Vibration Damping of Structural Elements Vibration Dalllping of Structural Elelllents C.T.Sun Y.P.Lu Prentice Hall PTR Library of Congress Cataloging-in-Publication Data Sun, C T (Chang-Tsan) Vibration damping of structural elements / C T Sun, Y P Lu p em Includes bibliographical references and index ISBN 0-13-079229-2 Structural dynamics Damping (Mechanics) I Lu, Y P (Yeh-Pei) II Title TA654.S831995 624.1'76-dc20 94-45861 CIP Acquisitions editor: Bernard Goodwin Cover designer: DEFRANCO, Inc Manufacturing buyer: Alexis R Heydt Compositor/Production services: Pine Tree Composition, Inc © 1995 by Prentice Hall PTR Prentice-Hall, Inc A Pearson Education Company Upper Saddle River, NJ 07458 All rights reserved No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Printed in the United States of America 10 ISBN: 0-13-079229-2 Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Pearson Education Asia Pte Ltd., Singapore Editoria Prentice-Hall Brasil, Ltda., Rio De Janeiro Contents Chapter1 Preface ix Fundamentalsof VibrationDamping 1.1 Introduction 1.2 Scope of the Book 1.3 Classification of Damping 1.4 Characterization of Viscoelastic Materials 10 1.5 Effects of Environmental Factors 1.6 Fundamentals of Damping Material Properties 20 1.7 Fundamentals of Vibration Control Techniques 1.8 Summary References Chapter2 2.3 29 37 37 Vibrationsof DiscreteSystems 2.1 Introduction 2.2 24 Free Vibrations of Single-Degreeof-Freedom Systems with Viscous Damping Forced Vibrations of Single-Degreeof-Freedom Systems with Damping v 40 40 45 48 vi CONTENTS 2.4 2.5 2.6 2.7 2.8 2.9 Chapter3 83 Vibrations of Multi-Degree-of-Freedom Systems with Damping 89 Summary References Dampingof Fiber-Reinforced Composite Materials 3.1 Introduction 3.2 3.3 3.4 Chapter4 Transient Vibrations of Single-Degreeof-Freedom Systems with Damping Free Vibrations of Two-Degree-of-Freedom Systems with Viscous Damping Forced Vibrations of Two-Degreeof-Freedom Systems with Damping Transient Vibrations of Two-Degreeof-Freedom Systems with Damping Damping of Aligned Discontinued Fiber Composites Damping of Aligned Off-Axis Discontinuous Fiber Composites Damping of Randomly Oriented ShortFiber Composites 54 66 77 102 104 105 105 106 120 128 3.5 3.6 Damping of Laminated Composites The Influence of Fiber-Matrix Interface on Damping 137 3.7 Summary References 156 156 Vibrations of Constrained Damped BeamStructures 4.1 Introduction 4.2 Vibrations of Three-Layered Damped Beam Structures 4.3 Other Vibration Analyses of ThreeLayered Damped Beam Structures 4.4 Vibration Characteristics of Constrained Three-Layered Damped Beam Structures 146 160 160 161 174 186 vii CONTENTS 4.5 4.6 Chapter Vibrations of Multilayered Damped Beam Structures 192 Summary References 193 194 Vibrations of Constrained Damped Plate Structures 5.1 Introduction 5.2 Vibrations of Three-Layered Damped Plate Structures 5.3 Vibration Solutions Using A General Purpose Finite Element Computer Program 5.4 Vibration Characteristics of Constrained Three-Layered Damped Plate Structures 5.5 Summary References Chapter Vibrations of Constrained Damped Ring Structures 6.1 6.2 6.3 6.4 6.5 6.6 Chapter Introduction Vibrations of Discontinuously Constrained Damped Ring Structures Vibrations of Continuously Constrained Damped Ring Structures Vibration Solutions Using General Purpose Finite Element Computer Program Continuously Versus Discontinuously Constrained Damped Ring Structures Summary References Vibrations of Constrained Damped Cylindrical Shell Structures 7.1 Introduction 7.2 Effectiveness of Damping Treatments of Constrained Beam-Damped Cylindrical Shell Structures 198 198 199 210 226 236 236 239 239 239 252 272 276 279 279 281 281 282 viii CONTENTS 7.3 Vibrations of Constrained Beam-Damped Cylindrical Shell Structures 284 7.4 Vibration Characteristics of Beam-Damped Cylindrical Shell Structures 293 7.5 Vibrations of Constrained Beam-Damped Titanium Cylindrical Shells 295 7.6 Vibrations Structures 7.7 Vibrations Structures 7.8 Summary of Constrained Damped Shell with Attached Mass Segments of Constrained Damped Shell with Curved Elements References Chapter Continuous and Discontinuous Constrained Viscoelastic Material on Structural Elements Using Finite Element Method 8.1 Introduction 8.2 Finite Element Formulation of Structures with Constrained Viscoelastic Damping Layer 8.3 Derivation of Elastic Stiffness Matrix of Plate Structures 8.4 Derivation of Elastic Stiffness Matrix of Beam Structures 303 312 313 316 318 318 321 327 332 8.5 Derivation of Mass Matrix 8.6 Evaluation of Loss Factors 8.7 Constrained Viscoelastic Damping Materials on Structural Elements With Prestress Effects 8.8 Summary References 334 336 349 362 362 Author Index 367 Subject Index 369 Preface Er years, vibration damping as a technology has been well received by various industries, however, very few books exist on this subject In 1985, Nashif, Jones, and Henderson published an excellent book entitled Vibration Damping which provides practical and detailed information on the research and development work ranging from the fundamentals of vibration damping to design aspects in many practical applications in this subject area Our intent is that the contents of this book be considered as a continuation of and complementary to the above work rather than competitive in the technical subject of vibration damping This book is intended as a reference book for aerospace, mechanical, civil, and acoustical engineers It should also serve as a valuable reference work for graduate students, professors, and researchers in the area of Aerospace Engineering, Mechanical Engineering, and Engineering Mechanics This book consists of eight chapters The first three chapters address the fundamentals of vibration damping, properties of viscoelastic damping materials, vibrations of discrete damped systems, and damping of fiber-reinforced composite materials Chapters four to seven present vibrations of damped structures for beams, plates, rings, and shells In chapter eight, a finite element numerical method is presented to solve vibration problems of beam and plate structures with a partialiy attached damping treatment on the surface of the structures The effect of initial loading is also included ix X PREFACE Originally, a chapter to address the vibrations of damped cylindrical shells with curved elements was planned A series of outstanding papers on this specific subject were published by Dr Michael El-Raheb and his colleague Mr Paul Wagner However, the research work was performed for their specific purposes and no numerical solutions on the vibrations of shell structures without incorporating the enclosed fluid medium was available We regret that these materials are unable to be included in this book Instead, we have added paragraphs pertaining to this subject topic at the end of Chapter which deals with vibrations of constrained damped cylindrical shell structures It should be pointed out that this book deals with vibration damping characteristics of structures or systems employing damping materials We use the properties of the viscoelastic materials in the vibration analysis, but the detailed analysis regarding their material behaviors will not be our primary concern The book addresses the vibration damping of structural elements, and is not a materials oriented book This book emphasizes analyses in the presentation of damped structural systems, their validations and verifications This is done because the authors feel that analyses are the tools which not only enable us to better understand the complicated physical phenomena, but also can help calculate the physical quantities which are useful in practical applications One might be critical of the fact that there are not enough tables and figures which may be used directly and readily for design purposes Our reply is that much of this information for damped beam structures may be found in the book by Nashif, Jones, and Henderson Additionally, because information pertinent to damped structures other than beams may not be available and because the vibration characteristics of damped structures depend strongly on the realistic (not assumed) properties of the damping materials employed as well as the geometrical parameters of the structures considered, we strongly believe that the presentation of "design data" should be reduced to· a minimum unless the geometrical and particularly the damping material parameters of a given damped structural system are specified Since it is technically difficult to develop and to manufacture viscoelastic materials, the damping material properties can not be assumed for materials not commercially available Though vibration damping technology is multi-disciplinary, the researchers and practitioners have, however, formed a rather close-knit community In preparing this book, we would be remiss if we did not acknowledge the fact that Dr Lynn Rogers, currently a vibration damping consultant and formerly with the Flight Dynamics Laboratory of the Air PREFACE xi Force Wright Aeronautical Laboratories, has for years not only devoted his efforts in promoting vibration technology tirelessly, but also has organized meetings and workshops on a regular basis to provide a forum for the exchange and dissimulation of the latest state-of-the-art technology His professional and enthusiastic efforts in advancing vibration damping technology certainly deserve our recognition We are very happy to take advantage of this opportunity to acknowledge the support and encouragement provided by the managers and individuals of our respective organizations in the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida and at the Carderock Division, and the Naval Surface Warfare Center (formerly David Taylor Research Center) Particularly, we would like to thank Dr J M Bai at the University of Florida, Dr Bruce Douglas, Director of Research, and Mr A J Roscoe, at the Carderock Division for their assistance, support, and invaluable comments Finally, the support and cooperation from the staff of Prentice Hall Publication Company, especially our editor Mr Michael Hays in all phases of the production process are also acknowledged C T Sun, Gainesville, Florida Y P Lu, Annapolis, Maryland The geometric stiffness matrix [KG] can be evaluated by using the Gaussian quadrature scheme, as in the case of the elastic stiffness matrix [K] 8.7.3 Numerical Examples2 As in the numerical examples of damped beams without prestress given in section 8.6.3, the base structure and the constraining layer are modeled by using the three-node shear deformation offset beam element, and the viscoelastic core is modeled by a six-node plane solid element The exact same thickness and aspect ratio for elements have been used (i.e., aspect ratios up to 200:1) To validate this formulation just presented, several calculations are made to determine natur~l frequencies and tip displacements of simple systems from which closed-form solutions are easily derived The loss factor is evaluated by using the direct frequency response technique and the modal strain energy method In the direct frequency response method a forced vibration at a given frequency is considered Displacements are obtained by solving a system of complex-valued linear equations This technique is used only to verify the results from the modal strain energy method Three different unidirectional glass-epoxy composite beams are tested, and the average measured loss factor is used as input data to the finite element model The same specimens are tested after application of damping tape Each specimen is made of 50 percent fiber volume fraction and is 203 mm long and 3.6 mm thick (total) The viscoelastic core is 0.127 mm thick and the aluminum constraining layer is 1.524 mm The width of each specimen is 25.4 mm The damping tape used is SJ2052X, a product commercially available from 3M Company Figures 8.20 through 8.22 show the effect of preload on damping 2The material in this section is taken from Rao, Sankar, and Sun (1992) and frequency for different lengths of damping tape In all cases the damping tape is applied starting from the fixed end of the beam Frequency dependent properties of the damping material at room temperature are used in the calculations The specimen used is a five-ply graphite epoxy beam of stacking sequence [90/90/0/90/90], length 0.2032 m, width 0.01905 m, and thickness 0.00064 m For the static part of the loading, the beam is fixed with an axial load acting at the free end; for the dynamic part of the transverse loading acting at the center span, the beam is fixed at both ends In all the cases damping decreases with increasing preload for each of the first three modes, but the stiffness of the structure has increased at the same time Decreased loss factor, however, does not necessarily result in increased vibration amplitude (see Fig 8.23) since dynamic displacement depends on both stiffness and damping of the structural system Fig 8.24 shows the results of loss factor and frequency of full taped and bare beams for different preloads compared with experimental re- Fig 8.24 Variation ofloss factor and frequency with preload (100 percent taped from the root) (From Rao, Sankar, and Sun, Constrained layer damping of initially stressed composite beams using finite elements, Journal of Composite Materials, 26(12), 1752-1766, 1992 Reprinted with permission ofTechnomic Publishing Co., Inc.) sults of Mantena (1989) At the preload of 45 N, the analytical and experimental first mode frequencies significantly differ At other preloads the results of both frequency and loss factor are in good agreement before any failure is initiated Failure is believed to be initiated at the point at which the loss factor shows an increase with increasing preload in the experimental result At high preloads, the difference between finite element results and experimentally measured data is due to the problems associated with the measurement of small levels of damping 362 8.8 C HAP T E R ContinuousandDiscontinuousConstrainedMaterials SUMMARY This chapter presents a finite element analysis formulation using offset technique for continuous and discontinuous constrained viscoelastic material on either monolithic or laminated fiber-reinforced composite base structural elements An efficient modification of the modal strain energy approach for the estimation of damping ability about prestressed structural configurations is also given Based upon all the numerical examples for damped beams with or without prestress effects, it can be concluded that the present finite element technique presented is capable of evaluating the loss factor, maximum displacement, and natural frequencies of the damped structural systems Results suggest that application of the damping material to the points of high strain energy density in the structure would yield best results This result can be seen by comparing the effect of different lengths of damping tape on loss factors of the three modes Stress stiffening can have a significant effect on the location of damping material as it redistributes the strain energy distribution, and should be considered in the design of a structure for maximum damping by using constrained layer damping treatments The effects of prestress on structural stiffness, natural frequencies, and loss factor are significant Since the geometric stiffness matrix does not dissipate energy, it will strengthen the system and decrease the loss factor However, the maximum displacement depends on both the stiffness as well as damping Therefore, the trend of variation of the maximum displacement as the preload increases (or decreases) is not as clear as the natural frequencies and the loss factor of the structure References Agarwal, B.D., and Broutman, L.J (1990) Analysis and performance of fiber composites (2nd 00.) New York: Wiley Interscience Barlow, J (1989) More on optical stress points-reduced element distortions and error estimate International Numerical Methods in Engineering, 28, 1487-1504 integration, Journal for Bishop, R.E.D (1955) The treatment of damping forces in vibrating theory Journal of the Royal Aeronautical Society, 59, 738-742 Bishop, R.E.D., and Johnson, D.C (1960) The mechanics of vibration London: Cambridge University Press REFERENCES 363 Brockman, R.A (1984, November) On vibration damping analysis using the finite element method Vibration damping 1984 workshop proceedings, AFWAL-TR-84-3064 (pp 111-1110) DiTaranto, R.A (1965, December) Theory of vibratory bending for elastic and viscoelastic layered finite length beams Journal of Applied Mechanics, 87,881-886 Hughes, T.R.J., Cohen, M., and Haroun, M (1978) Reduced and selective integration technique in the finite element analysis of plates Numerical Engineering and Design, 46, 203-222 Johnson, C.D., and Kienholz, D.A (1984, November) Finite element design of viscoelastically damped structures-methods Vibration damping 1984 workshop proceedings, AFWAL-TR-84-3064 (p FF-1) Johnson, C.D., Kienholz, D.A, Austin, E.M., and Schneider, M.E (1984, November) Finite element design of viscoelastic ally damped structures Vibration damping 1984 workshop proceedings, AFWAL-TR-84-3064 (pp HH1-HH28) Kerwin, E.M Jr (1959) Damping of flexural waves by a constrained viscoelastic layer Journal of the Acoustical Society of America, 31(7), 952-962 Kerwin, E.M Jr., and McQuillan, R.J (1960, June) Plate damping by a constrained viscoelastic layer: Partial coverage and boundary effects Report No 760 Bolt, Beranek and Newman, Inc Kienholz, D.A, Johnson, C.D., Austin, E.M., and Schneider, M.E (1984, November) Finite element design of viscoelastically damped structures-applications Vibration damping 1984 workshop proceedings, AFWAL-TR-84-3064 (p 00-1) Kimball, AL (1932) Vibration prevention in engineering New York: John Wiley & Sons, Inc Kimball, AL., and Lowell, D.E (1927) Internal Physics Review, 30(6), 954-959 friction in solids Kluesener, M.F (1!)84, November) Results of finite element analysis of damped structures Vibration damping 1984 workshop proceedings, AFWAL-TR-84-3064 (pp JJ1 JJ12) Lazan, B.J., Metherell, AF., and Sokal, G (1965, September) Multiple- 364 C HAP T E R band surface treatments OH ContinuousandDiscontinuousConstrainedMaterials for high damping AFML-TR-65-269, Dayton, Malvern, L.M (1969) Introduction to the mechanics of a continuous medium Englewood Cliffs, NJ: Prentice Hall, Inc Mantena, P.R (1989, August) Vibration control of composite structural elements with constrained layer damping treatments Ph.D dissertation, Mechanical Engineering Department, University of Idaho, Moscow, ID Mead, D.J., and Markus, S (1969) The forced vibration of three-layer damped sandwich beams with arbitrary boundary conditions Journal of Sound and Vibration, 10(2), 163-175 Mead, D.J., and Markus, S (1970) Loss factors and resonant frequencies of encastre damped sandwich beams Journal of Sound and Vibration, 12(1), 99-112 Mycklestad, N.O (1952) The concept of complex damping Journal of Applied Mechanics, 19, 284-288 Parfitt, G.G (1962, August) The effect of cuts in damping tapes 4th International Congress on Acoustics (pp 21-28), Copenhagen, Denmark Plunkett, R., and Lee, C.T (1969, July) Length optimization for constrained viscoelastic layer damping Technical Report AFML-TR-68-376, Dayton, OH Pugh, E.D.L., Hinton, E., and Zienkiewicz, O.C (1978) A study of quadrilateral plate bending elements with reduced integration International Journal for Numerical Methods in Engineering, 12(7), 1059-1079 Rao, D.K (1978) Frequency and loss factors of sandwich beams under various boundary conditions Journal of Mechanical Engineering Science, 20(5), 271-282 Rao, V.S (1991, December) Finite element analysis of viscoelastically damped structures Ph.D dissertation, Department of Aerospace Engineering, Mechanics & Engineering Science, University of Florida, Gainesville, FL Rao, V.S., Sankar, B.V., and Sun, C.T (1992) Constrained layer damping of initially stressed composite beams using finite elements Journal of Composite Materials, 26(12), 1752-1766 365 REFERENCES Rogers, L.C., Johnson, C.D., and Keinholz, D.A (1980, October) The modal strain energy finite element analysis method and its application to damped laminated beams 51st Shock and Vibration Symposium, San Diego, CA, October 21-23,1980 Snowdon, J.C (1968) Vibration and shock in damped tems New York: John Wiley & Sons, Inc mechanical sys- Soni, M.L (1980) Finite element analysis of vibration of initially stressed viscoelastic structures University of Dayton Report, UDR-TR80-107, Dayton, OH Soni, M.L (1981) Finite element analysis of viscoelastically damped sandwich structures Shock and Vibration Bulletin, 51(1), 97-109 Soni, M.L., and Bogner, F.K (1982) Finite element vibration analysis of damped structures AIAA Journal, 20(5), 700-707 Sun, C.T., Sankar, B.V., and Rao, V.S (1990) Damping and vibration control of unidirectional composite laminates using add-on viscoelastic materials Journal of Sound and Vibration, 139(2),277-287 Ungar, E.E (1962) Loss factors of viscoelastically damped beam structures Journal of the Acoustical Society of America, 34(8), 1082-1089 Ungar, E.E., and Kerwin, E.M Jr (1962) Loss factors of viscoelastic systems in terms of energy concepts Journal of the Acoustical Society of America, 34(7), 954-957 Zienkiewicz, O.C., Taylor, R.L., and Too, J.M (1971) Reduced integration technique in general analysis of plates and shells International Journal for Numerical Methods in Engineering, 5, 275-290 Author Index AbduIhadi, F., 214, 236 Adkin, R.L., 31, 38 Agarwal, R.D., 127, 156,327,362 Agbasiere, J A., 193, 194 Arnold, R.N., 292, 316 Asnani, N.T., 193, 194 Austin, E.M., 321, 363 Earles, S.W.E.,7, 38 El-Raheb, M., 313, 316 Everstine, G.C., 186, 195, 198,237,273, 280 Fliigge, W., 240, 241, 279, 284, 316 Fung, Y.C., 106, 111, 112, 157 Babington, W., Bake~R., 108, 157 Barlow,J., 331, 362 Barry, J.E., 198,238 Bishop,R.E.D., 85, 104,336,362 Blasingame, W., 188, 194,253, 280 Bogner F.K 213 214 238 321 341 365 ' , , , " Gibson, R.F., 106, 115, 117, 120, 124, 128, 137,138,157,158,159 Grootenhuis, P., 193, 194 Hammer, AN., 9, 37 Haroun, M., 331, 363 Henderson, J.P., 3, 34, 38, 50, 161, 195 Henrici, P., 267, 279 Bouchard, M.P., 213, 214, 236 Bowie,G.E., 9, 37 Brockman, R.A, 213, 363 Broutman, L.J., 127, 156,327,362 Hinton, E., 331, 364 Hughes, T.R.J., 331, 363 Hunt, F.V., 25, 38 lremonger, M.J., 108, 150, 158 Carrara, A.S., 108, 157 Chan, H.C:, 211, 236 Chaturvedi,S.K.,106,120,146,151,157,158 Johnson, C.D.,85, 104, 186, 194, 198,211, 213,214,236,276,279,321,336,362, 363,365 Chen, G.S., 314, 316 Cheung, Y.K.,211, 236 Chon, C.T., 117, 157 Christensen, R.M., 128, 157 Clemens, J.C., 203, 237 Cohen, M., 331, 363 Cox,H.L., 107, 109, 121, 157 Cramer, W.S., 219, 236, 306, 316 Jones, D.I.G.,3, 31, 34, 38, 50, 161, 193, 194, 195 Jones, W.J., 314, 316 Kelly,A, 107, 158 Kelly,T.J., 198,238 Kerwin, E.M Jr., 36, 38,161,175,176, 196, 198,213,236,237,238,319,320, 337,363,365 Kienholz,DA, 186, 194, 198,211,213, 214, 276, 279, 321, 363, 365 Killian, J.W., 198,222,237,276,279 Kimball,AL.,336,363 HJuesener,M.F., 321,339,363 Kulkarni, S.V., 119, 158 Davies, G.T., 108, 159 Den Hartog, J.P., 31, 38 DiTaranto, RA, 161,176, 180, 188, 194, 198,236,246,248,250,251,255,267, 271,278,279,288,316,319,363 Dolgin,B.P., 314, 316 Douglas,B.E., 161, 194,237,246,248, 253,279,280,283,295,317 Dow,N.F., 107, 157 Dowell,E.H., 161, 180, 196, 198,238 Drake, M.L., 27, 37, 38, 188, 196,213,214, 236, 296, 316 Lalanne,M., 213,237 Lazan, B.J., 320, 363 Lee, C.T., 114, 158,320, 364 Leissa, A.W.,284, 317 Lowell,D.E., 336, 363 367 368 Lu, Y.P., 186, 194, 195, 198, 203, 214, 219, 222,237,240,246,251,253,260,271, 273,276,278,279,280,283,295,300, 302,317 MacLaughlin, T.F., 108, 157, 158 Malvern, L.M.,364 Mantena, P.R., 361, 364 Markus,S., 161, 167, 175, 177, 180, 188, 189,190,195,314,317,319,364 Mau, S.T., 211, 237 Maurer, G.J., 214, 217, 237 McGarry, F.J., 108, 157 McGraw,J.R Jr., 198,236 McLean, D., 109, 158 McQuillan,RJ., 319, 363 Mead, D.J., 161, 167, 175, 177, 180, 187, 188,189,190,195,198,237,319,364 Meirovich,L., 50,104 Metherell, AF., 320, 363 Milewski,J.V., 117, 119, 158 Miller, H.T., 22, 39 Mycklestad,N.O., 336, 364 Nachman, J.F., 9, 37 Nakra,B.C., 161, 181, 184, 189, 193, 194, 195, 196,203,227,231,237,238 Nashif,AD., 3, 21, 27, 31, 33, 34, 36, 38, 50, 100,104,161,176,188,193,194,195 Neilson, H.C., 295, 317 Newland, D.E., 96, 98 Ormondroyd,J., 27, 31, 38 Paipetis, SA, 146, 158 Papanicolaou, G.C., 146, 147, 158, 159 Parfitt, G.G.,320, 364 Parin, M.L., 193, 194 Paulard, M., 213, 237 Pian, T.H.H., 211, 237 Pipes, R.B., 119, 158 Plunkett, R, 7, 38,114,124,138,157,158, 320,364 ' Pugh, E.D.L., 331, 364 Read, B.W., 109, 158 Rao, DK, 161, 166, 189, 195, 320, 364 Rao, S.S., 85,104 Rao, V.S., 321, 332, 338, 341, 342, 357, 364,365 Rao, Y.V.KS., 181, 184, 189, 198,203, 227,231,237,238 Reed, F.E., 31, 38 AUTHOR INDEX Rogers, L.C., 16, 38, 236, 321, 365 Roscoe,AJ III, 203, 237, 253,280,283, 317 Rosen, B.W.,107, 158 Ross, D., 36, 161, 175, 176, 196 Ruzicka, J.E., 161, 196 Salermo, V.L., 314, 316 Sankar, B.V.,341, 357, 364,365 Scanlan, RH., 65,104 Schlesinger,A, 7, 38 Schneider, M.E., 321, 363 Smith, G.E., 108, 158 Snowdon,J.C., 33, 38, 85,104,170,196, 336,365 Sokal, G., 320, 363 8oni, M.L.,213,214, 238, 321,339,341, 365 8oovere,J., 36, 39,188,196 Spencer, AJ.M., 108, 158 Sun, C.T., 106, 117, 120, 122, 128, 137, 157,158,159,341,357,364,365 Taylor, RL., 331, 365 Theocaris, P.S., 146, 147, 158, 159 Thomas,E.V., 237, 253,280 Thomas, J.M., 69,70, 104 Thomson,W.T.,32, 39, 44, 50, 85, 91, 104 Tong,P.,211,237 Too,J.M., 331, 365 Trompette, P., 213, 237 Tyson, W.R, 107, 108, 158, 159 Tzeng, G.Y.,146, 151, 157 Ungar, E.E., 8,38,39,161,175,176,196, 213,238,320,337,365 Waals, F.M., 128, 157 Wagner, P., 313, 316 Warburton, G.B.,292, 316 Wamaka, G.E., 22, 39 Weissman, G.F., Wood,W.G.,108, 150, 158 Wu, J.K, 122, 158, 159 Yan, M.J., 161, 180, 196, 198,238 Yang, J.C.S., 161, 194 Yau, A., 117, 157 Yin, T.P., 198,238 Young,D., 33, 36, 39 Zienkiewicz,O.C., 331, 364, 365 Zweben,C.H., 119, 158 , Subject Index Acceleration,171 in-plane, 178 transverse, 178 Alloy high damping, Sonoston,8 Amplitude at resonance, 49, 52 complex,52 cyclicstrain, 2, 22 dynamic strain, 23 maximum, 77 of response, 5, 81,348 radial displacement, 267 strain, 18 Anti-resonance,293 fiber-reinforcedcomposites,105 full surface treatment, 318 high alloy, hysteresis, 44, 65, 337 interface, 146 materials, 296 measures of, 24 nonmaterial, 4, nonproportional,2 partial surface treatment, 318 solid,44 specific,24 structural, 4,44, 294 tape, 308 treatment, 295, 313 viscous,4, 45, 48, 52, 65,66,77, 79,83 89,94 viscoelastic,10, 50, 52, 62, 65, 79, 86, 96, 97 viscoelasticshear, Damper (see also Vibration, control) shear tuned, 33 tuned, 32, 33 Dashpot, 4, 12, 40 Designcriteria, 267 Dilatation of viscoelasticmaterial, 11 DiTarantolMead-Markusequation, 181 Domain complex,213 frequency, 17, 18,62,64,86,178 time, 17, 18,62,64,86 Duhammel's integral, 57 Dynamicinteraction, 244 Bandwidth 171, 248 Biharmonicequation, 179 Causality condition,64 Characteristic equation, 68, 71, 74, 83, 259 Complexintegral, 65 Compositematerials aligned discontinued fiber, 2, 106 aligned off-axisdiscontinued fiber, 2, 120 continuousfiber, discontinuousfiber, fiber orientations, 10 laminated, 13, 137 polymermatrix, randomly oriented short-fiber, 3, 13, 128 Constitutive equation, 17 Continuous spectrum, 248 Convolutionintegral, 10, 57 Eigenmode,214 Eigenvector,90, 91 Elastic-viscoelasticcorrespondence principle, 106, 111 Elastomers, Encastre beam, 189, 191 Energy dissipation, 5, 11, 105, 108 kinetic, 200 peak potential, 20 potential, 200 stored, 108 Damping, 106, 120, 128, 137, 139, 146, 294, 302 acoustic radiation, Coulomb,6 critical viscous,25, 46 dry, effectsof, 152, 154 factor, 25, 46, 52 369 370 Energy model, 108, 148 Environment effectsof, 20 Finite element, structural elastic stiffness matrix, 327, 331, 332, 333 general purpose computer program, 184, 198,209,218,272 generalized complexstiffness, 231 generalized mass matrix, 231 generalized stiffness matrix, 231 globalgeometric stiffness matrix, 340, 354,357 hybrid stress element, 211 interpolation function ofbeam, 332 interpolation function of plate, 330 interpolation function of preload, 356 isoparametric element, 325 isoparametric quadrilateral brick element, 185, 219 isoparametric quadrilateral membrane element, 185 MAGNAcode, 321 mass matrix, 334, 335 MSC/NASTRAN,186,214, 276 NAS~, 184, 186,209,214,218,219, 220,222,224,225,226,273,276 off-setbeam element, 322, 349 off-setplate element, 323 prestress effects, 340, 349, 361 quadrilateral plate element, 219 rectangular plate element, 322 serendipity element, 326 shear locking, 331,332 shear deformable element, 322, 323 triangular plate element, 214 UFPACprogram, 321 Fl'iigge'sshell equation, 240, 284 Force arbitrary excitation, 63 Coulombdamping, impulsive, 63 inertia, 165 spring, viscousdamping, Force-balancemodel, 111, 120, 149 Fourier coefficient,289 Fourier inverse transform, 88 Fourier transform, 63, 64 Frequency, 2, 5, 7, 49, 52 circular, 166,241,308 damped beam, 170, 174, 177, 186, 189 SUBJECT INDEX damped cylindrical shell, 283, 291, 292, 293,294,295,297,298,299,300,301, 302, 309, 310, 312 damped plate, 209, 210, 215, 216, 217, 218, 220, 221, 222, 224, 225, 226, 228, 229,230,231 damped ring, 247, 248, 249, 250, 251, 252,259,260,266,267,268,269,270, 271,272,273 damped, 343 damped natural, 70, 90, 98 effectsof, 21 natural, 49,73,74,83,170,175 ofexcitation, 166,171 undamped shell, 293, 294, 295, 310 Function complextransfer, 78 Delta, 55, 242 Dirac, 13 frequency response, 100,217 harmonic response, 95, 100 Heaviside, 13 Rayleigh's energy dissipation, 94 unit impulse response, 86, 88 Gauss-Jordan elimination method, 173, 246,309 Halpin-Tsai equation, 121 Hamilton's principle, 201 Harmonic motion, 19 Hysteresis damping, 44, 65, 337 loop,6, 8, 19, 26, 336 Impedance spectrum, 171 Impulse, 55 Inertia force, 165 longitudinal, 165, 180 rotatory, 165, 199,220, 230, 240, 255 translator, 199, 220, 230 transverse, 199,220, 229, 231 transverse shear, 199,220,240 Initial condition, 11, 13,46 Interaction technique, 240 Interface, 146 bond, 178 damping, 146 fiber-matrix, 107 Isotropicmaterials, 11 Jordan matrix, 98 371 SUBJECT INDEX Linear air pump, Lobar frequency, 249 Lobar modes, 248,249, 259,264,267 Logarithmic decrement, 25, 47 Loss factor, 6, 10, 18, 19,20,22, 23, 24, 27, 44,52,62,111,123,150,298 at resonance, 273 damped beam, 175, 188, 189, 190,337, 339,343,345,359 damped cylindrical shell, 287, 291 damped plate, 211, 214, 215, 217, 218, 219,231,232,233,234,235 damped ring, 253, 270 Loss factor, evaluation complexeigenvalue method, 211, 213, 214 complexmodulus approach, 336 direct frequency response method, 99, 211, 218, 337 energy approach, 108, 148 force balance approach, 111, 120 modal strain energy method, 337 Matrix inversion, 246 Material damping, Mechanicalimpedance damped beam, 173, 174, 185, 187, 188 damped plate, 209, 220, 221, 222, 224, 225,227 dampedring, 246, 247, 248, 250,251,252, 262,263,264,265,266,274,275,277 damped cylindrical shell, 283, 291, 292, 294,295,297,298,299,300,301,302, 309, 312 method, 253 spectrum, 249 Mode shapes, 217, 218 Model generalized Maxwell,16, 17 generalized Voigt,16, 17 linear elastic, 15 linear viscous, 15 Maxwell, 12,14,16 rnicromechanical, 146 standard linear, 14, 16 Voigt, 13, 16 Modulus bulk., 11 complex, 17, 18,44,52,62,89,161,165, 167,336 complex dynamic Young's,223 complexshear, 219, 223, 252,261, 278, 291, 308 loss, 18, 43, 110, 113, 114, 123, 131, 149 shear, 161, 165, 170,175, 183,209,244, 298, 308 shear loss, 135, 169,263 shear storage, 168,262 storage,18,43, 110,113,123,131,149 Young's,27, 33, 43,178,183,216,244, 252,285,287,305 Neumann factor, 242, 286 Nyquistdiagrarn,26 Orthogonal properly of mass II}atrix,92 of stiffness matrix, 92 Packing geometry hexagonal array, 112, 115 square array, 112, 115 Parameters density, 179 frequency, 227, 260,267, 268 material, 269 natural frequency, 267, 269, 270, 271, 272 sttlIness,179 shear, 179, 189, 190,267,269 thickness, 268,270 viscoelasticthickness, 270 Phase angle, 5, 49, 52 Piezoelectric,171 Preload, prestress effectsof, 23, 340, 349 tensile static, 23 Quality factor, 25 Quotient-differencealgorithm, 267 Region flow,21, 22 glassy, 21, 22 rubberlike, 21, 22 transition, 21, 22 Resonant amplification factor, 25 Reverberation time, 26 Rubberlike materials, 21 Shear lag analysis, 107, 147 modified,147 Shear loss tangent, 263 Spatial decay, 26 Sonostonbeam, 372 Spring constant compresson-tension,243 equivalent compressiontension, 305 eqivalent shear, 305 shear,243 Static preload, 23 Stiffness extensional, 137 coupling, 137 flexural, 137 Strain deviatoric, 11 Green-Langrange tensor, 350 shear, 165 transverse, 165 Stress Cauchy, 351 deviatoric, 11 hydrostatic, 11 incremental, 352 Piola-Kirchhofftensor, 350, 351 Superposition integral, 57 Temperature effects of, 21, 22 nomogram, 27 transition, 21 Test creep, 13 DMA,27 relaxation,13 FUleovibron,27 Velocityimpedance,216 Vibration behavior, 186 damped natural mode, 189 extensional modes, 188,233 flexural, 180, 181,188 forced, 48, 77, 94, 99, 166 free, 45, 66, 76,89 multi-degree-of-freedom,89, 94 single-degree-of-freedom,45, 48, 54, 86 thickness shear mode, 188 transient, 54, 56,83,86 SUBJECT INDEX three-degree-of-freedom,98 two-degree-of-freedom,66, 77, 79, 83 Vibration, control absorber, 31, 83 active control, 30 controltechniques, 29 damper, 32 discrete damping devices-tuneddamper, 32 passive control, 30 shear tuned, 33 surface damping treatment, 49 tuned, 32, 33 tuned vibration absorber, 311 Vibration, structural beam-dampedcylindrical shell, 311, 312 beam-dampedtitanium shell, 295 cylindricalshell, 2, 282, 284 damped cylindrical shell, 281 damped beam, 174, 177, 184 damped ring, 239, 249, 276 mass-damped titanium cylindricalshell, 303 three-layered beam, 179, 180 three-layered plate, 199 titanium beam, 299 titanium cylindrical shell, 295,296, 297, 299,300, 301, 302 Viscoelastic core, 216 layer, 242,243,246,249,254,255,256, 275,277,278,281,287,290,299,305, 311 Viscoelasticmaterial, 42, 43,199,208,217, 219,227,239,241,243,246,249,253, 261,263,265, 276, 278,284,286,290, 293,298,299, 300, 303,305,306,318 characterization, 10 constitutive equation, 10, 16 dilatation, 11 distortion, 11 energy dissipation, 11 shear damping, 282 Viscoelasticity,12 Viscosity,12 ... Single-Degreeof-Freedom Systems with Damping Free Vibrations of Two-Degree -of- Freedom Systems with Viscous Damping Forced Vibrations of Two-Degreeof-Freedom Systems with Damping Transient Vibrations of. .. of vibration damping, properties of viscoelastic damping materials, vibrations of discrete damped systems, and damping of fiber-reinforced composite materials Chapters four to seven present vibrations... Vibrations of Two-Degreeof-Freedom Systems with Damping Damping of Aligned Discontinued Fiber Composites Damping of Aligned Off-Axis Discontinuous Fiber Composites Damping of Randomly Oriented ShortFiber