Forced damped two-degree-of-freedom system

Equations of motion

Take masses $m_1, m_2$, springs $k_1$ (wall to $m_1$), $k_2$ (between masses), dampers $c_1, c_2$ in parallel with them, and force $F_1\sin\omega t$ on $m_1$.

$$\begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}\begin{Bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \end{Bmatrix} + \begin{bmatrix} c_1+c_2 & -c_2 \\ -c_2 & c_2 \end{bmatrix}\begin{Bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{Bmatrix} + \begin{bmatrix} k_1+k_2 & -k_2 \\ -k_2 & k_2 \end{bmatrix}\begin{Bmatrix} x_1 \\ x_2 \end{Bmatrix} = \begin{Bmatrix} F_1\sin\omega t \\ 0 \end{Bmatrix}$$

or $[M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{F\}$.

Steady-state amplitudes

Write the force as $F_1e^{i\omega t}$ and the response as $\{x\} = \{X\}e^{i\omega t}$. Substituting:

$$\left[-\omega^2[M] + i\omega[C] + [K]\right]\{X\} = \{F\}$$

The bracket is the mechanical impedance matrix $[Z(\omega)]$ with entries $Z_{rs} = -\omega^2 m_{rs} + i\omega c_{rs} + k_{rs}$. Then

$$\{X\} = [Z(\omega)]^{-1}\{F\}, \qquad X_1 = \frac{Z_{22}F_1}{Z_{11}Z_{22} - Z_{12}^2}, \quad X_2 = \frac{-Z_{12}F_1}{Z_{11}Z_{22} - Z_{12}^2}$$

The amplitudes are complex: $|X_r|$ is the amplitude, and the argument gives the phase lag behind the force. Damping makes the denominator never vanish, so the resonance peaks are finite.

Coupling

• Static (elastic) coupling: the stiffness matrix has off-diagonal terms ($k_{12} \ne 0$); a static force on one coordinate displaces the other.
• Dynamic (inertia) coupling: the mass matrix has off-diagonal terms ($m_{12} \ne 0$), e.g. a vehicle body with the c.g. not midway between the axles.
• The coupling depends on the choice of coordinates, not on the system.

Principal (normal) coordinates

They are special coordinates in which both $[M]$ and $[K]$ become diagonal, so the equations are uncoupled and each is a single-degree-of-freedom equation:

$$\ddot{q}_r + \omega_r^2 q_r = 0, \qquad \{x\} = [\Phi]\{q\}$$

where $[\Phi]$ is the matrix of mode shapes. Each principal coordinate vibrates in one mode only.