Free oscillations of a dissipative oscillator with double quadratic nonlinearity

Abstract

Approximate formulas for calculating the amplitudes of free damped oscillations of the oscillator with a quadratic nonlinear elastic characteristic under the action of a resistance force that is proportional to the square of the velocity are derived by the energy balance method. Two variants of this method have been implemented. In the first, an approximate differential equation of the envelope graph of oscillations is composed and its analytical solution is constructed. As a result, for the calculation of amplitudes, iterative relations were obtained using the Lambert W function. The argument of this special function is positive for the hard power characteristic and negative for the soft one. Asymptotic approximations of the Lambert W function, which simplify the practical implementation of analytical solutions, are proposed, and the possibility of using known tables of this special function is indicated. In presenting the second variant of the energy balance method, the recurrent relation between the amplitudes of oscillations related to the analytical solution of the cubic equation is derived. Unlike the first option, it does not require iterations. It has been made a comparison of numerical results, which these methods of calculating the amplitudes lead to, and numerical computer integration of the differential equation of oscillations of the oscillator. Satisfactory consistency of the results obtained in different ways confirmed the suitability of the derived approximate formulas for engineering calculations. The main advantage of this method is that it does not involve the construction and use of an exact solution of the double nonlinear differential equation of motion of the oscillator. In addition to the direct problem, the inverse problem of determining the coefficient of quadratic resistance of the medium based on the results of measuring the amplitudes of free oscillations on the oscillogram is also analytically solved here. The obtained solution of the direct dynamics problem was used to check the accuracy of the coefficient identification.