Homework 4, due February 16, 2005.

Due Wednesday, February 16, 2005.
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goodwine
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Homework 4, due February 16, 2005.

Post by goodwine »

  1. Consider the system illustrated in the following figure. Assume k=m=1.
    • Image
    1. Derive the equations of motion for the system.
    2. Assuming two solutions of the form
      • Image
      determine two solutions for the frequency and the amplitude ratio of the two responses.
    3. What is the response of the system if
      • Image
    4. Write the equations of motion in the form
      • Image
    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
    • Image
    where
    • Image
    1. Compute a set of 4 vectors that span the null space of
      • Image
    2. For each of the four (generalized) eigenvectors you computed in part a, determine the corresponding homogeneous solution of the form
      • Image
    3. Compute a different set of 4 vectors (at least 3 of the 4 must be different) that span the null space of
      • Image
    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
      • Image
      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
    • Image
    where
    • Image
    by computing homogeneous solutions of the form
    • Image
Last edited by goodwine on Thu Jun 15, 2006 4:23 pm, edited 1 time in total.
asayers

Problem 1c

Post by asayers »

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?
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Re: Problem 1c

Post by goodwine »

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?
I don't have my notes with me, but since the initial velocities are assumed to be zero, there are only two constants to specify. I don't remember the specific answers, but assume a1/a2=2 for one answer and a1/a2=3 for the other one. You can leave the coefficient for the solution for the first mass unspecified for each mode (and call the coefficients a and b) and then for the second mass the coefficients would just be 2a and 3b, respectively.

Caveat: since I don't have my notes with me, I can't say whether I used a and b in the above answer in the same way I did in class, but if you can follow what I said, you should be able to get the idea.
Bill Goodwine, 376 Fitzpatrick
harrier03

HW5

Post by harrier03 »

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
goodwine
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Re: HW5

Post by goodwine »

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
Nope, this is fine. I'll make it available on Friday at the latest.
Bill Goodwine, 376 Fitzpatrick
mightyduck

Post by mightyduck »

Let's have the homework due Friday of next week instead of Wednesday of next week, please. Thanks!!
student

Homework 5 vote

Post by student »

I agree with mightyduck. Have it due on a Friday instead of Wednesday.
HugsyMcHugs

HW # 5

Post by HugsyMcHugs »

Due next wednesday as it normally would be.
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Post by goodwine »

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.
Bill Goodwine, 376 Fitzpatrick
Laplace

frequency response

Post by Laplace »

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.

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
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Re: frequency response

Post by goodwine »

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.
The answer is not 1 and -1. If you think about it intuitively, in at least one mode the second mass should move more than the first one. There are kind of crazy analyitcal expressions for them. If I recall correctly, in decimals one is something like 1.6xxx (it may be -1.6xxx).
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
Again, I don't have my notes with me, but what you got for omega^2 does not look familiar. Your expressions for a1/a2 do look familiar.

I'm not sure what you mean by "pick 2 and 3."
Bill Goodwine, 376 Fitzpatrick
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[color=red]TYPO[/color]

Post by goodwine »

The "i" in the exponent for the expression of the homogeneous solution for a vector in the null space of (A - lambda_i I )^m should be "lambda_i".

I fixed it, but I'm sure that by this time everyone has already printed the homework. Hopefully it hasn't caused any confusion. Since I haven't heard anything about it, I suppose it hasn't.

Sorry.
Bill Goodwine, 376 Fitzpatrick
Laplace

solutions

Post by Laplace »

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.
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Re: solutions

Post by goodwine »

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.
Yes you may. Since I have such a lenient extension policy, some people actually just submitted the homework today, so there was no way that I could post it sooner. It will appear soon.
Bill Goodwine, 376 Fitzpatrick
Laplace

Post by Laplace »

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.
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Post by goodwine »

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.
I just sent an email harassing the TA's for the solutions. I thought most of them were actually posted. I'll get them all up within a couple days, with the exception of the solutions for HW 5 which will be posted next Monday after class.
Bill Goodwine, 376 Fitzpatrick
Laplace

Post by Laplace »

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.
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.
goodwine
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Post by goodwine »

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.
Well, I hate to say it but you're probably going to have to get by with what's in the book, the solutions (I admit they've been late) and the notes. 90% of the material is actually in the book so a good starting point would be to work to reconcile your class notes with the book since you seem to see them more disconnected than they really are. What's not in the book is the vibrations stuff (which was really only a couple days) and the approach for repeated eigenvectors since the book treated it superficially. To the extent I can (office hours and/or a meeting with you) I'm happy to try to get it all together as much as possible.
Bill Goodwine, 376 Fitzpatrick
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