*F(t)*is harmonic. The reason we want to be able to do this is that such solutions describe the motion of the system illustrated in the following figure, which is obviously a decent model of a lot of shaking/vibrating mechanical systems.

- Last week you solved
- Using your plots from the previous problem, determine an approximate steady state solution for each of the following
- For the equation in part (b) of the previous problem write a computer program to determine an approximate numerical solution using Euler's method for the case when both initial conditions are zero. Determine an appropriate step size by continuing to decrease the size of the time step until the solution does not change.

Plot the approximate numerical solution and compare this with the approximate steady state solution you determined in the previous problem. Indicate from the plot of the two solutions- whether or not they match for all time, and if not explain the differences;
- the relationship between the phase angle determined from the graph from the first problem and the relationship between the two solutions you plotted here.

- If the damping ratio is greater than zero, are there any conditions under which the homogeneous solution will not decay? If so, what are they? If not, is it always then justified to consider the steady state response of the system to be the particular solution.
- Plot the solution to
*t=0*to*t=60.*You may use any method you want, including writing a computer program to determine an approximate numerical solution, but plot the whole solution, not just the steady state solution. Explain the various features of the problem, namely- what is happening between 0 and 10 seconds; and
- what is happening betwen 10 and 60 seconds.

- Consider
*M*'s and phase shifts are determined from the graphs separately? Justify your answer. Demonstrate whether or not it works for this problem by plotting the approximate steady state solution given by the previous equation and the actual solution (or actual steady state solution) on the same graph.