Book contents
- Frontmatter
- Preface
- Contents
- Introduction
- 1 Basic properties and definitions
- 2 Methods of solution
- 3 Auxiliary curves for analysis of non-linear systems
- 4 Analysis in the phase plane
- 5 Forced, parametric and self-excited vibrations
- 6 Vibrations of systems with many degrees of freedom
- 7 Investigation of stability in the large
- 8 Analysis of some excited systems
- 9 Quenching of self-excited vibration
- 10 Vibration systems with narrow-band random excitation
- 11 Vibration systems with broad-band random excitation
- 12 Systems with autoparametric coupling
- Appendix
- Bibliography
- Index
7 - Investigation of stability in the large
Published online by Cambridge University Press: 10 May 2010
- Frontmatter
- Preface
- Contents
- Introduction
- 1 Basic properties and definitions
- 2 Methods of solution
- 3 Auxiliary curves for analysis of non-linear systems
- 4 Analysis in the phase plane
- 5 Forced, parametric and self-excited vibrations
- 6 Vibrations of systems with many degrees of freedom
- 7 Investigation of stability in the large
- 8 Analysis of some excited systems
- 9 Quenching of self-excited vibration
- 10 Vibration systems with narrow-band random excitation
- 11 Vibration systems with broad-band random excitation
- 12 Systems with autoparametric coupling
- Appendix
- Bibliography
- Index
Summary
Fundamental considerations
Many non-linear systems of engineering interest possess more than one stationary solution stable against small disturbances, that is, more than one attractor. Such solutions are termed locally stable. In practical applications usually only one of these is desirable; the others, for the most part, are unwelcome because they signal danger to safe and reliable operation of the device whose model is being analyzed. The problem is to establish the conditions which lead to a particular steady state, or alternatively, to examine the disturbances which are apt to cause one stationary state to change to another (for example, small-amplitude stationary vibration to large-amplitude motion, a non-oscillatory state to an oscillatory condition, etc.).
The solution to the first aspect of the problem, i.e. establishing the domains of the initial conditions which lead to different stationary solutions, is well known. These domains are called the domains of attraction of a particular solution. For systems which are directly described by a set of two first-order differential equations of the type (4.1, 1) or whose set of such equations is identical with the original second-order differential equation the problem is solved by means of phase portrait analysis outlined in Chapter 4. The differential equations of motion of one-degree-of-freedom systems excited harmonically by an external force or parametrically can be converted, by application of known procedures (van der Pol or Krylov and Bogoljubov methods) to a system of two first-order differential equations of the type (4.1, 1) which, however, are not identical with the original equation of motion.
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- Non-linear Vibrations , pp. 199 - 246Publisher: Cambridge University PressPrint publication year: 1986