Question

The Equation of motion for the standard mass-spring-damper system is

Mẍ + Bẋ + Kx = f(t).

Given the parameters {M = 2kg, B = 67.882 N-s/m, K = 400 N/m}, determine the free response of the system to initial conditions { x₀ = -1m, v₀ = 40 m/s}. To help verify the correctness of your answer, a plot of x(t) should go through the coordinates {t, x(t)} = {.015, -0.5141} and {t, x(t)} = 0.03, -0.2043}.

determine the steady-state response of the system to sinusoidal inputs of unit amplitude at specified frequencies. You may use any initial conditions you wasn’t for this section. Use the following frequencies: ω = {0.2, 1, 6, sqrt (200), 20, 100, 1000} rad/s.

Modify your Matlab code to numerically simulate the response of the system to the frequencies listed above. Simulate one frequency at a time. Simulate for a sufficient time that the system will have attained steady state. Plot three periods of the systems’s response at steady state (so the initial time value on the plot will not be t = 0). Then estimate the amplitude of the response for each frequency; you should have seven amplitudes when you are done.

Make a second plot where you plot the amplitudes determined above against the frequencies (make a frequency response magnitude plot). The x-axis should be logarithmic and the y-axis should be in dB. CLEARLY LABEL YOUR AXES. For the plot, connect the points with solid black lines with a LineWidth of 3.

Evaluate │H(jω)│ for the frequencies: ω = {0.25, 1.5, 6.5, 15, 25, 150, 800} rad/s. Do this BY HAND. Clearly show your work. Then plot the resulting {ω, │H(jω)│} points as green diamonds using Markersize of 10 and LineWidth of 3.

Answer 1