Eigenvalues and eigenvectors - Your turn

Three blocks of mass \(m\) are placed in between four springs with constant \(k\). A system of equations can be set up such that

\[\begin{split} \begin{align*} m \ddot{x}_1 &= -2kx_1 + kx_2 \\ m \ddot{x}_2 &= kx_1 - 2kx_2 + kx_3 \\ m \ddot{x}_3 &= kx_2 - 2kx_3 \end{align*} \end{split}\]

If \(k = 1.5\), \(m = 2\), and the initial displacement of the blocks are \(x_1 = 0.1\), \(x_2 = -0.9\), \(x_3 = 0.3\), find the equation of the position of the blocks as a function of time. Plot the displacement of the blocks as a function of time with \(dt=0.01\) for the first thirty seconds. Draw a picture of the mode shapes of the blocks.

# TODO: Write your solution below

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