F-V and F-I analogy of two-mass system, X2 output

Displacements: $x_1$ for $M_1$ (where $F$ acts) and $x_2$ for $M_2$. Assume no ground friction other than the dampers shown.

Differential equations

• $M_1$: inertia, damper $B_1$ (relative to $M_2$), damper $B_2$ to the wall.
• $M_2$: inertia, damper $B_1$, spring $k$ to the wall.

$$\begin{aligned} M_1\ddot x_1 + B_1(\dot x_1 - \dot x_2) + B_2\dot x_1 &= F(t) \\ M_2\ddot x_2 + B_1(\dot x_2 - \dot x_1) + kx_2 &= 0 \end{aligned}$$

F-V analogous circuit (mesh)

$F\to e$, $M\to L$, $B\to R$, $k\to 1/C$, $\dot x\to i$, $x \to q$.

$$\begin{aligned} L_1\frac{di_1}{dt} + R_1(i_1 - i_2) + R_2 i_1 &= e \\ L_2\frac{di_2}{dt} + R_1(i_2 - i_1) + \frac{1}{C}\int i_2\,dt &= 0 \end{aligned}$$

($L_1 = M_1$, $L_2 = M_2$, $R_1 = B_1$, $R_2 = B_2$, $C = 1/k$)

     L1=M1   R2=B2         L2=M2
  +--LLLL--/\/\/--+--------LLLL--+
  |               |              |
 (~) e         R1 = B1        C = 1/k
  |      i1       |      i2      |
  +---------------+--------------+

F-I analogous circuit (node)

$F\to i$, $M\to C$, $B\to 1/R$, $k\to 1/L$, $\dot x\to v$.

$$\begin{aligned} C_1\frac{dv_1}{dt} + \frac{v_1 - v_2}{R_1} + \frac{v_1}{R_2} &= i \\ C_2\frac{dv_2}{dt} + \frac{v_2 - v_1}{R_1} + \frac{1}{L}\int v_2\,dt &= 0 \end{aligned}$$

($C_1 = M_1$, $C_2 = M_2$, $R_1 = 1/B_1$, $R_2 = 1/B_2$, $L = 1/k$)

 v1           R1     v2
 +----+----+--/\/\/--+----+
 |    |    |         |    |
(i)   C1   R2        C2   L
 |    |    |         |    |
 +----+----+---------+----+ gnd

Transfer function $X_2(s)/F(s)$

Laplace form:

$$\begin{aligned} \left[M_1s^2 + (B_1 + B_2)s\right]X_1 - B_1sX_2 &= F \\ -B_1sX_1 + (M_2s^2 + B_1s + k)X_2 &= 0 \end{aligned}$$

By Cramer's rule:

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

Cancelling the common factor $s$:

$$\frac{X_2(s)}{F(s)} = \frac{B_1}{M_1M_2s^3 + (B_1M_1 + B_1M_2 + B_2M_2)s^2 + (B_1B_2 + kM_1)s + (B_1 + B_2)k}$$