Overdamped and Gyroscopic Vibrating Systems

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Abstract
Some linear vibrating systems give rise to differential equations of the form Iẍ(t) + Bẋ(t) + C x(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 −1 C , 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 B abs >kI −k −1 C for some k >0 will ensure stability. In fact, this condition can be generalized to systems with B* = B, C> 0.

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