On the motion of a quadratically nonlinear oscillator with dry friction

Abstract

The work is devoted to the derivation and testing of formulas for calculating the displacement of an oscillator and determining the duration of a half-cycle of oscillations under conditions of dry friction. It is possible to derive an exact recurrence relation for calculating the ranges of damped free oscillations under conditions of dry friction without constructing a solution to the differential equation of motion, using the energy method. But determining the displacements of the oscillator in time requires decoupling the differential equation of motion.

The paper describes free damped oscillations of an oscillator with a symmetric quadratically nonlinear power characteristic, has a linear component. The cause of oscillations is the initial deviation of the system from the position of static equilibrium, and their damping is a consequence of the action of the force of dry friction. Variants of hard and soft elastic characteristics are considered. For both of them, exact solutions of the equation of motion are constructed. As a result, the movement of the oscillator in time is expressed in terms of Jacobi elliptic functions. The duration of a quarter and a half-cycle is expressed in terms of an elliptic integral of the first kind, requires the use of tables of these special functions. Approximate formulas for calculating the values ​​of elliptic functions are also given, where they are reduced to calculating elementary functions. Comparison of numerical results obtained using analytical solutions and numerical integration of the original differential equation of motion on a computer. Small differences in the values ​​of the displacements due to the approximate calculation of elliptic functions are revealed. Errors in the implementation of analytical solutions associated with the approximate calculation of the Jacobi function. Based on the comparison of the numerical results, the probability of the derived design formulas for the displacements and half-cycle durations was confirmed, depending on the range of fluctuations.

It was found that the differential equation of free oscillations of an oscillator with a quadratically nonlinear force characteristic and dry friction has exact analytical solutions expressed in terms of Jacobi elliptic functions, and the obtained approximate solutions have a fairly good agreement with the numerical integration of the equations of motion on a computer.