Bill Goodwine's Courses

The purpose of this homework is to develop the ability to properly use the equations associated with the solution(s) to

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in the case where the forcing function, 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.

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In class we determined a particular solution for the case where

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which was of the form

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1. Last week you solved
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   which, if you did it correctly would have a particular solution of the form
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   Determine a phase shift and static deflection scale factor that are a function of the damping ratio and ratio of the forcing frequency to the natural frequency that converts this to the form
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   Accurately plot the phase angle versus frequency ratio for various damping ratios. Similarly, accurately plot the static deflection scale factor as a function of frequency ratio for different damping ratios. Indicate whether or not they are exactly the same as for the case we solved in class. Make an extra copy of these graphs. They may be useful on the first exam!
2. Using your plots from the previous problem, determine an approximate steady state solution for each of the following
   1. 
   2. 
   3. 
   4. 
   5. 
   6. 
   For all three equations, plot the steady state solution and the forcing function on the same graph. If it makes sense to plot them on the same graph, do so. Regardless of whether or not they are plotted on the same graph, make sure that the phase shift is accurately represented.
3. 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
   1. whether or not they match for all time, and if not explain the differences;
   2. the relationship between the phase angle determined from the graph from the first problem and the relationship between the two solutions you plotted here.
4. 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.
5. Plot the solution to
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   for 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
   1. what is happening between 0 and 10 seconds; and
   2. what is happening betwen 10 and 60 seconds.
   Hopefully you know that "by explain" I do not mean to describe what you see, but explain what it is in the structure of the equation or solution that causes the various features you see in the solution.
6. Consider
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   If all we are interested in is the steady state response, is it appropriate to simply write
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   where the 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.

There should be a F multiplying the sine term in the solution.

For problem 2, "For all three equations" should say "For the first three equations"

For the first problem, when it says "accurately plot the static deflection scale factor as a function of frequency ratio for different damping ratios," do you mean to graph delta vs w/wn or M vs w/wn? Also, on problem 1, you said in a later post that there should be an F in front of sin for the solution, but isn't that F included within the delta (F/(m*wn^2))?

I now see what you mean by the F being in front of the sine term--it's for the particular solution to last week's problem, correct?

TAllen1 wrote:For the first problem, when it says "accurately plot the static deflection scale factor as a function of frequency ratio for different damping ratios," do you mean to graph delta vs w/wn or M vs w/wn?

The static deflection is delta. The scale factor is M, so it says to plot M.

Also, on problem 1, you said in a later post that there should be an F in front of sin for the solution, but isn't that F included within the delta (F/(m*wn^2))?

Yes, F is in delta. But, the given solution with both the sine and cosine term isn't correct without an F multiplying the sine term. You'll have a lot of trouble converting that to the solution with the delta in it without having F in each term.

TAllen1 wrote:I now see what you mean by the F being in front of the sine term--it's for the particular solution to last week's problem, correct?

Yes, that's right. I guess I should have read both posts before answering.

Someone asked me:

When I graph the frquency ratio versus phi, when the ratio equals one my plot
drops instead of continuing a pi/2 and continues at -pi/2 and approaches zero
instead of pi. I have the same problem just and inverted picute if I plot the
frequency ratio versus phi that we did in class. Is there something I need to
do in MatLab to tell it to use the other value of arctan(x/0)?

I think you want to use the atan2 function instead.