Coupled oscillator eigenvalues

In summary, the conversation revolved around finding the eigenvalues and eigenvectors for a system of 3 masses and 4 springs. The speaker encountered an error in their calculations and spent the day trying to find the mistake. Eventually, they discovered that they had made a mistake in replacing 2k/m with 3k/m in the determinant formula. This led to incorrect eigenvalues, which were eventually corrected.
  • #1
Redwaves
134
7
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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  • #2
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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  • #3
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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  • #4
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
 
  • #5
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.
 
  • #6
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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  • #7
That will do it. Thanks for the clarification.
 

Related to Coupled oscillator eigenvalues

1. What are coupled oscillator eigenvalues?

Coupled oscillator eigenvalues are the values that represent the natural frequencies of a system of coupled oscillators. They determine how the oscillators will behave and interact with each other.

2. How are coupled oscillator eigenvalues calculated?

Coupled oscillator eigenvalues can be calculated using mathematical equations that take into account the masses, spring constants, and damping coefficients of the oscillators in the system. These equations can be solved to find the eigenvalues.

3. What is the significance of coupled oscillator eigenvalues?

The eigenvalues of coupled oscillators are important because they determine the stability and behavior of the system. They can also provide insight into the dynamics of the system and how it will respond to different inputs.

4. Can coupled oscillator eigenvalues change?

Yes, coupled oscillator eigenvalues can change depending on the parameters of the system. For example, if the masses or spring constants are altered, the eigenvalues will also change, resulting in a different behavior of the system.

5. How are coupled oscillator eigenvalues used in real-world applications?

Coupled oscillator eigenvalues have many practical applications, such as in engineering, physics, and biology. They can be used to analyze and design systems with multiple oscillators, such as electronic circuits, musical instruments, and biological networks.

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