Analytical solution to the problem of free vibrations of a quadratic nonlinear oscillator with dry friction

Abstract

Elements of structures during operation can be subjected to dynamic loads, which due to oscillations that can lead to dangerous conditions. Dissipative oscillations are dangerous due to the asymmetry of the elasticity characteristic. This paper describes free damped oscillations of a dissipative oscillator with an asymmetric quadratically nonlinear characteristic of elasticity in the presence of Coulomb dry friction. The aim of the study is to derive exact formulas for calculating the oscillator displacements in time and to determine the amplitude and duration of the half-cycles, which depend on the amplitudes due to the nonlinearity of the system. The first integral of the equation of motion is expressed in elementary functions, and the second – in Jacobi elliptic functions. The first integral is associated with the calculation of the ranges of oscillations, and with the second - the movement of the oscillator in time. It is shown that, due to the asymmetry of the elastic characteristic, the process of free vibrations depends on the direction in which the starting deviation of the system from the position of static equilibrium was set. The durations of the half-cycles are expressed in terms of complete elliptic integrals of the first kind. The obtained solution retains its shape even in the absence of dry friction in the system, when the system is conservative, and the range of oscillations in the direction opposite to the initial deviation may be greater than the initial deviation. The derived formulas for the first cycle of oscillations can be easily extended to any cycle of oscillations. A compact approximate formula is proposed for calculating the values of the elliptic sine. Comparisons of displacements obtained with its use in analytical solutions and numerical integration of the original differential equation on a computer are made. Good consistency of the calculation results was obtained in two ways, thereby confirming the adequacy of the formulas. Their practical implementation requires the calculation of complete elliptic integrals of the first kind, which is not difficult to implement by interpolating tabular data published in many publications on special functions.