Homework 4, due March 1, 2017

Due at the beginning of class on Wednesday, March 1, 2017.
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goodwine
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Homework 4, due March 1, 2017

Post by goodwine »

Consider the system in Fig. 7.18 (Exercise 7.4) but make three changes
  1. instead of n masses, you may specifically assume there are 10 masses
  2. remove all the dampers
  3. on the right end after the 10th mass, put a wall with a spring connecting the wall to the spring.
Part of this homework is for you to decide the effective way to communicate your work. I will answer questions and give feedback along those lines, but won't specify what you should do. I would recommend doing the whole problem first, and then write your homework. Clarity in your understanding of what is going on will probably make the communication clearer and easier. This is still a homework, not a project report, so typing is not necessary. Clarity, however, is.

Do the following:
  1. Write the equation of motion for the ith mass.
  2. Express the equations of motion for the entire system as a system of first order linear differential equations in matrix form, i.e., \dot \xi = A \xi. You do not have to write out the entire A matrix, but if you use a "..." notation, your answer must be complete enough so that anyone could fill in those parts.
  3. Choose values for the masses and spring constants. You may have all the masses, m, be the same and all the springs, k, be the same, but do not chose either m=1 or k=1 and do not choose m=k. If you are working with a friend, chose different values. You will change one mass value or one spring value later.
  4. Compute the eigenvalues and eigenvectors. Of course you may use matlab or something similar. Use variables, not numbers, in your A matrix, because you may have to change the values later. If the frequencies of oscillation for the system are not in a physically-realistic range change m and/or k to make them so.
  5. It makes sense that if one of the spring constants or masses changes a little bit, all of the frequencies will change slightly. Is this the case? Vary one of them and plot how the frequencies change as it is varied. You decide the best manner in which to communicate this. If you are working with a friend, choose a different mass or spring to change.
  6. Put a force on the system in the form of \cos \omega t. If the frequency is one of the eigenvalues, does the solution blow up? Does the shape of the solution correspond to the eigenvector corresponding to that frequency? Does it matter to which mass the force is applied? Compare the magnitude of the motion when the frequency of the forcing is between two eigenvalues to when it is close to one of them. Does the size of the solution correspond to what you expect?
Bill Goodwine, 376 Fitzpatrick
amuelle1

Re: Homework 4, due March 1, 2017

Post by amuelle1 »

Do we need to report our eigenvalues and eigenvectors for part 4, or will those be graded based on application to parts 5 and 6?
goodwine
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Re: Homework 4, due March 1, 2017

Post by goodwine »

amuelle1 wrote: Mon Feb 27, 2017 4:40 pm Do we need to report our eigenvalues and eigenvectors for part 4, or will those be graded based on application to parts 5 and 6?
Since you are going to use or report them in 5 and 6, you can just say something like "the original eigenvalue is the value in this plot for k=10 (if k=10 was the original k).
Bill Goodwine, 376 Fitzpatrick
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