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This paper presents the first application of the direct parametrisationmethod for invariant manifolds to a fully coupled multiphysics probleminvolving the nonlinear vibrations of deformable structures subjected to anelectrostatic field. The formulation proposed is intended for model orderreduction of electrostatically actuated resonating Micro-Electro-MechanicalSystems (MEMS). The continuous problem is first rewritten in a manner that canbe directly handled by the parametrisation method, which relies upon automatedasymptotic expansions. A new mixed fully Lagrangian formulation is thusproposed which contains only explicit polynomial nonlinearities, which is thendiscretised in the framework of finite element procedures. Validation isperformed on the classical parallel plate configuration, where differentformulations using either the general framework, or an approximation of theelectrostatic field due to the geometric configuration selected, are compared.Reduced-order models along these formulations are also compared to full-ordersimulations operated with a time integration approach. Numerical results show aremarkable performance both in terms of accuracy and wealth of nonlineareffects that can be accounted for. In particular, the transition from hardeningto softening behaviour of the primary resonance while increasing the constantvoltage component of the electric actuation, is recovered. Secondary resonancesleading to superharmonic and parametric resonances are also investigated withthe reduced-order model.

Reduced order modelling of fully coupled electro-mechanical systems through invariant manifolds with applications to microstructures