Large-amplitude geometric nonlinear vibrations of plates are considered. The von Kármán equations are used as model for both unknowns : transverse displacement and Airy stress function. The key feature is the use of a modal approach for time-integrating the system with a very large number of degrees-of-freedom. The targeted application is the strongly nonlinear vibration of thin plates characterized by a turbulent state with a cascade of energy from the injection to the dissipative scale, involving a huge dimension of the phase space. More specifically, we are interested in the sound synthesis of cymbals and gong-like instruments, that show this wave turbulence regime when strongly beaten with a vigorous strike. A special emphasis is put on the derivation of an ad-hoc, conservative method. Thanks to symmetry properties of the nonlinear coupling coefficients, an energy-conserving scheme especially designed for the modal equations of the von Kármán perfect and imperfect plates is derived, for arbitrary boundary conditions.