Keywords: non-linear dynamics, quasi-periodic response, beating effect, stability of post-critical processes.

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) non-linear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs more or less from the system basic eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. The shape of the response amplitude envelope within one quasi-period changes from a constant (stationary process) dramatically when the frequency distance from the resonance is rising. The ratio of energy content of quasi-periodic and stationary component is decreasing in the same time. Starting a certain frequency the quasi-periodic part fully vanishes and the stationary part absorbs the whole response energy.

The above phenomena have been identified qualitatively in many papers including authors contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference between excitation and eigen-frequency is still missing. This paper is a contribution to fill this gap using qualitative analytical methods in combination with numerical procedures.

Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system. They lead to non-linear differential and algebraic equations serving as a basis for qualitative analytic estimation or numerical description of characteristics of quasi-periodic system response in zero, first and second level perturbation techniques. Bifurcation analysis and dynamic stability of relevant response types is discussed at various frequency domains. Appearance, stability and neighborhood of limit cycles is evaluated. Parametric evaluations are presented together with discussion concerning applicability of the presented approach. Some open problems are outlined.

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