The many models of Elmer may be utilized to solve the behavior of elastic plates under the influence of electrostatic and fluidic forces. A thin prestressed silicon plate of size 1mm x 1mm was studied with different models.

The computations were done on an computational mesh consisting of 1600 elements, gif. The mesh is quite sufficient since a further increase in the mesh density affected the results less than 1 %.

Eigenmodes

The first analysis usually done on resonators is the eigenmode analysis. It gives the natural frequencies that the resonator will take. The first mode often dominates as it is energetically favorable. The first ten modes are given below. Some are degenerated and are therefore omitted.

• 1st: 25031 Hz, jpg.
• 2nd and 3rd: 41941 Hz, jpg.
• 4th: 55727 Hz, jpg.
• 5th: 64462 Hz, jpg.
• 6th: 64529 Hz, jpg.
• 7th and 8th: 76516 Hz, jpg.
• 9th and 10th: 92661 Hz, jpg.

Pull-In Analysis

The pull-in analysis was performed in order to determine the pull-in voltage and critical displacement. Accurate determination of these parameters requires a coupled solution of the elastic and electrostatic problems. For this membrane the pull-in voltage was found to be voltage: 1.754 V and the corresponding capacitance was 8.332 pF. Here are some pictures of the solution:

• Deflection computed by the elasticity solver, jpg.
• Amplitude derived from the deflection, jpg.
• Electric energy density of the plate, jpg.
• Electric force that is used also used as load for the elasticity solver, jpg.

Dynamical pull-in

In the previous calculations the pull-in phenomena was treated as a stationary phenomena. However, in some cases it may be interesting to know what is the dynamical behavior. In order to investigate this case we set the potential difference to 3.0 V which clearly exceeds the critical pull-in voltage. We then performed a dynamical analysis with very small timesteps. Now the inertial effects of the resonator are also taken into account. The pull-in time was around 10.8 us. Unfortunately there is no contact model so the simulation crashes when the plates collide.

• The following animation shows the shape deformation of the membrane as a function of time, mpg.

Squeezed film damping

The squeezed film damping may be modeled with the Reynolds equation. Here we have taken the first eigenmode as a wake. When there are no holes the harmonic motion would result to large forces.

• Spring constants as a function of frequency with the given eigenmode, ps. The circles present the frequency of the 1st eigenmode.
• Pressure and displacements in the time-harmonic solution, mpg.

Dynamical pull-in with squeezed film damping

The solution of fully coupled squeezed film damping is not straight-forward when the damping is large. We cannot evaluate the damping without first solving for the displacements. This displacement field then gives damping values significantly larger than the electrostatic force. Therefore in the beginning of the iteration the damping must be presented implecitely to the elasticity solver. When the evaluation of the damping has settled it may be taken as a pressure acting on the structure.

• Here is a comparison of the damped and undamped case in the dynamical pull-in with by 3 Volts, ps.
• This animation shows the squeezed film pressure in the dynamical analysis. It can be noted that the pressure quickly rises and stays at almost constant value. The reason is that the electrostatic force is the main opposite force and its magnitude remains almost constant throughout the run, mpg.