A new development in the Bogolubov-Mitropolski method for solving nonlinear differential equations of motion for systems with finite degrees of freedom with coupled deflection is presented. A function is introduced which takes into account the influence of small nonlinearities on the amplitude ratio and on the perturbation of the phase angle. The results obtained by the method for several systems of two bodies exhibiting coupled deflection are compared with results obtained by the classical Bogolubov-Mitropolski method and with exact solutions (where possible). The comparison illustrates and confirms the effectiveness of the method.
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