運動方程式 \[ \begin{align} \ & \left\{\begin{array}{cc} m_1 x_1= - k_1 x_1 - k_2 x_1 + k_2 x_2 \\ m_2 x_2= - k_2 x_1 + k_3 x_3 - k_2 x_2 - k_3 x_2 \\ m_3 x_3= - k_3 x_3 - k_4 x_1 - k_3 x_2 \end{array}\right. \\ \\ &\left\{\begin{array}{cc} m_1 x_1 + \left( k_1 + k_2 \right) x_1 + k_2 x_2 =0\\ - k_2 x_1 + m_2 x_2 - \left( k_2 + k_3 \right) x_2 - k_3 x_3 =0 \\ - k_3 x_2 + m_3 x_3 + \left( k_3 + k_4 \right) x_3 =0 \end{array}\right. \\ \\ & \left. \begin{array}{cc} x_1 = X_1 \sin \omega T \\ x_2 = X_2 \sin \omega T \\ x_3 = X_3 \sin \omega T \end{array}\right) とおくと \left(\begin{array}{cc} x_1 = - X_1 \omega ^2 \sin \omega T \\ x_2 = - X_2 \omega ^2 \sin \omega T \\ x_3 = - X_3 \omega ^2 \sin \omega T \end{array}\right. \end{align} \] 運動方程式に代入する \[ \begin{align} \ & \left\{\begin{array}{cc} m_1 \left( -x_1 \omega ^2 \right) + \left( k_1 + k_2 \right)=0 \\ - k_2 X_1 +m_2 \left( -X_2 \omega ^2 \right) + \left( k_2 + k_3 \right) - k_2 X_3=0 \\ - k_3 X_2 +m_3 \left( - X_3 \omega ^2 \right) + \left( k_3 + k_4 \right)X_3=0 \end{array}\right. \\ \\ &\left\{\begin{array}{cc} \left\{ - m_1 \omega ^2 + k_1 +k_2 \right) X_1 - k_2 X_2 =0 \\ - k_2 X_1 \left( +m_2 \omega ^2 + k_1 + k_3 \right) + \left( k_2 + k_3 \right) - k_2 X_3=0 \\ - k_3 X_2 +m_3 \left( - m_3 \omega ^2 + k_3 + k_4 \right) X_3=0 \end{array}\right. \end{align} \] $ m_1 = m_2 = m_3 = m, \ k_1 = k_2 = k_3 = k_4 = k$なので， \[ \left|\begin{array}{cccc} -m \omega ^2 + 2 k & -k & 0 \\ -k & -m \omega ^2 + 2 k & -k \\ 0 & -k & -m \omega ^2 + 2 k \end{array}\right| =0 \] \[ \left( -m \omega ^2 + 2 k\right)^3 - k^2 \left( -m \omega ^2 + 2k\right) - \left( -m \omega ^2 + 2k\right) k^2 =0 \\ \left( -m \omega ^2 + 2 k\right)^3 - 2 k^2 \left( -m \omega ^2 + 2k\right) =0 \\ \left( -m \omega ^2 + 2 k\right) \left\{ \left( -m \omega ^2 + 2k\right)^2 - 2k^2 \right\} =0 \] $-m \omega ^2 + 2 k =0$より， \[ \begin{align} \omega &= \sqrt \frac{2 k}{m}\\ &= \sqrt \frac{2 \times 50 \times 10^3 \,\rm{N/m}}{3\,\rm{kg}} \\ &= 182.5\,\rm{rad/s} \end{align} \] $\left( -m \omega ^2 + 2 k^2 \right)^2 -2 k^2 =0$より， \[ m^2 \omega ^4 - 2 m \omega ^2 \times 2 k +4 k^2 - 2 k^2=0 \\ m^2 \omega ^4 - 4 m \omega ^2 k + 2 k^2=0 \\ \begin{align} \omega ^2 &= \frac{4 m k \pm \sqrt {16 m^2 k^2-4 m^2 \times 2 k^2} }{2 m^2} \\ & = \frac{4 m k \pm \sqrt {8 m^2 k^2}}{2m^2} \\ & = \left( 2 \pm \sqrt 2 \right) \frac{k}{m} \\ \omega & = \sqrt{2 \pm \sqrt 2} \cdot \sqrt \frac{k}{m} \\ & = \sqrt{2 \pm \sqrt 2} \cdot \sqrt \frac{50 \times 10^3\,\rm{N/m}}{3 \,\rm{kg}} \\ & = 238.5\,\rm{rad/s}, \ 98.80\,\rm{rad/s} \end{align} \] \[ \begin{align} \therefore \ &\omega _1=98.8\,\rm{rad/s}\\ &\omega _2=183\,\rm{rad/s}\\ &\omega _3=239\,\rm{rad/s} \end{align} \]