Advances in Acoustics and Vibration III: Proceedings of the Third International Conference on Acoustics and Vibration (ICAV2021), March 15-16, 2021 (Applied Condition Monitoring, 17) [1st ed. 2021] 3030765164, 9783030765163

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Applied Condition Monitoring

Nabih Feki · Mohamed Slim Abbes · Mohamed Taktak · Mohamed Amine Ben Souf · Fakher Chaari · Mohamed Haddar   Editors

Advances in Acoustics and Vibration III Proceedings of the Third International Conference on Acoustics and Vibration (ICAV2021), March 15–16, 2021

Applied Condition Monitoring Volume 17

Series Editors Mohamed Haddar, National School of Engineers of Sfax, Sfax, Tunisia Walter Bartelmus, Wroclaw, Poland Fakher Chaari, Mechanical Engineering Department, National School of Engineers of Sfax, Sfax, Tunisia Radoslaw Zimroz, Faculty of GeoEngineering, Mining and Geology, Wroclaw University of Science and Technology, Wroclaw, Poland

The book series Applied Condition Monitoring publishes the latest research and developments in the field of condition monitoring, with a special focus on industrial applications. It covers both theoretical and experimental approaches, as well as a range of monitoring conditioning techniques and new trends and challenges in the field. Topics of interest include, but are not limited to: vibration measurement and analysis; infrared thermography; oil analysis and tribology; acoustic emissions and ultrasonics; and motor current analysis. Books published in the series deal with root cause analysis, failure and degradation scenarios, proactive and predictive techniques, and many other aspects related to condition monitoring. Applications concern different industrial sectors: automotive engineering, power engineering, civil engineering, geoengineering, bioengineering, etc. The series publishes monographs, edited books, and selected conference proceedings, as well as textbooks for advanced students. ** Indexing: Indexed by SCOPUS, WTI Frankfurt eG, SCImago

More information about this series at http://www.springer.com/series/13418

Nabih Feki Mohamed Slim Abbes Mohamed Taktak Mohamed Amine Ben Souf Fakher Chaari Mohamed Haddar •

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Editors

Advances in Acoustics and Vibration III Proceedings of the Third International Conference on Acoustics and Vibration (ICAV2021), March 15-16, 2021

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Editors Nabih Feki Higher Institute of Applied Sciences and Technology University of Sousse Sousse, Tunisia Mohamed Taktak Faculty of Sciences University of Sfax Tunisia, Tunisia Fakher Chaari National School of Engineers University of Sfax Sfax, Tunisia

Mohamed Slim Abbes National School of Engineers of Sfax Sfax, Tunisia Mohamed Amine Ben Souf National School of Engineers University of Sfax Tunisia, Tunisia Mohamed Haddar National School of Engineers of Sfax Sfax, Tunisia

ISSN 2363-698X ISSN 2363-6998 (electronic) Applied Condition Monitoring ISBN 978-3-030-76516-3 ISBN 978-3-030-76517-0 (eBook) https://doi.org/10.1007/978-3-030-76517-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The Third International Conference on Acoustics and Vibration (ICAV) was organized by the Tunisian Association of Industrial Acoustics and Vibration (ATAVI) and held online on March 15–16, 2021. This conference was initially scheduled in March 2020 and postponed to 2021 due to COVID-19 pandemic situation. After two successful editions in 2016 and 2018 with proceedings published in the Springer Applied Condition Monitoring (ACM) book series, ICAV conference series has continued to promote high-level contributions in the fields of acoustics and vibrations and promoted communication and collaboration between international and local communities. The organizers of the 2021 edition were honored by the presence of the following eminent scientists, who kindly agreed to share their knowledge in the field of acoustics and vibration, and held very interesting keynotes: – Professor Weidong ZHU, Department of Mechanical Engineering, University of Maryland, USA. – Professor Philippus HEYNS, Mechanical and Aeronautical Engineering Department, University of Pretoria, South Africa. – Professor Pierre-Olivier MATTEI, Deputy Director of Mechanical and Acoustics Laboratory (LMA), CNRS, Marseille, France. – Professor Abdelkhalek ELHAMI, Mechanical Engineering Department, National Institute of Applied Sciences in Rouen (INSA de Rouen), France. – Professor Mabrouk BENTAHAR, Roberval Laboratory, University of Technology of Compiegne, France. – Professor Abderahim ELMAHI, Laboratory of Acoustics, University of Lemans, France. – Professor Jean Yves CHOLEY, Research Director at Supmeca, Groupe ISAE, Paris, France.

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During the 2 days of the congress, about 80 attendees discussed several topics relating to: – – – – – – –

structural and machines dynamics and vibrations fault diagnosis and prognosis nonlinear dynamics vibration control of mechatronic systems fluid–structure interaction and computational vibro-acoustics vibration field measurements dynamic behavior of materials.

This book includes 31 chapters, based on the contributions to the conference by active research teams, and accepted after a rigorous peer-review process. A lot of thanks are addressed to the organizing committee, program committee and to all participants from Tunisia, Algeria, Morocco, South Africa, France, China and Saudi Arabia. We would also like to thank Springer for the continuous support to the ICAV conference series. March 2021

Nabih Feki Mohamed Slim Abbes Mohamed Taktak Mohamed Amine Ben Souf Fakher Chaari Mohamed Haddar

Contents

Gearbox Fault Identification Under Non-Gaussian Noise and Time-Varying Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . Stephan Schmidt, Fakher Chaari, Radoslaw Zimroz, P. Stephan Heyns, and Mohamed Haddar A Multiple-Grid Technique–Based Finite Element Solution of Free-Surface Flows in a Trapezoidal Open Channel . . . . . . . . . . . . . Souad Mnassri and Ali Triki Rotor-Ball Bearings System Under Variable Regime . . . . . . . . . . . . . . . Mohamed Habib Farhat, Xavier Chiementin, Fakher Chaari, Taissir Hentati, and Mohamed Haddar The Efficiency of the Rayleigh-Ritz Method Applied to In-Plane Vibrations of Circular Arches Elastically Restrained at the Two Ends and Supporting Point Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmed Babahammou and Rhali Benamar Nonlinear Wind-Induced Response Analysis of Substation Down-Conductor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guoqing Yu, Zhitao Yan, and Xinpeng Liu Bending Fatigue Behaviour of a Bio-based Sandwich with Conventional and Auxetic Honeycomb Core . . . . . . . . . . . . . . . . . Khawla Essassi, Jean-luc Rebiere, Abderrahim El Mahi, Mahamane Toure, Mohamed Amine Ben Souf, Anas Bouguecha, and Mohamed Haddar Thermoplastic Elium Recycling: Mechanical Behaviour and Damage Mechanisms Analysis by Acoustic Emission . . . . . . . . . . . . . . . . . . . . . . Sami Allagui, Abderrahim El Mahi, Jean-luc Rebiere, Moez Beyaoui, Anas Bouguecha, and Mohamed Haddar

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Passive Control of Tensegrity Domes Using Tuned Mass Dampers, a Reliability Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Mrabet, M. H. El Ouni, and N. Ben Kahla Dynamic Analysis of a Pumping Station with Coupling Misalignment Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmed Ghorbel, Oussama Graja, Moez Abdennadher, Lassâad Walha, and Mohamed Haddar

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On the Unidirectional Free-Surface Flow Solution in a Rectangular Open Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Souad Mnassri and Ali Triki

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On the Numerical Solution of the Rapidly Varied Regime in Open-Channel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Souad Mnassri and Ali Triki

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Controlling of Steel-Pipe-Based Hydraulic Systems Using Dual In-Series Polymeric Short-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mounir Trabelsi and Ali Triki

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Large Strokes of a Piezocomposite Energy Harvester with Interdigitated Electrodes Accounting for Geometric and Material Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Ahmed Jemai, Sourour Baroudi, and Fehmi Najar Transient Comprehensive Modelling Due to Pump Failure . . . . . . . . . . 117 Badreddine Essaidi and Ali Triki Innovative In-Plane Converter Design for a Capacitive Energy Harvester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 M. A. Ben Hassena, H. Samaali, F. Najar, and Hassen M. Ouakad Dynamic Interaction Between Transmission Error and Friction Coefficients for FZG-A10 Spur Gears . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Maroua Hammami, Olfa Ksentini, Nabih Feki, Mohamed Slim Abbes, and Mohamed Haddar Optimal Linear Quadratic Stabilization of a Magnetic Bearing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Abdelileh Mabrouk, Olfa Ksentini, Nabih Feki, Mohamed Slim Abbes, and Mohamed Haddar Evaluation of the Acoustic Performance of Perforated Multilayer Absorber Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Amine Makni, Marwa Kani, Mohamed Taktak, and Mohamed Haddar

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An Anisotropic Model with Non-associated Flow Rule to Predict HCP Sheet Metal Ductility Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Mohamed Yassine Jedidi, Anas Bouguecha, Mohamed Taoufik Khabou, and Mohamed Haddar A Model for Simulating Transients in Looped Viscoelastic Pipe Systems. Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Oussama Choura, Silvia Meniconi, Sami Elaoud, and Bruno Brunone Analytical Approach in the Pre-design Phase for Vibration Analysis of a Flexible Multibody System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Ghazoi Hamza, Maher Barkallah, Hassen Trabelsi, Amir Guizani, Jamel Louati, and Mohamed Haddar Pressure Calculation and Fatigue of a Trans-tibial Prosthetic Socket Made from Natural Fiber Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Sofiene Helaili, Wahbi Mankai, and Moez Chafra Online Adaptive MFC for Nonlinear Active Half Car System . . . . . . . . 210 Maroua Haddar, Fathi Djemal, Riadh Chaari, S. Caglar Baslamisli, Fakher Chaari, and Mohamed Haddar The Interaction of Different Transmission System Using the Substructuring Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Marwa Bouslema, Taher Fakhfakh, Rachid Nasri, and Mohamed Haddar Combined Effect of Roughness Anisotropy and Roughness Parameters on the Friction Behavior Under Boundary Lubrication . . . . 227 F. Elwasli, S. Mzali, F. Zemzemi, and S. Mezlini A New Dynamic Model for Worm Drives . . . . . . . . . . . . . . . . . . . . . . . 235 Ala Eddin Chakroun, Ahmed Hammami, Ana De-Juan, Fakher Chaari, Alfonso Fernandez, Fernando Viadero, and Mohamed Haddar Shear-Normal Coupling Effects on Composite Shafts Dynamic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Safa Ben Arab, Slim Bouaziz, and Mohamed Haddar Experimental Contact Model Calibration for Computing a Vibrating Beam Coupled to a Granular Chain Impact Damper . . . . . . . . . . . . . . . 251 Chaima Boussollaa, Vincent Debut, Jose Antunes, Tahar Fakhfakh, and Mohamed Haddar Numerical Analysis of Transient Flow in a Hydraulic Tree Network: Zarroug Aqueduct Network in Gafsa, Southern Tunisia . . . . . . . . . . . . 261 Lazher Ayed and Zahreddine Hafsi

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Assessment of Temperature History When Abrasive Milling of Long Fiber Reinforced Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 F. Guesmi, M. Elfarhani, S. Ghazali, A. S. Bin Mahfouz, A. Mkaddem, and A. Jarraya Free Vibration of Sandwich Nanobeam . . . . . . . . . . . . . . . . . . . . . . . . . 277 Nouha Kammoun, Nabih Feki, Slim Bouaziz, Mounir Ben Amar, Mohamed Soula, and Mohamed Haddar Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Gearbox Fault Identification Under Non-Gaussian Noise and Time-Varying Operating Conditions Stephan Schmidt1(B) , Fakher Chaari2 , Radoslaw Zimroz3 , P. Stephan Heyns1 , and Mohamed Haddar2 1 Centre for Asset Integrity Management, University of Pretoria, Pretoria, South Africa

{stephan.schmidt,stephan.heyns}@up.ac.za

2 Laboratory of Mechanics, Modelling and Production, National School of Engineers of Sfax,

Sfax, Tunisia [email protected], [email protected] 3 Diagnostics and Vibro-Acoustics Science Laboratory, Faculty of GeoEngineering Mining and Geology, Wroclaw University of Technology, Wroclaw, Poland [email protected]

Abstract. The Synchronous Average of the Squared Envelope (SASE) is very useful to visualise the periodicities in the instantaneous power of the machine due to damage. However, the SASE is sensitive to impulsive noise and the presence of non-synchronous damaged components and therefore provide unreliable representations of the condition of the gearbox under these conditions. Also, the instantaneous power is adversely affected by time-varying operating conditions. Impulsive noise and/or time-varying operating conditions can be encountered in the power generation (e.g. wind turbines) and mining industries (e.g. bucket wheel excavators). Hence, a method is proposed for impulsive data that were acquired under time-varying operating conditions. This method firstly estimates and removes the instantaneous power changes caused by the time-varying operating conditions, whereafter the Synchronous Geometric Average of the Squared Envelope (SGASE) is applied. A more numerically stable calculation of the SGASE is performed, which also provides further insights into its suitability for impulsive noise environments. The methodology is investigated on a bevel gearbox model that was simulated under time-varying operating conditions and an experimental dataset also acquired under time-varying conditions. The results indicate that the SGASE is to be preferred to the SASE for performing fault diagnosis in the presence of non-Gaussian noise. Keywords: Gearbox fault diagnosis · Synchronous average of the squared envelope · Synchronous geometric average of the squared envelope

1 Introduction Developing robust gearbox condition monitoring methods is very important for expensive rotating machines such as wind turbines. This ensures that the correct condition is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 1–9, 2021. https://doi.org/10.1007/978-3-030-76517-0_1

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inferred from the condition monitoring data, which enables the appropriate maintenance decisions to be made. The instantaneous power of the vibration signal is rich with fault information that can be extracted by investigating its angle-domain statistics (e.g. synchronous average of the squared envelope) and its spectrum (e.g. squared envelope spectrum) (Borghesani and Antoni 2017). However, rotating machines such as wind turbines operate inherently under timevarying operating conditions and the noise is often non-Gaussian, which impedes the application of the conventional techniques. Borghesani and Antoni (2017) indicated that the squared envelope spectrum is not robust to non-Gaussian noise and therefore proposed that the spectrum of the logarithm of the envelope should be used instead of the squared envelope spectrum in non-Gaussian noise conditions. Schmidt and Heyns (2020) proposed a method to Normalise the Amplitude Modulation caused by time-Varying Operating Condition (NAMVOC) that could impede the fault detection process. The synchronous average of the squared envelope spectrum is especially powerful for gearbox condition monitoring, because it can be used for fault detection, fault trending and it is possible to visualise the modulation caused by the damaged components in the gearbox (e.g. it can help to distinguish between localised and distributed gear damage). However, the time-varying operating conditions and non-Gaussian noise impede its ability to provide a robust representation for the condition of the gearbox. Hence, the combination of two strategies is investigated to visualise the modulation caused by the damaged components in the gearbox. Firstly, the influence of the time-varying operating conditions is attenuated by using the NAMVOC method proposed by Schmidt and Heyns (2020), whereafter the synchronous geometric average of the squared envelope is used to present the modulation caused by the damaged components. The layout of this paper is as follows: In the next section, Sect. 2, an overview of the considered methods is given, whereafter the methods are investigated on numerical bevel gearbox data in Sect. 3 and on experimental gearbox data in Sect. 4. Finally, conclusions are drawn in Sect. 5.

2 Gearbox Fault Diagnosis Vibration-based fault diagnosis methods for gearboxes operating under time-varying conditions and non-Gaussian noise conditions are considered in this section. 2.1 The Influence of Time-Varying Operating Conditions Time-varying operating conditions result in simultaneous amplitude and phase modulation (Stander et al. 2002, Chaari et al. 2012). The phase modulation can be attenuated by interrogating the signal in the angle domain, since the cyclostationary content attributed to damage is periodic in the angle domain. This can be performed by measuring or estimating the rotational speed of the system and using this information to transform

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the signal from the angle to the time domain through order tracking. Stander and Heyns (2006) indicated that if the instantaneous phase of the signal is estimated from a source that has a different transmission path as the vibration signal, phase distortion can occur. However, since the instantaneous power is interrogated in this signal, the influence of the phase distortion induced by varying speed conditions is not considered. The amplitude modulation impedes the detection of gear damage as well as the estimation of the severity of the gear damage defects. Hence, a Normalisation of the Amplitude Modulation due to Varying Operating Condition (NAMVOC) method was proposed by Schmidt and Heyns (2020) to reduce the amplitude modulation due to varying operating conditions, while retaining the amplitude modulation caused by damage. The NAMVOC procedure assumes that the measured vibration signal x(t) can be decomposed as follows in terms of a modulation function N(t) and the raw signal q(t): x(t) = N (t) · q(t)

(1)

After N(t) is estimated, it can be used to calculate q(t) which is analysed for damage. The modulation function is estimated by using a moving median filter on the squared vibration signal as discussed by Schmidt and Heyns (2020). This filter ensures that the fault information is retained by the normalisation procedure. The modulation function N(t) is especially detrimental for vibration-based condition monitoring when it varies synchronously with the component-of-interest. The application of the NAMVOC procedure is shown in Fig. 1 on the numerical bevel gear dataset described in Sect. 3. The vibration signal x(t), the rotational speed and load are presented in Fig. 1 for one of the measurements. Firstly, the NAMVOC procedure is applied on the signal with the resulting normalisation function N(t) being shown in Fig. 1 as well. The vibration signal is thereafter normalised to obtain a new signal q(t) = x(t)/N(t), which can be analysed for damage. The amplitude of the normalised signal is much less influenced by the time-varying operating conditions when compared to the raw vibration signal and therefore it is better suited for fault detection. 2.2 The Influence of Non-gaussian Noise Non-Gaussian noise is attributed to the following two reasons in this work: 1. The operating environment of the rotating machine can result in non-Gaussian phenomena in the signal. For example, if the machine operates in an impulsive environment (e.g. mining industry (Wyłoma´nska et al. 2016)) or in the presence of electromagnetic interference, the signal would be leptokurtic. 2. Damaged machine components inherently result in non-Gaussian behaviour, because the signals tend to become more impulsive. Additionally, if the non-synchronous mechanical components are damaged, it would result in their corresponding signal components to be non-Gaussian with respect to the synchronous component under consideration.

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Fig. 1. Illustration of NAMVOC procedure for a signal x(t). The corresponding rotational speed ω(t), load and the estimated normalisation function N(t) are presented.

The Synchronous Average of the Squared Envelope, s(φ; ) =

Nx −1 1  |x(φ + k · )|2 Nx

(2)

k=0

is calculated for a specific cyclic period  (determined by the component under consideration) and is presented over a specific angular period range [0,ϕ). The number of rotations is denoted Nx and the angle variable is denoted ϕ. The SASE is especially adversely affected by non-Gaussian components in the vibration signal due to the fact that it utilises the sample average as an estimator of the central tendency of the instantaneous power. The sample average is unreliable for long-tailed distributions. This can potentially be solved by considering the Synchronous Geometric Average of the Squared Envelope (SGASE) g(φ; ) =

N −1 x 

1/Nx |x(φ + k · )|

2

(3)

k=0

instead of the SASE. If the SGASE is directly implemented with Eq. (3), it would be adversely affected by numerical underflow and lead to an erroneous estimation. Hence, the SGASE is rewritten as   Nx−1 1  log |x(φ + k · )|2 (4) g(φ; ) = exp Nx k=0

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which provides a numerically more stable implementation than Eq. (3). This representation also highlights why it is more suitable for non-Gaussian signals; the synchronous average of the logarithm of the squared envelope is calculated instead of the synchronous average of the squared envelope. According to Smith et al. (2019), the squared envelope of the signal that is estimated with the logarithm of the absolute of the signal, i.e.  (5) SEx (t) = log |x(t)|p(x(t))dx(t) where p(x) is the probability density function of the signal x(t), converges for a broader range of distributions than using the conventional squared signal. Hence, the logarithm of the squared signal is expected to be more robust to non-Gaussian noise than the squared signal. Therefore, calculating averages in the logarithmic domain, would be more beneficial for analysing non-Gaussian signals and therefore the SGASE is expected to be more robust than the SASE. The adverse influence of the time-varying operating and non-Gaussian noise conditions can therefore be attenuated in two sequential steps, namely, applying the NAMVOC procedure to remove the modulation functions due to the load and the speed and then by calculating the synchronous geometric average as opposed to the synchronous average. These methods are used in the next section on vibration data from a numerical gearbox model.

3 Bevel Gearbox Investigation A numerical bevel gearbox model, originally presented in Mahgoun et al. (2016), was used to generate data under time-varying operating conditions. The equation of motion of the eight degree-of-freedom model is solved for the operating conditions shown in Fig. 2 with Newmark’s algorithm.

Fig. 2. The load and rotational speed for three Operating Condition (OC) states.

Non-Gaussian noise is investigated by adding r(t) to the vibration response of the structure. The noise, r(t), is sampled from a symmetric α-stable distribution with a mean of 0, a standard deviation of 1, and a parameter α which specifies its impulsivity. If α = 2, r(t) is Gaussian distributed, i.e. Gaussian noise is added to the signal, and as α becomes smaller, r(t) becomes more impulsive. Two distinct noise cases are investigated here; Gaussian noise (α = 2) and very impulsive noise with α = 1.2.

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The measured signal is firstly normalised with the NAMVOC procedure whereafter the signal is order tracked. Thereafter, the SASE and SGASE are calculated for the case where the gear is healthy and the pinion contains localised damage. The results are presented in Fig. 3. (a) α = 2.0

(b) α = 1.2

Fig. 3. Synchronous Average of the Squared Envelope (SASE) and the Synchronous Geometric Average of the Squared Envelope (SGASE) for α-stable noise with different levels of impulsivity.

For the Gaussian noise case in Fig. 3(a), the location of the damage on the pinion can clearly be seen for the SASE and SGASE. The SASE of the gear, however, contains very impulsive information which is attributed to non-Gaussian noise induced by the presence of the pinion damage. This means that the SASE cannot separate the contributions of the gear and the pinion. In contrast, the SGASE of the gear is much more uniform, which indicates that it is more robust to damage on the pinion. For the non-Gaussian noise case in Fig. 3(b), it is impossible to infer the condition of gears by using the SASE. This is attributed to the fact that the SASE is very sensitive to non-Gaussian noise. In contrast, the SGASE is much more robust, with very similar results being obtained when compared to the Gaussian data case. Hence, the SGASE is capable of visualising the modulation caused by the damaged components with data that have non-Gaussian characteristics. The robustness of the SGASE and the SASE are further investigated by calculating their Root-Mean-Square (RMS) for different fault severity levels of the pinion. The results are presented in Fig. 4 for the three operating conditions shown in Fig. 2. The RMS of the SASE for the gear and the pinion have approximately the same values in Figs. 4(a) and (b) and are highly correlated, which means that it is impossible to effectively localise the damaged component. This has severe consequences for diagnosis as well as prognosis. In contrast, the RMS of the SGASE is much more robust, with the deteriorating pinion and the healthy gear being easily distinguished.

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Fig. 4. The RMS of the SASE and the SGASE for different fault severities.

4 Experimental Investigation The experimental setup presented in Fig. 5 is located in the Centre for Asset Integrity Management (C-AIM) laboratory at the University of Pretoria. It consists of an electrical motor, which drives the system, and an alternator which dissipates the rotational energy and three helical gearboxes. The centre gearbox, indicated as the test gearbox, is monitored for damage. The test gearbox, which consists of a gear and a pinion, was damaged by seeding the gear with root damage. This gear was tested under the time-varying operating conditions shown in Fig. 5 until the gear failed, whereafter the test was stopped. The pinion was healthy for the duration of the test. Regular condition monitoring measurements were taken from the gearbox, with the axial component of a tri-axial accelerometer (located on the back of the gearbox) being used for monitoring. The rotational speed of the input shaft is estimated with an optical probe and a zebra tape shaft encoder. The proposed procedure was implemented for four measurements with the results presented in Fig. 6. The vibration data from the gearbox contain much impulsive noise irrespective of the condition of the gearbox. The influence of this impulsive noise is clearly seen in the SASE of the gear in Fig. 6(a), with the gear damage at approximately 135 degrees not seen. In contrast the SGASE is robust to the impulsive noise, with the damage easily detected in Fig. 6(b). The SASE and SGASE perform similarly for the pinion in Fig. 6(a), because it is clear that the pinion is healthy.

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Fig. 5. The experimental setup, the gear before and after the test, and the operating conditions that are investigated in this work, are presented.

Fig. 6. The SASE and the SGASE are compared for four measurements from the experimental setup considered in this work. The damaged gear becomes progressively worse from measurements (a) to (d).

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As the damage progresses from Fig. 6(a) to Fig. 6(d), the damaged component becomes more prominent in the SASE and SGASE. The damaged gear component can be identified with the SASE in Figs. 6(b) and (c), however, the corresponding SASE of the pinion contains much impulsive information, which makes it difficult to infer its condition. Hence, the SASE is very sensitive to non-Gaussian noise, while the SGASE is much more robust.

5 Conclusion A procedure is proposed in this work for visualising the modulation caused by the damaged components of the gearbox from non-Gaussian data acquired under timevarying operating conditions. Firstly, the operating conditions are attenuated, whereafter the synchronous geometric average of the squared envelope is calculated. This method is investigated on a numerical and experimental dataset, with the results indicating that the synchronous geometric average of the squared envelope is much more robust to nonGaussian noise than the synchronous average of the squared envelope and therefore can provide a more reliable representation for the condition of the gearbox. Acknowledgements. The South African and Tunisian authors acknowledge the South African and Tunisia Research Cooperation Programme 2019 (SATN 180718350459) for partially supporting this research. The South African authors gratefully acknowledge the support that was received from the Eskom Power Plant Engineering Institute (EPPEI) in the execution of this research.

References Borghesani, P., Antoni, J.: CS2 analysis in presence of non-Gaussian background noise–Effect on traditional estimators and resilience of log-envelope indicators. Mech. Syst. Signal Process. 90, 378–398 (2017) Chaari, F., Bartelmus, W., Zimroz, R., Fakhfakh, T., Haddar, M.: Gearbox vibration signal amplitude and frequency modulation. Shock Vib. 19(4), 635–652 (2012) Mahgoun, H., Chaari, F., Felkaoui, A.: Detection of gear faults in variable rotating speed using variational mode decomposition (VMD). Mech. Ind. 17(2), 207 (2016) Schmidt, S., Heyns, P.S.: Normalisation of the amplitude modulation caused by time-varying operating conditions for condition monitoring. Measurement 149, 106964 (2020) Smith, W.A., Borghesani, P., Ni, Q., Wang, K., Peng, Z.: Optimal demodulation-band selection for envelope-based diagnostics: A comparative study of traditional and novel tools. Mech. Syst. Signal Process. 134, 106303 (2019) Stander, C.J., Heyns, P.S., Schoombie, W.: Using vibration monitoring for local fault detection on gears operating under fluctuating load conditions. Mech. Syst. Signal Process. 16(6), 1005–1024 (2002) Stander, C.J., Heyns, P.S.: Instantaneous angular speed monitoring of gearboxes under non-cyclic stationary load conditions. Mech. Syst. Signal Process. 19(4), 817–835 (2005) Wyłoma´nska, A., Zimroz, R., Janczura, J., Obuchowski, J.: Impulsive noise cancellation method for copper ore crusher vibration signals enhancement. IEEE Trans. Ind. Electron. 63(9), 5612– 5621 (2016)

A Multiple-Grid Technique–Based Finite Element Solution of Free-Surface Flows in a Trapezoidal Open Channel Souad Mnassri1 and Ali Triki2,3(B) 1 Research Unit: Mechanics, Modelling, Energy and Materials M2EM, Department of

Mechanical Engineering, National Engineering School of Gabès, University of Gabès, Gabes, Tunisia 2 Research Laboratory: Advanced Materials, Applied Mechanics, Innovative Processes and Environment 2MPE, University of Gabès, Gabes, Tunisia 3 Higher Institute of Applied Sciences and Technology of Gabès, University of Gabès, Gabes, Tunisia [email protected] Abstract. Open-channel hydraulic remains an attractive topic for scientists due to complexities of physical environmental processes associated therewith. These complexities arise from (i) the adequate selection of the correct mathematical model to describe the physical flow processes; and (ii) the numerical tool used to solve the derived model. Commonly, flows in open - channels are described using the St.- Venant equations; which approximates the unsteady velocity distribution and friction factors by their respective ones derived from the steady flow. This paper presents a Multiple - Grid technique - based Finite Element solver to discretize the one-dimensional St. - Venant equations. This technique was employed in order to reduce the computational time involved in the standard technique – based solver. The numerical solver uses a discontinuous Galerkin formulation for the space discretization, in conjunction with a Semi-Implicit Euler procedure being employed for time integration. Test case addressed a trapezoidal open channel supplied by a reservoir at the upstream extremity and connected with a hydraulic power plant placed at its downstream extremity. Results demonstrated the accuracy of the developed solver and evidenced the benefit of the Multiple Grid technique in terms of saving the consumed computational time. Finally, the model is verified by comparison with alternative numerical results. Keywords: Discontinuous Galerkin method · Finite element method · Free-surface · Multiple-Grid · Open-channel · Transient flow · Trapezoidal

1 Introduction Open-channel hydraulic remains an attractive topic for scientists and engineers due to its great importance in conveying water for human living. In particular, unsteady freesurface flow, which takes place in natural or man-made rivers, are usually considered as being complex physical environmental processes. These complexities arise from (i) the adequate selection of the correct mathematical model to describe the physical flow processes; and (ii) the numerical tool used to solve the derived model. Commonly, flows © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 10–18, 2021. https://doi.org/10.1007/978-3-030-76517-0_2

A Multiple-Grid Technique–Based Finite Element Solution

11

in open-channels are described using the St.-Venant equations. Of late, considerable progress has been observed in the development of numerical solvers; providing optimal performance for solving free-surface flow problems. Basically, these solvers are based on the Finite Difference Method (FD) (Fennema and Chaudhry 1987; Liang et al. 2000; Prasada 2002; Prasada and Miguel 2003; Triki 2014a, 2017, 2019a, b; Mnassri and Triki 2020a, 2021), the Finite Element Method (FEM) (Cooley and Moin 1976; Szymkiewicz 1991, 2010; Hervouet 2003; Triki 2013, 2014b; Mnassri and Triki 2020b), and the Finite Volume Method (FVM) (Hou and Le Floch 1994; Toro 2004; Wuyi et al. 2019). Incidentally, these solvers are high-resolution schemes, originally devoted to simulate high-speed compressible flows, and have been recognized as being efficient numerical tools to solve the shallow-water equations. In particular, the FEM solvers demonstrated high capacities in describing flow problems involving complex boundaries condition such as sudden shut off flow downstream or continuously increasing flow upstream (e.g. Triki 2013, 2014b for an informal overview). However, these solvers use small space-step value to allow accurate solution; and, hence, require extra computational time. One possible alternative to overcome the limitation of the FEM solver involved by the standard technique of grid distribution, referred to as Multiple-Grid technique (MG), was addressed by Triki (2014a). Incidentally, this technique was firstly applied by Yost and Prasada (2000a, b) and Prasada (2005) in order to reduce the computational time consumed in the McCormack –based FDM solver. Basically, this technique is a numerical tool used to alter the time- or space- discretization between coarse or fine meshes. Incidentally, this technique may be employed inherently to any discretization approach. In this line, the author exploited the benefit of the Multiple-Grid technique (MG) in order to reduce the computational time consumed in simulating supercritical flow regime in rectangular open channel. Interestingly, the author evidenced that such a technique reduced about 12% the computational time involved by the standard technique –based FE solver. Accordingly, the purpose of this study is to explore the benefit of the MultipleGrid–based Finite element solver within trapezoidal open-channel framework. This paper is organized as follows: In Sect. 2, the basic equations are presented and the numerical discretization steps are outlined. In Sect. 3, numerical example is investigated to highlights the benefit of the proposed solver. Concluding remarks are enumerated in Sect. 4.

2 Theory and Calculations As per Hou and Le Floch (1994), the basic equations, describing unsteady free-surface flow in open-channel, may be expressed as follows: ∂F(U ) ∂U + −H =0 (1) ∂t ∂x  T in this equation, U = A q denotes the vector of conserved variables; F =  T    T 2 is the flux vector; and H = 0 g s0 − sf is the source term vecq uq + gT d2 tor; A = T × d corresponds to the wetted cross-section of the channel, d designates the

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S. Mnassri and A. Triki

flow-depth measured  normal to the bottom; T is the free-surface width; q corresponds to the flow-rate; u = q A designates the flow velocity;  s0 denotes the slope of the channel4 3 2 2 bed; g refers to gravity acceleration; s = m u R / designates the flow resistance; f

H

RH is the hydraulic radius; m is the Manning’s roughness coefficient; t and x are time and distance, respectively. In this study, derivation relies upon the work of Triki (2014a, b). Briefly, the numerical solution of equations set (1) is obtained using a discontinuous Galerkin space discretization (Dhatt and Touzot 1984; Zienkiewicz and Taylor 1988). In this line, a linear shape function N with two nodes and two degrees of freedom per node (d , u) (Fig. 1) are employed. Conjointly, a Semi-Implicit Euler approach isimplemented to estimate time ¯ i − Uj−1 derivative terms (Dhatt and Touzot 1984) i.e. ∂U ∂t = U (θ t), wherein i ¯ i and Uj−1 are computed at the node i and the time steps (j − 1 + θ )t and (j − 1)t, U i respectively; and θ is a numerical parameter relying upon the stability of the solution (0 < θ ≤ 1,j = 1, 2, . . .). 1

0

(i)

(i-1)

xi-1

xi-1

x xi+1

Fig. 1. Linear shape interpolation functions.

Following these considerations, the discrete formulations for the shallow water Eqs. (1) can be summarized by a set of 2(n + 1) times partial derivative equations, which is solved basing on the Gauss substitution -based iterative algorithm:    

  K U¯ U¯ = f U¯ (2) wherein:

; n

is the number of finite elements; Mik is (2 ×  (Mik = 0 if k < i − 1

2) square-matrix ¯ i; U ¯ i+1 ); M11 , M12 , Mn+1n , and ¯ i−1 ; U or k > i + 1 else it relies upon the values: U Mn+1n+1 relate to the boundary conditions at the nodes 1, 2, n and n + 1, respectively; f is composed of the flow parameters evaluated at the time (j − 1)t; f1 and fn+1 include the boundary conditions of the problem.

A Multiple-Grid Technique–Based Finite Element Solution

13

Finally, the Multiple-Grid technique (Yost and Rao 2000a, b; Triki 2014a), is used in order to reduce the computational efforts. This technique is based on the implementation of coarse and fine space stepping of the open-channel, denoted as cG and fG , respectively (Fig. 2). In this condition, calculations are altered between the two space stepping types basing on the Froude number value.

Fig. 2. Schematic illustration of the Multiple-Grid technique [cG : coarse-; fG : fine-; and mG : multiple- mesh].

The stability and the convergence of the numerical algorithm are conditioned by    |u| + gd and θ ≥ 0.5, (Richtmeyer and Morton CFL condition, t x ≤ 1 max 1967).

3 Case Study The MG-FEM solver is applied to describe the free-surface wave behavior in the trapezoidal open-channel sketched in Fig. 3-a (length L = 1500 m; width b = 10 m; side slope side β = 0.5; Manning roughness m = 0.016 and bottom slope α = 0.0002). Under normal operating condition, the channelis connected to the upstream reservoir and supplying a uniform flow-rate Q0 = 40 m3 s to the head-works of a hydroelectric power plant, located at the channel outlet (Fig. 3-b). The transient event results from the shutdown of the hydraulic power plant and, hence, the  channelis filled  with water under 2 stationary flow regime at reservoir level (i.e.: d|x=0 + u|x=0 2g = dR if u|x=0 ≥ 0; else d|x=0 = dR ; wherein dR designates the reservoir level above the channel bottom). Thereupon, with zero initial flow in the system(i.e.: Q1 = 0), the discharge is increased linearly to the full flow-rate value: Qf = 40 m s3 , during  tf = 2 mn. In other words, the outlet flow may be expressed as: Q = Q1 + Qf − Q1 t tf or Q = Qf for 0 ≤ t ≤ tf or t ≥ tf , respectively. Numerical results presented below were performed using a set of variables: {θ = 0.6 ; n = 10 N = 10; and t = 0.1 s}, for the MG-FEM algorithm.

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Fig. 3. (a)- Definition sketch of hydraulic system layout (Wylie et al. 1993); (b)- illustration of a hydroelectric power plant (Balkan Green Energy News).

Figures 4-a and -b display, respectively, the depth and velocity signals computed by the MG-FEM at different sections of the open-channel. In general, the wave pattern of this figure indicates that the shutdown event of the hydraulic power plant induces a fluvial flow across the channel. Namely, Fig. 4-a indicates that the depth, at the downstream extremity, decreases firstly to the depth value d|x=L = 2.3 m during the shutdown event; and subsequently drops to a crest value d|x=L = 1.8 m at t = 8.4 mn. Conjointly, Fig. 4-b suggests that the flow velocity, at the downstream extremity, increases due to the increase of the flow rate at this extremity. Similarly, but less important depth drop, due to friction effects on the open-channel bed, are involved by the depth and velocity curves computed at the intermediate sections. Incidentally, the phase shift between these different signals (depth or velocity) is due to the duration of wave propagation. Thereafter, after reflection at the upstream reservoir, the depth increases continuously. Finally, after a series of wave propagation and reflection cycles the depth reaches the depth value of the upstream reservoir. As regard the robustness of the MG-FEM–based algorithm used in this study, results suggests that this algorithm offers about 7% less computational time than the standard algorithm -based FEM (Triki 2013).

A Multiple-Grid Technique–Based Finite Element Solution

d, m

15

(a)

3.2 2.8 2.4

x=0

2 1.6

x=0.2 L 0

4

8

12

u, m/s 2.5

16

20

24

28

x=0.4 L

t, min

(b)

x=0.6 L

2

x=0.8 L

1.5

x= L

1 0.5 0

0

4

8

12

16

20

24

28 t, min

Fig. 4. Depth and velocity signals at different cross sections of the open-channel.

Figure 5 compares downstream depth traces issued from the MG-FEM solver (θ = 0.6; N = 6 and n = 10) and those obtained by Wylie et al. (1993) employing the method of characteristics. Figure 5 reveals close correspondence between the numerical result issued from the finite element method and its counterpart performed by Wylie et al. (1993); however, it should be pointed out herein that the former approach is more general than the latter one. Indeed, as delineated by Wylie et al. (1993), the method of characteristics needs extra numerical treatment to handle flow discontinuities due to hydraulic bores or flow shocks.

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d, m 3.2 2.8 2.4

FEM

2 1.6

MOC (Wylie et al. 1993) 0

4

8

12

16

20

24

28 t, min

Fig. 5. Comparison between the MG-FEM and the MOC (Wylie et al. 1993).

4 Conclusion In this study, the numerical accuracy and the computational efficiency of the MG-FEM were explored, within fluvial flow regime in trapezoidal open channel. Overall, this study highlighted that: (i) The numerical solution issued from the multiple grid technique–based finite element solver compare well with its counterparts issued from method of characteristics. (ii) The proposed algorithm allows the direct computation of hydraulic parameter values, at the boundaries of the channel; were deduced without using characteristic equations. (iii) Differently from the method of characteristics, the MG-FEM can be extended for handling applications involving hydraulic bores. (iv) The multiple grid technique requires less computational efforts as compared to the classical finite element method. Although this study addressed fluvial flow regime, such a numerical solver can also be adjusted to produce results for rapidly varied flow regimes.

References Cooley, R.L., Moin, S.A.: Finite element solution of Saint-Venant equations. J. Hydraul. Div. 102(6), 759–775 (1976) Dhatt, G., Touzot, G.: Une présentation de la méthode des éléments finis- (A presentation of the finite element method). Edition Maloine SA, Paris (1984) Fennema, R.J., Chaudhry, M.H.: Simulation of one-dimensional dam-break flows. J. Hydraul. Res. 25(1), 41–51 (1987). https://doi.org/10.1080/00221688709499287 Hou, T.Y., Le Floch, P.G.: Why non conservative schemes converge to wrong solutions: error analysis. Math. Comput. 62, 497–530 (1994). https://doi.org/10.1090/S0025-5718-1994-120 1068-0

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Hervouet, J.M.: Hydrodynamique des Ecoulements à Surface Libre Modélisation Numérique avec la Méthode des Eléments Finis-(Hydrodynamics of Free Surface Flows Numerical Modeling with the Finite Element Method), Presse de l’école nationale des ponts et chaussées (2003) Liang, D., Ingram Falconer, R., Lin, A.: Comparison between TVD-McCormack and ADI-type solvers of the shallow water equation. Adv. Water Resour. 23, 545–562 (2000). https://doi.org/ 10.1016/j.advwatres.2006.01.005 Mnassri, S., Triki, A.: Numerical investigation towards the improvement of hydraulic-jump prediction in rectangular open-channels. ISH J. Hydraul. Eng. (2020a). https://doi.org/10.1080/ 097150.2020.1836684 Mnassri, S., Triki, A.: A finite element –based solver for simulating open-channel transient flows The gradually varied regime. ISH J. Hydraul. Eng. (2020b). https://doi.org/10.1080/09715010. 2020.1815090 Mnassri, S., Triki, A.: Investigating the unidirectional flow behavior in trapezoidal open-channel. ISH J. Hydraulic Eng. (2021). https://doi.org/10.1080/09715010.2021.1918027 Mnassri, S., Triki, A.: On the unidirectional free-surface flow behavior in trapezoidal crosssectional open-channels. Ocean Eng. 223, (2021). https://doi.org/10.1016/j.oceaneng.2021. 108656 Prasada, R.: Contribution of Boussinesq pressure and bottom roughness terms for open channel flows with shocks. Appl. Math. Comput. 133, 581–590 (2002). https://doi.org/10.1016/S00963003(01)00259-4 Prasada, R., Miguel, A.M.: Evaluation of V and W multiple grid cycles for modeling one and twodimensional transient free surface flows. Appl. Math. Comput. 138, 151–167 (2003). https:// doi.org/10.1016/S0096-3003(02)00226-6 Prasada, R.: Numerical modeling of open channel flows using a multiple grid ENO scheme. Appl. Math. Comput. 161, 599–610 (2005). https://doi.org/10.1016/j.amc.2003.12.051 Richtmeyer, R.D., Morton, K.W.: Difference Methods for Initial Value Problems. Intersciences Publishers, A Division of John Wiley and Sons, New York (1967) Szymkiewicz, R.: Finite-element method for the solution of the Saint Venant equations in an open channel network. J. Hydrol. 122(1), 275–287 (1991). https://doi.org/10.1016/0022-169 4(91)90182-H Szymkiewicz, R.: Numerical Modelling in Open Channel Hydraulics, Water Science and Technology Library. Springer, Amsterdam (2010). https://doi.org/10.1007/978-90-481-3674-2 Triki, A.: A finite element solution of the unidimensional shallow-water equation. J. Appl. Mech. ASME 80(2), (2013). https://doi.org/10.1115/1.4007424 Triki, A.: Resonance of free-surface waves provoked by floodgate maneuvers. J. Hydrol. Eng. 19, 1124–1130 (2014a). https://doi.org/10.1061/(ASCE)HE.1943-5584.0000895 Triki, A.: Multiple-grid finite element solution of the shallow water equations: water-hammer phenomenon. Comput. Fluids 90, 65–71 (2014b). https://doi.org/10.1016/j.compfluid.2013. 11.007 Triki, A.: Further investigation on the resonance of free-Surface waves provoked by floodgate maneuvers: negative surge waves. Ocean Eng. 133, 133–141 (2017). https://doi.org/10.1016/j. oceaneng.2017.02.003 Triki, A.: Investigating the free-surface flow behavior due to sluice-gate maneuvers. In: Advances in Mechanical Engineering, Materials and Mechanics - Selected Contributions from the 7th International Conference on Advances in Mechanical Engineering and Mechanics, ICAMEM 2019, Hammamet (2021) Toro, E.F.: Schok Capturing Methods for Free Surface Shallow Flows. John Wiley and sons, New York (2004) Wylie, E.B., Streeter, V.L., Suo, L.: Fluid Transients in System. Prentice Hall, Hoboken (1993) Wuyi, W., Awais, R., Xiaoyi, C.: Effect of height and geometry of stepped spillway on inception point location. Appl. Sci. 9(10), 2091 (2019). https://doi.org/10.3390/app9102091

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Yost, S.A., Rao, P.: A moving boundary approach for one-dimensional free surface flows. Adv. Water Resour. 23, 373–382 (2000). https://doi.org/10.1016/S0309-1708(99)00029-9 Yost, S.A., Prasada, R.: A multiple grid algorithm for one-dimensional transient open channel flows. Adv. Water Resour. 23, 645–665 (2000). https://doi.org/10.1016/S0309-1708(99)000 52-4 Zienkiewicz, C., Taylor, R.L.: The Finite Element Method. McGraw-Hill Book Company, New York (1988)

Rotor-Ball Bearings System Under Variable Regime Mohamed Habib Farhat1,2(B) , Xavier Chiementin2 , Fakher Chaari1 , Taissir Hentati1 , and Mohamed Haddar1 1 Laboratory of Mechanics, Modeling and Production (LA2MP), National School of Engineers

of Sfax, BP1173 3038 Sfax, Tunisia {Mohamed-habib.farhat,taissir.hentati}@enis.tn, [email protected] 2 Institute of Thermics, Mechanics and Material (ITHEMM), University of Reims, Moulin de la Housse, 51687 Reims Cedex 2, France [email protected]

Abstract. One of the most frequent events that can occur in rotating machinery is bearing damage. Diagnosis of bearing failure under varying operating conditions present a huge challenge for research and investigation. In this paper, a model based study of a rotor-bearing system is carried out under variable speed and load conditions. The model studied refers to a real test bench developed at the ITHIMM laboratory of the University of Reims in France. It consists of a line of rotating shafts supported by two identical ball bearings, excited by a synchronous electric motor, and radially preloaded by a hydronic actuator. Based on the Hertizian contact theory, the proposed model has been performed in the case of a crack defect in the outer ring of the load-side bearing. The equations of motion of the system are formulated and solved using Newton Raphson combined with Newmark algorithms. Because of the noise masking effect, the recorded time response signal does not clearly show the defect impulses. Therefore, wavelet denoising techniques are used to highlight the damage. A designed time-frequency envelope spectrum is also applied to the noise-free signal to study the effect of the speed and load variation on the defect frequency. Keywords: Numerical modelling · Rotating shaft · Bearing · Vibration analysis · Defects

1 Introduction Bearings can be found wherever there is rotation, they reduce friction between rotating bodies and provide a privileged path for mechanical vibration. In the healthy case, bearing excitations are low and are generally neglected [1]. However, in the presence of defects these excitations may become predominant. Until the 1960s, bearing studies were mainly carried out by experimental measurements. Subsequently, various empirical formulations of bearing performance were developed, as in the work of Stribeck et al. [2], who performed one of the earliest investigations © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 19–26, 2021. https://doi.org/10.1007/978-3-030-76517-0_3

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for a zero-clearance, radially loaded ball bearing system. Since the 1960s, with the rapid development of computers, several studies have been carried out to understand bearings dynamics. Tiwari et al. [3], developed a numerical model of a deep groove ball bearing which takes into consideration clearance and Hertzian contact properties. Hentati et al. [4], have developed a dynamic model of a rigid ball bearing that takes into consideration the non-linear contact between balls and raceways. Ghafari et al. [5], used the Hertzien contact deformation theory to develop a non-linear dynamic model for a balanced ball bearing system. As the clearance increased, they found that the bearing equilibrium point was underwent a supercritical pitchfork bifurcation. Further investigations have been conducted on the dynamic modeling of rotor-bearing systems. Kappaganthu and Nataraj. [6], developed a non-linear dynamic model of a rotor-bearings system presenting internal radial clearance. Tomovic et al. [7], examined the effect of the radial internal clearance and the number of rolling elements on the vibration response of an unloaded rotor. Metsebo et al. [8], suggested a non-linear dynamic model in order to predict the dynamic behavior of a rotor-bearing system. They considered the shafts as an association of Timoshenko beams and the contact between ball and ring as a non-linear spring. Wardle. [9], was the first to study bearing defects by expressing the relationship between the number of balls and the order of defects. Patel et al. [10], developed a dynamic model of rolling elements bearings including single and double raceway defects in the case of constant depth of balls sinking. In further work, Patel et al. [11], took into account the variation in the defect profile. This latter was expressed as a function of the defect size and the ball radius. Jun Fan et al. [12], simulated the dynamic behavior of a defective bearing subjected to unbalanced force excitation. Rafsanjani et al. [13], developed a dynamic model of rolling element bearings. They analyzed the vibration behaviors of localized defects on the balls, the inner ring and the outer ring. In this context, a numerical study focused on the dynamic behavior of a ball bearing system in variable time regime in the presence of localized outer race defect is done. The paper is divided in two sections without considering the introduction. Section 2 presents the modelled test bench and the system’s equation of motion and Sect. 3 presents the simulations performed as well as a discussion of its results.

2 Test Bench Modelling The numerical model presented in Fig. 1 refers to a real test bench developed at the ITHIMM laboratory of the University of Reims in France. It consists of: a synchronous electric motor, a rotating shaft line, two identical ball bearings, a hydraulic actuator to exerts a radial preload on the shaft, and a flywheel to reduces the shaft’s torsional vibrations. 5 nodes are selected to represent the system dynamics. Each node includes 3° of free→ → doms dofs: (v, w) which present the flexion in − x and − y directions, and (θ ) that present − → the torsion in z direction. The shaft line is assumed to a multi-nodes conventional Timoshenko beam. Both flywheel and motor are modeled by their masses (Mf , Mm ) and   their inertias If , Im , which are supposed to be concentrated at nodes 1 and 10 respectively. The hydraulic actuator is considered in the modelling by the radial load it exerts − → Fv which is assumed to be directed towards node 4 (see Fig. 1). Referring to the work

Rotor-Ball Bearings System Under Variable Regime

21

Fig. 1. Test rig model

of Harsha et al. [14], the contact between bearing’s balls and raceways is considered as Hertizian, this allows to express the bearings excitation efforts as follows: z 3 B= k(x cos θi + y sin θi − γ ) 2 i=0

z present the number of balls, θi present the relative angular location of each ball, γ present the radial internal clearance,(x, y) present the relsative displacements of the inner race, and k present Hertz coefficient. Figure 2 shows a typical schematic diagram of a rolling element bearing. Let’s Ts be the sampling period for the time axe. For t = t0 , the ball 1 is supposed to be located at θ1 = 0 rad. For a constant speed, Harsha et al. [14] expressed θi (t) in function of the angular velocity of the cage ωc by: θi (t) = ωc ∗ t +

2π z ∗ (z − 1)

When the drive speed varies, the angular velocity of the cage changes over time. Let’s ωc (tk ) be the cage’s angular velocity at the k th time step. The angle that locate the ith ball can be approximated at tk by: θi (tk ) ≈

k  j=0

  ωc tj .Ts +

2π z ∗ (z − 1)

The inner ring is supposed to be mounted rigidly to the shaft. The angular velocity of the cage can therefore be expressed as follows:   r ωc (tk ) = ωr (tk ). R+r r and R are the radius of the inner race and the outer race respectively. According to Lagrange’s formalism, The equation of motion of the developed model is expressed as: [M ]¨r + [C]˙r + [K]r = Fe (t) + B(t)

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Fig. 2. Rolling elements bearing

q present the dofs vector, C present the damping matrix, M present the mass matrix, K presents the shaft and bearing stiffness matrix, Fe (t) presents the vector of the excitation forces which contains the driving torque, the radial load of the hydraulic actuator and the gravity forces, and B(t) presents the bearing excitations. The damping matrix C is defined by [15]: [C] = α[M ] + β[K] α and β are Rayleigh damping coefficients: α = 0.05 and β = 10−5 , 16.

3 Simulations In this section, a rotor-bearing system is examined under variable operational regime in the case of bearing’s outer raceway defect. The equations of motion of the system are solved using Newton Raphson combined with Newmark algorithms. The accelerations are simulated and recorded in the 3th node in the vertical direction. The chosen sampling 1 , where fr presents the shaft rotation frequency. The technical frequency is Tm = 2000∗fr specifications of the system components are summarized in Table 1. 3.1 Defect Formulation A crack defect is introduced to the outer race of bearing 2 (see Fig. 1). This latter is modelled as square-shaped chip with a width e = 0.5 mm and a depth h = 5 μm. Referring to Patel et al. [10], this defect is inserted into the model by an increase in the radial clearance as each ball passes through the defective area. The instantaneous restoring force in the case of defected bearing is given by: B(t) =

z  i=0

k.[x(t).cos(θi (t)) + y(t).sin(θi (t)) − (γ + A(φi (t))

3 2

+

)

Rotor-Ball Bearings System Under Variable Regime

23

Table 1. Technical specifications of the system’s components. Motor Flywheel Bearing Mass (kg)

8

7

Moment of inertia (kg.m2 ) 0.0125 0.0108 Number of balls Nb

9

Inner race radius (mm)

31

Outer race radius (mm)

56

Backlash γ (μ m)

18

Bearing stiffness k (N/m)

8.5 · 109

A(φi (t)) presents the additional clearance expressed by:

 b) h. sin π (φφi −φ , if φb ≤ φi ≤ φd + φd [2π ] d A(φi ) = 0, else φi present the relative angle between the ball number i and the location of the defective area. It is defined as: φi (tk ) ≈

k 

  (ωc tj .Ts ) + φb − θb .(i − 1), aveci ∈ (1..Nb )

j=0

Theoretically, the vibration response of a healthy bearing is almost zero, therefore, a white noise will be added to all the simulated signals to approximate the actual response of the system. The amplitude of the noise is calculated directly from the real test bench. To be more realistic, a 5% fluctuation in the noise amplitude will be taken into account. 3.2 Results The system response is carried out under two different operating conditions: according to a change in driving speed and/or according to a change in the applied radial load. Figure (4a) illustrates the system’s acceleration response under the variable load shown in Fig. (3a) at 1500 rpm fixed drive speed. Figure (4b) illustrates the system’s acceleration response under the variable speed shown in Fig. (3b) at Fr = 150.N fixed radial force. Fault pulses are very difficult to identify from Fig. 4, so the time response of the noisy signals as such is limited in this case. Therefore, de-noising techniques must be used to remove noise before proceeding with the envelope analysis. In this paper, the wavelet shrinkage method (WT) is used in order to eliminate noise from the signals [17]. The wavelet transform (WT) offers a better time-resolution compared to the Short Term Fourier Transform (STFT), which makes it suitable for the analysis of non-stationary signals.

24

M. H. Farhat et al.

Fig. 3. Speed and load variations: a): Load pace; b) Speed pace

Fig. 4. System vibration responses: a): For variable load and constant speed, b): For variable speed and constant load

Figure (5a) and Fig. (5b) present the de-noised signals for variable radial load and variable drive speed respectively. Race-fault impact generates varying vibration levels according to the instantaneous rotational speed and the applied radial load at the moment of impact. The amplitude of impulses is more important as the hydraulic actuator’s radial load is important. At higher speeds, the fault pulses are much closely spaced and the vice versa, the slower the speed, the greater the pulse spacing. In order to highlight the variations as a function of time of the fault frequency, a time-frequency envelope spectrum is applied to the noisy signals. The latter calculates the envelope of the signal in specific intervals of time which gives the possibility to detect the frequency variation.

Rotor-Ball Bearings System Under Variable Regime

25

Fig. 5. System de-noised vibration responses: a): For variable load and constant speed; b): For variable speed and constant load

Due to the independence between speed and load in a synchronous motor, for constant speed, Fig. (6a) shows that the outer race defect frequency is invariable whatever the radial load applied and corresponds exactly to FPBO = 82 Hz. For variable speed case the time-frequency envelop spectrum given in Fig. (6b) shows clearly that the defect frequency varies according to the speed pace.

Fig. 6. Time-Frequency envelop spectrums: a): For variable load and constant speed; b): For variable speed and constant load

4 Conclusion This article deals with the model-based diagnosis of bearing defects. The system studied is a rotor bearing system, excited by a synchronous motor and subjected to a radial

26

M. H. Farhat et al.

loading. The influence of the variation in speed and load have been investigated in the case of outer race defect. The wavelet de-noising technique was used to filter the recorded signals and has proven its effectiveness in the processing of vibration signals even under variable operating conditions. The result indicates that the vibration level of the system varies according to the rotational speed and the applied radial load. Focusing on the defect frequency, it appears that when the system is excited by a synchronous motor, the defect frequency varies as a function of the shaft speed, independently of the applied radial load.

References 1. Randall, R.B.: Detection and diagnosis of incipient bearing failure in helicopter gearboxes. Eng. Failure Anal. 11(2), 177–190 (2004) 2. Stribeck, R.: Ball bearings for various loads. Trans. ASME 29, 420–463 (1907) 3. Tiwari, M., Gupta, K., Prakash, O.: Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. J. Sound Vib. 238(5), 723–756 (2000) 4. Hentati, T., Maatar, M., Dammak, F., et al.: Quasi static analysis of a ball bearing having geometrical imperfection (waviness). Int. J. Eng. Simul. 9(1), 22–29 (2008) 5. Ghafari, S.H., Abdel-Rahman, E.M., Golnaraghi, F., et al.: Vibrations of balanced fault-free ball bearings. J. Sound Vib. 329(9), 1332–1347 (2010) 6. Kappaganthu, K., Nataraj, C.: Nonlinear modeling and analysis of a rolling element bearing with a clearance. Commun. Nonlinear Sci. Numerical Simul. 16(10), 4134–4145 (2011) 7. Tomovic, R., Miltenovic, V., Banic, M., et al.: Vibration response of rigid rotor in unloaded rolling element bearing. Int. J. Mech. Sci. 52(9), 1176–1185 (2010) 8. Metsebo, J., Upadhyay, N., Kankar, P.K., Nana Nbendjo, B.R.: Modelling of a rotor-ball bearings system using Timoshenko beam and effects of rotating shaft on their dynamics. J. Mech. Sci. Technol. 30(12), 5339–5350 (2016). https://doi.org/10.1007/s12206-016-1101-x 9. Wardle, F.P.: Vibration forces produced by waviness of the rolling surfaces of thrust loaded ball bearings Part 1: Theory. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 202(5), 305–312 (1988) 10. Patel, V.N., Tandon, N., Pandey, R.K.: A dynamic model for vibration studies of deep groove ball bearings considering single and multiple defects in races. J. Tribol. 132(4) 11. Patel, V.N., Tandon, N., Pandey, R.K.: Vibration studies of dynamically loaded deep groove ball bearings in presence of local defects on races. Procedia Eng. 64, 1582–1591 (2013) 12. Fan, J., Cui, W., Han, Q.: Vibration signal modeling of a localized defective rolling bearing under unbalanced force excitations. J. Vibroengineering 19(7), 5009–5019 (2017) 13. Rafsanjani, A., Abbasion, S., Farshidianfar, A., et al.: Nonlinear dynamic modeling of surface defects in rolling element bearing systems. J. Sound Vib. 319(3–5), 1150–1174 (2009) 14. Harsha, S.P., Sandeep, K., Prakash, R.: The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings. Int. J. Mech. Sci. 45(4), 725–740 (2003) 15. Parker, R.G., Vijayakar, S.M., Imajo, T.: Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons. J. Sound Vib. 237(3), 435–456 (2000) 16. Khabou, M.T., Bouchaala, N., Chaari, F., et al.: Study of a spur gear dynamic behavior in transient regime. Mech. Syst. Signal Process. 25(8), 3089–3101 (2011) 17. Mishra, C., Samantaray, A.K., Chakraborty, G.: Rolling element bearing defect diagnosis under variable speed operation through angle synchronous averaging of wavelet de-noised estimate. Mech. Syst. Signal Process. 72, 206–222 (2016)

The Efficiency of the Rayleigh-Ritz Method Applied to In-Plane Vibrations of Circular Arches Elastically Restrained at the Two Ends and Supporting Point Masses Ahmed Babahammou(B) and Rhali Benamar Mohammadia School of Engineers, Mohamed V University in Rabat, BP 765 Rabat, Agdal, Morocco

Abstract. The natural frequencies and mode shapes of in-plane vibration of circular arches elastically restrained against rotation at their ends are determined using the Rayleigh-Ritz method (RRM) and trial functions obtained as particular solutions of the sixth order differential equation of arch vibrations corresponding to an opening angle equal to 1 rad. The investigations are made under the classical hypotheses: the effect of shear deformation and rotary inertia are neglected, the arch axis is inextensible, and the dimensions of the cross-section are constant and small in comparison with the radius. The first eight mode shapes and natural frequencies of arches with different opening angles and torsional spring stiffnesses are determined and shown to compare well with the available literature. Arches with added concentrated masses are then examined. The effect of the rotational stiffness and the added masses on the natural frequencies and mode shapes are determined and illustrated in the joint plots. The accuracy and relative simplicity of the RRM applied in a systematic way to such complicated problems is established, making it ready to use in more complex situations, such as those of arches with more added masses, with non-uniform cross section or with one or more point supports. Keywords: In-plane vibration · Circular arches · Rayleigh-Ritz method · Elastically restrained · Added masses

1 Introduction Arches are among the basic structural elements found in various real-world applications, such as aerospace structures, bridges, tunnels and roof structures. Free in-plane vibration has been the subject of numerous researches. In several works, the studies were restricted to arches with classical end conditions supporting added masses. In the present paper, the same topic is considered for arches elastically restrained in rotation at their both ends; the influences of spring stiffness at the ends, the arc opening angle and the added masses on the natural frequencies and mode shapes are studied. By assigning in the model the values zero or infinity to the spring stiffnesses at the arch two ends, one obtains the particular cases of simply-supported (SS) and clamped-clamped (CC) arches. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 27–34, 2021. https://doi.org/10.1007/978-3-030-76517-0_4

28

A. Babahammou and R. Benamar

Fig. 1. The circular arch under investigation

2 Arch Description and the Particular Solution of the In-Plane Equation of Motion Obtained by the Analytical Method (AM) The thin arch studied is shown in Fig. 1. It is elastically restrained against rotation, has a radius R, an opening angle θ, a Young’s modulus E, a mass per unit length of ring segment ρ and a total mass Ma = ρ.R.θ . The stiffnesses of the left and right elastically restrained ends Kl and Kr are supposed to be equal K = Kr = Kl . Considering that the effect of shear deformation and rotary inertia can be safely neglected and that the arch axis is inextensible, the radial displacement u and the tangential displacement v are related by [1]: u=

∂v ∂α

(1)

The AM is based on the motion’s differential equation of the arch in-plane displacement [2, 3]:   v(6) + 2v(4) + 1 − Ω 2 v(2) + Ω 2 v = 0 (2) Where v(n) indicates the nth derivative of v(α) with respect to, Ω 2 = ω2 ρ.R E.I is the square of the frequency parameter Ω, and ω is the natural frequency. The sixth degree characteristic polynomial, associated with Eq. (2) is:   (3) X6 + 2X4 + 1 − Ω 2 X2 + Ω 2 = 0 4

Using Cardan method [4] to solve the polynomial Eq. (2), one gets: v(α) = C1 sinh(X1 α) + C2 cosh(X1 α) + C3 sin(X2 α) + C4 cos(X2 α) + C5 sin(X3 α) + C6 cos(X3 α)

(4)

The Efficiency of the Rayleigh-Ritz Method

29

In which X1 toX3 are functions of Ω. The end conditions are [5, 6]: ∂v = 0 at α = 0 and α = 1 rad ∂α     M(0) = K v(2) (0) + v(0) , M(θ) = K v(2) (θ ) + v(θ ) u=

(5) (6)

  where M (α) = −EI /R2 v(3) + v(1) is the bending moment. The boundary conditions lead to the homogeneous system [S]{C} = 0 in which C T = (C1 ; C2 ; C3 ; C4 ; C5 ; C6 ) and [S] is a 6 × 6 square matrix. The solutions for Ω are deduced by stating the nullity of the matrix determinant. This leads to a transcendental equation solved by the NewtonRaphson method using the Matlab software to get the series of frequency parameters of the arch considered. Table (1) gives the first eight values of Ωi found for arches with an opening angle θ = 1 rad and different values of the rotational stiffness. The corresponding tangential and radial modes U (α) and V (α), α varying from 0 to 1, are plotted in Fig. 2, for K ∗ = KR EI = 24, as an illustration of the series of basic functions used in the RRM applied below to various values of θ and K. Table 1. First eight values of the frequency parameter Ωi obtained for circular arches with an opening angle θ = 1 rad and various values of K ∗ . K∗

Ω1

0

42.415

6

50.839

12

54.794

24

58.595

∞

65.724

Ω2

Ω3

Ω4

Ω5

Ω6

Ω7

Ω8

160.88

243.51

358.31

480.45

634.67

796.31

92.528

171.11

252.35

369.16

490.11

645.84

806.49

96.708

177.62

258.54

377.10

497.53

654.64

814.80

101.38

185.30

266.53

387.72

508.04

667.40

827.12

112.46

204.69

291.05

422.16

548.03

718.44

88344

85.394

Fig. 2. First six tangential and radial modes of an arch with an opening angle θ = 1 rad and a stiffness K = 100 obtained by the AM.

3 Application of the RRM to Arches with Added Point Masses For a given value of K, the particular solutions obtained in Sect. 2 by the AM for θ = 1 rad denoted as U(α) and V(α), for α = 0 to 1, are used to define the trial functions u(α) and

30

A. Babahammou and R. Benamar

v(α) implemented in the Rayleigh-Ritz formulation corresponding to an arch with the same value of K and a given value of θ as follows: α  α  , v(α) = θ V (7) u(α) = U θ θ The coefficient θ , added to the expression for v, ensures satisfaction of the compatibility Eq. (1). The arch is now assumed to carry two added masses m1 and m2 at the positions M1 and M2 defined by α = β1 , β2 respectively. The kinetic energy T of the arch and the two added masses and the strain energy V of the arch and the two torsional springs are given by [7]:   2

 l  2  2

2  ∂v ∂v 2 mk 1 ∂u ∂u T= ρ + + (8) ds + 2 0 ∂t ∂t 2 ∂t ∂t α=βk

k=1

V =

E.I 2

 l 0

∂ 2v ∂ v + ∂s2 ∂s R

2 ds +



Kl 2

∂ 2v ∂α 2

2 α=0

+

Kr 2



∂ 2v ∂α 2

2 (9) α=θ

Where ds = Rdα. Substituting Eq. (1) into Eqs. (7, 8) gives: ⎛ ⎛  

2 ⎞

2 ⎞    2  ∂v(1) ⎠ ∂v(1) ⎠ ρ.R θ ⎝ ∂v 2 mk ⎝ ∂v 2 T= + + dα + 2 0 ∂t ∂t 2 ∂t ∂t k=1

α=βk

(10) V =

E.I 2.R3



 2  2 θ K  (2) 2 (3) (1) v v +v dα + + v(2) α=0 α=θ 2 0

(11)

v(α, t), assumed to be harmonic in time, is expanded as a series of basic functions N  v(α, t) = ai vi (α) sin(ωt), vi being the ith trial function and ai its contribution coefi=1

ficient. After discretization of the energy expressions and application of Hamilton’s principle, one gets the matrix equation, written in dimensionless form:     2 K ∗ {A} − 2Ω 2 M ∗ {A} = {0} (12)     {A} = [a1 , . . . aN ]T , M ∗ and K ∗ are the dimensionless mass and rigidity matrices: m∗ij = θ

 0

1

vi∗ vj∗ d α ∗ +

1 θ

 0

1

∗(1) ∗(1) vj d α

vi

1 kij∗ = θ1 0 vi∗(1) vj∗(1) d α +  1 ∗(3) ∗(3) + θ15 0 vi vj d α + θ14



+

2  k=1

  ∗(1) ∗(1) μk vi∗ vj∗ + vi .vj

α=βk

(13)



 ∗(1) ∗(3) 1 1 vj + vi∗(3) vj∗(1) d α ∗ 3 0 vi θ      ∗(2) ∗(2) ∗(2) ∗(2) K.R v v + v v i j i j EI α=0 α=θ

μk = mk /Ma , k = 1, 2, are the ratios of the added masses to the arch mass.

(14)

The Efficiency of the Rayleigh-Ritz Method

31

4 Numerical Results Equation (12) presents the vibration equation of the arch with added masses based on the RRM. This linear eigen value problem is solved numerically using the MATLAB software, leading to the eigenvalues Ωi and their corresponding eigenvectors. Table 2 lists the values of the fundamental frequency parameters for elastically restrained circular arches without added mass, for several values of the opening angle θ and of dimensionless stiffnesses K ∗ = KR EI . The comparison with the results of Ref [8] shows that the difference percentages do not exceed 3.23%. It may be noted that the frequency parameter decreases when the arch opening angle increases but the rate of decrease is higher for small opening angles and very small for large opening angles. It increases strongly when the rotational stiffness K ∗ is small, but it does not change significantly for high values of K ∗ . It should be mentioned that the values of Ωi given in Ref [8] for the case K ∗ = ∞ (last line of Table 2), seem to be inappropriate as shown by the results given by several authors ([1, 2]) and also by the results obtained in the present work. Table 2. Fundamental frequency parameters of arches elastically restrained for several values of the opening angle θ and the end stiffness k without added masses. (a) Present results, (b) those of Ref [7]

The first eight frequency parameters Ω of a 20° elastically restrained arch are given in Table 3. The first and last couple of columns give the results obtained here and in Ref [1] for SS and CC arches and a good agreement is noticed. Table 4 gives the fundamental frequency parameters Ωθ 2 for two 20° elastically restrained arches with two added masses located at (β1 .β2 ) = (0.25, 0.75) and (0.3, 0.7) respectively. Calculations are made for several values of the rotational stiffness K and for a mass ratio μ equal to 0.2

32

A. Babahammou and R. Benamar

and 0.4. It is noted that the frequency parameter decreases when the mass ratio increases. In the extreme cases for K∗ i.e. 0 and infinity, relative to SS and CC arches, the present results compare well with those of Laura [7], as can be seen in the first and last couple of columns. Table 3. First eight frequency parameters Ω of arches elastically restrained for several values of stiffness without added masses. θ0 = 20◦ K∗

0

12

18

24

1.00E+07

Ω1

321.51

321.52*

378.17

397.50

412.27

503.55

503.55*

Ω2

690.04

690.04*

740.47

760.53

777.14

909.14

909.14*

Ω3

1293.51

1293.51*

1361.19

1390.59

1415.49

1637.27

1637.26*

Ω4

1987.98

1987.97*

2050.04

2078.27

2103.39

2373.79

2373.79*

Ω5

2913.52

2913.50*

2985.94

3021.15

3052.33

3419.18

3419.10*

Ω6

3932.70

3932.70*

4000.85

4033.91

4064.59

4482.38

4482.39*

Ω7

5181.53

5181.49*

5256.53

5296.59

5332.18

5849.47

5849.02*

Ω8

6525.09

6525.79*

6600.04

6633.42

6669.85

7234.08

7237.91*

Table 4. Fundamental frequency parameters Ω.θ 2 for 20° elastically restrained arches carrying two point masses located at (a) (β1 .β2 ) = (0.25.0.75) and (b) (0.3.0.7),with two values of the mass ratio (μ = 0.2, 0.4). The star indicates the results of Ref [7]. K∗ 0

6

12

18

24

100

1e10

μ = 0.2 (a) 34.569 33.990* 37.921 40.566 42.612 44.177 50.609 53.908 53.720* (b) 34.851 34.370* 38.137 40.709 42.683 44.184 50.268 53.329 53.590* μ = 0.4 (a) 31.269 30.910* 34.270 36.635 38.466 39.865 45.627 48.599 48.600* (b) 31.680 31.490* 34.579 36.837 38.561 39.869 45.135 47.771 48.440*

Figure 3 gives the variation of Ω with K for various values of θ . It shows that the frequency parameter increases with the rotational stiffness and decreases with the arch opening angle. The RRM doesn’t lead only to the frequencies but also to the associated mode shapes which are of a crucial importance in determining the structural response. As an illustration, the four first mode shapes of 90° arches with different values of stiffness K∗ are shown in Fig. 4.

The Efficiency of the Rayleigh-Ritz Method

33

Fig. 3. Fundamental frequency parameter as a function of the rotational stiffness

Fig. 4. Four mode shapes of a 90° arch elastically restrained with different stiffness values

5 Conclusion The free in-plane vibrations of inextensible circular arches elastically restrained in rotation with added point masses is investigated. Several end conditions are considered, starting from SS arches, and continuously increasing the stiffness to CC arches. The frequencies and mode shapes obtained by the RRM in the cases of arches with added masses are determined by the MATLAB software for several values of the arch opening angle, the mass ratios and the rotational stiffness. The rate increasing rate of the frequency parameter with θ is high for small opening angles and small otherwise. As may be expected, the frequency parameter increases with the rotational stiffness and decreases with the added mass.

34

A. Babahammou and R. Benamar

References 1. Auciello, N.M., De Rosa, M.A.: Free vibrations of circular arches: a review. J. Sound Vib. 176(4), 433–458 (1994) 2. De Rosa, M.A., Franciosi, C.: Exact and approximate dynamic analysis of circular arches using DQM. Int. J. Solids Struct. 37(8), 1103–1117 (2000) 3. Henrych, J.: The dynamics of arches and frames, vol. 2. Elsevier Science Ltd., (1981) 4. Poirier, S.: Nouvelle méthode de résolution des équations du 3eme degré1. SOMMAIRE DU No 96, p. 15 5. Benamar, R., Bennouna, M., White, R.G.: The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, part II: fully clamped rectangular isotropicplates. J. Sound Vib. 164(2), 295–316 (1993) 6. El Kadiri, M., Benamar, R.: Improvement of the semi-analytical method, based on Hamilton’s principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part iii: steady state periodic forced response of rectangular plates. J. Sound Vib. 264(1), 1–35, (2003) 7. Verniere De Irassar, P.L., Laura, P.A.A.: A note on the analysis of the first symmetric mode of vibration of circular arches of non-uniform cross-section. J. Sound Vib. 116, 580–584 (1987) 8. De Rosa, M.A.: The influence of the support flexibilities on the vibration frequencies of arches. J. Sound Vib. 146, 162–169 (1991)

Nonlinear Wind-Induced Response Analysis of Substation Down-Conductor System Guoqing Yu1(B) , Zhitao Yan1,2 , and Xinpeng Liu2 1 School of Civil Engineering, Chongqing University, Chongqing 400044, China 2 Department of Civil Engineering and Architecture, Chongqing University of Science

and Technology, Chongqing 401331, China

Abstract. Substation down conductor structure system is generally bundled conductor, and has the characteristics of small span, large height difference, short length and upper and lower connection. Substation lead-down accidents caused by strong wind often occurs. Down conductor accidents of substation caused by strong wind often occur. In order to study the nonlinear wind-induced response of the conductor under wind load, the AR model method is used to simulate the fluctuating wind load, and the modal analysis and nonlinear time history analysis of the wind-induced response of the flexible conductor system are carried out by using ANSYS software. The effects of aerodynamic damping, wind direction and Span-to-height ratio on wind-induced vibration response are studied. The analysis results show that the response of down conductor system has obvious nonGaussian characteristics; Compared with the conventional horizontal long-span transmission lines, the natural frequency of down conductor is higher, the influence of aerodynamic damping is smaller, and the response wind-induced vibration coefficient is larger. The wind load in plane will lead to larger reaction response, and the wind load out of plane will lead to larger displacement response. The increase of Span-to-height ratio will lead to the increase of reaction response and the decrease of displacement response of the structure. Keywords: Down conductor · Non-gaussian statistics · Wind-induced response

1 Introduction The down conductor structure of UHV substation is a typical flexible conductor structure, which is usually a multi split flexible conductor with spacer. It has the characteristics of small span, large height difference, long and short line, up and down connection, and is very sensitive to wind load. There have been many researches on wind-induced vibration response of horizontal and long-span split conductor. For example, scholars have conducted modal analysis on the split conductor, which shows that the loworder natural frequency order of the large-span transmission conductor is in the order of 10–2 –10–1 [1–3]. Yu Yongshuai [4] and Wang Shuliang [5] obtained the relationship between the aerodynamic damping and the natural frequency of the structure, indicating that the aerodynamic damping has a greater impact on the wind-induced vibration © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 35–43, 2021. https://doi.org/10.1007/978-3-030-76517-0_5

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G. Yu et al.

response of the horizontal long-span transmission line, and the aerodynamic damping plays a dominant role compared with the structural damping [6, 7]. With the increase of wind speed, the aerodynamic damping can significantly reduce the wind-induced vibration response of the structure. Due to the short length of the down lead system in the substation, its natural frequency is significantly different from that of the horizontal long-span transmission line [8], and the modal analysis of the small-span split down lead is also few. Zhang Xuesong et al. Studied the low-order mode of the soft bus and its electrical equipment [9]. Therefore, it is necessary to analyze the wind-induced vibration response characteristics of the down lead flexible system. In this paper, AR method [10] is used to simulate the fluctuating wind load, and ANSYS software is used to analyze the time history of the down lead system by time domain method [11, 12], so as to obtain the wind-induced vibration response of the structure under the wind load, which provides a reference for the wind load calculation of the down lead system of the substation.

2 Model Parameter The down conductor of Aksu UHV substation is used for analysis. The down lead is of four bundled structure. The type of the down lead is JGQNRLH55XK-700, and the type of tubular bus is 6063G-T6-∅200/184. basic parameters is shown in Table 1. In this paper, combined with the example of Aksu substation down conductor project system, the model is established by ANSYS software, and the whole model adopts BEAM189 unit. the upper end of down conductor is consolidated, the lower end is coupled with the tubular bus, and the two ends of the tubular bus are consolidated. Table 1. Parameter of the conductor Category Down conductor Tubular bus

M/kg/m R1/R2/mm E/MPa 1.927 16.29

L/m H/m

41/51

5.5 × 105 2

16

180/200

7.4 × 105 8

/

Where M is unit mass, R1 and R2 are inner diameter and outer diameter respectively. And E is young’s modulus, L is span, H is height difference.

3 Simulation of Wind Load In this paper, Davenport wind speed spectrum [13] is used to simulate the wind speed time history with randomness, time correlation and spatial correlation by using AR model under MATLAB workspace. The basic wind pressure is taken as 0.56 kN/m2 , the site category is B, the change of average wind speed along the height adopts exponential wind profile, and the wind speed profile index a is 0.15. See Table 2 for other main parameters.

Nonlinear Wind-Induced Response Analysis

37

Table 2. Parameters of simulated pulsating wind velocity time history Surface roughness coefficient K

Stage frequency Hz

Simulation interval/s

Number

Sample time/s

0.00327

4

1/16

21

256

According to the location coordinates of down conductor, the wind speed time history of all coordinate points is simulated. Limited to the space, only part of the wind speed time history curves of node 1 and node 10 are listed in this paper (Fig. 1). It can be seen from Fig. 1 that the AR model method is in good agreement with the target power spectrum.

Fig. 1. Wind speed time-history curve and simulated wind spectrum

4 Dynamic Characteristic Analysis The down conductor structure system of substation is generally four bundled conductor, which has the characteristics of small span, large height difference, long and short line, up and down connection. Before the dynamic time history analysis of the down conductor system, it is necessary to carry out modal analysis, the dynamic characteristics of the structure can be obtained after modal analysis. The main indexes include the natural frequency of the structure and the corresponding vibration mode. See Table 3 for the specific parameters. These indexes are important parameters for dynamics study. As shown in the Fig. 2, Model 1 is the parameter four bundled down conductor model in this paper; model 2 is a single wire model with the same parameter; model 3 is a four bundled down conductor model with horizontal layout; model 4 is from literature [14], which is a single span equal height four bundled wire model with a span of 353 m (representing general long-span transmission wire). In this paper, only the first four modes of the down conductor system are given, as shown in Fig. 3. In the figure, the first mode is the overall swing deformation along the yz plane of the down conductor; the second mode is swing deformation in the xy plane; the third mode is the torsion deformation along the y axis of the coupled system; the

38

G. Yu et al. Table 3. Natural frequencies of models Frequency

First mode

Second mode

Third mode

Fourth mode

Model 1

1.925

2.584

4.183

5.182

Model 2

0.937

1.687

2.579

3.724

Model 3

1.769

4.066

4.112

5.212

Model 4 [14]

0.172

0.343

0.344

0.438

Model 4: large span four bundled conductor model

Fig. 2. Schematic layout of model 1 to model 4

Fig. 3. The first four modes of Model 1

Nonlinear Wind-Induced Response Analysis

39

fourth mode is the double wave bending deformation in the xy plane, the performance of the first four modes is also related to the basic guide The linear modal form is consistent [4, 5]. It is worth noting that the vibration mode behavior of the down conductor model (model 1) in this paper is the same as that of model 3, and the natural frequency is also close. In addition to the second-order natural frequency, the average difference is 2.3%. Compared with model 4, the natural frequency of model 3 is one order of magnitude different. Literature [8] has studied the natural frequency of the suspension wire, and the natural frequency is mainly composed of span L and level under tension T control, when the span is small, the natural frequency of the conductor will obviously increase, which is basically consistent with the theory. And the average of the first four orders of model 1 is 1.6 times higher than that of the single conductor model 2. When the single conductor is extended to the four bundled conductor, T will also change significantly, which is also verified in reference [11]. The comparison of several models shows that the down lead system is different from the general large-span conductor, because of its short line length, small span but large height difference, its natural frequency is obviously different from that of the general large-span transmission line, and its natural frequency is large, so the dynamic response characteristics of them are also significantly different.

5 Parameter Analysis Based on the dynamic characteristics of the four bundled down lead system, the wind speed V, the wind direction angle θ, and the aerodynamic damping ξ a will affect the dynamic response of the structure. Therefore, the main three different parameters are set to study the influence of different parameters on the wind-induced vibration response of down conductor structure. For a highly nonlinear structural system, the response may show non-Gaussian distribution, such as some large-span membrane structures. Down lead system is also a typical nonlinear system, which will lead to the change of dynamic response characteristics and damping ratio. Therefore, this paper uses ANSYS software to carry out the finite element nonlinear time history analysis. The frequency distribution of its response is shown in Fig. 4. Its frequency distribution deviates from the Gaussian type. It is no longer applicable to directly use variance to express the dynamic performance of the response. Therefore, the maximum value of the response is directly used to represent the dynamic performance of the structure. 5.1 Influence of Aerodynamic Damping Ratio In the wind-induced vibration response of long-span transmission lines, the aerodynamic damping of conductors is dominant relative to the structural damping, and the wind-induced vibration response of transmission lines will be overestimated without considering the aerodynamic damping [4, 5]. According to the aerodynamic damping formula (1) in reference [4], it can be seen that the aerodynamic damping is mainly affected by the wind speed V, the natural frequency ƒ of the structure and the vibration mode. ζai =

ρa dCD V (1 + cos2 φ) 8π fi m

(1)

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Fig. 4. Response time history curve and frequency distribution of node 1

 φ=

π/2- arctan(f /mg) in-plane vibration mode out-plane vibration mode arctan(f /mg)

(2)

C D is the drag coefficient. In this paper, according to《Electrical Design of Electric Power Engineering Handbook》 , the value is 1.2; air density ρa is 1.293 g/L; d is the diameter of a single wire, the value is 0.051m; m is the wire density of a single wire, the value is 1.972 kg/m; V is the average wind speed of the wire; fi is the i-th order natural frequency of the model; and f is the average wind load acting on the unit length of the transmission wire. For the down lead structure, compared with the general long-span transmission line, the first four order natural frequency of the down lead structure is larger, and the aerodynamic damping of the down lead structure is smaller than that of the long-span transmission line structure when other parameters are the same. It is suggested in reference [14] that the damping ratio of cable net structure should be 1%, and in reference [5] that the first four damping ratio of six bundled conductor vibration mode should be 0.97% to 1.64% through wind tunnel test, so the damping ratio of down-lead system structure in this paper should be ξ s = 1%. According to the literature [15], the main influence of the damping ratio is the resonance response. Through the calculation and analysis, the smaller the damping ratio is, the larger the proportion of the resonance response

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component is, and the larger the structural response amplitude is. The damping ratio has great influence on the structure with resonance response as the main response, but has little influence on the down lead flexible system with background response as the main response. In the wind-induced vibration response of conductor, the influence of aerodynamic damping ratio is greater. When calculating the aerodynamic damping ratio of the down conductor, the influence of aerodynamic damping is directly considered in the damping matrix: ξ = ξ s + ξ a . therefore, the aerodynamic damping ratio ξ a under different wind speeds of the structure can be obtained by setting the damping matrix of different sizes to make its response consistent with that of the structure considering the aerodynamic damping. Through the above method, the aerodynamic damping ratios of model 1 and model 4 under different wind speeds are calculated and compared with formula (1). The specific results are shown in Fig. 5.

Fig. 5. Effect of aerodynamic damping on dynamic response of lead-down system

It can be seen from Fig. 5(a): the aerodynamic damping ratio calculated by ANSYS is relatively consistent with formula (1) (where: Fz, Fx is the direction of wind load, A is calculated by ANSYS, and E is calculated by formula (1)). It can be seen that the aerodynamic damping ratio of model 1 is significantly lower than that of model 4, that is, the aerodynamic damping of down conductor structure is smaller than that of large-span conductor; the aerodynamic damping under different wind directions is also different, the aerodynamic damping ratio out of plane is higher than that in plane, and the aerodynamic damping ratio out of plane is 40% higher than that in plane on average. Figure 5(b) shows the influence of the aerodynamic damping ratio on the response of model 1 in the plane. For the maximum reaction force of model 1, after considering the aerodynamic damping, the effect of aerodynamic damping is stronger with the increase of wind speed, and the difference is 5.3% with the wind speed of 30 m/s; for the maximum displacement response, the maximum difference is no more than 6%. The above results show that the wind-induced vibration response of the down lead system is different from that of the traditional horizontal long-span transmission line, and the aerodynamic damping has little influence on the wind-induced vibration response of the down lead system.

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5.2 Influence of Wind Direction Angle In Sect. 5.1, two wind direction angles of wind direction along z axis and wind direction along z axis are set to analyze the wind-induced vibration response of model 1. After considering the influence of aerodynamic damping, two groups of results under different wind speeds are shown in Fig. 6.

Fig. 6. The influence of wind direction angle on wind-induced response of lead-down system

It can be seen from Fig. 6 that the response of the wind direction angle to the reaction force and displacement of the down conductor system are different: the displacement response of the structure under the z-axis wind direction angle is significantly higher than that under the x-axis wind direction angle, and the greater the difference between the two is with the increase of the wind speed, the average and maximum difference between them is 33.4% and 35.9%; for the reaction, the response under the z-axis wind direction angle is significantly lower than that under the x-axis wind direction angle. The difference between the mean value and the maximum value of the reaction under the action of the two is 41.5% and 58.1%. The difference between them lies in the modal distribution of the structural system and the influence of its own stiffness distribution. The down conductor system is distributed in the x-y plane as a whole, and its stiffness in the plane is significantly higher than that out of the plane. Under the x-axis wind force, it will lead to the “tension” of the down conductor, while under the z-axis wind force, it will not. Therefore, the displacement of the z-axis wind direction angle is larger, while the reaction force is smaller. In order to ensure the safety of down conductor structure, wind load of x-axis wind direction angle is adopted.

6 Conclusion In this paper, through the simulation of fluctuating wind load, the analysis of dynamic characteristics of down conductor system and the dynamic time history response analysis of wind load, the following conclusions can be obtained:

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(1) due to the high nonlinearity of the down conductor system, the wind-induced vibration response is obviously non-Gaussian, so it is no longer suitable to directly use the response variance to express the dynamic performance, Therefore, the maximum value of the response is directly used to represent the dynamic performance of the structure. (2) for the wind-induced vibration response of the structure: because the length of the down lead structure is shorter than that of the previous large-span transmission line, and the natural frequency of the structure is larger, which also leads to the smaller aerodynamic damping of the down lead structure. the out of plane displacement response of the structure is larger under the out of plane wind force of the zaxis wind direction angle; while the in-plane reaction response of the x-axis wind direction angle is larger under the in-plane wind force.

References 1. Liu, X., Qinchang, Zhang, L., et al.: Study on the out-of-plane modes and frequencies of transmission lines. Chinese J. Appl. Mech. (3) (2017) 2. Liu, L., Zhao, Q., Liu, Z., et al.: Anti-galloping mechanism and modal analysis of clamp rotary conductor damper spacer. Electr. Power Constr. 35(3), 74–78 (2014) 3. Shao, Y., Zhu, K., Liu, L., et al.: Static computation and finite element modal analysis for triple-span bundled power lines. Proc. CSEE 33(13), 157–164 (2013) 4. Yu, Y., Wang, W.: Analysis of the influence of aerodynamic damping on the response of windinduced vibration coefficient of flexible structure based on time-domain analysis method. Urban Road Bridge Flood Control (1), 113–115 (2014) 5. Wang, S., Liang, S., Zou, L., et al.: Aerodynamic damping effects of a transmission conductor by wind tunnel tests. J. Vibr. Shock (20), 30–36 (2016) 6. Loredo Souza, A.M.: A novel approach for wind tunnel modelling of transmission lines. J. Wind Eng. Ind. Aerodyn. 89(11), 1017–1029 (2001) 7. Zou, L., Liang, S., Wang, S.: Study on aerodynamic damping of transmission tower based on aeroelastic model wind tunnel test. Vibr. Test Diagn. 35(2), 268–275(2015) 8. Irvine, H.M.: Cable structure. The MIT Press, Cambridge (1981) 9. Zhang, C., Lu, Z., Zhang, X., et al.: Seismic coupling effect of UHV electrical equipment with flexible bus connection. J. South China Univ. Technol. (Nat. Sci. Edn.) 44(8), 74–81 (2016) 10. Owen, J.S., Eccles, B.J., Choo, B,S., et al.: The application of auto-regressive time series modelling for the: time-frequency analysis of civil engineering structures. Eng. Struc. 23(5), 521–536 (2001) 11. Xiang, Y., Shen, S., Li, J.: Nonlinear wind-induced vibration response analysis of thin film structures. J. Archit. Struc. 20(06) (1999) 12. Wu, Y., Guo, H., Chen, X., et al.: Study on wind-induced vibration of a large-span dot point glazing supporting system. J. Build. Struc. 23(5), 49–55 13. Davenport, A.G.: The spectrum of horizontal gustiness near the ground in high wind. Q. J. Royal Meteorol. Soc. 87(372), 194–211 (1961) 14. Yang,Y.: Study on calculation of conductor swinging under fluctuating wind loads (2015) 15. Duan, C., Zhang, E.R.L.E., Han, J., et al.: Influence of damping ratio on wind load of transmission tower. Shandong Electr. Power Technol. 45(3), 34–38 (2018)

Bending Fatigue Behaviour of a Bio-based Sandwich with Conventional and Auxetic Honeycomb Core Khawla Essassi1,2(B) , Jean-luc Rebiere1 , Abderrahim El Mahi1 , Mahamane Toure1 , Mohamed Amine Ben Souf2 , Anas Bouguecha2 , and Mohamed Haddar2 1 Acoustics Laboratory of Mans University (LAUM) UMR CNRS 6613, Mans University,

Av. O. Messiaen, 72085 Le Mans Cedex 9, France {Jean-Luc.Rebiere,abderrahim.elmahi, Mahamane.Toure.Etu}@univ-lemans.fr 2 Laboratory of Mechanics Modeling and Production (LA2MP), National School of Engineers of Sfax, University of Sfax, BP N° 1173, 3038 Sfax, Tunisia [email protected]

Abstract. We are witnessing a great increase in the use of auxetic structure in comparison to their conventional counterparts in these recent years. They have exhibited numerous advantages such as a high strength, stiffness and energy absorption. On the other side, we are also witnessing the emergence of eco-composite materials that significantly have the upper hand when compared to synthetic alternatives. This paper describes the static and fatigue bending behaviour of a bio-based sandwich structure with conventional and auxetic honeycomb core. The eco-composite used is a tape of polylactic acid reinforced with flax fibers. Additive manufacturing technology is used to produce these structures. The quasi-static bending tests were performed to identify the mechanical properties of sandwich composites and the ultimate loading. The fatigue tests were conducted with displacement-controlled technique in order to study the effect of the core configuration on the stiffness, hysteresis loops, energy absorption and damping ratio of the sandwich composite. The interesting part is that we observed a great increase in the capacity of energy absorption for the auxetic structure when compared to the conventional ones, which results higher loss factors. This work is used to determine the static and fatigue properties and to give an opinion on the damage mechanisms of these composites during its lifetime. Keywords: Fatigue · Damping · Auxetic structure · Conventional honeycomb · 3D-printing

1 Introduction Due to environmental and ecological challenges, bio-composites have become the center of interest of several researches. These materials are biodegradable, recyclable and considered as a good replacement to their synthetic counterparts (Lefeuvre et al. 2014). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 44–52, 2021. https://doi.org/10.1007/978-3-030-76517-0_6

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On the other hand, sandwich composites have gained a wide popularity in the transportation industry, due to their interesting stiffness/mass ratio and energy absorption capacity (Schaedler and Carter 2016). These structures are well known to be of the best when it comes to their abilities to increase the bending resistance, shear stiffness and energy absorption (Allen 2013). Conventional honeycomb cellular core (Buitrago et al. 2010) is practically considered as the most core configurations studied in the literature. Recently, auxetic structures have been the center of attention thanks to their interesting properties. These structures have exhibited their abilities to improve numerous mechanical properties, such as indentation resistance (Lakes and Elms 1993), energy absorption (Hou et al. 2015) and acoustic properties compared to their conventional counterparts (Chen and Lakes 1996). As a result of their complex geometry, we are witnessing breakthroughs in manufacturing procedures that allowed to produce these structures. 3D printing technology (additive manufacturing) is considered one of the most efficient ways to produce these topologies. The main idea of the present study is to compare the damping fatigue properties of bio-sandwich with conventional and auxetic honeycomb core. The structures are manufactured with polylactic acid (considered as a matrix) reinforced with flax fibers using a 3D printing machine. The mechanical properties of the examined structure under static and fatigue 3-point bending tests were determined. This paper highlights the effect of the core topology on the stiffness, energy absorption and damping ratio of the biosandwich composites.

2 Materials and Method 2.1 Materials and Manufacturing The production of the sandwich composite was made using a filament of polylactic acid (the matrix) reinforced with micro-flax fibers (PFF) specific to 3D printing technology. It is biodegradable material and made from natural and renewable resources. The specimen topologies were designed using Solidworks software. The mechanical characteristics of the material are affected by the layer-by-layer 3D printing technique. Hence, the printing direction was selected in the way that ameliorate the mechanical properties of the structures. Conventional and auxetic honeycomb are used as the core material of the sandwich composite, as shown in Fig. 1. h and l are the length of the vertical and inclined cell walls, respectively. θ is the angle between the inclined cell walls and the X direction. t is the cell wall thickness. H and L are the height and width of the unit cells, respectively. The thickness of the core is 5 mm. The thickness of the sandwich is 7 mm. The length of the specimens is equal to 120 mm and the width is equal to 25 mm. Table 1 presents the average dimensions of the conventional and auxetic core cells.

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Fig. 1. Design of unit cell of: a) conventional and b) auxetic honeycomb

Table 1. Design parameters of conventional and auxetic core. Parameters

l (mm) h (mm) θ (°) t (mm)

Conventional honeycomb 3.32

4.26

20

Auxetic honeycomb

4.26

−20 0.6

3.32

0.6

2.2 Experimental Setup Experimental bending tests (static and fatigue) were realized using on a standard hydraulic machine INSTRON 8801 with ±10 kN capacity, as shown in Fig. 2. Tests were executed according to ASTM D790-86 standard. In static tests, the sandwich beams were loaded at a displacement rate of 5 mm.min−1 and an LVDT sensor is used to measure the displacement. The test consists of applying a load P to the beam supported by two pins (diameter 15 mm) with a span length of 110 mm. The load is applied in a single line using a 15 mm diameter pin too. The fatigue tests were realized using a load cell of 1 kN and sinusoidal type of waveform with a frequency of 5 Hz. Tests were implemented using displacement control technique. An average displacement dmoy of 4 mm is chosen with an amplitude damp of 1 mm. The machine was equipped with an anti-rotation device to prevent the rotation of the lower supports around the axis of the hydraulic cylinder, particularly during fatigue tests.

Fig. 2. Three-point bending test set up

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3 Results and Discussion 3.1 Static Results The bending performance of 3D printed sandwich composites with auxetic and conventional honeycomb is determined. Each configuration is tested in three-point bending with a span length of 110 mm. The effect of the core topology on the bending properties is tested. Figure 3 presents the load/deflection curves accomplished for the sandwich with auxetic and conventional honeycomb core. The core topology has an important effect on the obtained results. Sandwich with auxetic core exhibits the higher loading forces and stiffness. On the other hand, the sandwich with conventional honeycomb core exhibits the biggest bending deflection. Each curve shows three important parts. Firstly, a linear elastic domain extends up to a deflection of about 6 mm for the two configurations. Then, a shorter nonlinear behaviour that goes on until the break for sandwich with auxetic honeycomb core and a longer one that is presented for sandwich with conventional honeycomb core.

Fig. 3. Bending characteristic of sandwich beams with conventional and auxetic honeycomb core

To determine the bending and shear stiffness, an elastic investigation was affected. Under three-points bending tests, the relation that connect the applied load P to the deflection W is given by: d2 1 W = + Pd 48D 4N

(1)

where d is the span length, D the bending stiffness and N the shear stiffness of the beam. The span length was varied between 100 mm and 240 mm. Figure 4 depicts the evolution of W/(Pd) as a function of the square of the span length (d2 ). Experimental results are then fitted by a linear equation. With the use of Eq. (1), the bending and shear stiffness are inferred respectively, from the slope and the intercept of the linear fitting curve.

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Fig. 4. Evolution of the ratio (W/Pd) with respect to the span length in three-point bending for sandwich beams with conventional and auxetic honeycomb core

Table 2 presents the bending stiffness D and the shear stiffness N for the bio-sandwichs with auxetic and conventional honeycomb cores. Table 2. Elastic properties of the sandwich with auxetic and conventional honeycomb. Bending stiffness D (N.mm2 )

Shear stiffness N (N)

Conventional honeycomb

1041666

125000

Auxetic honeycomb

1493336

53348

3.2 Fatigue Results 3.2.1. Stiffness Degradation The fatigue bending behavior of the bio-sandwich structures with the two different core configurations are studied. Results are obtained for the evolution in stiffness loss (F/F0 ) according to the number of cycles, as shown in Fig. 5. It presents the transition of the mechanical properties using a semi-logarithmic scale. This method let us follow the damage proliferation in the material. The evolution of the stiffness loss (F/F0 ) can be discretized in three phases started with a rapid decrease in the stiffness loss (appears in the first few cycles) followed by a very slow one. These two phases correspond to the damage initiation and proliferation in the bio-sandwich beams. Also, it can be seen that the two configurations display the same behavior. 3.2.2 Hysteresis Curves The hysteresis curves (loops) are generally used to determine the energy absorption performance of the materials during fatigue tests. Each cycle is defined by 200 experimental points. Figure 6 presents the experimental load/displacement curves for loading and unloading cycles at 10 and 1000 cycles for bio-sandwiches with auxetic and conventional honeycomb core. The behavior of hysteresis curves is similar for the two sandwich configurations. On the other hand, the peak loads are different for each configuration. It

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Fig. 5. Stiffness loss versus the number of cycles of sandwich beams with conventional and auxetic honeycomb core

is clearly observed that the top load of sandwich with auxetic honeycomb core are larger than that with conventional one. With the augmentation in the number of cycles, the area under the hysteresis loops reduces. This result can be explained by the stiffness loss and damage development in the sandwich, which is determined by the degradation of the unit cells forming the core. Due to cyclic loading, the cell walls have totally collapsed. The total rupture of the unit cells and the fracture of cell edges give on to the appearance of irreversible damage in the sandwich and then failure.

Fig. 6. Hysteresis curves for different number of cycles of sandwich beams with: a) conventional and b) auxetic honeycomb core

3.2.3 Energy Analysis It seems very essential to differentiate the energy absorption capability between the sandwich composite with auxetic and conventional honeycomb core. Considering the hysteresis cycle illustrated in Fig. 6, the area covered by the hysteresis loops presents the dissipated energy E d for each cycle. The potential energy stored E p is the area beneath the upper part (loading part) of the hysteresis loop, as shown in Fig. 7. Using a trapezoidal summation of area, the potential energy stored E p and the dissipated energy E d are approximated. To decrease calculation error of area, the hysteresis curves are divided by a large number n. The potential and dissipated energy are given by Idriss et al. (2013): Ep =

  1 n (di+1 − di ) f (di+1 ) + f (di ) i=1 2

(2)

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Ed =

    1 n (di+1 − di ) f (di+1 ) + f (di ) − g(di+1 ) + g(di ) i=1 2

(3)

Fig. 7. Illustration of a hysteresis cycle

The energy dissipated according to the number of cycles for the two configurations of sandwich is given in Fig. 8. Results are set forth using a semi-logarithmic scale. Sandwich with auxetic honeycomb core dissipate more energy than that with conventional one. During the first cycles, the energy dissipated exhibits a short drop. Afterwards, it became constant for a high number of cycles. It is interesting to mention that the response to fatigue displays greater changes from the initial part of fatigue life to the end of life. This is due to the increase of damage in the bio-sandwich during fatigue tests. During the loading and unloading of the auxetic material, the structure manifests a synclastic deformation which squander a large quantity of energy compared to conventional one. This justifies the bigger amount of energy dissipated by sandwiches with auxetic core.

Fig. 8. Energy dissipation versus number of cycles in sandwich specimens with conventional and auxetic honeycomb core

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3.2.4 Damping Ratio The energy dissipation in sandwich composites is affected by the topology of the core. The damping factor in cyclic fatigue tests is represented by the ratio of the energy dissipated per cycle E d with the maximum potential energy per cycle E p : η=

Ed 2π Ep

(4)

Figure 9 depicts the evolution of the fatigue loss factor ï for the two studied configurations. A semi-logarithmic scale is used. It is obviously observed that the loss factor lowers with the number of cycles. During the first cycles, a rapid drop in the loss factor can be remarked for the two configurations. This result can be explained by the first damage mechanisms in the core caused by the first load applied to the sandwich. Then, the loss factor becomes stable. At the end of the tests, a slight rise in the loss factor can be observed. Sandwich with auxetic honeycomb core present the highest loss factor than those with conventional honeycomb.

Fig. 9. Evolution of loss factor versus number of cycles in sandwich specimens with conventional and auxetic honeycomb core

4 Conclusion An experimental assessment of the quasi-static and fatigue bending properties of a biosandwich composite with an auxetic and conventional honeycomb core is studied. These structures are generated using additive manufacturing technique (3D printing). The utilized material is a tape of polylactic acid (matrix) reinforced with flax fibers PFF. The bending and shear stiffness of the two different configurations were compared. Then, the fatigue behavior of the two different topologies of the bio-composites is investigated. The stiffness degradation during fatigue tests provides an insight about the cohesion of the sandwiches. Fatigue behaviour is characterized by energy dissipation and changes in damping ratios. These parameters are obtained from the measurements of the bending fatigue tests. Results proves that the auxetic honeycomb core possess the highest dissipated energy and damping ratio when compared to conventional counterpart.

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References Allen, H.G.: Analysis and design of structural sandwich panels: the commonwealth and international library: structures and solid body mechanics division. Elsevier (2013) Buitrago, B.L., Santiuste, C., Sánchez-Sáez, S., Barbero, E., Navarro, C.: Modelling of composite sandwich structures with honeycomb core subjected to highvelocity impact. Compos. Struct. 92, 2090–2096 (2010) Chen, C.P., Lakes, R.S.: Micromechanical analysis of dynamic behavior of conventional and negative Poisson’s ratio foams. J. Eng. Mater. Technol. 118, 285–288 (1996) Hou, S., Liu, T., Zhang, Z., Han, X., Li, Q.: How does negative Poisson’s ratio of foam filler affect crashworthiness? Mater. Des. 82, 247–259 (2015) Idriss, M., El Mahi, A., Assarar, M., El Guerjouma, R.: Damping analysis in cyclic fatigue loading of sandwich beams with debonding. Compos. B Eng. 44(1), 597–603 (2013) Lakes, R.S., Elms, K.: Indentability of conventional and negative poisson’s ratio foams. J. Compos. Mater. 27, 1193–1202 (1993) Lefeuvre, A., Bourmaud, A., Morvan, C., Baley, C.: Elementary flax fibre tensile properties: correlation between stress–strain behaviour and fibre composition. Ind. Crops Prod. 52, 762–769 (2014) Schaedler, T.A., Carter, W.B.: Architected cellular materials. Ann. Rev. Mater. Res. 46, 187–210 (2016)

Thermoplastic Elium Recycling: Mechanical Behaviour and Damage Mechanisms Analysis by Acoustic Emission Sami Allagui1,2(B) , Abderrahim El Mahi1 , Jean-luc Rebiere1 , Moez Beyaoui2 , Anas Bouguecha2 , and Mohamed Haddar2 1 Le Mans University, Acoustics Laboratory of Le Mans University LAUM, UMR CNRS 6613,

Av. O. Messiaen, 72085 Le Mans Cedex 9, France [email protected] 2 Department of Mechanical Engineering, National School of Engineering of Sfax, Laboratory of Mechanics, Modelling and Production, Route Soukra, 3038 Sfax, Tunisia

Abstract. This study aims to prove the recyclability of an innovative thermoplastic called Elium resin. It’s the first liquid thermoplastic resin that provides the production of composite materials with an interesting mechanical behavior. Until now, its recyclable character is unjustified. It appeared on the resin market in 2014. But until now, no studies were available concerning how it can be recycled and reused. For this study, a thermocompression recycling process was investigated and applied to Elium resin. Three levels of recycling were carried out on the initial material. The recycled specimens were investigated and tested by means of tensile tests. Also, the acoustic emission (AE) technique was used to characterize microstructural damage events leading to overall failure of the recycled Elium. This study has been further supported by scanning electron microscopy (SEM) micrographs of the fractured face. The results obtained show that the failure tensile properties of Elium resin as well as flax fiber reinforced composites decrease during recycling operations. Conversely, recycling induces a rise in the elastic modulus. Also, the acoustic emission technique associated with SEM observation showed two classes of damage mechanisms. A first “class A” related to “matrix cracking” and a second “class B” associated to matrix/matrix friction. Keywords: Liquid thermoplastic · Recycling · Thermocompression · Tensile properties · Damage mechanisms · Acoustic emission (AE)

1 Introduction Polymeric materials are often classified as thermosets and thermoplastics. Thermoset polymers are obtained by a three-dimensional assembly of macromolecules, characterized by strong bridging type bonds (Pascault et al. 2002). These connections result in the irreversible character of the structure even at high temperatures. This structure makes the thermosets solid, dimensionally stable and extremely resistant to heat. On the other © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 53–61, 2021. https://doi.org/10.1007/978-3-030-76517-0_7

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hand, thermoplastic resins have a linear structure characterized by weak Van der Waals type bonds. This means that the material’s structure can be altered by heating, hence the term “thermoplastic” (Brazel and Rosen 2012). There are three types of thermoplastic polymers: crystalline thermoplastics, amorphous thermoplastics and semi-crystalline polymers (Grigore 2017). Due to the ideal properties of the thermoplastic polymers such as corrosion resistance, low density, high strength, and user-friendly design, the production of thermoplastic composites has increased significantly which generate a large amount of composite waste. The question of how to dispose of end-of-life composite materials parts is growing in importance. Traditional disposal routes such as landfill and incineration are becoming increasingly restricted, and composites companies and their customers are looking for more sustainable solutions. The use of recyclable matrices during composite manufacturing is one of the best solutions to face the need for unfavorable methods of waste disposal. The purpose of this work is to study an innovative thermoplastic called ELIUM. It’s the first liquid thermoplastic resin that provides the production of recycled composite parts with an interesting mechanical behavior. But its recyclable character is not investigated and justified until now. Therefore, the main objective of this study is to justify the recyclability of the recent thermoplastic resin Elium. Recycling thermoplastic polymer can be accomplished in several ways (Oliveux et al. 2015) such as mechanical processes (mainly grinding), pyrolysis and other thermal processes, and solvolysis. An excellent overview on the recycling technologies was addressed in the handbook published by Goodship (2010). This study focuses specifically on a recycling method based on a thermomechanical process. This technique, which will be presented later, is inspired by the progress achieved in recent years on the processes of implementing cut thermoplastic prepregs. It consists of reshaping the waste into the required component by the combined action of temperature and pressure. The thermocompression recycling technique was used by Moothoo et al. (2017). They apply this process to bidirectional glass/polypropylene laminate waste. It is shown that the modulus values are slightly affected compared to the raw composite. However, a very significant decrease in the stress at break is observed. These results are explained by the presence of zones rich in resin and the concentration of stress on the grain edges. Also, we can identify other works concerning the recycling of polypropylene alone such as Aurrekoetxea et al. (2001) and Guerrica-Echevarria et al. (1996). They show that five cycles of recycling are necessary to observe significant loss of mechanical properties. They also show that the Young’s modulus increases with reprocessing. They explain this improvement in the properties by the increase in the crystallization rate with recycling.

2 Materials and Recycling Method Elium resin is a peroxide-activated acrylic resin. Due to its thermoplastic nature, the resin obtained after polymerization can theoretically be thermoformed and potentially recycled. It appeared on the market resin in 2014. But until now, no studies are currently available concerning its ability to be recycled and reused. The recycling of Elium waste was carried out using thermocompression process, which is a method of manufacturing composite materials. The recycling process consists

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55

firstly of cutting the initial material into small rectangular pieces. Then, placing these small pieces between two steel plate which forms the compression mold. Then, using a press machine, we can obtain recycled plates for specimens. This technique is carried out by compression of the materials between two hot plates. A 40-ton TCE press machine, manufactured by DK Technologies, was used for recycling composite waste. Three levels of recycling were carried out on the initial material. All specimens were cut in the compressed plate by a laser-cutting machine marketed by the Epilog laser company.

3 Experimental Setup At ambient temperature, static tensile experiments were performed on recycled specimens following the standard test method ASTM D3039/D3039M. The tests were performed with a typical hydraulic machine, INSTRON 8516, equipped with a 10 kN load cell. The machine is connected to a dedicated computer for controlling and data acquisition. The mechanical properties of recycled specimens were obtained on a minimum of 5 tests at a strain rate of 0,5 mm/min. During tensile testing, a 50 mm extensometer was associated to the data acquisition system and fixed on the gauge length section of the specimen to record the strain evolution. The strains in the transverse direction were measured with 5 mm strain extensometer. The recycled specimens have rectangular geometries (30 × 250 × 3 mm). During the static tensile tests, AE technique was used identify damage mechanisms progression in the tested specimens. AE experiments were carried out using devices provided by Physical Acoustics Corporation (PAC). Two sensors with a bandwidth of 100 kHz to 1 MHz and a resonance peak of 300 kHz, were fixed on the test specimen at a spacing of 50 mm. The collected AE signals were registered with a sampling frequency of 5 MHz. Each sensor was connected to a preamplifier with a 40dB gain in order to amplify the AE data. Before each tensile test, pencil lead break tests were carried out to define amplitude of acquisition. This parameter was set at 38 dB to filter background noise. The quality of the measured AE data depends mainly on the choice of the acquisition parameters, namely, peak definition time (PDT), hit definition time (HDT) and hit lockout time (HLT). After preliminary trials, the temporal parameters employed were set to the following values: PDT = 50 μs, HDT = 100 μs and HLT = 200 μs. After the tensile and acoustic tests, microscopic analyses of the failure modes and mechanisms were achieved. In order to detect smaller damage mechanisms, microscopic observation was carried out on small pieces (10 × 10 mm) of the fractured face by scanning electron microscopy (SEM).

4 Results and Discussion 4.1 Mechanical Properties Figure 1 presents a comparative study of Elium resin with and without recycling. These figures give the evolution of stress/strain curves for every kind of specimen: virgin specimen ER0 (not recycled) and recycled specimens (ERi ). The symbol “i” indicates

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the number of times they were recycled (i = [1, 2, 3]). Different properties of Elium (ER0) were examined by Monti et al. (2016). For all specimens, the mechanical behavior was similar: each stress/strain curve is divided into two zones: the first one is linear, which is elastic. The second zone is quasi-linear until the break of the specimens.

Fig. 1. Recycling effect on the stress-strain curves of the Elium resin

The averages of the different properties obtained are summarized in Table 1. For each specimen, the value of Young’s modulus E is obtained by measuring the slope of the axial stress/strain curve in the elastic region. The Poisson’s ratio ν was calculated for the same stress interval as the slope of the transverse strain-longitudinal strain curve. Stresses and strains at break, noted σmax and εmax , were also measured. The results show that the Young’s modulus E values increase as a function of levels of recycling. A progressive increase from 3,3 GPa to 3,7 GPa is observed. This evolution shows close correlation with the increase in density resulting from repeated recycling (Allagui et al. 2021). Due to the reprocessing conditions, the material becomes more and more compact. This can be explained by the thermocompression process which involves a compaction phase in which the polymer will crystallize at high pressure. Table 1. The mechanical properties of Elium resin (with and without recycling) ERi

Density

E [GPa]

σmax [MPa]

εmax [%]

ν

ER0

1130

3,3±0,32

44±4

2,03±0,38

0,4±0,03

ER1

1145

3,4±0,13

24±1,9

1,07±0,23

0,45±0,02

ER2

1154

3,6±0,1

22±1,7

0,71±0,09

0,44±0,03

ER3

1190

3,7±0,2

19±1,5

0,58±0,05

0,47±0,02

In addition, this manufacturing process requires, at each level of recycling, a cutting step which leads to a reduction in the particle size as well as a decrease in the porosity

Thermoplastic Elium Recycling

57

and consequently, increased density. Also, it should be noted that that the structure of a thermoplastic polymer is composed of two zones: the first is the crystalline zone in which the molecular chains are ordered and the second is the amorphous where the chains are disordered (Nguyen 2014). Therefore, the main properties of the polymer result from the crystalline and amorphous characteristics. The density of crystalline phases is generally higher than that of the amorphous phases which result consequently in an increase in polymer density. In conclusion, reprocessing during recycling results in an increase in the density and consequently an increase in the degree of crystallinity which involve an increase in rigidity of Elium resin. However, it can be seen that the recycling process has a negative effect on stress and strain failure. The stress failure curve decreases from 44 MPa (ER0 ) to 19 MPa (ER3 ) after three reprocessing cycles. Also, a significant decrease in the ultimate strain from 2,03% to 0,58% is observed. The negative effect of the recycling process on the failure’s properties can be explained by repeated cutting of Elium waste into small grains which results in multiple ruptures in the polymer chain and consequently the degradation of the ultimate properties (strain and stress). 4.2 Damage Mechanisms in the Recycled Elium AE tests were performed to examine the microstructural failure events of recycled Elium that appearing during tensile tests. Acoustic Emission data was processed with NOESIS. Five temporal parameters were selected for the classification of the acoustic emission signals: amplitude, duration, rise time, number of counts to peak and absolute energy. Monti et al. (2016) were used this method to classify the different acoustic signals detected during tensile tests carried out on Flax/Elium composite specimens. They show a good repeatability of the data clustering by using these five parameters of classification. For the classification processing, The K-mean algorithm (Likas et al. 2003) was used for the unsupervised pattern recognition. The objective of this method is to group signals with similar characteristics by an unsupervised classification method in order to identify the acoustic signature of different mechanisms that appear during tensile tests. Using this method, the accumulated acoustic data was separated into an optimal number of k clusters. K-mean algorithm constitutes the best separation of the data with regard to mathematical considerations, without really taking physics into account. The analysis of data signals by Noesis software returned the following results. Two classes were obtained for all recycled specimens suggesting the presence of two groups of damage. The damage growth detection and monitoring have been performed using the amplitude analysis of Acoustic Emission signals generated during the advent of damage and failure mechanisms. Figure 2(a) presents the amplitudes of these AE classes with respect to the time and the applied load. We can notice that the amplitude seems to properly split these classes for specimens. In addition of this parameter use of the cumulative number of hits will allows a better comprehension and thus the best followed mechanisms Green circles represent bursts for which the amplitudes are between 38 dB and 50 dB (class A), and blue circles represent amplitudes between 55 dB and 90 dB (class B). The number of recorded events is about 640, 818, 1771 for the recycled Elium ER1 , ER2 , ER3 respectively. The ER3 material involve the highest number of events. Figure 2(b) presents the cumulative number of hits for each class with respect to time.

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Fig. 2. AE classification: (a) Amplitude of the events and stress vs time, (b) chronology of appearance of different classes

This is a good indication of the chronology and evolution of the acoustic events. For all specimens, the number of hits presents a significant increase of every class just before failure. It can be noticed from that the acoustic activity and the evolution of the loading take place in two phases. First, no acoustic activity was recorded in the short linear elastic phase (characterized by an elastic modulus E). It shows that this loading not having any effect on the specimen’s integrity. Second, the slope of stress curve begins to decreases slightly, reflecting the onset of viscoelastic behavior. This phase is characterized by a steady increase in acoustic activity. Therefore, the AE data can be correlated to the stress/strain curve shape. The detected class A and B corresponds probably to the initiation and propagation of microcracks and matrix/matrix friction, respectively. Bravo et al. (2013) were applied

Thermoplastic Elium Recycling

59

the acoustic emissions (AE) technique on the matrix alone in order to understand failure mechanisms in bio-composites. The matrix used is a thermoplastic resin linear lowdensity polyethylene. The authors show that only damage events due to matrix microcracking and matrix/matrix friction can be observed. Concerning the recycled Elium, microscopics analyses were performed by scanning electron microscopy (SEM) of the failure profiles to confirm the effectiveness of the multivariable statistical analysis. The results show the presence of many surface of matrix/matrix friction (Fig. 3 label 1). Some matrix cracks can also be observed (Fig. 3 label 2). They appear to be propagations of microcracks in the defects zone which can be present in the matrix (inclusions, porosities). These microcracks will cause the propagation of damage in the direction perpendicular to the stress. In conclusion, regarding the observations of microscope (SEM) in Fig. 3 and the results of classification (Fig. 2), the damage mechanisms appearing in recycled Elium are matrix cracking “class A” (label 2) and matrix/matrix friction “class B” (label 1).

2 1 1 2

(b)

(a)

2

1

(c)

Fig. 3. SEM observations of different failure profiles of recycled Elium: (a) ER1 (b) ER2 (c) ER3

5 Conclusion In summary, the present study has demonstrated the feasibility of recyclability of a recent thermoplastic resin called Elium. An innovative recycling process, based on thermocompression process, was used during this work. Three levels of recycling were carried out on the initial material. Tensile tests associated with monitoring damage by

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acoustic emissions are carried out on the recycled material. For each sample, values of elastic modulus, failure strength, failure strain and Poisson’s ratio were obtained. An improvement in the elastic modulus was obtained. This effect is probably related to an increase in the crystallinity rate of the polymer. Increasing crystallization is revealed by the rise in density according to the recycling level since the density of crystalline phases is generally higher than that of amorphous phases. However, a decrease in the ultimate strength and the ultimate elongation with repeated to recycling was noticed. The negative effect of the recycling process on the failure’s properties can be explained by repeated cutting of Elium waste into small grains which results in multiple ruptures in the polymer chain and consequently the degradation of the ultimate properties (strain and stress). Acoustic emissions (AE) instrumentation was used for the identification and characterization of micro failure mechanisms in the recycled Elium. Also, microscopic analyses, associated with this approach, made it possible to identify the various damage mechanisms during the tests. The results obtained show two classes of events A and B which appear simultaneously for all recycled specimens. The damage mechanisms considered according to the collected AE signals and SEM observation are matrix cracking “class A”, matrix/matrix friction “class B”.

References Allagui, S., Mahi, A.E., Rebiere, J., et al.: Effect of recycling cycles on the mechanical and damping properties of flax fibre reinforced elium composite: experimental and numerical studies. J. Renew. Mater. (2021). https://doi.org/10.32604/jrm.2021.013586 ASTM D30309/D3039M-14: Standard test method for tensile properties of polymer matrix composite Aurrekoetxea, J., Sarrionanda, M.A., Urrutibeascoa, I., Maspoch, M.L.I.: Effects of recycling on the microstructure and the mechanical properties of isotactic polypropylene. J. Mater. Sci. (2001). https://doi.org/10.1023/A1017983907260 Bravo, A., Toubal, L., Koffi, D., Erchiqui, F.: Characterization of tensile damage for a short birch fiber-reinforced polyethylene composite with acoustic emission. Int. J. Mater. Sci. 3(3), 79–89 (2013) Brazel, C.S., Rosen, S.L.: Fundamental Principles of Polymeric Materials. Wiley, Hoboken (2012) Goodship V(2010) Management, Recycling and Reuse of Waste Composites. WP and CRC Press. Grigore, M.E.: Methods of recycling, properties and applications of recycled thermoplastic polymers. Recycling 2(4), 24 (2017) Guerrica-Echevarria, G., Eguiazabal, J.I., Nazabal, J.: Effects of reprocessing conditions on the properties of unfilled and talc-filled polypropylene. Polym. Degrad. Stab. 53(1), 1–8 (1996) Kattis, S.: Noesis-Advanced Data Analysis, Pattern Recognition & Neural Networks Software for Acoustic Emission Applications. In: Kolloquium Schallemission, Statusberichte zur Entwicklung und Anwendung der Schallemissionsanalyse, pp. 9–10, March 2017 Likas, A., Vlassis, N., Verbeek, J.J.: The global k-means clustering algorithm. Pattern Recogn. 36(2), 451–461 (2003) Monti, A., El Mahi, A., Jendli, Z., Guillaumat, L.: Mechanical behaviour and damage mechanisms analysis of a flax-fibre reinforced composite by acoustic emission. Compos. A Appl. Sci. Manuf. 90, 100–110 (2016)

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Moothoo, J., Garnier, C., Ouagne, P.: Valorisation matière de chûtes de production de composites thermoplastiques par le procédé de thermo-compression. In: Journées Nationales sur les Composites 2017 June 2017 Saccani, A., Manzi, S., Lancellotti, I., Lipparini, L.: Composites obtained by recycling carbon fibre/epoxy composite wastes in building materials. Constr. Build. Mater. 204, 296–302 (2019) Pascault, J.P., Sautereau, H., Verdu, J., Williams, R.J.: Thermosetting Polymers, vol. 64. CRC Press, Boca Raton (2002) Nguyen, T.L.: Approche multi-échelles dans les matériaux polymères: de la caractérisation nanométrique aux effets d’échelles (Doctoral dissertation, Compiègne) (2014)

Passive Control of Tensegrity Domes Using Tuned Mass Dampers, a Reliability Approach E. Mrabet1,2(B) , M. H. El Ouni3,4,5 , and N. Ben Kahla3,4,5 1 Laboratory of Mechanics, Modeling and Manufacturing of the National School of Engineers

of Sfax, University of Sfax, Sfax, Tunisia [email protected] 2 Higher Institute of Applied Sciences and Technologies of Kasserine, University of Kairouan, Kairouan, Tunisia 3 Department of Civil Engineering, College of Engineering, King Khalid University, Abha 61421, Kingdom of Saudi Arabia {melouni,bohlal}@kku.edu.sa 4 Laboratory of Systems and Applied Mechanics, Tunisia Polytechnic School, University of Carthage La Marsa, 2078 Tunis, Tunisia 5 Department of Mechanical Engineering, Higher Institute of Applied Sciences and Technologies of Sousse, University of Sousse, Sousse, Tunisia

Abstract. Tensegrity domes are space reticulated pre-stressed systems composed of bars in compression and cables in tension. They are lightweight flexible systems with low structural damping, and thus they are sensitive to vibrations induced by wind or earthquakes. This paper investigates the passive vibration control of a randomly driven tensegrity dome of Geiger’s type using a Tuned Mass Damper (TMD). The parameters of the TMD are optimized based on a reliability strategy. The optimization strategy consists in minimizing a failure probability characterized by the out-crossing, for the first time, of a given threshold value of the vertical displacement of a given node (of the structure) during a time interval. The proposed strategy is compared to other strategies from the literature and showed better capabilities in reducing the vertical vibrations of the cable dome. Indeed, it has been observed that significant reductions have been obtained especially in the vicinity of the off-targeted frequencies where the classical tunings showed low performances. The proposed passive control strategy using a TMD is efficient in reducing the excessive vibrations of tensegrity systems and significantly less expensive than active and semi-active control techniques that are usually used in such a case. Keywords: Tensegrity · Cable dome · Random vibrations · Reliability based optimization of TMD

1 Introduction Tensegrity domes, Geiger’s type (Ouni and Ben Kahla 2014) for instance, are innovative lightweight space reticulated pre-stressed systems composed of a network of continuous © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 62–70, 2021. https://doi.org/10.1007/978-3-030-76517-0_8

Passive Control of Tensegrity Domes

63

tensile cables and discontinuous compressive bars. They are employed as large span roofs of structures including arenas, stadiums and sport centers. These structures are very flexible and have low damping and thus they are sensitive to vibrations induced by wind or earthquakes. In order to control these induced vibrations, several studies have been carried out and the focus was particularly made on active control techniques (Ouni and Ben Kahla 2014). Although actives control techniques showed effectiveness, their implementation in real large scale structures remains of particular complexity. An alternative to these techniques is the use of passive techniques such as mounting TMD devices into vibrating structure (Mrabet et al. 2016). TMD devices are very reliable and efficient in vibration control of structures and have been widely used for their low cost and design simplicity. Nevertheless, the parameters of such devices should be carefully optimized in order to effectively reduce the vibrations. Several optimization strategies can be found in the literature and among them the reliability based optimization (RBO) (Mrabet et al. 2016) that has been efficiently used in the passive control (using TMD) of earthquakes induced vibrations, for instance. In the present work a TMD device, optimized using the RBO strategy, is used for the control of a Geiger’s structure. The RBO strategy has been compared to other strategies and the obtained results showed its superiority in reducing the global vibration of a Geiger’s structure.

2 Governing Equations, Optimization of the TMD Parameters Figure 1 shows the considered tensegrity dome of Geiger’s type with its components of strings and struts. In Fig. 1(a) the nodes are numbered whereas in Fig. 1(b) the location of the TMD device (on node 2) is showed in a typical section view. The TMD is characterized by its natural frequency ωT and its damping ratio ξT = cT /2mT ωT , where cT is the damping coefficient and mT = μ × ms is its mass; μ and ms are the mass ratio and the mass of the tensegrity structure, respectively. Let x = (x1 , y1 , z1 , .., x31 , y31 , z31 , zT )T be the (78 + 1) × 1 displacement vector of the nodes and the TMD device, where zT is the TMD displacement and the superscript notation T denotes transpose. Assuming that the deformations of the structure is linear elastic (Feng et al. 2018), the governing equation of motion, around an equilibrium configuration, of the cable dome equipped with a TMD can be written as follows, M¨x+C˙x+Kx = Wexcit

(1)

     M 0 C0 K0 ,C = +cT b.bT , K = +kT b.bT , Wexcit is the (78 + 0 mT 00 0 0 1) × 1 vector of the point forces that are assumed to be uncorrelated stationary zeros mean Gaussian white noise; the point forces are applied on the top nodes (1, 3, 5, 8, …., 28, 30) of the structure along the z-axis (vertical forces). Besides, b = (0, 0, 0, .., −1, 0, .., 1)T is the location vector of the TMD; M, C and K are the mass, damping and the tangent stiffness matrices, respectively. The expressions of these matrices are given in Feng et al. (2018). 

where M =

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Fig. 1. (a) The tensegrity dome of a Geiger’s type made up of 13 vertical struts and 60 strings (Ouni and Ben Kahla 2014), (b) typical section view with the TMD mounted on node2

Equation (1) can be recast in space state form and the covariance responses of the structure can be obtained, in stationary conditions, using the Lyapunov equation (Wijker 2009) as follows: AR + RAT + B =0

(2)

  where R = XXT is the covariance matrix, X = (x, P x)T , . denotes the mathematical  T  0 I expectation, A = , B = GS0 GT , G = 0 −M−1 and S0 is the −1 −1 −M K −M C constant power spectral density (PSD) matrix of the white noise excitations. In the present work the optimization strategy consists in minimizing the failure probability Pf characterized by the out-crossing, for the first time, of a given threshold value β of the vertical displacement of node 2 during some interval time T. The failure probability can be approximated, using the Rice’s formula based on the Poisson assumptions (Mrabet et al. 2016), as follows: Pf = 1 − exp[−T × (σz˙2 /π σz2 ) × exp(−β 2 /2σz22 )]

(3)

Passive Control of Tensegrity Domes

65

where σz2 and σz˙2 are the root mean square (RMS) displacement (z-axis) and velocity of node 2 of the dome structure, respectively; these quantities can be obtained using Eq. (2). Let d = (ωT , ξT )T be the design vector, the optimization problem can be formulated as follows: Find d = (ωT , ξT )T to minimize Pf

(4)

3 Numerical Example Tables 1 and 2 show the geometric characteristics of the Tensegrity dome. The elastic modulus and density of all the elements are equal to 160 GPa and 7.85 × 10−3 kg/cm3 , respectively. The cable elements have yield strength of 0.6 GPa and a breaking strength of 0.8 GPa. The struts are round tubes (diameter = 2.13 cm and thickness = 0.16 cm) and they have a cross section area equal to 1 cm2 . The natural damping of the structure is assumed equal to 1% for all modes. A modal analysis of the uncontrolled structure has been conducted and the first three flexural mode shapes that are sensitive to the vibrations induced by vertical random excitations are shown in Fig. 2 where their corresponding resonant frequencies are also presented. For the sake of comparisons, the performance of the TMD, optimized using the RBO strategy, has been compared with three other strategies from the literature. The first strategy is that established by Den Hartog (Warburton 1982), which is a deterministic strategy, the two other are that of Leung (Leung and Zhang 2009) and that of Warburton (Warburton 1982) which are stochastic optimization strategies, since they involve white noise excitations. The corresponding tuning formulas are shown in Table 3 where ξs is the modal damping ratio of the structure (the mode to be controlled),  is the target frequency and μeff is the effective mass ratio as defined by Warburton (Warburton 1982). For the RBO strategy, the mass ratio is taken μ = 2%, T = 10 s, β = 0.25 m and the PSD intensity of the forces is taken S0 = 1N2 /Hz. The targeted mode was the first flexure mode of the structure corresponding to the natural frequency  = 31.95 rad/s and the optimization (RBO) has been carried out. ∗ = (30.62, 0.26)T and Pf∗ = 9.910–4 whereas for The optimal parameters are found dRBO ∗ ∗ ∗ the other strategies dLeung = (29.86, 0.20)T , dWarburton = (29.90, 0.15)T and dDenHartog T = (29.22, 0.18) . The performances of the TMD are shown in Figs. 3, 4, 5 where the frequency response functions (FRF) are plotted for three different nodes. Indeed, because of the symmetry in the dome structure, the FRFs of nodes 4, 9, 14, 19, 24 and 29 are similar and those of nodes 6, 11, 16, 21, 26 and 31 are also similar.

66

E. Mrabet et al. Table 1. Initial coordinates of the Geiger’s dome.

Node N°

Y(cm)

Z(cm)

Node N°

X(cm)

Y(cm)

Z(cm)

0

0

21

17

−30

51.9615

0

2

0

0

15

18

−20

0

18.5

3

20

0

18.5

19

−20

0

4.5

4

20

0

4.5

20

−40

0

11.5

5

40

0

11.5

21

−40

0

−11.5

6

40

0

−11.5

22

−60

0

0

7

60

0

0

23

−10

−17.3205

18.5

8

10

17.3205

18.5

24

−10

−17.3205

4.5

9

10

17.3205

4.5

25

−20

−34.6410

11.5

10

20

34.6410

11.5

26

−20

−34.6410

−11.5

11

20

34.6410

−11.5

27

−30

−51.9615

0

12

30

51.9615

0

28

10

−17.3205

18.5

13

−10

17.3205

18.5

29

10

−17.3205

4.5

14

−10

17.3205

4.5

30

20

−34.6410

11.5

15

−20

34.6410

11.5

31

20

−34.6410

−11.5

16

−20

34.6410

−11.5

32

30

−51.9615

0

1

X(cm)

Table 2. Cross-section area and initial pre-tension of different cable families. Cable family

Cable number

Cross-section area (cm2 )

Initial pretension (KN)

1

14–19

0.01

0.20

2

20–25

0.01

0.144

3

26–31

0.02

0.519

4

32–37

0.02

0.519

5

38–43

0.04

1.304

6

44–49

0.04

1.304

7

50–55

0.01

0.15

8

56–61

0.02

0.48

9

62–67

0.01

0.15

10

68–73

0.04

1.13

Passive Control of Tensegrity Domes

67

31.95 rad/s

60.19 rad/s

68.54 rad/s

Fig. 2. Flexural mode shapes sensitive to vibrations induced by a vertical excitation on the upper nodes of the dome (Ouni and Ben Kahla 2014)

The inspection of Fig. 3 shows good performance of the RBO strategy where significant reduction has been recorded (19.5 dB) in the vicinity of the targeted frequency. In addition, although the Leung’s tuning provided better performance (20.5 dB), the RBO strategy has performed better in the vicinity of the off-target frequencies. Indeed, one can observe that for the third resonant mode, for instance, the RBO provided a reduction of 18.1 dB in the FRF magnitude whereas it has been 16.1 dB when the Leung’s tuning (which is the better among the other strategies) is applied. Figures 4 and 5 show the FRFs of nodes 4 and 6 where significant reductions of 20.6 dB and 20.4 dB are obtained, respectively, in the vicinity of the targeted frequency. This fact implies that the RBO strategy provide better global attenuation performances since all displacement nodes are better controlled. For the other strategies, the best performance is obtained when the Leung’s tuning has been applied.

68

E. Mrabet et al. Table 3. The tuning formulas of the TMD parameters. ωT∗ /

Den Hartog

ξT∗

Warburton



√

Leung



Tuning

3μeff 8(1+μeff ) μeff (1+3μeff /4) (1+μeff )(4+2μeff )

1 1+μeff 1+μeff /2 1+μeff

√ μeff (1+3μeff /4) (1+μeff )(4+2μeff )

1+μeff /2 + α1 α2 , 1+μeff

 α1 = (−0.5047 + 0.0764 μeff + 0.6023μeff )

Magnitude of FRF (dB)

Magnitude of FRF (dB)

α2 =

0

w/o TMD Reliability based Criterion Leung Tuning Warburton Tuning Den Hartog Tuning

-20 -30 -40 10

1.3

10

1.6

Frequency (rad/s)

10 -20

20.5dB

19.5dB

1.9

12.1dB

10.8dB

16.1dB

-25

18.1dB

-30

-10

-35

-15

-40

-20 10

μeff ξs + 0.3737μeff ξs2

-10

0 -5

√

-45 1.4

10

1.5

Frequency (rad/s))

10

1.8

Frequency (rad/s)

Fig. 3. FRF (observation node 2)

10

1.9

Magnitude of FRF (dB)

Passive Control of Tensegrity Domes

69

0

w/o TMD Reliability based Criterion Leung Tuning Warburton Tuning Den Hartog Tuning

-20 -40 -60 2

10

Frequency (rad/s)

Magnitude of FRF (dB)

0 -5

-35

19.1dB

-40

20.6dB

-10

-45

-15

-50

-20

-55

-25

-60

-30 25

30

35

40

Frequency (rad/s)

60

45

65

70

75

Frequency (rad/s)

80

Magnitude of FRF (dB)

Fig. 4. FRF (observation node 4) -10 -20

w/o TMD Reliability based Criterion Leung Tuning Warburton Tuning Den Hartog Tuning

-30 -40 -50 -60 -70 -80 2

10

Frequency (rad/s)

Magnitude of FRF (dB)

-10

-35

-15

18.7dB

-20

20.4dB

-40 -45

-25

-50

-30

-55

-35 25

30

35

Frequency (rad/s)

40

50

55

60

65

70

Frequency (rad/s)

75

Fig. 5. FRF (observation node 6)

4 Conclusion In the present work, a passive control of a tensegrity structure using TMD is investigated. The TMD parameters are obtained using a reliability based optimization and the

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TMD showed good performance in the vicinity of the targeted frequency as well as the off-targeted ones. Compared to other optimization strategies, the reliability based optimization strategy has shown its superiority especially in the global attenuation of structural vibrations. Although it needs to be investigated in real scale structures, such strategy is a promising tool for the TMD design for the passive vibration control of such types of spatial structures. Acknowledgements. The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

References Feng, X., Miah, M.S., Ou, Y.: Dynamic behavior and vibration mitigation of a spatial tensegrity beam. Eng. Struct. 171, 1007–1016 (2018) Leung, A.Y.T., Zhang, H.: Particle swarm optimization of tuned mass dampers. Eng. Struct. 31, 715–728 (2009) Mrabet, E., Guedri, M., Ichchou, M.N., et al.: A new reliability based optimization of tuned mass damper parameters using energy approach. J. Vib. Control 24, 153–170 (2016) Ouni, M.H.E., Ben, K.N.: Active tendon control of a Geiger dome. J. Vib. Control 20, 241–255 (2014) Warburton, G.B.: Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Eng. Struct. Dynam. 10, 381–401 (1982) Wijker, J.J.: Random Vibrations in Spacecraft Structures Design. Springer, Amsterdam (2009)

Dynamic Analysis of a Pumping Station with Coupling Misalignment Fault Ahmed Ghorbel(B) , Oussama Graja, Moez Abdennadher, Lassâad Walha, and Mohamed Haddar Laboratory of Mechanics, Modeling and Production, National School of Engineering of Sfax, University of Sfax, Sfax, Tunisia

Abstract. Misalignment fault is a common source that generates strong vibrations and even dysfunction in rotating machinery. However, the characteristics of the dynamic response and the modeling of this type of defect have not been fully understood and there remains today an attractive field of research. This paper sheds the light on the modeling of vibrations and the diagnosis of coupling misalignment in a water pumping station for a dam case. Firstly, the paper presents a detailed dynamic model of a pump station using lumped elements. The model includes the motor, pump and two elastic couplings. These components are connected by shafts supported by bearings. Secondly, an approach to modeling an alignment defect, specifically parallel misalignment has been developed and presented. Thirdly, motion equations of the model were formulated and solved using Newmark’ method. Finally, simulated results are presented and discussed using temporal and spectrum analysis. In parallel misalignment, it is found that multifrequency components are obvious, static components are shown in the dynamic response. 2x rotational frequency of the motor will appear and is dominated and 1x rotational frequency is weak in the spectrum. The vibration responses of the three supporting bearing according to the radial directions with misalignment defect are investigated. Keywords: Pumping station · Misalignment · Modeling · Dynamic analysis

1 Introduction Pumping stations play a vital role in the network for distributing dam water to consumers, which is why they must be protected against assembly and installation faults to ensure continuity of operation. Mechanical couplings are central elements in transmission systems like pumping stations, since they allow the motor and the driven shaft to be connected and provide mechanical flexibility so that the system tolerates a certain misalignment level. When two misaligned shafts are linked by a coupling, the structure of the system is subjected to additional forces. These efforts represent a new source of vibration. Signatures of vibration are widely used for studying machine malfunctions and detect coupling misalignment fault. The misalignment forces formulation and the comprehension of its © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 71–78, 2021. https://doi.org/10.1007/978-3-030-76517-0_9

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spectral content are needed to study the vibration response of the system and to provide a diagnosis of misalignment (Sawalhi et al. 2019). Vibration analysis of a coupling misalignment was investigated in a many studies. In order to analyze the behavior of this fault, Gibbons (1976) developed a mechanical model for various coupling types and calculated the misalignment forces and moments. The misalignment forces for a flexible coupling was formulated by Xu and Marangoni (1994). From their results, the spectrum of misalignment forces contains at multiples of the motor rotation frequency (two times 2x rotational speed and its harmonics). Using the finite element method, a simple misalignment model was developed by Bloch et al. (1999). This model was later used by Sekhar and Prabhu (1995) to investigate the dynamic behavior of misaligned couplings. Guan et al (2017) proposed two dynamic models types of shaft misalignment. Experiments and numerical simulation have been represented. Janusz (2011) studied the longitudinal vibration behavior of a rotor with large misalignment. In the work presented in (Patel and Darpe 2009), the authors used the spectrum analysis of the measured signal of the bearing to investigate the harmonics generated by misalignment. In other research works, Lees (2007), Bahaloo et al (2009) and Wang et al. (2019) found that vibrations responses due to parallel misalignment of rigidly coupling occurs at 2x rotational speed. The present work consists on pumping station study in the presence of misalignment defect. In the first part, we present the adopted kinematic model, while the second part describes a new modeling of a parallel misalignment. The misalignment is modeled by an external force applied to the bearings depends on the variation of their stiffness over time. Furthermore, we formulate the motion equations. Finally, the obtained results were presented and discussed.

2 Pumping Station Modeling Figure 1 represents the lumped parameter model of the pumping station. The proposed model can be divided into three main blocks: the first one composed by driving motor (inertia I 1 ) and the shaft connected to the first part of the first elastic coupling (I 12 ). The second block including the other half of coupling (I 21 ) connected to the second elastic coupling part (I 31 ). The third block is composed from the second part of the second coupling (I 32 ), shaft 3 and the pump (I p ). The mass of the three blocks are mi (i = 1, 2, 3). The shafts are assumed as massless with torsional stiffness k ti (i = 1, 2, 3). The three blocks are supported by bearings modeled by parallel springs (k xi , k yi ). The elastic coupling is approached by a Nelson and Crandall model with translation stiffness (k xc , k yc ) and torsional stiffness (k tc ). Using Lagrange formalism, the motion equation can be written as: [M ]¨q + [K]q = F(t) where q is the degree of freedom vector which describes the generalized coordinates of the system and is defined by   q = θ1 θ12 θ21 θ31 θ32 θp x1 y1 x2 y2 x3 y3

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[M] and [K] are the mass matrix and the stiffness matrix. F(t) is the external force applied on the system.

Coupling 2

Pump I p ,θ p

I 32 , θ32

Coupling 1

I 31 , θ31

I 21 , θ 21

I12 , θ12

x3 , y3

x2 , y2

x1 , y1

Bearing 3

Bearing 2

Bearing 1

Electric motor I1 , θ1

Fig. 1. Pumping station model

3 Misalignment Modeling Misalignment is one of the main causes of reducing the service life of equipment. It concerns either two shafts linked by a coupling, or two bearings supporting the same axis. It is said that there is a misalignment of the shaft lines when the motor shaft and the receiver shaft assembled by a coupling are not perfectly aligned. The bad lineage is a very common problem, and as important as the unbalance. The causes of misalignment are improper mounting, thermal expansion or presence of shear forces on the bearings. The axes of the two rotors may have an angular misalignment at the coupling or a radial misalignment (Parallel) or the combination of the two. In this paper, we are interested in the modeling and the study of radial misalignment.

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When the center lines of the shafts are mounted parallel without meeting, there is a parallel misalignment fault. Figure 2 shows the movement diagram of coupling misalignment fault. In this case, the bearing of the receiving shaft is stressed at compressive stresses in the plane (xz) and a bending stress in the plane (xy).

Fig. 2. Parallel misalignment

External forces are applied to the bearings caused by the misalignment. Therefore, the stiffnesses of the bearings and the first coupling will be modeled by a periodic variation in the plane (xy). This variation is expressed by the following equation k(t): k(t) = k0 + k sin(2ωt) k is the stiffness variation which depend on the amount of misalignment DE. Figure 3 represents the dynamic behavior of the bearing in the presence of misalignments. In this case, the force exerted by the bearing is periodic in the radial direction at the frequency 2 times the frequency of rotation of the 2 ω motor shaft.

Fig. 3. Bearings behavior for parallel misalignment

4 Characteristics Analysis of Parallel Misalignment In the parallel misalignment simulation, misalignment amount DE is 5 × 10−3 m. The parameters detail for the system under investigation is given in Table 1.

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Table 1. Model parameters values Parameters

Value

Rotational speed (RPM)

3000

Torsional shaft stiffness (Nm/rad)

105

Bearing stiffness (N/m)

108

Inertia of the coupling (kg m2 )

4.10–3

Mass of the coupling (kg)

4.5

Torsional stiffness of the coupling (Nm/rad) 352 Translation stiffness of the coupling (N/m)

462.104

The effect of parallel misalignment on the dynamic behavior of the second bearing is shown in Figs. 4 and 5. From the time signal (Fig. 4), the presence of a parallel misalignment generates an increase in vibration of the bearings. This increase is due to the additional force (induced by the fault) acting on the bearings. There is also a phenomenon of amplitude periodicity. Figure 5 shows the displacement spectra of the second level in the x and y directions for two cases; without and with misalignment. A misalignment is revealed by a peak of preponderant amplitude at 2 times the frequency of rotation. A vibration in the radial direction of component of order appears 2 times of the frequency of rotation, with amplitudes greater than the components of order 1. The other peaks presented in the spectra (in both cases) correspond to the system’s Eigen frequencies. The same phenomenon is manifested in the axial direction but less clear than the radial direction.

Displacement x2 (m)

-3

1

x 10

With misalignment Without misalignment

0.5 0 -0.5 -1

0

0.1

0.2

0.3

0.4 Time (s)

0.5

0.6

0.7

0.8

Fig. 4. Dynamic response of the second bearing without and with misalignment

Figure 6 shows the vibration levels of each bearing according to the radial directions x and y in the case with misalignment. The signals obtained show that the vibration of the bearing on the motor side has the greatest amplitude while the bearing on the pump side has the lowest amplitude. This result confirms the interpretation of the previous figure (Fig. 4) that the presence of the misalignment increases the vibration level in the near bearings.

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Fig. 5. Spectrum of the second bearing response (Blue: without misalignment, Red: with misalignment)

A useful tool in vibration analysis is the shaft centerline orbits (Fig. 7). It is shown that a change in the orbit appears with presence misalignment. The center of the tree describes the shape of the ellipse with the presence of parallel misalignment and a line in the case without defect. The width of the orbits increases with presence misalignment.

Fig. 6. Dynamic responses of the three bearings with presence misalignment

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Fig. 7. Orbits of the bearings vibration

5 Conclusion Embedding of the effect of misalignment in coupling into a dynamic simulation model of a pumping station was presented. Mass, stiffness matrices and external forces were calculated using the Lagrange method. The displacement and the stiffness of the bearing were used to calculate the misalignment forces, which were fed to the system as external excitation forces. The motion equations of the system were formulated and solved using the Newmark method. The main features were the increase in amplitude due to the misalignment. The appearance of a new frequency corresponding to 2 times the rotation frequency of the shaft is another characteristic, which was observed in the spectrum. The orbit of displacement of the shaft center shows a remarkable change between the two cases; without and with misalignment. In the future works, we will propose to improve this model to integrate the effect of damping matrix and some other defect such as blade and cavitation faults. Acknowledgements. This work is partially supported by Mechanics, Modelling and Manufacturing LAboratory (LA2MP). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

References Sawalhi, N., Ganeriwala, S., Tóth, M.: Parallel misalignment modeling and coupling bending stiffness measurement of a rotor-bearing system. Appl. Acoust. 144, 124–141 (2019)

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Gibbons, C.B.: Coupling misalignment forces. In: Proceedings of the fifth turbomachinery symposium. Gas Turbine Laboratories, Texas, pp. 111–116 (1976) Xu, M., Marangoni, R.: Vibration analysis of a motor-flexible coupling-rotor system subjected to misalignment and unbalance – part I: theoretical model and analysis. J. Sound Vib. 176, 663–679 (1994) Bloch, H.P., Geitner, F.K.: Practical machinery management for process plants. Gulf. Profess. Publish (1999). https://doi.org/10.1016/S1874-6942(12)60001-7 Sekhar, A.S., Prabhu, B.S.: Effects of coupling misalignment on vibration of rotating machines. J. Sound Vib. 185, 655–671 (1995) Guan, Z., Chen, P., Zhang, X., Zhou, X., Li, K.: Vibration analysis of shaft misalignment and diagnosis method of structure faults for rotating machinery. Int. J. Perform. Eng. 13, 337–347 (2017) Janusz, Z.: The analysis of the rotor’s longitudinal vibration with large misalignment of shafts and rotex type coupling. Diagn. Struct. Health Monitor 2, 19–23 (2011) Patel, T.H., Darpe, A.: Experimental investigations on vibration response of misaligned rotors. Mech. Syst. Signal Process 23, 2236–2252 (2009) Lees, W.: Misalignment in rigidly coupled rotor. J. Sound Vib. 305, 261–271 (2007) Bahaloo, H., Ebrahimi, A., Samadi, M.: Misalignment modeling in rotating systems. In: ASME Turbo Expo 2009: Power for Land, Sea, and Air. American Society of Mechanical Engineers Digital Collection, pp. 973–979 (2009) Wang, H., Gong, J.: Dynamic analysis of coupling misalignment and unbalance coupled faults. J. Low Frequency Noise, Vibr. Active Control 1461348418821582 (2019)

On the Unidirectional Free-Surface Flow Solution in a Rectangular Open Channel Souad Mnassri1 and Ali Triki2(B) 1 Research Unit: Mechanics, Modelling, Energy and Materials M2EM, Department of

Mechanical Engineering, National Engineering School of Gabès, University of Gabès, Gabes, Tunisia 2 Research Laboratory: Advanced Materials, Applied Mechanics, Innovative Processes and Environment 2MPE, Higher Institute of Applied Sciences and Technology of Gabès, University of Gabès, Gabes, Tunisia

Abstract. Spillways, sluice-gates, and steep chutes are common appurtenant structures for open-channels. These structures provide an estimate of the total height of the dam in order to be used for water storage year-round and to avoid overflow water and dam failure. Often, these open-channel transitions produce discontinuities in the flow variables; and, consequently, lead to large free-surface disturbance. This paper probes into the free-surface wave behavior involved in an open channel, including transition. The one-dimensional shallow-water model is used to describe the unsteady free-surface flow, along with the Multiple-Grid technique-based Finite element solver, being used for numerical computations. Test case relates to a rectangular open channel discharging over a spillway and subjected to a storm concentrated at its inlet. The findings of this study evidence the capacities of the proposed solver in describing accurately the wave propagation and reflection mechanisms within the gradually varied free-surface flow regime framework. Findings also delineate the gain of the Multiple-Grid technique upon the standard one in terms of saving consumed computational time. In principle, the proposed MGTFEM numerical solver could be extended and applied to other shapes of prismatic open-channel, such as trapezoidal or triangular cross-section. Keywords: Finite element method · Free-surface · Galerkin · Multiple-grid technique · Open-channel · Rectangular · Shallow-water · Transient flow

1 Introduction Spillways, sluice-gates, and steep chutes are common appurtenant structures for open channels. These structures provide an estimate of the total height of the dam in order to be used for water storage year-round and to avoid overflow water and dam failure (Fig. 1). Often, these open-channel transitions produce discontinuities in the flow variables; and, consequently, lead to large free-surface disturbances (Wuyi et al. 2018, 2019). From the design side, accurate estimates of the maximum magnitude and the arrival time characteristics of the free-surface wave are needed to determine the limits of potential damage. These characteristics may be predicted based on the one-dimensional © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 79–86, 2021. https://doi.org/10.1007/978-3-030-76517-0_10

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St.-Venant, model (also known as the shallow-water model); which yields acceptable solutions if the open-channel is relatively straight and prismatic. Generally, the one-dimensional shallow-water model is solved numerically using the Method of Characteristics (MOC) the Finite Difference (FDM) or Finite Element Method (FEM) (Fennema and Chaudhry 1987; Wylie et al. 1993; Szymkiewicz 1991, 2010; Chunbo and Liang 2009; Triki 2013, 2014a, b, 2016, 2017, 2019). On this point, the MOC has received special attention thanks to its ability to describe physical flow processes and boundary conditions; however, this method could not describe discontinuous flows. Contrarily, the FDM can reproduce supercritical and subcritical flows, but this model results in important numerical oscillations. On the other hand, the FEM presents several benefits in comparison to the abovementioned methods. In this line, the author demonstrated the suitability of the FEM–based solver to predict the free-surfacewave features for subcritical and supercritical flow regimes, simultaneously (Mnassri and Triki 2020a; Triki 2013). Nonetheless, this method requires important grid points and hence much computational efforts to allow a convergent and accurate solution. In order to address the aforementioned drawback, the authors embedded the Multiple Grid Technique (MGT) to the FEM solver to enhance the solution time and quality near high gradient regions of the flow domain (Triki 2014a, 2020). Accordingly, we planned in this paper to apply this technique to assess and analyze the free-surface behavior in rectangular open-channel equipped with a spillway.

Fig. 1. Illustration of open-channel discharging over a spillway (Table Rock Dam - The Branson Tri-Lakes News).

In the following section, the Finite Element procedure based upon the Multiple-Grid Technique (MGT-FEM) is outlined.

2 Theory and Calculations The free-surface wave behavior in a rectangular open-channel may be written as (Wylie et al. 1993; Mnassri and Triki 2020a; Triki 2013, 2014a): ∂W ∂W +B −H=0 ∂t ∂x

(1)

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     0 1 A ql    2 . wherein, W = ; and H = ;B= g TA − QA2 QA gA S0 − Sf Q A: cross-sectional area of the flow; Q: flow-rate; T : top width of the cross section for depth d ; d = A/T : flow depth; u = Q/A flow velocity; s0 : slope of the channel4/3 bed; sf = m2 u2 /RH : flow resistance; RH : hydraulic radius; m: Manning’s coefficient; g: acceleration due to gravity; ql : lateral inflow; t and x: time and distance along the channel. Referring to Zienkiewicz and Taylor (1988); Dhatt and Touzot (1984), and Triki (2013, 2014a), the discretization of the flow equations using the MGT-FEM, may be outlined as follows: (i)- The channel is discretized into N = L/x space steps. (ii)- The hydraulic variables are expressed as functions of nodal variables using linear Galerkine functions ϕ (Fig. 2). these functions make use of two nodes (i.e.: i − 1 and i) and two degrees of freedom per node (i.e. A and Q). Accordingly, the Galerkin procedure, applied to Eq. (1), leads to:

 () ∂W ∂W +B − H dx = 0 (2) ϕi ∂t ∂x 

 L 

()

where in, L is the channel reach within the element i and ϕi = 0 for all elements not bounded by a node i. Hence, the discretization of Eq. (1) leads to the following system of 2×(n + 1) times partial derivative equations; which is commonly solved using an iterative algorithm:        (3) K W W = f W wherein:

: stiffness matrix; W: vector  composed of unknown hydraulic parameters computed at t = (j − 1 + θ )t; and f W : flux vector. Mik is a (2 × 2) sub-square matrix located along the main diagonal of the global matrix, which isnull if k < i − 1 or k > i + 1, else, it depends on the unknowns of the problem Wi−1 Wi Wi+1 at the time (j − 1 + θ)t; fi is a second member vector which depends on the unknowns of the problem at the time (j − 1)t; M11 , M12 , MN +1N , MN +1N +1 : depends on the boundary conditions in the nodes 1 and N + 1; and f1 and fN +1 contain the boundary conditions. The variation in time of variables, not in a time derivative form, can be approximated using weighted averages over time:   j−1+θ j j−1 j−1 + Wi Wi = Wi = θ Wi − Wi (4)

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1

x=0 x1

ϕi −1

xi −1

ϕi +1

ϕi

Ω i −1

xi

Ωi

xi +1

x=l [ xk +1

Fig. 2. Function of linear interpolation.

(iii)- The semi-implicit Euler Method is employed to approximate the time derivative of fluid variables such as:  jt j j−1 j−1 W − Wi Wi − Wi 1 ∂Wi dt = i = (5) t (j−1)t ∂t t θt   Where in Wi = Ai Qi is the flow parameter values computed at the time (j − 1 + θ )t; θ is a weighting coefficient chosen in the range θ ≥ 0.5 (Zienkiewicz and Taylor 1988). (iii)- The MGT (Triki 2014a) is incorporated in the numerical solver basing on the Froude number value. This technique is based on an additive decomposition of the open-channel into coarse cG and fine fG scales (Figs. 2 and 3).

Fig. 3. Schematics of fine-grid (fG ), coarse-grid (cG ), and multiple-grid (mG ).

The stability of the numerical solved is ensured basing on the Courant-FreidrichsLewy stability condition (i.e.:t ≤ Cr × x/{max(|u| + c)}; in which, c: wave speed and Cr: Courant number, chosen in the range: Cr ≤ 1 (Szymkiewicz 1991, 2010). The next section implements the MG-FEM solver for simulating the transient flow behavior involved by a gradually varied regime.

3 Case Study The hydraulic system, studied in this section is sketched in Fig. 4a (Wylie et al. 1993). The apparatus consists of a rectangular open-channel (L = 3300 m; T = 3.658 m; s0 = 0.001; and m = 0.0185) discharging over a spillway at its downstream extremity.

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Initially, the channel is discharging Q0 = 23.6 m3 /s at the depth d0 (x) = 1.83 m under a steady-uniform flow regime. As per the authors (Wylie et al. 1993), the depth-velocity relationship over the downstream spillway may be expressed as: T × d × u = 132(d − 0.707)3/2

(6)

The transient regime is induced by a storm concentrated at the channel inlet during the 30 first minutes (Fig. 4b). Specifically, the flow is increased linearly from the initial value until it has doubled in 20 mn; and subsequently decreases linearly until it is one-half the original flow in 10 mn additional min. Next numerical simulation employs the set of MG-FEM parameters: {θ = 0.6; n = 10 x = 330 m; and t = 0.1 s}.

Fig. 4. Illustration of (a)- the hydraulic system layout (b)- the discharge signal at the channel inlet.

Figures 5a and b show, respectively, the depth and velocity signals versus time predicted using the MGT-FEM solver. The wave pattern of Fig. 5 indicates that the depth and velocity flow parameters increase due to water supply at the channel inlet. Additionally, Fig. 5 indicates that the maximum amplitudes reached by the hydraulic parameters decrease from the upstream to the downstream extremities. This is mainly attributed to the energy dissipation at the spillway. For example, the depth value at the upstream and downstream extremities of the open-channel are equal to: d |max x=0 = 2.3 m (at t ≈ 20 mn) and d |max x=L = 2.3 m (at t ≈ 27 mn), respectively.

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Subsequently, after the storm stop (i.e.: t ≈ 30 mn), the depth and velocity patterns illustrate a second steady-state behavior. In particular, the flow parameters computed at the upstream and downstream extremities of the open-channel decrease to d |min x=0 = = 1.4 m, respectively. 1.2 m and d |min x=L Figure 6 shows the free-surface profiles printed at 10 min time step. This figure shows a set of gradually free-surface-waves propagating downstream. For example, the freesurface wave profiles computed at t ≈ 10 mn or t ≈ 30 mn correspond to a significant rise of free-surface level; while the wave profiles computed at t ≈ 40 mn or t ≈ 60 mn illustrate a quasi-steady wave behavior. (a)

d, m 3 2.5 2

x=0

1.5 1

x=0.2 L 0

10

20

30

40

50

60

(b)

u, m/s

70 t, mn

x=0.4 L x=0.6 L

3

x=0.8 L

2.5

x= L

2 1.5 1

0

10

20

30

40

50

60

70 t, mn

Fig. 5. (a)- depth and (b)- velocity time-histories.

From a numerical standpoint, the MGT-FEM –based algorithm provides about 11% less computational time as compared with the conventional algorithm-based FEM (Mnassri and Triki 2020a; Triki 2013).

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85

d, m 3 t=10 mn

2.5

t=20 mn t=30 mn

2

t=40 mn t=50 mn

1.5

t=60 mn 1

0

500

1000

1500

2000

2500

3000 x, m

Fig. 6 Profiles of free-surface level.

4 Conclusion This paper applied a Multiple Grid technique–based Finite Element Method to describe the free-surface wave behavior in a rectangular open-channel equipped with a spillway at its outlet. As a result, this study showed that the developed algorithm could be successfully employed to predict gradually varied free-surface flow regime due to fluctuations of discharge into rectangular open-channel. Furthermore, this study demonstrated the attribute of the multiple grid technique in comparison to the standard one in terms of saving consumed computational time. In principle, the proposed MGT-FEM numerical solver could be extended and applied to other shapes of prismatic open-channel, such as trapezoidal or triangular cross-section.

References Chunbo, J., Liang, C.D.: Computation of shallow wakes with the fractional step finite element method. J. Hydraul. Res. 47(1), 127–136 (2009) Dhatt, G., Touzot, G.: Une présentation de la méthode des éléments finis-(A presentation of the finite element method). Edition Maloine SA, Paris (1984) Fennema, R.J., Chaudhry, M.H.: Simulation of one-dimensional dam-break flows. J. Hydraul. Res. 25(1), 41–51 (1987). https://doi.org/10.1080/00221688709499287 Mnassri, S., Triki, A.: A Finite Element solver for simulating open-channel transient flows Gradually varied regime. ISH J. Hydraul. Eng. (2020) Mnassri, S., Triki, A.: Numerical investigation towards the improvement of hydraulic-jump prediction in rectangular open-channels. ISH J. Hydraul. Eng. (2020a). https://doi.org/10.1080/ 097150.2020.1836684 Szymkiewicz, R.: Finite-element method for the solution of the Saint Venant equations in an open channel network. J. Hydrol. 122(1), 275–287 (1991). https://doi.org/10.1016/0022-169 4(91)90182-H Szymkiewicz, R.: Numerical Modeling in Open Channel Hydraulics. Springer Netherlands, Dordrecht (2010)

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Triki, A.: A Finite Element Solution of the Unidimensional Shallow-Water Equation. J. Appl. Mech - ASME 80(2), 021001 (2013). https://doi.org/10.1115/1.4007424 Triki, A.: Multiple-grid finite element solution of the shallow water equations: water-hammer phenomenon. Comput. Fluids 90, 65–71 (2014a). https://doi.org/10.1016/j.compfluid.2013. 11.007 Triki, A.: Resonance of Free-Surface Waves Provoked by Floodgate Maneuvers. J. Hydrol. Eng. 19(6), 1124–1130 (2014b). https://doi.org/10.1061/(ASCE)HE.1943-5584.0000895 Triki, A.: Erratum for “resonance of free-surface waves provoked by floodgate maneuvers” by Ali Triki. J. Hydrol. Eng. 21(7), 08216003 (2016). https://doi.org/10.1061/(ASCE)HE.1943-5584. 0001408 Triki, A.: Further investigation on the resonance of free-Surface waves provoked by floodgate maneuvers: Negative surge waves. Ocean Eng. 133, 133–141 (2017). https://doi.org/10.1016/ j.oceaneng.2017.02.003 Triki, A.: Investigating the free-surface flow behavior due to sluice-gate maneuvers. In: Advances in Mechanical Engineering and Mechanics- Proceeding of the 7th International Conference on Advances in Mechanical Engineering and Mechanics, ICAMEM 2019, December 16–18, Hammamet, Tunisia (2019) Toro, E.F.: Schok Capturing Methods for Free-surface Shallow Flows. John Wiley and sons (2004) Wuyi, W., Liu, B., Awais, R.: Numerical prediction and risk analysis of hydraulic cavitation damage in a high-speed-flow spillway. Shock Vibr. 1817307 (2018) https://doi.org/10.1155/ 2018/1817307 Wan, W., Raza, A., Chen, X.: Effect of height and geometry of stepped spillway on inception point location. Appl. Sci. 9(10), 2091 (2019). https://doi.org/10.3390/app9102091 Wylie, E.B., Streeter, V.L., Suo, L.: Fluid Transients in System. Prentice-Hall, Englewood Cliffs, New Jersey (1993) Yost, S.A., Prasada, R.: A multiple grid algorithm for one-dimensional transient open channel flows. Adv. Water Resour. 23, 645–665 (2000). https://doi.org/10.1016/S0309-1708(99)000 52-4 Zienkiewicz, C., Taylor, R.L.: The Finite Element Method. McGraw-Hill Book Company, U.K. (1988)

On the Numerical Solution of the Rapidly Varied Regime in Open-Channel Flows Souad Mnassri1 and Ali Triki2(B) 1 Research Unit: Mechanics, Modelling, Energy and Materials M2EM, Department of

Mechanical Engineering, National Engineering School of Gabès, University of Gabès, Gabes, Tunisia 2 Research Laboratory: Advanced Materials, Applied Mechanics, Innovative Processes and Environment 2MPE, Higher Institute of Applied Sciences and Technology of Gabès, University of Gabès, Gabes, Tunisia [email protected] Abstract. The prediction of free-surface flows in rivers or man-made open channels attracted great interests of hydraulic engineers and designers, in order to provide significant information for the design of these hydraulic utilities and the procedures of operational setting. In this line, the St.-Venant model; approximating the unsteady velocity distribution and friction factors by their respective ones derived from the steady flow, could not be employed for forecasting supercritical regimes and hydraulic shocks connected therewith. In this paper, a study on the modelling of free-surface transient flows succeeding a downstream sluice-gate closure was performed. The McCormack scheme was implemented for the numerical discretization of the extended 1-D St.-Venant model, based on the Prandtl power law as a momentum correction coefficient. The proposed solver was compared with numerical results quoted in the literature. This comparison demonstrated that the result obtained by the McCormack scheme complied closely with that obtained by the Method of Characteristics. Results evidenced that such a solver is simple and well suited to handle simultaneously subcritical and supercritical flow regimes. On the other hand, results suggested that, unlike the conventional St.-Venant model, the extended one allowed smooth signals without any numerical oscillations in the regions close to hydraulic shocks. Keywords: Extended St.-Venant model · McCormack · Hydraulic-shock · Open channel · Prandtl power-law

1 Introduction The prediction of free-surface flows in rivers or man-made open channels is of great interest for hydraulic engineers and designers, to provide significant information for the design of hydraulic structures and the procedures of operational setting. Basically, the earliest mathematical formulations adopted by investigators for modelling unsteady flows in open-channels, were based upon the shallow-water wave assumptions (also referred to as St.-Venant model). Besides, the one-dimensional (1-D) assumption was generally used for the numerical solution of these formulations. In essence, these equations are hyperbolic in structure, and thus, discontinuities may appear when solving these © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 87–94, 2021. https://doi.org/10.1007/978-3-030-76517-0_11

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equations for rapidly varied regimes involving flow or hydraulic -shocks (or jumps), and bores (Wylie et al. 1993; Chaudhry 2008; Wuyi et al. 2019; Triki 2013, 2014b, 2017, 2021; Mnassri and Triki 2020a,b, 2021a,b). For example, these regimes may be generated by abrupt change in flow conditions such as the sudden opening and/or closure of sluice-gates for emergency situation or dam failure cases (Fig. 1). Incidentally, the St.-Venant model is based on the assumption of a hydrostaticpressure distribution across the wetted section of the open-channel. This assumption is satisfactory whenever no significant difference is observed between the vertical velocity profile corresponding to the unsteady flow regime and its counterpart associated with the uniform flow; otherwise, two additional effects, arising from inertial forces wall friction stress (Fig. 2), should be considered (in the momentum equation of the onedimensional free-surface flow) in addition to the corresponding ones involved in the steady-state regime. In this regards, one alternative approach, based on a momentum correction coefficient add-on, was proposed by Chen (1991, 1992), to describe the vertical velocity profile and the Darcy-Weisbach friction factor involved during fast transient event. Specifically, the author addressed a 1/6th Prandtl power law as a correction coefficient for smooth concrete open-channel flows; and a logarithmic law for fully rough flows.

Fig. 1. Spill-gate failure of the Lake Dunlap Dam (San Antonio News-OnExpressNews.com).

From a numerical modelling stand point, numerous numerical solvers were proposed by investigators to describe rapidly varied free-surface flow regimes involving discontinuities. Principally, these solvers range from the Finite Difference (Fennema and Chaudhr 1987; Gharangik and Chaudhry 1991; Szymkiewicz 1991, 2010; Triki, 2010, 2013, 2014a, b, 2016), the Finite Element (Triki, 2013, 2014b, Mnassri and Triki, 2020, 2021), the Finite Volume (Toro 2000) to the Methods Of Characteristics (Wylie et al. 1993). In particular, under specific conditions, the McCormack scheme could

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capture these discontinuities and, thereby, can lead to acceptable solution of these equations. In addition this numerical approach is attractive thanks to its generality to describe simultaneously subcritical and supercritical flow regimes. In this study, the McCormack scheme is implemented to discretize the extended (1-D) Extended St.-Venant model. Investigations relate to a hydraulic shock formation in a rectangular channel.

d

9:

(d ) d

Q0

0

(d )

d0

[

R

Fig. 2. Schematic of vertical velocity distribution.

2 Theory and Calculations According to Chen (1992), the 1-D Extended St.-Venant model, describing unsteady free-surface flow, is given by: ∂U ∂G(U) + = H(U, x) ∂t ∂x

(1)

wherein, U = {Q, A}T : vector of hydraulic parameters; G(U) =      T T   Q λ Q2 A + gA2 (2T ) : flus vector; and H(U, x) = 0 Ag s0 − sf : source vector; Q: discharge; d = Q A: flow depth, measured normal to the channel bed; A = T × d : cross-section area of the channel;  T : channel width; g: gravity accelera4 3 2 2 tion; s0 : slope of the channel bed; sf = n u RH/ : flow resistance; u: flow velocity; RH : hydraulic-radius; n: Manning’s roughness; λ is the velocity-distribution coefficient. From practical stand point, λ = 1.02 corresponding to a 1/6th Prandtl power law, is acceptable for smooth concrete channel cases (Chen 1992); t: time; and x: distance measured parallel to the channel bed direction. Referring Jiménez and Chaudhry (1988), the vector of flow parameters, may be solved in two steps McCormack procedure, as follows: Predictor step:  (2) U∗i = Uki − σ G(U)ki − G(U)ki−1 + t A g Hki Corrector step:

  ∗ ∗ ∗ ∗ U∗∗ i = Ui − σ Gi+1 − Gi + tHi

(3)

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The new values for the vector U are obtained from:  1 k+1 Ui + U∗∗ Uik+1 = i 2

(4)

where in, i and k are grid point in the x and t axis. Incidentally, the McCormack scheme uses a backward space difference in the predictor step; while a forward space difference is utilized in the corrector step. The stability  of the McCormack scheme may be ensured by selecting a grid-mesh ratio σ = t x according to the CFL condition (Szymkiewicz 2010; Triki 2013, 2014a, b, 2017):       (5) σ ≤1 μ+ gA T max

Validation of the Numerical Solver In order to validate the developed numerical solver in reproducing the propagation of free-surface waves, the numerical tests performed by Wylie et al. (1993) on a rectangular open-channel (L = 304.8 m; T = 3.66 m; s0 = 0.001; and m = 0.014) was employed (Fig. 3). Under steady-state conditions, the channel is delivering a uniform flow-rate: Q0 = 20.388 m3 / s corresponding to the depth and velocity values equal to: d0 = 2.4 m and u0 = 2.32 m/ s, respectively.

d=d0

Q0

L

T

Fig. 3. Definition sketch of the applied open channel

Figure 4 compares downstream depth signal issued from the McCormack scheme –based algorithm and its counterpart obtained by Wylie et al. (1993) using the Method Of Characteristics. This figure suggests that such a transient event provokes a rapidly varied flow regime including the formation of hydraulic shock. From numerical stand point, Fig. 4 evidences close agreement between the two numerical solutions. However, it should be delineated that the Method Of Characteristics is no longer convergent after the reflection of the hydraulic shock at the upstream extremity of the open channel i.e.: t ≥ 70s). According to the authors (1992), additional relationship should be incorporated in the numerical code to describe the hydraulic-shock behaviour.

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d, m 4 3.5 3 2.5 2

0

15 Wylie et al. 1993

30

45 60 t, s McCormack scheme

Fig. 4. Comparison of depth signals at the downstream extremity of the open-channel.

3 Case Study Test case relates to the hydraulic system presented above. The transient regime is induced by a constant flow-rate at the channel inlet in conjunction with a switch-off maneuver of the spill-gate located at the channel outlet. Such a transient scenario may be described by the following boundary conditions (Chen 1983):   x = l x = 0   =0 (6) = Q0 ; and Q Q t > 0 t > 0 Concerning the numerical solution, the McCormack algorithm uses a parameters set: {N = 120 and t = 0.687 s (selected referring to the C.F.L. stability condition)}. Besides, the flow depth and   velocity are straightforwardly computes as: uik+1 = Qik+1 Aik+1 and dik+1 = Aik+1 T , respectively. Figures 5a and b display, respectively, depth and velocity signals according to the time, at different sections across the open-channel. Alongside, Fig. 6 illustrates the profiles of the free-surface level versus the axial position at different times. At first sight, these figures evidence that the McCormack scheme reproduces correctly the hydraulic-shock propagation. In general, this solver results in smooth curves especially in the regions close to the formed hydraulic shocks. On the other hand, Fig. 6 elucidates that the free surface profiles fluctuate and increase continuously. From hydraulic stand point, Figs. 5-a and -b show a series of propagation and reflection mechanisms of hydraulic-shocks induced by the transient scenario. Specifically, the depths illustrate an increasingly behaviour due to water supply at the upstream extremity of the open-channel; while the steepness of the wave front is attributed to the switch-off manoeuvre of the the spill-gate located at the channel outlet. On this point, succeeding to the switch-off manoeuvre of the the downstream spill-gate the depth fist, the upstream section, rises to a first pick value equal to: dx=L = 3.7 (Figs. 5-a and Fig. 6:

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see free-surface curve at t = 50 s). Meanwhile, the formed hydraulic-shock is propagated upstream during t = 60.245 s. Subsequently, this hydraulic-shock is reflected upon the upstream reservoir toward the downstream extremity of the open-channel (see free-surface curve at t = 100 s Fig. 6); and hence, the depth value is significantly amplified to: dx=0 4.5 m. The reflected hydraulic-shock reaches the upstream extremity of the open-channel at t = 60.245 s. These wave propagation and reflection mechanisms are continuously repeated due to continuous water supply at the open-channel inlet. d, m

(a)

8 7 6 5 4 3 2

0

50

100

150

200

150

200

250

t, s

(b)

u, m/s 2.25 1.75 1.25 0.75 0.25 -0.25

0

50 x=0

100 x=L/4

x=L/2

x=3L/4

250

t, s

x=L

Fig. 5. Time-histories of (a)- depths and (b)-velocities, at different cross-sections of the channel.

Inverse interpretations may be carried out for the velocity curves plotted in Fig. 5b. Indeed, the velocity is set to ux=L = 0 m/ s at the downstream extremity of the openchannel, due to switch-off manoeuvre of the the downstream spill-gate. Subsequently,

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when the hydraulic-shock reaches the upstream extremity of the open-channel, the upstream velocity at this extremity drops to ux=0 = 1.25 m/ s. This velocity value corresponds to the velocity of the water supplied to the upstream extremity of the open-channel. d,m 7 6 5 4 3 2

0

50 t = 50s t = 200s

100

150

200

250

t = 100s t = 250s

300 x, m

t = 150s

Fig. 6. profiles of free-surface levels

4 Conclusion Overall this study described the numerical discretization of (1-D) Extended St.-Venant model based on the 1/6 Prandtl power law using the McCormack scheme. The hydraulic parameter values at the boundary extremities of the channel were simulated using the characteristic equations. Results evidenced that the McCormack scheme is able to simulate correctly the surge wave propagation and reflection mechanisms induced by the fluctuations of discharge at either ends of the open-channel. Unlike the results issued from the discretization of the conventional St.-Venant model, the extended one resulted in smooth curves without any numerical oscillations in the regions close to the hydraulic-shock.

References Chaudhry, M.H.: Open-Channel Flow, Second Edition, Springer (2008) Chen, C.: Rainfall Intensity-Duration-Frequency Formulas. J. Hydraul. Eng. 199(12), 1571–1584 (1983). https://doi.org/10.1061/(ASCE)0733-9429(1983)109:12(1603) Chen, C.: Unified theory on power laws for flow resistance. J. Hydraul. Eng. 17(3), 371–389 (1991). https://doi.org/10.1061/(ASCE)0733-9429(1991)117:3(371) Chen, C.: Momentum and energy coefficients based on power-law velocity profile. J. Hydraul. Eng. 118(11), 1571–1584 (1992). https://doi.org/10.1061/(ASCE)0733-9429(1992)118:11(1571)

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Jiménez, O.F., Chaudhry, M.H.: Computation of supercritical free-surface flows. J. Hydraul. Res. 114(4), 41–51 (1988). https://doi.org/10.1061/(ASCE)0733-9429(1988)114:4(377) Fennema, R.J., Chaudhry, M.H.: Simulation of one-dimensional dam-break flows. J. Hydraul. Research 25(1), 41–51 (1987). https://doi.org/10.1080/00221688709499287 Gharangik, M.A., Chaudhry, M.H.: Numerical simulation of hydraulic jump. J. Hydraul. Eng. ASCE. 117(9), 1195–1211 (1991). https://doi.org/10.1061/(ASCE)0733-9429(1991) Mnassri, S., Triki, A.: Numerical investigation towards the improvement of hydraulic-jump prediction in rectangular open-channels. ISH J. Hydraulic Eng. (2020) https://doi.org/10.1080/097 15010.2020.1836684 Mnassri, S., Triki, A.: A Finite Element –based solver for simulating open-channel transient flows The gradually varied regime. ISH J. Hydraulic Eng. (2020). https://doi.org/10.1080/09715010. 2020.1815090 Mnassri, S., Triki, A.: Investigating the unidirectional flow behavior in trapezoidal open-channel. ISH J. Hydraul. Eng. (2021). https://doi.org/10.1080/09715010.2021.1918027 Mnassri, S., Triki, A.: On the unidirectional free-surface flow behavior in trapezoidal crosssectional open-channels. Ocean Eng. 223, 108656 (2021). https://doi.org/10.1016/j.oceaneng. 2021.108656 Szymkiewicz, R.: Finite-element method for the solution of the Saint Venant equations in an open channel network. J. Hydrol. 122(1), 275–287 (1991). https://doi.org/10.1016/0022-169 4(91)90182-H Szymkiewicz, R.: Numerical Modeling in Open Channel Hydraulics. Water Science and Technology Library Springer, Netherlands. (2010). https://doi.org/10.1007/978-90-481-3674-2 Toro, E.F.: Schok capturing Methods for Free surface Shallow Flows. John Wiley and Sons (2000) Triki, A.: A finite element solution of the unidimensional shallow-water equation. J. Appl. Mech - ASME 80(2), 021001 (2013). https://doi.org/10.1115/1.4007424 Triki, A.: Resonance of free-surface waves provoked by floodgate maneuvers. J. Hydrol. Eng. 19, 1124–1130 (2014a). https://doi.org/10.1061/(ASCE)HE.1943-5584.0000895 Triki, A.: Multiple-grid finite element solution of the shallow water equations: water-hammer phenomenon. Comput. Fluids 90, 65–71 (2014b). https://doi.org/10.1016/j.compfluid.2013. 11.007 Triki, A.: Erratum for resonance of free-surface waves provoked by floodgate maneuvers, by Ali Trik. J. Hydrol. Eng. 21(7), 1124–1130 (2016). https://doi.org/10.1061/(ASCE)HE.1943-5584. 0000895 Triki, A.: Further investigation on the resonance of free-Surface waves provoked by floodgate maneuvers: Negative surge waves. Ocean Eng. 133, 133–141 (2017). https://doi.org/10.1016/ j.oceaneng.2017.02.003 Triki, A.: Investigating the free-surface flow behavior due to sluice-gate maneuvers. In: Advances in Mechanical Engineering and Mechanics- Proceeding of the 7th International Conference on Advances in Mechanical Engineering and Mechanics, ICAMEM 2019, December 16–18, 2019, Hammamet, Tunisia (2021) Wylie, E.B., Streeter, V.L., Suo, L.: Fluid transients in system. Prentice Hall, Englewood Cliffs, New Jersey (1993) Wuyi, W., Awais, R., Xiaoyi, C.: Effect of height and geometry of stepped spillway on inception point location. Appl. Sci., 9(10), 2091 (2019). https://doi.org/10.3390/app9102091

Controlling of Steel-Pipe-Based Hydraulic Systems Using Dual In-Series Polymeric Short-Sections Mounir Trabelsi1 and Ali Triki2(B) 1 Department of Mechanics, National Engineering School of Sfax, University of Sfax,

B.P 1173, 3038 Sfax, Tunisia [email protected] 2 Research Laboratory: Advanced Materials, Applied Mechanics, Innovative Processes and Environment 2MPE, Higher Institute of Applied Sciences and Technology of Gabès, University of Gabès, Gabes, Tunisia [email protected]

Abstract. The in-series design measure, also known as the inline measure, was identified as being an efficient alternative water-hammer control tool into existing steel pipes-based hydraulic systems. Specifically, this design tool attenuated markedly the extent of pressure-wave surges. However, further evaluation revealed that such a measure was limited by an expansion effect of the period value of pressure-wave oscillations. Alternatively, this work addressed a dual-technique– based in-series control measure, in order to address the foregoing drawback of the conventional-technique-based in-series measure. Basically, the reported technique was devised upon the splitting of the single in-series section, used in the conventional technique, into two subsections placed upstream each of the steelpipe connections to the hydraulic system parts. The one-dimensional unconventional water-hammer model, including the Vitkovsky and Kelvin-Voigt formulations, was discretized using the Method of Characteristics, for numerical solution. Application assessed the efficiency of the dual technique for a reservoir-pipe-valve system. Two plastic material types were demonstrated in this study and including high- and low-density polyethylene (HDPE and LDPE). Results evidenced that the dual technique devised upon an (HDPE-steel-LDPE) configuration provided the best compromise between the attenuation and expansion effects of pressurewave oscillations. It was also found that the last two effects are sensitive to the employed dual in-series sections. Keywords: Design · Dual · In-series · HDPE · Kelvin-Voigt · LDPE · Viscoelasticity · Vitkovsky · Water-hammer

1 Introduction The control of water-hammer surges is a challenging issue pertaining to the design of urban water distribution and industrial hydraulic systems. If this phenomenon was not properly considered, the hydraulic installation may experience service disruption and/or unsafe operations for operators. These surges also referred to as flow shocks, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 95–104, 2021. https://doi.org/10.1007/978-3-030-76517-0_12

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induce a series of positive and negative pressure-waves, which may cause, in specific cases, significant damage to the piping system (e.g.: pipe-wall stretch or collapse) and hydraulic devices; and even risks to operators (Chen et al. 2015). In order to attenuate the extent of these pressure-wave surges, the designers resort to the change of the piping system layout and/or operating procedure parameters, or the installation of surge protection devices (such as pressurized vessels or air-relief valves) (Fig. 1). Conjointly, the use of plastic materials such as polyethylene has recently increased in hydraulic system networks. These plastic material types are distinguished by their ability to dampen high-pressure load in addition to their flexibility and great resistance to aggressive agents. Physically, this dampening effect is attributed to the mechanical behavior of plastic materials; which is characterized by temperature, time, and pressure gradient dependent deformations. In other words, these materials exhibit a viscous response to pressure-load; in addition to the elastic one, involved in steel materials (Brinson and Brinson 2008). This effect is quite interesting to dissipate gradually the energy caused by pressure-load. Basing on the aforementioned merits of plastic materials, several studies investigated the usefulness of substituting a short-section of existing steel-pipe–based hydraulic systems by another one made of plastic material, in order to improve the capacity level of these hydraulic systems (Triki 2016, 2017, 2018a, b; Ben Iffa and Triki 2019; Trabelsi and Triki 2019, 2020a, b, c; Triki and Fersi 2018; Fersi and Triki 2019a, b, 2020; Triki and Chaker 2019; Chaker and Triki 2020a, b, c, d). In these studies, the authors utilized High- and Low-Density PolyEthylene (HDPE and LDPE) materials for the replaced in-series short-section. These studies highlighted that such plastic materials types helped attenuation of excessive magnitudes of the initiated positive or negative pressure-wave. Although these studies delineated the benefit from such a handling technique of the piping system, hereinafter referred to as in-series design measure, it should be emphasized that the use of in-series plastic section leads to the expansion of pressure-wave oscillation period. Consequently, this design tool may cause an important increase in the duration of operational procedures (such as the critical time value of valve closure setting). (a)

(b)

Fig. 1. Illustration of ruptured (a)- check valve; (b)- Cast Iron Strainer (Wayne F. Kirsner Steam Accidents & Forensic Investigations).

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Accordingly, it is interesting to re-examine the conventional technique of implementation of the in-series design measure according to the two foregoing effects. Alternatively, we planned in this paper to address a dual technique–based in-series design measure; which is based on the splitting of the single in-series section, used in the conventional technique, into two subsections placed upstream each of the steel-pipe connections to the hydraulic system parts. Such a technique is expected to limit the expansion of the wave-oscillation period involved in the controlled system case based upon the conventional technique; while safeguarding acceptable attenuation of pressurewave magnitude. Special analyses are reported on the compromise between the two aforementioned effects. In the following, the method of characteristics (MOC) procedure used for discretizing the extended one-dimensional water-hammer Eqs. (1-D E-WH Eqs) based upon the Vitkovsky et al. and Kelvin-Voigt formulation, is briefly outlined.

2 Theory and Calculations Referring to Triki (2018a, b), the (1-D) extended water-hammer model incorporating the Vitkovsky et al. and Kelvin-Voigt formulations is given by:   a2 ∂Q a2 d εr ∂H 1 ∂Q ∂H + 0 +2 0 = 0 and +g + g hfs + hfu = 0 ∂t gA ∂x g dt A ∂t ∂x

(1)

wherein, H designates the pressure-head; Q denotes the flow-rate; A is the area of the pipe hfs cross-section; g stands for the accelerationdue gravity; a0 refers   to  wave-speed;  to  corresponds to pressure-head losses; hfu = kv gA +a0 Sgn(Q)∂Q ∂x hfu accounts for unsteady friction losses, evaluated according to Vitkovsky, in which, kv = 0.004 designates the Vitkovsky et al. (2000) decay coefficient; εr designates the retarded radialstrain of the pipe-wall; and x and t: axial and time coordinates, respectively. Incidentally, the retarded radial-strain of the pipe-wall may be evaluated according to the generalized Kelvin-Voigt linear-viscoelastic mechanical model sketched in Fig. 2 (Aklonis et al. 2006):   nkv (2) Jk 1 − e−t / τk J (t) = J0 + k=1

  wherein, Jk = 1 Ek and τk = μk Ek (k = 0 · · · nkv ) are associated with creep compliance of the spring and the retardation time of the dashpot, respectively; Ek and μk (k = 1 · · · nkv ) are the modulus of elasticity of the spring and viscosity of the dashpot of the k th Kelvin-Voigt element, respectively, nkv designates the number of Kelvin-Voigt elements. The numerical solution of the set of Eqs. (1) using the Method Of Characteristics (MOC) leads to (Triki 2018a, b; Trabelsi and Triki 2019, 2020a, b, c): Qi,t = cp ∓ ca∓ Hi,t along C± : j

j

j

j

j

a xj = ± 0j t cr

(3)

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in which, j: pipe number (np: number of pipes 1 ≤ j ≤ np); t: time-step increment; j j cr : Courant number;ns : number of sections of the jth pipe (1 ≤ i ≤ ns );

Fig. 2. Sketch of the generalized Kelvin-Voigt model.

For an in-series connection of multi-pipes, common flow-rate and pressure-head values are assumed (Wylie and Streeter 1993; Triki 2016; Ben Iffa and Triki 2019; Trabelsi and Triki 2019, 2020a, b, c): j−1

j

j−1

j

Qx=L = Qx=0 and Hx=L = Hx=0

(4)

The next section is devoted to exploring the effectiveness of the dual technique-based in-series control measure within upsurge initiated water-hammer waves framework.

3 Case Study Test case corresponds to a reservoir steel-pipe valve system. The specifications of the steel piping system are: L = 143.7 m; D = 50.6 mm; a0 = 1369.7 m/s; e = 3.35 mm;

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and J0 = 0.0049 GPa−1 . Initially, the steady-state flow regime corresponds to the values of discharge  and pressure-head, at the valve located downstream extremity, equal to: Q0 = 0.58 l s and H0|valve = 45 m, respectively. The transient regime relates to the abrupt switch-off of the downstream valve; jointly with a constant value condition, set for the water level in the upstream reservoir. Such a scenario may be coded as: t0 t0 Q|x=L = 0 and H|x=0 = H0|reservoir

(5)

In this case, the implementation of the dual technique- based in-series design measure is based on the substitution of an upstream and down-stream in-series subsections of the original steel piping system by other ones build of (HDPE) or (LDPE) plastic materials (Fig. 3). According to Keramat and Haghighi (2014), the coefficients

 HDPE of the linear-viscoelastic model of these materials are: Jk GPa−1 ; τk [s] k=0−5 =

 {0.8032; −/1.057; 0.05/1.054; 0.5/0.905; 1.5/0.262; 5/0.746; 10}; or Jk GPa−1 ; τk [s]}LDPE k=0−3 = {2.083; −/7.54; 0.00089/10.46; 0.022/12.37; 1.864}, respectively. Firstly, the length and diameter values of employed sub-short-sections are equal to: dual dual lsub short−section = 2.5 m and dsub short−section = 50.6 mm, respectively.

Fig. 3. Definition sketch of the controlled system using the in-series design measure based upon the dual-technique.

Incidentally, the results associated with the conventional technique-based in-series design measure are also addressed, in the following, in order to compare the capacities of the different control techniques. Consequently, for the consistency of comparison, the same plastic material volume is utilized in each technique of implementation of the in-series design measure. In other words, the diameter and length values of the shortsection, utilized in the in-series control measure based upon the conventional technique, conventional = 5 m and d conventional = 50.6 mm, respectively. are equal to: lshort−section short−section Figure 4 compares the downstream pressure-head associated with the original hydraulic system case, and its counterpart estimated into the controlled system cases based upon the dual or conventional technique of the in-series design measure. Jointly, Table 1 summarizes the main features of pressure-wave traces shown in Fig. 4.

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The general trend of the wave curves plotted in Fig. 4 illustrates magnitude attenuation and period expansion of pressure-wave oscillations. In order to address comprehensively the relationship between the two foregoing effects, the magnitude values of positive- and negative surge according to the period value of pressure-wave oscillation are traced in Fig. 5 for the hydraulic systems with and without using the control techniques. Figures 4 and 5 infer that the dual technique configurations involving an (HDPEsteel-LDPE) or (LDPE-steel-LDPE) in-series connections, illustrate remarkable attenuation of pressure-surge magnitude, as compared with the original system case. In particular, the values of positive-surge magnitudes, involved in the foregoing controlled HDPE−steel−LDPE LDPE−steel−LDPE = 30.4 m or Hup−surge = 30.1 m, system cases, are: Hup−surge respectively; while the corresponding value estimated in the original system case is equal steel−pipe to: Hup−surge = 40.6 m. H, m 80

60

40

20

0 0

1 2 non-protected system Steel-HDPE LDPE-Steel-LDPE

3

4 5 6 t, s blank Steel-LDPE HDPE-Steel-LDPE

Fig. 4. Pressure-wave time histories in the original and control hydraulic system cases.

Table 1. Main features of pressure-wave signals in Fig. 2. Parameters:

Original system

(Sub) short-section configurations of the controlled systems HDPE

LDPE

LDPE-LDPE

HDPE-LDPE

H max

(m)

85.6

79.0

66.6

75.1

75.4

H min

(m)

5.4

15.5

26.1

29.5

20.6

T1

(s)

0.42

0.756

1.18

0.918

0.876

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Besides, Figs. 4 and 5 display significant expansion of pressure-wave oscillation period, in the cases involved in the dual- or conventional- technique–based in-series design measure. In particular, the phase-shifts depicted between the pressure-wave signals involved in an (HDPE-steel-LDPE) or (LDPE-steel-LDPE) configuration of the dual-technique and their counterparts involved into original system case are equal to: T1HDPE−steel−LDPE − T1steel = 0.498 s or T1LDPE−steel−LDPE − T1steel = 0.456 s, respectively. Similarly, but less important phase-shifts are induced by the aforementioned configurations of the dual-technique as compared with the (steel-HDPE) configuration of the conventional technique-based in-series design measure: T1HDPE−steel−LDPE − T1steel−HDPE = 0.12 s or T1LDPE−steel−LDPE − T1steel−HDPE = 0.162 s. Contrarily, the (HDPE-steel-LDPE) or (LDPE-steel-LDPE) configurations of the dual-technique induce more important spreading of wave oscillation period as compared with the conventional one based on (steel-LDPE) configuration: T1HDPE−steel−LDPE − T1steel−LDPE = 0.304s or T1LDPE−steel−LDPE − T1steel−LDPE = 0.262 s, respectively.

Fig. 5. Variations of up- and down-surge amplitudes as a function of the period, for the first cycle of pressure-wave oscillation.

Basing on the preceding discussion, the (LDPE-steel-LDPE) configuration of the dual technique may be considered as the best configuration; providing a satisfactory compromise between surge attenuation and period spreading of pressure-wave oscillations. The variations of the first pressure-head peak depending on the diameter and length of utilized in-series LDPE sub-short-sections are illustrated in Figs. 6- a and-b, respectively. Basing on these curves slopes, it obvious that the first pressure-head values are significantly attenuated when increasing the length of the downstream in-series sub-shortdownstream section for the range values within: 1 ≤ lsub short−section ≤ 2.5 m; however, no remarkable effects are depicted for the variation of the upstream in-series sub-short-section length and diameter values. In addition, this figure suggests that the first pressure-head values are significantly attenuated when increasing the diameter value of the downstream inseries sub-short-section from 25 mm to the original value (i.e.: 50.6 mm); whereas, no

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Fig. 6. Dependencies of the first peak and period values, for the 1st cycle of pressurewave oscillations on the utilized up- and down-stream sub-short-sections: (a)- diameter (for lsub−short−section = 2.5 m), (b)- length (for dsub−short−section = 50.6 mm).

remarkable attenuations are depicted for the variation of the length and diameter values of the upstream in-series sub-short-section. Similar conclusions may be addressed for the period value of pressure-wave oscillations.

4 Conclusion Overall, this investigation evidenced that the dual technique-based in-series design measure could improve the conventional technique-based one; with regards to the attenuation of surge-magnitude and limitation of period-expansion criteria of pressure-wave oscillations. On this point, the specific configuration of the dual-technique based upon an upstream and downstream in-series LDPE sub-short-section presented a satisfactory compromise between the aforementioned two criteria. Additionally, the examination of the sensitivity of the first pressure-wave peak value according to the dimension of the utilized in-series plastic sub-short-sections, suggests that the pressure surge attenuation increases as the downstream in-series sub-short-section volume increases. Nonetheless, this relationship is not remarkable beyond the primitive diameter and length values of the utilized sub-short-section.

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References Aklonis, J.J., MacKnight, W.J., Shen, M.: Introduction to Polymer Viscoelasticity. WileyInterscience John Wiley & Sons Inc., NewYork (2006) Ben Iffa R., Triki, A.: Assessment of inline techniques-based Water-Hammer control strategy in water supply systems. J. Water Supply Res. Technol. – AQUA 68(7), 562–572 (2019). https:// doi.org/10.2166/aqua.2019.095 Brinson, H.F., Brinson, L.C.: Polymer Engineering Science and Viscoelasticity: an Introduction. Springer, New York (2008) Chaker, M.A., Triki, A.: Exploring the performance of the inline technique -based Water-Hammer design strategy in pressurized steel pipe flows, In: Aifaoui, N., Affi, Z., Abbes, M.S., Walha, L., Haddar, M., Romdhane, L., Benamara, A., Chouchane, M., Chaari, F. (eds.) Design and Modeling of Mechanical Systems - IV CMSM 2020 Lecture Notes in Mechanical Engineering, pp. 83–91. Springer, Cham (2020a). https://doi.org/10.1007/978-3-030-27146-6_10 Chaker, M.A., Triki, A.: Investigating the removal of hydraulic cavitation from pressurized steel piping systems, In: Aifaoui, N., Affi, Z., Abbes, M.S., Walha, L., Haddar, M., Romdhane, L., Benamara, A., Chouchane, M., Chaari, F. (eds.) Design and Modeling of Mechanical Systems - IV CMSM 2020 Lecture Notes in Mechanical Engineering, pp. 92–101. Springer, Cham (2020b). https://doi.org/10.1007/978-3-030-27146-6_11 Chaker, M.A., Triki, A.: The branching redesign technique used for upgrading steel-pipes-based hydraulic systems: Re-examined. J. Pressure Vessel Technol. Trans. ASME (2020c) Chaker, M.A., Triki, A.: Investigating the branching redesign strategy for surge control in pressurized steel piping systems. Int. J. Press. Vessels and Pip. 180, 104044 (2020d). https://doi. org/10.1016/j.ijpvp.2020.104044 Chen, T., Ren, Z., Xu, C., Loxton, R.: Optimal boundary control for water hammer suppression in fluid transmission pipelines. Comput. Math Appl. 69(4), 275–290 (2015). https://doi.org/10. 1016/j.camwa.2014.11.008 Fersi M., Triki A.: Investigation on re-designing strategies for water-hammer control in pressurized-piping systems. J. Press. Vessel Technol. Trans. ASME 141(2), 021301 (2019a). https://doi.org/10.1115/1.4040136 Fersi, M., Triki, A.: Alternative design strategy for water-hammer control in pressurized-pipe flow. In: Fakhfakh, T., Karra, C., Bouaziz, S., Chaari, F., Haddar, M. (eds.) Advances in Acoustics and Vibration II. ICAV 2018 Applied Condition Monitoring, vol. 13, pp. 157–165. Springer, Cham (2019b). https://doi.org/10.1007/978-3-319-94616-0_16 Fersi, M., Triki, A.: Investigating the inline design measure in existing pressurized steel piping systems. In: Aifaoui, N., et al. (eds.) CMSM 2019. LNME, pp. 74–82. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-27146-6_9 Keramat, A., Haghighi, A.: Straightforward transient-based approach for the creep function determination in viscoelastic pipes. J. Hydraul. Eng. 140(12) (2014). https://doi.org/10.1061/(ASC E)HY.1943-7900.0000929 Trabelsi, M., Triki, A.: Dual control technique for mitigating Water-Hammer phenomenon in pressurized steel-piping systems. Int. J. Press. Vessels Pip. 172, 397–413 (2019). https://doi. org/10.1016/j.ijpvp.2019.04.011 Trabelsi, M., Triki, A.: Water-Hammer control in pressurized pipe flow using dual (LDPE/LDPE) inline plastic sub-short-sections, In: Aifaoui, N., Affi, Z., Abbes, M.S., Walha, L., Haddar, M., Romdhane, L., Benamara, A., Chouchane, M., Chaari, F. (eds.) Design and Modeling of Mechanical Systems - IV CMSM 2020 Lecture Notes in Mechanical Engineering. Springer, Cham (2020a). https://doi.org/10.1007/978-3-030-27146-6_102

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Trabelsi, M., Triki, A.: Assessing the inline and branching techniques in mitigating water-hammer surge waves, In: Aifaoui, N., Affi, Z., Abbes, M.S., Walha, L., Haddar, M., Romdhane, L., Benamara, A., Chouchane, M., Chaari, F. (eds.) Design and Modeling of Mechanical Systems - IV CMSM 2020 Lecture Notes in Mechanical Engineering, pp. 157–165. Springer, Cham (2020b). https://doi.org/10.1007/978-3-030-27146-6_17 Trabelsi, M., Triki, A.: Exploring the performances of the dual technique -based Water-Hammer redesign strategy in water-supply systems. J. Water Supply Res. Technol. - AQUA 69(1) (2020c). https://doi.org/10.2166/aqua.2019.010 Triki, A.: Water-hammer control in pressurized-pipe flow using an in-line polymeric short-section. Acta Mech. 227(3), 777–793 (2016). https://doi.org/10.1007/s00707-015-1493-13 Triki, A.: Water-Hammer control in pressurized-pipe flow using a branched polymeric penstock’. J. Pip. Syst. – Eng. Pract. 8(4), 04017024 (2017). https://doi.org/10.1061/(ASCE)PS.19491204.0000277 Triki, A.: Dual-technique based inline design strategy for Water-Hammer control in pressurizedpipe flow. Acta Mech. 229(5), 2019–2039 (2018a). https://doi.org/10.1007/s00707-017-2085-z Triki, A.: Further investigation on water-hammer control inline strategy in water-supply systems. J. Water Supply Res. Technol. – AQUA 67(1), 30–43 (2018b). https://doi.org/10.2166/aqua. 2017.073 Triki, A., Fersi, M.: Further investigation on the Water-Hammer control branching strategy in pressurized steel-piping systems. Int. J. Press. Vessels Pip. 165, 135–144 (2018). https://doi. org/10.1016/j.ijpvp.2018.06.002 Triki, A., Chaker, M.A.: Compound technique -based inline design strategy for water-hammer control in steel pressurized-piping systems. Int. J. Press. Vessels Pip. 169, 188–203 (2019). https://doi.org/10.1016/j.ijpvp.2018.12.001 Vitkovsky, J.P., Lambert, M.F., Simpson, A.R., Bergant, A.: Advances in unsteady friction modelling in transient pipe flow. In: 8th International Conference on Pressure Surges, Amsterdam (2000) Wylie, E.B., Streeter, V.L.: Fluid Transients in Systems. Prentice-Hall, Hoboken (1993)

Large Strokes of a Piezocomposite Energy Harvester with Interdigitated Electrodes Accounting for Geometric and Material Nonlinearities Ahmed Jemai(B) , Sourour Baroudi, and Fehmi Najar Applied Mechanics and Systems Research Laboratory, Tunisia Polytechnic School, University of Carthage, BP 743 La Marsa 2078, Tunis, Tunisia [email protected]

Abstract. Piezocomposite materials are characterized by high flexibility and large displacements when used for energy harvesting applications. However, for beam-based designs, geometric and material nonlinearities are often neglected because of their complexity. Within this context an analytically model of the system is proposed including the geometric nonlinearities and relatively large electric fields. The proposed device is composed of unimorph cantilever beam used as energy harvester. A metallic layer is used as substrate for an AFC piezocomposite patch. The beam is excited transversely using a harmonic displacement at the clamped side. The Hamilton principle is applied to obtain the mathematical model of the proposed device. Galerkin procedure is used for space discretization and limit-cycle solutions are calculated using a continuation technique based on the finite difference approach. The simulated frequency-response of the harvested voltage across a purely resistive load shows a softening behavior. A hardening behavior is also observed if the material nonlinearity are taken into account. The results are compared and validated with a finite element model developed using ANSYS and accounting for large displacements. The influence of the number of electrode and the dimensions of the device are also analyzed and compared with previously published linear simulations. Keywords: Piezocomposite · Nonlinear vibration · Interdigitated electrodes · Piezoelectricity · Vibration energy harvesting

1

Introduction

Recently, a large interest has been given by researchers to the field of energy harvesting based on the transduction of the ambient motions into usable electrical energy. Thanks to its efficiency, transduction methods based on piezoelectric material are the most effective. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 105–116, 2021. https://doi.org/10.1007/978-3-030-76517-0_13

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Generally piezoelectric energy harvesters use monolithic ceramic piezoelectric material. For beam-based devices, the d31 mode is preferred due to fabrication restrictions. They are characterized by their brittle nature and delicate integration with curved and complex shapes (Ayed et al. 2009). On the other hand, piezocomposite fiber-based materials are characterized by their high flexibility and large displacements capabilities. The Active-Fiber Composite (AFC) (Bent et al. 1995) and the Macro-Fiber Composite (MFC) (Williams et al. 2004) are examples of such materials that in addition use Interdigitated Electrodes (IDE) (Wang et al. 1999) with the possibility to activate the d33 mode. The complex internal structure of the piezocomposite easily initiate the nonlinear behavior of the material even if only few research work tackled this aspect. For example, Mahmoodi and Jalili (2007) used an nonlinear model of a flexural vibrating non-homogenous piezoelectrically actuated microcantilever beam using monolithic piezoelectric material. Daqaq et al. (2009) did the same by considering a lumped parameter non-linear model that describes the first-mode dynamics of a parametrically excited cantilever-type harvester. Stanton et al. (2010) provided the ground work for the modeling of inherent piezoelectric material nonlinearities in energy harvesters. The proposed model dealt with uniform shapes. However, geometric nonlinearity were not considered in their model. Later, Abdelkefi et al. (2012) used a nonlinear distributed parameter model taking into account parametric excitation for an unimorph cantilever beam and accounted for nonlinear geometry in addition to material nonlinearities. Recently, Leadenham and Erturk (2015) proposed a unified nonlinear nonconservative constitutive relation with two-way coupling for actuators, sensors and energy harvesters. The model was validated with experimental data. The results show that softening behavior is dominant for the PZT-5A piezoelectric material. The purpose of this work is to derive a new model for a piezocomposite AFC energy harvester by taken into account the presence of IDE and accounting for geometric, electric field and material nonlinearities. In a previous work (Jemai et al. 2014b), the authors proposed a simpler model for cantilever beam energy harvested with an AFC patch where only geometric nonlinearities have been taken into account. In this work, the electrical voltage between two successive interdigitated electrodes is considered highly nonlinear due to the presence of both material and geometric nonlinearities. In other hand, the proposed model takes into account large displacements, as well as material nonlinearity for both substrate and piezoelectric layers.

2

Governing Equations

The proposed energy harvester, shown in Fig. 1, is composed of an aluminum cantilever beam partially covered by an AFC piezoelectric patch with interdigitated electrodes. A resistance load is linked to the electrodes where the harvested voltage will be measured.

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Fig. 1. Schematic of the AFC unimorph energy harvester bender using interdigitated electrodes.

2.1

Constitutive Relations

The set of equations of motions describing the behavior of the proposed AFC energy harvester, composed of PZT fiber and Epoxy matrix for the AFC (Fig. 2(a)) and an aluminum substrate, are obtained by using the Hamilton’s principle and the nonlinear enthalpy density associated to the proposed material. Assuming that the longitudinal strain is the only nonzero component, denoted by ε and the only nonzero electric field component is also along the same direction, it is denoted here by E. Therefore, the enthalpy density is reduced to H=

1 P 2 1 2 1 1 C ε − E − e1 Eε + C2P ε3 − e2 Eε2 2 1 2 6 2

Fig. 2. Comparison between the nonlinear FE model and the developed model.

(1)

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where e1 = 9.80 C/m2 and e2 are the piezoelectric constants, C1P = 29.75 GPa and C2P are the elastic stiffness and  = 4.7 × 10−9 F/m is the electric permittivity. All these parameters are effective quantities obtained from the homogenization of the AFC shown in Fig. 2(b). Letting the axial stress be defined by ∂H σP = ∂H ∂ε and the electrical displacement by D = − ∂E , we obtain the following constitutive relations σP = C1P ε − e1 E + 12 C2P ε2 − e2 Eε D = E + e1 ε + 12 e2 ε2 σA =

C1A ε

+

(2)

1 A 2 2 C2 ε

The scripts P and A denotes quantities associated to piezoelectric and aluminum layers. 2.2

Electric Field Assumptions Inside the Piezoelectric Layer

Since, most of the proposed models in the literature are derived by assuming that the voltage varies linearly within the piezoelectric material. These models do not locally satisfy Maxwell’s equations. Therefore, nonlinear function have been proposed to overcome this inconsistency (Krommer 2001, Zhou et al. 2007, Ke et al. 2012). The piezocomposite is replaced by an equivalent homogenized material, shown in Fig. 2(b) actuated by N transverse IDEs (Jemai et al. 2014a). Under the above assumptions, the longitudinal electric field component E can be approximated by the following expression: E(x, y) = E ∗ (x, y) ξ(x) [H(y − y1 ) − H(y − y2 )]

id ≤ x ≤ (i + 1)d

i ∈ {0, N − 2}

(3)

where H(x) is the Heaviside function, N is the total number of electrode, d is the gap separating two constitutive electrodes, yi is the layer position, the electric field between two consecutive IDEs E ∗ (x, y), which takes into account the self field component, is expressed as     (i+1)d  (i+1)d 2  E ∗ (x, y) = − V d(t) − e1 ε − d1 id εdx − 12 e2 ε2 − d1 id ε dx N −1   (4) 2  − 2H (x − (2i − 1) d) + H (x − 2id) + H (x − (2i − 2) d) ξ(x) = i=1

where V (t) is the electrical potential at the upper electrodes. 2.3

Nonlinear Equation of Motion

We assume that the beam can undergo relatively large deflection v(x, t) and longitudinal deformation u(x, t). As a result, the nonlinear axial strain ε can be expressed as Nayfeh and Pai (2008) 1 1 ε = e − yκ = u + v 2 − y(v  + v  v 2 ) 2 2

(5)

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where e = u + 12 v 2 is the extension and κ = v  + 12 v  v 2 is the nonlinear curvature. The fixed side of the beam is subjected to an harmonic displacement in the y-direction denoted by Y (t) = Y0 sin(Ωt), where Ω is the driving frequency and Y0 is the vibration amplitude. The geometric and physical properties of the harvester are given in Table 1. Table 1. Material and geometrical properties of the cantilever beam L (mm) L1 (mm) b (mm) hP (mm) hA (mm) ρP (kg/m3 ) ρA (kg/m3 ) C1A (GPa) νA 100

60

20

0.33

1

4475

2700

68

QA QP

0.35 500 50

Next, we use the Hamilton’s Principle to derive the equations of motion. To this end, we calculate the variation of the kinetic energy T and potential energy Π. They are given by 

L

δT = −

m(x) (¨ u δu + v¨ δv) dx

(6)

0

where m(x) = m1 H1 (x)+m2 H2 (x) and m1 = ρA bhA +ρP bhP and m2 = ρA bhA are the mass per unit length of two parts of the beam located between 0 and L1 and L1 and L, respectively. Also, H1 (x) = H(x) − H(x − L1 ) and H2 (x) = H(x − L1 ) − H(x − L). The variation of the total potential energy of the harvester ∂Π is expressed, in regions 1 and 2, as 

L1

δΠ = 0



 A

σA δεdAdx +

 δΠ1

0

L1

 A



L

σP δεdAdx + L1

 A

σA δεdAdx



(7)

δΠ2

The inextensibility condition is generally applied at this step by considering that the extension is e = 0, that is u = − 12 v 2 . Therefore, substituting Eqs. (2) and (5) into Eq. (7), integrating the outcome by parts, we end up with variation of Π1 expressed as     L1 1  2    δv dx N1 δu + N1 v + M1 + M1 v δΠ1 = − 2 0

    L1 1 1  2 2    (8) + N1 δu − M1 + M1 v δv + N1 v + M1 + M1 v δv 2 2 0

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  where M1 = A yσdA and N1 = A σdA are, respectively, the bending moment and axial force in region 1 of the beam, with    (i+1)d V (t) 1 M1 = − K1 κ + K2 κ2 + χ1 (x)V (t) − χ2 (x) κ − k1 (x) κ − κdx d d id  + k2 (x) κ2 − κ

1 d



(i+1)d

 κdx

id



1 1 − k3 (x) κ3 − κ 2 d





(i+1)d

(9)

κ2 dx id

where the different coefficients are given in the Appendix. The variation of the second partial potential energy Π2 is given by     L 1  2    δv dx N2 δu + N2 v + M2 + M2 v δΠ2 = − 2 L1

    L 1 1  2 2    (10) + N2 δu − M2 + M2 v δv + N2 v + M2 + M2 v δv 2 2 L1

where N2 and M2 are the axial force and bending moment for the second part of the beam, where     b b M2 = − C1A y1 3 − y0 3 κ + C2A y1 4 − y0 4 κ2 3 8

(11)

Applying Hamilton’s principle, we end up with the equation of motion describing the axial displacement, it is given by 

m¨ u + (N1 H1 + N2 H2 ) = 0 Next, we calculate the axial forces as follows:  x  x     v v¨ + v˙ 2 dxdx In region 1: N1 = m1 L 0  x1  x     v v¨ + v˙ 2 dxdx m2 In region 2: N2 = L

(12)

(13) (14)

0

Finally we calculate the nonlinear equations of motion of the transverse displacement and the associated boundary conditions, given by     2  1  2    m¨ v+ Mi v (15) (Ni v ) + Mi + Hi + mY¨ (t) = 0 2 i=1 v(0) = 0

v  (0) = 0 v  (L) = 0 v  (L) = 0

(16)

We also use the Gauss law to couple the displacement to the electrical potential, that is  d V (t) (17) D dA = dt A R

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Substituting Eqs. (2) and (3) into (17) keeping only terms up to the third order, we obtain the following relation:  −2  (i+1)d  3 3 N  e b y − y V (t) 2 2 1 = v  v˙  dx (18) CP V˙ (t) + R 3d id i=0  N −2     e1 b y1 2 − y2 2  (i+1)d  1   2    + v˙ + v˙ v + v v v˙ dx 2d 2 i=0 id where CP is the internal equivalent capacitance expressed as CP = (N − 1) b d hp .

3

Reduced-Order Model and Limit-Cycle Solutions

To discretize the derived governing equations given by Eqs. (15) and (18), we use the Galerkin procedure with an assumed mode approach. The obtained set of nonlinear ODEs are solved for limit-cycle solutions using a sequential continuation technique based on the finite difference method (Nayfeh and Balachandran 2008). 3.1

Discretization of the Equation of Motion

We start by assuming that the transverse displacement v(x, t) can be expressed as follows: v(x, t) =

∞  

 (2) φ(1) (x)H + φ (x)H 1 2 ηr (t), r r

i ∈ {1, 2}

(19)

r=1 (i)

where φr (x) is the rth normalized mode shape for region i of the free undamped linearized short-circuit (V = 0) eigenvalue problem associated to Eq. (15) and ηr (t) is the associated modal displacement. The details about the calculation of the mode shapes for the variable properties are presented in Jemai et al. (2014a). Therefore, the mode shapes are given by (i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

φr (x) = Ar sin(βr x) + Br cos(βr x) + Cr sinh(βr x) + Dr cosh(βr x)

i ∈ {1, 2}

(20)

4 4   (1) (2) m2 i i i i 1 = ωr2 m and β = ωr2 C where βr r A and Ar , Br , Cr and Dr are conK1 1

stants determined by the use of the boundary conditions in Equations (16). Substituting Eqs. (19) and (20) into the Eq. (15), multiplying the outcome by an arbitrary mode shape, integrating over the beam’s length, applying the orthogonality conditions and adding modal damping coefficients cr = 2ξr ωr Mr , the set of nonlinear ODEs describing the modal response of the beam are given by

112 (1)

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¨ (t) + cr η˙ r (t) + Λ(1) ηr (t) + Λ(2) ηr 2 (t) + Λ(3) ηr 3 (t) + Δr Y r r r

(1)

Vr (t) + Θr

η ¨r (t) + Γr

+ Θr

(2)

(3)

ηr (t)Vr (t) + Θr

2

(4)

ηr (t)Vr (t) + Θr

3



2

2



ηr (t)¨ ηr (t) + ηr (t)η˙ r (t)

ηr (t)Vr (t) = 0

(21)

where different modal time-independent coefficients are given in the Appendix. 3.2

Projection of the Electrical Equation

The reduced-order model given Eq. (21) is completed by the electrical equation in (18). Using the Galerkin decomposition, we rewrite Eq. (18) as follows: Vr (t) = ir (t) CP V˙ r (t) + R

(22)

where the different coefficients are given in the Appendix.

4 4.1

Results and Finite Element Validation Finite Element Model and Validation

The FE Method in ANSYS is used to validate the reduced-order model. A nonlinear FE model (18001 elements and 20853 nodes) is first proposed, it is based on the transient analysis and takes into account the geometric nonlinearity. It is used to calculate the generated voltage and the electrical potential. Second, a FE model (22800 elements and 26313 nodes) is proposed to extract the linear mode shapes and natural frequencies. Using the linear FE model we calculate the natural frequencies of the device and compare them the one obtained analytically for Open-Circuit (OC) and

Fig. 3. Comparison between the nonlinear FE model and the developed model (N = 17, Y0 = 1 mm and R = 106 Ω).

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Short Circuit (SC) conditions. The results, give 0.98% for SC and 1.15% for OC for the analytical first frequency (f1SC = 99.96 Hz and f1OC = 110.89 Hz). We compare in Figs. 3 the developed solution with the one obtained by FE. Figure 3(a) compares the transient responses of the energy harvester in terms of generated voltage. In Fig. 3(b), the frequency-response curve of the device is computed for both models. A good agreement is obtained between the results for the proposed model and FE simulations. Also, in Fig. 3(a) the simulation time for the FE method is three times higher than the proposed approach when the simulations are run in the same computer.

Fig. 4. Variation of the electrical potential for different configurations of nonlinear piezoelectric coefficients when i = 0, R = 106 Ω, Y0 = 0.1 mm, N = 7 and Ω is fixed at the short-circuit frequency.

4.2

Analysis of the Voltage Distribution

Using Eq. (4), the electric potential distribution between two consecutive electrodes is given by the following relation:    x V (t) (x − id) e1 y x − id (i+1)d − ϕi (x, y) = κ dx − κ dx (23) d  d id id    x e2 y 2 x − id (i+1)d 2 2 + κ dx − κ dx i ∈ {0, N − 2} 2 d id id Figure 4 represents the variation of electrical potential ϕ along the x and y-axis for different values of the nonlinear piezoelectric coefficient. From these figures, we can note that the electrical potential along the x-axis between two consecutive electrodes is highly non-linear. However, ϕ depicts a quasilinear variation along the y-axis especially for small values of e2 . The effect of material nonlinearity becomes clearer, when the coefficient e2 is not zero.

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Performance Analysis of the AFC Energy Harvester

In this section the performance of the AFC based energy harvester is tested for different configurations of the system’s parameters. In particular, we analyze the influence of the number of electrodes for the IDE, the length and thickness ration and the Fiber Volume Fraction (FVF). The performance of the energy harvester is tested by estimating the generated voltage. For each case the frequencyresponse and force response curves will be plotted in order to clearly analyze the effect of the nonlinearities on the performances. Limit-cycle solution are obtained by using the Finite Difference Method (FDM) (Najar et al. 2010). The discretized equations are solved using a NewtonRaphson technique. The frequency response curves are obtained by using a sequential continuation technique. Figure 5(a) shows the frequency-response curve representing the maximum value of the harvested voltage at steady state. One can observe that increasing the number of electrodes can increase the harvested voltage. However, for certain condition, and due to the presence of the dominating softening behavior, the order can be inversed. As the resistance load is increased to R = 108 Ω, an opposite effect can be shown in the Fig. 5(b), i.e. increasing N lowers the harvested voltage.

Fig. 5. Frequency-response curves of the maximum voltage output with the excitation frequency and excitation amplitude for different numbers of electrode, when C2A = −6 × 1014 Pa, C2P = −9 × 1014 Pa, e2 = 0 and Y0 = 0.1 mm and Ω/2Π = 90 Hz.

6

Conclusion

A nonlinear model for AFC energy harvester is derived including its interdigitated electrode configuration. The model is based on Euler-Bernoulli beam theory applied to a unimorph cantilever beam at which an AFC patch is partially covering the substrate layer made of aluminum. A reduced-order model

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is obtained using the Galerkin procedure associated to an assumed mode approach. The transverse excitation of the cantilever beam induces limit-cycle solutions computed using the presented model and validated with the finite element method using ANSYS. The nonlinearities of the model are produced by taken into account relatively large displacements and a quadratic electrical potential distribution between two constitutive interdigitated electrodes. The simulated results are first validated by the FEM findings. Therefore, a performance analysis of the energy harvester is presented including the effect of the number of electrodes, dimensions and fiber-volume fraction. The results also show that the nonlinear approach is essential to correctly understand the behavior of the device with AFC patch. In fact, when the proposed model is compared with previously published linear model, the discrepancies found fully justify the necessity of the proposed nonlinear approach for AFC based devices not only for energy harvesting applications but also for actuators and sensors.

Appendix     b CA y 4 − y 4 + b CP y 4 − y 4 K2 = 8 0 2 1 2  1 8 2  be be χ1 (x) = ξ(x) 2d1 (y2 2 − y1 2 ), χ2 (x) = ξ(x) 32 y2 3 − y1 3     be2 be2 e1 be2 2 3 3 1 y2 4 − y1 4 , k3 (x) = ξ(x) 5 y2 5 − y1 5 k1 (x) = ξ(x) 3 (y2 − y1 ), k2 (x) = ξ(x) 4     L L (1) (1) (2) (2) (1) (2) L L Γr = m1 0 1 φr (x)2 dx + m2 L φr (x)2 dx, Γr = m1 0 1 φr (x)dx + m2 L φr (x)dx   1 1 A   L1  bC1 (1) (1)  (1) (2)  (2) 3 3 L Λr = 0 (K1 + k1 (x)) φr y1 − y0 (x) φr (x)dx + 3 (x) φr (x)dx L1 φr  N −2  L (i+1)d (1)  (1)  φ (x)dx k (x)φ (x)dx − d(N1−1) 0 1 r r 1 id b CP K1 = 3 1



y2 3 − y1 3



  b CA y 3 − y 3 , + 3 1 0 1

i=0

       L  2k2 (x) (2)  (2) (1)  (1) L y1 4 − y0 4 K2 + φr (x)2 φr (x)dx − 0 1 (x)2 φr (x)dx L1 φr 3   N −2   L1 k2 (x)φ(1)  (x) (i+1)d (1) (1) r φr (x)dx φr (x)dx + 0 id d(N −1) i=0 

N −2 L   (i+1)d (1)  k2 (x) (1) φr (x)2 dx φr (x)dx + 0 1 2d(N id −1) i=0    (1) L x x (1)  (1)  m1 φr (x)2 dxdx φr (x)dx Δr = 0 1 φr (x) L 1 0   L1   L1  (1) (1) (2) (1)  (1) 1 χ2 (x)φr Θr = − 0 χ1 (x)φr (x)dx, Θr = d 0 (x) φr (x)dx     L1 (3) (1) (1) 2  (x) χ (x) φ (x)dx φ Θr = − 1 r r 1 2 0 (2)

Λr

=−

A bC2 8



 (1)  φr (x) (1)  (1)  (1)  χ2 (x)φr (x)φr (x) φr (x) dx d      A   bC1 (3) (2)   L 2 (x) φ2 φ2 y1 3 − y0 3 Λr = 3 r (x)φr (x)  r (x)dx L1 φr     L (1)  (1)  (1)  (1) + 0 1 φr (x) (K1 + k1 (x)) φr (x)φr (x) φr (x)dx  (1)  (x) N −2  L φr (x)k1 (i+1)d (1)  (1)  φr (x)φr (x)2 dx dx − 0 1 id 2d(N −1)  i=0 N −2 L   (i+1)d (1)  (1) φr (x)dx φr (x)dx − 0 1 id (4)

Θr

L = 0 1

i=0

(1)



(1) 



 (x)φ (x)2  φr (x) k1 r L  (1)  + 0 1 1 k3 (x)φr (x)3 dx 2 2d(N −1) 

 N −2   L1 k3 (x) (i+1)d (1)  (1)  (1) 2 φr (x) dx φr (x) φr (x)dx − 0 id 2d(N −1) i=0    (2) L x x (2)  (2)  + L φr (x) L (x)2 dxdx φr (x)dx 0 m2 φr 1  ∞  (0)  (1) (2) 2 κr + κr ηr (t) + κr ηr (t) η˙ r (t) ir (t) = r=1

= 2d1 (y2 2 − y1 2 )

(2)

=

κr

be

3be1 4d

(y2 2 −

N −2 

be (i+1)d (1)  (1) φr (x)dx, κr = 3d2 id i=0 N −2  (i+1)d (1)  (1) y1 2 ) φr (x)φr  (x)2 dx id i=0

(0)

κr

  N −2  (i+1)d 1  y2 3 − y1 3 φr (x)2 dx id i=0

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References Abdelkefi, A., Nayfeh, A., Hajj, M.: Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters. Nonlinear Dyn. 67(2), 1147– 1160 (2012) Ayed, S.B., Najar, F., Abdelkefi, A.: Shape improvement for piezoelectric energy harvesting applications. In: 2009 3rd International Conference on Signals, Circuits and Systems (SCS), pp. 1–6 (2009) Bent, A.A., Hagood, N.W., Rodgers, J.P.: Anisotropic actuation with piezoelectric fiber composites. J. Intell. Mater. Syst. Struct. 6(3), 338–349 (1995) Daqaq, M.F., Stabler, C., Qaroush, Y., Seuaciuc-Os´ orio, T.: Investigation of power harvesting via parametric excitations. J. Intell. Mater. Syst. Struct. 20(5), 545–557 (2009) Jemai, A., Najar, F., Chafra, M., Ounaies, Z.: Mathematical modeling of an activefiber composite energy harvester with interdigitated electrodes. Shock and Vibration (2014a) Jemai, A., Najar, F., Chafra, M., Ounaies, Z.: Modeling and nonlinear dynamics of an active-fiber composite energy harvester with interdigitated electrodes. In: 2014 International Conference on Composite Materials Renewable Energy Applications (ICCMREA), pp. 1–6 (2014b) Ke, L.L., Wang, Y.S., Wang, Z.D.: Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Compos. Struct. 94(6), 2038–2047 (2012) Krommer, M.: On the correction of the bernoulli-euler beam theory for smart piezoelectric beams. Smart Mater. Struct. 10(4), 668 (2001) Leadenham, S., Erturk, A.: Unified nonlinear electroelastic dynamics of a bimorph piezoelectric cantilever for energy harvesting, sensing, and actuation. Nonlinear Dyn. 79(3), 1727–1743 (2015) Mahmoodi, S.N., Jalili, N.: Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers. Int. J. Non-Linear Mech. 42(4), 577–587 (2007) Najar, F., Nayfeh, A., Abdel-Rahman, E., Choura, S., El-Borgi, S.: Nonlinear analysis of mems electrostatic microactuators: primary and secondary resonances of the first mode. J. Vib. Control 16(9), 1321–1349 (2010) Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. John Wiley & Sons, New York (2008) Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. John Wiley & Sons, New York (2008) Stanton, S.C., Erturk, A., Mann, B.P., Inman, D.J.: Nonlinear piezoelectricity in electroelastic energy harvesters: modeling and experimental identification. J. Appl. Phys. 108(7), 074–903 (2010) Wang, Q.M., Du, X.H., Xu, B., Cross, L.E.: Electromechanical coupling and output efficiency of piezoelectric bending actuators. IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 46(3), 638–646 (1999) Williams, R.B., Inman, D.J., Schultz, M.R., Hyer, M.W., Wilkie, W.K.: Nonlinear tensile and shear behavior of macro fiber composite actuators. J. Compos. Mater. 38(10), 855–869 (2004) Yg, Z., Ym, C., Hj, D.: Analytical modeling of sandwich beam for piezoelectric bender elements. Appl. Math. Mech. 28(12), 1581–1586 (2007)

Transient Comprehensive Modelling Due to Pump Failure Badreddine Essaidi1(B) and Ali Triki2,3(B) 1 Research Unit: Mechanics, Modelling, Energy and Materials M2EM, Department of

Mechanical Engineering, National Engineering School of Gabès, University of Gabès, Gabes, Tunisia 2 Research Laboratory: Advanced Materials, Applied Mechanics, Innovative Processes and Environment 2MPE, University of Gabès, Gabes, Tunisia 3 Higher Institute of Applied Sciences and Technology of Gabès, University of Gabès, Gabes, Tunisia [email protected] Abstract. After the pump shutdown in a piping system, the water hammer phenomenon is triggered in the form of propagating and refracting waves travelling, respectively, before and after the pump. Hence, special consideration should be addressed in the design stage of a hydraulic power plant to comprehensively describe the full model and optimize the total flow process. This paper describes part of ongoing research on modeling the transients following a pump failure in order to comprehensively address the flow process caused by this event. In this line, previous studies make use of the water-hammer model; which uses steadystate friction formulation in order to describe frictional effects. Nonetheless, this assumption could not be applied for fast transient scenarios. Alternatively, in this study, the extended one-dimensional water-hammer model, including the Vitkovsky et al. formulation, based upon one- decay coefficient, is discretized using the Method of Characteristics to describe the hydraulic system behavior; without considering the cavitation onset. Additionally, the handling of turbo machine characteristics is treated, and boundary conditions for a single pump station developed. Test case relates to a pumping station delivering water through steel pipeline to a storage reservoir. Results illustrate that the pump failure scenario leads to a severe pressure-wave behavior. Keywords: Water hammer analysis · Pressure · Pump failure · Method of characteristics · Vitkovsky et al. formulation

1 Introduction Transient analysis in pumping systems pertains to numerous scenarios within the design process of these systems. For example, this may be the startup or stoppage of a centrifugal pump, or it may be a control setting of their associated valves. Sudden stoppage of pump is often referred to as pump failure and means the inadvertent stoppage of the pump, without possibility for prior valve adjustments, as when power is interrupted or safety devices of the pump or motor are activated due to excessive heating and vibration. These changes of operating conditions may induce the water hammer phenomenon © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 117–124, 2021. https://doi.org/10.1007/978-3-030-76517-0_14

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in the attached hydraulic system. This phenomenon characterized by serious pressure surges, which need special considerations in the design stage of hydraulic power plants (Adamkowski et al. 2009; Guo et al. 2017, Guo et al. 2019a, b, 2020; Guo and Gao 2020). In fact, there are many theoretical and experimental researches on this event but most of them have been focused on the valve closure which is mostly theoretically important not practically, while studying the pump shut down or start up is very determinate and should be performed to make a safe and economic design meaning optimum selection and placement for the network and its devices for controlling transients. In this line, Wylie et al. (1993), Nourbakhsh et al. (2008), Keramat et al. (2009) and Chaudhry (2014) outlined crucial ideas to model the transients due to turbomachines. Besides, Thorley (2004) reported on safety operation of these systems. These investigations were based on the conventional water hammer model. Yet, this model is not suitable for modelling of fast transient events. Accordingly, we planned in this study to outline a numerical algorithm to predict the flow behavior in a pumping system using the extended water-hammer model including unsteady friction effects. Following this introduction, the water-hammer solver, based on the Method of Characteristics (MOC), is introduced in Sect. 2. The implementation procedure of the pump as boundary condition is also detailed in this section. Subsequently, the numerical solver is applied to a typical pumping system, and obtained results are discussed in Sect. 3. Ultimately, conclusions are summarized in Sect. 4.

2 Theory and Calculations The extended one-dimensional water-hammer model including unsteady friction effects, may be expressed in the following equations set (Wylie et al. 1993; Vitkovsky et al. 2000; Triki 2018):   a2 ∂Q 1 ∂Q ∂H ∂H + 0 = 0 and +g + g hfs + hfu = 0 ∂t gA ∂x A ∂t ∂x

(1)

wherein, H and Q: hydraulic-head and flow-rate; A area of the pipe cross-section; g: gravitational acceleration; a0 : wave-speed; hfs : quasi-steady head-loss component per unit length; hfs is the quasi-steady head loss component per unit length; hfu : unsteady friction losses, which is modelled referring to the Vitkovsky et al. (2000) formulation: hfu = (kv /gA){(∂Q/∂t) + a0 Sgn(Q)|∂Q/∂x|} (kv = 0.03); x and t are the coordinates along the pipe axis and time, respectively. Briefly, the discretization equations of set (1), in time domain, using the (MOC) leads to following compatibility equations (Wylie et al. 1993; Triki 2017, 2018): (2)     i−1 i−1  + c / 1 + c + c + c ; B = Qt−1 + (1/B)Ht−t + cp1 p p1 p2 p2     i−1 i+1     a0 /(gA); cn = Qt−1 + (1/B)Ht−t + cn1 + cn1 / 1 + cn + cn2 ; ca+ = 1 +         / B 1 + c + c  = Rt  i−1  c = Rt Q  ; c cp2 Q i−1,t−1 ; R = f /2DA; p t−1 ; n p2 p2 in which, cp =

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119

     i−1 i−1 i−1 i−1  = k θQ i i  − k Q cp1 − k (1 − θ ) Q − Q sgn Q − Q v v v t−1 t−2 t−1 t−1 ; cn1 = t−1 t−1      i+1 i+1 i+1 i+1 i i  = c = k θ; − kv sgn Qt−1 Qt−1 ; cp2 kv θQt−1 − kv (1 − θ ) Qt−1 − Qt−2 − Qt−1 v n2 (θ = 1 is a relaxation coefficient); and c0 = αγD/2e; t and x: time- and space-steps, respectively. For a piping system supplied by a pump, complementary boundary relations should be provided, in conjunction with the characteristic equations. In the following these particular boundary relations are described briefly following the concepts presented in (Wylie et al. 1993; Thanapandi and Prasad 1995; Tsukamoto et al. 1986; Frelin 2002). Principally, two assumptions are made throughout the transient analysis of pump: (i)-the steady-state characteristics hold for unsteady-state situations and (i)-the validity of the homologous relations. Commonly, the pump characteristics are defined graphically in the terms of four dimensionless-homologous relationships: h=

T Q N H ;β= ;υ= and α = HR TR QR NR

(3)

in which T denotes the shaft torque; N stands for the rotational speed and the subscript R corresponds to the rated condition (i.e. H , T , Q and N evaluated at the point of optimal efficiency). Generally, the characteristic θ = tan−1 (υ/α) as are graphed in terms   2 curves  2 of2  2 or wB (x) = β/ υ + α giving the head or abscissa against wH (x) = h/ υ + α torque (Fig. 2 (Wylie et al. 1993; Wan and Huang 2011)). 4 3 2 1 0 -1 -2 -3 -4

π/2

0

wH wB

Ns = 25 rpm

π Ns = 147 rpm

2π 3π/4 x = π + tan −1 (υ α ) Ns = 261 rpm

Fig. 1. The Complete pump characteristics for different rotational speeds.

Alternatively, wH (x) or wB (x) may also be plotted versus the angle x = π + tan−1 (υ/α). Then using a linear extrapolation, the following relations are respectively

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written to represent the characteristic pump head and torque according to their germane curves.  υ 

 υ 

β h   = A0 + A1 π + tan−1   = B0 + B1 π + tan−1 ; α α υ2 + α2 υ2 + α2 (4) where in A0 , A1 , B0 , B1 are constants which are determined using linear interpolation procedure between two adjoining data points of υ and α. For a pump failure particular scenario, the dynamic behavior of the pump depends on the instantaneous pressure-flow response in the immediate piping system, the moment of momentum equation for the rotating masses, and the pump characteristic curves described previously (Fig. 1). For exemplification of such a scenario, Fig. 2 shows a pump with discharge valve in a pipeline system. The first element equations are the torque-angular deceleration equation describing the speed change in response to the resisting torque, written as a differential equation for the pump, and a head balance equation across the pump. and (Eqs. 2) are applied to pipe-sections 1 and The compatibility equations 4, respectively. Assuming no flow storage between nodes and nodal flows are zero, the instantaneous flow-rate, is the same at the pipe ends and through the pump and valve (i.e.: Q1 = Q2 = Q3 ).

1 flow

2 pump

Qa

3 check-valve

Qb

Qc

4 a

b i −1

i

c i +1

Fig. 2. Pump and its control valve represented as node i.

The head balance applied between nodes a and b (i..e.: Ha + H = Hb ; wherein H is the total dynamic head across the pump) and nodes b and c may be expressed by the following dimensionless-homologous equations (Wylie et al. 1993): H1 + hHR = H2 and H2 −

Q|Q|H0 = H3 τ2 Q02

(5)

wherein H0 designates the head loss across the valve evaluated for Q0 and τ = 1. The change in rotational speed of the pump relies upon the torque applied: β + β0 − I −

NR π × (α0 − α) = 0 TR 30t

(6)

in which, t and I are the time step and the rotational inertia of the pump impeller; the subscript “o” stand for the values computed at the earlier time step. It is interesting to

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point out herein that the above relation is derived from the relation between the torque and angular velocity parameters (T = −I (d ω/dt)). Finally, the unknown parameters υ and α may be obtained from the following nonlinear equations set; which may be solved using the iterative Newton-Raphson numerical procedure:   υ|υ|H0 =0 cp − cn − (ca+ + ca− )Q + HR υ2 + α2 (A0 + A1 x) − τ2   NR π υ2 + α2 (B0 + B1 x) + β0 − I × (α0 − α) = 0 TR 30t υ N Q ;α= ; x = π+ tan−1 υ= QR NR α

(7)

In the solution procedure, for each time step, after arriving at the converged values for υ and α, the coefficients A0 , A1 , B0 , have to be recalculated and this must be iterated until their convergence. The steady operating condition is obtained using the Newton’s method to find the initial operating point that matches the pump and system characteristics.

3 Case Study Test case relates to the hydraulic system sketched in Fig. 3. The pump characteristics are summarized in Table 1 and curve data for NR = 147 rpm plotted in Fig. 2. The transient scenario is associated to the removal of power from the pump at time zero; while the valve begins to close after 1.5 s. The prescribed valve motion is specified in with tabular values identified at equal intervals of time.

Fig. 3. Pipeline system for pump failure.

Figure 4 illustrates the dimensionless pressure-head, flow, and speed change along with the prescribed valve motion. This figure suggests that the flow reverses at the pump before the rotational speed reverses, as would be anticipated when pumping against a gravity load. Another system response to observe in the output is the head-discharge

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B. Essaidi and A. Triki Table 1. Characteristics of the piping system in Fig. 3.

Parameters

N R [rpm]

T R [kg.m]

H R [m]

Loss coef. [-]

I pump [kg.m2 ]

QR [m3 /s]

Wave celerity [m/s]

Value

147

99

94.5

0.02

7.88

0.178

1067

relationship at the pump prior to the first reflection from the downstream reservoir. The results produce a straight line whose dope is the pipeline characteristic impedance. Additionally, referring to this figure it may be concluded that the valve closure time should be delayed to mitigate the accumulated effects of water hammer surges due the pump failure and valve closure; which may lead to serious damages. 1.5 1

τ 0.5

α v

0

h -0.5 -1 0

2

4

6

8

10

t, s

12

Fig. 4. Dimensionless head, velocity, speed, and valve position.

4 Conclusion The present study outlined the numerical solution procedure to predict the flow behavior following a pump failure event and particular valve closure. Numerical results were presented for a simple test case involving single pump system without considering cavitation; which is commonly considered by hydraulic piping systems designers taking some criteria into account such as the optimal valve closure procedure to control the transient mechanism. However, during the water hammer event, it seems likely that cavities form in the pump during the transient flow. In such a case, the role of the check valve especially if closed in a short period of time could be very destructive. The method outlined here may be extended to cover complex situations such as the assemblage of pumps in series or in parallel, with different characteristics.

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References Adamkowski, A., Krzemianowski, Z., Janicki, W.: Improved discharge measurement using the pressure-time method in a hydropower plant curved penstock. ASME. J. Eng. Gas Turbines Power 131(5), 053003 (2009). https://doi.org/10.1115/1.3078794 Bergant, A.: The behavior of Hydraulic Turbomachine during transients. J. Mech. Eng. Strojniski vestnik 150–160 (2003) Bergant, A., Simpson, A.R., Vitkovsky, J.: Developments in unsteady pipe flow friction modeling. J. Hydraul. Res. 39(3), 249–257 (2001). https://doi.org/10.1080/00221680109499828 Chaudhry, M.H.: Applied Hydraulic, Transients, 3rd edn. Van Nostrand Reinhold Co., New York (2014) Frelin, M.: Coups de bélier – (water hammer). Les techniques de l’Ingénieur, BM 4(176), 1–27 (2002) Guo, C., Gao, M., Lu, D., Wang, K.: An experimental study on the radiation noise characteristics of a centrifugal pump with various working conditions. Energies 10(12), 2139 (2017). https:// doi.org/10.3390/en10122139 Guo, C., Gao, M., He, S.: A review of the flow-induced noise study for centrifugal pumps. Appl. Sci. 10, 1022 (2020). https://doi.org/10.1063/5.0003937 Guo, C., Gao, M.: Investigation on the flow-induced noise propagation mechanism of centrifugal pump based on flow and sound fields synergy concept. Phys. Fluids 32, 035115 (2020). https:// doi.org/10.1063/5.0003937 Guo, C., Gao, M., Wang, J., Shi, Y., He, S.: The effect of blade outlet angle on the acoustic field distribution characteristics of a centrifugal pump based on Powell vortex sound theory. Appl. Acoust. 155, 297–308 (2019). https://doi.org/10.1016/j.apacoust.2019.05.031 Guo, C., Wang, J., Gao, M.: A numerical study on the distribution and evolution characteristics of an acoustic field in the time domain of a centrifugal pump based on Powell vortex sound theory. Appl. Sci. 9(23), 5018 (2019). https://doi.org/10.3390/app9235018 Ismaier, A., Schlücker, E.: Fluid dynamic interaction between water hammer and centrifugal pumps. Nucl. Eng. Design 41(30), 41082867 (2009). https://doi.org/10.1016/j.nucengdes.2009. 08.028 Keramat, A., Ahmadi, A., Majd, A.: Transient cavitating pipe flow due to pump failure. In: 3rd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Brno, Czech Republic, 14–16 October 2009 Nourbakhsh, S.A., Jaumotte, B.A., Hirsch, C., Parizi, H.B.: Turbopumps and Pumping Systems. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68214-1 Thanapandi, P., Prasad, R.: Centrifugal pump transient characteristics and analysis using the method of characteristics. Int. J. Mech. Sci. 37(1), 77–89 (1995). https://doi.org/10.1016/00207403(95)93054-A Thorley, A.R.D.: Fluid Transients in Pipeline Systems: A Guide to the Control and Suppression of Fluid Transients in Liquids in Closed Conduits, 2nd edn. Professional Engineering Publishing, London (2004) Triki, A.: Further investigation on water-hammer control inline strategy in water-supply systems. J. Water Supply Res. Technol. AQUA 67(1), 30–43 (2018). https://doi.org/10.2166/aqua.201 7.073 Triki A. (2017) Water-hammer control in pressurized-pipe flow using a branched polymeric penstock. J. Pipeline Syst. Eng. Pract. ASCE 8(4), 04017024. https://doi.org/10.1061/(ASCE)PS. 1949-1204.0000277 Tsukamoto, H., Matsunaga, S., Yoneda, H., Hata, S.: Transient characteristics of a centrifugal pump during the stopping period. ASME J. Fluids Eng. 108(4), 392–399 (1986). https://doi. org/10.1115/1.3242594

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Wan, W., Zhang, B., Chen, X.: Investigation on water hammer control of centrifugal pumps in water supply pipeline systems. Energies 12(1), 108 (2019). https://doi.org/10.3390/en12010108 Wan, W., Li, F.: Sensitivity analysis of operational time differences for a pump-valve system on a water hammer response. J. Pressure Vessel Technol. Trans. ASME 138(1), Article number 011303 (2016). https://doi.org/10.1115/1.4031202 Wan, W., Huang, W.: Investigation on complete characteristics and hydraulic transient of centrifugal pump. J. Mech. Sci. Technol. 25, 2583 (2011). https://doi.org/10.1007/s12206-0110729-9 Wylie, E.B., Streeter, V.L., Suo, L.: Fluid Transients in Systems. Prentice Hall (1993)

Innovative In-Plane Converter Design for a Capacitive Energy Harvester M. A. Ben Hassena1 , H. Samaali1 , F. Najar1(B) , and Hassen M. Ouakad2 1

Applied Mechanics and Systems Research Laboratory, Tunisia Polytechnic School, University of Carthage, BP 743 La Marsa 2078, Tunis, Tunisia [email protected] 2 Mechanical and Industrial Engineering Department, Sultan Qaboos University, P.O Box 33, Al-khoudh 123, Muscat, Sultanate of Oman

Abstract. This work presents an innovative design of a MEMS vibration energy harvester based on electrostatic transducer and allowing multidirectional in plane excitation. The proposed design uses a single variable capacitor axially attached to a shallow arched microbeam. The latter generates axial displacements when it is subject to external excitation. It is expected that the fundamental resonance frequency of the microbeam can be triggered by imposing a secondary DC voltage. The equations of motion are derived using the Hamilton’s principle for large displacements of the microbeam by taking into account the nonlinear form of the electrostatic force. Numerical simulations of the axial and transverse displacements of the microbeam are extracted and coupled to a conditioning circuit. The generated numerical simulations are used to calculate the harvested energy for several directions of the excitation. It was shown that, in contrast to the parallel-plates beam-based designs, the suggested design can generate energy even if the direction of excitation is changed, as a result the energy is augmented if the excitation is present in two, or even more, different in-plane directions.

Keywords: Energy harvester circuit

1

· e-KEH · 2D converter · Conditioning

Introduction

Nowadays, the use of integrated Microelectromechanical systems (MEMS) for smart systems design is highly increased thanks to their higher performance, low cost and small dimensions. MEMS are used as sensors (Jrad et al. 2016), actuators (Samaali and Najar 2017), energy harvesters (Baroudi et al. 2018), oscillators (Ben Sassi et al. 2018) and filters (Ouakad 2013) in many applications such as sensor networks, communication applications, industrial machines and medical systems (Li 2010; Zhang et al. 2005; Samaali et al. 2015). For instance, for portable microsystems, MEMS-based kinetic energy harvesters can be used to transform mechanical vibration into electric energy and c The Author(s), under exclusive license to Springer Nature Switzerland AG 2021  N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 125–135, 2021. https://doi.org/10.1007/978-3-030-76517-0_15

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avoid the use of battery or extend their lifetime. Several transduction mechanisms have been proposed for this conversion. However, electrostatic transduction turns out to offer interesting integration properties compared to other techniques such as piezoelectric or electromagnetic (Amri et al. 2018, Ben Ouanes et al. 2019, Yang et al. 2013). Most vibration-based electrostatic energy harvesters are designed as linear resonators working efficiently only near their resonant frequencies (Ben Hassana et al. 2019). To increase the efficiency of the energy harvester and provide maximum power density generation, several researchers emphasize on designing nonlinear resonator in order to increase the bandwidth of the device (Abdelkefi et al. 2012). Few of them tried to use the multi-directional aspect of ambient vibration to increase the performance of the device (Tao et al. 2014). Recently, the arched microbeam design has been proposed as nonlinear oscillator for several applications (Casals-Terre et al. 2008; Ouakad and Najar 2019). Tensile and compressive axial forces are used to control the dynamic behavior of the microbeam (Ramini et al. 2016; Alcheikh et al. 2017). In this work, we propose a new design of a nonlinear electrostatic energy harvester based on the use of an arched microbeams. The design is intended to harvest vibrations at any direction in the device in-plane motion. The design combines the advantages of nonlinear oscillator with large bandwidth and displacement.

2

The MEMS Device

The proposed design, shown in Fig. 1, consists of a clamped-guided shallow arched microbeam with simultaneous motions in the z and x directions. A rigid mass (M1 ) is attached to the midspan of microbeam to produce inertial forces in the vertical direction. Also, a DC voltage is applied to the microbeam to trigger the effective stiffness of the arched beam especially at its resonance frequencies. Also, a second rigid mass (M2 ) is attached to the guided end of the beam by means of a rigid vertical beam to convert external motions in the horizontal directions into inertial forces. This second mass forms also a movable electrode of a variable capacitor (Cvar ) used as an electrostatic transducer. The equations of motion of the proposed design are derived using the Hamilton’s principle. For this, the expression of the total kinetic energy K and the total potential energy P are given by:  L  2  1 1 1 w˙ + u˙ 2 dx + M1 w˙ 2L + M2 (β u˙ L )2 (1) K = ρA 2 2 2 2 0   2   1 L 1 2    2 P= EA u + w + w w0 + EIw dx (2) 2 0 2 where u(x, t) and w(x, t) are, respectively, the axial displacement and the bending of the shallow arched microbeam with respect to its initial deformation w0 and r(x, t) represents the horizontal displacement of the variable capacitor. We note, here, that:

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Rigid mass M1 Base acceleration z

θ

h w0(x)

x

L d

Insolator

VDC2

VDC1

Vvar

Vstore Insolator

Rigid mass M2

Fig. 1. A Schematic of 2D EH device

r(x, t) = −β u(x, t)|x=L = −β uL

(3)

and

do (1 − cos(2πx)) (4) 2d where β is the amplification factor of the axial displacement. The prime ( ) denotes the space derivative of each variable with respect to the space (x), whereas the dot (˙) is the time derivative. The expression of the total work done by external forces is given by: w0 (x) =

 W=

L

(−F1 w − c1 ww) ˙ dx − F2 βuL − c2 β 2 u˙ L uL

(5)

0

where c1 and c2 are the viscous damping coefficients, F1 and F2 are the electrostatic forces which are given by: F1 =

2 bVDC2

2(d + w + w0 )

2;

F2 =

2 App Vvar 2

2 (g − βuL )

(6)

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where d is the initial electrostatic gap distance of the microbeam and VDC2 is the corresponding applied DC voltage, g is the initial electrostatic gap associated to the mass M2 and App is the surface area of the variable capacitance. Vvar is the voltage across Cvar . The extended Hamilton’s principle is given by:  t2 (δK − δP + δW) dt =0 (7) t1

Applying the Hamilton’s principle in equation (7) and considering that for slender beam, the longitudinal natural frequency is much higher than the transverse one. Thus we have a constant axial force as described by Nayfeh and Pai (2008). We derived, in non-dimensional form, a set of equations of motion and associated boundary conditions, describing both the vibration of the shallow arched microbeam and the associated horizontal motion of the variable capacitor (Table 1). w ¨ + αc1 w˙ + w

iv

  + αM 1 w ¨ L − (w + w0 ) α1 



2

1



2

w + 2w



w0



dx + α3 β r

0

= −αM 1 Z¨ |x= 1 − Z¨ + α2 2

2 VDC1 2

(1 + w + w0 )  1

2 α5 2 ¨ + α6 VV ar w + 2w w0 dx = −X r¨ + αc2 r˙ + α4 r − 2 β 0 (1 + r)

(8) (9)

The associated boundary conditions are: w (0, t) = 0, w (0, t) = 0, w (1, t) = 0, w (1, t) = 0

(10)

where  2 d M1 L5 c1 L4 6L4  12gL , α , α = = 6 , α2 = , α3 = , c1 1 2 EIT EIT h Ed3 h3 h2 c2 T EA T 2 EAd2 T 2 App T 2 = , α4 = , α5 = ; α = (11) 6 M2 L M2 2 gM2 L2 2 g 3 M2

αM 1 = αc2

For the rest of this paper, the primes denotes derivatives with respect to the nondimensional length position x ˆ and the dots denotes derivatives with respect to the nondimensional time tˆ. The Differential Quadrature Method is used to discretize the equations of motion where n grid points are used for discretization (Ben Sassi and Najar, 2018), they are defined as:   1 i−1 π , i = 1, · · · , n 1 − cos xi = 2 n−1

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Letting w(xi , t) = wi , one can end-up with the following discretized set of equations: ⎡ ⎤ n n      (4) (2)  ¨ + αc1 w˙ i + 1 + αM 1 δ n+1 w ¨i + Z Aij wj − ⎣ Aij wj + w0 (xi )⎦ 2

× α1

n 

⎛ Cp ⎝

p=1

n 

j=1

⎞2 (1) Apq wq ⎠



+ 2w0 (xp )

q=1

j=1

n 

 (1) Apq wq

⎡

− α3 βr ⎣

q=1

n  j=1

⎤ (2) Aij wj

2 VDC2

= α2

(1 + wi + w0 (xi ))2 ⎛ ⎞2  n n n   α5  (1) (1)  ¨ ⎝ ⎠ r¨ + X + αc2 r˙ + α4 r − Cp Apq wq + 2w0 (xp ) Apq wq β p=1 q=1 q=1 = α6

2 Vvar

+ w0 (xi )⎦ 

(12)

(13)

(1 + r)2

with the boundary conditions w1 = 0,

n

(1)

A1j wj = 0, wn = 0,

j=1

where δ n+1 = 2

1 C n+1

if i =

2

n

j=1

n+1 2

(1)

Anj wj = 0

(14)

and 0 otherwise.

The DQM matrices are given by Bert and Malik (1996): n v=1;v=i (xi − xv ) (1) n Aij = j = i (xi − xj ) v=1;v=j (xj − xv )

Ar−1 ij (r) 1 j = i Aij = r Ar−1 A − ij ij xi − xj n

(r) (r) Aii = − Aiv

(15)

v=1;v=i

where i, j = 1, 2, ..., n The integrals term are discretized using the Newton-Cotes technique, where  Ci =

1

n 

0 j=1;j=i

x − xj dx xi − xj

i = 1, ..., n

(16)

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M. A. Ben Hassena et al. Table 1. Geometrical an physical properties for the microbeam Microbeam length (L)

1000 µm

Microbeam thickness (h) 2.4 µm

3

Microbeam width (b)

30 µm

Microbeam gap size (d)

10 µm

Cvar plate area (App )

30 × 1000 µm2

Young’s Modulus (E)

166 GPa

Density (ρ)

2332 kg/m3

Initial rise (do )

10 µm

Lateral gap size (g)

10 µm

Mass 1 (M1 )

28 µg

Mass 2 (M2 )

28 µg

Electric permittivity ()

8.854 × 10−12 F/m

Pump Charge Working Principle

In this section we introduce the conditioning circuit used to convert mechanical energy into electrical energy. In the proposed design we use the pump charge principle presented in Fig. 2. It is composed by a DC source VDC1 , an electrostatic transducer associated to the variable capacitor Cvar , two diodes D1 and D2 and a storage capacitor Cstore . To explain the pump principle, we assume initially that Cvar is at its maximum value and Vvar , VDC1 and Vstore have the same value. At this state both diodes D1 and D2 are OFF and charges are conserved at each capacitor. When the external excitation Cvar starts decreasing, and according to the charge conservation law, the voltage Vvar increases. As a result, D2 becomes ON leading to a partial discharge of the capacitor Cvar to Cstore with Vvar = Vstore . When the vibration reverses its respective direction, Cvar increases starting from its minimum value causing the decrease of Vvar and D2 turns OFF. When Vvar becomes less than VDC1 , D1 turns ON again which results in charge injection from VDC1 into Cvar until saturation occurs.

Fig. 2. Electrical pump charge circuit.

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The electrical circuit can be easily modeled using the classical Kirchoff low and the capacitance variation is calculated through the mechanical model developed in the next sections. i = i1 − i2 =

dq1 VDC1 − Vvar Vvar − Vstore dq2 dqvar ) − Exp( )] − = = Is [Exp( dt dt dt μVT μVT

(17) dqstore Vvar − Vstore ) − 1] = Is [Exp( dt μVT

(18)

Then VDC1 − Vvar Vvar − Vstore d(Cvar Vvar ) = Is [Exp( ) − Exp( )] dt μVT μVstore dVstore Vvar − Vstore = Is [Exp( Cstore ) − 1] dt μVT

(19) (20)

where Is = 3 nA is the reverse bias saturation current of the diodes, μ = 1 is their ideality factor and VT = 25 mV is the thermal voltage.

4 4.1

Mechanical Characterization The Static Analysis

In this study, a static analysis is first established for the developed equations of motion (Eqs. 12 to 14) by omitting the time derivative terms. The static response of the arched microbeam in presence of VDC2 and Vvar is presented in Fig. 3(a) and Fig. 3(b), respectively. In these figures, the stable solution are denoted by solid lines and the unstable solutions by dashed lines. Two different pull-in voltages can be identified in our case. The first is related to Vvar and corresponds to the collapse of the capacitance Cvar . In this case around 700 V are required to reach pull-in. The second one is related to the voltage VDC2 , which is directly applied to the arched microbeam (Fig. 1). In the latter case a voltage around 110V is required to have pull-in. In both cases the pull-in instability should be avoided to extend the harvesting capability of the device and avoid saturation of the harvested energy. 4.2

Natural Frequencies

A modal analysis of the MEMS device is proposed in this section where we extract the fundamental natural frequencies of the device at different applied DC voltages. In Fig. 4(a) and 4(b) the nondimensional fundamental bending frequency of the device as different VDC2 and Vvar is presented. As shown, the increase of VDC2 decreases the fundamental frequency of the device until pull-in is reached and the effective stiffness of the system is lowered due to the presence of the electrostatic force, forcing the fundamental natural frequency to be highly reduced. In the contrary, the increase of Vvar tends to increase the fundamental frequency of the system even if in this case very high voltages are needed to reach the pull-in.

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Nondimensional displacement

Nondimensional displacement

1.0 r wmax+w0(1/2)

r wmax+w0(1/2)

0.5

0.0

- 0.5

- 1.0 0

200

VDC2

400

600

800

Vvar (V)

(a) Vvar =0

(b) VDC2 =0

Fig. 3. Nondimensional static response at midpoint of the arched microbeam under different applied voltages (n = 9).

VDC2

(a) Vvar =0

Vvar

(b) VDC2 =0

Fig. 4. Nondimensional fundamental natural frequency of the arched microbeam under different applied voltages (n = 9).

4.3

Frequency Response Curves of the Device

In order to estimate the frequency response curves of the MEMS device for a set of excitation frequencies around the fundamental mode of the structure, we choose to simulate the equations of motion and calculate limit-cycle solutions obtained using a combination of finite difference method and a sequential continuation technique (Ben Sassi and Najar, 2018, Nayfeh and Balachandran, 2008). We denote by θ the tilt angle of the excitation. θ = 0 means that we have an excitation with respect to x-axis whereas if θ = π/2, we have an excitation with respect to z-axis. The results shown in Figs. 5(a), 5(b) and 5(c), where the maximum value of the deflection, at midpoint of the arched microbeam, for a given limit-cycle solution is shown with respect to the excitation frequency Ω. In these figures, we depict an in-plane motion of the device for different values of the tilt angle θ.

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0.20

0.15

0.10

0.05

0.00

... ... ... ... .... wMax ... ... ... ... .... ..... . . ... ... ... ... .. ... ... .... .. .. .... ... . ... ... ... ... . ... .... ... .... ..... ..... ..... .... ..... ..... .... . .. .. ..... ..... .... .... ... .... .... .... ... .... . ........ .. ..... ... ...... .. ........ ... uMax ........ .. . .. . . .. .......................... .. . . . . . . . . . . . . ...... . . . . .. ...... . . . . . . . . . . . . . .. .. .. .. .. .. .. . . . .............................

1.2

1.3

1.4

Nondimensional displacement

Nondimensional displacement

0.25

Nondimensional displacement

In Fig. 5(a), the frequency response depict a linear-like response because the amplitude of the response is too low to trigger the nonlinearity of the system. A larger amplitude is observed in the z-direction expressed by the microbeam deflection w. This is because the effective axial stiffness of the microbeam is too high to let the inertia of the lower mass M2 freely initiate large amplitude motions of the variable capacitor’s.

1.5

(a) θ = 0

(b) θ = Π/2

(c) θ = Π/4

Fig. 5. Frequency-response curves of the arched microbeam at midpoint for different values of θ (n =9).

On the other hand, in Fig. 5(b), the frequency response curve depict large amplitude of motion in both directions. Also, one can depict that jumps and snap-through behaviors can be obtained for some excitation frequencies. This fact can be suitable in energy harvesting applications because it enlarge the overall amplitude of response where bistability can be obtained. The same observations can be made for Fig. 5(c) except that frequency response curve has a lower bandwidth due to the variation of the direction of the excitation.

5

Energy Harvesting

For energy conversion, we used the pump charge circuit presented in Sect. 3 to convert capacitance variation into electrical energy. The simulated results, shown in Fig. 6, show that the proposed MEMS design is capable to harvest energy in 2D In-plane directions with some preferences for certain directions compared to others. The variation of the harvested energy is observed through the calculated voltage Vstor in the storage capacitance Cstor . In fact since the voltage can be instantaneously calculated it is preferred in this case to the stored energy. We observed also that, at saturation of the voltage, the maximum amount of generated energy is obtained at θ = Π/4 which correspond to 1.65μJ (49.5μJ/μm3 ) for Cstore = 3.3nF , when the excitation frequency is fixed at 1.7kHz.

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Vstor (V)

10 9 8

θ=0 θ = π/2 θ = π/4

7 6 5 0

0.085

0.17

0.26

0.34

t (s)

Fig. 6. Generated voltage Vstor as a function of the time at the storage capacitance Cstor , for different values of θ for VDC1 = 5 V, VDC2 = 0 V and Ω = 1.85 (Corresponding to 1.7 kHz).

6

Conclusion

In this study, we proposed to design, model and simulate a new electrostatic kinetic energy harvester working with multidirectional in-plane source of excitation. The proposed device converts vibration oscillations into electrical energy through an electrostatic nonlinear transducer integrated with a pump charge conditioning circuit. We showed that for arbitrary in-plane direction of excitation, the nonlinear transducer, based on the use of an arched microbeam, is stimulated provided that the direction the excitation is at the same plane as the device. The mechanical and electrical governing equations of the device are first derived. The nonlinear static and dynamic responses are solved. Large bandwidth of resonant frequencies are obtained. The harvested voltage is obtained for a different directions of the excitation and optimal performance are observed when the device is excited with a tilt angle of π/4. It was shown that the proposed device can harvest energy at any in-plane direction and that an optimal value of energy density was calculated at 49.5 µJ/µm3 . Acknowledgements. The last two authors are grateful for the support of the Internal Research Grant provided by the Deanship of Research at Sultan Qaboos University (SQU) through grant number IG/ENG/MIED/20/01.

References Abdelkefi, A., Nayfeh, A., Hajj, M.: Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters. Nonlinear Dyn. 67(2), 1147– 1160 (2012) Alcheikh, N., Ramini, A., Hafiz, M.A.A., Younis, M.I.: Tunable clamped-guided arch resonators using electrostatically induced axial loads. Micromachines 8(1), 14 (2017) Amri, M., et al.: Stiffness control of a nonlinear mechanical folded beam for wideband vibration energy harvesters. tm-Technisches Messen 85(9), 553–564 (2018)

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Baroudi, S., Najar, F., Jemai, A.: Static and dynamic analytical coupled field analysis of piezoelectric flexoelectric nanobeams: a strain gradient theory approach. Int. J. Solid Struct. 135, 110–124 (2018) Ben Hassana, M.A., et al.: Design of a mems electrostatic kinetic energy harvester and its bennet conditioning circuit in integrated technologies. In: Proceedings of the IEEE Symposium on Design, Test, Integration and Packaging of MEMS and MOEMS, Pars, France (2019) Ben Ouanes, M., Samaali, H., Galayko, D., Basset, P., Najar, F.: A new type of triboelectric nanogenerator with self-actuated series-to-parallel electrical interface based on self-synchronized mechanical switches for exponential charge accumulation in a capacitor. Nano Energy 62, 465–474 (2019) Ben Sassi, S., Najar, F.: Strong nonlinear dynamics of mems and nems structures based on semi-analytical approaches. Commun. Nonlinear Sci. Numer. Simul. 61, 1–21 (2018) Ben Sassi, S., Khater, M., Najar, F., Abdel-Rahman, E.: A square wave is the most efficient and reliable waveform for resonant actuation of micro switches. J. Micromech. Microeng. 28(5), 055,002 (2018) Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49(1), 1–28 (1996) Casals-Terre, J., Fargas-Marques, A., Shkel, A.M.: Snap-action bistable micromechanisms actuated by nonlinear resonance. J. Microelectromechan. Syst. 17(5), 1082– 1093 (2008) Jrad, M., Younis, M.I., Najar, F.: Modeling and design of an electrically actuated resonant microswitch. J. Vibr. Control 22(2), 559–569 (2016) Li, L.: Recent development of micromachined biosensors. IEEE Sensor J. 11(2), 305– 311 (2010) Nayfeh, A.H., Balachandran, B.: Applied nonlinear dynamics: analytical, computational and experimental methods. John Wiley & Sons (2008) Nayfeh, A.H., Pai, P.F.: Linear and nonlinear structural mechanics. John Wiley & Sons (2008) Ouakad, H.M.: An electrostatically actuated mems arch band-pass filter. Shock Vibr. 20(4), 809–819 (2013) Ouakad, H.M., Najar, F.: Nonlinear dynamics of mems arches assuming out-of-plane actuation arrangement. J. Vibr. Acoustics 141(4) (2019) Ramini, A., Alcheikh, N., Ilyas, S., Younis, M.I.: Efficient primary and parametric resonance excitation of bistable resonators. AIP Adv. 6(9), 095–307 (2016) Samaali, H., Najar, F.: Design of a capacitive mems double beam switch using dynamic pull-in actuation at very low voltage. Microsyst. Technol. 23(12), 5317–5327 (2017) Samaali, H., Ouni, B., Najar, F.: Design and modelling of mems dc-dc converter. Electron. Lett. 51(11), 860–861 (2015) Tao, K., Liu, S., Lye, S.W., Miao, J., Hu, X.: A three-dimensional electret-based micro power generator for low-level ambient vibrational energy harvesting. J. Micromech. Microeng. 24(6), 065,022 (2014) Yang, J., Wen, Y., Li, P., Yue, X., Yu, Q., Bai, X.: A two-dimensional broadband vibration energy harvester using magnetoelectric transducer. Appl. Phys. Lett. 103(24), 243,903 (2013) Zhang, G., Gaspar, J., Chu, V., Conde, J.: Electrostatically actuated polymer microresonators. Appl. Phys. Lett. 87(10), 104 (2005)

Dynamic Interaction Between Transmission Error and Friction Coefficients for FZG-A10 Spur Gears Maroua Hammami1,2(B) , Olfa Ksentini1,2 , Nabih Feki1,3 , Mohamed Slim Abbes1 , and Mohamed Haddar1 1 Laboratory of Mechanics, Modelling and Manufacturing (LA2MP), National Engineering

School of Sfax, Ministry of Higher Education and Research of Tunisia, BP 1173, 3038 Sfax, Tunisia [email protected], [email protected] 2 Higher Institute of Industrial Systems of Gabes (ISSIG), University of Gabes, 6029 Gabes, Tunisia {MarouaHammami123,olfa.ksountini}@issig.u-gabes.tn 3 Higher Institute of Applied Science and Technology of Sousse, University of Sousse, 4003 Sousse, Tunisia [email protected]

Abstract. A 12 degree-of-freedom dynamic model for FZG-A10 spur gears considering the actual time-varying gear mesh stiffness and the frictional effects between meshing gear teeth is investigated. The energetic Lagrange formulation was used to recover the equations of motion of the generalized translational torsional coupled dynamic system. Its dynamic response was computed by an iterative implicit scheme of Newmark. The main ameliorations achieved by this new model are evaluated in this work through a gear dynamics responses comparison with previously classical developed models which can be found in the literature. The simulation results are arranged using the listed dynamic models considering no coefficient of friction, an experimental constant coefficient of friction and an EHL based coefficient of friction formulation including an average surface roughness. The vibration responses are calculated based on the indicated previous models under several operating conditions (load and rotational speed). The transmission error (TE) parameter is evaluated domains since it is considered as an index for the vibration performance. The influence of constant and variable coefficient of friction formulation was studied and their corresponding transmission error were compared. The new dynamic model proves their refining results counter to other models in diagnostic and monitoring performance. Keywords: Dynamic model · Spur gears · Coefficient of friction · Vibration responses

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 136–144, 2021. https://doi.org/10.1007/978-3-030-76517-0_16

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1 Introduction The modern automotive engineering considers that the gear noise reduction in the ground and air vehicles like heavy-duty trucks and helicopters is a prominent challenge (He 2008). The transmission error (TE) parameter is quantified as it is a reliable indicator of the noise and vibration performance. To achieve such objective, various researchers carried out models aiming to simulate the gear dynamics (Beyaoui et al. 2016). They developed a large variety of dynamic models to predict the gear vibration response which improves gearbox monitoring and diagnostic. Most models process the cases of the damaged gearbox which is contaminated by different noises (Velex and Maatar 1996; Velex and Ajmi 2006; Chaari et al. 2008; Walha et al. 2009; Feki et al. 2013; SainteMarie et al. 2017; Feki et al. 2017). They reported that the main source of transmission error is the derivation from the ideal tooth profile which can be induced by spalling, tooth breakage, tooth surface pitting, wear or tooth crack. They studied these faults to show their effects on the gear dynamic behavior. They stated that the gear mesh frequency band presenting the sidebands around its harmonics is very sensitive to the defect degree which is useful mainly for tooth fault detection and localization. Therefore, the gear design engineers who prospect the reduction of transmission error variations are working on manufacturing processes and tooth modifications to conceive low noise gears. From most of the presented models, the transmission error is the primary excitation that generates noise and vibration in the gearing system. However, scarce models proposed that friction might also be a significant contributor in defining the vibration features (Tang et al. 2010; Li et al. 2013; Jiang et al. 2017; Park 2019). They introduced the effect of the friction coefficient between gear tooth contacts in more complex dynamic models which may lead to increase in the accuracy of diagnostic and monitoring results. In this study, the main effects of tooth friction on spur gear dynamics with their associated vibration responses are evaluated. The transmission error variations are predicted using a contact mechanics model coupling with and without tooth friction effects. A twelve degree-of-freedom dynamic model considering constant and local coefficient of frictions is investigated to predict the vibration characteristics. The analytical simulations are performed using different frictional cases and they are compared with experimental data to be validated. The proposed model predictions are expected to provide a better understanding of the mechanisms of noise generation in spur gear pairs and to establish an effective and accurate vibration detection for monitoring lubrication conditions.

2 MDOF Dynamic Gear Model 2.1 Frictional Model A twelve degree of freedom model is developed to quantify the frictional effects on gearbox-noise through transmission error prediction. The test gearbox of the FZG gear machine is modelled using a multi-degree of freedom (MDOF) model as shown in Fig. 1. Extra details about the experimental set up can be found in (Hammami et al. 2019a). Torsional (θi ) and translational (vi , wi ) coupled effects are considered through the following DOF’s vector where i = 1, g, p, 2 designate shaft 1, gear, pinion and shaft 2,

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respectively X = [v1 , w1 , θ1 , vg , wg , θg , vp , wp , θp , v2 , w2 , θ2 ]T . The studied system is composed of shaft 1 connected the gear g to the motor and shaft 2 related the pinion p to the static load which is applied through load lever and weights. Mi , Ji , and θi are the mass, polar mass moment of inertia and torsional motion of the system elements. The Kxi and Kyi are the bearings stiffness and the Cxi and Cyi are the damping elements. Km (t) is the time-varying mesh stiffness and Cm is the gears damper and e represents static transmission error excitation for the gear mesh.

Fig. 1. A 12 degrees of freedom dynamic model simulating FZG test gearbox

A computational frictional model of the FZG test gearbox, with integrated mass, damping, nonlinear stiffness and frictional stiffness matrices [M], [C], [K] and [Kμ ] is developed to characterize the spur gear dynamic responses. The governing equations for the proposed model considering friction coefficients, where the gyroscopic moments and centrifugal effects are neglected, are detailed as follow:        [M ] X¨ + [C] X˙ + [K(t, {X })] + Kμ (t, {X }) {X } = {F1 (μ, t, {X })} + {F2 (μ, t, e(M ))}

(1) Where F1 (μ, t, {X }) is a vector embedded the nominal input torque (Cm (t, X, μ) and the instantaneous output torque (Cr (T, X, μ) which is function of the toot friction (μ) and profile deviations (e) and F2 (μ, t, e(M )) is an external vector induced by tooth shape deviations and errors (e (M)).

Dynamic Interaction Between Transmission Error and Friction Coefficients

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2.2 Incorporation of the Coefficient of Friction The EHL based formulation for a time-dependent coefficient of friction proposed by Xu et al. (2007) is defined in Eq. (2). The μ coefficient issued from many linear regression analysis depends on several factors like SR, Ph , ν0 , S, Vr , μ0 , R are respectively the slide to roll ratio, maximum Hertzian pressure, dynamic viscosity, surface roughness, rolling velocity, kinematic viscosity at oil inlet temperature and effective radius of curvature. The constant coefficients bi for the selected axle gear oil are illustrated for i = 1–9 respectively −8.92, 1.03, 1.04, −0.35, 2.81, −0.10, 0.75, -0.39, and 0.62. b8 μ = ef (SR,Ph ,ν0 ,S) Phb2 |SR|b3 Vrb6 μb7 0 R

(2)

f (SR, Ph , ν0 , S) = b1 + b4 |SR|Ph log10 (ν0 ) + b5 e−|SR|Ph log10 (ν0 ) + b9 eS

(3)

2.3 Model Discretization The presented equations of motion are solved thanks to the Newmark iterative algorithm. The numerical simulations have been conducted by using a constant Newmark parameters λ = β = 0.5 which provide a stable and convergent method. Figure 2 shows the diagram for transmission error simulation using the dynamic model of spur gear pair considering friction of tooth surface. All the presented steps are given by authors previous work (Hammami et al. 2019b).

Fig. 2. Diagram for the transmission error prediction using the dynamic model

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3 Dynamic Results and Discussion An FZG A10 spur gear is taken as an example to apply the proposed MDOF model. Table 1 displayed the relative parameters of the spur gear pair. Table 1. FZG A10 gear geometric parameters (Hammami et al. 2019a). Parameters

Pinion Gear

Tooth number

16

24

Normal modulus

4.5

Pressure angle (°)

20

Helix angle (°)

0

Face width (mm)

10

20

Transverse contact ratio

1.333

RMS surface roughness (μm)

0.49

All simulation results illustrated subsequently were run under K8 load stage which corresponds to an input torque of 172 Nm and at a constant rotational speed of 1000 rpm. Based on experimental results, a constant coefficient is determined for these operating conditions where COF = 0.035 (Hammami et al. 2019b). The modelled test gearbox is lubricated with 75W90-A axle gear oil. The lubricant properties can be found in Table 2 (Hammami et al. 2017). Table 2. The selected axle gear oil physical properties Physical properties

75W90-A lubricant

Density at 15 °C (g/cm3 ) 0.87 Viscosity at 70 °C (cSt)

36.7

Dynamic viscosity (Pas)

0.0298

Viscosity Index

147

To provide an efficient evaluation of the dynamic interaction between transmission error and frictional effects, the simulated results are drawn afterwards. The dynamic responses under stabilized operating conditions and mainly the simulated transmission error results with and without coefficient of friction are presented in Fig. 3. Figures 3a, 3c and 3e show the time domain dynamic transmission error while Figs. 3b, 3d and 3f present the frequency domain of DTE. Only steady-state responses are presented. It is observed that the shape of the temporal DTE curve is modified as the coefficient of friction changed. Larger oscillations appear with constant COF against COF equal

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to zero. Slightly different behavior is noticed between constant and Local COF. The introduction of tooth contact friction enhanced the dynamic transmission error (see Figs. 3a, 3c and 3e). Figures 3b, 3d and 3f expose the TE spectra for the three listed cases: COF = 0, constant COF and local COF. The mesh frequency (fm = 266.7 Hz) and its harmonics (2fm , 3fm, 4fm, 5fm ) are shown in each spectrum and their amplitudes are displayed in Table 3. This highest amplitude is demonstrated at the fifth harmonic of the mesh frequency. These results are proved based on experimental TE signals which are measured using optical encoders mounted on FZG test rig as reported by Feki et al. (2013). According to their analysis, the frequency with significant amplitude corresponds

a)

b)

c)

d)

e)

f)

Fig. 3. Predicted transmission error and its corresponding spectrum under K8 load stage at 1000 rpm: subfigures (a-b) are for COF = 0, (c-d) are for constant COF and (e-f) are for Local COF

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to a critical frequency of the test gearbox system including shafts, bearings and test gears where the gear element contributes with a large portion about 84,19%. Table 3. The TE amplitudes values for the spur gear mesh frequency and its harmonics ET (×10–6 ) [m] fm

2fm

3fm

4fm

5fm

COF = 0

8.957 1.685 0.4873 5.531

Constant COF

9.104 1.232 0.5661 7.046 14.82

Local COF

9.064 1.098 0.5105 7.033 14.82

9.171

It can be noticed that an increase in COF has a significant effect mainly at the fifth harmonics (see Table 3). This reveals the influence of the lubrication regime related to the coefficient of friction on the dynamic transmission error spectra. For the sake of comparison, Fig. 4 illustrated the TE spectrum of each model in a logarithmic scale where the difference between the used dynamic models became more plausible. A new interesting frequency appears which corresponds to the first natural resonance frequency which is equal to 492 Hz. The constant and local COF increased the amplitude of DTE up to 11 times compared to the corresponding results without COF. It is observed that the local COF compared to constant COF can reduce slightly the gear noise and vibration.

Fig. 4. Spectra of the simulated TE signal with and without COF

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In particular, the proposed model quantifies the contribution of sliding friction which could be significant when the transmission error is minimized through a local COF. The sets of results concluded a critical analysis concerning the influence of COF on the transmission errors behavior inducing noise and vibration in gear dynamics.

4 Conclusion MDOF analytical models are investigated to study the frictional effects on dynamic transmission error of spur gears. On this basis, the effect of the friction coefficient is introduced through an effective mesh stiffness in the derived equations. Analytical models match well with a benchmark experimental published results thus validating the proposed model. The DTE responses of spur gears were analyzed with and without friction coefficient. The increase of the frictional force magnitude causes a change of DTE than that without friction, which indicates the increase of gear noise. The coefficient of frictions are considered with constant or variable values. A constant friction coefficient provides marginal results. However, a local COF is implemented to simulate realistic lubrication conditions where the COF values are smaller which can reduce the friction-induced vibration as expected. This methodology proved its efficiency in promoting a better estimation of the dynamic diagnostic.

References Beyaoui, M., Tounsi, M., Abboudi, K., Feki, N., Walha, L., Haddar, M.: Dynamic behaviour of a wind turbine gear system with uncertainties. C.R. Mec. 344(6), 375–387 (2016). https://doi. org/10.1016/j.crme.2016.01.003 Chaari, F., Baccar, W., Abbes, M.S., Haddar, M.: Effect of Spalling or tooth breakage on gear mesh stiffness and dynamic response of a one-stage spur gear transmission. Eur. J. Mech.-A/Solids 27(4), 691–705 (2008). https://doi.org/10.1016/j.euromechsol.2007.11.005 Feki, N., Cavoret, J., Ville, F., Velex, P.: Gear tooth pitting modelling and detection based on transmission error measurements. Eur. J. Comput. Mech./Revue Européenne de Mécanique Numérique 22(2–4), 106–119 (2013). 10(1080/17797179),pp. 820885 (2013) Feki, N., Karray, M., Khabou, M.T., Chaari, F., Haddar, M.: Frequency analysis of a two-stage planetary gearbox using two different methodologies. C.R. Mec. 345(12), 832–843 (2017) Hammami, M., Fernandes, C.M., Martins, R., Abbes, M.S., Haddar, M., Seabra, J.: Torque loss in FZG-A10 gears lubricated with axle oils. Tribol. Int. 131, 112–127 (2019a). https://doi.org/ 10.1016/j.triboint.2018.10.017 Hammami, M., Feki, N., Ksentini, O., Hentati, T., Abbes, M.S., Haddar, M.: Dynamic effects on spur gear pairs power loss lubricated with axle gear oils. In: Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science (2019b). https:// doi.org/10.1177/0954406219888236 Hammami, M., Martins, R., Abbes, M.S., Haddar, M., Seabra, J.: Axle gear oils: friction behaviour under mixed and boundary lubrication regimes. Tribol. Int. 116, 47–57 (2017) He, S.: Effect of sliding friction on spur and helical gear dynamics and vibro-acoustics. Dissertation, The Ohio State University (2008)

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Jiang, L., Deng, Z., Gu, F., Ball, A.D., Li, X.: Effect of friction coefficients on the dynamic response of gear systems. Front. Mech. Eng. 12(3), 397–405 (2017). https://doi.org/10.1007/ s11465-017-0415-4 Li, W., Wang, L., Chang, S.: Excitation prediction by dynamic transmission error under sliding friction in helical gear system. Trans. Tianjin Univ. 19(6), 448–453 (2013). https://doi.org/10. 1007/s12209-013-1971-2 Park, C.I.L.: Tooth friction force and transmission error of spur gears due to sliding friction. J. Mech. Sci. Technol. 33(3), 1311–1319 (2019). https://doi.org/10.1007/s12206-019-0232-2 Sainte-Marie, N., Velex, P., Roulois, G., Caillet, J.: A study on the correlation between dynamic transmission error and dynamic tooth loads in spur and helical gears. J. Vibr. Acoustics 139(1) (2017). https://doi.org/10.1115/1.4034631 Tang, J.Y., Wang, Q.B., Luo, C.W.: Study on effect of surface friction on the dynamic behaviours of cylindrical gear transmission. Adv. Mater. Res. 139, 2316–2321 (2010). https://doi.org/10. 4028/www.scientific.net/AMR.139-141.2316 Velex, P., Maatar, M.: A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour. J. Sound Vibr. 191, 629–660 (1996) Velex, P., Ajmi, M.: On the modeling of excitations in geared systems by transmission errors. J. Sound Vib. 290, 882–909 (2006). https://doi.org/10.1016/j.jsv.2005.04.033 Walha, L., Fakhfakh, T., Haddar, M.: Nonlinear dynamics of a two-stage gear system with mesh stiffness fluctuation, bearing flexibility and backlash. Mech. Mach. Theory 44(5), 1058–1069 (2009). https://doi.org/10.1016/j.mechmachtheory.2008.05.008 Xu, H., Kahraman, A., Anderson, N.E., Maddock, D.G.: Prediction of mechanical efficiency of parallel-axis gear pairs. J. Mech. Des. 10(1115/1), 2359478 (2007)

Optimal Linear Quadratic Stabilization of a Magnetic Bearing System Abdelileh Mabrouk1 , Olfa Ksentini1(B) , Nabih Feki1,2 , Mohamed Slim Abbes1 , and Mohamed Haddar1 1 Laboratory of Mechanics, Modeling and Manufacturing (LA2MP), National Engineering

School of Sfax, Ministry of Higher Education and Research of Tunisia, BP 1173, 3038 Sfax, Tunisia [email protected], [email protected] 2 Higher Institute of Applied Sciences and Technology of Sousse, University of Sousse, 4003 Sousse, Tunisia [email protected]

Abstract. Magnetic bearings’ modelling is performed using different methodologies. Generally, the linear modelling of these actuators, which is required for the use of linear controllers, gives an approximation of the nonlinear relation between the bearing load and the control current. However, this approach may have some disadvantages because the model is linearized around an equilibrium position. Thus, the performance of the linear design can decreases if there is a perturbation of the system. Although, when the variation of the rotor displacement is too small, a linearized control law can offer valid results. The main objective of this work is to study the stabilization of an electromechanical system based on the linear quadratic (LQ) control method. The stabilization of a high sped rotating shaft supported by four α-degree oriented magnetic bearings will be studied. The linearization around an equilibrium position is performed to adopt the linear control law. Some simulations results will be illustrated to evaluate the performances of the proposed controller. Keywords: Magnetic bearings · Dynamic behaviour · LQ control · High speed · Weight

1 Introduction Magnetic suspensions are efficient solutions for very different field. They can support small electric machines up to huge mechanics such as some compressors. The suspended parts can be in a stationary position (telescopes) and can be subjected to high speeds (centrifuges, turbines…). Mechanical suspensions have a limited rotational speed due to mechanical problems and overheating which causes unbalancing phenomenon at high speed and lead to a significant vibration. The balancing issues can be avoided by the use of magnetic bearings because the axis of inertia can be adjusted with the axis of rotation. A controlled magnetic bearing allows the rotor positioning with a great © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 145–154, 2021. https://doi.org/10.1007/978-3-030-76517-0_17

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precision. In addition, a high speed could be reached due to the absence of contact in a magnetic bearing. The reduction of mechanical wear leads to lower maintenance costs and a longer system life. In magnetic bearings, adaptive stiffness could be used in vibration isolation, for avoiding critical speeds and external disturbances. For that, it’s necessary to understand the behaviour of this system. Many techniques was been developed in literature for the magnetic bearing modelling. Rigid modelling method was used by some researchers (Toumi and Reddy 1992; Fan et al. 1992; Chen et al. 2007) to supervise the magnetic bearing system behaviour. Others (Hentati et al. 2013; Ding et al. 2015; Yanliang et al. 2006) presented a finite element model to study the dynamic behaviour of a spindle. Results showed that there is a difference between the rigid and the flexible model. The control of the magnetic bearing system was the objective of other researchers (Schweitzer and Lange 1976). Different methods was been adopted for the shaft stabilization. Zhuravlyov et al. (2000) had tested the validity of an LQ regulator to stabilize a MB system in a high speed motion and the ability of this controller to minimize copper losses in coils. Also, Barbaraci and Mariotti (Barbaraci and Mariotti 2012) kept a contactless motion during the increase of the velocity by varying the rotor angular speed. Moreover, a Linear Quadratic Gaussian (LQG) controller (Hutterer et al. 2014; Gu et al. 2004) was used to compensate the gyroscopic effect of a high angular velocity rotor. Results were compared to those obtained from a decentralized PID controller. The aim of this paper is to control a high-speed rotating shaft supported by four magnetic bearings with an LQ method. It will be organized as follows: a shaft supported by α-degree oriented symmetric magnetic bearings elements is modelled using rigid method, and then an LQ regulator is adopted for the stabilization of the system, finally different gain values are tested to identify the adequate one for the system control.

2 Model Presentation A four degrees of freedom model corresponding to the translational and the rotational components is studied. It’s composed from a shaft actioned by two magnetic bearings (see Fig. 1).

Fig. 1. The linear control of a shaft supported by two radial bearings

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Using Euler-Lagrange equation, and based on the linearized expression that binds the force and the current of the electromagnetic bearings (Schweitzer and Lange 1976; Toumi and Reddy 1992), the motion equation without (1) and with (2) control are obtained respectively as follows: M q¨ + G q˙ + KX q = fg

(1)

M q¨ + G q˙ + KX q = fg + KI U

(2)

Where, – M = diag (m, m, J, J) is the mass matrix with m is the rotor mass and J is the moment mg of inertia of the shaft and fg = cos α , g is the gravitational acceleration, α = 45°. ⎤ 00 0 0 ⎢0 0 0 0 ⎥ ⎥ G=⎢ ⎣ 0 0 0 −Jz ⎦ 0 0 Jz 0 ⎡

(3)

– KX : is the Displacement stiffness matrix. It depends on the bearing displacement stiffness kx and the distance d between the centre of the rotor and the bearing position as it’s shown in 4. ⎤ ⎡ −2kx 0 0 0 ⎢ 0 −2kx 0 0 ⎥ ⎥ (4) KX = ⎢ 2 ⎣ 0 0 ⎦ 0 −2kx d 0 0 0 −2kx d 2 – KI : is the Current stiffness matrix. It depends on the bearing current stiffness ki and the distance d. It’s defined in 5. ⎤ ⎡ 0 ki 0 ki ⎢ 0 ki 0 ki ⎥ ⎥ (5) KI = ⎢ ⎣ ki d 0 −ki d 0 ⎦ 0 ki d 0 −ki d – X = [xβ] is the DOF’s vector – I = [Ixa Iya Ixb Iyb ] is the current vector with Ia and Ib are respectively currents in bearings A and B in X and Y directions.

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3 LQ Control Electromagnets exerted attractive forces in order to maintain the rotor in an adequate position. The magnetic bearing rotor requires greater values of control parameters to be positioned accuracy . The aim from some control strategy is to minimize the control values to be in optimized conditions. Therefore, Linear-Quadratic (LQ) controller is suggested as a solution for an optimal AMB control (Lurie 1951; Kwakernaak and Sivan 1972) (Fig. 2).

Fig. 2. Block diagram for LQ control method

To adopt this control method a state space representation is required. So the equation of motion can be written as follows. • (6) X (t) = [A]{X } + [B]{u} {Y } = [C]{X }

(7)

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Where X(t) is a state variables vector, Y(t) is the vector of output variables, u(t) is the vector of input variables and A, B, C are constant matrices. These matrices and vectors are detailed in Eqs. (8) and (9). ⎡ ⎤ ⎡ ⎤ ⎤ 0 0 0 0 10 0 0 x˙ x ⎢ 0 ⎥ 0 0 0 0 1 0 0 ⎥⎢ y ⎥ ⎢ y˙ ⎥ ⎢ ⎢ ⎥ ⎢ ˙ ⎥ ⎢ 0 0 0 0 00 1 0 ⎥ ⎥⎢ βx ⎥ ⎢ βx ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 00 0 1 ⎥⎢ βy ⎥ ⎢ β˙y ⎥ ⎢ 0 ⎥⎢ ⎥ ⎢ ⎥ = ⎢ 2kx ⎥⎢ x˙ ⎥ ⎢ x¨ ⎥ ⎢ − 0 0 0 0 0 0 0 ⎥⎢ ⎥ ⎢ ⎥ ⎢ m ⎥⎢ y˙ ⎥ ⎢ y¨ ⎥ ⎢ 0 − 2kx 0 0 0 0 0 0 ⎥⎢ ⎥ ⎢ ⎥ ⎢ m 2 ⎥⎣ ˙ ⎦ ⎣ β¨x ⎦ ⎢ ⎣ 0 0 0 0 0 −Jz ⎦ βx 0 −2kx . dJ ¨ 2 β˙y βy 0 0 0 −2kx . dJ 0 0 Jz 0 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 ⎢ 0 ⎥ 0 0 0 ⎥ ⎢ ⎢0 ⎥ ⎢ ⎥⎡ ⎤ ⎢ ⎥ 0 0 0 ⎥ Ixa ⎢ 0 ⎢0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥⎢ Iya ⎥ ⎢ 0 ⎥ 0 0 0 ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ + ⎢ ki ki ⎣ ⎦ + ⎢ fg ⎥ 0 0 ⎥ ⎢ m ⎥ Ixb ⎢ ⎥ m ⎢ ⎢0 ⎥ ki ki ⎥ Iyb 0 ⎢ 0 ⎥ ⎢ ⎥ m m ⎢ d ⎥ ⎣0 ⎦ d ⎣ ki . J 0 −ki . J 0 ⎦ 0 0 ki . dJ 0 −k . d ⎡ ⎤ i J⎡ ⎤⎡ ⎤ x xa 1 0 0 d ⎥ ⎢ ya ⎥ ⎢ 0 1 d 0 ⎥⎢ ⎢ ⎢ ⎥=⎢ ⎥ x˙ ⎥ ⎥ ⎣ xa ⎦ ⎣ 1 0 0 −d ⎦⎢ ⎣β ⎦ yb 0 1 −d 0 β˙ ⎡

(8)

(9)

The adequate control command gives a minimized quadratic integral performance index J (see Eq. (10)) when it brings the system from an arbitrary to a zero-state position. An optimal motion control is also based on the selection of the optimum states Q (symmetric and positive matrix) and R (symmetric and positive matrix) which represent respectively the weighting and the control-weighting matrix. By this way, the quadratic cost function J which involves these two matrices is minimized.

 t (10) J = q Q q + P ut R u dt where: Q = 0.1 ·[In ] R = 10 ·[In ] with, In is the identity matrix. The command is optimized when an optimal gain K is chosen (see Eq. 10). u = −K X (t) = −P −1 Bt P X (t)

(11)

where, P is the solution of the steady-state matrix Riccati equation (Zhuravlyov et al. 2000) and K is the system gain.

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4 Simulation Results The simulation is carried out into two steps. In the first one, the command current is not introduced. The objective from this step is to follow the system behaviour without control and to see if it’s necessary to control the system by the LQ approach. If the system is not stabilized, it is crucial to adopt the LQ method in a second step to maintain the shaft at an equilibrium position. Figure 3 and 4 show the displacement in the X and Y directions and the rotation around X and Y respectively. They are obtained after the simulation of Eq. 1 where the command current is not taken into account (without control).

Fig. 3. Rotor displacement in the X and Y directions without control

Fig. 4. Rotor rotation around the X and Y directions without control

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It can be noticed that the command current has a great influence on the displacement and the rotation. In fact, if it is not taken into account the system responses increase rapidly. In fact, the response grows exponentially thus the rotor may fall down or touch the magnet. It is a critical phenomenon, which can affect the behaviour of the system. Therefore, it is necessary to introduce a control law to stabilize this system. As it has mentioned previously a linear method LQ is adopted for the system control. The Eq. 2 is solved and results of the displacement in x direction and the rotation around Y direction are presented respectively in Fig. 5 and 6. -6

x 10

10

8

Amplitude [m]

6

4

2

0

-2

-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [s]

5 -4

x 10

Fig. 5. Displacement of the rotor in the X and Y directions with LQ control.

-6

10

x 10

Amplitude [rad]

8

6

4

2

0

-2

0

0.1

0.2

0.3

0.4

0.5

Time [s]

0.6

0.7

0.8

0.9

1 -3

x 10

Fig. 6. Rotation of the rotor around the X and Y directions with LQ control.

Within 0.8 ms, the shaft is stabilized at the equilibrium position. This result proves the LQ regulator effectiveness. By examining Fig. 5, it is observed that the response fluctuates around the axes of rotation, which correspond to the transient step. This fluctuation is not noticeable in the rotation response. In fact, the weak displacement does

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not give an important rotation. So in order to ameliorate the rotor accuracy, it is indispensable to supervise the displacement behaviour by adjusting the gain of the system to eliminate vibration. To provide an efficient evaluation of LQ controller and to find the best conditions of stabilization, different gain values are tested. Results obtained after the simulation are shown in Fig. 7 and 8. From this parametric study, the optimal gain will be extracted. -6

x 10

10

K1 k2 k3

8

Amplitude [m]

6

4

2

0

-2

-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [s]

5 -4

x 10

Fig. 7. Control of the rotor displacement in the X and Y directions with different gain values -6

10

x 10

K1 k2 k3

Amplitude [rad]

8

6

4

2

0

-2

0

0.1

0.2

0.3

0.4

0.5

Time [s]

0.6

0.7

0.8

0.9

1 -3

x 10

Fig. 8. Control of the rotor rotation around X and Y directions with different gain values

It is observed that the first matrix gain K1 shows a damped behaviour with an undershoot of more than 3 μm and a reduced time of control. The behaviour is ameliorated in the case of gain K3 where the undershoot and the stabilization time are minimized. The system behaviour under the gain K2 shows the best result in term of absence of damped behaviour compared to the first and the second case with a short rise time (down to

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about 10 times compared to the gain K1). That’s why, it’s better to choose the adequate gain for the system stabilization to minimize vibration. In fact matrices Q and R, where Q defines the weights of the states and R defines the control signal weight in the cost function J are used to control the signal behaviour. The more their values are increased the more the gain values increased and the more the signal behaviour is penalized. A larger value of Q allows stabilizing the system with little changes in the states. So, because there is a trade-off between the two parameters, we kept usually Q and we altered R. In the case of a limited control current (saturation zone) we perform a less weighted energy strategy with a large R values (expensive control strategy) otherwise we choose a small R without penalizing the states behaviour (cheap control strategy). Finally, an LQ controller proves it ability to stabilize a rotor suspended by oriented symmetric MBs in high speed movement without touching magnets.

5 Conclusion In this paper, an electro-mechanical model of a high-speed rotating shaft actioned by an oriented magnetic bearing system is studied. The system diverges when the command current is not taken into account. The stabilization of this model is obtained using an LQ control method. A parametric study is elaborated to choose the adequate gain for the system stabilization with the minimum vibration. For all tested cases, the LQ controller proves its ability to stabilize a rotor supported by four magnetic bearings without touching magnets. Further works are in progress in order to introduce other methods of control and to compare them with this method. Also, asymmetrical bearings positions will be tested to prove the influence of bearings positions. Acknowledgement. The authors gratefully acknowledge the funding of Project 19PEJC10-09 financed by the Tunisian Ministry of Higher Education and Scientific Research.

References Barbaraci, G., Mariotti, G.V.: Sub-optimal control law for active magnetic bearings suspension. J. Control Eng. Technol. (JCET) 2(1), 1–10 (2012) Chen, W.J., Gunter, E.J., et al.: Introduction to Dynamics of Rotor-Bearing Systems, vol. 175. Trafford, Victoria (2007) Ding, G., Wang, H., Liu, J., Gao, B., Zhang, B.: Development of thin-slice fiber Bragg gratinggiant magnetostrictive material sensors used for measuring magnetic field of magnetic bearings. Opt. Eng. 54(10), 107102 (2015) Fan, Y.H., Chen, S.T., Lee, A.C.: Active control of an asymmetrical rigid rotor supported by magnetic bearings. J. Franklin Inst. 329(6), 1153–1178 (1992) Gu, H., Zhao, L., Zhao, H.: A LQG controller design for an AMB-flexible rotor system. In: Proceedings of 9th International Symposium on Magnetic Bearings, pp. 31–34 (2004) Hentati, T., Bouaziz, A., Bouaziz, S., Choley, J.Y., Haddar, M.: Dynamic behaviour of active magnetic bearings spindle in high-speed domain. Int. J. Mechatron. Manuf. Syst. 6(5–6), 474– 492 (2013)

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Hutterer, M., Hofer, M., Nenning, T., Schrodl, M.: LQG control of an active magnetic bearing with a special method to consider the gyroscopic effect. In: 14th International Symposium on Magnetics Bearings (2014) Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems, vol. 1, p. 608. Wiley-Interscience, New York (1972) Lurie, A.I.: On Some Nonlinear Problems in Nonlinear Control (Russian-English translation). H.M. Stationary Office, London (1951) Schweitzer, G., Lange, R.: Characteristics of a magnetic rotor bearing for active vibration control. In: Conference on Vibrations in Rotating Machinery, Cambridge, no. C239/76, pp. 301–306. Institution of Mechanical Engineers (1976) Yanliang, X., Yueqin, D., Xiuhe, W., Yu, K.: Analysis of hybrid magnetic bearing with a permanent magnet in the rotor by FEM. IEEE Trans. Magn. 42(4), 1363–1366 (2006) Youcef-Toumi, K., Reddy, S.: Dynamic analysis and control of high speed and high precision active magnetic bearings (1992) Zhuravlyov, Y.N.: On LQ-control of magnetic bearing. IEEE Trans. Control Syst. Technol. 8(2), 344–350 (2000)

Evaluation of the Acoustic Performance of Perforated Multilayer Absorber Materials Amine Makni1,2 , Marwa Kani2 , Mohamed Taktak1,2(B) , and Mohamed Haddar1 1 Laboratory of Mechanics, Modeling and Production (LA2MP), National School of Engineers

of Sfax, University of Sfax, BP N° 1173, 3038 Sfax, Tunisia [email protected] 2 Faculty of Sciences of Sfax, BP N° 1171, 3000 Sfax, Tunisia [email protected]

Abstract. Acoustic absorber based on perforated plate and air cavity presents a fundamental liner used in many acoustic applications as a liner applied at the wall of duct systems to reduce noise. The use of this kind of absorber in form of multilayer is generally used to improve their acoustic performance and the attenuation capacity is increased. Such absorber has an acoustic performance varying with their intrinsic and geometrical proprieties (perforation rate and the thickness of the plate, cavity length, etc…). To quantify and qualify the acoustic performance of these absorbers an acoustic indicator can be used which is the acoustic absorption coefficient. In this paper, the transfer matrix method is used to compute this coefficient in the case of the presence of plane wave. For this, several cases of multilayer configurations of multilayer configurations containing two and three layer are studied and investigated. After that, and for each studied configuration, a parametric study of the influence of the multilayer materials parameters on this coefficient for various configurations is carried on to deduce the more influent parameters on the acoustic performance of this kind of absorber. Keywords: Plane waves · Acoustic response · Perforated material · Multilayer absorber · Porous material · Helmholtz resonators · Transfer matrix · Absorber coefficient

1 Introduction Helmholtz resonators are generally used in many industrial fields (transport, building, etc…) for sound attenuation or impact energy dissipation. Thus, an accurate and thorough understanding of the acoustic behavior of these materials and the effect of their parameters is necessary. So a parametric study on the variation of these parameters was carried out by several researchers. In fact, (Nakai and Yoshida 2013) had made a simulation of the normal incidence absorption coefficient of perforated panels with and without glass wool. (Dengke et al. 2014) have studied the influence of the position of porous material on absorption of perforated plates. Also (Hai and Yadong 2014) studied the acoustic absorption coefficient for different perforated plate’s configurations. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 155–170, 2021. https://doi.org/10.1007/978-3-030-76517-0_18

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transfer matrix method is also used to the study of this kind of absorbers as presented in the work of Gerges and Jordan (2005), Campa and Camporeale (2012) and Campa and Camporeale (2010). To improve the acoustic performance of this kind of absorber, it can be used in form of multilayer, each layer is a resonator. For this case, an electric – acoustic analogy can be used as presented in Mina et al. (2013) and Congyun and Qibai (2005). Also, the air cavity can be replaced by a porous material to enlarge the frequency domain of the absorption. In this article, a parametric study of multilayer absorber materials composed by air, porous materials and perforated plates is presented based on the transfer matrix method. In the second section, we are interested by determining of the absorption coefficient expressed as a function of the impedance of the resonator surface. This is based on the Elnady model (Elnady and Boden 2003) to determine the impedance of the perforated plate, and the Johnson - Allard model (Sellen 2003; Henry 1997; Allard and Champoux 1992) to determine the surface impedance of the cavity. In the third section, the studied configurations are presented and in the fourth section, a parametric study of different studied configurations of multilayer materials is made to determine the influential parameters on this coefficient.

2 Theoretical Basis 2.1 The Transfer Matrix The transfer matrix method is used to model electric quadrupoles. This method can be adapted to model one-dimension acoustic system as presented in Lee and Kwon (2004). The acoustic transfer matrix links the pressures (pi ) and the acoustic velocities (ui ) in two different positions r and r + 1 in the same axe as presented in Eq. (1).        p T11 T12 pr+1 pr = [T ] r+1 = (1) ur ur+1 T21 T22 ur+1 With [T] is the transfer matrix. 2.2 Computation of Multilayer Absorber Transfer Matrix The studied multilayer materials are a succession of layers composed even by perforated plate and an air cavity (Helmholtz resonator) or by perforated plate and a porous material (absorber). So, to model multilayer absorbers, two types of matrices must be computed: the transfer matrix of the plate P and the transfer matrix of the cavity S. Then the total transfer matrix T of the layer is given by the Eq. (2):   T11 T12 (2) T =P·S= T21 T22

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2.2.1 Transfer Matrix of the Plate P1

The perforated plate (Fig. 1) is characterized by:

P2

d

– The plate thickness t, – The of perforation rate σ p , – The whole diameter d.

Impedance, ζ

P1

P2

x=0

Fig. 1. Perforated plate

The transfer matrix of the perforated plate is given by Lee and Kwon (2004) and Narayana and Munjal (1986):        1 ρ0 c0 ζ p p1 p2 = =P 2 (3) u1 u2 u2 0 1 ρ0 c0 are respectively the air density and the sound speed. ζ is the plate impedance expressed as follows (Elnady and Boden 2003):    δre ik t ζ = Re σp CD + F(ks d /2)  F(ks dp /2)

  t + F(kδsimd /2) +i Im σpikCD 

(4)

F(ks d /2)

with C D is the discharge coefficient, k = 2π c0 .f is the constant of propagation of air and f is the frequency, δ re and δ im are correction coefficients: δre = (0, 2).d + (200).d 2 + (16000).d 3 and δim = (0, 2856).d F( ks d /2) = 1 − Jd1 (ks d /2) ks 2 J0 (ks d /2) J k  d /2 and F( ks d /2 = 1 −  d1p ( s  ) ks 2 J0 (ks d /2)

  ks = −iω ν  and ks = −iω ν

(5)

(6)

(7)

With ω is an angular frequency, ks and ks are respectively the Stokes number of insulating and conductive walls, ν = μρ is the kinematic viscosity and ν =

(2, 179)μ μ = ρ ρ

μ is the dynamic viscosity and ρ is the material density.

(8)

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2.2.2 Transfer Matrix of an Air Cavity This matrix is analytically obtained by writing pressure p1 at x = 0 and pressure p2 at x = L out the form of plane wave as detailed in the work of (Beranek and Vér 1992) (Fig. 2). The matrix S of an air cavity is expressed as follows (with Z0 is the air impedance):   cos kl jZ0 sin kl (9) S= j with Z0 = ρ0 c0 Z0 sin kl cos kl

p1 u1

Ae − jkx Be x=0

jkx

p2 u2

x =l

Fig. 2. Construction of the transfer matrix of the cavity

2.2.3 Transfer Matrix of a Cavity Containing Porous Material The porous material is considered like an equivalent fluid. For this study, we use the Lafarge – Allard model (Lafarge et al. 1997) for computing the acoustic impedance (ZC ) in a porous material. This impedance is given in the following equation:  ZC = ρ(ω).KLA (ω) (10) ⎡

⎤  2 .ω.η .α σ.φ 4.j.ρ 0 ∞ ⎦ . 1+ ρ(ω) = α∞ .ρ0 .⎣1 − j ρ0 .α∞ .ω σ 2 .φ 2 . 2 ⎤

⎡ ⎢ KLA (ω) = γ .P0 .⎢ ⎣γ −

(11)

γ −1  1+

η.φ  . j.ω.ρ0 .Npr .k0

1+

 4.j.ω.ρ0 .Npr .k02  η.φ 2 . 2

⎥ ⎥ ⎦

(12)

In those equations, ω is an angular frequency, ρ(ω) is the effective density, KLA (ω) is the dynamic compressibility, α∞ is thematerial tortuosity, φ is the material porosity, σ is the material flow resistivity, γ = Cp Cv is the ratio of specific heats at respectively constant pressures C p and volumes C v , N pr is the Prandtl number, η is the dynamic viscosity, and ’ are the viscous and thermal characteristic lengths respectively, P0  is the atmospheric pressure and k0 is the thermal permeability. Finally, and when the pores of the material are saturated with air, the transfer matrix is given by:     j.ZC . sin(k(ω).L) cos(k(ω).L)  Sp = (13) (j ZC ) sin(k(ω).L) cos(k(ω).L)

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 k(ω) = ω.

With

ρ(ω) KLA (ω)

159

(14)

k(ω) is the wave number. 2.3 Computation of the Acoustic Absorption Coefficient The reflection coefficient is computed from the transfer matrix coefficients as follows: R=

T11 − ρ0 c0 T12 T11 + ρ0 c0 T21

(15)

The absorption coefficient α is given by Eq. (16) (Scarpato et al. 2013):   4Re Z ρ0 c0 α=   2    2 + Im Z ρ0 c0 1 + Re Z ρ 0 c0

(16)

Z is the system impedance.

3 Studied Configurations Firstly, the multilayer Helmholtz resonators are obtained with a series of perforated plates separated by air cavity (Fig. 3). Table 1 defines the geometric parameters of the different used layers. Secondly, the air cavity is replaced by a porous material. Three kinds of porous material were used. Table 2 gives the acoustic intrinsic parameters of these materials. This study was performed to determine the effect, on the absorber coefficient versus frequency, of the layers order, the plate and the cavity geometric parameters, the porous material kind and its intrinsic parameters. The parametric study was realized for the two and three layers materials. Table 1. Geometrical parameters of studied layers Thickness of the plate t (mm)

Diameter of the holes d (mm)

Rate of perforation σp

Length of the cavity Lc (mm)

Layer 1 (M1)

1

1

0.025

21

Layer 2 (M2)

0.8

0.3

0.05

20

Layer 3 (M3)

1

1

0.025

8

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α∞ : tortuosity

Rockwool

Glass wool

Acousticel

2.1

1

1.38

σ : resistivity (N.m−4 .s)

135000

9000

22000

∅: porosity

0.94

0.99

0.95

η: viscosity (Pa.s)

0.1

0.1

0.1

: viscous characteristic length (m)

0.49 e−4

1.92 e−4

1.7 e−4

: thermal characteristic length (m)

166 e−6

576 e−6

5.1 e−4

k0 : thermal permeability (m2 )

k0 = η/σ

k0 = η/σ

0.83 e−8

γ : report of the heat capacity and mass

1.4

1.4

1.4



Perforated plates

Perforated plates

Air cavity Air cavity

a)

Air cavity Air cavity Air cavity

b)

Fig. 3. (a) Two layers Helmholtz resonator, (b) Three layers Helmholtz resonator

4 Results and Discussion 4.1 Multilayer Resonators The curves of absorption coefficient versus frequency show a number of peaks of absorption equal to the number of the layers. The parametric study was performed firstly on the plate’s parameters (the plates order, the plate thickness, the holes diameter and the plate rate of perforation) and secondly on the cavity parameters (the cavity length). 4.1.1 Effect of the Plate’s Order According to the Fig. 4, the better absorption is obtained when the layer 2 is placed in the first position with values reaching 1. In terms of peaks frequencies, we note a slight modification between all the configurations, except in the second peak of the three layers’ system (Fig. 4b) when we place the third layer in the first place. This can be explained by the fact that the layer 2 has the perforated plate with the greatest perforation rate that increases the acoustic resistivity of the layer and therefore the increases the absorption phenomenon.

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Fig. 4. Effect of the plate’s order in a: (a) two layers resonator, (b) three layers resonator

4.1.2 Effect of the Plate’s Thickness Figure 5 shows that this parameter has a low effect on the first peak value and its frequency. In all configurations, when the thickness of the first plate increases (Figs. 5a and 5c), the absorption coefficient decreases and the peak’s appearance frequency remains constant (in the second and the third peak). But when the thickness of the second or third plate increases, the absorption coefficient increases (except in Fig. 5d) and the peak’s appearance frequency decreases (in the second and the third peak). 4.1.3 Effect of the Plate’s Holes Diameter According to Fig. 6a and Fig. 6c, we note that the evolution of absorption coefficient value is inversely proportional to that of the holes diameter of the plate 1 in all the absorption peaks. When the hole’s diameter of the plate 2 or those of the plate 3 increases, the absorption coefficient value increases, this is observed on the second and third peaks of Fig. 6. But on the first peaks, the effect of the plate’s 2 holes’ diameters and the plate’s 3 holes’ diameter are different: The increasing of the plate’s 2 holes’ diameters causes a decreasing of the absorption coefficient value (Fig. 6b and Fig. 6d), and the increasing of the plate’s 3 holes’ diameters have no effect on the absorption coefficient value (Fig. 6e). Also, the hole’s diameter variation has no effect on the peaks frequencies. 4.1.4 Effect of the Plate’s Perforation Rate In Fig. 7, the absorption coefficient values in the first peaks are modified only when the plate 1 perforation rate change (Fig. 7a and Fig. 7c). So when the plate 1 perforation rate increases, the peak value decreases and the peak appearance frequency increases. In the second and third peaks, the evolution of the absorption coefficient value is proportional to that of the plate 1 perforation rate, and is inversely proportional to that of the plate 2 and the plate 3 perforation rates. In those peaks, the plate’s 1 perforation rate has no effect on the peak’s occurrence frequency. The latter presents a proportional evolution with that of the plate’s 2 and plate’s 3 perforation rate.

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Fig. 5. Absorption coefficient versus frequency resulting from the thickness variation of: (a) plate 1 in two layers resonator, (b) plate 2 in two layers resonator, (c) plate 1 in three layers resonator, (d) plate 2 in three layers resonator, (e) plate 3 in three layers resonator

4.1.5 Effect of the Cavity Length Figure 8 shows that the cavity length evolution has a low effect on the absorption coefficient value and the peak’s frequencies. In the second and third peaks, the peak’s frequencies decrease when the layer’s 1, layer’s 2 or layer’s 3 cavity length increase. The absorption coefficient values increases when the layer’s 2 or layer’s 3 cavities lengths increases, and its decreases when the slayer’s 1 cavity length increases. 4.2 Multilayer System with Cavities Containing Porous Material When the cavity contains a porous material, the absorption peaks become wider and the absorption covers a larger frequency domain. The parametric study was performed firstly on the plate’s parameters with Rockwool porous material, and secondly on the

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Fig. 6. Absorption coefficient evolution versus frequency resulting from the holes diameter variation of: (a) plate 1 in two layers resonator., (b) plate 2 in two layers resonator, (c) plate 1 in three layers resonator, (d) plate 2 in three layers resonator, (e) plate 3 in three layers resonator

porous material’s parameters. The porous material’s parameters are material’s order, the tortuosity and the viscous characteristic length. 4.2.1 Effect of the Plate’s Order According to Fig. 9, we obtained the same curves shape between the two and three layers systems. The curves, in this figure, show two types of evolutions. The first type is when the system 1 or system 3 is placed in the first case; the absorption reaches the value of 30% of absorption, and then decrease. The second type, when the system 2 is placed in the first position, present a continuous increasing of the absorption, the maximum measured is 33% of absorption. As showed in the first study, when the layer 2 (having the greatest perforation ratio) is in the first place, the multilayer system is more absorbent.

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Fig. 7. Absorption coefficient evolution versus frequency resulting from the perforation rate variation of: (a) plate 1 in two layers resonator, (b) plate 2 in two layers resonator, (c) plate 1 in three layers resonator, (d) plate 2 in three layers resonator, (e) plate 3 in three layers resonator

4.2.2 Effect of the Plate’s Thickness Using Rockwool porous material and according to Fig. 10, only the variation of the plate’s thickness n°1 has an effect on the absorption coefficient. So in the two or three layer systems, the increasing of this thickness causes the decreasing of the maximum value of the absorption and his occurrence frequency. 4.2.3 Effect of the Plate’s Hole’s Diameter In Fig. 11, only the effect of the hole diameter variation of plate 1 was presented because there are the only parameters that affect the absorption coefficient. So when this parameter increases, the absorption coefficient value and his occurrence frequency decreases slightly.

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Fig. 8. Absorption coefficient evolution versus frequency resulting from the cavity length variation of: (a) system 1 in two layers resonator, (b) system 2 in two layers resonator, (c) system 1 in three layers resonator, (d) system 2 in three layers resonator, (e) system 3 in three layers resonator

Fig. 9. Influence of the plate’s order using the Rockwool porous material in: (a) two layers absorber, (b) three layers absorber

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Fig. 10. Absorption coefficient evolution versus frequency resulting from the thickness variation of: (a) plate 1 in two layers absorber, (b) plate 2 in two layers absorber, (c) plate 1 in three layers absorber, (d) plate 2 in three layers absorber, (e) plate 3 in three layers absorber

Fig. 11. Absorption coefficient evolution versus frequency resulting from the holes diameter variation of: (a) plate 1 in two layers absorbent. (b) plate 1 in three layers absorber

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4.2.4 Effect of the Plate’s Perforation Rate The same effect of the perforation rate variation of plate 1 is observed on the absorption coefficient evolution in the two and three layers absorbers. We obtained a proportional evolution of the absorption value and the absorption maximum occurrence frequency compared with the plate’s perforation rate evolution (Fig. 12).

Fig. 12. Absorption coefficient evolution versus frequency resulting from the perforation rate variation of: (a) plate 1 in two layers absorber, (b) plate 1 in three layers absorber

4.2.5 Effect of the Porous Material’s Order Using plate 3, the better absorption coefficient, in two and three layers absorber, is detected when the “rigid glass wool” is placed in the first position. This is observed at frequencies less than 3000 Hz. At higher frequencies (frequencies greater than 4000 Hz), the system with Rockwool porous material, placed in the first position, gives the maximum of absorption (Fig. 13).

Fig. 13. Influence of the porous material’s order in: (a) two layers absorber, (b) three layers absorber

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4.2.6 Effect of the Porous Material’s Tortuosity In the two layers absorber, the order of layers is: layer 1 backed by the layer 2. In the three layers absorber, the order of the systems is: layer 1 backed by layer 2 backed by the layer 3. The principal effects are observed when the variation of the tortuosity is made on the first porous material in front of the incident wave. This is the same using an absorber with two and three layers absorbers. In those absorbers and according to Fig. 14, the absorption coefficient evolution is proportional to that of the porous material’s tortuosity. This behavior is reversed when the frequency reaches the value of 5000 Hz in absorber with two layers absorber, and 2700 Hz in absorber with three layers absorber.

Fig. 14. Influence of the porous material’s tortuosity in: (a) two layers absorber, (b) three layers absorber

4.2.7 Effect of the Viscous Characteristic Length In the two layers absorber, the order of the layers is: layer 1 backed by the layer 2. In the three layers absorber, the order of layers is: layer 1 backed by layer 2 backed by the layer 3. The major effects are observed when the variation of the tortuosity is made on the first porous material in front of the incident wave. This is the same using an absorber with two (Fig. 15a) and three degrees of freedom (Fig. 15c). According to Fig. 15, the absorption coefficient evolution is proportional to that of the porous material’s viscous characteristic length. This behavior is reversed when the frequency reaches the value of 6000 Hz in absorber with two layers absorber, and 2800 Hz in absorber with three layers absorber.

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Fig. 15. Absorption coefficient evolution versus frequency resulting from the viscous characteristic length variation of: (a) material 1 in two layers absorber, (b) material 2 in two layers absorber (c) material 1 in three layers absorber, (d) material 2 in three layers absorber, (e) material 3 in three layers absorber

5 Conclusion In this paper, the transfer matrix method is used to determine the absorption coefficient of multilayer absorbers composed by perforated plates, air and porous materials. Then, a parametric study of the geometric and intrinsic parameters is made to determine the influence of geometrical and intrinsic parameters of these absorbers on the absorption coefficient. For multilayer resonators, it is observed that the parameters of the first plate have no influence on the first peak of the absorption. These parameters have an effect on the frequency and the values of the second and third peaks of absorption in the case of two or three layers’ resonators. For multi layers containing porous materials, the results showed that the variations in the tortuosity and the viscous characteristic length have an

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influence on the absorption coefficient in the opposite of other parameters like the flow resistivity and porosity.

References Nakai, T., Yoshida, K.: Simulation of normal incidence sound absorption coefficients of perforated panels with/without glass wool. In: Proceedings of Meetings on Acoustics, Montreal, Canada, vol. 19, p. 015043 (2013) Dengke, L., Daoqing, C., Bilong, L., Jing, T.: Improving sound absorption band width of microperforated panel by adding porous materials. In: Proceedings of Inter-Noise Congress (2014) Hai, X., Yadong, L.: Sound absorption characteristics of the perforated panel resonator with tube bundles. In: Proceedings of the 21st International Congress on Sound and Vibration, ICSV 21 (2014) Gerges, S.N.Y., Jordan, R.: Muffler modeling by transfer matrix method and experiment entail verification . J. Braz. Soc. Mech. Sci. Eng. 17(2), 132–140 (2005) Campa, G., Camporeale, S.M.: Eigenmode analysis of the thermo acoustic combustion instabilities using a hybrid technique based on the finite element method and transfer matrix method. Adv. Appl. Acoust. 1, 1 (2012) Campa, G., Camporeale, S.M.: Application of transfer matrix method in acoustics. In: Proceedings of the COMSOL Conference (2010) Elnady, T., Boden, H.: On semi-empirical liner impedance modeling with grazing flow. In: Proceedings of 9th AIAA/CEAS (2003) Sellen, N.: Changing the surface impedance of an active control material: application to the characterization and optimization of an acoustic absorber,. Ph.D. thesis, Lyon central school, France (2003) (in French) Henry, M.: Measurements of the parameters characterizing a porous medium. Experimental study of acoustic behavior of foams at low frequencies, Ph.D. thesis, Maine University, France (1997) (in French) Allard, J.F., Champoux, Y.: New empirical equations for sound propagation in rigid frame fibrous materials. J. Acoust. Soc. Am. 91(6), 3346–3353 (1992) Lee, D.H., Kwon, Y.P.: Estimation of the absorption performance of multiple layer perforated panel systems by transfer matrix method. J. Sound Vib. 278, 847–860 (2004) Narayana, R.K., Munjal, M.L.: Experimental evaluation of impedance of perforates with grazing flow. J. Sound Vib. 108(2), 283–295 (1986) Beranek, L., Vér, I.L.: Noise and Vibration Control Engineering: Principles and Applications. Wiley-Interscience, New York (1992) Lafarge, D., Lemarinier, P., Allard, J.F., Tarnow, V.: Dynamic compressibility of air in porous structures at audibles frequencies. J. Acoust. Soc. Am. 102(4), 1995–2006 (1997) Scarpato, A., Ducruix, S., Schuller, T.: Optimal and off-design operations of acoustic dampers using perforated plates backed by a cavity. J. Sound Vib. 332, 4856–4875 (2013) Mina, S., Nagamura, K., Nakagawa, N., Okamura, M.: Design of compact micro-perforated membrane absorbers for polycarbonate pane in automobile. Appl. Acoust. 74(4), 622–627 (2013) Congyun, Z., Qibai, H.: A method for calculating the absorption coefficient of a multi-layer absorbent using the electro-acoustic analogy. Appl. Acoust. 66, 879–887 (2005)

An Anisotropic Model with Non-associated Flow Rule to Predict HCP Sheet Metal Ductility Limit Mohamed Yassine Jedidi(B) , Anas Bouguecha, Mohamed Taoufik Khabou, and Mohamed Haddar Laboratoire de Mécanique, Modélisation et de Production (LA2MP), École Nationale D’Ingénieurs de Sfax (ENIS), Route Soukra Km 3.5, Sfax, Tunisia [email protected], [email protected], [email protected]

Abstract. In this paper, an anisotropic plasticity model is coupled with Marciniak and Kuczynski localized necking criterion to predict the ductility limit of hexagonal closed packed (HCP) crystal structure sheet metals submitted to uniaxial and biaxial stretching. The anisotropic plastic evolution at room temperature is in fact described by the constitutive model using the non-associated flow rule (non AFR). In the current investigation, the plastic potential and the yield function are independently defined by taking into account the quadratic anisotropic yield criterion Hill48 which leads to the accurate prediction of the progressive plastic anisotropy as well as the material behavior using a minimal set of material parameters. To predict forming limit diagrams (FLDs), An efficient implicit algorithm is developed to solve the necessary constitutive equations to provide the coupling between Marciniak and Kuczynski instability approach and the anisotropic plasticity model with non-AFR. The numerical simulations are compared with several reference results from the literature to validate the mechanical behavior of the particular HCP magnesium alloy AZ31. Then, the ductility limit of this particular HCP magnesium alloy is numerically predicted and compared with analytical results of forming limit diagrams from the literature based on the associated and non-associated flow rule. Keywords: Anisotropic plasticity · Hexagonal closed packed material · Non-associated flow rule · Plastic instability · Forming limit diagrams

1 Introduction Due to their lightweight, their high specific strength as well as their high fatigue and corrosion resistance, the materials with hexagonal closed packed (HCP) crystal structure are becoming an important matter for researchers and engineers of especially aircraft, automotive, aerospace and prostheses device industries. Despite their benefits, HCP materials such as zirconium, beryllium, titanium and magnesium alloys have a limited formability at room temperature. Despite the scarcity of basic information on their formability limit, many researchers have succeeded to model the mechanical behavior of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 171–182, 2021. https://doi.org/10.1007/978-3-030-76517-0_19

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HCP materials at room temperature (Cazacu et al. 2006; Gilles et al. 2011…) thanks to their plastic anisotropy and plastic deformation. However, predicting the ductility limit for HCP materials at room temperature is presented in limited works from the literature (Wu et al. 2015; Kondori et al. 2018; Abedini et al. 2018…). Many phenomenological yield functions (Hill 1948; Barlat et al. 1991; Cazacu et al. 2006…) are implemented in numerical tools to predict the isotropic or anisotropic behavior of a material, and most anisotropic yield functions are based on the associated flow rule (AFR) hypothesis where the equality of yield function and potential for plastic strain rate enforced by the AFR assumption describes both the elastic limit (yielding) and the flow direction (plastic strain rate). Due to the limitation of the AFR concept to predict material plasticity, many researchers have adopted the non-associated flow rule (non-AFR) using two independent functions to describe the elastic limit and the plastic strain rate direction. The non-AFR concept takes into consideration the strong plastic anisotropy and the large gradients on the curvature without any convergence problems due to the separation of yield and plastic potential functions (Stouthon and Yoon 2006) and the behavior in bi-axial tension was modeled by Yoon et al. (2007) using the aluminum alloy. More recently, Hippke et al. (2018) and Oya et al. (2018) justify the use of the non-associated flow rule concept to describe the plastic anisotropy of HCP materials such as AZ31B magnesium alloy which is studied at room temperature by Ahn and Seo (2018) using the simplest anisotropic yield and plastic potential function to predict the anisotropic evolution and the isotropic hardening effects. Ahn and Seo (2018) have used in their contribution the maximum shear stress diffuse necking as the necking theory to predict FLDs for AZ31b magnesium alloy. Actually in our investigation, we adopt the constitutive framework developed by Ahn and Seo (2018) due to the simplicity and reduced material parameters required by the Hill48 quadratic anisotropic yield function. A numerical tool based on the coupling between the constitutive framework and the localized necking criteria is developed to predict the occurrence of localized necking in the shape of forming limit diagrams. In fact several instability criteria (Swift 1952; Hill 1952, Marciniak and Kuczynski 1967; Rudnicki and Rice 1975; Dudzinski and Molinari 1991; Aretz 2004; Signorelli et al. 2019…) are developed to predict the occurence of plastic instability in thin metal sheets. In our investigation we adopt the (Marciniak and Kuczynski 1967) initial imperfection approach to predict the occurrence of the localized necking in HCP magnesium alloy AZ31b using a non-AFR hypothesis based on analytical framework and experimental application of Ahn and Seo (2018). A brief outline of this paper is presented by 5 sections. We have dedicated the Sect. 2 of this paper to the constitutive framework based on the non-AFR to model the mechanical behavior of HCP materials as well as the main equations which leads to define the M-K necking criterion whereas Sect. 3 outlines the numerical implementation of the M-K instability criterion and constitutive equations of Sect. 2. The Sect. 4 provides the validation of the developed model and the numerical investigation by comparing the numerical results with the analytical and experimental ones obtained by Ahn and Seo (2018). Finally, Sect. 5 closes the present contribution by some conclusions and perspectives.

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2 Theoretical Framework 2.1 Yield Criterion and Plastic Potential Function The yield criterion CPB06 (Cazacu et al. 2006) is the better adapted to the hexagonal closed packed (HCP) materials due to their strong plastic anisotropy and the yielding asymmetry between tension and compression. However, our investigation is based on the work of (Ahn and Seo 2018) and we use the simplest quadratic yield criterion Hill 48 (Hill 1948) under the plane stress condition assuming the coincidence of the direction of principal stresses with the symmetry axes of orthotropic sheet metal. Thus, the effective stress is described as below:    Y2 2 Y2 Y2 σ − 1 + 2 − 2 σ1 σ2 + σ22 , (1) σ˜ = X2 1 X B where X is the yield stress along the rolling direction, Y is the yield stress along the transverse direction, and B is the balanced bi-axial yield stress. F, G and H are the anisotropic coefficients determined by the yield stresses as following: 2G =

1 1 1 1 1 1 1 1 1 − 2 + 2 , 2F = − 2 + 2 + 2 , 2H = 2 + 2 − 2 , X2 Y B X Y B X Y B

(2)

According to Ahn and Seo (2018) we emphasize the use of σ2 ≥ σ1 ≥ 0, where σ1 and σ2 are the principle stresses described by the Cauchy stresses components in a plane stress condition σ33 = 0 with:    1 2 2 2 σ + σ22 − σ11 + 4σ12 − 2σ11 σ22 + σ22 σ1 = 2 11   (3)  1 2 2 2 σ + σ22 + σ11 + 4σ12 − 2σ11 σ22 + σ22 , σ2 = 2 11 The plastic flow is defined by the plastic potential and is obtained using different anisotropic coefficients which are determined by measuring the yield stress along the transverse direction as well as the Lankford coefficients (R-values) along rolling direction (R0 ) and transverse direction (R90 ). The plastic potential is given by:  1 + R0 2 1 + R90 2 φ= σ1 − 2σ1 σ2 + σ2 , (4) R0 R90 As mentioned by (Sect. 2.2) the anisotropy coefficients of Hill48 criterion for both yield function and potential function are distinguished by the case of associated flow rule where σ˜ = φ and coefficients parameters are the same for yield and potential function, and the case of non-associated flow rule where σ˜ = φ and the anisotropy coefficients of yield and plastic potential functions are independent.

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2.2 Associated and Non-associated Flow Rules For both rules of associated flow and non-associated one the yield criterion is: ψ = σ˜ (σ) − σ iso (γ ),

(5)

where the isotropic yield stress σ iso (γ ) is a function of the equivalent plastic strain given by the Swift equation as following: σ iso = σ˜ = K(γ0 + γ )n ,

(6)

where K, γ0 and n are the coefficients of the material hardening and are identified by tensile tests (Ahn and Seo 2018). Supplemented by the equivalence relationship of the plastic work rate (Eq. 7) and Euler’s theorem the AFR and non-AFR are respectively expressed by Eq. 8 and Eq. 9: γ˙ σ˜ = d : σ,

(7)

σ:

∂ σ˜ = σ˜ (σ), ∂σ

(8)

σ:

∂φ = φ(σ), ∂σ

(9)

where γ˙ is the equivalent plastic strain rate and the operator “:” is the double contraction of σ and ∂φ ∂σ . The plastic strain rate is given by the AFR and the yield function is regarded as the plastic potential function. In this case the plastic strain rate tensor is: d = λ˙

∂ σ˜ , ∂σ

(10)

where λ˙ is the plastic multiplier and the second order tensor ∂∂σσ˜ is the first gradient of the yield function. In the case of non-AFR concept, the plastic potential and the yield function are independently defined and the plastic strain rate is regarded as normal only to the plastic potential surface. The plastic strain rate is given in this case by: d = λ˙

∂φ , ∂σ

(11)

where the second order tensor ∂φ ∂σ is the first gradient of the plastic potential function. Using the non-AFR concept with the substitution of Eq. (11) in Eq. (7) and by applying Eq. (9) we obtain a relationship between the equivalent plastic strain rate and the plastic multiplier as following: γ˙ =

λ˙ ∂φ : σ φ d:σ = ∂σ = λ˙ . σ˜ σ˜ σ˜

(12)

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We should remind that the plastic potential function φ and yield criterion σ˜ equality in case of the AFR concept leads to the equivalence of the plastic strain rate γ˙ and the plastic multiplier λ˙ . Governed by the Kuhn-Tucker constraints and due to the activation of the plastic deformation for the two proposed flow rules the plastic flow is: ψ = σ˜ − σ iso ≤ 0; γ˙ ≥ 0; ψ γ˙ = 0

(13)

The Fischer-Burmeister constraints (Fischer 1992, 1997) are used by Akpama et al. (2016) to replace the Kuhn-Tucker constraints and we adopt also the FischerBurmeister constraints to be implemented in the elastoplastic prediction algorithm of our investigation as following:    2 (14) ψFB = σ˜ − σ iso + γ˙ 2 − σ˜ − σ iso + γ˙ = 0

2.3 Initial Imperfection Ductility Approach The theoretical M–K approach is described by two zones of the sheet, the homogeneous zone (H) and the band zone (B) as shown in Fig. 1. The band is characterized by its orientation θ with respect to the major strain direction, and its relative thickness with respect to the homogeneous zone. The relative thickness f = t B /t H is called the imperfection factor which depends on the current thicknesses t H in zone “H” and t B in zone “B”.

Fig. 1. Marciniak–Kuczy´nski (M–K) model.

The current band orientation θ is related to its initial counterpart θ0 by the following relation: tang(θ ) = tang(θ0 )e



H −ε H ε11 22



,

(15)

H and ε H are the major and the minor strain in the plane of the sheet (in the where ε11 22 homogenous zone).  N ˙ . e3 = 0 and g . e3 = 0),  Based on the plane stress condition (σ . e3 = 0, B H the linking of the velocity gradient g in the band zone with its counterpart g in the homogeneous zone is covered by the kinematic compatibility condition between zone

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“H” and zone “B”, and the balance between the band and the homogeneous zone are respectively: .

n, gB = gH + c ⊗

(16)

  B H  −ε33 ˙ B = n . N ˙ H ⇔ f0 e ε33 ˙H ˙ B = n . N n . N f n . N

(17)

.

where c is the jump vector, n is the normal vector to the band with θ ∈ [0°, 90°] and the operator “ ⊗ “ is the tensor product. f = t B t H is the current imperfection factor and H and ε B as it can be described by its initial counterpart f0 and the strain components ε33 33 H H ˙ ˙ following, N and N are respectively the nominal stress state for homogeneous zone and band zone.

3 Implicit Incremental Algorithm Actually a coupling between the constitutive framework and the M-K ductility approach is adopted to identify the unknowns using the following non-linear equations: Actually a coupling between the constitutive framework and the M-K ductility approach is adopted to identify the unknowns using the following non-linear equations: • The plane-stress condition: (18) (19) • The non-associated flow rule defined by Eq. 11: (20)

H = ε H = ε H = ε H = 0. with ε23 32 13 31

(21) B = ε B = ε B = ε B = 0. with ε23 32 13 31

• The Fischer-Burmeister conditions specified by Eq. 14: (22) (23)

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• For the M-K approach, the combination of the Eq. 16, the Eq. 17 is given by: (24) H B where L˜ and L˜ are the tangent modulus in homogeneous and band zones respectively. The unknowns are represented by a vector X such as:  H H H H B B B B (25) , ε11 , ε22 , ε12 , ε33 , ε11 , ε22 , ε12 , c˙ 1 , c˙ 2 X = ε33

and the above non-linear equations from Eq. 18 to Eq. 24 are reduced to the next equation: A(X) = 0

(26)

The implicit numerical solving of this equation is based on the predefined function “FindRoot” of Mathematica inspired of the Newton-Raphson algorithm.

4 Results and Discussions The many uses of the AZ31 magnesium HCP alloy in several engineering applications such as aircraft and automotive industries as well as the availability of its parameters in the literature are behind choosing this alloy for our numerical investigation. To characterize the anisotropic mechanical properties of this alloy, the parameters of Table 1 are obtained by Ahn and Seo (2018) using uniaxial tensile tests at room temperature for both rolling and transverse directions with an initial imperfection factor equals to 0.98. Table 1. Material parameters of the AZ31 magnesium alloy (stress-like parameters are expressed in MPa). E

υ

K

γ0

n

R0

R90

X

Y

B

45000

0.35

438.30

0.0107

0.16

2.222

2.2857

271

275.8

331.9

such as: • • • • •

K , γ0 , n

Young’s modulus Poisson’s ratio Hardening parameters

R0 , R90 X,Y, B

Lankford coefficients Yield stresses

E

The anisotropic properties of AZ31 magnesium sheet alloy are determined by Ahn and Seo (2018) by subjecting to uniaxial tensile tests the specimen cut in rolling and

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transverse direction of the sheet, and these anisotropic properties are used to review the accuracy and to validate our numerical simulations based on the anisotropic modelling with the non-AFR concept as shown by Fig. 2. A good correlation between numerical predictions and experiments is obtained for both rolling and transverse directions as shown by Fig. 2(a) and Fig. 2(b) and the small difference between hardening curves of Fig. 2(c) is due to the condition σ2 ≥ σ1 already outlined by Eq. 3 of Sect. 2.1.

400

400

350

350 300

True stress (MPa)

True stress (MPa)

300 250 200 150 100

200 150 100

Our prediction Ahn and Seo (2018)

50 0 0,00

250

0,05

0,10

0,15

0,20

0,25

50 0 0,00

0,30

True plastic strain

Our prediction Ahn and Seo (2018) 0,05

0,10

0,15

0,20

0,25

0,30

True plastic strain

(a)

(b) 400 350

True stress (MPa)

300 250 200 150 100

RD TD

50 0 0,00

0,05

0,10

0,15

0,20

0,25

0,30

True plastic strain

(c)

Fig. 2. Comparison by Ahn and Seo (2018) of rheological curves for AZ31 magnesium alloy: (a) Validation of the implementation of the yield surface in rolling direction (RD); (b) Validation of the implementation of the yield surface in transverse direction (TD); (c) Difference between the yield surface in rolling and transverse direction.

We have carried out a comparison between non-AFR and AFR concepts to validate our numerical developped tool and also to see the influence of these two concepts on the material mechanical behavior as sown by Fig. 3. We remind that the Hill48 quadratic anisotropic criterion is adopted to independently describe the yielding and potential functions to develop the non-AFR concept in the current investigation. Actually to describe the yield and the potential functions of the AFR concept in this paper we distinguish a case 1 using the parameters X, Y and B of Table 2 and a case 2 based on Lankford coeffecients. Justified by the strong influence of yield stresses parameters along rolling and

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transverse directions, a good agreement between the hardening curves of the non-AFR and the AFR (case 1) is shown by Fig. 3(a) while the AZ31 magnesium alloy Lankford parameters (R0 , R90 ) identified by Lou et al. (2007) lead to the case 2 hardening curve which is lower than the case 1. Similar comments can be made about the equivalent plastic strain curves of the Fig. 3(b) due to the Swift hardening relationship between the yield stress σiso and the equivalent plastic strain γ .

400 350

0,4

0,3

250 200

γ

True stress (MPa)

300

0,2

150 100

Non-AFR AFR (Case 1) AFR (Case 2)

50 0 0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,1

Non-AFR AFR (Case 1) AFR (Case 2)

0,0 0,0

0,1

0,2

True plastic strain

(a)

0,3

0,4

ε11

(b)

Fig. 3. Comparison between plasticity model with non-AFR and AFR for AZ31 magnesium alloy along the rolling direction (RD): (a) Hardening curves evolutions; (b) Evolution of the equivalent plastic strain γ as a function of ε11 .

Actually we intend to predict the onset of the localized necking in a AZ31 magnesium sheet alloy using the M-K ductility approach and the developed constitutive framework. Due to the deviation of magnesium sheet properties, the Fig. 4 shows a confusion in the area of separation of the strain values of the fail and the safe zones according to Ahn and Seo (2018). The analytical FLD is compared by this figure with numerical FLD predicted by our investigation. Let’s remind that analytical values and numerical prediction are based on the nonAFR using the quadratic Hill48 criterion. However, the necking approach used by Ahn and Seo to predict the ductility limit is the maximum shear stress criterion which is classified as diffuse necking. The difference between the predicted and the measured FLD using the M-K localized necking criterion and the maximum shear stress diffuse necking criterion respectively is shown by Fig. 4. In case of uniaxial strain path for negative strain, the analytical and predicted FLDs have a good agreement until 6% of major strain which corresponds to the numerically predicted major strain. Moreover, the analytical major strain given by Ahn and Seo (2018) for uniaxial strain path is almost twice our numerical predicted one. This result does not mean that diffuse necking (analytical results) precedes localized necking (our numerical predictions) for this strain path, but is justified by the boundary conditions of the M-K localized necking criterion and the continuity of calculation of minor and major strains using analytical equations of (Ahn and Seo 2018). The strains predicted numerically between and outstrip the measured strains by diffuse necking. this result is demonstrated by the imposition of

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ε11

0.08

0.06

safe zone fail zone Ahn and Seo (2018) Our prediction

0.04

0.02

-0.06

0.00 -0.03 0.00

ε22 0.03

0.06

Fig. 4. Analytical and numerical forming limit diagrams (FLDs).

the initial geometrical imperfection. For strain path upper than 0.5, strain localization predicted by the M-K approach is greater than strain predicted by the diffuse necking. Toward a better prediction of the ductility limit for magnesium alloy using the nonAFR, we must use the localized necking because if the diffuse necking is observed, we will obtain a localized necking until failure when the diffuse necking is complete. Furthermore, the diffuse necking is observed if our investigation is subjected to tensile tests with small strain rate. However, localized necking can take place under high speed loading path of the same used material. We focus on the following on the difference between the AFR and the non-AFR and their effect on the ductility limit as shown by Fig. 5. We use the same cases of the AFR introduced by Sect. 4. We conclude that the prediction based on AFR (case 1, case 2) and non-AFR leads to a similar shape FLD curves with different magnitude as shown in Fig. 5. For the negative strain paths and plane strain path, the second case of the associated flow rule leads to major strains higher than the ones obtained by the AFR (case 1) and the non-AFR due to the influence of limited slip system at room temperature which requires to choose large values of Lankford coefficients. In Fig. 5, we conclude that the minimum values of true stress can give a FLD with highest values of major strains. For positive strain paths, the magnitude of curve dedicated for the second case decreases, but it increases for the others. This can be explained by the influence of the yield stresses along rolling and transverse directions and especially the influence of the balanced biaxial yield stress. For equi-bi-axial strain path, the two cases of AFR have very close major strains due to the low deformation. This result is justified by increasing yield stress parameters of the first case and decreasing R-values of the second case when the band orientation increases.

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181

ε11

0.08 Non-AFR AFR (case 1) AFR (case 2)

0.06

0.04

0.02

-0.06

-0.04

0.00 -0.02 0.00

ε22 0.02

0.04

0.06

Fig. 5. Comparison between forming limit diagrams using AFR and non-AFR.

5 Conclusions A quadratic anisotropic Hill48 criterion is used in this paper to model the mechanical behavior of HCP materials taking into account both AFR and non-AFR hypothesis. A preliminary study is developed to assess and validate the mechanical behavior of the AZ31 magnesium sheet before predicting the ductility limit. This comparison is based on our numerical results and analytical values of Ahn and Seo (2018). It is also used to compare analytical FLD with the predicted ductility limit for HCP sheet metals using the non-AFR. It is well known that the analytical approach based on the diffuse necking and our numerical investigation based on the localized necking have different assumption and despite this difference, a good correlation is observed between our numerical FLD and the result of Ahn and Seo (2018). To predict the forming limit diagrams, a comparitive study between constitutive framework is conducted using AFR and non-AFR concepts. The method of M-K analysis with non-AFR is finally mentioned to provide a suitable framework to predict the ductility limit of HCP materials. One can note the implementation of the Rice bifurcation criterion with the non-AFR concept but it does not work for positive part of the FLD due to the limited assumptions of this criterion. Hence, the method of the linear stability analysis may allow to better predict the ductility limit of HCP materials taking into account the non-AFR.

References Aretz, H.: Numerical restrictions of the modified maximum force criterion for prediction of forming limits in sheet metal forming. Modell. Simul. Mater. Sci. Eng. 12(4), 677–692 (2004) Abedini, A., Butcher, C., Worswick, M.: Application of an evolving non-associative anisotropicasymmetric plasticity model for a rare-earth magnesium alloy. Metals 8(12), 1013 (2018)

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Ahn, K., Seo, M.H.: Effect of anisotropy and differential work hardening on the failure prediction of AZ31B magnesium sheet at room temperature. Int. J. Solids Struct. 138, 181–192 (2018) Barlat, F., Lege, D.J., Brem, J.C.: A six-component yield function for anisotropic materials. Int. J. Plast 7(7), 693–712 (1991) Cazacu, O., Barlat, F.: Application of the theory of representation to describe yielding of anisotropic aluminum alloys. Int. J. Eng. Sci. 41(12), 1367–1385 (2003) Cazacu, O., Plunkett, B., Barlat, F.: Orthotropic yield criterion for hexagonal closed packed metals. Int. J. Plast 22(7), 1171–1194 (2006) Dudzinski, D., Molinari, A.: Perturbation analysis of thermoviscoplastic instabilities in biaxial loading. Int. J. Solids Struct. 27(5), 601–628 (1991) Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992) Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Program. 76(3), 513–532 (1997) Gilles, G., Hammami, W., Libertiaux, V., Cazacu, O., Yoon, J.H., Kuwabara, T., Habraken, A.M., Duchêne, L.: Experimental characterization and elasto-plastic modeling of the quasi-static mechanical response of TA-6 V at room temperature. Int. J. Solids Struct. 48(9), 1277–1289 (2011) Hill, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. Lond. Ser. A Math. Phys. Sci. 193(1033), 281–297 (1948) Hill, R.: On discontinuous plastic states, with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1(1), 19–30 (1952) Hippke, H., Manopulo, N., Yoon, J.W., Hora, P.: On the efficiency and accuracy of stress integration algorithms for constitutive models based on non-associated flow rule. Int.J. Mater. Form. 11(2), 239–246 (2017). https://doi.org/10.1007/s12289-017-1347-6 Kondori, B., Madi, Y., Besson, J., Benzerga, A.A.: Evolution of the 3D plastic anisotropy of HCP metals: Experiments and modeling. International Journal of Plasticity (2018, in Press) Marciniak, Z., Kuczy´nski, K.: Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9(9), 609IN1613–612IN2620 (1967) Oya, T., Yanagimoto, J., Ito, K., Uemura, G., Mori, N.: Experimental and analytical investigations on a function for the non-associated flow rule model. Procedia Manuf. 15, 1908–1915 (2018) Rudnicki, J.W., Rice, J.R.: Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23(6), 371–394 (1975) Swift, H.: Plastic instability under plane stress. J. Mech. Phys. Solids 1(1), 1–18 (1952) Stoughton, T.B., Yoon, J.W.: Review of Drucker’s postulate and the issue of plastic stability in metal forming. Int. J. Plast 22(3), 391–433 (2006) Signorelli, J. W., Bertinetti, M. A., & Roatta, A. (2019). A review of recent investigations using the Marciniak-Kuczynski technique in conjunction with crystal plasticity models. Journal of Materials Processing Technology, 116517. Wu, S.H., Song, N.N., Pires, F.M.A., Santos, A.D.: Prediction of forming limit diagrams for materials with HCP structure. Acta Metall. Sin. 28(12), 1442–1514 (2015) Yoon, J.W., Stoughton, T.B., Dick, R.E.: Earing prediction in cup drawing based on non-associated flow rule. In: AIP Conference Proceedings, vol. 908, no. 1, pp. 685–690. AIP, May 2007

A Model for Simulating Transients in Looped Viscoelastic Pipe Systems. Preliminary Results Oussama Choura1(B) , Silvia Meniconi2 , Sami Elaoud1 , and Bruno Brunone2 1 Laboratory of Applied Fluids Mechanics, Process and Environment Engineering,

National Engineering School of Sfax, University of Sfax, Sfax, Tunisia 2 Water Engineering Laboratory, Department of Civil and Environmental Engineering,

University of Perugia, Perugia, Italy

Abstract. Because of their mechanical characteristics and durability, polymeric pipes use has grown, significantly, in Water Distribution Systems replacing elastic pipes (Steel, Concrete, …). In this paper, the study of transient flows in a polymeric looped network exhibiting a specific rheological behaviour compared to elastic pipes has been carried out. The transient flow is governed by a set of two hyperbolic partial derivative equations. The numerical simulation was conducted by solving continuity and momentum equations for uni-dimensional flow using the Method of Characteristic (MOC) for specified time intervals taking into account the viscoelastic behaviour of the pipe wall material. The numerically obtained results have been compared to the experiments carried out in a 6-loops network made of 19 high-density polyethene pipes of, relatively, small system-scale. The transient flow was generated by a valve closure. The manoeuver was simulated based on a linear closure law as the downstream boundary condition. The experimental setup was fed by a constant level reservoir. The results have proven the efficiency of the MOC for transient simulations even in complex networks. However, the accuracy of the MOC-based model depends on the calibration of the viscoelastic parameters describing the behaviour of the pipe wall during the hydraulic transients. Keywords: Transient flow · MOC · Viscoelasticity · Looped network · Polymeric pipes

1 Introduction The use of Polyethylene (PE) and Polyvinyl Chloride (PVC) pipes in the construction or renovation of Water Distribution System (WDS) has recently increased. This is owing to their ability to resist higher pressures, their considerable durability, their manufacturing procedure simplicity and their lower price comparing with steel or concrete pipes. Unlike elastic or rigid pipes, polymeric pipes exhibit viscoelastic behaviour. During transient flows, this rheological behaviour affects the transient wave speed and the shape of the pressure oscillation (Gally et al. 1979; Pezzinga 2002). In fact, the viscoelasticity of the pipe wall material results in residual stress and deformation which fasten the dampening and the wave energy dissipation (Covas et al. 2004, 2005; Soares et al. 2008; Meniconi et al. 2012). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 183–190, 2021. https://doi.org/10.1007/978-3-030-76517-0_20

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Due to the viscoelasticity of these pipes, common water hammer model was found to be inaccurate to describe the pressure variation and the transient wave propagation in PE and PVC pipes (Covas et al. 2004; Pezzinga et al. 2016). For this reason, the development of a reliable model for the evaluation of the viscoelastic effect on the transient flows in polymeric pipes was put under the scope in the past twenty years. Researchers dived into developing an accurate model for describing the viscoelastic behaviour of the pipe wall during transient flow. Covas et al. (2004, 2005) developed a numerical model for the simulation of water hammer in High-Density Polyethylene (HDPE). Moreover, Soares et al. (2008) tested the proposed model on PVC pipes. Found results showed a good agreement with experimental data. Furthermore, Meniconi et al. (2012, 2014) solved the set of 1-dimensional equations using the MOC to simulate the effect of the viscoelasticity of the pipe wall on the transient flow in a pipeline. The previously mentioned researches were devoted to analyzing the viscoelastic effect on transient in a single pipeline. However, Evangelista et al. (2015) studied this effect on a Y-shaped pipe system. The study included the experimental and numerical analysis of the phenomena in a branched system. Ferrante and Capponi (2017) tested the model proposed by Covas et al. (2004) in HDPE pipes and oriented polyvinyl chloride (PVC-O) pipes in a Y-shaped system. Covas et al. (2004, 2005), Pezzinga et al. (2014, 2016) and Ferrante and Capponi (2018) found that the Kelvin-Voight model is convenient to describe the instantaneous and retarded strain and its influence on the pressure curve during transient flows. These studies were oriented towards transient flow analysis in simple systems from a topological point of view. Few types of research were reported in the literature that studied transient flows in complex systems, i.e. looped networks. However, pressurized systems of WDS are mainly looped or branched pipes. Fathi-Moghadam and Kiani (2019) conducted experiments and simulated transient flows in HDPE 6-loops network. The authors aimed to measure experimentally the pressure variation in a complex polymeric pipe system during transient flow and simulate the rheological behaviour of the pipe wall using creep data resulting from the calibration of a 100 m long single pipe with an internal diameter of 50 mm. The studied system is composed of 3-by-3 m loops fed by a pressurized tank connected by a 14.5 m pipe to the looped system. The transient was generated using by means of a valve manoeuvre installed 0.5 m from the loop. The aim of this paper is the numerical validation of transient flow in the experimental setup of the Faculty of Water Science and Engineering, Shahid Chamran University of Ahvaz, Iran (Fathi-Moghadam and Kiani 2019).

2 Material and Methods 2.1 Water-Hammer Theory The governing equations of transient flow in viscoelastic pipes are continuity and momentum equations (Chaudhry 2014; Wylie and Streeter 1993): gA

∂Q ∂εr ∂H + a2 + 2Aa2 =0 ∂t ∂x ∂t

(1)

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gA

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dQ ∂H + + hf = 0 ∂x dt

(2)

where A is the cross-sectional area, H is the pressure head, a is the wave velocity, Q is the flow discharge, εr is the retarded strain, hf is the loss friction term and x is the distance along the pipe axis and t is the time. The viscoelastic behaviour of the pipe wall is incorporated into the continuity equation (Eq. 2) as the rate of change in time of the retarded strain. In fact, polymers have an immediate-elastic response, εe , and a retarded-viscous response, εr (Covas et al. 2005). The total strain, ε, is given by (Montgomery and MacKnight 2005; Ward and Sweeney 2012) and with accordance with Boltzmann superposition principle for small strains: t ε(t) = εe (t) + εr (t) = J0 σ (t) +

σ (t − t  )

0

∂J (t  )  dt ∂t 

(3)

being σ = the applied stress, t = the time and t = a dummy variable of the convolution integral. The elastic strain, J0 σ (t), is accounted for in the wave speed, a, in Eq. 1 as a function of the Young modulus of the first spring of the generalized Kelvin Voight (KV) model. The creep compliance function J(t) is expressed as follows (Covas et al. 2005): J (t) = J0 +

nkv 

−t Jk (1 − e /τk )

(4)

k=1

being τ k = ηk /E k the retardation time, and ηk and E k are the viscosity of the dash-pot and the elasticity modulus of the spring, respectively, of kth KV element. J k is the inverse of the Young moduli of the kth spring of the model and nkv is the number of elements, herein taken as three (Fig. 1).

Fig. 1. Generalized Kelvin Voight Model (nkv = 3) (Montgomery and MacKnight 2005).

2.2 Numerical Resolution (Method of Characteristics) In order to solve the set of hyperbolic partial differential equations governing the transient flow, the MOC was adopted because of its unconditional stability, computing-time and simplicity. The pair of equations were transformed into two ordinary equations along the characteristic lines (C ± ) and therefor into finite-difference equations:

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C ± : (H (x, t) − H (x ± x, t − t)) ± +

a (Q(x, t) − Q(x ± x, t − t)) gA

2a2 t ∂εr ( )(x, t) ± at hf = 0 g ∂t

(5)

In the above equation, the time derivative of the retarded strain is to be discretized and solved numerically (Covas et al. 2005). 2.3 Experimental Facility Experiments were held at the Faculty of Water Science and Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran by Fathi-Moghadam and Kiani (2019) on a 6 loops network made of 50 mm of diameter and 5.5 mm of thickness pipes. All pipes used are identical in terms of dimensions and nominal pressure supported. The network is connected to a pressurized tank providing a constant initial pressure of 40 m for the piping system through a 14.5 m long pipe (designated by P18). The network is made of 6 3-by-3 m square loops (19 nodes counting the feeding and end node). The downstream end is connected to a global valve for inflow adjustment and ball valve for transient generation distant of 0.5 m from the closest node as shown in the figure below (Fig. 2). Three Pressure transducers were mounted: one at the reservoir (designated by T3), another next to the inlet node of the network (designated by T2) and a third on outlet node (designated by T1) close to the transient generation point (Fig. 2). The experiments were carried out at two different flows of Q0 = 0.96 and 1.23 l/s as the authors stated (Fathi-Moghadam and Kiani 2019).

Fig. 2. Looped experimental setup (Fathi-Moghadam and Kiani 2019).

The studied system contained 19 pipes (designated as Pi in Fig. 2) and 14 nodes (designated by Ni). The network is composed of 12 nodes and 17 pipes and it is singlesourced.

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3 Results and Discussion The transient flow was generated by closing the ball valve at the node 13. The closing law was considered linear with a closing time equal to 140 ms. The manoeuvre starts after 100 ms. The flow evolution in time is shown in Fig. 3. The space step considered in the discretization was fixed to Δx = 0.5 m corresponding to time step Δt = 0.0011 s for a wave speed a = 450 m/s as the authors suggested (Fathi-Moghadam and Kiani 2019). Three KV elements were used to describe the viscoelastic behaviour. These parameters are summed in the table below (Table 1). These parameters were obtained by the calibration process in a single pipe system (Fathi-Moghadam and Kiani 2019) using the inverse method (Covas 2005). Table 1. Creep coefficients of the generalized KV model (Fathi-Moghadam and Kiani 2019). Element number k

Retardation time τ k (s)

Creep compliance coefficient J k (10–10 Pa−1 )

0

–

5.71

1

0.05

0.68

2

0.5

1.03

Fig. 3. Valve closure law: flow variation at the valve (node 13)

The simulated pressure variation showed a good agreement with the experimental data provided by Fathi-Moghadam and Kiani (2019) for the flow rates mentioned above. The head pressure variation in the measurement section (T1 and T2) are shown in the figures below (Fig. 4) for a flow rate Q = 0.96 l/s. The simulation was able to predict the pressure surge which is not the case for the wave energy dissipation and dispersion. That said; the creep compliance functions obtained by estimation using the inverse analysis for a single pipe may not apply for networks, at least of a small scale. The numerical results of transient for the 14 nodes are shown in Fig. 5.

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Fig. 4. Numerical results versus experimental data measured by pressure transducers T1 (at the left) and T2 (at the right) for a flow rate Q = 0.96 l/s.

Fig. 5. Numerical results at the nodes for a flow rate Q = 0.96 l/s.

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4 Conclusion A transient flow generated by the closure of a gate valve was simulated and validated using experimental data for polymeric pipes network available in the literature. The viscoelasticity of the pipe wall effect was taken into account in the model by means of Kalvin Voight model of three elements. The numerical model based on the method of characteristic was able to describe the transient wave travelling the system. However, the accuracy of the model depends on the calibration process of the creep compliance function responsible for the delay of the pressure peaks observed.

References Chaudhry, M.H.: Applied Hydraulic Transients. Springer, New York (2014). https://doi.org/10. 1007/978-1-4614-8538-4 Covas, D., Stoianov, I., Ramos, H., Graham, N., Maksimovic, C.: The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part I—experimental analysis and creep characterization. J. Hydraul. Res. 42(5), 517–532 (2004). https://doi.org/10.1080/00221686.2004.9641221 Covas, D., Stoianov, I., Mano, J.F., Ramos, H., Graham, N., Maksimovic, C.: The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part II—model development, calibration and verification. J. Hydraul. Res. 43(1), 56–70 (2005). https://doi.org/10.1080/002216805095 00111 Evangelista, S., Leopardi, A., Pignatelli, R., Marinis, G.D.: Hydraulic transients in viscoelastic branched pipelines. J. Hydraul. Eng. 141(8), 04015016 (2015). https://doi.org/10.1061/(asc e)hy.1943-7900.0001030 Fathi-Moghadam, M., Kiani, S.: Simulation of transient flow in viscoelastic pipe networks. J. Hydraul. Res. 1–10 (2019). https://doi.org/10.1080/00221686.2019.1581669 Ferrante, M., Capponi, C.: Comparison of viscoelastic models with a different number of parameters for transient simulations. J. Hydroinf. 20(1), 1–17 (2017). https://doi.org/10.2166/hydro. 2017.116 Ferrante, M., Capponi, C.: Viscoelastic models for the simulation of transients in polymeric pipes. J. Hydraul. Res. 55(5), 599–612 (2018). https://doi.org/10.1080/00221686.2017.1354935 Gally, M., Guney, M., Rieutord, E.: An investigation of pressure transients in viscoelastic pipes. J. Fluids Eng. 101(4), 495–499 (1979). https://doi.org/10.1115/1.3449017 Massey, B.: Mechanics of Fluids, 8th edn. Taylor and Francis (2006) Meniconi, S., Brunone, B., Ferrante, M.: Water-hammer pressure waves interaction at cross-section changes in series in viscoelastic pipes. J. Fluids Struct. 33, 44–58 (2012). https://doi.org/10. 1016/j.jfluidstructs.2012.05.007 Meniconi, S., Brunone, B., Ferrante, M., Massari, C.: Energy dissipation and pressure decay during transients in viscoelastic pipes with an in-line valve. J. Fluids Struct. 45, 235–249 (2014). https:// doi.org/10.1016/j.jfluidstructs.2013.12.013 Montgomery, T., MacKnight, W.: Introduction to Polymer Viscoelasticity, 3rd edn. Wiley, Hoboken (2005) Pezzinga, G., Brunone, B., Cannizzaro, D., Ferrante, M., Meniconi, S., Berni, A.: Two-dimensional features of viscoelastic models of pipe transients. J. Hydraul. Eng. 140(8), 04014036 (2014). https://doi.org/10.1061/(asce)hy.1943-7900.0000891 Pezzinga, G., Brunone, B., Meniconi, S.: Relevance of pipe period on kelvin-voigt viscoelastic parameters: 1D and 2D inverse transient analysis. J. Hydraul. Eng. 142(12), 04016063 (2016). https://doi.org/10.1061/(asce)hy.1943-7900.0001216

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Pezzinga, G.: Unsteady flow in hydraulic networks with polymeric additional pipe. J. Hydraul. Eng. 128(2), 238–244 (2002). https://doi.org/10.1061/(asce)0733-9429(2002)128:2(238) Soares, A.K., Covas, D.I., Reis, L.F.: Analysis of PVC pipe-wall viscoelasticity during water hammer. J. Hydraul. Eng. 134(9), 1389–1394 (2008). https://doi.org/10.1061/(asce)0733-942 9(2008)134:9(1389) Ward, I., Sweeney, J.: Introduction to the Mechanical Properties of Solid Polymers, 2nd edn. Wiley, Sussex (2012) Wylie, E., Streeter, V.: Fluid Transients in Systems, vol. 1. Prentice-Hall, Englewood Cliffs (1993)

Analytical Approach in the Pre-design Phase for Vibration Analysis of a Flexible Multibody System Ghazoi Hamza(B) , Maher Barkallah, Hassen Trabelsi, Amir Guizani, Jamel Louati, and Mohamed Haddar Mechanics Modeling and Production Research Laboratory (LA2MP), National School of Engineers of Sfax (ENIS), University of Sfax, B.P. 1173, 3038 Sfax, Tunisia

Abstract. The growing demand for fast and precise complex systems while decreasing the time to market will require the application of new methods in the product design process. This paper addresses an approach which includes a dynamic study into the conceptual design process. This includes the development of a flexible multibody model based on the object oriented modeling approach. An analytical method is considered in our study. Indeed, it is not greedy in computation time if we compare it with other methods of analysis. This characteristic is very interesting especially in the preliminary design. In fact, this phase doesn’t require a large amount of data and a fine calculation models. The vibrational behavior of an elementary mechanical system embedded in a multibody one in investigated. The dynamic behavior of a beam structure with simply supported boundary conditions, loaded in a variety of ways is investigated using an analytical approach with Modelica/Dymola language. The flexible beam undergoes small elastic deformations. Two types of loads are taken into consideration in this study, a concentrated load and uniformly varying load. This new approach allows designers to have early insights about the overall system in a very early state of the design process while lowering lead times. Different simulation tests are considered with the goal of performing a parametric study. The influences of some variables that affect the beam response to diverse excitation types are analyzed. Keywords: Predesign · Modelica · Vibrational behavior · Multibody

1 Introduction The performances of mechatronic systems result from the multiphysics interaction between mechanical, electrical and thermal components. Depending on the system approach, the designer attention may focus either at the component or at the system level.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 191–198, 2021. https://doi.org/10.1007/978-3-030-76517-0_21

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The challenge in the system level is to search efficiently and quickly for the right component parameters that achieve the best system performances (Samin et al. 2007). At the system level, modeling and simulation allow for the preliminary evaluation. Redesign of suggested solutions is possible, lowering then the risks (Khaled et al. 2014). Dynamics of multibody system can be regarded as the study of mechanical systems, which are assembled from different subsystems, interconnected with kinematics constraints and subjected to the acting of external forces (Shabana 2009; Flores et al. 2008). In a multibody system, bodies may be considered as rigid or deformable. In the investigation of the dynamic behavior of a flexible multibody system, often there is a need to study problems of transverse vibration of beams subjected to dynamic actions. Beam may be loaded in a variety of ways. For example, In Hamza et al. (2018), authors studied the vibrational behavior of a flexible multibody system. In fact, the vibration interaction between motors and electronics cards as exciters and receivers located on a flexible beam with diverse boundary conditions is investigated. The object oriented methodology with Modelica language is considered to create bodies. The proposed approach can support designers in the predesign phase, to choice system architect as well as to analysis the vibration behavior of the overall system. In practice, numerous structures are exposed to moving loads. Railways and overhead cranes are typical examples of such structures. In Hamza et al. (2020), authors proposed a new methodology for the predesign of a multibody system taken into account the vibrational behavior. The proposed method is illustrated to the railway system, considered as a multibody system. In fact, the railway is modeled as an elastic beam resting on an elastic foundation and subjected to an external moving excitation. The effects of several parameters on the system response are carried out. In many situations a surface zone of structure is subjected to a distributed load. Such forces are caused by winds, fluids, or excitations induced by motors, etc. In Hamza et al. (2015) authors studied the vibration transmission between electronic cards and motors using Modelica language. These components are supported by a flexible simply supported plate. The motor is modeled by uniformly distributed harmonic force acting over a rectangular subdomain of the plate. The electronic card model consists of a concentrated mass supported by a spring and damper in parallel. In literature, there are numerous numerical and analytical methods to study the dynamic behavior of a multibody system. Each method possesses its advantages and disadvantages (Da Silva et al. 2006; Skrinjar et al. 2018). In integrated approach is presented in this paper for the modeling of a multibody system. This paper is structured as follows: after the introduction, the theoretical background used in this study is presented in Sect. 2, Sect. 3 gives the implementation of the model in Modelica/Dymola, Sect. 4 presents the simulation results, and Sect. 5 concludes the paper.

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2 Model Formulation of the Flexible Multibody System In this section, the analytical formulation of the flexible multibody system is formulated. The system under consideration consists of a simply supported flexible beam loaded in a variety of ways. The system is shown in Fig. 1. Some assumptions and restrictions are taken into account. In this study, only transverse vibration is considered. The EulerBernoulli linear beam theory is used and the material of the beam is considered as linear elastic. In this work, the mathematical model will follow that exposed previously by Rao (2007).

Fig. 1. Simply supported beam (a) with varying distributed load, (b) with concentrated load

The dynamic of the beam is governed by the partial differential equation (PDE) given by Rao (2007): EI

∂ 2 u(x, t) ∂ 4 u(x, t) + ρA = f (x, t) ∂x4 ∂t 2

(1)

Where, u is the transverse vibration field of the beam, E denotes the Young’s modulus, ρ is the mass density of the beam material, I is the moment of inertia, A is the area of the beam cross-section, f(x,t) is the time varying force per unit length and l is the beam length. For a beam subjected to a varying distributed load, the dynamic excitation applied to the beam takes this form: nπ x sin ωt (2) f (x, t) = f0 sin l With, f 0 is the excitation amplitude, ω is the load frequency. The initial displacement and velocity of the beam are expected to be zero. u(x, 0) = 0,

∂u(x, 0) =0 ∂t

(3)

The boundary conditions of the beam are given by: u(0, t) = 0, u(L, t) = 0;

∂ 2u ∂ 2u t) = 0, (0, (L, t) = 0 ∂x2 ∂x2

(4)

The solution of the beam subjected to varying distributed load, becomes (Rao 2007):   ∞  ω f0 l 4 nπ x   sin sin ωt − sin ωn t (5) u(x, t) = l ωn EI (nπ )4 1 − (ω/ωn )2 n=1

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For a beam subjected to a concentrated load, the dynamic excitation applied to the beam takes this form: f (x, t) = F sin(t)δ(x − xk )

(6)

With,  is the load frequency, F is the excitation amplitude of the concentrated load. The beam response subjected to a concentrated force can be expressed as:   ∞   un (x)un (xk ) sin t − sin ωn t u(x, t) = F ωn2 − 2 ωn

(7)

n=1

ωn is the natural frequency of the beam, un (x) are the normal modes which take the form (Rao 2007):  nπ x 2 sin (8) un (x) = ρAl l

3 Modeling of the Multibody System with Modelica/Dymola Modelica language has been used in order to study the dynamic behavior of the proposed multibody system. The system model developed in Modelica/Dymola consists of two elements which are the load component and the beam component. Each subsystem is modeled according to its modeling equations developed in Sect. 2. In this approach, each sub-system is described separately in its most suitable formalism. The load component represents an excitation which can be concentrated or uniformly varying. The designer has then the ability to choose the load type as well as the load parameters, which are set in a graphical interface to define values. The beam component is a flexible structure and has simply supported boundary conditions. The beam is able to deform in its transversal direction. The beam parameters such as the Young modulus, the beam length and the section can be easily defined by the user through its graphical interface. The components connection is done through a new developed connector. The connector represents an interface with the rest of the system. In this paper, each connector is designed to have two variables. The multibody system developed in Modelica/Dymola is shown in Fig. 2.The exit of these components may be used as an input for other elements or it may be plotted in the simulation set up.

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Fig. 2. Multibody system model developed in Modelica/Dymola

4 Parametric Study In this section, some numerical computations have been carried out in order to simulate the system response in term of parameters. Table 1 presents the geometrical parameters used in this study. Table 1. System parameters Parameters

Value

Beam length

l=1m

Material density

ρ = 7800 kg/m3

Young’s modulus

E = 210000 MPa

Force amplitude

F = 10 N

Square cross section

S = 0.01 × 0.01 m2

Excitation natural frequency f = 60 Hz

In this paper, two combinations of loads types are considered which are uniformly varying load and a concentrated load. Figure 3 depicts a comparison of the beam midpoint displacement between the two excitations cases. It can be seen that the vibration amplitude of the beam in the case of concentrated load is more important than the other case. Accordingly, using this model, designer can examine the response of the structure to different loads types simultaneously depending on these choices and can also check whether the simulated behavior replicates the desired one. Figure 4 shows the vibrations amplitudes of some points of the beam which is subjected to uniformly varying load. It can be observed that displacements haves periodic functions shapes with an oscillation frequency identical to that of the excitation and vibration amplitude in the beam center is the greater.

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Fig. 3. Midpoint displacement according to two different load types

Fig. 4. Beam displacements

The influence of the excitation amplitude is shown in Fig. 5. It is visible that the amplitude of vibration increases with the growth of the excitation amplitude in a linear way.

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Fig.5. Displacement in the mid-point of the beam subjected to uniformly varying lad according to the excitation amplitude

5 Conclusions Flexibility impacts may be modeled with two ways, by flexible joints and by flexible subsystems. Flexibility in bodies is our interest in this paper. The dynamic behavior of a flexible multibody system depends largely on the mechanical configuration. In fact, based on the object oriented modeling language Modelica/Dymola, an efficient predesign method to investigate the vibrational behavior of a flexible multibody system is proposed. In this contribution, a special case of a system was analyzed. The dynamics of a simply supported beam loaded in a variety of ways. The proposed approach is based on an analytical method, which allows for fast analyses of complex models. All components models developed in this paper can be easily modified and extended. The influence of different parameters on the mutibody system response is investigated. Acknowledgements. The authors gratefully acknowledge the assistance and the financial support of the project 19PEJC10-03 by the Tunisian Ministry of Higher Education and Scientific Research.

References Samin, J.C., Brüls, O., Collard, J.F., Sass, L., Fisette, P.: Multiphysics modeling and optimization of mechatronic multibody systems. Multibody Sys.Dyn. 18(3), 345–373 (2007) Khaled, A.B., Gaid, M.B., Pernet, N., Simon, D.: Fast multi-core co-simulation of cyber-physical systems: application to internal combustion engines. Simul. Model. Pract. Theory 47, 79–91 (2014) Shabana, A.A.: Computational Dynamics. John Wiley & Sons (2009)

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Flores, P., Ambrósio, J., Claro, J. P., Lankarani, H.M.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies, vol. 34. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-74361-3 Hamza, G., et al.: Conceptual design decision support of a mechatronic system using analytical approach with Modelica. Mech. Ind. 19(1), 103 (2018) Hamza, G., et al.: An analytical approach for modeling a multibody system during pre-design with application to the railway system. In: Chaari, F., et al. (eds.) Advances in Materials, Mechanics and Manufacturing. LNME, pp. 150–157. Springer, Cham (2020). https://doi.org/10.1007/9783-030-24247-3_17 Hamza, G., et al.: Pre-designing of a mechatronic system using an analytical approach with dymola. J. Theor. Appl. Mech. 53(3), 697–710 (2015) Da Silva, M.M., Brüls, O., Paijmans, B., Desmet, W., Van Brussel, H.: Concurrent simulation of mechatronic systems with variable mechanical configuration. In: Proceedings of ISMA (2006) Skrinjar, L., Slaviˇc, J., Boltežar, M.: A review of continuous contact-force models in multibody dynamics. Int. J. Mech. Sci. 145, 171–187 (2018) Rao, S.S.: Vibration of Continuous Systems, vol. 464. Wiley, New York (2007)

Pressure Calculation and Fatigue of a Trans-tibial Prosthetic Socket Made from Natural Fiber Composite Sofiene Helaili1,2(B) , Wahbi Mankai4,5 , and Moez Chafra1,3 1 Tunisia Polytechnic School, LASMAP (LR03ES06), Carthage University,

Rue El-Khawarizmi, BP 743, 2078 La Marsa, Tunisia [email protected] 2 ISTEUB, Carthage University, 2 Rue de l’Artisanat Charguia 2, 2035 Tunis, Tunisia 3 IPEIEM, Tunis El Manar University, Campus Universitaire, B.P 244, 2092 Tunis, Tunisia 4 Al Ahsa College of Technology, Technical and Vocational Training Corporation, Riyadh, Saudi Arabia 5 Materials, Optimization, and Energy for Sustainability, National Engineering School, University of Tunis El Manar, Tunis, Tunisia

Abstract. The sockets are generally made from carbon fiber which makes them cost high. Replacing carbon fiber with a more ecological and socio-economical fiber such as Alfa fiber can considerably reduce the cost. The objective of this paper is to develop and validate a numerical finite elements virtual prototype that can be used for prosthesis design. An experimental bench is developed to measure the pressure between a residual limb and a trans-tibial prosthetic socket. Three finite element models are created and compared to experimental results. The numerical results from the virtual prototypes show a good correlation with the experimental results in fact the predictability is equal to 96.62%. Based on these results, the virtual prototype can be adopted to design resistant and comfortable trans-tibial prosthetic sockets. In the final part of the paper, a fatigue analysis is made. The fracture is observed on the first cycle and it is a ductile failure. The socket reinforced by ALFA fibers does not meet the static and fatigue requirements of ISO 10328 for the test failure, in fact required resisting force Fsu is equal to 3019N in the case of a ductile failure but the measured failure force is ~2700N. The failure occurred on the junction; no cracks appeared in the body of the socket. The junction can be studied and reinforced with a better strategy. Keywords: Pressure stump-socket · 3D finite element model · Natural fiber · Composites · Virtual prototyping

1 Introduction Several research studies [1, 2] have shown that the Alfa fiber can be used as reinforcement in composite materials. The advances in manufacturing techniques have allowed using Alfa fiber as reinforcement in several applications, like pedestrian’s security [3] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 199–209, 2021. https://doi.org/10.1007/978-3-030-76517-0_22

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and in biomechanics applications such as prosthetic sockets. The study of the interaction between the trans-tibial residual limb and the prosthetic socket is a widely discussed subject in biomechanics [4]. During the last 45 years, many studies have been conducted to understand the load transfer in the stump-socket interface [5]. Several clinical investigations have been undertaken to measure pressures and shear stresses [6]. Technological development of transducers and channel acquisitions has considerably improved the outcomes of these investigations [7]. Nevertheless, this experimental approach has several drawbacks. First, it requires the availability of several amputees to do measurements. Second, it is limited to measure pressure and shear stress at very specific points of the skin, which is not enough to understand the global mechanical behavior of the interface between the trans-tibial residual limb and the prosthetic socket.

2 Methods An experimental bench is developed to measure the pressure between a residual limb and a trans-tibial prosthetic socket. The socket made in this study was physically tested using a test bench shown in Fig. 1.

Fig. 1. Test bench

The developed bench is a static and dynamic test bench [8] developed according to the ISO 10328 specifications (see Fig. 2). A second method based on finite elements is developed in parallel with the experimental methods [9]. The advantage of this method is that it allows an examination of the stresses in the entire residual limb including the surface. The analysis by finite elements has been efficient to study several other problems that are related to this subject [10–12]. 2.1 Assumptions and Geometrical Parameters It has been shown in Sect. 2 of the ISO 10328 that the loading condition proposed by this regulation [13] can be modeled by a plane problem. The finite element models are based on the simplified geometry shown in Fig. 3.

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Fig. 2. 3D View and plan view of the test bench

The previous studies show the importance of the pressure inside the sockets. It confirms the non-uniform distribution of the pressure. From reference [14] the reference pressure Pref is equal to 65 kPa. This pressure value is considered the maximum pressure value to not exceed to be in a comfortable situation for the patient. The experiment was done until this pressure value is reached. The socket must show an elastic behavior during the experiment. 2.2 Numerical Modeling for Maximal Pressure Calculation In this section three numerical models are developed to see the effect of the following parameters on the accuracy of the pressure value: – the socket geometry (cylindrical and conical). – the modeling of attachment between the stump and the rear part of the socket. – the socket bottom contribution. For the three finite element models, the stem is considered rigid and can rotate at the bottom of the socket at the origin point O (see Fig. 3).

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Fig. 3. Model definition geometry

2.2.1 Socket Without a Bottom Plate and with Rear Stump Surface Free (Model 1) The bottom side of the stump is not attached (free), with the cylindrical socket rear surface. That means that the stump form will not be circular as seen in Fig. 4. The pressure value at the top of the stump is equal to 63 kPa (see Fig. 7). We obtain the pressure function over the height (see Fig. 7). It is not a perfect linear curve because the shear effect and the cohesion effects are considered. The obtained value is very close to the reference value Pref which is equal to 6.5 kPa. the difference is equal to 3%.

Fig. 4. Rear stump surface free and Pressure value in model 1

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2.2.2 Effect of the Bottom Part of the Socket (Model 2) With model 2, all parameters of model 1 are kept, only the bottom side of the socket is suppressed from the model (see Fig. 5). The effect of neglecting the socket bottom is studied.

Fig. 5. Bottom side and Pressure value model 2

The pressure value at the top of the stump is nearly equal to 6.28 kPa (see Fig. 5). Model 2 shows that the bottom side has no impact on the modeling of the socket. The difference in pressure between model 1 and model 2 is ~0%. 2.2.3 Conical Socket Form Without a Bottom Plate and with Rear Stump Surface Free (Model 3) In Model 3, the geometry of the socket is conical. The bottom side radius of the stump R0 is equal to 41 mm. The top radius Rn is equal to 59 mm (see Fig. 6). The rear surface of the stump is free (not attached to the socket). The pressure value is equal to 5.39 kPa (see Fig. 6).

Fig. 6. Conical model 3 and Pressure value model 3

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The pressure along Z-axis is shown in Fig. 9. Close to the boundary, the pressure value is not smooth compared to the pressure close to the maximum value of Z. 2.3 Experimental and Numerical Fatigue Analysis 2.3.1 Test Conditions The tests were made in the mechanical tests laboratory of the research company Tecnotessile (National Technology Research for the textile sector) Italy. The fatigue testing machine used is the Italsigma MTS 407 Controller which has a full capacity of 100 kN at 25 Hz. The average test temperature was between 20 and 25 °C. The force F is increased at a speed of 200 N/s until failure or until reaching the limit force Fsu, presented in Table 1. The test force F at which the test sample must withstand to meet the requirements of the standard ISO10328 depends on the type of failure that may occur brittle or ductile failure. The list of values corresponding to Fsu conditions for test load conditions I and II at the A100 level are shown in Table 1. Table 1. Loading condition and dimensions Condition of loading Cyclic procedure Test force load, Fsp [N]

2013

Test force limit, Fsu [N]

Ductile 3019

Test force limit, Fsu [N]

Brittle

4025

Range of test force cycle, FC [N]

1150

Maximum test load, Fmax = (Fmin + FC ) [N]

1200

Endurance (cycles)

3106

2.3.2 Experimental test for a Socket reinforced with nonwoven ALFA The result of the failure test for a socket reinforced with ALFA fibers is shown in Fig. 7. The fracture is observed on the first cycle and it is a ductile failure. The socket reinforced by ALFA fibers does not meet the requirements of ISO 10328 for the static test failure, in fact, the required force Fsu is equal to 3019 N in the case of a ductile failure but the failure force is ~2700 N.

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Fig. 7. Compression force function of deformation

From these tests, we find that the most stressed section is at the junction between the socket and the anchor (Fig. 8). No cracks appeared in the body of the socket. The junction can be studied and reinforced with a better strategy.

Fig. 8. Junction failure of socket reinforced with ALFA fibers.

2.3.3 Numerical Model In this section, the numerical model will be described. The numerical model aims to reproduce the results of the fatigue loading test bench. The finite element model (Fig. 9) is built to reproduce the condition of loading listed in Table 1. The loads are applied from 0 with an increase of 200 N/s. The mesh is regular and refined. The prosthesis junction was modeled as a shell-beam finite element connection. As a failure criterion, maximum stress failure criteria are used: σ(Failure stress) = 229.3 Mpa (in all directions). Failure occurs when the fiber fails in tension. Material modeled as orthotropic and elastic. An explicit solver was used.

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Fig. 9. Loading direction and boundary condition.

The mesh is a regular one, which is refined to have an accurate description of the results. The prosthesis junction was modeled as a shell-beam finite element connection. The shell finite element used is fully integrated. The model is highly nonlinear, large deformation and contact nonlinear problem. So, the Explicit solver scheme was used. Material properties are listed in Table 2. The homogenized properties of the ALFA/PMMA 60% composite are used in this model. The properties are taken from reference [2]. Table 2. Approximated properties of Woven ALFA/PMMA 60% Ex = Ey (GPa) vxy ALFA/PMMA 60% 7.865

Gxy (GPa)

0.285 6.1

The resistance of the prosthesis in the function of the endpoint displacement is shown in Fig. 10. and the failure of some elements in the junction is shown in Fig. 11. Compared to the results in Fig. 7, the numerical model for failure must be improved. Figure 10 shows a total loss of resistance at 40 mm which can be considered as correlated to test results in Fig. 7 that shows a losing resistance near 40 mm value. The force reached in the numerical model is more than 3000 N which is 10% greater than the value obtained in the physical test which is ~2700 N.

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Fig. 10. Loading-displacement curve

Fig. 11. Failure at the junction

3 Results and Discussion In this section, the three numerical models’ results are given (Fig. 12). Freeing up the rear contact between the socket and the stump provides a maximum pressure close to the pressure of experimental reference for the two geometries. Modeling the bottom of the socket does not affect the reference pressure. The predictability of model 1 and model 2 is 96,92% (see Table 3). The maximum pressure can be calculated using simplified finite elements models like model 1 or model 2. The socket material and thickness can be accurately chosen based on the geometrical properties of the tibia of the patient. On the other hand, the failure test of the socket reinforced with nonwoven ALFA is unsatisfactory, but the failure force ~2700 N not so far from the recommended value. If we divide this value by the recommended one, the percentage is 89.4%. It is an encouraging result for a socket reinforced with fibers ALFA.

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Fig. 12. Pressure function over the height for model 1, model 2, and model 3 Table 3. Virtual prototypes predictability compared to reference pressure value. Pressure (kPa) Predictability Experimental/Reference 65

–

FEM Model 1

63.0

96,92%

FEM Model 2

62.8

96,62%

FEM Model 3

53.9

82,92%

4 Conclusions The results of this study might have applications not only for designing the socket experimental device or dimensioning sockets; but also, for deducing certain similitude properties that could reduce the number of experiments related to the stump-socket interaction. Full contact between the stump and the socket was assumed. It will be interesting to repeat this study considering friction/slid conditions. By applying the superposition principle an analytical solution can be calculated for a more complex problem where the tube is applied to bending and twisting. The solution of this compound load can give the stump-socket interaction more realistic and accurate modeling. Acknowledgments. The authors would like to thank all the staff of the Research Laboratory of Biomechanics and Orthopedic Biomaterials of the National Orthopedics Institute M.T. KASSAB, Tunis, Tunisia, for their cooperation.

References 1. Paiva, M., Ammar, I., Campos, A., et al.: Alfa fibers: mechanical, morphological and interfacial characterization. Compos. Sci. Technol. 67, 1132–1138 (2007)

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2. Helaili, S., Chafra, M.: Anisotropic visco-elastic properties identification of a natural biodegradable ALFA fiber composite. J. Compos. Mater. Epub ahead of print 3 June 2013. https://doi.org/10.1177/0021998313488811. 3. Helaili, S., Chafra, M., Chevalier, Y.: Hybrid aluminum and natural fiber composite structure for crash safety improvement. In: Karaman, I., Arróyave, R., Masad, E. (eds.) Proceedings of the TMS Middle East — Mediterranean Materials Congress on Energy and Infrastructure Systems (MEMA 2015), pp. 249–258. Springer, Cham (2016). https://doi.org/10.1007/9783-319-48766-3_25 4. Pirouzi, G., Abu Osman, N.A., Eshraghi, A., et al.: Review of the socket design and interface pressure measurement for transtibial prosthesis. Sci. World J. 2014, 1–9 (2014) 5. Zhang, M., Roberts, C.: Comparison of computational analysis with clinical measurement of stresses on below-knee residual limb in a prosthetic socket. Med. Eng. Phys. 22, 607–612 (2000) 6. Zhang, M., Turner-Smith, A.R., Tanner, A., et al.: Clinical investigation of the pressure and shear stress on the trans-tibial stump with a prosthesis. Med. Eng. Phys. 20, 188–198 (1998) 7. Dumbleton, T., Buis, A.W.P., McFadyen, A., et al.: Dynamic interface pressure distributions of two transtibial prosthetic socket concepts. J. Rehabil. Res. Dev. 46, 405–415 (2009) 8. Wahbi, M., Brahim, B.S., Moez, C., Ridha, B.C., Jomah, A.: Modeling of a fatigue test performed on a trans-tibial prosthetic socket made of natural fiber. In: Aifaoui, N., et al. (eds.) CMSM 2019. LNME, pp. 204–213. Springer, Cham (2020). https://doi.org/10.1007/978-3030-27146-6_22 9. Jia, X., Zhang, M., Lee, W.C.C.: Load transfer mechanics between trans-tibial prosthetic socket and residual limb—dynamic effects. J. Biomech. 37, 1371–1377 (2004) 10. Lee, W.C.C., Zhang, M., Boone, D.A., et al.: Finite-element analysis to determine effect of monolimb flexibility on structural strength and interaction between residual limb and prosthetic socket. JRRD 41, 775 (2004) 11. Bonnet, X., Pillet, H., Fodé, P., et al.: Finite element modelling of an energy–storing prosthetic foot during the stance phase of transtibial amputee gait. Proc. Inst. Mech. Eng. H 226, 70–75 (2012) 12. Lenka, P.K., Choudhury, A.R.: Analysis of trans tibial prosthetic socket materials using finite element method. JBiSE 04, 762–768 (2011) 13. ISO 10328:2016: Prosthetics — Structural testing of lower-limb prostheses — Requirements and test methods 14. Kim, W.D., Lim, D., Hong, K.S.: An evaluation of the effectiveness of the patellar tendon bar in the trans-tibial patellar-tendon-bearing prosthesis socket. Prosthet. Orthot. Int. 27, 23–35 (2003)

Online Adaptive MFC for Nonlinear Active Half Car System Maroua Haddar1(B) , Fathi Djemal1 , Riadh Chaari1 , S. Caglar Baslamisli2 , Fakher Chaari1 , and Mohamed Haddar1 1 Mechanics, Modeling and Production Laboratory (LA2MP), Mechanic Department,

National Engineering School of Sfax (ENIS), BP 1173, 3038 Sfax, Tunisia [email protected], [email protected] 2 Department of Mechanical Engineering, Hacettepe University, 06800 Beytepe, Ankara, Turkey

Abstract. In this paper, an intelligent control technique with ameliorated performance for active half car systems with nonlinearities is presented. In fact, nonlinear characteristics of mechanical items and exogenous environmental disturbances that may deteriorate stiffness-damping parameters are neglected in some applications. However, this kind of parameter uncertainty are always present and can deteriorate the behavior of controller. In order to enhance the car body displacement and ensure a good compromise between the suspension constraints, a new model free controller MFC based on non-asymptotic observer is created. The intelligent-Proportional–Integral-Derivative strategy (i-PID) can update online the actuator force without getting precise information’s about unpredictable malfunction in suspension items. Considering, an auxiliary block is implemented to compensate the endogenous phenomenon’s of parameter uncertainties and is able to cope with all of these based on online compensator. Furthermore, the simplicity of implementation is an observed feature of the i-PID. Finally, the effectiveness of proposed controller is verified by a comparative analysis with optimal control Linear Quadratic Gausien (LQR) and the traditional Proportional–Integral-Derivative control (PID). A deterministic road profile was used to test satisfaction constraints: ride comfort, road holding and rattle space.The numerical simulations illustrate the effectiveness and the robustness of proposed scheme against nonlinearities. Keywords: Nonlinear stiffness · Nonlinear damper · Model free control · Ultra-local modelling

1 Introduction The active suspensions categories require an additive actuating force integrated in the closed loop control parts not separately conventional passive items. The control power depends greatly on the choice of the appropriate control schemes. Wider range of active controllers does not use mechanism for getting prior information about the environment. The suspension system receives the excitation of the road and then moves autonomously to track the imposed reference trajectory (Nagarkar et al. 2018). This kind of control is adaptable to simple cases without unexpected perturbations. However, it may have © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 210–217, 2021. https://doi.org/10.1007/978-3-030-76517-0_23

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some troubles such as no convergence to desired dynamic (Zhao et al. 2017). For thus, the researchers have proposed several solutions to enhance the active suspension system performance. There is different forms of nonlinearities that can affect the dynamic behaviour of vertical motion of suspension system. For example, a combination between linear and quadratic term of suspension deflection gives a nonlinear pneumatic force related the suspension spring. On the other hand, the nonlinear suspension damper can be modelled by a piecewise function related to different damping coefficients of the extension and compression travel. Furthermore, the friction phenomena due to the contact between shaft and sliding plate causes several oscillations that threaten the ride comfort of passenger. However, in real application, these nonlinearities are present but the knowledge of precise mathematical model of this kind of perturbation is difficult. Under this circumstance, recent development in control process try to deal with this un-modelled disturbances for obtaining a robust control scheme (Haddar et al. 2019a, b). Based on the previous analysis, this paper suggest an intelligent Model Free Control (MFC) technique to Multiple-Input Multiple-Output (MIMO) nonlinear active half car systems. Combined with an algebraic observer, we formulate an adaptive control law with improved performances. The organization of the paper is as follows: a nonlinear half car model and the description of its mathematical equations are presented in Sect. 2. Section 3 describes the MIMO MFC technique and its principle rules for implementation process. The effectiveness of the proposed controller is illustrated in Sect. 4 with a comparative scenario with classical controllers. Finally, the conclusion is given in Sect. 5.

2 Problem Formulation Previous works are concentrated on a model with linear viscoelastic characteristics. However, in this paper, we are concentrated in studying the impact of nonlinear-authorities to be closer to the real case. A mathematical model can be described by four equations that are the issue of many assumptions (Fig. 1): ms z¨s = −Fs,1 − Fs,2 − Fd ,1 − Fd ,2 + FA,1 + FA,2

(1)

Iyy θ¨ = Fs,1 a − Fs,2 b + Fd ,1 a − Fd ,2 b − FA,1 a + FA,2 b

(2)

mu1 z¨u1 = Fs,1 + Fd ,1 − Ft,1 − FA,1

(3)

mu2 z¨u2 = Fs,2 + Fd ,2 − Ft,2 − FA,2

(4)

   3 Fs,i (t) = ks,i zs,i − zu,i + kns,i zs,i − zu,i

(5)

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  Ft,i (t) = kt,i zu,i − zr,i

(7)

where the spring forces are given by Fs,i (t); the damping forces are given by Fd ,i (t) and the tire forces are described by Ft,i (t). i is the ith corner of the half car. The half car parameters and values are the following: ms = 580 kg, Iyy = 1100 kg.m2 , mu1 = mu2 = 40 kg, ks,1 = ks,2 = 23500 N/m, ds1 = 1500 N.s/m, ds2 = 1600 N.s/m, kt1 = kt2 = 190000 N/m, kns1 = kns2 = 2.35 × 106 N/m and dns1 = dns2 = 400 (Ns/m) a = 1 m and b = 1.5 m (León-Vargas et al. 2018).

Fig. 1. Active nonlinear half car system

3 MFC for Nonlinear Half-Vehicle Model The powerful impact of intelligent controller is its ability to deal with unpredictable perturbations online. In fact its parameters for calibration are variable and should be updated online under different running conditions (Haddar et al. 2019a, b). The proposed scheme is classified as an Active Disturbance Rejection Controller.However,the design steps of an i-PID (intelligent-Proportional-Integral-Derivative) are different to classical ADRC (Active-Disturbance –Rejection -Control) by the use of algebraic tools as was described in Haddar et al. (2019a, b). Two ultra-local model equations replace the total mathematical model of half car system. Only two inputs and outputs of the system are required for the implementation. (2)

g1 = φ1 + α1 FA,1

(8)

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g2(2) = φ2 + α2 FA,2

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(9)

where: g1 and g2 are the measured acceleration at each corner of the sprung mass. φ1 and φ2 are the internal-external motions that should be rejected automatically and independently to their sophisticated models. α1 and α1 are constant values that are chosen by the operator based on an empirical essays. The actuators forces in the closed loops (FA,1 and FA,2 ) are expressed as following: ⎧  ⎨ F = − φˆ1 + g¨ ref ,1 −(KP,1 e+KI ,1 e+KD,1 e˙ ) A,1 α1 α1  (10) ⎩ FA,2 = − φˆ2 + g¨ ref ,2 −(KP,2 e+KI ,2 e+KD,2 e˙ ) α2

α2

where: ei = gi − gref ,i is the tracking error. i = 1. The reference trajectories gref ,1 and gref ,2 are set to zero. ALIEN filters are the key elements for estimating φˆ 1 and φˆ 2 (Haddar et al. 2019a, b; Fliess and Join 2013). The implementation of the proposed scheme is easy in comparison with traditional techniques. In fact, only two sensors are required: an accelerometer in the left of the vehicle body and an accelerometer in the right side as depicted in Fig. 2.

Fig. 2. Block diagram of proposed scheme

The following chart describes the steps for getting the equations of algebraic estimators (Fig. 3):

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Fig. 3. Algebraic estimator steps

4 Numerical Investigations For a comparative study, two traditional controllers are used in the numerical simulation:  • PID control: FA PID,i = −KP,i ei −KP,i ei − KD,i e˙ i , where KP1 = 9450, KP2 = 4050, (León-Vargas et al. KD1 = 3000, KD2 = 2500, KI 1 = 550 and KI 2 = 250, 2018). FA PID,i is the actuator force in each corner of the half car system produced by PID controller. • LQR control: FA LQR = −Klqr x,where x is the state vector. x1 = z¨s , x2 = θ¨ , x3 = zs1 − zu1 , x4 = zs2 − zu2 , x5 = z˙s1 , x6 = z˙s2 , x7 = zu1 − zr1 , x8 = zu2 − zr2 . Klqr is the LQR gain (Nagarkar et al. 2018). FA LQR is the actuator force produced by LQR controller. 1.0e+04 *Klqr =

−0.2596 0.0739 1.4495 0.0373 0.9439 0.0418 −0.4197 −0.0164 −0.7628 0.0053 −2.8081 −0.2541 −0.4362 −0.0141 1.5129 0.0641



The controllers’ excitation is achieved by a deterministic input. The half car speed is set to 43 km/h and the height level of the bump is equal to 80 mm. The comparative analysis results are plotted in Figs. 4, 5 and 6.

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It is clear that the proposed controller acquires the small magnitude of the vertical body acceleration when compared with LQR and PID controllers (Fig. 4). As it is traded, the best ride comfort criteria is given by the best vibration reduction of the vehicle body oscillation. The results shown that the well ride comfort is accomplished by the i-PID controller. The RMS (Root-Mean-Square) values are the same in the case of linear and nonlinear systems with i-PID control (For linear case RMS(¨zs ) = 0.458 m/s2 ; For nonlinear case RMS(¨zs ) = 0.459 m/s2 ). Body acceleration 4 Passive Linear LQR Linear PID Linear i-PID Linear Passive Non-Linear LQR Non-Linear PID Non-Linear i-PID Non-Linear

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Fig. 5. Effect of nonlinearities on road holding

Figure 5 shows the suspension deflection in each corner of the half car system. In order to keep car components without damage, the suspension travel must not surpass a limit given by this equation:   zs,i − zu,i  ≤ zmax (11) The travel restriction zmax is set to ± 10 cm. This boundary condition is not exceeded by active controllers.On the other hand, i-PID control is able to avoid the impact of

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nonlinearities and produces more damping effort to be close to linear behavior. i-PID gives smoother suspension rattle space than other controllers. However, LQR is more sensitive to nonlinearities and its suspension deflection oscillations can damage car components. To investigate the impact of nonlinearities in road holding, the Normalized Tire Deflection (NTD) should be kept low than 1. This boundary condition implies that dynamic tire load never exceeds static ones. Therefore, the road contact is ensured. 

b.ms (12) + mu1 < 1 NTD1 = kt1 (zu1 − zr1 ) 9.81 a+b 

a.ms NTD2 = kt2 (zu2 − zr1 ) 9.81 (13) + mu2 < 1 a+b Results obtained after the simulation are shown in Fig. 6. The normalized tire deflections are also smaller in the case of i-PID controller. Therefore, it is concluded that the active half car system with the proposed control enhances the ride comfort performance while maintaining the road handling characteristics. That is why, it’s better to choose the intelligent controller for minimizing vibration and attenuating the effect of nonlinearities. In fact, classical schemes of control are sensitive to nonlinearities. They need a new recalibration for scaling parameters: such as the matrices Q and R in the case of LQR controller.

5 Conclusion In this work a new robust, active control process, based on online estimation and rejection of unknown nonlinearities in half car components. The introduced scheme exploits the vertical displacement of the suspension system and the actuator force signal to predict the exogenous and endogenous behaviors. Therefore, an amelioration of classical design purposes of active controller. The estimated disturbances due to nonlinear suspension spring and nonlinear suspension damper are rejected online without an accurate model of nonlinearities in a vehicle model. This new scheme is able to avoid the penalization of signal behavior.

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References Fliess, M., Join, C.: Model-free control. Int. J. Control 86(12), 2228–2252 (2013) Haddar, M., Baslamisli, S.C., Chaari, R., Chaari, F., Haddar, M.: Road profile identification with an algebraic estimator. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 233(4), 1139–1155 (2019) Haddar, M., Chaari, R., Baslamisli, S.C., Chaari, F., Haddar, M.: Intelligent PD controller design for active suspension system based on robust model-free control strategy. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 233(14), 4863–4880 (2019) León-Vargas, F., Garelli, F., Zapateiro, M.: Limiting vertical acceleration for ride comfort in active suspension systems. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 232(3), 223–232 (2018) Nagarkar, M.P., Bhalerao, Y.J., Patil, G.V., Patil, R.Z.: Multi-objective optimization of nonlinear quarter car suspension system–PID and LQR control. Procedia Manuf. 20, 420–427 (2018) Zhao, J., Wong, P.K., Ma, X., Xie, Z.: Chassis integrated control for active suspension, active front steering and direct yaw moment systems using hierarchical strategy. Veh. Syst. Dyn. 55(1), 72–103 (2017)

The Interaction of Different Transmission System Using the Substructuring Method Marwa Bouslema1(B) , Taher Fakhfakh1 , Rachid Nasri2 , and Mohamed Haddar1 1 Mechanical Modeling and Manufacturing Laboratory (LA2MP), National School of Engineers

of Sfax, University of Sfax, B.P 1173, 3038 Sfax, Tunisia {tahar.fakhfakh,mohamed.haddar}@enis.rnu.tn 2 Applied Mechanics and Engineering, University of El-Manar II, 1002 Tunis, Tunisia [email protected]

Abstract. The interaction of two different transmission systems was investigated using the substructure methodology based on the Frequency Response Functions (FRFs). The coupled approach has been simulated to investigate the studied system dynamic behavior. The formulation of the lumped-parameter model and the resolution of the eigenvalue problem are determined. The analysis of two subsystems are determined separately and their FRF are obtained. The response of the global system is based on the dynamic characteristics of each subsystem. A comparison of the coupling and not coupling subsystem FRFs is analyzed. Two models of coupling were investigated. A comparison of the FRFs between the response of a rigid and a joined system is discussed. The determination of the FRF of the flexible coupling system is very important in order to avoid the effect of coupling FRFs. In fact, the contribution of a flexible coupling on the simulation of different transmission systems obtained by the sub-structuring technique was discussed. Moreover, two value of helical angle was introduced for the studied parallel gear stage. The identification of modes that are excited by the helical gear is solved. Finally, the impact of helix angle on the simulation properties such as modes shapes and FRFs is studied. Keywords: Substructure technique · Transmission system · Helical gear · Frequency response function · Coupling

1 Introduction The behavior of connecting subsystems contribute significantly in the overall dynamic characteristics, such as natural frequencies, mode shapes, and response characteristics. Numerous research have used dynamically coupled substructure models to expect the vibratory behavior of complex structures (Choi 2002). In instance, Bishop and Johnson (1960) developed the receptance coupling method. Others like Ewins (2000) proposed the substructure synthesis technique for the determination of joints dynamic characteristic who coupled substructure-using FRFs. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 218–226, 2021. https://doi.org/10.1007/978-3-030-76517-0_24

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However, few research works were dedicated to dynamic of compound planetary system gears using substructring method. In this context, the interaction of different transmission system using substructuring method appears an important study. Gear transmissions are extensively used in machine, wind turbines and transportation such as helicopters. The research of vibration behavior of gearboxes shows as an attractive filed for machine design planetary. Many researchers are focused on a parallel gear stage (Velex and Maatar 1996). Kahraman (1993) discussed the investigation of a helical gear parallel stage. For planetary gears, natural frequencies were analyzed by Kahraman (1993) using a lumped-parameter model. Habib et al. (2005) examined the sensitivity of helical planetary gears frequencies to the variation of helix angle. Guo (2010) investigated the sensibility of dynamic characteristics and modes shapes in composed planetary gears. Sun et al. (2014) examined the coupled modes properties in a coupled planetary gear. Furthermore, Wei (2013) was established the coupled model composed of both planetary and parallel stages. In this chapter, a gearbox system formed by one parallel stage helical gear and another planetary gear is proposed to analysis the interaction properties. The substructure methodology was applied to characterize the dynamic behavior. The objective of this study is the determination of coupling system FRF for analyzing the interaction of different subsystem with rigid and flexible joint. In addition, the FRF coupling system was analyzed to explain the sensitivity of the helical gear. This paper is structured as followed: in the first section a brief presentation of the receptance method is presented. Then, a description of the studied system is carried out. Finally, the impact of coupling and the helix angle on the simulation results are performed.

2 Theoretical Formulation The determination of the FRF obtained by substructuring method is focused on a vibration behavior of a rigid-coupled system. This methodology is related to the FRFs of uncoupled subsystems. The coupled FRF is obtained by combining the independent subsystem FRFs. 2.1 FRF Formulation The FRFs subsystem was determined theoretically. The motion equation of a damped linear subsystem is expressed by:     (1) [M ] U¨ (t) + [C] U˙ (t) + [K]{U (t)} = {F(t)} where [M ], [K] and [C] represent the mass, stiffness, and damping matrices of studied system respectively. {U (t)} and {F(t)} represent the displacement and the excitation force, respectively.

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The FRF study system are expressed by realizing an FRF synthesis build on a mode shapes. The expression related the mode shapes with the FRF matrix Hjk (w) is defined by Hjk (w) =

n 

w2 r=1 r

r jr k − w2 + j2ξ

(2)

r wr w

where Hjk (w) is the displacement at coordinate j due to a force excitation at coordinate k, n is the number dofs. r j and r k are the mode shapes normalized of mass. 2.2 Coupling Method The developing of coupling problem is expressed mathematically in Liu and Ewins (2002). The subsystems receptance matrices are expressed in partitioned form as:      A Ui A fi A Hii A Hic = (3) H H U A ci A cc A c A fc      B Uj B fj B Hjj B Hjc = (4) B Hcj B Hcc B Uc B fc The subscripts “i” and “j” represent the uncoupled DOFs for subsystem A and B The subscript “c” denotes the degree of freedom of coupling. The rigid coupling was connected subsystem A to subsystem B at DoF c. Jetmundsen et al. (1988) developed the coupled system receptance: ⎤ ⎡ Haa Hac Hab ⎣ Hca Hcc Hcb ⎦ = Hab Hbc Hbb (5) ⎤ ⎡ ⎤ ⎡ ⎤T ⎡ A Hii A Hic 0 A Hic A Hic ⎣ A Hci A Hcc 0 ⎦ − ⎣ A Hcc ⎦[[A Hcc ] + [B Hcc ]]−1 ⎣ A Hcc ⎦ 0

0

B Hjj

B Hic

B Hic

Where [A H ] and [B H ] are the subsystems receptance matrices. For the flexible coupling, the kernel matrix [[A Hcc ] + [B Hcc ]] has been modified by 

[A Hcc ] + [B Hcc ] + [S K]−1 . Where the coupling stiffness matrix was defined by [S K]. Equation (5) is applied to calculate the coupling system FRF.

3 Studied System Description The present transmission system is formed by two subsystems, which are parallel gear stage joined to planetary gear stage via a rigid coupling. The present system was investigated based on the lumped parameter model (Fig. 1). Bouslema et al. (2017) established these models. The reducer second shaft is coupled to planetary sun shaft via a rigid coupling. The mesh stiffness is modeled by a linear spring on the meshing teeth. The bearings are considered by linear springs.

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Fig. 1. A reducer stage joined to a planetary gear

4 Numerical Applications The sensitivity of coupling different subsystems on the FRFs was discussed. The response of subsystem A is compared to the subsystem coupling FRF. The mechanical parameters of the studied system are given in Table 1. Table 1. Parallel and planetary gears model parameters

Wheel and pinion mass (kg) Wheel and pinion inertia moment (kg) Wheel and pinion inertia moment (kg) Bearings stiffness (N/m) Torsional shaft flexibilities (N/m) Helix angle Teeth module (m) Average mesh stiffness (N/m) Inertia coupling (kg m2)

Teeth number Module Teeth width (mm) Mass (Kg) I/r2 (Kg) Base radius (m) Helix angle Gear mech stifness Bearing stiffness (N/m) Torsional (N/m) Pressure angle (*)

mi = 2 Ii/(ri)2 = 0.58 Ji/(ri)2 = 1.16 kyi = 3.5 × 108, kzi = 108 kbρyi/(ri)2 = 108 β =0 β = 20 m = 4 × 10−3 k1moy = k2moy = 2 × 108 I A = 4.48 × 10−8

Sun Ring Carrier 30 70 1.7 1.7 25 25 0.46 0.588 3 0.272 0.759 1.5 0.024 0.056 24.6 24.6 Ksp = Krp = 2.108 Kp = Ksu,v = Kru,v = 108 krw = 109; Ksw =105; Kcw = 0 s = r = 21.34

Planet 20 1.7 25 0.177 0.1 0.016 0

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4.1 Effect of Coupling on Subsystem Response The dynamic response and the interaction of both subsystems will be established. The frequencies of parallel and planetary stage are presented in Table 2. Table 2. Frequencies of parallel and planetary gear stages

m=1

m=2

m=N-3

Natural Frequency (B) R1 R2 R3 R4 R5 R6 T1 T2 T3 T4 T5 T6 S1

N : Number of satellites (4) 0 1536.6 1970.6 2625.7 7773.6 13071.1 727 1091 1892.8 2342.5 7189.9 10437.6 1808.2

S2 S3

5963.8 6981.7

f2=1125 f3=1125 f1=1566 f1=2105 f1=2201 f1=2201 f1=3972

The frequencies of planetary gear characterized the modes of vibration into three types: planet modes, translational modes and rotational modes. Figure 2 shows the simulation of a coupled subsystem A in the shaft rotational direction. We can see the appearance of several peaks on these spectra. These peaks correspond to the two frequencies of the parallel gear and the planetary gear because it is closer to the connecting zone of two systems. The apparition of some frequencies around 0, 1536, 1970 and 2630 Hz at the rotational direction FRF is due to coupling. The subsystem B frequencies are transmitted to subsystem A FRF when they coupling. These frequencies correspond to the rotational

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modes of planetary gear. It is noted that the magnitudes of the FRF parallel stage gear is greater than these of planetary gear.

Fig. 2. FRF in rotational direction of the shaft

Figure 3 presents the subsystem B FRF under the sun rotational direction.

Fig. 3. FRF in rotational direction of the sun

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The frequencies of planetary gear were dominated. In rotational direction, the coupling and the non-coupling subsystem B FRF are superposed (show Fig. 3). The subsystem A response does not affect the FRF of subsystem B in rotational direction by reason of low inertia than these of planetary. Figure 4 show the response of subsystem A with rigid and flexible coupling under the shaft rotational direction.

Fig. 4. FRF in rotational direction of the shaft with flexible joint

Frequencies difference of subsystem A appears between two models of coupling under the rotational direction. The frequencies are decreased because of a diminution in coupling stiffness. 4.2 Effect of Helical Angle on Subsystem a FRF The sensitivity of the helical gear value of parallel stage on the eigen- frequencies values is investigated. The vibratory responses of subsystem A presented in Fig. 5. The FRFs in the case of β = 0° are supporposed to those of β = 20° in order to comparing. Helical gear affects the response of subsystem A. The difference between the modes shapes for the cases of β = 0° and β = 20° is discussed. Thus, It is noted that in the case of β = 20°, the modes shapes presents the coupled translational and rotational modes. In contrast to β = 0° we found uncoupled modes. From these figures, it appears that the translational modes appears in the subsystem A FRF in rotational direction. In fact, the spectral FRF appears a shift in magnitude.

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Fig. 5. Subsystem A FRF in shaft rotational direction for two case β = 0° and β = 20°

5 Conclusion This paper established the dynamic vibration of helical gear stage connected to a helical planetary gearbox. The simulation of vibration behavior is achieved in frequency domain by using substructure methodology. The interaction of two studied subsystems was discussed. The subsystem B frequencies are transmitted to the subsystem A FRF when they coupling. In addition, the elastic coupling affects the response of different subsystems. A parametric study of the helix angle was determined to investigate their effect on the subsystem response. It’s observed that the value of helical gear affects the FRF of the overall assembly, such as modes shapes, and response characteristics.

References Lin, J., Parker, R.G.: Analytical characterization of the unique properties of planetary gear free vibration. J. Vib. Acoust. 121(3), 316–321 (1999) Ozguven, H.N., Houser, D.R.: Mathematical models used in gear dynamics – a review. J. Sound Vib. 121, 383–411 (1988) Habib, R., Chaari, F., Fakhfakh, T., Haddar, M.: Three dimensional model for a helical planetary gear train vibration analysis. Int. J. Eng. Simul. 6(3), 32–38 (2005) Sun, W., Ding, X., Wei, J., Hu, X., Wang, Q.: An analyzing method of coupled modes in multistage planetary gear system. Int. J. Precis. Eng. Manuf. 15(11), 2357–2366 (2014). https://doi. org/10.1007/s12541-014-0601-9 Jetmundsen, B., Bielawa, R.L., Flannelly, W.G.: Generalized frequency domain substructure synthesis. J. Am. Helicopter Soc. 33(1), 55–64 (1988) Guo, Y., Parker, R.G.: Sensitivity of general compound planetary gear natural frequencies and vibration modes to model parameters. J. Vib. Acoust. 132(1), 011006 (2010)

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Ewins, D.J.: Modal Testing: Theory and Practice. Research Studies Press Ltd., Hertfordshire (2000) Bishop, R.E.D., Johnson, D.C.: The Mechanics of Vibration. University Press Edition, Cambridge (1960) Kahraman, A.: Effect of axial vibrations on the dynamics of a helical gear pair. Trans. ASME 115(33), 39 (1993) Choi, B.L., Park, J.M.: Application of the impedance coupling method and the equivalent rotor model in rotor dynamics. J. Finite Elem. Anal. Des. 39(2), 93–106 (2002) Bouslema, M., et al.: Effects of modal truncation and condensation methods on the Frequency Response Function of a stage reducer connected by rigid coupling to a planetary gear system. C.R. Mec. 345, 807–823 (2017)

Combined Effect of Roughness Anisotropy and Roughness Parameters on the Friction Behavior Under Boundary Lubrication F. Elwasli1(B) , S. Mzali1 , F. Zemzemi2 , and S. Mezlini1 1 LGM, Ecole Nationale d’Ingénieurs de Monastir, Université de Monastir,

Avenue Ibn Aljazzar, 5019 Monastir, Tunisia 2 LMS, Ecole Nationale d’Ingénieurs de Sousse, Université de Sousse, BP 264 Sousse Erriadh,

4023 Sousse, Tunisia

Abstract. This study set out to assess the effects of the surface properties on the friction behavior of AA5083 aluminum alloy counter AISI-52100 steel ball under lubricated sliding conditions. Using a linear reciprocating tribometer, it was possible to examine the combined influence of roughness parameters and the sliding direction (θ) on the apparent friction coefficient (μApp ). Five surfaces, G50 , G80 , G240 , G400 and G1000 , have been polished in a unidirectional manner using a belt grinder with five abrasive papers P 50 , P 80 , P 240 , P 400 and P 1000 , respectively. Reciprocating sliding tests were performed at room temperature under a loading pressure P of 200 MPa. The oscillatory stroke length L and the total sliding distance Ltot were fixed at 15 mm and 30 m, respectively. To investigate the influence of surface anisotropy, three sliding directions (0°, 45° and 90°) were considered for the experiments. Results did not identify a significant link between the apparent friction coefficient (μApp ) and the number of sliding cycles (N). μApp remains almost constant during sliding. By contrast, a close correlation was revealed between the roughness anisotropy and the apparent friction coefficient (μApp ). Moreover, the initial surface topography was an important determinant of the friction behavior, especially the mean absolute profile slope (Δa). Keywords: Roughness properties · Roughness anisotropy · Lubrication · Reciprocating sliding · Friction

1 Introduction Lubrication offers an effective way of controlling friction and wear between two surfaces in a sliding motion. Industries have utilized lubricants to improve systems efficiency for several applications, such as automotive and aerospace (Khaemba et al. 2020). A limitation of using lubrication is that the couple steel/aluminum alloy was known as extremely hard to be lubricated (Wan et al. 1996), even though sliding at a modest load. Consequently, their tribological properties may be influenced, which reduces their scope of application. This could be countered by the use of the surface modification approach. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 227–234, 2021. https://doi.org/10.1007/978-3-030-76517-0_25

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It is a widely held view that surface properties were a key factor to enhance tribological performance (Cha and Erdemir 2015; Dan et al. 2019; Khaemba et al. 2020). Several processes, such as grooving, honing, grinding, milling, polishing, and laser texturing, have been adopted to investigate the contribution of surface properties to the tribological behavior (Stickel et al. 2015a; Stickel et al. 2015b; Flegler et al. 2020). To gain further understanding of the correlation between surface preparation and tribological behavior, the analysis of different surface parameters is needed. Generally, the average roughness Ra was adopted to characterize a surface. This parameter is commonly used in engineering practice. However, Elwasli et al. have reported that Ra cannot provide sufficient information about the contact pair behavior (Elwasli et al. 2018a, b, c). In fact, Ra gives just a general description of the surface topography (Menezes et al. 2009). Some authors have mainly been interested in studying the effect of a further surface parameter; kurtosis Sku , skewness SSk , reduced valley depth Svk , mean absolute profile slope Δa . The sensitivity of friction to those parameters has been demonstrated (Sedlaˇcek et al. 2009, 2012; Elwasli et al. 2018a, b, c). Recently researchers have highlighted the relevance of roughness anisotropy to determining the tribological behavior (Menezes et al. 2006, 2008, 2011; Holmberg et al. 2015; Elwasli et al. 2018a, b, c). Menezes et al. (Menezes et al. 2006, 2008, 2011) have conducted experimental scratch tests on a steel surface with unidirectional grinding marks. They found that the rise of the angle of grinding marks leads to an increase in the friction coefficient. The purpose of this investigation was to explore the relationship between the surface properties and μApp under lubricated sliding conditions. The reciprocating sliding test was realized to investigate the friction behavior of AA5083 aluminum alloy counter AISI52100 ball. The combined effect of roughness anisotropy and roughness parameters on the friction behavior under boundary lubrication was considered.

2 Experimental Details: Specimen Preparation and Sliding Wear Test The initial sample consisted of an aluminum sheet with 40 × 40 × 8 mm3 in dimensions. Tests specimens, G50 , G80 , G240 , G400 and G1000 , have been polished in a unidirectional manner using a belt grinder with five abrasive papers P 50 , P 80 , P 240 , P 400 and P 1000 , respectively. More details of surface preparation were given in our previous paper (Elwasli et al. 2018a, b, c). Profile scan and roughness measurements were conducted perpendicular to the directional pattern of the surface via a Taylor-Hobson Surtronic S series tester and Talyprofile (Gold) software, respectively. To identify the surface properties, the following parameters were used; Ra and the mean absolute profile slope, Δa . The average roughness, Ra , and the mean absolute profile slope, Δa , of the prepared surfaces were given by Eqs. (1) and (2), respectively. 1 Ra = l



l o

|y(x)|dx

(1)

Combined Effect of Roughness Anisotropy and Roughness Parameters

Δa =

1 l

 l   dy   dx   o dx

229

(2)

Where l is the measurement length, x and y are the abscissa and ordinate of the roughness measurement space, respectively. Table 1 provides roughness parameter values obtained from the profilometry analysis. Table 1. Surface roughness values. Ra (μm) Δa (°) Rough surfaces

G50

4.98

3.9

G80

3.31

12.9

G240

0.81

6.25

Smooth surfaces G400

0.45

5.3

G1000 0.41

7

Surfaces could be divided into two groups, namely rough surfaces and smooth surfaces. It was interesting to note that, contrary to the average roughness Ra , Δa did not show any correlation with the abrasive grit size. Indeed, Δa is sensitive to the abrasive grain sharpness (Hisakado 1999). The reciprocating sliding tests with the ball on flat contact configuration is one of the more practical ways of investigating the tribological behavior during cyclic sliding. In this study, reciprocating sliding was conducted using the linear reciprocating tribometer (Fig. 1a) that was detailed by Guezmil et al. (Guezmil et al. 2016). AISI-52100 ball bearing steel of an average roughness Ra = 0.05 μm was mounted as the upper specimen in the ball holder of the tribometer. AA5083 specimen was fixed in the lubricant container and immersed by the lubricant. The table was animated with a reciprocating motion under a loading pressure P = 200 MPa, with an oscillatory stroke length of L = 15 mm and a total sliding distance of L tot = 30 m. The sliding velocity was imposed at V = 50 mm/s. Table 2 provides a summary of the experimental test conditions. Table 2. Experimental test parameters. Parameters

Notation Units Value

Loading pressure

P

MPa

200

Stroke length

L

mm

15

Total sliding distance L tot

m

30

Sliding velocity

V

mm/s 50

Sliding direction

θ

°

0, 45, 90

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Sliding tests were executed at room temperature on three directions: 90°, 45° and 0° to the ploughing marks (Fig. 1b). The lubricant chosen for the sliding tests was a commercially available engine oil lubricant, type Mobil1-0W-40, with a kinematic viscosity at 40 °C of 70.8 mm2 /s. The specific film thickness calculated for each surface did not exceed the unit. Hence, this study was performed under boundary lubrication regime. Experiments were replicated at least three times under the same sliding conditions. The recorded friction force was used to evaluate the apparent friction coefficient (μApp ). In fact, μApp was determined as the ratio between tangential-to-normal forces.

Fig. 1. a) Linear reciprocating tribometer, b) Sliding directions.

3 Results and Discussions 3.1 Effect of Roughness Anisotropy Figure 2 presents the evolution of μApp during lubricated sliding for the surface G400 . It is worth noting that μApp exhibits a steady variation with the number of sliding cycles (N). Those curves show that N has an insignificant influence on μApp . In fact, the maximum μApp gap doesn’t exceed 14%. However, the sliding direction has revealed a significant impact on μApp . The highest μApp was observed when sliding parallel to the polishing marks θ = 0° followed by 45° then 90°. Moreover, only a slight difference was found between the sliding direction 45° and 90°. When sliding 45° and 90°, polishing marks play the role of micro-reservoirs to improve lubricant retention and take away the debris from the contact zone. Nevertheless, when sliding 0° to the polishing marks, the ball pushes the lubricant away from the contact area. Consequently, the lubricant contribution decreases, resulting in a high μApp . Moreover, wears debris trapped between the ploughing marks along the sliding path provokes the third body wear and the increase of μApp .

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0.8 0.7

0°

45°

90°

0.6

μApp

0.5 0.4 0.3 0.2 0.1 0 0

200

400

600

800 1000

Number of sliding cycles N

Fig. 2. Apparent friction coefficient vs the number of sliding cycles under lubricated condition for G400 .

3.2 Effect of Roughness Parameters Figure 3 illustrates the evolution of μApp and Δa . The surface topography has a significant impact on the friction behavior irrespective of the sliding direction. Figure 3a and b present practically the same evolution. In fact, the increase of Δa gives rise to high μApp . Indeed, during sliding 90° and 45° to the ploughing marks, the ball sways between peaks and valleys. Consequently, surfaces that were distinguished by a high Δa need an additional tangential force to deform the profile asperities. This provokes the dissipation of further energy and the increase of μApp (Sedlaˇcek 2012). Therefore, a good correlation between μApp and Δa was found. This does not appear to be the case for Fig. 3c. In fact, for surfaces G80 and G1000 , a low μApp was detected. These surfaces have a high Δa . When sliding 0° to the ploughing marks over a surface characterized by high Δa , peaks carried the ball, thereby reducing the contact area and μApp . In contrast, regardless of Ra , surfaces characterized by low Δa generate a larger contact area since they exhibit almost the same behavior as a smooth surface.

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μ

Smooth Surfaces

∆a

Rough Surfaces

μ

∆a

Smooth Surfaces

0.8

20

0.6

15

0.6

15

μApp

μApp

∆a (°)

20

∆a (°)

0.8

0.4

10

0.4

10

0.2

5

0.2

5

0

0

G50

G80

G240

G400

G1000

Rough Surfaces

0 G50

μ

∆a

G80

G240

G400

G1000

Smooth Surfaces

20

0.6

15

μApp

0.8

0.4

10

0.2

5

∆a (°)

0

0

0 G50

G80

G240

G400

G1000

Fig. 3. Apparent friction coefficient vs the absolute mean slope of the profile under lubricated condition for G400 . a) 90°, b) 45° and c) 0°.

4 Conclusions This work contributes to the effort to understand the correlation between apparent friction coefficient and roughness properties under lubricated sliding conditions. A reciprocated sliding test was conducted on AA5083 aluminum sheet. Especial attention has been focused on the investigation of the effect of the sliding direction on the friction behavior. The finding of this investigation has led to the following conclusions. • For the surface G400 , no significant correlation was found between μApp and N. By contrast, μApp displays high dependency on the sliding direction. The highest μApp was observed when sliding in the direction 0° followed by 45° then 90°. • The initial surface topography was an important determinant of the friction behavior, especially Δa . i. When sliding 90° or 45° over a surface distinguished by a high Δa μApp increases. Surfaces with high Δa need an additional tangential force to run over high asperities. ii. When sliding 0° over a surface characterized by high Δa , high peaks maintain the ball, which reduces the contact area and improve lubrication performance, resulting in low μApp .

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References Cha, S.C., Erdemir, A. (eds.): Coating Technology for Vehicle Applications, 1st edn. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-14771-0 Dan, L., Xuefeng, Y., Chongyang, L., et al.: Tribological characteristics of a cemented carbide friction surface with chevron pattern micro-texture based on different texture density. Tribol. Int. 106016 (2019). https://doi.org/10.1016/j.triboint.2019.106016 Elwasli, F., Mzali, S., Zemzemi, F., et al.: Correlation between roughness anisotropy and tribological behavior of AA5083: experimental and numerical analysis. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 233(3), 372–382 (2018a). https://doi.org/10.1177/1464420718811360 Elwasli, F., Mzali, S., Zemzemi, F., et al.: Effects of initial surface topography and contact regimes on tribological behavior of AISI-52100/AA5083 materials’ pair when reciprocating sliding. Int. J. Mech. Sci. 137, 271–283 (2018b) Elwasli, F., Mzali, S., Zemzemi, F., Mkaddem, A., Mezlini, S.: Effects of pretextured surface topography on friction and wear of AA5083/AISI52100 materials’ pair. In: Haddar, M., Chaari, F., Benamara, A., Chouchane, M., Karra, C., Aifaoui, N. (eds.) CMSM 2017. LNME, pp. 771– 779. Springer, Cham (2018c). https://doi.org/10.1007/978-3-319-66697-6_75 Flegler, F., Neuh, S., Groche, P.: Influence of sheet metal texture on the adhesive wear and friction behaviour of EN AW-5083 aluminum under dry and starved lubrication. Tribiol. Int. 141, 105956 (2020). https://doi.org/10.1016/j.triboint.2019.105956 Guezmil, M., Bensalah, W., Mezlini, S.: Effect of bio-lubrication on the tribological behavior of UHMWPE against M30NW stainless steel. Tribol. Int. 94, 550–559 (2016). https://doi.org/10. 1016/j.triboint.2015.10.022 Hisakado, T.: Effect of abrasive particle size on fraction of debris removed from plowing volume in abrasive wear. Wear 236(1–2), 24–33 (1999) Holmberg, K., Laukkanen, A., Ronkainen, H., et al.: Topographical orientation effects on friction and wear in sliding DLC and steel contacts, part 1: experimental. Wear 330–331, 3–22 (2015). https://doi.org/10.1016/j.wear.2015.02.014 Khaemba, D.N., Azam, A., See, T., et al.: Understanding the role of surface textures in improving the performance of boundary additives, part I: experimental. Tribol. Int. 106243 (2020). https:// doi.org/10.1016/j.triboint.2020.106243 Menezes, P.L., Kishore, Kailas, S.V.: Effect of roughness parameter and grinding angle on coefficient of friction when sliding of Al–Mg alloy over EN8 steel. J. Tribol. 128(4), 697–704 (2006). https://doi.org/10.1115/1.2345401 Menezes, P.L., Kishore, Kailas, S.V.: On the effect of surface texture on friction and transfer layer formation—a study using Al and steel pair. Wear 265(11–12), 1655–1669 (2008). https://doi. org/10.1016/j.wear.2008.04.003 Menezes, P.L., Kishore, Kailas, S.V., et al.: Influence of inclination angle of plate on friction, stick-slip and transfer layer—a study of magnesium pin sliding against steel plate. Tribol. Int. 267(5–6), 476–484 (2009). https://doi.org/10.1016/j.triboint.2009.12.028 Menezes, P.L., Kishore, Kailas, S.V., et al.: Influence of inclination angle and machining direction on friction and transfer layer formation. J. Tribol. 133(1), 014501-8 (2011). https://doi.org/10. 1115/1.4002604 Sedlaˇcek, M., Podgornik, B., Vižintin, J.: Influence of surface preparation on roughness parameters, friction and wear. Wear 266(3–4), 482–487 (2009). https://doi.org/10.1016/j.wear.2008. 04.017 Sedlaˇcek, M., Podgornik, B., Vižintin, J.: Correlation between standard roughness parameters skewness and kurtosis and tribological behaviour of contact surfaces. Tribol. Int. 48, 102–112 (2012). https://doi.org/10.1016/j.triboint.2011.11.008

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Stickel, D., Fischer, A., Bosman, R.: Specific dissipated friction power distributions of machined carburized martensitic steel surfaces during running-in. Wear 330–331, 32–41 (2015a). https:// doi.org/10.1016/j.wear.2015.01.010 Stickel, D., Goeke, S., Geenen, K., et al.: Reciprocating sliding wear of case-hardened spheroidal cast iron against 100Cr6 under boundary lubrication. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 229(10), 1214–1226 (2015b). https://doi.org/10.1177/1350650115576245 Wan, Y., Liu, W., Xue, Q.: Effects of diol compounds on the friction and wear of aluminum alloy in a lubricated aluminum-on-steel contact. Wear 193(1), 99–104 (1996)

A New Dynamic Model for Worm Drives Ala Eddin Chakroun1,2(B) , Ahmed Hammami1 , Ana De-Juan1 , Fakher Chaari1 , Alfonso Fernandez2 , Fernando Viadero2 , and Mohamed Haddar1 1 Laboratory of Mechanics, Modeling and Production (LA2MP), National School of Engineers

of Sfax, BP1173, 3038 Sfax, Tunisia [email protected], [email protected] 2 Department of Structural and Mechanical Engineering, Faculty of Industrial and Telecommunications Engineering, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain {alfonso.fernandez,fernando.viadero}@unican.es

Abstract. Worm drives are commonly used for power transmission. It is therefore important to study their dynamic behavior. Dynamic models of gears proved that it is possible to detect multiple defects and various types of failures. The literature showed a lack of detailed worm drive models. The existing ones are simple and envisage a minimum of degrees of freedom, which is not the case in reality. This work proposes a new tridimensional model of this type of gear. It is composed of two blocks. One block contains a worm and the other block contains a worm gear. Each block has a bearing and inertia. One inertia represents a motor and the other represents a receiver. The gear mesh stiffness between the worm and the worm gear is introduced as a trapezoidal function. The deflexion is calculated in the contact line between the worm and the worm gear. The equations of motion are obtained by using the energetic method which is “Lagrange formulism”. Then, they are solved using Newmark solving method. From the results found only acceleration is taken into account. Its signal is presented in time domains. The spectrum of the acceleration is also plotted. Keywords: Worm drive · Worm gear · Stiffness · Dynamic behavior · Dynamic model

1 Introduction Many researchers have been interested in gear modeling due to the fact that it can predict the dynamic behavior of a gear system in a virtual interface. In the literature, we find multiple types of models for different types of gears. A single stage spur gear model is established by Chaari et al. (Chaari et al. 2008). Several other researches, such as, Khbou et al. (Khabou et al. 2011) and Chaari et al. (Chaari et al. 2012) have relied on this same model. Hmida et al. (Hmida et al. 2019) adapted this model to a system after adding an elastic couple. Similarly, a clutch has been used by Walha et al. (Walha et al. 2011) in a helical gear model. Moreover, bevel and planetary gears were modeled by Driss et al. (Yassine et al. 2014) and Hammami et al. (Hammami et al. 2015) respectively. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 235–242, 2021. https://doi.org/10.1007/978-3-030-76517-0_26

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The above mentioned models focused on different types of gears but avoided mentioning the worm drive model. Worm drives are widely used to transmit high power between non-intersecting shafts. They are also known for their high reduction ratio. Nevertheless, they have many drawbacks due to the fact that many errors can occur in the processes of assembling and manufacturing. The high coefficient of friction is also a major disadvantage of this gear type, which makes modeling it increasingly difficult. Despite these difficulties, many researchers have been able to introduce many worm drive models. Benabid and Mansouri (Benabid and Mansouri 2016) introduced an original eight-degrees-of-freedom model. They tried not to overload the model and to keep it as simple as possible. Chung and Shaw (Chung and Shaw 2007) solved a dynamic equation of transmission that predict the dynamic behavior of only a specific worm drive model, that with a flywheel. Liu et al. (Liu et al. 2014) attempted to use the spur gear dynamic model proposed by Tamminana et al. (Tamminana et al. 2007) in order to predict the dynamic performance of a worm drive. This study follows the procedure introduced by Chaari et al. (Chaari et al. 2008) in his study of the spur gear and adapts it to the specific geometry and parameters of the worm drive. Chaari et al.’s model consists of two blocks and each block contains an inertia wheel, a bearing and a gear. It solves the equation of motion using Lagrange formulism after taking into consideration the displacement in the line of action. The objective of this procedure is to represent the graph of the gearmesh stiffness in healthy and broken cases and then to use it in order draw conclusions. This study uses the same procedure to observe a worm drive model and arrive at different results. This study arrives at a general but reliable worm drive model that enables other researchers to use it to arrive at their own distinct results. The roadmap of this study is first to establish a worm drive model. Second, it represents the expression of deflexion. Then, it deduces the motion equations from the energetic equations. Finally, it solves these equations using Newmark solving method. This procedure aims at characterizing acceleration and spectrum.

2 Dynamic Model The dynamic model presented in Fig. 1 by two projected views is a kinematic chain of the worm reducer. The first block is composed of a Motor, a bearing block and a worm linked by a transmission shaft. The second block is composed of a receiver, a bearing block and a worm gear also linked by a transmission shaft. In Fig. 1, the motor and the receiver are considered as rigid bodies. The shafts are supposedly light weighted. Their torsionel and axial stiffness are kθi (i = 1, 2) and kzi (i = 1, 2) respectively. Gear meshing and bearings are modeled using linear springs. The translations of the two blocks are xi , yi , zi (i = 1, 2) and the rotations are ∅i , ψi , θ1i , θ2i (i = 1, 2). mi is the weight of each gears. It is measured using the following equation: mi = π bi ri2 ρi (i = 1, 2)

(1)

bi is the width of the worm gear or the length of the worm and ρi represents the material density of the gear.

A New Dynamic Model for Worm Drives

237

Block 1 Worm Motor

Worm gear

Worm

Worm gear

Receiver

Block 2 Fig. 1. The dynamic model of the worm drive

I11 is the inertia of the motor, I12 is the inertia of the worm, I21 is the inertia of the worm gear and I22 is the inertia of the receiver.

3 Transmission Error Calculation The two points Considered M1 and M2 belong to the active flank of the worm and the worm gear respectively. Their relative displacement is calculated using the following: −−−−−→ → −−−−−→ − n1 + U2R (M2 ).→ n2 δ(l, t) = U1R (M1 ).−

(2)

−−−−→ Ui (Mi ): the displacement of Mi (i = 1, 2), → → → − → n2 are the normal unitary outgoing vectors of Mi with − n2 = −− n1 n1 and − In the action plan, li is the distance separating Mi from the middle of the line of action.

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The gear pair is supposedly rigid. Both of the gears are rotating in the positive sense. By taking into consideration the abovementioned conditions, it becomes possible to write the following: (3) The displacement torsors of O1 and O2 are expressed against a fixed benchmark R(x, y, z ).  −−−−→ U1R (O1 ) = x1 x + y1 y + z1 z − → ω1R = ∅1 x + ψ1 y + θ12 z  −−−−→   U2R (O2 ) = x2 x + y2 y + z2 z R τ2 = − → ω2R = ∅2 x + ψ2 y + θ21 z   τ1R =

(4)

(5)

After the mathematical development of Eq. (3), it is then possible to write the expression of the deflexion that will be used afterwards.

4 Equation of Motion The differential equation system of the movement reaction is put in a usual matrix form: M {¨q} + C{˙q} + ([KA ]+[K(t)]){q} = {F0 }

(6)

It was deduced after the following steps: – the identification of all the energies of the system – using Lagrange formalism to find the equation of motion – solving it using Newmark solving method   With {q} = x1 , y1, z1 , x2 , y2 , z2 , ∅1 , ψ1 , ∅2 , ψ2 , θ11 , θ12 , θ21 , θ22 [M ] is a matrix of the total weight. It is time-independent:   ML 0 [M ] = 0 MA [ML ] is a matrix composed of weight terms. [MA ] is a matrix composed of inertia terms. The matrix of the average stiffness is written as follows:   Kp 0 [KA ] = 0 Kθ

Kp is composed of the stiffness of the bearing blocks.

(7)

(8)

(9)

A New Dynamic Model for Worm Drives

[Kθ ] is composed of: Kθi : the tensional stiffness of the shaft. K∅i and Kψi : the stiffness of the bearings. [K(t)] is the matrix of the mesh stiffness. It is time-dependent:   K11 (t) K12 (t) [K(t)] = K21 (t) K22 (t) With

[K21 (t)] = [K12 (t)]T

239

(10) (11)

[C] is the damping matrix. It is used following Rayleigh form: [C] = μ[K] + λ[M ]

(12)

μ and λ are constants of proportionality. The vector {F0 } of the external static force is: {F0 } = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Cm , 0, 0, Cr }T

(13)

5 Numerical Results The characteristics of the worm and worm gear used in this experiment are summarized in Table1. Table 1. Characteristics of the worm drive Worm Worm gear Teeth number

1

50

Turning speed (rpm)

1500

30

Modulus

1

Normal pressure angle αn 20° Lead angle γ

3.58°

Figure 2(a) presents the gearmesh stiffness of the worm drive. Its shape is rectangular. However, after zooming in (Fig. 2(b)), it becomes clear that the shape of the figure is trapezoidal, which makes more sense for worm drives. The choice of gears that have a very small lead angle contributes directly to the apparent rectangular shape. Solving the movement equation by using Newmark method provides the following results. Figure 3 shows the acceleration of a worm gear couple against a time frame. The signal fluctuates around zero. It is composed of periodic signals for each meshing period Tm . Tm relates to the mesh frequency fm using the following: Tm =

1 fm

(14)

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Fig. 2. (a) multiple and (b) single stiffness against a time frame.

fm is the mesh frequency in the contact of the teeth. It is equal to the frequency of each gear fr multiplied by its teeth number Z. f m = fr Z

(15)

The spectrum is presented in Fig. 4 the first peak is the frequency of the speed of the turning worm. The following frequencies are its harmonics. The natural frequencies are f1 = 0, f2 = 340, f3 = 870, f4 = 1040, f5 = 1980, f6 = 3942, f7 = 4010, f8 = 5251, f9 = 5717, f10 = 5863, f11 = 1.35 × 106 , f12 = 1.5 × 106 , f13 = 1.6 × 106 and f14 = 1.89 × 106 . The above mentioned results correspond to the functioning of an ideal worm drive.

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Fig. 3. The time-dependent acceleration of the worm.

Fig. 4. The spectrum of acceleration of the worm

6 Conclusion The results this paper present bring to the fore an original model of worm drives and their dynamic behavior. Lagrange formulism is used in order to obtain the motion equations. They are solved using Newmark method. The acceleration is observed against a time frame. It shows fluctuation around zero. The faultless spectrum highlights the fact that the worm drive is excited mainly by the variation of the gearmesh stiffness. The numerical results seem to be reasonable. Yet, these results still have to be validated by experimental measurements.

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Acknowledgements. This paper was financially supported by the Tunisian-Spanish Joint Project N° A1/037038/11. We thank Ms Safa Belhoul for her helpful review of the manuscript.

References Benabid, Y., Mansouri, S.: Dynamics study and diagnostics with vibration analysis from worm gear manufactured by reverse engineering techniques. J. Vibroengineering 18(7), 4458–4471 (2016) Chaari, F., et al.: Gearbox vibration signal amplitude and frequency modulation. Shock. Vib. 19(4), 635–652 (2012) Chaari, F., Baccar, W., Abbes, M.S., Haddar, M.: Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission. Eur. J. Mech. A/Solids 27(4), 691–705 (2008) Chung, M.Y., Shaw, D.: Parametric study of dynamics of worm and worm-gear set under suddenly applied rotating angle. J. Sound Vib. 304(1–2), 246–262 (2007) Hammami, A., et al.: Dynamic behaviour of back to back planetary gear in run up and run down transient regimes. J. Mech. 31(4), 481–491 (2015) Hmida, A., et al.: Effect of elastic coupling on the modal characteristics of spur gearbox system. Appl. Acoust. 144, 71–84 (2019). https://doi.org/10.1016/j.apacoust.2017.06.013 Khabou, M.T., et al.: Study of a spur gear dynamic behavior in transient regime. Mech. Syst. Signal Process. 25(8), 3089–3101 (2011). https://doi.org/10.1016/j.ymssp.2011.04.018 Liu, Z.Y., Huang, C.C., Hao, Y.H., Lin, C.C.: The mesh property of the steel involute cylindrical worm with a plastic involute helical gear. J. Mech. 30(2), 185–192 (2014) Tamminana, V.K., Kahraman, A., Vijayakar, S.: A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs. J. Mech. Des. 129(1), 75 (2007) Walha, L., et al.: Effects of eccentricity defect on the nonlinear dynamic behavior of the mechanism clutch-helical two stage gear. Mech. Mach. Theory 46(7), 986–997 (2011) Yassine, D., Ahmed, H., Lassaad, W., Mohamed, H.: Effects of gear mesh fluctuation and defaults on the dynamic behavior of two-stage straight bevel system. Mech. Mach. Theory 82, 71–86 (2014). https://doi.org/10.1016/j.mechmachtheory.2014.07.013

Shear-Normal Coupling Effects on Composite Shafts Dynamic Behaviour Safa Ben Arab(B) , Slim Bouaziz, and Mohamed Haddar Mechanics, Modelling and Production Laboratory (LA2MP), Mechanic Department, National School of Engineers of Sfax, University of Sfax, BP. 1173, 3038 Sfax, Tunisia [email protected], [email protected]

Abstract. This paper concerns the dynamic analysis of rotating composite shafts and focuses on the formulation of the shear-normal coupling effect using Equivalent Single Layer Theory (ESLT) and Layerwise Shaft Theory (LST). On the one hand, ESLT consists in considering the laminated shaft made of several orthotropic layers as an equivalent single layer having equivalent mechanical properties of all the layers. On the other hand, LST aims on modelling configurations where different layers may consist of materials with different properties of several orders of magnitude. Thus, different layers may have different slopes. In fact, the shear-normal coupling term is introduced directly in both formulations unlike the development available in the literature where the equivalent longitudinal Young’s modulus expression is modified to consider the shear-normal coupling effect. Results are compared with those available in the literature using different formulations. Obtained results prove that the developed theories can be efficiently used for rotating laminated shaft dynamic analysis. Indeed, obtained results prove that laminate parameters such as: fiber orientation and stacking sequence have an important influence on the shear-normal coupling and show the significant effect of the shear-normal coupling on dynamic behaviour of rotating composite shafts. Keywords: Dynamic analysis · Rotating shafts · Composite materials · Shear-normal coupling

1 Introduction The arrival of composite materials opened new ways by increasing the performance of industrial machines such as: drive shafts for aerospace and automotive industries (Bhajantri et al. 2014, Ravi 2014, Dattatray et al. 2015). In fact, using composite materials improved many properties like: stiffness, strength, weight reduction, corrosion resistance, lower vibration level and wear resistance. Besides, composite materials provide designers the ability to obtain required behaviours by changing the stacking of different layers: orientation, number and material of layers (Kaviprakash et al. 2014, Maheta and Patel 2015, Arab et al. 2018). Singh et al. (1997) and recently Gupta (2014) summarized many published works on rotating composite shafts dynamic problems. Several finite element formulations have been developed for the dynamic analysis of rotating laminated shafts based on homogenized beam and shell theories. Singh and Gupta (1996a) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 243–250, 2021. https://doi.org/10.1007/978-3-030-76517-0_27

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proved that Equivalent Modulus Beam Theory (EMBT) does not consider the stacking sequence effects. Indeed, in this approach the laminate characteristics for symmetric stacking sequence is adopted to identify the equivalent longitudinal Young’s modulus and the equivalent shear modulus. Therefore, EMBT considers only the symmetric configurations. By comparing EMBT and Layerwise Beam Theory (LBT), Singh and Gupta (1996b) showed that LBT is more efficient than EMBT because it takes into account the effects of stacking sequences, thickness shear deformation and bending-stretching coupling. Gubran and Gupta (2005) developed the Modified EMBT which considers the effects of stacking sequences and different mechanical couplings. Authors modified the equivalent longitudinal Young’s modulus expression to consider the shear-normal coupling effect present in the unbalanced configurations. More recently, Arab et al. (2017a) performed the Equivalent Single Layer Theory (ESLT) to avoid the main limitations associated with formulations that does not consider the stacking sequence effects. In fact, this theory considers the shear-normal coupling effect without modifying the expression of the equivalent longitudinal Young’s modulus as proposed by Gubran and Gupta (2005). Moreover, Arab et al. (2017b) developed the Layerwise Shaft Theory (LST) based on shaft finite element model unlike the LBT available in the literature which is reduced from shell finite element model. In fact, Moreira and Rodrigues (2006) proved that finite element formulations based on layerwise theory give more accurate results because of more realistic displacement field. The present work focuses on the analysis of the shear-normal coupling effects on the dynamic behaviour of rotating composite shafts. Furthermore, this work presents particularly the direct incorporation of the shear-normal coupling term in ESLT and LST developed by Arab et al. (2017a) and Arab et al. (2017b) and illustrates the capabilities of both theories to prove the significant influence of the shear-normal coupling.

2 Incorporation of Shear-Normal Coupling Effect The composite shaft consists of P orthotropic layers where the coordinate axis x coincides with the shaft axis as shown in Fig. 1 is considered.

Fig. 1. Composite shaft.

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2.1 Equivalent Single Layer Theory The continuous displacement field at material points along the shaft cross section is described as follows (Reddy 1997): ⎧ ⎨ ux (x, y, z, t) = −yθz (x, t) + zθy (x, t) {u(x, y, z, t)} = (1) uy (x, y, z, t) = v(x, t) ⎩ uz (x, y, z, t) = w(x, t) In fact, it is more appropriate to express the stress-strain relations using the cylindrical coordinate system because the shaft cross shape section is supposed circular. Therefore, the stress-strain relations can be expressed as follows: ⎧ ⎨ σxx = Q11 εxx + ks Q16 γxφ {σ } = (2) τxr = ks Q55 γxr ⎩ τxφ = ks Q16 εxx + ks Q66 γxφ Where ks is the transverse shear correction factor and Qij are the constitutive terms. In fact, they are related to the fiber orientation angle, Fig. 2, and the elastic constants Qij .

Fig. 2. Plane of the layer.

The shaft deformation energy is given as follows:   1  2 2 dV = Q11 εxx + 2ks Q16 γxφ εxx + ks Q55 γxr2 + ks Q66 γxφ 2

(3)

V

It can be noted that the shear-normal coupling is introduced directly using ESLT formulation by the term 2ks Q16 γxθ εxx unlike using Modified EMBT formulation, proposed by Gubran and Gupta (2005), where the equivalent longitudinal Young’s modulus expression is modified to take into account the shear-normal coupling effect. 2.2 Layerwise Shaft Theory The continuous displacement field of the kth layer of a composite shaft can be represented through a set of generalized variables as (Arab et al. 2017b): {U }k = [N ]k {d }

(4)

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Where {d } represents the generalized displacement field given by:  T {d } = u0 v w θy1 θz1 ... θyk θzk ... θyP θzk

(5)

The strain components in the cylindrical coordinate system of the kth layer is given as follows: {ε}xxk = [B]εk {d } γ

{γ }xφk = [B]k φ {d }

(6)

γ

{γ }xrk = [B]k r {d } γ

γ

Where [B]εk is the stretching-bending matrix and [B]k φ and [B]k r are the transverse shear deformation matrices of the kth layer. The kth layer stress-strain relations is expressed as follows: ⎧ ⎨ σxxk = Q11k εxxk + ks Q16k γxφk (7) τxrk = ks Q55k γxrk ⎩ τxφk = ks Q16k εxxk + ks Q66k γxφk Using the integral evaluated over the volume domain of the whole set of P individual layers, the deformation energy is obtained as: =

 P 

1  T {ε}xxk {σ }xxk + {γ }Txrk {τ }xrk + {γ }Txφk {τ }xφk dVk 2 k=1

(8)

V

Replacing the stress-strain relation expressions (7) in the deformation energy expression (8), one gets: =

   P 

1 {ε}Txxk Q11k {ε}xxk + {ε}Txxk ks Q16k {γ }xφk + {γ }Txφk ks Q16k {ε}xxk + {γ }Txφk ks Q66k {γ }xφk + {γ }Txrk ks Q55k {γ }xrk dAk dx 2 k=1

l Ak

(9) Where {ε}Txxk ks Q16k {γ }xφk and {γ }Txφk ks Q16k {ε}xxk account for the shear-normal coupling effect. Therefore, LST formulation incorporates directly the shear-normal coupling term without modifying the laminate material characteristics. The kinetic energy of the rotating composite shaft with rotational speed is obtained from the integral evaluated over the volume of the whole set of P individual layers as: T=

   P P



 1  ˙ T  ˙ 1 0 −1 U k ρk U k dVk + [N ]k d˙ dV ([N ]k {d })T ρk 1 0 2 2 k=1

Vk

k=1

Vk

(10)  Where U˙ k and ρk represent respectively the velocity field and the mass density of the kth layer.

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3 Results and Discussions 3.1 Equivalent Single Layer Theory Let consider a composite shaft clamped at an extremity and with a rigid steel disk at the free extremity. The stacking sequence of the carbon/epoxy shaft consists of one layer. The geometric and material properties of the rotor system are: • Shaft: L = 1.2 m, outer radius = 0.048 m, wall thickness = 0.008 m, E1 = 172.7 GPa, E2 = 7.2 GPa, Gij = 3.76 GPa, υ12 = 0.3, ρ = 1446.2 kg/m3 ; • Steel disk: inner radius = 0.048 m, outer radius = 0.15 m, thickness = 0.05 m, ρ = 7800 kg/m3 . In order to show the influence of the shear-normal coupling on the rotating structure dynamic behaviour, the variation of the equivalent shear-normal coupling term for different fiber orientations is presented in Fig. 3.

Fig. 3. Equivalent shear-normal coupling term for different fiber orientation.

It can be observed from Fig. 3 that the equivalent shear-normal coupling term is equal to zero for 0° and 90° fiber orientation angles. In this case, the shear-normal coupling effect has no influence and can be neglected. While, for other fiber orientation angles, the equivalent shear-normal coupling term is different from zero. In fact, the highest value of the equivalent shear-normal coupling term is associated to 30° fiber orientation angle. Indeed, an important variation is obtained when comparing the different values of the shear-normal coupling term for different fiber orientation angles. It can be deduced that shear-normal coupling effect should not be neglected for fiber orientation angles different from 0° and 90°. A comparison between the equivalent Young’s modulus and the equivalent shearnormal coupling for different fiber orientation angles is shown in Fig. 4.

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Fig. 4. Comparison between the equivalent Young’s modulus and the equivalent shear-normal coupling term.

It can be observed from Fig. 4 that the shear-normal coupling effect is more significant for the fiber orientation angle region between 30° and 75°. Moreover, the equivalent shear-normal coupling term and the equivalent Young’s modulus have almost the same values for 60 ± 5° fiber orientation angle. Therefore, this comparison confirms that the shear-normal coupling term have an important influence on the dynamic behavior of rotating laminated shafts. 3.2 Layerwise Shaft Theory In order to illustrate the capability of LST to analyse the influence of the shear-normal coupling, a simply supported composite shaft made of one single layer is considered. The composite shaft presents the following geometrical and material properties: • L = 1 m, mean radius = Rm = 0.05 m, wall thickness = 4 × 10−3 m. • E1 = 130 GPa, E2 = 10 GPa, Gij = 7 GPa, υ12 = 0.25, ρ = 1500 kg/m3 ;

• Shear correction factor: ks = 1 2. The finite element model data are given as: • Number of degrees of freedom = 5; • Number of elements = 20; • Number of nodes for the finite element mesh = 21. Table 1 gives a comparison between the natural frequencies obtained using LST excluding shear-normal coupling effects and those obtained using LST including shearnormal coupling effects. Percentages shown in Table 1 are those of the comparison between both methods.

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Table 1. Shear normal coupling effects on natural frequencies (Hz). Angles 0°

LST excluding shear-normal

LST including shear-normal

1st

2nd

3rd

1st

2nd

3rd

428.89

1234.94

2079.08

428.89

1234.94

2079.08

0%

0%

0%

1259.10

2237.20

15°

431.83

1349.89

2377.92

397.39 8%

6.7%

5.9%

30°

384.13

1341.71

2571.78

329.72

1165.37

2267.72

14.2%

13.1%

11.8%

250.55

936.74

1931.01

13.8%

13.3%

12.6%

702.79

1488.11

45°

290.75

1081.71

2209.27

60°

199.56

762.37

1609.35

183.63 8%

7.8%

7.5%

75°

150.74

577.09

1221.48

148.86

570.03

1207.12

1.2%

1.2%

1.2%

90°

140.98

534.41

1117.32

140.98

534.41

1117.32

0%

0%

0%

It can be observed from Table 1 that, using LST including shear-normal coupling effects, natural frequencies of the first three bending modes decrease when the fiber orientation increases from 0° to 90°. In fact, the largest longitudinal modulus corresponds to the layer with 0° fiber orientation, consequently, it has the highest natural frequencies. Furthermore, it can be observed from the given percentages that for 0° and 90° fiber orientation, the shear-normal coupling has no influence on natural frequencies. In fact, the highest percentages correspond to 30° and 45° fiber orientation angle as explained using Figs. 3 and 4.

4 Conclusion The shear-normal coupling is introduced directly using Equivalent Single Layer Theory (ESLT) and Layerwise Shaft Theory (LST) unlike using Modified Equivalent Modulus Beam Theory (EMBT) where the equivalent longitudinal Young’s modulus is modified to consider the shear-normal coupling effect. Obtained results from ESLT and LST show that laminate parameters like fiber orientation, number of layers and stacking sequence have an important effect on shear-normal coupling. Moreover, obtained results prove that shear-normal coupling should not be neglected and have a significant influence on the dynamic behaviour of rotating composite shafts.

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References Bhajantri, V., Bajantri, S., Shindolkar, A.M., Amarapure, S.: Design and analysis of composite drive shaft. Int. J. Res. Eng. Technol. 3, 738–745 (2014) Ravi, A.: Design, comparison and analysis of a composite drive shaft for an automobile. Int. Rev. Appl. Eng. Res. 4, 21–28 (2014) Dattatray, G., Kale, K.B., Dongare, A.D.: Design and analysis of composite drive shaft for automobile application. Int. J. Res. Advent Technol. 161–165 (2015) Kaviprakash, G., Kannan, C.R., Lawrence, I.D., Regan, A.P.: Design and analysis of composite drive shaft for automotive application. Int. J. Eng. Res. Technol. 3, 429–436 (2014) Maheta, V.V., Patel, A.B.: Design, analysis and optimization in automobile drive shaft. Int. J. Innov. Res. Sci. Technol. 1, 432–439 (2015) Arab, S.B., Rodrigues, J.D., Bouaziz, S., Haddar, M.: Stability analysis of internally damped rotating composite shafts using a finite element formulation. C.R. Mec. 346, 291–307 (2018) Singh, S., Gubran, H., Gupta, K.: Developments in dynamics of composite material shafts. Int. J. Rotating Mach. 3, 189–198 (1997) Gupta, K.: Composite shaft rotor dynamics: an overview. In: Sinha, J. (ed.) Vibration Engineering and Technology of Machinery, pp. 79–94. Springer (2014) Singh, S., Gupta, K.: Dynamic analysis of composite rotors. Int. J. Rotating Mach. 2, 179–186 (1996a) Singh, S., Gupta, K.: Composite shaft rotordynamic analysis using a layerwise theory. J. Sound Vib. 191, 739–756 (1996b) Gubran, H., Gupta, K.: The effect of stacking sequence and coupling mechanisms on the natural frequencies of composite shafts. J. Sound Vib. 282, 231–248 (2005) Arab, S.B., Rodrigues, J.D., Bouaziz, S., Haddar, M.: A finite element based on equivalent single layer theory for rotating composite shafts dynamic analysis. Compos. Struct. 178, 135–144 (2017a) Arab, S.B., Rodrigues, J.D., Bouaziz, S., Haddar, M.: Dynamic analysis of laminated rotors using a layerwise theory. Compos. Struct. 182, 335–345 (2017b) Moreira, R., Rodrigues, J.D.: A layerwise model for thin soft core sandwich plates. Comput. Struc. 84, 1256–1263 (2006) Reddy, J.: Mechanics of Laminated Composite Plates - Theory and Analysis. CRC Press, Boca Raton (1997)

Experimental Contact Model Calibration for Computing a Vibrating Beam Coupled to a Granular Chain Impact Damper Chaima Boussollaa1,2(B) , Vincent Debut1,3 , Jose Antunes1 , Tahar Fakhfakh2 , and Mohamed Haddar2 1 Centro de Ciências e Tecnologias Nucleares (C2TN), Instituto Superior Técnico (IST),

Universidade de Lisboa, 2695-066 Bobadela LRS, Lisbon, Portugal [email protected], {vincentdebut, jantunes}@ctn.tecnico.ulisboa.pt 2 Laboratory of Mechanics Modeling and Production (LA2MP), National School of Engineers of Sfax, University of Sfax, BP N°1173, 3038 Sfax, Tunisia {tahar.fakhfakh,mohamed.haddar}@enis.rnu.tn 3 Polytechnic Institute of Castelo Branco (IPCB/ESART), Avenida do Empresário - Campus da Talagueira, 6000-767 Castelo Branco, Portugal

Abstract. Impact dampers of various designs are currently used for mitigating vibrations of industrial components. Although they may generate disturbing noise, these devices offer many advantages, as they are quite robust, exempt from finetuning and stand severe temperature and environmental conditions. Moreover, they can be applied to systems subjected to wideband excitations, which is a major advantage. However, they are highly nonlinear, hence the numerical modelling issues and the difficulties to predict their behaviour and to optimize their design. This work addresses some basic steps to increase our understanding of such devices, focusing on the family of “simple” chain impact dampers, composed of one or several spheres impacting inside a guide-tube. Damping of such 1-D device is related to the visco-elasticity of the impacting spheres, with only residual friction phenomena. In practice, this device can easily be coupled to any structure, such as the clamped-free vibrating beam as used in the present work. In this paper, we report the detailed dynamical modelling of the beam-damper coupled system and present preliminary simulation results obtained from a model calibrated based on experimental data stemming from modal analysis and drop test experiments. Keywords: Nonlinear dynamics · Impact damper · Computational model · Nonlinear time-domain numerical simulations · Vibrations

1 Introduction Since the pioneering work by Masri [1], a long tradition has been developed on the use of impact dampers to mitigate the vibratory responses of buildings and industrial components to seismic and other types of excitations. As discussed in [2], a large range of impact © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 251–260, 2021. https://doi.org/10.1007/978-3-030-76517-0_28

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damper designs are currently in use, ranging from single comparatively large impacting mass to highly complex granular dampers. In 1937, Paget developed the impact damper when he was looking for a solution for the vibrations of turbine blades. In 1969, Masri studied the multi-unit impact damper, and found that compared to the impact damper mass equivalent total mass, multi-unit impact dampers are more effective for reducing the noise generated by impacts. In the 80s, Popplewell et al. [3] addressed the problem of vibration attenuation of the boring bar using the bean bag impact damper. Binoy et al. [4] showed that piston-based particle dampers are more effective than impact damper in case of small external excitation. Li et al. [5] developed the buffered impact damper, made of a rubber material covering the inner wall of the particle damper which can alter the stiffness and leads to considerable improvements on the working performance of the damper by decreasing the impact force and noise levels over a wide range of frequencies. Chen et al. [6] adopted the tuned particle damper obtained by a particle damper attached to the main structure. Fairly recently, Ben Romdhane [7] studied the Non-Obstructive Particle Damping in order to measure the loss factor of these systems. Even if impact dampers usually generate disturbing noise, their interest stems from many advantageous features. They are mechanically robust, exempt of fine-tuning, and can stand severe temperature and harsh environmental conditions. Moreover, they can be applied to mechanical systems subjected to wideband excitations. However, these devices are highly nonlinear, which leads to difficult numerical modelling issues [8, 9]. Contrary to the standard linear dynamical damper, it is quite challenging to predict their actual dynamical behaviour and optimize their efficiency. This work addresses some basic steps to increase our understanding of the behaviour of these devices. More specifically, we focus on the family of “simple” chain impact dampers, composed of one or several spheres impacting inside a guide-tube (see [10, 11]). The damping properties of these 1-D devices are essentially related to the visco-elasticity of the impacting spheres, with only some residual (but unavoidable) friction phenomena. Such device can be easily coupled to any structure, either simple or complex. In this paper, we report the detailed dynamical modelling of a cantilever vibrating beam coupled to a sphere-chain impact damper, which may be thought as half-way between a conceptual academic toy and a real-life application. To be more realistic, our computational model is calibrated using parameter values obtained from measurements, including drop test experiments and an experimental modal analysis of the cantilever beam. Finally, we illustrate the computational method with time-domain numerical simulations, showing the relevance and interest of the developed approach in order to understand and later optimize their dissipative efficiency.

2 Computational Model 2.1 Beam Model The physical model consists on a vibrating clamped-free beam (with length L, crosssection area A and moment of inertia I , mass density ρ and Young’s modulus E), and a column of s = 1, 2, · · · , Ns translating and impacting spheres (with constant diameter

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D, mass density ρs and Young’s modulus Es ). A classical modal formulation is used for the vibrating beam, written as: mn q¨ n (t) + cn q˙ n (t) + kn qn (t) = Fn (t)

(1)

where qn (t) are the modal responses of the n = 1, 2, · · · , N beam modes, with modal properties mn , cn = 2mn ωn ζn , kn = mn ωn2 and mode shapes φn (x), while Fn (t) are the generalized forces obtained by modal projection of the physical forces F(x, t), calculated classically as: ⎞ ⎛ L L  P P  Fp (t)δ(x − xp )⎠φn (x)dx = Fp (t)φn (xp ) Fn (t) = F(x, t)φn (x)dx = ⎝ 0

0

p=1

p=1

(2) where Fp (t) are a set of localized contact forces acting at locations xp (p = 1, 2, · · · , P). Finally, from the knowledge of the modal amplitudes, the beam physical response Y (x, t) can be computed by modal summation as: Y (x, t) =

N 

φn (x)qn (t)

(3)

n=1

2.2 Impact Damper Model Concerning the spheres of the impact damper, a 1-D particle model consisting of a number of s = 1, 2, · · · , Ns spheres is used, for which the dynamics of each sphere is governed by: ms Y¨ s (t) = −ms g + F(s−1),s (t) − Fs,(s+1) (t)

(4)

where ms = ρs π D3 /6 is the mass of the sphere s, g is the gravity acceleration, and F(s−1),s (t) and Fs,(s+1) (t) are the interaction forces between sphere s and the adjacent spheres s − 1 and s + 1, respectively. For s = 1, the index s − 1 stands for the interaction force with the vibrating bar through the damper base. Conversely, for s = Ns , the index s + 1 stands for the opposite interaction force with the bar through the damper body/cup (if the damper is open, this force term is nil). 2.3 Contact Interaction Model A very important aspect of the computational model concerns the formulation of the interaction forces between the impacting spheres within the damper and with the beam. In this work, we use a quite general compliant contact force model [6] for which the contact force F ij (t) between spheres i and j is given by:  

α

β Fij (t) = ± Kij δij (t) + Cij δij (t) δ˙ij (t) (5)

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with δij (t) = Yi (t) − Yj (t) and δ˙ij (t) = Y˙ i (t) − Y˙ j (t) are the relative displacement and velocity between the contacting spheres respectively. The first term stands for the conservative Hertz model, with α = 1.5, for which the stiffness parameter Kij is an explicit function of Di , Dj , Ei and Ej . The second term is dissipative, with an exponent typically in the range 0.25 ≤ β ≤ 1.5 and the damping parameter Cij changing accordingly, depending on the specific model used, see [7]. Other options might be used for modelling the interaction, in particular inspired by impulse-based techniques [4–6]. However, if these approaches are computationally faster, they remain quite delicate to implement under multiple collisions. Moreover, they are ill suited for a detailed description of the interaction forces, under general conditions, in particular steady contact. 2.4 Time-Domain Solutions In order to integrate and solve the coupled equations of motions (1), (4) and (5), a number of time integration algorithms can be used, which are generally classified into two categories: explicit and implicit methods. In this paper, an explicit time integration algorithm is used. For a multiple-degree-of-freedom (MDOF) non-linear structure, the equation of motion can be expressed as: M x¨ i+1 + C(˙xi+1 ) + K(xi+1 ) = Fi+1

(6)

where M is the mass matrix, C(˙xi+1 ) , K(xi+1 ) and Fi+1 are the (eventually nonlinear) damping force, restoring force and external force vectors computed at the time step (i + 1) respectively. A family of integration algorithms, for which the velocity vector (˙xi+1 ) and displacement vector (xi+1 ) are dependent on structural response only (displacement, velocity and acceleration) at the time step i is assumed as: x˙ i+1 = x˙ i + α1 t x¨ i

(7)

xi+1 = xi + t x˙ i + α2 t 2 x¨ i

(8)

where α1 and α2 are two matrices of integration parameters to be determined and t is the time step size. Equation (6) can be explicitly integrated with Eqs. (7) and (8) once α1 and α2 are known. If the MDOF system is nonlinear, with α1 = α2 = α [12] and α is assumed to be invariant in the entire integration procedure, and can be determined from the initial damping matrix C0 and the initial stiffness matrix K0 as: α1 = α2 = α = 2λ(2λM + λ tC0 + 2 t 2 K0 )−1 M

(9)

where λ is considered as the positive parameter determining the numerical properties. The subfamily with λ is originated from the Newmark family with γ = 21 , β = λ1 . Hence, Eq. (6) can be solved using an explicit algorithm, following the steps: the substitution of Eq. (9) into Eqs. (7) and (8) result in the velocity vector (˙xi+1 ) and displacement vector (xi+1 ); the nonlinear damping force C(˙xi+1 ) and restoring force K(xi+1 ) are then calculated, and finally the acceleration vector x¨ i+1 is obtained from Eq. (6).

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3 Contact Modal Calibration 3.1 Experimental Rig Identification of the empirical dissipation parameters β and Cij of the contact model (see Eq. (5)) was based on the comparison between computed and measured contact forces, the latter being obtained through drop test experiments. The sphere was dropped inside a tube, from a height of 82 mm, making successive rebounds on a force sensor (B&K 8200) attached to a rigid base. The force signals were acquired using a dynamic signal analyser B&K Photon+, during 4 s, at a sample rate of 96 kHz. A schematic view of the experimental set up is shown in Fig. 1.

Fig. 1. Left plot: drop test experimental set up; Right plot: measured impact force.

3.2 Contact Model Calibration The calibration of the contact force model was done manually, by testing a large range of values for the two parameters included in the model, and then by comparing the modelled impact force with a set of measured signals. Figure 2 shows the comparison between several measurements and the best fit obtained, using values of β = 1; Cij = 9.106 . As seen, the model globally reproduces the general trends of the measurements regarding the timing and decay of the successive bounces, especially the second measurement.

Fig. 2. Comparison between measured and simulated contact forces.

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4 Application to a Vibrating Beam with an Impact Damper 4.1 Experimental Rig Our computational model simulates the experimental setup illustrated in Fig. 3. It includes a beam of steel, of L = 0.48 m and cross-Sect. 50.5 × 8.2 mm2 , excited by a shaker, and one impact damper located close to the free end of the beam, at Xd = 0.445 m. The impact damper consists of a guiding tube in aluminium closed by two stops at the tube extremities. The spheres are made of steel, with D = 18.25 mm and up to five spheres can be included in the damper. An electrodynamic shaker, operated by a power amplifier, is connected to the beam using a stinger at Xe = 0.15 m. The beam response is measured through two small accelerometers located at the shaker and damper locations, while three piezoelectric force transducers are used for measuring the input force (provided by the shaker) and the two contact forces between the beam and the spheres at the damper location.

Fig. 3. Left: global view of the experimental set-up. Right: details of the chain impact damper.

4.2 Modal Identification of the Beam Impact tests were performed along the beam length, when no damper is applied, resulting in a set of transfer functions. The modal identification was based on processing the force and acceleration time-domain signals. From the impulse responses, the modal parameters were extracted using a parametric multi-degree-of-freedom technique called the Eigen system Realization Algorithm (ERA) [13]. Based upon concepts of control theory, the idea behind ERA is to reconstruct the system responses from the experimental data using the minimum order of the state space formulation, which is obtained by SVD filtering of the zero-order Hankel matrix, built from the system outputs. For the identification, we used the first 0.15 s of each impulse response, and limited our analysis within the frequency range 0–6000 Hz. The model reduction was obtained by the analysis of a stabilization diagram built by repeating the identification process with an increasing number of modes each time. Such a stability diagram makes possible to differentiate between the physical and non-physical modes of the actual system. The identified modal

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frequencies and modal damping values for the cantilever beam are given in Fig. 4. To show the reliability of the modal extraction technique, a typical impulse response and corresponding transfer function of the reduced-order model (using 8 identified modes) are compared with the measurement in Fig. 5.

Fig. 4. Experimentally identified modal frequencies and modal damping

Fig. 5. Measured and synthesized impulse response (top) and transfer functions (bottom).

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4.3 Illustrative Computations 4.3.1 Dynamics Behavior of the Beam Without Impact Damper Figure 6 shows the computed velocity response at the free end of the beam for an increasing linear sweep in the frequency range 1–1000 Hz, when applying a force of constant amplitude of 10 N. As the excitation frequency increases, one can see the successive excitation of the modes of the beam occurring when the excitation frequency is close to the resonant frequency of one of the beam mode. As seen, each mode responds differently according to its characteristics of modal mass, modal damping and modeshape values at the excitation and response locations.

Mode 1

Mode 2

Mode 3

Fig. 6. Beam velocity at the free end for a sinusoidal linear force sweep in the range 1–1000 Hz.

4.3.2 Dynamics Behavior of the Beam Coupled to the Impact Damper Figures 7–8 and Table 1 present simulation results stemming from parametric computations consisting of varying the number of spheres, considering zero, one, two and five spheres. The plots in Figs. 7 and 8 display the time histories of the total contact forces acting on the beam, including the contributions of the first and last spheres of the damper, and the beam velocity response for the four considered configurations. A quick view at both figures reveals that increasing the number of spheres has a positive effect on the reduction of the beam vibration. As anticipated, the presence of a single sphere strongly reduces the beam vibratory motion but note that increasing the number of spheres also intensifies this trend. However, it seems that the relation between the number of spheres and the reduction of the beam vibration does not follow a linear behavior. The second noticeable and important consequence of increasing the number of spheres is to reduce the peak amplitude of the global forces acting on the beam. Impacts are less intense but globally produce more dissipation in the system, certainly because the number of impacts is larger compared to the case of a single sphere. Quantitatively, Table 1 presents some statistics computed from the simulated force and responses confirming the trends discussed. Notably, the global interaction forces between the beam and the impacting spheres at both ends of the damper increases as the number of spheres increases, thus providing more dissipation of vibrational energy from the beam, while both the displacement and velocity of the beam decrease, thus confirming the reduction of the beam vibration.

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Table 1. Influence of the number of spheres on the dynamics of the system: average of the absolute of the total impact forces acting on the beam, including the contributions of the first and last spheres of the damper, and RMS values of the beam displacement and velocity at the damper location. Interaction damper force (N)

Displacement at the damper Velocity at the damper location RMS (m) location RMS (m/s)

0 sphere

–

3.349.10−4

3.26.10−1

1 sphere

3.431.10−1

2.15.10−4

1.538.10−1

2 sphere

5.988.10−1

2.144.10−4

1.349.10−1

1.332

1.817.10−4

1.177.10−1

5 sphere

Fig. 7. Influence of the number of spheres on the dynamics of the system. Time history of the total contact force acting on the beam, including the contributions of the first and last spheres of the damper. Top: global view of the full sweep; bottom: zoom around the excitation of the second beam mode.

Fig. 8. Influence of the number of spheres on the dynamics of the system: time history of the beam velocity response close to the free end. Same computations as in Fig. 7.

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5 Conclusions As a step forward the understanding of the physical behavior and damping efficiency of impact dampers, we presented in this study some modelling aspects when coupling a flexible multi-body structure to a chain impact damper. The model was built based on current modelling and numerical techniques developed in the field of particle dynamics and using a modal framework for the flexible structure. Before performing any timedomain simulations, the computational model was calibrated using experimental data stemming from drop test experiments and modal analysis, in order to feed the model with realistic values for the contact model and the beam modal parameters. Some preliminary parametric simulation results were presented and discussed, illustrating the relevance and interest of the developed approach for gaining a deeper understanding of the behavior of such impact dampers and ultimately, for designing of optimal dampers. The analysis of the energy exchanges occurring in the system are currently being investigated by the authors through extensive parametric simulations, and results will be presented in a forthcoming paper. Also, experiments using the test rig presented in Sect. 4.1 will be performed and results will be compared with simulation results in order to validate the proposed model.

References 1. Masri, S.F.: On the stability of the impact damper. J. Appl. Mech. 33(3), 586–592 (1967) 2. Lu, Z., Wang, Z., Masri, S.F., Lu, X.: Particle impact dampers: past, present, and future. Struct. Control Health Monit. 25(1), 1–25 (2017) 3. Popplewell, N., Semercigil, S.E.: Performance of the bean bag impact damper for a sinusoidal external force. J. Sound Vib. 133(2), 193 (1989) 4. Shah, B.M., Pillet, D., Bai, X.-M., Keer, L.M., Wang, Q.J., Snurr, R.Q.: Construction and characterization of a particle-based thrust damping system. J. Sound Vib. 326(3–5), 489 (2009) 5. Li, K., Darby, A.P.: A buffered impact damper for multi-degree-of-freedom structural control. Earthq. Eng. Struct. Dyn. 37(13), 1491–1510 (2008) 6. Chen, L.A., Semercigil, S.E.: A beam-like damper for attenuating transient vibrations of light structures. J. Sound Vib. 164(1), 53–65 (1993) 7. Romdhane, M.B.: The loss factor experimental characterization of the non-obstructive particles damping approach. Mech. Syst. Sig. Process. 38, 585–600 (2013) 8. Haile, J.M.: Molecular Dynamics Simulation: Elementary Method, vol. 7, pp. 625. Wiley, New York (1993). https://doi.org/10.1063/1.4823234 9. Hinrichsen, H., Wolf, D.E. (eds.): The Physics of Granular Media. Wiley, Weinheim (2004) 10. Nguyen, N.S., Brogliato, B.: Multiple Impacts in Dissipative Granular Chains, vol. 72, pp. 11– 29. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-39298-6 11. Flores, P., Lankarani, H.M.: Contact Force Models For Multibody Dynamics, vol. 226. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-30897-5 12. Gui, Y., Wang, J.-T., Jin, F., Chen, C., Zhou, M.-X.: Development of a family of explicit algorithms for structural dynamics with unconditional stability. Nonlinear Dyn. 77(4), 1157– 1170 (2014). https://doi.org/10.1007/s11071-014-1368-3 13. Juang, J.: Applied System Identification. PTR Prentice-Hall Inc., Englewood Cliffs (1994)

Numerical Analysis of Transient Flow in a Hydraulic Tree Network: Zarroug Aqueduct Network in Gafsa, Southern Tunisia Lazher Ayed(B) and Zahreddine Hafsi Laboratory of Applied Fluids Mechanics, Process and Environment Engineering, National Engineering School of Sfax, University of Sfax, Sfax, Tunisia

Abstract. The numbers of long-distance transport pipelines constructed in areas with relatively complex terrain and large topographic changes have increased in recent years. Hydraulic transients are characterized by an abrupt change in the flow condition leading the pressure and the velocity of water to undergo rapid changes over time. Such transient events are likely to damage water distribution systems if not equipped with protection devices. The sudden variation in pressure propagates throughout the hydraulic system and it may result in the breakdown to the network pipelines. The aim of this paper is to numerically analyze the transient water flow in Zarroug aqueduct network. This network feeds the Tunisian Chemical Group in the city of Mdhila, Gafsa. After establishing the steady state conditions, the transient regime is studied using the method of characteristics. The numerical model is solved via a Matlab code and the temporal pressure evolution is followed up. The phenomenon of cavitations appears in a repetitive manner for which the pressure drop reaches very risky values that can collapse the pipe network. Through this modest work we want to study in our future works the entire network in order to protect it against degradations. Keywords: Transient flow · Water hammer · Pipe network · Method of characteristics · Pressure wave

1 Introduction The transport of fluids from the production site to the operating site presents a delicate sector for industries. This is why the most effective method to transport is via a pipe connection which built a pipe network, by working with this large scale, the disruption of the initial flow state causes enormous pressure fluctuations which generate a very dangerous transient phenomenon called water hammer. Indeed, water hammer is an oscillatory phenomenon created following a sudden change in the initial steady state conditions of the flow parameters. It generates enormous pressure distributions creating a pressure wave propagating along pipe. The latter phenomenon is due to transients occurring in a sudden manner in pressurized conduits and this viz. abrupt stoppage of pumps to power failure for example, load changes in hydropower plants, rapid closure of shut-off valves, etc. Due to the elasticity of the medium (the fluid and the pipe), the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 261–268, 2021. https://doi.org/10.1007/978-3-030-76517-0_29

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disturbance resulted from water hammer, created in a pipe section, propagates in both sides of this section in the form of waves. In practice, the occurrence of these phenomena can have hazardous consequences and cause serious material damage such as the rupture of pipes, disorders in pumping installations, and sometimes even loss of life. The problem of transient flows was first studied by Menabera in 1885, and then in 1913, Allievi demonstrated the possibility of neglecting convective terms in the momentum equation without greatly affecting the accuracy of obtained results. Later the validity of the one-dimensional approach in the case of pipes was confirmed under both laminar and turbulent conditions (Vardy and Hwang 1991). Transient one dimensional flow in a pipeline is governed by a set of two partial differential equations describing mass and momentum conservations. This system of equations, being unsuitable to be solved analytically, has been numerically solved using diverse numerical techniques. Based on the method of characteristics, in 1960, Lister used a fixed grid for a fixed time step however; the pipes may have different lengths and wave velocities causing a problem of discretization. In 1975, Trikha suggested the use of a different time step for each pipe. Later, this problem of discretization has been overcome by interpolation techniques (Fox 1977; Ouragh 1994), artificial adjustment of wave celerity (Wylie and Streeter 1978) or their combination. In 1983, Goldberg and Wylie developed a retrospective interpolation technique. In 2008, Afshar and Rohani have proposed an implicit MOC. This research aims to control the pressure wave in the water distribution network at the two first successive branches. This pressure wave control provides Informations on the state of the flow and the above-mentioned transport structure. It serves as a diagnostic tool and helps in prevention.

2 Mathematical Formulations The transient flow of incompressible fluids along a pipeline is described by conservation of mass and momentum equations. Written under matrix form, both equations yield:        2 H ∂ H ∂ 0 CgA 0 (1) = + ∂t Q ∂x gA 0 − λQ|Q| Q 2DA where H, Q, D, A, C, g and λ stand respectively for the pressure head, the flow rate, the pipe diameter, the pipe’s cross section area, the celerity of pressure waves, the gravitational acceleration and the friction factor. The frictional factor is given by Colebrook-White equation (Colebrook and White 1937):   e 1 2.51 (2) + √ √ = −2 log10 3.7D Re λ λ where e is the internal roughness of the pipe and Re is the Reynolds number. For the determination of λ, by knowing Re and e, approximate implicit as well as explicit solutions of Eq. (2) are encountered in literature (Brkic and Praks 2019; Hafsi 2021). Additionally λ can be determined using Moody chart.

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Following the principle of the method of characteristics, one can convert the two partial differential Eqs. 1 and 2 into ordinary differential equations that can be numerically solved using finite difference techniques (Gray 1954). The application of the latter technique yields the following system of equations: λQ|Q| dQ gA dH ± + =0 dt C dt 2DA

(3)

dx = ±C dt

(4)

Equations (3) are called compatibility equations while Eqs. (4) are called characteristics equations. Since for incompressible fluids flow the celerity of pressure waves C is constant, the characteristic lines having for slopes ±C are straight in the (x, t) plane as represented in Fig. 1.

Fig. 1. Characteristic grid

By integrating the compatibility equations respectively on the characteristic lines C + and C − , the flow rate and the pressure head at the point P are written: QPi =

(CP + Cn ) 2

(5)

HPi =

(CP − Cn ) 2M

(6)

where: CP = Qi−1 +

gA λ Hi−1 − tQi−1 |Qi−1 | C 2DA

(7)

Cn = Qi+1 −

gA λ Hi+1 − tQi+1 |Qi+1 | C 2DA

(8)

M =

gA C

(9)

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3 Applications and Results 3.1 Description of Pipe Network Our real model of study is the Zarroug aqueduct network that supplies the Tunisian Chemical Group with water located in Mdhila Gafsa. The studied hydraulic system is a 17 km long network. Feeding Tank

TK5602 P1

P2

P3 and P4

P5

P6, P7 and P8

P9

P10 , P11 and P12

P13

P14 , P15 and P16

Table 1. Characteristics of the pipes network table. Pipe ID

Length (m)

Diameter (m)

Material

Young modulus (GPa)

Poisson ratio

Relative roughness

1

7420

0.4

Concrete

30

0.2

0,005

2

10

0.3

Concrete

30

0.2

0,0067

3

120

0.4

Concrete

30

0.2

0,005

4

120

0.4

Concrete

30

0.2

0,005

5

10

0.3

Concrete

30

0.2

0,0067

6

120

0.4

Concrete

30

0.2

0,005

7

1440

0.4

Concrete

30

0.2

0,005

8

120

0.4

Concrete

30

0.2

0,005

9

10

0.3

Concrete

30

0.2

0,0067

10

120

0.4

Concrete

30

0.2

0,005

11

1860

0.4

Concrete

30

0.2

0,005

12

2800

0.4

Concrete

30

0.2

0,005

13

400

0.4

HDPE

0.15

0.29

0.025e−3

14

925

0.4

Concrete

30

0.2

0,005

15

875

0.4

Concrete

30

0.2

0,005

16

80

0.4

Concrete

30

0.2

0,005

The source node is “Z3bis” that feeds the network from a concrete cylindrical tank having a volume of 100 m3 , delivering water with a flow rate of 360 m3 /h. This network is composed of 16 pipes connected in series. The majority of the pipes are made up of concrete except a 400 m long portion which is made up of high density polyethylene (HDPE), the downstream end of the network is a cylindrical steel tank TK5602 with a diameter of 20 m, a height of 10.75 m. The main characteristics of the network are reported in Table 1.

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3.2 Results and Discussion For the numerical modeling of transient flow in the considered network, the latter was subdivided into eight main branches. In this paper, only results related to the flow within the first and the second branches are presented. The first branch is made up of pipes 1 and 2 having the same wall thickness e = 0.03 m, two different diameters and a total length L = 7830 m as illustrated in Fig. 2. Pipe 1 is delimited by the tank that feeds the network from Zarroug in the upstream side and the beginning of pipe 2 with a section decrease (shrinkage) in the downstream side. Pipe 2, connected to the outlet of pipe 1, ends up with a section increase (expansion), then it is connected to the following pipe.

Tank feeding the network

First-Pipe

Second-Pipe

Fig. 2. The first branch of the network

In Fig. 3, the numerically obtained pressure head evolutions with in the outlet sections of each pipe of the first branch are plotted. It is observed from Fig. 3 that the shrinkage in the downstream side of pipe 1 results in an abrupt decrease in pressure head. The latter can be explained by the the Joukowski formula H = ± CV/g (Joukovosky 1904). In fact, the shrinkage causes an increase in the wave velocity and consequently a decrease in the pressure head. The other branch chosen as a network diagnostic tool is the second branch which ends with a valve, an origin of excitation and a source of wave propagation. A schematic representation of the second branch in presented in Fig. 4. In Fig. 5, obtained piezometric head pressure evolution with time at the end section of the second branch is illustrated. According to Fig. 5, it is obvious that just upstream of the valve (the end of the third pipe), high pressures are maintained for a longer time-period. Therefore, wave dampers should be placed on the pipe joints to avoid breakdowns as they are less sensitive to high pressures. Furthermore, a resonance phenomenon is observed at the valve that means a maximum of the potential mechanical energy of the wave. The overpressure is suddenly established at time origin and it remains constant for about 0.1 s, which is the characteristic period of the pipe. The pressure wave amplitude undergoes damping due to energy losses by friction. The hazardous phenomenon observed from the first two branches of network is not reduced to overpressure, but also to a depression that can lead to negative pressure values susceptible to result in the vapor cavitations’ phenomenon.

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Fig. 3. Time history of the pressure head in the outlet sections of pipes 1 and 2. a and b Enlargement of the Fig. 3 for two time scales

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the second branch of the aqueduct network

Fig. 4. The second branch of the network

Fig. 5. Piezometric head pressure evolution at the end section of branch 2 (valve)

4 Conclusion The analysis of transient flows in the water pipeline network which supplies the company of the Tunisian chemical group located in Gafsa was performed numerically using the method of characteristics. The obtained results were discussed in order to find the adequate solutions for the reduction of harmful effects of water hammer on hydraulic equipments in order to maintain the integrity of pipeline network. The transient analysis of the first two branches of the pipe network makes easy a preventive measure that can effectively control the water hammer pressure and reduce the occurrence of incidents.

References Menabrea, L.F.: Note sur les effets du choc de l’eau dans les conduites. C. R. Hebd. Seances Acad. Sci. 47, 221–224 (1885) Allievi, L.: Teoria generale del moto perturbato dell’acquali nei turbi in pressione. Annali della societa degli Ingeneri ed architetti italiani. Milano (1903) Vardy, A.E., Hwang, K.L.: A characteristics model of transient friction in pipes. J. Hydraul. Res. 29(5), 669–684 (1991). https://doi.org/10.1080/00221689109498983 Lister, M.: The numerical solution of hyperbolic partial differential equations by the method of characteristics. In: Ralston, A., Wilf, H.S. (eds.) Mathematical Methods of Digital Computers. Wiley, New York, pp. 165–179 (1960)

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Trikha, A.K.: an efficient method for simulating frequency-dependent friction in transient liquid flow. ASME J. Fluids Eng. 97, 97–105 (1975). https://doi.org/10.1115/1.3447224 Fox, J.A.: Hydraulic Analysis and Unsteady Flow in Pipe Networks. MacMillan Press, London (1977) Ouragh, Y.: Ecoulement forcé en hydraulique. Tome 2, Edition O.P.U. Alger (1994) Wylie, E.B., Streeter, V.L.: Fluid Transients. MacGraw-Hill (1978) Goldberg, D.E., Wylie, E.B.: Characteristics method using time-line interpolations. J. Hydraul. Eng. 109:5(670) (1983). https://doi.org/10.1061/(ASCE)0733-9429 Afshar, M.H., Rohani, M.: Water hammer simulation by implicit method of characteristic. Int. J. Press. Vessels Pip. 85, 851–859 (2008) Colebrook, C.F., White, C.M.: Experiments with fluid friction in roughened pipes. Proc. R. Soc. A Math. Phys. Eng. Sci. 161(906), 367–381 (1937). https://doi.org/10.1098/rspa.1937.0150 Gray, C.A.M.: Analysis of water hammer by characteristics. Trans. Am. Soc. Civ. Eng. 119, 1176–1194 (1954) Brkic, D., Praks, P.: Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function. Mathematics 7(34) (2019). https://doi.org/10.3390/ math7010034 Hafsi, Z.: Accurate explicit analytical solution for Colebrook-White equation. Mech. Res. Commun. 111 (2021). https://doi.org/10.1016/j.mechrescom.2020.103646 Joukovosky, N.: Water hammer. Proc. Am. Waterworks Assoc. vol. 24 (1904)

Assessment of Temperature History When Abrasive Milling of Long Fiber Reinforced Polymers F. Guesmi1 , M. Elfarhani1 , S. Ghazali2 , A. S. Bin Mahfouz3 , A. Mkaddem2 , and A. Jarraya1,2(B) 1 LA2MP, National School of Engineering of Sfax, University of Sfax, PO Box 1173,

3038 Sfax, Tunisia [email protected], [email protected], [email protected] 2 Department of Mechanical and Materials Engineering, FOE, University of Jeddah, PO Box 80327, Jeddah 21589, Saudi Arabia {sghazali,amkaddem}@uj.edu.sa 3 Department of Chemical Engineering, Faculty of Engineering, University of Jeddah, PO Box 80327, Jeddah 21589, Saudi Arabia [email protected]

Abstract. This attempt addresses an investigation on temperature evolution owing to tool-material interfaces when milling glass fiber reinforced polymers (GFRP). An experimental set-up was built for determining cutting temperatures during cutting period. The composite panels were prepared using unidirectional glass fibers and Epoxy matrix basing on appropriate curing cycle. After curing, the obtained panels were post-cured in autoclave at constant pressure and temperature. The tests were conducted on vertical CNC milling machine and monoblock diamond wheel. A measuring device equipped with 8 type-K thermocouples (TCs) was connected to an acquisition card capable of recording the temperature history during cutting. The TCs were placed inside the workpiece through beforehand prepared holes. Focus was specially put on the analysis of temperature evolution versus the tool advance along the cut surface. The obtained plots exhibit typically two different phases: (i) relatively sudden heating phase to reach a pick temperature value, and (ii) a much more slightly cooling phase up to room temperature. The pick temperature values show a continuous increase with the tool advance reflecting heat localization process due to cutting time increase. The heat inevitably yields at the interfaces between the moving tool and the specimen due to contact conditions was also discussed. Keywords: GFRP · Cutting temperature · TCs · Abrasive milling

1 Introduction Glass Fiber reinforced polymers (GFRP) are widely used in many industries due to their low cost and high specific mechanical properties. Machining of GFRP composites is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 269–276, 2021. https://doi.org/10.1007/978-3-030-76517-0_30

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an important activity in engineering applications. While composite manufacturing processes are not completely mastered, they make it possible to manufacture close to thread forms. In many cases, machining operations are still remain necessary to achieve the finish. Several issues are in queue for solving when cutting these materials if compared to conventional materials. Fiber-reinforced composites consists of different phases with different mechanical properties i.e. fiber, matrix, and interfaces, which induces anisotropy in their structure and, hence, complicate their manufacturing. Cutting-induced damage is closely related to thermal deformation involved at the tool-material interfaces. Typically, milling of composites is a fairly complex task due to the heterogeneity nature of the cut material, fiber abrasiveness, etc. This enhances delamination, promote in-depth interface failure and tool wear which plays to substantially reduce the structure integrity and tool lifetime. GFRP composite parts are generally subjected to corrective plain milling operations. The machinability of fiber-reinforced composites is strongly influenced by the type and properties of fiber. Thermal properties have an extremely importance on machining GFRP since they control sensitively the polymeric matrix behavior and, consequently, the interfaces resistance. Direct contact between the tool, the material and the formed chip during machining operation yields thermal stresses in the different components. The control of these particular thermal effects becomes challenging when processing composite parts. Besides revealing the physical significance of thermal effects, temperature analysis might also display the heat flux distribution around the tool-material contact area and, hence, helps to optimize the cutting parameters. By aforementioned reasons, thermal conductivity and thermal shock resistance are of great importance when cutting at high cutting speeds, where the thermal load applied is significant. Thermal matrix degradation such as delamination and cracks are also observed as the result of tool wear or inappropriate machining conditions (Ramulu et al. 2001; Davim and Reis 2005). As it is commonly known, polyester resin has relatively low glass transition temperature (Pelivanov et al. 2016). This makes the Glass/Polyester composites highly sensitive to the variation of temperature, particularly, within the range of 40– 200 °C. Therefore, the cutting temperature was assumed among the key parameters that should be monitored during machining of GFRP. Irreversible mechanical and chemical degradation could be occurred once the machining temperature exceeds the glass transition temperature of the matrix (Mullier and Chatelain 2015; Juan et al. 2012). Machining experiments have been conducted by (Wang et al. 2016) showed prominent thermal degradation throughout the matrix when exceeding the glass transition temperature. This yields a loss in the mechanical stiffness of the resin resulting in fiber-matrix interface fail, and consequently, leads to inadequate chip formation mechanisms. The influence of generated temperature on machining quality of carbon fiber reinforced polymers (CFRP) was also discussed in literature (Brinksmeier et al. 2011; Pecat et al. 2012). In order to achieve the desired quality of the finish surface, it is necessary to understand the material removal process, and the kinetics of machining processes affecting the performance of the cutting tools (Sreejith et al. 2000). Indeed, Machining parameters

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depend upon several morphological and structural factors such as matrix volume fraction, reinforcement architecture, and fiber type (Teti 2002). Several researches (Ramulu et al. 2001; Davim and Reis 2005; Juan et al. 2012; Uhlmann et al. 2016) were dedicated to highlight the mechanisms of chip formation in orthogonal cutting since it consists in underlying base of most machining operations. (Hosokawa et al. 2014) investigated the ability of Diamond-Like Carbon (DLC) coated carbide end mills with different helix angles in milling CFRP. (Azmi et al. 2013) evaluated the machinability of GFRP using end milling tests. They based on machining forces, surface roughness, and tool life for achieving their conclusions. (He et al. 2017) studied both unidirectional and multidirectional CFRP laminates. They developed a mechanistic milling force model by considering cutting speed, fiber orientation, and instantaneous chip thickness. To clarify the nature of fundamental changes occurring in the mechanisms of chip formation, (Uhlmann et al. 2016) investigated the influence of high cutting speed in processing CFRP composites. (Hintze and Hartmann 2013) who studied the composites integrity during machining, proved that the main milling-induced damages are fiber protrusions and laminates’ delamination. From the investigation of (Liu et al. 2014), the optimization of CFRP cutting process is closely associated with the temperature distribution throughout the cut part. Guesmi et al. (2020) proposed an experimental approach to analyze the thermal behavior of composite matrices during milling. The temperature history in three resinous matrices, namely, epoxy, polyester, and polyamide are typically investigated using same cutting conditions. This investigation covers the scrutinizing of the temperature evolution along the machined surface of unidirectional glass fiber reinforced epoxy laminates when dry milling perpendicular to the fiber. Peak values and temperature distribution obtained were specially discussed.

2 Experimental Procedures 2.1 Cutting Test Set-up The milling tests were performed on 5-axes High Speed CNC Machine Type Spinner U-620 with a maximum spindle speed of 12000 rpm and maximum power of 19 kW. Cutting tests were conducted using a unique diamond grinding – metal bonding wheel of 10 mm in diameter. Figure 1 reports the experimental set-up including the specimen and the different components of data acquisition.

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Fig. 1. (a) Test set-up showing temperature acquisition system, and (b) tool-specimen pair.

The temperature control system allows measuring the in-process temperature during the tool advance. This device was built based on an Arduino board and Type-K TCs. Table 1. Specifications of data acquisition instrumentation. Components

Specifications

8 TCs type K (Nickel-Chromium/Nickel-Alumel)

Measurement range: 270 ◦ C− + 1370 ◦ C/ − 454 ◦ F− + 2498 ◦ F Diameter of conductor: 0.2 mm Diameter of cable: 2 mm Precision: 2.2% to 0.75% Seebeck Coefficient at 0 ◦ C: 39.45 µV/°C

8 Max6675 TCs Type K Sensor Module For Arduino

Voltage: 5 V DC Intensity: 50 mA Accuracy: ±1.5 ◦ C Resolution: 0.25 ◦ C

Arduino board based on ATMega2560 with USB connection

16 MHz 54 Inputs/Outputs including: 14 PWM (Pulse Width Modulation) input channels, 16 analog input channels, and, 4 UART (Universal Asynchronous Receiver-Transmitter) channels

Dupont male/female connection cables

Suitable cable model

Computer with Arduino user interface

Suitable interface

The TCs being supplied by ELECTRONIC SHOP Co. were placed into the composites samples through eight pre-drilled holes 2 mm in diameter. Table 1 summarizes the specifications of different components constituting the acquisition device.

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During processing, each sensor produced a voltage signal relative to the sampling point temperature. This signal was acquired directly by the Arduino Mega board which is connected to the computer via a USB link. The data acquisition software (PLX-DAQ) was employed together with an Arduino application for ensuring the data storing and signals acquisition. The temperature outputs will be recorded at 4 Hz. 2.2 Specimen Preparation The epoxy 1050 pre-blended with a 100:35 ratio of the hardener 1055S belonging to RESOLTECH Co. – France was considered for laying the required specimens. The hardener and epoxy resin were mixed based on 1:10 weight ratio. The two phases were firstly placed on the mold using hand lay-up method of 18 layers of unidirectional Eglass cloth of 530 g/m2 belonging to CASTRO COMPOSITES Co.–Spain, particularly, recommended for advanced structural reinforcement for wide composites parts with high mechanical properties. According to the curing cycle, the composite laminates were kept in the mold for 24 h at room temperature, constant pressure of 3 kPa, and maximum humidity level of 70%. Then, they were post-cured in autoclave at same pressure level and 60 ◦ C for 15h. 8 mm-thickness GFRP laminate was finally obtained after cooling phase of 8 h and mold released. The composite laminate was then laser cut resulting hence in multiple specimens of dimensions 80 × 40 × 8 mm3 (Fig. 2) which fit to the milling test requirements. Table 1 shows the typical characteristics of epoxy resin used for preparing the composite laminate.

Fig. 2. Dimensions and TCS location within the GFRP specimen.

All milling operations were conducted without lubrication. Each test was repeated three times under same conditions. The cutting speed, feed rate and depth of cut were kept constants at 9000 rpm, 150 mm.mn−1 and 0.2 mm, respectively (Table 2).

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Property

Epoxy 1050

Appearance

Opalescent neutral liquid

Density

1.27 ± 0.05 g/cm3

Thermal conductivity at 20 ◦ C

0.2 W/m × ◦ C

Dry extract in volume

33 ± 2%

Viscosity

1.043 N.s/m2

Hardener type

1055S

Gel time

210 min at 23 ◦ C with 1055 S hardener

3 Results and Discussion The acquired data provided by the thermocouples leads to plot the temperature history over time in the sampling points, regularly distributed through the machined surface of the composite specimen. Figure 3 shows the typical tendency of the temperature versus time obtained during the milling period. Figure 2a depicts the temperature curves obtained within the median plan of the machined specimen. It can be seen that the temperature increases sharply for reaching a maximum value at a too short period. Then, it falls with a very slow rate during longer step. This can be attributed to the fact that the cutting temperature reaches a thermal level after a transition period occurring when the tool started the machining process. During milling period, the peak temperature was found ranging in 23–28.25 ◦ C, recorded by TC1 and TC8, respectively. The obtained tendencies prove that temperature resulting mainly in the interaction between tool-specimen system is sensitive to the cutting time. This means that any change in cutting speed will play to variate sensitively the peak temperature. It should be also mentioned that the heat generated at the beginning of cutting step influences increasingly heat localization at the following TCs. Cumulative effect involved by tool advance seems governing significantly heat generation at the trim plan. In spite the distance separating the TCs location along the specimen thickness is relatively small i.e. 2 mm, compared to TCs enter-axes i.e. 12 mm in longitudinal direction, the temperature vs. time obtained through specimen thickness exhibit neat difference in peak values. From Fig. 2b, one can point out a gap approximating 1 ◦ C between when the passes from TC3 to TC4 and from TC4 to TC5. This confirms that temperature distribution is also nom-uniform within the thickness due to boundary conditions when milling.

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Fig. 3. Temperature history (a) through cutting length, and (b) through the thickness.

4 Conclusions This investigation addresses the cutting temperature using localized measurement resulting in TCs outputs. Dry milling operation on GFRP was studied while feed rate, cutting speed and depth of cut are kept constant. The temperature history was recorded through both the thickness and length of the composite specimen. Correlation between machining variables and cutting temperature was discussed. Sensitivity of temperature to measurement location was proved and attributed mainly to the cumulative effect transmitting toward the heat quantity when the tool advances. The heating step was found much faster compared with cooling step that lasts more than 90% of the time period. Controlling the generated heat seem to be sensitively related to the cutting parameters, especially, tool speed.

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References Ramulu, M., Branson, T., Kim, D.: A study on the drilling of composite and titanium stacks. Comp. Struct. 54, 67–77 (2001) Davim, J.P., Reis, P.: Damage and dimensional precision on milling carbon fiber-reinforced plastics using design experiments. J. Mater. Process. Technol. 160, 160–167 (2005) Pelivanov, I., Ambrozinski, Ł., O’Donnell, M.: Heat damage evaluation in carbon fiber reinforced composites with a kHz A-scan rate fiber-optic pump-probe laser-ultrasound system. Compos. A Appl. Sci. Manuf. 84, 417–427 (2016) Mullier, G., Chatelain, J.F.: Influence of thermal damage on the mechanical strength of trimmed CFRP. World Acad. Sci. Eng. Technol. Int. J. Mech. Aerospace Ind. Mech. Manuf. Eng. 9(8), 1540–1547 (2015) Juan, M.S., Martín, O., Santos, F.J., Cabezudo, J.A., Sánchez, A.: Study of temperature and workpiece damage in drilling of carbon fiber composites. AIP Conf. Proc. 1431, 495–501 (2012) Wang, H., Sun, J., Li, J., Lu, L., Li, N.: Evaluation of cutting force and cutting temperature in milling carbon fiber-reinforced polymer composites. Int. J. Adv. Manuf. Technol. 82(9–12), 1517–1525 (2015). https://doi.org/10.1007/s00170-015-7479-2 Brinksmeier, E., Fangmann, S., Rentsch, R.: Drilling of composites and resulting surface integrity. CIRP Ann. Manuf. Technol. 60(1), 57–60 (2011) Pecat, O., Rentsch, R., Brinksmeier, E.: Influence of milling process parameters on the surface integrity of CFRP. Proc. CIRP 1, 466–470 (2012) Sreejith, P.S., Krishnamurthy, R., Malhota, S.K., Narayanasamy, K.: Evaluation of PCD tool performance during machining of carbon/phenolic ablative composites. J. Mater. Process. Technol. 104, 53–58 (2000) Teti, R.: Machining of composite materials. CIRP Ann. 51, 611–634 (2002) Hosokawa, A., Hirose, N., Ueda, T., Furumoto, T.: High-quality machining of CFRP with high helix end mill. CIRP Ann. 63, 89–92 (2014) Azmi, A.I., Lin, R.J.T., Bhattacharyya, D.: Machinability study of glass fibre-reinforced polymer composites during end milling. Int. J. Adv. Manuf. Technol. 64, 247–261 (2013) He, Y., Qing, H., Zhang, S., Wang, D., Zhu, S.: The cutting force and defect analysis in milling of carbon fiber-reinforced polymer (CFRP) composite. Int. J. Adv. Manuf. Technol. 93(5–8), 1829–1842 (2017). https://doi.org/10.1007/s00170-017-0613-6 Uhlmann, E., Richarz, S., Sammler, F., Hufschmied, R.: High speed cutting of carbon fibre reinforced plastics. Procedia Manuf. 6, 113–123 (2016) Hintze, W., Hartmann, D.: Modeling of delamination during milling of unidirectional CFRP. Procedia CIRP 8, 444–449 (2013) Liu, J., Chen, G., Ji, C., Qin, X., Li, H., Ren, C.: An investigation of workpiece temperature variation of helical milling for carbon fiber reinforced plastics. Int. J. Mach. Tools Manuf. 86, 89–103 (2014) Guesmi, F., Mkaddem, A., Beyaoui, M., Al-Zahrani, A., Jarraya, A., Haddar, M.: Preliminary analysis of temperature history when milling polymer matrices for fibre reinforced composites. In: Chaari, F., et al. (eds.) Advances in Materials, Mechanics and Manufacturing. LNME, pp. 175–181. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-24247-3_20

Free Vibration of Sandwich Nanobeam Nouha Kammoun1(B) , Nabih Feki1,2 , Slim Bouaziz1 , Mounir Ben Amar3 , Mohamed Soula4 , and Mohamed Haddar1 1 Laboratory of Mechanical Modeling and Production (LA2MP), National School of Engineers

of Sfax, Sfax, Tunisia [email protected] 2 Higher Institute of Applied Sciences and Technology of Sousse, University of Sousse, 4003 Sousse, Tunisia [email protected] 3 Laboratoire des Sciences des Procédés et des Matériaux (LSPM), Centre National de la Recherche Scientifique (CNRS), Université Paris 13 Sorbonne Paris Cité, Villetaneuse, France [email protected] 4 Laboratory of Applied Mechanics and Engineering (LMAI-ENIT), DGM ENSIT, High National School of Engineering of Tunis, Tunis, Tunisia

Abstract. Free vibration of sandwich nanobeam is presented in this chapter. The upper and lower sheets are modeled considering Euler-Bernoulli beam formulation and for the core the Timoshenko beam formulation is used taking into account the shear effect. The Non-dimensional procedure is used in order to simplify the analysis of governing equations. The sandwich nanobeam is modeled using Generalized Differential Quadrature (GDQ) method. The influence of the nonlocal parameter (NP) on the first three natural frequencies of Simply Supported - Simply Supported (SS-SS) sandwich nanobeam is discussed . For different edge conditions, mode shapes of the present sandwich nanobeam are discussed. It can be noted that deflection shapes are touched by increasing the NP. For the C-C sandwich nanobeam, natural frequencies are higher than the SS-SS one. Variation of the nonlocal parameter (NP) on the frequencies of the sandwich nanobeam is also investigated for different length . It can be observed that the first frequency have not an important effect for different length, for that, we move to present the second frequency. With increasing the NP the natural frequency decreases clearly. Explication can be regarded as the dispersive behavior of frequencies. The present model can be used as guideline for applications of sandwich nanostructures. Keywords: Nanobeams · Size effect · Free vibration · GDQ

1 Introduction Nanotechnology has attracted considerable interest in many engineering applications such as microactuators, microswtiches, nanosonsers, Micro Electro Mechanical System (MEMS) and Nano Electro Mechanical System (NEMS) studied in works of (Li et al. 2003; Moser and Gijs 2007; Najar et al. 2010). Numerous researchers have studied the mechanical characteristics of structures based on the Eringen’s nonlocal elasticity © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 277–284, 2021. https://doi.org/10.1007/978-3-030-76517-0_31

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theory (NET) (Eringen 1972). Based on the superiority of their mechanical and electrical properties, carbon nanotubes (CNTs) (Wang and Wang 2007) and graphene nanoribbons (GNRs) (Nazemnezhad and Hosseini-Hashemi 2014) are applied for the design of the nanostructures (Geim 2009). The frequencies of these nanostructures rise to the terahertz order. In fact, vibration of Multi Layered Graphene Nanoribbons MLGNRs with the shear effect into the layers was studied by (Nazemnezhad and Hosseini-Hashemi 2014). Furthermore, (Kammoun 2017) discussed the free vibration of 3 layered nanobeams using the NET. This chapter investigates the vibration of the sandwich nanobeam. The GDQ method is used to solve the governing equations which are then solved to obtain the frequencies of sandwich nanobeam with different edge conditions. The influence of the small scale effect on the vibration of the sandwich nanobeam are discussed.

2 Nonlocal Sandwich Nanobeam Model Figure 1 presents a sandwich nanobeam with a length L. Each layer has thickness hi and width bi (so that area Ai = hi bi ).

Fig. 1. Schematic configuration for a sandwich nanobeam.

We propose to model upper and lower sheets (1–3) by considering Euler-Bernoulli beam formulation and the core (layer 2), is captured using Timoshenko beam formulation taking into the shear effect. According to the Euler-Bernoulli beam theory, the displacement of an arbitrary point of the top and bottom layers of the sandwich nanobeam along the x and z axes denoted by uU , wU , uL , and wL respectively, are:   h2 h2 2 ∂w1 (x,t) uU (x, z, t) = u1 (x, t) − z − h1 +h 2 ∂x ; 2 ≤ z ≤ 2 + h1 (1) wU (x, z, t) = w1 (x, t)   h2 h2 2 ∂w3 (x,t) uL (x, z, t) = u3 (x, t) − z + h3 +h 2 ∂x ; −h3 − 2 ≤ z ≤ − 2 (2) wL (x, z, t) = w3 (x, t) where u1 , u3 are the axial displacements w1 , w3 and transverse displacements of an arbitrary point located on mid-axis of the top and bottom layers, respectively and t is the

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time. It is further assumed for the transverse displacement that uU = uL = uc = w(x, t). For the core, the displacement is expressed using the TBT follows: uc (x, z, t) = u2 (x, t) + zφ(x, t); − h22 ≤ z ≤ wc (x, z, t) = w(x, t)

h2 2

(3)

where φ is the rotation of beam cross-section. The strain energy U of the sandwich nanobeam is expressed as:   L   (1) (1) (2) (2) (3) (3) U = 21 0 H σxx εxx + σxz γxz + σxx εxx dxdz    L  (1) 1 (2) ∂u2 (3) ∂u3 (1) (3) ∂ 2 w = 21 0 Nx ∂u + N + N − M + M x x x x ∂x ∂x ∂x ∂x2  ∂w  (2) ∂φ + Mx ∂x + Q ∂x + φ dx

(4)

where Nx(i) , Mx(i) and Q are the normal resultant force, the bending moment and the transverse shear force for the layer (i), respectively. The kinetic energy T can be given by:      

 2 2 2   ∂w 2  ∂w 2 ∂u1 ∂u2 1 L + I2 + I3 ∂φ + ∂t + ∂t T = 2 0 I1 ∂t ∂t ∂t  

(5) 2   2 ∂w 3 dx + I4 ∂u + ∂t ∂t  bh3 where {I1 , I2 , I3 , I4 } = ρ bh1 , bh2 , 122 , bh3 .   t Using the Hamilton’s principle 0 (δT − δU ) dt = 0 , we obtain the classical governing equations of the sandwich nanobeam. The nonlocal constitutive relations developed by Eringen can be rewritten to a one-dimensional form and expressed as follows: σxx − (e0 a)2

∂ 2 σxx = Eεxx ; ∂x2

σxz − (e0 a)2

∂ 2 σxz = Gγxz ∂x2

(6)

where σxx and σxz are the normal and shear nonlocal stresses, respectively. μ = (e0 a)2 is the nonlocal parameter. E and G are the elastic and shear modulus of the beam. Non-dimensionalization procedure has important applications in the analysis of differential equations (Kammoun 2017). The governing equation can be written taking into account the dimensionless parameters as: 

2 ∂ 2 U1 ∂ 2 U1 2 ∂ U1 1 − μ (7a) A11 = I 1 ∂ξ 2 ∂τ 2 ∂ξ 2 

2 ∂ 2 U2 ∂ 2 U2 2 ∂ U2 1 − μ (7b) A12 = I 2 ∂ξ 2 ∂τ 2 ∂ξ 2 

2 ∂ 2 U3 ∂ 2 U3 2 ∂ U3 1−μ (7c) A13 = I4 ∂ξ 2 ∂τ 2 ∂ξ 2

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2   ∂ 4W  ∂ 2W  ∂φ ∂ 2W 2∂ W 1−μ B11 + B13 + ks C 12 +η = I1 + I2 + I4 ∂ξ 4 ∂ξ 2 ∂ξ ∂τ 2 ∂ξ 2 (7d)

 

∂ 2φ ∂ 2φ ∂ 2φ ∂W + ηφ = I 3 2 1 − μ2 2 B12 2 − ks C 12 η (7e) ∂ξ ∂ξ ∂τ ∂ξ in which the parameters of governing equations of motions are changed to the dimensionless form as follows: ξ = Lx ;η =

L H ;μ

=



e0 a t L ;τ = L  u1 u2 u3 H ,H , H ,

(U1 , U2 , U3 , W ) =   B11 , B11 , B11 , C 12 =

At It ; At  ∗ w H ;φ

= A11 + A12 + A13 ;It = I1 + I2 + I4 ;H = h1 + h2 + h3 ;     ; = φ; C12 = Ac12 ; A11 , A12 , A13 = AA11 , AA12 , AA13 12 11 12 13



   B11 , B12 , B13 , C12 ; I 1 , I 2 , I 3 , I 4 = II11 , II22 , I Ih3 2 , II44 A h2 A h2 A h2 A12 11 1

12 2

13 3

2 2

(8) To solve the governing Eq. (7a–7e), the GDQ method is used since it is a reduced order method. In fact, the main idea of using the GDQ method is to solve the problem with high order derivatives (Ke and Wangn 2012). Hence, U1 , U2 , U3 , W and φ ∗ and their k th derivatives are according to ξ can be given as:  N   U (ξ , t), U2m (ξm , t), U1 , U2 , U3 ,W,φ ∗ = lm (ξ ) 1m m U3m (ξm , t),Wm (ξm , t),φm∗ (ξm , t) m=1   N   ∂k  (k) U1m (ξm , t), U2m (ξm , t), ∗  U1 , U2 , U3 ,W,φ  = Cim  U3m (ξm , t),Wm (ξm , t),φm∗ (ξm , t) ∂ξ k 

ξ =ξi

(9a)

(9b)

m=1

where N is the number of grid points, the sandwich nanobeam deflection at the Chebyshev–Gauss–Lobatto grid points ξi (Ke and Wangn 2012), is written as: 

i−1 1 1 − cos π , i = 1, 2, ...., N (10) ζi = 2 N −1

3 Results and Discussion Numerical results are depicted in this section for vibration of the clamped–clamped (C– C), simply supported– simply supported (SS–SS) and clamped– simply supported (C– SS) sandwich nanobeam. The material of sandwich nanobeam is the Bilayer Graphene Nanoribbons BLGNR, the length is L = 14 nm, thicknesses h1 = h3 = 0.3 nm, h2 = 0.1 nm and shear correction factor ks = 0.563. Decreasing thickness of the core of the beam comparing to the upper and lower layers thicknesses can be the case of neglecting the shear effect, so, our model can be considered as the nonlocal Euler-Bernoulli beam model. Moreover, increasing thickness

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Fig. 2. The first three mode shapes of SS-SS Euler Bernoulli (EB) nanobeam.

Fig. 3. The first three mode shapes of C-C Timoshenko nanobeam.

of the core of the beam comparing to the top and bottom layers thicknesses lead to taking into account the shear effect, so, our model can be reduced to the nonlocal Timoshenko beam model. Figure 2 shows the first three mode shapes of SS-SS Euler Bernoulli nanobeam and Fig. 3 gives first deflection shape of C-C Timoshenko nanobeam of the single-walled carbon nanotubes. Parameters used in this example are taken as (Ba˘gdatlı 2015; Wang and Wang 2007). Solutions obtained by the sandwich nanobeam based on NET are in

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good agreement with the analytical results given by (Ba˘gdatlı 2015; Wang and Wang 2007) using nonlocal Euler-Bernoulli and Timoshenko beam theory as well. In fact, it can be observed that deflection shapes are touched by increasing the NP. Note that when μ = 0 this is corresponding to the local beam. For the C-C sandwich nanobeam, natural frequencies are higher than the SS-SS one. In fact, for the C-C sandwich nanobeam, the end support is the more rigid than the SS-SS.

(a)

(b) Fig. 4. The first three modes shapes of sandwich nanobeam (a) SS-SS, (b) C-SS

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Deflection graphs of nonlocal C-C and C-SS sandwich nanobeams incorporating mid-layer shear effect are plotted in this study for different scaling effect parameters to be useful for benchmarking. In fact, by understanding the modes of vibration, we can better design structures in accordance with the need. Table 1. The effect of the nonlocal parameter on the first three natural frequencies (THz) of SS-SS sandwich nanobeam. Frequencies

μ=0

μ = 2.1

μ = 5.5

μ = 6.95

μ = 11

μ = 20

1

0.7734

0.6996

0.4869

0.4174

0.2904

0.1682

2

1.5469

1.5469

0.5808

0.4722

0.3071

0.1712

3

2.3203

1.3399

0.6050

0.4849

0.3105

0.1718

Figure 4 presents the first three modes shapes of the proposed model sandwich nanobeam for different boundary conditions. The effect of the NP on the first three natural frequencies of the sandwich nanobeam is presented in Table 1. We can conclude that with increasing the nonlocal parameter the frequencies decrease. + First frequency µ=0

Second frequency µ=0

First frequency µ=2.1

Second frequency µ=2.1

2

Frequency (THz)

1.8 1.6 1.4 1.2 1 0.8 0.6 11

12

13

14

15

16

17

Length (nm)

Fig. 5. Natural frequencies (THz) of SS-SS sandwich nanobeam with different NP

Figure 5 shows the natural frequencies of SS-SS sandwich nanobeam with different NP, while considering the influence of length. The dispersive behavior of frequencies in this model is clearly observed. However, when the length of the nanobeam is large, the gap

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between curves (different values of NP) decreases. This is normal while the length of the nanobeam increase the thicknesses reduce of same the NP. Also, we can conclude from this figure that it have not an important effect on the fundamental frequency. Therefore, we move to present the second frequency. With increasing the NP the natural frequency decreases clearly. Explication can be regarded as the dispersive behavior of frequencies.

4 Conclusion Free vibration of sandwich nanobeam is studied. The GDQ method is used to obtain the frequencies of the sandwich nanobeam with different end supports. Results show that increases in the NP produce the decreasing of the frequencies. Also, the NP has an influence on the deflection mode shapes of vibration of the sandwich nanobeam.

References Ba˘gdatlı, S.M.: Non-linear vibration of nanobeams with various boundary condition based on nonlocal elasticity theory. Compos. B Eng. 80, 43–52 (2015) Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972) Geim, A.K.: Graphene: status and prospects, AK Geim Manchester Centre for Mesoscience and Nanotechnology. University of Manchester Oxford Road M13 9PL Manchester UK Prospects 324:1–8 (2009) Kammoun, N.: Vibration analysis of three-layered nanobeams based on nonlocal elasticity theory. J. Theor. Appl. Mech. 55, 1299–1312 (2017) Kordkheili, S.A., Sani, H.: Mechanical properties of double-layered graphene sheets. Comput. Mater. Sci. 69, 335–343 (2013) Li, X., Bhushan, B., Takashima, K., Baek, C.W., Kim, Y.K.: Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 97, 481–494 (2003) Moser, Y., Gijs, M.A.M.: Miniaturized flexible temperature sensor. J. Microelectromech. Syst. 16, 1349–1354 (2007) Najar, F., Nayfeh, A.H., Abdel-Rahman, E.M., Choura, S., El-Borgi, S.: Global stability of microbeam-based electrostatic microactuators. J. Vib. Control 16, 721–748 (2010) Nilsson, J., Neto, A.C., Guinea, F., Peres, N.: Electronic properties of bilayer and multilayer graphene. Phys. Rev. B 4(78), 405–434 (2008) Nazemnezhad, R., Hosseini-Hashemi, S.: Free vibration analysis of multi-layer graphene nanoribbons incorporating interlayer shear effect via molecular dynamics simulations and nonlocal elasticity. Phys. Lett. A 44, 3225–3232 (2014) Nazemnezhad, R., Zare, M.: Nonlocal Reddy beam model for free vibration analysis of multilayer nanoribbons incorporating interlayer shear effect. Eur. J. Mech. A/Solids 55, 234–242 (2016) Wang, Q., Wang, C.M.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18(7) (2007) Ke, L.L., Wangn, Y.S.: Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater. Struct. (2012). https://doi.org/10.1088/0964-1726/21/2/ 025018

Author Index

A Abbes, Mohamed Slim, 136, 145 Abdennadher, Moez, 71 Allagui, Sami, 53 Antunes, Jose, 251 Ayed, Lazher, 261 B Babahammou, Ahmed, 27 Barkallah, Maher, 191 Baroudi, Sourour, 105 Baslamisli, S. Caglar, 210 Ben Amar, Mounir, 277 Ben Arab, Safa, 243 Ben Hassena, M. A., 125 Ben Kahla, N., 62 Ben Souf, Mohamed Amine, 44 Benamar, Rhali, 27 Beyaoui, Moez, 53 Bouaziz, Slim, 243, 277 Bouguecha, Anas, 44, 53, 171 Bouslema, Marwa, 218 Boussollaa, Chaima, 251 Brunone, Bruno, 183 C Chaari, Fakher, 1, 19, 210, 235 Chaari, Riadh, 210 Chafra, Moez, 199 Chakroun, Ala Eddin, 235 Chiementin, Xavier, 19 Choura, Oussama, 183

D Debut, Vincent, 251 De-Juan, Ana, 235 Djemal, Fathi, 210 E El Mahi, Abderrahim, 53 El Ouni, M. H., 62 Elaoud, Sami, 183 Elfarhani, M., 269 Elwasli, F., 227 Essaidi, Badreddine, 117 Essassi, Khawla, 44 F Fakhfakh, Tahar, 251 Fakhfakh, Taher, 218 Farhat, Mohamed Habib, 19 Feki, Nabih, 136, 145, 277 Fernandez, Alfonso, 235 G Ghazali, S., 269 Ghorbel, Ahmed, 71 Graja, Oussama, 71 Guesmi, F., 269 Guizani, Amir, 191 H Haddar, Maroua, 210 Haddar, Mohamed, 1, 19, 44, 53, 71, 136, 145, 155, 171, 191, 210, 218, 235, 243, 251, 277 Hafsi, Zahreddine, 261

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 N. Feki et al. (Eds.): ICAV 2021, ACM 17, pp. 285–286, 2021. https://doi.org/10.1007/978-3-030-76517-0

286 Hammami, Ahmed, 235 Hammami, Maroua, 136 Hamza, Ghazoi, 191 Helaili, Sofiene, 199 Hentati, Taissir, 19 Heyns, P. Stephan, 1 J Jarraya, A., 269 Jedidi, Mohamed Yassine, 171 Jemai, Ahmed, 105 K Kammoun, Nouha, 277 Kani, Marwa, 155 Khabou, Mohamed Taoufik, 171 Ksentini, Olfa, 136, 145 L Liu, Xinpeng, 35 Louati, Jamel, 191 M Mabrouk, Abdelileh, 145 Mahfouz, A. S. Bin, 269 Mahi, Abderrahim El, 44 Makni, Amine, 155 Mankai, Wahbi, 199 Meniconi, Silvia, 183 Mezlini, S., 227 Mkaddem, A., 269 Mnassri, Souad, 10, 79, 87 Mrabet, E., 62 Mzali, S., 227

Author Index N Najar, F., 125 Najar, Fehmi, 105 Nasri, Rachid, 218 O Ouakad, Hassen M., 125 R Rebiere, Jean-luc, 44, 53 S Samaali, H., 125 Schmidt, Stephan, 1 Soula, Mohamed, 277 T Taktak, Mohamed, 155 Toure, Mahamane, 44 Trabelsi, Hassen, 191 Trabelsi, Mounir, 95 Triki, Ali, 10, 79, 87, 95, 117 V Viadero, Fernando, 235 W Walha, Lassâad, 71 Y Yan, Zhitao, 35 Yu, Guoqing, 35 Z Zemzemi, F., 227 Zimroz, Radoslaw, 1