On the calculation by the method of linearization of the interaction of parametric and self-oscillations at delay and limited excitation

The aim of the study is to develop methods for calculating nonlinear oscillatory systems with limited excitation on the basis of direct linearization of nonlinearities. Unlike the known methods of nonlinear mechanics, direct linearization methods simplify obtaining of the finite calculating formulas regardless of the particular type of characteristics and require less labor and time. The results obtained using the known methods of nonlinear mechanics and methods of direct linearization coincide qualitatively, but there are some insignificant quantitative differences which disappear in some cases. The interaction of self-oscillations and parametric oscillations in the presence of delay, nonlinear elasticity, and an energy source of limited power, is considered. A well-known model of mechanical friction self-oscillating system is used, in which selfoscillations occur under the nonlinear friction force action with a delay. The solution of the system of nonlinear differential equations of motion is obtained using direct linearization methods. Applying these methods, the linearization of nonlinear functions is first performed, and then the equations of unsteady and steady motions in the main parametric resonance region are derived. The stability conditions of steady-state oscillations are further considered using the Routh-Hurwitz criteria. To gain information about dynamics of the system, calculations are performed. Amplitude-frequency characteristics are plotted with determination of their stable and unstable regions both at an ideal source of energy and under limited excitation.