SOUND AND VIBRATION ANALYSIS OF COMPLIANT PIEZOELECTRIC ENABLED STRUCTURES BY CAI CHAO (B. Eng. JU, M. Eng. JU and NUS) DEPARTMENT OF MECHANICAL ENGINEERING A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NATIONAL UNIVERSITY OF SINGAPORE 2005 i ACKNOWLEDGEMENTS It has been a great pleasure and inspiration to work with my supervisors, Associate Professor Liu Gui Rong (Dr) and Professor Lam Khin Yong (Dr). I am grateful to them for their extensive encouragement, advice, guidance and patience during the years I worked on this research and dissertation at National University of Singapore. This research has been undertaken with the support of my employer, Institute of High Performance Computing (IHPC) which is gratefully acknowledged. I could not fulfill my dream to pursue this advanced study without this sponsor. I wish to acknowledge the help and assistance of my colleagues and friends at Computational Multiphysics Modelling Programme in IHPC. Special thanks go out to Dr. Zheng Hui for the many productive and challenging discussions. Also thank Dr. Cui Fangsen for spending one whole afternoon picking apart the presentation. It made the actual presentation go much more smoothly. Finally, I would like to express my deep gratitude to my parents whose caring and support have been the important factors which let me seek this degree without worries, Last, but certainly not least, I would like to express my deepest appreciation to my wife, Wu Jian, and my children, Zhang Junbo, Jack and Cai Junyao, Stefanie, who have been there throughout the years, encouraging me to go further than I thought possible. The sacrifices, patience and love from my wife have been my guiding light. I hope I make all of you proud. ii TABLE OF CONTENTS Acknowledgements i Table of contents ii Summary viii Nomenclature x List of figures xiv List of tables xx Chapter Introduction and literature review 1.1 Overview 1.2 Literature review 1.2.1 Wave interactions with structures of passive damping 1.2.2 Wave interactions with piezoelectric enabled structures 1.2.3 Passive damping treatment for vibration control 11 1.2.4 Active damping treatment for vibration control 16 1.3 Outline of the research 21 1.3.1 Objective and scope of the research 21 1.3.2 Synopsis of the research 22 1.4 Thesis organization 26 iii Chapter Overview of theory on elastic and piezoelectric materials 29 2.1 Elastic materials 29 2.2 Material symmetry 30 2.3 Engineering constants of orthotropic materials 34 2.4 Viscoelastic effects in elastic materials 36 2.5 Transformation of field parameters 37 2.6 Piezoelectric materials 41 2.6.1 Linear piezoelectric constitutive relations 42 2.6.2 Commonly used piezoelectric materials 43 2.6.3 Direct piezoelectric effect 44 2.6.4 Converse piezoelectric effect 45 2.6.5 Basic relations of piezoelectric materials 46 2.7 Boundary conditions 51 2.8 Plane stress assumption 52 2.9 Plane strain assumption 53 Chapter An exact method for wave interactions with infinite plates 58 3.1 Introduction 58 3.2 Formulation 58 3.2.1 Basic equations for a layer in plate 60 3.2.2 Wave field in plate 61 3.2.3 Wave field in fluid 64 3.2.4 Interaction between plate and fluid 65 iv 3.3 Application examples 69 3.4 Remarks 73 Chapter An exact method for wave interactions with an infinite bi-layer piezoelectric plate 86 4.1 Introduction 86 4.2 Governing equations for an infinite bi-layer plate 86 4.2.1 Formulations for elastic layer 87 4.2.2 Formulations for piezoelectric layer 87 4.3 Wave interactions with a bi-layer piezoelectric enabled plate 88 4.3.1 Surface impedance matrix for elastic layer 88 4.3.2 Surface impedance matrix for piezoelectric layer 93 4.3.3 Boundary conditions 101 4.3.4 Sound wave incidence from a bottom-free bi-layer plate 105 4.4 Distribution of displacement and traction 106 4.4.1 Distribution of displacement and traction in elastic layer 106 4.4.2 Distribution of displacement and traction in piezoelectric layer 107 4.5 Application examples 109 4.5.1 Sound reflections and radiation 109 4.5.2 Effects of the thickness ratios on active sound control 110 4.5.3 Sound cancellation properties 111 4.5.4 Field parameter distributions along the thickness of plate 112 4.6 Remarks 113 v Chapter An exact method for wave interactions with an infinite multilayered piezoelectric structure 131 5.1 Introduction 131 5.2 Wave propagation in an elastic layer 131 5.3 Wave propagation in a piezoelectric layer 134 5.4 Wave propagation in fluid media 138 5.4.1 Wave propagation in the fluid above structure 140 5.4.2 Wave propagation in the fluid below structure 141 5.5 Transfer matrix for a layer 142 5.6 Transfer matrix for multiple layers 145 5.7 Reflection and transmission from a piezoelectric substrate 145 5.8 Reflection and transmission from an infinite layered elastic plate 149 5.9 Reflection and transmission from an infinite layered piezoelectric plate 150 5.10 Formulation for wave normal incidence 153 5.10.1 Transfer matrices for layers with different types 157 5.10.2 Boundary conditions 158 5.10.3 Application examples 159 5.11 Remarks 165 Chapter Interaction of waves with piezoelectric structures at oblique incidence 178 6.1 Introduction 178 6.2 Sound radiation from a point source 179 6.3 Linear array 181 vi 6.3.1 Linear array in phase 181 6.3.2 Linear array with phase shift 184 6.3.3 Linear array of piecewise line sources with phase shift 186 6.4 2-D finite array 188 6.4.1 2-D array of discrete point sources in phase 188 6.4.2 2-D array with continuous point sources in phase 189 6.4.3 2-D array with continuous point sources and a phase shift 191 6.5 Sound reflection from a finite passive reflector 192 6.5.1 Sound reflection from a 1-D reflector 192 6.5.2 Sound reflection from a 2-D reflector 194 6.6 Suppressing of sound reflection using a piecewise bi-layer plate 195 6.6.1 Excitations of the bi-layer plate 196 6.6.2 Pressure field radiated from the bi-layer plate 197 6.6.3 Application Examples 208 6.7 Remarks 209 Chapter Vibration response of an elastic structure with passive damping treatment 226 7.1 Introduction 226 7.2 Governing equations 227 7.2.1 Basic kinematic relation 227 7.2.2 Energies of a PCLD system 228 7.2.3 Assumed modes method 229 7.2.4 Lagrange’s equation 231 vii 7.3 Selection of admissible functions 233 7.4 Application examples 235 7.5 Remarks 238 Chapter Vibration response of an elastic structure with active damping treatment 246 8.1 Introduction 246 8.2 Governing Equations 247 8.2.1 Basic kinematic relationships 248 8.2.2 Energies of a ACLD system 249 8.2.3 Assumed modes method 252 8.2.4 Lagrange’s Equation 254 8.3 Governing equations for conventional analytical mode 257 8.4 Application examples 263 8.5 Remarks 264 Chapter Conclusions and recommendations 276 9.1 Summary and conclusions 276 9.2 Contributions of the research 279 9.3 Recommendations for further study 280 References 281 Appendix 295 viii SUMMARY It is imperative that the accurate prediction of the interactions of waves with fluid-loaded passive or piezoelectric-enabled structures for suppression of sound reflection and transmission. The work in the thesis concentrates on using different exact methods to predict sound transmission and reflection by an infinite compliant plate-like structure immersed in fluids, subjected to an incident plane wave in Chapter 3. The coupling between the fluid and structure is taken into account in a rigorous manner. A matrix formulation for the submerged plate subject to a plane sound wave excitation, which can have a stack of arbitrary number of isotropic or anisotropic layers, is derived to obtain the transmission and reflection coefficients for waves in the frequency domain. In Chapter 4, the technique of transformation of the second-order ordinary differential equation to a first-order ordinary differential equation is employed to study sound wave interactions with an infinite bi-layer piezoelectric enabled plate. Mode superposition method is applied to obtain solutions of the first-order ordinary differential equation with consideration of the upward and downward waves’ separation. For multilayered piezoelectric plate-like structures, the transfer matrix approach is proposed as well to investigate the sound wave interaction with the structure in Chapter 5. For piecewise piezoelectric layers or infinitely periodic structures subject to oblique wave incidence, both incident wave and piecewise electric potential excitations undergo the Fourier transforms respectively which convert the problem in piecewise x coordinate system into the problem in the wave-number space in Chapter 6. The ix upward pressure waves in the x coordinate system are obtained through inverse Fourier transformations. In Chapter 7, a refined analytical method is proposed to analyze the vibration analysis of the beam with passive constraining layer damping treatments. The refined approach differs from the conventional analytical approach in that the third admissible function is introduced to represent the longitudinal displacements of constraining layer in passive damping treatments. The longitudinal vibration mode shape function of a free-free beam is used for this third admissible function. Finally in Chapter 8, the refined approach working well with passive constraining layer damping treatment is extended for the vibration analysis of the beam with active constraining layer damping treatment under both mechanical and electric excitations. The contributions of this thesis are: (1) the derivation of governing equations that consider all the couplings among elastic, piezoelectric and acoustic combinations; (2) the proposal of a refined analytical method for vibration analysis of a beam with passive and/or active constraining layer damping treatments. The proposed analytical method can achieve much more accurate prediction of damping effects for a damped beam compared to the conventional analytical method, thereby rendering it more practical. References 283 Cai C., G. R. Liu and K. Y. Lam 2001b A technique for modelling multiple piezoelectric layers, Smart Material and Structures 10 pp. 689-694 Chandrashekhara K. and A. Agarwal 1993 Active vibration control of laminated composite plates using piezoelectric device: a finite element approach, Journal of Intelligent Material Systems and Structures pp. 496-508 Chen C. Q., X. M. Wang and Y. P. Shen 1996 Finite element approach of vibration control using self-sensing piezoelectric actuators, Computers & Structures 60(3) pp. 505-512 Corsaro R. D. and R. M. Young 1995 Influence of backing compliance on transducer performance, Journal of the Acoustical Society of America 97(5) pp. 2849-2854 Corsaro R. D., B. Houston and J. A. Bucaro 1997 Sensor-actuator tile for underwater surface impedance control studies, Journal of the Acoustical Society of America 102(3) pp. 1573-1581 Crawley E. F. and de Luis J. 1987 Use of piezoelectric actuators as elements of intelligent structures, AIAA Journal 25 pp. 1373-1385 Dimitriadis E. K., C. R. Fuller and C. A. Rogers 1991 Piezoelectric actuators for distributed vibration excitation of thin plates, Transactions of the ASME 113 pp. 100-107 DiTaranto R. A. 1965 Theory of vibratory bending for elastic and viscoelastic layered finite-length beams, ASME Journal of Applied Mechanics 32 pp. 881-886 Dong S. B. and R. B. Nelson 1972 On natural vibrations and waves in laminated orthotropic plates, ASME Journal of Applied Mechanics 39 pp 739 Dong S. B. 1977 A block-Stodola eigensolution technique for large algebraic systems with non-symmetrical matrices, International Journal for Numerical Methods in Engineering 11 pp. 247-267 References 284 Douglas B. E. and J. C. S. Yang 1978 Transverse compressional damping in the vibratory response of elastic-viscoelastic-elastic beams, AIAA Journal 16 pp. 925-930 Fasana A. and S. Marchesiello 2001 Rayleigh-Ritz analysis of sandwich beams. Journal of Sound and Vibration 241 pp. 643-652 Gao J. and Y. Shen 1999 Dynamic characteristic of a cantilever beam with partial selfsensing active constrained layer damping treatment, ACTA MECHANICA SOLIDA SINICA 12 pp. 316-327 Gentry C. A., C. Guigou and C. R. Fuller 1997 Smart foam for application in passiveactive noise radiation control, Journal of the Acoustical Society of America 101(4) pp. 1771-1778 Gibbs G. P. and C. R. Fuller 1992 Excitation of thin beams using asymmetric piezoelectric actuators, Journal of the Acoustical Society of America 92 pp. 3221-3227 Gopinathan S. V., V. V. Varadan, V. K. Varadan 2000 A review and critique of theories for piezoelectric laminates, Smart Material and Structures (1) pp 24-48 Hagood N. W., W. H. Chung and A. Von Flotow 1990a Modelling of piezoelectric actuator dynamics for active structural control. AIAA Pap, AIAA-90-1087-CP pp. 2242-2256 Hagood N. W., W. H. Chung and von Flotow A. 1990b Modelling of piezoeletric actuator dynamics for active structural control, Journal of Intelligent Materials, Systems, and Structures 1(3) pp. 327-354 He J-F and B-A Ma 1988 Analysis of flexural vibration of viscoelastically damped sandwich plates, Journal of Sound and Vibration 126 pp. 37-47 References 285 Honein B., A. M. B. Braga, P. Barbone and G. Herrmann 1991 Wave propagation in piezoelectric layered media with some applications, Journal of Intelligent Material Systems and Structures pp. 542-557 Honein B. and G. Herrmann 1992 Wave propagation in nonhomogeneous piezoelectric materials, ASME Active Control of Noise and Vibration, DSC-Vol. 38, pp. 105112 Howarth T. R., V. K. Varadan, X. Q. Bao and V. V. Varadan 1992a Piezocomposite coating for active underwater sound reduction, Journal of the Acoustical Society of America 91(2) pp. 823-831 Howarth T. R., V. K. Varadan and V. V. Varadan, 1992b Intelligent sensor systems for underwater acoustic applications, Intelligent Structural Systems, Kluwer Academic Publishers (edited by H. S. Tzou and G. L. Anderson), pp. 285-304 Huang S. C., D. J. Inman and E. M. Austin 1996 Some design considerations for active/passive constrained layer damping treatments, Smart Materials and Structures pp. 301-313 Hwang Woo-Seok Park, Hyun Chul 1993 Finite element modelling of piezoelectric sensors and actuators, AIAA JOURNAL 31 (5) pp. 930-937 Johnson C. D. and D. A. Kienholz 1982 Finite element prediction of damping in structures with constrained viscoelastic layers, AIAA Journal 20 pp. 1284-1290 Jones R. M. 1975 Mechanics of composite materials, Washington, DC: Scripta Book Company. Kausel E. 1986 Wave propagation in anisotropic layered media, Int. J. Numer. Methods Eng. 23 pp 1567 References 286 Kerwin E. M. Jr. 1959 Damping of flexural waves by a constrained viscoelastic layer, Journal of the Acoustical Society of America 31 pp. 952-962 Kerwin Jr. E. M. and Smith Jr. P. W. 1984 Segmenting and mechanical attachment of constrained viscoelastic layer damping treatments for flexural and extensional waves, Damping '84, kk1-kk24. Kim J., V. V. Varadan and V. K. Varadan 1997 Finite element modelling of structures including piezoelectric active devices, International Journal for Numerical Methods in Engineering 40(5) pp. 817-832 Koval L. R. 1980 Sound transmission into a laminated composite cylindrical shell, Journal of Sound and Vibration 71 pp.523-530 Kraut E. A. 1969 New mathematical formulations for piezoelectric wave propagation, Physical Review 188(3) pp. 1450-1455 Kung S. W. and R. Singh 1998 Vibration analysis of beams with multiple constrained layer damping patches, Journal of Sound and Vibration 212 pp. 781-805 Lafleur L. D., F. D. Shields and J. E. Hendrix 1991 Acoustically active surfaces using piezorubber, Journal of the Acoustical Society of America 90 (3) pp. 1230-1237 Lall A. K., N. T. Asnani and B. C. Nakra 1987 Vibration and damping analysis of rectangular plate with partially covered constrained viscoelastic layer, Journal of vibration, Acoustics, Stress and Reliability in Design, Transactions of the ASME 109 pp. 241-247 Lall A. K., N. T. Asnani and B. C. Nakra 1988 Damping analysis of partially covered sandwich beams, Journal of Sound and Vibration 123 pp. 247-259 Lam K. Y., X. Q. Peng, G. R. Liu and J. N. Reddy 1997 A finite element model for piezoelectric composite laminates, Smart Materials and Structures pp. 583-591 References 287 Lam K. Y. and T. Y. Ng 1999 Active control of composite plates with integrated piezoelectric sensors and actuators under various dynamic loading conditions, Smart Materials and Structures 8(2) pp. 223-237 Lam M. J. 1997 Hybrid active/passive models with frequency dependent damping. Ph.D Dissertation. Virginia Polytechnic Institute and State University Lee C. K. and F. C Moon 1989 Laminated piezopolymer plates for torsion and bending sensors and actuators, Journal of the Acoustical Society of America 85 pp. 24322439 Lee C. K. 1990 Theory of laminated piezoelectric plates for the design of distributed sensors and actuators. Part I: governing equations and reciprocal relations, Journal of the Acoustical Society of America 87 pp. 1144-1158 Levy C. and Q. Chen 1994 Vibration analysis of a partially covered, double sandwichtype cantilever beam, Journal of Sound and Vibration 177 pp. 15-29 Liu G. R., J. Tani, K. Watanabe and T. Ohyoshi 1990a Lamb wave propagation in anisotropic laminates, ASME J. Appl. Mech. 57 pp. 923-929 Liu G. R., J. Tani, K. Watanabe and T. Ohyoshi 1990b A semi-exact method for the propagation of harmonic waves in anisotropic laminated bars of rectangular cross section, Wave Motion 12 pp. 361-371 Liu G. R. and J. Tani 1991a Characteristics of wave propagation in functionally gradient piezoelectric material plates and its response analysis, part 1: theory, Transactions of the Japan Society of Mechanical Engineers 57(A) pp. 180-185 Liu G. R., J. Tani, T. Ohyoshi and K. Watanabe 1991b Characteristic wave surfaces in anisotropic laminated plates, ASME Journal of Vibration and Acoustics 113 pp. 279-285 References 288 Liu G. R. and J. Tani. 1992a SH surface waves in functionally gradient piezoelectric material plates, Transactions of the Japan Society of Mechanical Engineers 58(A) pp. 504-507 Liu G. R., J. D. Achenbach, J. O. Kim and Z. L. Li 1992b A combined finite element method/boundary element method for V(z) curves of anisotropic-layer/substrate configurations, Journal of the Acoustical Society of America 92 pp. 2734-2740 Liu G. R. and J. Tani 1994 Surface waves in functionally gradient piezoelectric plates, Transactions of the American Society of Mechanical Engineers 116 pp. 440-448 Liu G. R. and J. D. Achenbach 1994a A strip element method for stress analysis of anisotropic linearly elastic solids, ASME Journal of Applied Mechanics 61 pp. 270-277 Liu G. R. and J. Tani 1994b Surface waves in functionally gradient piezoelectric plates, Transactions of the ASME Journal of Vibration and Acoustics 116 pp. 440-448 Liu G. R. and J. D. Achenbach 1995a A strip element method to analyze wave scattering by cracks in anisotropic laminated plates, ASME Journal of Applied Mechanics 62 pp. 607-613 Liu G. R., K. Y. Lam and J. Tani 1995b An exact method for analysing elastodynamic response of anisotropic laminates to line loads, Mechanics of Composite Materials and Structures pp. 227-241 Liu G. R., K. Y. Lam and E. S. Chan 1996 Stress waves in composite laminates excited by transverse plane shock waves, Shock and Vibration pp. 419-433 Liu G. R., Z. C. Xi, K. Y. Lam and H. M. Shang 1999 A strip element method for analyzing wave scattering by a crack in an immersed composite laminates, ASME Journal of Applied Mechanics 66 (4) pp. 898-903 References 289 Liu G. R., C. Cai and K. Y. Lam 2000 Sound reflection and transmission of compliant plate-like structures by a plane sound wave excitation, Journal of Sound and Vibration 230(4) pp. 809-824 Liu G. R. and Y. T. Gu 2001a A point interpolation method for two-dimensional solids, Int. J. Muner. Meth. Engrg, 50 pp. 937-951 Liu G. R and Xi Z. C. 2001b Elastic waves in anisotropic laminates, CRC Press. pp. 235-253 Liu G. R. 2002 Mesh free methods: moving beyond the finite element method, CRC Press. Liu G. R., K. Y. Dai, K. M. Lim and Y. T. Gu 2002 A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures, Computational Mechanics 29 (6) pp. 510-519 Liu Y. and K. W. Wang 2000 Active-passive hybrid constrained layer for structural damping augmentation, Transactions of the ASME Journal of Vibration and Acoustics 122 pp. 254-262 Mead D. J. and S. Markus 1969 The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, Journal of Sound and Vibration 10 pp.163-175 Mechel F. P. 2001 Panel absorber, Journal of Sound and Vibration 248(1) pp.43-70 Mitchell J. A. and J. N. Reddy 1995 A refined hybrid plate theory for composite laminates with piezoelectric laminate, International Journal of Solids and Structures 32 pp. 2345-2367 Mooney R. L. 1945 An exact theoretical treatment of reflection-reducing optical coating, Journal of the Optical Society of America 35 pp. 574-583 References 290 Moffett M. B., J. M. Powers, J. C. McGrath 1986 A ρc hydrophone, Journal of the Acoustical Society of America 80 (2) pp 375-381 Nayfeh A. H. and D. E. Chimenti 1988a Ultrasonic reflection from liquid-coupled orthotropic plates with application to fibrous composite, Journal of Applied Mechanics 55 pp. 863-870 Nayfeh A. H. and D. E. Chimenti 1988b Propagation of guided waves in fluid-coupled plates of fiber-reinforced composite, Journal of the Acoustical Society of America 83 pp. 1736-1743 Nayfeh A. H. and T. W. Taylor 1988 Surface wave characteristics of fluid-loaded multilayered media, Journal of the Acoustical Society of America 84 pp. 21872191 Nayfeh A. H. 1995 Wave propagation in layered anisotropic media with applications to composites, North-Holland Series in Applied Mathematics and Mechanics. Elsevier Nokes D. S. and F. C. Nelson 1968 Constrained layer damping with partial coverage. Shock and Vibration Bulletin 38 pp. 5-10 Olson H. F. 1956. Electronic control of mechanical noise, vibration reverberations, The Journal of the Acoustical Society of America 28(5) pp. 966-972 Photiadis D. M., J. A. Bucaro and R. D. Corsaro 1994 Double layer actuator, Journal of the Acoustical Society of America 96(3) pp. 1613-1619 Plantier G., C. Guigou, J. Nicolas, J. B. Piaud and F. Charette 1995 Variational analysis of a thin finite beam excitation with a single asymmetric piezoelectric actuator including bonding layer and dynamical effects, Acta Acustica pp. 135151 References 291 Plunkett R. and C. T. Lee 1970 Length optimization for constrained viscoelastic layer damping, Journal of the American Society of Acoustics 48 pp. 150-161 Ramakrishnan J. V. and L. R. Koval 1987 A finite element model for sound transmission through laminated composite plates, Journal of Sound and Vibration 112 pp. 433-446 Rao D. K. 1978 Frequency and loss factors of sandwich beams under various boundary conditions, International Journal of Mechanical Engineering Science 20 pp. 271278 Rao K. S. and B. C. Nakra 1973 Theory of vibratory bending of unsymmetrical sandwich plates, Archives of Mechanics 25 pp. 213-225 Rao S. S. and M. Sunar 1993 Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures, AIAA Journal 31(7) pp. 1280-1286 Rao S. S. and M. Sunar 1994 Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey, ASME Journal of Applied Mechanics Review 47(4) pp.113-121 Reddy J. N. 1997 Mechanics of laminated composite plates: theory and analysis, Boca Raton: CRC Press Reddy J. N. 1999 On laminated composite plates with integrated sensors and actuators, Engineering Structures 21 (7) pp.568-593 Ross D., Ungar, Eric and Kerwin E. M. Jr. 1959 Damping of plate flexural vibrations by means of visoelastic lamine, Structural damping edited by Ruzicka J. E., American society of mechanical Engineer, New York Roussos L. A., C. R. Powell, F. W. Grosveld and L. R. Koval 1984 Noise transmission characteristics of advanced composite structure materials, Journal of Aircraft 21 pp. 528-535 References 292 Ruppel T. H. and F. D. Shields 1993 Cancellation of air-borne acoustic plane waves obliquely incident upon a planar phased array of active surface elements, Journal of the Acoustical Society of America 93(4) pp. 1970-1977 Saravanos, Dimitris A. and Heyliger, Paul R. 1995 Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators, Journal of Intelligent Material Systems and Structures pp. 350-363 Saravanos, Dimitris A., Heyliger, Paul R., Hopkins, Dale A. 1997 Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates, International Journal of Solids and Structures 34(3) pp. 359-378 Scandrett C. L., Y.S. Shin, K. C. Hung, M. S. Khan, C. C. Lilian 2004 Cancellation techniques in underwater scattering of acoustic signals Journal of Sound and Vibration 272 pp. 513–537 Shen I. Y. 1994 Bending-vibration control of composite and isotropic plates through intelligent constrained-layer treatments, Smart Materials and Structures pp. 5970 Shen M. H. H. 1994 Analysis of beams containing piezoelectric sensors and actuators, Smart Materials and Structures pp. 439-447 Shen M. H. H. 1995 A new modeling technique for piezoelectrically actuated beams, Computers and Structures 57 pp. 361–366 Shields F. D. and L. D. Lafleur 1997 Smart acoustically active surfaces, Journal of the Acoustical Society of America 102(3) pp. 1559-1566 Skelton E. A. and J. H. James 1992 Acoustic of anisotropic planar layered media, Journal of Sound and Vibration 152 pp. 154-174 Swallow W. 1939 An improved method of damping panel vibrations, British Patent Specification 513 pp. 171 References 293 Tani J. and G. R. Liu 1993 SH surface waves in functionally gradient piezoelectric plates, JSME International Journal 36 pp. 152-155 Tiersten H. F. 1969 Linear piezoelectric plate vibrations, Plenum, New York Ting R. Y., T. R. Howarth and R. L. Gentilman 1996 Underwater evaluation of piezocomposite panels as active surfaces, Proceedings of SPIE – The International Society for Optical Engineering (Smart Structures and Materials 1996—Industrial and commercial Applications of Smart Structures Technologies 27-29 Feb, 1996 San Diego, California), 2721, pp. 214-221 Torvik, P. J. (ed). 1980 The analysis and design of constrained layer damping treatments, damping applications for vibration control, presented at the winter annual meeting of the American Society of Mechanical Engineers, Chicago, Illinois, November 16-21, 1980 / sponsored by the Shock and Vibrations Committee, the Applied Mechanics Division, ASME 38, pp. 85- 112 Tzou H. S. and M. Gadre 1989 Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls, Journal of Sound and Vibration 132 pp. 433–450 Van Nostrand W. C. and D. J. Inman 1994 Finite element model for active constrained layer damping, SPIE North American Conference on Smart Structures and Materials 2427 pp. 124-139 Waas G. 1972 Linear two-dimensional analysis of soil dynamics problems in semiinfinite layer media, Ph.D. Thesis, University of California, Berkeley, California Yan M. J. and E. H. Dowell 1972 Governing equations of vibrating constrained-layer damping sandwich plates and beams, Journal of Applied Mechanics 94 pp.10411047 References 294 Yang S. K. and S. Y. Hsia 2000 Chiral composites as underwater acoustic attenuators, IEEE Journal of Oceanic Engineering 25 (1) pp. 139-145 Yin T. P., T. J. Kelly and J. E. Barry 1967 A quantitative evaluation of constrainedlayer damping, ASME Journal of Engineering for Industry 89 pp. 773-784 Yu Y. Y. 1960 Forced flexural vibrations of sandwich plates in plane strain, Journal of Applied Mechanics 27 pp. 535-540 Zheng H. 1992 A study on the sound transmission loss characteristics and vibratory damping properties of elastic-viscoelastic layered damping panels, Ph D Thesis, Shanghai Jiaotong University Zhou Y. L. 1999 Finite element models of piezoelectric composite laminates. A Thesis for the degree of master of engineering, National University of Singapore Appendix The following is a list of publications originating from thesis work Journal papers Cai C., G. R. Liu, K. Y. Lam 2000 An exact method for analysing sound reflection and transmission by anisotropic laminates submerged in fluids, Applied Acoustics 61 pp. 95 -109 G. R. Liu, C. Cai and K. Y. Lam 2000 Sound reflection and transmission of compliant plate-like structures by a plane sound wave excitation, Journal of Sound and Vibration 230(4) pp. 809-824 C. Cai, G. R. Liu and K. Y. Lam 2001 A technique for modelling multiple piezoelectric layers, Smart Material and Structures 10 pp. 689-694 C. Cai, G. R. Liu and K. Y. Lam 2001 A transfer matrix approach for acoustic analysis of a multilayered active acoustic coating, Journal of Sound and Vibration 248(1) pp. 71-89 M S Khan, C. Cai, K C Hung and Vijay K Varadan 2002 Active control of sound around a fluid-loaded plate using multiple piezoelectric elements, Smart Material and Structures 11 pp. 346–354 C. Cai, K. C. Hung and G. R. Liu 2003 A survey of recent developments in numerical modelling of piezoelectric enabled structures for vibration control, Research Trends: Current Topics in Acoustical Research pp. 127-136 Appendix 296 H. Zheng and C. Cai 2004 Minimization of sound radiation from baffled beams through optimization of partial constrained layer damping treatment, Applied Acoustics 65 (5) pp. 501-520 H. Zheng, C. Cai, G. S. H. Pau and G. R. Liu 2004 Minimizing vibration response of cylindrical shells through layout optimization of passive constrained layer damping treatments, in press, Journal of Sound & Vibration C. Cai, H. Zheng and G. R. Liu 2004 Vibration analysis of a beam with PCLD patch, in press, Applied Acoustics Conference papers G. R. Liu, C. Cai, X. M. Zhang and K. Y. Lam 1999 Effects of piezocomposite coating on sound radiation and scattering by a submerged cylindrical structure, 70th shock and Vibration Symposium Nov. 15-19, 1999, USA (Albuquerque, NM) G. R. Liu, C. Cai, and K. Y. Lam 2000 Application of piezoelectric layers for active underwater acoustic control, Separated Paper of ICAPV2000 Proceedings, June 19-22, 2000, Xi’an China X. M. Zhang, C. Cai, G. R. Liu and K. Y. Lam 2000 Acoustical control of sound radiation and scattering from plate-like structures with piezoelectric sensors and Actuators, Proceedings of the 29th International Congress on Noise Control Engineering, 28-30 August 2000, Nice, France, 2000 H. Zheng, C. Cai, G. R. Liu and K. Y. Lam 2000 FEM/BEM Simulation of Vibroacoustics of a Fluid-loaded, Stiffened Plate Under Combined Force-moment Excitation, ASME DAS’2000 Appendix 297 C. Cai, G. R. Liu, H. Zheng and K. Y. Lam 2000 A Study on Boundary Integral Equation for Acoustic Radiation from Axisymmetric Bodies, ASME DAS’2000 G. R. Liu, C. Cai and V. K. Varadan 2000 Wave propagation in composite laminates with piezoelectric layers and related inverse problems, SPIE’s 2000 Symposium on Smart Materials and MEMS, 13-15 Dec. 2000, Melbourne, Australia. (Invited paper) Zheng H, Cai C, K. C. Hung, K. L. Lee & C. A. Teh 2001 Simulation Based Design for Naval Vessels with Low Acoustic Signature, May 09, NPTS 2001 Liu G R, C. Cai and K. Y. Lam 2001 Computational simulation of active piezoelastic coating with sensors and actuators applied in underwater acoustics, Proceedings of the SPIE: Smart Structures & Devices, 13-15 December 2000, Melbourne, Australia, edited by D K Sood, R A Lawes & V V Varadan, pp. 48-62. SPIE, 2001 M. S. Khan, C. Cai and K. C. Hung 2002 Modeling of acoustics field and active structural acoustic control in ANSYS, 2002 Ansys User’s Conference and Exhibition, April 22-24, 2002, Pittsburgh, Pennsylvania U.S.A. (Best of Session Technical Paper Award) C. Cai, H. Zheng, Md. Shahiduzzaman Khan and K. C. Hung 2002 Modeling of Material Damping Properties in ANSYS, Ansys User’s Conference and Exhibition, April 22-24, 2002, Pittsburgh, Pennsylvania U.S.A. G. R. Liu, C. Cai, K. Y. Lam and V. K. Varadan 2002 A Review of Simulation Methods for Smart Structures with Piezoelectric Materials. IUTAM Symposium on Dynamics of Advanced Materials and Smart Structures, May 20– 24, 2002, Japan (Keynote lecture) Appendix 298 C. Cai, K. C. Hung, H. Zheng and M. S. Khan 2002 Damping Considerations in Modal Analysis, 4th ASEAN ANSYS User conference 2002 C. Cai, H. Zheng, K. C. Hung and G. R. Liu 2003 An Improved Approach for Vibration Anslysis of Beam with Partial VEM Patch treatment, Internoise 2003 Jeju Internaltional Convention Center, Korea, August 25-28, 2003 [...]... interaction of acoustic waves with fluidloaded passive compliant structures; 2) interaction of acoustic waves with fluid-loaded piezoelectric enabled compliant structures; 3) vibration analysis of the beam with a partial covered PCLD patch; and 4) vibration analysis of the beam with a partial covered ACLD patch 1.2 Literature review In order to investigate the interaction of acoustic waves and structures. .. the area of in situ fibre optic sensors for measurement of strain and vibration, and for structural health monitoring As one of the smart materials, piezoelectric material can transform mechanical energy into electric energy and vice versa Correspondingly, there are two kinds of piezoelectric effects: the converse and direct piezoelectric effects (Auld 1973) The former allows one to use the piezoelectric. .. the piezoelectric enabled structures One uses distributed piezoelectric devices that cover or embed the entire structure (laminated-type smart structures) The other uses discrete piezoelectric devices that occupy a relatively small area of structures (discrete-type smart structures) The modeling approaches and analysis techniques differ considerably between the laminated-type and discrete type smart structures. .. theory for sandwich plates and solved for generally forced vibration in case of plane-strain Yin et al (1967) evaluated CLD quantitatively based on experiments Most earlier theoretical work on sandwich beams with viscoelastic cores can be traced to DiTaranto (1965) and Mead and Markus (1969) for the axial and bending vibrations of beams resulting in a sixth-order equation of motion Their analysis was... Transmission loss and reflection coefficients of a sandwich plate for case 3 in Table 3-1 84 Figure 4 - 1 A bi-layer piezoelectric enabled plate 115 Figure 4 - 2 L An elastic layer with end impedances IM e and IMU e 115 Figure 4 - 3 A piezoelectric layer with end impedances IM L and IMU p p 115 Figure 4 - 4 Sound reflection with different incidence angles at p in = 1 Pa and ϕ =0 116 Figure 4 - 5 Sound radiation... of a beam section 266 Figure 8 - 2 Schematics of two types of beam with a partial ACLD patch 266 Figure 8 - 3 Frequency response of a simply-supported beam with “soft” VEM and ϕ0 = 0 (x1=0.19 m, x2=0.21 m) 267 xix Figure 8 - 4 Frequency response of a simply-supported beam with “soft” VEM and ϕ0 = 200+ j 200 Vol (x1=0.19 m, x2=0.21 m) 267 Figure 8 - 5 Frequency response of a cantilever beam with “soft”... row i and column j cCE Stiffness matrix of piezoelectric material at constant electric field strength cw1, cw2 Sound speeds in lower and upper fluids D Electric displacement vector Dx, Dy, Dz Components of electric displacement in x, y and z directions d Piezoelectric strain matrix Ep Electric field vector Epx, Epy, Epz Components of electric field in x, y and z directions Ex, Ey, Ez Moduli of elasticity... Wave number components in x and z directions k f1 , k f 2 Wave number vector in lower and upper fluids kxf1, kzf1 Components of kf1 in x and z directions xi kxf2, kzf2 Components of kf2 in x and z directions respectively N Number of layers p in Amplitude of incident wave p re Amplitude of reflected wave p tr Amplitude of transmitted wave && q Volume acceleration per unit length of linear array && Q Volume... discussed the use of single and double layers of piezoelectric material to form acoustically active surfaces for the elimination of reflected and transmitted waves from a theoretical standpoint Lafleur’s paper was declared to be the first to give out the basic equations relating the reflection or transmission coefficients of a layer of piezoelectric material to the driving voltage applied across, and furthermore,... direction of piezoelectric 57 Figure 3 - 1 A sketch of physical model of a multilayered anisotropic laminate in fluids 75 Figure 3 - 2 Geometric schematic of bi-layer plates 75 Figure 3 - 3 Geometric schematic of a sandwich plate 75 Figure 3 - 4 Transmission loss and reflection coefficients of the plates with a coating above when thickness of the coating changes 76 Figure 3 - 5 Transmission loss and reflection . SOUND AND VIBRATION ANALYSIS OF COMPLIANT PIEZOELECTRIC ENABLED STRUCTURES BY CAI CHAO (B. Eng. JU, M. Eng. JU and NUS) DEPARTMENT OF MECHANICAL ENGINEERING. imperative that the accurate prediction of the interactions of waves with fluid-loaded passive or piezoelectric- enabled structures for suppression of sound reflection and transmission. The work in the. Distribution of displacement and traction in piezoelectric layer 107 4.5 Application examples 109 4.5.1 Sound reflections and radiation 109 4.5.2 Effects of the thickness ratios on active sound control