Estimation of the Accuracy of the Averaging Method for Systems with Multifrequency Perturbations

We consider systems of ordinary differential equations whose right-hand sides contain timeperiodic functions with some frequencies. An averaged system is constructed by introduction of additional variables and by step-by-step averaging over these variables. An upper estimate of the deviation of the solution to the initial system from the solution to the averaged system is given. Examples are given of mechanical systems in which vibrations with several frequencies occur and for the analysis of which the statements obtained are applied.