Study of the quasilinear oscillations of mechanical systems with distributed and lumped parameters

Abstract

The averaging method is used to study a class of complex oscillatory systems which are described by vector integrodifferential equations with oscillating kernels. These equations arise when analysing mechanical objects which contain elements with distributed and lumped inertial and elastic parameters. Two physically distinct cases of the oscillation of rigid bodies are considered: “resonant” and “non-resonant”, as determined by the properties of the mean values of the kernels of the integral terms. In the first case, it is shown that the equations of the first approximation are equivalent to a system of ordinary second-order differential equations, i.e., the order of the system of equations of the motion of a rigid body is doubled. In the second case, sufficient conditions are found for the oscillating initial variables to be slow in the usual sense of the averaging method; the order of the system is then preserved. The conditions are stated, under which the averaging method can be shown to be strictly applicable in asymptotically long time intervals and constructive error estimates are obtained. On the basis of this approach the perturbed horizontal oscillations of a rigid body containing a rectangular cavity with a two-layer heavy fluid which is elastically connected with a fixed base are investigated and qualitative effects are discovered and examined.