Rotational system θ2/T with T-V and T-I analogies

Assumptions: zero initial conditions; $\theta_3$ is the angle of the junction between damper $D$ and spring $K$ (D on the $J_1$ side). The series $D$–$K$ coupling is massless.

Equations (Laplace form)

Inertia $J_1$:

$$T(s) = (J_1s^2 + D_1s)\theta_1 + Ds(\theta_1 - \theta_3)$$

Junction: $\;Ds(\theta_3 - \theta_1) + K(\theta_3 - \theta_2) = 0$

Inertia $J_2$: $\;(J_2s^2 + D_2s)\theta_2 + K(\theta_2 - \theta_3) = 0$

Solving

Damper and spring in series transmit the same torque, so they act as one element with "stiffness"

$$K_e(s) = \frac{DsK}{Ds + K}$$

Then $(J_1s^2+D_1s+K_e)\theta_1 - K_e\theta_2 = T$ and $-K_e\theta_1 + (J_2s^2+D_2s+K_e)\theta_2 = 0$, giving

$$\frac{\theta_2}{T} = \frac{K_e}{(J_1s^2+D_1s)(J_2s^2+D_2s) + K_e\left[(J_1+J_2)s^2 + (D_1+D_2)s\right]}$$

Multiply top and bottom by $(Ds+K)$ and cancel the common $s$:

$$\frac{\theta_2(s)}{T(s)} = \frac{DK}{s\left[(Ds+K)(J_1s+D_1)(J_2s+D_2) + DK\{(J_1+J_2)s + D_1 + D_2\}\right]}$$

Expanded:

$$\frac{\theta_2(s)}{T(s)} = \frac{DK}{s\left[a_3s^3 + a_2s^2 + a_1s + a_0\right]}$$ $$\begin{aligned} a_3 &= DJ_1J_2, \quad a_2 = D(D_1J_2 + D_2J_1) + KJ_1J_2 \\ a_1 &= DD_1D_2 + K\left[D(J_1+J_2) + D_1J_2 + D_2J_1\right] \\ a_0 &= K\left[D(D_1+D_2) + D_1D_2\right] \end{aligned}$$

Torque–voltage (T-V) analogy

Torque → voltage, $J$ → $L$, $D$ → $R$, $K$ → $C = 1/K$, angular velocity → loop current. The series $D$–$K$ pair becomes $R \parallel C$ shared by the two loops.

  +--[L1]--[R1]--+-----+--[L2]--[R2]--+
  |              |     |              |
 (T)  i1 ->      R     C     <- i2    |
  |              |     |              |
  +--------------+-----+--------------+
 L1=J1, R1=D1, R=D, C=1/K, L2=J2, R2=D2

Torque–current (T-I) analogy

Torque → current source, $J$ → $C$, $D$ → conductance ($R = 1/D$), $K$ → $L = 1/K$, angular velocity → node voltage.

      ω1          ω3           ω2
  +----o----+--[R]--o--[L]--+----o
  |    |    |               |    |
 (T)   C1   R1              C2   R2
  |    |    |               |    |
  +----+----+---------------+----+
              ground
 C1=J1, R1=1/D1, R=1/D, L=1/K, C2=J2, R2=1/D2