Abstract

We consider in this paper the following system of coupled nonlinear oscillators
x..+x-k(y-x)=εf(x,x.),
y..+(1+δ)y-k(x-y)=εf(y,y.).
In this system we assume ε to be a small parameter, i.e. 0 < ε ≪ 1. A coupling between the two oscillators is established through the terms involving the positive parameter k. The coupling may be interpreted as a mutual force depending on the relative positions of the two oscillators. For both ε and k equal to zero the two oscillators are decoupled and behave as harmonic oscillators with frequencies 1 and 1+δ, respectively. The parameter δ may therefore be viewed as a detuning parameter. Finally, the term ε f represents a small force acting upon each oscillator. Note that this force depends only on the position and velocity of the oscillator upon which the force is acting.

To analyse the system’s dynamic behaviour we use the method of averaging. When k and δ are choosen such that no internal resonance occurs, one typically observes the following behaviour. If the trivial solution is unstable, solutions asymptotically tend to one of the two normal modes or to a mixed mode solution.

For the special case with δ = 0 a system of two identical oscillators is found. If in addition k is O(ε) we obtain a 1 : 1 internal resonant system. The averaged equations may then be reduced to a system of three coupled equations — two for the amplitudes and one for the phase difference. Due to the fact that we consider identical oscillators there is a symmetry in the averaged equations. The normal mode solutions, as found for the non-resonant case, are still present. New mixed mode solutions appear. Moreover, Hopf bifurcations in the averaged system lead to limit cycles that correspond to oscillations in the original system with periodically modulated amplitudes and phases.

We also consider the case with δ = O(ε), i.e. the case with nearly identical oscillators. If k = O(ε) again a 1 : 1 internal resonant system is found. Contrary to the previous cases the normal mode solutions no longer exist. Moreover, different bifurcations are observed due to the disappearance of the symmetry present in the system for s = 0.

We apply some of the results obtained to a model describing aeroelastic oscillations of a structure with two-degrees-of-freedom.

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