This paper presents a set of novel physical models for sound synthesis of membrane percussion instruments. First, a model for tension modulation in a struck circular membrane is discussed, which simulates the dynamic variations of partial frequencies occurring at large amplitude vibrations of the membrane. It is shown that the model, which is derived from a more general theory of nonlinear elastic plates, can be efficiently integrated into a modal synthesis engine. Novel models for two relevant sound production mechanisms in membrane percussions are then proposed, i.e., coupling between two membranes through enclosed air in two-headed percussions, and string-membrane coupling. Both are based on a lumped modeling approach and can be straightforwardly connected to the nonlinear membrane model. By virtue of this modular approach, individual elements (circular linear/nonlinear membranes, impact force, membrane coupling through air, string-membrane coupling) can be combined to form different instruments. The acoustic results of the proposed models are demonstrated by means of analysis of numerical simulations.