Program Description: Purpose of the problem is to solve the initial value problem m x ″ + c x ′ + k x = f ( t ) , x ( 0 ) = x ′ ( 0 ) = 0 and construct the graph of the position function x ( t ) .

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Chapter 7.5, Problem 33P

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Program Description: Purpose of the problem is to solve the initial value problem mx″+cx′+kx=f(t),x(0)=x′(0)=0 and construct the graph of the position function x(t) .

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PROBLEM 24 - 0589: A forced oscillator is a system whose behavior can be described by a second-order linear differential equation of the form: ÿ + Ajý + A2y (t) = (1) where A1, A2 are positive %3D E(t) constants and E(t) is an external forcing input. An automobile suspension system, with the road as a vertical forcing input, is a forced oscillator, for example, as shown in Figure #1. Another example is an RLC circuit connected in series with an electromotive force generator E(t), as shown in Figure #2. Given the initial conditions y(0) = Yo and y(0) = zo , write a %3D FORTRAN program that uses the modified Euler method to simulate this system from t = 0 to t = tf if: Case 1: E(t) = h whereh is %3D constant Case 2: E(t) is a pulse of height h and width (t2 - t1) . Case 3: E(t) is a sinusoid of amplitude A, period 2n/w and phase angle p . E(t) is a pulse train Case 4: with height h, width W, period pW and beginning at time t =

2. calculates the trajectory r(t) and stores the coordinates for time steps At as a nested list trajectory that contains [[xe, ye, ze], [x1, y1, z1], [x2, y2, z2], ...]. Start from time t = 0 and use a time step At = 0.01; the last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when z(t) ≤ 0; 3. stores the time for hitting the ground (i.e., the first time t when z(t) ≤ 0) in the variable t_contact and the corresponding positions in the variables x_contact, y_contact, and z_contact. Print t_contact = 1.430 X_contact = 0.755 y contact = -0.380 z_contact = (Output floating point numbers with 3 decimals using format (), e.g., "t_contact = {:.3f}" .format(t_contact).) The partial example output above is for ze = 10. 4. calculates the average x- and y-coordinates 1 y = Yi N where the x, y, are the x(t), y(t) in the trajectory and N is the number of data points that you calculated. Store the result as a list in the variable center = [x_avg, y_avg]…