Bill Goodwine's Courses

1. Consider the system illustrated in the following figure. Assume k=m=1.
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   1. Derive the equations of motion for the system.
   2. Assuming two solutions of the form
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      determine two solutions for the frequency and the amplitude ratio of the two responses.
   3. What is the response of the system if
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   4. Write the equations of motion in the form
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   5. Compute the eigenvalues and eigenvectors of A (using a computer is allowed).
   6. Determine the general solution of the system in terms of real-valued functions.
   7. Verify your answer from part c using your solution from this problem.
2. Consider
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   where
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   1. Compute a set of 4 vectors that span the null space of
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   2. For each of the four (generalized) eigenvectors you computed in part a, determine the corresponding homogeneous solution of the form
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   3. Compute a different set of 4 vectors (at least 3 of the 4 must be different) that span the null space of
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   4. Determine the homogeneous solution corresponding to each (generalized) eigenvector from part c.
   5. Determine the general solution for the system, including the -2 eigenvalue, using your 4 homogeneous solutions from part b.
   6. Determine the general solution for the system, including the -2 eigenvalue, using your 4 homogeneous solutions from part d.
   7. The two general solutions shold be different. If
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      are the solutions to the initial value problem the same? If they are, must they be? Are the two general solutions actually equivalent?
3. Find the general solution to
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   where
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   by computing homogeneous solutions of the form
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asayers wrote:I found the amplitude ratio and frequncies, and they weren't 1 and -1. So when I add the two modes in the solution, is it the first ratio times the a and the other ratio times the b constant?

harrier03 wrote:Sorry if this was supposed to be posted elsewhere. Regardless of when the homework is due, if it could be posted at the end of this week, like it normaly would, it would be nice. thanks

Is it possible to give us a little guidance on where the material in the homework can be found in the text book? My notes alone seem to be not enough information to solve the homework problems. Perhaps the other students are encountering similar problems.

Laplace wrote:I am confused about the amplitude ratio.

Is it supposed to be -1 and 1? Hartog shows this is the case when k1 = k2, but he is using a system where k2 is actually a k3.

I did it out myself and got something other than -1 and 1.

Laplace wrote: For omega^2 I got + or - 1/(2^.5). I'll double check that later.

for a1/a2, after I substitute in for k and m equal 1 I get the ratio equals 1/(-w^2 + 2) or -w^2/1 +1. I'll check that later, but if I'm wrong on these let me know.

I'm not sure why we can pick 2 and 3.

Thanks

Laplace wrote:I cannot find the solutions, I assume they are not posted yet.

Could we get these as soon as possible? It would be beneficial for studying for the upcoming exam.

Laplace wrote:Thank you.

It would be good to get solutions for HW 2-5 before the test, the earlier the better. All I can find is 1. Old exams with solutions would be good also.

goodwine wrote:

Is it possible to give us a little guidance on where the material in the homework can be found in the text book? My notes alone seem to be not enough information to solve the homework problems. Perhaps the other students are encountering similar problems.

As I mentioned in class, the vibrations stuff is from Den Hartog. However, since that book isn't required, I made sure that I gave basically all the information in class.

If you came to all the classes, your notes are complete with regard to what you need to know for the material on repeated roots. The corresponding book section is the "Repeated Roots" one. However, apparently Boyce and Di Prima don't think that undergraduates are smart enough to handle repeated roots for the case where the matrix is bigger than 2x2. I disagree; hence, I did the full nxn case in class. You'll have to rely on your notes for this. It hasn't seemed to be a problem for classes in the past, so I expect that the notes will suffice.

Laplace wrote:I would like to politely agree. This semester it is a lot more difficult because we have no solution tree and cannot reference the text. I have notes for each class, but sometimes it is hard to see where the notes apply. I am not saying that class is useless. I come to every class and try to get the most out of it.

I have found that typeset class handouts are very helpful such as the one for HW3. It would also be good if we could get a summary of what we have learned so far and where we can find it in our notes. Studying is a little harder this semester because we have so few examples, little book backup, and no solutions for the homework.