Overdamped and Gyroscopic Vibrating Systems

Abstract

Some linear vibrating systems give rise to differential equations of the form Ix¨(t) + Bx˙(t) + Cx(t) = 0, where B and C are square matrices. Stability criteria involving only the matrix coefficients I, B, C, and a single parameter are obtained for some special cases. Thus, if B* = B> 0, C* = C>0 and B>kI + k−1C, then the system will be overdamped (and hence stable). Gyroscopic systems also have the above form where B is real and skew symmetric. The case where C>0 is well understood and for the case −C>0 we show the condition Babs>kI−k−1C for some k>0 will ensure stability. In fact, this condition can be generalized to systems with B* = B, C>0.