--- title: "Rotational system θ/T and T-I analogy" subject: "Control Systems" chapter: 2 key: "semester_3/control_systems:tf-ti-rot-j1-j2-k1k2" source: https://www.bctengineer.com/q/semester_3/control_systems/tf-ti-rot-j1-j2-k1k2 site: BCT Engineering --- # Rotational system θ/T and T-I analogy Control Systems · Chapter 2: Component Modeling - Asked in: 2075 Chaitra, 8 marks - Video: [Problem on Mechanical Rotational System](https://www.youtube.com/watch?v=qHbUeuVFwLE) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/tf-ti-rot-j1-j2-k1k2 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2 ## Question Find the transfer function θ(s)/T(s) for the mechanical rotational system shown below. Also develop T-I analogous circuit. [Figure: torque T(t) applied to inertia J1; J1 coupled through torsional spring k1 in parallel with damper b1 to a shaft point, which has viscous damper b2 to the fixed frame; that point is coupled through torsional spring k2 to inertia J2, whose angle is θ(t)] ## Short answer Introduce angles θ₁, θ₃, θ, write three equations, eliminate θ₁ and θ₃ to find θ(s)/T(s) = k₂(b₁s+k₁)/Δ, where Δ = J₁J₂(b₁+b₂)s⁵+[J₁J₂(k₁+k₂)+J₂b₁b₂]s⁴+[(J₁+J₂)b₁k₂+J₁b₂k₂+J₂b₂k₁]s³+[(J₁+J₂)k₁k₂+b₁b₂k₂]s²+b₂k₁k₂s. T–I analogy: T→i, J→C, b→1/R, k→1/L; circuit: C₁=J₁, C₂=J₂, L₁∥R₁ between nodes v₁ and v₃, L₂ from v₃ to v₂, R₂ from v₃ to ground. ## Answer Let $\theta_1$ = angle of $J_1$, $\theta_3$ = angle of the massless shaft point (where $b_2$ is attached) and $\theta$ = angle of $J_2$. Bearings of $J_1$, $J_2$ are taken as frictionless (only the shown elements act). #### Differential equations (Laplace form) $$ \begin{aligned} J_1s^2\theta_1 + (b_1s + k_1)(\theta_1 - \theta_3) &= T \\ (b_1s + k_1)(\theta_3 - \theta_1) + b_2s\,\theta_3 + k_2(\theta_3 - \theta) &= 0 \\ J_2s^2\theta + k_2(\theta - \theta_3) &= 0 \end{aligned} $$ In matrix form: $$ \begin{bmatrix} J_1s^2 + b_1s + k_1 & -(b_1s+k_1) & 0 \\ -(b_1s+k_1) & (b_1+b_2)s + k_1 + k_2 & -k_2 \\ 0 & -k_2 & J_2s^2 + k_2 \end{bmatrix} \begin{bmatrix}\theta_1\\ \theta_3\\ \theta\end{bmatrix} = \begin{bmatrix}T\\0\\0\end{bmatrix} $$ #### Transfer function By Cramer's rule, the numerator for $\theta$ is the product of the off-diagonal couplings, $(b_1s+k_1)k_2$: $$ \frac{\theta(s)}{T(s)} = \frac{k_2(b_1s + k_1)}{\Delta} $$ where $$ \Delta = (J_1s^2 + b_1s + k_1)\left\{\left[(b_1+b_2)s + k_1 + k_2\right](J_2s^2 + k_2) - k_2^2\right\} - (b_1s + k_1)^2(J_2s^2 + k_2) $$ Expanded: $$ \begin{aligned} \Delta =\; & J_1J_2(b_1 + b_2)s^5 + \left[J_1J_2(k_1 + k_2) + J_2b_1b_2\right]s^4 \\ & + \left[(J_1 + J_2)b_1k_2 + J_1b_2k_2 + J_2b_2k_1\right]s^3 \\ & + \left[(J_1 + J_2)k_1k_2 + b_1b_2k_2\right]s^2 + b_2k_1k_2\,s \end{aligned} $$ The factor $s$ in $\Delta$ shows a free integration: nothing ties the system to the frame by a spring. #### T-I analogy Torque $\to$ current, $J \to C$, $b \to 1/R$, $k \to 1/L$, angular velocity $\omega \to$ node voltage. | Mechanical | Electrical | |---|---| | $T$ | current source $i$ | | $J_1$, $J_2$ | $C_1 = J_1$, $C_2 = J_2$ | | $b_1$, $b_2$ | $R_1 = 1/b_1$, $R_2 = 1/b_2$ | | $k_1$, $k_2$ | $L_1 = 1/k_1$, $L_2 = 1/k_2$ | | $\omega_1, \omega_3, \omega$ | $v_1, v_3, v_2$ | ```text v1 L1∥R1 v3 L2 v2 +----+---+--LLLL--+---+--LLLL--+ | | +-/\/\/--+ | | (i) C1 R2 C2 | | | | +----+----------------+--------+ gnd ``` ($L_1$ and $R_1$ are in parallel between nodes 1 and 3.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why are three equations needed? A. There are three angles: J1, the massless shaft point and J2 B. There are three springs C. Three torques act D. There are three dampers Answer: A. theta1, theta3 (shaft point) and theta each need a torque balance. (Key point: Number of equations) 2. The numerator of theta/T is: A. J2 s^2 + k2 B. b2 s k2 C. k1 k2 D. k2 (b1 s + k1) Answer: D. The path from T to J2 passes through the coupling (b1 s + k1) and then k2. (Key point: Numerator) 3. The factor s in the denominator indicates: A. A friction-free system B. A zero C. Instability D. No spring ties the system to the frame, so a free integration exists Answer: D. Only damper b2 connects to the frame, so there is a pole at the origin. (Key point: Free integration) 4. In the T-I analogy, J maps to: A. Inductance L = J B. Voltage source C. Capacitance C = J, torque to current, and angular velocity to node voltage D. Resistance R = J Answer: C. Torque-current: T is current, J is capacitance, b is 1/R, k is 1/L. 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