OSCILLATIONS DESCRIBED BY ANALOGUES OF VAN DER POL AND RAYLEIGH EQUATIONS

Abstract

The paper deals with the modes of steady-state quasilinear self-oscillations described by the analogues of Van der Pol and Rayleigh differential equations. The differential formulation of the energy balance method is applied for studying the motion. The conditions for the equations to describe quasilinear self-oscillations with the amplitude independent of the initial conditions are derived in the form of inequalities. The formulae for computing this amplitude using the table of gamma functions are proposed. The steady-state mode of the self-oscillations is proved to be stable in contrast to the static equilibrium which appears unstable. The inequalities are also obtained which guarantee the equations of the type considered to describe the damped free oscillations about the zero equilibrium or the oscillations which build up resulting in the loss of system stability. These forms of motion depend on the initial conditions. For small initial deflections, which are less than the threshold value, the oscillations decay whereas for large once they build up. The dynamical system, which is stable in small, is unstable in large. The impact of the constant component of the resistance force on the oscillatory process is also studied. It is shown to cause the shift of the position about which the steady-state self-oscillations occur but not to influence their amplitude or frequency, which is the result of the linear elasticity of the system. The special cases are separated, when the computational formulae proposed become the results previously known. The analytical studies are followed by numerical solution of the respective Cauchy problem. By comparing the results obtained by the two methods we substantiate the adequacy of the computational formulae obtained.