A perturbation solution for the flexible rotating disk: non-linear equilibrium and stability under transverse loading

Abstract

A singular perturbation solution is developed for the transversely loaded, flexible rotating disk using the von Kármán large deflection theory of thin plates. No solution exists for these equations in the limit of vanishing dimensionless bending stiffness, ϵ. This fact, along with the desire to have the terms describing non-linear membrane stretching appear in the leading order equations, motivates a perturbation expansion for displacement of the form ω ≈ ϵ −1/2 ω 0 + ω 1 + …. The dimensionless transverse loading term is retained in the leading order equations only if it is rescaled by ϵ 1/2 . This rescaling of transverse load magnitude constitutes a similitude prediction for unifying the equilibrium and stability measurements of different flexible, rotating disks. Experimental measurements of the stability of flexible rotating disks under transverse point loading support this prediction. In addition, the numerically computed stability of the perturbation solution agrees quantitatively with the experimental measurements.