- #1

Redwaves

- 134

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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.

##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}##

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.

##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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