The dynamics of flexible bodies spinning at rates near or above their first natural frequencies is a notoriously difficult area of analysis. Recently, a method of analysis, tentatively referred to as a method of quadratic modes, has been developed to address this sort of problem. This method restricts consideration to configurations in which all kinematic constraints are automatically satisfied through second order in deformation. Besides providing robustness, this analysis method reduces the problem from one that would otherwise require the reformulation of stiffness matrices at each time step to one of solving only a small number of nonlinear equations at each time step. A test of this method has been performed, examining the vibrations of a rotating, inflated membrane.