Contents:

Summary: Mechanical Vibrations: Theory and Application to Structural Dynamics, Third Edition is a comprehensively updated new edition of the popular textbook. It presents the theory of vibrations in the context of structural analysis and covers applications in mechanical and aerospace engineering. Key features include: A systematic approach to dynamic reduction and substructuring, based on duality between mechanical and admittance concepts An introduction to experimental modal analysis and identification methods An improved, more physical presentation of wave propagation phenomena A comprehensive presentation of current practice for solving large eigenproblems, focusing on the efficient linear solution of large, sparse and possibly singular systems A deeply revised description of time integration schemes, providing framework for the rigorous accuracy/stability analysis of now widely used algorithms such as HHT and Generalized-α Solved exercises and end of chapter homework problems A companion website hosting supplementary material.
Cover -- TItle Page -- Copyright -- Contents -- Foreword -- Preface -- Introduction -- Suggested Bibliography -- List of main symbols and definitions -- Chapter 1 Analytical Dynamics of Discrete Systems -- Definitions -- 1.1 Principle of virtual work for a particle -- 1.1.1 Nonconstrained particle -- 1.1.2 Constrained particle -- 1.2 Extension to a system of particles -- 1.2.1 Virtual work principle for N particles -- 1.2.2 The kinematic constraints -- 1.2.3 Concept of generalized displacements -- 1.3 Hamilton's principle for conservative systems and Lagrange equations -- 1.3.1 Structure of kinetic energy and classification of inertia forces -- 1.3.2 Energy conservation in a system with scleronomic constraints -- 1.3.3 Classification of generalized forces -- 1.4 Lagrange equations in the general case -- 1.5 Lagrange equations for impulsive loading -- 1.5.1 Impulsive loading of a mass particle -- 1.5.2 Impulsive loading for a system of particles -- 1.6 Dynamics of constrained systems -- 1.7 Exercises -- 1.7.1 Solved exercises -- 1.7.2 Selected exercises -- References -- Chapter 2 Undamped Vibrations of n-Degree-of-Freedom Systems -- Definitions -- 2.1 Linear vibrations about an equilibrium configuration -- 2.1.1 Vibrations about a stable equilibrium position -- 2.1.2 Free vibrations about an equilibrium configuration corresponding to steady motion -- 2.1.3 Vibrations about a neutrally stable equilibrium position -- 2.2 Normal modes of vibration -- 2.2.1 Systems with a stable equilibrium configuration -- 2.2.2 Systems with a neutrally stable equilibrium position -- 2.3 Orthogonality of vibration eigenmodes -- 2.3.1 Orthogonality of elastic modes with distinct frequencies -- 2.3.2 Degeneracy theorem and generalized orthogonality relationships -- 2.3.3 Orthogonality relationships including rigid-body modes.

2.4 Vector and matrix spectral expansions using eigenmodes -- 2.5 Free vibrations induced by nonzero initial conditions -- 2.5.1 Systems with a stable equilibrium position -- 2.5.2 Systems with neutrally stable equilibrium position -- 2.6 Response to applied forces: forced harmonic response -- 2.6.1 Harmonic response, impedance and admittance matrices -- 2.6.2 Mode superposition and spectral expansion of the admittance matrix -- 2.6.3 Statically exact expansion of the admittance matrix -- 2.6.4 Pseudo-resonance and resonance -- 2.6.5 Normal excitation modes -- 2.7 Response to applied forces: response in the time domain -- 2.7.1 Mode superposition and normal equations -- 2.7.2 Impulse response and time integration of the normal equations -- 2.7.3 Step response and time integration of the normal equations -- 2.7.4 Direct integration of the transient response -- 2.8 Modal approximations of dynamic responses -- 2.8.1 Response truncation and mode displacement method -- 2.8.2 Mode acceleration method -- 2.8.3 Mode acceleration and model reduction on selected coordinates -- 2.9 Response to support motion -- 2.9.1 Motion imposed to a subset of degrees of freedom -- 2.9.2 Transformation to normal coordinates -- 2.9.3 Mechanical impedance on supports and its statically exact expansion -- 2.9.4 System submitted to global support acceleration -- 2.9.5 Effective modal masses -- 2.9.6 Method of additional masses -- 2.10 Variational methods for eigenvalue characterization -- 2.10.1 Rayleigh quotient -- 2.10.2 Principle of best approximation to a given eigenvalue -- 2.10.3 Recurrent variational procedure for eigenvalue analysis -- 2.10.4 Eigensolutions of constrained systems: general comparison principle or monotonicity principle -- 2.10.5 Courant's minimax principle to evaluate eigenvalues independently of each other.

2.10.6 Rayleigh's theorem on constraints (eigenvalue bracketing) -- 2.11 Conservative rotating systems -- 2.11.1 Energy conservation in the absence of external force -- 2.11.2 Properties of the eigensolutions of the conservative rotating system -- 2.11.3 State-space form of equations of motion -- 2.11.4 Eigenvalue problem in symmetrical form -- 2.11.5 Orthogonality relationships -- 2.11.6 Response to nonzero initial conditions -- 2.11.7 Response to external excitation -- 2.12 Exercises -- 2.12.1 Solved exercises -- 2.12.2 Selected exercises -- References -- Chapter 3 Damped Vibrations of n-Degree-of-Freedom Systems -- Definitions -- 3.1 Damped oscillations in terms of normal eigensolutions of the undamped system -- 3.1.1 Normal equations for a damped system -- 3.1.2 Modal damping assumption for lightly damped structures -- 3.1.3 Constructing the damping matrix through modal expansion -- 3.2 Forced harmonic response -- 3.2.1 The case of light viscous damping -- 3.2.2 Hysteretic damping -- 3.2.3 Force appropriation testing -- 3.2.4 The characteristic phase lag theory -- 3.3 State-space formulation of damped systems -- 3.3.1 Eigenvalue problem and solution of the homogeneous case -- 3.3.2 General solution for the nonhomogeneous case -- 3.3.3 Harmonic response -- 3.4 Experimental methods of modal identification -- 3.4.1 The least-squares complex exponential method -- 3.4.2 Discrete Fourier transform -- 3.4.3 The rational fraction polynomial method -- 3.4.4 Estimating the modes of the associated undamped system -- 3.4.5 Example: experimental modal analysis of a bellmouth -- 3.5 Exercises -- 3.5.1 Solved exercises -- 3.6 Proposed exercises -- References -- Chapter 4 Continuous Systems -- Definitions -- 4.1 Kinematic description of the dynamic behaviour of continuous systems: Hamilton's principle -- 4.1.1 Definitions.

4.1.2 Strain evaluation: Green's measure -- 4.1.3 Stress-strain relationships -- 4.1.4 Displacement variational principle -- 4.1.5 Derivation of equations of motion -- 4.1.6 The linear case and nonlinear effects -- 4.2 Free vibrations of linear continuous systems and response to external excitation -- 4.2.1 Eigenvalue problem -- 4.2.2 Orthogonality of eigensolutions -- 4.2.3 Response to external excitation: mode superposition (homogeneous spatial boundary conditions) -- 4.2.4 Response to external excitation: mode superposition (nonhomogeneous spatial boundary conditions) -- 4.2.5 Reciprocity principle for harmonic motion -- 4.3 One-dimensional continuous systems -- 4.3.1 The bar in extension -- 4.3.2 Transverse vibrations of a taut string -- 4.3.3 Transverse vibration of beams with no shear deflection -- 4.3.4 Transverse vibration of beams including shear deflection -- 4.3.5 Travelling waves in beams -- 4.4 Bending vibrations of thin plates -- 4.4.1 Kinematic assumptions -- 4.4.2 Strain expressions -- 4.4.3 Stress-strain relationships -- 4.4.4 Definition of curvatures -- 4.4.5 Moment-curvature relationships -- 4.4.6 Frame transformation for bending moments -- 4.4.7 Computation of strain energy -- 4.4.8 Expression of Hamilton's principle -- 4.4.9 Plate equations of motion derived from Hamilton's principle -- 4.4.10 Influence of in-plane initial stresses on plate vibration -- 4.4.11 Free vibrations of the rectangular plate -- 4.4.12 Vibrations of circular plates -- 4.4.13 An application of plate vibration: the ultrasonic wave motor -- 4.5 Wave propagation in a homogeneous elastic medium -- 4.5.1 The Navier equations in linear dynamic analysis -- 4.5.2 Plane elastic waves -- 4.5.3 Surface waves -- 4.6 Solved exercises -- 4.7 Proposed exercises -- References -- Chapter 5 Approximation of Continuous Systems by Displacement Methods -- Definitions.

5.1 The Rayleigh-Ritz method -- 5.1.1 Choice of approximation functions -- 5.1.2 Discretization of the displacement variational principle -- 5.1.3 Computation of eigensolutions by the Rayleigh-Ritz method -- 5.1.4 Computation of the response to external loading by the Rayleigh-Ritz method -- 5.1.5 The case of prestressed structures -- 5.2 Applications of the Rayleigh-Ritz method to continuous systems -- 5.2.1 The clamped-free uniform bar -- 5.2.2 The clamped-free uniform beam -- 5.2.3 The uniform rectangular plate -- 5.3 The finite element method -- 5.3.1 The bar in extension -- 5.3.2 Truss frames -- 5.3.3 Beams in bending without shear deflection -- 5.3.4 Three-dimensional beam element without shear deflection -- 5.3.5 Beams in bending with shear deformation -- 5.4 Exercises -- 5.4.1 Solved exercises -- 5.4.2 Selected exercises -- References -- Chapter 6 Solution Methods for the Eigenvalue Problem -- Definitions -- 6.1 General considerations -- 6.1.1 Classification of solution methods -- 6.1.2 Criteria for selecting the solution method -- 6.1.3 Accuracy of eigensolutions and stopping criteria -- 6.2 Dynamical and symmetric iteration matrices -- 6.3 Computing the determinant: Sturm sequences -- 6.4 Matrix transformation methods -- 6.4.1 Reduction to a diagonal form: Jacobi's method -- 6.4.2 Reduction to a tridiagonal form: Householder's method -- 6.5 Iteration on eigenvectors: the power algorithm -- 6.5.1 Computing the fundamental eigensolution -- 6.5.2 Determining higher modes: orthogonal deflation -- 6.5.3 Inverse iteration form of the power method -- 6.6 Solution methods for a linear set of equations -- 6.6.1 Nonsingular linear systems -- 6.6.2 Singular systems: nullspace, solutions and generalized inverse -- 6.6.3 Singular matrix and nullspace -- 6.6.4 Solution of singular systems -- 6.6.5 A family of generalized inverses.

6.6.6 Solution by generalized inverses and finding the nullspace N.

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Cover -- TItle Page -- Copyright -- Contents -- Foreword -- Preface -- Introduction -- Suggested Bibliography -- List of main symbols and definitions -- Chapter 1 Analytical Dynamics of Discrete Systems -- Definitions -- 1.1 Principle of virtual work for a particle -- 1.1.1 Nonconstrained particle -- 1.1.2 Constrained particle -- 1.2 Extension to a system of particles -- 1.2.1 Virtual work principle for N particles -- 1.2.2 The kinematic constraints -- 1.2.3 Concept of generalized displacements -- 1.3 Hamilton's principle for conservative systems and Lagrange equations -- 1.3.1 Structure of kinetic energy and classification of inertia forces -- 1.3.2 Energy conservation in a system with scleronomic constraints -- 1.3.3 Classification of generalized forces -- 1.4 Lagrange equations in the general case -- 1.5 Lagrange equations for impulsive loading -- 1.5.1 Impulsive loading of a mass particle -- 1.5.2 Impulsive loading for a system of particles -- 1.6 Dynamics of constrained systems -- 1.7 Exercises -- 1.7.1 Solved exercises -- 1.7.2 Selected exercises -- References -- Chapter 2 Undamped Vibrations of n-Degree-of-Freedom Systems -- Definitions -- 2.1 Linear vibrations about an equilibrium configuration -- 2.1.1 Vibrations about a stable equilibrium position -- 2.1.2 Free vibrations about an equilibrium configuration corresponding to steady motion -- 2.1.3 Vibrations about a neutrally stable equilibrium position -- 2.2 Normal modes of vibration -- 2.2.1 Systems with a stable equilibrium configuration -- 2.2.2 Systems with a neutrally stable equilibrium position -- 2.3 Orthogonality of vibration eigenmodes -- 2.3.1 Orthogonality of elastic modes with distinct frequencies -- 2.3.2 Degeneracy theorem and generalized orthogonality relationships -- 2.3.3 Orthogonality relationships including rigid-body modes.

2.4 Vector and matrix spectral expansions using eigenmodes -- 2.5 Free vibrations induced by nonzero initial conditions -- 2.5.1 Systems with a stable equilibrium position -- 2.5.2 Systems with neutrally stable equilibrium position -- 2.6 Response to applied forces: forced harmonic response -- 2.6.1 Harmonic response, impedance and admittance matrices -- 2.6.2 Mode superposition and spectral expansion of the admittance matrix -- 2.6.3 Statically exact expansion of the admittance matrix -- 2.6.4 Pseudo-resonance and resonance -- 2.6.5 Normal excitation modes -- 2.7 Response to applied forces: response in the time domain -- 2.7.1 Mode superposition and normal equations -- 2.7.2 Impulse response and time integration of the normal equations -- 2.7.3 Step response and time integration of the normal equations -- 2.7.4 Direct integration of the transient response -- 2.8 Modal approximations of dynamic responses -- 2.8.1 Response truncation and mode displacement method -- 2.8.2 Mode acceleration method -- 2.8.3 Mode acceleration and model reduction on selected coordinates -- 2.9 Response to support motion -- 2.9.1 Motion imposed to a subset of degrees of freedom -- 2.9.2 Transformation to normal coordinates -- 2.9.3 Mechanical impedance on supports and its statically exact expansion -- 2.9.4 System submitted to global support acceleration -- 2.9.5 Effective modal masses -- 2.9.6 Method of additional masses -- 2.10 Variational methods for eigenvalue characterization -- 2.10.1 Rayleigh quotient -- 2.10.2 Principle of best approximation to a given eigenvalue -- 2.10.3 Recurrent variational procedure for eigenvalue analysis -- 2.10.4 Eigensolutions of constrained systems: general comparison principle or monotonicity principle -- 2.10.5 Courant's minimax principle to evaluate eigenvalues independently of each other.

2.10.6 Rayleigh's theorem on constraints (eigenvalue bracketing) -- 2.11 Conservative rotating systems -- 2.11.1 Energy conservation in the absence of external force -- 2.11.2 Properties of the eigensolutions of the conservative rotating system -- 2.11.3 State-space form of equations of motion -- 2.11.4 Eigenvalue problem in symmetrical form -- 2.11.5 Orthogonality relationships -- 2.11.6 Response to nonzero initial conditions -- 2.11.7 Response to external excitation -- 2.12 Exercises -- 2.12.1 Solved exercises -- 2.12.2 Selected exercises -- References -- Chapter 3 Damped Vibrations of n-Degree-of-Freedom Systems -- Definitions -- 3.1 Damped oscillations in terms of normal eigensolutions of the undamped system -- 3.1.1 Normal equations for a damped system -- 3.1.2 Modal damping assumption for lightly damped structures -- 3.1.3 Constructing the damping matrix through modal expansion -- 3.2 Forced harmonic response -- 3.2.1 The case of light viscous damping -- 3.2.2 Hysteretic damping -- 3.2.3 Force appropriation testing -- 3.2.4 The characteristic phase lag theory -- 3.3 State-space formulation of damped systems -- 3.3.1 Eigenvalue problem and solution of the homogeneous case -- 3.3.2 General solution for the nonhomogeneous case -- 3.3.3 Harmonic response -- 3.4 Experimental methods of modal identification -- 3.4.1 The least-squares complex exponential method -- 3.4.2 Discrete Fourier transform -- 3.4.3 The rational fraction polynomial method -- 3.4.4 Estimating the modes of the associated undamped system -- 3.4.5 Example: experimental modal analysis of a bellmouth -- 3.5 Exercises -- 3.5.1 Solved exercises -- 3.6 Proposed exercises -- References -- Chapter 4 Continuous Systems -- Definitions -- 4.1 Kinematic description of the dynamic behaviour of continuous systems: Hamilton's principle -- 4.1.1 Definitions.

4.1.2 Strain evaluation: Green's measure -- 4.1.3 Stress-strain relationships -- 4.1.4 Displacement variational principle -- 4.1.5 Derivation of equations of motion -- 4.1.6 The linear case and nonlinear effects -- 4.2 Free vibrations of linear continuous systems and response to external excitation -- 4.2.1 Eigenvalue problem -- 4.2.2 Orthogonality of eigensolutions -- 4.2.3 Response to external excitation: mode superposition (homogeneous spatial boundary conditions) -- 4.2.4 Response to external excitation: mode superposition (nonhomogeneous spatial boundary conditions) -- 4.2.5 Reciprocity principle for harmonic motion -- 4.3 One-dimensional continuous systems -- 4.3.1 The bar in extension -- 4.3.2 Transverse vibrations of a taut string -- 4.3.3 Transverse vibration of beams with no shear deflection -- 4.3.4 Transverse vibration of beams including shear deflection -- 4.3.5 Travelling waves in beams -- 4.4 Bending vibrations of thin plates -- 4.4.1 Kinematic assumptions -- 4.4.2 Strain expressions -- 4.4.3 Stress-strain relationships -- 4.4.4 Definition of curvatures -- 4.4.5 Moment-curvature relationships -- 4.4.6 Frame transformation for bending moments -- 4.4.7 Computation of strain energy -- 4.4.8 Expression of Hamilton's principle -- 4.4.9 Plate equations of motion derived from Hamilton's principle -- 4.4.10 Influence of in-plane initial stresses on plate vibration -- 4.4.11 Free vibrations of the rectangular plate -- 4.4.12 Vibrations of circular plates -- 4.4.13 An application of plate vibration: the ultrasonic wave motor -- 4.5 Wave propagation in a homogeneous elastic medium -- 4.5.1 The Navier equations in linear dynamic analysis -- 4.5.2 Plane elastic waves -- 4.5.3 Surface waves -- 4.6 Solved exercises -- 4.7 Proposed exercises -- References -- Chapter 5 Approximation of Continuous Systems by Displacement Methods -- Definitions.

5.1 The Rayleigh-Ritz method -- 5.1.1 Choice of approximation functions -- 5.1.2 Discretization of the displacement variational principle -- 5.1.3 Computation of eigensolutions by the Rayleigh-Ritz method -- 5.1.4 Computation of the response to external loading by the Rayleigh-Ritz method -- 5.1.5 The case of prestressed structures -- 5.2 Applications of the Rayleigh-Ritz method to continuous systems -- 5.2.1 The clamped-free uniform bar -- 5.2.2 The clamped-free uniform beam -- 5.2.3 The uniform rectangular plate -- 5.3 The finite element method -- 5.3.1 The bar in extension -- 5.3.2 Truss frames -- 5.3.3 Beams in bending without shear deflection -- 5.3.4 Three-dimensional beam element without shear deflection -- 5.3.5 Beams in bending with shear deformation -- 5.4 Exercises -- 5.4.1 Solved exercises -- 5.4.2 Selected exercises -- References -- Chapter 6 Solution Methods for the Eigenvalue Problem -- Definitions -- 6.1 General considerations -- 6.1.1 Classification of solution methods -- 6.1.2 Criteria for selecting the solution method -- 6.1.3 Accuracy of eigensolutions and stopping criteria -- 6.2 Dynamical and symmetric iteration matrices -- 6.3 Computing the determinant: Sturm sequences -- 6.4 Matrix transformation methods -- 6.4.1 Reduction to a diagonal form: Jacobi's method -- 6.4.2 Reduction to a tridiagonal form: Householder's method -- 6.5 Iteration on eigenvectors: the power algorithm -- 6.5.1 Computing the fundamental eigensolution -- 6.5.2 Determining higher modes: orthogonal deflation -- 6.5.3 Inverse iteration form of the power method -- 6.6 Solution methods for a linear set of equations -- 6.6.1 Nonsingular linear systems -- 6.6.2 Singular systems: nullspace, solutions and generalized inverse -- 6.6.3 Singular matrix and nullspace -- 6.6.4 Solution of singular systems -- 6.6.5 A family of generalized inverses.

6.6.6 Solution by generalized inverses and finding the nullspace N.

Mechanical Vibrations: Theory and Application to Structural Dynamics, Third Edition is a comprehensively updated new edition of the popular textbook. It presents the theory of vibrations in the context of structural analysis and covers applications in mechanical and aerospace engineering. Key features include: A systematic approach to dynamic reduction and substructuring, based on duality between mechanical and admittance concepts An introduction to experimental modal analysis and identification methods An improved, more physical presentation of wave propagation phenomena A comprehensive presentation of current practice for solving large eigenproblems, focusing on the efficient linear solution of large, sparse and possibly singular systems A deeply revised description of time integration schemes, providing framework for the rigorous accuracy/stability analysis of now widely used algorithms such as HHT and Generalized-α Solved exercises and end of chapter homework problems A companion website hosting supplementary material.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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