Forced Vibration of a Two-Degree-of-Freedom System with Combined Coulomb and Viscous Damping

Abstract

An exact solution for steady forced vibration of a two-degree-of-freedom system with two viscous dampers and one Coulomb damper subjected to a simple harmonic ground excitation is presented. The solution is valid for motions without standstill. The results are expressed in terms of nondimensional parameters that allow evaluations of response amplitudes and phase angles for various frequency ratios, mass ratios, and amounts of viscous and Coulomb damping. In the limits, the resulting expressions agree properly with those for a system without the Coulomb damper and with Den Hartog's expressions for a single-degree-of-freedom system with a Coulomb damper. Formulations for the special case of a two-degree-of-freedom system with a Coulomb damper (without viscous dampers) are also obtained. Numerical results are presented to compare the exact and approximate solutions for a single-degree-of-freedom system with combined Coulomb and viscous damping. The exact solution obtained in this paper can be used to estimate the accuracy of approximate methods for analyzing more-complex systems containing Coulomb damping.