Vibration Characteristics of Rotating Thin Disks—Part II: Analytical Predictions

Abstract

This paper is concerned with the geometric nonlinear analysis of the lateral displacement of thin rotating disks when subjected to a space fixed stationary force. Of particular interest is the development of the stationary wave and the effect of this wave on the frequency response of the disk as a function of its rotational speed. The predictions of this analysis are compared with experimental data obtained in a companion paper (Khorasany and Hutton, “Vibration Characteristics of Rotating Thin Disks—Part I: Experimental Results,” ASME J. Appl. Mech., 79(4), p. 041006). The governing equations are based on Von Kármán plate theory. A Galerkin solution of the governing non linear equations is developed. The eigenfunctions derived from the linear analysis of a stationary disk are used as approximations to the spatial response of the disk, and the eigenfunctions of the biharmonic equation as approximations for the stress function. Using the developed solution, the equilibrium configuration of the disk under the application of a space fixed force is found. In order to facilitate the prediction of the frequency response, as a function of disk rotational speed, the governing nonlinear equations are linearized around the equilibrium solution. The linearized equations are then used to find the eigenvalues of the spinning disk under the application of a space fixed force. The effect of different levels of nonlinearity on the disk frequencies is studied and compared with experimental results. The analysis is shown to produce an accurate representation of the measured response. Of particular interest is the disk response at speeds close to and above the linear critical speed. In this region, both the analysis and the experimental results display frequency “lock-in” behavior in which the frequency of backward travelling waves becomes constant for supercritical speeds. No speed exists for which backward travelling waves have zero frequency. Thus, critical speeds do not exist in the presence of geometric nonlinearities.