Evaluation of multi-modes for finite-element models: systems tuned into 1:2 internal resonance

A non-linear multi-mode of vibration arises from the coupling of two or more normal modes of a non-linear system under free-vibration. The ensuing motion takes place on a 2M-dimensional invariant manifold in the phase space of the system, M being the number of coupled linear modes; the manifold contains a stable equilibrium point of interest, and at that point is tangent to the 2M-dimensional eigenspace of the system linearised about that equilibrium point, which characterises the corresponding M linear modes. On this manifold, M pairs of state variables govern the dynamics of the system; that is, the system behaves like an M-degree-of-freedom oscillator. Non-linear multi-modes may therefore come about when the system exhibits non-linear coupling among generalised co-ordinates. That is the case, for instance, of internal resonance of the 1:2 or 1:3 types, for systems with quadratic or cubic non-linearities, respectively, in which a four-dimensional manifold should be determined. Evaluation of non-linear multi-modes poses huge computational challenges, which is the explanation for very limited reports on the subject in the literature so far. The authors developed a procedure to determine the non-linear multi-modes for finite-element models of plane frames, using the method of multiple scales. This paper refers to the case of quadratic non-linearities. The results obtained by the proposed technique are in good agreement with those coming out from direct integration of the equations of motion in the time domain and also with those few available in the literature.