Why Are My Coupled Oscillator Eigenvalues Incorrect?

AI Thread Summary
The discussion revolves around a user struggling to find the correct eigenvalues for a system of three masses and four springs, ultimately discovering an error in their determinant calculation. They initially derived an incorrect polynomial due to mistakenly substituting terms in the matrix, specifically replacing ##(-\omega_n^2 + \frac{3k}{m})^2## with ##(-\omega_n^2 + \frac{2k}{m})^2##. After multiple checks, they realized this substitution led to the wrong eigenvalues. Other participants provided insights on expanding the determinant and suggested looking for common factors in the characteristic equation to simplify the process. The user expressed frustration over the time spent on the error but ultimately thanked the community for the clarification.
Redwaves
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Homework Statement
What are the eigenvalue and eigenvectors
##m_a = m_b = m_c##
Relevant Equations
##\frac{d^2x_a}{dt^2} + \frac{2kx_a}{m} - \frac{kx_b}{m}##

##\frac{d^2x_b}{dt^2} + \frac{3kx_b}{m} - \frac{kx_a}{m} - 2\frac{kx_c}{m}##

##\frac{d^2x_c}{dt^2} + \frac{2kx_c}{m} - \frac{2kx_b}{m}##
Hi,
I have to find the eigenvalues and eigenvectors for a system of 3 masses and 4 springs. At the end I don't get the right eigenvalues, but honestly I don't know why. Everything seems fine for me. I spent the day to look where is my error, but I really don't know.
zOPIHiX.png

##m_a = m_b = m_c##

I got these motion equations

##\frac{d^2x_a}{dt^2} + \frac{2kx_a}{m} - \frac{kx_b}{m} = 0##

##\frac{d^2x_b}{dt^2} + \frac{3kx_b}{m} - \frac{kx_a}{m} - 2\frac{kx_c}{m} = 0##

##\frac{d^2x_c}{dt^2} + \frac{2kx_c}{m} - \frac{2kx_b}{m} = 0##

Then, after plugging the solution ##x(t) = X_{ni} cos(\omega_n + \alpha_n)##

I get this matrix

##\begin{pmatrix}
-\omega_n^2 +\frac{2k}{m} & -\frac{k}{m} & 0\\
- \frac{k}{m} & -\omega_n^2 +\frac{3k}{m} & -\frac{2k}{m} \\
0 & -\frac{2k}{m} &-\omega_n^2 +\frac{2k}{m}
\end{pmatrix}##

Next, I have to find that ##det(A)## = 0
##(-\omega_n^2 + \frac{2k}{m})[(-\omega_n^2 + \frac{3k}{m})^2 -(\frac{2k}{m})^2 - (\frac{k}{m})^2 ] = 0##

thus,
I have ##\omega_1^2 = \frac{2k}{m}, \omega_2^2 = -\frac{k}{m}(-3 -\sqrt{5}), \omega_3^2 = -\frac{k}{m}(-3 +\sqrt{5})##

However the correct eigenvalues are
I have ##\omega_1^2 = \frac{2k}{m}, \omega_2^2 = \frac{5-\sqrt{21}}{2}\frac{k}{m}, \omega_3^2 = \frac{5+\sqrt{21}}{2}\frac{k}{m}##

 
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I don't see how you got that polynomial from expanding the determinant. In particular, how do you get a term
##(-\omega_n^2 + \frac{3k}{m})^2##
?
I get the book answer, except I get 24s instead of 21s.
 
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haruspex said:
I don't see how you got that polynomial from expanding the determinant. In particular, how do you get a term
##(-\omega_n^2 + \frac{3k}{m})^2##
?
I get the book answer, except I get 24s instead of 21s.
I agree that the ##(-\omega_n^2 + \frac{3k}{m})^2## term does not belong. However, the determinant I got gives the book's eigenvalues.
 
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kuruman said:
I agree that the ##(-\omega_n^2 + \frac{3k}{m})^2## term does not belong. However, the determinant I got gives the book's eigenvalues.
My bad after checking for 100 times I see my error. I replaced ##(-\omega_n^2 + \frac{3k}{m})^2## with ##(-\omega_n^2 + \frac{2k}{m})^2##
I'm so frustrated I spent all day just for that.
Thanks guys
 
Redwaves said:
My bad after checking for 100 times I see my error. I replaced ##(-\omega_n^2 + \frac{3k}{m})^2## with ##(-\omega_n^2 + \frac{2k}{m})^2##
I'm so frustrated I spent all day just for that.
Thanks guys
I don't see how one gets ##(-\omega_n^2 + \frac{2k}{m})^2## out of this.

When I am faced with the eigenvalues of a matrix larger than 2×2, I try to find common factors in the characteristic equation before multiplying out any terms in parentheses. In this case, expanding the determinant along the top row gives $$\left(\frac{2 k}{m}-\omega_n^2 \right)\left[\left(\frac{3 k}{m}-\omega_n^2 \right)-\frac{4 k^2}{m^2}\right] +\frac{k}{m} \left[-\frac{k}{m}\left(\frac{2k}{m}-\omega_n^2\right)\right]=0.$$There is an obvious common factor ##\left(\frac{2 k}{m}-\omega_n^2 \right)## which provides the first eigenvalue ##\omega_1^2=2k/m##. To find the other two eigenvalues, we assert that ##\omega_n^2\neq2k/m## and divide the equation by the common factor to get $$\left(\frac{3 k}{m}-\omega_n^2 \right)-\frac{5 k^2}{m^2} =0.$$Solving the quadratic is trivial.
 
kuruman said:
I don't see how one gets ##(-\omega_n^2 + \frac{2k}{m})^2## out of this.

When I am faced with the eigenvalues of a matrix larger than 2×2, I try to find common factors in the characteristic equation before multiplying out any terms in parentheses. In this case, expanding the determinant along the top row gives $$\left(\frac{2 k}{m}-\omega_n^2 \right)\left[\left(\frac{3 k}{m}-\omega_n^2 \right)-\frac{4 k^2}{m^2}\right] +\frac{k}{m} \left[-\frac{k}{m}\left(\frac{2k}{m}-\omega_n^2\right)\right]=0.$$There is an obvious common factor ##\left(\frac{2 k}{m}-\omega_n^2 \right)## which provides the first eigenvalue ##\omega_1^2=2k/m##. To find the other two eigenvalues, we assert that ##\omega_n^2\neq2k/m## and divide the equation by the common factor to get $$\left(\frac{3 k}{m}-\omega_n^2 \right)-\frac{5 k^2}{m^2} =0.$$Solving the quadratic is trivial.
I meant, in the matrix at position 2ij, I replaced 2k/m with 3k/m in the determinant formula.
 
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That will do it. Thanks for the clarification.
 
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