Abstract
If a contact of two purely elastic bodies with no sliding (infinite coefficient of friction) is subjected to superimposed oscillations in the normal and tangential directions, then a specific damping appears, that is not dependent on friction or dissipation in the material. We call this effect “relaxation damping”. The rate of energy dissipation due to relaxation damping is calculated in a closed analytic form for arbitrary axially-symmetric contacts. In the case of equal frequency of normal and tangential oscillations, the dissipated energy per cycle is proportional to the square of the amplitude of tangential oscillation and to the absolute value of the amplitude of normal oscillation, and is dependent on the phase shift between both oscillations. In the case of low frequency tangential oscillations with superimposed high frequency normal oscillations, the dissipation is proportional to the ratio of the frequencies. Generalization of the results for macroscopically planar, randomly rough surfaces as well as for the case of finite friction is discussed.
Highlights
In our analysis we use the method of dimensionality reduction, MDR11
In the framework of the Method of Dimensionality Reduction (MDR), two preliminary steps are performed[11]: First, the three-dimensional elastic half-space is replaced by a one-dimensional linearly elastic foundation consisting of an array of independent springs, with a sufficiently small separation distance ∆x and normal and tangential stiffness ∆kz and ∆kx defined according to the rules
We have shown that a superposition of normal and tangential oscillation leads to a specific damping, which we call “relaxation damping”
Summary
If the MDR-transformed profile g (x) is indented into the elastic foundation and is moved normally and tangentially according to an arbitrary law, the contact radius and the force-displacement relations of the one-dimensional system will exactly reproduce those of the initial three-dimensional contact problem (proofs have been done in[18] and[11]). Effect Since the springs of elastic foundation in the MDR model are independent, it is sufficient to analyze the energy dissipation of a single spring (Fig. 2), and to sum over all springs which come into contact during an oscillation cycle. In the shape-invariant form, the energy dissipation per cycle of tangential oscillation is: W
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