Three-mass system X2/F with F-V and F-I analogy

Displacements $X_1, X_2, X_3$ of $M_1, M_2, M_3$; no ground friction ($f_{v1} = f_{v2} = f_{v3} = 0$).

Equations of motion (Laplace)

$$\begin{aligned} (M_1s^2 + D_1s + K_1 + K_2)X_1 - (D_1s + K_2)X_2 &= 0 \\ -(D_1s + K_2)X_1 + (M_2s^2 + D_1s + K_2 + K_3)X_2 - K_3X_3 &= 0 \\ -K_3X_2 + (M_3s^2 + K_3)X_3 &= F \end{aligned}$$

Transfer function by Cramer's rule

$$\frac{X_2(s)}{F(s)} = \frac{K_3(M_1s^2 + D_1s + K_1 + K_2)}{\Delta}$$ $$\begin{aligned} \Delta =\; & (M_1s^2 + D_1s + K_1 + K_2)\left[(M_2s^2 + D_1s + K_2 + K_3)(M_3s^2 + K_3) - K_3^2\right] \\ & - (D_1s + K_2)^2(M_3s^2 + K_3) \end{aligned}$$

($\Delta$ is a 6th-order polynomial; its constant term is $K_1K_2K_3$.)

F-V analogy (mesh)

$F\to e$, $M\to L$, $D\to R$, $K\to 1/C$, velocity $\to$ mesh current. Elements between two masses are shared by their meshes.

  L1=M1       L2=M2          L3=M3
 +-LLLL-+-----LLLL----+------LLLL----+
 |      |             |              |
C1     R1=D1        C3=1/K3         (~) e
=1/K1   |             |              |
 |     C2=1/K2        |              |
 | i1   |      i2     |      i3      |
 +------+-------------+--------------+

Mesh 1: $L_1, C_1$ plus shared ($R_1 + C_2$); mesh 2: $L_2$ plus shared branches; mesh 3: $L_3$, source $e$, shared $C_3$.

F-I analogy (node)

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

 v1         L2∥R1      v2  L3   v3
 +----+---+--LLLL--+---+--LLLL--+----+
 |    |   +-/\/\/--+   |        |    |
 C1   L1               C2       C3  (i)
 |    |                |        |    |
 +----+----------------+--------+----+ gnd

($C_1 = M_1$, $C_2 = M_2$, $C_3 = M_3$, $L_1 = 1/K_1$, $L_3 = 1/K_3$.) $L_2 = 1/K_2$ and $R_1 = 1/D_1$ are in parallel between nodes 1 and 2.