CESSATION OF FREE VIBRATIONS OF A NONLINEAR ELASTIC SYSTEM DUE TO VISCOUS RESISTANCE

Abstract

The paper deals with free vibrations of a system with power-law nonlinear elasticity subjected to power-law viscous resistance. The relation between the nonlinearity indices is determined when the impact of the viscous resistance force causes the vibrations to die away. In this case the vibrations are limited in time i.e. consist of a finite number of cycles analogous to a system with Coulomb dry friction. The research exploits the energy balance method. The periodic Ateb-functions are used to obtain an approximate formula for the work of dissipative force over a semi-cycle of vibrations. A recursive power-law equation for the vibration swings is derived from the condition of equality of the work to the potential energy change. By analyzing the change of the coefficient in the equation, which is related to the change of the semi-cycle number as well as the vibration swings, the condition for the equation to have no positive root is determined, which means that the vibrations die away. The condition is formulated in the form of an inequality. It is shown to generalize the results previously known. The theoretical inferences are verified by numerical integration of the nonlinear differential equation of motion. It is shown that under the conditions proposed in the paper the free vibrations consist of a finite number of cycles even if dry friction is absent from the system. Special cases are highlighted, when the approximate energy balance method results into exact computational formulae. The length of the cycles increases during the motion since it depends on the swing of damped vibrations in the essentially nonlinear system with rigid force characteristics considered.