EBOOK - Mechanical Vibrations Theory and Application to Structural Dynamics (Michel Geradin)


This monograph results from a complete recasting of a book on Mechanical Vibrations, initially written in French and published by Masson Éditions in 1992 under the title Théorie des vibrations, Application à la dynamique des structures.
The first edition in English was issued shortly after, thanks to the support of DIST (French Ministry of Scientific Research and Space) and published by John Wiley & Sons in 1994. The book was indubitably felt to fill a gap since both editions were a success in France as well as internationally, so that both versions were almost immediately followed by a second edition by the same publishers: in French in 1996, and in 1997 for the English version. Due to the short delay between editions, only minor changes – essentially corrections – took place between the first and second versions of the manuscript.
The numerous constructive comments received from readers – university colleagues, students and practising engineers – during the following decade convinced both of us that a deep revision of the original manuscript was definitely needed to meet their expectations.
Of course there were still remaining errors to be corrected – and the very last one will never be discovered, error-making being a common trait of human beings – and more rigor and accuracy had to be brought here and there in the presentation and discussion of the concepts.
But the subject of mechanical vibration has also rapidly evolved, rendering the necessity of the addition of some new important topics. Proposed exercises to help, on the one hand, teachers explain the quintessence of dynamics and, on the other hand, students to assimilate the concepts through examples were also missing.

We were already planning to produce this third edition in French in the early 2000s, but the project could never be achieved due to overwhelming professional duties for both of us.The necessary time could finally be secured from 2010 (partly due to the retirement of the first author).
However, priority has now been given to the English language for the writing of this third, entirely new edition since our perception was that the demand for a new, enhanced version comes essentially from the international market. We are indebted to Éditions Dunod for having agreed to release the rights accordingly.
We are thus pleased to present to our former readers a new edition which we hope will meet most of their expectations, and to offer our new readers a book that allows them to discover or improve their knowledge of the fascinating world of mechanical vibration and structural dynamics.

Introduction 1
Suggested Bibliography 7
List of main symbols and definitions 9
1 Analytical Dynamics of Discrete Systems 13
Definitions 14
1.1 Principle of virtual work for a particle 14
1.1.1 Nonconstrained particle 14
1.1.2 Constrained particle 15
1.2 Extension to a system of particles 17
1.2.1 Virtual work principle for N particles 17
1.2.2 The kinematic constraints 18
1.2.3 Concept of generalized displacements 20
1.3 Hamilton’s principle for conservative systems and Lagrange equations 23
1.3.1 Structure of kinetic energy and classification of inertia forces 27
1.3.2 Energy conservation in a system with scleronomic constraints 29
1.3.3 Classification of generalized forces 32
1.4 Lagrange equations in the general case 36
1.5 Lagrange equations for impulsive loading 39
1.5.1 Impulsive loading of a mass particle 39
1.5.2 Impulsive loading for a system of particles 42
1.6 Dynamics of constrained systems 44
1.7 Exercises 46
1.7.1 Solved exercises 46
1.7.2 Selected exercises 53
References 54
2 Undamped Vibrations of n-Degree-of-Freedom Systems 57
Definitions 58
2.1 Linear vibrations about an equilibrium configuration 59
vi Contents
2.1.1 Vibrations about a stable equilibrium position 59
2.1.2 Free vibrations about an equilibrium configuration corresponding
to steady motion 63
2.1.3 Vibrations about a neutrally stable equilibrium position 66
2.2 Normal modes of vibration 67
2.2.1 Systems with a stable equilibrium configuration 68
2.2.2 Systems with a neutrally stable equilibrium position 69
2.3 Orthogonality of vibration eigenmodes 70
2.3.1 Orthogonality of elastic modes with distinct frequencies 70
2.3.2 Degeneracy theorem and generalized orthogonality relationships 72
2.3.3 Orthogonality relationships including rigid-body modes 75
2.4 Vector and matrix spectral expansions using eigenmodes 76
2.5 Free vibrations induced by nonzero initial conditions 77
2.5.1 Systems with a stable equilibrium position 77
2.5.2 Systems with neutrally stable equilibrium position 82
2.6 Response to applied forces: forced harmonic response 83
2.6.1 Harmonic response, impedance and admittance matrices 84
2.6.2 Mode superposition and spectral expansion of the admittance matrix 84
2.6.3 Statically exact expansion of the admittance matrix 88
2.6.4 Pseudo-resonance and resonance 89
2.6.5 Normal excitation modes 90
2.7 Response to applied forces: response in the time domain 91
2.7.1 Mode superposition and normal equations 91
2.7.2 Impulse response and time integration of the normal equations 92
2.7.3 Step response and time integration of the normal equations 94
2.7.4 Direct integration of the transient response 95
2.8 Modal approximations of dynamic responses 95
2.8.1 Response truncation and mode displacement method 96
2.8.2 Mode acceleration method 97
2.8.3 Mode acceleration and model reduction on selected coordinates 98
2.9 Response to support motion 101
2.9.1 Motion imposed to a subset of degrees of freedom 101
2.9.2 Transformation to normal coordinates 103
2.9.3 Mechanical impedance on supports and its statically
exact expansion 105
2.9.4 System submitted to global support acceleration 108
2.9.5 Effective modal masses 109
2.9.6 Method of additional masses 110
2.10 Variational methods for eigenvalue characterization 111
2.10.1 Rayleigh quotient 111
2.10.2 Principle of best approximation to a given eigenvalue 112
2.10.3 Recurrent variational procedure for eigenvalue analysis 113
2.10.4 Eigensolutions of constrained systems: general comparison
principle or monotonicity principle 114
2.10.5 Courant’s minimax principle to evaluate eigenvalues independently of each other 116
Contents vii
2.10.6 Rayleigh’s theorem on constraints (eigenvalue bracketing) 117
2.11 Conservative rotating systems 119
2.11.1 Energy conservation in the absence of external force 119
2.11.2 Properties of the eigensolutions of the conservative rotating system 119
2.11.3 State-space form of equations of motion 121
2.11.4 Eigenvalue problem in symmetrical form 124
2.11.5 Orthogonality relationships 126
2.11.6 Response to nonzero initial conditions 128
2.11.7 Response to external excitation 130
2.12 Exercises 130
2.12.1 Solved exercises 130
2.12.2 Selected exercises 143
References 148
3 Damped Vibrations of n-Degree-of-Freedom Systems 149
Definitions 150
3.1 Damped oscillations in terms of normal eigensolutions of the undamped system 151
3.1.1 Normal equations for a damped system 152
3.1.2 Modal damping assumption for lightly damped structures 153
3.1.3 Constructing the damping matrix through modal expansion 158
3.2 Forced harmonic response 160
3.2.1 The case of light viscous damping 160
3.2.2 Hysteretic damping 162
3.2.3 Force appropriation testing 164
3.2.4 The characteristic phase lag theory 170
3.3 State-space formulation of damped systems 174
3.3.1 Eigenvalue problem and solution of the homogeneous case 175
3.3.2 General solution for the nonhomogeneous case 178
3.3.3 Harmonic response 179
3.4 Experimental methods of modal identification 180
...


This monograph results from a complete recasting of a book on Mechanical Vibrations, initially written in French and published by Masson Éditions in 1992 under the title Théorie des vibrations, Application à la dynamique des structures.
The first edition in English was issued shortly after, thanks to the support of DIST (French Ministry of Scientific Research and Space) and published by John Wiley & Sons in 1994. The book was indubitably felt to fill a gap since both editions were a success in France as well as internationally, so that both versions were almost immediately followed by a second edition by the same publishers: in French in 1996, and in 1997 for the English version. Due to the short delay between editions, only minor changes – essentially corrections – took place between the first and second versions of the manuscript.
The numerous constructive comments received from readers – university colleagues, students and practising engineers – during the following decade convinced both of us that a deep revision of the original manuscript was definitely needed to meet their expectations.
Of course there were still remaining errors to be corrected – and the very last one will never be discovered, error-making being a common trait of human beings – and more rigor and accuracy had to be brought here and there in the presentation and discussion of the concepts.
But the subject of mechanical vibration has also rapidly evolved, rendering the necessity of the addition of some new important topics. Proposed exercises to help, on the one hand, teachers explain the quintessence of dynamics and, on the other hand, students to assimilate the concepts through examples were also missing.

We were already planning to produce this third edition in French in the early 2000s, but the project could never be achieved due to overwhelming professional duties for both of us.The necessary time could finally be secured from 2010 (partly due to the retirement of the first author).
However, priority has now been given to the English language for the writing of this third, entirely new edition since our perception was that the demand for a new, enhanced version comes essentially from the international market. We are indebted to Éditions Dunod for having agreed to release the rights accordingly.
We are thus pleased to present to our former readers a new edition which we hope will meet most of their expectations, and to offer our new readers a book that allows them to discover or improve their knowledge of the fascinating world of mechanical vibration and structural dynamics.

Introduction 1
Suggested Bibliography 7
List of main symbols and definitions 9
1 Analytical Dynamics of Discrete Systems 13
Definitions 14
1.1 Principle of virtual work for a particle 14
1.1.1 Nonconstrained particle 14
1.1.2 Constrained particle 15
1.2 Extension to a system of particles 17
1.2.1 Virtual work principle for N particles 17
1.2.2 The kinematic constraints 18
1.2.3 Concept of generalized displacements 20
1.3 Hamilton’s principle for conservative systems and Lagrange equations 23
1.3.1 Structure of kinetic energy and classification of inertia forces 27
1.3.2 Energy conservation in a system with scleronomic constraints 29
1.3.3 Classification of generalized forces 32
1.4 Lagrange equations in the general case 36
1.5 Lagrange equations for impulsive loading 39
1.5.1 Impulsive loading of a mass particle 39
1.5.2 Impulsive loading for a system of particles 42
1.6 Dynamics of constrained systems 44
1.7 Exercises 46
1.7.1 Solved exercises 46
1.7.2 Selected exercises 53
References 54
2 Undamped Vibrations of n-Degree-of-Freedom Systems 57
Definitions 58
2.1 Linear vibrations about an equilibrium configuration 59
vi Contents
2.1.1 Vibrations about a stable equilibrium position 59
2.1.2 Free vibrations about an equilibrium configuration corresponding
to steady motion 63
2.1.3 Vibrations about a neutrally stable equilibrium position 66
2.2 Normal modes of vibration 67
2.2.1 Systems with a stable equilibrium configuration 68
2.2.2 Systems with a neutrally stable equilibrium position 69
2.3 Orthogonality of vibration eigenmodes 70
2.3.1 Orthogonality of elastic modes with distinct frequencies 70
2.3.2 Degeneracy theorem and generalized orthogonality relationships 72
2.3.3 Orthogonality relationships including rigid-body modes 75
2.4 Vector and matrix spectral expansions using eigenmodes 76
2.5 Free vibrations induced by nonzero initial conditions 77
2.5.1 Systems with a stable equilibrium position 77
2.5.2 Systems with neutrally stable equilibrium position 82
2.6 Response to applied forces: forced harmonic response 83
2.6.1 Harmonic response, impedance and admittance matrices 84
2.6.2 Mode superposition and spectral expansion of the admittance matrix 84
2.6.3 Statically exact expansion of the admittance matrix 88
2.6.4 Pseudo-resonance and resonance 89
2.6.5 Normal excitation modes 90
2.7 Response to applied forces: response in the time domain 91
2.7.1 Mode superposition and normal equations 91
2.7.2 Impulse response and time integration of the normal equations 92
2.7.3 Step response and time integration of the normal equations 94
2.7.4 Direct integration of the transient response 95
2.8 Modal approximations of dynamic responses 95
2.8.1 Response truncation and mode displacement method 96
2.8.2 Mode acceleration method 97
2.8.3 Mode acceleration and model reduction on selected coordinates 98
2.9 Response to support motion 101
2.9.1 Motion imposed to a subset of degrees of freedom 101
2.9.2 Transformation to normal coordinates 103
2.9.3 Mechanical impedance on supports and its statically
exact expansion 105
2.9.4 System submitted to global support acceleration 108
2.9.5 Effective modal masses 109
2.9.6 Method of additional masses 110
2.10 Variational methods for eigenvalue characterization 111
2.10.1 Rayleigh quotient 111
2.10.2 Principle of best approximation to a given eigenvalue 112
2.10.3 Recurrent variational procedure for eigenvalue analysis 113
2.10.4 Eigensolutions of constrained systems: general comparison
principle or monotonicity principle 114
2.10.5 Courant’s minimax principle to evaluate eigenvalues independently of each other 116
Contents vii
2.10.6 Rayleigh’s theorem on constraints (eigenvalue bracketing) 117
2.11 Conservative rotating systems 119
2.11.1 Energy conservation in the absence of external force 119
2.11.2 Properties of the eigensolutions of the conservative rotating system 119
2.11.3 State-space form of equations of motion 121
2.11.4 Eigenvalue problem in symmetrical form 124
2.11.5 Orthogonality relationships 126
2.11.6 Response to nonzero initial conditions 128
2.11.7 Response to external excitation 130
2.12 Exercises 130
2.12.1 Solved exercises 130
2.12.2 Selected exercises 143
References 148
3 Damped Vibrations of n-Degree-of-Freedom Systems 149
Definitions 150
3.1 Damped oscillations in terms of normal eigensolutions of the undamped system 151
3.1.1 Normal equations for a damped system 152
3.1.2 Modal damping assumption for lightly damped structures 153
3.1.3 Constructing the damping matrix through modal expansion 158
3.2 Forced harmonic response 160
3.2.1 The case of light viscous damping 160
3.2.2 Hysteretic damping 162
3.2.3 Force appropriation testing 164
3.2.4 The characteristic phase lag theory 170
3.3 State-space formulation of damped systems 174
3.3.1 Eigenvalue problem and solution of the homogeneous case 175
3.3.2 General solution for the nonhomogeneous case 178
3.3.3 Harmonic response 179
3.4 Experimental methods of modal identification 180
...

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