---
title: "Two-rotor torsional system frequency"
subject: "Theory of Mechanism and Machine II"
chapter: 6
key: "semester_7/theory_of_mechanism_and_machine_ii:torsional-two-rotor-frequency-node"
source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/torsional-two-rotor-frequency-node
site: BCT Engineering
---

# Two-rotor torsional system frequency

Theory of Mechanism and Machine II · Chapter 6: Vibration of Single Degree of Freedom Systems

- Asked in: year not given, 6 marks
- Video: [Torsionally Equivalent Shaft- Two rotor System - Problem | DOM |  Module 6 | KTU S6 ME](https://www.youtube.com/watch?v=_vjCswbBOMk) — Mohith Dadu
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/torsional-two-rotor-frequency-node
- Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-6

## Question

Two rotors A and B of mass moments of inertia $I_A$ and $I_B$ are attached to the ends of a uniform shaft of torsional stiffness $k_t = GJ/l$. Derive the expression for the natural frequency of torsional vibration and locate the node.

## Short answer

The torsional natural frequency is ω_n = √[(GJ/l)(I_A+I_B)/(I_A I_B)]. The node is at distances l_A = l I_B/(I_A+I_B) from rotor A and l_B = l I_A/(I_A+I_B) from rotor B, nearer the rotor with larger inertia.

## Answer

#### Model
```text
  [ A ]=========shaft (l, GJ)=========[ B ]
   I_A         k_t = GJ/l             I_B
   theta_A                            theta_B
```
Let $\theta_A$ and $\theta_B$ be the angular displacements of the rotors, both measured in the same sense. (In the vibrating mode they turn in opposite directions, so the result will show $\theta_A$ and $\theta_B$ with opposite signs.)

#### Equations of motion
The shaft torque is $k_t(\theta_A - \theta_B)$ (relative twist):

$$
I_A\ddot\theta_A + k_t(\theta_A - \theta_B) = 0, \qquad I_B\ddot\theta_B - k_t(\theta_A - \theta_B) = 0
$$

Take harmonic motion $\theta = \Theta\sin\omega t$:

$$
(k_t - I_A\omega^2)\Theta_A - k_t\Theta_B = 0, \qquad -k_t\Theta_A + (k_t - I_B\omega^2)\Theta_B = 0
$$

For a non-trivial solution, the determinant is zero:

$$
(k_t - I_A\omega^2)(k_t - I_B\omega^2) - k_t^2 = 0 \;\Rightarrow\; \omega^2\left[I_AI_B\omega^2 - k_t(I_A+I_B)\right] = 0
$$

The root $\omega = 0$ is rigid-body rotation. The other root is the natural frequency:

$$
\omega_n = \sqrt{\frac{k_t(I_A+I_B)}{I_AI_B}} = \sqrt{\frac{GJ}{l}\cdot\frac{I_A+I_B}{I_AI_B}}, \qquad f_n = \frac{\omega_n}{2\pi}
$$

#### Position of the node
At the node the shaft does not twist (its angular displacement is zero). From the first equation, $\dfrac{\Theta_B}{\Theta_A} = \dfrac{k_t - I_A\omega_n^2}{k_t} = -\dfrac{I_A}{I_B}$, so the amplitudes are inversely proportional to the inertias and are of opposite sign.

Let $l_A$ and $l_B$ be the distances of the node from A and B ($l_A + l_B = l$). The twist per unit length is uniform, so $|\Theta_A|/l_A = |\Theta_B|/l_B$:

$$
l_A = \frac{I_B}{I_A+I_B}\,l, \qquad l_B = \frac{I_A}{I_A+I_B}\,l
$$

The node is nearer the rotor with the larger inertia. Each rotor vibrates as if fixed to a shaft of length $l_A$ or $l_B$, with $\omega_n = \sqrt{\dfrac{GJ}{l_A I_A}} = \sqrt{\dfrac{GJ}{l_B I_B}}$.

## Check yourself

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

1. The torsional natural frequency of two rotors on a shaft is:
   A. sqrt(k_t (I_A + I_B)/(I_A I_B))
   B. sqrt(k_t/(I_A + I_B))
   C. sqrt(k_t I_A I_B/(I_A+I_B))
   D. sqrt(k_t/ I_A)
   Answer: A. Setting the determinant to zero and discarding w = 0 (rigid body) gives this result. (Key point: Natural frequency)

2. What does the root w = 0 of the frequency equation represent?
   A. Rigid-body rotation of both rotors together
   B. The fundamental vibration
   C. A damped mode
   D. Resonance
   Answer: A. w = 0 means no relative twist; the system just rotates as a rigid body. (Key point: Zero root)

3. In the vibrating mode the amplitude ratio Theta_B/Theta_A is:
   A. -I_A/I_B
   B. I_A/I_B
   C. -I_B/I_A
   D. 1
   Answer: A. The amplitudes are inversely proportional to inertias and of opposite sign. (Key point: Amplitude ratio)

4. The node distance from rotor A is:
   A. l_A = I_B l /(I_A + I_B)
   B. l_A = I_A l /(I_A + I_B)
   C. l/2
   D. l_A = I_A l / I_B
   Answer: A. The node lies nearer the rotor with the larger inertia, so l_A = I_B l/(I_A + I_B). (Key point: Node location)

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