The Analysis Of Critical Speed And Dynamics For Thin Circular Plate

Spinning circular plate is widely used in many important engineering fields, such as turbines, circular saws, computer hard drive, CD drive device and so on. Whenever the rotating speed of plate is close to the critical speed of a certain mode, the plate can occur enormous transverse displacement, which could cause instability or catastrophic result to the device. So it is extremely necessary to study the critical speed and the dynamic behavior of the spinning plate.This paper studies a linear elastic thin circular plate with uniform thickness rotating about its axis at a constant angular speed, formulates a large amplitude transverse free vibration equation of a spinning thin axisymmetric circular plate based on the von Karman's plate theory considering the effect of centrifugally force, then obtains the equation of linear vibration through simplifying the large deformation equation, finally assumes transverse vibration mode of circular plate with clamped inner-boundary and free outer-boundary that is substituted into the simplified equation with Galerkin method used twice to gain the frequency of forward and back traveling wave. A critical speed for certain mode is rotating speed which causes the corresponding mode's frequency of the back traveling wave to be zero. The critical speed of lower modes (the number of nodal diameter less than or equals to five) are calculated and critical speed versus inner-to-outer radius ratio and Poisson's ratio are investigated.A non-linear forced vibration of the spinning is acquired by adding a space-fixed point external transverse load into the above large amplitude equation . In order to simplify this question the forced oscillation equation is rewritten in a rotate reference frame. It is noted that viewed from the rotate reference frame the space-fixed point load rotates opposite to the plate. Near a critical speed the transverse displacement and stress function are expanded in their eigenfunctions, and then substitute them into the governing equation. Making use of the orthogonality of these eigenfunctions, the original equation is simplified and the non-linear coupling coefficient of the sine and cosine modes is calculated explicitly. So the equation describing the resonant dynamics of the sine and cosine modes in the close vicinity of a critical speed resonance is derived. Using the method of first-order averaging of the non-linear oscillation, the averaged equations in mode-based coordinates and traveling wave-based coordinates are formulated by transforming the above equation.Two cases absence and presence damping are considered for the averaged equations. The conditions which can induce the backward traveling waves with equal and unequal amplitudes, forward traveling waves, standing waves and mixed waves and existence of them are investigated firstly. Then the stead-state motions are solved and the stability of these solutions are concluded primarily through the linear Jacob matrix's eigenvalue of the equilibrium on these motion. During this process, the bifurcation points are calculated according to definition and the bifurcation type is determined, which can use to explore the complicated dynamic behavior of this non-linear system.