Vibration of the Liénard Oscillator with Quadratic Damping and Constant Excitation

In this paper, the Liénard oscillator with nonlinear deflection, quadratic damping, and constant excitation is considered. In general, there is no analytic solution for the Liénard equation. However, for certain parameter values, the exact analytic solution exists and has the form of the Ateb function. In addition, for the oscillator with weakly perturbed parameters, the approximate analytic solution is obtained. For the considered Liénard equation, independently of parameter values, the first integral is found. The main advantage of the first integral is that after simple analysis and without solving the equation of motion, it gives important data about oscillation: the dependence of vibration on initial conditions and on the variation of the constant of excitation. In addition, by integration of the first integral, the period of vibration follows. The results of the research on the Liénard equation are applied for optimization of the properties of a sieve in the process industry. For the sieve with mass variation, dependent on the displacement function, the influence of excitation force on the system vibration is analyzed, and the optimal value is suggested.