Matlab Project: Dynamical Systems

Matlab Project: Dynamical Systems
Goal: Compute the eigenvalues and eigenvectors of a dynamical system
Problem:
Consider a system of five masses m 1 , m 2 , m 3 , m 4 , m 5 connected in a straight line by four springs
with spring constants k 1 , k 2 , k 3 , k 4 as shown below:
k 1 k 2 k 3 k 4
✉ ✉ ✉ ✉ ✉ ✲ x
m 1 m 2 m 3 m 4 m 5
Such a system could serve as a simple model for various physical systems, such as a linear molecule
composed of five atoms, or a string of five railroad cars loaded on a barge. We want to determine
the natural frequencies of vibration of the system: for a molecule, these will correspond to the
frequencies of light which it absorbs; for a string of railroad cars, forcing at any of these frequencies
(by wave motion under the barge) will cause resonance and possible destruction of the train. The
purpose of this project is to illustrate how you can solve such a problem using MATLAB to compute
eigenvalues and eigenvectors.
Details:
Assuming that the motion is confined to the x-direction only and that the springs obey Hooke’s
Law, the motion of the system is described by the equations
m 1
d 2 x 1
dt 2 = k 1 (x 2 − x 1 )
m 2
d 2 x 2
dt 2 = −k 1 (x 2 − x 1 ) + k 2 (x 3 − x 2 )
m 3
d 2 x 3
dt 2 = −k 2 (x 3 − x 2 ) + k 3 (x 4 − x 3 )
m 4
d 2 x 4
dt 2 = −k 3 (x 4 − x 3 ) + k 4 (x 5 − x 4 )
m 5
d 2 x 5
dt 2 = −k 4 (x 5 − x 4 )
Here x i (t) denotes the position of mass m i at time t, with each x i = 0 if all springs are unstretched.
(a) Assuming for simplicity that all masses are equal (m 1 = · · · = m 5 = m), the system (1) can be
written in matrix form as
m d2 x
= −Ax, (2)
dt2 where x(t) = [x 1 , x 2 , x 3 , x 4 , x 5 ] T . On the answer sheet, write out the matrix A explicitly in terms
of k 1 , k 2 , k 3 , k 4 . Take care to account for the minus sign in (2). Note that A is tridiagonal (i.e.,
only three diagonals are nonzero) and symmetric.
(1)
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(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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Matlab Project: Dynamical Systems
ANSWER SHEET
Name:
(a) Matrix A (in terms of k 1 , k 2 , k 3 , and k 4 ):
(b) Formula relating λ, ω, and m:
(c) Eigenvalues λ of the matrix A (in increasing order, rounded to four decimal places):
(d) Highest natural frequency ω of the system (assuming m = 100):
Corresponding eigenvector u:
Draw a picture describing the motion of the system for this mode of vibration:
(e) Put your answer for this part on the back of this sheet.
Also attach a copy of your MATLAB input and output to this answer sheet.