Vibration of rings on a general elastic foundation

The free vibration eigensolutions of a thin ring on a general elastic foundation are obtained by perturbation and Galerkin analyses. Natural frequencies and vibration modes are determined as closed-form expressions for a ring having a circumferentially varying foundation of very general description. The elastic foundation consists of two orthogonal distributed springs oriented at an arbitrary inclination angle. The foundation stiffnesses vary circumferentially. The simple eigensolution expressions explicitly show the parameter dependencies, lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation, and identify which nodal diameter Fourier components contaminate a given n nodal diameter base mode of the free ring. Discrete spring supports are treated as a special case where the natural frequencies are determined by five parameters: nondimensional spring stiffness, stiffness angle, support angle, number of springs, and location of the springs. The predicted effects of these parameters on the natural frequencies are verified numerically. As an application and as the motivating problem for the study, the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated.