Acoustics and Vibration of Mechanical Structures [1 ed.] 9783038262589, 9783037858776

Citation preview

Acoustics & Vibration of Mechanical Structures

Edited by Nicolae Herişanu Vasile Marinca

Acoustics & Vibration of Mechanical Structures

Selected, peer reviewed papers from the XII-th International Symposium Acoustics & Vibration of Mechanical Structures (AVMS 2013), May 23-24, 2013, Timişoara, Romania

Edited by

Nicolae Herişanu and Vasile Marinca

Copyright  2013 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Kreuzstrasse 10 CH-8635 Durnten-Zurich Switzerland http://www.ttp.net

Volume 430 of Applied Mechanics and Materials ISSN print 1660-9336 ISSN cd 1660-9336 ISSN web 1662-7482

Full text available online at http://www.scientific.net

Distributed worldwide by

and in the Americas by

Trans Tech Publications Ltd Kreuzstrasse 10 CH-8635 Durnten-Zurich Switzerland

Trans Tech Publications Inc. PO Box 699, May Street Enfield, NH 03748 USA

Fax: +41 (44) 922 10 33 e-mail: [email protected]

Phone: +1 (603) 632-7377 Fax: +1 (603) 632-5611 e-mail: [email protected]

Preface The Proceedings of the XII-th International Symposium „ACOUSTICS & VIBRATION OF MECHANICAL STRUCTURES” – AVMS-2013, contains papers contributed to the Symposium held on 23‐24th May 2013 in Timişoara, Romania. The first AVMS was successfully taken place in Timişoara, May 1991. Since then, the AVMS has targeted the creation of a platform for researchers, engineers, academicians as well as industrial professionals to share their research results and experiences in the field of acoustics and vibration. At every two years, the AVMS offered a broad overview on the latest advances and future perspectives in this exciting field of research. The Symposium covers broad areas of topics related to acoustics and vibration, among them: Noise and vibration control; Noise and vibration generation and propagation; Effects of noise and vibration; Condition monitoring and vibration testing; Nonlinear acoustics and vibration; Analytical, numerical and experimental techniques for noise and vibration; Modelling, prediction and simulation of noise and vibration; Environmental and occupational noise and vibration; Noise and vibration attenuators; Biomechanics and bioacoustics. All papers included in this volume had undergone the careful peer-review by the experts of the International Scientific Committee and external reviewers before publication. We are very grateful to them for spending their valuable time to review the papers. We would like to express our sincere appreciation to keynote speakers and all contributors of the presented papers for sharing their knowledge and experiences with all the participants. We are also expressing our sincere thanks to the members of the International Scientific Committee and also to the members of the Organizing Committee for ensuring the success of this Symposium which would not have been possible without their support. We hope this book will inspire future developments in the field of acoustics and vibration of mechanical structures, stimulating and accelerating the advancement in knowledge in this field. Finally, special thanks are given to Trans Tech Publications for producing this volume.

Editors, Nicolae HERIŞANU Vasile MARINCA

Organized by: "Politehnica" University of Timisoara – Acoustics and Vibration Laboratory University of Nis, Faculty of Occupational Safety – Noise and Vibration Laboratory Romanian Academy,Timisoara branch-Center for Fundamental and Advanced Technical Research Romanian Acoustical Society

International Scientific Committee VASILE MARINCA – (chair) – “ Politehnica” University of Timişoara, Romania DRAGAN CVETKOVIC – (co-chair) – University of Nis, Serbia JAN AWREJCEWICZ – Lodz University of Technology, Poland VASILE BACRIA – “Politehnica” University of Timişoara, Romania MALVINA BAICA – University of Wisconsin, U.S.A. LIVIU BERETEU – “Politehnica” University of Timişoara, Romania POLIDOR BRATU – ICECON Bucharest, Romania FLORIN BREABAN – University d’Artois, France VETURIA CHIROIU - Institute of Solid Mechanics, Bucharest, Romania LIVIJA CVETICANIN – University of Novi Sad, Serbia GHEORGHE DRAGANESCU – “Politehnica” University of Timişoara, Romania GILBERT-RAINER GILLICH – “Eftimie Murgu” University of Resita, Romania NICOLAE HERISANU – “Politehnica” University of Timişoara, Romania METIN O. KAYA - Istanbul Technical University, Turkey IVANA KOVACIC – University of Novi Sad, Serbia DAN B. MARGHITU – Auburn University, USA KALE OYEDEJI - Morehouse College, Atlanta, GA, USA CRISTIAN PAVEL – Technical University of Civil Eng. Bucharest, Romania MEHMET PAKDEMIRLI – Celal Bayar University, Manisa, Turkey ZORAN PETROVIC – University of Kragujevac, Serbia MOMIR PRASCEVIC – University of Nis, Serbia PETAR PRAVICA - University of Belgrade, Serbia ZLATAN SOSKIC – University of Kragujevac, Serbia SORIN VLASE – “Transilvania” University Braşov, Romania JERZY WARMINSKI – Lublin University of Technology, Poland LJILJANA ZIVKOVIC – University of Nis, Serbia NENAD ZIVKOVIC – University of Nis, Serbia

Organising Committee Chairman NICOLAE HERIŞANU – “Politehnica” University of Timişoara, Romania

Co-Chairman MOMIR PRASCEVIC – University of Nis, Serbia

Members of the organising committee VIOREL-AUREL ŞERBAN – Rector - “Politehnica” University of Timişoara, Romania VASILE MARINCA – “Politehnica” University of Timişoara, Romania LIVIU BERETEU – “Politehnica” University of Timişoara, Romania DARKO MIHAJLOV – University of Nis, Serbia RAMONA NAGY – “Politehnica” University of Timişoara, Romania KAROLY MENYHARDT – “Politehnica” University of Timişoara, Romania COSMINA VIGARU – “Politehnica” University of Timişoara, Romania ALEXANDRU PERESCU – “Politehnica” University of Timişoara, Romania DORIN SIMOIU – “Politehnica” University of Timişoara, Romania

Table of Contents Preface and Committees

Chapter 1: Analytical Approaches to Nonlinear Vibrations On the Van Der Pol Oscillator: an Overview L. Cveticanin Nonlinear Oscillators with a Power-Form Restoring Force: Non-Isochronous and Isochronous Case I. Kovacic Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method V. Marinca, N. Herişanu and T. Marinca Optimal Homotopy Asymptotic Approach to Self-Excited Vibrations N. Herişanu and V. Marinca Physical Instability and Functional Uncertainties of the Dynamic Systems in Resonance P. Bratu Approximate Analytical Solutions to Nonlinear Vibrations of a Thin Elastic Plate R.D. Ene, V. Marinca and B. Marinca The Study of the Pendulum with Heavy Neo-Hookean Rod N.D. Stanescu and D. Popa The Vibrations of the Engine with Neo-Hookean Suspension N.D. Stanescu and D. Popa

3 14 22 27 32 40 45 53

Chapter 2: Damage Assessment of Structures Multiple Fault Identification Using Vibration Signal Analysis and Artificial Intelligence Methods N. Zuber, D. Cvetkovic and R. Bajrić Rolling Element Bearings Fault Identification in Rotating Machines, Existing Methods of Vibration Signal Processing Techniques and Practical Considerations N. Zuber and D. Cvetkovic Recent Advances in Vibration Signal Processing Techniques for Gear Fault Detection-A Review R. Bajrić, N. Zuber and S. Isić Methods of Interpreting the Results of Vibration Measurements to Locate Damages in Beams P.F. Minda, Z.I. Praisach, A.A. Minda and G.R. Gillich Evaluation of Crack Depth in Beams for Known Damage Location Based on Vibration Modes Analysis Z.I. Praisach, G.R. Gillich, C. Protocsil and F. Muntean Assessment of Corrosion Damages with Important Loss of Mass and Influences on the Natural Frequencies of Bending Vibration Modes G.R. Gillich, Z.I. Praisach, D. Bobos and C. Hatiegan Vibration Tests for Determination of Longitudinal Elasticity Modulus and Shear Modulus of some Structures Welded with Tubular Wire R. Moisa, T. Medgyesi, L. Bereteu, G. Drăgănescu, D. Simoiu and M. Sava Monitoring the Behaviour of Fireworks to Vibrations and the Establishment of the Mechanical Conditioning Influence E. Gheorghiosu, E. Ghicioi, A. Kovacs, C. Jitea, S. Ilici and C. Cioara Acoustical Methods Used in the Study of Concrete Durability E. Jebelean, C. Badea, L. Iures, C. Jiva and I. Borza Testing of Level of Vibration and Parameters of Bearings in Industrial Fan D. Jovanovic, N. Zivkovic, M. Raos, L. Zivkovic, M. Jovanovic and M. Praščevič

63 70 78 84 90 95 101 108 113 118

b

Acoustics & Vibration of Mechanical Structures

Chapter 3: Modeling and Simulation Techniques with Applications Influence of Heavy Data Transmission Losses on Spectra of Signals Z. Soskic, J. Tomić, N. Bogojević and S. Ćirić Kostić Compensations of the Discontinuous Nonlinearities in the Independent Joints Control of an Articulated Robot D. Receanu Modeling and Testing of a New Dynamic Balancing System Based on Magnetic Interaction C. Ilie, D.C. Comeaga and O.G. Donţu Analytic-Experimental Method for Determining the Eccentricity of a Cantilever Rotor A. Costache, A. Craifaleanu and N. Orăşanu Study on Vibration Transmission, with Application to the Calibration of a Measuring Stand A. Craifaleanu, C. Dragomirescu, N. Orăşanu and A. Costache Vibrations Modeling and Simulation Using Stochastic Bondsim Elements B.M. Zlatkovic and B. Samardzic Considerations on Kinematics and Dynamics of Gravitational Separators with NonBalanced Eccentric Masses Used for Cereal Seeds Cleaning C. Bracacescu, I. Pirna and S. Popescu Impact of a Kinematic Link with MATLAB and SolidWorks J. Ragan and D.B. Marghitu Active Synthesis of Machine Drive Systems T. Dzitkowski and A. Dymarek Random Excitation of a Car Component from the Road C. Razvan and I. Lupea Study of a Half Car Suspension Model R. Nagy and K. Menyhardt An Analysis of Forced Vibrations to Railway Vehicles E. Ghita and L. Bucur

125 135 143 148 153 158 165 170 178 184 191 195

Chapter 4: Biomechanics Particularities of Upper Limb Movements of Healthy and Pathologic Subjects M. Toth-Taşcău, F. Bălănean and D.I. Stoia Plantar Pressure Improvement after Correction of Hallux Valgus Condition — Static and Dynamic Approach D.I. Stoia, M. Toth-Taşcău, O. Pasca and C. Vigaru The Internal Fixator Principle Applied to Proximal Tibial Fractures - An Early Gait Analysis Study D. Crisan, D.I. Stoia, R. Prejbeanu, D. Vermeşan and H. Haragus Biomechanical Modeling of Human Finger P.L. Radu Mechanical Properties of Hydroxyapatite Doped with Magnesium, Used in Bone Implants O. Suciu, T. Ioanovici and L. Bereteu Dynamic Analysis of a Lower Limb Prosthesis L. Rusu, M. Toth-Taşcău and C. Toader-Pasti

203 208 213 217 222 230

Chapter 5: Environmental and Occupational Noise and Vibrations Application of the Prediction Model "SCHALL 03" for Railway Noise Calculation in Serbia M. Praščevič, A. Gajicki, D. Mihajlov, N. Zivkovic and L. Zivkovic Acoustic Zoning and Noise Assessment M. Praščevič, D. Mihajlov, D. Cvetkovic, A. Gajicki and N. Holecek Noise Control in an Industrial Hall V. Bacria and N. Herişanu

237 244 251

Applied Mechanics and Materials Vol. 430

Some Effects of Rubberized Asphalt on Decreasing the Phonic Pollution N. Herişanu and V. Bacria Noise Source Monitoring in Industrial and Residential Mixed Areas R. Nagy, D. Simoiu, K. Menyhardt and L. Bereteu Analysis of the Sound Power Level Emitted by Portable Electric Generators (Outdoor Powered Equipment) Depending on Location and Measuring Surface E. Postelnicu, V. Vladut, C. Sorica, P. Cardei and I. Grigore Model for Forecasting the Exposure Risk of Workers to Hand-Arm Occupational Vibrations G.D. Vasilescu, E. Ghicioi, S. Simion and V. Pasculescu Exposure of Workers to Noise in Mining Industry S. Simion, C. Vreme, M. Kovacs and L. Toth Experimental Research and Modeling of the Underwater Sound in Marine Environment D. Arsene, C. Borda, L. Butu, M. Marinescu, V. Popovici and M. Arsene Influence of Vibrations on Grain Harvesters Operator V. Vladut, S. Biris, S.T. Bungescu and N. Herişanu Troubleshooting Technics to Identify the Airborne and Structure-Borne Noise Content inside an Electric Vehicle M.C. Morariu, I. Lupea and C. Anderson

c

257 262 266 276 281 285 290 297

Chapte 6: Structural Vibration, Attenuators and Isolation Corrective Analysis of the Parametric Values from Dynamic Testing on Stand of the Antiseismic Elastomeric Isolators in Correlation with the Real Structural Supporting Layout P. Bratu Modification of the Dual Kelvin-Voigt/Maxwell Rheological Behavior for Antiseismic Hydraulic Dampers P. Bratu Analysis of the Dynamic Behavior of the Antiseismic Elastomeric Isolators Based on the Evaluation of the Internal Dissipated Energy C. Alexandru Experimental Evaluation of the Hysteretic Damping of Elastomeric Systems at Low-Cyclic Harmonic Kinematic Displacements V. Ovidiu Experimental Evaluation of the Damping Variation of an Elastomeric Device Harmonically Excited V. Ovidiu Calibration of the Ground Motion Model Using a Simplified Stochastic Model in the Case of the Central Exhibition Pavilion ROMEXPO P. Murzea The Computer Aided Passive Reduction of Vibration to Desired Vibration Amplitude A. Dymarek and T. Dzitkowski Theoretical and Experimental Studies on Magnetic Dampers N. Orăşanu, A. Craifaleanu and C. Dragomirescu Some Models of Elastomeric Seismic Isolation Devices V. Iancu, G.R. Gillich, C.M. Iavornic and N. Gillich Influence of Hysteretic Behavior on Seismic Strength Demands: An Analysis for Romanian Vrancea Earthquakes I.G. Craifaleanu Strength Reduction Factors: A Unified Analytic Expression for Narrowband and Broadband Ground Motion Records I.G. Craifaleanu

305 312 317 323 329 335 342 351 356 362 367

CHAPTER 1: Analytical Approaches to Nonlinear Vibrations

Applied Mechanics and Materials Vol. 430 (2013) pp 3-13 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.3

On the Van der Pol oscillator: An overview Livija Cveticanin1,a 1

Faculty of Technical Sciences, 21000 Novi Sad, Trg D. Obradovica 6, Serbia a

[email protected]

Keywords: Van der Pol oscillator, Limit cycle motion, Rayleigh oscillator, Self-sustained vibrations.

Abstract. In this paper an overview of the self-sustained oscillators is given. The standard van der Pol and the Rayleigh oscillators are considered as basic ones. The cubic nonlinear term of Duffing type is included. The special attention is given to the various complex systems based on the Rayleigh’s and van der Pol’s oscillator which are extended with the nonlinear oscillators of Duffing type and also excited with a periodical force. The connection is with the linear elastic force or with linear damping force. The objectives for future investigation are given in this matter. Van der Pol oscillator The van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol (1889-1959) while he was working at Philips. Namely, at that time the radio and the vacuum tube technology were developed. Diode, triode or tetras were involved into electrical circuits forming an oscillatory system. In Fig.1 the electrical circuits with triode (a) and diode (b) are shown.

a)

b)

Fig.1. a) Electrical circuit with a triode, b) Electrical circuit with a tunnel diode and the currentvoltage diagram of the vacuum tube. The electrical circuit in Fig.1.a contains: a triode, a resistor R, a capacitor C, a coupled inductorset with self inductance L and mutual inductance M. In the serial RLC circuit there is a current i, and towards the triode anode ("plate") a current ia, while there is a voltage ug on the triode control grid. The system is forced by an AC voltage source Es. Van der Pol considered the oscillations of the electrical circuit and published the experimental results in 1926 [1]. The oscillator is named the van der Pol oscillator. After Reona (Leo) Esaki (1925 - ) invented the tunnel diode in 1957, making the van der Pol oscillator with electrical circuits has become much simpler (Fig.1.b). The mathematical model of the electrical circuit with the tunnel diode with input-output relation i   (V ) , shown in Fig.1.b, is as follows

1 1 V  ( (V )  W ), W  V C L i.e.,

(1)

4

Acoustics & Vibration of Mechanical Structures

1 1 V   ' (V )V  V  0, C LC

(2)

where  ' (V )  d / dV . For the case when the input-output relation of the tunneled diode is a cubic function i   (V )  V 3  V , the relation (2) transforms into

1 3  V  V  ( V 2  )V  0 . LC C C

(3)

Introducing a new variable x  V 3 /  , the relation (3) transforms into

x  c12 x   ( x 2  1) x  0 ,

(4)

where c12  1 / LC and    L / C  0 are positive parameters. The model (4) is often called “the standard van der Pol equation”. It is worthy to say that the same mathematical model (4) corresponds to the electrical circuit shown in Fig.1.a, but also to a significant number of systems in biology, chemistry, economics, and mechanics. In dynamics, the Van der Pol oscillator is a nonconservative system with the non-negative damping: x is the position coordinate which is a function of the time, and  is a parameter indicating the nonlinearity and the strength of the damping. The following interesting regimes of the unforced oscillator are: 1. When  = 0, i.e. there is no damping function, the equation becomes: x  c12 x  0 . This is a form of the simple harmonic oscillator and there is always conservation of energy. 2. When x(0) and x are small, the quadratic term is negligible and the system becomes a linear differential equation with negative damping. The fixed point x=0, x  0 is an unstable focus. 3. When x is large, the quadrate term is dominant and the damping becomes positive. Two cases may occur depending on the value of the initial value x(0). Introducing the new variable y  x  x 3 / 3  x /  , the Liénard transformation [2] of Eq. (4) is:

x   ( x 

x3  y ), 3

y 

c12 x



.

(5)

Analyzing the two coupled first order differential equations (5) we conclude that for significant values of x the limit cycle motion appears [3]. Namely: a) If x(0) is far from zero but in the neighborhood of the amplitude of the limit cycle independently of the initial conditions, the motion tends to self-sustained vibrations. In Fig.2 the x-t diagram (a) and also the limit curve (b) in x-y plane are plotted. b) If the system is away from the limit curve, the velocity in x direction is much higher than in y one, and the system moves quickly in horizontal direction while the motion along the curve is slow. The system has stable limit cycle (Fig.3). The period of oscillation is approximately . For the electric circuit of van der Pol type with the triode the parameter is =RC. The RC is the time constant of relaxation in the circuit. Van der Pol found this stable oscillations, which he called relaxation oscillations and are now known as limit cycles, in electrical circuits employing vacuum tube. When these circuits are driven near the limit cycle they become entrained, i.e. the driving signal pulls the current along with it. The properties of this oscillation are the slow asymptotic behavior and then the sudden jump from one to other value.

Applied Mechanics and Materials Vol. 430

5

a) b) Fig.2. a) x-t diagram and b) y-x curve for =0.1 and initial conditions x(0)=0.5 and x (0)  0 . Analyzing equation (4), it is obvious that the parameter c12 controls how much voltage is injected into the system. The parameter  controls the way in which voltage is established across the elements of the system. Think of the oscillator as having two distinct phases: a slow recovery phase and a fast relaxed phase (that is in fact where the term relaxation comes from: vacuum tubes quickly release or relax their voltage after slowly building up tension.) The variable  controls the rate at which the slow build up occurs. As the parameter is decreased, the voltage builds up more slowly, making the slow phase take longer and hence slowing down the oscillator. The van der Pol oscillator is believed to be the first relaxation oscillator.

a) b) Fig.3. a) x-t diagram and b) x-y curve for =10 and initial conditions x(0)=0.5 and x (0)  0 . Balthazar van der Pol and his colleague van der Mark were the first to model the electric activity of the heart and of the human heartbeat based on the relaxing oscillations (see [4]). Using these results the cardiac pacemakers are modeled by a minor change in the van der Pol equation [5]. In Fig.4, the evolution of the limit cycle in the phase plane is plotted. Notice, the limit cycles begin as a circle and, with varying , becomes increasingly sharp.

Fig.4. Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle and the direction field.

6

Acoustics & Vibration of Mechanical Structures

Rayleigh’s oscillator The famous British mathematical physicist Lord Rayleigh (John William Strutt, 1842–1919) was one of the first scientists who were dealing with the problem of limit cycle motion. Namely, he introduced the nonlinear velocity damping to model of oscillations of a clarinet reed [6]. The so called Rayleigh model has the form [7]

y   ( y 2  1) y  c12 y  0

(6)

where  is the coefficient of damping and c12 is a positive constant. Introducing the new variable x  y / 3 it is evident that the relation equation (6) transforms into the van der Pol equation (4). Forming the Rayleigh electrical circuit with model equation (6), it is seen that it is an oscillator much like the van der Pol one. The key difference between the two circuits is that as voltage increases, the van der Pol oscillator increases in frequency while the Rayleigh oscillator increases in amplitude. Like the van der Pol oscillator, c12 controls how much voltage is injected into the system.  controls the way in which the voltage is established across the elements of the system. In Fig.5. we see outward and inward spiral trajectories converging to a “limit cycle” solution that corresponds to periodic oscillations of the reed.

Fig.5. Trajectories in the Rayleigh oscillator The Rayleigh oscillator is of real physical interest, since it might be useful to model many physical and engineering systems, and in the context of chemical and biological oscillators. Forced van der Pol oscillators The van der Pol and also the Rayleigh oscillators are often driven by an excitation force. The forced, or driven, van der Pol oscillator takes the 'original' function and adds a driving function Fsin(t) to give a differential equation of the form:

x  c12 x   ( x 2  1) x  F sin(t ),

(7)

where F is the amplitude and  is the frequency of the excitation force. This oscillator was modeled by an electrical circuit and was investigated by van der Pol and his colleague, van der Mark, and they reported in September 1927 in [8] that at certain drive frequencies  an irregular noise was heard. This irregular noise was always heard near the natural entrainment frequencies. This was one of the first discovered instances of deterministic chaos [9]. In Fig.6. the time history diagram for chaotic motion is plotted.

Fig.6. Chaotic behavior in the van der Pol oscillator with sinusoidal forcing for F=1.2, =/5 and =8.53.

Applied Mechanics and Materials Vol. 430

7

Since that time a significant number of investigations have been done on the problem. The extensive studies given in [3] and [10] – [13], devoted to the van der Pol oscillator have revealed that it possesses a rich dynamic behavior especially when submitted to a sinusoidal excitation: frequency entrainment, chaotic behavior with period doubling, etc. Non-linear van der Pol and Rayleigh oscillators As it is well known, the most of the phenomena in physics, biology, economics, etc., have nonlinear properties. For example, the elements of the electrical circuit have nonlinear characteristics which can be realized using operational amplifiers. They are proper elements to study nonlinear phenomena in physical systems experimentally. The aforementioned van der Pol i.e. Rayleigh oscillators are with nonlinear damping but with a linear displacement. Unfortunately, the assumption of the phenomenological model of the elastic restitution force to be linear with respect to the displacement is not correct. This is at this point where the words of Pippard referred in [14] have to be recalled: There is something of a tendency among physicists to try to reduce everything to linearity….., reality may not always conform to what might wish, rather more so with the damping forces than with the restoring force in small- amplitude vibrations. According to this suggestion, the non-linear restoring force model is included into the standard van der Pol and Rayleigh oscillators. The Rayleigh oscillator equation (6), which is one canonical example of self-excited systems, is simply generalized by addition of cubic displacement term and of the excitation force. The resultant equation of motion is given by

x   (1  x 2 ) x  x  x 3  F cos t

(8)

where  is a real parameter, F and  are respectively the amplitude and the frequency of the external perturbation. The model is analyzed by Siewe et al. [15]. The present approach can be used to generalize model of magneto-rheological dampers in novel studies of their influence on vehicle dynamics [16]. To reduce harmful vibrations one can consider application of dampers composed of Duffing oscillator with the Rayleigh damping. In the [17] the standard van der Pol oscillator is extended with a Duffing type cubic restoring force

x  x(1  x 2 )  x  x 3   11 (t ) x   2 2 (t )

(9)

where the excitation forces 1(t) and 2(t) are stochastic. The dissipative force in equation (9) is nonlinear, and the main feature of the oscillator is to be a self-excited system like the van der Pol oscillator. In [18] the attention is focused on the Duffing-Rayleigh oscillator subject to harmonic and stochastic white noise excitations which is described by x   (1 

 2 1

c

x 2 ) x  c12 (1  x 2 ) x  c12 F sin t  c12  2

(10)

where c1>0 is the natural circular frequency and  is the coefficient of linear damping,  and  are the nonlinearity parameters, F and  are intensity and frequency of the harmonic excitation, respectively, and 2 is the intensity of the additive white noise . However, Alfred - Marie Liénard (1869-1958) made the most wide generalization of the van der Pol equation. The Liénard - van der Pol equation is a second order differential equation [2]

8

Acoustics & Vibration of Mechanical Structures

x  f ( x) x  g ( x)  0

(11)

where f and g are continuously differentiable functions on R, with g an odd function and f an even function. Liénard gave the theorem which guarantees the uniqueness and the existence of a limit x

cyclic solution for equation (11). If g(x)>0 for all x>0, lim F ( x)  lim  f ( x)dx   , and the x  x  0

function F(x) has exactly one positive root at some value p which satisfies the requirement that F(x) is negative for 0 0 is the self-sustained factor of the self-sustained oscillations. Akulenko et al. also investigated in detail the self-sustained oscillations of the equation [22] 2

x   (1  x ) x  x x

 1

 0,

(17)

for  = 3 and 5. Cveticanin [23] treated the damped van der Pol oscillator

x  kx  c2 x x

 1

2

  (1  x ) x,

(18)

Applied Mechanics and Materials Vol. 430

9

where  is a positive rational number (integer or non-integer). The same author treated the special type of van der Pol equation with time variable parameter [24] m( ) x  c2 x x

 1

 ( ) x,   (b  cx 2 ) x  m

(19)

where =t is the slow time and  is a small positive parameter. The influence of the reactive force due to mass variation is treated. Nagumo, et al. [25] and Fitzhugh and [26] described the self-sustained oscillations in a neuron combining the van der Pol and Rayleigh damping terms. The obtained equation is usually called Fitzhugh – Nagumo equation

   w   v(v   )(v  1)  w   w

(20)

where

v  w   w.

(21)

Equation (20) describes the action potentials of neurons [25], [26]. A neuron, a nerve cell, communicates with other neurons by sending them pulse-like electrical signals. When a neuron receives enough electrical input from its neigbours it will fire, sending off its own electrial pulse. Once fired, the neuron must recover before firing again. This entire sequence (integrate, fire, rest) is given by equations (20) and (21). v is voltage and w is recovery of voltage. This equation is applied in physical as well as biological sciences. The same model has also been utilised in seismology to model the two plates in a geological fault. Like all relaxation oscillators, this oscillator has a slow accrual phase and a fast release phase. The variable parameter  represents the coupling between the slow and fast phases. As epsilon increases, so does frequency. Coupled oscillators To give a more realistic description of the systems and their behavior, the models have to be more complex and to include various types of nonlinear oscillators. Three of them are fundamental: the Duffing, the van der Pol and the Rayleigh oscillators. These oscillators are available in the extensive examination of a number of dynamical features which are embedded in the physical systems. Thus, the van der Pol oscillator and damped Duffing oscillator stand as paradigms of nonlinear oscillators: the former is the limit cycle prototype and the later is the strange attractor prototype [27]. At the other side, the Duffing - Rayleigh systems can occur in physical and electromechanical devices [28], [29] or for some chemical or biological oscillators and also in engineering [16] for example vehicle vibrations [30]. Concerning the coupling between a van der Pol oscillator and a Duffing oscillator, three basic schemes can be listed [27]: - Gyroscopic coupling – coupling through accelerations, - Dissipation coupling – coupling through velocity - Elastic coupling – coupling through solutions as it is discussed in [31] and [32]. The case of elastic coupling is given in the form

x   (1  x 2 ) x  x  k ( y  x)  0 y  y  y  y 3  k ( x  y )  0 and

,

(22)

10

Acoustics & Vibration of Mechanical Structures

x  x  12 x  x 3  k ( y  x)  0, y   (1  y 2 ) y   22 y  k ( x  y )  0

(23)

where  and  are positive parameters controlling the nonlinearities of the model,  accounts for the dissipation, 1 and 2 are the linear frequencies of the Duffing resonator, while k represents the coupling strength discussed in [31] and [32], respectively. If the coupling stiffness k diminishes to zero in the above equations, the equations uncouple to an autonomous Duffing resonator and a van der Pol oscillator. Comparing the two systems of equations (22) and (23) it is evident that in [32] the Duffing oscillator with both positive coefficients is discussed, while in [31] the linear term is negative. Such an assumption gives the significant difference in the behavior of the system especially for the chaotic motion. In the paper [32] the residue harmonic balance analysis is assumed for solving the system of two coupled differential equations. In [33] the dissipative coupled standard van der Pol oscillator with a Duffing oscillator with another oscillator of the same type is considered. The mathematical model is

x  x  x 3  (1  x 2 ) x   ( x  y )  0, y  (1   ) y  y 3  (2  y 2 ) y   ( x  y )  0,

(24)

where 1 and 2 are parameters of bifurcation,  is the frequency mismatch between the autonomous second and first oscillators and  is the coefficient of dissipative coupling.  is the addition coefficient of the nonlinearity. The synchronization for the parameter  is investigated. Qian et al. [34] extended the model by assuming a linear elastic and a linear damping coupling for two oscillators: van der Pol and Duffing ones. The extended homotopy analysis method is applied to derive the accurate approximate analytical solutions for the two degree of freedom coupled van der Pol –Duffing oscillator:  x  (1  x 2 ) x  (1  ) x  x 3   ( x  y )   ( x  y )  0 2  y  ( 2  y 2 ) y  (1  ) y  y 3   ( y  x)   ( y  x )  0 2

(25)

1 and 2 are bifurcation parameters,  is the detuning parameter that is proportional to the difference of the oscillator frequencies,  is the non-isochronisms parameter that relies on the oscillation frequency and amplitude,  and  are the coupling inertial and dissipative parameters. The system of van der Pol –Duffing oscillators connected with a nonlinear elastic force of cubic order is also investigated (see [35]):

x   (1  x 2 ) x  ( x  x 3 )  x 2 (x  y )  y 2 (x  y), y   (1  y 2 ) y  ( y  y 3 )   x 2 (x  y )  y 2 (x  y ),

(26)

In the paper the extended homotopy analysis method is applied which is also used for analytical consideration of the limit cycle motion of the system with delayed amplitude limitation [36]

x  x  2 (1  z ) x  zx

z  z  x 2

(27)

where  is the delay time,  is a constant and x(t) and z(t) are two unknown functions to be determined.

Applied Mechanics and Materials Vol. 430

11

Future investigation Based on the overview on the van der Pol and Rayleigh oscillators and their extension by connecting them with other types of oscillators (mainly Duffing’s one), it is evident that it gives an improvement in comparison to the standard van der Pol and Rayleigh systems but fail to explain the real systems. It requires the extension of the model with pure nonlinear terms of any rational order and also the introduction of the damping terms of nonlinear polynomial type of integer or noninteger order. It is believed that the model would give more accurate description of most of the problems in mechanics, electro techniques (engineering), including biology, physics, chemistry and economics. The improvement of the solution procedures for the nonlinear differential equations is also necessary. Summary In this paper the standard van der Pol and Rayleigh oscillators are considered. The historical development of the knowledge in self-sustained oscillations and limit cycle motion is given. The two standard oscillators are coupled with Duffing ones and various excitation forces are added. The obtained models are suitable for description and explanation of a significant number of phenomena in engineering, physics, biology, chemistry, and economic. Acknowledgement The investigation was supported by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina (Proj. No 114-451-2094/2011) and Ministry of Science of Serbia (Proj. No. ON 174028 and IT 41007). References [1] B. van der Pol, On relaxation-oscillations. The London, Edinburgh and Dublin Philosophical Magazine & Journal of Science 2 (7) (1926) 901-912. [2] A. Lienard, Etude des oscillations entrenues. Rev. Generale de l’Electricite 23 (1928) 946-954. [3] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations. New York: Wiley, 1979. [4] B. van der Pol, J. van der Mark, The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Phil. Mag. Suppl. 6 (1928) 763-775. [5] H.G. Narm, A new state observer for two coupled van der Pol oscillators, Int. J. of Control. Automation and Systems 9 (2011) 420-414. [6] J. Rayleigh, The Theory of Sound. New York: Dover, 1945. [7] M.R. Navabi, M.S.M. Deghghan, Numerical solution of the controlled Rayleigh nonlinear oscillator by the direct spectral method. J. of Vibr. and Control 14 (6) (2008) 795-806. [8] B. van der Pol, J. Van der Mark, Frequency demultiplication. Nature 120 (1927) 363-364. [9] J. M. Thompson, H.B. Stewart, Nonlinear Dynamics and Chaos. New York: John Wiley and Sons, 1986. [10] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Field. New York: Springer, 1983. [11] C. Partliz, W. Lauterborn, Period doubling cascade and devil staircases of the driven van der Pol oscillator. Phys. Rev. A 36 (1987)1428-1434. [12] A. Venkatesan, M. Lakshmanan, Bifurcation and chaos in the double –well Duffing-van der Pol oscillator: numerical and analytical studies. Phys. Rev. E 56 (1997) 6321-6330.

12

Acoustics & Vibration of Mechanical Structures

[13] J.C. Chedjou, H.B. Fotsin, P. Woafo, S. Domngang, Analog simulation of the dynamics of a van der Pol oscillator coupled to a Duffing oscillator. IEEE Trans. Circuits Syst. I Fudam. Theory Application 48 (2001) 748-756. [14] M.S. Siewe, H. Cao, M.A.F. Sanjuan, Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh-Duffing oscillator. Chaos, Solitons & Fractals 39 (2009) 1092-1099. [15] M.S. Siewe, C. Tchawoua, P. Woafo, Melnikov chaos in a periodically driven RaylegyDuffing oscillator. Mech. Research Commun. 37 (2010) 363-368. [16] S. Li, S. Yang, W. Guo, Investigation of chaotic motion in histeric non-linear suspension system with multi-frequency excitations. Mech. Research Commun. 31 (2004) 229-. [17] Q. He, W. Xu, H. Rong, T. Fang, Stochastic bifurcation in Duffing-Van der Pol oscillators. Physica A 338 (2004) 319-334. [18] W.X. Xie, W.Xu, L. Cai, Path integration of the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations. Appl. Math. Comput. 171 (2005) 870-884. [19] Q. Ding, A.Y.T. Leung, The number of limit cycle bifurcation diagrams for the generalized mixed Rayleigh-Liénard oscillator. J. of Sound and Vibr. 322 (2009) 393-400. [20] S. Lynch, C.J. Christopher, Limit cycles in highly non-linear differential equations. J. of Sound and Vibr. 224 (1999) 505-517. [21] L.D. Akulenko, L.I. Korovina, S.A. Kumakshev, S.V. Nesterov, Self-sustained oscillations of Rayleigh and van der Pol oscillators with moderately large feedback factors. J. of Appl. Math. Mech. 68 (2004) 241-248. [22] L.D. Akulenko, L.I. Korovina, S.V. Nesterov, Self-sustained oscillations of a highly non-linear system. Izvestija Rossija Akademia Nauk MTT 3 (2002) 42-48. [23] L. Cveticanin, On the truly nonlinear oscillator with positive and negative damping. Submitted for publication [24] L. Cveticanin, Van der Pol oscillator with time-variable parameters. Acta Mech. 224 (5) (2013) 945-955. [25] J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating 1214nerve axons. Proceedings IRL 50 (1960), pp. 2061-2070. [26] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membranes. 1182-Biophysics Journal 1 (1961), pp. 445-466. [27] J. Kengne, J.C. Chedjou, G. Kenne, K. Kyamakya, G.H. Kom, Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator. Nonlinear Dynamics 70 (2012) 2163 -2173 [28] R. Yamapi, P. Woafo, Dynamics and synchronization of coupled self-susteained electromechanical devices. J. of Sound and Vibr. 285 (2005) 1151-1170. [29] J.C. Chedjou, K. Kyamakya, I. Moussa, H.-P. Kuchenbecker, W. Mathis, Behavior of a self sustained electromechanical transducer and routes to chaos. J. Vibr. Acoustics 128 (2008) 282-293. [30] G. Litak, M.I. Borowiec, K. Friswell, Chaotic vibration of a quarter-car model excited by the road surface profile. Commun. in Nonlinear Sci. and Numer. Simul. 13 (2008) 1373-1383. [31] A.P. Kuznetsov, J.P. Roman, Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization, Physica D 238 (2009) 1499-1506.

Applied Mechanics and Materials Vol. 430

13

[32] A.Y.T. Leung, Z. Guo, H.X. Yang, Residue harmonic balance analysis for the damped Duffing resonator driven by a van der Pol oscillator. Int. J. of Mech. Sci. 63 (2012) 59-65. [33] A.P. Kuznetsov, N.V. Stankevich, L.V. Turukina, Coupled van der Pol-Duffing oscillators: phase dynamics and structure synchronization. Physica D 238 (2009) 1203-1215. [34] Y.H. Qian, W. Zhang, B.W. Lin, S.K. Lai, Analytical approximate periodic solutions for twodegree-of-freedom coupled van der Pol-Duffing oscillators by extended homotopy analysis method. Acta Mech. 219 (2011) 1-14. [35] Y.H. Qian, C.M. Duan, S.M. Chen, S.P. Chen, Asymptotic analytical solutions of the twodegree-of-freedom strongly nonlinear van der Pol oscillators with cubic couple terms using extended homotopy analysis method. Acta Mech. 223 (2012) 237-255. [36] A.K. Eigoli, M. Khodabakhsh, A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude limiting. Appl. Math. Comput. 217 (2011) 9404-9411.

Applied Mechanics and Materials Vol. 430 (2013) pp 14-21 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.14

Nonlinear Oscillators with a Power-Form Restoring Force: Non-Isochronous and Isochronous Case Ivana Kovacic1, a 1

Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia a

[email protected]

Keywords: nonlinear oscillator, frequency, amplitude, isochronicity.

Abstract. This work is concerned with single-degree-of-freedom conservative nonlinear oscillators that have a fixed restoring force that comprises a linear term and an odd-powered nonlinear term with an arbitrary magnitude of the coefficient of nonlinearity. There are two cases of interest: i) nonisochronous, when the system has an amplitude-dependent frequency and ii) isochronous, when the frequency of oscillations is constant (amplitude-independent). The first case is associated with the constant coefficient of the kinetic energy, while the frequency-amplitude relationship and the solution for motion need to be found. To that end, the equation of motion is solved by introducing a new small expansion parameter and by adjusting the Lindstedt-Poincaré method. In the second case, the condition for the frequency of oscillations to be constant is derived in terms of the expression for the position-dependent coefficient of the kinetic energy. The corresponding solution for isochronous oscillations is obtained. Numerical verifications of the analytical results are also presented. Introduction Nonlinearity is unambiguous in nature, which implies that all physical systems found in nature are, at their core, nonlinear. This holds for many engineering systems as well. Despite the initial attempts to linearise nonlinear systems whenever possible, last several decades have been characterized by a profound increase in the number of studies of nonlinear systems, accompanied by an impressive number of analytical, numerical and experimental methods that resulted in the identification and deep understanding of phenomena associated with nonlinearity [1, 2], features which are often perceived as inconvenient and undesirable. One of the features that is seen as inconvenient is related to the fact that nonlinear systems, in general, have an amplitude-dependent frequency and, consequently, their backbone curve bends, yielding hysteresis and jump phenomena. However, the question is if there are nonlinear systems whose frequency is amplitude-independent so that their backbone curve is a straight line. In the attempt to answer this question, we consider the conservative nonlinear oscillators completely described by the Lagrangian L =T ( x, x ) − V ( x ) , where the kinetic energy is ~ ~ T ( x, x ) =T ( x )x 2 / 2 , with the coefficient T ( x ) being dependent on the non-dimensional generalized coordinate x , while V ( x ) stands for the potential energy. The corresponding equation of motion is ~ T′ V′ x + ~ x 2 + ~ = 0, 2T T .

where ( ) = d ( ) dt , ( )' = d ( power-form nonlinearities V′ Fr = ~ = x + εx 2 k +1 , T

(1)

) dx ,

and where the restoring force Fr has a form that includes

(2)

Applied Mechanics and Materials Vol. 430

15

with ε being an arbitrary positive real number (not necessarily small) and k being a positive integer. Note that the term proportional to x 2 in Eq. 1 stems from the coordinate-dependent kinetic energy, and is not related to the mechanism of quadratic damping. Two cases are of interest here: i) Case I, when the coefficient of the kinetic energy is constant (it is assumed here as being ~ equal to unity, i.e. T = 1 ) and the frequency-amplitude relationship needs to be found, as well as the solution for motion; Case II, when the frequency of oscillations is assumed to be constant (not amplitudeii) ~ dependent) and the form of T ( x ) needs to be determined for this to be achieved; the corresponding solution for motion is also sought. Thus, this study concerns a family of conservative nonlinear oscillators that have a fixed powerform nonlinear restoring force. Many methods have been developed to determine their frequencyamplitude relationship (see, for example, [1-3]). The system analysed here in Case I does not assume weak nonlinearity as ε is not necessarily small. In order to solve the equation of motion with a general odd nonlinear restoring force and with the arbitrary magnitude of the coefficient of nonlinearity, Burton [4] modified the Lindstedt-Poincaré method by introducing a new small expansion parameter. In [5], Burton and Rahman dealt with the same system but added a linear viscous damping force. They developed a new multiple scales approach by using again the concept of a new expansion parameter. Cheung et al. [6] presented a new modified Lindstedt-Poincaré method by following the concept of the introduction of a new small parameter, enabling a strongly nonlinear system corresponding to the original parameter ε to be transformed into a weakly nonlinear system. A newly constructed small expansion parameter was also used in [7], where viscously damped oscillators with a purely nonlinear non-negative-real power-form restoring force were considered, as well as in [8], where the restoring force involves a non-negative real powerform nonlinear term and a negative, zero or positive linear term. The approach presented in this paper relies on the procedure presented in [7], but herein unforced conservative oscillators are dealt with, and the frequency-amplitude expression and the solution for motion are derived both in the first and second approximation. Then, in Case II, assuming the same form of the restoring force, the ~ coefficient T ( x ) is found so that the oscillator has a constant (amplitude-independent) frequency, i.e. that it has the property of isochronicity [9].

Case I: Nonlinear Oscillators with an Amplitude-Dependent Frequency The frequency-amplitude relationship and the solution for motion will be found by adjusting the Lindstedt-Poincaré method. First, a new time τ = ωt is defined, where ω is the unknown angular ~ frequency. Thus, Eqs. 1 and 2 with T = 1 transform into

d 2x + x + εx 2 k +1 = 0. (3) 2 dτ Given the fact that ε is not necessarily small, a new small expansion parameter p needs to be introduced. Based on the results presented in [7], it is defined as ω2 −ω2 p= a 2 e , (4)

ω2

ωa

where

ω a2 = 1 + εb1a 2 k ,

3  Γ + k  2 2 , b1 = π Γ(k + 2)

(5a,b)

16

Acoustics & Vibration of Mechanical Structures

with Γ being the Euler gamma function. Note that b1 represents the first Fourier coefficient of the

power function (cos θ ) . More details about this Fourier series expansion are given in the Appendix. Further, knowing that b1 is just the first Fourier coefficient, and that the power form nonlinearity brings the influence through the coefficients whose exact value c is known, the square of the exact angular frequency ω e of the conservative oscillator considered can be expressed as 2 k +1

ω e2

( (

) )

Γ 2k(k++21) = 1 + εc a , c = π (k + 1) . Γ 2(k1+1) 2

2k

(6a,b)

Thus, the new expansion parameter becomes

εb1a 2 k − εc 2 a 2 k . 1 + εb1a 2 k

p=

(7)

When εa 2 k → ∞ , this parameter is such that p → 1 − c 2 b1 . For example, when k=1, p=0.0430 and when k=2, p= 0.1076, which confirms that its value stays small. By using Eq. 7, the original ordering parameter ε can now be expressed as

p . (8) a (1 − p ) b1 − c 2 Further, to follow the methodology of the Lindstedt-Poincaré method, the square of the frequency is expanded as

ε=

2k

(

)

(

)

n ω 2 = 1 + εb1a 2 k 1 + ∑ p i ω i , i =1

(9)



where ωi are to be determined. By using Eq. 8, the square of the frequency can be expressed in terms of the new parameter p as follows

b1 − c 2

n   i 1 + ∑ p ω i .  (1 − p )b1 − c  i=1 By substituting Eqs. 8 and 10 into Eq. 3, one obtains

ω2 =

(10)

2

 b1 1 2 k +1  . − p x x 2 2k 2  b − c a ( b − c )  1 1  Now, x is expanded in a power series in p 2 n  d x i ω +x= 1 + p  ∑ i  i =1  dτ 2

(11)

x(τ ) = x0 (τ ) + px1 (τ ) + p 2 x2 (τ )... (12) As a result of this, Eq. 11 is transformed into the following succession of differential equations obtained by equation like powers of p to zero d 2 x0 dτ 2 d 2 x1 dτ 2 d 2 x2 dτ

2

+ x0 = 0, + x1 = −ω1 + x 2 = −ω1

d 2 x0 dτ 2 d 2 x1 dτ

2

+ x0 − ω2

b1 b1 − c 2 d 2 x0 dτ

2

−

1 x02 k +1 , 2k 2 a (b1 − c )

+ x1

b1 b1 − c

2

−

2k + 1 x1 x02 k . 2 a (b1 − c ) 2k

(13a-c)

Applied Mechanics and Materials Vol. 430

17

The solution of Eq. (13a) x0 = a cosτ is substituted into the right-hand side of Eq. (13b). Note that

the initial conditions are assumed as x(0) = a , x(0) = 0 , and associated only with x0 , while those for xi , i=1,2,… are zero. Expanding the power-form function x02 k +1 into the Fourier series as shown in the Appendix, the secular terms cancel each other, as a result of which it follows that ω1 = 0 . Truncating the Fourier series for x02 k +1 to the fifth harmonic, the following solution for x1 is obtained ab3

ab5

(cos 5τ − cosτ ), 8(b1 − c ) 24(b1 − c 2 ) where the coefficients b3 and b5 are given in the Appendix. Based on Eqs. 12 and 7, the solution at this level of approximation is found to be x1 =

2

(cos 3τ − cosτ ) +

εb3 a 2 k +1 εb5 a 2 k +1 (cos 3ωt − cos ωt ) + (cos 5ωt − cos ωt ), x = a cos ωt + 8 1 + εb1a 2 k 24 1 + εb1a 2 k

(

)

(

)

(14)

(15)

where the frequency-amplitude relationship is given by

ω 2 ≡ ω a2 = 1 + εb1a 2 k .

(16)

This frequency-amplitude relationship is plotted in Fig. 1 for ε=1 and k=2 as a solid line. It represents the hardening backbone-curve of this oscillator, which is bent to the right-hand side with respect to ω = 1 .

Fig. 1 Backbone curves corresponding to for ε=1 and k=2: Eq. 16 (solid line) and Eq. 18 (dashed line) and ω = 1 (dotted line) Fig. 2 shows the solution for motion given by Eq. 15 (solid line) as well as numerically obtained time response from the equation of motion (dots). Although being a low-level approximation, the analytically obtained solution agrees reasonably well with the numerical solution. If needed, the higher level of approximation can be calculated by proceeding in a similar way. Thus, if one would treat Eq. 13c, while approximating x02k with the Fourier series truncated to the sixth harmonic as given in the Appendix, the following expression for ω 2 would be obtained from the requirement of removal of secular terms 2b (3b + b ) − (3b3 (d 0 − d 4 ) + b5 (d 0 + d 2 − d 4 − d 6 ))(2k + 1) ω2 = 1 3 5 , (17) 2 48 b1 − c 2 where the coefficients d0-d6 are listed in the Appendix.

(

)

18

Acoustics & Vibration of Mechanical Structures

The frequency-amplitude relationship has the form

(

)

ω 2 = 1 + εb1a 2 k 1 + p 2ω 2 .

(18)

The frequency-amplitude relationship given by Eqs. 18, 17 and 7 is also plotted in Fig. 1 for ε=1 and k=2 as a dashed-line backbone curve. It is seen that this approximation coincides with the one given by Eq. 16 until a ≈ 0.8 and then slightly deteriorates from it. The solution for x2 is obtained by solving what is left in Eq. 13c, as x 2 = ∑ A2 j +1 (cos(2 j + 1)τ − cos τ ), 5

(19)

j =1

where

A3 = − A5 = − A7 = A11 =

a (6b1b3 + (b5 d 4 + 3b3 (− d 0 + d 2 + d 4 − d 6 ))(1 + 2k ) ) 384(b1 − c 2 ) 2

,

a (2b1b5 − (b5 (d 0 − d 4 − d 6 ) + 3b3 (d 2 − d 4 − d 6 ))(1 + 2k )) 1152(b1 − c 2 ) 2

a (b5 d 2 + 3b3 d 4 − (3b3 + b5 )d 6 )(1 + 2k ) 2304(b1 − c 2 ) 2 ab5 d 6 (1 + 2k ) 5760(b1 − c 2 ) 2

,

A9 =

,

a (b5 d 4 + 3b3 d 6 )(1 + 2k ) 3840(b1 − c 2 ) 2

(20a-e)

,

.

~ Fig. 2 Time response obtained numerically from Eq. 1, 2 with T = 1 (dots) and analytically from Eq. 15 (solid line) and Eqs. 12, 7, 14, 17-20a-e (dashed line) for ε=1, k=2, x(0) = 0 and three sets of the initial amplitude x(0) = 0.1; 0.5; 1 The solution defined by Eqs. 12, 7, 14, 17-20a-e is also plotted in Fig. 2 as a dashed-line. It is seen that it matches the numerical solution. It should be emphasized that the technique presented here can be used to derive the frequencyamplitude relationship and the solution for motion of the oscillators whose restoring force contains power-form nonlinearities with a non-integer power α, i.e. for Fr = δx + ε sgn ( x ) x . In addition, α

the value of the coefficient δ in front of the linear term can be negative, zero or positive. The technique can also be applied to find steady-state response of such nonlinear oscillators to harmonic excitation [7].

Applied Mechanics and Materials Vol. 430

19

Case II: Nonlinear Oscillators with a Constant Frequency To find conservative oscillators with a nonlinear restoring force that have a constant frequency, the equivalence between the energy of the oscillators under consideration defined by Eqs. 1 and 2, and the energy of the simple harmonic oscillator, which is known to have a constant (amplitudeindependent) frequency is established. Thus, the kinetic energy of the simple harmonic oscillator TSHO = X 2 / 2 should correspond to the kinetic energy of the nonlinear oscillators under ~ consideration T =T ( x ) ⋅ x 2 / 2 , and this should also hold for their potential energies VSHO = X 2 / 2

and V ( x ) . This yields

X2 ~ X = T ( x ) x, V ( x ) = . 2 Based on Eq. 21a, b, the restoring force is found to be

(

(21a,b)

)

dX ~ x ~ x dx = ∫0 T ( s )ds ⋅ T ( x) = ∫0 ~ ~ T T ( x)

~ T ( s )ds . ~ T ( x) ~ Equating this expression with Eq. 2 and solving it for T ( x ) , one can obtain V′ Fr = ~ = T

X

(22)

~ − 2 k +1 T ( x ) = (1 + εx 2 k ) 2 k . (23) This expression indicates how the coefficient of the kinetic energy should change so that the resulting oscillations are isochronous, with the frequency being equal to unity. Note that this can stem from the mass that changes with the displacement or it can be the consequence of the existence of a geometric/kinematic constraint. Eq. 1 now gives the equation of motion of the nonlinear oscillator with isochronous motion x 2 k −1 2 x + x + εx 2 k +1 = 0. (24) x − ε (2k + 1) 2k 1 + εx ~ ~ Note that Eq. 21a gives X = ∫0x T ( s )ds , where T is defined by Eq. 23. Now, knowing that the solution for motion of the simple harmonic oscillator can be expressed as X = A cos(t + α ) , where A and α are constants, and equating these two forms of X, one follows x

(

)

1 2k 2k 1 + εx

= A cos(t + α ).

(25)

(

For the initial conditions x(0) = a and x(0) = 0 , one calculates α =0 and A = a 1 + εa 2 k that the implicit solution for motion is as follows x

(

)

1 2k 2k 1 + εx

=

a

(

)

1 2k 2k 1 + εa

cos t .

)

1 2k

, so

(26)

Fig. 3 shows numerically obtained time response from Eq. 24 (dots) and the analytical solution given by Eq. 26 (solid line) for ε=1 and k=2. These solutions confirm that the period stays constant regardless of the amplitude, unlike the one in Fig.2. They also confirm the validity of the analytical results obtained. Note that this solution has the period equal to 2π, which implies that the corresponding backbone curve is a straight-line, as shown in Fig. 1.

20

Acoustics & Vibration of Mechanical Structures

It should be emphasized that the approach presented above can be applied to find other oscillators with a specified nonlinear restoring force and with a constant frequency. To that end, the ~ coefficient of the kinetic energy T ( x) should be such that Eq. 22 is satisfied for a given form of the restoring force Fr . Note that additional consideration should be carried out to detect if the solution obtained is global or not [10].

Fig. 3 Time response obtained numerically from Eq. 24 (dots) and analytically from Eq. 26 (solid line) for ε=1, k=2, x(0) = 0 and three sets of the initial amplitude x(0) = 0.1; 0.5; 1 Summary This study has been concerned with oscillators having a restoring force that includes a linear term and a positive odd-powered nonlinear term. It has been shown that the frequency of these oscillators can be amplitude-dependent (non-isochronous), which holds for the constant coefficient of the kinetic energy, but that it can also be amplitude-independent (isochronous), which occurs when the coefficient of the kinetic energy changes with the displacement in a particular way. In the former case, the frequency-amplitude relationship yields a backbone curve that is bent to the right-hand side, while in the latter case, the backbone curve is a straight vertical line. The solution for motion has been obtained in both cases: in the former case as an approximation and in the latter one as the exact solution.

The Appendix: On Some Fourier Series Expansions The power-form function f = x 2 k +1 , where k is a positive integer and can be developed into the following Fourier series expansion f = (cos θ )

2 k +1

∞

= ∑ b(2 n−1) ⋅ cos(2n − 1)θ , n =1

(A.1)

where the coefficients b(2n −1) are given by b(2 n−1) =

4π

π

2

∫ (cos θ )

2 k +1

⋅ cos(2n − 1)θ dθ .

(A.2)

0

The first few of them are 3  3  3  Γ + k  Γ + k  Γ + k  2 1 2 2 2 k k ( k − ) , b = 2 , b = 2 . b1 = 3 5 Γ(k + 4 ) π Γ(k + 2) π Γ(k + 3) π

(A.3a-c)

Applied Mechanics and Materials Vol. 430

21

Another function of interest here is the power-form function g = x 2 k , where k is a positive integer and x = cos θ . This function has the following Fourier series ∞ d0 + ∑ d 2 N cos(2 N )θ , 2 N =1 where the Fourier coefficients are defined by

g = (cos θ )

d (2 N − 2 ) =

2k

4

π

=

π

∫02 (cos θ ) cos(2 N − 2 )θ dθ . 2k

(A.4)

(A.5)

The first few of them are

1  1  1  Γ + k  Γ + k  Γ + k  2 2  , d = 2k  2  , d = 2k (k − 1)  2 , d0 = 2 4 ( ) ( ) ( ) 1 Γ 2 Γ 3 Γ k + k + k + π π π 2k (k − 1)(k − 2 ) Γ(1 + k ) . d6 = Γ(k + 4 ) π

(A.6a-d)

Acknowledgment: Ivana Kovacic acknowledges support received from the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina (Project No. 114451-2094). References [1] I. Kovacic, M.J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, John Wiley and Sons, 2011. [2] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, NewYork, 1979. [3] R.E. Mickens, Truly Nonlinear Oscillations: Harmonic Balance, Parametric Expansions, Iteration, and Averaging Methods, World Scientific, Singapore, 2010. [4] T.D. Burton, A perturbation method for certain non-linear oscillators, Int. J. Non Linear Mech. 19 (1984) 397-407. [5] T.D. Burton, Z. Rahman, On the multiple-scale analysis of strongly non-linear forced oscillators, Int. J. Non Linear Mech. 21 (1986) 135-146. [6] Y.K. Cheung, S.H. Chen, S.L. Lau, A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators, Int. J. Non Linear Mech. 26 (1991) 367-378. [7] I. Kovacic, Forced vibrations of oscillators with a purely nonlinear power-form restoring force, J. Sound Vib. 330 (2011) 4313-4327. [8] I. Kovacic, The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force, Chaos Solitons Fract. 44 (2011) 891–901. [9] F. Calogero, Isochronous Systems, Oxford University Press, Oxford, 2008. [10] M. Sabatini, On the period function of x ′′ + f ( x )x ′ 2 + g ( x ) = 0 , J. Diff. Equ. 196 (2004) 151168.

Applied Mechanics and Materials Vol. 430 (2013) pp 22-26 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.22

Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method Vasile Marinca1,2, a, Nicolae Herişanu1,2,b, Traian Marinca3,c 1,2 1,2

3

“Politehnica” University of Timişoara, Bd.Mihai Viteazu, 1, 300222, Timişoara, Romania Center for Advanced Research in Engineering Sciences, Romanian Academy-Timişoara Branch, Bd. Mihai Viteazu, 24, 300223, Timişoara, Romania

Technical University of Cluj-Napoca, Str. C. Daicoviciu nr 15, 400020 Cluj-Napoca, Romania a

b

c

[email protected], [email protected], [email protected]

Keywords: nonlinear parametric vibration, Optimal Homotopy Asymptotic Method.

Abstract. The response of a cantilever beam with a lumped mass attached to its free end subject to harmonical excitation at the base is investigated by means of the Optimal Homotopy Asymptotic Method (OHAM). Approximate accurate analytical expressions for the solutions and for approximate frequency are determined. This method does not require any small parameter in the equation. The obtained results prove that our method is very accurate, effective and simple for investigation of such engineering problems. Introduction Many engineering structures can be modeled as cantilever beam with a lumped mass attached to its free end and with support motion. This includes for example mast antennas, towers, flexible robot manipulators and space structures. When the beam support undergoes motion, the beam might be subject to external or parametric excitation. Since the resonance is considered, the nonlinearities begin to affect the motion and hence they cannot be ignored. The problems of forced vibrations of beams have received considerable attention in the past few decades. Comprehensive reviews on this subject have been presented by Hoppman [1], Timoshenko [2], Laura et al [3], Esmailzadeh and Jazar [4] or Nayfeh and Mook [5] and so on. The most common and widely used methods for determining analytical approximate solutions of a nonlinear system are perturbation methods [5]. There exist some analytical approaches for determining analytical approximations to nonlinear equations, such as the Lindstedt-Poincare method [5], the KBM method [6], iteration procedure [7], Adomian decomposition method [8], modified method of equivalent linearization [9], the harmonic balance method [10], but sometimes these approaches lead to inconsistencies, especially for strongly nonlinear problems. In this paper we apply the Optimal Homotopy Asymptotic Method (OHAM) for damped forced oscillations of a nonconservative nonlinear system consisting of a cantilever beam with a lumped mass attached to its free end while being excited harmonically at the base. The method proposed in this paper avoids some limitations identified within the above mentioned classical perturbation methods, since this method is valid not only for small, but also for large parameters. The results demonstrate the applicability of the proposed method to strongly nonlinear oscillations and its accuracy, since the obtained results are quasiidentical with those obtained by numerical simulations. Fig.1. Cantilever model with end mass and base excitation

Applied Mechanics and Materials Vol. 430

23

Application of OHAM to the cantilever beam Fig.1 shows a slender bar with length l, and end mass m, being studied in the coordinate reference frame X-Y. The mass of the beam is assumed to be small and negligible compared with the lumped mass and therefore the beam is considered as a purely flexural element. The support of the beam undergoes a harmonic motion f(t), while the beam s restricted to oscillate in the X-Y plane. The governing equation in this case is [1-4] u ′′ + ω 02 u + u 2 u ′′ + uu ′ 2 + a cos η = 0

(1)

with the initial conditions u (0) = A , u ′(0) = 0

(2)

where η = λt , ω0, λ and a are known parameters. We apply the OHAM [11], [12]. For Eq.(1) we construct a family of equations (1 − p)L(u (η) ) = H (p, C i ) N (u (η), u ′(η), u ′′(η) )

(3)

where p ∈ [0,1] is an embedding parameter, L(u (η)) is the linear operator L(u (η)) = u ′′(η) + ω 2 u (η)

(4)

where ω the frequency of the system; N is the nonlinear operator N (u, u ′, u ′′) = u ′′ + ω 02 u + u 2 u ′′ + uu ′ 2 + a cos η

(5)

and H (p, C i ) is a nonzero auxiliary function for p ≠ 0 and H (0, C i ) = 0 . We choose the auxiliary function H of the form H (p, C i ) = pH 1 (η, C i ) + p 2 H 2 (η, C j ) + ...

(6)

Let us consider the approximate solution of the order m in the form u (η) = u 0 (η) + pu 1 (η) + p 2 u 2 (η) + ... + p m u m (η)

(7)

Substituting Eq.(7), (6) and (4) into Eq.(3) and equating the coefficients of like powers of p we obtain the governing equations of u0(η) and ui(η), i=1,2,…,m. In this way, for p=0 into Eq.(3) we obtain u ′′(0) + ω 2 u 0 = 0 , u 0 (0) = A , u ′(0) = 0 (8) Eq.(8) has the solution u 0 (η) = A cos η . If we consider only m=1, the governing equation of u1(η) is obtained from Eq.(3): u 1′′ + ω 2 u 1 = H 1 (η, C i )(u ′0′ + ω 02 u 0 + u 02 u ′0′ + u 0 u ′02 + a cos η

(9)

There are many possibilities to choose the auxiliary function H1(η,Ci). The convergence of the solution u1(η) and consequently the convergence of the approximate solution u (η) depend on the auxiliary function H1.

24

Acoustics & Vibration of Mechanical Structures

Substituting the solution of Eq.(8) into (9), we obtain for the last term the following expression: u ′0′ + ω 02 u 0 + u 02 u ′0′ + u 0 u ′02 + a cos η = A (ω 02 − ω 2 −

1 2 2 1 ω A ) cos ωτ + a cos τ − A 3 ω 2 cos 3ωτ 2 2

(10)

Basically, the shape of H1(η,Ci) must follow the terms appearing in Eq.(10). Therefore, we choose H1 as a combination of functions cos ωη , cos 3ωη ,…: H 1 (η, C i ) = C1 + C 2 cos 2ωη + C 3 cos 4ωη + C 4 cos 6ωη + C 5 cos 8ωη

(11)

where C i are unknown parameters which will be determined later. Eq.(9) can be written as  1 1 1  u 1′′ + ω 2 u 1 = − A 3 ω 2 (C1 + C 3 ) + A(ω 02 − ω 2 − ω 2 A 2 )(2C1 + C 2 )  cos ωη + ω 2 A 2 cos 3ωη + 2 2   4 + ω 2 A 3 cos 5ωη + ω 2 A 4 cos 7ωη + ω 2 A 5 cos 9ωη + ω 2 A 6 cos 11ωη + B1 cos(2ω + 1)η +

(12)

+ B 2 cos(2ω − 1)η + B 3 cos(4ω + 1)η + B 4 cos(4ω − 1)η + B 5 cos(6ω + 1)η + B 6 cos(6ω − 1)η + + B 7 cos(8ω + 1)η + B 8 cos(8ω − 1)η + B 9 cos η

where A2 =

A 3 [(C 2 + C 3 ) 2 − ( 2C1 + C 2 )(2C1 + C 4 )] A 3 [(C 2 + C 3 )(C 3 + C 5 ) − (2C1 + C 2 )(C 2 + C 5 )] ; A3 = 4( 2C1 + C 2 ) 4( 2C1 + C 2 )

A4 =

A 3 [(C 2 + C 3 )(C 4 + C 5 ) − C 3 (2C1 + C 2 )] A 3 [(C 2 + C 3 )C 5 − (2C1 + C 2 )C 4 ] ; A5 = ; 4( 2C 1 + C 2 ) 4( 2C 1 + C 2 )

A6 =

− A 3C5 aC aC 2 aC 4 ; B 4 = 5 ; B 5 = aC1 ; B1 = ; B3 = 4 2 2 2

(13)

No secular terms in Eq.(12) requires that ω2 =

2ω 02 (2C1 + C 2 ) 4C1 + 2C 2 + A 2 (2C1 + 2C 2 + C 3 )

(14)

The initial conditions for Eq.(12) are u 1 (0) = u 1′ (0) = 0 , such that the solution of Eq.(12) becomes A2 A A (cos ωη − cos 3ωη) + 3 (cos ωη − cos 5ωη) + 4 (cos ωη − cos 7ωη) + 8 24 48 A5 A6 B1 + (cos ωη − cos 9ωη) + (cos ωη − cos 11ωη) + [cos ωη − cos(2ω + 1)η] + 80 120 3ω 2 + 4ω + 1 B1 B2 + [cos ωη − cos(2ω − 1)η] + [cos ωη − cos(4ω + 1)η] + 2 2 3ω − 4ω + 1 15ω + 8ω + 1 B3 B2 [cos ωη − cos(4ω − 1)η] + [cos ωη − cos(6ω + 1)η] + 2 2 15ω − 7ω + 1 35ω + 12ω + 1 B3 B4 [cos ωη − cos(6ω − 1)η] + [cos ωη − cos(8ω + 1)η] + 2 2 35ω − 12ω + 1 69ω + 16ω + 1 B B4 + [cos ωη − cos(8ω − 1)η] + 2 5 (cos η − cos ωη) 2 69ω − 16ω + 1 ω −1 u 1 (η) =

From Eq.(9), (15) and (7) we obtain the first order approximate solution in the form

(15)

Applied Mechanics and Materials Vol. 430

A2 A A A A cos 3ωλt − 3 cos 5ωλt − 4 cos 7ωλt − 5 cos 9ωλt − 6 cos 11ωλt − 8 24 48 80 120 B1 B1 B2 − cos(2ω + 1)λt − cos(2ω − 1)λt − cos(4ω + 1)λt − 3ω 2 + 4ω + 1 3ω 2 − 4ω + 1 15ω 2 + 8ω + 1 B3 B3 B2 − cos(4ω − 1)λt − cos(6ω + 1)λt − cos(6ω − 1)λt − 2 2 2 15ω − 8ω + 1 35ω + 12ω + 1 35ω − 12ω + 1 B B4 B4 − cos(8ω + 1)λt − cos(8ω − 1)λt − 2 5 2 2 35ω + 12ω + 1 35ω − 12ω + 1 ω −1

25

u ( t ) = A 1 cos ωλt −

(16)

where A1 = A +

A3 ( −120C12 − 80C1 C 2 − 10C1 C 3 − 66C1 C 4 − 24C1 C 5 + 20C 22 + 65C 2 C 3 − 960(2C1 + C 2 )

 C1 (15ω 2 + 1)C 3 (3ω 2 + 1)C 2 − 28C 2 C 4 + 6C 2 C 5 + 40C 32 + 5C 3 C 4 + 18C 3 C 5 ) + a  + + + 4 2 4 2 1 − ω 2 9ω − 10ω + 1 225ω − 34ω + 1 +

(35ω 2 + 1)C 4 1225ω − 74ω + 1 4

2

+

(17)

  3969ω − 130ω + 1 (63ω 2 + 1)C 5 4

2

The parameters Ci, i=1,2,…,5 can be optimally identified from the system: R ( t 1 , C i ) = R ( t 2 , C i ) = .... = R ( t 5 , C i ) = 0

 2π  , t i ∈  0, , i = 1,2,...,5  ω

(18)

where R is the residual of the Eq.(1) obtained using the approximate analytical solution Numerical examples In the following, for a=0.1 and ω0=1.0709680693, we consider two cases: Case 3.1. For A=1, from Eqs.(19) we obtain ω= 1.3645826050978114 and C1 = −0.746577572; C 2 = 0.396564476; C 3 = −0.550707704; C 4 = 0.258696563; C 5 = −0.009647211

(19)

Case 3.2. For A=2, we obtain ω= 1.0243147906542391 and C1 = −0.350078184; C 2 = 0.551423747; C 3 = −0.535988983; C 4 = 0.161973152; C 5 = −0.018956307

(20)

At this moment, the first order approximate solution (16) is well determined, using the optimal values of the convergence-control parameters (19) and (20), respectively.

Fig.1. Comparison between the solutions of Eq.(1) for A=1: ____ numerical; _ _ _ _ analytical solution (17)

Fig.2. Comparison between the solutions of Eq.(1) for A=2: ____ numerical; _ _ _ _ analytical solution (17)

26

Acoustics & Vibration of Mechanical Structures

Figures 1-2 show a comparison between the present analytical solution and the numerical integration results obtained using a fourth-order Runge-Kutta method in the considered cases. One can be observed that the first-order approximate analytical results obtained by OHAM are nearly identical with numerical results in all considered cases for various values of the initial amplitude. Conclusions In this paper we have studied a massless beam with a lumped mass attached to its free end being excited at the base, which is a model for many engineering systems. The nonlinear ordinary differential equation is solved using OHAM. Our procedure has been proved to be effective and has some distinct advantages over some usual approximate methods in that it is valid for strongly nonlinear equations. The method is optimized to control the convergence of solutions through the auxiliary function such that the accuracy is always guaranteed. This approach proves to be very rapid, effective and accurate and this is proved by comparing the analytical solution with numerical integration results. By means of OHAM we obtained accurate approximate solutions after only one iteration, which proves that our procedure is very efficient in practice. References [1] W.H. Hoppman, Forced lateral vibration of beam carrying a concentrated mass, J. Appl. Mech. 19 (1952) 301-307. [2] S.P. Timoshenko, Vibration problems in engineering, Mc.Graw-Hill, New York, 1954. [3] A.A. Laura, J.L. Porubo, E.A. Susemihl, A note on the vibration of a clamped-free beam with a mass at the free end, J. Sound Vibr. 37 (1974) 161-168. [4] E. Esmailzadeh, G. Nakhaie-Jazar, Periodic behavior of a cantilever beam with end mass subject to harmonic base excitation, Int. J. of Non-Linear Mechanics, 33 (4) (1998) 567-577. [5] A.H. Nayfeh, D.T. Mook, Nonlinear oscillations, Willey, New York, 1979. [6] N. Bogoliubov, J.A. Mitropolsky, Asymptotic methods in the theory of nonlinear oscillations, Hindustan Publishing, Delhi, 1962. [7] R.E. Mickens, Iteration procedure for determining approximate solutions to nonlinear oscillator equations, J. Sound Vibr. 116 (1987) 185-188. [8] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal Appl. 135 (1988) 501-544. [9] H. Hu, A modified method of equivalent linearization that works even when the nonlinearity is not small, J. Sound Vibr. 276 (2004) 1145-1149. [10] A.V. Rao, B.N. Rao, Some remarks on the harmonic balance method for mixed-parity nonlinear oscillations, J. Sound Vibr. 170 (1994) 571-576. [11] N. Herişanu, V. Marinca, T. Dordea, G. Madescu, A new analytical approach to nonlinear vibration of an electrical machine, Proc. Rom. Acad. 9 (2008) 229-236. [12] V. Marinca, N. Herişanu, Nonlinear dynamical systems in engineering. Some approximate approaches, Springer, Berlin, 2011.

Applied Mechanics and Materials Vol. 430 (2013) pp 27-31 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.27

Optimal Homotopy Asymptotic Approach to self-excited vibrations Nicolae Herisanu1,2, a, Vasile Marinca1,2,b 1,2 1,2

“Politehnica” University of Timişoara, Bd.Mihai Viteazu, 1, 300222, Timişoara, Romania Center for Advanced Research in Engineering Sciences, Romanian Academy-Timişoara Branch, Bd. Mihai Viteazu, 24, 300223, Timişoara, Romania a

b

[email protected], [email protected]

Keywords: self-excited vibration, OHAM, nonlinear damped vibration.

Abstract. This paper is concerned with analytical treatment of nonlinear oscillation of a self excited system. An analytic approximate technique, namely OHAM is employed for this purpose. Our procedure provides us with a convenient way to optimally control the convergence of solutions, such that the accuracy is always guaranteed. An excellent agreement of the approximate solutions with the numerical ones has been demonstrated. Three examples are given and the results reveal that the procedure is very effective and accurate, demonstrating the general validity and the great potential of the OHAM for solving strongly nonlinear problems. Introduction Friction-induced self-excited vibration is known as a strongly nonlinear physical phenomenon encountered by some engineering structures [1]. From the point of view of practical applications, the variable friction force generates a periodic motion, which can cause serious problems to mechanical devices. Some examples of such self-excited systems are the automotive brake system [2], a mass-spring system placed on a belt moving along its length [3], machine-tool chatter [4]. The phenomenon is quite complex since the friction force depends on many parameters, such as the normal load, the nature of surfaces in contact, the materials properties, and obviously the slip velocity. In most of cases, this kind of self-excited vibration could be detrimental for the mechanical structures causing excessive wear of some parts, surfaces damage, undesirable positioning inaccuracies, fatigue failure and even high level noise. Therefore, if elimination of the vibration caused by friction force is not possible, then this phenomenon should be controlled and kept to such minimum amplitude which does not affect the integrity of the structure and does not cause degradation of system performances or other undesirable effects. Hence, from engineering viewpoints, a deep analysis having both theoretical and practical significance is needed in order to explore the characteristics of these phenomena to further understand and manage these problems aiming at finding means to control such unwanted vibration. For this purpose, the best way to perform such an analysis is an analytical approach, which should be valid for strongly nonlinear phenomena. However, prior to such an analysis, a friction model should be employed, but this is a relatively easy task since many classical and modern friction models are available in the literature [3], usually the friction force appearing as a nonlinear function of slip velocity up to cubic order. This paper is mainly focused on providing an analytical approach to the study of friction-induced self-excited vibration based on one of the newest analytical techniques, namely the Optimal Homotopy Asymptotic Method (OHAM). Alternative approaches applicable in such cases are presented in [5-7]

28

Acoustics & Vibration of Mechanical Structures

Formulation and solution approach We consider the model of a mass-spring system placed on a moving belt depicted in Fig.1. The corresponding equation for this system exhibiting self-excited vibrations may be written as [1]: x + 2cx + ω2 x + αx 2 + β x 3 = 0

(1)

with ω>c and with the initial conditions: Fig.1. Mechanical system

x (0) = 0, x (0) = v 0

(2)

where the dot denotes the derivative with respect to variable t, and c, ω, α, β, v0 are arbitrary parameters. Under the transformation x ( t ) = e − ct y( t ) and introducing a new independent variable τ = ωt , Eqs.(1) and (2) become, respectively y ′′ +

ω2 − c 2 y + αe − cτ ( y ′ − cy) 2 + βΩe − 2 cτ ( y ′ − cy) 3 = 0, 2 Ω

y(0) = 0, y ′(0) =

v0 Ω

(3)

where Ω is the frequency of the system, the prime denotes the derivative with respect to τ, and c = c / Ω . By the homotopy technique, we construct a homotopy in a more general form [6-8]:

 ∂ 2 φ ω2 − c 2 H(φ(τ, p, C i ) = (1 − p)L(φ(τ, p, C i ) ) − h (τ, p, C i )  2 + φ+ Ω2  ∂τ 2 3  ∂φ   ∂φ   + α e − c τ  − c φ  + β Ωe − 2 c τ  − c φ   = 0  ∂τ   ∂τ  

(4)

∂ 2φ where h is a convergence-control function, Lφ = 2 + φ and φ(τ, p, C i ) = y 0 (τ) + py1 (τ, C j ) . Eq.(4) ∂τ becomes  ω2 − c 2 2 ′ ′ Ly 0 (τ) + p{Ly1 (τ) − Ly 0 (τ) − h (τ, C i )  y 0 + y 0 + αe − cτ (y′0 − cy 0 ) + 2 Ω (5) 

(

+ β Ωe − 2 cτ y ′ − c y ) 3

)]} = 0.

By equating the coefficients of like powers of p to zero, from Eq.(5) we obtain respectively: y ′0′ + y 0 = 0, y 0 (0) = 0, y ′0 (0) = v 0 / Ω

(6)

 ω2 − c 2 2 3 y1′′ + y1 = h 1 (τ, C i )  y′0′ + y 0 + αe − cτ (y′0 − cy 0 ) + βΩe − 2 cτ (y′0 − cy 0 ) , y1 (0) = y1′ (0) = 0 (7) 2 Ω   Eq.(6) has the solution v y 0 (τ) = 0 sin τ. Ω

(8)

This result is substituted into Eq.(7) and it is assumed that

h 1 (τ, C i ) = C1 + 2C 2 cos τ + 2C 3 sin τ + 2C 4 e − cτ cos τ + 2C 5 e − cτ sin τ

(9)

Applied Mechanics and Materials Vol. 430

29

where Ci, i=1,2,3,4,5 are unknown parameters. No secular terms in y1 (τ) requires Ω 2 = ω 2 − c 2 . In this case, the solution of Eq.(7) becomes y1 (τ, C i ) = A cos τ + B sin τ + e − cτ [k 11C1 +(k 12 C 2 + k 13 C 3 ) cos τ + (k 14 C 2 + k 15 C 3 ) sin τ + k 16 C1 cos 2τ + k 17 C1 sin 2τ + (k 18 C 2 + k 19 C 3 ) cos 3τ + (k 20 C 2 + k 21C 3 ) sin 3τ] + + e − 2 cτ [k 22 C 2 + k 23 C 3 + (k 24 C 4 + k 25 C 5 ) cos τ + (k 26 C1 + k 27 C 4 + k 28 C 5 ) sin τ + + (k 29 C 2 + k 30 C 3 ) cos 2τ + (k 31C 2 + k 32 C 3 ) sin 2τ + (k 33 C1 + k 34 C 4 + k 35 C 5 ) cos 3τ +

(10)

+ (k 36 C1 + k 37 C 5 + k 38 C 5 ) sin 3τ + (k 39 C 2 + k 40 C 3 ) cos 4τ + (k 41C 2 + k 42 C 3 ) sin 4τ] + + e −3cτ [(k 43 C 4 + k 44 C 5 ) cos 2τ + (k 45 C 4 + k 46 C 5 ) sin 2τ + (k 47 C 4 + k 48 C 5 ) cos 4τ + + (k 49 C 4 + k 50 C 5 ) sin 4τ] where k 11 =

k 15 =

αv 0 2Ω 2

, k 12 =

α ( c 2 − 1) v 02 2( c 2 + 4) Ω 2

α (3c 2 + 5) v 02 2( c 2 + 4) Ω 2

k 18 = k 21 = − k 23 = − k 29 = − k 32 = −

2( c 4 + 20 c 2 + 64)Ω 2

(16 c 4 + 40 c 2 + 9)Ω 2

2(16 c 4 + 40 c 2 + 9)Ω 2

k 35 = − k 37 = −

k 43 = −

8( c 2 + 4)Ω 2

8Ω 2

, k 30 =

(4 c 4 + 5c 2 + 3)β v 30 cαv 02

αv 02

3(9 c 2 + 1)Ω 2

k 47 = k 50 = −

, k 44 =

( c 2 + 5)β v 30 12(9 c 2 + 25)Ω 2

, k 26 = −

16cΩ 2

8( c 2 + 4)Ω 2

16( c 2 + 4)Ω 2

3(1 + c 2 )βv 30 4(1 + 4 c 2 )Ω 2

, k 27 = −k 28 = −

, k 31 = −

3αv 02 8c Ω 2

,

(4 c 5 + 9 c 3 + 7 c )β v 30 2(16 c 4 + 40 c 2 + 9)Ω 2 ( c 2 + 2)αv 02 8( c 2 + 4)Ω 2

,

,

,

,

, t k 45 = −

, k 48 = − k 49 = −

, k 22 =

, k 34 = − k 38 = −

, k 40 = −k 41 = −

3(9 c 2 + 1)Ω 2

,

,

( c 2 + 9) Ω 2

3( c 2 + 1)β v 30

βv 30

(3c 3 + 2 c )β v 30

c ( c 2 + 4) Ω 2

( c 4 + 20 c 2 + 64)Ω 2

c ( c 2 + 3)β v 30

4(16 c 4 + 136 c 2 + 225)Ω 2

(2 c 2 + 1)β v 30

, k 17 =

(16 c 4 + 40 c 2 + 9)Ω 2

(4 c 4 + c 2 − 15)βv 30

α (3 + 2 c 2 ) v 02

cαv 02

(4 c 5 + 9 c 3 + 4 c )β v 30

, k 33 = −

, k 36 =

, k 14 = −

(2 c 2 + 5) cαv 02

, k 19 = k 20 =

, k 24 = k 25 =

(4 c 4 + 8c 2 + 3)βv 30

c ( c 2 + 4) Ω 2

2( c 4 + 10 c 2 + 9)Ω 2

( c 4 + 3c 2 + 8)αv 02

4(1 + 4 c 2 )Ω 2

α(1 + 2 c 2 ) v 02

( c 4 + 2 c 2 + 3)αv 02

, k 16 = −

3c (1 + c 2 )β v 30

k 39 = k 42 =

, k 13 =

(4 c 5 + 21c 3 + 29 c )β v 30 4(16 c 4 + 136 c 2 + 225)Ω 2 (3c 3 + 5c )β v 30 3(9 c 2 + 1)Ω 2

, k 46 =

,

( c 2 − 1)β v 30 6(9 c 2 + 1)Ω 2

(3c 3 + 7 c )β v 30 12(9 c 2 + 25)Ω 2

, (11)

A = −C1 (k 11 + k 16 + k 33 ) − C 2 (k 12 + k 18 + k 22 + k 38 + k 39 ) − C 3 (k 13 + k 29 + k 23 + k 30 + k 40 ) − − C 4 (k 24 + k 31 + k 45 + k 47 ) − C 5 (k 25 + k 35 + k 44 + k 48 ) B = C1 ( ck 11 + ck 10 + 2 ck 33 − 2k 17 − k 26 − 3k 36 ) + C 2 ( ck 12 + ck 18 + 2 ck 21 + 2 ck 29 + + 2 ck 39 − k 14 − 3k 20 − 2k 31 − 4k 41 ) + C 3 ( ck 13 + ck 19 + 2 ck 23 + 2 ck 30 + 2 ck 40 − k 15 − − 3k 21 − 2k 32 − k 42 ) + C 4 (2 ck 24 + 2 ck 34 + 3ck 43 + 3ck 47 − k 27 − 3k 37 − 2k 45 − 4k 49 ) + + C 5 (2 ck 25 + 2 ck 35 + 3ck 44 + 3ck 48 − k 28 − 3k 38 − 2k 46 − 4k 50 ).

(12)

30

Acoustics & Vibration of Mechanical Structures

The first-order approximate solution is obtained as y(τ) = y 0 (τ) + y1 (τ) where y 0 (τ) is given by Eq.(8) and y1 (τ) is given by Eq.(10). In this way, Eq.(1) has the first-order approximate solution: x ( t ) = e − ct [ y 0 (Ωt ) + y1 (Ωt )].

(14)

At this moment, the m-th order approximation depends on the parameters C1, ….,Cm, which can be identified via the least square method, the Galerkin method, the collocation method or by minimizing the residual error. Our procedure contains the auxiliary functions h1, h2,…which provide us with a simple way to control the convergence of solutions. More details are presented in [8-10].

Results and discussions We illustrate the accuracy of our procedure by comparing previously obtained approximate solutions with the numerical integration results obtained by means of a fourth-order Runge-Kutta method choosing ω=1, α=0.3, β=0.2, v0=1. Case 4.1. In the first case we consider c=0.1. The obtained convergence-control parameters are C1 = −0.615645245; C 2 = −0.014405822; C 3 = 0.004469578; C 4 = −0.023909716; (15) C 5 = −0.0032313661.

Case 4.2. For c=0.15, we obtain: C1 = −0.782970697 ; C 2 = 0.050267369 ; C 3 = 0.074509591; C 4 = −0.033367363;

C 5 = −0.021901109 . Case 4.3. In the last case we consider c=0.2 and therefore C1 = −0.846937369; C 2 = 0.040996181; C 3 = 0.053505381; C 4 = −0.034823395; C 5 = −0.027939191.

Fig.2. Comparison between analytical and numerical solutions for c=0.1

(16)

(17)

Fig.3. Comparison between analytical and numerical solutions for c=0.15

Fig.4. Comparison between analytical and numerical solutions for c=0.2

Applied Mechanics and Materials Vol. 430

31

From Figs.2-4 it can be seen that the solutions obtained by the proposed procedure (dashed line) are nearly identical with numerical solutions (continuous line), which proves the accuracy of the method.

Conclusions In this paper we employed OHAM to propose analytic approximate solutions to nonlinear selfexcited vibration. This procedure is optimized to control the convergence of solutions through some convergence-control functions such that the accuracy is always guaranteed, even if the nonlinear equation does not contain any small or large parameters. Excellent agreement between the analytical and numerical solutions has been demonstrated on several examples. This approach proves to be very rapid, effective and accurate, since excellent results were achieved after only one iteration. The proposed procedure can easily be used to find analytical approximate solutions to other strongly nonlinear dynamical systems. The obtained analytical results could be of considerable practical value from engineering point of view for further controlling friction-driven self-excited vibration and for designing better devices.

References [1] Gh.Silas, M.Rădoi, L.Brîndeu, M.Klepp, A.Hegedus, Culegere de probleme de vibraţii mecanice (in Romanian), Ed. Tehnică, Bucureşti, 1973. [2] J.-J. Sinou, Transient non-linear dynamic analysis of automotive disc brake squeal – On the need to consider both stability and non-linear analysis, Mech. Res. Commun. 37 (2010) 96–105. [3] Albert C.J. Luo, Jianzhe Huang, Discontinuous dynamics of a non-linear, self-excited, frictioninduced, periodically forced oscillator, Nonlinear Anal. Real World Appl. 13 (2012) 241–257. [4] M. Siddhpura, R.Paurobally, A review of chatter vibration research in turning, Int. J. Mach. Tools Manufact. 61 (2012) 27–47. [5] J. Awrejcewicz, P. Olejnik, Analysis of dynamic systems with various friction laws, Appl. Mech. Rev. 58 (2005) 389-411. [6] C.W. Lim, B.S. Wu, A new analytical approach to the Duffing-harmonic oscillator, A new analytical approach to the Duffing-harmonic oscillator, Phys. Let. A. 311 (2003) 365-373. [7] B.S. Wu, C.W. Lim, Large amplitude nonlinear oscillations of a general conservative system, Int. J. Non-Linear Mech. 39 (2004) 859-870 [8] V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of Optimal Homotopy Asytmptotic Method, J. Sound Vibr. 329 (2010) 1450-1459. [9] N. Herisanu, V. Marinca, Accurate analytical solutions to nonlinear oscillators with discontinuities and fractional-power restoring force by means of the Optimal Homotopy Asymptotic Method, Comp. Math. Appl. 60 (2010) 1607-1611. [10] V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering. Some Approximate Approaches, Springer, Berlin, 2011.

Applied Mechanics and Materials Vol. 430 (2013) pp 32-39 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.32

Physical Instability and Functional Uncertainties of The Dynamic Systems In Resonance BRATU Polidor1, 2, a 1

ICECON S.A., Bucharest, Sos. Pantelimon 266, sector 2, 021652, CP 3-33, Romania

2

Technical University “Dunărea de Jos”, Galati, Str. Domnească, nr. 47, 800008, Romania a

[email protected]

Keywords: dynamic system, instability, uncertainty, resonance

Abstract. Depending on the type of systems, the resonance states are either short and without inducing structural changes, or lead to forced variations of the elasticity and damping parameters with the additional energy absorption. For the material physical systems, that simulate the dynamic behavior of equipments, industrial plants and constructions, the operation in resonance mode is unstable being characterized by pulse variations in the band-pass. In this case, the necessary energy input, from outside, when the operation state is maintained within the band-pass, inevitably leads to the modification of the elastic and damping parameters and to structural degradation processes appearance. Thus the resonance band, considered as a significant parameter of the excitation, and the specific relaxation duration, considered as an intrinsic system parameter, are in a functional correlation with the dynamic system. When the dissipation increases, the relaxation duration drops in a way similar to the Heisenberg indeterminacy formula for subatomic particles. In the ideal case, the resonance interval is reduced to a single value that corresponds to its own pulsation. The systems dynamic behavior generated by periodic functions resulting in either forced vibrations or wave propagation processes in viscoelastic media is characterized by doubtful operation and physical instability within resonance. Introduction The need of forced, stationary and stable regimes for achieving high-performance technologies has imposed the technical solution of parametric adjustments for functioning in resonance. The goal of attaining and maintaining the technological resonance, using vibrating equipment modeled as second-order dynamic systems, has led to numerous studies, models and analyses without converging to an undoubted solution. The previous experimentations of the author, performed in the framework of researches meant to optimize the technological performances depending on the vibratory regime, have led to the conclusion that no system remains in resonance by itself. In case the system is maintained at the resonance point by special interventions on the engine, after a number of cycles the system “degrades” either by successive reductions of the damping coefficients c, or by changes of the stiffness k. The tests performed on several equipments, using appropriate real time measuring instrumentation, have shown an uncertain behavior at resonance, either concerning the displacement of the functioning resonance point (towards pre or post-resonance), or concerning the degradation of some “sacrifice” elements until their total damage. The “sacrifice” elements most likely to be damaged are the electric motor drive, the transmission gear, the belts drive, the viscoelastic links, etc. In these cases the equipments show a physical functional instability at resonance, which is unacceptable for a guaranteed, performant system. The technological equipments that have been tested are characterized by an important installed power, corresponding to actuation forces of over 10 kN, while the dynamical subassemblies are excited by forces of 10 ... 200 daN, in the frequency range of 8 ... 50 Hz. The tests have been

Applied Mechanics and Materials Vol. 430

33

performed on site, on the following technological equipments: vibrating sieves, vibrating grinding mills, vibratory rollers, vibratory pile driving machines, concrete vibrators, etc Dynamic Regime at Resonance Two dynamic excitation methods for vibrating equipments are considered here: 1. using applied forces of the form F (t ) = F0 sin ωt , with F0 = const. ; 2. using rotating eccentric masses, the applied force being F (t ) = m0 r ω 2 sin ωt , where the force amplitude is proportional to the square of the angular velocity ω. In this paper, the operating element of the vibrating equipment is modeled as an one-degree-offreedom system, with mass m and with the viscoelastic links k,c specific to the external medium (material) with which the operating element comes into contact for the required operating process. The following modeling and analysis hypothesis is used: the viscoelastic external medium can modify its parameters k = k (t ) or c = c(t ) , as well as both parameters k (t ) and c(t ) simultaneously. Thus, in the resonance vicinity the functioning point may “slip”, more precisely ωn = ω0 1 − 2ζ 02 is the natural angular frequency in the presence of damping, where ω 0 =

k0 is the angular frequency m

in the absence of damping, k 0 is the initial stiffness of the system, m is the mass of the operating element, ζ 0 =

c0 2 k 0m

is the fraction of the critical damping in the ideal state of maintaining

resonance, while ∆ω n is the variation of the angular frequency due to the modification of characteristic parameters c and k. This study is performed for the two types of external excitations listed above, for which are determined the system responses in terms of displacements, velocities, accelerations, specific energies and powers. Another issue is to determine the influence of the unstable functioning in the resonance vicinity as a consequence of the modification of parameter c = c(t ) , due to the technological process as expression of the parametric modification and of the completion of the technological process. In order to evidence the state of resonance, in the analyzed frequency band, two types of perturbing forces are used, with the same pulsation ω, harmonic, but with different amplitudes with respect to ω. Thus, in the first case the amplitude of the perturbing force F0 is constant (Fig 1.a) and in the second one the amplitude of the perturbing force is variable with ω, meaning F0= m0r ω2. (Fig 1.b). Linear Dynamic Models. Parametric Variations For the two types of external excitations listed above, Fig. 1 shows the dynamic models with variable c = c(t ) .

a) b) Fig 1. Dynamic models of the operating element of the vibrating equipment a) The force amplitude is F0=ct b) The force amplitude is variable with ω2

34

Acoustics & Vibration of Mechanical Structures

Stationary Response For the two dynamic models in Fig. 1, the functional regime can be characterized by harmonic functions in the time domain and by amplification functions in the frequency domain as amplitude continuous spectra, as follows: 1. For the dynamic excitation using applied forces of the harmonic form F (t ) = F0 sin ωt : A1 ( Ω, ζ ) = ∆ 0 H ( Ω, ζ ) ,

(1)

where A1 is the response function in the domain of the relative frequency Ω , H ( Ω, ζ ) is the amplitude function in dynamic regime for Ω > 0 and ∆ 0 =

F0 is the equivalent static strain of the k

elastic system for the case when the force F0 is applied slowly, quasi-statically. 2. For the dynamic excitation using rotating eccentric masses, the applied force is F (t ) = m0 r ω 2 sin ωt and the response with respect to time is as follows: A2 ( Ω, ζ ) =

m0 r 2 Ω H ( Ω, ζ ) , m

(2)

m0 r = A∞stab is the amplitude in post-resonance, which has an asymptotically stable m character only if Ω > ( 4… 6 ) .

where

L ( Ω, ζ ) = Ω 2 H ( Ω, ζ )

(3)

is the amplitude function in dynamic regime with respect to Ω > 0 , with spectral character. By taking into account the previous notations, the amplitude A2 can be written as follows: A2 ( Ω, ζ ) = A∞stab L ( Ω, ζ ) .

(4)

Representation of the Curves Let us consider the influence of the variation of the viscosity coefficient of the material (external medium) on which operates the operating element of the equipment, in vibratory regime. In this case, one must search the variation curves of functions H (ζ ) and L(ζ) for discrete values of  2 Ω1 = 0,8… 0,95 and Ω 2 = 1,05…1,2 , respectively, and for the continuous variation of ζ ∈ 0; .  2 

The families of curves are plotted in Fig 2.

a) b) Fig. 2 Variation Curves of Functions H (ζ ) and L(ζ) with respect to the continuous variation of ζ and the discreet parameter Ω a) For Ω 1;

Applied Mechanics and Materials Vol. 430

35

Resonance State For each system and for the given dynamic excitations, the resonance of the system response in instantaneous displacement can be attained when simultaneously ω = ωn and A ( ω, ζ ) = max A ( ω = ω n ) , i.e., when the displacement amplitude attains a maximum value.

rez For the system excited by F (t ) = F0 sin ωt , the maximum response to resonance H max ( ζ ) with respect to the variation of ζ is as follows: rez H max (ζ) =

1

,

2ζ 1 − ζ 2

(5)

rez while the maximum response to resonance H max ( Ω ) with respect to Ω is given by: rez H max (Ω) =

1

.

1− Ω4

(6)

The graphical representations of functions 5 and 6 are shown in Fig. 3

a) b) Fig.3 Response to Resonance at Dynamic Excitation with F0=ct a) The curve which describes the locus of the points of maxima for the function Hmax(Ω), in anteresonance; b) Response curves H=H(Ω,ζ) for the continuous variation of Ω and of the discrete variation of ζ; For the system with inertial excitation, where the applied force is F (t ) = m0 r ω 2 sin ωt , the rez maximum responses to resonance Lrez max (ζ ) with respect to the variation of ζ and L max (Ω) with respect to Ω are as follows: Lrez max (ζ ) =

Lrez max (Ω) =

1 2ζ 1 − ζ 2 Ω2 Ω4 −1

,

.

Fig. 4 presents the graphical representations of functions 7 and 8.

(7)

(8)

36

Acoustics & Vibration of Mechanical Structures

a) b) Fig. 4 Response to Resonance at Dynamic Excitations with F0= m0r ω2 a) The curve which describes the locus of the points of maxima for the function Lmax(Ω), in postresonance; b) Response curves L=L(Ω,ζ) for the continuous variation of Ω and of the discrete variation of ζ; Dissipated Energy For the two cases of dynamic excitation studied in this paper, since the responses in instantaneous displacements are different, it results that the dissipated energy has the same mathematical expression. It will be different under numerical aspect due to the different values of the amplitudes of the excitation forces. Thus, the distinct final parametric relations of the dissipated energy for the two dynamic excitation cases are as follows: Wd1 = 2π∆ 02 k ζ Ω H 2( ζ, Ω )

(9)

Wd 2 = 2π ( A∞stab ) k ζ Ω L2( ζ, Ω ) 2

(10)

For the cases Ω < 1 (pre-resonance) and Ω > 1 (post-resonance), the variations of the dissipated energy are given by:

(

)

(

)

WdΩ1 Ω, ζ j = 2π∆ 02 k ζ j Ω H 2 Ω, ζ j , 2

(11)

m r WdΩ2 Ω, ζ j = 2π  0  k ζ j Ω L2 Ω, ζ j ,  m 

(

)

(

)

(12)

where ζ j , j =1,2,3,4 is the dissipative parameter with discrete variations, while Ω is the current value of the relative frequency, with values in the range Ω ∈ [0,6] . The study of the case considered here gives the following numerical values of the system parameters: 1 m0 = 10 kg ; m 0 r = 1 kg ⋅ m ; 2 k = 8 ⋅ 10 4 kN m ; m = 2 ⋅ 10 3 kg ; F0 = 50 kN ; ζ 1 = 0.225 ; ζ 2 = 0.163 ; ζ 3 = 0.105 ; ζ 4 = 0.051 . The functions Wd1 and Wd 2 , parameterized by ζ j , j =1,2,3,4, are plotted in Fig 5. r = 0.05 m ;

Applied Mechanics and Materials Vol. 430

37

a) b) Fig.5 Variation curves of the dissipated energy function of the relative pulsation Ω and the discrete parameter ζ; a) For dynamic excitation with the force F0=ct; b) For dynamic excitation with the force F0= m0r ω2. The dissipated energy with respect to the variation of ζ and parameterized by Ω i , i = 1,6 , for the two cases of systems, can be expressed as follows: Wdζ1 ( ζ, Ω i ) = 2π∆ 02 k ζ Ω i H 2 ( ζ, Ω i ) ,

(13)

2

m r Wdζ2 ( ζ, Ω i ) = 2π  0  k ζ Ω i L2 ( ζ, Ω i ) ,  m 

(14)

with the following Ω i , i = 1,6 : Ω1 = 0.2 ; Ω 2 = 0.5 ; Ω 3 = 0.8 ; Ω 4 = 0.85 ; Ω 5 = 0.9 ; Ω 6 = 0.95 . The graphical representation of the families of curves corresponding to the parametric relations 13 and 14 is given in Fig 6.

a) b) Fig.6 Variation Curves of the dissipated energy function of Ω and ζ; a) For dynamic excitation with the force F0=ct; b) For dynamic excitation with the force F0= m0r ω2. Uncertainty and Instability The resonance bandwidth is given by: 1 ∆ω = , τ

where τ =

(15) m is the relaxation time for a system defined by m and c. c

38

Acoustics & Vibration of Mechanical Structures

If c = c(t ) is variable, more precisely increasing from c 0 to c(t ) , i.e., from c 0 = 2ζ 0 k m to c = 2ζ k m , then it results ∆c = c − c 0 = 2∆ζ km , also ∆τ =

m 1 and ∆ω = ∆c , and eq. 25 becomes ∆c m

∆ω⋅ ∆τ = 1 .

(16)

Equation 16 represents the uncertainty relation at resonance, being similar to Heisenberg's uncertainty relation for subatomic particles. This uncertainty equation 16 points out the following cases: for ∆τ → 0 , we have ∆c → ∞ , which means increasing values of c, respectively ∆τ > 0 , τ − τ 0 > 0 or τ >> τ 0 . In this case, it results: ∆ω =

1 →∞. ∆τ

(17)

This means that the frequency bandwidth is very large, thus the system is able to function on any of the frequencies in the range [ω1 , ω 2 ] with ∆ω = ω 2 − ω1 for the amplitude A1 = A2 =

2 Arez . 2

for ∆τ → ∞ , we have ∆c → 0 , i.e. ζ − ζ 0 = 0 , ζ → ζ 0 (the smallest value), which can also be written as: ∆ω =

1 → 0. ∆τ

(18)

This means that the frequency bandwidth is very narrow, with ω − ω 0 = 0 or ω = ω 0 , the oscillator being unique and ideal when ζ 0 → 0 . Conclusions For oscillating systems with technological motions maintained by dynamic excitations, the resonance regime cannot be maintained due to the following reasons: if the parameters m, c, k are rigorously constant, then the energy required at resonance has high values, thus overloading the energy source, which usually breaks down, leading to the “slip” of the functioning point; if the parameters are variable due to the functioning process, usually the damping increases, which enlarges the working bandwidth on possible frequencies, leading to an unstable functioning of the system on any intermediate frequency composed between ω1 and ω 2 with ω 0 =

k . m

The resonance of the systems cannot be rigorously maintained due to the occurrence of malfunctions, instabilities and major defects caused by structural damage. References [1] JP. Bandstra, Comparison of equivalent viscous damping and nonlinear damping in discret and continuous vibrating system, Journal of Mechanisms, Transmissions and Automation in Design, J.V.A.S.R.D., vol I 105, 1983 [2] P. Bratu, Elastic bearing systems for machines and equipment, 260 pages, Technical Publishing House, 1990. [3] P. Bratu, Evaluation of the dissipation energy capacity inside damping systems in neoprene elements, The 16th International Congress on Sound and Vibration ICSV16, Kraków, Poland, July 5-6, 2009.

Applied Mechanics and Materials Vol. 430

39

[4] P. Bratu, Analysis of the antivibrating elastic systems having amplified static deflection, Proceedings of the National Conference THE ACADEMIC DAYS OF the Academy of Technical Sciences in Romania, Timişoara, May, 2005. [5] P. Bratu, Rigidity and damping characteristics in case of composite neoprene systems due to passive vibration isolation, Proceedings of the Annual Symposium of the Institute of Solid Mechanics SISOM’06, Bucharest, Romania, May, 2006. [6] TK. Caughey, MEJ. O’Kelly, Effect of damping on the natural frequencies on linear dynamic systems, Journal of the American Statistical Associating, J.A.S.A., vol. 33, 1961 [7] I. Cochin, Analysis and Deisign of Dynamic Systems, Harper&Raw, New York, 1980 [8] RW. Fox, AT. McDonald, Introduction to Fluid Mechanics (4th ed.), Wiley, New York, 1992 [9] E. Kreyszig, Advanced Engineering Mathematics (7th ed.), Wiley, New York, 1993 [10] S. Rao, Mechanical Vibrations, Addison-Wesley, New York, 1995

Applied Mechanics and Materials Vol. 430 (2013) pp 40-44 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.40

Approximate analytical solutions of nonlinear vibrations of a thin elastic plate R.-D. Ene1,a, V. Marinca2,3b and B. Marinca4,c 1

Department of Mathematics, “Politehnica” University of Timişoara, Pta Victoriei, No.2, 300006, Timisoara, Romania . 2 Department of Mechanics and Vibration, “Politehnica” University of Timişoara, Bd. M. Viteazu, No.1, 300222, Timişoara, Romania, 3 Center for Advanced and Fundamental Technical Research, Romanian Academy, Timişoara Branch, Bd.Mihai Viteazu, Nr.24, 300223, Timişoara, Romania. 3

Department of Electronics and Telecommunications, “Politehnica” University of Timişoara. a

[email protected], [email protected], [email protected]

Keywords: Optimal homotopy asymptotic method, nonlinear partial differential equations, thin elastic plate.

Abstract. In this paper we consider the propagation equation of the longitudinal elastic wave in the presence of the volume forces, taking into account the shear phenomena of a thin elastic plate. In order to find approximate analytical solutions of the governing system we apply Optimal Homotopy Asymptotic Method (OHAM). This technique combines the features of the homotopy concept with an efficient computational algorithm which provides a simple and rigorous procedure to control the convergence of the solution. An excellent agreement is found between the results obtained using OHAM and numerical integration results. Introduction Thin elastic plates have become more and more important in the latest engineering technologies. Thus silicon plates are used in nanotechnology researchers [1], alumina and zirconia plates [2] find use in the manufacture of fuel cell electrolyte substrates [2]. The mathematical modeling of most of the natural and physical processes leads to such nonlinear partial differential equations whose analytic solutions are hard to find. Many researchers are interested in the domain of elasticity. Rajagopal and Wineman [3] established several new exact solutions to boundary value problems in nonlinear elasticity. Apostol [4] showed that anharmonic corrections to the elastic energy may lead to unphysical solutions for longitudinal deformations such that the third-order anharmonic terms are defined as the continuum limit of the Fermi-PastaUlam equation. Bokhari et al. [5] showed that a similarity analysis of a nonlinear wave equation in elasticity can be tacked using the group theoretical method. Mustafa and Masood [6] applied the Lie symmetry method to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Approximate symmetries and prolongation technique were used to carry out symmetry analysis of some cases of such nonlinear wave equations by Alfinito et al. in [7]. In this paper, OHAM [8], [9], [10] is proposed to analytically solve nonlinear problems with particular emphasis on stress and deformation states of a thin elastic plate. The efficiency of our procedure, which does not require a small parameter in the equation, is based on the construction and determination of the auxiliary function combined with a convenient way to optimally control the convergence of the solution. In this paper, we consider the longitudinal elastic wave equation [3], [11], [12], [13], [14], [15]: u − 2πv12 (sin 4πx cos 4πv 2 t + sin 8πx ) − v12 (u′′ + u ′u′′) = 0.

(1)

Applied Mechanics and Materials Vol. 430

41

λ + 2µ µ , v 22 = , λ and µ are the Lame’s elastic constant, ρ is density of medium, ρ ρ ∂u ∂u u = , u′ = , t ∈ [0, 1], x ∈ [0, 1] and the initial and boundary conditions: ∂t ∂x

where v12 =

u ( x , 0) =

1 cos 4πx; u ( x , 0) = 0; u′(0, t ) = u′(1, t ) = 0 4π

(2)

The main results using OHAM In order to apply the OHAM technique we rewrite Eq. (1) in the form L(u ( x , t )) + N (u ( x , t )) = 0 where the linear and nonlinear operators are respectively

(3)

L(u ( x , t )) = u ( x , t )

(4)

N(u ( x , t )) = −2πv12 (sin 4πx cos 4πv 2 t + sin 8πx ) − v12 (u ′′ + u′u′′)

(5)

By using OHAM, we construct the following family of equations corresponding to Eq. (3):

(1 − p)L(u ( x, t )) + H( x, t , p, Ci )[L(u ( x, t )) + N(u ( x, t ))] = 0 where p is an embedding parameter p ∈ [0, 1] with the auxiliary function H of the form

(6)

H( x , t , p, Ci ) = pH1( x , t , p, Ci ) + p 2H 2 ( x , t , p, Ci ) + … (7) where the component H1, H 2 , … ,depend of unknown parameters Ci , i = 1, 2, 3, … . By means of the OHAM, we seek the formal approximate solution of the m th-order of the Eq. (1) of the form ~ u ( x , t ) = u 0 ( x , t ) + u1( x , t , Ci ) + u 2 ( x , t , Ci ) + … + u m ( x , t , Ci ).

(8)

Substituting Eqs. (4), (5), (7) and (8) into Eq. (6) and identifying the coefficients of the p0 and p1 ( m = 1 ), we obtain the following linear set of equations: u 0 ( x , t ) = 0, u 0 ( x ,0) =

1 cos 4πx , u 0 ( x ,0) = 0, u′0 (0, t ) = 0, u′0 (1, t ) = 0 4π

u1( x, t , Ci ) + H1( x , t , Ci )[u 0 − 2πv12 (sin 4πx cos 4πv 2 t + sin 8πx ) − v12 (u′0′ + u′0u′0′ )] = 0, u1( x,0) = u1( x,0) = u1′ (0, t ) = u1′ (1, t ) = 0. From Eq. (9) we obtain u 0 (x, t ) =

1 cos 4πx 4π

(9)

(10)

(11)

Substituting Eq. (11) into Eq. (10) and of we choose the auxiliary function H1( x, t , Ci ) of the form H1( x , t , Ci ) = 2C1 sin 2πx + 4C 2 sin 2πx cos 2πv 2 t + 4C3 sin 2πx cos 6πv 2 t + + 4C4 sin 2πx cos 8πv 2 t

(12)

42

Acoustics & Vibration of Mechanical Structures

where C1, C2 , C3 and C4 are unknown parameters. In this case Eq. (10) has the solution: u1( x , t , Ci ) = 2πv 22C1(cos 6πx − cos10πx ) t 2 + (1 − cos 2πv 2 t )[(C2 + C3 ) cos 2πx + 1 + (C 2 − C3 ) cos 6πx − 2C2 cos10πx ] + (C1 + C4 )(1 − cos 4πv 2 t )(cos 2πx − cos 6πx ) + 4 1 + (1 − cos 6πv 2 t )[C2 cos 2πx + (2C3 − C2 ) cos 6πx − 2C3 cos10πx ] + 9 1 1 + C4 (1 − cos 8πv 2 t )(cos 6πx − cos10πx ) + C4 (1 − cos12πv 2 t )(cos 2πx − cos 6πx ). 8 36

(13)

The first-order approximate solutions of Eq. (1) with the initial and boundary conditions (2) is obtained from Eq. (8) for m = 1 : ~ u ( x , t , Ci ) = u 0 ( x , t ) + u1( x , t , Ci )

(14)

where u 0 and u1 are given by Eqs. (11) and (13) respectively. The parameters C1, C2 , C3 and C4 can be optimally identified via various methods, such as the least square method, the Galerkin method, the collocation method and so on. With these parameters known (namely convergence-control parameters) the first-order approximate solution is well determined. Our procedure contains the auxiliary function H( x, t , Ci ) which provides us with a simple way to adjust and control convergence of the solution.

Numerical results We illustrate the accuracy of this approach by comparing previously obtained approximate solution (14) with numerical integration results computed by means of the Wolfram Mathematica 6.0 software. In Tables 1 and 2 and Figs. 1 and 2 we present comparison between the approximate solution (14) and numerical results in the cases of three values of t: t1 = 0.15 , t 2 = 0.75 .

x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Table 1. Numerical date for the time moment t1 = 0.15 Numerical Approximate Relative error solution solution ε = u numeric − u OHAM u numeric ( x ,0.15) u OHAM ( x ,0.15) 0.0794589 0.0795775 0.000118563 0.0247573 0.0245908 0.000166524 -0.0642142 -0.0643795 0.000165284 -0.0642547 -0.0643795 0.000124816 0.0243142 0.0245908 0.000276648 0.0793993 0.0795775 0.000178205 0.0247573 0.0245908 0.000166508 -0.0642142 -0.0643795 0.000165308 -0.0642547 -0.0643795 0.000124792 0.0243141 0.0245908 0.000276635 0.0793385 0.0795775 0.000238988

Applied Mechanics and Materials Vol. 430

43

Fig. 1 Comparison between approximate solution (14) , and numerical results for t1 = 0.15 ―― numerical results ······ approximate results given by Eq. (14)

x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Table 2. Numerical date for the time moment t 2 = 0.75 Approximate Relative error Numerical solution solution ε = u numeric − u OHAM u numeric ( x ,0.75) u OHAM ( x ,0.75) 0.0766455 0.0795775 0.00293194 0.0287429 0.0245901 0.0041528 -0.0602664 -0.0643791 0.00411267 -0.0614451 -0.06438 0.00293494 0.0178819 0.0245915 0.00670959 0.0750879 0.0795775 0.0044896 0.0287419 0.0245915 0.00415048 -0.0602664 -0.06438 0.00411362 -0.0614451 -0.0643791 0.00293399 0.0178814 0.0245901 0.00670874 0.0733952 0.0795775 0.00618228

Fig. 2 Comparison between approximate solution (14) , and numerical results for t 2 = 0.75 ―― numerical results ······ approximate results given by Eq. (14) The parameters C1, C2 , C3 and C4 were determined by means of the least square method: C1 = −10,2082163915435; C2 = 5,4429668642543; C3 = −0,0000333375486425; C4 = −0,338849005021. It is easier to emphasize the accuracy of the obtained result: solution obtained by the present method is nearly identical with that given by the numerical ones. The relative error between the approximate and numerical solutions varies between 1.1 *10− 4 and 6.7 *10−3 .

44

Acoustics & Vibration of Mechanical Structures

Conclusions In this paper, are used OHAM to propose an analytic approximate solution to a nonlinear problem related to the stress and deformation states of a thin elastic plate. The proposed construction of the homotopy is different from other approaches, especially referring to the auxiliary function H( x , t , p, Ci ) and the presence of some parameters Ci which ensure a rapid convergence of the solutions. Our procedure is valid even if the nonlinear equation does not contain any small or large parameters. The OHAM provides us with a simple and rigorous way to control and adjust the convergence of the solution through a number of parameters Ci which are optimally determined. The main strength of the OHAM consists in its fast convergence since after only one iteration, the solution converges to the exact one, which proves that our procedure is very rapid, effective and accurate. References [1] R. Fu, Q. Y. Long, C. W. Lung, Relevance elastic exponent on the thickness of porous plate, J. of Phy. Condensed Matter, 4 (1), 1992, 49-52. [2] A. Pompei, M. A. Rigano, On the bending of micropolar plates, Int. J. of Eng. Sci., 44 (2006), 1324-1333. [3] K.R. Rajagopal, A.S. Wineman, New exact solutions in nonlinear elasticity, Int. J, Eng. Sci. 23(2) (1985), 217-234. [4] B.-F. Apostol, On a non-linear wave equation in elasticity, Physics Lett. A 318 (2003), 545-552. [5] A.H. Bokhari, A.H. Kara, F.D. Zaman, Exact solutions of some general nonlinear wave equations in elasticity, Nonlinear Dyn. DOI 10.2007/s11071-006-9050-8, 6 pages. [6] M.T. Mustafa, K. Masood, Symmetry solutions of a nonlinear elastic wave equation with thirdorder anharmonic corrections, Appl. Math. Mech. – Engl. Ed. 30(8) (2009), 1017-1026. [7] E. Alfinito, M.S. Causo, G. Profilo, G. Soliani, A class of nonlinear wave equations containing the continuous Toda case, J. Phys. A 31 (1998), 2173-2189. [8] V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering. Some Approximate Approaches, Springer Verlag, Berlin, Heidelberg, 2011. [9] V. Marinca, N. Herisanu, An optimal homotopy asymptotic approach applied to nonlinear MHD Jeffery – Hamel flow, Mathematical Problem in Engineering, article ID 169056 (2011) DOI: 10.1155(20111). [10] V. Marinca, N. Herisanu, C. Bota, B. Marinca, An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl. Math. Lett. 22 (2009), 245-251. [11] S. Miroslav, The Mechanics and Thermodynamics of Continuous Media (Theoretical and Mathematical Physics), Springer, 2002. [12] L. Landau, E. Lifschitz, Theorie de l’Elasticite, Nauka, Moskow, 1967. [13] Y. A. Kosevich, Nonlinear sinusoidal waves and their superposition in anharmonic latticesreply, Phys. Lett. 71 (1993), 2058-2062. [14] C. W. Lim, L. H. He, Exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting, Int. J. Mech. Sci., 43(11) (2001), 2479-2492. [15] C. W. Lim, Z. R. Li, G. W. Wei, DSC-Ritz method for high-mode frequency analysis of thick shallow shells, Int. J. Num. Meth. Eng., 62(2) (2005), 205-232.

Applied Mechanics and Materials Vol. 430 (2013) pp 45-52 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.45

The Study of the Pendulum with Heavy neo-Hookean Rod Nicolae–Doru STĂNESCU1, a, Dinel POPA2, b 1

University of Piteşti, Târgul din Vale, 1, Piteşti, 110040, Romania

2

University of Piteşti, Târgul din Vale, 1, Piteşti, 110040, Romania a

[email protected], b [email protected]

Keywords: pendulum, neo-Hookean, stability, small oscillations

Abstract. The present paper is a generalization of the problem of a rubber spring pendulum discussed by Bhattacharyya in 2000 and Stănescu in 2011, which studied the case of the neoHookean rod without mass. In our paper we consider that the mass of the neo-Hookean rod is not negligible and its deformation is realized such that at any moment of time the rod can be treated as a homogeneous rigid bar of variable length. Using the second order Lagrange equations we obtained the equations of motion in the most general case and we identified as particular cases the situations presented in the bibliography. We also performed a study of the equilibrium positions and their stability. A study of the small oscillations about the stable equilibrium positions is realized too. The theoretical results are finally compared to those obtained by numerical simulation. Introduction The pendulum is a mechanical system which is much studied. The specific literature can be divided in a few categories: – those which studies the mathematical pendulum which is hanged by an element (rod, cable) of negligible mass; the pendulum can be excited or not, with or without damping [1, 2, 3, 4]; – those which studies the physical pendulum with or without damping, excited or not [5, 6, 7, 8]; – those which treats the pendulum hanged by an elastic element of negligible mass; the pendulum is excited or not, with or without damping [9, 10, 11, 12]. Our study purposes a model in which the ball of mass M and negligible dimensions is linked at the end B of the rod AB of mass m and length in non-deformed status equal to l 0 (Fig. 1). The rod AB suffers deformations during the motion, its length at an arbitrary moment of time being l . The deformations are uniform, such that at any moment of time, the rod A has the behavior of a homogeneous bar of mass m and length l , its center of weight being the point C , the middle of the segment AB . y

A θ

l

Y

C mg O

x B Mg

Fig. 1. Mathematical model.

46

Acoustics & Vibration of Mechanical Structures

The elastic force in the rod is considered to be given by a potential U = U (l ) , the expression of which will be detailed further. Moreover, the end A of the rod AB has a vertical displacement after the law Y = Y (t ) which is assumed to be known. The mechanical system considered here has, therefore, two degrees of freedom: the length l of the rod AB and the angle θ made by this rod with the Oy -axis. The Equations of Motion We will determine the equations of motion using the second order Lagrange equations. Since xC = (l 2 ) sin θ , yC = Y − (l 2 ) cos θ , x B = l sin θ , y B = Y − l cos θ , J C = ml 2 12 , one obtains for the kinetic energy T the expression 1 1! 1! 1 1 l2 1 m  mvC2 + J C θ 2 + Mv B2 = (m + M )Y 2 + (m + 4 M ) +  + M l 2 θ 2 2 2 2 2 2 4 2 3  (1) 1 1 − (m + 2 M )Yl cos θ + (m + 2 M )Ylθ sin θ , 2 2 where J C is the moment of inertia with respect to an axis perpendicular to the plan Oxy and passing through C , while the potential energy reads T =

m  V = −U + mgy C + mgy B = −U + gY (m + M ) −  + M  gl cos θ . 2  The Lagrange equations read

(2)

1 (m + 4M )l −  m + M lθ 2 − ∂U =  m + M (g + Y ) cos θ , 4 ∂l 3  2 

(3)

∂U m  m  m  2 + M llθ +  + M l 2 θ − = − + M (g + Y )l sin θ . ∂l 3  3  2 

(4)

With the changes of variables l = l 0 λ , l = l 0 λ , l = l 0 λ , Y = Y l 0 , Y = Y l 0 , Y = Y l 0 , denoting f (λ ) = ∂U ∂λ and assuming that U = U (λ ) , hence ∂U ∂θ = 0 , we obtained the equations we are looking for

 m  m  m  g  + M λ −  + M λθ 2 − f (λ ) =  + M  + Y  cos θ , 4  3  2  l 0 

(5)

 m  m  m  g  + M λθ + 2 + M λθ = − + M  + Y  sin θ . 3  3  2  l 0 

(6)

Particular Cases Case 1. If the rod AB is a neo-Hookean one, then U (λ ) = − A0 G[λ2 2 + λ1 − k (k − 1)] l 0 , with k ∈ N , k > 1 , A0 is the area of the transversal section of the non-deformed rod, while G is the shear modulus. It results f (λ ) = − A0 G (λ − λ− k ) l 0 and the equations of motion

 AG 1  m m  m   g  + M λ −  + M λθ 2 + 0  λ − k  =  + M  + Y  cos θ , l0  λ  2 4  3   l 0 

(7)

Applied Mechanics and Materials Vol. 430

 m  m  m  g  + M λθ + 2 + M λθ = − + M  + Y  sin θ . 3  3  2  l 0  Case 2. If the rod AB is a neo-Hookean one of mass m = 0 , then it results the equations g  A0 G  1   λ − k  = M  + Y  cos θ , λθ + 2λθ = l0  λ   l0  In this situation the equilibrium positions result from the system Mλ − Mλθ 2 +

g  − + Y  sin θ .  l0 

g  g  AG 1  −  + Y  sin θ = 0 , 0  λ − k  = M  + Y  cos θ . l0  λ   l0   l0 

47

(8)

(9)

(10)

If Y = − g l 0 , then the angle θ is arbitrary, while λ results from the equation λk + 1 − 1 = 0 , with the unique solution λ = 1 . If Y ≠ − g l 0 , then it results sin θ = 0 , θ1 = 0 , θ 2 = π , while λ

(

)

is obtained from λk + 1 ± M g l 0 + Y l 0 λk A0 G − 1 = 0 , the superior sign being for θ1 = 0 , while the inferior one corresponds to θ 2 = π . The Descartes theorem assures the existence and uniqueness of the equilibrium position in the case θ1 = 0 if g l 0 + Y > 0 , or the existence and uniqueness of the same position in the case θ 2 = π if g l 0 + Y < 0 . A more particular case studied in the references is the case defined by Y = BHλµ , in which B is a real constant, µ ∈ N , while H denotes the parameter H = A0 G (Ml 0 ) . Eqs. 10 transform in

g  1  g  λ − λθ 2 + H  λ − B cos θλµ − k  = cos θ , λθ + 2λθ +  + BHλµ  sin θ = 0 . λ  l0   l0  This case is studied in [11, 12].

(11)

The Case of the neo-Hookean Pendulum with Heavy Rod The Equations of Motion. We will study the case described by Eqs. 8 and 9. We denote H = A0 G [(m 2 + M )l 0 ] , ω02 = g l 0 and we consider Y = BHλµ , where B is a real constant, and µ ∈ N , µ ≥ 1 . It results the equations of motion m m +M +M 1   3 4 λ− λθ 2 + H  λ − B cos θλµ − k  = ω02 cos θ , m m λ   +M +M 2 2 m m +M +M 3 3 λθ + 2 λθ + (ω 02 + BHλµ ) sin θ = 0 . m m +M +M 2 2 The Equilibrium Positions. These positions are deduced from the system

(ω02

1   + BHλµ ) sin θ = 0 , H  λ − B cos θλµ − k  = ω02 cos θ , λ  

(12)

(13)

(14)

48

Acoustics & Vibration of Mechanical Structures

wherefrom it results sin θ = 0 , θ1 = 0 , θ 2 = π and λ ∓ Bλµ − λ− k = ± ω02 H , in which the superior sign corresponds to θ1 = 0 , while the inferior one corresponds to θ 2 = π . The last equation takes the form ω02 k λ − 1 = 0. (15) H Further on, we will limit ourselves to the classic case defined by µ = 1 , such that Eq. 15 may be written as ∓ Bλµ + k + λk + 1 ∓

(1 − B )λk + 1

−

for θ1 = 0 , and

ω02 k λ −1 = 0 H

(16)

ω 02 k λ −1 = 0 (17) H for θ 2 = π , respectively. If θ1 = 0 , then the Descartes theorem assures that if B < 1 , than Eq. 16 has only one positive root, and if B > 1 , then the same equation has no positive roots. If θ 2 = π , then for B > −1 the Descartes theorem states that Eq. 17 has only one positive root, and if B < −1 , then the same equation has zero or two positive roots. Finally, if B = 1 and θ1 = 0 , then Eq. 16 has no positive root, while if B = −1 and θ 2 = π , then Eq. 17 has only one positive root. The Small Oscillations. We will study the small oscillations around the equilibrium position defined by θ1 = 0 and λ = λ static . We may write

(1 + B )λk + 1

+

l + z static + z l 0 + z static l z = 0 = + = λ static + u , (18) l0 l0 l0 l0 in which z static and z are the static elongation of the neo-Hookean rod and the elongation relative to the static equilibrium of the same rod AB , respectively, λ static = (l 0 + z static ) l 0 , u = z l 0 . Since u is small with respect to λ static , one may write λ =

  1 1  k  µ µ −1    . ≈ λ − B λ − + u 1 − B µλ + (19) static static static    k k +1  λk λ λ static  static    Assuming now that θ is small enough to be neglected, θ ≈ 0 , Eqs. 12 and 13 take the form λ − Bλµ −

m +M 1   4 λ + H  λ − Bλµ − k  = ω02 , m λ   +M 2 where we made the approximation sin θ ≈ For λ = λ static , Eqs. 20 offer H = ω 02 the first Eq. 20 becomes

m +M 3 λθ + (ω02 + BHλµ )θ = 0 , (20) m +M 2 θ. k (λ static − Bλµstatic − λ−static ) . Keeping into account Eq. 19,

m +M  1 4 u + H  λ static − Bλµstatic − k m λ static  +M 2

  k −1  + u 1 − Bµλµstatic + k +1 λ static  

  = ω02 , 

(21)

Applied Mechanics and Materials Vol. 430

49

−1 k −1 )H (m 2 + M ) (m 4 + M ) . defining a periodical motion of pulsation ωλ = (1 − Bµλµstatic + kλ−static Analogically, the second Eq. 20 defines a harmonic motion the pulsation of which is given by the

relation ωθ = negligible ωθ =

(ω

(ω

2 0

mass,

+ BHλµstatic )(m 2 + M ) [(m 3 + M )λ static ] . For m = 0 , that is, the rod AB is of the

last

two

relations

become

ωλ =

+ BHλµstatic ) λ static , which are the values reported in [12].

2 0

(1 − Bµλ

µ −1 static

k −1 )H , + kλ−static

Stability of the Equilibrium Positions. We will study the case defined by µ = 1 , k = 2 , for which the equations of motion read m  m  m H + M  ω 02  + M  +M 2  (1 − B cos θ)λ − 1  = 2  cos θ , λ− 3 λθ 2 +   2  m m m λ  +M +M  +M 4 4 4

(22)

m  + BHλ ) + M  2  sin θ = 0 . λθ + 2λθ + (23) m +M 3 Denoting ξ1 = λ , ξ 2 = θ , ξ 3 = λ , ξ 4 = θ , one obtains a system of four first-order differential equations. The equilibrium positions are situated at the intersection of the nullclines of this system [13],

(ω02

m  M  H + M  ω02  + M 2  (1 − B cos ξ )ξ − 1  +  2  cos ξ = 0 , ξ3 = 0 , ξ 4 = 0 , −  2 1 2  2  m m ξ1  +M  +M 4 4

m  + BHξ1 ) + M  2  sin ξ = 0 . (24) 2 m +M 3 The last Eq. 24 leads us to sin ξ 2 = 0 , such that at the equilibrium we have ξ 2 = 0 or ξ 2 = π . If ξ 2 = 0 , then the third Eq. 24 becomes

(ω02

ω 02 2 ξ1 − 1 = 0 . (25) H If B = 1 , then there exists no equilibrium position. If B > 1 , then there exists no equilibrium position. If B < 1 , then, according to the Descartes theorem, Eq. 25 has only one root ξ1 > 0 and therefore we have only one position of equilibrium. For ξ 2 = π , the third Eq. 24 reads

(1 − B )ξ13

−

ω 02 2 ξ1 − 1 = 0 . H For B = −1 , Eq. 26 has two roots

(1 + B )ξ13

ξ1 = ±

+

(26)

H , ω 02

(27)

50

Acoustics & Vibration of Mechanical Structures

one positive and one negative. We deduce the existence of one equilibrium position defined by the positive root in Eq. 27. If B > −1 , then 1 + B > 0 , such that the Descartes theorem states that Eq. 26 has exactly one positive root which offers the equilibrium position. If B < −1 , then using the Descartes theorem, one deuces that Eq. 26 has zero or two positive roots, that is, there exist zero or two equilibrium positions. We will write Eq. 26 in the form ω 02 1 (28) ξ12 − =0 H (1 + B ) 1+ B and will denote α = ω 02 [H (1 + B )] , β = − 1 (1 + B ) . With the change of variable ξ1 = ζ 1 − α 3 , Eq. 28 becomes ξ13 +

α2 2α 3 ζ1 + + β = 0. (29) 3 27 Let a = − α 2 3 , b = α 3 + β , and the discriminant of Eq. 29 takes the form [13] ζ 13 −

[

][

]

∆ = 4a 3 + 27b 2 , ∆ = 27 H 3 (1 + B ) − 8ω 60 (1 + B ) H 3 . To have equilibrium positions (which is equivalent to say that Eq. 29 has three real roots) it is necessary and sufficient that ∆ ≤ 0 , that is 2 27 H 3 (1 + B ) ≤ 8ω 60 . If ∆ = 0 , then Eq. 29 has two real roots, one of them being double. This double root is a solution for Eq. 29 and for its derivative too, 3ζ 12 + a = 0 . Considering the system of equations ζ 13 + aζ 1 + b = 0 , 3ζ 12 + a = 0 , we obtain the common root ζ 1 = − 3b (2a ) . 2

4

[

Keeping into account the previous relations, we find ζ 1 = 2ω 60 − 27 H 3 (1 + B )

2

] [6H (1 + B )ω ] , 4 0

ξ1 = ζ 1 − α 3 = − 9 H 2 (1 + B ) (2ω04 ) > 0 . If ∆ < 0 , then Eq. 29 has three distinct real roots which means that Eq. 28 has also three distinct real roots, two positive and one negative. Moreover, Eq. 29 has two roots greater than α 3 and one lesser than α 3 . For the study of stability of the equilibrium positions we will denote by f i (ξ1 , ξ 2 , ξ 3 , ξ 4 ) , i = 1, 4 , the right-hand terms of the Eqs. 24, and by j kl , k , l = 1, 4 , their partial derivatives, j kl = ∂f j ∂ξ l . Because at the equilibrium ξ 3 = 0 , ξ 4 = 0 , sin ξ 2 = 0 , the characteristic equation becomes [13]

−λ 0 1 0 0 −λ 0 1 = ( j31 − λ2 )( j 42 − λ2 ) = 0 . j31 0 − λ 0 0 j 42 0 − λ An equilibrium position is stable (in fact, simply stable) [13] if and only if

(30)

j31 < 0 , j 42 < 0 . (31) Let us analyze each position. If ξ 2 = 0 and B < 1 , then we have only one equilibrium position. The expressions of j31 and

j 42 , and the conditions in Eq. 31 lead us to 1 − B + 2 ξ13 > 0 , ω02 + BHξ1 > 0 . Since ξ1 > 0 , the

second condition is obviously true, while the first one leads us to ξ1 > 3 2 (B − 1) . Replacing in Eq. 25, we have to find in the left-hand side of Eq. 25 a negative expression,

− ω 02 3 4 (B − 1) H < 3 , an obviously true relation. The equilibrium position is always simply stable. If ξ 2 = π , then the expressions for j31 and j32 and the conditions in Eq. 31 lead us to 2

Applied Mechanics and Materials Vol. 430

1+ B +

51

2 > 0 , ω02 + BHξ1 < 0 . ξ13

(32)

H ω 0 , then Eqs. 32 become 2 ξ13 > 0 ,

We have a few sub-cases. If B = −1 and ξ1 =

ω02 − H H ω 0 < 0 , wherefrom ω30 < H H , which is exact the condition we search for a simply stable equilibrium position. If B > −1 , then we deduce 1 + B > 0 and since ξ1 > 0 it results that the first Eq. 32 holds always true. The second Eq. 32 leads to Bξ1 < − ω 02 H , such that we get ξ1 < − ω 02 (BH ) for B > 0 , and ξ1 > − ω 02 (BH ) for B > 0 , respectively. Let us observe that for B > 0 we obtain ξ1 < 0 , which is absurd. It results that the only possibility for the equilibrium position be simply stable is that given by ξ1 > − ω 02 (BH ) , with B < 0 . Moreover, replacing ξ1 by − ω 02 (BH ) in Eq. 26, the left-hand side of this expression must be negative, that is B < H ω02 . In conclusion, the equilibrium position is simply stable if and only if B < H ω02 , B > −1 . If ∆ = 0 , ξ1 = − 9 H 2 (1 + B ) (2ω 04 ) , B < −1 , then we obtain ω60 < 27(1 + B ) H 3 4 , ω60 < 9 H 3 B(1 + B ) 2 . 2

Since ∆ = 0 , we have ω60 = 27 H 3 (1 + B ) 8 , such that the previous expressions lead to 2

(1 + B )(3 − B ) < 0 .

On the other hand, B < −1 , such that 1 + B < 0 , 3 − B > 0 ; hence the

− (1 + B ) 2 , ξ1 > − ω (BH ) , the equilibrium position defined by one of the positive roots of Eq. 28 being simply stable if and only if the last conditions are fulfilled.

previous condition is always true. If ∆ < 0 , ξ 2 = π , then Eq. 32 leads us to ξ1
0 .  l 0  l 0   The theorem of momentum offers the equation

(1)

mz = − FA − FB − FD − FE − mg + F0 cos ωt , while the theorem of moment of momentum gives us the relations J x ψ = (FA − FB − FD + FE )l 2 , J y θ = (FA + FB − FD − FE )l1 .

(2)

(3)

The Equilibrium Positions for the Case without Excitation These equilibrium positions are determined from the system [7]

 l + z − l ψ − l θ  −2  l + z + l ψ − l θ  −2 l0 + z 2 1 2 1  +  0  − 4k + k  0 l0 l l  0 0     l + z + l 2 ψ + l1θ   +  0 l 0  

−2

−2  l 0 + z − l 2 ψ + l1θ     − mg = 0 , +  l 0   

 l + z − l ψ − l θ  l ψ 2 1  − 4k 2 − k  0 l0 l0    l + z + l 2 ψ + l1θ   −  0 l0  

−2

−2

 l + z + l 2 ψ − l1θ   −  0 l 0  

 l + z − l 2 ψ + l1θ   +  0 l0  

−2

−2

  = 0, 

 l + z − l ψ − l θ  −2  l + z + l ψ − l θ  −2 l1θ 2 1 2 1  +  0  − 4k − k  0 l0 l l  0 0    −2

 l + z + l 2 ψ + l1θ   l + z − l 2 ψ + l1θ   −  0  −  0 l l 0 0     Subtracting and summing Eqs. 5 and 6, we obtain

8l 02 (l 0 + z ) 4 −  l 0 (l0 + z )2 − (l 2 ψ − l1θ)2 

(l 2 ψ − l1θ)−

[

−2

 = 0, 2  

]

  = 0. 

(4)

(5)

(6)

(7)

Applied Mechanics and Materials Vol. 430

55

 8l 02 (l 0 + z ) 4 − (8)  = 0, 2 2 2 l (l0 + z ) − (l 2 ψ + l1θ)   0 We deduce from here that (the second parentheses in Eqs. 7 and 8 define strict negative expressions) l 2 ψ − l1θ = 0 , l 2 ψ + l1θ = 0 ; hence, at equilibrium, ψ = 0 , θ = 0 . These values will 

(l 2 ψ + l1θ)−

[

]

be used in Eq. 7, obtaining 4(l 0 + z ) + mgl0 (l 0 + z ) k − 4l 03 = 0 . We denote now ξ = l 0 + z , and the previous equation becomes 3

2

mg (9) l 0 ξ 2 − l 03 = 0 . 4k We make ξ −ξ in the last relation obtaining ξ 3 − mgl 0 ξ 2 (4k ) + l 03 = 0 . The Descartes theorem states that Eq. 9 has one positive root and zero or two negative roots. With the change of variable ξ = ζ − mgl 0 12k Eq. 9 takes the form ξ3 +

ζ3 − Let

m 2 g 2 l 02 m 3 g 3l 03 ζ + − l 03 = 0 . 48k 2 864k 3 be a = m 2 g 2 l 02 (48k 2 ) , b = m 3 g 3 l 03 (864k 3 ) − l 03

(10) and

[7]

∆ = 4a 3 + 27b 2 ,

∆ = l 06 (27 k 3 − m 3 g 3 16 ) k 3 . The solving of the third degree equation [7] implies that: if ∆ < 0 ,

( )

( ) (6 2 ), then

that is k < mg 63 2 , then Eq. 10 has three distinct real roots; if ∆ = 0 , that is k = mg 63 2 , then Eq. 10 has two real roots, one of them being double, and if ∆ > 0 , that is k > mg 3 Eq. 10 has only one real root. From the previous discussion it results that Eq. 9 has always a positive root and that root is a simply one. If we keep into account the change of variable, we may say that Eq. 10 has a single positive root and, moreover, this root is greater than mgl 0 (12k ) . In addition, the root z of the equation in z , which corresponds to the positive root of Eq. 9 is strictly grater than − l 0 .

Stability of the Equilibrium We will write the equations of motion (Eqs. 2 and 3) as a system of six first-order nonlinear differential equations ξ1 = ξ 4 = f 1 (ξ ) , ξ 2 = ξ 5 = f 2 (ξ ) , ξ 3 = ξ 6 = f 3 (ξ ) , ξ 4 = f 4 (ξ ) , ξ 5 = f 5 (ξ ) , ξ 6 = f 6 (ξ ) ,(11)

in which ξ = (ξ1 , … , ξ 6 ) , ξ1 = z , ξ 2 = ψ , ξ 3 = θ , ξ 4 = z , ξ 5 = ψ , ξ 6 = θ .

But the partial derivatives of these functions, calculated for ψ = ξ 2 = 0 , θ = ξ 3 = 0 , read ∂

(l 0

l 02

+ ξ1 − l 2 ξ 2 − l1ξ 3 ) ∂ξ1

2

= −2(l 0 + ξ1 ) l 02 , −3

(12)

ξ 2 = ξ3 = 0

and similar, so that

∂f 4 ∂ξ1

=− ξ 2 = ξ3 = 0

 ∂f 4 8l 02 k 4  + , m  l 0 (l 0 + ξ1 )3  ∂ξ 2

= 0, ξ 2 = ξ3 = 0

∂f 4 ∂ξ 3

=0 ξ2 = ξ3 = 0

(13)

56

Acoustics & Vibration of Mechanical Structures

∂f 5 ∂ξ1 ∂f 6 ∂ξ1

= 0, ξ 2 = ξ3 = 0

= 0, ξ 2 = ξ3 = 0

∂f 5 ∂ξ 2 ∂f 6 ∂ξ 2

= ξ 2 = ξ3 = 0

 ∂f 5 8l 02 kl 22  4 − , − 3 J x  l 0 (l 0 + ξ1 )  ∂ξ 3

= 0, ξ 2 = ξ3 = 0

∂f 6 ∂ξ 3

= ξ 2 = ξ3 = 0

kl12 Jy

= 0,

(14)

 4  8l 02 − − ,  3  l 0 (l 0 + ξ1 ) 

(15)

ξ2 = ξ3 = 0

Denoting j kl = ∂f k ∂ξ l , k = 4, 6 , l = 1, 3 , one obtains the characteristic equation

−λ 0 0 1 0 0 0 −λ 0 0 1 0 0 0 −λ 0 0 1 = 0, j 41 0 0 − λ 0 0 0 j52 0 0 − λ 0 0 0 j 63 0 0 − λ

(16)

the roots being λ21, 2 = j 41 , λ23, 4 = j52 , λ25, 6 = j 63 . The equilibrium is stable (in fact, simply stable) [7] if and only if the expressions j 41 , j52 and j 63 are negative; keeping into account the previous relations, we find the conditions 2l 02 2l 02 1 1 + > 0 , + > 0. (17) l 0 (l 0 + ξ1* )3 l 0 (l 0 + ξ1* )3 The relations in Eq. 17 are redundant because, as we have seen before, at equilibrium l 0 + ξ1* > 0 , where ξ1* is the equilibrium position. The equilibrium is thus simply stable, no matter the values taken by the parameters. Small Oscillations around the Equilibrium Position

These oscillations are harmonic motions of periods given by [7], [5] T1 = 2π

T2 = 2π

j52 , T3 = 2π 2π

T1 =

j 41 ,

j 63 , or, equivalently, 2π

, T2 = 2 0

2 0

2 2

8l k 4 + m l 0 (l1 + ξ1* )3

2 1

8l kl 4 + J x l 0 (l1 + ξ1* )3

Let us remark that T2 T3 = l1 J y

(l

2

2π

, T3 =

.

(18)

2 0

8l kl 4 + J y l 0 (l1 + ξ1* )3

)

J x , T1 T2 = l 2 m

J x ; hence, the ratios of the

eigenpulsations do not depend on the characteristic of the neo-Hookean springs, but they depend on only the geometrical and mass characteristics of the shell. Obviously, the small oscillations are described by the equations ξ1 = z = A1 cos

ξ 2 = ψ = A2 cos

(

)

j52 t + ϕ 2 , ξ 3 = θ = A3 cos

(

)

(

)

j 41 t + ϕ1 ,

j 63 t + ϕ 3 , where A1 , A2 , A3 , ϕ1 , ϕ 2 and ϕ 3

are constants which will be determined from the initial conditions.

Applied Mechanics and Materials Vol. 430

57

Numerical Simulation

The results obtained before will be validated by numerical simulation. -3

2.44

-3

x 10

1.5

2.42

x 10

1

2.4

ψ [rad]

l0+ ξ1 [m]

0.5 2.38

2.36

0

-0.5 2.34 -1

2.32

0

0.1

0.2

0.3

0.4

0.5 t [s]

0.6

0.7

0.8

0.9

-1.5

1

0

0.1

0.2

0.3

0.4

0.5 t [s]

0.6

0.7

0.8

0.9

1

Fig. 3. Time history for ψ , 0 ≤ t ≤ 1 .

Fig. 2. Time history for l 0 + ξ1 , 0 ≤ t ≤ 1 . -3

1.5

x 10

1

θ [rad]

0.5

0

-0.5

-1

-1.5

0

0.1

0.2

0.3

0.4

0.5 t [s]

0.6

0.7

0.8

0.9

1

Fig. 4. Time history for θ , 0 ≤ t ≤ 1 . We select the following values for different parameters:

l 0 = 0.01 m ,

m = 150 kg ,

g = 10 m s 2 , k = 20 N , J x = 2 kgm 2 , J y = 8 kgm 2 . With these values, Eq. 9 becomes now ξ 3 + 0.1875ξ 2 − 10 −6 = 0 , with the solution (the positive one), ξ = 0.002296 m . The periods given by Eq. 18 are T1 = 0.066729 s , T2 = 0.019263 s , T3 = 0.077052 s . The integration method was the fourth order Runge–Kutta one [8]. Time histories for different variables are plotted in Fig.2 (for l 0 + ξ1 = ξ ), Fig. 3 (for ψ ) and Fig. 4 (for θ ). The initial conditions are as follows: (l 0 + ξ1 )0 = ξ 0 = 0.0023 m , ψ 0 = 0.001 rad , θ 0 = 0.001 rad , ξ 0 = 0 m s , ψ 0 = 0 rad s , θ 0 = 0 rad s (that is, very closed to the equilibrium position given by ξ = 0.002296 m , ψ = 0 rad and θ = 0 rad . The diagrams are realized for free vibrations of the shell (no exciting force, that is, F0 = 0 ) and the reader can easily observe the quasi-periodical character of the vibrations. The quasi-periods corresponding to ξ , ψ and θ are in very good agreement with the periods T1 , T2 and T3 , respectively, obtained by analytical calculation. For the period T1 we obtain an eigenpulsation given by ω1 ≈ 94.15 s -1 . In the Figs. 5, 6 and 7 we maintain the same parameters as previous and we add ω = 94 s -1 (that is, very closed to the eigenpulsation ω1 ) and F0 = 300 N . The reader can easily observe the beat

phenomenon presented now. The beat phenomenon has a period Tbeat ≈ 0.6 s , which is approximately ten times greater than the period T1 . We can also observe that the amplitude of the beat phenomenon tends to decrease in the case of the variable ψ , but it remains almost constant for the rest of the variables.

58

Acoustics & Vibration of Mechanical Structures

For the last diagrams plotted in Figs. 8, 9 and 10, the parameters remain unchanged, excepting ω which has the value ω = 30 s -1 (that is far away from ω1 . In this case there is not beat phenomenon, the diagrams presenting a quasi-periodical shape. -3

7

-3

x 10

2

x 10

1.5

6

1

5

ψ [rad]

l0+ ξ1 [m]

0.5 4

3

0 -0.5

2

-1

1

0

-1.5

0

0.5

1

1.5 t [s]

2

2.5

-2

3

0

0.5

1

1.5 t [s]

2

2.5

3

Fig. 6. Time history for ψ , 0 ≤ t ≤ 3 , -1 ω = 94 s . The beat phenomenon is now present. ω = 94 s . The beat phenomenon is now present. Fig. 5. Time history for l 0 + ξ1 , 0 ≤ t ≤ 3 , -1

-3

2.5

-3

x 10

3.4

2

x 10

3.2

1.5

3

1

2.8 l0+ ξ1 [m]

θ [rad]

0.5 0

2.6 2.4

-0.5 2.2

-1

2

-1.5

1.8

-2 -2.5

0

0.5

1

1.5 t [s]

2

2.5

1.6

3

0

0.5

1

1.5 t [s]

2

2.5

3

Fig. 7. Time history for θ , 0 ≤ t ≤ 3 , Fig. 8. Time history for l 0 + ξ1 , 0 ≤ t ≤ 3 , -1 ω = 94 s . The beat phenomenon is now present. ω = 30 s -1 . The beat phenomenon is now absent. -3

-3

x 10

1.5

1

1

0.5

0.5

θ [rad]

ψ [rad]

1.5

0

0

-0.5

-0.5

-1

-1

-1.5

0

0.5

1

1.5 t [s]

2

2.5

3

x 10

-1.5

0

0.5

1

1.5 t [s]

2

2.5

3

Fig. 9. Time history for ψ , 0 ≤ t ≤ 3 , Fig. 10. Time history for θ , 0 ≤ t ≤ 3 -1 -1 ω = 30 s . The beat phenomenon is now absent. ω = 30 s . The beat phenomenon is now absent. Conclusions

In our paper we presented a three degrees of freedom system for the vibrations of the engine. The model is highly non-linear because of the expressions of forces that appear in the neo-Hookean springs. For this model we proved the existence and uniqueness of the equilibrium position and it’s simply stability in the case without excitation. The small oscillations around this equilibrium position were described as harmonic vibrations. The theoretical analysis was validated by numerical

Applied Mechanics and Materials Vol. 430

59

simulation, showing a very good agreement between the two methods. The case with external excitation was study in two conditions: when the pulsation of the external force is much closed to an eigenpulsation of the system highlighting the beat phenomenon and when the pulsation of the external force is far away from any eigenpulsation of the system. References

[1] N. Pandrea, S. Pârlac, D. Popa, Modele pentru studiul vibraţiilor automobilelor, Tiparg, Piteşti, 2001. [2] B.H. Tongue, Principles of Vibrations, Oxford University Press, Oxford, 2001. [3] T.L. Schmitz, K.S. Smith, Mechanical Vibrations: Modeling and Measurement, Springer, New York, 2011. [4] P.H. Wirsching, T.L. Paez, K. Ortiz, Random Vibrations: Theory and Practice, Dover Books on Physics, Dover, 2006. [5] L. Meirovitch, Fundamental of Vibrations, Waveland Press, Inc, Long Grove, 2010. [6] N.–D. Stănescu, Mechanical Systems with neo-Hookean Elements, Stability and Behavior, LAP, Saarbrücken, 2011. [7] N.–D. Stănescu, L. Munteanu, V. Chiroiu, N. Pandrea, Sisteme dinamice. Teorie şi Aplicaţii, Editura Academiei Române, Bucureşti, 2007, 2011, 2013. [8] P. Teodorescu, N.–D. Stănescu, N. Pandrea, Numerical Analysis with Applications in Mechanics and Engineering, Wiley, Hoboken, 2013.

CHAPTER 2: Damage Assessment of Structures

Applied Mechanics and Materials Vol. 430 (2013) pp 63-69 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.63

Multiple fault identification using vibration signal analysis and artificial intelligence methods Ninoslav Zuber1,a, Dragan Cvetković2,b , Rusmir Bajrić3,c 1

Faculty of Technical Sciences – University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia 2

Faculty of Occupational Safety – University of Niš, Čarnojevića 10a, 18000 Niš, Serbia

3

Public enterprise Elektroprivreda BIH, Coal Mine Kreka, Mije Keroševica 1, 75000 Tuzla, Bosnia and Herzegovina a

[email protected], [email protected], b [email protected]

Keywords: Vibration analysis, artificial neural network, self-organized feature map, bearing fault, gearbox fault

Abstract. Paper addresses the implementation of feature based artificial neural networks and selforganized feature maps with the vibration analysis for the purpose of automated faults identification in rotating machinery. Unlike most of the research in this field, where a single type of fault has been treated, the research conducted in this paper deals with rotating machines with multiple faults. Combination of different roller elements bearing faults and different gearbox faults is analyzed. Experimental work has been conducted on a specially designed test rig. Frequency and time domain vibration features are used as inputs to fault classifiers. A complete set of proposed vibration features are used as inputs for self-organized feature maps and based on the results they are used as inputs for supervised artificial neural networks. The achieved results show that proposed set of vibration features enables reliable identification of developing bearing and gear faults in geared power transmission systems. Introduction An unscheduled stop of a rotating machine in a plant generally leads to high maintenance and production risks and costs. High costs are initiated through production stops, losses and urgent procurements of spare parts. High risks are associated with the possibilities of workers’ injuries and secondary damages of other machines. To avoid this, several maintenance strategies have been developed. The implementation of condition-based maintenance implies monitoring of machine operating condition based on the physical parameter that is sensitive to machine degradation. Among many possible parameters, mechanical vibration acquired at the bearing’s housing is one of the best parameter for early detection of a developing fault inside a machine. Methods of vibration signal analysis enable the extraction of type and severity of a fault. Despite the fact that the information on type and severity of a fault is contained within the vibration signal, due to the: • existence of multiple faults on a machine, • dependence of vibration signal content on operating conditions (load and speed), • existence of vibration components from neighboring machines, derivation of inaccurate vibrodiagnostical conclusions and wrong estimation of machine criticality in the plant, is a very common situation. To avoid this, there are two approaches: • engagement of highly skilled and trained vibration analysts or • implementation of artificial intelligence (AI) methods for reliable extraction of an existing fault. Engagement of certified vibration analysts can be an issue due to several reasons: there are not many of them, in many cases they don’t have a substitution when absent and they are often engaged in other maintenance tasks so they cannot be fully focused on the analysis of acquired data from the

64

Acoustics & Vibration of Mechanical Structures

machine. In such an environment, implementation of AI methods through previously developed and validated fault identification algorithm has a huge potential. Methods of AI, which can be used for automatic fault identification of rotating machine, are: artificial neural networks (ANN), fuzzy logic, expert systems and hybrid intelligence systems. The most applied are ANN ([18], [1]). One of the reasons for that is due to their ability to learn - adopt novelties. This adaptability of ANN results in a possibility for detection of an occurrence of a new condition (fault) based on the existing data ([17], [9]). A review of existing literature ([10] [6] [16]) shows that several types of ANN are successfully implemented in automatic fault identification: back propagation feed forward network (BPFF), multiple layer perceptron network (MLP), back propagation multiple layer perceptron (BPMLP), radial basis function network (RBF), selforganized feature map (SOFM) and principal component’s analysis (PCA). An excellent review of different types of ANN and training algorithms implementation for different types of rotating machinery can be found in [14]. From the data presented, the increasing trend of implementation of MLP with back propagation training algorithm, with the number of neurons in hidden layers taken as a variable, is evident. A successful implementation of BPMLP and SOFM for the identification of rotating machinery faults can be found in [2], [3], [8], [12], [13], [15], [7]. The authors used different scalar features obtained from vibration data as inputs for neuron classifiers. Review of the most of the papers describing the implementation of AI methods in rotating machinery fault identification reveals two conclusions: • they deal with the single machinery fault identification • if methods of AI algorithms for multiple fault identification are developed, they are reliable for a specific machine. The research conducted in this paper resulted in the AI algorithm for identification of different types of rolling elements bearing and gearbox faults. The AI algorithm is developed on the basis of the vibration signals acquired from the specially designed test rig while the universality of the algorithm is tested and proved on the drive systems of the mine strip bucket wheel excavator where several bearing and gearbox faults are identified. In this paper, the authors used vibration scalar features obtained in both frequency and time domains. Definition of vibration features is done based on an assumption that these parameters are sensitive to failures tested in this paper. Experimental part Experimental part of the work described in this paper is done on a specially designed test rig shown on Fig.1.

Fig. 1. Test rig used for data acquisition The test rig consists of a 0.37kW variable frequency drive connected over the universal joint shaft to the single stage gearbox with spur gears. Rotors are supported by UC201 type roller elements bearings. In reality and especially in mining industry, gearboxes often operate under the

Applied Mechanics and Materials Vol. 430

65

conditions of unsteady load and speed. As a result, their behavior and acquired vibration signals are highly dependent on the current operating regime ([4], [5], [19], [20]). For the purpose of load detection and control, the output shaft is connected to the friction brake. The resulting torque is measured through the bending force registered with a platform type load cell. For the purpose of load control, the stranded wire is connected to the friction pads and over the pulley; the other end is loaded by the mounted weight. This assures a constant torque for different level of brake pads wornness. Bearing housing vibrations are measured in radial directions, using an industrial type IEPE accelerometers mounted at the roller element bearing housings using mounting studs. Input shaft speed is measured using a non-contacting laser sensor and a reflective mark. Also equivalent noise levels with A weighting were measured using an IEPE based microphone. For simultaneous multichannel data acquisition, 16 channel acquisition system OneproD MVX with a corresponding software for measurement setup and analysis OneproD XPR300 ( 01dB-Metravib, France) is used.

Fig. 2. Gears with missing, worn teeth and bearings with inner race fault

Data analysis, results and industry case studies During the test, different roller elements bearing and gearbox faults are introduced. Three sets of gears were tested: gears in a new condition (“OK”), worn gears (“PZ”) and gears with two missing teeth (“NZ”) on the input gear. Four sets of roller elements bearings are tested: bearings in good condition (“OK”), bearings with serious looseness (“ZZ”), bearings with inner race fault (“BI”) and bearings with outer race fault (“BO”). This resulted in total of 12 machine states i.e. 12 output labels. Table. 1. Output labels definition and description Description, bearing and gear state Number Label Gear Bearing 1. “OKOK” good condition good condition 2. “OKBI” good condition inner race fault 3. “OKBO” good condition outer race fault 4. “OKZZ” good condition serious looseness 5. “NZOK” missing teeth good condition 6. “NZBI” missing teeth inner race fault 7. “NZBO” missing teeth outer race fault 8. “NZZZ” missing teeth serious looseness 9. “PZOK” worn teeth good condition 10. “PZBI” worn teeth inner race fault 11. “PZBO” worn teeth outer race fault 12. “PZZZ” worn teeth serious looseness Vibration acquisition included the measurement of: raw time waveforms, narrow band FFT in different frequency ranges with 3200 lines of resolution (2Hz-2kHz, 2Hz-5kHz, 2Hz-20kHz), envelope spectra, time waveform obtained by TSA1 technique with 100 averages, Cepstrum and 1

Time syncronized averaging technique

66

Acoustics & Vibration of Mechanical Structures

autocorrelation functions of the raw and TSA time waveforms. Based on these measurements 58 scalar features were extracted: RMS values of vibration velocity, RMS values of acceleration in several frequency bands (10Hz–20kHz, 2Hz–300Hz, 2Hz-2kHz, 1kHz-2kHz, 2kHz-6kHz, 6kHz10kHz, 10kHz-20kHz), 01dB bearing defect factor2, Kurtosis values obtained from raw, band pass filtered (700Hz-1400Hz) and TSA time waveforms, peak to peak values obtained from raw and TSA time waveforms, amplitudes of first five harmonics of GM3, overall accelerations obtained from narrow bands around first five harmonics of GM (with bandwidth equals to five sidebands from each side of the central frequency - GM), amplitude extractions for first four harmonics of the roller elements bearing defect frequencies obtained from FFT and envelope spectra and equivalent A weighted noise levels. .

Figure 3. 1xGM extraction and width for overall energy calculation in the frequency band SOFM [11] is an excellent tool for the visualization of high dimensional data. In this paper, the idea of using SOFM is the analysis of the applicability of the chosen parameters, since the success of ANN pattern recognition is highly dependent on the choice of input features. SOFM consists of neurons organized in a low dimensional grid where each neuron has a dimension that equals to the number of the input features. The map topology is dictated through neighboring relations between the adjacent neurons. During the SOFM training, the weight vectors move across the data, the map gets organized and, in result, the neighboring neurons have similar weight vectors. SOFM testing was performed in SOM toolbox for Matlab environment [21]. The quality of clustering is analyzed using distance matrix, which visualizes the distances between adjacent neurons on the map: low values indicate clusters while higher values indicate the borders between existing clusters. On the other hand, component planes for each input feature show the values of that feature for each unit on the map. This makes them convenient for analyzing the influence of each input feature on the clustering. For every introduced fault combination, 100 measurements were acquired with 5 minutes of delay between them. This resulted in the matrix of input features with 1200 rows. Input matrix with 58 scalar features was labeled and introduced to SOFM algorithm. As a results, a SOFM with quantization error 0.8469, shown on figure 4, is generated. The quantization error is a measure of map resolution and is defined as an average distance between each data vector and its best matching unit. Figure 4 shows the map topology with the projections of input vectors and output labels superimposed over the map. A clear border between the clusters gives an evidence that proposed input features are good candidates as input features for the unsupervised ANN. Figure 5 shows the component planes for 2 (of total 58) input features – the borders between the different clusters are clearly seen. MLP ANN utilized in this research had a classification task – to detect an exact defect type. Several architectures of MLP ANN were tested by the means of choosing the optimal network 2 3

Linear combination of peak and RMS values of acceleration. GM stands for gearmesh frequency

Applied Mechanics and Materials Vol. 430

67

architecture from the point of the number of neurons in the hidden layer, type of activation functions and type of the learning algorithm. For building, testing and training, Statistica Automatic Neural Networks package has been used. 840 input vectors (70% of the dataset) were used for training while 180 input vectors were used for cross verification and testing, respectively. The software automatically determined network complexity. 20 networks were tested. The best network is the one with 12 neurons in hidden layer (MLP 58-21-12) and with excellent classification – 100% for each output case.

Figure 4. Distance matrix for SOFM with 58 input features, output labels superimposed over the map – test rig

Figure 5. Component planes for overall acceleration and Kurtosis parameter – test rig

Driven by excellent classification results obtained on the laboratory test rig, authors tested the developed algorithm on a vibration dataset obtained from the online vibration monitoring system installed on the mine strip bucket wheel excavator SRs 1300 ([19], [20]). Two machine faults were successfully identified: 1. Roller elements bearing failure on the bucket wheel gearbox input stage. The success rate of the fault classification was 93%. The roller element bearing fault was a very progressive looseness developed in the very short period (three weeks). The vibration dataset was labeled with 3 output labels where an assumption on linear relationship between the looseness value and time of bearing exploitation is made. The SOFM topology is shown on the figure 6. 2. Tooth crack (missing teeth) of the 3rd transmission stage of the first belt conveyor gearbox. The moment of teeth crack occurrence was evident in the recorded time waveforms and frequency spectra – this was a basis for labeling the input dataset. The success of fault classification was 100%. The SOFM topology is shown on the figure 7.

Figure 6. SOFM topology with color coded output labels superimposed over the map – bearing failure of the bucket wheel gearbox. Quantization error: 0.3497.

Figure 7. SOFM topology with color coded output labels superimposed over the map – gearbox failure of the conveyor belt drive. Quantization error: 0.7273.

68

Acoustics & Vibration of Mechanical Structures

Conclusion For complex machines, such as gearboxes, evaluation of machine condition based on vibration measurements could be a hard task and implementation of AI can help. In this paper, we demonstrated the use of SOFM and ANN for automatic identification of different roller elements bearings and gearbox faults. Excellent classification of existing faults was obtained by the use of ANN. The applicability of the developed ANN was successfully verified on drives of the mine strip bucket wheel excavator. References [1] Baillie D, Mathew J, Diagnosing Rolling Element Bearing Faults with Artificial Neural Networks. Acoustics Australia; 1994; 22(3); 79-84 [2] Bartelmus W, Zimroz R, Batra H, Gearbox vibration signal pre-processing and input values choice for neural network training. Conference proceedings - AI-METH 2003 Artificial Intelligence Methods, Gliwice Poland; 2003; 21-14 [3] Bartelmus W, Zimroz R, Application of self-organised network for supporting condition evaluation of gearboxes. Conference proceedings - Methods of Artificial Intelligence AIMETH Series, Gliwice Poland; 2004; 17-20 [4] Bartelmus W, Zimroz R, A new feature for monitoring the condition of gearboxes in nonstationary operating conditions. Mechanical Systems and Signal Processing; 23, 2009; 1528-1534 [5] Bartelmus W, Zimroz R. Vibration condition monitoring of planetary gearbox under varying external load. Mechanical Systems and Signal Processing; 23, 2009; 246-259 [6] Bishop C. Neural Networks for Pattern Recognition. Oxford University Press 1995 [7] Czech P, Łazarz B, Wilk A. Application of neural networks for detection of gearbox faults. WCEAM CM 2007. Harrogate, United Kingdom, 2007 [8] Guanglan L , Tielin S, Weihua L, Tao H, Feature Selection and Classification of Gear Faults Using SOM. Advances in Neural Networks – ISNN 2005, Springer Berlin / Heidelberg; 2005; 556-560 [9] Hoon S, Worden K, Farrar C, Novelty Detection under Changing Environmental Conditions, LA-UR-01-1894. SPIE's 8th Annual International Symposium on Smart Structures and Materials, Newport Beach, CA; 2001 [10] Jyh-Shing R.J, Chuen-Tsai S, Mizutani E, Neuro-fuzzy and soft computing : a computational approach to learning and machine intelligence. MATLAB curriculum series. Prentice Hall, 1997. [11] Kohonen T. Self-Organizing Maps. Springer, Berlin, 1995. [12] Rafiee J, Arvani F, Harifi A, Sadeghi M.H, Intelligent condition monitoring of a gearbox using artificial neural network. Mechanical Systems and Signal Processing; 21(4); 2007; 17491754 [13] Sadeghi M.H, Rafiee J, Arvani F, Harifi A, A Fault Detection and Identification System for Gearboxes using Neural Networks. Neural Networks and Brain, ICNN&B '05. International Conference. 2005 [14] Sick B., Review On-Line And Indirect Tool Wear Monitoring In Turning With Artificial Neural Networks: A Review Of More Than A Decade Of Research, Mechanical Systems and Signal Processing; 16(4); 2002; 487-546 [15] Staszewski W. J, Worden K, Classification of faults in gearboxes — pre-processing algorithms and neural networks, Neural Computing and Applications; Vol. 5; 1997; 160-183 [16] Veelenturf L. P. J., Analysis and Applications of Artificial Neural Networks, Prentice Hall, 1995 [17] Worden, K.; Sohn, H.; Farrar, C. R., Novelty Detection in a Changing Environment: Regression and Interpolation Approaches, Journal of Sound and Vibration; Volume 258; Issue 4; 2002; 741-761

Applied Mechanics and Materials Vol. 430

69

[18] Zhong, B., Developments in intelligent condition monitoring and diagnostics, System Integrity and Maintenance , 2nd Asia-Pacific Conference(ACSIM2000) Brisbane Australia; 2000; 1-6 [19] Zuber N. Automation of rotating machinery failures by the means of vibration analysis. PhD Thesis, Faculty of Technical Sciences – University of Novi Sad, 2012 [20] Zuber N, Ličen H, Bajrić R. An innovative approach to the condition monitoring of excavators in open pits mines, TECHNICS TECHNOLOGIES EDUCATION MANAGEMENT-TTEM; 5(3); 2010; 841-847 [21] http://www.cis.hut.fi/projects/somtoolbox.

Applied Mechanics and Materials Vol. 430 (2013) pp 70-77 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.70

Rolling element bearings fault identification in rotating machines, existing methods of vibration signal processing techniques and practical considerations Ninoslav Zuber1,a, Dragan Cvetković2,b 1

Faculty of Technical Sciences – University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia 2

Faculty of Occupational Safety – University of Niš, Čarnojevića 10a, 18000 Niš, Serbia a

[email protected], [email protected]

Keywords: vibration analysis, bearing faults

Abstract. This paper addresses the suitability of vibration monitoring and analysis techniques to detect different types of defects in roller element bearings. Processing techniques are demonstrated on signals acquired from the test rig with defective bearings. As a result it is shown that there is no reliable universal method for bearing failure monitoring from its early occurence up to bearings failure. Two real life case studies with different types of bearing failures are presented with practical considerations on methods used for failure identification. Introduction Roller elements bearings (REB) are the most common components in rotating machines and they are claimed to be the most responsible elements for machine’s unplanned stoppages and failures. Unplanned stops leads to loss of production, high maintenance costs and sometimes to losses of human lives. As a result, development of special vibrodiagnostical methods and methods of signal processing, together with the analysis of their applicability in REB diagnostics is very important. Today, we can use several nondestructive testing methods for the evaluation of REB state: vibration measurement and analysis, measurement of acoustic emission, measurement of REB temperatures and infrared thermography, oil analysis and particle counting. Among them, vibration measurement and analysis is the most universal one. Under the assumptions that we use an appropriate transducer (accelerometer with appropriate amplitude and frequency range), that the mounting method used is good enough to adequately transfer the vibration signals from REB housing to transducer and that the appropriate methods of signal analysis is used, acquired vibration signal contains all the necessary information required for reliable identification of REB defect. Premature defects in REB can occur due to many factors where the most common are: fatigue crack, wearing, plastic deformation under the components of the bearing, corrosion, brinelling, bad lubrication, improper mounting methods, inadequate bearing choice etc. Often, the mentioned mechanisms are overlapped and it is very common that one mechanism excites other mechanisms, which leads to total failure of REB. In case of existence of indentation or external particle inside a REB, everytime the roller element passes over the indentation on a race, a decrease of internal strain will be produced. On the other hand, when a roller element passes over the metal particle that is on the race due to the flaking of the components, there will be an increase of REB internal stresses. This transient forces result in rapid change of accelerations of REB components – shock phenomena are present in the vibration signal. Due to the nature of this mechanism, techniques of applied vibration signal processing used in REB diagnostics should be appropriate to identify transient components in acquired signal. The most common used methods of vibration signals analysis applied to identification of defected REB are: time waveform analysis, analysis in the frequency domain, techniques of spectral enveloping and demodulation and time-frequency transformations. Due to the high frequency content of impacts, by default, vibration signals are acquired and analysed in the units of acceleration. Techniques of time waveform analysis are focused on extraction of statistical parameters of raw or

Applied Mechanics and Materials Vol. 430

71

filtered time waveform signals in order to quantify the transient phenomena. The most common universal parameters are: root mean square - RMS, peak-peak value - PP, peak factor – PF and Kurtosis – Kurt. Transformation of time waveform into frequency domain using the FFT alghoritm enables identification of possible typical REB defect frequencies in the signal (BPFO and BPFI– ball pass frequency on outer and iner race, respectively, BS – ball spin frequency, CF – cage frequency). Demodulation / enveloping technique is the advanced signal processing method with its wide application in REB condition monitoring in the early stages of defect development. Repetitive impulses caused by roller elements passage over the indentation could excite REB housing natural frequencies. As a result, frequency spectrum contains peak at the natural frequency modulated by the REB defect frequency. After band pass filtering and signal rectification, envelope spectrum is generated with the sidebands shown as harmonic family. The success of REB fault detection using envelope spectra is highly dependent on filter cut off frequencies. Methods of time-frequency transformations are widely used in REB vibrodiagnostics due to the nonstationary nature of the impact phenomena. As a result, time-frequency transformations result in signal presentation in both domains – time and frequency. This makes possible to analyze frequency contents nonstationarity over the time. The most common used methods for time-frequency transformations in REB diagnostics are: short time frequency transform - STFT, Wigner – Ville distribution – WVD and wavelet analysis - WA. In the last three decades many papers dealing with the diagnostics of REB using vibration measurement and analysis were published. Tandom and Nakra in [10] presented an excellent review of several possible techniques that could be applied for the monitoring of the REB state: analysis of vibration signal in frequency and time domain, sound pressure measurement and analysis, shock pulse measurements and measurement of acoustic emission. Kim and Lowe in [2] presented some alternative techiques for REB monitoring like shock pulse measurement and analysis and methods of spectrographic and ferrography oil analysis. In [4] Mathew and Alfredson presented some of the basic guidelines for REB vibration signal processing. Also McFadden and Smith in [5] and Kim in [3] gave a very detailed review of existing signal processing techniques that could be applied in identification of REB defects using vibration signals. There are several studies published which covers the mechanism of noise and vibration generation in REB ([1], [6], [7], [8], [9], [11]). According to these papers the main cause of noise and vibration generation is the variable stifness in REB due to the fact that the support of the REB is done using a finite number of roller elements. The total number of the roller elements as well as the influence of the loading zone are the main parameters that influence the frequency content and the levels of the vibration and noise signals generated in REB ([8], [9]). Experimental part and results Experimental part of the work described in this paper is done on a specially designed test rig shown on Fig.1.

Fig. 1. Test rig used for data acquisition

72

Acoustics & Vibration of Mechanical Structures

Test rig is consisted of one AC motor 0.37 [kW], which is over an elastic coupling connected to the rotor with two rotating discs where unbalanced masses can be mounted. Rotor is supported by two UC201 REB. Four IEPE accelerometers type AC201-1A (CTC, USA) are mounted in horizontal and vertical radial directions on the REB housings with mounting studs. For speed measurement a non-contact laser sensor (Monarch instruments, USA) is used with a reflecting tape mounted on rotating discs. For simultaneous multichannel data acquisition a 16 channel acquisition system OneproD MVX with a corresponding software for measurement setup and analysis OneproD XPR300 (01dB-Metravib, France) is used. For the experimental purposes, several damages on REB are introduced: bearing in healthy condition, bearing with a defect on the outer race out of the loading zone, bearing with a defect on the outer race in the loading zone, bearing with small defect on the inner race, bearing with a larger defect on the inner race, bearing with small looseness, bearing with large looseness. Local defects on inner and outer races were introduced using smal dental grinders. Loseness was introduced by contamination of bearing’s lubricant with grinding paste and leaving the bearing in operation for some time. REB defect frequencies can be calculated using REB dimension data according to the following formulae: 1 d (1) BPFI  n (1  cos  ) 2

D

1 d BPFO  n (1  cos  ) 2 D 1 d FT  (1  cos  ) 2 D 1D d BS  (1  ( ) 2 cos 2  ) 2d D

(2) (3) (4)

where D stands for the REB pitch diameter, d is the roller element diameter, n is the total number of roller elements and  is the REB contacts angle. In reality these frequencies could differ slightly from calculated due to the sliding inside the REB. Using equaitions 1-4 the applied REB defect frequencies in terms of rotating order are: BPFI = 4.9, BPFO = 3.1, BS = 2.1, CF = 0.39. All measurements are acquired at 22 [Hz] speed of the drive. Figure 2 shows time waveforms for different REB defects obtained on the test rig. It can be seen that for the localised type defects, like indentations on bearing components, time waveforms analysis can be sucesfully used in identification of transients that relate to the REB defect frequencies. In cases where a localised defect is rotating inside a bearing, an amplitude modulation can be present since the amplitudes of transients are dependent on the loading of the roller elements. This can be seen on the figure 2 for the BPFI case. In cases where uniform defects are present in REB – like rotating loseness, time waveform analysis is generaly not very efficient, since the signal contains wideband noise which is not very easy to identify in the time domain.

Fig. 2. Time waveforms for different REB defects under the test

Applied Mechanics and Materials Vol. 430

73

Figure 3 shows frequency spectra, with 3200 lines of resolution in 0 [Hz] – 200 [Hz] frequency band, for different REB defects obtained on the test rig. For the localised type defects, like indentations on bearing components, frequency spectra will contain components of the REB defect frequencies and their harmonics, especially in cases where the indentations on the bearings’ are large enough to produce transients with amplitudes which are visible along with the components from imbalance, misalignment and defects in the low frequency region. In cases where loseness is present in the bearing a raised level of wide band noise is present in the frequency spectra. As the loseness grows inside the REB, level of noise is raised too. It should be noted that measurement and analysis of time waveforms and frequency spectra should be done in the units of acceleration rather than in the units of vibration velocity due to the higher senitivity of acceleration to high frequency phenomena.

Fig. 3. Frequency spectra for different REB defects under the test Figure 4 shows STFT in the form of a sonogram (x and y axis are frequency and time, respectively while the amplitude is given using color coding) for two chosen cases from the test rig - looseness and defect at the inner race of the REB. From the right figure a nonstationarity of amplitudes around a natural frequency of the REB housing (at 4400 [Hz]) can be seen. This is due to the amplitude modulation of the transients that with the periodicity that equals to the BPFI excite the REB natual frequency.

Fig. 4. STFT for different REB defects under the test In order to get reliable information from a defective REB, it is necessary to known how fault development affects the recorded vibration signal in time and frequency domain. The most common case of a bearing fault that we need to analyze is the fault due to the fatigue crack on one of the four possible REB components.

74

Acoustics & Vibration of Mechanical Structures

Initial stage of fault development begins with the occurence of fatigue microcrack just bellow the contact surface of the faulty component. It is necessary to note that the level of degradation is small. Components of the vibration signals that originate from a REB are in the form of low amplitude and high frequency impacts (shock impulses) and friction that occurs due to the abnormal lubrication. These components are located in the high frequency region (from 20 [kHz] and above). In this stage, the signal processing techniques that will possibly give information on a presence of a defect are high frequency methods (ultra sound measurements and shock pulse based methods) and enveloping / demodulation techniques while standard time waveform analysis and FFT will not be very usefull. In the second stage of fault development, cracks on the faulty components become larger and wider and therefore the generated impulses could have enough energy to excite the REB natural frequencies. In this stage the acceleration envelope should give very clear information on the present defect in the REB. Also, high frequency methods and their scalar representatives will rise together with the fault development. Frequency spectra in units of acceleration will possibly give clear information on a fault presence. Raw time waveform will not show any usefull information. In the third stage, REB faults are larger and there could be more of them in comparison with a previous stage. In this phase it is possible to get the bearing seizure. Typical for this stage is a strong excitement of REB natural frequencies. Generaly the wideband noise will be raised, both, in FFT and envelope spectra. High frequency methods will still give an indication on the fault presence. Enveloping / demodulation of acceleration vibration signal will detect the existence of fault inside the REB but the raised noise will start to mask the REB defect frequencies. Frequency spectra – FFT of the vibration velocity and acceleration will give a very reliable and clear information on the bearing fault. Time waveform analysis is very useful for this stage. The third stage is generaly a point where the REB should be replaced, if possible. If not, the vibration analyst should monitor the vibration parameters of the REB on weekly basis. If the REB is still left in operation, initial faults on one component will damage the other REB components. Due to the indentations smearing, it is possible to get lower high frequency components, compared to the previous stage. Due to the material flaking, loosenes in the bearing grows. This directly affects the REB geometry and results in noisy operation and changing the REB defect frequencies. High frequency methods are very unreliable in this stage due to the lowering of high frequency components. The REB faults develops while the high frequency parameters are decreasing! Enveloping / demodulation of acceleration vibration signal could detect the existence of fault inside the bearing but the raised noise very often masks useful components. Frequency spectra – FFT of the vibration velocity and acceleration will give a very reliable and clear information on the REB fault. Time waveform analysis, is very useful for this stage since the impacts and amplitude modulation are visible. Case studies First case presented in the paper deals with the REB fault on a recirculation fan (Figure 5, REB number 4) in the local can manufacture plant. Machine consists of one 30 [kW] motor that drives the fan using a belt transmission. This machine has been monitored once per year in the last 5 years. The upward trend of overall acceleration in last three measurements shows the fan REB fault development (Fig 7). On the other hand, trend of overall values of vibration velocities doesn’t show much information that an ongoing REB fault development is present.

Applied Mechanics and Materials Vol. 430

Fig. 5. Recirculation fan under the test

75

Fig. 6. Screw compressor under the test

Fig. 7. Trend lines for overall acceleration (left) and overal vibration velocity (right) The REB type is 2218 (manufacturer SKF), so using formulae (1)-(4) and REB dimension data the characteristic defect frequencies in terms of rotating orders are: BS=7.02, BPFO=8.19, BPFI=10.81 and FT=0.43. A detailed analysis of time waveforms and frequency spectra reveals the nature of a fault (Fig. 8). The overal amplitude of last acquired time waveform compared to the time waveform acquired at the same location three years ago is much higher. The unit of time waveform is acceleration so this indicates a presence of strong metal to metal impacts in the signal. Comparison of frequency spectra shows two typical sympthoms of a faulty REB: 1. Frequency spectrum is raised by the wide frequency noise which comes from the excessive lossenes in the bearing caused by the REB with local faults left in operation. 2. Frequency spectrum shows the nature of the localized fault – there is a modulation around 3998 [Hz] (REB housing natural frequency) with the frequency of 187.5 [Hz]. Since the fan is rotating at 22.9 [Hz], this is the BPFO component of the bearing. Therefore, the nature of the initial localized fault of the REB is the defective outer race of the bearing.

76

Acoustics & Vibration of Mechanical Structures

Fig. 8. Years 2009 and 2013 measurement comparison on a faulty REB: time waveforms (left) and frequency spectra (right) According to the REB fault development stages explained in the previous section, this REB is considered to work in the last 4th stage of the REB life. As a result of leaving this faulty REB in operation, a failure of frequency inverter and a failure of drive occurred 6 months before the last measurement! Rotating machine in the previous case study has been monitored systematicaly in the last 5 years so the development of a REB fault could be observed over time. Very often the vibration analyst is left in front of the machine that has no historical data and the identification of the REB state could be a real engineering challenge. This was a case for the ammonia screw compressor in a local brewery (Fig. 6). Rated power of the compressor is 375 [kW]. The suspicion in the presence of the faulty REB located at the compressor input stage was a high level of overall acceleration value: 7.47 [g] in RMS and 56.6 [g] in peak-peak values. The REB type is 7317BDB (manufacturer NSK) and its characteristic defect frequencies are: BS=4.26, BPFO=4.95, BPFI=7.05 and FT=0.41.

Fig. 9. Frequency spectrum with the harmonic family of BPFI (left) and BPFO (right) components Frequency spectrum acquired in axial direction shows two asynchronous harmonic families: one at the 4.95X and the second at the 7.05X, where X stands for the harmonic order. The first one relates to the BPFO and the second to BPFI component. Their sum equals to 12 which is the total number of the roller elements in the bearing – this is a very usefull practical tip in relating the asyncronous components to the bearing frequencies. After disassembling the REB from the machine a visual inspection showed the presence of faults on both races of the REB.

Applied Mechanics and Materials Vol. 430

77

Conclusion Vibration analysis is the proven and the most reliable method for identification of faults inside the REB. The success and the content of the information we get from the measurement are highly dependent on the signal processing methods that were applied. The paper demonstrated that there is no unique universal method for monitoring the development of a fault inside the bearing from its mounting up to bearing’s wreck. The default measurement setup should include all the explained signal processing techniques in order to identify the REB fault in all developing stages. References [1] Choudhury A, Tandon N. A theoretical model to predict vibration response of rolling bearings to distributed defects under radial load. Trans ASME, J Vibr Acoust 1998;120(1):pp 214-220. [2] Kim PY, Lowe IRG. A review of rolling element bearing health monitoring. In: Proceedings of Machinery Vibration Monitoring and Analysis Meeting, Vibration Institute, Houston, TX, 1921 April, 1983. pp 145-154. [3] Kim PY. A review of rolling element bearing health monitoring (II): preliminary test results on current technologies. In: Pro-ceedings of Machinery Vibration Monitoring and Analysis Meeting, Vibration Institute, New Orleans, LA, 26-28 June, 1984. pp 127-137. [4] Mathew J, Alfredson RJ. The condition monitoring of rolling element bearings using vibration analysis. Trans ASME, J Vibr, Acoust, Stress Reliab Design 1984;106: pp 447-453. [5] McFadden PD, Smith JD. Vibration monitoring of rolling element bearings by the high frequency resonance technique - a review. Tribol Int 1984;17(1):pp 3-10. [6] Meyer LD, Ahlgren FF, Weichbrodt B. An analytic model for ball bearing vibrations to predict vibration response to distrib-uted defects. Trans ASME, J Mech Design 1980;102: pp 205-210. [7] Sunnersjo CS. Rolling bearing vibrations Ð geometrical imper-fections and wear. J Sound Vibr 1985;98(4): pp 455-474. [8] Sunnersjo CS. Varying compliance vibrations of rolling bearings. Journal of Sound and Vibration 1978;58(3): pp 363-373 [9] Tallian TE, Gustafsson OG. Progress in rolling bearing vibration research and control. ASLE Trans 1965;8(3):pp 195-207. [10] Tandon N, Nakra BC. Vibration and acoustic monitoring tech-niques for the detection of defects in rolling element bearings - a review. Shock Vibr Digest 1992;24(3):pp 3-11. [11] Wardle FP, Poon SY. Rolling bearing noise - cause and cure. Chartered Mech Engr July/ August 1983:pp 36-40. [12] Zuber N., “Automation of rotating machinery failure identification by the means of vibration analysis”, PhD Thessis, Faculty of Technical Sciences, University of Novi Sad, 2010. [13] Zuber N, Licen H., Bajric R.,: An innovative approach to the condition monitoring of excavators in open pits mines, Technics technologies education managment, Volume 5, number 1, 2010, pp 3-10, ISSN 1840-1503 [14] Mobius Institute Vibration training manual – category III

Applied Mechanics and Materials Vol. 430 (2013) pp 78-83 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.78

Recent advances in vibration signal processing techniques for gear fault detection-A review Rusmir Bajrić1, a, Ninoslav Zuber2, b, Safet Isić3, c 1

Public enterprise Elektroprivreda BiH, Coal Mine Kreka, Mije Keroševica 1, Tuzla, BiH

2

Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, Serbia

3

Faculty of Mechanical Eng., University Džemal Bijedić, University Campus, Mostar, BiH a

[email protected], [email protected], [email protected]

Keywords: vibration signal processing, cyclostationarity, non-stationary signal

gear

fault

detection,

time–frequency

analysis,

Abstract: This paper provides a review of the literature, progress and changes over the years on fault detection of gears using vibration signal processing techniques. Analysis of vibration signals generated by gear in mesh has shown its usefulness in industrial gearbox condition monitoring. Vibration measurement provides a very efficient way of monitoring the dynamic conditions of a machine such as gearbox. Various vibration analysis methods have been proposed and applied to gear fault detection. Most of the traditional signal analysis techniques are based on the stationary assumption. Such techniques can only provide analyses in terms of the statistical average in the time or frequency domain, but can not reveal the local features in both time and frequency domains simultaneously. Frequency/quefrency analysis, time/statistical analysis, time-frequency analysis and cyclostationarity analysis are reviewed in regard for stationary and non-stationary operations. The use of vibration signal processing detection techniques is classified and discussed. The capability of each technique, fundamental principles, advantages and disadvantages and practical application for gear faults detection have been examined by literature review. Introduction Gears are major transmission elements of the most of rotating machines used in the industry today. Their applications include production, manufacturing, mining and other industries. Therefore, they can be considered as one of the critical components of most rotating machinery where the availability of the entire system depends on the proper function of gears or gearboxes. The detection of specific gear faults is a critical task which is often difficult, especially in cases of complex machines like multistage gearboxes. In general, vibration signal processing has been found effective in the detection and diagnosis of gear faults. There are many vibration-based monitoring techniques currently available for the detection of gear faults [1,2]. Many vibration signal processing detection techniques are able to detect different faults. However most of these techniques suffer from low detection quality and/or slow response time, which restrict their use in online or real time fault detection applications [3]. To detect gearbox defects as early as possible to avoid fatal breakdowns of machines and to reduce the secondary damage caused by failures, researchers were forced to pay more attention on vibration signal processing techniques. The existence of the large body of literature published over the past few decades makes it difficult to review each and every article published in this field. Therefore, we will focus on recent advances in vibration signal processing techniques and their applications in gear fault detection. This review is structured as follows. We first review the traditional techniques including frequency/quefrency analysis techniques representation in Section 2. Then a review of the time/statistical analysis techniques is given in Section 3. In Sections 4 we present an overview of the newly developed time–frequency analysis based on spectral kurtosis, customized multiwavelet lifting schemes with sliding window for denoising, wavelet transform and empirical mode decomposition. In the next section, we present some cyclostationarity analysis examples to illustrate the applications of a few recently developed methods in gear fault diagnosis. Finally, Section 5 provides the conclusion and point out some application prospects in gear fault detection.

Applied Mechanics and Materials Vol. 430

79

Frequency/quefrency Analysis Techniques Traditional vibration signal analysis relies upon the spectrum analysis via the Fourier Transform (FT). In the presence of a fault in gearboxes, amplitude modulation and frequency modulation might usually emerge simultaneously. The meshing frequency and its harmonics are usually modulated by the frequency of the shaft rotation speed and its harmonics. When the gear is in good condition, the amplitude of the sidebands is usually small and difficult to detect in the noise [4]. With the development of fault, the energy of meshing signals and sidebands will increase simultaneously; if the sidebands are detected in the signals, serious faults in the gearbox are indicated [4] but it is highly related with fault type which does not have to be a gear fault. When the shaft rotation speed is time-varying, the modulating frequency and carrier frequency will both vary and, in that case, it is difficult to recognize the sidebands and to diagnose the gear faults based on the spectra of vibration signals [4]. The number and amplitude of such sidebands may indicate a fault condition and spacing of the sidebands is related to their source. Driven by its huge success in processing of stationary signals in a wealth of application areas, a FT technique has found other interesting extensions. One such extension is in the particular area of vibrations and machine-health monitoring, called the fast Fourier transform (FFT)-based order analysis (OA) technique, including its order-tracking capability [5]. The OA technique tries to overcome the effect of frequency change on the FFT and hence allows for a better tracking of speed-driven harmonics in rotating machinery. However, the assumption in this technique is that the frequency change within a single time interval is small, so that the necessity of a stationary signal for frequency transformation is not largely violated. If the frequency changes significantly within this time interval, then the FFT will yield an error in the actual value of the signal [6]. An important deficiency of the FFT is its inability to provide any information on the time dependence of the spectrum of the signal analyzed, as results are averaged over the entire duration of the signal. This feature becomes a problem when analyzing non-stationary signals [7]. The spectral analysis may be unable to detect gear failures at an early stage of fault development, above all in the case of local faults, which primarily affect sidebands; in fact, it may be very difficult to evaluate the spacing and evolution of sideband families in the spectrum, taking into account that several gear pairs and other mechanical components usually contribute to the overall vibration. For this reason, many researchers have proposed the application of other vibration analysis techniques [8]. Cepstrum is another signal processing technique, suitable for gear fault diagnostics purposes. The cepstrum signal, defined in the quefrency domain, can be obtained by inverse Fourier transform of the logarithmic auto-spectrum. The families of rahmonics identifiable in the new, quasi time domain are directly related to the frequency characterizing the phenomenon and in addition, since the cepstrum estimates the average sideband spacing over a wide frequency range, it is applicable to both detection and diagnosis of gear faults [9,10]. In the early development stage of the localized faults, the sideband components are not easy to recognize from the spectrum because they are distributed in broad frequency bands and may overlap with some background components [11]. However, in the industrial practice with varying operating conditions gearboxes it is difficult to identify and much more difficult to localize faults from the spectrum or the cepstrum because of the large number of components involved, in particular in the early stage. Time/statistical Analysis Techniques Local faults in gears produce impacts and consequently, transient modifications in vibration signals. Therefore, these signals have to be considered as non-stationary. However, most of the widely used signal processing techniques are based on the assumption of stationarity and globally evaluate signals. Thus, they are not fully suitable for detecting short-duration dynamic phenomena-the time localization of transient events is impossible [8]. Early papers about this topic are those of McFadden [12-14], in which the time synchronous average (TSA) signal is introduced. TSA is suitable for both diagnostic purposes and generation of the reference signal for other and more complex signal processing techniques, since it is based on the periodicity and the stationarity of the original signal. But even though TSA of the signal may contain all information about the smoothly

80

Acoustics & Vibration of Mechanical Structures

operating gearbox, it is often hard to detect clear symptoms of any defect in the gear from the use of TSA only, especially if the defect is at an early stage [15]. The technique may also fail to detect and differentiate between faults, particularly if multiple faults are present simultaneously in multiple gears within the gearbox. A wide variety of different techniques have been explored over the years to further process the TSA to make it more sensitive to early fault detection [16]. Another methodology has been proposed for estimation of the TSA without speed sensor, which requires only an a priori estimation of the shaft speed and the number of teeth of the meshing gears [17]. One drawback of the proposed method is obviously that no more angular reference is available for the TSA for the localisation of a fault on the gear [17]. In the process of diagnosis by many scalar indicators, kurtosis is found suitable in identifying the defect signatures of rotating machinery; however, kurtosis is not sufficient to localize the defects in the case of multiple damaged gear teeth. Time-frequency Analysis Limitation of the FFT has led to time–frequency signal processing tools, such as the Short-Time Fourier Transform (STFT), the Wigner-Ville Distribution (WVD) and others. Combet and Gelman have proposed optimal denoising, using Wiener filter based on the spectral kurtosis (SK) methodology, to enhance the small transients in gear vibration signals, in order to, detect local tooth faults such as pitting at early stage [18]. The problem of local fault detection in gears can be related to the more general problem of transient detection in a signal. In that purpose, a SK detection technique has been proposed [19]. The SK is a tool sensitive to non-stationary patterns in a signal and that can indicate at which frequencies those patterns occur. Furthermore, the SK can be used to design detection filters that adaptively extract the fault signal from the noisy background [18]. From the SK-based filtered residual signal, called the SK-residual, it is possible to define the local power as the smoothed squared envelope, which can be interpreted as the sum of the time-frequency energy distribution weighted by the values of the SK at each frequency, and so by the degree of non-stationarity of the transients [18]. Wang [20] proposed to apply the resonance demodulation technique which was based on envelope analysis of the residual signal after band-pass filtering within an excited resonance. Multiwavelet denoising techniques suffer from such main drawbacks as the fixed basis functions independent of the input dynamic response signals and the universal threshold denoising [21]. This may lead to the loss of some critical but relatively weak information in the fault feature detection [21]. In order to overcome the above limitations for effective gear fault detection, a novel method incorporating the customized multiwavelet lifting schemes with sliding window denoising is proposed by the same authors. Proposed method outperforms various wavelet methods as well as SK [21]. When the shaft rotation speed is time-varying, the modulating frequency and carrier frequency will both vary and, in that case, it is difficult to recognize the sidebands and to diagnose the gear faults based on the spectra of vibration signals [22]. At present, to deal with non-stationary and non-linear signals, time frequency analysis techniques such as STFT, wavelet transform (WT) [23,24], WVD [25,26] and Hilbert–Huang transform [27] are widely used. The time frequency analysis involves three-dimensional functions that allow for visualizing the frequency and amplitude variations of the spectral components [23]. WT is a relatively new and powerful tool in the field of signal processing, which overcomes problems that other techniques face, especially in the processing of non-stationary signals. WT [28] and empirical mode decomposition (EMD) [29] are the two methods most commonly used in the diagnosis of gearbox faults. WT is not a self-adaptive signal decomposition method essentially [30]. In the EMD method, signals are adaptively decomposed into several sums of intrinsic mode functions (IMFs) whose instantaneous frequencies have physical meaning. IMF’s instantaneous frequency and instantaneous amplitude are both obtained through the Hilbert Transform. In practice, the IMF is always multi-component, rather than a single component, which leads to an unexplainable irregularity in its instantaneous frequency. Therefore, the EMD method is not suitable for decomposition of signals with multiple components in a narrow band [31]. To overcome this limitation sparse signal decomposition method based on multi-scale chirplet has been developed. Favorable simultaneous resolutions in the time domain and the frequency domain and

Applied Mechanics and Materials Vol. 430

81

comparatively strong immunity to noise make the method highly suitable for non-stationary signal decomposition, especially the decomposition of signals from gearboxes with time-varying shaft rotation speeds [22]. An interesting innovative dynamic technique for gear damage monitoring within noisy environment that is based on wavelet multi-resolution analysis (WMRA) is presented in Refs. [32]. Authors emphasize that the gearbox vibration signal is first passed though Kaiser’s window with a certain sliding rate. The technique applies the WMRA on the windowed versions of the signal, evaluates the maximum resolution coefficients, and associates them with the window center location in time domain. The peak of the maximum coefficients value and its location are collected for a number of consecutive gear cycles and assessed against reference values. The windowing parameters and the mother wavelet are optimally selected while the machine is running. The optimally selected parameters are then utilized on-line to detect early stage damages within noisy environment. To overcome the inherent deficiencies of conventional time–frequency analysis a novel scheme named as the time–frequency data fusion (TFDF) is developed by extending the idea of data fusion technique. TFDF has potential to render a significantly improved time-frequency representation and greatly facilitates extracting time–frequency features of target signals [33]. Maximum correlated Kurtosis deconvolution (MCKD) is another novel state-of-the-art method which takes advantage of the periodicity of the faults. Simulation and experimental results indicate that the MCKD method is the most successful in deconvolving the periodic faults and has significantly better gear chip fault detection results, often on the order of 10 or more times better at fault detection on the experimental data [34]. Root mean square of the filtered signal (FRMS) and the normalized summation of the positive amplitudes of the difference spectrum between the unknown signal and the healthy signal (NSDS) are two new diagnostic parameters particularly developed for planetary gearbox fault detection [35]. Above proposed parameters perform better fault identicifaton, compared with the FM0, FM4, energy ratio (ER), sideband index (SI) and sideband level factor (SLF) in detecting and diagnosing planetary gearbox faults [34]. There is an excellent bibliography of time–frequency analysis methods with application examples presented in Reference [36]. To date, various types of time-frequency analysis technique have been proposed. However, the question of how to choose a suitable one among them to match the signal structure remains an open issue. There is still a lack of generally accepted effective methods to address this issue with particular industrial application regard. Cyclostationarity Analysis Industrial gearboxes usually operate under fluctuation load meaning gears are under variable motion. Fluctuating operating conditions tend to induce amplitude, frequency and phase modulation in vibration signals, hence destroying the wide-sense stationarity of the signal, which is implicitly assumed when applying Fourier analysis [37]. Gear vibration signal periodicity may be detected in the signal statistics. For instance, a gear meshing generates first-order cyclostationary vibration signals [37], i.e. signals whose a mean is periodic. Local faults in gears are likely to produce second-order cyclostationarity [38]. The use of cyclo-stationary properties of signals to identify and characterize sources of modulation has been approved in mining industrial gearbox [39]. Nowadays the subject has probably reached its state of maturity, as evidenced by the two excellent bibliographies in Refs. [40,41]. Raad et al. [42] introduced indices of cyclostationarity of orders one to four that generalise the degree of (second-order) cyclostationarity, and proposed a fast algorithm to compute them together with statistical thresholds for use in automatic monitoring. Cyclostationarity offers an elegant and powerful solution when non-stationary originates from periodical mechanisms, such as gearboxes. Conclusion This paper provides a review of the literature, progress and changes over the years on fault detection of gears using vibration signal processing techniques. Statistical and classical timefrequency domain techniques try to offer more direct approach; however all of them do some sort of averaging on the signal, which might suffer loss of time information. This is a disadvantage when

82

Acoustics & Vibration of Mechanical Structures

operating with signals that have very short duration or suddenly occurring component, like a signal generated from faulty gears. Time-frequency technique are more advanced in localization of nonstationary gear fault feature from one point but from another point they are more complicate to implement in practice. In particular, the capability of new approaches based on time-frequency and cyclostationarity analysis are literally reviewed against those obtained by means of the well accepted cepstrum analysis and amplitude and phase demodulation of meshing harmonics. To date, various types of time-frequency analysis technique have been proposed. However, the question of how to choose a suitable one among them to match the signal structure remains an open issue. Further work should literally investigate the vibration signal processing fault feature extraction methods for gear fault detection and classification. References [1] Wilson Q. Wang, Fathy Ismail, M. Farid Golnaraghi, Assessment of gear damage monitoring techniques using vibration measurements, Mech. Syst. Signal Process. 15-5 (2001) 905-922. [2] Andrew K.S. Jardine, Daming Lin, Dragan Banjević, A review on machinery diagnostics and prognostics implementing condition-based maintenance, Mech. Syst. Signal Process. 20-7 (2006) 1483–1510. [3] Sajid Hussain, Hossam A. Gabbar, A novel method for real time gear fault detection based on pulse shape analysis, Mech. Syst. Signal Process. 25-4 (2011) 1287-1298. [4] Fuqiang Peng, Dejie Yu, Jiesi Luo, Sparse signal decomposition method based on multi-scale chirplet and its application to the fault diagnosis of gearboxes, Mech. Syst. Signal Process. 25-2 (2011) 549-557. [5] C.W. de Silva, Vibration: Fundamentals and Practice, second ed., Taylor & Francis, 2007. [6] A. Brandt, T. Lago, K. Ahlin, J. Tuma, Main principles and limitations of current order tracking methods, Sound and Vibration, 39-3 (2005) 19–22. [7] F. Al-Badour, M. Sunar, L. Cheded, Vibration analysis of rotating machinery using timefrequency analysis and wavelet techniques, Mech. Syst. Signal Process. 25-6 (2011) 2083-2101. [8] G. Dalpiaz, A. Rivola, R. Rubini, Effectiveness and sensitivity of vibration processing techniques for local fault detection in gears, Mech. Syst. Signal Process. 14-3 (2000) 387-412. [9] M.E.Badaoui, J.Antoni, F.Guillet, J.Daniere, P.Velex, Use of the moving cepstrum integral to detect and localize tooth spalls in gears, Mech. Syst. Signal Process. 15-5 (2001) 873–885. [10] N.J.Wismer, Gearbox analysis using cepstrum analysis and comb-liftering, Application Note Bruel&Kjær,1994. [11] B. Liu, S. Riemenschneider, Y. Xu, Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum, Mech. Syst. Signal Process. 20-3 (2006) 718-734. [12] P.D. McFadden, Interpolation techniques for time domain averaging of gear vibration, Mech. Syst. Signal Process. 3-1 (1989) 87–97. [13] P.D. McFadden, Examination of a technique for the early detection of failure in gears by signal processing of the time domain average of the meshing vibration, Mech. Syst. Signal Process. 1-2 (1987) 173–183. [14] P.D. McFadden, J.D. Smith, A signal processing technique for detecting local defects in gears from the signal average of the vibration, Proceedings of the Institute of Mechanical Engineers 199 (C4) (1985) 287–292. [15] W.J. Wang, P. McFadden, Application of wavelets to gearbox vibration signals for fault detection, Journal of Sound and Vibration 192-5 (1996) 927–939. [16] Z. Peng, F. Chu, Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography, Mech. Syst. Signal Process. 18-2 (2004) 199–221. [17] F. Combet, L. Gelman, An automated methodology for performing TSA of a gearbox signal without speed sensor, Mech. Syst. Signal Process. 21-6 (2007) 2590-2606. [18] F. Combet, L. Gelman, Optimal filtering of gear signals for early damage detection based on the spectral kurtosis, Mech. Syst. Signal Process. 23-3 (2009) 652–668.

Applied Mechanics and Materials Vol. 430

83

[19] J. Antoni, The spectral kurtosis: a useful tool for characterizing non-stationary signals, Mech. Syst. Signal Process. 20-2 (2006) 282–307. [20] W. Wang, Early detection of gear tooth cracking using the resonance demodulation technique, Mech. Syst. Signal Process. 15-5 (2001) 887–903. [21] Jing Yuan, Zhengjia He, Yanyang Zi, Gear fault detection using customized multiwavelet lifting schemes, Mech. Syst. Signal Process. 24-5 (2010) 1509-1528. [22] Fuqiang Peng, Dejie Yu, Jiesi Luo, Sparse signal decomposition method based on multi-scale chirplet and its application to the fault diagnosis of gearboxes, Mech. Syst. Signal Process. 25-2 (2011) 549-557. [23] J. Lin, L. Qu, Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis, Journal of Sound and Vibration 234 -1 (2000) 135–148. [24] Ruqiang Yan, Robert X. Gao, Xuefeng Chen, Wavelets for fault diagnosis of rotary machines: A review with applications, Signal Processing(2013) http://dx.doi.org/10.1016/j.sigpro.2013.04.015 [25] Y.S. Shin, J.J. Jeon, Pseudo Wigner–Ville time-frequency distribution and its application to machinery condition monitoring, Journal of Shock and Vibration 1-4 (1993) 65–76. [26] W.J. Staszewski, K. Worden, G.R. Tomlinson, Time-frequency analysis in gearbox fault detection using the Wigner–Ville distribution and pattern recognition, Mech. Syst. Signal Process. 11-5 (1997) 673–692. [27] Hui Li, Yuping Zhang, Haiqi Zheng, Wear detection in gear system using Hilbert–Huang transform, Journal of Mechanical Science and Technology 20-11 (2006) 1781–1789. [28] S.G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, New York, 1999. [29] B. Liu, S. Riemenschneider, Y. Xu, Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum, Mech. Syst. Signal Process. 20-3 (2006) 718-734. [30] J.S. Cheng, D.J. Yu, J.S. Tang, Y. Yu, Application of frequency family separation method based upon EMD and local Hilbert energy spectrum method to gear fault diagnosis, Mechanism and Machine Theory 43 (2008) 712–723. [31] W.X. Yang, Interpretation of mechanical signals using an improved Hilbert–Huang transform, Mech. Syst. Signal Process. 22-5 (2008) 1061–1071. [32] Farag K. Omar, A.M. Gaouda, Dynamic wavelet-based tool for gearbox diagnosis, Mech. Syst. Signal Process. 26 (2012) 190-204. [33] Z.K Peng, W.M Zhang, Z.Q Lang, G. Meng, F.L Chu, Time–frequency data fusion technique with application to vibration signal analysis, Mech. Syst. Signal Process. 29 (2012) 164-173. [34] Geoff L. McDonald, Qing Zhao, Ming J. Zuo, Maximum correlated Kurtosis deconvolution and application on gear tooth chip fault detection, Mech. Syst. Signal Process. 33 (2012) 237-255. [35] Yaguo Lei Yaguo Lei, Detong Kong, Jing Lin, Ming J Zuo, Fault detection of planetary gearboxes using new diagnostic parameters, Meas. Sci. Technol. 23-5 (2012) 055605. [36] Zhipeng Feng, Ming Liang, Fulei Chu, Recent advances in time–frequency analysis methods for machinery fault diagnosis:A review with application examples, Mech. Syst. Signal Process. 38-1 (2013) 165–205 [37] C.J. Stander, P.S. Heyns, Instantaneous angular speed monitoring of gearboxes under noncyclic stationary load conditions, Mech. Syst. Signal Process. 30-4 (2005) 817-835. [38] J. Antoni, F. Bonnardot, A. Raad, M. El Badaoui, Cyclostationary modelling of rotating machine vibration signals, Mech. Syst. Signal Process. 18-6 (2004) 1285-1314. [39] Zimroz, R., Bartelmus, W., Gearbox condition estimation using cyclo-stationary properties of vibration signal, Key Engineering Materials 413-414 (2009) 471–478. [40] E. Serpedin, F. Panduru, I. Sari, G.B. Giannakis, Bibliography on cyclostationarity, Signal Processing 85-12 (2005) 2233–2303. [41] W. Gardner, A. Napolitano, L. Paura, Cyclostationarity: half a century of research, Signal Processing 86-4 (2006) 639–697. [42] A. Raad, J. Antoni, M. Sidahmed, Indicators of cyclostationarity: theory and application to gear fault monitoring, Mech. Syst. Signal Process. 22-3 (2008) 574–587.

Applied Mechanics and Materials Vol. 430 (2013) pp 84-89 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.84

Methods of interpreting the Results of Vibration Measurements to locate Damages in Beams Minda Petru Florin1,a, Praisach Zeno Iosif2,b, Minda Andrea Amalia3,c and Gillich Gilbert-Rainer1,d 1

Department of Mechanics, “Eftimie Murgu” University of Resita, 320085 Resita, Romania 2

Hydro-Engineering, 320168 Resita, Romania

3

Department of Economical Engineering, “Eftimie Murgu” University of Resita, 320085 Resita, Romania a

b

c

d

[email protected], [email protected], [email protected], [email protected]

Keywords: Damage, severity assessment, natural frequency, vibration, fracture mechanics.

Abstract. This paper deals with methods of interpreting the results of vibration measurement to identify structural changes in beam-like structures. We briefly presented an own developed damage assess method, that consider a large number of frequencies for the weak-axis banding vibration modes; it allows first a precise localization and afterwards evaluation of the damages. For the first step, recognition of the damage position, we introduce an algorithm implemented in C++ with the interface done using EXCEL features, indicating by one number the damage probable position, based on the Minkowski metrics. To avoid uncertainties, a graphic representation of all results is also presented. The method is tested for values determined by calculus for a randomly selected location, with and without measurement results debased by noise, proving its reliability. Introduction Damage assessment of structures, especially based on global methods, present great interest for engineering practitioners. They quantify the integrity of a structure by examining changes in its dynamic response to excitations or the static behavior under load. Most dynamic methods presented in literature [1,2] bases on parameters such as the natural frequencies, mode shapes, mode shape curvature, flexibility matrix and stiffness matrix. The main idea of these methods is to find some damage indicators that are sensitive to structural changes and use these data either to compare the features of healthy state with the damaged one by means of recognition techniques [3-6], or to change features of the model to fit its response to that identified by measurement on the damaged structure [7-10]. The most vibration based methods require measuring responses at several locations on the structure, not always possible due to operational and technical constrains. However, these methods dose not consider the physical phenomenon in deep, being just oriented to fitting features or cost reduction; consequently no scientific feedback regarding their reliability is possible. Prior researches developed by the authors of this paper lead to a mathematical relation expressing the frequency changes due to damage in respect to damage position and severity. This finding was used to develop a two-step method to assess damages in beams, first being found the position and afterwards the severity of damage. This paper presents a method based on the Minkowski metrics, implemented in C++ with a MS Excel interface, able to automatically identify damage locations. Method to detect damage locations In previous researches [11,12] we derived the exact solution for frequency changes due to damage in beam-like structures, for any transversal vibration mode and beam support type. It makes possible to express the frequency for mode i of the damaged beam with a crack of depth a placed at distance x from one end, denoted fi_D(x,a), considering the frequency of the undamaged beam fi_U in that mode and two terms controlling the depth and location of the damage. This relation is:

Applied Mechanics and Materials Vol. 430

85

f i _ D ( x, a ) = f i _ U 1 − γ ( xB max , a ) ⋅ (φi′′( x))2 

(1)

where γ(xBmax,a) is the term representing the stiffness reduction calculated on the location where the bending moment attend maxima (for the cantilever beam at the fixed end xBmax = 0) and φi′′( x) the normalized mode shape curvature, weighting the influence of stiffness reduction according to the damage position and mode shape. Obviously, this last squared tem takes values between 0 and 1. ∆f ( x, a ) fi _U − fi _ D ( x, a) as: From Eq. 1 one can derive the relative frequency shift δ fi ( x, a) = i = fi _ U fi _U

δ fi ( x, a) = γ (0, a) ⋅ (φi′′( x))

2

(2)

For any location xj on the beam we can derive the values of the relative frequency shift for n bending vibration modes. Normalizing the values obtained with the right term, by dividing these one by one to the highest value of the series, we obtain the damage location coefficients DLC, as:

Φ1 ( x j ) =

(φ1′′( x j )) 2

max {(φi′′( x j ))2 }

, Φ2 ( x j ) =

(φ2′′( x j )) 2

max {(φi′′( x j )) 2 }

, ... , Φ n ( x j ) =

(φn′′( x j ))2

max {(φi′′( x j ))2 }

, j = 1.. k

(3)

One observe that the DLC depend only on the mode shape curvature squares (φi′′( x)) 2 , as the term γ(0,a) is independent of location x and thus eliminated by normalization. A series of DCL specific for a damage location are called damage location index DLI; it can be represented as a histogram (e.g. Fig. 1.b-e) and characterize uniquely the dimensionless damage position x/L on any cantilever. Imagine now that we obtain the relative frequency shift by processing data from measurements for n vibration modes. This series of n values determined with the left term of Eq. 2 can be normalized by dividing them to the maximum value of the series, obtaining:

Ψ1 =

δ fn δ f1 δ f2 , Ψ2 = , ... , Ψ n = max(δ f i ) max(δ fi ) max(δ f i )

(4)

Comparing now the resulted series from (4) with every DLI from (3) the xj coordinate indicating the damage location is found. The approach is described in Fig. 1.a.

b.

c.

e. d. a. Fig. 1. Damage location algorithm (a) and examples of resulting histograms for damages in a cantilever beams placed at: (b) x/L = 0.3; (c) x/L = 0.55; (d) x/L = 0.67 and (e) x/L = 0.8.

86

Acoustics & Vibration of Mechanical Structures

Automated recognition of damage location A histogram Ψ={Ψi} is a graphical representation of non-negative data Ψi corresponding to n bins, where i = 1…n. Numerous measures are proposed for the dissimilarity between two histograms Ψ ={Ψi} and Φ={Φi}. The bin-by-bin dissimilarity measures only compare contents of corresponding histogram bins, i.e. they compare Ψi and Φi for all i, but not Ψi and Φk for i ≠ k. The dissimilarity between the two histograms is a combination of all the pairwise comparisons; a ground distance is used by these measures [15]. Some available estimators are presented in Table 1. Table 1 Dissimilarity estimators Histogram Intersection ∑i min ( Ψ i , Φi ) d∩ ( Ψ, Φ ) = 1 − ∑ Ψi

Minkowski-Form Distance 1

 r r d Lr (Ψ , Φ ) =  ∑ Ψ i − Φ i   i 

Kullback-Leibler Divergence d Ψ ,Φ = ∑ Ψ i log i

Ψi Φi

i

In this paper we use a method similar to the bin-by-bin approach involving the Minkowski distance dL2, where the bins are represented by vibration modes and the content represents the normalized relative frequency shift. Additionally, due to the possibility that the frequencies of some modes cannot be read accurately, we introduced a weighting factor wi. Thus, the estimator becomes:

d L 2 (Ψ, Φ) =

∑w

i

Ψ i − Φi

2

(5)

i

For normal conditions the weighting factor wi is the unit, while for total uncertainty it has credited with the null value. From previous expertise one can set intermediate values for the weight factor wi , increasing the performance of the recognition algorithm.

Results and discussions To validate the method described above, we realized a series of experiments base on a developed application able to an automated recognition of the damage location on the beam which follows the steps described in a previous section. As specimens, steel beams were used, having quadratic crosssection b = h = 0.012m, but different lengths. The material parameters of the specimen are: mass density ρ = 7850 kg/m3, Young’s modulus E = 2·1011 N/ m2 and Poisson’s ratio υ = 0.3. During the tests, the beams where fixed on one end in a milling machine vise, which assured optimal fixing conditions. The damages were made by saw cutting, the cuts being approximately 2 mm wide and reducing the cross-section by around 25%. The structure excitation for determining the natural frequency was realized by pushing the beam out of the equilibrium position. We measured the first 9 frequencies for every beam supposed to be undamaged and introduce the information, together with that concerning the beam geometry and characteristics in a database. This happens by pushing the button Save. Afterwards it is possible to call the information back from an interface realized in MS Excel, as shown in Fig. 2. This interface permit to introduce the support types; it automatically evaluate if the Euler-Bernoulli theory is adequate to model the beam. After creating the crack, we have measured again the natural frequencies of the studied beam and put them in the third column of our application. This information can be also be saved. We compare now the actual state with the initial state by using the dL2 estimator. If it takes values higher than 0.05, the beam is supposed to be damaged (obvious in our case); else it is considered undamaged. As example we have selected the beam 1, with set to L = 1.4m and one end clamped, the other free. The application displays the initial frequencies (if always saved for this length and support types) and a message will appear indicating that the Euler-Bernoulli model is proper to be used.

Applied Mechanics and Materials Vol. 430

87

If damage is presumed, the application calculates a series of DLIs for imposed number of location situated equidistant along the beam by selecting this number and acting the Calculate button. The DLI is compared with the series Ψ obtained from the measurement results by means of the Minkowski distance dL2. The DLI best fitting the Ψ series (i.e. smallest value of dL2) is represented as a histogram (central diagram in Fig. 2) as well as the Ψ series (left diagram in Fig. 2). Additionally, a diagram representing al calculated dL2 values is presented on the right bottom side of Fig. 2. Finally, the location providing the dL2 values is identified as the damage location. The role of the three diagrams is to permit a visual interpretation and validation of the results analytically determined. In our case the response of the application was that the probable location of the damage is at x/L =0.4 from the left end. At this point it is possible to store the frequency result as the Last known state and next evaluations can take as benchmark the Initial state or the Last known state.

Fig. 2 An application for automated damage location recognition To prove the method’s robustness, the measurement results have been altered artificially by considering three cases of noise contaminated measurements. As we can see in Table 2 and Fig. 3.a, in the first noisy case only 2 frequencies are affected by noise (case N1), in the second and third cases all frequencies are affected differently by noise. The second case (N2) is affected by errors in the positive and negative domain as well, while in the third case (N3) only negative errors occur. The relative frequency shifts of all measured data for the four analyzed cases are calculated with Eq. 2 and included in Table 3. For these results the dL2 is applied to find the DLI best fitting to the relative frequency shift obtained from measurement.

88

Acoustics & Vibration of Mechanical Structures

Table 2 Errors affecting the measured frequencies Mode i 1 2 3 4 5 6 7 Err fi_D (N1) 9.51% 0 7.27% 0 0 0 0 Err fi_D (N2) -9.53% 3.34% -1.37% 5.82% 2.35% -6.16% 5.95% Err fi_D (N3) -1.93% -6.9% -3.1% -4.55% -7.07% -2.55% -5.97%

Mode i fi_U fi_D δ fi f i_D (N1) δ fi (N1) fi_D (N2) δ fi (N2) fi_D (N3) δ fi (N3)

8 0 1.17% -3.2%

9 0 9.64% -0.8%

Table 3 The measured frequencies and the relative frequency shifts 1 2 3 4 5 6 7 8 4.993 31.285 87.552 171.431 283.094 422.356 589.014 782.831 4.918 30.545 86.297 170.141 274.578 422.103 574.359 771.923 0.0152 0.0236 0.0143 0.0075 0.03 0.0006 0.0248 0.0139 4.45 30.545 80.02 170.141 274.578 422.103 574.359 771.923 0.1089 0.02367 0.086 0.00752 0.03008 0.0006 0.02488 0.01393 5.387 29.524 87.482 160.234 268.123 448.135 540.142 762.856 -0.078 0.0563 0.00081 0.065 0.052 -0.061 0.082 0.0255 5.013 32.653 88.975 177.892 294.007 432.873 608.652 796.672 -0.003 -0.0437 -0.0162 -0.0377 -0.0385 -0.0249 -0.0333 -0.0177

9 1003.53 997.905 0.005 997.905 0.00561 901.678 0.1015 1005.9 -0.0024

a.

b.

Fig 3. The errors introduced in the relative frequency shift for the three noisy cases and the Minkowski distance for all four analyzed cases One observes the good results obtained using this method, even for strongly altered measurement results, qualifying the method as reliable and applicable in industrial applications.

Conclusions This paper presents an application based on a method able to determine the damage location for beam-like structures. The recognition program is realized in C++ and use the Minkowski distance to compare histograms constructed on measurement result performed on the real beam on one hand, and analytically on models. It uses a friendly interface to communicate with the users, done using EXCEL features. The probable damage location is presented as one-number indicator, being also possible to confirm the result based on a visual recognition method, implying the histograms and the dL2. The damage locations obtained with this application are quite accurate and have been experimentally confirmed. The program was proved in practice and found as very helpful, while the any beam can be introduced in the database and afterwards monitored in a very simple way. As future work, we intend to develop the application in order to make it able to estimate the damage depth.

Applied Mechanics and Materials Vol. 430

89

References [1] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage identification methods, Shock and Vibration Digest, 30(2), 1998, 91-105. [2] G.R. Gillich, E.D. Birdeanu, N. Gillich, D. Amariei, V. Iancu, C.S. Jurcau, Detection of damages in simple elements, Annals of DAAAM for 2009 & Proceedings of the 20th International DAAAM Symposium, Book Series: Annals of DAAAM and Proceedings 20 (2009) 623-624 [3] X.Q. Zhu, S.S. Law, M. Jayawardhan, Experimental study on Statistical Damage Detection of RC Structures based on Wavelet Packet Analysis, Journal of Physics: Conference Series 305 (2011) Paper: 012107 [4] J.H. Chou, J. Ghaboussi, Genetic algorithm in structural damage detection, Computers & Structures, 79(14) (2001) 1335–1353 [5] A. Cheung, C. Cabrera, P. Sarabandi, K.K. Nair, A.Kiremidjian, H. Wenzel, The application of statistical pattern recognition methods for damage detection to field data, Smart Mater. Struct. 17 (2008) 065023 [6] S.M. Pourhoseini Nejad, Gh.R. Ghodrati Amiri, A. Asadi, E. Afsharid, Z. Tabrizian, Damage detection of skeletal structures using particle swarm optimizer with passive congregation (PSOPC) algorithm via incomplete modal data. Comp. Meth. Civil Eng., 3(1) (2012) 1-13 [7] J.-J. Sinou, A Review of Damage Detection and Health Monitoring of Mechanical Systems from Changes in the Measurement of Linear and Non-linear Vibrations, in: Robert C. Sapri (Ed.), Mechanical Vibrations: Measurement, Effects and Control, (2009) 643-702. [8] C.R. Farrar, D.A. Jauregui, Comparative Study of Damage Identification Algorithms Applied to a Bridge: I Experiment, Smart Materials and Structures, 7 (1998) 704–719. [9] P.K. Jena, D.N. Thatoi, J. Nanda, D.R.K. Parhi, Effect of damage parameters on vibration signatures of a cantilever beam, Procedia Engineering 38 ( 2012 ) 3318 – 3330. [10] C.G. Koh, M.J. Perry, Structural Identification and Damage Detection using Genetic Algorithms, Structures and Infrastructures Book Series, Vol. 6, CRC Press, 2009. [11] G.R. Gillich, Z.I. Praisach, Robust method to identify damages in beams based on frequency shift analysis, Conference on Health Monitoring of Structural and Biological Systems, San Diego, CA, March 12-15, 2012, Article Number: 83481D. [12] G.R. Gillich, Z.I. Praisach, Damage patterns based method to locate discontinuities in beams, Conference on Health Monitoring of Structural and Biological Systems, San Diego, CA, March 1114, 2013, Article Number: 8695-1100. [12] G.R. Gillich, Z.I. Praisach, D.M. Onchis, About the effectiveness of damage detection methods based on vibration measurements, 3rd WSEAS International Conference on Engineering Mechanics, Structures, Engineering Geology/International Conference on Geography and Geology Corfu Island, Greece, Jul 22-24, 2010, 204-209. [14] D. Onchis-Moaca, G.R. Gillich, R. Frunza, Gradually improving the readability of the timefrequency spectra for natural frequency identification in cantilever beams, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO) Book Series: European Signal Proceedings Conference (2012) 809-813 [15] Y. Rubner, C. Tomasi, L. J. Guibas, The Earth Mover’s Distance as a Metric for Image Retrieval, International Journal of Computer Vision 40(2), (2000) 99–121. [16] P. Bratu, Vibratiile sistemelor elastice, Editura Tehnica, Bucuresti, 2000

Applied Mechanics and Materials Vol. 430 (2013) pp 90-94 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.90

Evaluation of Crack Depth in Beams for known Damage Location based on Vibration Modes Analysis Praisach Zeno Iosif1,a, Gillich Gilbert-Rainer2,b, Protocsil Carla2,c and Muntean Florian2,d 1

Hydro-Engineering, 320168 Resita, Romania

2

Department of Mechanics, “Eftimie Murgu” University of Resita, 320085 Resita, Romania a

b

c

d

[email protected], [email protected], [email protected], [email protected]

Keywords: Damage, severity assessment, natural frequency, vibration, fracture mechanics.

Abstract. The paper presents a method to assess the crack depth in beams for which the damage location is known. Previous researches lead us to a method to identify damage locations in beams, based on a relation providing frequency changes in respect to damage location and depth. Separating the two variables it is possible to find first the damage location, and afterwards to derive the term controlling the severity. Comparing it with values indicating the frequency shift in respect to damage depth, the severity can be assessed. The paper presents a relation reflecting this dependency for any cross-section type, involving the static deflection for the healthy and damaged beam alone; it has a general character, being not influenced by the cross-section shape. Introduction There are two approaches in using frequency as a candidate for damage detection: numerical and analytical. In the numerical approach, where finite element models are used, the stiffness of the model is changed systematically until the natural frequencies of the numerical model matches those measured on the real structure [1,2]. In the analytical approach, the damage is modeled as a spring, assuming that it produces a discontinuity in stiffness. Rizos et al. [3] modeled the crack as a rotational spring whose stiffness was proportional to the influence of damage at a given location. Liang et al. [4] adopted the same methodology but solved the inverse problem by separating the matrix of the characteristic equation, avoiding any nonlinear solvers. Ostachowicz and Krawczuk [5] derived the characteristic equation for beams containing multiple cracks. The concept of modeling the crack as a massless rotational spring was later used in different references for various support conditions, numbers of cracks and loading of the beam [6]. Later, closed-form expressions of the mode shapes and resonant frequencies of a vibrating cracked beam were formulated for different boundary conditions and crack specifications [7-10]. This paper presents a method to assess damage severity in cracked beams, for which the damage location is known, by considering the relation between natural frequency changes and damage depth. Opposite to other authors, we consider the stiffness reduction at the location where the bending moment is maximum, afterwards weighting it for damages placed on other locations with the influence of energy stored in that location. Damage location In accordance with developed method of authors [11-15], for damage location there is no necessary to know the damage severity. The relation which expressing the natural frequencies fi-D for a beam with a discontinuity in form of closed crack of depth a placed at distance xc from the beam end is:

[

f i − D ( x C , a ) = f i −U 1 − γ (0, a ) ⋅ ( φ i′′ ( x C )) 2

]

or

fi − D ( xC , a ) = 1 − γ (0, a ) ⋅ ( φi′′( xC )) 2

(1)

Applied Mechanics and Materials Vol. 430

91

Here fi-U is the frequencies of the undamaged beam for the i-th bending vibration mode, γ (0, a ) is a term depending on the damage severity representing the stiffness loss and φi′′ ( x C ) is the normalized mode shape curvature of the i-th mode, given by:

φi′′ ( x C ) =

φ ′i′ ( x C ) max(φ ′i′ ( x))

(2)

and its square reflect the way how the effect of stiffness loss is diminished in respect to the local curvature. Dividing for any location xC the n values of the relative frequency shifts to the highest value of the series, the term γ (0, a ) is eliminated and the normalized relative frequency shifts for that location are obtained as:

Φ 1 ( xC ) =

( φ1′′ ( x C )) 2 max((φi′′ ( x C )) 2 )

, Φ 2 ( xC ) =

( φ 2′′ ( x C )) 2 ( φ n′′ ( x C )) 2 , ... , Φ ( x ) = n C max((φ i′′ ( x C )) 2 ) max((φi′′ ( x C )) 2 )

(3)

The series presented in Eq. 3 is now independent of depth a, consequently it characterize the crack location xC alone. Since patterns for any damage location are known, it is easy to implement a method to detect and locate damages in beam-like structures. Suppose that by starting monitoring a beam the measured frequencies for undamaged beam are FUm : { f 1m−U ,... f i −mU ,... f nm−U } and for damaged beam we obtain FDm : { f 1m− D ,... f i m− D ,... f nm− D } , with i = 1…n. Applying Eq. 3 from the two series FUm and FDm we obtain the relative frequency shift for n modes, i.e. n dimensionless values: F *m : {∆f 1*m ,...∆f i*m ,...∆f n*m } These values can be normalized by dividing them to the highest value

of the series; the mathematical formulation is presented below: Ψ1 =

∆f 1*m max(∆f i*m )

, Ψ2 =

∆f 2*m max(∆f i*m )

, ... , Ψn =

∆f n*m

(4)

max(∆f i*m )

Comparing the series Ψ obtained using Eq. 4 with numerous series Φ obtained from Eq. 3 by considering a large number of locations x along the beam, one can find the location xC where the terms of the two series match together. Therefore, damage location is reduced to a pattern recognition problem.

Evaluation of crack depth For evaluation of crack depth it will be use a steel cantilever beam (Fig. 1) with L = 600mm length and different types of cross sections: rectangular, square, circular, hexagonal and I profile (Fig. 2). vUmax vU L Fig. 1. Cantilever beam

b

h

H

D

h

h

a

a

a

a

Acoustics & Vibration of Mechanical Structures

a

92

H

h

Fig. 2. The analyzed cross sections The maximum deflection of the undamaged beam is noted with vUmax, and the maximum deflection for the damaged beam is vDmax. The cross section dimensions are: rectangular cross section with b = 50mm width and h = 5mm height; square cross section with h = 16mm width and height; circular cross section with D = 18mm diameter; hexagonal cross section with H = 17mm key opening and I profile cross section with H = 40mm width, h = 10mm height and 5 mm thickness, so that all the analyzed cases have the same mass approximately. The damage depth is noted a and has values between 8% and 58% from the height of the cross section. The term γ (0, a ) in Eq. 1 represents the stiffness loss at the location where the curvature achieves maxima and has, for a given beam, the same value for all bending vibration modes. It is: γ (0, a ) = 1 −

vU max vD max (a )

(5)

where vU max is the maximum deflection of the undamaged beam and vD max (a) the maximum deflection of the damaged beam under own load, the damage of depth a being located at the place where the bending moment has the maximum value. The deflection values can be calculated for particular cases by means of the finite element method. It is also possible to use, in stand of Eq. 5, some relations derived from the fracture mechanics theory. For rectangular cross-sections in literature are presented: Eq. 6 determined by Rizos et al. [3], Eq. 7 by Ostachowicz and Krawczuk [5], Eq. 8 by Bilello [16] and Eq. 9 by Chondros et al. [17]. In these relations we denoted a/h=d.

γ (d ) = 5.346[1.8624d 2 − 3.95d 3 + 16.375d 4 − 37.226d 5 + 76.81d 6 − 126.9d 7 + 172d 8 − − 143.97 d 9 + 66.56d 10 ] γ (d ) = 6πγ 2 [0.6384 − 1.035d + 3.7201d 2 − 5.1773d 3 + 7.553d 4 − 7.332d 5 + 2.4909d 6

γ (d ) =

]

d (2 − d ) 0.9(d − 1) 2

(

)[

γ (d ) = 6π 1 − µ 2 0.6272d 2 − 1.04533d 3 + 4.5948d 4 − 9.9736d 5 + 20.2948d 6 − 33.0351d 7 + + 47.1063d 8 − 40.7556d 9 + 19.6d 10 ]

(6)

(7)

(8)

(9)

where µ is the Poisson’s ratio. However, opposite to the relation proposed by this paper (see Eq. 5), it is necessary to scale Eq. 6 to 9 to find the real stiffness loss due to damage; for the rectangular cross-section the scale factor is equal to πh. The reliability of Eq. 1, respectively Eq. 5 is proved by means of the finite element method. Table 1 present the maximum deflection and the natural frequencies for the first vibration mode obtained for cantilever undamaged beam and damaged with various damage ratios. In all cases Eq. 1 accurate predict the frequency of the damaged beam by considering the effect of stiffness reduction calculated with Eq. 5 and using the deflection values from Table 1.

Applied Mechanics and Materials Vol. 430

Depth a/h [%] 0 8 17 25 33 42 50 58

93

Table 1. Deflections and natural frequencies for different type of cross sections Rectangular Square Circular Hexagonal I profile vmax f vmax f vmax f vmax f vmax f [mm] [Hz] [mm] [Hz] [mm] [Hz] [mm] [Hz] [mm] [Hz] 2.962 11.384 0.292 36.263 0.307 35.332 0.310 35.174 1.354 16.835 2.978 11.353 0.295 36.046 0.309 35.225 0.313 35.001 1.363. 16.782 3.012 11.288 0.304 35.532 0.316 34.862 0.321 34.580 1.380 16.679 3.061 11.197 0.316 34.823 0.326 34.290 0.332 33.993 1.398 16.569 3.135 11.065 0.335 33.820 0.344 33.411 0.349 33.141 1.439 16.331 3.268 10.836 0.369 32.219 0.377 31.896 0.388 31.731 1.529 15.837 3.466 10.520 0.419 30.218 0.429 29.868 0.430 29.846 1.688 15.068 3.815 10.022 0.505 27.485 0.527 26.908 0.522 27.045 2.017 13.767

Fig. 3 present the frequency change due to damage of dimensionless depth a/h, for beams with rectangular cross section, plotted using the Gillich-Praisach relation (Eq. 5) and using the scaled functions presented in Eq. 6 to 9; one observes the good fit between the proposed relation and especially that proposed by Rizos et al. [3]. Additionally, our relation is valid for any cross-section shape, proved using data from Table 1, extending the availability of the frequency-based damage detection methods to beams with any cross-section shape and even made by composites. Obviously, to assess the damage depth it is sufficient to calculate from Eq. 5 the value of γ(0,a), possible if the frequency shift is measured and the damage location identified using the algorithm presented in [11], and finally to evaluate the severity using nomograms like that presented in Fig. 3.

Fig. 3. Graphical representation for function γ(a/h) proposed by different authors Conclusions This paper introduces a mathematical relation who expresses the stiffness reduction in respect to damage depth, applicable to any beam structure. It can be involved in evaluating damage severity in a two-step damage assessment algorithm developed by the authors, permitting first to find the crack location and afterwards to evaluate its severity. Opposite to other approaches, the contrived relation presents a large degree of generality, being applicable irrespective to boundary conditions and crosssection shape. It totally bases on the (simple) physical phenomenon, providing accurate results. As future research, we intend to find the method’s sensitivity for various cross-section shapes.

94

Acoustics & Vibration of Mechanical Structures

References [1] P. Cawley, R.D. Adams, Location of defects in structures from measurements of natural frequencies, Journal of strain analysis, 14(2) (1979) 49-57. [2] L. Kannappan, S. Krishnakumar, Frequency measurement based damage detection methods applied to different cracks configuration, 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 23-26 April 2007, Honolulu Hawaii, 1-12. [3] P. F. Rizos, N. Aspragathos, A. D. Dimarogonas, Identification of crack location and magnitude in a cantilever beam from the vibration modes, Journal of Sound and Vibration, 138(3) (1990) 381388. [4] R.Y. Liang, J. Hu, F.K. Choy, Detection of cracks in beam structures using measurements of natural frequencies, Journal of Franklin Institute, 328(4) (1991) 505. [5] W.M. Ostachowicz, M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam, Journal of Sound and Vibration, 150(2) (1991) 191-201. [6] M. Ashfari, D.J. Inman, Continuous crack modeling in piezoelectrically driven vibrations of an Euler-Bernoulli beam, Journal of Vibration and Control, 19(3) (2012) 341-355. [7] N. Khiem, T. Lien, A simplified method for natural frequency analysis of a multiple cracked beam, Journal of Sound and Vibration, 245(4) (2001) 737-751. [8] J. Fernandez-Saez, C. Navarro, Fundamental frequency of cracked beams in bending vibrations: an analytical approach, Journal of Sound and Vibration, 256(1) (2002) 17-31. [9] Q. Li, Vibratory characteristics of Timoshenko beams with arbitrary number of cracks, Journal of Engineering Mechanics, 129(11) (2003) 1355-1359. [10] K. Aydin, Vibratory characteristics of Euler-Bernoulli beams with an arbitrary number of cracks subjected to axial load, Journal of Vibration and Control, 14(4) (2008) 485. [11] G.R. Gillich, Z.I. Praisach, Damage patterns based method to locate discontinuities in beams, Conference on Health Monitoring of Structural and Biological Systems, San Diego, CA, March 1114, 2013, Article Number: 8695-1100. [12] G.R. Gillich, Z.I. Praisach, D.M. Onchis, About the effectiveness of damage detection methods based on vibration measurements, 3rd WSEAS International Conference on Engineering Mechanics, Structures, Engineering Geology/International Conference on Geography and Geology Corfu Island, Greece, Jul 22-24, 2010, 204-209. [13] G.R. Gillich, Z.I. Praisach, Robust method to identify damages in beams based on frequency shift analysis, Conference on Health Monitoring of Structural and Biological Systems, San Diego, CA, March 12-15, 2012, Article Number: 83481D. [14] G.R. Gillich, Z.I. Praisach, I. Negru, Damages influence on dynamic behavior of composite structures reinforced with continuous fibers, Materiale Plastice, 49(3) (2012) 186-191. [15] G.R. Gillich, Z.I. Praisach, D.E. Birdeanu, Considerations on natural frequency changes in damaged cantilever beams using FEM, 3rd WSEAS International Conference on Engineering Mechanics, Structures, Engineering Geology/International Conference on Geography and Geology, Corfu Island, Greece, Jul 22-24, 2010, 214-219. [16] C. Bilello, Theoretical and experimental Investigation of Damaged Beams under Moving System, Ph.D. Thesis, Universita degli Studi di Palermo, Italy, 2001. [17] T.J. Chondros, D.A. Dimarogonas, J. Yao, A continuous cracked beam vibration theory, Journal of Sound and Vibration, 215(1) (1998) 17-34. [18] P. Bratu, O. Vasile, Modal Analysis of the Viaducts Supported on the Elastomeric Insulators within the Bechtel Constructive Solution for the Transilvania Highway, Romanian Journal of Acoustics and Vibration, 9(2) 2012 77-82

Applied Mechanics and Materials Vol. 430 (2013) pp 95-100 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.95

Assessment of corrosion damages with important loss of mass and influences on the natural frequencies of bending vibration modes Gillich Gilbert-Rainer1,a, Praisach Zeno Iosif2,b, Bobos Daniel1,c and Hatiegan Cornel3,d 1

Department of Mechanics, “Eftimie Murgu” University of Resita, 320085 Resita, Romania 2

Hydro Engineering S.A., 320168 Resita, Romania

3

Department of Electrical Engineering, “Eftimie Murgu” University of Resita, 320085 Resita, Romania a

[email protected], [email protected], [email protected], [email protected]

Keywords: Damage detection, corrosion, mass loss, stiffness, natural frequency, vibration.

Abstract. Corrosion as material destruction affects the safety of structures, leading to more serious consequences than the simple loss of a mass. The effect of corrosion on the dynamic behaviour of structures act in two ways: the loss of mass increases the natural frequencies, opposite to the effect of stiffness loss. This paper present the results of researches made to extend the mathematical relation presenting the influence of damage location and severity on the natural frequency changes, by adding the effect mass loss. Therefore we modeled the beam once with the discontinuity and loss of mass, afterwards the damaged segment is replaced by an intact one but having the mass similar to that of the damaged segment. This permitted to plot frequency shift curves for both cases and to contrive the relations defining that curves. Finally a relation summarizing the both effects was contrived; it was confirmed both by numerical simulations and experiments. Introduction Corrosion, as a consequence of interaction between materials and their environment, determine changes in the materials properties and geometry with negative impact upon the performances of the technical system to which they belong [1]. This process always triggers from the surface, but sometimes penetrates deep inside the materials; consequently, it is difficult to evaluate its impact by visual or local methods. On the other hand, global methods presented in literature reviews [2-4] are not proper to be used for such kind of damage, because beside the stiffness, the mass suffer important changes, making impossible to use the frequencies of the healthy structure as a benchmark [5]. Models of corroded structures Several criteria are used to classify the different corrosion types; among them the corrosion mechanism [6], the characterization of corrosion phenomena according to the appearance of the corroded area are the classical one [7-8] and the influence upon the mechanical characteristics as stiffness, fatigue and hardness [9]. Regarding the appearance of the corroded area, the basic forms of corrosion are: (1) generalized uniform corrosion, which can be both uniform, where the metal surface is affected at the same rate on large areas; (2) generalized non-uniform corrosion, characterized by different corrosion rates in different zones of the surface; (3) localized corrosion, which is restricted to reduced areas and takes the form of pits, crevices or cavities. The importance of accurate modelling of corrosion damage is essential for a correct analysis of its influence upon the dynamic behaviour of structures, a series of approaches being presented in literature, i.e. [10]. Some typical damage geometries are presented in Fig. 1, in which also the equivalent models are presented.

96

Acoustics & Vibration of Mechanical Structures

a.

b.

c.

d.

Fig.1. Typical corrosion geometries and the equivalent geometrical models: (a) and (b) generalized uniform corrosion; (c) and (d) local corrosion One remarks that for generalized uniform corrosion loss of mass and cross-section reduction occurs, hence this damage is best represented by a stepwise modelled beam with two moments of inertia: for the undamaged section IU and for the corroded ID (IDαz ,(czzco, an effect which is amplified

with the reduction of Dz.Within the domain v1€(0, 2 ) there run, in fact, rigid-suspension freight wagons, so that the presence of dampers is justified, particularly for the vehicles which run inside the resonance domain. ξ6

ψ/ π

Dz=0,0

5 4

1 0,9

Dz=0,0

0,8

Dz=0,1

0,7

Dz=0,1

0,6

3

Dz=0,2 Dz=0,3

2

Dz=0,2

0,5

Dz=0,3

0,4

Dz=0,5

0,3

Dz=1,0 1

Dz=0,5

0,2

Dz=1,0

0,1

0

0

0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50

v1

Fig. 2a. The amplification factor of the amplitude A2 function of the normal non-dimensional pulsation

0

0,25

0,5

0,75

1

1,25

1,5

1,75

2

2,25

v1

Fig. 2.b. The initial phase ψ function of the normal non-dimensional pulsation

2,5

Applied Mechanics and Materials Vol. 430

ξ' 5

Ψ' /π

Dz=0,0

4,5 4

1

Dz=0,1

0,6

2,5

0,5

2

0

0,25

0,5

0,75

1

1,25

1,5

1,75

2

Dz=0,5

0,2

Dz=0,5

Dz=1,0

Dz=1,0

0,3

Dz=0,3

1

Dz=0,3

0,4

Dz=0,2

1,5

0

Dz=0,1 Dz=0,2

0,7

3

0,5

Dz=0,0

0,9 0,8

3,5

199

0,1

2,25 2,5 v1

0 0

0,25 0,5 0,75

1

1,25 1,5 1,75

2

2,25 2,5 v1

Fig. 3.a. The amplification factor of the Fig. 3.b. The initial phase ψ` function of the ` amplitude A2 function of the normal normal non-dimensional pulsation non-dimensional pulsation In the case of the dynamic perturbations (Figure 3-a), for v1€(0, 2 ) and at low values of Dz , it results ξ' > 1 , therefore A2>zst. The critical speed corresponding to the resonance pulsation is:

λ λ kz ⋅ ωz = (27) 2π 2π m In case of the critical speed regime, or close to it, the amplitudes may decrease by increasing the factor Dz. Within the domain v1 > 2 , the phenomenon takes place in a reverse manner. Here the dampers reduce the amplitudes only for the dynamic excitation. The decrease of “ ξ ” is essential for low values of Dz. It represents the domain of passenger trains circulation. If the railways are of a high quality, D z= 0,3...0,35 must be chosen, whereas in the case of a low-quality railways, with a lot of level differences, Dz ≈ 0,25 is preferred. The function Ψ(v1) (Figure 2-b) indicates the lagging behind of the function z(t) as compared to zc(t) for different values Dz. For the dynamic perturbations (Figure 3-b) the delay increase.After a certain amount of time, the free (self-induced) vibrations are damped, while the vehicle is subjected only to the vibration caused by the disturbing forces with the pulsation “p”, having the amplitude ξ(p)zco or ξ’(p) zst. It may be concluded that for a transient state the oscillation depends both on the initial conditions and on the perturbation forces and for the stationary state, the oscillation depends only by the disturbed forces, function of the pulsation “p”. For the case of the double levels suspension vehicles, the forced vibrations should be approached in a similar manner, nevertheless two areas of resonance are to be distinguished:for the k1 + k 2 c + c2 self-induced pulsation of the bogies → ω1 = and α1 = 1 and for the self-induced m1 2 m1 v cr =

pulsation of the axle box → ω 2 =

k2 c and α 2 = 2 . m1 2 m2

References [1] V.Marinca, N.Herisanu, Nonlinear Dynamical Systems in Engineering, Some Approximate Approaches,Springer,Berlin,Heidelberg,2011 [2] E.Ghita,Gh. Turos-Dynamics of Railway Vehicles,Ed.Eurostampa,Timisoara,2006,pp.156-167 [3] E .Ghita-Strength on wheel-rail contact,Ed.Mirton,Timisoara,1998,pp.13-36. [4] T.Mazilu,I.Sebesan,Vibrations of Railway Vehicles, Ed.Matrix Rom,Bucuresti,2010,pp.143-146

CHAPTER 4: Biomechanics

Applied Mechanics and Materials Vol. 430 (2013) pp 203-207 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.203

Particularities of upper limb movements of healthy and pathologic subjects Mirela Toth-Taşcău1,a, Flavia Bălănean1,b and Dan-Ioan Stoia1,c 1

“Politehnica” University of Timisoara, Romania, 300006 Timisoara, P-ta Victoriei no. 2 a

mirela @cmpicsu.upt.ro, [email protected], [email protected]

Keywords: upper limb movement, Zebris measuring system, joint angle, range of motion, lateral distal humerus implant.

Abstract. The paper presents a comparative study of the kinematic parameters of the upper limbs of one healthy subject and one patient with lateral distal humerus implant. This study aims to identify the movement patterns of the upper limb joints, having a particular interest in elbow joint, due to the patient’s pathology. Both subjects have been recorded in identical conditions, performing the same exercise what simulates one of the common daily activities. The kinematic parameters which have been analyzed were flexion-extension and abductionadduction in shoulder joint and flexion-extension of the elbow joint. The joint angles have been averaged per each valid trial and exercise. The comparison of the joint angle variation was performed in terms of normalized time. Standard deviation was computed to evaluate the variability of joint angles. Movement symmetry between left and right arms was evaluated by computing the p-values of the averaged series. Introduction Functionality of the upper limb joints is usually assessed clinically using various types of scoring systems [1]. Assessment of the upper limb mobility has two main drawbacks: no standardized movements of the upper limb, and subjectivity of the clinical assessing based on different scoring systems. Most studies dealing with biomechanical behavior of the elbow implants are based on numerical simulations and in vitro experiment, simulating the fracture osteosynthesis [2], [3]. Study of the elbow joint movements is a component of the overall study of the movements in upper limb joints in most of the published work. The reported experimental motion analyses had different objectives [4]: − determining the maximum range of motion (ROM) of the shoulder and elbow joints; − determining the functional range of motion of joints related to most common activities of daily life [5], [6], with and without any restriction [7]; − comparison of the pathological movements with normal movements [8], [9], [10] to assist the physician in diagnosing or evaluation of the rehabilitation progress; − development of a database with kinematic (angular amplitudes) and dynamic (forces and moments of the joints) parameters to further develop computerized biomechanical models of the upper limb that can be used in clinical applications [5], [11]; − design prostheses of the upper limb joints; − assessing of shoulder and elbow functionality following Arthroplasty [1], [12], [13] or implantation; − determining the loads responsible for implants or prosthesis failure. There is a great lack of data related to assessment of shoulder and elbow functionality following Arthroplasty or implantation. We found in literature only few reports on functional outcome following the implantation based on in vivo three-dimensional kinematics of artificial elbow joint [1], [12], [13], but no report on elbow mobility after implantation with certain stabilization system.

204

Acoustics & Vibration of Mechanical Structures

This study aims to identify the movement patterns of the shoulder and elbow joints, having a particular interest in elbow joint, due to the patient’s pathology. Based on comparison of the pathological and normal pattern the patient rehabilitation can be assessed. Materials and Methods Kinematic analysis of the movements in shoulder and elbow joints was performed using Zebris CMS-HS measuring system, in Motion Analysis Laboratory at the Research Centre in Medical Engineering from "Politehnica" University of Timisoara. The measurements and data processing have been developed based on a specific protocol. The measuring system consists in two ultrasound emitters and two pairs of ultrasound markers that are applied on the body of the investigated subjects [14]. The recordings were conducted at a sampling rate of 25 Hz. Two persons voluntarily involved in the study: one healthy subject S1 (male, 59 years old) who stated that there is no known pathology of the upper limb or spine and a patient S2 (male, 56 years old) having an elbow implant (lateral distal humerus locking plate code SHBEP8TDS - Stryker), due to an open comminuted fracture of the right distal end of the humerus (Fig. 1). The condition of the patient at the time of recording was after 2 months of intensive rehabilitation of the right upper limb. Both investigated subjects provided informed consent before participating in the experiment.

Fig. 1. Radiographic images of right elbow joint The subject calibration position was set at a distance of about 1 m from each of the two ultrasound emitters. Two special ultrasound marker sets (one set on the upper limb and one set on the lower limb) were attached on each side of the subject body using Velcro patches, according to Zebris Operating Instructions [14]. Both subjects have been recorded in identical conditions, performing the same exercise that simulates a movement that is similar to eating or drinking. Both the patient and the subject executed a 5 minutes training session before the effective recording, in order to accommodate with the measuring requirements and ensure the movement repeatability. Also, the training session consists of various arm exercises that lead to a good muscular and ligament warm up, aiming the achievement of the maximum amplitudes in movements. After the training session ten anatomical landmarks have been marked on each subject side using the system pointer [14]. The movement cycle consists in lifting the upper limb from the neutral position, targeting the tip of the nose (or up to the pain symptoms, in the case of the patient), and returning to neutral position. For both subjects, both upper limbs have been investigated. Each investigation involved three trails of six movement cycles. The data acquired by the measuring system are the spatial coordinates of the anatomical landmarks. WinGait software generates the geometrical model of the subject and allows determining of the joint angles variation during the selected cycles. The raw data have been exported from the WinGait software for further processing.

Applied Mechanics and Materials Vol. 430

205

Results and Discussions The kinematic parameters which have been analyzed were flexion-extension and abductionadduction in shoulder joint and flexion-extension of the elbow. Average duration of the movement cycle was about 1.6 sec for the healthy subject and approx. 2.4 sec for the patient. The joint angles have been averaged per each trial and exercise. Standard deviations have been computed to evaluate the variability of joint angles. The movement symmetry between left and right arms was evaluated by computing the P values of the averaged series. A comparative kinematical analysis of the angle variations of flexion-extension and abductionadduction in shoulder joint and flexion-extension in elbow joint was performed for both left and right upper limbs. To compare the movement patterns of the investigated subjects, a normalized time representation was performed. Comparison of kinematic parameters recorded for two investigated subjects is supported by the graphs presented in Fig. 2 to Fig. 7: − In case of the flexion-extension movement of the left shoulder joint (Fig. 2), the healthy subject performed movements with a maximum amplitude (-42.517°) lower than the maximum amplitude of the patient which recorded -57 350°. It may be noted also that the range of motion is reached by both subjects at mid-cycle moment. The start position of subject S2 shows an extension of a few degrees (7.35°). − Graph in Fig. 3 presents the comparison of flexion-extension movement of the right shoulders of the two subjects and reveals major differences of the two movement patterns: ROM in right shoulder of subject S2 is much lower than that achieved by the right shoulder of subject S1. − Comparing the movement of abduction - adduction of the left shoulder joint (Fig. 4) of the two subjects is observed that subject S1 shows a greater ROM than subject S2. Also, in the reference position, the left upper limb of both subjects amounts to 9.05° (S2), and 10.633° (S1) abduction. − ROM of the abduction-adduction in right shoulder (Fig. 5) of subject S2 is lower than that of subject S1 (16.137° as against 25.383°). Major differences consist in a much lower range in case of subject S2, the angle of the neutral position, and shape of the graph which contains no longer two cycles of abduction-adduction, but one. − Comparison of flexion-extension movements of the left elbow joint (Fig. 6) shows the smallest differences, meaning approximately equal ROM (125.600° for subject S1 as against 106.933 ° for subject S2). − Comparison of flexion-extension movements of the right elbow joints (Fig. 7) shows large differences of ROM: 80.463° for subject S2, and 125.283° for subject S1, respectively. It also notes that in the neutral position, right elbow is in flexed position of about 45°, due to the existence of plate stabilization. An overview of the determined range of motion and variability of the studied movements is presented in Table 1. The movement symmetry between the left and right upper limbs of each of the investigated subject was analyzed using a paired t-test, two samples, and unequal variances (Table 2). The threshold p-value chosen for statistical significance was 0.05. As expected, there were found significant differences (p-value 15 2.27 2.27

2010. day night 41.00 7.00 38.64 59.09 9.09 34.09 0.00 2.27 2.27 2.27

2011. day night 41.00 7.00 52.27 25.00 18.18 47.73 6.82 15.91 4.55 9.09

Conclusion The absence of the international standard or internationally recognized methodology for the creation of acoustic zones has determined the authors of this paper to develop a concept of acoustic zoning which is in accordance with the current regulations in Serbia and the specificities of urban areas. The developed conception of acoustic zoning is based on the application of two approaches for acoustic zoning and the identification and assessment of different zones – the qualitative and quantitative approaches. Acoustic zoning is carried out in 6 different phases which are described in detail in the paper. The adopted conception of acoustic zoning has been applied to the example of the territory of the city of Nis and its municipalities. The territory of the city of Nis and its municipalities have been acoustically divided into 6 acoustic zones with the dominant representation of a residential zone.

250

Acoustics & Vibration of Mechanical Structures

Acoustic zoning of the territory of the city of Nis and the results of a noise monitoring which were collected in the previous period during the procedure of a systematic noise monitoring which has been conducted in the city of Nis since 1996 have created the conditions for the assessment of the condition of noise levels. The collected results of the assessment of the condition of noise levels point out that limit exceedances are present at 59% of the measurement spots during the day and at 93% of the measurement spots at night, which points to the fact that Nis can be treated as a very noisy city. Acknowledgement This research is part of the project “Development of methodology and means for noise protection from urban areas” (No. TR-037020) and “Improvement of the monitoring system and the assessment of a long-term population exposure to pollutant substances in the environment using neural networks“ (No. III-43014. The authors gratefully acknowledge the financial support of the Serbian Ministry for Education, Science and Technological Development for this work. References [1] Guiding Principles for Sustainable Spatial Development of the European Continent, Guiding Principles for Sustainable Spatial Development of the European Continent, 2010 [2] SILENCE (Suistinable Development Global Change and Ecosystems), Practionitioner Handbook for Local Noise Action Plans, 2008 [3] SMILE (Sustainable Mobility Intitatoves fro Local Environment), Guidlines for Rod Traffic Noise Abatment, 2004 [4] J. L.Cueto at all, Decision-making tools for action plans based on GIS: A case study of a Spanich agglomeration", Proceedings of Inter-noise, Lisabon, 2010 [5] M.Prascevic, Cvetkovic D, Environmental noise (book), University of Nis, Faculty of occupational safety of Nis, 2005. [6] S.Barbaro, R. Caracausi, Noise in hospital areas - measure and evaluation, Proceedings of 14th International Congress on Sound and Vibration, 2007, Australia [7] G. Brambilla, Noise and Soundscape in Rome, Proceedings 147th Meeting of the Acoustical Society of America, 2004, New York [8] L. Cosimo, G. Umberto, Acoustic zoning and noise map of the territory of Taranto (Italy), Tecni Acustica, 2000, Madrid [9] Law on the Environmental Noise Protection, "Official Gazette of RS", no. 36/2009 and 88/2010 [10] Regulation on Noise Indicators, Limit Values, Assessment Methods for Indicators of Noise, Disturbance and Harmful Effects of Noise in the Environment, "Official Gazette of RS", no. 75/2010 [11] Rulebook on Methodology of Acoustic Zoning, "Official Gazette of RS", no. 72/2010

Applied Mechanics and Materials Vol. 430 (2013) pp 251-256 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.251

Noise Control in an Industrial Hall Vasile Bacria1,a , Nicolae Herisanu1,b 1,2

“Politehnica” University of Timişoara, Bd.Mihai Viteazu, 1, 300222, Timişoara, Romania a

[email protected], [email protected]

Keywords: noise control, industrial hall, mitigation methods.

Abstract. Inside the industrial halls one can find different noisy machines and equipments. During their work these ones generate noise and vibrations which affect human beings inside or outside the hall. In this paper we present the results obtained in the investigation of the acoustic field generated by sources within an industrial hall emphasizing the frequency spectra, characteristic parameters, propagation way and some effects generated. A description of measurements is included together with an analysis of obtained results as well as the establishment of noise mitigation methods consisting in acoustical arrangement of the hall in order to eliminate the unpleasant effects. Introduction Machines and equipments working in the industrial environment are characterized by a high noise level and a complex spectrum which affect human beings inside or outside the hall. Having in view this problem, we proposed to investigate the noise sources from an industrial hall of bottling juices and from another industrial hall equipped with machines used in automotive device production, by determining noise levels and characteristic spectra, the propagation way and some effects generated, identifying also some noise mitigation methods based on acoustical arrangement of the halls. Within the industrial hall of bottling juice, the most important sources of noise are the installation of bottling, the movement of the conveyor band, the functioning of some vibrators for driving the conveyor band, the washing machine for bottles, the set for fixing caps on bottles, collisions between bottles, the system of moving empty bottles towards the bottling installation etc. The second industrial hall is intended for manufacturing some automotive devices and is equipped with cutting machines, grinders, assembling machines and devices etc. The noise generated by these ones originates in mechanical, electromagnetical, aerodynamic and hydrostatic processes which arise during the working regimes. Thus, the causes of noise appearance during the mentioned processes are: the shock interaction of two or more bodies, the friction of interacting surfaces, the aerodynamic turbulences, the forced oscillations of the rigid bodies, the action of variable electromagnetic forces, the vibration of membrane-shaped parts, and pulsating pressure in hydrostatic operated devices. The noise generated inside of the industrial hall can be characterized by the equivalent noise level

1 L Aeq,Te = 10 lg   Te

∫

Te

0

 10 0.1L ( t ) dt , 

(1)

where Te is the time of daily exposure to noise., It can be observed that for determining LAeq,Te it is necessary to know the instantaneous noise level L(t). Effectively, this noise level can be also obtained by direct measurements using sound level meters. Using the value of LAeq,Te one can determine the daily exposure of workers:

252

Acoustics & Vibration of Mechanical Structures

L EP ,d = L Aeq,Te + 10 lg

Te , T0

(2)

where T0=8h=28800s. With the help of this variable one can compute the weekly average exposure to noise by:  1 5 0.1( L )  L EP ,w = 10 lg  ∑10 EP ,d k ,  5 k =1 

(3)

where (LEP,d)k are the values of LEP for each of the 5 working days from the considered week. Control of the acoustical field Taking account of the shape and size of perturbation sources, in the closed space of the industrial hall one forms a very complex acoustical field by overlapping direct and reflected waves which could be spherical and cylindrical waves, or even plane waves at long distance from the source. The acoustic pressure from a point in the field is obtained by adding the acoustic pressures corresponding to each type of waves. In the case when the hall has a parallelepiped shape and the surfaces are perfectly reflective, the number of vibration modes where the frequencies are inferior to certain frequency f is given by the relation [1]: N=

4πV 3 πS 2 L f + 2 f + f, 3c 3 4c 8c

(4)

where V is the total volume of the hall, S is the total surface and L is the perimeter. The number of normal modes of vibration included in a certain frequency band which extends from f to f+df, when the hall is excited by a certain sound is given by [1] πS L  4πV dN =  3 f 2 + 2 f + df . 8c  2c  c

(5)

This expression is valid when df is small compared to f. The closed space inside of the hall behaves as a resonator which resonates on the proper frequencies. The more regulate the distribution of proper frequencies is, the more uniform the acoustic filed will be. The sound pressure level in a point of the acoustic field is given by L = 20 lg

p , p0

(6)

where p is the acoustic pressure in a point of the acoustic field and p 0 = 2 ⋅10 −5 N/m2 is the reference pressure. In order to evaluate the acoustic pressure one must take into account that the boundary surfaces of the hall are not perfectly reflective and when the acoustic waves collide with these surfaces, a dissipation of energy takes place and the stationary waves generated by the closed space from inside the hall will be damped. Therefore, in the equation of the stationary waves it must be introduced the damping term e − δt by replacing the wave number k with a constant of propagation, which is a complex number of the form k + δ / c ⋅ j , where δ is the damping constant of the waves. In the wave equation, also the phase angles must be included since they are non-vanishing as a result of the fact that the boundary surfaces of the hall are not rigid and perfectly reflective.

Applied Mechanics and Materials Vol. 430

253

Concerning the situation outside of the industrial hall, the difference is that the propagation happens in free space. In the case of propagation of spherical waves in elastic, homogeneous and isotropic environment, the acoustic pressure in a point of the acoustic field is given by [1] A p = ρ 0 ω sin(ωt − kr + α ), (7) r where r is the radial coordinate, A is the amplitude of the spherical wave which propagates from the source with the speed c, and k is the wave number. In the case of propagation as cylindrical waves traveling uniformly, the acoustic pressure can be expressed as [1] p = A[J 0 (kr ) + jY0 (kr )]e − jωt ,

(8)

where r is the cylindrical coordinate, A is a constant, J0 is the Bessel function of the first degree and zero-th order and k is the Bessel-Neumann function of the second degree and zero-th order. Similarly, in the case of plane waves, considering the case of divergent wave, the acoustic pressure in a certain point of the acoustical field is given by p = ρ 0 ωA sin(ωt − kx + ϕ).

(9)

By computing the acoustical pressure in a point of the acoustic field for inside or outside the hall, Eq.(6) allows the determination of the acoustic pressure level from a point of the considered acoustic field.

Measurement of the industrial noise Taking into account the large number and variety of sources which participate in generating the noise, as well as the nature of the generated acoustic waves, the acoustic field which forms inside or outside the industrial halls is extremely complex and its control is recommended to be performed experimentally. The measurements were performed using the Bruel & Kjaer Investigator 2237 and the hand-held analyser type 2250. In the industrial hall of bottling juice the measurements have been made in the points 1 and 2 in the place where the worker stays during his activity right in front of his ear at 1.6 m from the ground and at distances from the walls mentioned in the figure, according to STAS 7150-77. These measurements were performed over a whole working week for the point 1 and a day for the point 2, over an interval of 8 hours every day in normal working conditions.

Fig. 1. Position of the measurement points

254

Acoustics & Vibration of Mechanical Structures

The results of the measurements for the points 1 and 2 are presented in tables 1 and 2, respectively. The values of the noise levels LAeq shown in tables 1 and 2, which correspond to formula (1) are obtained by measurement over a daily exposure Te=T0=8 h and therefore it overlaps the personal daily exposure LEP,d.

No Freq. Time weight weight 1 2 3 F A 4 5

Table 1. Results of the measurements in point 1 Dom. Time interval LAeq,Te LEP,d MaxP [dB] [dB] [dB] [dB] 8.12-16.12 82.7 82.7 120.8 8.15-16.15 90.7 90.7 114.1 506.58-14.58 90.7 90.7 115.4 120 6.45-14.45 91.0 91.0 118.5 6.40-14.40 92.9 92.9 117.5

MaxL [dB] 111.5 108.2 104.4 107.6 103.2

Table 2. Results of the measurements in point 2 No Freq. Time Dom. Time interval LAeq,Te LEP,d MaxP weight weight [dB] [dB] [dB] [dB] 1 F A 50-120 6.45-14.45 91.2 91.2 116.2

MinL [dB] 63.2 74.4 76.8 75.4 77.8

MaxL [dB] 126.1

LEP,w [dB]

90.6

MinL [dB] 76.2

For the measurement point 1, located in the place where the employee works at the bottling station, determining of daily personal exposure through measurements for the 5 days of a working week, Eq.(3) allows the calculus of a weekly average LEP,w. In the case when LEP|,d or LEP,w exceed the admissible limit of 87 dB(A), then Eq.(2) allows the determination of the time allowed for so that the daily exposure will not exceeded. For the measurement point 1, Te=3h29’32” and for the point 2, Te=3h2’29”. In order to investigate the noise from inside of the industrial hall intended for manufacturing some automotive parts, 12 measuring points were chosen, each one being located near to working places on the production flow, in the mechanical shop and in the storage space. Measurements were taken during the production process in normal operating conditions during the shift which worked between 6.00-14.00 hours. The microphone was mounted in the place where the worker’s ear is located, during the normal activity, near the machine. The acoustical pressure subjecting the workers does not present fluctuations on a large range of noise levels and does not have irregular time-characteristics. The recorded data were downloaded on a computer and then printed in the laboratory. Simultaneously with the noise measurements, climate conditions were monitorized. The values recorded for the equivalent noise levels in the 12 measured points from inside the industrial hall are presented in table 3.

Measured point LAeq [dB]

Table 3. Equivalent noise levels inside of the hall 3 4 5 6 7 8 9

1

2

82.7

80.8

85.9

83.5

81.2

82.4

82.4

78.6

77.1

10

11

12

74.7

75.8

80.8

In the same time the diagrams for time-variation of the parameters LCpeak and LAeq were recorded and a spectral analysis was also performed and a statistical distribution was obtained. Figures 2 and 3 show these diagrams recorded for the point no.6 from inside the hall.

Applied Mechanics and Materials Vol. 430

Fig. 2. Time-history of the noise level

255

Fig. 3. Spectral distribution of the noise

The results of measurements were stored in a database designed for the study of the noise climate in the industrial environment in the frame of Acoustics and Vibration Laboratory at “Politehnica” University of Timisoara. Noise mitigation methods Using the results of the measurements, the noise climate in the industrial halls can be characterized. The maximum admisible limit at working places for the daily noise exposure level is 87 dB(A). From the obtained data it results that the equivalent noise level recorded inside the industrial hall of bottling juice exceeds this limit and inside an industrial hall equipped with machines used in automotive device production does not exceed this limit. In the benefit of improving the working conditions for the employees by limiting the effects of noise, it is imperative the implementation of some mitigation measures. Thus, if the noise mitigation at source is harder to be done and the attenuation of the noise in the industrial hall of bottling juice is very expensive, it is reccomendable to perform an attenuation of the noise over the propagation way or at the receiver. It would be necessary to reduce the time of exposure for the employees or mounting noise barriers. It would be also recommended to build some sound insulated cabins for the protection of employees during working. From the analysis of the daily exposure of workers to noise in the industrial hall for automotive devices production, it results the followings: - In the measuring points 8,9,10,11, the daily exposure to noise is lower than 80 dB(A). - In the measuring points 1,2,4,5,6,7 and 12 there are found values which imply the action of the employer concerning safety problems, which means that the employer must provide individual protection equipment against noise. - In the measuring point no.3, the daily noise exposure exceeds 85 dB(A) and bearing of the protection equipment is mandatory, and this working place must be properly signalized. For the attenuation of the noise level in the industrial hall equipped with machines used in automotive devices production, it is required the application of reduction measures adequate for each particular machine including the assembling of some acoustic screens and acoustical absorbing treatment for the walls. Conclusions In order to establish the ways the industrial hall generates noise which affects the environment, it is necessary to investigate the noise from inside and outside of the hall. The values obtained for the characteristic parameters of the noise are compared with admissible limits. From the realized comparison one can conclude if the environment is affected or not by the noise and some noise mitigation methods can be established. The presented method of investigation can be easily applied in every practical situation concerning the industrial noise.

256

Acoustics & Vibration of Mechanical Structures

References [1] N. Enescu, I. Magheti, M.A. Sarbu, Acustica Tehnica, Ed. I.C.P.E., Bucuresti, 1998. [2] M. Grumazescu, A. Stan, N. Wegener, V. Marinescu, Combaterea zgomotului si vibratiilor, Ed. Tehnica, Bucuresti, 1964. [3] N. Herisanu, V. Bacria, Considerations concerning the investigation of the noise inside an industrial hall, Proc. XXI Conf. „Noise and Vibration”, Tara, 7-9.10.2008. [4] V. Bacria, N. Herisanu, Considerations upon the noise generated by some Diesel engines used in agriculture, Proc. Int. Symp. USAMVB, Timisoara, 2008

Applied Mechanics and Materials Vol. 430 (2013) pp 257-261 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.257

Some effects of rubberized asphalt on decreasing the phonic pollution Nicolae Herisanu1,a, Vasile Bacria1,b 1,2

“Politehnica” University of Timişoara, Bd.Mihai Viteazu, 1, 300222, Timişoara, Romania a

b

[email protected], [email protected]

Keywords: rubberized asphalt, noise decreasing.

Abstract. An important contribution to the noise generated by the road transportation means in the environment has the noise produced by the tire-road contact. This paper is concerned with the investigation of the noise produced by the contact between the tire and the rolling surface covered by rubberized asphalt. Initial test performed in the past were performed again in identical conditions in order to assess the durability and absorbent properties of the rubberized asphalt over time in the climate specific to Timisoara. It was found that the rubberized asphalt has not changed its acoustical properties over time, since practically the same values were obtained for the equivalent noise level as well as for the spectral distribution of the noise, which confirm the efficiency of rubberized asphalt to decreasing phonic pollution in urban areas. Introduction Noise and vibration pollution is nowadays an issue which received a special attention from engineers around the world, with the aim to obtain efficient solutions intended to reduce as much as possible unpleasant effects upon the human beings and environment. Large urban agglomerations exhibit noise pollution phenomena, especially due to the multiplication of noise sources and also due to their increased powers, which lead to a noisy life climate generating quite important problems in the framework of human health care. In most cases, the road traffic is the main source of noise in the urban environment. As it is known, the noise due to the road traffic primarily depends on the traffic intensity and composition, as well as on the speed of vehicles, but also on the nature of the rolling surface. The noise produced by a vehicle is mainly generated by sources such as the engine, the transmission system, the exhaust of gases and the contact between tires and the rolling surface, the later strongly depending on the nature and state of the road surface texture. This paper will focus upon an analysis of the noise generated by the interaction of the tire with the pavement, which is proposed as rubberized asphalt, having better mechanical properties such as flexibility, durability, strength and resistance to cracking, beside the main property related to noise reduction. Results of experimental measurements performed in order to assess the efficiency of the rubberized asphalt are presented and analyzed. Initial tests were performed two years ago, when the rubberized asphalt was mounted on the test-road and new tests were performed after two years in identical conditions in order to verify the behavior of this material in time, concerning its acoustical properties and its capability to ensure quieter surface transport in urban areas. Noise generated by tire-road contact At it is known, more than 70% of the traffic noise generated on a highway originates at the tirepavement interface [1]. In the actual stage of development of more silent vehicle power units, the interaction between the tire and the road is the main source of noise at vehicles in normal driving conditions for velocities over 30-40 km/h [2]. Generally, this interaction is nonlinear because of the time-varying nature of the contact area and thus the noise emitted is hard to be predicted, but relatively easy to be measured in real conditions [3].

258

Acoustics & Vibration of Mechanical Structures

Basically, the mechanism of noise production at the tire-road contact is compound of two parts: the mechanical and the aerodynamical mechanism. The mechanical mechanism is based on the radial and tangential vibration of the profile elements, radial vibrations of the tire carcass, stick-slip and adhesion stick-snap. The aerodynamical mechanism produce noise due to the cavity resonance in tire tube, air-pumping due to deforming of the tire in the contact region, air resonant radiation and pipe resonance. Noise mitigation using rubberized asphalt The potential of pavement types to reduce traffic noise has been taken into account and used since the 1980s, when the road traffic noise has been recognized to represent a problem. There are known some alternatives based on active and passive methods for road traffic noise mitigation, and among them, the use of rubberized asphalt seems to be one of the most promising, but unfortunately insufficiently exploited until now. Rubberized asphalt consists of regular asphalt mixed with crumb rubber obtained from waste tires and thus using it lead to a double advantage: it contributes significantly to the reduction of waste tires, while crumb rubber is a material relatively inexpensive which has notable sound energy absorbing characteristics. Therefore this would be a solution to the waste tire problem combined with a solution for a durable and long term performance road coating which capture the flexible nature of the rubber in a longer lasting paving surface. Even if such a solution apparently involves higher initial costs, this will be balanced by significantly reduced maintenance costs since the pavemets resist cracking and less material can be used due to increased durability and strength. As a rezult, we will have quiter, safer, and longer lasting roadways. Moreover, rubberized asphalt could be only a part of a complex practical solution intended to reduce noise in a particular situation. Experimental investigations In order to investigate the noise generated by vehicles on different rolling surfaces, measurements were performed using two hand-held analyzers type Bruel & Kjaer 2250 mounted as shown in fig.1, according to the recommendation of STAS 6161/3-82, where lg represents the length of the vehicle under test, lm is the length of the path between the starting and ending point of the measurement, A-A is the starting line, B-B is the arrival line, C-C is the circulation axis, D-D is the line of microphones mounted at 1.3 meters high from the ground.

Fig. 1. Sketch of the measurements place for measuring noise generated by all sources within a vehicle

Applied Mechanics and Materials Vol. 430

259

We note that the standardised arrangement of microphones depicted in fig.1 has been used only for determination of global noise emitted by all sources within a vehicle (engine, gases exhaust, tireroad contact etc.). For tire-road contact noise only, a simpler scheme has been used. During measurements, the vehicle has been driven with constant speed, successively fixed to 30, 40, 50, 60 and 70 km/h, in two different situations: displacement with and without the functioning of the vehicle’s engine. For comparison purposes, the dual road from the testing location was coated with two types of surface textures: a half of the road was coated with rubberized asphalt (R.A.) and the other one with regular concrete asphalt (C.A.16). Therefore, it was obtained a test-road where two types of surfaces can be tested in similar environmental conditions. The road has been arranged as described above by a road company located in Timisoara, Romania, which prepared also the rubberized asphalt within its own facilities. Two vehicles were used for tests: a Peugeot 307 equipped with summer tires and a VW Golf 4D Diesel equipped with Pirelli all-seasons tyres. The initial tests were performed in 2011 [1] and two years later, the same tests were performed again in order to assess the behaviour of the road coating in time. The most eloquent tests for tire-road noise were performed with the Peugeot 307 without the functioning of the engine (pure tire-road contact noise) at successive displacement speeds of 30, 40, 50, 60 and 70 km/h, respectively, with the microphone located at 2 meters distance from the line of the tire-road contact. The results of these tests are presented in Table 1. Table 1. Equivalent noise levels obtained without engine for the Peugeot 307 Speed [km/h] 30 40 50 60 70

Road coating R.A. C.A.16 R.A. C.A.16 R.A. C.A.16 R.A. C.A.16 R.A. C.A.16

Leq [dB] Initial test Repeated test 63.0 63.1 64.3 64.1 62.7 62.5 64.7 64.6 64.8 64.9 66.2 66.4 65.1 65.0 68.6 68.9 66.9 66.8 73.0 72.6

We can remark from table 1 that the values obtained for the equivalent noise level in the case of the repeated test are almost the same as those obtained initially, which prove that the coating texture did not change its absorbent properties over time. There are found deviations between 0.1 and 0.4 dB, which are supposed to appear due to the variation of the environmental conditions, and however, they are of the range of the measurement uncertainty. Moreover, the same remark applies also to the spectral distribution. Fig. 2 shows the variation of the equivalent noise level generated by the contact between the tire and the rolling surface in two investigated cases depicted above: for R.A. (dashed line) and CA16 (solid line). Analysing the values from table 1 and the diagram from fig.2, we observe that the rubberized asphalt ensure a mitigation of the rolling noise between 1.0 dB and 6.1 dB, depending on the speed of displacement. This mitigation is growing along with increasing the speed. This conclusion recommends the use of the rubberized asphalt for acoustical arrangement of the road superstructure, being very efficient especially for high speeds. It must be mentioned that the results of the measurements performed in the circumstances described above are influenced by the configuration of the testing place, since the location is a canyon-type street. However, both sets of measurements were identically influenced.

Acoustics & Vibration of Mechanical Structures

Leq [dB(A)]

260

74 72 70 68 66 64 62 60 58 56

RA CA16

30

40

50

60

70

speed [km/h]

Fig. 2. Comparison between the noise levels obtained for regular and rubberized asphalt In the following diagrams we present the spectral distribution of the noise levels for displacement on the road coated with concrete asphalt CA16 (fig.3) and for rubberized asphalt (fig.4) for the speed of 70 km/h. From figures 3 and 4 we observe the mitigation of the noise level for every frequency band, but the most significant mitigation of the noise level is remarked for the frequency bands corresponding to 1 kHz and 2 kHz.

Fig. 3. Spectral distribution of the noise on CA16 at 70 km/h

Fig. 4. Spectral distribution of the noise on RA at 70 km/h

The tests performed in the conditions depicted in fig.1 to determinate the total noise levels on R.A. and CA16, including the noise generated by the engine and gases exhaust led to the results presented in table 2, for different speed of displacement for the vehicle VW Golf 4T Diesel with allseasons tyres Pirelli 195/65 R16. Table 2. Values of the total noise (tire-road and engine noise) Speed Road [km/h] coating 30 40 50 60 70

R.A. C.A.16 R.A. C.A.16 R.A. C.A.16 R.A. C.A.16 R.A. C.A.16

Leq Mic-1 59.8 58.7 61.7 62.2 63.4 64.7 65.9 66.8 69.9 71.2

Initial test Leq Mic-2 57.5 60.0 61.1 62.1 63.1 64.1 65.8 67.7 69.6 71.4

Av. [dB] 58.65 59.35 61.4 62.15 63.25 64.4 65.85 67.25 69.75 71.3

Repeated test Leq Leq Av. Mic-1 Mic-2 [dB] 59.9 57.6 58.75 58.6 60.2 59.4 61.9 61.1 61.5 62.4 62.0 62.2 63.0 63.2 63.1 64.5 64.0 64.25 65.8 65.8 65.8 66.9 67.3 67.1 69.5 69.4 69.45 71.0 71.7 71.35

Applied Mechanics and Materials Vol. 430

261

Leq [dB(A)]

Fig. 5 shows the dependence of the total noise on the speed during displacements of the vehicle on the road coated with R.A. (dashed line) and CA16 (solid line). We underline that these noise levels include the noise generated by the engine, which increases along with increasing the speed. 75 73 71 69 67 65 63 61 59 57 55

R.A. CA16

30

40

50

60

70

v [km/h]

Fig. 5. Comparison between the total noise levels obtained for the VW Golf 4T Conclusions In this paper the influence of the rubberized asphalt on noise mitigation is experimentally investigated in real conditions. Comparisons were made between regular asphalt (C.A.16) and rubberized asphalt which coated a double road within the urban area. Two types of tests were performed. The first one was intended to reveal only the tire-road contact noise, hence the displacements took place at several speeds without engine, and the second one was intended to reveal the influence of all noise sources within a vehicle, hence displacements took place with working engine. Initial tests were performed two years ago, when the rubberized asphalt has been mounted in the testing place by a local company, and the same tests, in the same environmental conditions were repeated after two year of service, in order to find out if acoustic properties of the rubberized asphalt were changed or not. It was found that these properties were not changed over time, demonstrating excellent durability and stability of the employed rubberized asphalt. Practically, after two years of service, the same values were obtained for the equivalent noise level as well as for the spectral distribution of the noise, which confirm the efficiency of rubberized asphalt to decreasing phonic pollution in urban areas It was demonstrated that the sound absorbent performances of the rubberized asphalt are better emphasized at higher speeds. References [1] V. Bacria, N. Herisanu, Considerations concerning the effects of rubberized asphalt on decreasing the phonic pollution in urban environment, Int. Conf. Energy, Environment, Ecosystems and Sustainable Development EEESD '10, Timisoara, 2010, 311-315. [2] U. Sandberg, Tyre/road noise—myths and realities, Proceed. of Inter-Noise 2001, pp. 35–56. [3] M.J. Crocker, Z. Li, J.P. Arenas, Measurements of tyre/road noise and of acoustical properties of porous road surfaces, Int. J. of Acoustics and Vibr. 10 (2005) 52–60.

Applied Mechanics and Materials Vol. 430 (2013) pp 262-265 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.262

Noise Source Monitoring in Industrial and Residential Mixed Areas Ramona Nagy1,a, Dorin Simoiu1,b, Karoly Menyhardt1,c, Liviu Bereteu1,d 1

Mechanics and Materials Strength Department, “Politehnica” University of Timişoara, Bd. MihaiViteazu, nr. 1, 300222, Timişoara, Romania

a

[email protected]; [email protected]; [email protected]; [email protected];

Keywords: noise source, industrial noise monitoring

Abstract. Today, reduction of industrial noise is a widely used concept for improving the life standard in vicinity of industrial and residential mixed areas. The human ear has a very poor auditory acuity in perceiving sounds produced by sound pressure variations in the very low frequencies or very high frequency range. Noise is, in general interpretation, disturbing sound, annoying or even dangerous, although this perception is subjective. There are at least two kinds of people, one that "adapted" psychological noise and learned to work with it and another which gradually became sensitive to noise. In the current situation of industrial development, and housing, many locations of manufacturing facilities are near residential areas, so it is very important to realize noise levels monitoring. Legislation provides for annual measurements of noise levels to limit industrial properties. This paper describes the monitoring, over a period of several years, the noise sources within a company at the limits of Timisoara. Introduction Noise is perceived as an unwanted sound, being in fact the human feeling due to changes in air pressure in the inner ear. Noise levels are expressed in decibels (dB) on a logarithmic scale, where 0 dB is the threshold of hearing, and 120dB is the threshold of pain. As a result of using a logarithmic scale, a 3dB change in noise level is barely perceived, while an increase of 10 dB is perceived as a doubling of sound intensity. Frequency is an important feature of sound, measured in Hz and is given by the number of oscillations per second. A young, healthy person has an audible band from about 20Hz to 20,000 Hz (20 kHz). The sensitivity of the human ear at different frequencies in the audible range is not uniform. In order to reproduce and simulate this phenomenon and to obtain objective acoustic measurements, the measurement instruments have built-in filters for weighted integration. The most used frequency weighting is the proper weighting curve which corresponds to the frequency response of the ear. In determining acceptable noise levels the existing noise levels are taken into account, the character of the area and the nature of their development. In 1996, the European Commission issued a Green Paper which stated that around 20% of EU citizens were exposed to noise levels that scientists and health experts consider to be unacceptable, where most people become irritable, have disrupted sleep and health may be in danger. In 2000, the Commission issued a proposal for a directive on the assessment and management of environmental noise [1]. It is about traffic, air, rail and industrial noise. It focuses on the impact of noise on individuals and it completes EU law, which sets standards for noise emissions from existing specific sources. Although currently there are no imposed limits, they could be compared with noise exposure recommendations of the World Health Organization [2]. This paper aims to analyze the results of monitoring noise levels, over several years, at an industrial society (Coca-Cola HBC), located at 6 km from Timisoara within 10 m of DN 6, near a growing residential area. Although Timisoara has a noise map from 2008, the monitored area is not included in the measured and mapped area [3].

Applied Mechanics and Materials Vol. 430

263

Noise levels calculation method Romania has not adopted calculation methods for road traffic noise, rail traffic noise and industrial emissions noise, and it was decided to use the NMPB-96 [4,5,6], the French Method as recommended by the Environmental Noise Directive (END). In END according to Article 5 the formula for calculation of noise indicator: Lden (day-evening-night noise indicator) is: Lday

Lden =10lg

12*10 10 +4*10

Levening Lnight 10 +8*10 10

24

(1)

where Lday (day-noise indicator), Levening (evening-noise indicator) and Lnight (night-time noise indicator) are the long term average sound level as defined in ISO 1996-2: 2007, determined over all the day periods or evening periods or night periods of a year . In Romania day is considered as the interval between 7 am and 7 pm, 12 hours. The 4 hours between 7 pm and 11 pm is considered evening and the range of 8 hours between 11 pm and 7 am is considered night. Measurement site description According to the Romanian standard SR ISO 1996-1:2008 [7] that describes environmental noise, measurements were performed to determine the noise sound pressure, level A continuous, equivalent LAeq in a few points at the property limits of the company. The perimeter of this company is considered as an industrial one, but it is located in close proximity to a residential area. Therefore noise levels have a reference of 70 dB (WHO Community Noise Guidance) for industrial area, and 50-55 dB for outdoor area. Measuring point P1 is located 20 m from the entrance gate on the right. All cars enter and leave through this area. The company’s parking lot for the loading bay is also located on the platform. Therefore the area is expected to have high noise levels. Measurement point P2 is located in front of the loading bay. It is an area where there is heavy traffic of cars and forklifts. Measuring point P3 is behind the unit, about 20 m from the building. It is a discharge area for containers, thus it has an intense traffic area for forklifts. There are times when this activity is reduced therefore may be low levels of noise. P4 measuring point is in front of the compressor station. In this area there are a number of compressors that contribute to high levels of noise. P5 measuring point is located in the left of the entrance in the company, about 100 m. It is an area near DN6 but also entrance for company and transport vehicles. Therefore noise is affected by traffic. Measurement results Measurements were performed according to ISO 1996-2:2007 [8] fulfilling the conditions for the location of microphone, away from obstacles, the wind direction, not opposing, a wind speed lower than 5 m/s, with a microphone set at 1.4 m above the ground. The device used is a type 2250 sound level meter. For signal processing software was used to measure sound levels BZ 7222 and BZ program for analyzing frequency 7223 Bruel & Kjaer. Measurements were made in five years: 2007, 2009, 2010, 2011 and 2012 In Fig. 1-4 the measured noise levels are presented during the five year period, for the day interval (Lday), evening interval (Levening), night interval (Lnight) and that for the entire day Lden.

264

Acoustics & Vibration of Mechanical Structures

80

Lday [dB]

70 60

2007

50

2009

40

2010

30 2011

20

2012

10 0 First point

Second point

Third point

Fourth point

Fifth point

Fig. 1 – Lday noise level at the measurement points 80

Levening [dB]

70 60

2007

50

2009

40

2010

30 2011

20

2012

10 0

First point

Second point

Third point

Fourth point

Fifth point

Fig. 2 - Levening noise level at the measurement points 100 80 Lnight [dB]

2007 60

2009 2010

40

2011 20 0

2012 First point

Second point

Third point

Fourth point

Fifth point

Fig. 3 - Lnight noise level at the measurement points 100

Lden [dB]

80

2007

60

2009

40

2010 2011

20 0

2012 First point

Second point

Third point

Fourth point

Fifth point

Fig. 4 - Lden noise level at the measurement points

Applied Mechanics and Materials Vol. 430

265

Conclusions After analyzing the measurements we found that the noise level in P1 ranges over the years during the day interval by 6 dB, between the evening with more than 9 dB, during night with over 11 dB and Lden with 10 dB. At this point the noise level did not exceed 70 dB Lden. In the area allocated to P2 was a single 70 dB level overcome Lden. Also there was a difference of more than 16 dB for noise at night. In the area of measurement point P3 the conclusions are similar to the previous area, finding one exceeding of the 70 dB level. Most of the threshold level exceeding recommended by the WHO Community for Noise Guidelines were found in the compressor station P4, and this only after 2009, when it was expanding its capacity. Relative to the allocated area for point P5, where the noise level should have been the lowest in the company area, high levels were found, some even more than 70 dB, which is justified by its location near a curve on the national road DN6. Therefore an important contribution has traffic noise. Measurement results show that in the compressor station noise levels Lday, Levening, Lnight and Lden are above those permitted in industrial areas. It is recommended the use of noise barriers starting from the roof and down to the compressor drive motor cases. The pumping station and the sewage treatment plant are a source of noise that makes the readings higher at measuring point P2. It is recommended that the tanks, pump housings and pipes to be wrapped in noise absorbent material. References [1] Directive 2002/49/EC of the European Parliament and Council. [2] World Health Organisation, WHO Guidelines for Community Noise (1999). [3] T. Neda, M. Bite, P. Bite, I. Dombi, Noise Mapping in Hungary and Romania, RJAV IX (2012) 61-65 [4] J. Lang, Calculation method for road traffic noise. Description of the calculation method NMPB Contract: B4-3040/329/2001/329750/MAR/C1. [5] G. Dutilleux, J. Defrance, D. Ecotière, B. Gauvreau, M. Bérengier, F. Besnard, E. Le Duc, NMPB-Routes-2008: The Revision of the French Method for Road Traffic Noise Prediction, Acta Acustica united with Acustica, 96, (2010), 452-462 [6] G.Dutilleux, J. Defrance, B. Gauvreau, F.Besnard,The revision of the French method for road traffic noise prediction, Acustics’08 (2008), 875-880 [7] SR ISO 1996-1:2008 Caracterizarea şi măsurarea zgomotului din mediul înconjurător. Partea 1: Mărimi şi procedee de bază, 1995 [8] ISO 1996-2:2007 Caracterizarea şi măsurarea zgomotului din mediul înconjurător, Partea 2: Obţinerea de date corespunzătoare pentru utilizarea terenurilor, 1995

Applied Mechanics and Materials Vol. 430 (2013) pp 266-275 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.266

Analysis of the sound power level emitted by portable electric generators (outdoor powered equipment) depending on location and measuring surface Postelnicu Elena1,a, Vladut Valentin1,b, Sorica Cristian1,c, Cardei Petru1,d and Grigore Ion1,e 1

Ion Ionescu de la Brad Blv. No. 6, Sector 1 Bucharest, Romania

a

[email protected], [email protected], [email protected], d [email protected], [email protected]

Keywords: sound power level, noise measurement methods, portable power generator

Abstract: Acoustic power is a measure which must be specified on the outdoor used equipments and its determination depends on several factors: the place where the equipment works (indoor or outdoor), the placement of the microphones for its determination (the distance less or greater from the noise source), the shape of the measurement surface (parallelepiped or hemispherical). This paper aims to analyze the values obtained in these situations and interpret the data to determine the influence that each factor has on the acoustic power compared with the values obtained (permissible) according to Directive regarding noise emission D 2000/14/EC. Introduction The 2000/14/EC directive is one of those regulations that lead to improved conditions for reducing noise in crowded urban areas. The fifth action program in the field of environment, identifies noise as one of the priority environmental problems in congested urban areas and emphasizes the necessity to act upon various sources of noise. The equipment subjected to the provisions and regulations of 2000/14/EC directive are set out in art. 2 and 13 of this Directive: • equipment subjected to noise limitations; • equipment not subjected to noise limitations, only marking of this level. [1] Under the 2000/14/EC directive, a number of technical equipments are summarized, subjected to limitation or marking sound power level, which include power and portable generators for domestic purposes to supply electricity at places where electricity grid does not exist or where a temporarily electricity cut off is present. According to the directive, power generators with electric power 50 dB). [8] Calibration of measuring channels is performed at the beginning of each set of measurements required to measure a noise source. In order to do that, one must first activate the calibration program, that will automatically sense the presence of the microphone calibrator connected to the PULSE system measurement channels. When the calibrator is detected, the required correction is performed to the amplification channel and then, the calibrator is searched on other microphones connected at the system and the procedure repeats again. After the last measurement channel calibration, the calibration program is closed and the data acquisition program is opened, according to the number of measuring channels. After its execution, we need to run another program that takes the data from the previous one and processes them according to SR EN ISO 3744 which refers to the determination of sound power levels of noise sources using sound pressure. In order to determine the sound power level, the following operations were performed, necessary for preparing the product for tests: • noise source dimensions were measured; • measuring surface area was calculated. According to 2000/14/EC directive up to 12 microphones may be used to determine the sound power level using hemispherical measurement surface or up to 9 microphones for parallelepiped surface measurement. The number of the microphones may be reduced to six, but positions 2, 4, 6, 8, 10 and 12 are mandatory in all cases, as required by clause 7.4.2 of EN ISO 3744:2010. [2] In this case we used six microphones (2, 4, 6, 8, 10, 12) positioned on a hemispherical measurement surface (Fig. 2) according to D 2000/14/EC directive.

Fig. 2 Microphone positions for hemispherical measurement surface with 6 microphones Coordinates of the 6 microphones positions are shown in the table below:

Applied Mechanics and Materials Vol. 430

Table 1 Coordinates of the 6 microphones positions for r=0.5 m and r=1 m r = 0.5 [m] x y z r = 1 [m] x y z 2 0.35 0.35 1.5 2 0.7 0.7 1.5 4 -0.35 0.35 1.5 4 -0.7 0.7 1.5 6 -0.35 -0.35 1.5 6 -0.7 -0.7 1.5 8 0.35 -0.35 1.5 8 0.7 -0.7 1.5 10 -0.135 0.325 0.355 10 -0.27 0.65 0.71 12 0.135 -0.325 0.355 12 0.27 -0.65 0.71 Table 2 Coordinates of the 6 microphones positions for r=2 m and r=4 m r=2m x y z r=4m x y z 2 1.4 1.4 1.5 2 2.8 2.8 1.5 4 -1.4 1.4 1.5 4 -2.8 2.8 1.5 6 -1.4 -1.4 1.5 6 -2.8 -2.8 1.5 8 1.4 -1.4 1.5 8 2.8 -2.8 1.5 10 -0.54 1.3 1.42 10 -1.08 2.6 2.84 12 0.54 -1.3 1.42 12 1.08 -2.6 2.84

Fig. 3 Placing microphones around the generator for indoor and outdoor measurements (in accordance with D 2000/14/EC)

Results and discussions Initial conditions of the measurements are presented in the following table: Table 3 Initial conditions of the measurements Environment conditions Temperature [°C] 26 Atmospheric pressure [mmHg] 735 Relative humidity [%] 52,5 Wind speed [m/s] 1,6 Correction for background noise K1 [dB] 0 Correction for the reflected sound K2 [dB] 0

269

270

Acoustics & Vibration of Mechanical Structures

Following the measurements, for outdoor, were obtained the results presented in Table 4:

Radius [m] 0,5

1

2

4

Lp2 [dB] 88,2 88,7 88,8 87,5 88,5 88,1 85,6 86,2 86,3 81,1 85,7 83,7

Table 4 The results obtained for outdoor measurements Average Lp6 Lp4 Lp8 Lp10 Lp12 Lw background [dB] [dB] [dB] [dB] [dB] [dB] noise [dB] 88,1 87,3 86,5 93,5 92,9 92,3 88,4 87,4 87,1 94 93,3 52,96 92,7 88,5 87,4 87,2 94,2 93,4 92,8 87,6 86,4 85,5 94,1 90,6 97,7 88,4 86,7 85,8 95,4 90,7 51,45 98,6 88,3 86,8 85,8 95,2 90,7 98,4 85,9 85,9 83,2 86,6 85,3 99,5 53,98 86,5 86,7 83,9 87,6 85,7 100,2 86,7 87,1 84,1 88 85,8 100,5 80,5 81,6 78 81 79 100,4 60,92 81,3 82,7 79 82,3 79,5 102,4 81,5 83,4 79,2 82,6 79,3 102

Measurement uncertainty

0,41

0,60

0,65

1,19

Lp2, Lp4, Lp6, Lp8, Lp10 si Lp12 - represent the determined sound pressure level in each of the 6 measurement points marked in red in Fig. 2. Lw – represents the sound power level that is calculated by the PULSE system using the formula (Eq. 4) The uncertainty of measurement was estimated with a probability of 95%, according to SR ISO / IEC 98-3:2010. Uncertainty of measurement. Part 3: Guide to the expression of uncertainty in measurement (GUM: 1995). [9] Flat distribution of the measurement points and the sound pressure level for outdoor measurements is shown in the following figure:

Fig. 4 Distribution of the measurement points of the sound pressure level for outdoor measurements

Applied Mechanics and Materials Vol. 430

271

For a height of 1.8 m, the interpolation of the flat distribution of the values of sound pressure level for outdoor measurements is shown in Fig. 5.

Fig. 5 Distribution of the sound pressure level for outdoor measurements, interpolated for a height of 1.8 m, represented by izocline For outdoor measurements it may be tried an elementary interpolation as: L p (r ) = a + b ⋅ ln(r )

(5)

where:

r ( x, y , z ) = ( x − x s ) 2 + ( y − y s ) 2 + ( z − z s ) 2

(6)

The function r(x,y,z) is the distance from a certain point of coordinates x,y,z, to the point in which the source is located, xs, ys, zs. For this case, a minimization calculation of the functional corresponding to the least squares method leads to the following values: a=108.023, b=-439.264, xs=0.015 m, ys=0.021 m, zs=0.092 m. This model corresponds to the source point that emits from the point of coordinates xs, ys, zs. Graphic representation of the relation (Eq. 5) for z=1.8 m, is given in Fig. 6.

272

Acoustics & Vibration of Mechanical Structures

Fig. 6 Representation of sound pressure level depending on the flat coordinates of the recording location, interpolated for a height of 1.8 m by theoretical function (Eq. 5). For the indoor mesurements we have obtained the following results:

Radius [m] 0,5

1

2

4

Lp2 [dB] 90,8 90,8 90,7 90,7 91,3 91,3 89,8 90,1 90,4 88,4 88,6 88,5

Table 5 The results obtained for indoor measurements Average Lp4 Lp8 Lp10 Lp12 Lw Lp6 background [dB] [dB] [dB] [dB] [dB] [dB] noise [dB] 90,6 90,2 90,3 96,4 95,5 95,1 90,7 90,3 90,5 96,7 95,7 40,11 95,3 90,7 90,3 90,4 96,8 95,7 95,3 90,2 90,1 89,5 95,6 94,1 100,3 90,8 90,6 90 96,1 94,2 38,68 100,8 91 90,7 90,2 96,2 94,2 100,8 90,1 90,2 89,4 90,3 90,9 104,1 90,3 90,3 89,7 90,8 91,1 38,78 104,4 90,6 90,6 89,9 90,9 91,2 104,6 88,1 90,2 87,3 88,8 88,7 108,7 88,3 90,4 87,6 88,8 88,9 40,67 108,9 88,3 90,4 87,7 88,8 89 109

Measurement uncertainty

0,32

0,41

0,41

0,35

Applied Mechanics and Materials Vol. 430

273

Flat distribution of the measurement points and the sound pressure level for indoor measurements is shown in the following figure:

Fig. 7 Distribution of the measurement points of the sound pressure level for indoor measurements For a height of 1.8 m, the interpolation of the flat distribution of the values of sound pressure level for indoor measurements is shown in Fig. 8.

Fig. 8 Distribution of the sound pressure level for indoor measurements, interpolated for a height of 1.8 m, represented by izocline

274

Acoustics & Vibration of Mechanical Structures

Fig. 9 presents the variation of sound power levels depending on the radius of the hemispherical measurement surface.

Fig. 9 Variation of sound power levels depending on the radius of the hemispherical measurement surface

Conclusions After analyzing and interpreting data obtained by measurements, the following observations were highlighted: - Sound pressure level decreases with the increasing of the radius of the hemispherical measurement surface; - Sound power level increases with the increasing of the radius of the hemispherical measurement surface; - The difference in [dB] between sound power level and sound pressure level, it positively accentuates with the increasing of the radius of hemispherical measurement surface; - In the indoor measurements, both sound pressure level and sound power level, have higher values compared with outdoor measurements, for equal-rays of the hemispherical measurement surface. This phenomenon occurs because of the existence of reflective planes that fosters reflection and amplify the two physical quantities analyzed; - Using the experimental data of the kind presented in this paper, together with the theoretical interpolation function adapted to the specific measurements, may give clues on identifying areas with high levels of pressure and sound power, helping to optimize noise reduction solutions in these areas.

Applied Mechanics and Materials Vol. 430

275

References [1]. V. Vlăduţ, S. Biriş, S. Bungescu, At. Atanasov, D. Gafiţianu, Analysis of the methods and regulations in force, harmonized with EU legislation regarding the noise emission of equipment with outdoor operation, INMATEH. I (2006) 143-152. [2]. *** Directive 2000/14/EC of the European Parliament and of the Council of 8 May 2000 on the approximation of the laws of the Member States relating to the noise emission in the environment by equipment for use outdoors, Official Journal. L 162 (2000) 0001-0078. [3]. S. Băjenaru, M. Ganga, V. Vlădut, S. Biriş, A. Dimitrijevic, Noise level determination of an equimpment for use outdoors – motor hoe, INMATEH. I (2007) 26-32. [4]. M. Ganga, S. Băjenaru, V. Vlădut, S. Bungescu, Bikić S., Measurement equipment for the acoustic power level and the evaluation of the uncertainty of the measurement, INMATEH. I (2007) 20-25. [5]. V. Vlăduţ, D. Manea, S. Biriş, S. Bungescu, At. Atanasov, Test of an equipment which works in open air according to the requirements of D 2000/14/EC, PROCEEDINGS OF THE 35 INTERNATIONAL SYMPOSIUM ON AGRICULTURAL ENGINEERING "Actual Tasks on Agricultural Engineering", 35 (2007) 405-414. [6]. SR EN ISO 3744:2010, Acoustics – Determination of sound power levels of noise sources using sound pressure – Engineering method in an essentially free field over a reflecting plane. [7]. *** PULSE Multi-analyzer System Type 3560 – technical documentation. [8]. *** Sound Level Calibrator type 4231 – technical documentation. [9]. SR GHID ISO/CEI 98-3:2010, Uncertainty of measurement. Part 3: Guide to the expression of uncertainty in measurement (GUM: 1995). [10]. *** Bruel & Kjaer apparatus used for measurement – documentation.

Applied Mechanics and Materials Vol. 430 (2013) pp 276-280 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.276

Model for forecasting the exposure risk of workers to hand-arm occupational vibrations VASILESCU GABRIEL-DRAGOȘ, Phd. Eng.1, a, GHICIOI EMILIAN, Phd. Eng.2,b , SIMION SORIN, Phd. Eng.3,c and PĂSCULESCU VLAD, Phd. Student Eng.4,d 1,2,3,4

National Institute for Research and Development in Mine Safety and Protection to ExplosionINCD INSEMEX Petrosani, 32-34 G-ral Vasile Milea Street, Postcode:332047, Petrosani, Hunedoara County, Romania a

b

c

[email protected], [email protected], [email protected], d [email protected]

Keywords: diagnosis, distribution, forecast, hand-arm vibration, occupational risk

Abstract: This paper work presents the model for forecasting the exposure risk of workers to handarm occupational vibrations, which has been achieved in the PN 07 45 01 18 Project from within the framework of the “NUCLEU/2012-2013” Program [4]. This project is of national and European interest, in order to increase occupational health and safety level and to ensure sustainable environmental quality and comfort at work. The scientific novelty degree is given by the complex and interdisciplinary aspect of the research results regarding the analysis and assessment of exposure risk to hand-arm vibrations, as a viable and certain solution for promoting the sustainable development of these vibrations management at work. In order to estimate and assess the risk of exposure to hand-arm vibrations, there has been designed a diagnosis and forecasting mathematical model, based on the Gumbel exponential decay distribution function, which has variables that can be written depending on the weighted acceleration parameter values or depending on points of exposure. 1. Introduction Starting from theory to practice, this paper tries to conceptualize objectively, through researches based on modern mathematical models in the field of the risk of exposure to hand-arm occupational vibrations, the structural and process components that allow the determination with sufficient confidence of a relative share of hazards identified through the harmful values of the vibration parameters, as well as the implications of these results on the occupational health and safety of exposed workers, while providing alternatives and viable solutions for ensuring the sustainable safety in operation of the work systems which are equipped with sources that generate vibrations. For establishing the potential causes of injuries and/or occupational disease, the term “risk factor” is more and more frequently used in specialized analyses. The orientation of specialists for the further study of risk factors (including risk factors generated exposure to mechanical vibrations), is perfectly justified by the possibility offered to establish prevention methods starting from the potential risks of injury or occupational disease (following the exposure to this type of noxious). 2. General considerations on the legal basis for the assessment of risks arising from the exposure to hand-arm vibrations Law 319/2006 on occupational health and safety with further modifications and completions establishes the general principles regarding the prevention and control of occupational risks, as well as the general directions for implementing these principles [7].

Applied Mechanics and Materials Vol. 430

277

The European Directive 2002/44/EC has been transposed in the national law by Government Decision 1876/2005 on the minimum occupational health and safety requirements regarding the exposure of workers to the risks arising from vibrations [5]. Within this regulation there are established “exposure limit values” and “exposure action values”, at the same time specifying the obligations of employers regarding the determination and assessment of risks, the provisions aimed at avoiding or reducing exposure, as well as the details for ensuring the worker information and training. According to these regulations, for hand-arm vibrations, the daily exposure limit value standardised to an eight-hour reference period shall be 5 m/s² and the daily exposure action value standardised to an eight-hour reference period shall be 2.5 m/s². Also, the risks arising from exposure to mechanical vibration shall be eliminated at their source or reduced to a minimum. The exposure of workers, determined in compliance with the legislation in force, shall not exceed, in any event, the exposure limit value. If the exposure limit value is exceeded, the employer shall take immediate action to reduce exposure below the exposure limit value. He shall identify the reasons why the exposure limit value has been exceeded, and shall amend the protection and prevention measures accordingly in order to prevent it being exceeded again. 3. Negative influence of hand-arm vibrations upon the workers The severity of biological effects of vibrations transmitted to the hand-arm system during the work process depends on several factors: Exposure time and work method (the duration, frequency and sequencing of work periods and breaks; the depositing or holding the switched off equipment in hands during work breaks). The daily exposure and the cumulative duration per working day is also important; The direction of the vibration transmitted to the hand as well as the size and direction of the forces applied by the operator to his hands, tool, or body during the exposure (the angles of the articulations of the fingers, hand, fist, elbow, and shoulder); The surface and the part of the hands/arms exposed to vibrations; The type and state of the vibrating mechanisms, hand-tools or of the processed part; The work method and qualification of the operator are factors which have to be taken into account. The main factors which may affect the circulation changes caused by hand-arm vibrations are the following: - Factors from the work environment: especially microclimate, noise and chemical agents as well as several individual medical problems, the presence of agents affecting the peripheral blood circulation such as smoking or some drugs. - The Raynaud Syndrome – is usually related hand-arm vibrations, but is also involved in other general or occupational diseases. It may have several causes, some of them being related to the work place. The Raynaud Syndrome is a disorder of the blood circulation which affects the fingers and their reaction to the cold, generating the “white fingers” crisis by the strong reduction of the blood circulation. 4. Design and achievement of a model for forecasting the risk of exposure to hand-arm vibrations In order to quantify the exposure to different values of hand-arm mechanical acceleration in the work process, we have the following analytical equations [6]: 2

 A(8)  PEi =   100  2.5 

(1)

278

Acoustics & Vibration of Mechanical Structures

respectively a PEi =  hvi  2.5

2  Tej 100   T0

(2)

where: A(8) – daily exposure to hand-arm vibrations (m/s2); PE – points of exposure which quantify the hand-arm mechanical vibrations values; ahv – vibration magnitude (m/s2); Tej –daily duration of exposure to hand-arm vibrations (hours); T0 – reference period (8 hours). Based on these mathematical relations is obtained, for different values of exposure to handarm mechanical vibrations during the work process, as well as for their quantification through points of exposure: Following the application of the ahvi=f(Tej) mathematical relation (from the last column of Table 1), the ahvi grid of values is obtained, grid which corresponds to hand-arm mechanical vibrations generated during the work process for different values of the A(8) parameter. Following the application of the PEi=f(ahvi,Tej) mathematical relation, the PE points of exposure grid of values is obtained, grid which corresponds to hand-arm mechanical vibrations generated during the work process for different values of the A(8) parameter. Below, there is presented the differential equation which quantifies the risk of exposure to different levels of hand-arm mechanical vibrations generated during the work process, depending on its levels or on the value of the points of exposure [2,3,6]:

g ( xi ) = G ( xi )′

(3)

which accepts as solution the following G(ahvi) repartition function: G ( xi ) =

e

−

e

−

2 ln i ( xi − µ − σ

2 ln i )

σ

(4)

where: xi - variable of the function which can be explicated in form of ahvi values or PEi points of exposure; g(xi) - probability density function for the xi variable values; G(xi) – repartition (probability) function for the xi values; µ - average value of the xi variable; σ - standard deviation of the xi variable values; i – order index of the xi variable. Depending on the ahvi parameters or on the PEi points, the functions may be put in the following forms [2,3,6]:

g ( a hvi ) = G ( a hvi )′

(5)

which accepts as solution the following G(ahvi) repartition function:

G ( PEi ) = e or

−

−

e

2 ln i ( PEi − µ −σ

2 ln i )

σ

(6)

g ( PEi ) = G ( PEi )′

(7)

which accepts as solution the following G(PEi) repartition function: −

G ( PEi ) = e

−

e

2 ln i ( PEi − µ −σ

σ

2 ln i )

(8)

Applied Mechanics and Materials Vol. 430

279

Based on the G(xi) repartition function, there is determined the average risk of exposure to different values of hand-arm mechanical vibrations, respectively [2,3]: Ri (xi ) = xiG(xi )dxi , for a continuous domain of xi values (9)

∫

Ri (xi ) =

∑x G(x ), i

i

for a discrete domain of xi valuesi

(10)

This risk can be made explicit: Depending on the ahvi parameter:

∫

∫ e

Ri (ahvi ) = ahvi G(ahvi )dahvi = ahvi

R i ( a hvi ) =

∑a

hvi G ( a hvi

)=

∑a e

σ

e

−

hvi

2 ln i ( ahvi − µ −σ 2 ln i )

−

−

e

dahvi , for a continuous domain of ahvi values

2 ln i ( a hvi − µ − σ

−

(11)

2 ln i )

σ

for a discrete domain of ahvi values

,

(12)

Depending on the PEi points:

∫

∫ e

Ri ( PEi ) = PEi G ( PEi )dPEi = PEi

R i ( PEi ) =

∑

PEi G ( PEi ) =

∑

−

PEi

e

−

e

2 ln i ( PEi − µ − σ

2 ln i )

σ

−

e

−

dPEi , for a continuous domain of PEi values

2 ln i ( PEi − µ − σ

(13)

2 ln i )

σ

,

for a discrete domain of PEi values

(14)

Taking into account the results obtained, there are achieved the following grids (Table 1) for assessing the risk of exposure to different values of hand-arm mechanical acceleration during the work process, depending on its levels or on the value of the points of exposure [2]: Table 1. Assessment of the risk of exposure to different values of hand-arm mechanical acceleration during the work process Values of the handarm mechanical acceleration , ahvi m/s2

Estimation of the risk of exposure to different values of the hand-arm mechanical acceleration ahvi*G(ahvi)

ahvi ≤ 2.5

ahvi*G(ahvi ≤ 2.5)

Assessment of the risk of exposure to different values of hand-arm mechanical acceleration during the work process LOW

2.5 < ahvi ≤ 5.0 5.0 < ahvi

ahvi*G(2.5 < ahvi ≤ 5.0) ahvi*G(5.0 < ahvi) Estimation of the risk of exposure depending on the resulted value for The points of exposure, PEi*G(PEi) PEi *G(PEi ≤ 100.00)

MEDIUM HIGH Assessment of the risk of exposure to different values of hand-arm mechanical acceleration during the work process LOW

PEi *G(100.00 < PEi ≤ 400.00)

MEDIUM

PEi *G(400.00 < PEi)

HIGH

Resulted value for the points of exposure, PEi PEi ≤ 100.00 100,00 < PEi ≤ 400.00 400.00 < PEi

280

Acoustics & Vibration of Mechanical Structures

5. Conclusions Law 319/2006 on occupational health and safety with further modifications and completions establishes the general principles regarding the prevention and control of occupational risks, as well as the general directions for implementing these principles. The minimum occupational health and safety requirements regarding the exposure of workers to the risks arising from vibrations are enforced in the national law by Government Decision 1876/2005 which transposes the European Directive 2002/44/EC and establishes responsibilities of employers for ensuring, eliminating or reducing to minimum the risk In order to estimate and assess the risk of exposure to hand-arm vibrations, there has been designed a diagnosis and forecasting mathematical model, based on the Gumbel exponential decay distribution function, which has variables that can be written depending on the weighted acceleration parameter values or depending on points of exposure. By applying the analytic mathematical relations there are obtained databases which are specific for the grids of values that represent the results of the statistic functions which quantify the risk of exposure to hand-arm vibrations. So, the assessment of risk of exposure to hand-arm vibrations may be carried out both by using the grids of values which quantifies the average objective risks, based on the weighted acceleration values and their related probabilities, as well as by using the grids of values obtained according to the points of exposure values and the related probabilities. References [1] Desroches A., (1995), Probabilistic Methods and Concepts for Basic Safety, (in French), Lavoisier Tec & Doc, Paris, France [2] Vasilescu G.D., (2008), Unconventional methods for occupational risk assessment and analysis, (in Romanian), INSEMEX Publishing House, Petrosani, Romania [3] Vasilescu G.D., (2008), Probabilistic calculation methods used in the diagnosis and prognosis of industrial risk, (in Romanian), INSEMEX Publishing House, Petrosani, Romania [4] NUCLEU Project, (2010), Advanced system for the assessment of environmental conditions for operators dealing with the extraction of mineral substances from underground or from surface, from the point of view of risks arising from noise and vibrations, (in Romanian), Nucleu program, Project PN 07 45 01 15, INCD INSEMEX, Petrosani, Romania [5] Directive 2002/44/EC of the European Parlament and of the Council on the exposure of workers to the risks arising from physical agents (vibrations) [6] Guide to good practice on Hand-Arm Vibration [7] Law 319/2006 on occupational health and safety

Applied Mechanics and Materials Vol. 430 (2013) pp 281-284 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.281

EXPOSURE OF WORKERS TO NOISE IN MINING INDUSTRY SIMION SORIN1;a, VREME CIPRIAN 1;b, KOVACS MARIUS1;c, TOTH LORAND1;d 1

INCD-INSEMEX Petrosani, ADDRESS: G-ral Vasile Milea street, no. 32-34, Petrosani, Hunedoara County, Romania

a

[email protected], b [email protected], c [email protected], d [email protected]

Keywords: noise, hearing protection, occupational disease

Abstract. The troubles generated by noise represent an issue, unanimously agreed by everybody, and this have numerous psychological and social implications. The effect of sound hindrance during the carrying out of different working tasks shall tired up the spirit of attention diminishing the efficiency of one of the most important function of hearing, decreasing the efficiency of the work process. Consequently, the working capabilities shall diminish and there shall appear a state of tension, a diminution of focusing, less coordinated movements made during the working process, a diminution of the capabilities for a critical evaluation of certain situations, as a result of occupational stress. The issues related to noise are very important from an economic point of view; sometimes, the diminution of the working capability shall decrease up to 60%, the reasoning being strongly affected; there occurs a low output due to fatigue and over-stress of the workers, the mistakes made during performing certain operations, especially due to less attention moments, all these being produced by noise. The noise is a breakdown of the engineering process representing one of the most important factors leading to troubles for the personnel working in the underground coal mining. These paper analyses the way where the noise generated during the engineering process in underground mining shall influence the working capability.

1. Introduction The development of modern technology by increasing power and speed of the machinery, contributed to the diversification and increased sources of noise and vibration, and thus to increase the number of people exposed. Noise is one of the most important industrial emissions that generate hazards to workers' health are forming harmful factors arising from damage and / or malfunction of equipment work [1]. Industrial activities carried out at the workplace in the Jiu Valley in the presence of noise risks affecting the health and safety of workers [2]. By stimulating measures for the improvement of health and safety were issued directives referring to the specific areas which specifies minimum requirements for ensuring ensure an optimal level of worker protection [3, 4]. 2. The purpose of underground noise measurements The technological process of coal extraction is performed using drilling operations, shooting, exhaust horizontal and vertical transport, ventilation partially main drainage, air and

282

Acoustics & Vibration of Mechanical Structures

compression processes of processing and separation of the mass of coal mining that used machinery and equipment. For improvement in health and safety and achieve the objective of reducing noise exposure should consider the following general principles relating to: Reducing high noise. Prevention of occupational risks. Health and safety at work. Elimination of risk factors for injury and disease. Informing, consultation, balanced participation in accordance with national legislation and / or practice and training of workers and their representatives. [3, 4] 3. Presenting features underground noise measurements Noise emitted technological processes of extraction of coal have a continuous (pneumatic energy production in groundwater discharge to discharge harmful gases) or intermittently (in the perforation, blasting, systems for the transport of minerals and tailings). Each noise source is characterized by sound pressure level which depends on external factors such as distance, orientation, receiver, temperature and velocity gradients in the medium. Peculiarities of underground activity is characterized by two aspects: Harmful effects of physiological noise. Ensure that staff exposed to perceive pressure noises mining and gas eruptions fines etc. Noise levels produced by the plant and machinery used in the extraction of coal include all types of noise including impulse that produce a number of effects on the body. Noises in the process of extracting coal have numerous consequences of the human body. Depending on the intensity of the noise, there are several categories of harmful effects on humans, namely: The effect of masking. Auditory fatigue. Acoustic trauma. Acute hearing loss. Occupational deafness. The effects of noise on the nervous system. The influence of noise on visual function. In coal extraction processes most appropriate methods to control noise from existing sources consist of active and passive protection measures. The aim is to achieve a noise reduction optimum acoustic comfort at a fair price. The apparatus used Direct measurement of sound pressure levels equivalent continuous A-weighted, used integrative sonometers have complied with IEC 804, Class 2 or better. Acoustic pressure measurement to determine the A-weighted sound exposure and / or level of equivalent continuous A-weighted sound pressure were performed with the microphone placed in the position (s) occupied (filled) normally by worker’s head . If that person is present or that the person to move around, the microphone should be placed at 0.10 m ± 0.01 m from the entrance to the ear canal that receives high value weighted sound exposure or level equivalent continuous a-weighted sound pressure. [5] Important details of the measuring apparatus and the conditions prevailing during the measurements shall be recorded and archived for your reference. [5]. In reporting the results of measurements include an estimate of the overall uncertainty of the measurements, taking into account the influence of factors such as equipment measurement microphone position, number of measurements, the variation in time and space of the noise source.

Applied Mechanics and Materials Vol. 430

283

4 Results The results are presented in average measurements performed on noise-generating equipment used in mines in the Jiu Valley, presenting and analyzing the bands performance. [2]

Pumps water collection

101,6

109,1

93,9

88,9

90,6

86,7

93,3

108,4

45 52,9 77,2 68,5 102,0 90,7 86 88 83,3

46,3 56,5 62,1 67,9 74,7 77,7 78,1 77,2 77,5

73,5 75,3 89,3 88,3 93,5 89,2 90,6 88,8 86,5

64,6 78,4 87,4 89,9 90,1 105,5 94,9 89,2 82,9

73,2 82,6 79,7 80,3 80,6 83,5 84,4 79,6 70,1

74,9 83,8 84,1 87,2 83,1 80,9 75,7 67,9 61,7

58,3 61,1 63,5 73,1 88,2 82,3 73,2 69,3 61,2

67,9 65,6 69,0 68,9 76,4 77,8 72,8 66,0 64,1

48,5 56,3 86,4 79,1 80,6 82,8 78,6 83,7 83,4

78,2 81,3 88,8 95,7 97,8 90,9 94,5 98,6 96,9

Perforation P 90 sterile

Belt conveyor

90,7

pressure pumps

Flight scraper

High Pressure Pump

CA14

104,6

PR8

P90

Frequency [ Hz]

Noise Level [dB(A)] 31,5 63 125 250 500 1000 2K 4K 8K

APET pump

Job title

Table 1 Noise-generating equipment

Knowledge intensity noise in coal extraction processes allow to quantify effects on humans such as masking, auditory fatigue, noise trauma, acute hearing loss, occupational deafness, nervous system and circulatory visual. The analysis of noise sources from underground was found that noise reduction solutions most appropriate are: The application of active and reactive silencers specific machines. Replacing the machine with a new type silent. Installation in the vicinity of the noise source of acoustic screens and change controls. Installing the machine. Vibroinsulator elements. Acoustic treatment. Use of individual hearing protection. Excess exposure. 5 Conclusions From the above materialized following conclusions: A big influence on noise has conditions of work, nature of surrounding rock and acoustic properties (coal is better than sterile absorbent acoustic). Also, an important role in increasing the level of noise in the underground rocks surrounding it have hardness and narrow spaces. Tests conducted to determine the sound power at stands at the highest noise emission at hammer drills CA 14 and P-90 equipments that showed that the surface value obtained under standardized conditions are different from those due to underground narrow spaces (which creates a large resonant sound waves), a growth level between 6 ÷ 8 dB. Absence markings of noise emission level make difficult to assess noise exposure of underground workers by indirect methods. Apply markings on acoustic power equipment and information to workers leads to creating a safer work by reducing exposure to these pollutants. Regarding the protection of individuals with type external ear, there was a positive trend by equipping workers with EIP noise, given the financial difficulties on the acquisition of equipment and appropriate facilities that generate noise emissions to acceptable levels.

284

Acoustics & Vibration of Mechanical Structures

To check the efficiency of noise reduction, regardless of their nature (technical, organizational) is proposed periodical audio diagrams workers because hearing loss is an irreversible disease that affects quality of life both at work and leisure. References [1] A. Darabont, Prevention and control of noise in mining. Bucharest 1973. [2] Simon Sorin, Research on noise and vibration in the basin of Jiu Valley mines 2012. [3] *** Law 319/2006 . Law on occupational safety and health. [4] *** Directive 2003/10/EC of 6 February 2003 on the minimum health and safety requirements regarding the exposure of workers to the risks arising from physical agents (noise) [Seventeenth individual Directive within the meaning of Art. 16 para. (1) of Directive 89/391/EEC]. [5] *** SR ISO 1999:1996 ver.eng. Acoustics. Determination of occupational noise exposure and estimation of noise - induced hearing impairment.

Applied Mechanics and Materials Vol. 430 (2013) pp 285-289 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.285

Experimental research and modeling of the underwater sound in marine environment Delicia Arsene 1, a, Claudia Borda1,b , Larisa Butu 1, c, Marinela Marinescu 1, d, Victor Popovici 1, e, Mihai Arsene 2, f 1

Universitatea “Politehnica” Bucureşti, Splaiul Independentei nr. 313, sector6, Bucuresti, România 2

SC AFICO SA, Str. Ion Urdăreanu, nr.32, sector 5,România

a

[email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected]

d

Keywords: underwater noise, acoustic field modeling, acoustic project, marine habitat.

Abstract. The paper presents a modeling of the acoustic field emitted by a rig tender located offshore exploration in the Black Sea. Are presented measuring system, experimental context, the types of noise that participate in the overall noise budget and the conditions in which measurements were made, also are mentioned and the project established through were measured the parameters and were defined devices which these measurements were made. Measurement system used is adaptable, allowing dynamic measurements and operative analysis of the acquired data. Measurement methodology was determined by an acoustic project, which was set to be purchased both: sizes and ways of interpreting them. Modeling acoustic field shows that the maximum acoustic impact area, below 120 dB re 1μPa criterion (proposed and accepted by Marine Directive) was extended to 3-5Km distance around the ship. It is expected to produce adverse effects on the marine habitat and on the underwater life. 1. Introduction Sound in the ocean environment is an important aspect of marine life habitat; they elect and change their location and behavior, in part, and depending on anthropogenic noise [1]. At the European Community level there are efforts to establish a legal framework designed to protect marine biodiversity. Community policies keeping a good underwater environment are considering several directives, which define the legal framework suitable for the preservation of the good state of the ecosystem. Assessing the impact of human activities on the marine environment by integrating environmental concepts and its sustainable use is one of the main objectives of the Marine Strategy Framework Directive of the European Union. Unintentional underwater noise introduced by man or outcome as a result of anthropogenic activities is a major source of noise pollution. Noise radiated by offshore drilling rigs is part of the high acoustic power sources [2]. In order to assess the level of noise pollution introduced during an offshore operation, were performed measurements of the acoustic field parameters produced by a rig tender. Using these measurements and previous recordings of ambient noise, were performed a modeling of the sound field support vessel [3]. In the paper are described both experimental context used, procedures and sound project developed for performing measurements. 2. Measurement Methodology To achieve measurements must be described procedures proposing to undertake them. They target specific data acquisition from the environment, describing the location at which measurements are made, describing equipment which are taken the necessary data, and describe experimental context in which research is done [4].

286

Acoustics & Vibration of Mechanical Structures

Description of the location of the platform position The study area was located 21 km offshore 44º21’LatN, 28º72’LongE, 110m water depth, relatively constant. Rig tenders are designed to keep in touch with the coast, to make supplies, transport personnel and, in general, to provide logistical support for the drilling rigs. The rig tender had the following characteristics: - 30 tons, 450 HP and a length of 20 m. Description environmental conditions Environmental conditions were almost totally calm during the study period so there was no masking of the ship signatures by sea or wind noise. There was recorded a level of broadband noise of 90 dB re 1μPa, which is considered to be close to the lowest level possible in the region. Description of the measuring system For radiated noise measurements were used a hydrophone array, which consists of three hydrophones placed at different depths and maintained in vertical alignment with a buoy and a ballast. System is located at the measurement location and then recovered with a motor boat. The acoustic field Measurement Mobile Polygon (Fig.1) provides the following performance parameters: - dynamic measurement and operative analysis of the rig tender acoustic radiation values - high reliability, embedded test, easy operation by a single operator, - high degree of automation of measurement, - the minimum level of the measured field: 80 dB/Pa.

Acoustic project description To perform signal acquisition it has been made an acoustic project with the PULSE software (Fig.2). In the project are described: - organizing project configuration, measurement organization, functions organization, - system acquisition properties: Input Module LAN-XI 51, 2kH. Fig. 1. Measuring stand: testing procedure Source levels are reported as 1 – multichannel rack for data acquisition; measured sound pressure levels at 2 – reinforced laptop; a distance of 1 m from the source, 3 – signal amplifier; 4 – anechoic tank. so it was necessary to correct the recorded level of the measurements, taking into account the distance between the source and the measurement site. Propagation in shallow water is complicated, because sound can be reflected several times from the surface and from the deep, during propagation from source to receiver.

Applied Mechanics and Materials Vol. 430

287

a)

b)

c)

Fig. 2. Properties of the acquisition system and of the signals to be purchased: a - project „mare” 14.051645. pls : project configuration organizing, b - organization measurement functions, organization; c properties acquisition system: Input Module LAN-XI 51.2kH.

3. Measurement noise level generated by a rig tender We presented further signal acquisition done for the rig tender at speed of 22.24km/h (12 knots) (Fig.3). The highest value of sound pressure was recorded by the hydrophone measuring at medium depth, 32Pa. This record was used to model the acoustic field accompanying the ship. Each ship noise was composed of installations noise and cavitation. Hydrophones placed vertically, during the study period, showed that there are strong currents throughout the water column. To verify the acoustic field modeling of noise, recordings were made at several locations at distances up to 10Km. This is consistent with broadband noise modeling shown in Fig. 4, where the signal must be at least 3 dB above the background noise level to be charged. 4. Acoustic modeling of the underwater noise emitted by a rig tender Sound field or ship noise vector patterns described using: Noise levels recorded during transitions; Propagation models derived from transmission loss and frequency dependence, taken from previous recordings [5];

288

Acoustics & Vibration of Mechanical Structures

The lowest level of noise experienced at the site, available in previous recordings. To configure broadband noise levels as vector functions we choused a grid xy and it was calculated distances and angles to the central point for bands 1/3 octave frequencies ranging from 25 Hz to 10 kHz. Received level at each point was scaled by adding noise (which led to resize points for which the noise level was below the level of ambient noise) and the resulting matrix was converted into intensity. Intensities of each Fig. 3. Presentation of acquired signals and processed ship accompanying band 1/3 octave were traveling at a speed of 22.24 km/h (12 knots) :a- signals acquired from the three then summed and the hydrophones; result converted back to dB signal to get broadband in all grids. Points outside the ship trajectory were removed. The matrix was then configured as broadband curves level contour. Fig. 4 shows the evolution of predictive modeling vector and broadband noise for ship trajectory on a grid of 2.5 x 2.5 km. It was used the lowest ambient noise intensity with location, encountered during the Fig. 4. Acoustic modeling of the underwater noise emitted by a rig tender: measurements. Acoustic modeling of the underwater noise emitted by a rig tender, presented as The ambient noise broadband sound level (dB re 1μPa) in the frequency range 25Hz - 10 kHz band level was 90 dB in the 1/3 octave, with reference to the lowest level of noise on a grid of 2.5 km x 2.5 frequency range km. Ship (source) is considered located in the center of plotting and moves to 25 Hz ... 10 kHz band the right with a speed of 12 22.24 km/h (12knots). Color Coding: 90 dB re 1μPa is pure blue and 180 dB re 1μPa is pure red. 1/3 octave.

Applied Mechanics and Materials Vol. 430

289

Due to modeling is observed that the high levels of noise - 120 dB criterion - is produced in the 250m contour accompanying Ship (depending on location and sea conditions). Sound pressure values obtained by modeling were verified by measurements made by moving the measuring system at various distances from the source. There was a concordance between the measured and modeled values within ± 3dB re 1μPa what is acceptable as a variation of 6dB sound pressure level is not likely to produce changes in behavior shellfish. Accompanying ship only exceeded by 3dB noise. In record, a ton ship product can be detected near the 5.5 km distance but is distinct from the 5.2 km. 5. Conclusions In a study it was found that the acoustic pressure field accompanying vessel does not exceed the level of 120dB, considered to be the level of disturbance to marine life than a 250m contour around the ship. Since the ship is moving, and thus the influence of increased pressure occurs only during the passage of ships through the area, we believe that the effect of sound on marine habitat is not likely to produce permanent changes. Modeling shows a high degree of fidelity to sound pressure values measured at increasing distances from the source. The paper presents: - a methodology for measuring underwater sound pressure, - description of an underwater acoustic pressure measuring, - description of a stand designed for signal acquisition, - presentation of the draft prepared for acoustic measurements and subsequent data processing, - Predictive modeling of the evolution of a sound field for a rig tender Predicting the level of noise pollution is a tool that should be used in all anthropogenic activities that occur offshore in order to take measures for its mitigation. It is highlighted the need to draw up national databases that include records and hydro acoustic modeling parameters. 6. References [1] AMZA Gh., “Ecotehnologie şi Dezvoltare Durabilă”, Vol II, Ed. Printech, Bucuresti, 2009, ISBN 978-606-521-318-0, p. 478. [2] AMZA Gheorghe, Mihai ARSENE, Delicia ARSENE, Claudia BORDA, Marinela MARINESCU, Larisa BUTU, Underwater measurements concerning high frequency sounds radiated by ships, The Annual Symposium of the Institute of Solid Mechanics, SISOM, and Session of the Comission of Acoustics, Bucharest, May, 2009. [3] Ovidiu Radu, Mihai Arsene, Underwater Acoustic Noise Measurement of Vessels. Case Study, Simpozionul Aniversar cu participare Internationala: 40 de ani de Cercetare Stintifica in Domeniul Naval, Centrul de Cercetare Stiintifica pentru Fortele Navale, Bucuresti, Mai 2010. [4] Noise Measurement Procedures Manual, Environment Division, Department of Primary Industries, Water and Environment, GPO Box 44, Hobart Tasmania 7001, Australia, First edition, July 2004, ISBN 0 7246 63150. [5] Mihai Arsene, Victor Popovici, Delicia Arsene, Claudia Borda, Marinela Marinescu, Larisa Butu, Transmission Loss Measurement in the Black Sea, The Annual Symposium of the Institute of Solid Mechanics, SISOM, and Session of the Comission of Acoustics, Bucharest, May, 2012.

Applied Mechanics and Materials Vol. 430 (2013) pp 290-296 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.290

Influence of Vibrations on Grain Harvesters Operator Valentin Vlăduţ1,a, Sorin-Ştefan Biriş2,b, Sorin Bungescu3,c, Nicolae Herişanu4,d 1

INMA Bucharest, Bd. I,I, de la Brad, 6, Bucharest, Romania, ”Politehnica” University of Bucharest, 313 Splaiul Independentei, Bucharest, Romania, 3 U.S.A.M.V. of Banat, Timişoara, Romania, 4 ”Politehnica” University of Timişoara, Bd. Mihai Viteazu, 1, Timişoara, Romania

2

a

[email protected], [email protected], [email protected], [email protected]

Key words: vibrations, harvester, accelerations, nomographs, frequency

Abstract. The paper proposes an analysis of vibrations within many Romanian and foreign harvesters provided with tangential threshing unit, analysing the influence which these ones have on the human operator's health. The determination of vibrations has been performed on the basis of accelerations measured at various frequencies, on the three directions: x, y and z (by means of the accelerometers mounted on the main working parts of the harvester), tracing the nomographs representing the limits up to which these frequencies are not dangerous for the operator's health. Introduction Vibrations are dynamic phenomena [1] which appear in elastic or quasi-elastic media after a local excitation and which are manifested by propagating the excitation within the medium under the form of some elastic oscillations. Determination of the vibrations influence on the human body is an important goal, because the vibration duration and intensity can lead to serious operator health disorders: kidney, liver, column disorders, etc. and even to operator lifetime diminishing. A lot of researches have been conducted on operator driving characteristics in vibration exposure [3,7], influence of vibration direction [5], vibration energy absorption [9], influence o handle size and hand–handle contact force on the hand-arm system [11, 18], influence of tyre inflation pressure [4], reducing vibration exposure [8], variation in exposure to vibration [10], assessment of the influence of the eccentricity of tires [12], vibration in heavy equipment operators [13], health effects of long-term occupational exposure [14], vibration transmitted to the framesaw operator [15], effects of translational whole-body vibration [17]. In the last years there have also been studied the response of a standing operator to vibration stress [6], the influence of damping on waves and vibrations [19] or the influence of vibration on people in general [16], in order to identify the possibilities of attenuating these vibrations and their negative influence on operators health. Materials and method Vibration measurements [2] were performed on the following cereal harvesting machines with a tangential threshing unit: SEMA 140 M, NEW HOLLAND TC 56, TOPLINER 4075, NEW HOLLAND TX 66, LAVERDA L 6261, BIZON RECORD Z 058, in two conditions: in stationary position and during work, by using transducers for each of the following working parts of the harvester: thresher, header, chassis and operator's chair. On the basis of the values of longitudinal, cross and vertical accelerations measured to various frequencies in the two operating conditions of the harvester, there are established the effect of the vibrations on each of these parts and, finally, the vibrations sent to the operator's chair, the nomographs being traced and the limits range up to which they are not dangerous for the operators health being established. The results of the determinations on the vibrations sent to the chair of the harvester SEMA 140M are shown in fig. 1 and 2. On the basis of the values of ax, ay, az, accelerations measured at various frequencies within three possible conditions of the harvester operation (at stationary, during transport and work conditions), some graphical representations have been performed. At stationary position, the accelerations which have appeared were very low and from the nomographs within the admissible limits (ISO 2631-78) it results that the operator can work without any danger for over 24 hours. During the effective work (the coupled thresher) the highest accelerations were registered on

Applied Mechanics and Materials Vol. 430

291

a longitudinal direction, respectively 0.18 m/sq.s at 2.5 Hz. It results that the operator could work ceaselessly, without any danger, for maximum 16 hours. During transport the vertical accelerations were the highest, reaching 1.05 m/sq.s at the frequency of 3.15 Hz, so the danger for the operator occurs only if the harvester runs ceaselessly for over 1.5 hours.

Fig. 1 - Longitudinal and cross accelerations at stationary, during work and transport (SEMA 140M)

Fig. 2 - Vertical accelerations at stationary, during work and transport (SEMA 140M)

The results of the determinations on the vibrations sent to the chair of the harvester NEW HOLLAND TC 56 are shown in fig. 3 and 4. Both at stationary and during the effective work, the accelerations were low, resulting in the operator's work without any danger for over 20 hours. During transport the vertical accelerations were the highest, reaching 0.18 m/sq.s at 8 Hz frequency, so the danger for the operator occurs only if the harvester runs ceaselessly for over 16 hours.

Fig. 3 - Longitudinal and cross accelerations at stationary during work and transport (NH TC 56)

292

Acoustics & Vibration of Mechanical Structures

Fig. 4 - Vertical accelerations at stationary, during work and transport (NH TC 56)

The results of the determinations on the vibrations sent to the chair of the harvester DEUTZ-FAHR TopLiner 4075 are shown in fig. 5 and 6. At stationary position, the operator can work without any danger for over 24 hours and during effective work - for maximum 20 hours (0.13 m/sq. s at 1.25 Hz). During transport the maximum vertical accelerations were 0.64 m/sq.s at 2.5 Hz and the cross accelerations 0.39 m/sq.s at 1.25 Hz, resulting in a danger for the operator if the harvester runs ceaselessly for over 3 hours.

Fig. 5 - Longitudinal and cross accelerations at stationary, during work and transport (DF Top Liner 4075)

Fig. 6 - Vertical accelerations at stationary, during work and transport (DF Top Liner 4075)

Applied Mechanics and Materials Vol. 430

293

The results of the determinations on the vibrations sent to the chair of the harvester LAVERDA L 626I are shown in fig. 7 and 8. At stationary position, the operator can work without any danger for over 24 hours and during effective work for maximum 18 hours (0.13 m/sq.at 2.0 Hz). During transport the maximum vertical accelerations were 1.44 m/sq.s at 20 Hz and the cross ones - 0.74 m/sq.s at 2.0 Hz resulting in a danger for the operator if the harvester runs ceaselessly for over 1.5 hours.

Fig. 7 - Longitudinal and cross accelerations at stationary, during work and transport (LAVERDA L626 I)

Fig. 8 - Vertical accelerations at stationary, during work and transport (LAVERDA L626 I)

The results of the determinations on the vibrations sent to the chair of the harvester NEW HOLLAND TX66 are shown in fig. 9 and 10. At stationary position, the operator can work without any danger for over 24 hours and during effective work for max. 6 hours (0.16 m/sq.s at 1.25 Hz - cross accelerations). During transport the maximum accelerations were the vertical ones, reaching the value 0.55 m/sq.s at 2.5 Hz and they result in the operator's risk if the harvester runs ceaselessly for over 4 hours.

Fig. 9 - Longitudinal and cross accelerations at stationary, during work and transport (NEW HOLLAND TX

294

Acoustics & Vibration of Mechanical Structures

Fig. 10 - Vertical accelerations at stationary position, during work and transport (NEW HOLLAND TX 66)

Fig. 11 - The vertical acceleration spectrum in operating mode for the five combines

Analysing the vertical acceleration spectrum (which has the highest values) for all five combines, it can be seen that New Holland TC 56 is the combine providing a working regime without affecting the operator's health. Using an energetic estimation of the type: E=

1 m 2

n

a zi2

∑ 4π i =1

2 2 fi

,

(1)

Energy [J]

for each combine type (where n is the number of frequencies from the selected spectrum, and m is the typical weight of a combine operator, 75 daN), for the same type of acceleration and the same type of operating regime, one can obtain a ranking by quality (figs.12-14).

Fig. 12 - The energy transmitted by the combine to operator in normal working regime

Applied Mechanics and Materials Vol. 430

295

Fig. 13 - Energy variation transmitted to the operator by direction of acceleration and the type of combine, on the longitudinal direction in various types of operation (in normal working regime)

Fig. 14 - Energy variation transmitted to the operator by direction of acceleration and the type of combine, on the lateral direction in various types of operation (in normal working regime)

Conclusions Analyzing the vibrations sent from the working parts of the five investigated harvesters to the chair, one can conclude the followings: • the maximum values of the longitudinal accelerations - ax were registered for the harvester SEMA 140M during work, and they were 0.18 [m/sq.s] at 2.5 Hz; this means that the operator can work ceaselessly, without any danger, for maximum 16 hours. • the maximum values of the cross accelerations - ay were registered for the harvester LAVERDA L 6261, during transport, and they were 0.74 [m/sq.s] at 2.0 Hz; this means that the operator can work ceaselessly, without any danger, for maximum 1,5 hours. • the maximum values of the vertical accelerations - az were registered for the harvesters SEMA 140M (1.44 m/sq.s at 20 Hz) and LAVERDA L 6261 (1.05 m/sq.s at 3.15 Hz), during transport, this means that the operator can work ceaselessly, without any danger, for max. 1.5 hours. REFERENCES [1] Brüel & Kjaer, Mechanical Vibration and Shock Measurements, K. Larsen&Søn (1984). [2] V. Vlăduţ, a.o., Testing of the Cereal Harvesting Machine New Holland TC 56, New Holland TX 66, Deuty-Fahr Top Liner 4075, Laverda L 626I, Sema 140 M, Test Reports (1998-2002). [3] N. Costa, P.M. Arezes, The influence of operator driving characteristics in whole-body vibration exposure from electrical fork-lift trucks, Int. J. Industrial Ergonomics 39 (2009) 34-38. [4] L.M. Sherwin, P.M.O. Owende, C.L. Kanali, J. Lyons, S.M. Ward, Influence of tyre inflation pressure on whole-body vibrations transmitted to the operator in a cut-to-length timber harvester, Appl. Ergonomics 35 (2004) 253-261. [5] A.J. Besa, F.J. Valero, J.L. Suñer, J. Carballeira, Characterization of the mechanical impedance of the human hand–arm system: The influence of vibration direction, hand–arm posture and muscle tension, Int. J. Industrial Ergonomics 37 (2007) 225-231.

296

Acoustics & Vibration of Mechanical Structures

[6] M. Fritz, Simulating the response of a standing operator to vibration stress by means of a biomechanical model, J. of Biomechanics 33 (2000) 795-802. [7] K.N. Dewangan, V.K. Tewari, Characteristics of hand-transmitted vibration of a hand tractor used in three operational modes, Int. J. Industrial Ergonomics 39 (2009) 239-245. [8] T.H. Langer, T.K. Iversen, N.K. Andersen, O.Ø. Mouritsen, M.R. Hansen, Reducing wholebody vibration exposure in backhoe loaders by education of operators, Int. J. Industrial Ergonomics 42 (2012) 304-311. [9] K.N. Dewangan, V.K. Tewari, Vibration energy absorption in the hand–arm system of hand tractor operator, Biosystems Eng. 103 (2009) 445-454. [10] B. Rehn, R. Lundström, L. Nilsson, I. Liljelind, B. Järvholm, Variation in exposure to wholebody vibration for operators of forwarder vehicles - aspects on measurement strategies and prevention, Int. J. of Industrial Ergonomics 35 (2005) 831-842. [11] Y. Aldien, P. Marcotte, S. Rakheja, P.-É. Boileau, Influence of hand forces and handle size on power absorption of the human hand–arm exposed to zh-axis vibration, J. of Sound and Vibr. 290 (2006) 1015-1039. [12] M. Cutini, E. Romano, C. Bisaglia, Assessment of the influence of the eccentricity of tires on the whole-body vibration of tractor drivers during transport on asphalt roads, J. of Terramechanics 49 (2012) 197-206. [13] R.P. Blood, P.W. Rynell, P.W. Johnson, Whole-body vibration in heavy equipment operators of a front-end loader: Role of task exposure and tire configuration with and without traction chains, J. of Safety Research 43 (2012) 357-364. [14] B.O. Wikström, A. Kjellberg, U. Landström, Health effects of long-term occupational exposure to whole-body vibration: A review, Int. J. of Industrial Ergonomics 14 (1994) 273-292. [15] V. Goglia, I. Grbac, Whole-body vibration transmitted to the frame saw operator, Appl. Ergonomics 36 (2005) 43-48. [16] J. Sandover, Vibration and people, Clinical Biomechanics 1 (1986) 150-159. [17] R.W. McLeod, M.J. Griffin, Review of the effects of translational whole-body vibration on continuous manual control performance, J. of Sound and Vibr. 133 (1989) 55-115. [18] P. Marcotte, Y. Aldien, P.-É. Boileau, S. Rakheja, J. Boutin, Effect of handle size and hand– handle contact force on the biodynamic response of the hand–arm system under zh-axis vibration, J. of Sound and Vibr. 283 (2005) 1071-1091. [19] L. Gaul, The influence of damping on waves and vibrations, Mech. Syst. and Signal Processing 13 (1999) 1-30.

Applied Mechanics and Materials Vol. 430 (2013) pp 297-302 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.297

Troubleshooting Technics to Identify the Airborne and Structure-Borne Noise Content inside an Electric Vehicle Marius-Cristian Morariu1, a, Iulian Lupea1,b and Colin Anderson2,c 1

Technical University of Cluj-Napoca, B-dul Muncii, nr. 103-105, 400641 Cluj-Napoca, Romania 2

LMS International N.V., Interleuvenlaan 68, B-3001 Leuven, Belgium

a

[email protected], [email protected], [email protected]

Keywords: Transfer Path Analysis, Airborne Source Quantification, Electric Vehicle Noise

Abstract. An electric vehicle was subjected to on-road acoustic tests. A specific high frequency tone was perceived in a sound field dominated by wind and road noise. The car was instrumented with microphones which measured the noise inside the passenger compartment and with tachometers to record the motor’s rotational velocity with respect to time. Waterfall diagrams were generated by tracking the spectrums of noise from fixed time samples against the rpm of the motor. The analysis of the diagrams revealed that high orders, like the 24th and 48th were responsible for the sound. These orders represent the acoustic response of the electromagnetic interaction between the stator and the rotor of the electric motor. To analyze the propagation of noise from the source (motor) to the target (driver), a transfer path analysis (TPA), respectively an airborne source quantification (ASQ) were proposed. The TPA focused on the structure borne noise generated by the forces transmitted into the body through the powertrain supports, and the ASQ, on the airborne noise radiated by the surface of the motor and gearbox casing. The conclusion was that the airborne noise is the main contributor to the total pressure level in the passenger compartment, but at lower speeds a strong structure borne noise content is present. 1 Introduction The electric vehicle, though an old concept, is being reintroduced nowadays on the market as an alternative to the classical combustion engine vehicles. One of its main advantages is the fact that it reduces a lot the interior noise, by using an electric motor for propulsion. A study made by Eckstein in 2011 [1] revealed a maximum sound pressure level (SPL) of 68 dB(A) measured inside the passenger compartment of a Mitsubishi i-MiEV, during a run-up. The autors of this article obtained a 69 dB(A) maximum SPL, while performing similar tests on a Nissan Leaf [2]. In the same study, a particularity of the noise was outlined, namely the existance of sweeping tones, corresponding to the 24th and 48th orders of the electric motor and others related to the gear meshing. The level of the orders was rather low and constant, with some high peaks in the low frequency range. The question that remained unanswered was how much of the order’s level measured at the target location is the consequence of the forces transmitted from the motor to the structure through the rubber mounts and how much is the contribution of the noise radiated through air by the motor’s casing. In other words, which is the percentage of structure borne noise vs. airborne noise? To answer it, this paper proposes two troubleshooting techniques: Transfer Path Analysis (TPA) and Airborne Source Quantification (ASQ). The TPA method was first described by Wyckaert and Auweraer in 1995 [3]. According to Plunt [4,5] there are four TPA approaches: a classic TPA for low frequencies; a mid and high frequency range TPA; an old fashioned “Physical TPA” which was used before any commercial software and it consisted in actual physical decoupling of certain suspected mounts to sort out structure-borne noise paths; a combination of the classical TPA with a decoupling program. Knapen applied the first two methods to find the booming noise source inside the passenger compartment of an internal combustion engine vehicle at certain engine rotational speeds [6]. Kim and Lee went even further

298

Acoustics & Vibration of Mechanical Structures

into the problem and tried to predict vehicle interior noise by combining TPA with FE, resulting a new method called Hybrid TPA [7,8]. The classic method was chosen for the analysis of the noise inside the electric vehicle and it is described further in this article. 2 Problem description and assumptions

dB(A)

MotorTacho (DT1) rpm

2.1 Interior Noise Analysis. As noted in the introduction, a study on the Nissan Leaf interior noise was already performed by the authors. The car was accelerated on the road from 0 to 100 Km/h while measuring the sound pressure at the 6500.00 24.00 48.00 50.00 target location, chosen near the driver’s right ear. The rotational velocity of the electric motor was also acquired with a tacho sensor. Fixed time samples were transformed into frequency domain spectra and tracked against the rpm of the motor, forming a 3D waterfall diagram. A colormap of the waterfall diagram is presented Fig. 1. The brighter area on the left of the graph is assumed to be noise generated by the tire-road interaction and by the aerodynamic turbulences. Orders 24 and 0.00 0.00 0.00 Hz 5500.00 48 are visible on the colormap as diagonal FRLE:IN:S (CH95) lines. They are associated to the number of Figure 1 Collormap diagram of the SPL measured at the level of the right ear of the driver magnets of the rotor respectively the number of poles of the stator from the electric motor. Order 24 sweeps on a frequency range from 0 Hz to 2600 Hz, and order 48 to 5200 Hz, but the first is covered by noise above 1000 Hz. The perceived sound is therefore a combination of noise and high frequency tones. The assumption is made that the noise generated by the structural vibrations (that are transmitted through the mounts from the motor) and the noise radiated by the surface of the motor casing contribute together to the total SPL of the orders at the target location. 2.2 Problem approach. The vehicle can be considered a mechanical system composed by an active side (electric motor) and a passive side (body) linked together by rubber mounts. A schematic representation is shown in Fig. 2. The mounts represent paths for the transmission of vibrations from the motor to the body. For a pressure target at the driver’s ear level, the total contribution could be written as a combination of k partial pressures from structural loads at the level of the paths and n-k pressures from the acoustic loads [9]. n

k

p 1

p 1

Pt ( )   Pt , p ( )   H tP/ /pF  Fp   

n

H

p k 1

P / Q t/ p

 Q p   .

(1)

Pt ( ) is the operational pressure at target ‘t’ (Pa). Pt , p ( ) are the partial pressures at target ‘t’(Pa)

Figure 2 Schematic representation of the active and passive sides of the vehicle

H tP/ /pF represents the transfer function between the force at path ‘p’ and pressure at target ‘t’(Pa/N).  H tP/ /pQ is the transfer function between volume acceleration at path ‘p’ and pressure at target ‘t’ ( m3  s 2  N 1 ). Fp   (N) are the operational forces at path p. Q p   ( m3  s 2 ) are the volume accelerations at source p.

Applied Mechanics and Materials Vol. 430

299

The motor-gearbox assembly is sustained on the subframe of the car by three supports with rubber bushings named mounts. We assume that the structural loads transmitted through the mounts are only forces, without taking in consideration any torques. The acoustic loads are volume accelerations of the air in the vicinity of the motor-gearbox casing. The further investigations of the noise will be performed only on orders sections. 3 Transfer Path Analysis (TPA) The partial pressures at the target that result from the contribution of each structural path can be calculated by multiplying the operational force at the path location with the transfer function from the path to the target as in Eq. 1. The operational forces are considered to act in the three orthogonal directions X, Y, Z at the level of each mount. As a consequence, nine transfer functions were measured directly by exciting the mounts with an impact hammer and measuring the response sound pressure at the target location. 3.1 Load identification method. Measuring the operational forces which act at the level of each path is very difficult. This is the reason why it was decided to estimate the forces through the method called matrix inversion [9]. The assumption is made that the structure has a linear behavior. The method consists in assembling frequency response functions H Xi / Fp in a matrix, where p=1,…,k and i=1,...,m. They are measured between the indicators X i and the forces Fp at the transfer paths and obtained by impact testing. The indicators are in this case accelerations from sensors placed close to each path’s location. The obtained matrix is then inverted and multiplied by the operational accelerations to obtain the vector of operational forces. The corresponding equations are shown below:

 F1   H X1 / F1         Fk   H Xm / F1

H X1 / F2  H Xm / F2

1

 H X1 / Fk   X1           H Xm / Fk   X m 

(2)

3.2 Instrumentation. To limit the errors in the process of load identification, an overdetermination of the system of equations up to a factor 2 is advised [9]. The number of forces to identify was k=9 so a minimum number of 18 indicators was needed. There were 18 3D available accelerometers mounted all on the subframe close to the paths, which resulted in a number of m=42 indicators. They were all used within a least square approximation method to obtain a good estimation of the operational forces. A 1D accelerometer was attached to the motor casing so that all the operational data will contain phase information relative the vibration of the motor. 3.3 Limitations. The impact testing limits the FRF bandwidth from 0 to 2000 Hz. Above this value, the signal to noise ratio becomes low according to the coherence function and the results could be compromised.

300

Acoustics & Vibration of Mechanical Structures

3.4 Results. The resulted partial pressures were summed together with respect to phase information. Hz 1000.00

2600.00

50.00

0.00 180.00

0.00 180.00

-44.5e-12 380.80 840.00

°

950.00

-180.00

2500.00

0.00

rpm

-180.00 0.00

6500.00

MotorTacho (DT1) Figure 4 Measured SPL (_) vs. estimated structural contribution (...) on order 24

Hz

5200.00

rpm

6500.00

1050.00 476.00

°

dB(A)

-22.3e-12 380.00

dB(A)

50.00

MotorTacho (DT1) Figure 3 Measured SPL (_) vs. estimated structural contribution (...) on order 48

The SPL of the orders measured during the run up was compared with the total estimated sum of the structural contributions. The sums were calculated in a frequency range limited to 2000 Hz for the reasons mentioned above. In Fig. 3, the measured level of order 24 was plotted with continuous line and the total sum with dotted line. The calculated level of the order is close to the measured level in the lower frequency band, being similar at the 380 Hz peak. Above 1000 Hz the calculated level drops considerably, but the broadband noise is masking the measured order at those frequencies. In Fig. 4, the measured and the calculated levels of order 48 were plotted in the same way. Like in the case of order 24, the estimated level of order 48 fits the measured level on some low frequency peaks like the one at 380 Hz. 4 Airborne Source Quantification (ASQ) For ASQ, the made assumption was that the radiating casing of the motor and gearbox can be substituted by independent sound sources. They are considered equally distanced between them and situated on the surface of the casing, like seen in the picture from Fig. 5. The idea is that the sound energy radiated by the closest corresponding patch from the surface is concentrated in the virtual point of the acoustic source. 4.1 Load identification method. The acoustic loads to be identified were the volume accelerations Q p of

Figure 5 Location of the acoustic sources on the casing of the motor and gearbox

the assumed independent sources. The casing’s surface was divided into 13 equal area patches giving the number of sources n-k=13. The small assumed number of sources was chosen like this for the ease of the measurements but it was big enough to cover all the faces of the motor. To calculate the loads, the matrix inversion method was used.

Q k 1   H P / Q    1 k 1      Q n   H P / Q    r k 1

H P1 / Q k 2  H Pr / Q k 2

 H P1 / Q n      H Pr / Q n 

1

 P1      Pr 

(3)

In order to apply this method, the transfer functions H Pj / Q p were measured between the volume accelerations Q p at source p and the acoustic pressure Pj at indicator j, where p=k+1,...,n and

Applied Mechanics and Materials Vol. 430

301

j=1,...,r. The transfer functions were grouped into a transfer matrix and its inverse multiplied by the vector of operational pressures from the microphones obtaining the acoustic loads as in Eq. 3. -22.3e-12

Hz

2600.00

960.00

-44.5e-12

Hz

5200.00

rpm MotorTacho (DT1)

6500.00

50.00

0.00 180.00

0.00 180.00

°

°

dB(A)

400.00

dB(A)

50.00

1000.00

-180.00 0.00

Figure 7

2400.00

rpm MeasuredMotorTacho SPL (_)(DT1) vs. estimated

contribution (...) on order 24

-180.00

6500.00

airborne

0.00

Figure 6 Measured SPL (_) vs. estimated airborne contribution (...) on order 48

4.2 Instrumentation. To measure the transfer functions, a mid-high frequency volume acceleration source was used for excitation, which operated on a bandwidth between 200 Hz and 8000 Hz. The nozzle of the source was placed successively at the position of each source and the responses were measured with r=29 microphones placed around the motor. This number of microphones was chosen in order to have an overdetermination of the system as explained previously in the TPA section. 4.3 Results. The estimated airborne contributions to the total pressure at the target were summed using an energetic method, namely an RMS sum (the square root of the sum of the squared amplitudes was taken). In Fig. 6, the total airborne contribution on order 24 is represented with dotted line and the measured SPL with continuous line. It is observed that the sum fits the measured signal between 400 Hz and 960 Hz, which is in upper part of the frequency range of interest. Under this frequency band the structural contribution is dominant and above it the order is masked by noise. In Fig. 7, the SPL of the order 48 is clearly a consequence of the airborne contribution. The calculated sum fits quite well the measured level. Only in the lower frequency band, below 500 Hz, the estimated level is lower than the measured one. It is proved in this way that the higher frequencies part of the order is a consequence of the airborne noise, while the lower frequency part is a consequence of the structure borne noise. 5 Conclusions and future work Acoustic measurements were performed on an electric car during several run-ups. The electric motor was considered an important source of noise inside the passenger compartment, and two troubleshooting techniques were proposed to analyze the generated noise. The TPA and ASQ were performed on orders 24 and 48 of the motor and revealed that the corresponding SPLs were mainly a result of the airborne noise, but bellow 1000 Hz a strong structure borne noise contribution was present. Further investigations can be made on the basis of this study, like the correlation of some high pressure level peaks with mount resonances, motor casing resonances or high force levels. Also, an estimation of the contribution of each path/acoustic source to the total SPL is possible now. This study proved that classic TPA and ASQ can be used as traditional NVH troubleshooting tools for the electric vehicles. Still, there is place for improvements in the higher frequencies domain, where the proposed ASQ technique brought overestimation of the measured pressure levels.

302

Acoustics & Vibration of Mechanical Structures

Acknowledgement: This paper was supported by the project "Improvement of the doctoral studies quality in engineering science for development of the knowledge based society-QDOC” contract no. POSDRU/107/1.5/S/78534, project co-funded by the European Social Fund through the Sectorial Operational Program Human Resources 2007-2013 and by LMS International, which with by their amiability provided the software [10], facilities and technical support. References [1]. L. Eckstein, R. Gobbels and R. Wohlecker, Benchmarking of the Electric Vehicle Mitsubishi IMIEV, ATZ. 113 (2011), 48-54 [2] M. Morariu, I. Lupea and C. Anderson, Interior Noise and Discrete Tones of Electric Vehicles, Acta Tehnica Napocensis, Cluj-Napoca, (forthcomming 2013) [3] K. Wyckaert, H. Van der Auweraer, Operational Analysis, Transfer Path Analysis, Modal Analysis: Tools to Understand Road Noise Problems in Cars, SAE Noise & Vibration Conference, Traverse City (1995), 139-143 [4] P.L. Knapen, Transfer Path Analysis Related to Booming, Performed on a Car, WFW Report No. 99-018, NedCar Report No. 52233/99-0125, Netherlands Car B.V. Vehicle Dynamics Department Steenovenweg 1, 5708 HN Helmond, Netherlands (1999) [5] J. Plunt, Finding and Fixing Vehicle NVH Problems with Transfer Path Analysis, Acoustical Publications, Inc., November, ProQuest Information and Learning Company (2005) [6] J. Plunt, Strategy for transfer path analysis (TPA) applied to vibro-acoustic systems at medium and high frequencies. Proc. ISMA 23, 16-18 sempember, Leuven, Belgium (1999). [7] S.J Kim, and S.K. Lee, Prediction of structure-borne noise caused by the powertrain on the basis of the hybrid transfer path, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, Professional Engineering Publishing (2009). [8] S.K. Lee, Identification of a vibration transmission path in a vehicle by measuring vibrational power flow, Proceedings of the Institute of Mechanical Engineers Part D, Journal of Automobile Engineering, 218(2) (2004), 167-175 [9] H. Van der Auweraer, P. Mas, S. Dom, A. Vecchio, K. Janssens, P. Van de Ponseele, Transfer Path Analysis in the Critical Path of Vehicle Refinement: The Role of Fast, Hybrid and Operational Path Analysis, SAE Technical Paper 2007-01-2352 (2007) [10]*LMS Test.Lab, LMS International, Leuven, Belgium, http://www.lmsintl.com

CHAPTER 6: Structural Vibration, Attenuators and Isolation

Applied Mechanics and Materials Vol. 430 (2013) pp 305-311 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.305

Corrective Analysis of the Parametric Values from Dynamic Testing On Stand of the Antiseismic Elastomeric Isolators in Correlation with the Real Structural Supporting Layout BRATU Polidor1, 2, a 1

ICECON S.A., Bucharest, Sos. Pantelimon 266, sector 2, 021652, CP 3-33, Romania

2

Technical University “Dunărea de Jos”, Galati, Str. Domnească, nr. 47, 800008, Romania a

[email protected]

Keywords: dynamic and kinematic excitation, laboratory testing, isolators

Abstract. The paper presents the result of experimental research of the viscoelastic behavior of the antiseismic elastomeric isolators on stand, in laboratory, where the excitation is given only through harmonic instantaneous displacements. Based on the dynamic response under the form of the elastic and dissipation forces, in the time domain, the hysteresis loops and the stiffness, damping and durability to imposed repeated cycles parameters are determined. In this case, the order I dynamic system is free of added mass which makes the evaluation of its own damping to be estimated as “equivalent damping” with that of a complete system of 2nd order with viscous damping. Using elastomeric isolators on site, for a base isolation project, either building or viaduct, imposes corrections of the experimental laboratory values considering the real conditions, function of the dynamic inertial excitation (earthquake, wind gusts, etc.) and of the response in instantaneous displacements. Introduction Antiseismic elastomeric devices are the fundamental elements in the composition, adjustment and performance capability of the base isolation systems during seismic actions. Until the 90s the design and use of hyper elastic supports made of elastomers in various configurations were limited due to execution technologies. Presently, the updated industry is capable of efficiently making quality products at significantly large geometrical shapes and dimensions. The evolution of the testing methods lead to both experimental system on the stand with displacements imposed by achievable laws, as well as precise measurements of the response function, including the hysteretic loop representation in real time. In accordance with the European Standard EN 15129:2010 [11], the conformity assessment and CE marking of the anti-seismic elastomeric isolators, used as components of the base isolation systems, are done. Experimental research carried out on specialized testing facilities, under dynamic regime, is conducted. The testing should reproduce the loading conditions equivalent to the operation specific parameters mainly defined by the geometrical and mechanical characteristics determining the damping and stiffness parameters [1,2]. In order to determine the damping capacity, the elastomeric isolators are subjected to shearing by means of kinematic harmonic excitations defined under the form x = A0 sin ω1t , where A0 is the absolute displacement amplitude of the loaded plate edge, in respect to the fixed edge and ω1 is the kinematic excitation pulsation [3]. In this case, considering of the hysteresis loop for the instantaneous viscoelastic force Q = kx + cx depending on the instantaneous deflection x, the loss factor of the internal energy η, representing the dissipation effect, as well as the equivalent critical damping fraction ζ eq are determined.

306

Acoustics & Vibration of Mechanical Structures

The loading conditions showed that the isolator has no attached concentrated mass, meaning that m ≡ 0 , and the excitation is kinematic exclusively with the harmonic displacement externally applied [4]. ∆W In this context, the damping expressed by the system parameter ζ eq , defined by ζ eq = , 4πWelmax differs as compared with the parameter ζ related to a linear viscoelastic system having the mass c m ≠ 0 , expressed under the form of ζ = [2]. 2 km Thus, the damping parameter ζ eq could be determined by laboratory testing only. Thus, the actual structural analysis using the supporting and the base passive isolation system, having appropriate configuration, the values of parameter ζ eq in correlation with the effective parameter ζ for the actual system, should be taken into account [5]. Experimental results of the elastomeric isolators were obtained in laboratory in Romania as well as in Italy and USA with the direct participation of the author, based on the testing program established by him, in some joint projects. Exterior Harmonic Actions Applied on the Elastomeric Isolator The angular excitation frequencies of the elastomeric dampers are different, meaning that in laboratory the frequency is ω1 with the amplitude of the displacement A0 and for a structural system which is going to be isolated under the conditions of a harmonic excitation with the amplitude F0=ct, the angular frequency corresponding to the first mode is ω2 with the response amplitude A, different from the first situation. In the end, the case of a structural element supported on elastomeric elements and tested in laboratory is presented, for which the dissipated energy in dynamic seismic regime must be equal to the energy dissipated on stand, thus for the first mode of the instantaneous seismic displacement, including that of the acceleration a0 of this vibration mode, the resulting amplitude is As. This correlation between ζ eq obtained under harmonic elastic deflection actions kinematic applied and ζ d under dynamic actions could be performed, so that the energy dissipated under the kinematic regime ∆Wc would be equal to the energy dissipated under dynamic regime ∆Wd . Fig.1 represents a physical system with a symmetric excitation system. The linear viscoelastic system characteristics are c and k, without an added mass.

Fig.1 Physical Model with a Symmetric Excitation System (Kinematic and Dynamic)

Kinematic Excitation Under the kinematic excitation, of the form x(t ) = A0 sin ω1t , the dynamic response related to the

viscoelastic connection force Q(t ) is obtained as [6]:

Applied Mechanics and Materials Vol. 430

Q(t ) = kx + cx ≡ Q0 sin(ω1t − ϕ1 )

307

(1)

and further: Q0 = kA0 1 + η12

(2)

cω1 = η1 k

(3)

tgϕ1 =

The dissipated energy ∆Wc can be written as:

∆Wc = πcω1 A02

(4)

If one introduces cω1 = kη1 , the previous relation becomes:

∆Wc = πkη1 A02

(5)

in which: η1- is the internal loss factor for the experimental system in kinematic regime. Dynamic Excitation Under the dynamic excitation of the form F (t ) = F0 sin ω2t , where F0 represents the amplitude of

the action, the displacement response u = u (t ) is obtained using the following instantaneous dynamic equilibrium equation: cu + ku = F0 sin ω2t

(6)

Equation (6) underlines that one works with a Ist order physical system, without mass, so its resonance is less. The solution u (t ) = A sin (ω2t − ϕ 2 ) should verify equation (6), leading to the following relation [5]:

A=

F0 1 , k 1+η 2 2

(7)

cω 2 = η2 k

(7’)

tgϕ 2 =

In which: η2- is the internal loss factor for the experimental system in dynamic regime. Taking into consideration relation (7), the dissipated energy ∆Wd is given by the equation ∆Wd = πkη 2

F02 1 k 2 1 + η 22

(8)

308

Acoustics & Vibration of Mechanical Structures

and using the equivalence condition that the dissipated energies in the two systems are equal ∆Wc = ∆Wd , results in:

η1η 22 − Ψ 2η 2 + η1 = 0

(9)

F0 being the dynamic multiplication factor. The physical significance of the imposed kA0 condition consists in the capacity to dissipate energy of the same elastomeric element in the two situations of dynamic loading. From (9) one obtains η2 under dynamic regime, depending on η1 under kinematic regime, as follows:

with Ψ =

η2 =

[

1 Ψ 2 ± Ψ 4 − 4η12 2η1

]

(10)

a) the single solution of equation (10) is possible only under the following condition: Ψ 4 − 4η12 = 0

and further Ψ =

(11) F0 = 2η1 A0 k

(12)

In this case, the solution of equation (10) is under the form:

η2 =

Ψ 2 2η1 = =1 2η1 2η1

(13)

1 Thus, the loss factor has a unitary value and ζ 2 = η 2 = 0,5 . ζ 2max = 0,5 represents the 2 maximum value. b) the actual and distinct solutions of equation (10) are possible only for Ψ 4 − 4η12 > 0 , or

Ψ > 2η1 . In case of actual parametric values 0,2 ≤ η1 ≤ 1,6 , the range for Ψ is obtained as 0,63 ≤ Ψ ≤ 1,78 . As an example, an elastomeric isolator without additional mass, with k = 1,5 ⋅ 10 6 N / m , ζ eq = 0,20 , η1 = 0,40 is tested in laboratory under harmonic cycles having the linear amplitude

A0 = 0,08 m. Under harmonic dynamic regime, characterized by F0 = 120 kN, the damping η2 is obtained as follows: Ψ=

F0 105 = = 1 > 2 ⋅ 0,4 = 0,89 kA0 1,5 ⋅ 106 ⋅ 8 ⋅ 10 − 2

and from relation (10) it results in η ' '2 = 2,0 leading to ζ '2 = 0,25 and ζ "2 = 1,0 , respectively, both values being higher than ζ eq = 0,20 .

Applied Mechanics and Materials Vol. 430

309

In which: η’’2 - is the equivalent loss factor in dynamic regime; ζeq – is the equivalent critical damping ratio of the order I system, namely the ratio obtained on the stand testing; ζ2’, ζ2’’ – represent the critical damping ratios on the stand test with force excitation. Harmonic Seismic Actions on the Elastomeric Isolator The dynamic model for the linear viscoelastic base isolation system, with m, k and c considered as system parameters, is represented in Fig. 2, for two distinct positions: under instantaneous translation motion and under the seismic action with the instantaneous acceleration u = a0 sin ωt [5].

Fig. Dynamic Model of the Passive Isolation System The equation of motion for the mass m, related to the fixed reference system O1X1Y1 is of the form:

mx1 + cx + kx = 0

(14)

in which x1 = u + x represents the absolute displacement. The relative displacement of the mass m, with respect to the moving reference system (having the acceleration u ) Oxy is x(t ) . Thus, one has:

m(u + x ) + cx + kx = 0

(15)

or mx + cx + kx = −mu

(16)

If one introduces u = a0 sin ωt , the previous relation becomes: mx + cx + kx = − ma0 sin ωt

(17)

The final solution is x = A sin (ωt − ϕ )

(18)

in which A and ϕ are obtained from the condition that verifies equation (17). Thus, it results: =

; tgϕ =

2ζ Ω k ω and Ω = ; ω n2 = . 2 ωn m 1− Ω

(19)

310

Acoustics & Vibration of Mechanical Structures

in which: ωn – is the natural pulsation of the order II system. The energy dissipated under seismic dynamic excitation regime, is [4,5]: m

∆Wd = 2πζ

ω n2

a02

(Ω

2

)

Ω

(20)

2

− 1 + 4ζ 2 Ω 2

and the condition ∆Wc = ∆Wd leads to the following relation

ζ eq =

α 2ζΩ

(Ω − 1)

2

2

+ 4ζ 2 Ω 2

.

(21)

Further we have the second order equation in ζ , of the form

(

)

2

4Ω 2ζ eqζ 2 + Ω 2 − 1 ζ eq = α 2 Ωζ

(22)

and having the solution:

ζ =

1 8Ω ζ eq 2

(

At resonance, for Ω = 1 , one obtains ζ rez = where α =

)

(23)

1 α2 4 ζ eq

(24)

2 2   2 4 2 2 2 α Ω ± α Ω − 16Ω Ω − 1 ζ eq   

a0 is the acceleration multiplication factor. A0ωn2

As an example, an elastomeric isolator having k = 1,5 ⋅ 10 6 N/m, ζ eq = 0,2 is tested at A0 = 0,088 m. For a structural system with ωn = 2π subjected to a maximum acceleration a0 = 0,25 g , the result is:

α=

0,25 ⋅ 10 = 0,707 0.088 ⋅ 4π 2

ζ =

1 0,707 2 = 0,62 4 0,2

meaning that ζ for the system is three times higher than ζ ech determined in the laboratory.

Applied Mechanics and Materials Vol. 430

311

Conclusions The following conclusions can be synthesized based on the analysis intended to equalize the dissipated energy under different excitation regimes, kinematic and dynamic: a) the testing method with kinematic excitation only, with harmonic displacement of the form x(t ) = A0 sin ω1t allows the determination of the equivalent damping ζ eq . This is specific to the isolation system under laboratory experimental configuration, only. b) the damping evaluation under dynamic excitation for systems having actual dynamic behavior puts into evidence the necessity to determine the system parameter ζ as a function of the experimental parameter ζ eq ; c) ζ for the system under actual configuration depends both on the type of the dynamic excitation as well as the existence of the mass with inertial effect. d) that a parametric correlation is required between the parametric values resulted from the product testing on stand and the necessary values which must be established as functional parameters in real experimental conditions of the antiseismic isolated building. References [1] EI. Rivice, Stiffness and Damping in Mechanical Design, Marcel Dekker Inc., New York, 1999 [2] P. Bratu, Analiza structurilor elastice. Comportarea la actiuni statice si dinamice., Editura IMPULS, Bucuresti, 2011 [3] MJ. Kelly, AD. Konstantinidis, Mechanics of Rubber Bearings for Seismic and Vibration Isolation, Wiley, 2011 [4] P. Bratu, N. Drăgan, L'analyse des mouvements désaccouplés appliquée au modèle de solide rigide aux liaisons élastiques, Analele Universitatii “Dunarea de Jos” of Galati, Fascicula XIV, 1997 [5] P. Bratu, O. Vasile, Modal analysis of viaducts supported on elastomeric insulation within the Bechtel constructive solution for the Transilvania highway, Romanian Journal of Acoustics and Vibration, 9(2012) 77-82 [6] M. Dolce, FC. Ponzo, A. DiCesare, G. Arleo, Progetto di Edifici con Isolamento Sismico, Iuss Press, Pavia, Italy, 2010. [7] JP. Bandstra, Comparison of equivalent viscous damping and nonlinear damping in discret and continuous vibrating system, Journal of Mechanisms, Transmissions and Automation in Design, J.V.A.S.R.D., vol I 105, 1983 [8] TK. Caughey, MEJ. O’Kelly, Effect of damping on the natural frequencies on linear dynamic systems, Journal of the American Statistical Associating, J.A.S.A., vol. 33, 1961 [9] I. Cochin, Analysis and Deisign of Dynamic Systems, Harper&Raw, New York, 1980 [10] S. Rao, Mechanical Vibrations, Addison-Wesley, New York, 1995 [11] SR EN 15129:2010, Dispozitive antiseismice, 2010

Applied Mechanics and Materials Vol. 430 (2013) pp 312-316 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.312

Modification of the Dual Kelvin-Voigt/Maxwell Rheological Behavior for Antiseismic Hydraulic Dampers BRATU Polidor1, 2, a 1

ICECON S.A., Bucharest, Sos. Pantelimon 266, sector 2, 021652, CP 3-33, Romania

2

Technical University “Dunărea de Jos”, Galati, Str. Domnească, nr. 47, 800008, Romania a

[email protected]

Keywords: dampers, rheological models, shock

Abstract. The hydraulic damper can be adjusted to realize the dissipation function depending on the viscous component of the silicone oil at an exterior excitation type shock, applied in the time domain in two manners: at maximum displacements with relatively small values (below 15 cm) having at its basis the Voigt-Kelvin rheological model; at maximum displacements with large values (above 30cm) having at its bases the Maxwell rheological model. For the same constructive solution, silicone hydraulic oil, the same structural mechanic elements, but with different settings/adjustments, one can model the rheological system Voigt-Kelvin, as well as the Maxwell system. The dual behavior of a hydraulic damper was verified in laboratory for 5 products made in companies from Italy, Germany and USA through the corresponding modification of the internal circulation of the silicone oil based on the adjustments of the commutation blocks with the purpose of highlighting the functional scheme and the response values at the same laboratory made shock, measuring the response as maximum displacement and the dissipated energy with hydraulic measuring devices. In this context, the main behavioral differences in dynamic response will be presented, as well as the stiffness, dissipation and displacement during the applied shock parameters. Introduction The hydraulic dampers with silicon oil are made in various constitutive solutions regarding the oil transfer between the operating rooms. Essentially, an efficient method of changing the way of the instantaneous transfer of the oil from one room to the other was established in such a manner that the same device is able to function differently, either as a rheological Voigt-Kelvin model or as a Maxwell one. Due to this reason, the dampers with viscous oil can have the functional behavior in an established duality for a structural solution or for each individualized rheological model there can only be one type of device. The two situations resulted from the performed adjustments are characterized through different values of the stiffness and viscosity. By establishing a range of parameters as discrete values for each adjustment/setting step, the dissipated energy as well as the maximum displacements were determined, choosing the pair of values, c and k, to be the same one in both situations. Response Capacity at Seismic Shocks Thus, the layout presented below suggests the function of a hydraulic cylinder with two separated chambers by a piston with adjustable orifices in terms of cross-section. At the exterior, a control/adjustment and command unit is located, which communicates with the two vehicular chambers of the silicon oil. Function of the established adjustment the damper can be modeled such that the stiffness component of the compressed oil to be dominant or to highlight the viscous component. Fig. 1 presents the principle scheme for a damper with oil, made from a cylinder (1), a piston with calibrated passing orifices for the oil (2) and a hydraulic block (3) with command, functional commutation and energy dissipation roles.

Applied Mechanics and Materials Vol. 430

313

Function of the adjustment model, the damper can be designed as Voigt-Kelvin, also called viscous arc, for which the stiffness of the oil is significant for its sudden compression. The Maxwell model is characterized through a significant viscous damping given by the forced flow of the oil without a high elastic compression. The two models presented in Fig. 2. Due to the fact that the stiffness of the oil is very high, meaning the compressibility is extremely reduced but still exists, the eigenpulsation of the system can be very large for the Voigt-Kelvin model, resulting a damper with oil for which the stiffness component is predominant ( ). For the case when the adjustment is performed for the functioning of the damper after the Maxwell model the total stiffness is , meaning the viscous character is predominant.

Fig. 1 The Hydraulic Damper Scheme: General layout

a)

b)

Fig. 2 Dynamic Models a) Dynamic Scheme with Rheological Voigt-Kelvin Model; b) Dynamic Scheme with Rheological Maxwell Model Based on the schematizations from Fig. 2 and based on the fact that , the transmissibility of the force F(t) to the fixed point is estimated through the transmitted force Q(t). Thus, one has [1,2,3]: √ √

(1) (2) (3)

in which: represents the dynamic transmissibility for the Voigt-Kelvin model; represents the dynamic transmissibility for the Voigt-Kelvin model; is the dissipation factor;

314

Acoustics & Vibration of Mechanical Structures

is the stiffness coefficient of the oil; is the viscous damping coefficient. When the eigenpulsation is rad/s with the frequency Hz of the damper-mass system then for excitations of the action at points between Hz, and , namely being in postresonance no forces are transmitted. From practical point of view, this case is unfeasible because the stiffness of the oil in compression is high meaning p will be characterized by large values. When the eigenpulsation is rad/s or Hz and the dominant spectrum of the earthquake is between Hz, namely then the system responds in anteresonance. For this situation and , meaning that the shock wave is entirely transmitted because . Thus, for the Kelvin-Voigt model one has and for the Maxwell one , with the ratio . In this case, the next relationships may be written [8]: (4)

√

√

(5)

in which: and are displacement amplitudes due to the shock given by the force F, with the system in anteresonance. √ and represents the eigenpulsation; is the pulsation of the perturbing force; represents the relative pulsation; - the excitation force with being its amplitude; - the reaction force with being its amplitude. The ratio of the amplitudes,

, may be written as: (6)

Response Variation Curves of Δ In figures 3, 4 and 5, the variations of with respect to the specific parameters and , and , and are presented under the form of families of curves, for , in which Hz, for a damper with N/m, Ns/m, where The variation curves were determined considering 5 real cases; afterwards the family of curves was parametrically extended at lower, respectively higher, values than the parametric interval specific to the 5 types of dampers [4], [5].

Applied Mechanics and Materials Vol. 430

315

Case 1

200

c1

180

c2

160

c3

140

c4

120

c5 c6



100

c7

80

c8

60

c9

40

c 10

20 0

0

0.5

1

1.5

2

2.5



Fig. 3 Variation of the ratio

with respect to

3

3.5

4

4.5

5

for distinct values of the parameter

It is observed that for values of the evolution of the curves is characterized by a monotonously decreasing tendency towards zero. Case 2

150

k1 k2 k3 k4

100

k5



k6 k7 k8

50

k9 k 10

0

0

1

2

Fig. 4 Variation of the ratio

3

4

5 c

with respect to

6

7

8

9

10 6

x 10

for distinct values of the parameter

Case 3

11



1

10



9



2 3



8

4



5

7



6





6 7



5

8



9

4



10

3 2

0

1

Fig. 5 Variation of the ratio

2

3

4

5 k

with respect to

6

7

8

9

10 6

x 10

for distinct values of the pulsation

316

Acoustics & Vibration of Mechanical Structures

Conclusions 1.

The characteristics of the antiseismic dampers are determined in laboratory testing regime, at harmonic cycles generated by given laws. For the elastomeric elements given kinematic excitations are applied. The hysteretic loops are drawn and based on them the elastic and dissipation parameters are determined. For the hydraulic dampers the dynamic shock regime is established. The significant harmonic force is determined leading to the dynamic action represented by the significant force . Based on real time pressure and flow measurements as well as instantaneous displacements, c, k and are determined. Function of the adjustment and geometrical, position and shape configurations of the calibration orifices, the damper can be modeled as either Kelvin-Voigt or Maxwell.

2.

For tall buildings, dampers best described by Maxwell models are recommended, while for buildings with large contact area with the ground dampers respecting the behavior of the Kelvin-Voigt model are recommended.

3.

The damper devices at seismic actions are useful both for buildings equipped with base isolation systems as well as for bridges (viaducts) located in seismic areas.

4.

The functionality of the damper, either as a Voigt-Kelvin model or a Maxwell one, is realized through switching systems for the circulation of the hydraulic oil [7] through orifices with calibrated diameters or variable through adjustable valves [9], systems whose description constitutes a separate subject to be approached in a different paper.

References [1] EI. Rivice, Stiffness and Damping in Mechanical Design, Marcel Dekker Inc., New York, 1999 [2] P. Bratu, Analiza structurilor elastice. Comportarea la actiuni statice si dinamice., Editura IMPULS, Bucuresti, 2011 [3] P. Bratu, AM. Mitu, V. Serban, M. Giuclea, Analytical models for anti-seismic devices with hysteretic characteristics, Anual Symposium of the Institute of Solid Mecanics SISOM, Bucuresti, 2011 [4] TK. Caughey, MEJ. O’Kelly, Effect of damping on the natural frequencies on linear dynamic systems, Journal of the American Statistical Associating, J.A.S.A., vol. 33, 1961 [5] E. Kreyszig, Advanced Engineering Mathematics (7th ed.), Wiley, New York, 1993 [6] JP. Bandstra, Comparison of equivalent viscous damping and nonlinear damping in discret and continuous vibrating system, Journal of Mechanisms, Transmissions and Automation in Design, J.V.A.S.R.D., vol I 105, 1983 [7] RW. Fox, AT. McDonald, Introduction to Fluid Mechanics (4th ed.), Wiley, New York, 1992 [8] S. Rao, Mechanical Vibrations, Addison-Wesley, New York, 1995 [9] CY. Hou, DS. Hsu, YF Lee, HY Chen, JD Lee, Shear-Thinning Effects in Annulas-Orifice Viscous Fluid Dampers, Journal of the Chinese Institute of Engineers, vol. 30, no.2, 2007

Applied Mechanics and Materials Vol. 430 (2013) pp 317-322 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.317

Analysis of the Dynamic Behavior of the Antiseismic Elastomeric Isolators Based on the Evaluation of the Internal Dissipated Energy ALEXANDRU Carmen1, 2, a 1

ICECON S.A., Bucharest, Sos. Pantelimon 266, sector 2, 021652, CP 3-33, Romania

2

Technical University “Dunărea de Jos”, Galati, Str. Domnească, nr. 47, 800008, Romania a

[email protected]

Keywords: kinematic and dynamic excitation, elastomers, viscoelastic model, loss factor, dissipated energy

Abstract The results of the testing method of the elastomeric isolators at shearing, in a system formed of two elements in parallel, are presented. The exterior actions can be represented by harmonic functions defined as kinematic excitation through the instantaneous displacement and dynamic excitation through the instantaneous force. In the end, the dynamic stiffness as well as the equivalent damping can be evaluated, with the mention that the stand testing individualizes the dynamic system only through viscous, elastic and inertial forces. Due to this reason, the 1st order differential equation eliminates the definition of the critical damping ratio as for a second order system, which leads to the introduction of the concept of equivalent damping, originating in the hysteretic damping or the loss factor. Introduction Presently tests on antiseismic elastomeric elements are realized on specialized stands at imposed displacement values, in harmonic regime with amplitudes of approximately 300-500 mm at frequencies of 0.5-5Hz [4,7]. In this manner the specialized stands must achieve on one hand instantaneous displacements with relatively high amplitudes which lead to sliding in horizontal plan and correspond to a sliding angle of at least 45o, on the other hand the stand must be capable of large testing forces in horizontal plane of approximately 20000-50000 kN and in vertical plane of compression forces of 30000-60000 kN [1,12]. For antivibrating elastomeric elements made at much smaller dimensions than the antiseismic elements, the tests are performed on specialized stands with harmonic forces with constant amplitude or with the force amplitude variable with . The present paper addresses the experimental methods for obtaining the dissipation specific values either under the form of loss factor or the form of dissipated energy in relation with the loss factor and the frequency. The families of curves are evidenced with respect with one specific parameter for a system of order I as well as for a system of order II. It is mentioned that for both categories the experimental determinations were performed in the ICECON laboratory. The laboratory tests are performed in order to determine, on stand, stiffness and internal damping (loss factor) for dynamic excitation points values of the excitation ω. This method can put into evidence, fairly, elastomeric bearings dissipation function in dynamic applications through actions that can be modeled as F=F0sinωt or P=P0sinω [9, 10]. This paper shows the internal energy dissipation in the elastomer due to the dynamic actions of forces F or P with constant amplitude F0=const, or variable amplitude ω2, corresponding to an inertial excitation P0=m0rω2, where m0r is the static moment of the off-balance system in rotative motion, with r≠0 [11].

318

Acoustics & Vibration of Mechanical Structures

Dissipated Energy for F=F0sinωt and a Massless Linear Viscoelastic System The linear viscoelastic force under steady regime is given by relation [2]: = + =

(1)

with the instantaneous displacement response under the form: = where

(

≡

=

− )

(2)

stands for the displacement amplitude of the viscoelastic system given by: (3)

√

φ- phase difference between force and displacement under the form: =

=!

(4)

η is loss factor and it means the hysteretic damping for constant values of η. From eq. 1 we have: "

=#; % $

"

= &1 + #

$

(5)

The dissipated energy () in the viscoelastic element Voigt-Kelvin is under the form [2,5]: () = * in which

(6) represents the displacement amplitude of the viscoelastic system.

Fig. 1 and Fig. 2 present the variation of A and Wd with respect to the loss factor.

Fig. 1 Family of curves for the variation of the amplitude

Applied Mechanics and Materials Vol. 430

319

Fig. 2 Family of curves for the variation of the dissipated energy in the viscoelastic element Dissipated Energy for F=F0sinωt and a Linear Viscoelastic System with Mass The dynamic model for the viscoelastic system having the parameters m, k, c, is represented in Fig. 2, at the static steady equilibrium position with ≡ 0.

x, x

Fig. 2 Dynamic Model For the dynamic equilibrium, under steady regime with viscoelastic model we have the equation under the form: - .+

+

= %

, in case of the linear

=

(7)

The dynamic response for the viscoelastic system in terms of the displacement amplitude Av is: =

=

#$

&( /0 )

6 =

where in which

7

/0

(8)

23 5 4

1

;8=

9

;

=:

is the pulsation of the perturbing force and

(9) ;

is the pulsation of the system.

The dissipated energy () is of the form [3]: () =




Taking into account = = > and the viscous damping ratio @A = ?

(FE =

2* 0 2

H8

B

√CD

we have: (11)

2 2

I1−8 J +(2H8)2

for viscoelastic behavior. Dissipated Energy for P=m0rω2sinωt The dynamic model presented in Fig. 2 has the excitation force P=m0rω2sinωt leading to the displacement amplitude and the dissipated energy expressed by the following relations [5,6]: (8, !) =

:$ L

0

: M( /0 )

7

(12)

Applied Mechanics and Materials Vol. 430

() (8, !) =

;*

(:$ L) :

0N 7 ( /0 ) 7

321

(13)

The figures below, Fig. 5 and Fig. 6, represent the variation of A and Wd.

Fig. 5 Curve Family A(η,f)

Fig. 6 Curve Family Wd(η,f) Conclusions All experimental results are based on the Voigt-Kelvin model which best characterizes the rheological behavior - linear viscoelastic - of the elastomeric compositions used [8]. The test results, presented under the form of parameterized functions for the real experimented cases, represent a new and a more complete manner of approach of the dynamic behavior of elastomeric materials, when highlighting the dissipated energy with respect to the previously mentioned parameters (loss factor, frequency, etc.). The dissipation capacity of the damping systems basing on elastomeric elements is expressed by the internal dissipated energy () representing a system characteristic and not a material characteristic. This parameter is very important for the evaluation of the damping system capability.

322

Acoustics & Vibration of Mechanical Structures

Thus, depending on the excitation, the dynamic response in displacement changes with the amplitude (O, !) having as result the dissipated energy dependence on the excitation parameters, , , under steady dynamic balanced regime. Having in regard the above mentioned items, one can conclude the following: - in case of dynamic excitation with steady harmonic forces with constant amplitude = % or inertial amplitude P = - Q , the dissipated energy is a function of the dynamic response amplitude in displacement (O, !) as a result of the dynamic equilibrium for the whole system; - the internal energy dissipation capacity for the damping elements in elastomer, in various constructions represents a system characteristic and not a material characteristic. As material, the elastomer is characterized by parametric values only stated on reduced scale samples and not for the whole damping element natural scaled: - all the plotted curves are for ! = 2H for stable values at resonance, 8 = 1. References [1] W. Baterma, R. Kohler, Elastomere Federung, Verlag von Wilhelm Ernst & Son, Berlin. Mϋnchen, 1982 [2] P. Bratu, Vibration transmissivity in mechanical systems with rubber elements using viscoelastic models, Proceedings of 5th European Rheology Conference, University of Ljubljana, Slovenia, 1998 [3] P. Bratu, “Evaluation of the dissipation effect while performing harmonic dynamic tests upon materials having variable damping characteristics”, Proceedings of INTER-NOISE 2007, Istanbul, Turkey, 2007 [4] Y. Jullien, Réduction des vibrations mècaniques sues aux machines, Colloque RILEM Budapest, 1963 [5] P. Bratu, N. Drăgan, L'analyse des mouvements désaccouplés appliquée au modèle de solide rigide aux liaisons élastiques, Analele Universitatii “Dunarea de Jos” of Galati, Fascicula XIV, 1997 [6] P. Bratu, O. Vasile, Modal analysis of viaducts supported on elastomeric insulation within the Bechtel constructive solution for the Transilvania highway, Romanian Journal of Acoustics and Vibration, 9(2012) 77-82 [7] M. Dolce, FC. Ponzo, A. DiCesare, G. Arleo, Progetto di Edifici con Isolamento Sismico, Iuss Press, Pavia, Italy, 2010. [8] JP. Bandstra, Comparison of equivalent viscous damping and nonlinear damping in discret and continuous vibrating system, Journal of Mechanisms, Transmissions and Automation in Design, J.V.A.S.R.D., vol I 105, 1983 [9] TK. Caughey, MEJ. O’Kelly, Effect of damping on the natural frequencies on linear dynamic systems, Journal of the American Statistical Associating, J.A.S.A., vol. 33, 1961 [10] I. Cochin, Analysis and Deisign of Dynamic Systems, Harper&Raw, New York, 1980 [11] S. Rao, Mechanical Vibrations, Addison-Wesley, New York, 1995 [12] SR EN 15129:2010, Dispozitive antiseismice, 2010

Applied Mechanics and Materials Vol. 430 (2013) pp 323-328 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.323

Experimental Evaluation of The Hysteretic Damping of Elastomeric Systems at Low-Cyclic Harmonic Kinematic Displacements VASILE Ovidiu1, 2, a 1

ICECON S.A., Bucharest, 266 Pantelimon, sector 2, 021652, CP 3-33, Romania

2

Department of Mechanics, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042, Romania a

[email protected]

Keywords: elastomer, hysteretic damping, kinematic displacements

Abstract. The paper presents the experimental results regarding the hysteretic behavior of elastomeric systems made in conformity with the European Standards SR EN 1337-3 and SR EN 15129. It is also mentioned that the hysteretic damping was determined by experimentally raising the hysteresis loop in a low-cycle harmonic regime, with a kinematic exterior excitation defined by a harmonic displacement law. In this context, it is mentioned that the area of the hysteretic loops was determined instrumentally (in analogical representation of the signals) as well as under digital form, through a sampling of the physical signal, which allows a high precision of the determination. Introduction The antiseismic elastomeric devices with functional role in the isolation base system of buildings which must be protected against earthquakes are: elastomeric isolators and fluidic dampers based on silicone oil. Integrating these devices in the support and dissipation structural system referred to as “base isolation system” is a requirement of the new and more modern design concepts. These specialized products must meet certain fundamental demands based on performance functions. Elastomeric devices are designed as viscoelastic isolation elements meant to resist to vertical loading and to lateral deformations with values of the slip angle greater than 450. Due to this reason the structure of the dynamic isolation device on horizontal direction is composed of multiple elastomeric layers alternating with metallic reinforcing, which, through vulcanization, form a multilayer elastomeric ensemble parametrically defined through stiffness, internal-loss and dissipation. The rheological model can be Kelvin-Voigt or hysteretic, depending on the internal dissipation mechanism. The dampers with viscous oil are devices especially designed in such a manner that the two chambers of the cylinder, separated by the division of the flow piston, are connected for two distinct situations: the dissipation function through the viscous fluid arc rheological system, modeled as Kelvin-Voigt; the dissipation function through the fluid-viscous rheological system, modeled as Maxwell. Conceptually, the two functions are fundamentally defined, being represented by the distinct dynamic response at initial shock, with the same energy dissipation. The Dynamic Response of Elastomeric Systems to Calibration Loadings The dynamic schemes from Fig.1 are specific to the kinematic excitation loading method given by the instantaneous displacement x=x(t)=Aosinωt, where ω=2πf, in which f represents the cycle frequency.

324

Acoustics & Vibration of Mechanical Structures

c

x(t )

QV (t )

k

a) Voigt-Kelvin model [V] η

x(t )

Q H (t )

k

b) Histeretic model [H] Fig. 1. Schematization of the order I model for the elastomeric systems test on stand The elastomeric systems are coupled in parallel at slip loadings under norm conditions, at harmonic cycles as kinematic excitations of the form: x(t)=Aosinωt, made in conformity with the European Standards SR EN 1337-3 and SR EN 15129. The dynamic response Q(t ) of the order I system is given by the differential equation for the Kelvin-Voigt model [1-4]:

cx + kx = Q(t ) and under the form: k

ω

ηx + kx = Q(t ) for the hysteretic model, in which:

c - represents the viscoelastic force coefficient; k - represents the stiffness coefficient to shearing; η – internal loss coefficient; ω – pulsation of the excitation; Q(t ) - reaction force at applied harmonic excitation. For the Kelvin-Voigt model the reaction force Qv ( x ) , is presented with respect to the instantaneous displacement x=x(t) or with respect to the pulsation of the excitation, ω , meaning Qv (ω ) as well as the dissipated energy ∆Wvcin , the dissipation power Pvcin and the equivalent

critical damping ratio ζ v,eqeq of a viscoelastic damping or a IInd order system. Thus, one obtains [4,5]: Qvcin ( x ) = kx ± cωA0 1 −

x2 A02

representing a family of ellipses which can be parameterized with and size of the excitation.

(1) or

considering the nature

Applied Mechanics and Materials Vol. 430

Q0v = A0 k 2 + c 2ω 2

325

(2)

is the maximum value of the viscoelatic reaction force of the form: Q(t ) = Q0v sin (ωt − ϕ ) . The maximum dissipated energy which corresponds to the area of an ellipse defined through ω and A0 is:

∆Wvcin = πcωA02

(3)

The dissipation power of the elastomeric element is given by:

Pvcin =

1 2 2 cω A0 2

(4)

The equivalent critical damping ratio of a complete IInd order system, with the elements m, c, k is [4,6,7]:

ζ veq,eq =

1 cω 2 k

(5)

For the hysteretic behavior of the elastomeric isolator in concordance with the previous model, one has the following parametric quantities of interest [4,8]:

 x2  QHcin ( x ) = k  x ± ηA0 1 − 2  A0  

(6)

Q0H = kA0 1 + η 2

(7)

∆WHcin = πkηA02

(8)

1 2

ζ Hcin,eq = η

(9)

The interest parameters for an elastomeric element are A0 = 0.16 m , f = [0;...;5,0] Hz and the viscoelastic or hysteretic characteristics are c = [0,25;...;1,0] ⋅ 10 5 Ns m , η = [0,2;...;0,8] and k = 1,5 ⋅ 10 6 N m . In figures 2 to 4 the resulted graphs for the interest parameters, considering ω = 2πf , are presented.

326

Acoustics & Vibration of Mechanical Structures

Qcin = Qv (x,c) v

5

1.5

x 10

Qcin = Qh(x, η) h

5

2

x 10

1.5

1 1

0.5 Qcin [N] h

Qcin [N] v

0.5

0

0 -0.5

-0.5

η1=0,2

c 1=0.25*105 Ns/m

-1

η2=0,4

-1

c 2=0.50*105 Ns/m c 3=0.75*105 Ns/m

η3=0,6 η4=0,8

-1.5

c 4=1.0*105 Ns/m -1.5 -0.08

-0.06

-0.04

-0.02

0 x [m]

0.02

0.04

0.06

η5=1,0 -2 -0.08

0.08

-0.06

-0.04

-0.02

a)

0 x [m]

0.02

0.04

0.06

0.08

b)

Fig. 2. Hysteresis curve: a) Qvcin = Qv (x, c ) ; b) Qhcin = Qh (x,η ) Qcin = Qv(x,c) v

5

2.5

x 10

Qcin = Qv (x,c); (f= 1 Hz); For c= 25000 Ns/m v

5

2.5

x 10

S ellipse = 3156.44; S triangle =4817.65; Damping: ζ =0.05214

2 2

1.5 1.5

1 1 0.5

0

c 1=0.25*105 Ns/m

-0.5

c 2=1.00*105 Ns/m

-1

c 3=1.75*105 Ns/m

-1.5

c 4=2.50*105 Ns/m

Qcin [N] v

Qcin [N] v

0.5

-1 -1.5

c 5=3.25*105 Ns/m

-2

-2

c 6=4.00*105 Ns/m

-2.5 -0.08

-0.06

-0.04

-0.02

0 x [m]

0.02

0.04

0.06

0 -0.5

-2.5 -0.08

0.08

-0.06

-0.04

-0.02

a) 5

5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

-0.5

-1

-1

-1.5

-1.5

-2

-2 -0.06

-0.04

-0.02

0 x [m]

c)

0.06

0.08

0.02

0.04

0.06

0.08

x 10

S ellipse = 22095.05; Striangle =5951.24; Damping: ζ =0.2954

0

-0.5

-2.5 -0.08

0.04

Qcin = Qv (x,c); (f= 1 Hz); For c= 175000 Ns/m v

S ellipse = 12625.74; S triangle =5203.95; Damping: ζ =0.1931

Qcin [N] v

Qcin [N] v

x 10

0.02

b)

Qcin = Qv (x,c); (f= 1 Hz); For c= 100000 Ns/m v

2.5

0 x [m]

-2.5 -0.08

-0.06

-0.04

-0.02

0 x [m]

0.02

0.04

0.06

d)

Fig. 3. Hysteretic curves for f = 1 Hz (a) and separately calculation for some curves (b, c, d)

0.08

Applied Mechanics and Materials Vol. 430

ζ= ζ(c)

0.45

327

X: 4e+05 Y: 0.4291

0.4 X: 3.25e+05 Y: 0.4027

0.35

X: 2.5e+05 Y: 0.3614

ζ

0.3 X: 1.75e+05 Y: 0.2954

0.25 0.2 X: 1e+05 Y: 0.1931

0.15 0.1 0.05

X: 2.5e+04 Y: 0.05214

0

0.5

1

1.5

2 c [Ns/m]

2.5

3

3.5

4 5

x 10

Fig. 4. Damping ratio variation for f = 1 Hz

Conclusion The characteristics of the antiseismic dampers are determined in laboratory testing regime, at harmonic cycles generated by given laws. For the elastomeric elements given kinematic excitations are applied. The hysteretic loops are drawn and based on them the elastic and dissipation parameters are determined. Based on hysteretic curves in Fig. 3 was drawn critical damping variation on Fig. 4. Research on rheological models by numerical simulation and tests bench to harmonic forces, where the effect of the mass of the elastomeric element is negligible, highlights the following: a) is the first-order dynamic system; b) hysteretic loops are ellipses; c) each loop area changes relative to c amortization so that for each case was represented damping ration variation ζ with discrete variation of c.

References [1] P. Bratu, Vibration of elastic systems, 600 pages, Technical Publishing House, ISBN 973-311418-9, Bucharest, 2000. [2] P. Bratu, Elastic bearing systems for machines and equipment, 260 pages, Technical Publishing House, 1990. [3] P. Bratu, Evaluation of the dissipation energy capacity inside damping systems in neoprene elements, The 16th International Congress on Sound and Vibration ICSV16, Kraków, Poland, July 5-6, 2009. [4] P. Bratu, Analysis of the antivibrating elastic systems having amplified static deflection, Proceedings of the VIII-th Symposium "Acoustics and Vibration of Mechanical Structures", AVMS-2005, Timişoara, May 26-27, 2005, pp.30-37.

328

Acoustics & Vibration of Mechanical Structures

[5] P. Bratu, Rigidity and damping characteristics in case of composite neoprene systems due to passive vibration isolation, Proceedings of the Annual Symposium of the Institute of Solid Mechanics SISOM’06, Bucharest, Romania, May, (2006). [6] P. Murzea, Methodological aspects regarding the experimental determination of the vibration eigenmodes of the Central Exhibition Pavilion Romexpo, Buletin Stiintific UTCB (2), (2012). [7] G.R. Gillich, P. Bratu, D. Frunzaverde, D. Amariei, V. Iancu, Identifying Mechanical Characteristics of Materials with Non-linear Behavior using Statistical Methods, 4th WSEAS International Conference on Computer Engineering and Applications, Harvard Univ, Cambridge, MA, (2010). [8] V. Iancu, O. Vasile, G.R. Gillich, Modelling and Characterization of Hybrid Rubber-Based Earthquake Isolation Systems, MATERIALE PLASTICE, Vol. 49 (4), pp. 237-241, (2012).

Applied Mechanics and Materials Vol. 430 (2013) pp 329-334 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.329

Experimental Evaluation of The Damping Variation of an Elastomeric Device Harmonically Excited VASILE Ovidiu1, 2, a 1

ICECON S.A., Bucharest, 266 Pantelimon, sector 2, 021652, CP 3-33, Romania

2

Department of Mechanics, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042, Romania a

[email protected]

Keywords: damping ratio, elastomeric insulators, dynamic system

Abstract. The results of the experimental tests, in dynamic regime, on antiseismic elastomeric devices, evidenced the fact that the internal damping represents a significant parameter tightly related to the specific shearing deformation of the elastomeric elements. Consequently, based on the final obtained results, correlation curves were raised meant to represent the variation of the equivalent critical damping ration (dynamic system of order I). Introduction The test methods, under laboratory conditions, to certify the conformity of the elastomeric systems, constitute the experimental assessment basis. These have the purpose to define and express the parametric values of the physico-mechanical characteristics for the finite elastomeric products, i.e. on the full-scale. From the elastomeric products category, with role of visco-elastic or hysteretic supporting for bridges, viaducts, buildings, are one of the bearings and anti-seismic devices, known in the speciality literature as elastomeric insulators LDRB, HDRB. The test cycles have low frequencies, from 0,15 Hz to 3,0 Hz as a rule, with significant values to 0,5 Hz and 1,0 Hz. The applied exterior action is the momentary displacement laid down by the law x  t   A0 sin t which constitutes the kinematic excitation through harmonic displacement. The response of the visco-elastic or hysteretic system, without mass, is given by the momentary reaction Q  t   k  x  t   c  x  t  , where k is the coefficient of constant stiffness, and c is the constant viscous damping factor. In this study the calculus relations of the response parameters, both for the Voigt-Kelvin viscoelastic model and for the hysteretic model, are presented. Thus, the characteristic curves are plotted highlighting the differences between the two models and in case of the appropriate elastomeric insulators. Dynamic Response of Elastomeric Device Harmonically Kinematic Excitations Since the insulators behaviour depending on the elastomers composition can be Voigt-Kelvin visco-elastic or hysteretic for the same bench test system, both types of dynamic responses will be presented. Thus, physical quantities will be marked by neutral superscripts or subscripts in relation to the mathematical operations. The elastomeric insulators are made of elastomeric layers between steel reinforcements so that the final configuration is of “sandwich” type with several intermediate reinforcements. Two steel reinforcements are provided on the outside, which have the role to carry out the connection with the rest of the system by an appropriate fixing, being known the fact that the elastomeric insulators are fastening visco-elastic elements into a structural construction (bridge, viaduct, foundation insulated building). Thus, an elastomeric insulator is presented in Fig. 1 (see Fig. 1.a) with several elastomeric structures 1 and intermediate reinforcements 2, stressed to compression (see Fig. 1.b), and shearing (see Fig. 1.c).

330

Acoustics & Vibration of Mechanical Structures

The bench test is carried out by means of two identical insulators mounted symmetrically, parallel for the shearing stress. Fz

x

Fz

Fx

z

H

H0

H

H0

Fig 1. Elastomeric insulator 0

Fz

2 1

t

Fx* x

*x

M

3 N

x  A0 sin t

4

O

OM  MN 

1 A0 2

0

Fz

a) c 2

Qt 

k 2

x  A0 sin t

k 2

c 2

b)

Fig 2. Assembly scheme for the kinematic excitation test In Fig. 2.a) the assembly comprises two identical insulators 1, mounted between the cups of a 0

press, by means of two reinforcement plates 2 for each insulator, with controlled compression F z . The kinematic displacement is given by a special kinematic system (connecting rod-crank mechanism) OMN with the angular frequency  , and the displacement is sent through the rigid element 3 which is guided through the sliding couples 4. The reaction force Qt  equal and counter-

Fx of the rigid element 3, is measured by a force cell Fx* , and the * deformation  x is measured by an inductive transducer  x . The scheme of the symmetrical current with the force

Applied Mechanics and Materials Vol. 430

331

Voigt-Kelvin system is shown in the Fig. 2.b), where k 2 , c 2 are the constants of one insulator, and Q  t  is the total reaction force, equivalent to the two insulators which emerges at the applied kinematic excitation. Significant Parameters The momentary visco-elastic force results from the linear differential equation where the solution x  A0 sin t is known [1,2]. Thus, we have:

c  x t   k  x t   Q t  . where:

(1)

x  A0 cos t is the momentary velocity.

If we substitute cos t  1  sin 2 t   1  will lead to the final form for Q  t  , thus:

Q  t   kx  t   c A0 1 

x x2 , then we will have x   A0 1  2 , what 2 A0 A0

x2 A02

(2)

The expression as against the trigonometric functions from the relation (1) can be written as follows: c A0 cos t  kA0 sin t  Q0 sin t    or c A0 cos t  kA0 sin t  Q0 sin t cos   Q0 cos t sin  whence by the identification of the trigonometric functions factors with the same name, we obtain:

Q0 cos   kA0  Q0 sin   c A0

(3)

From relation (3) the following relations result:

c   tan   k  Q  A k 2  c 2 2 0  0

(4)

i.e. the momentary visco-elastic force, according to the Voigt-Kelvin model can be expressed as follows: c   Q  t   A0 k 2  c 2 2 sin  t  arctan  k  

(5)

In the case of the harmonic kinematic excited elastomeric system [3], in the absence of mass, the differential equation is linear. This leads to the adopting of an equivalent damping factor  eqVK , similar to the fraction of the critical damping  , defined for a complete quadratic system, consisted c from m , c , k , which has the expression   . 2 km

332

Acoustics & Vibration of Mechanical Structures

Dissipated energy WdVK can be written as follows: WdVK   c A02

(6)

VK The equivalent damping  eq of a quadratic visco-elastic system is established applying the relation of definition, as follows [4,5]: WdVK  c A02 VK  eq   2 1 4 kA02 2 kA0 2 or

 eqVK 

1 c 2 k

(7)

i.e. the equivalent damping  eqVK depends on the excitation by the pulsation  . Graphical and Numerical Parametric Analysis In Table 1 the significant geometrical and physico-mechanical characteristics for three types of insulators are shown. The following symbols were used: Fz - normal force; x - horizontal displacement; k x - horizontal stiffness; d - elastomer hardness; G - shearing modulus;  eq - equivalent damping; Table 1. Significant characteristics of the elastomeric insulators No.

Elastomeric insulator

Significant characteristics Strength, elasticity, damping

Fzmax

x

k x*

d

G

 eq*

[kN]

[mm]

[kN/mm]

h

[MPa]

[%]

1

IAS / I

1200

250

0,75

40±3

0,4±0,05

12 *1

2

IAS / II

2200

250

1,50

60±3

0,8±0,15

22 *2

3

IAS / III

2150

250

2,50

75±3

1,3±0,20

31 *3

x  1, 0 Te *1 for 0,5 Hz and c1 = 60 kN.s/m; *2 for 0,5 Hz and c2 = 200 kN.s/m; *1 for 0,5 Hz and c3 = 450 kN.s/m; The kinematic excitation parameters for the digital simulation are: A0 = 0,08 m; f = 0,5 Hz … 4 Hz The parameters of elasticity and dissipation are appropriate to the anti-seismic device (insulator) IAS/II and they were established as follows: kx = 1,5 kN/mm; c = 0,25.105 … 4,0.105 Ns/m. The Fig. 3 shows the equivalent damping modification  eq depending on the viscosity factor, c , *

for slip movement  q 

or frequency f , as well as the stiffness k .

Applied Mechanics and Materials Vol. 430

333

Conclusions The visco-elastic behaviour of the elastomeric anti-seismic dissipators and insulators can be simulated numerically and graphically through the equivalent damping parameterization  eq , i.e. the modification of the viscosity factor, c, as well as through the parameterization of the frequency and/or kinematics excitation amplitude. This analysis method, allows the easy finding of the values of the dissipative and visco-elastic parameters which individualize a type of elastomeric insulator among the significant values and/or the variation curves of the parameterized families. eq = eq(c); for k=1.5*106 N/m

3.5

1=1 rad/s

eq [%]

2.5 2

c 1=0.25*105 Ns/m

2=5 rad/s

3

3=10 rad/s 4=15 rad/s



5=20 rad/s 6=25 rad/s

1.5

0.5

0.5

1

1.5

2 c [Ns/m]

2.5

3

3.5

c 5=3.25*105 Ns/m c 6=4.00*105 Ns/m

0 0

4

c

c 4=2.50*105 Ns/m

1.5 1

0.5

c 3=1.75*105 Ns/m

2

1

0 0

c 2=1.00*105 Ns/m

2.5

eq [%]

3

eq = eq(f); for k=1.5*106 N/m

3.5

0.5

1

1.5

2 f [Hz]

5

x 10

a)

2.5

3

3.5

4

b)

eq = eq(k); for c=0.5*105 Ns/m

30

1=1 rad/s 2=5 rad/s

25

3=10 rad/s 4=15 rad/s

eq [%]

20

5=20 rad/s 6=25 rad/s



15

10

5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

k [N/m]

c) Fig 3. Equivalent damping modification

2 5

x 10

Can be noticed that in the case of kinematics excitation, equivalent damping  eq is modified in relation to f , c and k and also in relation to the parameters c and  according to data from Fig. 3. This is actually a help because for choosing certain values of parameters can call on graphs above. Thus, on the basis of the plotted curves, the equivalent damping factor of a complete inertial visco-elastic system  eq is represented depending on the variable c, f or k with the parameters of the curves families ω or c. It is found that the increase influence of ω and c is established in the damping significantly increase. The stiffness increase leads also to the damping decrease.

334

Acoustics & Vibration of Mechanical Structures

References [1] M.J. Kelly, A.D. Konstantinidis, Mechanics of Rubber Bearings for Seismic and Vibration Isolation, J. Wiley & Sons, Ltd (2011). [2] I. Politoporelos, Le rȏle l’amortissement dans l’isolation sismique, lu 7ème Colloque National AFPS 2007, Ecole Centrale Paris. [3] P. Bratu, A. Mihalcea, O. Vasile, Evaluation of the dissipation capacity in case of structural system consisting of elastomeric anti-seismic devices tested under laboratory conditions, Proceedings of The annual Symposium of the Institute of Solid Mechanics and Session of the Commission of Acoustics, The XXII-th SISOM 2011, Bucharest, May 25-26, (2011), pp. 342-348, ISSN 2068-0481. [4] P. Bratu, Analysis of the antivibrating elastic systems having amplified static deflection, Proceedings of the VIII-th Symposium "Acoustics and Vibration of Mechanical Structures", AVMS-2005, Timişoara, May 26-27, 2005, pp.30-37. [5] V. Iancu, O. Vasile, G.R. Gillich, Modelling and Characterization of Hybrid Rubber-Based Earthquake Isolation Systems, MATERIALE PLASTICE, Vol. 49 (4), pp.: 237-241, (2012).

Applied Mechanics and Materials Vol. 430 (2013) pp 335-341 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.335

Calibration of the Ground Motion Model Using a Simplified Stochastic Model in the case of the Central Exhibition Pavilion ROMEXPO MURZEA Patricia1, 2, a 1

ICECON S.A., Bucharest, Sos. Pantelimon 266, sector 2, 021652, CP 3-33, Romania

2

Technical University of Civil Engineering, Bucharest, Bd. Lacul Tei 122-124, sector 2, 020396, Romania a

[email protected]

Keywords: stochastic models, ambient vibrations, velocigrams, calibration

Abstract. The aim of the paper is to present the results of applying the formulae of a simplified stochastic model for the calibration of some macroscopic parameters of the ground motion, on the basis of rather rough estimates. For this purpose the basic records combinations results obtained with the aid of computer programs (DaisyLab and LABView) are used, after digital recordings on the large span and rather isolated structure of the ROMEXPO Pavilion in Bucharest were performed. A stochastic model of stationary, low intensity, ground motion (referred to in literature as ”microtremors” or “ambient vibrations”) is proposed. Introduction Due to the complexity of the problem: variety of soil characteristics, wave propagation phenomena, ground-structure interaction and seismic events, the thematic has to be treated considering stochastic models or random vibration functions. Parametric models, such as the Kanai-Tajimi Spectrum [1] or its modified version [2], are often used when describing the spectral density of a random field. There are empirical and semi-empirical models, models made by seismologists, models related to coherence like Luco&Wong’s [3], etc. For engineering applications like random vibrations or Monte Carlo simulations, a spatial parametric model of the random field of the seismic ground motion is necessary. Its parametric description is generally given by its cross spectral density [4]. When developing such a model, the ground is considered as a 3D continuum and the motion is non-synchronous in different points, along different directions (because of the characteristics of the seismic waves’ propagation, finite velocities). Such a model raises many important questions among which “how are the degrees of freedom of the ground-structure interface chosen?” or “how to consider, anticipatively, the simultaneous and non-synchronous time dependence along these different degrees of freedom?“. A possible answer to these questions could be the use of a stationary (for simplification) stochastic model in order to account for the random character of the event, whose detailed characteristics correspond to the spectral density matrix used in the classical Wiener-Khincin relations [5,6,7]. Because data are either insufficient, like in the case of strong-motion events, or too detailed and varied, like in the case of geological composition at different site locations, a possible calibration attempt can be based on multichannel, digital recordings of ambient vibrations (micro-tremors), in order to evidence the non-synchronous character of the motion at different points The Structure The structure used for the calibration of the ground motion model is the main ROMEXPO Pavilion from Bucharest. A view and the ETABS model of the structure are given in Fig. 1.

336

Acoustics & Vibration of Mechanical Structures

Fig.1 The Main ROMEXPO Exhibition Hall Built in 1959, the Pavilion is formed of three galleries situated at different elevations (3.20 m, 7.70 m and 17.30 m) supported on two rows of reinforced concrete columns. On the 32 principal columns rest 32 arched trusses forming the structural system of the dome [12]. The dome, as well as the entire building, was designed to allow the use of prefabricated materials [8]. Digital and Structural Results The sensors SS-1 (Ranger) of the data acquisition system VSS-3000 are placed on the superior ring, at 17.30m, according to the layout presented in Fig. 2 for the simultaneous recording on 8 channels, based on ambient vibrations. Sensor „C1"

Column no. 1 Sensor „C2"

Sensor „C7"

Sensor „C3"

N Sensor „C8"

Sensor „C4"

94.20 m

Column no. 8

Column no. 24

Sensor „C5"

Superior ring level: +17.30 m Sensor „C6" Column no. 16 94.20 m

Fig.2 Sensor Layout The data acquisition system records velocities which are afterwards transformed into displacement recordings and then processed into Fourier spectra [8]. After the recordings are done, the objective is combining these records into vectors which are the representative components for the degrees of freedom corresponding to the vibration eigenmodes of ring translation along the E-W and N-S directions, rotation around the vertical symmetry axis, dilation, second and superior order ovalizations [12]. These results complete the already existing databases on the modification of the dynamic characteristics of the Pavilion due to successive strong earthquakes and to post-earthquake rehabilitation work [9]. Some formulae used for the basic records combination are presented below [8,12] for: a. Symmetrical axis dilatation of the ring Because there are only four points at which the seismoteres are placed, the information is limited and the dilatation oscillations overlap with those of higher order ovalization (4, 8, etc.) The identification of the modes should be done based on the spectral analysis. (1)

Applied Mechanics and Materials Vol. 430

337

b. Rigid translations of the ring along the two directions E-W and N-S, equivalent to 1st order ovalization: (2) (3) c. Ring rotations: (4) d. 2nd order ovalization: (5) e. 4th order ovalization: (6) The main results which will be needed for the calibration phase are presented in Table 1. Table 1, Obtained Frequencies and Periods Corresponding to the Modal Oscillations Spectral Frequency Spectral peak, Type of Oscillation Period (s) peak, (Hz) Displacements Velocities Rotation with respect to symmetry 1.34 0.746 0.52 4.2 axis NS Translation 1.28 0.781 0.35 2.8 EW Translation 1.28 0.781 0.32 3.2 Lowest order (2) proper 2.56 0.391 0.42 7.2 ovalization The values for which the motions' amplitudes ratio along the degrees of freedom is approaching the ratio determined experimentally are to be identified. The Stochastic Model To account for the random character of the motion, non-stationary random functions should be used in general. However, for micro-tremor analysis, stationary random functions can be used. Authors like Horea Sandi in 2005 [5], Aspasia Zerva in 2009 [4] and Izuru Takewaki in 2007 [10], have presented models for the spatial ground motion under stationary and non-stationary assumptions. A convenient starting point when determining a model for transient motions is the canonic expansion of a random, non-stationary, vectorial, accelerogram [5,6,7]. The model proposed concerns the motion of a half-space of a classical continuum, related to the orthogonal Cartesian axes x, y (horizontal) and z (vertical). The half-space may consist of a sequence (even continuous) of constant thickness, parallel, horizontal, isotropic and homogeneous layers. The local motion is characterized by three translation components, u, v and w, and by three rotation components of a horizontal elementary facet,  = w/x,  = w/y and  = (v/x u/y)/2 respectively. The components referred to build together a vector ug(P,t), having a transpose:

338

Acoustics & Vibration of Mechanical Structures

[

]

(7)

where P represents a point of coordinates (x, y, z). The ground motion is assumed to be random, homogeneous with respect to the horizontal axes and stationary [11]. Its basic characteristic is the coherence matrix S [wg; , x, y, z0] of the acceleration vector wg(P, t) corresponding to the displacement vector ug(P, t), having the circular frequency . The coherence matrix concerns the motion at two points P1 and P2, located at a same depth z0 and at a relative position in the horizontal plane defined by the components x and y. The isotropy and homogeneity assumptions mentioned lead to the conditions: (8) for some of the components of the matrix S [wg; , x, y, z0]. Further consequences are given by the expressions: [

]

[

]

[

]

[

[

]

(9)

]

(10)

where sh[wg; , z0] and sv[wg; , z0] mean the classical spectrum density characteristics of horizontal and vertical motions at a point of depth z0 respectively. In order to postulate analytical expressions for the non-zero components of the coherence matrix, following starting points were accepted: a. the similitude criteria of wave propagation s (for phase lag ) and s (for rotation to translation amplitudes), where c means a wave propagation speed and d means a relative distance: ⁄

(11)

⁄

(12)

b. a space – time similitude assumption of correlation - coherence characteristics, which makes it possible to take the well-known Kanai – Tajimi expressions of correlation – coherence of local motion at a point [

| | {

]

[

]

(

| |

⁄

) {

(

| | } )

(

(13) ) }

(14)

as a model in order to specify spatial coherence. The coherence characteristic for two different points (for illustration, for the first component of the coherence matrix) is postulated as [

]

[

] {

(

)}

(15)

where the spatial coherence factor { , , T*(cp, cs, x, y)} has the expression 



{



⁄



}

(16)

Applied Mechanics and Materials Vol. 430

339

while the argument T*(cp, cs, x, y), which has a T (time) physical dimension, will be (

)

(

)

{

}

(17)

(cp and cs mean here conventional wave propagation speeds for P- and S- waves respectively, while dl and dt mean relative distances along corresponding directions). A first attempt of calibration of the parameter , parameter equivalent with from the KanaiTajimi relations for the filtering of white noise through a system with one degree of freedom, was  0.5 . The coherence characteristics concerning the rotation components have expressions in which partial derivatives of terms for translation components, with respect to spatial coordinates, intervene. Given previous developments, the task of calibration of the stochastic ground motion model reduces to that of calibration of the conventional wave propagation speeds cp and cs. Results of Computations For computations, the previously presented formulae for the stochastic model in the general case are adapted to the particular case of the ROMEXPO structure. Table 2 presents the data used for the case of conventional velocities of 1000m/s for the longitudinal direction and 500m/s for the transversal one, considering only 4 support points (columns) displayed according to Fig.2. Table 2, Initial Data fr t

Symbol

r0

cp

cs

ft

Value

48 m

1000 m/s

500 m/s

1.28 Hz

1.34 Hz

r

1

2

3

4

4.0192 4.2076 0 1.5708 3.1416 4.7124 rad/s rad/s rad rad rad rad

In the table above the symbols represent:  r0: main ring radius;  cp: conventional (chosen) value of the propagation velocities for P waves;  cs: conventional (chosen) value of the propagation velocities for S waves;  ft: experimental (obtained) frequency for translation, from Table 1;  fr: experimental (obtained) frequency for rotation, from Table 1;  t: coefficient based on the Kanai-Tajimi relations for translation;  r: coefficient based on the Kanai-Tajimi relations for rotation;  i: azimuthal orientation angle of the 4 chosen columns. Table 3 presents the calculus results obtained for the translation case considering 4 support points, cp= 1000 m/s and a conventional velocities ratio of 0.5. The numbers 1 to 4 represent the columns, which paired 1-1, 1-2,…, 4-3, 4-4, allow the computations of the azimuthal angle. Table 3, Translation Motion Main Results (4 columns) T*ij 1 2 3 4

1 0.0000 0.1073 0.0960 0.1073

2 0.1073 0.0000 0.1073 0.1920

ij trans

3 0.0960 0.1073 0.0000 0.1073

4 0.1073 0.1920 0.1073 0.0000

1 2 3 4

1 1.0000 0.6692 0.7243 0.6692

2 0.6692 1.0000 0.6692 0.2437

3 0.7243 0.6692 1.0000 0.6692

4 0.6692 0.2437 0.6692 1.0000

340

Acoustics & Vibration of Mechanical Structures

In which: √〈[  ( 

( )

〉

]

)

{

(

〈[ )

( ) ⁄

] 

〉

(18)

}

(19)

Similar computations are performed for the rotation case obtaining results for ij rot. A trial and error approach was adopted in order to estimate the most appropriate values for the conventional parameters cp and cs, varying velocities, velocities ratio and the number of support columns. The approach was quite simple and it benefited from the fact that the dominant frequencies of oscillation for ring translation and for ring rotation with respect to its center were very close. The calculations used in this connection are starting from the relation between the vector of acceleration of ground-structure interface, wg (t), and the vector of acceleration along the degrees of freedom of the upper part of the structure, we (t). Below a synthesis table, Table 4, of the results obtained for the stochastic model of the ground considering propagation velocities of the waves P which vary from 1000 to 250 m/s and a conventional velocities ratio of the P and S waves of 0.5, using 4 columns. Table 4, Synthesis cp/cs=0.5 1000/500 500/250 400/200 300/150 250/125 200/100

ij trans

ij rot

avg.

avg.

0.7056 0.3262 0.2127 0.1309 0.1311 0.1829

0.6013 0.3966 0.4327 0.5079 0.5212 0.5023

rot/trans rot/trans (avg.) 1.1668 0.8056 -0.0568 -1.4870 1.0442 0.8700

0.8522 1.2156 2.0342 3.8792 3.9769 2.7463

rot/trans (displ. recordings.)

rot/trans (vel. recordings)

1.5522

1.4000

Conclusions The paper was concerned with an attempt of calibration of a stochastic model of ground motion that may seem to be quite sophisticated, even if it relies on quite strongly idealizing assumptions. This model was developed in order to provide as far as possible a consistent approach to the specification of seismic input for a spatial analysis of structural oscillations due to earthquakes. The goal adopted implied the use of quite complex calculations, that could be presented only quite briefly in the paper. The estimates of cp and cs, given in previous section of the paper, are quite close to the estimates obtained in a more empirical way, on the basis of full scale experimental recording of several buildings in Bucharest. Of course, the estimates obtained in this way may be not the most appropriate for strong ground motion and that is why it is recommended to equip the Pavilion with strong motion sensors. In order to calibrate the model through trials (an explicit algorithm not being prescribed in professional literature) ample calculations were done in Excel to determine what conventional propagation velocities of the waves and what ratio approached the real conditions the most. It was concluded that the pair 500m/s for P waves and 250m/s for S waves was the searched one, comparing the recorded results with the computed ones. It was also observed that the obtained results when considering only 4 support points (columns) are very close to the ones obtained for the case of 8 supports, thus proving that the number of columns does not represent a determinant factor for the model calibration and that the results can be extended for the entire structure (32 columns).

Applied Mechanics and Materials Vol. 430

341

Acknowledgement The author would like to thank dr.Horea Sandi, dr.h.c. of TUCEB for his permanent guidance throughout the entire process of research, and to prof. Ion Vlad and dr. Nausica Vlad for providing help in the instrumental investigation program. Their support is gratefully acknowledged. References [1] J. Kanai, Semi-empirical formula for the seismic characteristics of the ground, University of Tokyo, Bulletin of the Earthquake Research Institute, 35 (1957) 309-325. [2] H. Tajimi, A statistical method of determining the maximum response of a building structure during an earthquake, in 2nd World Conf. on Earthquake Engineering., Tokyo and Kyoto, Japan (1960) [3] J.L Luco, H.L. Wong, Response of a rigid foundation to a spatially random ground motion, Earthquake Engineering and Structural Dynamics, 14 (1986) 891-908 [4] A. Zerva, Spatial Variation of Seismic Ground Motions-Modeling and Engineering Applications, CRC Press, Taylor&Francis Group, 2009 [5] H. Sandi, On the seismic input for the analysis of irregular structures, Proc. IASS Annual Symp. “IASS 2005 – Theory, Technique, Valuation, Maintenance” Bucharest, September 2005. [6] H. Sandi, Random vibrations in some structural engineering problems, Studies in Applied Mechanics 14, Random Vibration – Status and Recent Developments,, Elsevier, 1986 [7] H. Sandi, Considerations on the updating of earthquake resistant design codes, Proc. 10th International Conf. on Structural Safety and Reliability, ICOSSAR 2009. Osaka, Sept. 2009, Balkema [8] P. Murzea, Aspecte metodologice privind determinarea pe bază experimentală a modurilor proprii de vibrație ale Pavilionului Expozițional Central Romexpo, Technical University of Civil Engineering, Buletin Stiintific UTCB, 2, 2012 [9] H. Sandi, M. Stancu, O. Stancu, I.S. Borcia, A biography of a large-span structure, pre- and post- earthquake, after the provisional and final strengthening. Proc. 8-th European Conf. on Earthquake Engineering, Lisbon, 1986. [10] I. Takewaki, Critical Excitation Methods in Earthquake Engineering, Elsevier, 2007 [11] H. Sandi, Stochastic models of spatial ground motions. Proc. 7-th European Conf. on Earthquake Engineering, Athens, 1982. [12] Murzea, P., Vlad, M., - A study on the non-synchronous character of the seismic motion in the case of the Central Exhibition Pavilion Romexpo using experimental data, Technical University of Civil Engineering, Scientific Journal, Mathematical Modelling in Civil Engineering, 4, 2012

Applied Mechanics and Materials Vol. 430 (2013) pp 342-350 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.342

The Computer Aided Passive Reduction of Vibration To Desired Vibration Amplitude Dymarek Andrzeja, Dzitkowski Tomaszb Silesian University of Technology, Faculty of Mechanical Engineering, Institute of Engineering Processes Automation and Integrated Manufacturing Systems, Konarskiego 18A, Gliwice, Poland a

b

[email protected], [email protected]

Keywords: dynamic flexibility, mechanical systems, vibration reduction.

Abstract. The paper presents the problem of vibration reduction in designed discrete mechanical systems. The passive vibration reduction based on the synthesis method by using the Synteza application. The presented application has been developed by performing the algorithmization of formulated and formalized synthesis methods provided by the authors. Introduction The passive suppression of vibrations within a mechanical system can be achieved by use of members that are referred to as vibration dampers. The characteristic attribute of vibration dampers is the fact that their parameters remain invariable in time and they supply no energy to the system. Such members for passive suppression of vibrations need no external source of energy since they only benefit from motions of the system itself and transform the received energy into forces necessary for vibration suppression. Before the passive suppression of vibrations is commenced it is necessary to determine dynamic properties of the desired system in the form of characteristic functions that define the system mobility (mechanical admittance) or immobility (mechanical impedance) [1,2]. s∏ s 2 + 2hzj s + ω zj2

)

∏ (s

2

)

∏ (s 2 + 2hbi s + ω bi

2

n −1

V (s ) = H

j =1 n

(

2

i =1

+ 2hbi s + ωbi

n

U (s ) = H

i =1 n −1

(

s ∏ s + 2hzj s + ω j =1

2

2 zj

,

(1)

) )

,

(2)

where: H- constant of proportionality, s - Laplace operator, ω bi - resonance frequencies, ω zj - antiresonance frequencies. When the analytic form of the foregoing functions is to be determined one has to specify both resonance and anti-resonance frequencies for the system to be identified as well as conditions or passive suppression of vibrations [1 – 5]. In this study the conditions for passive suppression of vibrations are determined as the values of factors specified by drop of the free vibration frequency. The presented work extends the task of synthesis with new methods, with particular focus on the method of passive synthesis of vibrating mechanical systems.

Applied Mechanics and Materials Vol. 430

343

Passive suppression of vibrations by means of a damping member proportional to a resilient two - terminal mechanical circuit Let the dynamical properties in the form of resonance and anti-resonance values for the identified (synthesized) system are known, i.e.:

ω b1, ω b 2,..., ω bi ω z1, ω z 2,..., ω zj

(i = 1,2,3… , n ) - the poles of the mobility function, ( j = 1,2,3…, n ) - the zeroes of the mobility function,

and the desired vibration amplitude Amax of the inertial two-node element by assuming that these are Kelvin-Voigt model damping elements that constitute the vibration reduction elements of the searched system. In this case the value of two-node damping element is directly proportional to the elastic element value, which can be expressed by the following equation (3).

(k = 1,2,3… , n ) ,

bk = λ ck

(3)

where: bk - the value of two-node damping element, ck - the value of two-node elastic element, λ = idem - the constant of proportionality. The value of the constant of proportionality λ depends on the value of determined free vibration frequency drop coefficient h bi in the following way:

λ=

2h bi

ω 2bi

(4)

,

where: ω bi - the value of resonance frequency at the desired amplitude Amax , h bi - the value of free vibration frequency drop coefficient corresponds to the value ω bi . In the case of the λ coefficient determined that way, it is necessary to calculate subsequent drop coefficients hbi ( zj ) for remaining resonance and anti-resonance frequencies in the following way: hbi ( zj ) =

λ ω bi2 ( zj )

(5)

,

2

Finally, the dynamical characteristics of the searched passive vibration reduction system adopt the form of immobility (2) or mobility (1). The determined characteristic functions describing the restrained systems take the form of immobility:

∏ (s 2 + 2hbi s + ωbi n

U (s ) = H

i =1 n −1

(

s∏ s + 2hzj s + ω j =1

2

2

2 zj

) )

,

(6)

or mobility n −1

1 V (s ) = =H U (s )

(

s∏ s 2 + 2hzj s + ω zj2 j =1 n

∏ (s i =1

2

+ 2hbi s + ωbi

2

)

)

,

(7)

344

Acoustics & Vibration of Mechanical Structures

to be presented in the form making it possible to perform the process of inertial and elastic parameter identification of the system subjected to passive vibration reduction. In the abovepresented case, the immobility (6) and mobility (7) take the analytic form taking into account solely the dynamical properties in the form of resonance and anti-resonance frequency sequence.

∏ (s 2 + ωbi n

U (s ) = H

i =1 n −1

(

s∏ s + ω j =1

2

2

2 zj

) )

,

(8)

At first immobility (8) is distributed into partial fraction in form: U ( s) c n −1 B(2 j −1) n −1 B(2 j ) = m∞ s + 0 + ∑ +∑ , H s j =1 s − jω zj j =1 s + jω zj

(9)

where: m∞ , c0 , B1 , B2 , ... , B(2 j −1) , B(2 j ) - values of residuum pole suitably equal ∞ , 0, jω zj , -jω zj .

The residua can be calculated as follows U (s )  , m∞ = lim s →∞ s  c0 = lim U (s )s s →0  (s − jωz1 )U (s ), B2 = s →lim (s + jωz1 )U (s ),  B1 = slim → jω − js   B = lim (s − jω zn )U (s ), B(2 n ) = lim (s + jω zn )U (s ).  ( 2 n −1) s → jω s → − jω  1

(10)

2

zn

zn

Out of the equations (9) and (10) it is evident that B1 , B2 , ..., B2 k −1 , B2 k are conjugate numbers, but analyzing the qualities of real positive rational function it is obvious that all residua on the imaginary axis are real and positive i.e. B1 = B2 = b2 , B3 = B4 = b4 , ... ,B(2 n-1) = B(2 n ) = b(2 n ),

(11)

B2 2b2 s  B1  s-jω + s+jω = s 2+ω 2 , z1 z1 z1  B B 2 b s 3 4 4 + = ,   s-jω z 2 s+jω z 2 s 2+ω z22   B(2 n-1) B 2b s  + (2 n ) = 2 (2 n )2 .  s-jω zn s+jω zn s +ω zn

(12)

Applying the results of (12), equation (9) can be written as n 2b U (s) c ( 2 j )s = m∞ s + 0 + ∑ 2 , H s j=1 s + ω zj2

where: m∞ > 0 , c0 > 0 , b(2 j ) > 0 , j = 1,2, ... , n.

(13)

Applied Mechanics and Materials Vol. 430

345

Equation (13) is presented as the sum of measurable functions in form

U (s ) c 2b s A s 2 n − 4 + A s 2 n −6 + … + A0 s = m∞ s + 0 + 2 2 2 + 2 2 2 2 3 2 , H s ( s + ω z1 ) ( s + ω z 2 )( s + ωz 3 ) … ( s 2 + ωzn2 )

(14)

n

U (s ) = m∞ s + H

c0 = ∑ c0 j j =1

s n

c0 = ∑ c0 j

U (s ) = m∞ s + H

j =1

s

+

C3 s 2 n − 6 + C 4 s 2 n −8 + … + C0 s 2b2 s 2b4 s + 2 + + , ( s + ω z21 ) ( s 2 + ω z22 ) ( s 2 + ω z23 )( s 2 + ω z24 ) … ( s 2 + ω zn2 )

(2b2 + 2b4 )s3 + (2b2ω42 + 2b4ω22 )s + … + ( s 4 + (ω z21 + ω z22 ) s 2 + ω z21ω z22 )

… C2 s 5 + C1s 3 + C0 s , … ( s 2 + ω z2( n − 2 ) )( s 2 + ω z2( n −1) )( s 2 + ω zn2 )

(15)

(16)

where: A, A1 , A2 , B, B1 , B2 , C1 , C2 , C0 - are parameters appointed with system of equations, on basis comparing of characteristics (14 ÷ 16) to (10). By using the mixed method, the dynamical characteristics in the form of immobility (8) is analyzed, giving:

U (s ) = m∞ s +

c10 + s s + c11 m s + 11

1

+

1 1 s + c12 m s 12

1 1

+

s c1(n −1)

…+

+

1 m1(n −1) s +

c20 s

1 s c(n −1)1

+

,

1 m(n −1)1 s +

(17)

1 s c(n −1)2

1

+ m(n −1)2 s

+

1 s c(n −1)(n −1)

+

1 m(n −1)(n −1) s +

cn 0 s

where: m∞ , m11 , m1(n−1) , m1n , m(n−1)(n−1) - the values of inertial elements, c10 , c20 , cn 0 , c(n+1)0 - the values of restrained elastic elements, c11 , c(n−1)1 , c1n , c(n−1)(n−1) - the values of elastic elements determined by using a mixed method. Taking into account the determined constant of proportionality λ (12) it is possible to calculate the values of vibration reduction elements using the following formula: bci = λ ci ,

(i = 1,2,3… , n ) ,

where: ci - the values of rigidity for the identified system.

(18)

346

Acoustics & Vibration of Mechanical Structures

As a result of the passive vibration reduction performed, it is possible to obtain a system with the dynamical characteristics compliant with the adopted dynamical properties in the form of immobility (2), (6) or mobility (1), (7). The systems obtained as a result of using the mixed synthesis method and the passive vibration reduction is presented in figures 1. c2(n-1),b2(n-1)

c1(n-1),b1(n-1) m∞ m∞

m∞

c(n-1)1,b (n-1)1 m

(n-1)1

c(n-1)(n-1),b(nc(n-1)2,b (n-1)2 m (n-1)2 m∞

m∞

m∞

…

m∞

c10,b10

cn0,bn0 m(n-1)(n-

1)(n-1)

m∞

m∞ m∞

m∞

c11,b11

m11

m∞

c12,b12

m12

c1(n-1),b1(n-1)

m1(n-1)

c20,b20

m∞ m∞ m∞

m∞

m∞

… c2(1),b2(1)

c1(1),b1(1) m∞

m∞

Fig.1. Structure of mechanical system obtained as a result of applying the passive vibration reduction in the case of properties described in the forms (6), (7)

Computer-aided numerical example The Synteza program is designed to determine a discrete mechanical system by taking into account the desired dynamical properties. The calculations performed provide a sequence of values for individual system components determined by using the distribution of characteristic functions (1), (2). The characteristics decomposition is performed by using the mechanical network methods [1÷8]. In addition, the application determines the parameters of passive and active vibration reduction reducing the system to a pre-set amplitude value of the selected resonance frequency. The results are generated in the graphic form as dynamical characteristics for the system obtained and in the numerical form included in the report. After launching the program the start window of the Synteza application appears. The first step when using the program is to choose the type of system searched (fixed, free) and the type of synthesis method (decomposition into partial fractions, decomposition into continued fractions) is defined in the window triggered by pressing the ‘Parameters’ button (fig.2). Here it should be noted that the selection of method affects the structure of the system searched, obtained as a result of the synthesis performed (branched or cascaded).

Fig. 2. Synteza application window

Applied Mechanics and Materials Vol. 430

347

In the next step in ‘Enter parameters’ window (fig.3) he operator enters the values of resonance and anti-resonance frequencies to be met by the searched system.

Fig. 3. Entering the dynamical properties of the system searched By confirming the assumed data, the values of inertial and elastic parameters of the fixed branched system, presented in the figure, are obtained (fig.4). c0 m0=m∞ c1

c2

m1

c3

m2

m3

Fig. 4. The results of the first stage of synthesis performed To modify the structure obtained choose the ‘Mixed method’ tab (fig.5) election makes it possible to transform the system obtained into a structure with a desired number of branches and restraints assumed for the system.

m2 c2 m0=m∞ c1 m1 c11

c3 m3 c31

Fig. 5. Modifying the system in the Mixed method window As a result of the synthesis performed by using the mixed method we can obtain a sequence of values for elastic and inertial components (fig.6) corresponding to the desired resonance and antiresonance frequencies for the modified system structure [4, 6, 9].

348

Acoustics & Vibration of Mechanical Structures

Fig. 6. The results of synthesis performed by using the mixed method The passive vibration reduction requires defining the value of proportionality constant depending on the vibration damping passive component models applied. For the damping components proportional to elastic components the λ value of the constant is entered (fig.7).

Fig. 7. Choosing a type of damping In the next step after confirming the value entered it is necessary to generate the dynamical characteristics in order to check the correctness of amplitude values assumed (fig.8).

Applied Mechanics and Materials Vol. 430

349

Fig. 8. The curve of dynamical characteristics and the amplitude of system deflection To determine the passive vibration reduction values to the desired amplitude value, choose the ‘Mixed method’ tab. The determined passive vibration reduction component values correspond to the damping components of the structure presented in the figure 9. m2 c2,b2 m0=m∞ c3 b 3

b1,c1 m3

m1 b11c11

c31 b31

Fig. 9. The result of passive vibration reduction performed

Conclusion The paper presents a passive vibration reduction based on the synthesis method by using the Synteza application. The presented application has been developed by performing the algorithmization of formulated and formalized synthesis methods provided by the authors. Passive synthesis is understood as a search for parameters and structure of dynamic systems in based on the requirements put forward. These requirements apply to obtaining the set dynamic properties of systems with control as characteristic functions (impedance, mobility). In this work the method of reduction of mechanical systems in accordance with the desired frequency spectrum has been formulated and formalised. Passive suppression of mechanical system has been performed in accordance with the method formulated.

Acknowledgements This work has been conducted as a part of the research project N 502 447239 supported by the Ministry of Science and Higher Education of Poland in 2010 – 2013.

350

Acoustics & Vibration of Mechanical Structures

References [1]. A. Dymarek, T. Dzitkowski, Method of active synthesis of discrete fixed mechanical systems, Journal of Vibroengineering. 14/2 (2012) 458-463. [2]. A. Dymarek, T. Dzitkowski, Active synthesis of discrete systems as a tool for reduction vibration, Solid State Phenomena. 198 (2013) 427-433. [3]. A. Dymarek, T. Dzitkowski, Reduction vibration of mechanical systems, Applied Mechanics and Materials. 307 (2013) 257-260. [4]. M. C. Smith, Synthesis of mechanical networks: the Inerter, IEEE Trans. Autom. Control. 47/10 2002. [5]. M. Płaczek, Dynamic characteristics of a piezoelectric transducer with structural damping, Solid State Phenomena 198 (2013) 633-638. [6]. A. Dymarek, The sensitivity as a criterion of synthesis of discrete vibrating fixed mechanical system, Journal of Materials Processing Technology. 157-158 (2004) 138-143. [7]. T. Dzitkowski, A. Dymarek, Active synthesis of machine drive systems using a comparative method, Journal of Vibroengineering. 14/2 (2012) 528-533. [8]. T. Dzitkowski, A. Dymarek, Active synthesis of discrete systems as a tool for stabilisation vibration, Applied Mechanics and Materials. 307 (2013) 295-298. [9]. C. Grabowik, K. Krenczyk, K. Kalinowski, The hybrid method of knowledge representation in a CAPP knowledge based system, Lecture Notes in Computer Science. 7209 (2012) 284-295. [10]. R. C. Redfield, Dynamic system synthesis with a bond graph approach, part II: Conceptual design of an inertial velocity indicator, J. Dyn. Sys., Meas. Control. 115/3 (1993) 364-369.

Applied Mechanics and Materials Vol. 430 (2013) pp 351-355 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.351

Theoretical and Experimental Studies on Magnetic Dampers Nicolaie ORĂŞANU1,a, Andrei CRAIFALEANU1,b and Cristian DRAGOMIRESCU1,c 1

University “Politehnica” of Bucharest, Department of Mechanics, 313 Splaiul Independentei, 060042, Bucharest, ROMANIA a

[email protected], [email protected] b

[email protected], [email protected]

c

[email protected], [email protected]

Keywords: magnetic damper, cantilever beam, vibration, meshing method.

Abstract. The proper operation and reliability of numerous mechanical devices depends on reducing the vibrations of embedded components, modeled as multi-degree-of-freedom dynamic systems. This can be achieved by using various types of absorbers. The design of such devices is conditioned by the understanding of the variation of their technical characteristics with the constructive parameters. The present paper is dedicated to a special type of dampers, based on the use of the magnetic phenomenon. However, available empirical formulae for the damping characteristics of magnetic dampers do not reveal all significant dependencies. Based on theoretical, numerical and experimental methods, the paper brings a number of contributions concerning the relation between the damping characteristics of a magnetic damper and some of its constructive parameters. Variation curves, useful both for theoretical studies and practical applications, are plotted. Introduction Magnetic dampers with permanent magnets have been used in recent years in order to eliminate the resonance. The parameters of these dampers were studied by K. Seto and his co-workers, in the seventies. In 1978, they published a paper which presented researches regarding the magnetic dampers [1]. In the paper, they proposed a relation for the calculation of the damping coefficient value. H. Nagaya (1987) and his co-workers studied the technical characteristics of magnetic dampers, using various types of permanent magnets and conductors [2]. The dampers with permanent magnets are based on Faraday’s law of induction, which states that a variable magnetic flux produces electrical currents, which, in their turn, generate an opposite magnetic flux. The magnetic flux in the neighborhood Acc.1 Acc.2 of the magnetic poles can produce swirl currents in any Magnetic conductor, if a relative motion exists. The damper must damper be designed with a gap (without contact) between the . Acc.3 Actuator parts in relative motion. The opposite magnetic streams interact by forces proportional with the A.C. relative velocity. Energy dissipation by Joulle effect Function Amplifier accompanies generation of swirl currents. The Generator P.C. characteristic of the damper is determined by the shape of the magnets, the geometry and the material of the Fig. 1. Experimental setup conductors, as well as by the width of the gap between the fixed and the mobile parts. The present study was performed on a cantilever beam, which was used as an experimental setups also in previous research of the authors [2,3], but, this time, it was equipped with a magnetic damper, attached at the free end (Fig. 1). The transverse vibrations of the beam were generated by an actuator, connected to the beam by means of a spring. The harmonic motion of the actuator was

352

Acoustics & Vibration of Mechanical Structures

controlled by a function generator connected to an amplifier. The measuring chain consisted of an acquisition card (A. C.) connected to three accelerometers and to a personal computer (P. C.). Theoretical considerations The theoretical study of the forced damped vibrations of the cantilever beam was based on the meshing method, which consists in approximating the continuous body with a relatively large number of concentrated masses. The beam is replaced by a finite number N of equidistantly placed material points (Fig. 2), with masses

m=M N ,

(1)

where M is the mass of the beam. In section 1, the mass ma of the mobile part of the damper is added to the mass m. m m m m m m+ma m m The actuator is connected to the beam in section r. x vi Fa Fp The resulting system has a finite number of degrees of freedom and its vibrations can Fig. 2. Meshing of the beam be studied with the specific methods, such as the method of the influence coefficients [6]. The differential equations of the transverse vibrations of the system are i

1 2

r

j

N-1 N

v1 = −δ11 (m1v1 + Fa ) − δ12 m2 v2 − ... − δ1r (mr vr − Fp )... − δ1N m N v N  v2 = −δ 21 (m1v1 + Fa ) − δ 22 m2 v2 − ... − δ 2 r (mr vr − Fp )... − δ 2 N m N v N  ................................................................ v N = −δ N 1 (m1v1 + Fa ) − δ N 2 m2 v2 − ... − δ Nr (mr vr − Fp )... − δ NN m N v N , 

(2)

where vi is the displacement in a section i, while the influence coefficient δij is the displacement in section i, produced by a unit force applied in section j [6]. It is known from the literature [6] that, for a cantilever beam,

δ ij = δ ji =

x 2j  x   xi − j  if 2 EI  3

x j < xi .

(3)

If the damper is linear, the damping force has the classical form: Fa = cv1 .

(4)

The perturbation motion of the actuator is presumed harmonic: xe = A cos(ωt ) .

(5)

The steady-state vibration of the beam is also harmonic, with the same frequency as the actuator, but with a phase shift. It follows that the displacement of the beam in section r, where the perturbation force acts, has the form

v r = B cos(ωt + ϕ) .

(6)

Applied Mechanics and Materials Vol. 430

353

The excitation force is F p = ( xe − v r )k ,

(7)

where k is the rigidity of the spring. Using the complex representations of the displacements (Fig. 3), the amplitude of the elongation xe − vr of the spring is

C=

A 2 + B 2 − 2 AB cos ϕ .

(8)

The vector triangle in Figure 3 rotates with the angular velocity ω and the elongation of the spring is a harmonic function: xe − v r = C cos(ωt − γ ) .

Im

B −ω A 2

ϕ

C γ

−ω B 2

C sin γ = B sin ϕ .

(10)

Re

xe

− ω2 C

Phase shift angle γ can be calculated from the relation

ωt A

vr

(9)

xe-vr

Fig. 3. Complex representation

Since the accelerations of the two ends of the spring, a2(t) and a3(t), are proportional with the corresponding displacements, a 2 = −ω 2 v r ,

a 3 = −ω 2 x e ,

(11)

the elongation is proportional with the acceleration difference, ∆a = a 2 − a3 .

(12)

It follows that the amplitude C of the elongation can be determined from the spectrum of the acceleration difference:

C=

1 Ampl(∆a )ω . ω2

(13)

System (2) can be written in the matrix form

[D]{v}+ [E]{v} + {v} = −

1 {δ}r k Ampl(∆a )ω cos(ωt − γ ) , ω2

(14)

where  δ 11 m1 δ12 m 2 ... δ 1N m N   δ 11 c 0 ... 0   δ m δ m ... δ m   δ c 0 ... 0  2N N  , [D] =  21 1 22 2 , [E] =  21  .......................  .........................................      δ N 1 c 0 ... 0 δ N 1 m1 δ N 2 m 2 ... δ NN m N 

 v1  v  {v} =  2  ,  ...  v N 

{δ}r

 δ1r  δ    =  2r  .  ...  δ Nr 

(15)

The steady state solution in complex form is:

{v} = {A}ei (ωt − γ ) , {v} = iω{A}ei ( ωt − γ ) , {v} = −ω2 {A}ei (ωt − γ ) .

(16)

354

Acoustics & Vibration of Mechanical Structures

The vibration amplitudes of the material points that approximate the beam are the moduli of the elements of the complex vector {A} , which is the solution of the algebraic system

[− ω [D]+ iω[E] + [ I ]]{A} = − ω1 {δ} k Ampl(∆a ) 2

2

ω

r

.

(17)

In matrix [E], the damping coefficient c can be determined using Seto’s formula,

c=k

VB 2 ⋅ 10 −14 [ Ns/m] , ρ

(18)

where the following notations have been used: B – magnetic flux [Gauss]; V – volume of the conductor material corresponding to the screening area in the neighborhood of the magnetic pole [cm3]; ρ – resistivity of the conductor material [Ω m]; k – a constant which depends on the ratio between the conductor area and the magnetic pole area. Experimental results A set of about 400 samples were obtained as a result of the measuring experiments. More than 50 perturbation frequencies, between 1 Hz and 100 Hz, were investigated, with two conductor materials – aluminium and copper. The plates of aluminium were rectangular, with the sizes of 67 x 100 mm and the thicknesses of 1.15 mm and 2.66 mm, respectively. The copper plates were cylindrical, with the diameter of 37 mm and the thicknesses of 2.66 and 5.32 mm, respectively. Two neodymium permanent magnets were used. The magnets are cylindrical, with the diameter of 20 mm and with the height of 10 mm. The magnets were placed symmetrically, on the sides of beam, close to the free end. The conductor material has been fixed on a nonmagnetic support, which allowed modifying the width of the gap between the magnets and the conductor materials. The measured and the calculated amplitude of the free end of the beam, for a perturbation force with the modulus equal to 1 N and without damping, are presented in Figure 4. −3

3×10

analytical experimental

−3

3×10

analyticaltrace 1 trace 2 experimental

−3

−3

2×10

2×10

c = 1.75 Nm/s c = 20.00 Nm/s c = 30.00 Nm/s

A [m]

A [m] −3

1×10

−3

1×10

0

20

40

60

80

f [Hz]

Fig. 4. Amplitude of the free end for a perturbation force with the modulus equal to 1 N and without damping

0

20

40

f [Hz]

60

80

Fig. 5. Amplitude curves illustrating the method for determining the damping coefficient

The damping coefficient, for a given material, conductor thickness and gap width, has been determined from the condition that the theoretical amplitude at the fundamental resonance coincides with the corresponding experimental one (Fig. 5).

Applied Mechanics and Materials Vol. 430

355

The damping coefficients estimated in this manner for the two thickness values of the aluminium plates, as well as for the two thickness values of the copper plates, each for various gap widths, are plotted in Figure 6. 6

3

c [Ns/m]

c [Ns/m]

4

2 1 0

4 2 0

0

2

4

6

0

1

2

gap width, x [mm]

3

4

5

6

7

8

gap width, x [mm]

a) Copper, with thickness of 2.66 mm

b) Copper, with thickness of 5.32 mm 3

2

c [Ns/m]

c [Ns/m]

3

1 0 0

1

2

3

4

gap width, x [mm]

c) Aluminium, with thickness of 1.15 mm

5

2 1 0 0

1

2

3

4

5

gap width, x [mm]

d) Aluminium, with thickness of 2.66 mm

Fig. 6. Variation of damping coefficient with conductor thickness and gap width, for two materials Conclusions

As can be seen from Figure 6, a simple law of variation of the damping coefficient with the width of the gap between the fixed and the mobile parts of the damper was not found. Also, even though formula (18) predicts, for the same width, a significantly higher damping coefficient c for the copper, than for the aluminium, since their resistances ratio is about 0.6, the experiment found close values. For the copper conductor, the damping attains its maximum value at the minimum value of the gap width x, while as x exceeds the conductor thickness, the damping coefficient tends to become constant. For the aluminium conductor, the damping coefficient also tends to become constant for large values of x, but it attains its minimum value at the minimum gap width. References

[1] K. Seto, M. Yamanouchi, On the Effect of a Variable Stiffness-type Dynamic Absorber with Eddy-Current Damping, Bulletin of the JSME, Vol. 21, No. 160, 1978, pp.1482-1489. [2] K. Nagaya, Y. Karube, A rotary eddy-current brake or damper consisting of several sector magnets and a plate conductor of arbitrary shape, Magnetics, IEEE Transactions on Volume: 23, Issue: 2, 1987, pp. 1819 – 1826. [3] N. Orăşanu, A. Craifaleanu, Theoretical and experimental analysis of the vibrations of an elastic beam with four concentrated masses, Proceedings SISOM 2011, Bucharest, pp. 471-480. [4] N. Orăşanu, A. Craifaleanu, Experimental study of the forced vibrations of a system with distributed mass and four concentrated masses, Proceedings SISOM 2012, Bucharest, pp. 200-205. [5] R. Voinea, D. Voiculescu, V. Ceauşu, Mechanics (in Romanian), Ed. Didactică şi Pedagogică, Bucureşti, 1989. [6] Gh. Silaş, Mechanics. Mechanical vibrations (in Romanian), Ed. Didactică şi Pedagogică, Bucureşti, 1986.

Applied Mechanics and Materials Vol. 430 (2013) pp 356-361 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.356

Some Models of Elastomeric Seismic Isolation Devices Vasile Iancu1,a, Gilbert-Rainer Gillich1,b, Claudiu Mirel Iavornic1,2,c and Nicoleta Gillich3,d 1

Departemnt of Mechanics, “Eftimie Murgu” University of Resita, 320085 Resita, Romania

2

CFD & Propulsion Laboratory, Colorado State University, Fort Collins, CO 80523-1374, USA 3

Departemnt of Electrical Engineering, “Eftimie Murgu” University of Resita, 320085 Resita, Romania a

[email protected], [email protected], [email protected], [email protected]

Key words: base isolation, elastomeric bearing, lead core, hybrid devices, mathematical model.

Abstract For the study and design of the elastomeric seismic devices is essential to know the mathematical relation between the horizontal displacement and the force leading it. In this paper we present mathematical models for three types of devices: (i) natural rubber bearings, (ii) lead rubber bearings and (iii) hybrid device combining the two first mentioned bearings. For all devices the specific domains of operation are determined and for each domain the relations connecting horizontal displacement and stiffness is contrived, highlighting the hysteretic behaviour in respect to ground excitation. Finally we present numerical results and a comparison between the three devices, defining the opportunity to involve them in specific applications, in function of the type and nature of the isolated structure. Introduction Seismic base isolation is one of the most popular means of protecting a structure against earthquake forces; it consists essentially of the installation of devices which decouple the structure from earthquake induced ground motion [1]. These devices are generally located between the foundation and the main structure in the case of buildings and between the pillars and the superstructure in the case of bridges [2-3]. Among them, as main categories we can nominate sliding and elastomeric bearings. This second category includes many types, all based on elastomersteel sheet sandwiches, potentially accompanied by external or internal damping or stiffening elements (e.g. special compounds added to the elastomer, energy absorbers, lead core etc.). Comprehensive reviews are presented in literature, see for example [4,6], concerning various aspects like manufacturing, design and use of elastomeric bearings. Mathematical models of these devices are crucial to describe their behaviour under various external actions, being an important support to correctly design and involve elastomeric bearings in base isolation systems. Numerous approaches are known, considering diverse aspects [7-11]; this paper presents mathematical models for three types of elastomeric bearings, highlighting their effective stiffness under diverse horizontal load ranges. Natural rubber bearings Natural rubber bearings (NRB) are manufactured in alternating layers of rubber and steel sheets, vulcanized under heat and pressure in a single process to produce the composite bearing. They have one upper and one bottom rigid fixing plate (Fig. 1.a). The steel sheets provide stiffness in the vertical direction and prevent the bulging when important axial loads are applied to the NRB. Its model is a spring-mass system actuated by a horizontal force F given by the ground motion (Fig. 1.b). The stiffness coefficient kech is supposed to be unchanged in respect to displacement d , and is: F (1) k ech = k N = d

Applied Mechanics and Materials Vol. 430

357

To reduce the weight and cost of the isolators, the steel plates (SREI) can be substituted by fibre reinforcement (FREI) [4]. It was proved that FREI-NRBs using different kinds of fibres such as carbon, glass, nylon and polyester present, under equal vertical loading, higher deformation than similar SREI-NRBs, indicating a lower vertical loading capacity of the first one [5]. F Rubber

Upper plate SREI

F

k NNRB

FREI Steel shims

Bottom plate

d

a.

b.

c.

Fig. 1. Natural rubber bearing: (a) schematic representation, (b) physical model and (c) displacement vs. horizontal force diagram for SREI and FREI Researches presented in [8] show that, for similar bearings under equal vertical and horizontal loads, the FREI-NRB present higher deformations than SREI-NRB, as presented in Fig. 1.c. Lead rubber bearings The lead rubber bearing (LRB), shown in Fig. 2.a, was developed in response to the inadequate damping characteristics of the NRB [9]. This bearing is essentially identical to the NRB, excepting the existence of the lead core located usually in the middle of the bearing. The lead core induces a bilinear response by high levels of lateral excitation, where the stiffness of the system decreases after the plug deforms non-elastically in shear. The LRBs work like a histeretic shock absorber, for low horizontal forces the device will have high lateral stiffness, provided by the rubber (stiffness coefficient kR) and the lead core (stiffness coefficient kPb). For horizontal forces exceeding the lead’s yielding point the lead offer no longer rigidity to the device [10,11]. For the entire operational domain, the forces acting on the LRB are distributed among the rubber and the lead as: FR ( d X ) = k R ⋅ d X

and

k Pb ⋅ d X FPb ( d X ) =  0

for d X ≤ d1 for d X > d1

(2)

While the LRB can be modelled as two in parallel linked elastic elements, the total force applied on it can be divided in components acting on the rubber and lead respectively. Thus: (k R + k Pb ) ⋅ d X F (d X ) =  k R ⋅ d1 + k Pb ⋅ d1 + k R ⋅ ( d X − d1 )

for d X ≤ d1 for d X > d1

(3)

Thus, in the first domain (dx < d1) the stiffness coefficients of the rubber, lead and LRB are: k R = k1el =

F1 ; d1

k Pb = k2el =

F2 d1

and

k1ech =

F (d X ) F1 + F2 = = k R + k Pb dX d1

(4)

where F1 and F2 are the force distributed to the rubber and lead core respectively, at which the device achieve displacement d1.

358

Acoustics & Vibration of Mechanical Structures

Rubber

Lead core Upper plate

F

F1 + F2 Steel sheet

Bottom plate

kPb = 0

F2

a.

k2el = kPb k1el = kR

F1

F

k NLRB k Pb

d1 d1

c.

d

k Pb

b.

Fig. 2. Lead rubber bearing: (a) schematic representation, (b) physical model and (c) displacement vs. horizontal force diagram

If the horizontal displacements exceed this limit, the lead yields (i.e. kPb = 0), consequently the device becomes more flexible. The stiffness coefficient k2ech of the LRB for the domain dX > d1 will depend on the total horizontal displacement achieved by the LRB, and is: k2ech =

F (d X ) (k R + k Pb ) ⋅ d1 + k R (d X − d1 ) d = = k R + k Pb 1 dX dX dX

(5)

Hybrid lead rubber bearings The hybrid lead rubber bearing (HLRB) is vertically composed of a LRB and a NRB with a stopper, as presented in Fig. 3a. The lateral stiffness of the NRB is designed to be lower than that of the LRB; therefore the NRB operate in weak earthquakes, while the LRB act when the ground motion becomes stronger [12,13]. Its model is presented in Fig. 3.b. Excited by horizontal force F1, the LRB achieves displacement d T, while the NRB achieves displacement d1. The relation between the force F1 and displacenets d1 and d T (see Fig. 3.c) can be written F1 = k RB ⋅ d1 and F1 = ( k RT + k Pb ) ⋅ d T respectively. By eliminating F1 from these relations one obtains ( k RT + k Pb ) ⋅ d T = k RB ⋅ d1 which permits finding the displacement d T of the LRB in respect to the admissible displacement d1 of the NRB, presented in the relation below: k RB d = T ⋅ d1 k R + k Pb T

(6)

From Eq. 6 one observes that, for low horizontal forces, the LRB’s displacement is some times smaller than that of the NRB. The cumulative displacement of the two components for a horizontal force F1 (Fig. 3.c), is: k RB + k RT + k Pb d t1 = ⋅ d1 k RT + k Pb

(7)

Applied Mechanics and Materials Vol. 430

Median plate Lead core

Upper plate

359

F

k Tech F2 + F2 Stopper ring Stopper pin Bottom plate

a. k NLRB

F

kRT

FRT

k Pb

k NNRB

k Pb

F2

d1

d2

k Pb

kRB

F1 dT

b.

d1 d t1

F

d1 +d2

d2

d

d. S k ech

kPb

kCS kCI

F1

k1ech

dS

c.

Fig. 3. Hybrid rubber bearing: (a) schematic representation, (b) physical model and (c) displacement vs. horizontal force diagram

d1 dt1

Consequently, for the domain in which the lateral force does not exceed the value F1, the device’s stiffness coefficient k1ech is: k1ech =

F1 F ⋅ ( k T + k Pb ) (k T + k ) = B 1 TR = k RB B R T Pb d t1 ( k R + k R + k Pb ) ⋅ d1 k R + k R + k Pb

(8)

One observes that k1ech is definitely lower than the NRB’s stiffness coefficient k RB (see Fig. 3) If the lateral force takes higher values than F1, it will produce displacement just in the LRB, because the stopper blocks the NRB. Thus we can identify two distinct intervals: F1 ≤ F (d X ) ≤ F2 = FRT + FPb and F (d X ) > F2 respectively. For the first interval the displacement of F F (d ) the NRB’s is d1 = B1 , while for the LRB it is d 2 X = T X . Thus, the total displacement kR k R + k Pb achieved by the device is: d X = d1 + d 2 X =

F1 F (d X ) F1 ⋅ ( k RT + k Pb ) + F ( d X ) ⋅ k RB + = k RB k RT + k Pb k RB ⋅ ( k RT + k Pb )

(9)

360

Acoustics & Vibration of Mechanical Structures

One observe that for F(dX)= F1, the total displacement of the device is dX = dt1, while for F(dX)=F2 it is dX = d1+d2. Thus, for the first interval the HLRB’s stiffness coefficient k2ech is: k 2ech =

F (d X ) F (d X ) F ( d X ) ⋅ k RB ⋅ ( k RT + k Pb ) = = dX F1 ⋅ ( k RT + k Pb ) + F ( d X ) ⋅ k RB F1 ⋅ ( k RT + k Pb ) + F ( d X ) ⋅ k RB k RB ⋅ ( k RT + k Pb )

(10)

For the second interval, the total displacement of the device can be written as sum of the displacement imposed to the NRB by force F1, the displacement imposed to the LRB by force F2 when the stiffness is assured by the rubber and lead simultaneously, and the displacement imposed to the LRB by force F(dX)–F2 when just the rubber contribute to the device’s stiffness. It is: d X = d 2 + d3 X =

F1 ⋅ ( k RT + k Pb ) + F2 ⋅ k RB F ( d X ) − F2 + k RB ⋅ ( k RT + k Pb ) k RT

(11)

hence F1 ⋅ ( k RB + k RT + k Pb ) ⋅ k RT + ( F2 − F1 ) ⋅ k RB ⋅ k RT + ( F ( d X ) − F2 ) ⋅ k RB ⋅ ( k RT + k Pb ) k RB ⋅ k RT ⋅ ( k RT + k Pb )

dX =

(12)

Consequently, for the second interval, the HLRB’s stiffness coefficient k3ech is: k 3ech =

F ( d X ) ⋅ k RB ⋅ k RT ⋅ ( k RT + k Pb ) F1 ⋅ ( k RB + k RT + k Pb ) ⋅ k RT + ( F2 − F1 ) ⋅ k RB ⋅ k RT + ( F ( d X ) − F2 ) ⋅ k RB ⋅ ( k RT + k Pb )

(13)

From Fig. 3.d. and Eq. 13 one observe that the stiffness coefficient k3ech decrease with the increase of the lateral force producing the device’s deformation.

Results and conclusions The behaviour of elastomeric isolation bearings is defined by their geometrical and mechanical characteristics. This paper present some mathematical models contrived by the authors, constructed with consideration of these operational subdomains, the final aim of the research being to provide guidelines in design and involvement of such devices in base isolation. 4

Natural Rubber Bearing Lead Rubber Bearing Hybrid Lead Rubber Bearing

S tiffn ess coeficien t kech [N /m ]

3.5 3

2.5 2

1.5 1

0.5 0 0

1

2

3

4

5

6

7

8

9

Displacement [mm]

10

11

12

13

14

15

Fig. 4. Value of the equivalent stiffness coefficient kech for the three analysed base isolation devices: - Natural Rubber Bearing - Lead Rubber Bearing - Hybrid Lead Rubber Bearing

Applied Mechanics and Materials Vol. 430

361

Numerical simulations made on the three analysed devices, involving the developed models in similar load cases [12], pointed out the differences between them: while the NRB has a relative constant stiffness for a large domain of lateral forces, the LRB is rather rigid for small displacements, opposite to the HLRB whose rigidity increase by larger displacement. The simulations where confirmed for the NRB and LRB by experiments [14-15]. As further work we intend to implement the models in codes, usable to define optimal values for the geometric dimensions and mechanical characteristics of base isolation parts, permitting the design of devices with desired behaviour under various excitation types and intensities.

References [1] T.K. Datta, Seismic Analysis of Structures, John Wiley and Sons, 2010. [2] P. Bratu, O. Vasile, Modal Analysis of the Viaducts Supported on the Elastomeric Insulators within the Bechtel Constructive Solution for the Transilvania Highway, Romanian Journal of Acoustics and Vibration, 9(2) 2012 77-82 [3] G.R. Gillich, D. Amariei, V. Iancu, C.S. Jurcau, Aspects behavior of bridges which use different vibration isolating systems, 10th WSEAS International Conference on Automation and Information, Prague, March 23-25, 2009. [4] J.M. Kelly, Aseismic base isolation: review and bibliography, Soil Dynamics and Earthquake Engineering, vol. 5, no. 4, pp. 202-216, 1986. [5] Gordon P. Warn, Keri L. Ryan, A Review of Seismic Isolation for Buildings: Historical Development and Research Needs, Buildings 2012, 2, 300-325. [6] M.C. Kunde, R.S. Jangid, Seismic Behavior of isoalted Bridges: A State-of-the-Art Review, Electronic Journal of Structural Engineering, vol. 3, 2003 pp. 140-170. [7] B.S. Kang, Le Li, T.-W. Ku, Dynamic response characteristics of seismic isolation systems for building structures, Journal of Mechanical Science and Technology 23 (2009) 2179-2192 [8] G.J. Kang, B.S. Kang, Dynamic analysis of fiber-reinforced elastomeric isolation structures, Journal of Mechanical Science and Technology 23 (2009) 1132-1141. [9] J.M. Kelly, R.I. Skinner, A.J. Heine, Mechanisms of energy absorption in special devices for use in earthquake resistant structures, Bulletin of the New Zealand national society for earthquake engineering, Vol. 5, 78-89, 1972. [10] R.I. Skinner, J.M. Kelly, A.J. Heine, Hysteretic dampers for earthquake resistant structures, Earthquake engineering and structural dynamics, Vol. 3, 287-296, 1975. [11] J.M. Kelly, D.F. Tsztoo, Energy absorbing devices in structures under earthquake loading, 6th World conference on the earthquake engineering, Vol. 2, 1369-1374, New Delhi, India, 1977. [12] V. Iancu, O. Vasile, G.R. Gillich, Modelling and Characterization of Hybrid Rubber-Based Earthquake Isolation Systems, Materiale Plastice, Volume: 49 Issue: 4 (2012) 237-241. [13] F. Naeim, James M. Kelly, Design of Isolated Structures from Theory to Practice, New York, John Wiley & Sons, 1999. [14] G.R. Gillich, G. Samoilescu, F. Berinde, C. Chioncel, Experimental determination of the rubber dynamic rigidity and elasticity module by time-frequency measurements, Materiale Plastice, Volume: 44 Issue: 1 (2007) 18-21 [15]G.R. Gillich, P. Bratu, D. Frunzaverde, V. Iancu, Identifying Mechanical Characteristics of Materials with Non-linear Behavior using Statistical Methods, 4th WSEAS International Conference on Computer Engineering and Applications, Cambridge, MA, Jan. 27-29, 2010

Applied Mechanics and Materials Vol. 430 (2013) pp 362-366 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.362

Influence of hysteretic behavior on seismic strength demands: an analysis for Romanian Vrancea earthquakes Iolanda-Gabriela CRAIFALEANU1,2 a 1

Technical University of Civil Engineering Bucharest, Department of Reinforced Concrete Structures, Bd. Lacul Tei, nr. 122-124, Sector 2, Bucharest, Romania 2

National Institute for Research and Development in Construction, Urban Planning and Sustainable Spatial Development “URBAN-INCERC”, INCERC Bucharest Branch, Sos. Pantelimon, nr. 266, Sector 2, Bucharest, Romania a

[email protected], [email protected]

Keywords: nonlinear response spectra, hysteretic model, frequency content, seismic ground motion, Vrancea earthquakes

Abstract. The paper presents the results of a study performed on a large ground motion database, containing records obtained during the three strongest earthquakes that occurred during the past four decades in the Vrancea seismogenic zone. In order to express strength demands imposed by these earthquakes, constant-ductility nonlinear acceleration spectra were computed for two sets of seismic records, selected as representative for narrow frequency band and broad frequency band ground motions, respectively. The spectra, determined for various types of bilinear hysteretic models, were normalized with respect to peak ground acceleration and mean values, as well as coefficients of variation, were computed for each analysis case. The sensitivity of spectral values to the variation of strength hardening and stiffness degradation parameters was determined, with reference to the elastic-perfectly plastic model. Conclusions were drawn, separately for the two distinct types of ground motion frequency content, on the significance of the considered hysteretic model parameters for the assessment of seismic strength demands. Introduction The design of buildings located in earthquake-prone areas requires the quantification, as accurate as possible, of seismic strength demands, in the form of lateral forces for use in analysis. Frequently, seismic strength demands are expressed by means of acceleration response spectra, which provide the advantage of displaying the maximum amplitudes of the acceleration response over the entire range of vibration periods considered as relevant for building structures. Seismic design codes provide such spectra in a schematized form; however, their development involves the application of probabilistic and statistical procedures, which use as input the available seismological information, as well as ground motion records obtained during past earthquakes. The assessment of seismic demands also requires taking into account the characteristics of building structures, regarded as oscillating systems. According to the current design philosophy, nonlinear behavior of building structures is accepted during strong earthquakes; this implies that models used in analysis should be able to reflect the potential incursions in the nonlinear range. To accommodate this concept, seismic design codes provide elastic acceleration spectra, which are affected by specific coefficients that account for nonlinear behavior. Nonlinear acceleration response spectra provide a more rational alternative to this approach, as their values are computed by directly assuming a hysteretic behavior law that governs the oscillations of the system. However, selecting the type and the parameters of this law for various applications is a difficult task, due to the complexity of real phenomena. A frequently used approach is that of determining the sensitivity of results to the modification of the hysteretic law, in order to retain in the analysis only those parameters that have a significant influence. Various studies, most of which performed during the last two decades, investigated the influence of hysteretic models on

Applied Mechanics and Materials Vol. 430

363

key quantities of the seismic response, as relative displacement and input energy, strength reduction factors, absolute acceleration etc. ([1]...[7] et al.). It was shown that certain parameters of the hysteretic model could considerably affect the values of nonlinear spectral response, while others could be neglected. A detailed state-of-the art of various studies in the field is presented in [6]. The paper presents the results of a research aimed to determine the influence of the hysteretic model on seismic strength demands expressed by nonlinear acceleration response spectra. As a starting point, previous studies of the author [8...11] were used. According to the author's knowledge, these were the first studies focused on Vrancea earthquakes to deal with the above aspects. A different procedure for the statistical processing of spectral values is used in the present study, in order to acquire compatibility with other studies in the literature, while the investigated period range is extended as well, as compared to previous studies. The sensitivity of spectral acceleration values to the variation of strength hardening and stiffness degradation parameters is analyzed, separately, for narrow frequency band and broad frequency band motions, and the significance of the considered hysteretic model parameters for the evaluation of seismic strength demands is assessed. Method The ground motion record database used for the study consisted of accelerograms obtained during the three strong Romanian Vrancea earthquakes with moment magnitude Mw > 6.9 that occurred in the last four decades. Of these, two sets of accelerograms were selected, according to their frequency content, as representative for narrow frequency band records and for broad frequency band records, respectively. The two record sets were also used in other studies of the author, concerning nonlinear spectra [8...11]. The selection criteria, briefly described in [12], were presented into more detail in other publications of the first author of the cited reference. A full-dissipative bilinear hysteretic model with strength hardening and stiffness degradation was considered in the analyses (Fig. 1). Strength hardening was expressed by the post-yield stiffness ratio, p, defined as the ratio between the post-yield stiffness (kh in Fig. 1) and the initial stiffness, k0. For the present study, it was assumed that p ≥ 0 . The degraded stiffness is given by the relation F Fy

2 1 5

kh =pk0

k deg = k 0 ⋅ (µ max )− a

(1)

where a is the stiffness degradation parameter and µmax is the maximum displacement ductility attained during the previous k"deg x loading cycles. Seismic strength demands were expressed in terms of 5% kh damping constant-ductility acceleration spectra, SA. These were 3 4 computed by an iterative procedure, which implies finding the yield strength required to limit the displacement to a specified Fig. 1. Bilinear hysteretic model ductility ratio and, subsequently, determining the corresponding with strength hardening and spectral acceleration of the SDOF system. Alternatively, various stiffness degradation indices based directly on yield strength can be used [13]. The acceleration spectra, normalized with peak ground acceleration, were computed for the considered ground motions, as follows: I. (p0a0) – for p = 5% and a = 0 (bilinear elastic-perfectly plastic hysteretic model) II. (p5a0) – for p = 5% and a = 0 (strength hardening only) III. (p0a5) – for p = 0 and a = 0.50 (strength degradation only) IV. (p5a5) – for p = 5% and a = 0.50 (combined strength hardening and stiffness degradation). For each ground motion set, mean values, standard deviations and coefficients of variation were computed for spectral values. k0 xy

k'deg < k0 x'max

364

Acoustics & Vibration of Mechanical Structures

By considering case I as a reference, ratios between mean spectral values obtained for cases II, III and IV and for the reference case were also determined.

Results The resulting diagrams are shown in Figs. 2...6.

a) b) Fig. 2. Mean values of normalized acceleration spectra

a) b) Fig. 3. Ratios between mean values of normalized SA obtained for p5a0 and p0a0 cases (p5a0/p0a0)

a) b) Fig. 4. Ratios between mean values of normalized SA obtained for p0a5 and p0a0 cases (p0a5/p0a0)

a) b) Fig. 5. Ratios between mean values of normalized SA obtained for p5a5 and p0a0 cases (p5a5/p0a0)

Applied Mechanics and Materials Vol. 430

365

a) b) Fig. 6. Coefficients of variation of normalized acceleration spectra It can be observed that the influence of stiffness degradation is larger than that of strength hardening (Figs. 2...5). This influence is much diminished, however, in the long period range. As a general observation, the sense and the amplitude of the influence vary non-monotonically with period and ductility, for both analyzed parameters. From the quantitative point of view, the observations are summarized in the following. a) Influence of the strength hardening parameter (post-yield stiffness ratio), p (Figs. 2 and 3): − for the considered value, p = 5%, the influence varies between +10% and -10...-15% of the reference case; for broad frequency band motions, the increase of p from 0 to 5% leads to the decrease of spectral values, practically over the entire period range. b) Influence of the stiffness degradation parameter, a (Figs. 2 and 4): − for the considered value, a = 0.50, the influence varies between +25% and -14% of the reference case, the largest values being recorded for the narrow-band record set, for periods shorter than 2 seconds. c) Combined influence of p and a (Figs. 2 and 5): − due to the fact that the influence of stiffness degradation is prevalent, as compared to that of the strength hardening, the spectral values, as well as computed ratios, are very close to each other, for cases III and IV. In what concerns the influence of the analyzed parameters on the coefficients of variation (Fig. 6), it can be observed that, even though the overall shape of their curves is roughly preserved, large variations of values occur, in both senses, with the increase of p and/or a. The magnitude of the influence of the parameters of the hysteretic model appears to be, for all cases, larger in the case of narrow frequency records, for periods below 2 s.

Concluding remarks The influence of the hysteretic model parameters on seismic strength demands was investigated, for narrow frequency band and broad frequency band ground motions recorded from strong earthquakes generated, during the last four decades, by the Vrancea seismogenic source in Romania. Strength demands were expressed in terms of normalized constant-ductility acceleration spectra, computed for two sets of records with distinct types of frequency content, i.e. narrow and broad frequency band motions, respectively. The nonlinear spectral values were statistically processed, and the influence of strength hardening (post-yield stiffness ratio) and of stiffness degradation was assessed, with reference to the bilinear elastic-perfectly plastic hysteretic model. It was concluded that, for the analyzed hysteretic models and parameter values, the influence of strength degradation prevails over that of strength hardening. The variation of the strength degradation parameter from 0 to 0.50 resulted in variations of spectral values of less than 25%, as compared to those obtained by using the elastic-perfectly plastic hysteretic model. The variation due to the increase of the post-yield stiffness ratio from 0 to 5% resulted of maximum 10...15%. The sense and the magnitude of the influence vary non-monotonically with ductility and period.

366

Acoustics & Vibration of Mechanical Structures

The largest influences appear to be correlated with the range of maximum spectral acceleration amplitudes, for both analyzed types of frequency content. The above conclusions are in general agreement with studies in the literature and confirm observations made by various authors, according to which, for full-dissipative hysteretic models, as those considered in the present study, the influence of degradation on strength demands is important, while that of strength hardening is rather low [1, 6 et al.]. The observations apply not only to constant-ductility spectra, but also to spectra determined for other parameters. In other words, for stiffness degrading systems, larger strength capacities are needed to keep at the required level the maximum ductility attained by the system. The influence of stiffness degradation is particularly important in the case of narrow frequency band motions, where the period range of affected systems is large. In the context of the present study, this conclusion has a special relevance for the capital city of Bucharest (with about two million inhabitants), where such motions were recorded during the strong Vrancea earthquakes of 1977 and 1986. From a general point of view, given that stiffness degradation occurs frequently in building structures, the quantification of strength demands should always consider the effect of this phenomenon as well, especially for soft soil conditions, where narrow frequency band motions are likely to occur.

References [1] M. Rahnama, H. Krawinkler, Effects of soft soil and hysteresis model on seismic demands, Report No. 108, The John A. Blume Earthquake Engineering Center, USA, 1993. [2] T. Vidic, P. Fajfar, M. Fischinger, Consistent inelastic design spectra: strength and displacement, Earthquake Engng. Struct. Dyn. 23:5 (1994) 507-521. [3] Fajfar, P. and Vidic, T. (1994), Consistent inelastic design spectra: Hysteretic and input energy. Earthquake Engng. Struct. Dyn., 23: 523–537 [4] B. Borzi, G.M. Calvi, A.S. Elnashai, E. Faccioli, J. J. Bommer, Inelastic spectra for displacement-based seismic design, Soil Dyn. Earthq. Eng. 21:1 (2001) 47-61. [5] E. Miranda, J. Ruiz-Garcia, Influence of stiffness degradation on strength demands of structures built on soft soil sites, Eng. Struct. 24:10 (2002) 1271-1281. [6] G.D. Corte, G.D. Matteis, R. Landolfo, Influence of different hysteretic behaviors on seismic response of SDOF systems, Proc. 12 WCEE, Auckland, New Zealand, Paper No. 2402 (2000). [7] Arroyo, D. and Ordaz, M. (2007), On the estimation of hysteretic energy demands for SDOF systems. Earthquake Engng. Struct. Dyn., 36: 2365–2382. [8] I.G. Craifaleanu: Contributions to the study of the inelastic seismic response of reinforced concrete structures. PhD Thesis. Techn. Univ. Civ. Eng. Bucharest (1996) (in Romanian). [9] I.G. Craifaleanu: Studies on response modification factors for Vrancea earthquakes. Buletinul AICPS, No. 2 (1998), p. 34-40 (in Romanian). [10] I.G. Craifaleanu, Studies on inelastic response spectra for Vrancea earthquakes. Buletinul AICPS, No. 3 (1999), p. 62-68 (in Romanian). [11] I.G. Craifaleanu: Nonlinear single degree-of-freedom models in earthquake engineering: Bucharest: Matrix Rom (2005) (in Romanian). [12] D. Lungu, T. Cornea, I. Craifaleanu, S. Demetriu: Probabilistic seismic hazard analysis for inelastic structures on soft soils. Proc. 11 WCEE, Acapulco, Mexico, Paper. No. 1768 (1996). [13] E. Miranda, Evaluation of Site-Dependent Inelastic Seismic Design Spectra, J. Struct. Eng. (ASCE) 119:5 (1993) 1319–1338.

Applied Mechanics and Materials Vol. 430 (2013) pp 367-371 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.430.367

Strength reduction factors: a unified analytic expression for narrowband and broadband ground motion records Iolanda-Gabriela CRAIFALEANU1,2 a 1

Technical University of Civil Engineering Bucharest, Department of Reinforced Concrete Structures, Bd. Lacul Tei, nr. 122-124, Sector 2, Bucharest, Romania 2

National Institute for Research and Development in Construction, Urban Planning and Sustainable Spatial Development “URBAN-INCERC”, INCERC Bucharest Branch, Sos. Pantelimon, nr. 266, Sector 2, Bucharest, Romania a

[email protected], [email protected]

Keywords: strength reduction factors, frequency bandwidth, narrowband records, broadband records, Vrancea earthquakes

Abstract. Strength reduction factors are an important component of the force modification factors used in the calculation of seismic code forces, as they account for the effect of the nonlinear hysteretic behavior of the structure. They are expressed as the ratio between the elastic strength demand and the inelastic strength demand induced in a structure by the earthquake. The evaluation of strength reduction factors has applications in the substantiation of seismic force modification factors in building design codes and in the direct application of the capacity spectrum method. The paper presents a study performed on two sets of accelerograms having different spectral content types, i.e. narrow frequency band records and broad frequency band records. Based on constant-ductility spectra of strength reduction factors, computed with 50% and 10% probability of exceedance, a unified analytic formula is developed, to describe the variation with ductility and vibration period of these factors. By modifying the values of the coefficients of the variables, the formula can be adapted for each type of spectral content. The formula is further refined, by determining a new set of coefficients, to also account for the influence of the predominant period of the ground motion in the case of narrow frequency band records. Introduction According to current code practice, seismic forces used in building design are lower than the lateral strength required to maintain the structure in the elastic range during severe earthquakes. The reduction of seismic forces is performed by using seismic force modification factors or, as termed by European norms, behavior factors. Seismic force modification factors account for energy dissipation capacity, overstrength, damping [1] and other effects that characterize the seismic response of building structures. The influence of the energy dissipation capacity on seismic force modification factors is generally expressed by the strength reduction factors, Rµ [2]. These are defined as the ratio between the lateral yielding strength, Fp(µ), required to limit the displacement ductility demand, µ, to a specified value, and the lateral yielding strength required to maintain the system in the elastic range (µ = 1), Fel,max: Rµ = Fel , max Fp (µ) .

(1)

The displacement ductility, µ, is defined as the ratio between the maximum displacement attained during the ground motion, xmax, and the yield displacement, xp. For a given system, subjected to a given ground motion, the displacement ductility demand will typically increase as the lateral yielding strength of the system decreases. However, as the dependency is not always monotonic, the same ductility could be obtained for multiple yielding strength values [5].

368

Acoustics & Vibration of Mechanical Structures

The quantification of R is important not only for the substantiation of seismic force modification factors in building design codes, but also for the direct application of the capacity spectrum method. Even though a significant number of formulas for strength reduction factors are available in the literature [3-6], some of the earliest, as those in [3], do not account for the influence of the vibration period of the structure and of the spectral content of ground motions. More recently, this deficiency was corrected by the development of R relations differentiated according to soil conditions, as those in [2]. A state-of-the-art of the most significant studies on strength reduction factors is presented in [7]. The paper presents a research aimed to develop a unified relation for R, which would express the variation of this factor with period and ductility and, at the same time, would be adequate for both narrowband and broadband ground motions, i.e. for motions that are characteristic to soft soil and to firm soil conditions, respectively. Using as starting point previous studies of the author [8...11], the base of computation is extended to a larger range of ductility and period values, in order to obtain a new improved formula, with a superior degree of generality and easier to use in current computations. Method For the study, a large ground motion record database was used, consisting of accelerograms obtained during the strong Romanian Vrancea earthquakes of March 4, 1977 (moment magnitude Mw = 7.4), August 30, 1986 (Mw = 7.1) and May 30, 1990 (Mw = 6.9). Of these, two sets of accelerograms were selected as representative for narrow frequency band records and for broad frequency band records, respectively, by using the criteria described in [12]. These criteria were detailed elsewhere by the first author of the cited reference. The first set consists of eight narrow frequency band accelerograms recorded in the soft soil conditions of Bucharest and its surroundings during the strong Vrancea earthquakes of March 4, 1977 and August 30, 1986. For the records in this set, the predominant period of the ground motion, Tg, was estimated, according to previous studies in the literature, to about 1.5...1.6 s. The second set consists of 23 broad and intermediate frequency band accelerograms (termed in the following, for brevity, “broad frequency band motions”) recorded in the northeastern part of Romania during the strong Vrancea earthquakes of August 30, 1986 and May 30, 1990. The studies were performed for the inverse of the Rµ factor, i.e. 1/Rµ, as it was considered that 1/Rµ reflects more intuitively the reduction of seismic forces, providing, at the same time, compatibility with previous studies of the author [8...11] and with the corresponding edition of the Romanian seismic design code. However, to underline its relation with the Rµ factor, more frequently used in the international literature, the notation "1/Rµ" was used. For all accelerograms, the spectral values of the 1/Rµ factor were determined, for a damping ratio ξ = 5% and for constant values of displacement ductility. The nonlinear behavior was modeled by using an elastic-perfectly plastic hysteretic model. By assuming a lognormal distribution of the spectral values, spectra with 50% and 10% probability of exceedance were computed. Regression analyses were performed on these values, in order to determine an analytic expression that fits both types of spectral content. Results According to previous studies performed by the author [8...11], the variation of the 1/Rµ factor with ductility and period can be approximated by an expression of the form: 1 / Rµ = µ − (T

2

+ C1T + C 2 ) ( C 3T 2 + C 4 T + C5 )

,

which can be easily inverted to express the variation of Rµ with the considered parameters.

(2)

Applied Mechanics and Materials Vol. 430

369

It was shown that the above formula provides a good prediction of the values of 1/Rµ, for both sets of ground motion records considered in the analysis [9]. However, the expression is rather complicated and, consequently, further simplification was sought in the present study. A new set of values of the coefficients C1...C5 was proposed (Table 1), for narrowband and broadband records, respectively. In order to better account for the decrease of 1/Rµ values in the vicinity of the predominant period, Tg, observed for narrow frequency band motions, two distinct sets of coefficients C1...C5 were used. The first set applies for the 0.01...1 s period range (where the upper limit, T = 1 s, is approximately equal to 2/3 Tg), while the second set applies for T = 1...34 s. The value T = 0 was not included, in order to allow the C5 coefficient to vanish, thus simplifying the formula. The resulting value ranges of the coefficient of determination, R2, for the considered ductility values, is also given in the table, showing a good fit for all cases. Table 1. Values of the coefficients in Eq. 2 Probability of exceedance Narrowband records

50% 10%

Broadband records

50% 10%

T

C1

C2

C3

C4

C5

0.01...1 s 1...4 s 0.01...1 s 1...4 s 0.01...4 s 0.01...4 s

0 -2.24 0 -2.96 0 0

0 1.69 0 3.02 0 0

1 1.22 1 1.58 1 1.27

0.24 -3.29 0.37 -5.23 0.05 0.17

0 2.64 0 5.11 0 0

R2 0.90...0.99 0.89...0.98 0.95...0.98 0.90...0.98

As it can be observed from the above table, the exponent in Eq. 2 has a simpler form in most cases, as many of the coefficients in the formula equal zero. Fig. 1 displays comparisons between the actual values of 1/Rµ (broken line) and the values resulting from Eq. 2 (continuous line), with the coefficients in Table 1.

a) 50 % probability of exceedance

b) 50 % probability of exceedance

c) 10 % probability of exceedance d) 10 % probability of exceedance Fig. 1. Comparison between the 10% and 50% probability of exceedance values of the 1/Rµ factors and the corresponding values computed with Eq. 2

370

Acoustics & Vibration of Mechanical Structures

For narrowband records (Fig. 1 a and c), a marked decrease of 1/Rµ values in the vicinity of the predominant period, Tg, can be noticed. This observation was made previously for other narrowband records [5], and represents a distinctive feature of this type of ground motions. The cited reference also mentions the period corresponding to the maximum value of the elastic 5%-damped velocity spectrum, as a good estimator of the predominant period of the ground motion. As an alternative, the maximum value of the elastic 5%-damped input energy spectrum is used, by the same author, in [2]. The first Tg assessment method was used in the present paper, for the considered set of narrowband ground motions. The 1/Rµ spectral values with 10% and 50% probability of exceedance were computed, this time as a function of the ratio between the vibration period of the system and the predominant period of the ground motion, χ = T/Tg. The regression analysis was applied to these values, seeking for an expression with the same form as Eq. 2, but in which T is replaced by χ, i.e. 1 / Rµ = µ − ( χ

2

+ C1χ + C 2 ) ( C3 χ 2 + C 4 χ + C5 )

.

(3)

It was found that the variation of 1/Rµ with χ could be satisfactorily approximated by such equation. In this case as well, the value T = 0 was not included in the first period range, in order to allow the C5 coefficient to vanish and to obtain a simpler formula. The new coefficients are shown in Table 2, together with the corresponding ranges of the coefficients of determination, R2, resulting for the considered ductility values. Table 2. Values of the coefficients in Eq. 3 Probability of exceedance 50% 10%

χ= T/Tg

C1

C2

C3

C4

C5

0.01...0.75 0.75...5 0.01...0.75 0.75...5

0 -2.184 0 -1.829

0 1.323 0 0.929

1 1.408 1 1.348

0.112 -3.126 0.254 -2.668

0 1.836 0 1.403

R2 0.64...0.83 0.63...0.96

A comparison between the actual values of 1/Rµ (broken line) and the values resulting from Eq. 3 (continuous line) is shown in Fig. 2. It can be observed that, for values with 50% probability of exceedance, both proposed equations give overestimations of 1/Rµ; however, these are on the conservative side.

a) 50 % probability of exceedance b) 10 % probability of exceedance Fig. 2. Comparison between the actual values of the 1/Rµ factors and the values resulting from Eq. 3

Applied Mechanics and Materials Vol. 430

371

Conclusions A unified analytic expression was developed, in order to express the variation of strength reduction factors with ductility, µ, and vibration period, T. By using different sets of values for the coefficients of the period terms, it was shown that the expression could satisfactorily predict the values of strength reduction factors both in the case of narrow frequency band and in the case of broad frequency band ground motions. The same formula, but with different coefficients, can be used, as well, for narrow frequency band records, to predict the variation of strength reduction factors with the T/Tg ratio, in order to take into account also the influence of the predominant period of the ground motion, Tg. The use of a single, adaptable formula for both considered types of frequency content and for values with different probabilities of exceedance represents a significant advantage, by comparison with other formulas in the literature. From the quantitative point of view, a direct assessment is difficult to make, as most of these formulas are determined based on mean values. However, previous comparisons have shown that the values for 50% probability of exceedance predicted by the proposed formula are relatively close to those obtained with other formulas, as those in [1, 2] The resulting analytic expression has applications in the substantiation of seismic force modification factors in building design codes, as well as in the direct application of the capacity spectrum method. References [1] E. Miranda, V.V. Bertero, Evaluation of Strength Reduction Factors for Earthquake-Resistant Design, Earthq. Spectra, 10: 2 (1994) 357-379. [2] E. Miranda, Site-Dependent Strength Reduction Factors, J. Struct. Eng. (ASCE), 119:12 (1993) 3503-3519. [3] N.M. Newmark, W.J. Hall, Earthquake Spectra and Design, Earthquake Engineering Research Institute, Berkeley, California (1982). [4] H. Krawinkler, A.A. Nassar, Seismic design based on ductility and cumulative damage demands and capacities, in: P. Fajfar, H. Krawinkler (Eds.), Nonlinear Seismic Analysis and Design of Reinforced Concrete Buildings, Elsevier Applied Science, London, 1992, pp. 95-104. [5] E. Miranda, Evaluation of Site-Dependent Inelastic Seismic Design Spectra, J. Struct. Eng. (ASCE), 119:5 (1993) 1319-1338. [6] T. Vidic, P. Fajfar, M. Fischinger, Consistent inelastic design spectra: strength and displacement, Earthq. Eng. Struct. D., 23:5 (1994) 507-521. [7] A.M. Mwafy, A.S. Elnashai, Calibration of force reduction factors of RC buildings. J. Earthq. Eng., 6:2 (2002) 239-273. [8] I.G. Craifaleanu, Contributions to the study of the inelastic seismic response of reinforced concrete structures, PhD Thesis, Tech. Univ. Civ. Engrg. Bucharest (1996) (in Romanian). [9] I.G. Craifaleanu, Studies on response modification factors for Vrancea earthquakes, Buletinul AICPS, 2 (1998) 34-40 (in Romanian). [10] I.G. Craifaleanu, Studies on inelastic response spectra for Vrancea earthquakes, Buletinul AICPS, 3 (1999) 62-68 (in Romanian). [11] I.G. Craifaleanu, Nonlinear single degree-of-freedom models in earthquake engineering, Matrix Rom, Bucharest, 2005 (in Romanian). [12] D. Lungu, T. Cornea, I. Craifaleanu, S. Demetriu, Probabilistic seismic hazard analysis for inelastic structures on soft soils, Proc. 11 WCEE., Acapulco, Mexico, Paper. No. 1768 (1996).

Keywords Index A Acceleration Acoustic Field Modeling Acoustic Methods Acoustic Project Acoustic Zoning Air-Conditioning Airborne Source Quantification Ambient Vibration Amplitude Articulated Robots Artificial Neural Network (ANN)

290 285 113 285 244 118 297 335 14 135 63

B Balancing System Base Isolation BCU Index Bearing Fault Bond Graph Bone Implants Broadband Records

143 356 118 63, 70 158 222 367

C Calibration Cantilever Beam Climate Chamber Concrete Durability Corrosion Cyclostationarity

335 351 118 113 95 78

D Damage Damage Detection Damper Damping Damping Ratio Data Processing Protocol Data Transmission Losses Diagnosis Digital Signal Processing Dissipated Energy Distribution Dynamic Excitation Dynamic Flexibility

84, 90 95 195, 312 217 329 208 125 276 125 317 276 305, 317 342

Dynamic Model Dynamic System

143, 217 32, 329

E Eccentricity Eigenfrequency Eigenmode Elastic Force Elastomer Elastomeric Bearing Elastomeric Insulators Electric Motovibrator Electric Vehicle Noise Engine

148 153 153 170 317, 323 356 329 165 297 53

F FFT Analysis Finite Element Analysis (FEA) Fireworks Fixing System Flux Cored Wire Forecast Fracture Mechanics Frequency Frequency Bandwidth Frequency Content Frequency Response Function Frictional Force

118 184, 230 108 153 101 276 84, 90 14, 290 367 362 184 170

G Gait Analysis Gait Parameters Gear Fault Detection Gearbox Fault Ground Reaction Forces

213 208 78 63 208

H Half Car Hallux Valgus Condition Hand-Arm Vibration Harvester Hearing Protection Human Lower Limb Hybrid Devices

191 208 276 290 281 230 356

374

Acoustics & Vibration of Mechanical Structures

Hydraulic Pump Hydroxyapatite (HA) Hysteretic Damping Hysteretic Model

158 222 323 362

I Independent Joint Control in a Robot Industrial Hall Industrial Noise Monitoring Instability Internal Fixator Isochronicity Isolators

135 251 262 32 213 14 305

J Joint Angle

203, 230

K Kinematic Displacements Kinematic Excitation

323 305, 317

L Laboratory Testing Lateral Displacement Lateral Distal Humerus Implant Lead Core Limit Cycle Motion Locked Plating Loss Factor

305 148 203 356 3 213 317

M Magnesium Doping Magnetic Damper Magnetic Interaction Marine Habitat Mass Mass Loss Mathematical Models MATLAB Measuring Stand Mechanical Conditioning Mechanical Impedance Mechanical Properties Mechanical Shock Apparatus Mechanical System Meshing Method Mitigation Methods

222 351 143 285 217 95 356 170, 217 153 108 178 222 108 178, 342 351 251

Modeling

191, 230

N Narrowband Records Natural Discontinuous Nonlinearities Natural Frequency Neo-Hookean Noise Noise Assessment Noise Calculation Noise Control Noise Decreasing Noise Measurement Noise Measurement Methods Noise Source Nomogram Non-Balancedeccentrics Masses Non-Destructive Method Non-Uniform Sampling Nonlinear Damped Vibration Nonlinear Oscillator Nonlinear Parametric Vibration Nonlinear Partial Differential Equations Nonlinear Response Spectra Nonstationary Signal

367 135 84, 90, 95 45, 53 281 244 237 251 257 237, 244 266 262 290 165 113 125 27 14 22 40 362 78

O Occupational Disease Occupational Risk OHAM Optimal Homotopy Asymptotic Method

281 276 27 22, 40

P Pendulum Portable Power Generator Power Spectral Density Prosthesis Proximal Tibial Plateau Fracture

45 266 184 230 213

R Railway Railway Vehicle Random Vibrations Range of Motion Rayleigh Oscillator Resonance

237 195 184 203 3 32

Applied Mechanics and Materials Vol. 430 Rheological Models Road Excitation Robot Dynamics Rubberized Asphalt

312 184 135 257

Vibration Reduction Vibration Signal Processing Vibration Test Viscoelastic Model Vrancea Earthquakes

375 342 78 101 317 362, 367

S SCHALL 03 Seismic Ground Motion Self-Excited Vibration Self-Organized Feature Map Self-Sustained Vibrations Severity Assessment Shock Simplified Car Models Small Oscillations SolidWorks Sound Power Level Spectral Analysis Stability Stiffness Stochastic Bondsim Elements Stochastic Models Strength Reduction Factors Suspension

237 362 27 63 3 84, 90 312 184 45, 53 170 266 125 45, 53 95, 217 158 335 367 53, 191

T Thin Elastic Plate Time-Frequency Analysis Torsional Vibration Transfer Path Analysis

40 78 135, 178 297

U Ultrasonic Impulse Methods Unbalanced Rotor Uncertainty Underwater Noise Upper Limb Movement

113 148 32 285 203

V Validation Van Der Pol Oscillator Velocigrams Vibrating Frame Vibration

Vibration Analysis Vibration Level

237 3 335 165 84, 90, 95, 108, 148, 158, 191, 195, 290, 351 63, 70 118

W Welded Joint Wet Precipitation

101 222

Y Young's Modulus

101

Z Zebris Measuring System

203

Authors Index A Alexandru, C. Anderson, C. Arsene, D. Arsene, M.

317 297 285 285

B Bacria, V. Badea, C. Bajrić, R. Bălănean, F. Bereteu, L. Biris, S. Bobos, D. Bogojević, N. Borda, C. Borza, I. Bracacescu, C. Bratu, P. Bucur, L. Bungescu, S.T. Butu, L.

251, 257 113 63, 78 203 101, 222, 262 290 95 125 285 113 165 32, 305, 312 195 290 285

C Cardei, P. Cioara, C. Ćirić Kostić, S. Comeaga, D.C. Costache, A. Craifaleanu, A. Craifaleanu, I.G. Crisan, D. Cveticanin, L. Cvetkovic, D.

266 108 125 143 148, 153 148, 153, 351 362, 367 213 3 63, 70, 244

D Donţu, O.G. Drăgănescu, G. Dragomirescu, C. Dymarek, A. Dzitkowski, T.

143 101 153, 351 178, 342 178, 342

E Ene, R.D.

40

G Gajicki, A. Gheorghiosu, E. Ghicioi, E. Ghita, E. Gillich, G.R. Gillich, N. Grigore, I.

237, 244 108 108, 276 195 84, 90, 95, 356 356 266

H Haragus, H. Hatiegan, C. Herişanu, N. Holecek, N.

213 95 22, 27, 251, 257, 290 244

I Iancu, V. Iavornic, C.M. Ilici, S. Ilie, C. Ioanovici, T. Isić, S. Iures, L.

356 356 108 143 222 78 113

J Jebelean, E. Jitea, C. Jiva, C. Jovanovic, D. Jovanovic, M.

113 108 113 118 118

K Kovacic, I. Kovacs, A. Kovacs, M.

14 108 281

Applied Mechanics and Materials Vol. 430

L Lupea, I.

S 184, 297

M Marghitu, D.B. Marinca, B. Marinca, T. Marinca, V. Marinescu, M. Medgyesi, T. Menyhardt, K. Mihajlov, D. Minda, A.A. Minda, P.F. Moisa, R. Morariu, M.C. Muntean, F. Murzea, P.

170 40 22 22, 27, 40 285 101 191, 262 237, 244 84 84 101 297 90 335

N Nagy, R.

191, 262

O Orăşanu, N. Ovidiu, V.

148, 153, 351 323, 329

P Pasca, O. Pasculescu, V. Pirna, I. Popa, D. Popescu, S. Popovici, V. Postelnicu, E. Praisach, Z.I. Praščevič, M. Prejbeanu, R. Protocsil, C.

208 276 165 45, 53 165 285 266 84, 90, 95 118, 237, 244 213 90

R Radu, P.L. Ragan, J. Raos, M. Razvan, C. Receanu, D. Rusu, L.

377

217 170 118 184 135 230

Samardzic, B. Sava, M. Simion, S. Simoiu, D. Sorica, C. Soskic, Z. Stanescu, N.D. Stoia, D.I. Suciu, O.

158 101 276, 281 101, 262 266 125 45, 53 203, 208, 213 222

T Toader-Pasti, C. Tomić, J. Toth-Taşcău, M. Toth, L.

230 125 203, 208, 230 281

V Vasilescu, G.D. Vermeşan, D. Vigaru, C. Vladut, V. Vreme, C.

276 213 208 266, 290 281

Z Zivkovic, L. Zivkovic, N. Zlatkovic, B.M. Zuber, N.

118, 237 118, 237 158 63, 70, 78