Vibration of Discrete and Continuous Systems

1 Introduction.- 1.1 Kinematics of Rigid Bodies.- 1.2 Dynamic Equations.- 1.3 Single Degree of Freedom Systems.- 1.4 Oscillatory and Nonoscillatory Motion.- 1.5 Other Types of Damping.- 1.6 Forced Vibration.- 1.7 Impulse Response.- 1.8 Response to an Arbitrary Forcing Function.- Problems.- 2 Lagrangian Dynamics.- 2.1 Generalized Coordinates.- 2.2 Virtual Work and Generalized Forces.- 2.3 Lagrange’s Equation.- 2.4 Kinetic Energy.- 2.5 Strain Energy.- 2.6 Hamilton’s Principle.- 2.7 Conservation Theorems.- 2.8 Concluding Remarks.- Problems.- 3 Multi-Degree of Freedom Systems.- 3.1 Equations of Motion.- 3.2 Undamped Free Vibration.- 3.3 Orthogonality of the Mode Shapes.- 3.4 Rigid-Body Modes.- 3.5 Conservation of Energy.- 3.6 Forced Vibration of the Undamped Systems.- 3.7 Viscously Damped Systems.- 3.8 General Viscous Damping.- 3.9 Approximation and Numerical Methods.- 3.10 Matrix-Iteration Methods.- 3.11 Method of Transfer Matrices173 Problems.- 4 Vibration of Continuous Systems.- 4.1 Free Longitudinal Vibrations.- 4.2 Free Torsional Vibrations.- 4.3 Free Transverse Vibrations of Beams.- 4.4 Orthogonality of the Eigenfunctions.- 4.5 Forced Vibrations.- 4.6 Inhomogeneous Boundary Conditions.- 4.7 Viscoelastic Materials.- 4.8 Energy Methods.- 4.9 Approximation Methods.- 4.10 Galerkin’s Method.- 4.11 Assumed-Modes Method259 Problems.- 5 The Finite-Element Method.- 5.1 Assumed Displacement Field.- 5.2 Comments on the Element Shape Functions.- 5.3 Connectivity Between Elements.- 5.4 Formulation of the Mass Matrix.- 5.5 Formulation of the Stiffness Matrix.- 5.6 Equations of Motion.- 5.7 Convergence of the Finite-Element Solution.- 5.8 Higher-Order Elements.- 5.9 Spatial Elements.- 5.10 Large Rotations and Deformations323 Problems.- 6 Methods for the Eigenvalue Analysis.- 6.1 Similarity Transformation.- 6.2 Polynomial Matrices.- 6.3 Equivalence of the Characteristic Matrices.- 6.4 Jordan Matrices.- 6.5 Elementary Divisors.- 6.6 Generalized Eigenvectors.- 6.7 Jacobi Method.- 6.8 Householder Transformation.- Appendix A Linear Algebra.- A.1 Matrices.- A.2 Matrix Operations.- A.3 Vectors.- A.4 Eigenvalue Problem.- Problems.- References.