Internal combination resonance in a Micro Electro-Mechanical three-axis gyroscope: Experimental results and reduced-order modelling via invariant manifolds

This work investigates internal combination resonance in a Micro-Electro-Mechanical System (MEMS) three-axis gyroscope, where energy couples among three vibrational modes via the relationship ω 1 + ω 2 ≈ ω 3 . The phenomenon is first characterised experimentally through frequency response measurements and spectrograms, revealing paired resonance peaks that emerge only under dual-frequency excitation. A theoretical framework grounded in complex normal form theory is then developed, deriving analytical expressions for the frequency-response curves which are shown to replicate the experimental findings. A reduced-order model of the fully coupled electromechanical problem is constructed via Direct Parametrisation of Invariant Manifolds applied to the full finite element mesh (1.3 million degrees of freedom), demonstrating quantitative agreement with experiments and validating the predicted mechanisms. By combining invariant manifold parametrisation with complex normal form theory, the methodology bridges the gap between analytical understanding and practical finite element reduction, yielding explicit solutions for nonlinear phenomena whilst maintaining computational efficiency for large-scale systems. The results validate invariant manifold methods as a robust tool for MEMS design and control applications.