Two-mass system X1/F, X2/F with F-V analogy

Displacements $x_1$ ($M_1$, ground friction $D_1$, force $f$) and $x_2$ ($M_2$, ground friction $D_2$). Coupling: $k_1 \parallel b_1$; $M_2$ to wall: $k_2 \parallel b_2$.

Differential equations

$$\begin{aligned} M_1\ddot x_1 + D_1\dot x_1 + b_1(\dot x_1 - \dot x_2) + k_1(x_1 - x_2) &= f(t) \\ M_2\ddot x_2 + D_2\dot x_2 + b_2\dot x_2 + k_2x_2 + b_1(\dot x_2 - \dot x_1) + k_1(x_2 - x_1) &= 0 \end{aligned}$$

Laplace form:

$$\begin{aligned} \left[M_1s^2 + (D_1 + b_1)s + k_1\right]X_1 - (b_1s + k_1)X_2 &= F \\ -(b_1s + k_1)X_1 + \left[M_2s^2 + (D_2 + b_1 + b_2)s + k_1 + k_2\right]X_2 &= 0 \end{aligned}$$

Transfer functions (Cramer's rule)

$$\Delta = \left[M_1s^2 + (D_1+b_1)s + k_1\right]\left[M_2s^2 + (D_2+b_1+b_2)s + k_1 + k_2\right] - (b_1s + k_1)^2$$ $$\frac{X_1(s)}{F(s)} = \frac{M_2s^2 + (D_2 + b_1 + b_2)s + k_1 + k_2}{\Delta}, \qquad \frac{X_2(s)}{F(s)} = \frac{b_1s + k_1}{\Delta}$$

Expanded:

$$\begin{aligned} \Delta =\; & M_1M_2s^4 + \left[M_1(D_2 + b_1 + b_2) + M_2(D_1 + b_1)\right]s^3 \\ & + \left[D_1D_2 + (D_1 + D_2)b_1 + D_1b_2 + b_1b_2 + M_1(k_1 + k_2) + M_2k_1\right]s^2 \\ & + \left[(D_1 + D_2)k_1 + D_1k_2 + b_1k_2 + b_2k_1\right]s + k_1k_2 \end{aligned}$$

F-V analogous circuit

$f\to e$, $M\to L$, $D,b\to R$, $k\to 1/C$, $\dot x\to i$.

   L1   R_D1             L2   R_D2   R_b2
 +-LLLL-/\/\-+---------+-LLLL-/\/\--/\/\-+
 |           |                           |
(~) e      R_b1                       C2=1/k2
 |           |                           |
 |   i1    C1=1/k1          i2           |
 +-----------+---------------------------+

$$\begin{aligned} L_1\frac{di_1}{dt} + R_{D1}i_1 + R_{b1}(i_1 - i_2) + \frac{1}{C_1}\int(i_1 - i_2)dt &= e \\ L_2\frac{di_2}{dt} + (R_{D2} + R_{b2})i_2 + \frac{1}{C_2}\int i_2dt + R_{b1}(i_2 - i_1) + \frac{1}{C_1}\int(i_2 - i_1)dt &= 0 \end{aligned}$$