Forced oscillations in poorly nonlinear system with one degree of freedom at power affecting

The lecture purpose – acquaintance with crude methods of the analysis of oscillatory processes in poorly nonlinear systems.

Unlike linear systems, there is no general method of mathematical research of these difficult oscillations at any nonlinearity of oscillatory system. However for poorly nonlinear systems there are research crude methods. In poorly nonlinear system the oscillations originating under the influence of a harmonious superposed force, poorly differ from harmonic oscillations. It allows to simplify essentially research of forced oscillations in poorly nonlinear systems, making the assumption that forced oscillations are harmonious and can be written down in an aspect:

. (1)

Let these oscillations exist under the influence of a harmonious superposed force

. (2)

Forced oscillations under the influence of this harmonious force will be presented by the nonlinear equation

(3)

Where - poorly nonlinear function.

Let's observe poorly nonlinear conservative system most simply giving in to the analysis with one degree of freedom, being under the influence of a harmonious superposed force. Weak nonlinearity of the equation is expressed by poorly nonlinear restoring force f (x).

In this case the differential equation looks like

. (4)

Let's substitute in this equation the prospective solution in the form of harmonic oscillations:.

Thus there can be a question on how the information on nonlinearity of system if oscillations in it are considered harmonious ­ in the same way, as well as in linear system remains­. The answer: this information is expressed in amplitude and, forced oscillations in nonlinear system­; namely, the amplitude and oscillations in poorly nonlinear system will be another in comparison about amplitude of forced oscillations in linear system.

Lecture 7

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