Vibration and Stability of Frictional Sliding of Two Elastic Bodies With a Wavy Contact Interface

Abstract

The stability of steady sliding, with Amontons-Coulomb friction, of two elastic bodies with a rough contact interface is analyzed. The bodies are modeled as elastic half-spaces, one of which has a periodic wavy surface. The steady-state solution yields a periodic set of contact and separation zones, but the stability analysis requires consideration of dynamic effects. By considering a spatial Fourier decomposition of the vibration modes, the dynamic problem is reduced to a singular integral equation for determining the eigenvectors (modes) and eigenvalues (frequencies). A pure imaginary root for an eigenvalue corresponds to a standing wave confined to the interface, while a positive/negative real part of the eigenvalue indicates instability/dissipation. A complex eigenvector indicates a complex mode of vibration. Two types of modes are considered—periodic symmetric modes with period equal to the surface waviness period and periodic antisymmetric modes with the period equal to twice the surface waviness. The singular integral equation is solved by reducing it to a system of linear algebraic equations using a Jacobi polynomial series and a collocation method. For the limit of zero friction it can be demonstrated analytically that the problem is self-adjoint and the eigenvalues, if they exist, are pure imaginary (no energy dissipation). These roots are found for a wide range of material properties and ratios of separation to contact zones lengths. For the limiting case of complete contact, the solution found corresponds to a superposition of two slip waves (generalized Rayleigh waves) traveling in opposite directions and forming a standing wave. With increasing separation zone length, the vibration frequency decreases from the slip wave frequency to the smaller surface wave frequency of the two bodies. With a nonzero separation zone, solutions can exist for material combinations which do not allow slip waves. For nonzero friction and sliding velocities, unstable solutions are found. The degree of instability is proportional to the product of the friction coefficient and the sliding velocity. These instabilities may contribute to the formation of friction-induced vibrations at high sliding speeds.