Table Of ContentNonlinear Targeted Energy Transfer in Mechanical and
Structural Systems I
Nonlinear Targeted Energy Transfer in Mechanical and
Structural Systems II
SOLID MECHANICS AND ITS APPLICATIONS
Volume 156
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A.F. Vakakis • O.V. Gendelman • L.A. Bergman •
D.M. McFarland • G. Kerschen • Y.S. Lee
Nonlinear Targeted Energy
Transfer in Mechanical and
Structural Systems I
A.F. Vakakis • O.V. Gendelman • L.A. Bergman •
D.M. McFarland • G. Kerschen • Y.S. Lee
Nonlinear Targeted Energy
Transfer in Mechanical and
Structural Systems II
Alexander F. Vakakis Oleg V. Gendelman
Department of Mechanical Science Faculty of Mechanical Engineering
and Engineering Technion – Israel Institute of Technology
University of Illinois Haifa, Israel
Urbana, Illinois, USA
and D. Michael McFarland
Mechanics Division Department of Aerospace Engineering
National Technical University of Athens University of Illinois at Urbana-Champaign
Athens, Greece Urbana, Illinois, USA
Lawrence A. Bergman Young Sup Lee
Department of Aerospace Engineering Department of Mechanical and Aerospace
University of Illinois Engineering
Urbana, Illinois, USA New Mexico State University
Las Cruces, New Mexico, USA
Gaëtan Kerschen
Department of Aerospace and Mechanical
Engineering
University of Liège
Liège, Belgium
ISBN-13: 978-1-4020-9125-4 e-ISBN-13: 978-1-4020-9130-8
Library of Congress Control Number: 2008940435
© 2008 Springer Science+Business Media, B.V.
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Contents Volume 1
Preface vii
Abbreviations xi
1 Introduction 1
2 Preliminary Concepts, Methodologies and Techniques 15
2.1 Nonlinear Normal Modes (NNMs) 16
2.2 Energy Localization in Nonlinear Systems 28
2.3 Internal Resonances, Transient and Sustained Resonance Captures 38
2.4 Averaging, Multiple Scales and Complexification 54
2.5 Methods of Advanced Signal Processing 70
2.5.1 Numerical Wavelet Transforms 71
2.5.2 Empirical Mode Decompositions and Hilbert Transforms 77
2.6 Perspectives on Hardware Development and Experiments 81
3 Nonlinear Targeted Energy Transfer in Discrete Linear Oscillators
with Single-DOF Nonlinear Energy Sinks 93
3.1 Configurations of Single-DOF NESs 93
3.2 Numerical Evidence of TET in a SDOF Linear Oscillator with a
SDOF NES 98
3.3 SDOF Linear Oscillators with SDOF NESs: Dynamics of the
Underlying Hamiltonian Systems 108
3.3.1 Numerical Study of Periodic Orbits (NNMs) 108
3.3.2 Analytic Study of Periodic Orbits (NNMs) 124
3.3.3 Numerical Study of Periodic Impulsive Orbits (IOs) 135
3.3.4 Analytic Study of Periodic and Quasi-Periodic IOs 137
3.3.5 Topological Features of the Hamiltonian Dynamics 157
3.4 SDOF Linear Oscillators with SDOF NESs: Transient Dynamics of
the Damped Systems 165
3.4.1 Nonlinear Damped Transitions Represented in the FEP 166
3.4.2 Dynamics of TET in the Damped System 171
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vi Contents
3.5 Multi-DOF (MDOF) Linear Oscillators with SDOF NESs: Resonance
Capture Cascades and Multi-frequency TET 233
3.5.1 Two-DOF Linear Oscillator with a SDOF NES 237
3.5.2 Semi-Infinite Chain of Linear Oscillators with an End
SDOF NES 269
4 Targeted Energy Transfer in Discrete Linear Oscillators with
Multi-DOF NESs 303
4.1 Multi-Degree-of-Freedom (MDOF) NESs 303
4.1.1 An Alternative Way for Passive Multi-frequency Nonlinear
Energy Transfers 304
4.1.2 Numerical Evidence of TET in MDOF NESs 309
4.2 The Dynamics of the Underlying Hamiltonian System 317
4.2.1 System I: NES with O(1) Mass 320
4.2.2 System II: NES with O(ε) Mass 325
4.2.3 Asymptotic Analysis of Nonlinear Resonant Orbits 328
4.2.4 Analysis of Resonant Periodic Orbits 336
4.3 TRCs and TET in the Damped and Forced System 347
4.3.1 Numerical Wavelet Transforms 347
4.3.2 Damped Transitions on the Hamiltonian FEP 352
4.4 Concluding Remarks 365
Index 369
Contents Volume 2
5 Targeted Energy Transfer in Linear Continuous Systems with Single-
and Multi-DOF NESs 1
5.1 Beam of Finite Length with SDOF NES 1
5.1.1 Formulation of the Problem and Computational Procedure 1
5.1.2 Parametric Study of TET 6
5.2 Rod of Finite Length with SDOF NES 12
5.2.1 Formulation of the Problem, Computational Procedure and
Post-Processing Algorithms 13
5.2.2 Computational Study of TET 18
5.2.3 Damped Transitions on the Hamiltonian FEP 39
5.3 Rod of Semi-Infinite Length with SDOF NES 66
5.3.1 Reduction to Integro-differential Form 67
5.3.2 Numerical Study of Damped Transitions 75
5.3.3 Analytical Study 86
5.4 Rod of Finite Length with MDOF NES 99
5.4.1 Formulation of the Problem and FEPs 99
5.4.2 Computational Study of TET 109
5.4.3 Multi-Modal Damped Transitions and Multi-Scale Analysis 117
5.5 Plate with SDOF and MDOF NESs 132
5.5.1 Case of a SDOF NES 142
5.5.2 Case of Multiple SDOF NESs 147
5.5.3 Case of a MDOF NES 150
5.5.4 Comparative Study with Linear Tuned Mass Damper 153
6 Targeted Energy Transfer in Systems with Periodic Excitations 161
6.1 Steady State Responses and Generic Bifurcations 162
6.1.1 Analysis of Steady State Motions 162
6.1.2 Numerical Verification of the Analytical Results 175
6.2 Strongly Modulated Responses (SMRs) 177
6.2.1 General Formulation and Invariant Manifold Approach 177
6.2.2 Reduction to One-Dimensional Maps and Existence Conditions
for SMRs 187
6.2.3 Numerical Simulations 194
6.3 NESs as Strongly Nonlinear Absorbers for Vibration Isolation 202
6.3.1 Co-existent Response Regimes 202
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vi Contents
6.3.2 Efficiency and Broadband Features of the Vibration Isolation 206
6.3.3 Passive Self-tuning Capacity of the NES 213
7 NESs with Non-Smooth Stiffness Characteristics 229
7.1 System with Multiple NESs Possessing Clearance Nonlinearities 229
7.1.1 Problem Description 230
7.1.2 Numerical Results 233
7.2 Vibro-Impact (VI) NESs as Shock Absorbers 241
7.2.1 Passive TET to VI NESs 242
7.2.2 Shock Isolation 251
7.3 SDOF Linear Oscillator with a VI NES 259
7.3.1 Periodic Orbits for Elastic Vibro-Impacts Represented on
the FEP 260
7.3.2 Vibro-Impact Transitions in the Dissipative Case: VI TET 279
8 Experimental Verification of Targeted Energy Transfer 311
8.1 TET to Ungrounded SDOF NES (Configuration II) 311
8.1.1 System Identification 312
8.1.2 Experimental Measurements 314
8.2 TET to Grounded SDOF NES (Configuration I) 320
8.2.1 Experimental Fixture 322
8.2.2 Results and Discussion 324
8.3 Experimental Demonstration of 1:1 TRCs Leading to TET 330
8.3.1 Experimental Fixture 331
8.3.2 Experimental TRCs 333
8.4 Steady State TET under Harmonic Excitation 342
8.4.1 System Configuration and Theoretical Analysis 344
8.4.2 Experimental Study 347
9 Suppression of Aeroelastic Instabilities through Passive Targeted
Energy Transfer 353
9.1 Suppression of Limit-Cycle Oscillations in the van der Pol Oscillator 353
9.1.1 VDP Oscillator and NES Configurations 355
9.1.2 Transient Dynamics 359
9.1.3 Steady State Dynamics and Bifurcation Analysis 381
9.1.4 Summary of Results 394
9.2 Triggering Mechanism for Aeroelastic Instability of an In-Flow Wing 397
9.2.1 The Two-DOF Aeroelastic Model 399
9.2.2 Slow Flow Dynamics 402
9.2.3 LCO Triggering Mechanism 428
9.2.4 Concluding Remarks 447
9.3 Suppressing Aeroelastic Instability of an In-Flow Wing Using a
SDOF NES 453
9.3.1 Preliminary Numerical Study 453
9.3.2 Study of LCO Suppression Mechanisms 463
