Two-mass system X2/F by free body diagram, F-V analogy

Displacements: $x_1$ ($M_1$), $x_2$ ($M_2$), and $x_3$ = junction of $K_1$ (on the $M_1$ side) and $B_1$ (on the $M_2$ side). $K_2$ connects the masses directly.

Free body equations

• $M_1$: $F$ applied; opposed by $M_1\ddot x_1$, $f_{c1}\dot x_1$, $K_1(x_1 - x_3)$, $K_2(x_1 - x_2)$.
• Junction (massless): $K_1(x_3 - x_1) + B_1(\dot x_3 - \dot x_2) = 0$.
• $M_2$: opposed by $M_2\ddot x_2$, $(f_{c2} + B_2)\dot x_2$, $B_1(\dot x_2 - \dot x_3)$, $K_2(x_2 - x_1)$.

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

Eliminating $x_3$

The series $K_1$–$B_1$ branch acts as $K_{eq} = \dfrac{K_1B_1s}{K_1 + B_1s}$, in parallel with $K_2$. Total coupling:

$$Z_c = K_2 + \frac{K_1B_1s}{K_1 + B_1s} = \frac{K_1K_2 + B_1(K_1 + K_2)s}{K_1 + B_1s}$$

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

$$\begin{aligned} (a + Z_c)X_1 - Z_cX_2 &= F \\ -Z_cX_1 + (b + Z_c)X_2 &= 0 \end{aligned}$$ $$\frac{X_2}{F} = \frac{Z_c}{ab + Z_c(a + b)}$$

Transfer function

Multiplying by $(K_1 + B_1s)$ and writing $P = K_1K_2 + B_1(K_1 + K_2)s$:

$$\frac{X_2(s)}{F(s)} = \frac{K_1K_2 + B_1(K_1 + K_2)s}{s\left\{(K_1 + B_1s)\,s\,(M_1s + f_{c1})(M_2s + f_{c2} + B_2) + P\left[(M_1 + M_2)s + f_{c1} + f_{c2} + B_2\right]\right\}}$$

The pole at $s = 0$ appears because neither mass is tied to the wall by a spring.

F-V analogy

$F\to e$, $M\to L$, $B, f_c\to R$, $K\to 1/C$. Mechanical elements in parallel (same velocity difference) become series elements, and elements in series (same force) become parallel. So the shared branch between mesh 1 ($M_1$) and mesh 2 ($M_2$) is $C = 1/K_2$ in series with the parallel pair $1/K_1 \parallel B_1$; the small loop inside that pair is mesh 3 (junction $x_3$).

   L1=M1  R=fc1           L2=M2  R=fc2+B2
 +-LLLL---/\/\--+--------LLLL---/\/\--+
 |              |                     |
(~) e        C=1/K2                   |
 |    i1        |           i2        |
 |          +---+---+                 |
 |        C=1/K1  R=B1                |
 |          | i3    |                 |
 |          +---+---+                 |
 |              |                     |
 +--------------+---------------------+

Mesh equations ($R_{c1} = f_{c1}$, $R_{c2} = f_{c2}$, $R_{B1} = B_1$, $R_{B2} = B_2$, $C_1 = 1/K_1$, $C_2 = 1/K_2$):

$$\begin{aligned} L_1\frac{di_1}{dt} + R_{c1}i_1 + \frac{1}{C_2}\int(i_1 - i_2)\,dt + \frac{1}{C_1}\int(i_1 - i_3)\,dt &= e \\ \frac{1}{C_1}\int(i_3 - i_1)\,dt + R_{B1}(i_3 - i_2) &= 0 \\ L_2\frac{di_2}{dt} + (R_{c2} + R_{B2})i_2 + \frac{1}{C_2}\int(i_2 - i_1)\,dt + R_{B1}(i_2 - i_3) &= 0 \end{aligned}$$