Application of Qualitative Methods to Research of Polyharmonic Oscillations
The development of the qualitative methods of investigation of dynamic systems, suggested by the authors, is the effective means for identification of dynamic systems. The results of the extensive investigations of the behaviour of linear dynamic systems and symmetrical system with double well potential under polyharmonic excitation are given in the paper. Phase space of dynamic systems is multi-dimensional. Each point of this space is characterized by not less than four co-ordinates. In particular: displacement, velocity, acceleration and time. Real space has three dimensions. It is more convenient for the analysis. We consider the phase space as limited to three dimensions, namely displacement, velocity and acceleration. Another choice of parameters of phase planes is also possible [1, 2]. Phase trajectory on a plane is of the greatest interest. It is known that accelerations of points are more sensitive to deviations of oscillations from harmonic ones. It is connected with the fact that power criteria on it are interpreted most evidently. Besides, dependence is back symmetric relative to axis of the diagram of elastic characteristic. Only the phase trajectories allow establishing a type and a level of non-linearity of a system. The results of the extensive investigations of the dynamic systems behaviour under polyharmonic excitation are given in the paper. The use of the given phase trajectories enables us to determine with a high degree of reliability the following peculiarities: - presence or absence of non-linear character of behaviour of a dynamic system; - type of non-linearity; - type of dynamic process (oscillations of the basic tone, combinative oscillations, chaotic oscillations.). Unlike existing asymptotic and stochastic methods of identification of dynamic systems, the use of the suggested technique is not connected with the use of a significant amount of computing procedures, and also it has a number of advantages at the investigation of complicated oscillations.
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