C245 Homework 3

Due 2/23/05


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      1. Calculate the deflection of a rectangular cantilever under it's own weight.  In class, we calculated an upper bound on the deflection by putting the entire mass of the cantilever at the end.  What is the error in this estimate (should be a ratio)? 
      2. You are designing a suspension using two constant-cross-section beams.  One beam is anchored to the substrate and has length Lx, the other is attached to the end of the first beam at a 90 degree angle and has length Ly.  When you apply a force along the x axis and measure x deflection you want to get the same result as when you apply a force along the y axis and measure a y deflection.  Calculate the ratio of the lengths of the two beams. (Hint: Write down the compliance matrix for the whole suspension, and make the diagonals equal). 
      3. Run the sugar tutorial and check your answer to the problem above.
      4. Run the intellisuite tutorial and check your answer to the problem above.
      5. Assuming a DRIE aspect ratio S, a minimum feature size l, a material density r, and a maximum voltage V,
        1. calculate the force per unit area of an ideal gap-closing actuator array.  (Use the zero-deflection force, not the force after pull-in; you may assume some magical sub-lithographic gap stop if you like).
        2. Calculate the force output per kilogram, assuming a film thickness t.  Will micro robots with gap closing electrostatic actuators be able to lift themselves without massive gearing or levers?
        3. What is the optimum value for t, assuming we want to make minimum sized gaps and the maximum aspect ratio possible in our beams?
        4. Calculate the maximum work done per cycle of the actuator against a constant force load (what is the load?).
        5. Calculate the electrical energy input per cycle, assuming that some smart control circuit limits the total charge on the capacitor plates to twice the charge applied initially to the actuator (the zero-deflection charge).
        6. What is the energy efficiency of this actuator?
        7. Assuming that you choose the support spring for your gap closing actuator such that when the gap closer has pulled in completely that the spring force is equal to the zero-deflection force of the electrostatic actuator, calculate the resonant frequency of your actuator in terms of only V , l, and r. (remember sqrt(K/m)?)
        8. Ignoring damping/resonance, assume that we can run the actuator close to its resonant frequency and estimate the maximum power output per kilogram of an electrostatic gap closing actuator.
        9. Insect flight muscle has a peak power density of 100W/kg, about as high as it gets for animals.  Will micro robots based on gap closing electrostatic actuators ever fly?


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(Kris Pister)