Free oscillations of an elastic nonlinear oscillator with dry friction

Abstract

Using the energy balance method, recurrence relations are derived for calculating the sequence of falling amplitudes of the ranges of a dissipative oscillator with dry Coulomb friction. Various variants of nonlinear elasticity of the oscillatory system are considered. The article discusses cases of oscillations of an oscillator with the following non-linearity coefficients of elasticity: nonlinear-power with options for a linear elastic oscillator, an oscillator with a soft characteristic of elasticity at its various values, quadratic and cubic nonlinearity; oscillations of the oscillator in the presence in the expression of the elastic force of the linear component with different nonlinearity indices. The recurrence relations between the amplitudes of the ranges are compiled and their numerical solution is proposed by the Newton iteration method. Cases are distinguished when the derived recurrence relations have closed analytical solutions, and their graphs are constructed. A comparison is made of the numerical results that such solutions and computer integration of the differential equation of motion lead to. The full consistency of the results obtained in various ways is shown. Compact formulas are proposed for calculating the stagnation zone width for various types of nonlinearity. The described calculation method is simple to implement, because it does not require solving the nonlinear differential equation of free oscillations of the oscillator. In deriving the recurrence relations, the exact analytical solution of the cubic equation proposed by Cardano is involved. Variants of nonlinearities are revealed when the calculation reduces to using explicit recurrence relations related to the solutions of quadratic and cubic equations. In the case of an arbitrary indicator of nonlinearity, the amplitudes are calculated by the iteration method.