The effectiveness of a dynamic absorber for self-excited vibration, such as parametric resonance, has been known for several years. However, in the case of self-excited vibration, the operating mechanism of a dynamic absorber is still poorly understood. In this paper, two simple and fundamental models with two degrees of freedom (DOFs), that is, a single-DOF parametric excitation system with one dynamic absorber, are considered and the operating mechanism is investigated from the viewpoint of the energy balance. As an analytical tool to clarify the operating mechanism, a new type of complex modal analysis is developed for accurately evaluating the energy for each mode. By applying the new method, the damping matrix can be diagonalized exactly in addition to the mass and stiffness matrices, and the equation of motion for a multi-DOF system can be converted into modal equations for a single-DOF system in the form of real second-order differential equations. Using approximate solutions obtained from the modal equation, the energy generated due to parametric excitation and that dissipated due to damping can be estimated accurately. The results show that the appropriate decentralization of the excitation energy to each mode and the increase of the dissipation energy, caused by the dynamic absorber, play dominant roles in the stabilization of the system, and the effects of the dynamic absorber on the stabilities of two models differ from each other. In addition, the very effective optimization procedure of the dynamic absorber is formulated based on the different effects. The validity of the analytical results is verified by comparing with very accurate numerical results.