Keywords: auto-parametric response, flow-induced vibration, van der Pol equation, vortex shedding, random load, combination of deterministic and stochastic excitation, Galerkin-Petrov method, Fokker-Planck equation.

Aero-elastic processes at large slender engineering structures are closely related with nonlinear interaction of a stream and a vibrating structure. Although more sophisticated models can be adopted, a commonly used single-degree-of-freedom (SDOF) system represents a reasonable compromise between accuracy and simplicity. Experiments in a wind tunnel show that the regime of the vortex shedding is typical by quasi-periodic beatings that are encountered in the lock-in regimes. Here the vortex shedding frequency becomes close to the SDOF eigen-frequency with a small positive or negative detuning. Experimental and also theoretical investigations indicate three regimes can be encountered that in the lock-in area, provided a combined deterministic (harmonic) and random excitation is applied: (i) small detuning: the response falls into synchronization and no beating effect occurs; (ii) a quasi-periodic response of the SDOF system emerges consisting of self-excited and forced components; (iii) self-excited oscillations do not occur and only (nearly) mono-harmonic forced vibration can be observed. The work proved an existence of a frequency detuning interval where the system response has a quasi-periodic character. The main difference from conventional approaches consists in a possibility that the random excitation component can change qualitatively the response portrait. The detuning interval, where the quasi-periodic response occurs, seems to be larger than in the deterministic case. On the other hand, the variability of the response amplitude within one quasi-period is not so dramatic as in the purely deterministic case. The particular form of the excitation spectral density is important and depends predominantly on its value in frequency which coincides with the eigen-frequency of the adjacent linear system and its integer multiples.

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