This study deals with a methodology for increasing the efficiency of dynamic process calculations in elastic elements of complex engineering constructions. We studied the complex dynamic processes in a simple engineering construction, a mechanical system of an elastic body–continuous flow of homogeneous medium. The developed methodology is based on the use of a priori information on some of the vibrations forms, the construction of a “simplified” mathematical model of system dynamics, and the obtaining of an analytical relationship that describe the overall range of factors on the elastic vibrations of system. The methodology is used for cases of complex vibrations of elastic bodies, and the obtained results can serve as a basis for choosing the main technological and operational parameters of elastic elements of mechanisms and machines that perform complex vibrations. The results obtained in this work are the basis for calculating the blast effect on the elements of protective structures in order to increase their protective capacity by improving the method of their attachment or by using additional reinforcement, buff load effects on the elements of drilling strings and dynamic processes that occur during surface strengthening by work hardening in order to avoid resonance phenomena, and technological processes of vibration displacement or vibration separation of granular media.

The effect of relative motion of cushion system on dynamic pressure force of steered wheeled vehicles with non-linear elastic characteristics of suspension was evaluated in the article under consideration. The problem is to find the dynamic force of system cushion-non-cushion part pressure on front wheel. The dependence for the limiting value of dynamic turning angle of steered wheels on longitudinal-and-angular oscillations amplitude and parameters describing elastic characteristics of elastic suspension was obtained. It was demonstrated that for more rigid suspensions the value of dynamic turning angle is smaller at small amplitudes of longitudinal-and-angular oscillations and bigger at big amplitudes of longitudinal-and-angular oscillations.

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A combination of asymptotic methods in nonlinear mechanics with basic techniques of perturbation theory to study a mathematical model of the nonlinear oscillation system is proposed in the paper. The system under consideration describes the torsional vibrations of an elastic body, where its elastic properties are under the nonlinear law. The relationships presented as the ordinary differential equations are obtained due to the proposed procedure. Therefore, the main parameters of the single-frequency oscillations and the resonance conditions can be determined. There are proposed applications of the obtained results to the optimization problem concerning the processing equipment.