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.

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