The method of study of the influence of torsional oscillations of one-dimensional models of nonlinear elastic bodies, along which moves with a constant velocity continuous flow of inelastic homogeneous medium, into bending, is developed. It is believed that information on torsional oscillations is known from empirical studies. Based on the latter, the refined model of the dynamics of the process of the investigated object is constructed. The latter is a boundary value problem for nonlinear nonautonomous differential equations with partial derivatives. The imposed restrictions on power factors and the main parameters of torsional oscillations allow for the analytical study of the dynamics of the process to use the basic ideas of the asymptotic integration of equations with partial derivatives. With their help, we obtain a two-parameter set of solutions that describe the determinant parameters of bending vibrations of an elastic body. It is established that for the considered elastic body there can be resonance oscillations, which are caused not only by external factors, but also by internal - torsional oscillations. Regarding the law of the change in the basic parameters of the dynamics of the bending motion of an elastic body, its rotation around the vertical axis reduces the frequency of its own flexural oscillations of the body, and even small torsional oscillations cause an additional periodic action on the transverse. In connection with the above bending vibrations of the elastic body, which performs complex oscillations (torsion and bending), resonances are possible both at the frequency of the external periodic perturbation and at the frequencies of the torsional oscillations (internal resonances). The amplitude of the transition through the resonance: a) at the basic frequency of external perturbation takes less value for elastic bodies of greater flexural rigidity and for higher values of the relative relative motion of the medium; b) at the frequency of torsional oscillations for larger valuesof the angular velocity takes more importance; c) with "fast" transition through resonance at the frequency of external or internal perturbation is less than with "slow". The obtained results can serve as the basis for the choice of operating parameters of elastic elements of machines that carry out complex oscillations.

The main provisions of the methodology of the study of complex oscillations f elastic bodies are outlined. Its main idea is as follows: a) on the basis of empirical studies, the change of the basic parameters of some forms of oscillation (usually smaller amplitude) is approximated by their analytical relations; b) these relationships are taken into account when constructing a mathematical model of the elastic body; c) for constructing and studying the solution of the obtained mathematical model of the process dynamics, the main ideas of the asymptotic integration of equations with partial derivatives are that take into account the influence on the dynamics of the process of external and internal factors. The methodology is illustrated by the example of an elastic body, which simultaneously performs longitudinal and transverse vibrations. With the its aid it is established that resonant processes can exist in an elastic body not only by external actions, but also by the mutual influence of some forms of oscillation on others. The obtained results can serve as the basis for the choice of operating parameters of elastic elements of machines that carry out complex oscillations.

The methodology of the studying of dynamic processes in two-dimensional systems by mathematical models containing nonlinear equation of Klein-Gordon was developed. The methodology contains such underlying: the concept of the motion wave theory; the single - frequency fluctuations principle in nonlinear systems; the asymptotic methods of nonlinear mechanics. The aggregate content allowed describing the dynamic process for the undisturbed (linear) analogue of the mathematical model of movement. The value determining the impact of nonlinear forces on the basic parameters of the waves for the disturbed analogue is defined.

Проілюстровано можливості поєднання асимптотичних методів нелінійної механіки та основних положень теорії збурень для дослідження математичної моделі нелінійної коливальної системи. Розглянута система описує крутильні коливання пружного тіла, пружні властивості якого описуються нелінійним законом. За допомогою розробленої методики дослідження динамічної системи отримано співвідношення у вигляді звичайних диференціальних рівнянь для визначення основних параметрів одночастотних коливань та умови виникнення резонансу. Вказано на практичні застосування отриманих результатів у задачах оптимізації параметрів технологічного обладнання.