Natural frequencies of a two-mass system

Data: $m_1 = 2$ kg, $m_2 = 1$ kg, $k_1 = 3000$ N/m, $k_2 = 2000$ N/m.

 wall |--k1--[ m1 ]--k2--[ m2 ]
          x1       x2

Equations of motion

$$m_1\ddot{x}_1 + (k_1+k_2)x_1 - k_2x_2 = 0, \qquad m_2\ddot{x}_2 - k_2x_1 + k_2x_2 = 0$$

In matrix form with $x = X\sin\omega t$:

$$\begin{bmatrix} 5000 - 2\omega^2 & -2000 \\ -2000 & 2000 - \omega^2 \end{bmatrix}\begin{Bmatrix} X_1 \\ X_2 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix}$$

Frequency equation

$$(5000 - 2\omega^2)(2000 - \omega^2) - (2000)^2 = 0$$ $$2\omega^4 - 9000\omega^2 + 6\times10^6 = 0 \;\Rightarrow\; \omega^4 - 4500\omega^2 + 3\times10^6 = 0$$ $$\omega^2 = \frac{4500 \pm \sqrt{4500^2 - 4(3\times10^6)}}{2} = \frac{4500 \pm 2872.28}{2}$$ $$\omega_1^2 = 813.86 \Rightarrow \omega_1 = 28.53\ \text{rad/s}\ (f_1 = 4.54\ \text{Hz})$$ $$\omega_2^2 = 3686.14 \Rightarrow \omega_2 = 60.71\ \text{rad/s}\ (f_2 = 9.66\ \text{Hz})$$

Mode shapes

From the second row, $\dfrac{X_2}{X_1} = \dfrac{2000}{2000 - \omega^2}$ (equivalently $\dfrac{5000 - 2\omega^2}{2000}$).

 Mode 1: m1 -> , m2 ---> (same direction)
 Mode 2: m1 -> , m2 <-   (node between m1 and m2)

Answer: $\omega_1 = 28.53$ rad/s, $\omega_2 = 60.71$ rad/s; mode shapes $\{1,\ 1.686\}$ and $\{1,\ -1.186\}$.