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Membrane deflection square membrane of 12 inch by 12 inch (no bending or shear stresses) is fastened at the outside boundaries. If a highly stretched membrane is subject to a pressure P creating a tension T, the partial differential equation for the deflection u in the x-y direction is Here, a - 10in2/s. Compute the 2-D (x-y) deflection profile for the following cases. What is the maximum and minimum deflection in each case? Provide detailed comments about your approach and results. a) Assume steady state. The pressure and tension are uniform and constant. P=5 psi, T = 100lb/in. Function f does not depend on x or y, and has the following form f T b) Same as part a, but the function f has the following form (here, L = membrane length = membrane width = 12 inch): c) Same as part a, but the membrane has a square hole in the center with dimensions of 4 inch by 4 inch. The inside boundaries of the membrane are also fastened (zero deflection). d) Same as part c, but use the function f from part b. e) Same as part a, but with unsteady state. Initial deflection = 0 in. How long does it take i.e.; what is the t required in seconds) to reach the same maximum deflection as in part a? f) After achieving steady state in part e, pressure on the membrane is relieved (P=0) and the membrane is allowed to vibrate freely. The initial velocity is zero. The governing equation is as follows: azu 0t² - Here, B - 0.6 in2/s². Conduct the simulation for t=60 seconds.

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