A reduced-order model for von Karimin plates, using a method of quadratic components, is presented in this paper. The method of quadratic components postulates the full kinematics of the plate as consisting of a combination of linear and quadratic components. This reduced model is then discretized in space with a Galerkin approximation and in time with an implicit integration scheme. The plate is coupled with a fluid flowing over it at a supersonic speed using a quasi-steady pressure model commonly referred to as piston theory. A static loading example is used to validate the model reduction of the plate with respect to other numerical and approximate solutions. The limit cycles of the plate coupled with piston theory are calculated via a cyclic method, and the results from parameter studies are compared to classical results. A previously unexplored regime of the limit cycle amplitudes is investigated; a second, coexisting, limit cycle is found that yields a discontinuity in the limit cycle amplitudes at a critical dynamic pressure.