The significance of higher modes for evolution of chaos in structural mechanical systems

Abstract

Even though chaotic vibrations have been observed in many structural mechanics systems, their analysis has almost always been limited to single-degree-of-freedom (SDOF) approximations. A typical example is the magnetoelastic beam studied by Moon and Holmes [1], which is reported to be the first experimental evidence of chaotic vibrations in structural mechanics. However, the authors have not come across any detailed structural analysis of the system. The present paper reports a structural dynamic analysis of the problem through a finite element formulation and the integration of the resulting equations of motion by a variable time stepping Newmark method (trapezoidal rule). The solution scheme has built-in algorithms for equilibrium interaction of the non-linear forces and check of the temporal solution trajectory. It is shown that the direct integration and mode superposition schemes are equally applicable for problems with chaotic response. The authors have the following conclusions: (1) the SODF approximation with a high accuracy integration scheme may not reveal the regime of chaos even coarsely; (2) the manifestation of chaos is significantly influenced by the higher modes; (3) a spatially discrete model which represents the beam accurately could reveal regimes of chaos reasonably well even with second order schemes such as the trapezoidal rule, but it is essential for the model to be fine enough to represent the motion in higher modes accurately; (4) computationally efficient methods such as the mode superposition method, with an adequate number of modes included, could give accurate solutions to vibration problems involving chaos.