Integration

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The motion of a mass suspended from a spring without friction is governed by md2x/ dt 2 + kx = f, where f = f(t) is the applied force acting on the mass.
a. Find the transfer function H(s) = X(s)/F(s) in terms of the natural frequency ωn = k m/ and the constant c = 1/m. Where are the poles located?
b. Use explicit Euler, implicit Euler, and trapezoidal integration to obtain discrete-time approximations, that is, find H(z) = X(z)/F(z). Leave your answers in terms of c, ωn, and the integration step size T
c. Find the poles for each z-domain transfer function H(z) in part (b). Comment on the stability in each case.
d. Let m = 1 slug, k = 0.5 lb/in., x(0) = 2 in., x(0) = 0 in./s, and f(t) = 0, t ≥ 0. Find and graph the continuous-time response x(t).
e. Choose the integration step T, so that ωnT = 0.01. Find the poles of each transfer function H(z) and the discrete-time responses xk, k = 0, 1, 2, 3, … for the same conditions in part (d). Plot the discrete-time responses on the same graph as x(t).

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