Analytical investigation on the dynamics of the smooth and discontinuous oscillator

Abstract

© 2019, Springer-Verlag GmbH Austria, part of Springer Nature. In this paper, the dynamics of a nonlinear smooth and discontinuous oscillator, modeled as a string–mass structure, is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquakes. The considered string–mass structure contains a translator movable mass connected with two strings. The motion of the mass is oscillatory and perpendicular to the string’s position. Usually, in strings preloading forces act. Due to geometric and physical properties of the system, the restitution force of the string is nonlinear. The model of the mass motion is a strong nonlinear second-order differential equation. The nonlinearity is of power type, and the order of nonlinearity is any positive real number. An exact solution of the truly nonlinear equation is introduced in the form of the Ateb function (inverse Beta function). Based on the exact solution, the approximate solving procedure of the nonlinear equation of motion is developed. The method is suitable for dynamic analyses of the system. The influence of the preloading force on the nonlinear vibrations of the string–mass system is considered. It is concluded that variation of the string force has influence on the velocity of the amplitude decrease in the system.