Modeling the impact of elastic bodies taking into account dry positional friction and the coefficient of velocity restoration

Considering dry positional friction in the model of collinear elastic impact, the stiffness of the system during compression is greater than its stiffness during decompression.The coefficients of the relative motion equation of the bodies involved in the impact have different values at these stages.They depend not only on the geometry of the bodies in the interaction region and the mechanical properties of their materials, but also on the velocity restitution coefficient, which is one of the input parameters of the model, determined a priori through experiments.In the case of power-law nonlinearities in the stiffness of the dynamic system, which corresponds to the solutions of the contact problem in the theory of elasticity, the equation of relative motion of the bodies has closed analytical solutions.During the compression phase, the solution is expressed with an Ateb-sine and its powers, while during the decompression phase, it is expressed with an Ateb-cosine and its powers.To simplify the numerical implementation of the solutions, an approximation of these special functions by trigonometric functions is proposed.This approximation ensures a calculation accuracy of three significant digits after the decimal point.Examples of calculations are provided, demonstrating that accounting for positional friction leads to a decrease in the maximum compression of the bodies, an increase in the peak impact force, and an extension of its duration over time.Variations of impacts between bodies with surfaces of second-and fourth-order in the dynamic interaction region are calculated.The model has much in common with the classical model, but additionally considers the actual value of the velocity restitution coefficient, which is less than one, in collinear impacts.The presented theory applies only to elastic impacts with low initial collision velocities, where no plastic deformations occur.