Combination instabilities and non-linear vibratory interactions in beam systems
As an extension of previously reported work on effects of internal resonance on non-linear vibration of beams, it has been shown that for blade-like beams excited parametrically by support motion in the plane of maximum stiffness, complex combination instabilities are observed. In addition to the well-known sum-type combination instability existing between the fundamental out-of-plane bending and the torsional modes of the beam, investigation has revealed the occurrence of higher order instabilities producing detectable bending and torsional vibrations which are not synchronous with external excitations. These effects are subsequently shown to exist in a related way in coupled beam configurations shown to exist in a related way in coupled beam configurations under forced vibration when specific internal resonance conditions exist between the natural frequencies of the various modes, and to produce visible patterns of non-linear energy flow between modes. This study considers one such effect both experimentally and theoretically, consisting predominantly of a coupling between the fundamental and second nonplanar bending modes, and torsion mode. This combination resonance was modelled by taking the perturbation analysis to second order and including other contributory terms in the system governing equations. An expression for the transition curve for this resonance has been derived which shows the regions of stable and unstable solutions in a two parameter plane. Very close agreement is obtained between theoretical and experimental results for different beam lengths. It is also shown that if the geometry of the system is such that theses two combination resonances can be excited simultaneously, very small alternations to the internal tuning of the system can generate noticeable intermodal energy exchange effects. This system is then examined in the context of non-linear forced vibration and to this end an arrangement of coupled beams is studied. The vertical blade-like beam is coupled to the free end of a horizontal cantilever beam which is externally excited at a frequency in the region of its second bending mode frequency. This allows for the possibility of four mode interaction between the three nonplanar modes described above and also the second planar bending mode. A four-degree-of-freedom model was formulated and perturbation analysis revealed that complex multimodal responses could occur for a single-frequency excitation. Steady-state solutions were derived by means of numerical integration techniques. A reasonable degree of agreement was observed between theoretical and experimental results.