Stability analysis for the flapwise motion of a cantilever beam with rotary oscillation

The flapwise motion of a cantilever beam with rotary oscillation is analyzed to investigate the dynamic stability of the beam. The cantilever beam is regarded as a system subjected to parametric excitation because the angular speed of the cantilever beam varies harmonically. To consider the stiffening effect due to the centrifugal force, this study adopts the linear partial differential equation of flapwise motion, which is derived by a modelling method using the stretch deformation instead of the conventional longitudinal deformation. After the partial differential equation is discretized by the Galerkin method, the method of multiple scales is applied. Using this method, the stability of the beam is analyzed for the variations of the oscillating frequency and the maximum angular speed. In addition, to verify the stability results, the time responses of flapwise motion are computed by the generalized-α time integration method.