Two-mass system X2/F with F-V analogy

Displacements: $x_1$ for $M_1$ (force $F$ applied, ground friction $B_3$) and $x_2$ for $M_2$ (ground friction $B_2$, spring $k_1$ to wall). $B_1 \parallel k_2$ couples the masses.

Equations of motion

$$\begin{aligned} M_1\ddot x_1 + B_3\dot x_1 + B_1(\dot x_1 - \dot x_2) + k_2(x_1 - x_2) &= F \\ M_2\ddot x_2 + B_2\dot x_2 + k_1x_2 + B_1(\dot x_2 - \dot x_1) + k_2(x_2 - x_1) &= 0 \end{aligned}$$

Laplace form:

$$\begin{aligned} \left[M_1s^2 + (B_1 + B_3)s + k_2\right]X_1 - (B_1s + k_2)X_2 &= F \\ -(B_1s + k_2)X_1 + \left[M_2s^2 + (B_1 + B_2)s + k_1 + k_2\right]X_2 &= 0 \end{aligned}$$

Transfer function

$$\frac{X_2(s)}{F(s)} = \frac{B_1s + k_2}{\Delta}$$ $$\Delta = \left[M_1s^2 + (B_1+B_3)s + k_2\right]\left[M_2s^2 + (B_1+B_2)s + k_1 + k_2\right] - (B_1s + k_2)^2$$

Expanded:

$$\begin{aligned} \Delta =\; & M_1M_2s^4 + \left[B_1(M_1 + M_2) + B_2M_1 + B_3M_2\right]s^3 \\ & + \left[B_1B_2 + B_1B_3 + B_2B_3 + M_1(k_1 + k_2) + M_2k_2\right]s^2 \\ & + \left[B_1k_1 + B_2k_2 + B_3(k_1 + k_2)\right]s + k_1k_2 \end{aligned}$$

F-V analogy

$F\to e$, $M\to L$, $B\to R$, $k\to 1/C$, $\dot x \to i$. Mesh 1 ($i_1$) for $M_1$, mesh 2 ($i_2$) for $M_2$; the coupling $B_1 \parallel k_2$ becomes the common branch $R_1$ in series with $C_2$ (parallel mechanical elements carry the same velocity difference, so they become series elements in the shared branch).

    L1=M1  R3=B3            L2=M2   R2=B2
 +--LLLL--/\/\/--+----------LLLL---/\/\/--+
 |               |                        |
(~) e         R1 = B1                  C1 = 1/k1
 |               |                        |
 |    i1      C2 = 1/k2        i2         |
 +---------------+------------------------+

$$\begin{aligned} L_1\frac{di_1}{dt} + R_3i_1 + R_1(i_1 - i_2) + \frac{1}{C_2}\int(i_1 - i_2)dt &= e \\ L_2\frac{di_2}{dt} + R_2i_2 + \frac{1}{C_1}\int i_2dt + R_1(i_2 - i_1) + \frac{1}{C_2}\int(i_2 - i_1)dt &= 0 \end{aligned}$$