Preface. 1. SINGLE DEGREE OF FREEDOM SYSTEMS. Introduction. Spring-Mass System. Spring-Mass-Damper System. Forced Response. Transfer Functions and Frequency Methods. Measurement and Testing. Stability. Design and Control of Vibrations. Nonlinear Vibrations. Computing and Simulation in Matlab. Chapter Notes. References. Problems. 2. LUMPED PARAMETER MODELS. Introduction. Classifications of Systems. Feedback Control Systems. Examples. Experimental Models. Influence Methods. Nonlinear Models and Equilibrium. Chapter Notes. References. Problems. 3. MATRICES AND THE FREE RESPONSE. Introduction. Eigenvalues and Eigenvectors. Natural Frequencies and Mode Shapes. Canonical Forms. Lambda Matrices. Oscillation Results. Eigenvalue Estimates. Computational Eigenvalue Problems in Matlab. Numerical Simulation of the Time Response in Matlab. Chapter Notes. References. Problems. 4. STABILITY. Introduction. Lyapunov Stability. Conservative Systems. Systems with Damping. Semidefinite Damping . Gyroscopic Systems. Damped Gyroscopic Systems. Circulatory Systems. Asymmetric Systems. Feedback Systems. Stability in the State Space. Stability Boundaries. Chapter Notes. References. Problems. 5. FORCED RESPONSE OF LUMPED PARAMETER SYSTEMS. Introduction. Response via State Space Methods. Decoupling Conditions and Modal Analysis. Response of Systems with Damping. Bounded-Input, Bounded-Output Stability. Response Bounds. Frequency Response Methods. Numerical Simulations in Matlab. Chapter Notes. References. Problems. 6. DESIGN CONSIDERATIONS. Introduction. Isolators and Absorbers. Optimization Methods. Damping Design. Design Sensitivity and Redesign. Passive and Active Control. Design Specifications. Model Reduction. Chapter Notes. References. Problems. 7. CONTROL OF VIBRATIONS. Introduction. Controllability and Observability. Eigenstructure Assignment. Optimal Control. Observers (Estimators). Realization. Reduced-Order Modeling. Modal Control in State Space. Modal Control in Physical Space. Robustness. Positive Position Feedback Control. Matlab Commands for Control Calculations. Chapter Notes. References. Problems. 8. VIBRATION MEASUREMENT. Introduction. Measurement Hardware. Digital Signal Processing. Random Signal Analysis. Modal Data Extraction (Frequency Domain). Modal Data Extraction (Time Domain). Model Identification. Model Updating. Chapter Notes. References. Problems. 9. DISTRIBUTED PARAMETER MODELS. Introduction. Vibrations of Strings. Rods and Bars. Vibration of Beams. Membranes and Plates. Layered Materials. Viscous Damping. Chapter Notes. References. Problems. 10. FORMAL METHODS OF SOLUTION. Introduction. Boundary Value Problems and Eigenfunctions. Modal Analysis of the Free Response. Modal Analysis in Damped Systems. Transform Methods. Green's Functions. Chapter Notes. References. Problems. 11. OPERATORS AND THE FREE RESPONSE. Introduction. Hilbert Spaces. Expansion Theorems. Linear Operators. Compact Operators. Theoretical Modal Analysis. Eigenvalue Estimates. Enclosure Theorems. Oscillation Theory. Chapter Notes. References. Problems. 12. FORCED RESPONSE AND CONTROL. Introduction. Response by Modal Analysis. Modal Design Criteria. Combined Dynamical Systems. Passive Control and Design. Distribution Modal Control. Nonmodal Distributed Control. State Space Control Analysis. Chapter Notes. References. Problems. 13. APPROXIMATIONS OF DISTRIBUTED PARAMETER MODELS. Introduction. Modal Truncation. Rayleigh- Ritz-Galerkin Approximations. Finite Element Method. Substructure Analysis. Truncation in the Presence of Control. Impedance Method of Truncation and Control. Chapter Notes. References. Problems. APPENDIX A: COMMENTS ON UNITS. APPENDIX B: SUPPLEMENTARY MATHEMATICS. Index.

Vibration With Control

The decentralized robust servomechanism problem with constant disturbances/set points is considered in this paper for large flexible space structures (LFSS). It is shown that for LFSS which have colocated, mutually dual sensors and actuators, the decentralized fixed modes [1] of the system are precisely equal to the centralized fixed modes [2] of the system. Simple necessary and sufficient conditions are then obtained for a solution to exist for the robust decentralized servomechanism problem [3] for the system. A controller is demonstrated which, for this class of LFSS systems, eliminates the "spillover problem." A two-hundreth-order numerical example of an LFSS control problem using the Purdue model [4] is included to illustrate the results.

The decentralized control of large flexible space structures

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Lyapunov stability, semistability, and asymptotic stability of matrix second-order systems

This article describes how a large number of practical feedback design problems can be reduced to the study of quantified multivariate polynomial inequalities (MPIs). However, the computation required to solve quantified MPI problems is very intensive. As defined here, most practical control problems do not have analytical solutions. Three approaches for the study of this class of mathematical problems are reviewed: symbolic quantifier elimination methods, Bernstein branch-and-bound methods, and probabilistic (Monte Carlo) methods. The three approaches are listed in order of computational complexity required for a solution, with symbolic computation the most computationally complex and probabilistic methods the least.

W. E. Schmitendorf

Papers

A Simple Test for Asymptotic Stability in Partially Dissipative Symmetric Systems