Nonlinear Dynamics of Circular Plates under Electrical Loadings for Capacitive Micromachined Ultrasonic Transducers (CMUTs)


TR Number



Journal Title

Journal ISSN

Volume Title


Virginia Tech


We created an analytical reduced-order model (macromodel) for an electrically actuated circular plate with an in-plane residual stress for applications in capacitive micromachined ultrasonic transducers (CMUTs). After establishing the equations governing the plate, we discretized the system by using a Galerkin approach. The distributed-parameter equations were then reduced to a finite system of ordinary-differential equations in time.

We solved these equations for the equilibrium states due to a general electric potential and determined the natural frequencies of the axisymmetric modes for the stable deflected position. As expected, the fundamental natural frequency generally decreases as the electric forcing increases, reaching a value of zero at pull-in. However, strain-hardening effects can cause the frequencies to increase with voltage. The macromodel was validated by using data from experiments and simulations performed on silicon-based microelectromechanical systems (MEMS). For example, the pull-in voltages differed by about 1% from values produced by full 3-D MEMS simulations.

The macromodel was then used to investigate the response of an electrostatically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode. The method of multiple scales was used to derive a semi-analytical expression for the equilibrium amplitude of vibration. The plate was found to always transition from a hardening-type to a softening-type behavior as the DC voltage increases towards pull-in.

Because the response of CMUTs is highly influenced by the boundary conditions, an updated reduced-order model was created to account for more realistic boundary conditions. The electrode was still considered to be infinitesimally thin, but the electrode was allowed to have general inner and outer radii. The updated reduced-order model was used to show how sensitive the pull-in voltage is with respect to the boundary conditions. The boundary parameters were extracted by matching the pull-in voltages from the macromodel to those from finite element method (FEM) simulations for CMUTs with varying outer and inner radii. The static behavior of the updated macromodel was validated because the pull-in voltages for the macromodel and FEM simulations were very close to each other and the extracted boundary parameters were physically realistic.

A macromodel for CMUTs was then created that includes both the boundary effects and an electrode of finite thickness. Matching conditions ensured the continuity of displacements, slopes, forces, and moments from the composite to the non-composite regime of the CMUT. We attempted to validate this model with results from FEM simulations. In general, the center deflections from the macromodel fell below those from the FEM simulation, especially for relatively high residual stresses, but the first natural frequencies that accompany the deflections were very close to those from the FEM simulations. Furthermore, the forced vibration characteristics also compared well with the macromodel predictions for an experimental case in which the primary resonance curve bends to the right because the CMUT is a hardening-type system.

The reduced-order model accounts for geometric nonlinear hardening, residual stresses, and boundary conditions related to the CMUT post, allows for general design variables, and is robust up to the pull-in instability. However, even more general boundary conditions need to be incorporated into the model for it to be a more effective design tool for capacitive micromachined ultrasonic transducers.



Primary Resonance, Pull-in, Macromodel, CMUT, Nonlinear Dynamics