Vibration of a two-mass system with non-integer order nonlinear connection

Abstract

In this paper a generalization of a two-mass oscillatory system is done. Connection between two masses is in general of the visco-elastic type where the elastic and damping properties are of the nonlinear type. Motion of the system is described with a system of two coupled second order differential equations (TDE) where the nonlinearity is of any order (integer and/or non-integer). An approximate solution of the TDE is obtained by introducing the intermediate variables which give a single uncoupled differential equation for which the solution is already known. Cveticanin's solution procedure developed for a single second order nonlinear differential equation is extended for solving the TDE. The procedure suggested in the paper is based on the exact analytically determined frequency and period of vibration. The obtained solutions show that for the case of the pure elastic connection the masses oscillate around the same position which is the averaged value of the initial deflections. For both masses the amplitudes and periods of vibration are equal but their motion is in opposite directions. The amplitude of vibration is a linear and frequency a nonlinear (of integer or non-integer order) function of the difference between initial deflections of masses. If in the system damping acts amplitudes of vibration for both masses decrease. The amplitude decrease depends not only on the coefficient of damping (as it was the case for linear systems) but on the initial properties of the system, coefficients of elasticity and order of nonlinearities of the elastic and damping forces. The periods of vibration increase if the damping acts. The frequency of vibration is a complex function of initial displacements of masses, coefficients of elasticity and damping and of the order of nonlinearities of the connection. Two numerical examples illustrate the suggested procedure and results.