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Using a Lagrangian framework, we study overdamping phenomena in gyroscopicsystems composed of two components, one of which is highly lossy and the otheris lossless. The losses are accounted by a Rayleigh dissipative function. As wehave shown previously, for such a composite system the modes split into twodistinct classes, high-loss and low-loss, according to their dissipativebehavior. A principal result of this paper is that for any such system a ratheruniversal phenomenon of selective overdamping occurs. Namely, first of all thehigh-loss modes are all overdamped, i.e., non-oscillatory, as are an equalnumber of low-loss modes. Second of all, the rest of the low-loss modes remainoscillatory (i.e., the underdamped modes) each with an extremely high qualityfactor (Q-factor) that actually increases as the loss of the lossy componentincreases. We prove that selective overdamping is a generic phenomenon inLagrangian systems with gyroscopic forces and give an analysis of theoverdamping phenomena in such systems. Moreover, using perturbation theory, wederive explicit formulas for upper bound estimates on the amount of lossrequired in the lossy component of the composite system for the selectiveoverdamping to occur in the generic case, and give Q-factor estimates for theunderdamped modes. Central to the analysis is the introduction of the notion ofa "dual" Lagrangian system and this yields significant improvements on someresults on modal dichotomy and overdamping. The effectiveness of the theorydeveloped here is demonstrated by applying it to an electric circuit with agyrator element and a high-loss resistor.

On overdamping phenomena in gyroscopic systems composed of high-loss and lossless components