Simultaneous combination resonances in an autoparametrically resonant system

There have been many reports to date of the effects of seemingly small, non-linear, autoparametric interactions in certain structures undergoing forced vibration, and it is known that large intermodal exchanges of energy can occur under several prescribed internal resonance conditions. This paper purports to illustrate the highly complex responses that can be generated within a system of coupled cantilever beams when two such internal resonances, each in the form of a combination resonance, exist in very close proximity to each other. The resulting four mode interaction can exhibit non-synchronous large amplitude responses of the indirectly excited modes, in addition to the well known phenomenon of saturation of the directly excited mode, and the stationary system responses are also shown to be highly susceptible to very small shifts in system tuning. Thus it is proposed that the non-linear response of the system can be dependent on the dominance, or otherwise, of one of the internally generated combination resonances over its near neighbour. A four degree of freedom model is proposed and a multiple scales treatment of the governing equations is presented. A selection of experimental results are presented which highlight the above phenomen

Numerical modelling of elastic space tethers

In this paper the importance of the ill-posedness of the classical, non-dissipative massive tether model on an orbiting tether system is studied numerically. The computations document that via the regularisation of bending resistance a more reliable numerical integrator can be produced. Furthermore, the numerical experiments of an orbiting tether system show that bending may introduce significant forces in some regions of phase space. Finally, numerical evidence for the existence of an almost invariant slow manifold of the singularly perturbed, regularised, non-dissipative massive tether model is provided. It is also shown that on the slow manifold the dynamics of the satellites are well-approximated by the finite dimensional slack-spring model