Matching of local expansions in the theory of non-linear vibrations

Abstract

Abstract -linear systems are a generalization of normal (principal) vibrations of linear systems [1-3]. In this case all position co-ordinates can be defined well from any one of them. R. M. Rosenberg is credited with being the first to introduce broad classes of conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space. In systems of a more general type, trajectories of normal vibrations are curvilinear. Assume that in a conservative system the potential energy is a positively definite polynomial in the co-ordinates. At small amplitudes a linear system is to be selected as the initial one, while at large amplitudes a homogeneous non-linear system allows normal vibrations with rectilinear trajectories. In the vicinity of a linear system, trajectories of normal vibrations can be determined as power series in the amplitude; while in the vicinity of a homogeneous non-linear system, they can be determined as power series in the inverse amplitude. In order to join together local expansions and to investigate the behavior of normal vibration trajectories at arbitrary amplitude values, fractional rational diagonal Pade approximants are used. Necessary conditions for the convergence of a succession of Pade approximants have been obtained, and that allows one to establish relations between quasi-linear and essentially non-linear expansions: that is, to decide which of them correspond to the same solution and which to different ones. Additional modes of vibrations exist only in a non-linear systems; as the amplitude decreases, they vanish at a certain limiting point.