---
title: "Gear-train rotational system transfer function"
subject: "Control Systems"
chapter: 2
key: "semester_3/control_systems:tf-gear-train-rotational"
source: https://www.bctengineer.com/q/semester_3/control_systems/tf-gear-train-rotational
site: BCT Engineering
---

# Gear-train rotational system transfer function

Control Systems · Chapter 2: Component Modeling

- Asked in: 2078 Bhadra, 8 marks
- Video: [Rotational System with Gears to Transfer Function](https://www.youtube.com/watch?v=14aXX9rvGWk) — RE-Lecture
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/tf-gear-train-rotational
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2

## Question

Find the transfer function θL(s)/T(s) of the mechanical rotational system shown below. [Figure: torque T(t) applied to gear N1 = 11, which drives gear N2 = 33; N2 is on a shaft with inertia 1 kg-m² and a viscous damper 2 N-m-s/rad, followed by a torsional spring 3 N-m/rad connected to gear N3 = 50; N3 drives gear N4 = 10, whose shaft (angle θL(t)) has viscous damping 0.04 N-m-s/rad to the frame]

## Short answer

Reflect all impedances to the N₃ shaft: input torque becomes 3T, load damping becomes 1 N·m·s/rad. Write equations for J=1, D=2, K=3, D_load=1, solve for θ₃, then θ_L = 5θ₃, giving θ_L(s)/T(s) = 45/[s(s²+5s+9)].

## Answer

Reflect everything to one shaft. Mechanical impedances move across a gear pair by $(N_{dest}/N_{source})^2$, and torques by $N_{dest}/N_{source}$.

#### Step 1: Reflect the input torque to the $N_2$ shaft
$$
T_2 = T\,\frac{N_2}{N_1} = T\cdot\frac{33}{11} = 3T
$$

#### Step 2: Reflect the load damping to the $N_3$ shaft
$$
D_{eq} = 0.04\left(\frac{N_3}{N_4}\right)^2 = 0.04\left(\frac{50}{10}\right)^2 = 1\ \text{N-m-s/rad}
$$

#### Step 3: Equivalent system
Let $\theta_2$ = angle of the $N_2$ shaft (with $J = 1$, $D = 2$) and $\theta_3$ = angle of gear $N_3$ (end of the spring, $K = 3$, carrying the reflected damping 1):
```text
 3T ─► [J=1, D=2] ──spring K=3── (θ3) ── D=1 ─► frame
        θ2
```
Equations (Laplace):
$$
\begin{aligned}
(s^2 + 2s + 3)\,\theta_2 - 3\,\theta_3 &= 3T \\
-3\,\theta_2 + (s + 3)\,\theta_3 &= 0
\end{aligned}
$$

#### Step 4: Solve
From the second equation $\theta_2 = \dfrac{(s+3)}{3}\theta_3$. Substituting:
$$
\begin{aligned}
\left[\frac{(s^2+2s+3)(s+3)}{3} - 3\right]\theta_3 &= 3T \\
\left[(s^3 + 5s^2 + 9s + 9) - 9\right]\theta_3 &= 9T \\
\frac{\theta_3(s)}{T(s)} &= \frac{9}{s(s^2 + 5s + 9)}
\end{aligned}
$$

#### Step 5: Output shaft
$$
\theta_L = \theta_3\,\frac{N_3}{N_4} = 5\,\theta_3
$$

**Answer:**
$$
\frac{\theta_L(s)}{T(s)} = \frac{45}{s(s^2 + 5s + 9)}
$$
The pole at $s=0$ appears because no spring ties the system to the frame, so a constant torque gives a steadily increasing angle.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The input torque T on N1 = 11 is reflected to the N2 = 33 shaft as:
   A. T
   B. 3T
   C. T/3
   D. 9T
   Answer: B. Torque scales by N_dest/N_source = 33/11 = 3. (Key point: Reflecting torque)

2. The 0.04 N-m-s/rad damping on the N4 shaft, reflected to the N3 shaft, is:
   A. 0.2
   B. 5
   C. 0.008
   D. 1
   Answer: D. Impedances scale by the square of the ratio: 0.04 x (50/10)^2 = 1. (Key point: Reflecting damping)

3. Given theta3/T = 9/(s(s^2+5s+9)), how is theta_L found?
   A. theta_L = theta3, no change
   B. theta_L = 5 theta3 (N3/N4), so the answer is 45/(s(s^2+5s+9))
   C. theta_L = theta3 / 5, so the answer is 9/(5s(...))
   D. theta_L = 25 theta3
   Answer: B. Angles scale by N3/N4 = 5, so multiply 9 by 5 to get 45. (Key point: Output shaft angle)

4. Why does the result contain a pole at s = 0?
   A. No spring ties the system to the frame, so constant torque gives continuously increasing angle
   B. The damper is ideal
   C. The gears have backlash
   D. The inertia is zero
   Answer: A. The only link to the frame is damping, so there is a free integration (type 1 behavior). (Key point: Pole at origin)

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