The vibration of a spinning body will cause relative circumferential motions, which will change the direction of the centrifugal load which, in turn, will tend to destabilize the structure. As a small deflection analysis cannot directly account for changes in geometry, the effect can be accounted for by an adjustment of the stiffness matrix, called spin softening. The spin softening contribution is included if any of the following criteria are met:
For the first two criteria, spin softening is an additional contribution to the tangent matrix (Equation 3–32). For the last criteria, it is part of the equations of a rotating structure when expressed in a rotating reference frame (Equation 14–12).
In the following sections, equations are first developed for a spring-mass system, and then a general system equation is formed:
Consider a simple spring-mass system, with the spring oriented radially with respect to the axis of rotation, as shown in Figure 3.7: Spinning Spring-Mass System. In the rotating reference frame, the equilibrium of the spring and centrifugal forces on the mass using small deflection logic requires:
| u = radial displacement of the mass from the rest position along the rotating reference frame direction X' |
| r = radial rest position of the mass along the rotating reference frame direction X' |
| = angular velocity of rotation |
Figure 3.7: Spinning Spring-Mass System
However, to account for large-deflection effects, Equation 3–68 must be expanded to: