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}.

Numerically simulate the response of the system using the Matlab function ode45. Plot the displacement of the mass as a function of time for the first 0.5s. Use a time-resolution of 0.001s (have ode45 return values of x(t) at 1ms intervals). This plot should be a solid black line with a LineWidth of 3.

“Hold” the plot and superimpose the two points listed above (the two {t, x(t)} pairs). Use red circles to show those points. Make the circles’ Markersize of 10 and LineWidth of 3.

Determine the exact analytic (“hand”) solution to the problem. Use Matlab to calculate the values of that solution at 0.05s time steps. Superimpose that data on top of the numerical solution (and the two red circles) using blue diamonds with the same Markersize and LineWidth parameters as for the circles.

Answer 1

Solution of given damped free vibration system is uploaded as images.

time	x(t)
0	-1
0.05	0.0351682638864572
0.1	0.216712845028194
0.15	0.199868267077817
0.2	0.150567255340181
0.25	0.106718799685943
0.3	0.0740082879560693
0.35	0.0509020963517529
0.4	0.0348984564210324
0.45	0.0238966756472850
0.5	0.0163552842386907