JAM 2004 Abstract
Vibration and Stability of Frictional Sliding of Two Elastic Bodies With a Wavy Contact Interface
Mikhail Nosonovsky and George G. Adams
Summary
he 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 non-zero
separation zone, solutions can exist for material combinations which do not allow slip
waves.
For non-zero 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.
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