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Damping Models for Structural Vibration

Damping Models for Structural VibrationCambridge UniversityEngineering DepartmentA dissertationsubmitted to the University of Cambridgefor the Degree of Doctor of PhilosophybySondipon AdhikariTrinity College, CambridgeSeptember, 2000iiTomyParentsivDeclarationThis dissertation describes part of the research performed at the Cambridge University EngineeringDepartment between October 1997 and September 2000. It is the result of my own work andincludes nothing which is the outcome of work done in collaboration, except where stated. Thedissertation contains approximately 63,000 words, 120 figures and 160 AdhikariSeptember, 2000vviDECLARATIONA bstractThis dissertation reports a systematic study onanalysisandidentificationof multiple parameter dampedmechanical systems.

Classical modal analysis is extended to deal with general non-viscously damped multiple degree-of- freedom linear dynamic systems. The new method is similar to the existing method with some modifications

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Transcription of Damping Models for Structural Vibration

1 Damping Models for Structural VibrationCambridge UniversityEngineering DepartmentA dissertationsubmitted to the University of Cambridgefor the Degree of Doctor of PhilosophybySondipon AdhikariTrinity College, CambridgeSeptember, 2000iiTomyParentsivDeclarationThis dissertation describes part of the research performed at the Cambridge University EngineeringDepartment between October 1997 and September 2000. It is the result of my own work andincludes nothing which is the outcome of work done in collaboration, except where stated. Thedissertation contains approximately 63,000 words, 120 figures and 160 AdhikariSeptember, 2000vviDECLARATIONA bstractThis dissertation reports a systematic study onanalysisandidentificationof multiple parameter dampedmechanical systems.

2 The attention is focused on viscously and non-viscously damped multiple degree-of-freedom linear vibrating systems. The non-viscous Damping model is such that the Damping forces dependon the past history of motion via convolution integrals over some kernel functions. The familiar viscousdamping model is a special case of this general linear Damping model when the kernel functions have concept of proportional Damping is critically examined and a generalized form of proportionaldamping is proposed. It is shown that the proportional Damping can exist even when the Damping mechanismis modal analysis is extended to deal with general non-viscously damped multiple degree-of-freedom linear dynamic systems. The new method is similar to the existing method with some modificationsdue to non-viscous effect of the Damping mechanism.

3 The concept of (complex)elastic modesandnon-viscous modeshave been introduced and numerical methods are suggested to obtain them. It is furthershown that the system response can be obtained exactly in terms of these modes. Mode orthogonalityrelationships, known for undamped or viscously damped systems, have been generalized to non-viscouslydamped systems. Several useful results which relate the modes with the system matrices are theoretical developments on non-viscously damped systems, in line with classical modal analy-sis, give impetus towards understanding Damping mechanisms in general mechanical systems. Based on afirst-order perturbation method, an approach is suggested to the identify non-proportional viscous dampingmatrix from the measured complex modes and frequencies.

4 This approach is then further extended to iden-tify non-viscous Damping Models . Both the approaches are simple, direct, and can be used with incompletemodal is observed that these methods yield non-physical results by breaking the symmetry of the fitteddamping matrix when the Damping mechanism of the original system is significantly different from what isfitted. To solve this problem, approaches are suggested to preserve the symmetry of the identified viscousand non-viscous Damping Damping identification methods are applied experimentally to a beam in bending Vibration withlocalized constrained layer Damping . Since the identification method requires complex modal data, a gen-eral method for identification of complex modes and complex frequencies from a set of measured transferfunctions have been developed.

5 It is shown that the proposed methods can give useful information aboutthe true Damping mechanism of the beam considered for the experiment. Further, it is demonstrated thatthe Damping identification methods are likely to perform quite well even for the case when noisy data work conducted here clarifies some fundamental issues regarding Damping in linear dynamic sys-tems and develops efficient methods for analysis and identification of generally damped linear am very grateful to my supervisor Prof. Jim Woodhouse for his technical guidance and encour-aging association throughout the period of my research work in Cambridge. I would also like tothank Prof R. S. Langley for his interest into my would like to express my gratitude to theNehru Memorial Trust, London, theCambridgeCommonwealth Trust,The Committee of Vice-chancellors and Principals, UK andTrinity College,Cambridge for providing the financial support during the period in which this research work wascarried wish to take this opportunity to thank The Old Schools, Cambridge for awarding me the JohnWibolt Prize 1999 for my paper (Adhikari, 1999).

6 I am thankful to my colleagues in the Mechanics Group of the Cambridge University Engineer-ing Department for providing a congenial working atmosphere in the laboratory. I am particularlythankful to David Miller and Simon Smith for their help in setting up the experiment and JamesTalbot for his careful reading of the also want to thank my parents for their inspiration, in spite of being far away from me. Finally,I want to thank my wife Sonia without her constant mental support this work might not comeinto this of Undamped Systems.. of Motion.. analysis .. of Damping .. Degree-of-freedom Systems.. Systems.. Degrees-of-freedom Systems.. Studies.. analysis of Viscously Damped Systems.. State-Space Method.. in Configuration Space.

7 Of Non-viscously Damped Systems.. of Viscous Damping .. Degree-of-freedom Systems Systems.. Degrees-of-freedom Systems.. of Non-viscous Damping .. Problems.. of the Dissertation..232 The Nature of Proportional .. Damped Systems.. of Classical Normal Modes.. of Proportional Damping .. Damped Systems.. of Classical Normal Modes.. of Proportional Damping ..35xixiiCONTENTS3 Dynamics of Non-viscously Damped .. and Eigenvectors.. Modes.. Modes.. and Special Cases.. Function.. of the Dynamic Stiffness Matrix.. of the Residues.. Cases.. Response.. of the Method.. Examples.. System.. 1: Exponential Damping .. 2: GHM Damping ..584 Some General Properties of the .. of the Eigensolutions.. of the Eigenvectors.

8 Of the Eigenvectors.. Between the Eigensolutions and Damping .. in Terms ofM 1.. in Terms ofK 1.. Matrices in Terms of the Eigensolutions.. for Viscously Damped Systems.. Examples.. System.. and Eigenvectors.. Relationships.. With the Damping Matrix..725 Identification of Viscous .. of Complex Modes.. of Viscous Damping Matrix.. Examples.. for Small .. for Larger ..92 CONTENTS xiii6 Identification of Non-viscous .. of Complex Modes.. of the Relaxation Parameter.. Method.. Results.. the Value of .. of the Coefficient Matrix.. of the Identification Method.. Results..1207 Symmetry Preserving .. of Viscous Damping Matrix.. Examples.. of Non-viscous Damping .. Examples..1408 Experimental Identification of.

9 Of modal Parameters.. Least-Square Method.. of the Residues.. Least-Square Method.. of the Method.. Beam Experiment.. Set-up.. Procedure.. Theory.. and Discussions.. and Fitted Transfer Functions.. Data.. of the Damping Properties.. analysis .. analysis for Viscous Damping Identification.. analysis for Non-viscous Damping Identification..183xivCONTENTS9 Summary and of the Contributions Made.. for Further Work..187A Calculation of the Gradient and Hessian of the Merit Function191B Discretized Mass Matrix of the Beam193 References195 List of of modal Damping ratios (simulated).. degree-of-freedom non-viscously damped system,mu= 1kg,ku= 1N/m.. plot showing the locus of the third eigenvalue (s3) as a function of.

10 Spectral density function of the displacement at the third DOF (z3).. functionH33(i ).. array ofNspring-mass oscillators,N= 30,mu= 1Kg,ku= 4 103N/m.. viscous Damping matrix for the local case, = , Damping model 2.. viscous Damping matrix using first 20 modes for the local case, = , dampingmodel 2.. viscous Damping matrix using first 10 modes for the local case, = , dampingmodel 2.. viscous Damping matrix for the non-local case, = , Damping model 2.. Q-factors, = , Damping model 2.. viscous Damping matrix for the local case, = , Damping model 1.. viscous Damping matrix for the local case, = , Damping model 2.. viscous Damping matrix for the local case, = , Damping model 1.. Fitted viscous Damping matrix for the non-local case, = , Damping model 1.


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