Vibration of Discrete and Continuous Systems

Mechanical engineering, an engineering discipline borne of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research monographs intended to address the need for information in con­ temporary areas of mechanical engineering. The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research. We are fortunate to have a distinguished roster of consulting editors on the advisory board, each an expert in one of the areas of concen­ tration. The names of the consulting editors are listed on the next page of this volume. The areas of concentration are: applied mechanics; bio­ mechanics; computational mechanics; dynamic systems and control; energetics; mechanics of materials; processing; thermal science; and tribology. Professor Marshek, the consulting editor for dynamic systems and control, and I are pleased to present the second edition of Vibration of Discrete and Continuous Systems by Professor Shabana. We note that this is the second of two volumes. The first deals with the theory of vibration.

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.