Free Undamped Non-Linear Vibrations of Imperfect Circular Disks

Abstract

Previous investigations of the vibrations of circular disks confined themselves to small amplitudes when it can be assumed that the strain energy of the disk is due to pure bending only. At large amplitudes this assumption ceases to be valid and the stretching of the middle surface has to be taken into consideration. As a result, the equations of motion become non-linear. The investigation is concerned with the vibration of ‘real’ disks, that is, disks which contain small imperfections. It is shown that, in general, the imperfections eliminate the indeterminancy of the angular position of the radial nodal lines which is characteristic of perfect disks. For those modes which contain at least one radial nodal diameter, there are two fixed nodal configurations which possess slightly different natural frequencies. At small amplitudes these nodal configurations are normal modes of vibration which vibrate independently of each other. At large amplitudes, when the equations of motion become non-linear, these configurations become coupled. The coupling is unsymmetrical in the sense that the configuration with the higher natural frequency can vibrate on its own at all amplitudes. On the other hand, above a certain amplitude, the configuration with the smaller natural frequency cannot vibrate on its own since it also excites, by parametric excitation, the other configuration. Under such conditions the phase difference between the two configurations is fixed to 90 deg. and as a result the motion of the disk corresponds to a beating travelling wave.