We study the energy dissipation features of systems comprised of two components one of which is highly lossy and the other lossless. One of the principal results is that all the eigenmodes of any such system split into two distinct classes, high-loss and low-loss,according to their dissipative properties. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition to that, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping. An exhaustive analytical study of the energy, dissipated power, and quality factor for such composite systems is given.