u u Question 2: Vibrating Membrane In this problem you will examine the natural modes of vibration of a rectangular membrane of length a and width [7. fixed along all four edges. Vibrating plates or membranes are commonly encountered in various fields such as structural engineering. mechanical engineering and acoustics. Understanding their mode shapes is crucial for analyzing their behavior and resonance frequencies. Assuming the membrane obeys the two-dimenional wave equation, the (normalized) amplitude f of the (m, it) natural mode of vibration is isms): where x and y are coordinates along the membrane and m and n are positive integer mode indices. f(x,y,a,b;m,n) = sin(x Question 2.0 mil" (1 Writeatunction modeishape(x,y,a,b,m,n) which takes in: 'x: float or np. ndarray x coordinates along the membrane, x E (O, a) : float or np.ndarray y coordinates along the membrane. y E (0, b) : float membrane length. a > 0 : float membrane width, b > O : int mode index along x, m = 1, 2, 3, : int mode index along y, n = 1,2,3, and returns the mode amplitude f(x, y, a, b ; m, n) shown in Eq._(i_ at each point defined by x and y . Once you are done, test your modeishape function for x assign it to the variable q2_0 . # ANSWER CELL # TEST YOUR FUNCTION HERE q2_0 np.linspace(0, 1) ,y np.linspace(0,1) . a : 1 , b 1 . m 1,n 1 and (3) Question 2.1 Let us now compute the m = 3, n = 2 natural mode of vibration for a membrane of length a = 3 and width b = 2 on a discrete two-dimensional grid. To this end, first create two linear ranges of 1000 equally spaced points each for the x and y coordinates, named x and y , respectively. Then, define combined two-dimensional coordinate arrays X and Y for each point on the grid. The documentation of gpmteshgm may be helpful. Finally, using your mode_shape function. compute the mode amplitude on the LIE grid and assign the output array to the variable Z . In In In # ANSWER CELL a? Length a = # Width b = ... 9'? Parameter m m: ... 9'? Parameter n n= ... 51' Create x and yranges x= y= 9? Create combined X and Y grid X,Y= # Compute Z z = modeishapetx,Y,a,b,m,n) # Do not modify and run this cell for grading purposes x_q2_1 = np.copy(x} Y_q2_1 = nP-COPY(Yl X_q2_1 = np.copy(X} Y7q271 = np.copy(Y} ziq271 = np.copy(Z} grader.check( "q2 . 1 "i