Two-mass system X2/F and F-I analogy

Let $x_1$, $x_2$ be the displacements of the two masses ($M_1$, $M_2$) and $x_3$ the displacement of the massless junction between $K_2$ and $B_1$ (spring on the $M_1$ side).

Differential equations

$$\begin{aligned} M_1\ddot x_1 + f_{c1}\dot x_1 + K_1 x_1 + K_2(x_1 - x_3) &= f \\ K_2(x_3 - x_1) + B_1(\dot x_3 - \dot x_2) &= 0 \\ M_2\ddot x_2 + (f_{c2} + B_2)\dot x_2 + B_1(\dot x_2 - \dot x_3) &= 0 \end{aligned}$$

Eliminating the junction

From the second equation, $X_3 = \dfrac{K_2X_1 + B_1sX_2}{K_2 + B_1 s}$. The force through the series spring–damper is then

$$K_2(X_1 - X_3) = K_{eq}(X_1 - X_2), \qquad K_{eq}(s) = \frac{K_2 B_1 s}{K_2 + B_1 s}$$

So, writing $a = M_1s^2 + f_{c1}s + K_1$ and $b = M_2s^2 + (f_{c2}+B_2)s$:

$$\begin{aligned} (a + K_{eq})X_1 - K_{eq}X_2 &= F \\ -K_{eq}X_1 + (b + K_{eq})X_2 &= 0 \end{aligned}$$

Transfer function

$$\frac{X_2(s)}{F(s)} = \frac{K_{eq}}{(a+K_{eq})(b+K_{eq}) - K_{eq}^2} = \frac{K_{eq}}{ab + K_{eq}(a+b)}$$

Multiplying through by $(K_2 + B_1 s)$ and cancelling the common factor $s$:

$$\frac{X_2(s)}{F(s)} = \frac{B_1K_2}{(K_2 + B_1s)(M_1s^2 + f_{c1}s + K_1)(M_2s + f_{c2} + B_2) + B_1K_2\left[(M_1+M_2)s^2 + (f_{c1}+f_{c2}+B_2)s + K_1\right]}$$

The denominator is a 4th-order polynomial in $s$.

F-I analogy

$f \to i$, $M \to C$, friction/damper $B \to$ conductance $1/R$, $K \to 1/L$, velocity $\to$ node voltage. Nodes $v_1, v_3, v_2$ correspond to $\dot x_1, \dot x_3, \dot x_2$.

 v1                L2    v3  RB1   v2
 +----+----+----+--LLLL--+--/\/\/--+----+----+
 |    |    |    |                  |    |    |
(i)   C1   R1   L1                 C2   R2   R3
 |    |    |    |                  |    |    |
 +----+----+----+------------------+----+----+ gnd

($C_1 = M_1$, $R_1 = 1/f_{c1}$, $L_1 = 1/K_1$; $L_2 = 1/K_2$, $R_{B1} = 1/B_1$; $C_2 = M_2$, $R_2 = 1/f_{c2}$, $R_3 = 1/B_2$.) Node equation at node 1: $C_1\dfrac{dv_1}{dt} + \dfrac{v_1}{R_1} + \dfrac{1}{L_1}\int v_1dt + \dfrac{1}{L_2}\int (v_1 - v_3)dt = i$, and similarly for nodes 3 and 2 (matching the three mechanical equations).