Two-mass system X2/F with F-V and F-I analogy (D3)

Displacements $x_1$ ($M_1$, force $F$, ground friction $D_1$) and $x_2$ ($M_2$, ground friction $D_2$). $K_1$ ties $M_1$ to the left wall, $K_3$ ties $M_2$ to the right wall, and $K_2 \parallel D_3$ couples them.

Equations of motion (Laplace)

$$\begin{aligned} \left[M_1s^2 + (D_1 + D_3)s + K_1 + K_2\right]X_1 - (D_3s + K_2)X_2 &= F \\ -(D_3s + K_2)X_1 + \left[M_2s^2 + (D_2 + D_3)s + K_2 + K_3\right]X_2 &= 0 \end{aligned}$$

(Sum of impedances at each mass on the diagonal; coupling impedance off the diagonal.)

Transfer function

$$\frac{X_2(s)}{F(s)} = \frac{D_3s + K_2}{\Delta}$$ $$\Delta = \left[M_1s^2 + (D_1+D_3)s + K_1 + K_2\right]\left[M_2s^2 + (D_2+D_3)s + K_2 + K_3\right] - (D_3s + K_2)^2$$

Expanded:

$$\begin{aligned} \Delta =\; & M_1M_2s^4 + \left[D_1M_2 + D_2M_1 + D_3(M_1 + M_2)\right]s^3 \\ & + \left[D_1D_2 + D_1D_3 + D_2D_3 + M_1(K_2 + K_3) + M_2(K_1 + K_2)\right]s^2 \\ & + \left[D_1(K_2 + K_3) + D_2(K_1 + K_2) + D_3(K_1 + K_3)\right]s \\ & + K_1K_2 + K_1K_3 + K_2K_3 \end{aligned}$$

F-V analogy (mesh)

$F\to e$, $M\to L$, $D\to R$, $K\to 1/C$, velocity $\to$ current.

  L1=M1 R=D1  C=1/K1        L2=M2 R=D2  C=1/K3
 +-LLLL-/\/\--||---+-------+-LLLL-/\/\--||---+
 |                 |                         |
(~) e           R3 = D3                      |
 |                 |                         |
 |    i1        C2 = 1/K2         i2         |
 +-----------------+-------------------------+

F-I analogy (node)

$F\to i$, $M\to C$, $D\to 1/R$, $K\to 1/L$, velocity $\to$ voltage.

 v1                   L2∥R3      v2
 +----+----+----+---+--LLLL--+---+----+----+
 |    |    |    |   +-/\/\/--+   |    |    |
(i)   C1   R1   L1               C2   R2   L3
 |    |    |    |                |    |    |
 +----+----+----+----------------+----+----+ gnd

($C_1 = M_1$, $R_1 = 1/D_1$, $L_1 = 1/K_1$; $L_2 = 1/K_2$ in parallel with $R_3 = 1/D_3$; $C_2 = M_2$, $R_2 = 1/D_2$, $L_3 = 1/K_3$.) The mesh (F-V) and node (F-I) equations have exactly the same form as the two mechanical equations above.