Free vibrations of Duffing oscillator with dry friction

Abstract

Free damped oscillations of an oscillator with dry Coulomb friction are described in the presence of linear and cubic terms in the expression of the restoring force, which depends on the displacement of the system. Using the first integral of the nonlinear differential equation of motion, determining the exact values of the amplitudes of the ranges is reduced to calculating the corresponding real roots of the cubic equation, which has an analytical implementation. For an approximate calculation of the amplitudes, an additional compact iterative formula is also proposed. The satisfactory convergence of iterations on it is confirmed by numerous calculations. The problem of determining the duration of the range is also solved. For this, a transition to new integration variables reduces the improper integral of the second kind to the sum of two eigen integrals expressing the duration of the range in time. They have to be integrated by numerical methods on a computer. Therefore, the inequalities of the two-sided estimation of the durations of the ranges are additionally presented and a compact formula is proposed that makes it possible to approximately calculate this duration. The numerical results obtained in various ways are compared. It is shown that the damping of the oscillation amplitudes, the duration of the amplitude over time, their number until the oscillator stops completely depend on the characteristics of nonlinearity. It is established that, due to nonlinear elasticity, the duration of the sweep of the Duffing oscillator depends on the amplitudes of the oscillations. The width of the oscillator stagnation region also depends on the nonlinearity characteristics, the definition of which is reduced to calculating the real root of the cubic equation using the Cardano formulas. Formulas are derived that make it possible, without using
the second integral of the nonlinear equation of oscillation of the oscillator, to calculate its basic kinematic characteristics that change during movement.