Abstract
The purpose of this study is to determine the effect of retarded forces in elasticity and damping on the dynamics of mixed forced, parametric, and self-oscillations in a system with limited excitation. A mechanical frictional self-oscillating system driven by a limited-power engine was used as a model. Methods. In this work, to solve the nonlinear differential equations of motion of the system under consideration, the method of direct linearization is used, which differs from the known methods of nonlinear mechanics in ease of use and very low labor and time costs. This is especially important from the point of view of calculations when designing real devices. Results. The characteristic of the friction force that causes self-oscillations, represented by a general polynomial function, is linearized using the method of direct linearization of nonlinearities. Using the same method, solutions of the differential equations of motion of the system are constructed, equations are obtained for determining the nonstationary values of the amplitude, phase of oscillations and the speed of the energy source. Stationary motions are considered, as well as their stability by means of the Routh–Hurwitz criteria. Performed calculations obtained information about the effect of delays on the dynamics of the system. Conclusion. Calculations have shown that delays shift the amplitude curves to the right and left, up and down on the amplitude–frequency plane, change their shape, and affect the stability of motion.
Highlights
The purpose of this study is to determine the effect of retarded forces in elasticity and damping on the dynamics of mixed forced, parametric, and self-oscillations in a system with limited excitation
A mechanical frictional self-oscillating system driven by a limited-power engine was used as a model
To solve the nonlinear differential equations of motion of the system under consideration, the method of direct linearization is used, which differs from the known methods of nonlinear mechanics in ease of use and very low labor and time costs
Summary
Представленная на рис. 1 модель фрикционной автоколебательной системы хорошо описывает автоколебания под действием трения, которые могут возникать в целом ряде объектов машиностроения [23,24,25,26,27,28,29,30,31,32]. Тело находится под действием параметрического возбуждения (c + b cos νt)x и вынуждающей силы λ sin ν1 t. Где m – масса тела, лежащего на ленте, k0 = const и c0 = const – соответственно коэффициенты демпфирования и жесткости демпфера 3 и пружины 2, λ = const, b = const, ν1 = const, ν = const – соответственно амплитуды и частоты вынуждающей силы и параметрического возбуждения, η и τ – постоянные временные факторы запаздывания в силах демпфирования и упругости, kη = const, cτ = const, M ( ̇ ) и – моментная характеристика и скорость вращения ротора двигателя, I – суммарный момент инерции вращающихся частей, r0 = const – радиус шкива, вращающего ленту. 2. Графики численных коэффициентов методом усреднения (кривые 1) и МПЛ (кривые 2) Fig. 2.
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