Prediction of energy dissipation by analytical solution to combined viscous and Coulomb damping

Damping in mechanical vibration systems primarily falls under (i) linear viscous and (ii) non-linear Coulomb damping. Practically, most systems possess characteristics of both linear viscous and non-linear Coulomb damping. In this work, the method of phase space coordinate averaging is used to obtain an approximate analytical solution for the system's response when it is subjected to both viscous and Coulomb damping. The accuracy of the proposed solution is examined by comparing it with the numerical simulations and experimental results. The energy dissipated by both damping mechanisms is quantified using semi-analytical integrals. The analysis of energy dissipation quantifies the conditions for natural frequency, damping ratio, and coefficient of friction, at which the energy dissipated by viscous damping is equal to the energy dissipated by Coulomb damping. This has led to the introduction of a non-dimensional quantity called the dissipation number (D). With certainty, it can be stated that if D > 1, friction is dominant, and if D < 1, viscosity is dominant.