Two-mass system with numeric values: F-V analogy and X1/F

Assumptions: damper $B_2$ and spring $K_3$ are in series and this pair is in parallel with $K_2$ between $M_1$ and $M_2$; $X_3$ is the junction between $B_2$ and $K_3$; zero initial conditions. Values: $M_1 = M_2 = 1$ kg, $K_1 = K_2 = 1$ N/m, $K_3 = 2$ N/m, $B_1 = 2$, $B_2 = 1$, $B_3 = 1$ N·s/m.

F-V analogous circuit

Force → voltage, mass → $L$, damper → $R$, spring → $C = 1/K$, velocity → loop current. The coupling ($K_2$ parallel to $B_2$–$K_3$ series) becomes $C_2$ in series with ($R_2 \parallel C_3$).

  +--[L1]--[R1]--[C1]--+--[L2]--[R3]--+
  |                    |              |
  |     <- i1         C2      i2 ->  (F)
  |                    |              |
  |                 +--+--+           |
  |                 R2    C3          |
  |                 +--+--+           |
  |                    |              |
  +--------------------+--------------+
 L1=M1=1 H, R1=B1=2 Ω, C1=1/K1=1 F
 C2=1/K2=1 F, R2=B2=1 Ω, C3=1/K3=0.5 F
 L2=M2=1 H, R3=B3=1 Ω, source = F(t)

Loop currents $i_1 = \dot x_1$, $i_2 = \dot x_2$; the shared branch carries $(i_2 - i_1)$.

Equations of motion

Equivalent of series $B_2$–$K_3$: $\dfrac{B_2sK_3}{B_2s+K_3} = \dfrac{2s}{s+2}$. Total coupling:

$$K_c(s) = 1 + \frac{2s}{s+2} = \frac{3s+2}{s+2}$$ $$\begin{aligned} M_1: &\quad (s^2 + 2s + 1)X_1 + K_c(X_1 - X_2) = 0 \\ M_2: &\quad (s^2 + s)X_2 + K_c(X_2 - X_1) = F(s) \end{aligned}$$

Transfer function $X_1/F$

From the first equation, $X_2 = \dfrac{s^2+2s+1+K_c}{K_c}X_1$. Substituting in the second:

$$\frac{X_1}{F} = \frac{K_c}{(s^2+2s+1)(s^2+s) + K_c(2s^2+3s+1)}$$

Using $(s^2+2s+1)(s^2+s) = s(s+1)^3$ and $2s^2+3s+1 = (2s+1)(s+1)$, then multiplying top and bottom by $(s+2)$:

$$\begin{aligned} \frac{X_1}{F} &= \frac{3s+2}{s(s+2)(s+1)^3 + (3s+2)(2s+1)(s+1)} \\ &= \frac{3s+2}{(s+1)\left[s(s+2)(s+1)^2 + (3s+2)(2s+1)\right]} \\ &= \frac{3s+2}{(s+1)(s^4 + 4s^3 + 11s^2 + 9s + 2)} \end{aligned}$$

Answer:

$$\frac{X_1(s)}{F(s)} = \frac{3s+2}{s^5 + 5s^4 + 15s^3 + 20s^2 + 11s + 2}$$