Dynamic Bending Stress in a Disk-Type Gyroscope Rotor Under Steady Precession

This paper derives the equations which govern the cyclic bending stresses in the web of a precessing gyro rotor, and discusses methods of solution. These stresses are important because they contribute to fatigue failure. Starting from the well-known partial differential equation describing the free lateral vibration of a thin variable thickness plate in the presence of initial centrifugal stresses, an ordinary differential equation for the mode displacement as a function of radius is obtained. Boundary conditions consist of a light, flexible shaft at the inside diameter of the web and a rigid, heavy rim at the outside diameter of the web. Three methods of solving for the modal functions and resonant frequencies are described. These are 1 Reduction to a matrix-eigenvalue problem by collocation, 2 Reduction to a matrix-eigenvalue problem by finite differences, and 3 An iterative solution based on numerical integration of the differential equation. Newton-Raphson interpolation against the eigenvalue is used to satisfy the boundary conditions. The forced vibration response to steady precession rate is evaluated from the Lagrange equation governing excitation of the fundamental normal coordinate. This coordinate corresponds to the lowest “fan” vibration made of the system, i.e., a mode in which the web has one diametral nodal line and no interior nodal circles. Numerical results show the variation of fan mode frequency with rotor spin rate, using web thickness as a parameter. Maximum radial and tangential bending stresses in the web are plotted against radius, using spin rate as a parameter. The numerical results indicate existence of an optimum rotor spin-rate, at which the allowable precession torque, based on web fatigue, is maximum for a given rotor structure.