The consistent numerical treatment of the balance laws in multi-physical systems requires specialized time integration schemes. These integrators must preserve the invariants of the system, as dictated by Noether’s theorem, while adhering to the fundamental laws of thermodynamics: conservation of linear and angular momentum, conservation of energy, and positive dissipation.The GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) formalism provides the foundation for constructing such integrators for discrete systems. A thermo-visco-elastic pendulum under large deformations is considered as a model problem. It incorporates heat exchange and viscous dissipation as challenging features for the numerical integrators. Energy-Momentum-Entropy schemes, based on discrete gradients as proposed by Gonzalez , are developed for the model problem and their energy and entropy consistency is shown. The use of temperature, entropy, internal energy or total energy as independent variables is discussed. A focus is put on the special form of GENERIC according to Mielke and the introduction of auxiliary variables like strain and its consequences on the structural properties of the discrete system. Extending the model problem to include electromagnetic forces enables a broader applicability to coupled electro-mechanical or magneto-mechanical systems, which are critical in modern engineering applications. The development of GENERIC-based integrators for discrete systems can be seen as a first step towards the development of GENERIC-based methods for continuum problems.