Rotational system θ1/T and electrical analogies

Angles: $\theta_1$ ($J_1$, torque $T$), $\theta_2$ ($J_2$) and $\theta_3$ = junction of $B_1$ (on the $J_1$ side) and $K_2$ (on the $J_2$ side). $K_3$ couples $J_1$ and $J_2$ directly; $K_1$ and $B_2$ go to the frame.

Differential equations (Laplace form)

$$\begin{aligned} J_1s^2\theta_1 + K_1\theta_1 + B_1s(\theta_1 - \theta_3) + K_3(\theta_1 - \theta_2) &= T \\ B_1s(\theta_3 - \theta_1) + K_2(\theta_3 - \theta_2) &= 0 \\ J_2s^2\theta_2 + B_2s\theta_2 + K_2(\theta_2 - \theta_3) + K_3(\theta_2 - \theta_1) &= 0 \end{aligned}$$

Reducing the series branch

$B_1$ in series with $K_2$ is equivalent to $\dfrac{B_1sK_2}{B_1s + K_2}$; with $K_3$ in parallel the total coupling is

$$Z_c = K_3 + \frac{B_1K_2s}{B_1s + K_2} = \frac{Q}{B_1s + K_2}, \quad Q = K_3(B_1s + K_2) + B_1K_2s$$

With $a = J_1s^2 + K_1$ and $b = J_2s^2 + B_2s$:

$$\begin{aligned} (a + Z_c)\theta_1 - Z_c\theta_2 &= T \\ -Z_c\theta_1 + (b + Z_c)\theta_2 &= 0 \end{aligned} \qquad\Rightarrow\qquad \frac{\theta_1}{T} = \frac{b + Z_c}{ab + Z_c(a + b)}$$

Transfer function

Multiplying by $(B_1s + K_2)$:

$$\frac{\theta_1(s)}{T(s)} = \frac{(B_1s + K_2)(J_2s^2 + B_2s) + Q}{(B_1s + K_2)(J_1s^2 + K_1)(J_2s^2 + B_2s) + Q\left[(J_1 + J_2)s^2 + B_2s + K_1\right]}$$

Expanded:

$$\frac{\theta_1}{T} = \frac{B_1J_2s^3 + (B_1B_2 + J_2K_2)s^2 + (B_1K_2 + B_1K_3 + B_2K_2)s + K_2K_3}{\Delta}$$ $$\begin{aligned} \Delta =\; & B_1J_1J_2s^5 + (B_1B_2J_1 + J_1J_2K_2)s^4 \\ & + \left[B_1J_1(K_2 + K_3) + B_1J_2(K_1 + K_2 + K_3) + B_2J_1K_2\right]s^3 \\ & + \left[B_1B_2(K_1 + K_2 + K_3) + J_1K_2K_3 + J_2K_2(K_1 + K_3)\right]s^2 \\ & + \left[B_1K_1(K_2 + K_3) + B_2K_2(K_1 + K_3)\right]s + K_1K_2K_3 \end{aligned}$$

Analogous electrical networks

T-I network (nodes $v_1, v_3, v_2$):

 v1                RB1    v3  L2   v2
 +----+----+----+--/\/\/--+--LLLL--+----+----+
 |    |    |    |                  |    |    |
 |    |    |    +-------LLLL-------+    |    |
 |    |    |             L3             |    |
(i)   C1   L1                           C2   R2
 |    |    |                            |    |
 +----+----+----------------------------+----+ gnd

($C_1 = J_1$, $L_1 = 1/K_1$, $R_{B1} = 1/B_1$, $L_2 = 1/K_2$, $L_3 = 1/K_3$, $C_2 = J_2$, $R_2 = 1/B_2$.) T-V network (mesh currents $i_1, i_2$ and inner loop $i_3$):

   L1=J1  C1=1/K1          L2=J2  R2=B2
 +-LLLL----||----+---------LLLL---/\/\-+
 |               |                     |
(~) e         C3=1/K3                  |
 |    i1         |          i2         |
 |           +---+---+                 |
 |         R1=B1  C2=1/K2              |
 |           | i3    |                 |
 |           +---+---+                 |
 |               |                     |
 +---------------+---------------------+

$K_3$ (parallel coupling) becomes a series element $C_3$ in the shared branch, and the series pair $B_1$–$K_2$ becomes the parallel pair $R_1 \parallel C_2$. The mesh equations are exactly the three equations above with $\theta \to q$.