Non-linear vibration, non-linear characteristics and basic definitions

As mentioned in the Introduction, vibration of linear systems is only a special case of vibration of non-linear systems. Two important principles, which are valid for linear systems, do not apply to non-linear systems:

1. (I) The principle of superposition,

2. (II) The principle of proportionality.

In linear systems described by linear differential equations (all terms are functions of dependent variables and their derivatives) the response to the various components of excitation can be added up; if the amplitude of harmonic excitation is increased n times, the amplitudes of steady vibration increase n times. It has been found advantageous to divide linear vibration into two large groups:

1. (a) Vibration of systems with constant parameters described by linear differential equations with constant coefficients. The term linear vibration is usually understood to mean vibration of systems of just this type. The theory of this vibration is essentially complete and will not be discussed in the present book.

2. (b) Vibration of systems with non-constant parameters described by linear differential equations whose coefficients are generally functions of a dependent variable, chiefly time. An important separate group is formed by systems whose parameters are periodic functions of a dependent variable. Such systems are called rheo-linear and their theory is usually studied together with that of non-linear vibration. One of the reasons for this is the fact that an investigation of stability for small disturbances of steady periodic solutions of non-linear differential equations generally leads to an investigation of stability of the trivial solution of systems of differential equations with periodically variable coefficients.