Thermal stresses in multiring MEMS gyroscopes: Mathematical modeling with experimental validation

Temperature is, arguably, the predominant environmental variable impacting the performance of MEMS gyroscopes. Nevertheless, examination and formulation of the solid mechanics underlying the effects of temperature, especially regarding thermal stresses, remains largely unexplored in the literature. Motivated to address this issue, we lay out a novel framework to mathematically model the effects of thermal stresses on a multiring gyroscope’s stiffness matrix. We also take into account the temperature dependency of material properties, including Young’s modulus and coefficient of thermal expansion (CTE). Adhering to the variational principles of solid mechanics and linear thermoelasticity, we formulate the displacement field calculation of the gyroscope considering ring-beam continuity/boundary constraints. We use the Ritz method to convert and solve the subsequent optimization problems as quadratic programs (QPs). We obtain analytical expressions for the stiffness matrix variations under thermal stresses. These results distinguish terms induced by nonhomogeneous boundary conditions from those caused by thermal deformations in the gyroscope’s moving structure. Such boundary conditions account for 1) expansion/contraction of the internal suspension and 2) thermo-mechanical effects due to the different CTEs across MEMS layers, including the glass substrate and the die-attach material. We compare our method against finite element simulations and validate it using experimental data from our 58 kHz, 3.2 mm-diameter gyroscope.