Homework #5

Assigned: Feb. 17, 2004
Due: Feb. 24, 2004
Points: 50

1. Points: 20
   You have now seen three levels of modeling of coupled electrostatics-elastostatics, viz., lumped 1D, distributed beam, and plane-stress/3D in FEMLAB. Only lumped 1D, we know how to estimate the pull-in voltage analytically. Now, come up with such a model for the beams. That is, obtain a formula for the fixed-fixed beam's pull-in voltage. It should be in terms of the beam's dimensions, the gap, and the material properties. Compare your model's validity by running Matlab scripts in emstatic for various cases. Indicate the % error in each case. You can consider various cases by varying the length and thickness of the beam, the gap, etc.
   Hint: The easiest thing is to introduce one or two unknown coefficients in the 1-D model's V_PI formula and try to obtain a best fit with the data generated by running the beam-model based Matlab program.
2. Points: 30
   The figure below shows the schematic of the micro-mirror used in the optical Lucent's optical cross-connect. It is a two-axis mirror that can reflect off the light rays incident on it to a desired location. There are four electrodes underneath the mirror separated by a gap, g. The details of the serpentine spring are shown below. The width of the beams that make up the serpentine structure is b. The four springs enable the mirror to rotate about the two axes. Any of the four actuators can be individually activated to apply electrostatic force on the mirror. The thickness of the structural layer (with which the mirror and springs are made) is t.
   
   • First, obtain equivalent torsional (rotational) spring constant of the serpentine spring using either analytical calculation or FEMLAB.
   • Using the equivalent spring constant, write down the lumped-model dynamic equations with electrostatic force.
   • Simulate the dynamics of the lumped model using Matlab. Take the case of applying voltage V on the two electrodes on the right side for five time periods corresponding to the lowest natural frequency. Then, turn them off for five time periods. Then, actuate the top two electrodes for five time periods by aplying the same voltage V.
   Use the following data:
   d = 500 um; c = 20 um; a = 200 um;
   t = 3 um; g = 25 um;
   V = 200 V (if it is too high, that is beyond pull-in, try a smaller value);
   p = 25 um; q = 2 um; b = 1 um;
   Young's modulus = 150 GPa; Poisson's ratio = 0.25; density = 2800 kg/m³