On the effects of a general form of the initial runout on the oscillation frequencies and critical speeds of a spinning disk

This paper is concerned with the effect of the lack of flatness on the dynamics of spinning disks. Of particular interest is the study of the effect of an unsymmetrical initial runout on the oscillation frequencies and the critical speeds of the spinning disks. It is assumed that a spinning disk is initially warped in an initial asymmetric shape. Using Von Karmanʹs plate theory, the equations of motion are derived in a rotating frame. Taking advantage of the orthogonal properties of the eigenfunctions of a stationary disk, the lateral displacement and the stress function are expressed as their summation. Since these eigenfunctions produce an orthonormal space, any shape and level of initial runout can be written as their summation also. Using Galerkinʹs method, the equations of motion are discretized and a set of coupled linear equations are found taking into account the effect of an initial arbitrary runout. The numerical results are found to have a good agreement with those obtained using ANSYS. It is found that the natural frequencies of the spinning disk calculated in the space fixed frame increases due to an initial runout. When the initial runout is assumed to be asymmetric, it is found that, due to asymmetric stress distributed in the disk, a frequency split between the backward and forward traveling waves of a given preferential mode of a stationary disk occurs. It is also seen that in some cases the initial runout in the form of the (n,m) mode has the least effect on the natural frequencies of the backward traveling waves of the same mode. As such it is observed that in the case the critical speed the (n,m) mode is less sensitive to an initial runout in the form of the (n,m) mode. To verify the accuracy of the numerical predictions, experiments were conducted on two disks, one which was flat and the other that had an initial nonflatness in the form of the (0,2) mode. The numerical and experimental results indicate a close correspondence.