High-frequency natural oscillations of mechanical systems

Higher modes of natural oscillations of a wide class of mechanical systems, described by general boundary-value problems with various types of boundar y conditions, are investigated. An effective method of determining the osci llation frequencies and shapes, based on the use of asymptotic methods of n on-linear mechanics (the averaging, accelerated convergence and asymptotic expansion methods) is developed. Expressions for the eigenvalues (frequenci es) and eigenfunctions (shapes) are obtained in an explicit form with the r equired degree of accuracy in negative powers of the order number of the mo de, and a justification of the method is given. The eigenvalues for specifi c mechanical systems, which perform free or parametric oscillations, are ca lculated. The oscillations of a homogeneous rod and the transverse vibratio ns of a tightly stretched inhomogeneous string are considered. The higher r esonance zones in Hill's problem of parametric oscillations and in the prob lem of small spatial oscillations of a dynamically symmetrical satellite wh ose polar axis performs non-linear oscillations in the plane of a circular orbit are investigated. Some mechanical effects are detected and described. (C) 2001 Elsevier Science Ltd. All rights reserved.