Parametric Resonance of a Spinning Disk Under Space-Fixed Pulsating Edge Loads

Abstract

The parametric resonance of a spinning disk under a space-fixed pulsating edge load is investigated analytically. We assume that the radial edge load can be expanded in a Fourier series. With use of the orthogonality properties among the eigenfunctions of a gyroscopic system, the partial differential equation of motion is discretized into a system of generalized Hill’s equations in the first-order form. The method of multiple scale is employed to determine the conditions for single mode as well as combination resonances to occur. For any two modes, with s and υ nodal diameters, respectively, combination resonance occurs only when there exists a specific Fourier component cos kθ in the edge load, where s + υ = ±k. Sum type resonance occurs when both modes are reflected or both modes are nonreflected. On the other hand, difference type resonance occurs when one mode is reflected and the other is nonreflected. In applying this rule, the number of nodal diameters of a forward and a reflected wave is considered as negative. Several typical loadings are discussed, including uniform and concentrated edge loads.