Modifications to the response of a parametrically excited cantilever beam by means of smart active elements

This article is concerned with applying active smart material elements for modifying parametric vibration in a flexible composite beam structure. The glass epoxy beam is bonded to two theoretically prestrained shape memory alloy (SMA) strips and fitted with a lumped end mass. In this study, the components of the recovery force generated during the SMA activation are derived with respect to a three-dimensional frame when the structure is undergoing combined bending and torsional motions. In order to employ Lagrangian dynamics, the generalized forces are formulated and the equations of motion are then derived. Three different parametric resonances for the structure are predicted by using the multiple scales perturbation method. In addition, the effects of the SMA strips on the natural frequencies, the mode shapes, and the instability regions of the structure are all investigated. It is shown that the different thresholds of instability for parametric resonances within a composite structure of this sort may be influenced by smart active elements.