- #1
BOAS
- 552
- 19
Hello,
I have a problem regarding the characteristic frequencies of a coupled mass-spring system. I have made some relevant progress, but I'm unsure of where to go from here.
1. Homework Statement
Find the characteristic frequencies and the two characteristic modes of vibration if the central spring constant in the following diagram (attached) is 2k.
Using Newton's second law, I have written the following two equations for x and y.
##m \ddot{x} = - kx - 2k(y - x) = kx - 2ky##
##m \ddot{y} = - ky - 2k(y - x) = ky + 2kx##
Rearranging for accelerations;
##\ddot{x} = \frac{k}{m} x - \frac{2k}{m} y##
##\ddot{y} = \frac{2k}{m} x + \frac{k}{m} y##
I'm not sure how to do matrices properly in LaTeX, but I write the above as a matrix and solve for the Eigen values. If I call the coefficient matrix ##A## then ##A_{11} = 1##, ##A_{12} = -2##, ##A_{21} = 2##, ##A_{22} = 1##.
Using the fact that ##\det (A - \mu I) = 0## I find that ##\mu = 1 \pm 2i##
Using this fact I find my eigen vectors to be ##\vec e_{1} = \frac{1}{\sqrt{2}}
\left(\begin{array}{c}1\\-i\end{array}\right)## and ##\vec e_{2} = \frac{1}{\sqrt{2}}
\left(\begin{array}{c}1\\i\end{array}\right)##
I think the next step is for me to write out the diagonalised matrix, with the eigen values as the entries top left and bottom right.
##\left(\begin{array}{c}\ddot{X}\\\ddot{Y}\end{array}\right) = \frac{k}{m} \left(\begin{array}{c}1 - 2i & 0\\\ 0 & 1 + 2i\end{array}\right) \left(\begin{array}{c}X\\ Y\end{array}\right)##
This gives me two differential equations;
##\ddot{X} = (1-2i)X##
##\ddot{Y} = (1+2i)Y##
I'm concerned about my complex eigenvalues and think I have gone wrong somewhere, but I can't find an obvious mistake.
Please can somebody take a look and help me to understand where to go from here.
Thank you very much,
BOAS.
Edit - Figured out how to write matrices
I have a problem regarding the characteristic frequencies of a coupled mass-spring system. I have made some relevant progress, but I'm unsure of where to go from here.
1. Homework Statement
Find the characteristic frequencies and the two characteristic modes of vibration if the central spring constant in the following diagram (attached) is 2k.
Homework Equations
The Attempt at a Solution
Using Newton's second law, I have written the following two equations for x and y.
##m \ddot{x} = - kx - 2k(y - x) = kx - 2ky##
##m \ddot{y} = - ky - 2k(y - x) = ky + 2kx##
Rearranging for accelerations;
##\ddot{x} = \frac{k}{m} x - \frac{2k}{m} y##
##\ddot{y} = \frac{2k}{m} x + \frac{k}{m} y##
I'm not sure how to do matrices properly in LaTeX, but I write the above as a matrix and solve for the Eigen values. If I call the coefficient matrix ##A## then ##A_{11} = 1##, ##A_{12} = -2##, ##A_{21} = 2##, ##A_{22} = 1##.
Using the fact that ##\det (A - \mu I) = 0## I find that ##\mu = 1 \pm 2i##
Using this fact I find my eigen vectors to be ##\vec e_{1} = \frac{1}{\sqrt{2}}
\left(\begin{array}{c}1\\-i\end{array}\right)## and ##\vec e_{2} = \frac{1}{\sqrt{2}}
\left(\begin{array}{c}1\\i\end{array}\right)##
I think the next step is for me to write out the diagonalised matrix, with the eigen values as the entries top left and bottom right.
##\left(\begin{array}{c}\ddot{X}\\\ddot{Y}\end{array}\right) = \frac{k}{m} \left(\begin{array}{c}1 - 2i & 0\\\ 0 & 1 + 2i\end{array}\right) \left(\begin{array}{c}X\\ Y\end{array}\right)##
This gives me two differential equations;
##\ddot{X} = (1-2i)X##
##\ddot{Y} = (1+2i)Y##
I'm concerned about my complex eigenvalues and think I have gone wrong somewhere, but I can't find an obvious mistake.
Please can somebody take a look and help me to understand where to go from here.
Thank you very much,
BOAS.
Edit - Figured out how to write matrices