SUMMARYThe incremental harmonic balance method with multiple time variables is developed for analysis of almost periodic oscillations in multi-degree-of-freedom dynamical systems with cubic non-linearities, subjected to the external multi-tone excitation. The method is formulated to treat non-autonomous as well as autonomous dynamical systems. The almost periodic oscillations, which coexist with periodic oscillations in a rotating system model with cubic restoring force and an electromagnetic eddy-current damper are analysed. The closed form solutions based on generalized Fourier series containing two incommensurate frequencies are obtained in the case of small non-dimensional sti ness ratio. Almost periodic oscillations of a rotating system model in dependence on variable parameters are also analysed, where solutions are computed through an augmentation process including a greater number of harmonics and combination frequencies involved.

This paper treats the control of nonstationary oscillations of the pressure which arise in the combustion process of a Rijke tube. The low order model of a nonstationary combustion process in a Rijke tube with mul tiplicative feedback control function is considered, which is investigated by means of two-DOF coupled van der Pol oscillator. The self-excited oscillations in the combustion process are analysed by means of the Extended Lindstedt-Poincare (EL-P) method with multiple time scales. By applying the EL-P method, it is ascertained, that nonstationary oscillations are almost periodic with amplitude and phase, which are functions of the slow time scale. Without control of the fuel influx, the self excited oscillations can approach limit cycles. By applying the fuel control, the influence of control parameters on the quenching of the self excited oscillations is investigated in a phenomenon of competitive quenching and mutual synchronization with close frequencies and multiple harmonics, respectively.

This paper presents the Extended Lindstedt-Poincare (EL-P) method with multiple time scales to treat nonstationary vibrations of the electromechanical system, which are forced by a nonideal energy source. The subject of research are two electromechanical systems consisting from rotor system with rotating disc mounted on an elastic shaft and a system with a rotating eccentric mass coupled by a nonlinear shock absorber, which are driven by a D.C. motor as a nonideal energy source. By using extended Hamilton principle, governing nonlinear differential equations of the system are derived. By using multiple time scales, which correspond to the nonlinear frequencies of the system in addition to the slow time scale, which corresponds to the slowly varying parameter, the system of partial differential equations is obtained, which is successively solved by using the proposed EL-P method. The results of computation of the nonstationary vibrations in the passage through fundamental resonance in both systems are presented.

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