Nonlinear responses of symmetrically laminated composite cantilever beams

We investigate the nonlinear responses of symmetrically laminated composite canttilever beams. A two-to-one internal resonance between the chordwise bending motion and the flapwise bending and torsional motions is considered. That is, rho approximate to 2 omega where rho is the fundamental natural frequency for the chordwise bending motion and omega is the fundamental natural frequency for the linearly coupled flapwise bending and torsional motions. We investigate the nonlinear responses of the beams to parametric excitations and consider two cases: (i) principal parametric resonance of the flapwise bending motion (i.e., Omega approximate to 2 omega) and (ii) principal parametric resonance of the chordwise bending motion (i.e., Omega approximate to 2 rho). We apply the method of multiple scales directly to the governing partial-differential system to determine a set of nonlinear ordinary-differential equations that govern the slow dynamics of the interacting modes. In addition, we use the method of time-averaged Lagrangian and virtual work to determine a similar set of modulation equations, We show that both sets of modulation equations are the same. Then, we determine fixed points and limit cycles of the modulation equations and examine their stability and the bifurcations they undergo.