  Using a Lagrangian framework, we study overdamping phenomena in gyroscopic
systems composed of two components, one of which is highly lossy and the other
is lossless. The losses are accounted by a Rayleigh dissipative function. As we
have shown previously, for such a composite system the modes split into two
distinct classes, high-loss and low-loss, according to their dissipative
behavior. A principal result of this paper is that for any such system a rather
universal phenomenon of selective overdamping occurs. Namely, first of all the
high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal
number of low-loss modes. Second of all, the rest of the low-loss modes remain
oscillatory (i.e., the underdamped modes) each with an extremely high quality
factor (Q-factor) that actually increases as the loss of the lossy component
increases. We prove that selective overdamping is a generic phenomenon in
Lagrangian systems with gyroscopic forces and give an analysis of the
overdamping phenomena in such systems. Moreover, using perturbation theory, we
derive explicit formulas for upper bound estimates on the amount of loss
required in the lossy component of the composite system for the selective
overdamping to occur in the generic case, and give Q-factor estimates for the
underdamped modes. Central to the analysis is the introduction of the notion of
a "dual" Lagrangian system and this yields significant improvements on some
results on modal dichotomy and overdamping. The effectiveness of the theory
developed here is demonstrated by applying it to an electric circuit with a
gyrator element and a high-loss resistor.
