Matlab Project: Dynamical Systems

(b) Free vibrations of the system correspond to solutions of the form
x(t) = e iωt u, (3)
where u is a constant vector, ω is the (angular) frequency, and i = √ −1 is the imaginary unit. By
plugging (3) into (2), derive a matrix eigenvalue problem of the form
Au = λu, (4)
where the eigenvalue λ is related to the frequency ω and mass m. Write the formula which relates
λ, ω, and m on the answer sheet.
(c) Use the MATLAB command eig to computer all eigenvalues λ and corresponding eigenvectors u
of the matrix A as defined above. Type “help eig” for details of how to use this command. Use the
last four digits of your student number for k 1 , k 2 , k 3 , and k 4 . For example, if your student number
is 0124659 use k 1 = 4, k 2 = 6, k 3 = 5, and k 4 = 9. Write the eigenvalues on the answer sheet
(rounded to four decimal places).
(d) From equation (3), each eigenvalue/eigenvector pair (λ, u) corresponds to a distinct mode of
vibration. The eigenvalue λ gives a corresponding frequency ω via your formula, and the components
of the eigenvector u describe the motions of the individual masses m 1 , . . . , m 5 . In particular,
the relative sizes of the components show the relative amplitudes of these motions, and the signs
indicate which masses are moving in the which direction at a given time. For example, if we had
u = [3, −2, 5, 1, −4] T then the corresponding motion would look like:
✉ ✲ ✛ ✉ ✉ ✲ ✉✲ ✛ ✉
m 1 m 2 m 3 m 4 m 5
Here, the arrows show the velocities at a time t when all masses are at their equilibrium positions
x i = 0 (they oscillate back and forth around these with velocities as shown). On the answer sheet,
write the largest natural frequency ω (use the value m = 100), and draw a similar picture describing
the corresponding motion of the masses for that mode of vibration.
(e) Show (using pencil and paper) that λ = 0 is always an eigenvalue of A (for any choice of
the k i ), find a corresponding eigenvector, and explain the corresponding motion. Hint: try different
(positive) values for the k i ’s and see what the eigenvector corresponding to λ = 0 looks like—then
show it always works.
Hand in the following:
1. The completed answer sheet. DO NOT hand in the instructions (I already have a copy).
2. A printed copy of your MATLAB session (input and output): for example, use File—Print
to print the MATLAB command window, or use the diary command to capture the input and
output to a file and then print that file. If you use an M-file (MATLAB script) to do these
calculations, also attach a copy of it.
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