Consider the following spring system.
$c_1 = 20,\quad c_2 = {\frac{2}{3}},\quad c_3 = 4 \\ m_1 = 4,\quad m_2 = {\frac{2}{3}}$

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Write the stiffness matrix $K =$$\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
Write the matrix $M^{-1}K =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$

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Find the eigenvalues and eigenvectors of $M^{-1}K$
$\qquad \bullet$ Smaller eigenvalue = with eigenvector $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
$\qquad \bullet$ Larger eigenvalue = with eigenvector $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$

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If this spring system oscillates without any external forces present, then the position of each mass satisfies the following general formula: If the system begins oscillation with initial position $\mathbf{u}(0) = \left[\begin{array}{c} 2\cr 0 \end{array}\right]$ and initial velocity $\mathbf{u}'(0) = \left[\begin{array}{c} -1\cr 1 \end{array}\right]$ then the position of the masses at time $t$ is given by