A method is disclosed for analyzing the vibrational characteristics of rotating devices, such as hard disk drives and jet engines, that are coupled through bearings to flexible supports. In the method, the rotating device is discretized, for example, using a mesh suitable for finite element analysis. The support is also discretized. The natural frequency of elastic vibration modes for the rotating device is calculated and the natural frequency of vibration modes for the support is calculated. The mode shapes are then calculated and a set of modal basis corresponding to the mode shapes is utilized. Bearing stiffness and damping matrices are input, and the Lagrangian equations of motion are integrated numerically in the modal space, to calculate the vibrational response of the rotating device and support. The modal space results are then transformed into ordinary space. The vibrational analysis may be used to design devices having desired vibrational characteristics.