In the last decade the development in vibration analysis was char­ acterized by increasing demands on precision and by the growing use of electronic computers. At present, improvements in precision are obtained by a more accurate modelling of technical systems. Thus, for instance, a system with one degree of freedom is often not accepted, as it used to be, as a model for vibration analysis in mechanical engineering. As a rule, vehicles and machines have to be modelled as systems with many degrees of freedom such as multibody systems, finite element systems or con­ tinua. The mathematical description of multi-degree-of-freedom systems leads to matrix representations of the corresponding equations. These are then conveniently analyzed by means of electronic computers, that is, by the analog computer and especially by the digital machine. Hence there exists a mutually stimulating interaction between the growing require­ ments and the increasing computational facilities. The present book deals with linear vibration analysis of technical systems with many degrees of freedom in a form allowing the use of computers for finding solutions. Part I begins with the classification of vibrating systems. The main characteristics here are the kind of differential equation, the time depen­ dence of the coefficients and the attributes of the exciting process. Next it is shown by giving examples involving mechanical vibrating systems how to set up equations of motion and how to transform these into state equations.

Inhalt

I Mathematical description of vibrating systems.- 1. Classification of vibrations.- 2. Mechanical vibrating systems.- 2.1. Modeling of technical systems.- 2.2. Kinematics of multibody system.- 2.3. Lagrange's equations of motion.- 2.4. Newton-Euler equations of motion.- Problems.- 3. State equations of linear vibrating systems.- 3.1. Ordinary mechanical systems.- 3.2. General linear systems.- 3.3. Transformation of linear state equations.- Problems.- II Time-invariant vibrating systems.- 4. The general solution of time-invariant vibrating systems.- 4.1. Fundamental matrix.- 4.2. General solution.- 4.3. Eigenvalues, eigenvectors.- 4.4. Theorem of Cayley and Hamilton.- Problems.- 5. Stability and boundedness.- 5.1. Definitions.- 5.2. Stability.- 5.2.1. Stability criteria by eigenvalues.- 5.2.2. Stability criteria by characteristic coefficients.- 5.2.3. Stability criteria by the Ljapunov matrix equation.- 5.2.4. Stability criteria for mechanical systems.- 5.3. Boundedness.- Problems.- 6. Free vibrations.- 6.1. Natural modes of vibrations.- 6.1.1. Nongyroscopic conservative systems.- 6.1.2. Lightly damped mechanical systems.- 6.1.3. Ordinary mechanical systems.- 6.2. Optimal natural vibrations.- 6.2.1. Cost functionals.- 6.2.2. Calculation of the cost functionals.- 6.2.3. Optimal parameters.- Problems.- 7. Forced vibrations.- 7.1. Impulsive excitation.- 7.2. Step excitation.- 7.3. Periodic excitation.- 7.4. Harmonically excited mechanical systems.- Problems.- 8. Resonance and absorption.- 8.1. Elementary frequency response matrix.- 8.2. Elementary frequency responses.- 8.3. Resonance and pseudoresonance.- 8.4. Absorption.- 8.5. Pseudoresonance and absorption in mechanical systems.- 8.6. Fixpoints.- 8.7. Optimal frequency responses.- 8.8. Parameter identification.- Problems.- 9. Random vibrations.- 9.1. Stochastic vector processes.- 9.2. Stochastic excitation.- 9.3. Spectral density analysis.- 9.4. Covariance analysis.- 9.5. Colored noise excitation processes.- Problems.- III Time-variant vibrating systems.- 10. General solution and stability.- 10.1. General time-variant systems.- 10.2. Periodic time-variant systems.- 10.3. Stability of periodic time-variant systems.- 10.4. Mechanical systems.- 10.4.1. Mathieu's differential equation.- 10.4.2. Single-degree-of-freedom mechanical systems.- 10.4.3. Multi-degree-of-freedom mechanical systems.- Problems.- 11. Parameter-excited and forced vibrations.- 11.1. Parameter-excited vibrations.- 11.2. Impulsive excitation.- 11.3. Step excitation.- 11.4. Periodic excitation.- 11.5. Stochastic excitation.- Problems.- IV Mathematical background.- 12. Controllability and observability.- 13. Matrix equations.- 13.1. The linear vector equation.- 13.2. The Ljapunov matrix equation.- 13.3. Stein's matrix equation.- 14. Numerical aspects.- 14.1. The linear vector equation.- 14.2. The Ljapunov matrix equation.- 14.3. Eigenvalues, eigenvectors.- 14.4. Frequency response.- 14.5. Fundamental matrix.- 14.6. Analog integration of the fundamental matrix.- 14.7. Digital integration of the fundamental matrix.- References.