# Coupled Harmonic Oscillator (Forced Vibration)

I derived two equations for a 2DOF harmonic oscillator system, declared state variable equations, and placed them into matrix form: $$Ax' + Bx = C$$. I have a Matlab script to determine the constants ($$K$$'s, $$m$$'s, & $$R$$'s). I'll be seeing how the system responds from 40 - 1000 Hz.

How can I manipulate these matrices to find the solutions to $$x'$$: $$\begin{array}{c} \dot{x}_s , \dot{x}_m , \dot{v}_s , \dot{v}_m \end{array}$$

$$\begin{split} m_s \ddot{x}_s + R_{s1}\dot{x}_s + R_{2s}\left( \dot{x}_s - \dot{x}_m\right) + K_{s1}x_s + K_{s2} \left( x_s - x_m \right) &=& P_0 \\ m_m \ddot{x}_m + R_{m}\dot{x}_m - R_{2s}\left( \dot{x}_s - \dot{x}_m\right) + K_{m}x_m - K_{s2} \left( x_s - x_m \right)&=&0 \\ \dot{x}_s - v_s&=& 0 \\ \dot{x}_m - v_m &=& 0 \end{split}$$

$$\left[ \begin{array}{cccc} 1 &0 & 0 &0 \\ 0 &1 & 0 & 0 \\ 0 & 0 & m_s &0 \\ 0 &0 & 0 & m_m \\ \end{array} \right] \left[ \begin{array}{c} \dot{x}_s \\ \dot{x}_m \\ \dot{v}_s \\ \dot{v}_m \end{array} \right] + \left[ \begin{array}{cccc} 0 &0 & - 1& 0 \\ 0& 0 &0 & -1 \\ K_{s1} + K_{s2} & -K_{s2} & R_{s1} + R_{s2} & -R_{s2} \\ -K_{s2} & K_{m} + K_{s2} & - R_{s2} & R_m + R_{s2} \end{array} \right] \left[ \begin{array}{c} {x}_s \\ {x}_m \\ {v}_s \\ {v}_m \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 1 \\0 \end{array} \right] e^{i\omega t}$$

• If you eliminate the "duplicate" $v = \dot x$ variables, you just have a standard undamped 2DOF system with equation of motion $M\ddot x + Kx = Fe^{i\omega t}$. See any textbook or web site on MDOF systems. – alephzero Jun 6 at 9:34