An approximate model to compute resulting wrench of the dry friction tangent forces in frame of the Hertz contact problem is built up. An approach under consideration develops in a natural way the contact model constructed earlier. Generally an analytic computation of the integrals in the Contensou-Erismann model leads to the cumbersome calculation, decades of terms, including rational functions depending in turn on complete elliptic integrals. To implement the elastic bodies contact interaction computer model fast enough one builds up an approximate model in the way initially proposed by Contensou. To verify the model built results obtained by several authors were applied. First the Tippe-Top dynamic model is used as an example under testing. It turned out the top revolution process is identical to one simulated with use of the set-valued functions approach. In addition, the ball bearing dynamic model was also used to verify different approaches to the tangent forces computational implementation in details. A model objects corresponding to contacts between balls and raceways were replaced by ones of a new class developed here. Then the friction model of the approximate Contensou type embedded into the whole bearing dynamic model was thoroughly tested.
A method of computational reduction of an elastic contact model for rigid bodies in frame of the Hertz contact model is considered. An algorithm to transform outer surfaces' geometric properties to the local contact coordinates system is described. It tracks permanently in time the surfaces of the bodies which are able to contact. An approach to compute the normal elastic force is represented. That one deals with the reduction to one transcendental scalar equation that includes the complete elliptic integrals of the first and second kinds. Simulation of the Hertz model was accelerated essentially due to use of the differential technique to compute the complete elliptic integrals and due to the replacement of the implicit transcendental equation by the differential one. Based on the Hertz contact problem classic solution, an invariant form for the force function which depends on the geometric properties of an intersection for undeformed rigid bodies' volumes, so-called volumetric model, is proposed then. The resulting force function reduced expression is supposed to be in use in cases when the classical contact theory hypotheses are broken. The expression derived has been applied to several cases of the elastic bodies contacting, and in particular back to the source Hertz model itself.
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