Solution to homework #6

1. The static pull-in voltage of the distributed model of the beam is 8.06 V. The error relative to the one dof model is 35.86% (note that one dof model has a pull-in voltage of 5.17 V)
   Maximum displacement, which occurs at the center, is 0.3840 microns for 8.05 V. Notice that it is more than the 1/3 of the gap that we saw in the one dof model.
   With bent supports added (see the data files nodebent.dat, elembent.dat, dispbcbent.dat in meam550/fea directory/folder), the static pull-in voltage comes to 7.49 V. This differs by 7.61% from the value obtained without the bent supports. Big difference with those tiny bends!
   See the following figure. Note that y-axis is exaggerated and so you don't get a realistic-looking picture.
   
   See the close-up of the bend on the left side to see how it is contributing to the downward displacement. That little perturbation is enough to cause the beam to become unstable much sooner (that is, the pull-in voltage is lower by 7.61 %).
   
2. For 12 V, the time for pull-in is about 4.6 microSec by extrapolating from the figure below that shows the displacement at the center vs. time.
   
   See also the figure of modal amplitudes vs. time for the first four modes. As you can see, compared to the first mode, the others have very small values.
   
3. See figure below to understand the symbols.
   
   The transfer function for the circuit on the left side is
   
   
   The transfer function for the circuit on the right side is