φ1 = φ1 (x, y, t) are a ∂y ∂x differentiable function. Differentiating Eq. (8) with respect to y and Eq. (9) with respect to x then adding the obtained equation, we have

Propagation of Edge Wave in Homogeneous Viscoelastic Sandy Media

 ∂ 4 φ1 ∂ 4 φ1 ∂ 4 φ1 1 ∂ 2 ∂ 2 φ1 ∂ 2 φ1 . + +2 2 2 = + ∂ y4 ∂x4 ∂y ∂x ∂ y2 η1 β¯12 ∂t 2 ∂ x 2

485

(10)

For the harmonic wave propagation along the x-axis, the solution of Eq. (10) may be written as φ1 (x, y, t) =

1 f 1 (ky) exp[i(kx − αt)], k2

(11)

where α(= kc) is frequency of oscillations and k is wave number. Now substituting Eq. (11) in Eq. (10), the function f 1 (ky) can be determined in the form as f 1 (ky) + 2R1 f 1 (ky) + S12 f 1 (ky) = 0.

(12)

The solution of Eq. (12) is written as f 1 (ky) = c1 cosh(a1 ky) + c2 sinh(a1 ky) + c3 cosh(a2 ky) + c4 sinh(a2 ky), (13) where ⎫ a12 = −R1 + R12 − S12 , a22 = −R1 − R12 − S12 , ⎬    2  2 ⎭ R12 = 2k 2αβ¯ 2 η − η11 , S12 = η11 − 2k 2αβ¯ 2 η . 1 1

(14)

1 1

and c1 , c2 , c3 , c4 are arbitrary constants. In light of Eqs. (11) and (13), we have 1 [c1 cosh(a1 ky) + c2 sinh(a1 ky) k2 + c3 cosh(a2 ky) + c4 sinh(a2 ky)] exp[i(kx − αt)].

φ1 (x, y, t) =

(15)

With the aid of Eq. (15), the displacements in the upper plate M1 are written as 1 u 1 (x, y, t) = − [a1 c1 sinh(a1 ky) + a1 c2 cosh(a1 ky) k +a2 c3 sinh(a2 ky) + a2 c4 cosh(a2 ky)] exp[i(kx − αt)]

(16)

i [c1 cosh(a1 ky) + c2 sinh(a1 ky)+ k + c3 cosh(a2 ky) + c4 sinh(a2 ky)] exp[i(kx − αt)].

(17)

and v1 (x, y, t) =

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2.2 Solution for Lower Homogeneous Incompressible Viscoelastic Sandy Plate M2 For the viscoelastic sandy incompressible homogeneous plate M2 , choosing j = 2 and using the stress-displacement relations, the non-vanishing equation of motion with the help of Eqs. (3) and (4) results in  ∂ 2u2  ∂ 2 v2  ρ2 ∂ 2 u 2 ∂ 2u2  = + μ¯ 2 2 + λ¯ 2 + μ¯ 2 λ¯ 2 + 2μ¯ 2 2 ∂x ∂y ∂ x∂ y η2 ∂t 2

(18)

  ∂ 2 v2  ∂ 2u2 ∂ 2 v2  ρ2 ∂ 2 v2 λ¯ 2 + 2μ¯ 2 + μ¯ 2 2 + λ¯ 2 + μ¯ 2 . = 2 ∂y ∂x ∂ x∂ y η2 ∂t 2

(19)

and

The condition of incompressibility for the propagation of edge wave is defined as ex(2)x + e(2) yy = 0,

(20)

∂v2 2 where ex(2)x = ∂u and e(2) yy = ∂ y . ∂x In view of Eq. (20), Eqs. (18), and (19) yield

1 ∂ 3 φ2 ∂ 3 φ2 ∂ 3 φ2 = + ∂ y3 ∂ x 2∂ y η2 β¯22 ∂t 2 ∂ y

(21)

∂ 3 φ2 1 ∂ 3 φ2 ∂ 3 φ2 = , + 2 3 ∂x ∂y ∂x η2 β¯22 ∂t 2 ∂ x

(22)

and

∂φ2 ¯ μ¯ 2 2 where u 2 (x, y, t) = − ∂φ , v , β = and φ2 = φ2 (x, y, t) is a y, t) = (x, 2 2 ∂y ∂x ρ2 differentiable function. Differentiating Eq. (21) with respect to y and Eq. (22) with respect to x then adding the obtained equations, we get

 ∂ 4 φ2 ∂ 4 φ2 1 ∂ 2 ∂ 2 φ2 ∂ 2 φ2 ∂ 4 φ2 . + + 2 = + ∂ y4 ∂x4 ∂ y2∂ x 2 ∂ y2 η2 β¯22 ∂t 2 ∂ x 2

(23)

For the harmonic waves propagation along the x-axis, the solution of Eq. (23) may be written as φ2 (x, y, t) =

1 f 2 (ky) exp[i(kx − αt)]. k2

(24)

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Now substituting Eq. (24) in Eq. (23), the function f 2 (ky) can be determined in the formed f 2 (ky) + 2R2 f 2 (ky) + S22 f 2 (ky) = 0.

(25)

The solution of Eq. (25) may be written as f 2 (ky) = d1 cosh(b1 ky) + d2 sinh(b1 ky) + d3 cosh(b2 ky) + d4 sinh(b2 ky), (26) where ⎫ b12 = −R2 + R22 − S22 , b22 = −R2 − R22 − S22 , ⎬    2  2 ⎭ R22 = 2k 2αβ¯ 2 η − η12 , S22 = η12 − 2k 2αβ¯ 2 η 2 2

(27)

2 2

and d1 , d2 , d3 , d4 are arbitrary constants. In view of Eqs. (24) and (26), we obtain φ2 (x, y, t) =

1 [d1 cosh(b1 ky) + d2 sinh(b1 ky) k2 + d3 cosh(b2 ky) + d4 sinh(b2 ky)] exp[i(kx − αt)].

(28)

With the help of Eq. (28), the displacements in the lower plate M2 are written as 1 u 2 (x, y, t) = − [b1 d1 sinh(b1 ky) + b1 d2 cosh(b1 ky) k + b2 d3 sinh(b2 ky) + b2 d4 cosh(b2 ky)] exp[i(kx − αt)]

(29)

i v2 (x, y, t) = [d1 cosh(b1 ky) + d2 sinh(b1 ky) k + d3 cosh(b2 ky) + d4 sinh(b2 ky)] exp[i(kx − αt)].

(30)

and

3 Boundary Conditions and Frequency Equation The boundary conditions of the problem are identified with the bounding planes of viscoelastic, incompressible, and sandy plates, which are supposed to be free from tractions and characterized as  H1  f x(1) = 0 , (31) at y = (1)  fy = 0 2

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 u 1 = u 2 , v1 = v2 at y = 0,  f x(1) =  f x(2) ,  f y(1) =  f y(2)  H2  f x(2) = 0 at y = − ,  f y(2) = 0 2

(32)

(33)

where  f x(1)  f x(2)

   ∂v1 ∂u 1 ∂u 1 ∂v1 ,  f y(1) = η1 λ¯ 1 = η1 μ¯ 1 + + η1 λ¯ 1 + 2μ¯ 1 , ∂y ∂x ∂x ∂y

   ∂v2 ∂u 2 ∂v2 ∂u 2 + ,  f y(2) = η2 λ¯ 2 + η2 λ¯ 2 + 2μ¯ 2 . = η2 μ¯ 2 ∂y ∂x ∂x ∂y

With the aid of Eqs. (16), (17), (29), and (30) along with prescribed boundary conditions Eqs. (31)–(33), we have the following relations:  

H1 H1 +c2 sinh a1 k c1 cosh a1 k 2 2  



A2 H1 H1 c3 cosh a2 k + c4 sinh a2 k = 0, + A1 2 2   

H1 H1 H1 a1 c1 sinh a1 k +a1 c2 cosh a1 k + a2 c3 sinh a2 k 2 2 2 

H1 = 0, + a2 c4 cosh a2 k 2  

H2 H2 d1 cosh b1 k −d2 sinh b1 k 2 2  



B2 H2 H2 d3 cosh b2 k − d4 sinh b2 k = 0, + B1 2 2   

H2 H2 H2 b1 d1 sinh b1 k −b1 d2 cosh b1 k + b2 d3 sinh b2 k 2 2 2 

H2 = 0, + b2 d4 cosh b2 k 2

(34)

(35)

(36)

(37)

a1 c2 + a2 c4 − b1 d2 − b2 d4 = 0,

(38)

c1 + c3 − d1 − d3 = 0,

(39)

A1 c1 + A2 c3 − B1 d1 − B2 d3 = 0,

(40)

Propagation of Edge Wave in Homogeneous Viscoelastic Sandy Media

T1 (a1 c2 + a2 c4 ) − T2 (b1 d2 + b2 d4 ) = 0,

489

(41)

where       A1 = η1 μ¯ 1 a12 + 1 , A2 = η1 μ¯ 1 a22 + 1 , B1 = η2 μ¯ 2 b12 + 1 , B2 = η2 μ¯ 2 b22 + 1 , T1 = 2η1 μ¯ 1 , T2 = 2η2 μ¯ 2 , and c1 , c2 , c3 , c4 , d1 , d2 , d3 , d4 are arbitrary constants. Eliminating the arbitrary constants c1 , c2 , c3 , c4 , d1 , d2 , d3 , d4 from Eqs. (34)– (41), we obtained the frequency equation of edge wave propagation as Q 1 Q 2 (A1 − B1 ) − Q 2 (A2 − B1 ) − Q 1 (A1 − B2 ) + (A2 − B2 ) = 0,

(42)

where   (a2 /a1 ) (a1 A2 )(1 − sec h(a1 γ1 ) sec h(a2 γ1 )) − (a2 A1 ) tan h(a1 γ1 ) tan h(a2 γ1 )   Q1 = , (a1 A2 ) tan h(a1 γ1 ) tan h(a2 γ1 ) − (a2 A1 )(1 − sec h(a1 γ1 ) sec h(a2 γ1 ))   (b2 /b1 ) (b1 B2 )(1 − sec h(b1 γ2 ) sec h(b2 γ2 )) − (b2 B1 ) tan h(b1 γ2 ) tan h(b2 γ2 )   Q2 = , (b1 B2 ) tan h(b1 γ2 ) tan h(b2 γ2 ) − (b2 B1 )(1 − sec h(b1 γ2 ) sec h(b2 γ2 )) γ1 = k H1 /2, γ2 = k H2 /2.

Equation (42) is the frequency equation of edge wave propagation in two uniform isotropic homogeneous viscoelastic incompressible sandy plates of finite thickness and infinite length associated with phase velocity and damped velocity.

4 Special Case When both homogeneous plates are isotropic without viscoelasticity and sandy parameter, i.e., μ¯ 1 = μ1 , μ¯ 2 = μ2 , λ¯ 1 = λ1 , λ¯ 2 = λ2 , η1 = η2 = 1, the frequency equation reduces to             μ1 1 + 1 − c2 β12 − Q 1 (1 − Q 1 ) − μ2 1 + 1 − c2 β22 − Q 2 (1 − Q 2 ) = 0,

(43) where

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Q1 =

 

     1 − c2 β12 2 − c2 β12 1 − sech(γ1 )sech γ1 1 − c2 β12 

  − 2 1 − c2 β12 tanh(γ1 )tanh γ1 1 − c2 β12

  ,     2 − c2 β12 tanh(γ1 )tanh γ1 1 − c2 β12



  − 2 1 − c2 β12 1 − sech(γ1 )sech γ1 1 − c2 β12

and

Q2 =

β1 =



μ1 ρ1

 

     1 − c2 β22 2 − c2 β22 1 − sech(γ2 )sech γ2 1 − c2 β22

   − 2 1 − c2 β22 tanh(γ2 )tanh γ2 1 − c2 β22  

,   2  2 2 2 2 − c β2 tanh(γ2 )tanh γ2 1 − c β2



 2  2 2 2 − 2 1 − c β2 1 − sech(γ2 )sech γ2 1 − c β2 and β2 =



μ2 .. ρ2

5 Numerical Calculation and Discussion The numerical simulation and graphical demonstration have been carried out for the deduced frequency equation of Edge wave propagation in a composite structure comprised of two uniform homogeneous viscoelastic incompressible sandy plates of finite width and infinite extent. For the sake of numerical simulation, the following data [14] have been taken into the account: (i) For upper homogeneous viscoelastic sandy plate μ1 = 6.77 × 1010 N/m2 , ρ1 = 3323 kg/m3 . (ii) For lower homogeneous viscoelastic sandy plate μ2 = 6.54 × 1010 N/m2 , ρ2 = 3409 kg/m3 . The frequency equation of Edge wave propagation relates the dimensionless phase velocity and damped velocity to wave number with affecting dimensionless viscosity

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and sandy parameters. The effects of aforementioned affecting parameters on the phase and damped velocities of Edge wave are plotted through Figs. 2, 3, 4, and 5. Variations of dimensionless phase velocity against dimensionless wave number (dispersion curves) are depicted in Figs. 2a, 3a, 4a, and 5a while the variations of damped velocity against dimensionless wave number (damping curves) are shown in Figs. 2b, 3b, 4b, and 5b associated to the sandy parameter and viscosity parameter, respectively.

Fig. 2 Effect of sandy parameter (η1 ) of upper plate on the a dimensionless phase velocity and b dimensionless damped velocity against dimensionless wave number

Fig. 3 Effect of sandy parameter (η2 ) of lower plate on the a dimensionless phase velocity and b dimensionless damped velocity against dimensionless wave number

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   Fig. 4 Effect of viscosity parameter αμ1 μ1 of upper plate on the a dimensionless phase velocity and b dimensionless damped velocity against dimensionless wave number

   Fig. 5 Effect of viscosity parameter αμ2 μ2 of lower plate on the a dimensionless phase velocity and b dimensionless damped velocity against dimensionless wave number

The effects of sandy parameters corresponding to the upper and lower viscoelastic plates on the phase and damped velocities of Edge wave have been depicted in Figs. 2 and 3. The variation of phase and damped velocities increases with the increase of value of sandy parameters (η1 ) and (η2 ) in viscoelastic plates (M1 ) and (M2 ) accordingly. It has been observed from Fig. 2a, b for fixed value of sandy parameters, the variation of phase velocity favors more than the damped velocity of Edge wave, and the damped velocity is more as compared to the phase velocity. Also, it has been revealed through Fig. 3a, b, the variation of phase velocity favors more as compare

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to the damped velocity of Edge wave with the increment of sandy parameter (η2 ) in viscoelastic plate (M2 ). The effects of viscosity parameters corresponding to the upper and lower plates on the phase velocity and damped velocity of Edge wave have been shown in Fig. 4 and Fig. 5, respectively. In Fig. 4, of phase velocity increases with the   thevariation increase of viscosity parameter αμ1 μ1 , while the variation of damped velocity decreases. Damped velocity of Edge wave propagation is higher than the phase velocity and increment of viscosity parameter favors the phase velocity while disfavors the damped velocity of edge wave propagation. In Fig. 5, the variation of dispersion     and damping curves increases with the increase of viscosity parameter αμ2 μ2 of viscoelastic plate (M2 ) in sandy media. Damped velocity of edge wave propagation favors more than the phase velocity. The damped velocity of Edge wave propagation disfavors substantially with the variation of viscosity parameter in viscoelastic sandy plates.

6 Conclusion An analytical approach is used to study the propagation of edge wave in two uniformly homogeneous, viscoelastic, and incompressible sandy plates with distinct finite width. A closed form of frequency equation relates the phase velocity and damped velocity to wave number with affecting dimensionless viscosity and sandy parameter. The effects of viscosity parameters and sandy parameters of both plates on the phase and damped velocities of edge wave have been depicted graphically. More precisely, the following outcomes are accomplished through study: • The sandy parameters of both plates give the significant favoring effect on phase velocity and damped velocity of edge wave. • Viscosity parameters of plates show the remarkable impact on the phase velocity and damped velocities of edge wave propagation. • The phase and damped velocities of edge wave increase with the increment in sandy parameter of the plates accordingly. • Viscosity parameter of upper plate favors the phase velocity while disfavor the damped velocity of edge wave propagation and the viscosity parameter of lower plate favors the phase as well as damped velocities of edge wave. • Phase velocity as well as damped velocity of edge wave decreases with the increase of wave number. It is worth mentioning that edge waves help to determine a possible approach to diagnose the presence of a problematic defect and corroded area through nondestructive testing (NDT). From a non-destructive evaluation (NDE) point of view, the scatter and randomness in size, shape, location, growth rate, and maximum depth of corrosion patches require an understanding of the relationships between the geometrical properties of the defect and the resulting reflection coefficient (RC) from them for correct interpretation and evaluation. The edge wave may be suitable

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for surface damage detection because it concentrates energy on the top surface, and the wave number selection depends on the sensitivity and penetration power in NDT. Acknowledgments The authors convey their sincere thanks to the Indian Institute of Technology (ISM), Dhanbad, India for granting access to its best research facility and providing a Senior Research Fellowship to Mr. Pulkit Kumar.

References 1. Reddy PM, Tajuddin M (2003) Edge waves in poroelastic plate under plane stress conditions. J Acoust Soc Am 114(1):185–193. https://doi.org/10.1121/1.1569258 2. Pichugin AV, Rogerson GA (2012) Extensional edge waves in pre-stressed incompressible plates. Math Mech of Solids 17(1):27–42. https://doi.org/10.1177/1081286511412440 3. Debnath L, Roy PP (1988) Propagation of edge waves in a thinly layered laminated medium with stress couples under initial stresses. Int J Stoch Anal 1(4):271–286. https://doi.org/10. 1155/S1048953388000206 4. Singh AK, Lakshman A, Chattopadhyay A (2016) The plane waves at the edge of a uniformly pre-stressed fiber-reinforced plate. J Vib Control 22(10):2530–2541. https://doi.org/10.1177/ 1077546314548087 5. Dey S, De PK (2010) Propagation of channel wave in an incompressible anisotropic initially stressed plate of finite thickness. Tamkang J Sci Eng 13(2):127–134 6. Ewing WM, Jardetzky WS, Press AB (1957) Elastic waves in layered media. Phys Today 10. https://doi.org/10.1063/1.3060203 7. Ben-Menahem A (1995) A concise history of mainstream seismology: origins, legacy, and perspectives. Bull Seismol Soc Am 85(4):1202–1225. https://doi.org/10.1130/0091-7613 8. Chattopadhyay A, Gupta S, Sharma V, Kumari P (2010) Propagation of shear waves in viscoelastic medium at irregular boundaries. Acta Geophys 58(2):195–214. https://doi.org/ 10.2478/s11600-009-0060-3 9. Chattopadhyay A, Gupta S, Kumari P, Sharma V (2012) Effect of point source and heterogeneity on the propagation of SH-waves in a viscoelastic layer over a viscoelastic half space. Acta Geophys 60(1):119–139. https://doi.org/10.2478/s11600-011-0059-4 10. Weiskopf WH (1945) Stresses in soils under a foundation. J Franklin Inst 239(6):445–465 11. Zakharov DD (2012) Surface and edge waves in solids with nematic coating. Math Mech Solids 17(1):67–80. https://doi.org/10.1177/1081286511412445 12. Fu YB (2003) Existence and uniqueness of edge waves in a generally anisotropic elastic plate. Q J Mech Appl Math 56(4):605–616. https://doi.org/10.1093/qjmam/56.4.605 13. Pal J, Ghorai AP (2015) Propagation of Love wave in sandy layer under initial stress above anisotropic porous half-space under gravity. Transp Porous Media 109(2):297–316. https://doi. org/10.1007/s11242-015-0519-4 14. Gubbins D (1990) Seismology and plate tectonics. https://doi.org/10.1007/978-0-387-77994-2

Love-Type Wave Propagation in Functionally Graded Piezomagnetic Material Resting on Piezoelectric Half-Space J. Baroi and S. A. Sahu

Abstract The propagation of Love-type waves in functionally graded piezomagnetic layer lying on a piezoelectric half-space is studied. We have considered the interface between the Functionally Graded Piezomagnetic Material (FGPM) layer and piezoelectric half-space as imperfect and the imperfection is taken in linear form by spring model. The exponential variation is taken for the material parameters of the layer along the thickness direction. The dispersion relation in determinant form has been obtained for both magnetically open and short cases. Numerical computation and graphical demonstration have been carried out to observe the effect of gradient factor, layer’s width, and the interfacial constant on the phase velocity of Love-type waves. For validation, the present study is matched with classical Love wave result. The application of the present work may be found in designing and optimization of Surface Acoustic Wave (SAW) devices and sensors. Keywords Love-type waves · Functionally graded piezomagnetic material · Piezoelectric material · Imperfect interface

1 Introduction The study of surface wave propagation through piezoelectric layered structure has some interesting applications for manufacturing of seismic devices. As a medium for wave transference the piezoelectric materials are of great significance due to their piezoelectric effect (e.g., exhibiting electric charge in contact with mechanical stress and inducing mechanical stress in response to electric charge). It is found that by such piezoelectric effect a linear electromechanical coupling induces which enhances the medium [1].

J. Baroi (B) · S. A. Sahu Department of Mathematics & Computing, IIT (ISM), Dhanbad 826004, Jharkhand, India e-mail: [email protected] S. A. Sahu e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_40

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The electromechanical coupling can be enhanced by constructing composite materials. Functionally Graded Materials are materials with variation in composition and structure, which results in changing properties. FGPM are functionally graded materials with piezomagnetic effect. Piezomagnetic effect is same as piezoelectric effect, i.e., in contact with mechanical force, it induces magnetic field and vice versa. In this context, Kong et al. [2] studied the propagation characteristics of SH waves in a functionally graded piezomagnetic layer on PMN-0.29PT single crystal substrate. Li et al. [3] studied the Love waves in functionally graded piezoelectric materials. Saroj et al. [4] have discussed love-type waves in Functionally Graded Piezoelectric Material (FGPM) sandwiched between initially stressed layer and elastic substrate. In the process of designing devices, the material–material interface is not always perfect. Sometimes there are imperfection at the interface (e.g., distortion of interface, fracture at the interface, erosion of the materials, etc.). Transference of wave through such interfaces results on the sensitivity of the considered device [5]. The present paper deals with the significance of imperfect interface, gradient factor and layer’s width on the Love-type wave propagation in composite structure with imperfect boundary. Dispersion relation in determinant form is obtained applying appropriate boundary conditions for both magnetically open and short cases using the separation of variables method.

2 Formulation of the Problem Propagation of Love-type waves in a composite structure with imperfect interface is considered (Fig. 1). The composite structure consists of a Functionally Graded Magnetic Material (FGPM) over a piezoelectric substrate. The width of the FGPM layer is taken as h. The wave is propagating along y-axis and the substrate is taken along the positive x-axis direction. The polarization direction of the materials is taken along z-axis. Along x-axis the material properties of the layer are varying. Fig. 1 Geometry of the problem

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2.1 For the Layer In the absence of external force the basic governing equation is σi j, j = ρ f g

∂ 2ui ∂t 2

Bi, i = 0

(1)

where the symbols Bi , ρ f g , σi j , u i represents magnetic induction, density of the layer, stress tensor and displacement component respectively, i, j = 1, 2, 3 and differentiation with respect to position is indicated by comma. The FGPM fundamental relations are σi j = ci jkl Rk l − h ki j Hk B j = h jkl Rk l + μ jk Hk

(2)

where σi j , h jkl , Rkl , Hk , B j , μ jk are stress tensor, FGPM coefficients, strain tensor, magnetic field, magnetic induction and magnetic permeability respectively. The magnetic field Hk and strain Rkl are written in terms of displacement u i and magnetic potential φ as Rk l =

 1 u k, l + u l, k 2

Hi = −φ,i For FGPM layer the constitutive equations are given by σx = c11 Rx + c12 R y + c13 Rz − h 31 H3 σ y = c12 Rx + c11 R y + c13 Rz − h 31 H3 σz = c13 Rx + c13 R y + c33 Rz − h 33 H3 σ y z = c44 R yz − h 15 H2 σx z = c44 Rx z − h 15 H1 σx y =

(c11 − c12 ) Rx y 2

(3) (4)

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B1 = h 15 Rx z + μ11 H1 B2 = h 15 R yz + μ11 H2 B3 = h 31 Rx + h 31 R y + h 33 Rz + μ33 H3

(5)

Since the wave is propagating along y-direction, so magnetic potential and displacement components can be written as u = v = 0, w = w(x, y, t) φ = φ 1 (x, y, t)

(6)

From Eqs. (3), (4), (5), and (6), we get Rx = 0 Ry = 0 Rz = 0 Rx y = 0 σx = 0 σy = 0 σz = 0 σx y = 0 H1 = −φ1, 1 H2 = −φ1, 2 H3 = 0 With the help of Eqs. (6), (7), and (5) reduces to

(7)

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σx z = c44 w, 1 + h 15 φ 1 , 1 σ yz = c44 w, 2 + h 15 φ 1 , 2 B1 = h 15 w, 1 − μ 11 φ 1 ,1 B2 = h 15 w, 2 − μ 11 φ 1 ,2

(8)

Using Eq. (8) into Eq. (1), we have 

∂ 2w ∂ 2w + 2 c44 (x) ∂x2 ∂y





∂ 2φ1 ∂ 2φ1 + h 15 (x) + ∂x2 ∂ y2



∂c44 (x) ∂w ∂h 15 (x) ∂φ 1 ∂ 2w + = ρ f g (x) 2 ∂x ∂x ∂x ∂x ∂t    2  ∂ w ∂ 2w ∂ 2φ1 ∂ 2φ1 h 15 (x) + 2 − μ 11 (x) + ∂x2 ∂y ∂x2 ∂ y2 +

+

∂h 15 (x) ∂w ∂μ 11 (x) ∂φ 1 − =0 ∂x ∂x ∂x ∂x

(9)

The variation of FGPM material properties are given as follows: h 15 (x) = h 15 e(ζ F x) c44 (x) = c44 e(ζ F x) μ11 (x) = μ11 e(ζ F x) ρ f g (x) = ρ e(ζ F x)

(10)

where h 15 , c44 , μ11 and ρ are the values of h 15 , c44 , μ11 and ρ f g at x = 0 and ζ F is the material gradient. From Eqs. (9) and (10), we get     ∂w ∂φ 1 ∂ 2w 2 2 1 + h 15 ∇ φ + ζ F c44 ∇ w + ζ F =ρ 2 ∂x ∂x ∂t     ∂w ∂φ 1 = μ11 ∇ 2 φ 1 + ζ F h 15 ∇ 2 w + ζ F ∂x ∂x

(11)

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2.2 For the Half-Space The motion of equation for the piezoelectric half-space is σiPj, j = ρ P

∂ 2 u iP ∂t 2

Di,P i = 0

(12)

where the symbols σiPj , ρ P , u iP , DiP symbolize the stress tensor, density of the half-space, displacement component, electrical displacement respectively and i, j = 1, 2, 3, differentiation with respect to position is indicated by comma. The well-known piezoelectric constitutive equations are σiPj = ciPjkl SklP − ekPi j E kP D Pj = e Pjkl SkPl + ε Pjk E kP

(13)

where e Pjkl , ε Pjk , SklP , σiPj , E kP , D Pj indicates the piezoelectric and dielectric coefficients, strain and stress tensor, electric field, electrical displacement, respectively. The relation between elastic displacement u iP , electric potential φ 1 , strain SklP , and electric field E kP is written as SklP =

 1 P P u + u l,k 2 k,l p

E i = −φ,i1

(14) (15)

For the considered problem, the electric potential and displacement components are given by u 1P = u 2P = 0, u 3P = w1P (x, y, t) φ 1 = ψ1 (x, y, t)

(16)

Using Eqs. (13), (14), (15), and (16), we get SxP =S yP = SzP = SxPy = 0, σxP = σ yP = σzP = σxPy 1 1 = 0, E 1P = −φ,1 , E 2P = −φ,2 , E 3P = 0

(17)

With the help of Eqs. (16) and (17) the constitutive relations for piezoelectric half-space (Eq. 13) can be rewritten as

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P P P σ yzP = c44 w1, 2 + e15 ψ1, 2 P P P σxPz = c44 w1, 1 + e15 ψ1, 1 P P P D1P = e15 w1, 1 − ε11 ψ1, 1 P P P D2P = e15 w1, 2 − ε11 ψ1, 2

(18)

Substituting Eq. (18) into Eq. (12), we obtain P c44

2 P 2 2 ∂ 2 w1P ∂ 2 w1P P ∂ w1 P ∂ ψ1 P ∂ ψ1 + c44 + e 15 + e15 = ρP 2 2 2 2 ∂x ∂y ∂x ∂y ∂t 2

(19a)

2 P 2 2 ∂ 2 w1P P ∂ w1 P ∂ ψ1 P ∂ ψ1 + e − ε − ε =0 15 11 11 ∂x2 ∂ y2 ∂x2 ∂ y2

(19b)

P e15

3 Solution of the Problem 3.1 Solution for the FGPM Layer Let us assume η = φ1 −

h 15 w μ11

(20)

Using Eqs. (20) and (11) becomes   2  h 15 ∂w ∂ 2w 2 ζF +∇ w =ρ 2 c44 + μ11 ∂x ∂t   ∂η ζF + ∇2η = 0 ∂x



(21)

To solve the Eq. (21), let us assume the solution in the following form: w = W (x) exp[ik(y − ct)]

(22a)

η = η(x) exp[ik(y − ct)]

(22b)

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With the help of above assumption (Eqs. 22a, 22b, and 21) reduces to η (x) + ζ F η (x) − k 2 η(x) = 0 





W (x) + ζ F W (x) + k

2

(23a)

 c2 − 1 W (x) = 0 2 csh

(23b)

 2 2 where csh = ρ1 c44 + hμ1511 and the single and double derivative(s) with respect to x are denoted by ( ) and ( ), respectively. Solving Eq. (23a), we have η(x) = c1 em 1 x + c2 em 2 x

(24)

√ −ζ ± ζ 2 +4k 2 where m 1,2 = F 2 F The solution of Eq. (23b) is given by     W = c3 es1 x cos μ f x + c4 es1 x sin μ f x where s1 =

− ζ2F

(25)

 2 , μf = 4k 2 cc2 − 1 − ζ F2 1 2

sh

From Eqs. (20), (22a), (22b), (24), and (25), we get      w = c3 es1 x cos μ f x + c4 es1 x sin μ f x exp[ik(y − ct)]

(26a)



φ = φ 1 = c1 e m 1 x + c2 e m 2 x +

    h 15  s1 x exp[ik(y − ct)] c3 e cos μ f x + c4 es1 x sin μ f x μ11

(26b) Using the Eqs. (8), (26a), and (26b), we obtain   ⎤ 2      m 1 x + c h m em 2 x + c c + h 15 s1 x s cos μ x − μ sin μ x + h m e c e 2 15 2 3 44 1 f f f ⎢ 1 15 1 ⎥ μ11 ⎥   =⎢ 2  ⎣     ⎦ h 15 c4 c44 + μ11 es1 x s1 sin μ f x + μ f cos μ f x ⎡

σx z

exp[ik(y − ct)]

 B1 = −μ11 m 1 em 1 x c1 − μ11 m 2 em 2 x c2 exp[ik(y − ct)]

(27a) (27b)

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3.2 Solution for the Half-Space Assume the transformation χ = ψ1 − w1P

P e15 ε11P

(28)

With the help of above Eqs. (27a), (19a), (19b) becomes c P ∇ 2 w1P = ρ P

∂ 2 w1P ∂t 2

(29a)

∇2χ = 0

(29b)

w1P (x, y, t) = W1P (x) exp[ik(y − ct)]

(30a)

χ = χ (x) exp[ik(y − ct)]

(30b)

2 eP P where c P = c44 + ( ε15P) 11 Consider

Under the assumption (30a) Eq. (29a) becomes  W1P (x) − k 2 λ2f W1P = 0

(31)

 P where λ2f = 1 − ρc P c2 Solving Eq. (31) and using Eq. (30a) and the condition x → ∞, we have   w1P = A1 exp −kλ f x exp[ik(y − ct)]

(32)

Using Eqs. (16), (29b), (30b), (28) and the condition x → ∞, we get     eP  φ 1 = ψ1 = A2 exp(−kx) + 15 A exp −kλ x exp[ik(y − ct)] 1 f P ε11

(33)

From Eqs. (18), (32), and (33), we obtain

σxPz

= −kλ f A 1

 P c44

  P 2  e15 −kλ f x P −k x + P − A2 e15 ke e exp[ik(y − ct)] ε11

 P ke−kx exp[ik(y − ct)] D1P = A2 ε11

(34a) (34b)

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4 Boundary Conditions and Dispersion Relations At the upper surface the traction free boundary condition is given by σx z (−h, y) = 0

(35)

At the upper surface the magnetically open and short conditions are described by B1 (−h, y) = 0

(36)

φ(−h, y) = 0

(37)

and

For the imperfect interface, the spring model by Rokhlin and Wang [6] is considered. According to this model, the stresses are continuous but the displacements has jump discontinuity at the interface, i.e.,  σx z (0, y) = σxPz (0, y) = α f w(0, y) − w1P (0, y)

(38)

where α f is spring constant, reflects the interface property, α f → ∞ signifies to a perfectly bonded interface, while α f → 0 gives a slip interface. In addition, the magnetic potential and electric potential are zero at the interface. So, we have φ(0, y) = 0

(39)

φ 1 (0, y) = 0

(40)

Using the solutions derived above [Eqs. (26a), (26b), (27a), (27b), (32), (33), (34a), (34b)] and the boundary conditions in Eqs. (35)–(40), we get M A = 0 and N A = 0 where A = [A1 , A2 , A3 , A4 , A5 , A6 ]T and M and N is the non-zero 6 × 6 matrix whose non-zero components are given below h 15 m 1 e−m 1 h = N11 , M12 = h 15 m 2 e−m 2 h = N12 , M11 =  2      M13 = c44 + hμ1511 e−s1 h s1 cos μ f h + μ f sin μ f h = N13 ,  2      = N14 , M21 = M14 = c44 + hμ1511 e−s1 h −s1 sin μ f h + μ f cos μ f h −μ11 m 1 e−m 1 h , M22 = −μ11 m 2 e−m 2 h ,M31 = h 15 m 1 = M41 = N31 = N41 , M32 = h 15 m 2 = M42 = N32 = N42 ,  2 2 M33 = c44 + hμ1511 s1 = N33 , M34 = c44 + hμ1511 μ f = M44 = N34 = N44 ,

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  2 eP P P M35 = kλ f c44 + ( ε15P) = N35 , M36 = e15 k = N36 , 11   2 c44 + hμ1511 s1 − α f = N43 , M45 = α f = N45 , M51 = 1 = N51 , M43 = M52 = 1 = N52 , eP M53 = μh 1511 = N53 , M65 = ε15P = N65 , M66 = 1 = N66 , N21 = e−m 1 h , 11     N22 = e−m 2 h , N23 = μh 1511 e−s1 h cos μ f h , N24 = − μh 1511 e−s1 h sin μ f h . For non-trivial solution, the determinant of the coefficients matrix has to be zero, i.e., det(M) = 0 and det(N ) = 0

(41)

Equation (41) gives the dispersion equation for the considered problem for both magnetically open and short cases, respectively.

4.1 Particular Case If the medium reduced to homogeneous isotropic medium, i.e., when h 15 = μ11 = P P P = ε11 = 0 and c44 = c and c44 = c1 , then we have e15  tan kh



 ρP 2 c 1 1 − c1 c ρ 2 c −1 =  c c ρc c2 − 1 

(42)

Equation (42) is the classical Love wave equation.

5 Numerical Examples and Discussions To demonstrate the effect of imperfect interface, FGPM layer’s width, gradient factor numerical example is presented and graphs are plotted. For FGPM layer, we have taken Terefenol-D and for the half-space PZT-5H is taken. 1. The material constants for Terefenol-D are [2] c44 = 5.99×109 (N/m2 ), h 15 = 167.665(H/m), μ11 = 3.976×10−6 (N S 2 /C 2 ), ρ = 9230(kg/m3 ) 2. For PZT-5H material  constants  P are [4]   P = 2.30 × 1010 N /m 2 , e15 = 17 c/m 2 , c44  P = 277 × 10−10 (F/m), ρ P = 7.50 × 103 kg/m 3 ε11 s=

c44 hα f

For graphical representation the interfacial constant is taken as

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Figure (2) is plotted to highlight the impact of FGPM layer’s width for both the cases. It is observed that the phase velocity of the Love-type waves increases for increasing values of FGPM layer’s width for both the cases. The increment is more significant in open case than short one. Figure (3) reveals that the phase velocity increases with increasing values of gradient factor for both the cases. Comparison of the figures [Fig. 4a, 4b] reveals that increament in the considered structure is remarkable for magnetically open case more than magnetically short case.

Fig. 2 Variation of dimensionless phase velocity against dimensionless wave number for different values of layer’s width (h) for magnetically (a) open case (b) short case

Fig. 3 Variation of dimensionless phase velocity against dimensionless wave number for different values of gradient factor (ζ F ) for magnetically (a) open case (b) short case

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Fig. 4 Variation of dimensionless phase velocity against dimensionless wave number for different values of interfacial constant (s) for magnetically (a) open case (b) short case

From Fig. 4, it is noticed that with the increasing values of interfacial constant the phase velocity decreases for both cases. The effect is more prominent in magnetically open case than short case.

6 Conclusions The present work may lead to the following conclusions: • The phase velocity of Love-type waves increases with the increasing values of width of the layer. • Increment in gradient factor results in increment in phase velocity. • Phase velocity of the considered wave is inversely proportional to interfacial constant. • The particular case leads to the validation of the current work. • The application of the present study may be found in designing and implementation of SAW devices and sensors.

References 1. Wang Q, Quek ST, Varadan VK (2001) Love waves in piezoelectric coupled solid media. Smart Mater Struct 10(2):380–388 2. Kong Y, Nie G (2016) Propagation Characteristics of SH waves in a functionally graded piezomagnetic layer on PMN-0.29PT single crystal substrate. Mech Res Commun 73:107–112

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3. Li XY, Wang ZK, Huang SH (2004) Love waves in functionally graded piezoelectric materials. Int J Solid Struct 41:7309–7328 4. Saroj PK, Sahu SA, Chaudhary S, Chattopadhyay A (2015) Love-type waves in functionally graded piezoelectric material (FGPM) sandwiched between initially stressed layer and elastic substrate. Waves in Random Complex Media 25:608–627 5. Nie G, Liu J, Fang X, An Z (2012) Shear horizontal (SH) waves propagating in piezoelectricpiezomagnetic bilayer system with an imperfect interface. Acta Mech 223:1999–2009 6. Rokhlin SI, Wang YJ (1991) Analysis of boundary conditions for elastic wave interaction with an interface between two solids. The Journal of the Acoustical Society of America 89(2):503–515

Fluid–Body Interactions in Fish-Like Swimming Dipanjan Majumdar, Chandan Bose, Prerna Dhareshwar, and Sunetra Sarkar

Abstract The present study focuses on formulating a fluid–structure interaction (FSI) framework by coupling a finite element analysis (FEA) based structural solver and a lumped vortex method (LVM) based potential flow solver to study the coupled dynamics involved in the undulatory and oscillatory swimming of fishes. The caudal fin of a carangiform fish is modelled as a continuous cantilever beam with a periodic support motion. The effect of the actuation frequency on the thrust coefficient is investigated. A significant increase in the aerodynamic thrust is noticed for the support motion frequencies nearing to the structural natural frequencies of the beam. Next, the whole fish body, considering the full-body undulations, is modelled as a continuous free-free beam. This model incorporates a time-dependent actuating moment varying along the length of the body which can be attributed to the muscle moments generated by the fish. A parametric study is carried out to obtain maximum thrust output for the muscle power input in terms of the actuation moment. It is observed that the generated thrust increases significantly when the frequency of the actuation moment approaches towards the natural frequencies of the free-free beam. A comparative study of the average thrust coefficient is carried out for these two cases.

D. Majumdar · P. Dhareshwar · S. Sarkar (B) Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] D. Majumdar e-mail: [email protected] P. Dhareshwar e-mail: [email protected] C. Bose Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_41

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1 Introduction Recently, a significant amount of research interest is geared towards the efficient design of aquatic propulsive devices like automated underwater vehicles (AUV) for futuristic applications like underwater monitoring. Most of these AUVs are primarily inspired by the natural propulsion of various underwater animals like fishes and thereby a proper understanding about the associated physics of fish locomotion is crucial. One can find substantial details about different aquatic propulsions in the recent review articles [1, 2]. Most of the natural swimmers generate propulsive forces by oscillating flexible flapping appendages inside the fluid. The main two mechanisms evolved in swimming of fishes are body and/or caudal fin (BCF) and median and/or paired fin (MPF) locomotion used for carangiform and anguilliform swimming, respectively, [3]. Both BCF and MPF propulsion mechanisms consist of mainly two types of body movement: in undulatory kinematics, a wave travels along the propulsive structure; whereas in oscillatory kinematics, the propulsive structure swivels on its base without exhibiting a wave formation. A carangiform fish produces thrust by primarily oscillating its caudal fin while its anterior part of the body moves relatively lesser and remains almost stationary. The posterior portion of the body or the caudal fin in this swimming mode is commonly studied as a canonical model of a continuous cantilever beam [4]. On the other hand, anguilliform movement involves the entire body of the fish to generate forward thrust unlike other species of fishes. Muscular contractions play a key role in anguilliform swimming and these contractions manifest as bending moment in the fish body. Cheng et al. [5], Toki´c and Yue [6] explored the possibility of modelling the fish body as a continuous dynamic beam with muscular bending moment acting on it. Williams et al. [7] have studied the muscle contractions and the nature of the force generated through EMG data obtained from direct experimentation. During the steady swimming, the fish body undergoes deformation due to the actuating moment induced by the muscles and thus disturbs the surrounding flow field. On the other hand, the motion of the body is manifested through the forces exerted on the body by the ambient fluid thereby making the combined system as a two-way coupled FSI problem. Due to the complexity of the FSI problems, Katz and Weihs [8] and Michelin and Llewellyn Smith [9] have employed a simplified potential flow solver to investigate the FSI dynamics of a flexible wing in an inviscid flow field. Alben et al. [10] have used a ‘body-vortex sheet’-based flow model coupled with an elastic structure in their numerical study. On the other hand, a reduced-order model has been used for the structural part by Vanella et al. [11]. Chen et al. [12] have explored a body–fluid interaction model in undulatory swimming, modelling the fish body as rigid links and the hydrodynamic forces were calculated based on Lighthill’s reactive theory. Recently, Alben [13] has shown through two-dimensional numerical simulations that the propulsive efficiency is maximised when the flapping and the structural natural frequencies are resonant. A number of theoretical [14] and

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experimental [15] studies have also been carried out to investigate the propulsive efficiency in various swimming modes. This paper employs a low-order FSI model comprising of a potential flow solver [16] coupled with an FEM-based structural solver. The primary objective of this paper is to develop a computational framework to represent different fish swimming modes by two different species-specific models (caudal fin model and full-body model). The present study aims to model the displacement and the orientation of the fish body in these two different modes of swimming as the deformation of a continuous beam with species-specific end-conditions, subjected to an external actuation. Finally, this study attempts to carry out a comparative parametric study between these two models to obtain optimum swimming efficiency in terms of maximum thrust output.

2 Computational Methodology In the present work, two different methodologies FEA and LVM [16], used for the structural and the flow parts, respectively, are weakly coupled in a staggered manner to simulate the combined FSI response. The structure is assumed to be an Euler–Bernoulli beam, and the fluid forces acting on the beam are modelled as distribution of circulatory force along the length of the body. The governing equation of motion of the continuous beam is given by ρs A

∂ 2 y(x, t) ∂ 4 y(x, t) 1 2 + E I = g(x, t) + ρ f U∞ LC L (x, t) , ∂t 2 ∂x4 2

(1)

where ρs is the density of the beam material, A is the area of cross section, L is the length of the beam, E and I are Young’s modulus and area moment of inertia of the cross section of the beam, respectively. The first term in the right-hand side g(x) corresponds to the actuation which causes the structure to oscillate inside the fluid. This term will be explained in detail for the two different models studied in this work in the later section. The second term in the right-hand side represents the lateral fluid load acting on the structure. Here, ρ f is the density of the fluid, U∞ is the free stream velocity and C L is the lift coefficient. For numerical simulation, the continuous beam is divided into N number of FEA elements and the PDE in Eq. (1) is converted to a set of ODEs. In FEA formulation, the governing equation for a single element in terms of the local coordinate system is given as (e) (e) } + { f fluid }. (2) [m (e) ]{ y¨ (e) } + [k (e) ]{y (e) } = { f act Thereafter, a global formulation for the entire beam is constructed by assembling all the local systems. The final form after the assembly is given by [M]{ y¨ } + [K ]{y} = {Fact } + {Ffluid } ,

(3)

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Fig. 1 Schematic of the flexible beam, the FE model and the flow model

where [M] and [K ] are the global mass and stiffness matrices, respectively, and {y} is the vector of unknown displacements and rotations at the finite element nodes; (e) } computed by the LVM {Ffluid } is the assembled force and moment matrix of { f fluid (e) and {Fact } is the assembled force and moment matrix of { f act } evaluated from the actuation term g(x, t). In LVM, the solid body is replaced by a distribution of singularity elements along the shape of the body, and the load calculation involves evaluation of the strength of these singularity elements. Each FEA element is further divided into n sub-elements for the application of LVM to determine the lateral fluid load on the beam; see Fig. 1. Each sub-element has a lumped vortex (singularity element), called the body bound vortex, at one-fourth position of its length, and the zero normal flow boundary condition is satisfied at the collocation points (three-fourth position of a sub-element). Each sub-element is assumed to move in the same way as the beam at every time step (elongation in the axial direction is neglected for the beam). The pitch velocity and the heave velocity of the sub-elements are taken into account while imposing the zero normal flow boundary condition at the collocation points. An unsteady vortex wake behind the oscillating body, created due to time-dependent movement of the body inside the fluid, also needs to be modelled. At every computational time step, a new wake vortex element is shed from the trailing edge of the body as shown in Fig. 1c. Kelvin’s circulation theorem is used to generate an additional equation along with the boundary conditions on the collocation points. The strengths of the body bound vortices and the new wake vortex are determined at every time step by solving a set of simultaneous linear algebraic equations. Unlike the body bound vortices, the free wake vortex elements are allowed to move with velocity induced by the surrounding flow field. The pressure difference on each sub-element is assumed to be constant over its length. Once the strengths of the vortex elements are calculated, the pressure difference (Δp j ) between the two surfaces for each sub-element is

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calculated using the unsteady Bernoulli’s equation [16]. The force per unit length in the lateral direction over any jth sub-element (q j ) is obtained by multiplying the pressure difference with the span (S) of the beam as q j = Δp j cos(α j )S ,

(4)

where α j is the clockwise angle made by the j-th sub-element with the positive x-axis. The distribution of force over an FEA element is converted to equivalent external forces and moments on its nodes to generate the vector {Ffluid } and is updated at every time step. The net thrust (T ) at every time step is evaluated as the summation of the forces in the longitudinal direction over the sub-elements. The structural Eq. (3) are numerically integrated using a fourth-order explicit Runge–Kutta method and {y} is updated at every time step using the structural solver. Therefore, a partitioned approach-based weak coupling has been implemented between the structural and flow solver. In this work, two different models have been studied using the developed coupled FSI solver: caudal fin model and full-body model. In carangiform locomotion mode, during steady swimming only the caudal fin undergoes an oscillatory motion whereas the anterior part of the body does not deform much. Therefore in this case it is sufficient to model only the posterior part of the fish body, i.e. the caudal fin to study the fluid–body interactions. On the other hand, in the anguilliform mode where the entire body participates in swimming, it becomes necessary to model the entire body instead of only the caudal fin. Even for the sub-carangiform fishes, which are the transition species in between the anguilliform and the carangiform; the entire body needs to be taken into consideration for fluid dynamic studies.

2.1 Model I: Caudal Fin Model The oscillatory motion of the caudal fin is modelled as a cantilever beam with a harmonic support motion. The fixed end of a cantilever beam is subjected to a timedependent motion as (5) y(0, t) = a(t) = a f sin(ωt) , where a f and ω are the amplitude and frequency of the support motion respectively. In this case, the solution (displacement in the vertical direction and the angle of rotation in the anti-clockwise direction) is assumed to consist of two parts (linear displacement due to support movement and dynamic displacement) as y(x, t) = ys (x, t) + w(x, t) ,

(6)

θ (x, t) = θs (x, t) + φ(x, t) ,

(7)

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where ys (x, t) = a(t)L 1 (x), θs (x, t) = 0 and L 1 (x) = 1 − Lx , 0 ≤ x ≤ L. Applying this assumptions, the governing Eq. (1) for Model I takes the form of ρs A

∂ 2 w(x, t) ∂ 4 w(x, t) ∂ 2 ys 1 2 + E I = −ρ A + ρ f U∞ LC L (x, t) . s 2 4 2 ∂t ∂x ∂t 2

(8)

In this case, the actuation term g(x, t) mentioned in Eq. (1) becomes g(x, t) = −ρs A

∂ 2 ys . ∂t 2

(9)

The above-mentioned governing equation (8) can be non-dimensionalized as ∂ 2 w¯ ∂ 4 w¯ ∂ 2 y¯s 1 CL + R1 4 = − 2 + , 2 ∂ t¯ ∂ x¯ ∂ t¯ 2 R2 R3 where R1 = A , L2

w¯ =

w , L

EI 2 L2 ρs AU∞ x¯ = Lx ,

a representative non-dimensional rigidity, R2 = y¯s = yLs and t¯ = tUL∞ .

(10) ρs ρf

, R3 =

2.2 Model II: Full-Body Model In Model II, instead of considering only the caudal fin, the entire fish body is modelled as a flexible structure. This is a realistic model to represent the swimming modes where the whole body undergoes an undulatory motion due to the actuation moment coming from the muscles. Both the sub-carangiform and the anguilliform mode of swimming can be studied with this model since the entire body is taken in consideration. In this model, the entire fish body is mathematically modelled as a continuous free-free beam and muscle actuation is modelled as a time-dependent bending moment acting along the length of the beam as suggested by Cheng et al. [5]. In this case, the governing equation of motion for the beam is given as ρs A

∂ 2 y(x, t) ∂ 4 y(x, t) ∂ 2 Mm (x, t) 1 2 + EI = + ρ f U∞ LC L (x, t) , 2 4 ∂t ∂x ∂x2 2

(11)

where Mm (x, t) is the muscle actuation moment varying along the length of the 2 body. Therefore, in this case, g(x, t) = ∂ M∂mx(x,t) . A model for this actuation term, 2 given by Hess and Videler [17], shows that the amplitude of the muscle moment is the maximum near the mid portion of the body whereas it becomes zero towards the ends and a wave of muscle moment travels along the length of the body. In this study, muscle moment is modelled as

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 Mm (x, t) = M0 1 − 4



1 x − L 2

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2 

  2π x , cos ωt − L

(12)

where M0 is the maximum value of the actuation moment supplied by the muscle and ω is the muscle actuation frequency. Equation (12) models a parabolic variation of the amplitude such that maximum moment can be attained at the midpoint of the body and also there is a linearly varying phase along the length. After non-dimensionalization, Eq. (11) becomes ∂ 4 y¯ ∂ 2 M¯ m 1 CL ∂ 2 y¯ + R = − + , (13) 1 2 4 2 ¯ ¯ ∂t ∂ x¯ ∂t 2 R2 R3 Mm where M¯ m = ρs AU represents the non-dimensional form of the actuation moment. 2 ∞L A parametric study, by varying the frequency of the actuation, is performed next using the above two models.

3 Results Since the present FSI dynamics is primarily dominated by the time scale of the oscillatory motion of the flexible structure, the time step for the coupled solver is chosen according to the frequency of the actuation through a time convergence study with respect to the system response. The number of FEA elements is decided to be 30 based on convergence test by comparing natural frequencies of the beam with the analytical results. The flow and the structural solvers have been validated separately. The peak lift coefficient obtained from the present flow solver shows a good match with Young [18] at low non-dimensional plunge velocities (as applicable to the present study) as shown in Fig. 2.

3.1 Model I The thrust coefficients (C T = 1 ρ TU 2 L ) obtained by varying the non-dimensional 2 f ∞ parameter R1 , a representative of non-dimensional bending rigidity of the cantilever beam, are presented in Table 1. The amplitude of the support motion (a f ) has been kept constant to a low value as the potential theory-based inviscid flow solvers are limited to small deformation assumption. The value of the non-dimensional parameter R2 is chosen to be sufficiently high so that the effect of viscosity becomes negligible. Comparison of the thrust coefficients for case 1 − 4 shows that the thrust coefficient decreases with an increasing bending rigidity. Case 5–6 depicts the effect of actuating frequencies in the first and second resonant mode with the cantilever beam.

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Fig. 2 Comparing the peak lift coefficients obtained from the present LVM-based potential flow solver with that of a Navier–Stokes solver (Young [18])

Table 1 Thrust coefficients for different test cases of caudal fin model Case R1 = R2 = ρρsf R3 = LA2 ω EI 2 L2 ρs AU∞

1 2 3 4 5 6

1.736 × 10−3 8.683 × 10−3 0.0174 0.0868 1.736 × 10−3 1.736 × 10−3

800 800 800 800 800 800

7.73 × 10−3 7.73 × 10−3 7.73 × 10−3 7.73 × 10−3 7.73 × 10−3 7.73 × 10−3

CT = T

1 2 2 ρ f U∞ L

20 20 20 20 3.66 22.95

1.6 × 10−3 2.701 × 10−5 2.454 × 10−5 1.380 × 10−6 0.0834 0.0770

Thrust coefficients corresponding to the first mode case (case 5) and the second mode case (case 6) show that the thrust produced in these cases is significantly higher than that of case 1. This implies that the support motion with an actuating frequency equal to one of the natural frequencies of the cantilever beam results in higher thrust. The corresponding deflection envelopes of the beam are presented in Fig. 3. The deflection envelopes observed in case 5 & 6 are similar to the first and second mode shapes of a cantilever beam, respectively. Alben [13] has also observed that the propulsive efficiency is maximised when the flapping and the structural natural frequencies are resonant. Although the natural frequencies of the cantilever beam are different from those of the coupled fluid-elastic system, significant increase in thrust coefficient has been observed around the structural natural frequencies.

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Fig. 3 Envelopes of beam deflections for different bending rigidity and actuation frequencies

3.2 Model II In this case, the actuation frequency is varied keeping R1 , R2 and R3 constant at 0.278, 240 and 0.005, respectively. The variation of the average C T with the muscle actuating frequency is shown in Fig. 4. Here, the frequency of the actuation is varied in between the fourth and sixth natural frequencies of the free-free beam. Even in this Model II, it is observed that significantly higher thrust is produced when the frequency of actuation approaches the structural natural frequencies of the free-free beam. On the other hand, the thrust drops when the actuation frequency is considerably different from the natural frequencies. Also, the thrust coefficients corresponding to the lower mode natural frequencies are seen to be higher than those obtained for the higher modes. The oscillation envelopes for this model at two different actuation moment frequencies (ω = 75 rad/s and ω = 155 rad/s) that are nearer to the fourth and fifth natural frequencies of a free-free beam are presented in Fig. 5. The deflection envelope (having only three nodes), shown in Fig. 5a, resembles the swimming mode of a sub-carangiform fish whereas the envelope in Fig. 5b is approximately similar to that of anguilliform fish swimming mode. Figure 4 shows that more thrust is produced

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Fig. 4 Variation of the thrust coefficient C T with the actuation frequency

Fig. 5 Beam deflection envelope of the free-free beam model at two different actuation muscle frequencies

at ω = 75 rad/s than at ω = 155 rad/s. Sub-carangiform swimming mode is seen to generate more thrust than anguilliform mode during steady swimming in real-life experiments as well [3]. The corresponding flow fields, presented in Fig. 6, show the presence of a vortex wake similar to the reverse Kármán wake behind the oscillating body. This can be a possible explanation for the generation of higher thrust at these frequencies. In Fig. 6, red dots represent an anti-clockwise vortex whereas the blue dots represent the clockwise vortex elements. Such well-organised wake behaviour is not observed for the actuation frequencies away from the natural frequencies.

Fig. 6 Flow field in the downstream of the beam at different actuation frequencies

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4 Conclusions This paper presents a coupled FSI solver developed by integrating an FEA-based flexible beam model to an inviscid LVM-based potential flow solver. Two different structural models are studied. Model I replicates the oscillating motion of the caudal fin and it shows that the chord-wise flexibility is a boon for getting higher thrust. On the other hand, the full-body model (Model II) shows that the thrust is maximised at the actuating frequencies nearer to the structural natural frequencies. The authors are currently working on improving the proposed FSI model to incorporate the effect of viscosity to investigate the present problem in the low Reynolds number regime.

References 1. Lighthill MJ (1969) Hydromechanics of aquatic animal propulsion. Ann Rev Fluid Mech 1:413–446 2. Müller UK, Van Leeuwen JL (2006) Undulatory fish swimming: from muscles to flow. Fish Fisheries 7(2):84–103 3. Sfakiotakis M, Lane DM, Davies JBC (1999) Review of fish swimming modes for aquatic locomotion. IEEE J Oceanic Eng 24(2):237–252 4. Zhang Z, Philen M, Neu WL (2010) A biologically inspired artificial using flexible matrix composite actuators: analysis and experiment. Smart Mat Struct 19(9):094017 5. Cheng JY, Pedley TJ, Altringham JD (1998) A continuous dynamic beam model for swimming fish. Phil Trans Royal Soc London B: Biol Sci 353(1371):981–997 6. Toki´c G, Yue DKP (2012) Optimal shape and motion of undulatory swimming organisms. Proc Royal Soc B: Biol Sci 279(1740):3065–3074 7. Williams T, Bowtell G, Curtin NA (1998) Predicting force generation by lamprey muscle during applied sinusoidal movement using a simple dynamic model. J Exp Biol 201(6):869–875 8. Katz J, Weihs D (1978) Hydrodynamics propulsion by large amplitude oscillations of an airfoil with chordwise flexibility. J Fluid Mech 88(3):485–497 9. Michelin S, Llewellyn Smith SG (2009) Resonance and propulsion performance of a heaving flexible wing. Phys Fluids 21:071902 10. Alben S, Witt C, Baker TV, Anderson E, Lauder GV (2012) Dynamics of freely swimming flexible foils. Phys Fluids 24:051901 11. Vanella M, Fitzgerald T, Predikman S, Balaras E, Balachandran B (2009) Influence of flexibility on the aerodynamic performance of a hovering wing. J Exp Biol 212(1):95–105 12. Chen J, Friesen WO, Iwasaki T (2011) Mechanisms underlying rhythmic locomotion: bodyfluid interaction in undulatory swimming. J Exp Biol 214(4):561–574 13. Alben S (2008) Optimal flexibility of a flapping appendage in an inviscid fluid. J Fluid Mech 614:355–380 14. Lighthill MJ (1970) Aquatic animal propulsion of high hydromechanical efficiency. J Fluid Mech 44:265–301 15. Triantafyllou MS, Triyantafyllou GS, Gopalkrishnan R (1991) Wake mechanics of thrust generation in oscillating foils. Phys Fluids A 3(12):2835–2837 16. Katz J, Plotkin A (2001) Low-speed aerodynamics. Cambridge University Press, 13 17. Hess F, Videler JJ (1984) Fast continuous swimming of saithe (Pollachius virens): a dynamic analysis of bending moments and muscle power. J Exp Biol 109(1):229–251 18. Young J (2005) Numerical simulation of the unsteady aerodynamics of flapping airfoils. PhD Thesis, The University of New South Wales

Comparison of Stochastic Responses of Circular Cylinder Undergoing Vortex-Induced Vibrations with One and Two Degrees of Freedom M. S. Aswathy and Sunetra Sarkar

Abstract Vortex-induced vibration of a circular cylinder is a major research topic due to the immense applications they have in daily and industrial scenarios. Large numbers of studies have been conducted in this area in numerical and experimental domains with focus on understanding the response types, understanding the range of lock-in, the flow behavior, etc. However, most of the studies till date have been done in a deterministic environment; on the pretext that all factors about the incoming flow and input parameters are exactly known. In real-time flows, there can be a significant amount of uncertainties associated with various system parameters, which are traditionally not taken into consideration for the system. For example, randomness associated with the incoming flow might have significant effect on the associated dynamics. In this study, we do a stochastic modeling on a circular cylinder exhibiting free vibrations with one and two degrees of freedom. For this, we use Duffing Van der Pol combined system and impose fluctuations at every time step in the input flow by modeling them through a uniform distribution. The transverse oscillations of each of the cases under the presence of noise are individually studied. It is seen that noise brings in new dynamical states to the cylinder response compared to the deterministic cases. It is observed that there is a considerable difference between the responses of the single degree of freedom and two-degree of freedom cylinder. These qualitative differences are investigated in detail in the current study. Keywords Vortex-induced vibrations · Uniformly distributed noise · Stochastic fluctuations

M. S. Aswathy (B) · S. Sarkar Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai, India e-mail: [email protected] S. Sarkar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_42

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1 Introduction Vortex-induced vibration of circular cylinder is a well-studied research area with its applications ranging from riser tubes to power transmission lines. Analytical formulations have been found useful to study such scenarios to understand the underlying physics of the problem. The dependence of structural response on the flow and other parameters can be better understood through such formulations. Numerous amounts of experimental results have been performed for cylinders with low and high mass to damping ratios [1, 2]. Bishop and Hassan [3], Hartlen and Curie [4] have formulated analytical models based on wake oscillators based on Van der pol or Rayleigh equations. Facchinetti et al. [5] formulated an accelerationcoupling model, which could effectively characterize the qualitative features of the VIV system. Later numerous studies came up stressing the importance of accounting the inline oscillations along with the transverse oscillations [6, 7]. Although numerous studies discussed the system responses in the deterministic scenario, not much of emphasis has been made to understand the role of stochastic fluctuations on the system and its dynamics. Noise, which might be inherently present in the system, or externally induced, can alter the characteristic features of the system. So it is important that we have an idea on the impact they will have on the system dynamics, which would be different from our deterministic studies. It is also important to extend our study to the stochastic bifurcations, which can occur in the VIV system. In the present study, we make attempts to capture the role of noise in the presence of a uniform incoming flow, on the response dynamics of a rigid cylinder, which is elastically mounted. We analyze the problem as two separate cases: In the first case, the cylinder is subjected to oscillations along only the transverse direction. In the second case, the cylinder oscillating with two degrees of freedom is addressed, that is, the cylinder is subjected to move in both the transverse and stream-wise directions. The transverse responses of each of these cases are expected to alter from their deterministic behaviors in the presence of noise. We analyze the responses at the prelock-in, lock-in, and post-lock-in cases and observe how the system behaves under the two different cases, while gradually increasing the bifurcation parameter.

2 Governing Equations and Mathematical Formulation There are two cases addressed in the present study: Case 1 addresses a single degree of freedom elastically mounted cylinder and Case 2 addresses a two-degree of freedom cylinder. The formulation of each of the cases is given as below: Case 1: The following model consists of a single degree of freedom elastically mounted circular cylinder subjected to incompressible constant flow. This model is based on the formulation by Facchinetti et al. [5]. The Strouhal number is assumed to be 0.2,

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as suggested by Blevins [8]. The structural dynamics is captured using a spring mass equation of the Duffing type and the wake dynamics has been captured using a Van der pol equation. The non-dimensional equations of motion in their final forms are y¨ + λ y˙ + α y y3 = M L 2 q

(1)

  q¨ + ε q 2 − 1 q˙ + 2 q = A y y¨

(2)

where y is the non-dimensional structural amplitude, q is the wake variable related to a fluctuating lift coefficient, λ is the sum of structural viscous damping and fluid added damping, that is, λ = 2ζ + Υ Ω μ. Here ζ is the structural reduced damping coefficient, ϒ is a stall parameter with a value of 0.8 [5].μ is the mass − C m ; m∗ = 248; m∗ ζ = 0.013;  is the ratio of the ratio, defined as m∗ = 4μ π vortex shedding frequency and the natural frequency of the cylinder. It is also defined as  = 1/(St ∗ U r ), with U r = f UD being the reduced velocity, where U is the n flow velocity, D is the diameter of the cylinder, and f n is the natural frequency in vaccum; ε and A y are empirical coefficients whose values are determined as 0.3 and 12, respectively, [5]. M is the mass number, a function of the mass ratio [5]. The acceleration-coupling model is used as the coupling model [5]. Case 2: The model incorporates a two degree of freedom elastically mounted circular cylinder undergoing oscillations both in transverse and inline directions and is based on the formulation of Srinil et al. [9]. The non-dimensional equations of motion in their final forms are as follows: ¨y + λ ˙y + α y y3 + β y yx 2 = M L 2 q − 2π M D 2

p˙y Ur

  q¨ + ε y  q 2 − 1 q˙ + 2 q = A y y¨  q˙x 2 x¨ + λ x˙ + f ∗ α x x 3 + β x x y2 = M D 2 p − 2π M L 2 Ur   2 p¨ + 2 εx p − 1 p˙ + 42 p = A x x¨

(3) (4) (5) (6)

Here, y and x are non-dimensional structural displacements in the transverse and stream-wise directions, respectively, q and p are the wake variables related to the fluctuating lift and drag coefficients. λ is a damping term (sum ofstructural viscous damping and fluid added damping), with λ x = 2ζ x f ∗ + Υ Ω μ, λ y =  2ζ y + Υ Ω μ; f ∗ is the cylinder natural frequency ratio taken as 1; μ is called a mass ratio, defined as m∗ = 4μ − C m , M D and M L are respectively defined as π C L0 D0 M D = 16πC St M = . A , 2 L x A y , ε x are empirical parameters with values 12, μ 16π St 2 μ

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12, 0.3, respectively. α x , α y ,β x , β y are taken as 0.7. The parameter ε y = b1 e b2 m with b1 = 0.00234 and b2 = 0.238. The values of ζ x , ζ y , m∗ are 0.023, 0.015, 3.5, respectively. More details on the formulation and the parameter values can be found in [9]. Now, we impose stochastic fluctuations on the input deterministic system and investigate the effects of noise on the overall system dynamics. Accounting for the flow uncertainties associated with the incoming flow velocity, we choose the reduced velocity as the fluctuating parameter. This has been mathematically done by modeling the input fluctuations through a uniform distribution. Hence the total reduced velocity becomes U r,total = U r + σ u , where σ is the noise intensity taken as 1 in the current case and u is a randomly fluctuating value from a uniform distribution and can vary between 0 and 1.

3 Results and Discussion We analyze the single and two-degree of freedom cases separately by gradually varying the bifurcation parameter. The deterministic responses of the system exhibit single frequency low amplitude sinusoidal oscillations in the non-lock-in regimes and single frequency high-amplitude oscillations during lock-in. We compare the transverse responses at three different regimes: pre-lock-in, lock-in and post-lock-in for both the cases. First of all, consider case 1, where the transverse responses of the single degree of freedom cylinder are observed. Figure 1 shows the responses for the one degree of freedom case for (a) Ur = 3 [pre-lock-in] (b) Ur = 4.1 [just before lock-in] (c) Ur = 4.6 [lock-in] (d) Ur = 5 [just after lock-in] (e) Ur = 6 [post-lockin] (f) Ur = 10 [post-lock-in]. It can be seen that in the pre-lock-in regions, there are heavy fluctuations for the responses. The responses seem to be heavily fluctuating across two orders of magnitude as we approach the lock-in region, as seen in Fig. 1b. This, we call as an intermittent response, since it is an intermediate stage showing signatures of both low and high amplitudes. At lock-in, the response behaves as a periodic high amplitude signal. Post-lock-in, again the response becomes a heavily fluctuating signal Fig. 1d, e. As the parameter is further increased and as we are away from the lock-in regime, the effect of noise in making the responses disorganized, subsides and the response settles down toward a periodic behavior as seen in Fig. 1f. Now, we analyze the time responses for a two-degree of freedom cylinder which exhibited single frequency periodic deterministic responses. Figure 2 shows the transverse response behaviors at four different parameters corresponding to different regimes such as (a) Ur = 2.7 [pre-lock-in] (b) Ur = 7.4 [lock-in] (c) Ur = 7.5 [just after lock-in] (d) Ur = 14 [post-lock-in]. It can be seen hat compared to the pre-lock-in case of the previous case, the responses are much more organized in this case. The rate of fluctuation of the response owing to the presence of noise has decreased in this case. Though there are fluctuations across the response, they are not as random and intermittent as in the pre-lock-in cases of case 1. Again, as the

Comparison of Stochastic Responses of Circular …

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Fig. 1 Time histories for single degree of freedom cylinder at a Ur = 3 [pre-lock-in] b Ur = 4.1 [just before lock-in] c Ur = 4.6 [lock-in] d Ur = 5 [just after lock-in] e Ur = 6 [post-lock-in] f Ur = 10 [post-lock-in]

Fig. 2 Time histories for two-degree of freedom cylinder at Ur = 2.7 [pre-lock-in] b Ur = 7.41 [lock-in] c Ur = 7.5 [just after lock-in] d Ur = 14 [post-lock-in]

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parameter is further increased, lock-in occurs at (b) Ur = 7.41. As in the previous case, lock-in region is not majorly influenced due to the presence of noise. Further, as the parameter is increased, there is an intermittent fluctuating state where amplitudes of two orders of magnitude are present. However, the qualitative appearance of this state is far different in comparison with the intermittent responses of Case 1. Unlike in the intermittent states of case 1, the high-amplitude signals occur as spiked responses occurring at random times in between the lower amplitude signals. Further as the parameter is increased, the responses settle down to a nearly periodic signal with very less fluctuations. From the observations above, it can be confirmed that the presence of inline oscillations has altered the behavior of the transverse responses significantly. First of all, the heavy fluctuating behavior of the responses in the non-lock-in states dies down substantially in the case of two-degree of freedom cylinder. Also, the intermittent state in case 2 is manifested itself as a burst-like behavior in between low amplitude signals compared to the random switching cross two zones of response behaviors as seen in case 1. These differences in the behavior of the transverse responses occur mainly due to the fact that the input fluctuations are along the stream-wise direction and impact of noise on the system is manifested mainly through the inline responses. In case 1, since only transverse oscillations were present, all the influence of input uncertainties along the flow direction were manifested as fluctuating and intermittent responses in the transverse oscillations themselves. But in case 2, it should be the inline oscillations which get largely influenced by the flow uncertainties. Therefore, in comparison with the inline responses, qualitative changes in the transverse responses are comparatively lesser and they exhibit more of deterministic like behavior. In other words, the presence of inline oscillations, in one way, dampens out the influence that noise has on the transverse dynamics and plays a significant role in determining the resultant dynamics.

4 Conclusions In this work, we analyze the response time histories of a circular cylinder in the presence of noise. The transverse responses of a freely vibrating cylinder having single degree of freedom and two degrees of freedom are separately analyzed. It has been observed that noise alters the deterministic cases considerably in both the cases. In the case of single degree of freedom cylinder, the non-lock-in states are characterized by heavy fluctuations and randomness. Prior and post-lock-in responses exhibit an intermittent switching across low and high amplitudes. But in the case of the two-degree of freedom cylinder, the transverse responses exhibit less rate of fluctuations. The qualitative appearance of the intermittent responses also looks different in this case compared to the single degree of freedom case. The non-lock-in responses resemble more of their deterministic counterparts instead of showing rapid fluctuations. The reason for these might be as follows: since the input uncertainties are along the stream-wise direction, their influence is also reflected more along the

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inline direction, leaving the transverse oscillations less sensitive to the presence of noise. A detailed investigation on this based on the analysis of inline responses, stochastic bifurcations, and time series analysis of the responses is being carried out by the authors currently.

References 1. Govardhan R, Williamson CHK (2000) Modes of vortex formation and frequency response of a freely vibrating cylinder. J Fluid Mech 420:85–130 2. Feng CC (1968) The measurement of vortex-induced effects in flow past a stationary and oscillating circular and D-section cylinders, PhD thesis, University of British Columbia, Vancouver 3. R.E.D. Bishop., A.Y. Hassan., The lift and drag forces on a circular cylinder oscillating in a flowing fluid, Proceedings of the Royal Society of London A, 277 (1964), pp. 51–75 4. Hartlen RT, Currie IG (1970) Lift-oscillator model of vortex-induced vibration. J Eng Mech Div EM5:pp. 577–591 5. Facchinetti ML, De Langre E, Biolley F (2004) Coupling of structure and wake oscillators in vortex-induced vibrations. J Fluids Struct 19.2:123–140 6. Jauvtis N, Williamson C (2003) Vortex-induced vibration of a cylinder with two degrees of freedom. J Fluids Struct 17(7):1035–1042 7. Jauvtis N, Williamsonc C (2004) The effect of two degrees of freedom on vortex-induced vibration at low mass and damping. J Fluid Mech 509:23–62 8. Blevins RD (1990) Flow-induced vibration. Van Nostrand Reinhold Co, Inc, New York, NY (USA) 9. Srinil N, Zanganeh H, Day A (2013) Two-degree-of-freedom viv of circular cylinder with variable natural frequency ratio: Experimental and numerical investigations. Ocean Eng 73:179–194

An Exact Solution for Magnetogasdynamic Shock Wave Generated by a Moving Piston Under the Influence of Gravitational Field with Radiation Flux: Roche Model G. Nath and Sumeeta Singh Abstract An exact solution for the propagation of shock waves in an ideal gas with radiation heat flux and magnetic field under the impact of gravitational field is obtained. A piston in motion with time obeying power law drives out the shock wave. The unsteady Roche model is comprised of a gas dispersed with spherical symmetry around a nucleus consisting of a large mass. The density and magnetic field are presumed to vary according to power law in the undisturbed medium. The flow variables fluid velocity, pressure, density, magnetic field, and radiation flux tend to zero as the piston is approached. The effects of change in values of Alfven Mach number, gravitational parameter, and initial density variation exponent on the flow variables are worked out in detail. The increase in value of Alfven Mach number or gravitational parameter has a decaying effect on shock strength. Keywords Shock waves · Radiation flux · Magnetogasdynamics · Roche model

1 Introduction Shock occurrence, such as global shock emerging from a supernova explosion passing outside through a stellar envelope or stellar pulsation or a shock originating from a point source like fabricated (man-made) explosions in the Earth’s atmosphere, has immense importance in astrophysics and space science. Many authors, Greifinger and Cole [1], Greenspan [2], Christer and Helliwell [3] have studied shock waves produced by abrupt explosion. Carrus et al. [4] took the energy release of the instant center explosion into consideration and studied propagation of explosion waves in stellar models. Similarity solutions in radiation-gas-dynamics have been obtained by Marshak [5], Elliott [6] and Wang [7] in which the flow is headed by a shock wave. G. Nath · S. Singh (B) Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, India e-mail: [email protected] G. Nath e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_43

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Wang [7] has discussed piston problem with radiation energy transfer. Many authors have studied different problems on shocks using the method of exact solutions based on the work of Vittie [8] (see [9–13]). Roche model is a model in which the whole mass of the star is supposed to be concentrated at the center and this point mass is surrounded by an evanescent envelope in which density is assumed to be varying inversely as some positive power of the distance from the center. In the case of actual stars, the greater part of their mass is concentrated very near to the center. Therefore, their structure comes much closer to the Roche model. Magnetic fields extend throughout the universe and have pivotal roles in a number of astrophysical affairs. Probably magnetic fields affect all astrophysical plasmas. Magnetic fields and radiation could rapidly release energy in flares and play a crucial role in energy and momentum transport [14]. The industrial applications of shocks in presence of magnetic field include—design of efficient coolant blankets in tokamak fusion reactors, drag mitigation in duct flows, control of turbulence of immersed jets in the steel casting process, and flow control plans for hypersonic vehicles (see Hartmann [15], Balick and Frank [16], Nath [17]). The significance of effects of radiation can be found in fields such as space research and nuclear power. A qualitative behavior of the gaseous mass can be estimated with the help of fundamental equations and equilibrium considering gravitational forces. For self-similar adiabatic flows, numerical solutions were obtained by Sedov [18] and Carrus et al. [4], independently, in self-gravitating gas. In Roche model, the gravitating effect of the gas itself can be disregarded compared with the attraction of the heavy nucleus at the center. Ashraf and Sachdev [9] have found the exact similarity solutions in radiation gasdynamics. Verma and Srivastava [19] have given exact solutions for magnetoradiative shock with increasing energy. Both used the product solutions of Vittie [8]. To the best of authors’ knowledge the problem of magnetogasdynamic shock wave generated by a moving piston under the influence of gravitational field with radiation flux using method of Vittie [8] has not been studied yet. Thus, the present work is an extension to the work of Ashraf and Sachdev [9] with consideration of magnetic field and Roche model and to the work of Verma and Srivastava [19] by considering the gravitational force, i.e., the Roche model. In our work the shock is considered to be driven out by a piston and different similarity transformations are also used. The “piston” is used to replicate blast waves and other similar phenomena in a model in order to simulate actual explosions and their effects, usually on a smaller scale. Thus, “piston” problem can be applied to quantify an estimate for the outcome from supernova explosions, sudden expansion of the stellar corona or detonation products, central part of star burst galaxies, etc. The radiation pressure and radiation energy are considered to be small in comparison to gas pressure and energy, respectively, and therefore only radiation flux is taken into account. It is necessary to consider a few simplifying idealizations in order to tender the problem tractable to obtain an exact similarity solution. The medium is presumed to be ideal. The density and magnetic field are assumed to be varying in the undisturbed medium. The flow variables fluid velocity U, density D, pressure P,

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magnetic field H, and radiation flux Q tend to zero as the piston is approached. The effects of variation in values of Alfven Mach number M A−2 , gravitational parameter G 0 , and density variation exponent δ on the flow variables are studied. The present study is represented in different sections. Section 2 comprises of the mathematical formulation of the problem. The flow is considered to be onedimensional, spherically symmetric, and electrically conducting perfect gas with radiation heat flux under the influence of magnetic field and gravitational field. In Sect. 3, the self-similarity transformations are used to transform the fundamental partial differential equations into ordinary differential equations, and the Vittie’s product solutions are used to obtain the solution of the present problem. Section 4 presents the results obtained and the discussion of the observations made. Section 5 is the brief summary of the results obtained and the significance of the study.

2 Fundamental Equations and Boundary Conditions The fundamental equations governing the one-dimensional spherically symmetric flow of an electrically conducting perfect gas with radiation heat flux in the presence of an azimuthal magnetic field in the gravitational field have the form [9, 20] ∂ρ ∂u 2uρ ∂ρ +u +ρ + = 0, ∂t ∂r ∂r r   ∂u ∂u 1 ∂p ∂h μh 2 Gm +u + + μh + + 2 = 0, ∂t ∂r ρ ∂r ∂r r r ∂h ∂h ∂u hu +u +h + = 0, ∂t ∂r ∂r r     ∂e ∂e p ∂ρ ∂ρ 1 ∂ Fr 2 +u − 2 +u + 2 = 0, ∂t ∂r ρ ∂t ∂r ρr ∂r

(1)

(2) (3)

(4)

where r and t symbolizes the independent coordinates of space and time; u, ρ, p, h, e, and F are the fluid velocity, density, pressure, azimuthal magnetic field, internal energy per unit mass, and radiation heat flux; m denotes the mass contained in a sphere of radius r; G denotes the gravitational constant; μ denotes the magnetic permeability. The equations of motion should be augmented with the equation of state and the internal energy per unit mass p = RρT ; e =

p , (γ − 1)ρ

(5)

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where R denotes the gas constant, γ denotes the adiabatic index, and T denotes the temperature of the gas. The flow variables immediately ahead of the shock front are u = u a = 0, ρ = ρa = ρ0 R δ , h = h a = h 0 R −α ,

(6)

where ρ0 , δ, h 0 , and α are constants, R denotes the shock radius and ‘a’ denotes the undisturbed medium. Using momentum Eq. (2), we obtain  Gmρ0 μ(α − 1)h 20 − R δ−1 , pa = (δ − 1) (δ − 1) 

(7)

where δ − 1 + 2α = 0. The jump conditions at the shock front under the consideration that there is transparency of radiative heat flux across the shock front are given by, namely [12, 20],

ea +

pa ρa

ρa C = ρn (C − u n ), h a C = h n (C − u n ), pa + 21 μh a2 + ρa C 2 = pn + 21 μh 2n + ρn (C − u n )2 , μh 2 μh 2 + 21 C 2 + ρaa − ρFa aC = en + ρpnn + 21 (C − u n )2 + ρnn − Fa = Fn ,

(8) Fn ρa C

where subscript “n” denote the conditions just behind the shock front. Using (8), the boundary conditions at the shock front are obtained to be ρn =

ρa ha , hn = , u n = (1 − β)C, β β

Fn = Fa ,

pn = Lρa C 2 ,

(9)

   1 1 1 where L = (1 − β) + γ M 1 − . The density ratio β (0 < β < 1) 2 2 + 2 β 2M A across the shock front is obtained by the quadratic equation

(γ + 1)β − 2

   2 −2 + γ 1 + M A − 1 β + (γ − 2)M A−2 = 0. M2

(10)

The fundamental equations of motion considered (1)–(4) are coupled non-linear partial differential equations. The continuity Eq. (1), momentum Eq. (2), and energy Eq. (4) arises from the conservation laws for mass, momentum, and energy, respectively, which govern the flow. Equation (3) is the magnetic field equation that is considered due to the presence of magnetic field. To obtain the solutions these basic equations of motion should be augmented with an equation of state which is given by (5) and also boundary conditions are required which are given by (9).

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3 Self-similarity Transformation The internal boundary of the flow-field at the rear of the shock is presumed a piston. dr For self-similar flow (Sedov [18]), the velocity of the piston C p = dtp is presumed to obey a power law (Wang [7]; Steiner and Hirschler [21]; Nath and Vishwakarma [22]) Cp =

dr p = V0 t n ; (n > −1) dt

(11)

where r p symbolizes the radius of piston, V0 symbolizes a dimensional constant, and n is a constant. With reference to the boundary conditions at the shock, for self-similarity it is required that the velocity of the shock C = ddtR and the velocity of the piston are proportional to each other, C=

dR = BV0 t n , dt

(12)

where B denotes a dimensionless constant. By making use of (12), the space and time coordinates could be presented in a dimensionless self-similarity variable η as η=

(n + 1)r r = . R BV0 t n+1

(13)

For self-similar solutions, the unknown flow variables are then written in the form as 2

u = rt U (η), ρ = ρa D(η), p = rt 2 ρa P(η), e = 3 √ √ μh = ρa rt H (η), F = ρa rt 3 Q(η).

r2 t2

E(η),

(14)

The expressions for Mach number and the Alfven Mach number are given as M2 =

C2 ρa C2 ρa ; M2A = . γpa μh2a

(15)

Condition for M and M A to be constant is n = − 13 . Using self-similarity transformations (14), the Eqs. (1)–(3) will be transformed into δ(n + 1)D(η) + [U (η) − (n + 1)]ηD  (η) + D(η)U (η) + D(η)ηU  (η) + 2U (η)D(η) = 0, 1 [U (η) − (n + 1)]ηH  (η) + δ(n + 1)H (η) − H (η) 2

(16)

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+ 3U (η)H (η) + ηH (η)U  (η) = 0,

(17)

[U (η) − (n + 1)]ηU  (η) − U (η) + U 2 (η)

G 0 (n + 1)3 1 + 2P(η) + P  (η)η + 2H 2 (η) + ηH (η)H  (η) + = 0, D η3 (18)  where G 0 = BGm 3 V 3 is the gravitational parameter and prime ( ) represents derivative 0 with respect to η. We have obtained a relation between M, M A and G 0 as

M2 =

(δ − 1)/(n + 1) .  (α−1)M −2 γ (n+1)A − G 0

The shock conditions (9) transform to (19), using self-similarity transformations (14), as U (1) = (1 − β)(n + 1), D(1) =

1 1 , H (1) = (n + 1)M A−1 , P(1) = L(n + 1)2 . β β (19)

If we employ “product solution” of the “progressive wave,” we pursue the solution of this problem in the form (see Vittie [8]) u=

a0 (t) r, t

(20)

ρ = (ϕ + 1) f (t)t −2ξ ψ κ−2 ,

(21)

p = ξ 2 f (t)t −2 b(t)ψ κ ,

(22)

κ

h = ξ f 2 (t)t −1 c0 (t)ψ 2 , 1

(23)

where ψ = r t −ξ , κ, and ξ are constants, and a0 , f, b, c0 are functions of t which satisfies the following equations   1 t (c0 )t t 2 + ξκ  , − − (κ + 1) 2 2f c0 +2 2    Gmt 2 μc02 μc02 (ϕ + 1)  2 b(t) + , + = a0 − a0 − (a0 )t t − 2 k ξ 2κ r3 a0 (t) =

ξκ −

t ( f )t f

, = κ

(24)

(25)

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535

where subscript “t” denotes differentiation w.r.t. t. Evidently these equations satisfy the Eqs. (1)–(3). After converting this solution to a similarity one, following which ‘a0 ’ ought to be constant, such that a0 = (n + 1)(1 − β), and we apply the boundary conditions (19) on (20)–(23) and obtain the solution as U (η) = (1 − β)(n + 1)η, D(η) =

1 κ−2 η , β

P(η) = (n + 1)2 Lηκ−2 , H (η) =

1 −1 κ M A (n + 1)η 2 −1 . β

(26) (27) (28) (29)

From Eq. (4) and using (14) we obtain δ P(η)(n + 1) D(η)  ∂  (n + 1)P(η)D  (η)η P(η)U (η)D  (η)η 1 Q(η)η5 = 0, + − + 2 2 4 D(η) D(η) D(η)η ∂η (30)

−2E(η) − (n + 1)ηE  (η) + 2U (η)E(η) + ηE  (η)U (η) −

which on integration gives Q(η) = L(n + 1)3 ηκ−2

 2 2(1 − β)η δ (κ − 2) (1 − β)(κ − 2)η . − + − + (γ − 1)(n + 1)(κ + 3) (γ − 1)(κ + 4) (κ + 3) (κ + 3) (κ + 4)

(31) Substituting Eqs. (26)–(29) into Eqs. (16)–(17), we get κ = 4δ + 14.

(32)

4 Results and Discussion An inequality obtained from Eqs. (27) and (28) that is essential for density to remain finite at the center and that the pressure is not negative anywhere, should hold κ > 2.

(33)

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Table 1 Values of the problem parameters at γ =

4 3

and δ = −0.8 for different

−2 MA

G0

M2

β

0.03

0.1

19.378

0.239954

values of M A−2 and G 0 0.06

Table 2 Values of the problem parameters at M A−2 = 0.03, G 0 = 0.2 and γ = δ

4 3

for different values of

0.3

6.65025

0.316013

0.5

4.01388

0.395232

0.1

18.578

0.283705

0.3

6.5534

0.35608

0.5

3.97839

0.432252

δ

M2

β

K

−0.2

6.19266

0.324802

13.2

−0.4

7.37705

0.304353

12.4

−0.6

8.61244

0.289166

11.6

−0.8

9.9022

0.277454

10.8

For the purpose of obtaining the variation of flow variables, the values taken for the physical parameters are γ = 43 ; M−2 A = 0, 0.03, 0.08 and G 0 = 0.1, 0.2, 0.3, 0.5 (see Nath [17, 22]). For relativistic gases the value of γ is taken to be 43 , which is significant for the interstellar medium [14]. The influence of magnetic field on the flow behind the shock is remarkable when M−2 A ≥ 0.01 [23], due to this reason are taken. The value of G above values of M−2 0 = 0 represents the case without A gravitational field which corresponds to the solution of Verma and Srivastava [19] in the case of strong shock. The attained solutions demonstrate that the radial fluid velocity, density, pressure, magnetic field, and radiation flux decrease as we move inward from the shock front and tend to zero as the piston is approached. Table 1 shows values of the problem parameters at γ = 43 and δ = −0.8 for different values of M A−2 and G 0 . Table 2 shows values of the problem parameters at M A−2 = 0.03, G 0 = 0.2 and γ = 4/3 for different values of δ. Figure 1 exhibits the variation of the fluid velocity U(η), density D(η), pressure P(η), magnetic field H(η), and radiation flux Q(η) for γ = 43 , δ = −0.8 and different values of M A−2 and G 0 . Figure 2 exhibits the variation of the fluid velocity U(η), density D(η), pressure P(η), magnetic field H(η), and radiation flux Q(η) for γ = 43 , M A−2 = 0.03, G 0 = 0.2 with δ = −0.2, −0.4, −0.6, −0.8. From Tables 1 and 2, it is observed that with increase in M A−2 or G 0 , the density ratio β increases, i.e., the shock strength decreases. This infers that consideration of magnetic field or gravitational field has decaying effect on shock wave. From Figs. 1 and 2, it is observed that the flow variables fluid velocity U, density D, pressure P, magnetic field H, and radiation flux Q decrease as we move inward from the shock to the piston. Also these flow variables tend to zero as the piston is approached. The consideration of constant Mach number and Alfven Mach number sets a value of piston velocity index n as n = −1/3.

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Fig. 1 Distribution of the flow variables for γ = 43 and δ = −0.8: (1) M A−2 = 0.03, G 0 = 0.1; (2) M A−2 = 0.03, G 0 = 0.3; (3) M A−2 = 0.03, G 0 = 0.5; (4) M A−2 = 0.08, G 0 = 0.1; (5) M A−2 = 0.08, G 0 = 0.3 (6) M A−2 = 0.08, G 0 = 0.5

538

Fig. 2 Distribution of the flow variables for γ = δ = −0.4; (3) δ = −0.6; (4) δ = −0.8

G. Nath and S. Singh

4 3,

M A−2 = 0.03, G 0 = 0.2:(1) δ = −0.2; (2)

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4.1 Analysis of Influence of Increase in Strength of the Magnetic Field, i.e., Increase in Value of M −2 A on Flow Variables and Shock Strength As in case of increase of strength of the magnetic field, there is an increase in density ratio β (see Table 1), i.e., shock strength decreases. As the value of M A−2 increases, the flow variables fluid velocity U, density D, pressure P, and radiation flux Q decrease (see Fig. 1a–c, e).

4.2 Analysis of Influence of Increase in Value of G 0 on Flow Variables and Shock Strength As the value of G 0 increases, the density ratio β also increases (see Table 1), i.e., shock strength decreases. As in increase in value of G 0 , the flow variables fluid velocity U, density D, and magnetic field H decreases (see Fig. 1a, b, d); however, pressure increases with increase in G 0 (see Fig. 1c). Radiation flux Q decrease in general for M A−2 = 0.03 but it increases in general for M A−2 = 0.08 (see Fig. 1e).

4.3 Analysis of Influence of Increase in Value of Initial Density Variation Exponent δ on Flow Variables All the flow variables fluid velocity U, density D, pressure P, magnetic field H, and radiation flux Q decreases (see Fig. 2a–e) with increase in value of density exponent δ.

5 Conclusions The problem of shock waves with radiation flux and magnetic field under the impact of gravitational field is considered with the supposition that the shock to be driven out by a piston in motion. The significance of such type of study could be observed in a number of astrophysical affairs as well as in problems relevant to blast waves. For calculations, the value of adiabatic exponent is taken to be 4/3 which is applicable to relativistic gases and thus significant for the interstellar medium. The influence of magnetic field on the flow behind the shock is remarkable when M A−2 ≥ 0.01 [23], thus in the present problem values of M A−2 are taken to be 0.03 and 0.08. The value G 0 = 0 (i.e., the case without gravitational field) corresponds to the solution of Verma and Srivastava [19] for strong shock. The consideration of constant Mach

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number and Alfven Mach number sets a value of piston velocity index n as n = − 1/3. The structure of the actual stars comes much closer to the Roche model as the substantial part of their mass is dense very near the center. Shocks are pervasive all over the observed universe. Radiation plays notable role in energy transport over the vast distances encountered between stellar objects, and can significantly modify the dynamics of a shock or blast wave. The analysis of radiative shocks has been a dynamic area of theoretical, experimental, and numerical research for many years [24]. Actual explosions and their influences can be simulated using “piston,” which replicates the blast waves and other alike events in a model, generally on a smaller scale. Hence “piston” problem could be applied to quantify an approximation of the consequences from abrupt expansion of the stellar corona or the detonation products, supernova explosions, etc. [14]. From the present study, we may conclude the following (i) The flow variables U, D, P, H, and Q tend to zero as the piston is approached. (ii) The consideration of magnetic field or gravitational field have decaying effect on shock. (iii) The increase in value of M A−2 and G 0 decreases the flow variables U and D. The increase in value of δ has same effect on these variables. (iv) P and Q decrease in both the cases of increase in value of M A−2 and δ. (v) The consideration of constant Mach number and Alfven Mach number sets a value of piston velocity index n as n = −1/3. Acknowledgments Sumeeta Singh gracefully acknowledges DST, New Delhi, India to provide INSPIRE fellowship, IF No.: 150736, for pursuing research work.

References 1. Greifinger C, Cole JD (1962) Similarity solution for cylindrical magnetohydrodynamic blast waves. Phys Fluids 5:1597–1607 2. Greenspan HP (1962) Similarity solution for a cylindrical shock-magnetic field interaction. Phys Fluids 5:255–259 3. Christer AH, Helliwell JB (1969) Cylindrical shock and detonation waves in magnetogasdynamics. J Fluid Mech 39:705–725 4. Carrus P, Fox P, Hass F, Kopal Z (1951) The propagation of shock waves in a stellar model with continuous density distribution. Astrophys J 113:496–518 5. Marshak RE (1958) Effect of radiation on shock wave behavior. Phys Fluids 1:24–29 6. Elliot LA (1960) Similarity methods in radiation hydrodynamics. Proc R Soc Lond 258:287– 301 7. Wang KC (1964) The piston problem with thermal radiation. J Fluid Mech 20:447–455 8. Mc Vittie GC (1953) Spherically symmetric solutions of the equations of gasdynamics. Proc R Soc Lond A220:339–355 9. Ashraf S, Sachdev PL (1970) An exact similarity solution in radiation gasdynamics. Proc Indian Acad Sci A71:275–280

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10. Nath Onkar, Ojha S, Thakar HS (1991) A study of stellar point explosion in a self-gravitating radiative magneto-hydrodynamic medium. Astrophys Space Sci 183:135–145 11. Srivastava SK, Singh RK (1983) An exact similarity solution for a spherical shock wave in a self-gravitating system. Astrophys Space Sci 92:365–372 12. Vishwakarma JP, Patel Nanhey (2015) Magnetogasdynamic cylindrical shock waves in a rotating nonideal gas with radiation heat flux. J Eng Phys Thermodyn 88:521–530 13. Vishwakarma JP, Srivastava RC, Kumar Arun (1987) An exact similarity solution in radiation magneto-gas-dynamics for the flows behind a spherical shock wave. Astrophys Space Sci 129:45–52 14. Nath G, Singh S (2017) Flow behind magnetogasdynamic exponential shock wave in selfgravitating gas. Int J Non-Linear Mech 88:102–108 15. Hartmann L (1998) Accretion processes in star formation. Cambridge University Press, Cambridge 16. Balick B, Frank A (2002) Shapes and shaping of planetary nebulae. Annu Rev Astron Astrophys 40:439 17. Nath G (2011) Magnetogasdynamic shock wave generated by a moving piston in a rotational axisymmetric isothermal flow of perfect gas with variable density. Adv Space Res 47:1463– 1471 18. Sedov LI (1959) Similarity and dimensional methods in mechanics. Academic Press, New York 19. Verma BG, Srivastava RC (1981) A model of flare-produced magneto-radiative shock with increasing energy. Astrophys Space Sci 78:95–103 20. Verma BG, Srivastava RC, Khan AH (1982) Propagation of blast waves in Roche model. Astrophys Space Sci 85:465–476 21. Steiner H, Hirschler T (2002) A self-similar solution of a shock propagation in a dusty gas. Eur J Mech B/Fluids 21:371–380 22. Nath G, Vishwakarma JP (2016) Magnetogasdynamic spherical shock wave in a non–ideal gas under gravitational field with conductive and radiative heat fluxes. Acta Astronaut 128:377–384 23. Rosena P, Frankenthal S (1976) Equatorial propagation of axisymmetric magnetohydrodynamic shocks I. Phys Fluids 19:1889–1899 24. Nath G, Vishwakarma JP (2014) Similarity solution for the flow behind a shock wave in a non-ideal gas with heat conduction and radiation heat-flux in magnetogasdynamics. Commun Nonlinear Sci Numer Simul 19:1347–1365

Isogeometric Collocation for Time-Harmonic Waves in Acoustic Problems M. Dinachandra, S. S. Durga Rao, and R. Sethuraman

Abstract Isogeometric Analysis introduced by Hughes et al. [1] has gained importance in the recent times due to its ability to capture accurate solutions and exact geometry representations. In the present study, IGA-Collocation is extended for oscillatory problems in acoustics. Numerical solutions of oscillatory problems often suffer from numerical dispersion errors, which demands use of minimum of ten nodes per wavelength or higher order bases. But employing higher order bases in IGA based on Galerkin approach (IGA-G) is computationally expensive. To overcome this issue, we employed IGA based on Collocation (IGA-C) [2] which is often regarded as a rank sufficient one-point quadrature scheme and has the potential to reduce the computational cost. In the present study, IGA-C with higher order bases is employed for solving rectangular waveguide and oscillating cylinder problems with different wave numbers. The performance of IGA-C is compared with the IGA-G in terms of efficacy and computational time. From the results, the potential of IGA-C is clearly observed in solving wave problems. Keywords Isogeometric analysis · Collocation · Acoustic waves · Time-harmonic · Waveguides · NURBS

1 Introduction Many industrial applications, like automobile, medical industry, etc., use sound wave propagation. With the growing interest in quieter products, new products are designed with concerns regarding noise production. Numerical simulations are often used to understand the performance of the product during its design cycle. The finite element method is often considered as a standard tool for these simulations. During the product development cycle, often many changes or iterative design procedures are carried out. In order to simulate the behavior, the Computer-Aided-Design (CAD) model is M. Dinachandra · S. S. Durga Rao · R. Sethuraman (B) Department of Mechanical Engineering, IIT Madras, Chennai 600036, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_44

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brought into the analysis environment. In Finite Element Analysis (FEA), a CAD model is first meshed into a number of elements and the analysis is done over the meshed computational domain. Geometry errors are also present while converting the CAD geometry to a meshed domain. With the iterative procedures in product design, the task of meshing and re-meshing takes almost 80% of the time in analysis. On top of the time consumption, the numerical solutions often contain many errors like discretization, geometry and dispersion errors, etc. In acoustic problems, Ihlenburg et al. [3] reported the error estimates under H1 semi-norm given as ||e|| H 1 ≤ C1 ( p)θ p + C2 ( p)kθ 2 p

(1)

where θ = kh/(2 p); k, p, and h represent the wavenumber, polynomial degree, and element size; and the first and second term in the right-hand side represent the approximation and dispersion errors, respectively. Based on this estimate, one can refer that in wave propagation, higher order bases are more suitable than low order bases as the convergence is poor for low orders. Moreover, low order elements in FEM show slow convergence rate and become more demanding at high frequencies, i.e., require more number of degrees of freedom (dof). In FEM, higher order methods are available but are seldom used due to their poor performance in predicting higher modes and optical branching in the eigenspectra. Recently, Isogeometric methods based on Galerkin weak forms (IGA-G) proposed by Hughes et al. [1] in 2005 are gaining importance in the field of engineering analysis. In IGA, the geometry basis is again used to approximate the field variables. This relaxes the need of converting the CAD model into a meshed structure and geometry approximation is not present. Non-Uniform Rational B-Splines (NURBS) are often considered as standard tools in the CAD industry. The NURBS basis has certain properties like higher continuity, positive basis, Partition of Unity, etc. which makes NURBS more suitable for CAD than other bases available in the literature. A higher continuity is also obtained in using NURBS bases in IGA. IGA has been considered to study the behavior of many engineering systems, viz. structural mechanics [4], fluid mechanics [5], electromagnetics [6], etc. In IGA, the direct link to CAD with analysis makes it more suitable for use in product design. In doing so, many other gains like higher accuracy, smoother field variables, etc. are observed. Isogeometric Analysis (IGA-G) based on weak form for acoustic problems is reported to have many advantages over traditional FEM by many researchers. On the other hand, employing higher order NURBS basis is costly as they require more quadrature points when used in the weak form. Cottrell et al. [7] studied structural vibrations using IGA and reported the absence of optical branching in the frequency spectra even with higher order NURBS bases. They reported that NURBS bases show lesser dispersion error when compared to the Lagrangian bases often considered in FEM. Simpson et al. [8] proposed Isogeometric Boundary Element Method (BEM) using NURBS bases for acoustics and reported higher convergence compared to BEM based on Lagrangian bases. Peake et al. [9] proposed plane wave enriched eXtended Isogeometric Boundary Element Method (XIBEM) in two-dimensional acoustic wave scattering problems and reported higher convergence compared to conventional Lagrangian based

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BEM. Wu et al. [10] employed IGA for modeling interior acoustic problems and reported better performances of IGA over FEM. Coox et al. [11] studied the performance of IGA for two-dimensional interior acoustic problems and demonstrated that the errors shown in the eigenfrequencies and the eigenvectors with IGA are lesser than conventional FEM. Isogeometric Collocation (IGA-C) proposed by Auricchio et al. [2] in 2010, makes use of the advantages of Collocation techniques and the features of NURBS. IGA-C is often treated as a one-point quadrature based method as it needs to evaluate the functions at only n points to form the system matrices for a domain with n control points. IGA-C is reported to have less computational cost compared to FEM and IGA-G when the computational time is accounted. The cost comparison of IGA-C with IGA-G and FEM is reported by Schillinger et al. [12]. In the present study, IGA-C is extended for time-harmonic waves and its performance is studied. Higher order polynomials are considered which wave problems often demand and the results obtained are compared with IGA-G results in terms of efficiency and computational time. In Sect. 2, the methodology is explained. A brief introduction to NURBS is presented in Sect. 3. In Sect. 4, the results obtained are reported and discussed.

2 Methodology The governing equation for time-harmonic waves in acoustics is the Helmholtz equation with the primary variable as the pressure or the velocity potential. A schematic of the acoustic problem is shown in Fig. 1. In Fig. 1, Ω − represents the solid domain which radiates or scatters any incoming wave; Ω + represents the unbounded domain where the acoustic waves propagates; φ represents the scattered or radiated potential; Γ D and Γ N represent the Dirichlet and Neumann boundary conditions; and n is the unit normal to Γ N considering Ω + . For the well posedness of the problem, Sommerfield boundary condition is applied at infinity given as ∇φ · n − ikφ = o

  1 as R → ∞ on Γ∞ R

(2)

From the computational aspect, as an unbounded problems cannot be solved, the unbounded problem is truncated at Γ A and the domain Ω is considered as the computational domain. Thus, the bounded problem can be posed as Problem Statement: Find φ ∈ C 2 (Ω) such that Δφ + k 2 φ = 0 in Ω

(3)

φ = g on Γ D ∇φ · n = h on Γ N

(4) (5)

∇φ · n − ikφ = f on Γ A

(6)

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Fig. 1 Schematic of the exterior acoustic problem. a Unbounded domain. b Truncated computational domain

In the above equations, k denotes the wave number; the imaginary number is rep√ resented as i = −1 . Equations (4) and (5) represent the Dirichlet and Neumann boundary conditions. Equation (6) represents the Robin boundary condition, where the function f can be computed from analytical solution. The weak form of the above problem is stated as   Finite-dimensional weak form: Find φ h ∈ S h ⊂ S = φ ∈ H1 (Ω); φ|Γ D = g  such that ∀ ψ h ∈ V h ⊂ V = ψ ∈ H1 (Ω); ψ|Γ D = 0 , the below holds a(ψ h , φ h ) = b(ψ h )

(7)

where  a(ψ h , φ h ) =  b(ψ ) =



 Ωh

h

∇ψ h · ∇φ h dΩ − k 2  ψ h dΓ + h

Γ Nh

Γ Ah

Ωh

ψ h f dΓ

ψ h φ h dΩ −

Γ Ah

ikψ h φ h dΓ (8) (9)

where φ h and ψ h are the trial and test functions, respectively; S and V are the corresponding trial and test function spaces, respectively, and H1 (Ω) represents the Sobolev space where the function and its first derivatives are square-integrable.

3 B-Splines and NURBS B-Spline curves are piecewise polynomials connected through a certain continuity requirement. They are formed by a set of linear combination of B-Spline bases. Given a finite set of parametric coordinates Ξ = {ξ1 , ξ2 , ξ3 , ..., ξn+ p+1 }, a set of n B-Spline bases of polynomial degree p is given by De Boor’s algorithm as

Isogeometric Collocation for Time-Harmonic Waves in Acoustic Problems



1 if ξ ∈ [ξi , ξi+1 ) 0 otherwise

(10)

ξi+ p+1 − ξ ξ − ξi p−1 p−1 N (ξ ) + N (ξ ) ξi+ p − ξi i ξi+ p+1 − ξi+1 i+1

(11)

Ni0 (ξ ) = p

Ni (ξ ) =

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The bivariate B-spline basis is constructed by taking tensor product of two univariate B-Splines basis functions defined through the knot vectors Ξ1 = {ξ1 , ξ2 , ξ3 , ..., ξn+ p+1 } and Ξ2 = {η1 , η2 , η3 , ..., ηm+q+1 } as given below pq

p

q

Ni j (ξ, η) = Ni (ξ )N j (η)

(12)

where p and q represent the polynomial degree along the two parametric coordinates, respectively; n and m represent the number of bases along the two parametric coordinates, respectively. Even though, B-Splines can model object easily, they cannot model conic sections accurately. A weighted B-Spline known as NURBS can model conic sections efficiently. A univariate set of n NURBS basis and a bivariate set of n × m NURBS bases are given as p

wi Ni (ξ ) p Ri (ξ ) = n p a=1 wa Na (ξ ) p

pq Ri j (ξ, η)

(13)

q

wi j Ni (ξ )N j (η) = n m p q a=1 b=1 wab Na (ξ )Nb (η)

(14)

In the above equations, wi are basically the weights associated with each basis function which gives a much better control over the shapes. The corresponding NURBS curve and NURBS surface are given as C(ξ ) =

n

p

Ri (ξ ) P i

(15)

i=1

S(ξ, η) =

m n

pq

Ri j (ξ, η) P i j

(16)

i=1 j=1

where P i and P i j are the control points of the NURBS curve and surface, respectively.

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4 Isogeometric Collocation In Isogeometric Collocation, the test functions are considered as Dirac-delta functions and collocated at specific points known as collocation points (ξi ). In the present study, the collocation points are considered as the Greville abscissae points given as ξ=

ξi+1 + ξi+2 + · · · + ξi+ p p

(17)

In Isogeometric Collocation method, the trial functions are considered to be spanned by the NURBS basis. Thus, the discrete trial and test functions are considered as φh =

m n

pq

Ri j φi j and ψ h = δ(ξ − ξi )δ(η − η j )ψi j

(18)

i=1 j=1

Thus, on a global numbering basis, we have the trial and test functions as φh =

nd

Ri (ξ )φi and ψ h =

nd

i=1

δ(ξ − ξ i )ψi

(19)

i=1

where n d = n × m represents the total number of basis functions; ξ = (ξ, η) and ξ z=( j−1)n+i = (ξi , η j ) represent the parametric coordinates and the Greville points in the bivariate parametric space, respectively. Performing integration by parts and using the above trial and test functions in the weak form, we have 

 Ωh

(Δφ h + k 2 φ h )ψ h dΩ +

 Γ Nh

(∇φ h · n − h)ψ h dΓ +

Γ Ah

(∇φ h · n − ikφ h − f )ψ h dΓ = 0

(20)

The above equations are simplified further using  Ω

Θ(ξ )δ(ξ − ξ i ) dΩ = Θ(ξ i )

(21)

where Θ(ξ ) is a function of ξ and ξ i is the collocation point. Thus, on discretization, we get a set of algebraic equation given as [K ] {φ} = {F}

(22)

where the corresponding row entries of K can be computed for each collocation points.

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5 Numerical Results Two case studies are considered viz., (i) wave propagation in a rectangular duct and (ii) an oscillating cylinder in its first mode. NURBS of polynomial degree p = 4, 6, and 8 are considered and solved using IGA-C and IGA-G. The performances of both methods are shown accounting the accuracy and computational time. The codes are written in MATLAB V16. The machine used for the computation is a 32 GB RAM with 4-core i7-4770 3.4 GHz. To measure the error, Relative L 2 error norm is used which can be computed as shown below Relative L 2 Error =

||φ − φ h || L 2 (Ω) ||φ|| L 2 (Ω)

(23)

where ||φ|| L 2 (Ω) represents the L 2 norm of φ over Ω given as  ||φ|| L 2 (Ω) =

Ω

|φ|2 dΩ

 21 (24)

5.1 Wave Propagation in a Rectangular Duct A wave propagation problem in a rectangular duct is considered as shown in Fig. 2 with the described boundary conditions and domain size. The analytical solution is given in [13]. Three studies are considered by employing NURBS polynomial of degree p = 4, 6, and 8 for wave number k = 40 and m = 12. The obtained results are listed in Tables 1, 2, and 3 respectively. In Table 1, we observe that initially IGA-C gives lesser accuracy and requires more degrees of freedom compared to IGA-G. On the other hand, if we fixed the accuracy

Fig. 2 Schematic of the wave propagation in a duct

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Table 1 Results obtained with IGA-G and IGA-C with NURBS of polynomial degree p = 4 for wave propagation problem in a duct Sl. no. Method Mesh Rel. L 2 error Total time (s) 1 2 3 4 1 2 3 4 5 6

IGA-G

IGA-C

44 × 24 54 × 29 64 × 34 74 × 39 44 × 24 64 × 34 84 × 44 104 × 54 124 × 64 144 × 74

0.2534 0.0259 0.0044 0.0011 1.129 0.9312 0.6664 0.3314 0.1637 0.0884

16.16 23.03 34.01 43.53 0.54 1.12 1.98 3.41 6.39 11.06

Table 2 Results obtained with IGA-G and IGA-C with NURBS of polynomial degree p = 6 for wave propagation problem in a duct Sl. no. Method Mesh Rel. L 2 error Total time (s) 1 2 3 4 1 2 3 4 5 6

IGA-G

IGA-C

26 × 16 36 × 21 46 × 26 66 × 36 46 × 26 66 × 36 86 × 46 106 × 56 126 × 66 146 × 76

1.0131 0.7099 0.0084 0.00006 0.9277 0.1868 0.0325 0.0084 0.0028 0.0011

19.11 31.89 47.95 92.42 0.66 1.37 3.26 6.77 12.44 39.85

Table 3 Results obtained with IGA-G and IGA-C with NURBS of polynomial degree p = 8 for wave propagation problem in a duct Sl. no. Method Mesh Rel. L 2 error Total time (s) 1 2 3 4 1 2 3 4 5

IGA-G

IGA-C

28 × 10 38 × 23 48 × 28 68 × 38 48 × 28 68 × 38 88 × 48 108 × 58 128 × 68

1.003755 0.182076 0.000495 0.000004 0.4367 0.013 0.0011 0.000181 0.000041

58.57 94.81 137.69 257.44 0.58 2.37 5.77 11.71 21.87

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as 25%, the time consumed by IGA-G is 16.16 s while in IGA-C, we observe an accuracy of 16% with a computational time of 6 s. In Table 2, we observe that IGA-G gives an accuracy of 0.84% with 1196 dof and consumes 47.95 s. On the other hand, IGA-C gives an accuracy of 0.28 % with 8316 dof and consumes 12.44 s. Similar results are also obtained for p = 8 as listed in Table 3. For an accuracy of 0.0495%, IGA-G requires 1344 dof and consumes 137.69 s while IGA-C gives an accuracy of 0.0181% with 6264 dof and consumes 11.71 s. This clearly shows the performance of IGA-C in comparison to IGA-G when computational time is accounted.

5.2 Radiation by an Oscillating Cylinder in the First Mode An oscillating cylinder in the first mode is considered as shown in Fig. 3. A unit cylinder is considered, i.e., Ri = 1 unit and the domain is truncated at Ro = 2 units. As the problem is symmetric about x-axis, only half domain is considered using symmetry boundary conditions. A wave number k = 50 is considered and solved using IGA-G and IGA-C. The analytical solution is given in [14]. The results obtained with IGA-G and IGA-C with NURBS polynomial p = 4, 6 and 8 are listed in Tables 4, 5 and 6, respectively. The mesh details are given for a quarter domain only. From Table 4, we observe that IGA-C initially consumes lesser time in assembling the system matrices compared to IGA-G. For a fixed accuracy, IGA-C gives lesser error but consumes more degrees of freedom compared to IGA-G. In Table 5, the same trend as mentioned above is observed. In the last entry of the table, we observe that IGA-G shows an accuracy of 0.0071% with 4005 dof and consumes 243.13 s. On the other hand, IGA-C shows an accuracy of 0.0071% with 21945 dof and consumes 165.72 s which is around 32% reduction in the computational cost. In Table 6, we observe that for an accuracy of around 0.0004%, IGA-C consumes lesser time of 133.35 s while IGA-G consumes 673.3 s which is close to five times reduction in the simulation time.

Fig. 3 Schematic of radiation by an oscillating cylinder in the first mode

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Table 4 Results obtained with IGA-G and IGA-C with NURBS of polynomial degree p = 4 for the radiation by an oscillating cylinder in the first mode Sl. no. Method Mesh Rel. L 2 error Total time (s) 1 2 3 4 1 2 3 4 5 6 7 8

IGA-G

IGA-C

13 × 25 23 × 45 33 × 65 43 × 85 33 × 65 43 × 85 53 × 105 63 × 125 73 × 145 83 × 165 93 × 185 103 × 205

0.98455 0.687426 0.012558 0.001212 0.833685 0.275251 0.112389 0.053667 0.028713 0.016714 0.010378 0.00678

6.85 19.47 40.36 74.43 1.42 2.52 4.41 9.61 17.89 30.62 53.17 74.06

Table 5 Results obtained with IGA-G and IGA-C with NURBS of polynomial degree p = 6 for the radiation by an oscillating cylinder in the first mode Sl. no. Method Mesh Rel. L 2 error Total time (s) 1 2 3 4 1 2 3 4 5 6 7 8

IGA-G

IGA-C

15 × 29 25 × 49 35 × 69 45 × 89 35 × 69 45 × X89 55 × 109 65 × 129 75 × X149 85 × 169 95 × 189 105 × 209

1.083516 0.203158 0.001075 0.000071 0.146459 0.022531 0.005471 0.001752 0.000675 0.000297 0.000145 0.000076

25.3 67.28 141.83 243.13 2.06 5.27 12.96 29.34 45.89 69.67 103.84 165.72

6 Conclusions Isogeometric Collocation is extended for time-harmonic waves in acoustic problems. Two problems are considered viz., wave propagation in a rectangular duct and radiation by an oscillating cylinder in the first mode. The case studies are solved using IGA-C and the performance is compared to IGA-G accounting the accuracy and computational time. We observed that IGA-C convergence is lower compared to IGA-G and requires more degrees of freedom to maintain the same amount of accuracy. On the other hand, if computational time is accounted, we observe that

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Table 6 Results obtained with IGA-G and IGA-C with NURBS of polynomial degree p = 8 for the radiation by an oscillating cylinder in the first mode Sl. no. Method Mesh Rel. L 2 error Total time (s) 1 2 3 4 1 2 3 4 5 6

IGA-G

IGA-C

17 × 33 27 × 53 37 × 73 47 × 93 37 × 73 47 × 93 57 × 113 67 × X133 77 × 153 87 × 173

0.981731 0.066995 0.000148 0.000005 0.022198 0.001613 0.000229 0.000049 0.000013 0.000004

86.32 213.43 416.67 673.3 4.17 11.25 22.87 44.39 73.81 133.35

IGA-C gives the same amount of accuracy at a lesser computational time in all the cases considered. The gain in computational time to achieve the same level of accuracy between IGA-C and IGA-C increases with higher polynomial degrees. As with increase in polynomial degree, the number of Gauss quadrature points in IGA-G increases quadratically while in IGA-C, it remains the same as the number of degrees of freedom. Moreover, the link to CAD model by using NURBS basis makes it easy to incorporate any geometry updates in the model and analyze faster in a product design cycle.

References 1. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194(39– 41):4135–4195 2. Auricchio F, Beiro Da Veiga L, Hughes TJR, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math Models Methods Appl Sci 20(11):2075–2107 3. Ihlenburg F, Babska I (1997) Finite element solution of the Helmholtz Equation with high wave number Part II: the h-p version of the FEM. SIAM J Numer Anal 34(1):315–358 4. Elguedj T, Bazilevs Y, Calo VM, Hughes TJR (2008) B-bar and F-bar projection methods for nearly incompressible linear and nonlinear elasticity and plasticity using higher-order NURBS elements. Comput Methods Appl Mech Eng 197:2732–2762 5. Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43:3–37 6. Buffa A, Sangalli G, Vzquez R (2010) Isogeometric analysis in electromagnetics: B-splines approximation. Comput Methods Appl Mech Eng 199(17):1143–1152 7. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR, Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195(41–43):5257–5296 8. Simpson RN, Scott M, Taus M, Thomas DC, Lian H (2014) Acoustic isogeometric boundary element analysis. Comput Methods Appl Mech Eng 269:265–290

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9. Peake MJ, Trevelyan J, Coates G (2013) Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems. Comput Methods Appl Mech Eng 259:93– 102 10. Wu H, Ye W, Jiang W (2015) Isogeometric finite element analysis of interior acoustic problems. Appl Acoust 100:63–73 11. Coox L, Deckers E, Vandepitte D, Desmet W (2016) A performance study of NURBS-based isogeometric analysis for interior two-dimensional time-harmonic acoustics. Comput Methods Appl Mech Eng 305:441–467 12. Schillinger D, Evans JA, Reali A, Scott MA, Hughes TJR (2013) Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput Methods Appl Mech Eng 267:170–232 13. Huttunen T, Gamallo P, Astley RJ (2009) Comparison of two wave element methods for the Helmholtz problem. Commun Numer Methods Eng 25:35–52 14. Guddati MN, Yue B (2004) Modified integration rules for reducing dispersion error in finite element methods. Comput Methods Appl Mech Eng 193(3):275–287

Effect of Seismic Excitation on Bubble Behavior in Liquid Between Fuel Rods of Boiling Water Reactors S. P. Chauhan, M. Eswaran, and G. R. Reddy

Abstract Behavior of gas–liquid two-phase flow of liquid–vapor behavior is unknown under the seismic conditions. Mainly, fluctuation of void faction is an important factor for the safety operation of the nuclear reactor. In this work, the bubble behavior in between two fuel rods under seismic excitation has been investigated through numerical simulation. Initially water-sloshing problem is solved for the purpose of validation. Then, bubble behavior is analyzed with and without seismic excitation. It is found that the bubble coalescence time with free surface and pressure exert on fuel rod, which is considered as important parameters in the safety of the reactor, depends on bubble depth and seismic excitation. Keywords Seismic excitation · Two-phase flow · Bubble behavior · Thermal hydraulics

1 Introduction Earthquake is one of the most serious phenomena for safety of a nuclear reactor in worldwide. Therefore, structural safety of nuclear reactors is studied and reactors are constructed with structural safety as paramount. As the safety system is concerned, the core is automatically stopped when the vibration crosses the design safety limit. On the other hand, characteristics of two-phase flows especially coolant in the core are intensively studied both experimentally and numerically under wide variety of thermal–hydraulic conditions concerning with nuclear reactor safety in nuclear engineering field. However, combined effect of thermal hydraulic and seismic oscillating condition became interesting after the nuclear accident at Fukushima Daiichi power

S. P. Chauhan (B) Homi Bhabha National Institute, Mumbai, India e-mail: [email protected] M. Eswaran · G. R. Reddy Reactor Safety Division, BARC, Mumbai, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_45

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plant in 2011. Zeng et al. [1, 2] investigated bubble behavior under nonlinear oscillation and concluded that frequency of bubble oscillation frequency intensified by system frequency. However, the liquid natural frequency is not considered as excitation frequency which may induce large liquid displacement. Watanade [3] concluded that bubble behavior is affected by nonlinear excitation, particularly a rising bubble without considering the gravitational effect. Satau et al. [4] investigated the nuclear plant behavior under seismic excitation by developed analysis code TRAC-BF1 and found that vertical component of excitation strongly influences the core power. In the view of above, the objective of the present work is taken as the investigation of various bubble behaviors in a channel with varying frequency ratios (ωr ). For this, bubble rising, coalescence time with interface, and local pressure during coalescence are studied under various excitation frequencies. Width of the channel is equal to the distance between two adjacent fuel rods. The open source, free CFD software OpenFOAM released by the OpenFOAM Foundation which is based on finite volume approach is used to simulate the seismic excitation of two-phase flow.

2 Numerical Methodology In this work, the bubble behavior in liquid between fuel rods of boiling water reactors under seismic excitation is studied in detail. For this purpose, liquid and bubble properties are taken closure to BWR operating pressure.

2.1 Governing Equation for Fluids Here, both the fluids are assumed as Newtonian and incompressible. The twodimensional, unsteady differential equations based on mass and momentum equations are continuity equation, ∇ · u = 0

(1)

and momentum equation, D u = −∇ P + ∇ · τ + ρg + f s (2) Dt   where u, ρ, and τ = μ ∇ u + ∇ u T are fluid velocity, density, and stress tensor term, respectively. In viscous tensor term, μ is dynamic viscosity. ρg and f s are force due to gravity and the surface tension force, respectively. To capture the liquid D represents the free surface Volume of Fluid (VOF) technique is used. The term Dt substantial derivative. The VOF equation, ρ

Effect of Seismic Excitation on Bubble Behavior in Liquid …

ρ

∂α Dα = + u∇ · α = 0 Dt ∂t

557

(3)

where represents the volumetric fraction between gas and liquid, which is taken as 1 for liquid and 0 for gas and interface in between 0 and 1. In order to improve the accuracy, a compression term is included in Eq. (3) as suggested by Rusche [6]. The modified equation is as follows, ∂α + ∇ · ( u α) + ∇( u r · α(1 − α)) = 0 ∂t

(4)

where ur is a velocity field suitable to compress the interface. Term α(1 − α) deactivates the artificial compression term in all cells except the interface. Density and viscosity are taken as ρ = αρ f + (1 − α)ρg

(5)

μ = αμ f + (1 − α)μg

(6)

The surface tension f s defined by Brackbill et al. [7] is adopted as f s = σ k∇α

(7)

where σ and k are surface tension and curvature interface, respectively. To attain this oscillatory motion, moving grid ALE method is used in this study. That is, present computational grid is moving with the same velocity U as the oscillating container. The two-phase interDyMFoam solver of OpenFoam is developed for dynamic mesh handling. InterDyMFoam combines the VOF method and a mesh deformation solver. Pressure–velocity linking is done by PIMPLE algorithm. The numerical schemes are as follows: Euler scheme used for the time derivative, the Gauss linear scheme for the pressure gradient term, the Gauss VanLeer scheme for the convection term, the Gauss cubic scheme for Laplacian scheme. Maximum Courant number as 0.2 is fixed to control time step automatically.

3 Results and Discussion The bubble behavior is simulated under seismic excitation. Frequency ratio and bubble depth are varied to observe the effect on bubble coalescence and free surface. Frequency ratios are 0.7 and 0.85. Bubble depth (D) is the distance from free surface to bubble center and varies from 0 to 10 mm. Zero bubble depth denotes that the bubble is not present.

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(a) Schematic sketch with location A and B

(b) Schematic sketch of channel with bubble

Fig. 1 Schematic sketch for validation sloshing problem (a) and (b) for bubble problem

2500

Free surface displacement (m)

Present work Analytical Ozdemir et al (Numerical)

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0.05

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1

Time (Sec)

(a) Total Pressure at location B

2

3

4

5

Time (Sec)

(b) Surface displacement at location A

Fig. 2 Validation for total pressure and surface displacement

3.1 Validation The physical domains for validation and bubble problem are depicted in Fig. 1a, b. For validation, the water-sloshing problem has been taken. The pressure is plotted as Fig. 2a at point B in Fig. 1a and free surface displacement is plotted as shown in Fig. 2b at point A highlighted in Fig. 1a. Results are compared with numerical and experimental data (Ozdemir et al. [5]) and are in good agreement.

3.2 Bubble Behavior Under Seismic Excitation The bubble behavior is analyzed in 4 mm width channel which is equivalent to width of two fuel rods. Liquid column height is 20 mm under seismic load. Schematic

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559

Table 1 Fluid properties closure to BWR operating pressure 7.1 MPa [3] Fluid

Fluid properties ρ (Kg/m3 )

μ (Pa.s)

υ (m2 /s)

σ (N.m) 0.0175

Bubble

36.9

1.9E-5

0.05149E-5

Water

738.4

9.5 E-5

0.01286 E-5

sketch of channel with bubble is shown in Fig. 1b. Liquid and bubble properties that are closure to BWR operating pressure 7.1 MPa is shown in Table 1. The first mode natural frequency (ωn ) of liquid column is 87.8 rad/sec. The dimensionless ωr is varied from 0.5 to 1, to observe the effect on pressure variation along left wall and coalescence time. Sinusoidal excitation is applied with amplitude (A) of 0.003 m and excitation frequency on the walls of channel in horizontal direction. The ωr = 0.7 case corresponds to the maximum velocity and acceleration observed on May 26, 2003, at Rikuzentakada city in Japan [3].

3.2.1

Pressure Variation Along the Left Wall

The pressure distribution lines along the left wall are plotted for each coalescence time corresponding to ωr as shown in Figs. 3a–f and 4a–f. The pressure along the left wall for each case (D = 2 to 10 mm) are compared with pressure corresponds to D = 0 mm case. For ωr = 0.7, pressure plots are similar at time 0.016 s and gradually decreases along the left wall of channel from bottom to top as shown in Fig. 3a. Bubble coalesce does not intensify pressure for D = 2 and 4 mm at 0.016 s and 0.039 s because it does not occur near the wall. For other D values at t = 0.039 s as shown in Fig. 3b, small peaks appear because the bubble is moving toward the wall during excitation. In Fig. 3c, the bubble coalesce on the sidewall as results a sharp peak seems for D = 6 mm. Small peaks appear for D = 8 and 10 mm because the bubbles are attaching with left wall and it is well known that inside the bubble pressure is more. As Fig. 3d, two sharp peaks appear for D = 8 mm due to bubble coalesces, and there is liquid breakage on the left wall. Single small peak also appears for D = 10 mm due to the bubble attachment on the wall; at time t = 0.134 s, a tallest peak of pressure appears for D = 10 mm when the bubble coalescence on the wall is seen as shown in Fig. 3e and corresponding pressure contour is also shown in Fig. 3f. When ωr = 0.85, as shown in Figs. 4a–f, at t = 0.017 s pressure lines are similar for all D (2–10 mm) including 0 mm as per Fig. 4a. At t = 0.047 s as shown in Fig. 4b and corresponding snapshots in Fig. 6a–f, pressure line of D = 2 mm is similar to D = 0 mm whereas pressure line of D = 4 mm shows pressure drops near y/H = 0.4 and then a sharp peak of pressure due to bubble coalesces occurs on the left wall. The pressure lines for D = 6,8,10 mm show pressure peaks whereas single peak for D = 6 and 8 mm. Two peaks appear from D = 10 mm pressure line because bubble separates into two small bubbles and attaches to the left wall and

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100

60

140 120 100 80

40 20

20 0

-20

-20

0.4

0.6

-40 0.0

1.0

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0.6

(a) t = 0.016 sec

(b) t = 0.039 sec

Liquid still level

100 80 60

D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

40 20

140 120 100 80

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60 40 20

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y/H

(c) t = 0.0565 sec

(d) t = 0.102 sec

Liquid still level

140 120 100 80

P (Pa)

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0

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D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

60

Liquid still level

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D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

Liquid still level

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120

P (Pa)

140

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1.0

D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

60 40 20 0 -20 -40 0.0

0.2

0.4

0.6

y/H

(e) t = 0.134 sec

0.8

1.0

(f) VOF and pressure contours for t = 0.134 sec

Fig. 3 Pressure along the left wall with varying depth D and time step for ωr = 0.7

move along the wall due to adhesion force which works between the bubble and the rod. Corresponding snapshot is shown in Fig. 7f. As shown in Fig. 4c, pressure peaks are observed for D = 8 and 10 mm due to the bubble’s attachment to the left wall. One peak is also observed for D = 4 mm, and it is because of violent sloshing in fluid and a new bubble is generated as shown in Fig. 8b. A small peak appears due to bubble coalesces for D = 6 mm. After the liquid is still in level line, many small peaks appear for each D and it is because

Effect of Seismic Excitation on Bubble Behavior in Liquid …

120 100

120 100 80

P (Pa)

60 40 20

20 0

-20

-20

0.2

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-40 0.0

1.0

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(b) t = 0.047 sec

Liquid still level

D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

140 120 100 80

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80 60 40 20

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D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

60 40 20

0

0

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0.4

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0.2

0.8

1.0

-40 0.0

0.2

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0.6

y/H

y/H

(c) t = 0.0835 sec

(d) t = 0.111 sec

Liquid still level

200 180 160 140 120 100 80 60 40 20 0 -20 -40 0.0

0.4

(a) t = 0.017 sec

100

-40 0.0

0.2

y/H

120

P (Pa)

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y/H

140

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40

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D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

60

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140

D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

Liquid still level

Liquid still level

140

561

0.4

0.6 y/H

(e) t = 0.1195 sec

0.8

1.0

D=2mm D=4mm D=6mm D=8mm D=10mm No Bubble

0.8

1.0

(f) VOF and pressure contours for t = 0.1195 sec

Fig. 4 Pressure along the left wall with varying depth D and time step for ωr = 0.85

water is sliding with left wall due to violent excitation; corresponding snapshots are as shown in Fig. 8a–f. As shown in Fig. 4d, two sharp peaks of pressure appear, one is for D = 10 mm due to bubble attachment to the wall and the other is for D = 4 mm due to the big water portion that is separated and slides along the wall. At time t = 0.1195 s as shown in Fig. 4e, a sharp peak of pressure line is observed for D = 10 mm and it is because of bubble coalescence on the left wall which creates high local pressure and simultaneously new bigger bubble is created as per Fig. 9f.

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Fig. 5 Free surface snapshots for ωr = 0.85 at t =0s

(a) D=0

(b) D=2

(c) D=4

(d) D=6

(e) D=8

(f) D=10

(a) D=0

(b) D=2

(c) D=4

(d) D=6

(e) D=8

(f) D=10

(a) D=0

(b) D=2

(c) D=4

(d) D=6

(e) D=8

(f) D=10

(a) D=0

(b) D=2

(d) D=6

(e) D=8

(f) D=10

Fig. 6 Free surface snapshots for ωr = 0.85 at t = 0.017 s

Fig. 7 Free surface snapshots for ωr = 0.85 at t = 0.047 s

Fig. 8 Free surface snapshots for ωr = 0.85 at t = 0.0835 s

(c) D=4

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Fig. 9 Free surface snapshots for ωr = 0.85 at t = 0.1195 s

(a) D=0

(b) D=2

(c) D=4

(d) D=6

(e) D=8

(f) D=10

Two small peaks also appear one for D = 8 mm and one for D = 0 mm. These small peaks appear because fluid is moving toward creating a new big bubble. In Fig. 4f, pressure contour is shown and it is observed at the coalesced on the left wall; high pressure generates on the left wall.

3.2.2

Free Surface Response with and Without Bubble for ω r = 0.85

Liquid VOF snapshots with bubble different bubble depth (D) corresponding to various coalescence time for ωr = 0.85 are shown in Figs. 5a–f to 9a–f. The bubble coalescence at free surface creates pressure fluctuation in the liquid. A liquid oscillation due to seismic excitation along with this fluctuation creates a huge pressure on side walls as shown in Figs. 3 and 4. Amplification of pressure on the left wall due to bubble coalescence is not desirable especially for fuel rod. For higher values of D, the bubble is stuck to the wall by excitation and after that the bubble does not leave the wall due to adhesion force, which is also not desirable for heat transfer between rods and liquid (water).

3.2.3

Coalescence Time

Figure 10 shows the variation of coalescence time with frequency ratios and for different bubble depth (D) values. When the bubble depth is 2 mm, coalescence time decreases with increasing frequency ratio. For D = 4 and 6 mm, at lower frequency (0.5 and 0.7) coalescence occurs earlier compared to the case where the frequency ratio is zero, whereas for higher frequency ratio (0.85 and 1) the coalescence time increases, due to increase in travel path for the bubble. For D = 8 mm, time increases as frequency ratio increases because the bubble attaches to the wall and this process due to adhesion force is delayed by the bubble. For D = 10 mm, time increases at lower frequency ratio (0.5, 0.7) comparative to at zero frequency ratio whereas for higher frequency ratio (0.85 and 1), water slide along the wall and it helps bubble coalescence earlier.

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D =2m m D =4m m D =6m m D =8m m D =10m m

0 .1 8

t (s)

0 .1 5 0 .1 2 0 .0 9 0 .0 6 0 .0 3 0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

wr

Fig. 10 Coalescence time for different frequency ratios

4 Conclusions Bubble behavior is analyzed under seismic excitation with varying frequency ratio and bubble depth and is compared to no bubble case. It is simulated in open source CFD package OpenFOAM. The interDyMFoam solver is used for two-phase water vapor problem under excitation. It is observed from this study that pressure amplification depends on the bubble depth and the location of bubble coalescence. It is found that coalescence time depends on the excitation frequency ratio. When the frequency ratio increases toward one, violent oscillation of bubble is observed due to resonance. Owing to bubble oscillations, the chance of bubble attachment on the wall of fuel rod is more. If bubble attaches to the wall, the bubble slide on the wall and not separate from the wall due to adhesion force between the wall and the bubble, which may not be desirable for effective heat transfer. Bubble coalescence occurs near the wall that creates more pressure on the wall and generally it appeared in excited fluid which may also not be desirable for fuel rod.

References 1. Zeng Q, Cai J, Watanabe T (2013) Simulation of free interface behavior by using OpenFOAM. In Proceeding of 21st international conference on nuclear engineering (ICONE21), China 2. Zeng Q, Cai J (2014) Three-dimensional simulation of bubble behavior under nonlinear oscillation. Ann Nucl Energy 63:680–690 3. Watanabe T (2012) on the numerical approach for simulating reactor thermal hydraulics under seismic condition. Ann Nucl Energy 49:200–206 4. Satou A, Watanabe T, Maruyama Y, Nakamura H (2010) Neutron-coupled thermal hydraulic calculation of BWR under seismic acceleration. Proc Joint lnt Conf. Supercomput Nucl Appl Monte Carlo 2:120–124 5. Ozdemir Z, Souli M, Fahjan YM (2010) FSI methods for seismic analysis of sloshing tank problems. Mech Ind 11:133–147

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6. Rusche H (2002) Computational fluid dynamics of dispersed two-phase flowsat high phase fractions. PhD thesis, Imperial College, University of London, 2002 7. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. Comput Phys 100:335–354

Experimental Study on Shallow Water Sloshing Saravanan Gurusamy and Deepak Kumar

Abstract Sloshing of liquid in a partially filled container, subjected to higher amplitude of dynamic load, is a complex phenomenon. In shallow water conditions, the natural frequency of sloshing depends on the amplitude of excitation. Sloshing frequency tends to change with increase in amplitude of excitation. The change in natural frequency is critical if we use the sloshing tank as a passive damping device, such as Tuned Liquid Damper (TLD) for offshore structures or onshore structures. A small change in sloshing frequency in TLD may affect the structural vibration control significantly. Therefore, it is essential to comprehend the natural frequency of shallow water sloshing. Experimental study is one of the best ways to understand the physical insights of change in sloshing frequency. Experimental studies are conducted to study the jump in sloshing frequency at different excitation amplitudes. Several rectangular tanks (1163, 1064, 951, and 844 mm) under different water depths (60, 50, and 40 mm) are taken for the study to generalize the results. The liquid tank is mounted on a uni-directional horizontal shake table, which is subjected to simple harmonic motion. The amplitude of excitation varied from 5 to 50 mm. A single capacitance-type wave probe is used at the end of the tank wall to measure the wave surface elevation. The wave elevation increases as the excitation frequency reaches toward the natural frequency of sloshing. The measured liquid sloshing frequency, at the resonance condition, is considered as actual sloshing frequency of liquid in tank. This sloshing frequency changes with the amplitude of excitation and shows the sudden jump in frequency from a particular amplitude of excitation. The objective of this paper is to generalize the relation between the jump frequency ratio (ratio of jump frequency to linear frequency) and the non-dimensional amplitude of excitation. Keywords Sloshing · Tuned liquid damper (TLD) · Jump frequency ratio

S. Gurusamy (B) · D. Kumar Indian Institute of Technology Madras, Chennai, India e-mail: [email protected] D. Kumar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Dutta et al. (eds.), Advances in Structural Vibration, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-15-5862-7_46

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1 Introduction Sloshing is a phenomenon of free surface motion in a partially filled liquid container subjected to dynamic loads. It is naturally associated with several fields like fuel motions in missile tanks and spacecraft, liquid motion in ships with large ballast tanks and Floating Protection Storage and Offloading (FPSO) units, tank trucks on road transport, and liquid metal cooled reactor vessels in nuclear industry. Because of diverse applications, sloshing problem has been studied for the past six decades. Since the early 1950s, the sloshing problem was the main concern to aerospace scientists who have studied the effects of liquid propellant on the dynamics of spacecrafts. Many researchers including [1–4] have studied linear sloshing using mechanical models. Later, in 1966, a series of experimental studies for spacecraft applications has been conducted by NASA, USA. Since then, the studies [5–7] on nonlinear sloshing have emerged gradually. Researchers have found that liquid sloshing tank can be utilized as dampers for suppressing the vibrations of lightly damped structures. In Tuned Liquid Dampers (TLDs), the natural frequency of sloshing motion is tuned to the structural frequency so that the liquid motion will be enhanced. Lee and Reddy [8], Modi and Welt [6], Fujino et al. [9], and Sun et al. [10] have developed different analytical and numerical models of TLD. Several studies by Warnitchai and Pinkaew [11], Kaneko and Ishika [12], Tait et al. [13], Tait [14], and Wei et al. [15] have been attempted to increase the TLD damping ratio by inserting energy dissipating devices, such as screens and nets inside the sloshing tank. In the study of Tuned Liquid Damper for structural control or sloshing of liquid in a tank carried by ship, the natural frequency of shallow water sloshing plays a key role. Unfortunately, the theoretical and computational studies are limited to estimate the natural frequency of shallow water sloshing with accuracy. Hence, there is a need for an extensive experimental study to comprehend the behavior of sloshing frequency. In shallow water, sloshing frequency is a function of excitation amplitude. As the amplitude of excitation increases, the sloshing frequency also increases. This phenomenon is known as “Hardening-spring type” nonlinear behavior of shallow water sloshing and is reported in many literatures such as Ockendon and Ockendon [16], Lepelletier and Raichlen [17], Sun et al. [10, 18], Khosropour et al. [19], Ikeda and Nakagawa [20], Reed et al. [21], Hill [4], Frandsen [22, 23], Gardarsson and Yeh [24], Forbes [25]. Yu and Reed [26] have obtained an empirical relation for nonlinear frequency of shallow water TLD. In this study, sloshing frequency under different excitation amplitude is experimentally studied.

2 Experimental Method The experiments are performed using a horizontal shake table. The motion is a simple harmonic motion. It works on the principle of crank mechanism which converts the rotary motion of the motor into the linear motion of the table. The shake table has

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569

Fig. 1 Sloshing experimental tank mounted on the shake table

a 1000 mm by 1000 mm platform. The shake table has mechanical capacities of ±50 mm maximum displacement, 3 g maximum acceleration, and 10 Hz maximum frequency. The displacement amplitude and frequency have been controlled manually. Figure 1 shows the photo of the sloshing experimental facility in the Department of Ocean Engineering, IIT Madras. The rectangular tanks of dimension 1163 mm × 288 mm, 1064 mm × 288 mm, 951 mm × 288 mm, and 844 mm × 288 mm have been taken. A single capacitancetype wave probe is used at the end of the tank wall to measure the wave surface elevation and the probe is placed 15 mm away from the wall. The tank is fabricated using clear acrylic plates. The thickness of the plate is 12 mm. In a sloshing tank, plain water has been used and in order to capture the wave patterns blue ink has been mixed with the water. For acquiring data, the “spider 8,” the dynamic data acquisition system is used. The sampling frequency is kept as 100 Hz.

2.1 Linear Sloshing Frequency Linear sloshing motion consists of smooth and non-breaking water waves. Because of small amplitude, the linear sloshing waves are of either standing wave form or harmonic wave form. Figure 2 displays the relationship between the linear sloshing frequency and the water depth and tank length. The frequency changes as the parameters vary. Therefore, these two parameters are playing an important role in the sloshing dynamics. Based on the linear potential flow theory, the natural frequency for the n-th sloshing mode is given by the Eq. (1). In particular, f 1 denotes the fundamental sloshing frequency.

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Fig. 2 Linear sloshing frequency

 1 fn = 2π

  nπ h gnπ tanh for n = 1, 2, 3, . . . L L

(1)

Let f denote the excitation frequency of the shake table. Let Frequency ratio = f/f 1 . The analysis will be done using this frequency ratio.

2.2 Shallow Water Sloshing Frequency From the shallow water wave theory, one can express the propagation speed, C, of a smooth wave as a function of water depth, h, and surface elevation, η.   (2) C = g(h + η) = gh(1 + ηι ) In Eq. (2), ηι = hη . The wave length of the fundamental mode (anti-symmetric mode) of sloshing motion in a rigid rectangular tank of length L is 2L; and hence the fundamental sloshing frequency, f shallow , is calculated as given below. f shallow =

C = 2L

√

gh(1 + ηι ) 2L

(3)

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In the shallow water limits, one gets the frequency relation as a function of free surface elevation. Therefore, the experimental study is also essential to comprehend the behavior of frequency of shallow water sloshing. f shallow ∼  = 1 + ηι f1

(4)

2.3 Measured Jump Frequency of Shallow Water Sloshing The natural frequency of the shallow water sloshing is experimentally measured by identifying the resonance forcing frequency. From the wave probe measurements, one can easily obtain the maximum free surface elevation. The free surface elevation gets peak value at the forcing frequency higher than the linear sloshing frequency. The amplitude of the free surface deformation grows gradually as the forcing frequency nears to sloshing frequency. Once forcing frequency crosses the sloshing frequency the amplitude of free surface elevation drops substantially. This particular forcing frequency is referred as the “Jump frequency.” Table 1 shows the measured jump frequency for the sloshing tank of length: 1163 mm, water depth: 50 mm. The amplitude of shake table displacement varies from 5 to 50 mm. The experiments are performed from low exciting frequency to high exciting frequency. The table also relates the non-dimensional amplitude of excitation to the jump frequency ratio. The increment in amplitude of 5 mm interval does not always increase the jump frequency. One can see from the table that there are the pairs of amplitudes {15 & 20 mm}, {25 & 30 mm}, and {45 & 50 mm} which share the same jump frequency. The increase in amplitude does not increase the jump frequency monotonically as Table 1 Measured Jump frequency in sloshing tank Tank Length: 1163 mm, Water Depth: 50 mm, Linear Frequency: 0.3002 Hz Amplitude (mm)

Non-dimensional amplitude

Measured Jump frequency (Hz)

Jump frequency/linear sloshing frequency

5

0.004299226

0.3287

1.094936709

10

0.008598452

0.3586

1.194536975

15

0.012897678

0.3785

1.260826116

20

0.017196905

0.3785

1.260826116

25

0.021496131

0.4034

1.343770819

30

0.025795357

0.4034

1.343770819

35

0.030094583

0.3986

1.327781479

40

0.034393809

0.4185

1.39407062

45

0.038693035

0.4484

1.493670886

50

0.042992261

0.4484

1.493670886

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in the amplitude of 35 mm where the jump frequency is decreased in comparison to 30 mm amplitude. The other cases of tank length: 1064 mm with water depth: 40, 50, and 60 mm; tank length: 951 mm with water depth: 50 and 60 mm; tank length: 844 mm with water depth: 50 mm are not presented here.

3 Results and Discussion In the present experimental investigations, the steady-state motion of shallow water sloshing is studied. In order to comprehend the wave characteristics and the participation of dominant frequencies in the sloshing motion, both the time-series analysis and spectral analysis are carried out. For brevity, the analysis of sloshing tank length: 1163 mm with water depth: 50 mm for the amplitude of excitation 10 and 40 mm is presented here. However, all the measured jump frequencies of all sloshing tanks with different water depth are used to obtain a relation between the amplitude and the jump frequency ratio.

3.1 Time-Series Analysis of Free Surface Elevation The experimental measurements of liquid free surface elevation in a rectangular tank subjected to harmonic excitation are discussed in this section. The time histories of free surface motion are measured at a predefined location. Figures 3 and 4 show the evolution of sloshing motion with respect to forcing frequency. Case 1 and 2 differ by forcing amplitude. Case (1): Tank length: 1163 mm with water depth: 50 mm, and forcing amplitude: 10 mm Figure 3 presents the steady-state time-series responses of the free surface elevation, for the case of the tank length: 1163 mm and water depth: 50 mm at various forcing frequencies (from lower to higher frequencies) for displacement excitation amplitude: 10 mm. Figure 3a–f represents the evolution of sloshing wave for increasing excitation frequency. A very small amplitude wave develops gradually up to a resonating wave. On this evolution, different wave patterns, which cause distinctive dynamic forces on the vertical walls of the tanks, are identified in this evolution process. At the lower frequency ratio 0.8299, the surface elevation is a small amplitude progressive wave with negligible contribution from higher frequency waves as shown in Fig. 3a. Due to higher harmonics, the crest amplitudes of this wave system are slightly lesser than the trough amplitudes. In case of the frequency ratio 0.9295 corresponds to Fig. 3b, the free surface elevation develops sufficiently and induces the multiple wave system in the sloshing tank. The vertical rise and fall of free surface due to reflection on the end walls cause a multiple wave system in the tank. When

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Fig. 3 Time-series records of free surface elevation: Tank length: 1163 mm, Water depth: 50 mm, Amplitude: 10 mm

the frequency ratio is 1.0290, as shown in Fig. 3c, a two-wave system is observed. In the evolution process, the two-waves merge and form a single soliton-like wave and it is noticed for the frequency ratio 1.0954 as seen in Fig. 3d. Further increment in the frequency ratio leads to a topological change in the free surface wave motion. A smooth and small amplitude standing wave system exists at the frequency ratios 1.1950 and 1.2614 and the amplitude of wave motion decreases drastically as in Fig. 3e, f. From time-series records, it is also found that the frequency ratio 1.0954 exhibits the resonance phenomenon as one noticed in Fig. 3d. The forcing frequency corresponding to the resonance condition is taken to be the natural frequency of the sloshing oscillation for the given tank length, water depth, and excitation amplitude. The resonance of sloshing wave motion occurs at the excitation frequency larger than the fundamental sloshing frequency. This implies that sloshing undergoes a strong nonlinear behavior known as “Stiffening or Hardening-spring type” behavior. One may also observe that the free surface elevation increases gradually up to a resonance condition and then at a particular excitation frequency ratio, 1.1950, the

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Fig. 4 As in Fig. 3 but Amplitude: 40 mm

wave elevation decreases significantly as shown in Fig. 3e. The excitation frequency corresponds to the frequency ratio, 1.1950, which is the “Jump frequency”. Case (2): Tank length: 1163 mm with water depth: 50 mm and forcing amplitude: 40 mm Figure 4 displays the time history responses of water surface elevation for the case of the tank with length: 1163 mm and water depth: 50 mm at various excitation frequencies for the excitation amplitude: 40 mm. On comparison of the time-series Figs. 3 and 4, one may appreciate the effects of amplitude of excitation on the sloshing motion. The higher amplitude of excitation yields different wave patterns with large amplitude and breaking wave system. Hence, one may clearly conclude that the water sloshing motion is sensitive to both the excitation amplitude and frequency and it is evident from the evolution process of sloshing oscillation as discussed below. For the frequency ratio 0.8229, Fig. 4a shows the asymmetric progressive wave trains with large oscillations of high-frequency waves. As the frequency ratio increases to 0.9295, in Fig. 4b, a strong traveling wave develops and a plunging-type wave breaking occurs and the wave front of traveling waves starts breaking nearly at the middle of the tank. In case of the frequency ratios, 1.0290 and 1.0954, the wave breaking becomes stronger and Fig. 4b–d shows the saw-tooth wave patterns, which are rich in harmonics. For the frequency ratio 1.1950, Fig. 4e shows a two-wave system and saw-tooth form disappears gently. Figure 4f, g, for the frequency ratio

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1.2614 and 1.3610, respectively, displays a single impact wave system, which travels back and forth in the sloshing tank. The impact wave system includes the collision between the end walls of the tank and wave front of the traveling water mass, high wave run-up on the vertical walls, impact sound due to collision, and formation of air-bubbles in water domain. From the time series shown in Fig. 4b, c, one can clearly note that the period of sloshing oscillation decreases as the excitation frequency increases. From the experiments, it is also identified that while increasing the frequency ratio as in the order 0.9295, 1.0290, 1.0954, and 1.1950, the breaking point of wave front shifts toward the end walls of the tank. This process of shift in breaking point finally leads to the formation of impact wave system in the tank. Moreover, wave breaking is apparent over wide range forcing frequencies. The frequency ratio 1.3610 results in the resonance phenomenon and on the other hand, the frequency ratio 1.4274, as in Fig. 4f, leads to the “Jump frequency” of the sloshing oscillation.

3.2 Fourier Analysis In addition to the time-series analysis, the Fourier analysis is also performed to unfold the contribution from different sloshing modes. The Fourier amplitude spectra are plotted for the free surface elevation. The Fourier amplitude spectrum is a plot of Fourier amplitude versus frequency showing the distribution of the amplitude of the surface elevation with respect to frequency. The magnitude of the spectrum indicates the strength of frequency components. In other words, the spectrum shows how the sloshing energy is distributed over a range of frequencies. The Fast Fourier Transform technique is adapted to obtain the amplitude spectra. Case (a): Amplitude spectrum of time history records in Fig. 3 Figure 5 presents the Fourier amplitude spectra of the time history responses of the free surface elevation given in Fig. 3. For relatively small values of frequency ratio, the spectrum contains a single dominant peak that indicates the smooth and small amplitude sloshing flow without breaking. As the frequency ratio increases toward the neighborhood of unity, the high-frequency spectral peaks emerge due to the nonlinear effects and wave breaking in the sloshing tank. Moreover, it is worth to mention that the spectral peaks appear in the vicinity of both the forcing frequency “f ” and sloshing natural frequencies f n . Figure 5a shows the spectrum with one spike at 0.2540 Hz which is nearer to the excitation frequency. A secondary peak is noticed at 2f and is very closer to the third modal frequency of sloshing. Figure 5b displays that the dominant peak is at 0.2838 Hz, which is closer to both the excitation frequency and the first modal frequency. The other subsequent peaks are observed at frequencies which are positive integer multiple of 0.2838 Hz. The spectral peaks at 0.8516, 1.9870, and 2.2709 Hz are comparable. This implies that those frequencies have almost the same sloshing energy content within them. Figure 5c exhibits an interesting feature in the spectrum

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Fig. 5 Fourier amplitude spectrum of time-series records of Fig. 3

that the dominant peak is not present in the neighborhood of either the first modal frequency 0.3002 Hz or the excitation frequency 0.3088 Hz. The predominant peak is observed at 2f , closer to the fourth modal frequency that is closer to 2f 1 . The next dominant peak is noticed at 6f . The peaks in the spectrum occur at those frequencies which are positive integer multiple of the excitation frequency. Figure 5d represents the resonance case where peaks appear at f (0.3288), 2f , 3f , …, 12f , and 13f . In this case, the considerable amount of energy sharing among the frequencies up to 3 Hz is apparent. The maximum spectral peak exists in the neighborhood of 3f , closer to f 11 . As displayed in Fig. 5e, f, one can observe that the spectral peaks are visible only in the neighborhood of excitation frequency. It is interesting to note that in most of the cases, the sloshing frequencies of odd modes are excited rather than frequencies of even modes. Further, it is notable that there is no monotonic-energy distribution among the frequencies. Some frequencies can have significant sloshing energy; some may have reasonable amount of energy; whereas few may have insignificant energy content; and rest of them can have only zero energy. This fact can be realized from Fig. 5c where the envelope of the amplitude

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spectrum increases initially and then decreases substantially and further it increases significantly before decreasing. This envelope also illustrates that there is a frequency zone, which lies between the low-frequency region and the high-frequency region, which has lesser sloshing energy in comparison to the low- and high-frequency regions. Case (b): Amplitude spectrum of time-series records of Fig. 4 Figure 6 presents the Fourier amplitude spectra of the time-series responses of the free surface elevation given in Fig. 4. The details run in the similar lines as in the case (a). As expected, the spectral peaks are present at the exciting frequency and sloshing natural frequencies. In all spectral plots, Fig. 6a through Fig. 6h, it is found that the dominant peak occurs at the excitation frequency. The successive spectral speaks appear at those frequencies which are the positive integer multiple of the excitation frequency. In Fig. 6a, the second dominant peak is present at 2.9497 Hz and it is in the neighborhood of 7f 2 and 4f 6 . As shown in Fig. 6b, the second dominant peak occurs at 2f that is

Fig. 6 Fourier amplitude spectrum of time-series records of Fig. 4

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closer to the third modal frequency. In Fig. 6c, the dominant peak is also closer to the first modal frequency. The second dominant peak is at 2f , closer to the fourth modal frequency. Figure 6d displays that the dominant peak is also closer to the first modal frequency. The second dominant peak is exactly at 2f , which is in the vicinity of the second modal frequency. In Fig. 6e, the second dominant peak is at 2f ; the third dominant peak is at 3f , closer to 2f 3 . The peaks at each distinct pair {2f & 3f }, {3f & 4f }, and {6f & 7f } are comparable. As noticed in Fig. 6f, the appreciable peaks exist at f , 2f , 3f , 4f , 5f , 6f , 7f , 8f, and 9f . Figure 6g shows that the spectral peaks decrease as the frequency increases. The second dominant peak presents at 2f , closer to seventh modal frequency. Figure 6h indicates that the dominant peak is closer to f 2 , second modal frequency. The second dominant peak exists at 2f , very close to f 8 , eighth modal frequency. Figure 6g corresponds to the resonance phenomenon where the envelope of the Fourier amplitudes monotonically decreases as it moves toward the high-frequency zone. One can also observe that the sloshing energy spreads over more than 3 Hz. The spectral plot for the “jump frequency” phenomenon of surface elevation is shown in Fig. 6h. The frequency band under which the sloshing energy is distributed in the case of Fig. 6h is much narrower than in the case of Fig. 6g. Moreover, from Fig. 6, it is easy to infer that the dominant spectral peak occurs at the excitation frequency when the excitation frequency is away from the sloshing natural frequency.

3.3 Jump Frequency Ratio Figure 7 shows the values of the jump frequency ratio for all cases versus the nondimensional amplitude of excitation. The plot has appropriately two regions for jump ratio. The first region of lower amplitude causes weak wave breaking. On the other hand, the high amplitude region leads to stronger wave breaking in the sloshing tank. At the low amplitude zone, the jump frequency ratio increases almost linearly; this means that the slope is almost a constant; whereas at the high amplitude region, a bifurcation point lies between 0.03 and 0.04, at which there is a shift in the frequency ratio. After that point, the change in slope of the curve becomes larger. This means that as the amplitude increases beyond 0.03 the sloshing natural frequency increases drastically. The increase in sloshing frequency may change the damping of shallow Tuned Liquid Dampers. Hence, the dynamics of Structure-TLD will be affected and the structural control will be altered.

4 Conclusions Experiments are conducted on several tank length for obtaining its sloshing frequency. Only forward frequency increase is used to obtain the resonance frequency. From the experimental study on shallow water sloshing, the following salient features are concluded.

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Fig. 7 Relation between the jump frequency ratio and the non-dimensional amplitude of excitation

1. The dominant wave system in the sloshing tank are the pure standing wave system, small amplitude progressive wave, the traveling wave with breaking at the vertical walls of the tank, solitary wave without wave breaking, Impact wave with plunging breaking and impact wave with breaking at the vertical walls with impact sound. 2. The breaking point of impact wave front moves toward the end walls of the sloshing tank. 3. Both odd and even modes of sloshing are excited. When the excitation frequency is closer to the first modal sloshing frequency, the primary spectral peak occurs in the first modal frequency or in the neighborhood of the first modal frequency. 4. In general, the higher harmonics of the forcing frequency, f , 2f , 3f , … are excited. 5. The jump frequency ratio is purely amplitude dependent and there is a bifurcation point, which lies between non-dimensional amplitude 0.03 and 0.04.

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