There are inequalities matrix derived which, except for some special cases, are necessary and sufficient for asymptotic stability of asymmetric, time-invariant linear mechanical systems. The results are applicable to systems with gyroscopic and hysteresial effects. Two examples of asymmetric systems include flexural vibration of a pipe conveying a fluid and an actively controlled vibrating system with asymmetric gain matrices. Well-known conditions are recovered for conventional symmetric positive definite systems. The inequalities are then used to establish asymptotic stability, marginal stability and instability in various examples of asymmetric systems. As further applications, a Lyapunov function is derived, response bounds are presented using the one-sided Lipschitz constant, and bounds are derived on the real and imaginary parts of the system eigenvalues.

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