Parametric vibration of composite beams with integrated shape memory alloy elements

This research is concerned with parametric vibration in composite beam structures with shape memory alloy elements. As a precursor to this investigation, a flexible steel beam of rectangular uniform cross-section is considered with a lumped end mass under a parametric excitation. A single frequency harmonic excitation in the vertical direction is applied to the system. As an extension of previsouly developed model by Cartmell (1990) and Forehand and Cartmell (2001), three nonlinear equations of motion, representing the first and second bending modes and the first torsion modes, are derived by recourse to the Lagrangian formulation. The variables in the equations of motions are , and respectively. They are coupled together and various nonlinearities appear in the equations. The three equations are used to predict different parametric resonances of the form , , by application of the perturbation method of multiple scales. Expressions for the transition curves for the three resonances have been derived which show the regions of stable and unstable solutions in a detuning parameter-excitation amplitude plane. Very close agreement is obtained between theoretical and experimental results for all the three resonance conditions. Laboratory tests confirm that these instabilities are bounded in practice by nonlinear effects. To investigate the effects of shape memory alloy on the dynamical properties of a composite material beam structure, two shape memory alloy strips are centrally-bonded to a glass epoxy beam with a lumped end mass. The two SMA strips are theoretically pre-strained and heated up to their full austenitic phase, and shown to generate large recovery forces due to this phase transformation. The forces are considered as compressive forces, and a theoretical model is introduced to evaluate the influences of the forces on the natural frequencies and the bending modes of the composite beam structure. The results show that the increase of the forces decrease the natural frequencies and reduce the excursion of the first and second bending modes. The beam system is then subjected to a vertical excitation. In order to utilize the Lagrangian formulation once again, the generalised forces corresponding to the generalised coordinates , and are derived in terms of the SMA recovery force. The three equations of motion of the free lateral vibration of the beam system are then derived. Three different parametric resonances are also predicted. Further study shows that the increase of the magnitude of the recovery force results in an increase of the instability region. An experimental investigation is conducted on two composite beam structures and each with an end mass, one with two centrally-bonded shape memory alloy (SMA) strips and the other with two diagonally-bonded SMA strips. The study suggests that when the strips are activated, the central-strip configuration can increase the natural frequencies of the bending modes noticeably more than the diagonal-strip one under certain circumstances, whilst the diagonal-strip configuration can easily be seen to change the frequencies of the torsion modes than the central-strip set-up.