In this paper, we analyze the transverse nonlinear vibration of a rotating flexible disk with a periodically varying rotating speed, subjected to a rotating point force. Based on a small-stretch, moderate-rotation flexible disk theory of the Nowinski and the von Karman type field equations, the nonlinear governing equations of motion are derived for the rotating flexible disk, which are coupled equations among the radial, tangential and transverse displacements. According to the Galerkin approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes are derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, the stabilities of the steady state nonlinear responses are analyzed. Using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including multi-pulse type chaotic motions, periodic and period-n motions for the spinning disk with a varying rotating speed. It is also found that among all parameters the damping and excitation have important influence on the nonlinear responses of the spinning disk with a varying rotating speed.
Droplet size and liquid-liquid mass transfer with reaction in a rotor-stator Spinning Disk Reactor