IB Design Technology Topic B3.3 Mechanical Systems…

B3.3.4 Gear and belt-driven system applications

Gear and Belt Systems: Working Out the Output Speed

Gear and belt systems do not just move things. They trade speed against torque, and the ratio decides the exchange rate.
Designers use them to:
1. Increase or decrease speed
2. Increase torque (rotational force)
3. Reverse the direction of rotation
4. Transfer motion between different axes

Understanding the System

$$\text{Gear ratio} = \frac{\text{Driven gear teeth}}{\text{Driver gear teeth}}$$

Common Mistake

Get the ratio the right way up: it is driven ÷ driver, not driver ÷ driven.
Invert it and every conclusion flips: you would predict a speed increase where the machine actually slows down.
Sanity-check with the physical intuition: a small gear driving a big one must turn the big one slowly. So a 12-tooth driver on a 48-tooth driven gives a ratio of 4, and the output runs at a quarter of the speed.

A high gear ratio → more torque, less speed
A low gear ratio → more speed, less torque

Belt Drives work similarly to gears, but:
1. Allow flexible distance between components
2. Can slip, which may be desirable in safety systems
3. Can reverse motion depending on belt twist

Gear and Belt-Driven System Applications

Product	Function of Gears or Belts
Bicycle gear system	Alters pedaling resistance and speed. Large front gear + small rear gear = fast, low-torque ride.
Cordless drill gearbox	Low gear for high torque when driving screws; high gear for speed when drilling.
Lathe belt system	Switches between belt positions to vary spindle speed for different materials.
Robotic arm	Gear trains reduce speed and increase precision torque for movement control.

Direction Change Example

Two meshed gears rotate in opposite directions.
Add a third (idler) gear between them to maintain original rotation direction while adjusting spacing.

Calculating Speed of Rotation in Gear Systems

To calculate the speed of rotation at different points in a gear system:
1. Identify the Input Speed: The speed of the drive gear.
2. Apply the Gear Ratio: Multiply the input speed by the gear ratio to find the output speed.
Formula: Speed of driven gear = Speed of driver gear ÷ Gear ratio. A ratio above 1 therefore slows the output down.

Example question

Calculate the output speed of a gear system with a drive gear rotating at 100 RPM and a gear ratio of 2:1. What is the speed of the driven gear?

Solution

$$\text{Speed of driven gear} = \frac{\text{Speed of driver gear}}{\text{Gear ratio}}$$

$$ = \frac{100}{2} = 50 \text{ RPM}$$

Speed at Several Points in a Compound Train

In a compound train, work through it one stage at a time. The output speed of one stage becomes the input speed of the next.
Two gears on the same shaft always turn at the same speed. That is the fact that carries the speed from one stage into the next.
The overall ratio is the ratios of the stages multiplied together, never added.
Ignore any idler gear when calculating the ratio. It reverses the direction but has no effect on the speed.

Example question

A compound gear train drives a conveyor. The motor turns gear A (15 teeth) at 1200 rpm. Gear A meshes with gear B (45 teeth). Gear C (20 teeth) is fixed to the same shaft as gear B, and meshes with gear D (60 teeth), which drives the conveyor roller.

(a) Calculate the speed of gear B. (2 marks)

(b) Calculate the output speed of gear D. (3 marks)

(c) State the overall gear ratio of the train. (1 mark)

Solution

(a) Award [1] for the stage ratio, [1] for the speed.

Stage 1 ratio = driven / driver = 45 / 15 = 3. [1]
Speed of B = 1200 / 3 = 400 rpm. [1]

(b) Award [1] for recognising C turns at B's speed, [1] for the stage ratio, [1] for the answer.

Gear C is on the same shaft as gear B, so it also turns at 400 rpm. [1]
Stage 2 ratio = 60 / 20 = 3. [1]
Speed of D = 400 / 3 = 133 rpm (to 3 significant figures). [1]

(c) Overall ratio = 3 x 3 = 9. [1] Check: 1200 / 9 = 133 rpm, which matches part (b). Ratios multiply, they do not add, so an answer of 6 scores zero.

Active recall

Write the formula for gear ratio. Which gear goes on top?
How do you find the output speed of a driven gear from the input speed?
In a compound train, why do two gears on the same shaft turn at the same speed?
Do the ratios of successive stages add or multiply?
What does an idler gear change, and what does it not change?

Understanding the System

Common Mistake

Get the ratio the right way up: it is driven ÷ driver, not driver ÷ driven.
Invert it and every conclusion flips: you would predict a speed increase where the machine actually slows down.
Sanity-check with the physical intuition: a small gear driving a big one must turn the big one slowly. So a 12-tooth driver on a 48-tooth driven gives a ratio of 4, and the output runs at a quarter of the speed.

A high gear ratio → more torque, less speed

A low gear ratio → more speed, less torque

Belt Drives work similarly to gears, but:

Allow flexible distance between components
Can slip, which may be desirable in safety systems
Can reverse motion depending on belt twist

Gear and Belt-Driven System Applications

Product	Function of Gears or Belts
Bicycle gear system	Alters pedaling resistance and speed. Large front gear + small rear gear = fast, low-torque ride.
Cordless drill gearbox	Low gear for high torque when driving screws; high gear for speed when drilling.
Lathe belt system	Switches between belt positions to vary spindle speed for different materials.
Robotic arm	Gear trains reduce speed and increase precision torque for movement control.

Direction Change Example

Two meshed gears rotate in opposite directions.

Add a third (idler) gear between them to maintain original rotation direction while adjusting spacing.

Calculating Speed of Rotation in Gear Systems

To calculate the speed of rotation at different points in a gear system:

Identify the Input Speed: The speed of the drive gear.
Apply the Gear Ratio: Multiply the input speed by the gear ratio to find the output speed.

Formula: Speed of driven gear = Speed of driver gear ÷ Gear ratio. A ratio above 1 therefore slows the output down.

Example question

Calculate the output speed of a gear system with a drive gear rotating at 100 RPM and a gear ratio of 2:1. What is the speed of the driven gear?

Solution

$$\text{Speed of driven gear} = \frac{\text{Speed of driver gear}}{\text{Gear ratio}}$$

$$ = \frac{100}{2} = 50 \text{ RPM}$$

Speed at Several Points in a Compound Train

In a compound train, work through it one stage at a time. The output speed of one stage becomes the input speed of the next.

Two gears on the same shaft always turn at the same speed. That is the fact that carries the speed from one stage into the next.

The overall ratio is the ratios of the stages multiplied together, never added.

Ignore any idler gear when calculating the ratio. It reverses the direction but has no effect on the speed.

Example question

(a) Calculate the speed of gear B. (2 marks)

(b) Calculate the output speed of gear D. (3 marks)

(c) State the overall gear ratio of the train. (1 mark)

Solution

(a) Award [1] for the stage ratio, [1] for the speed.

Stage 1 ratio = driven / driver = 45 / 15 = 3. [1]
Speed of B = 1200 / 3 = 400 rpm. [1]

(b) Award [1] for recognising C turns at B's speed, [1] for the stage ratio, [1] for the answer.

Gear C is on the same shaft as gear B, so it also turns at 400 rpm. [1]
Stage 2 ratio = 60 / 20 = 3. [1]
Speed of D = 400 / 3 = 133 rpm (to 3 significant figures). [1]

(c) Overall ratio = 3 x 3 = 9. [1] Check: 1200 / 9 = 133 rpm, which matches part (b). Ratios multiply, they do not add, so an answer of 6 scores zero.

Active recall

Write the formula for gear ratio. Which gear goes on top?
How do you find the output speed of a driven gear from the input speed?
In a compound train, why do two gears on the same shaft turn at the same speed?
Do the ratios of successive stages add or multiply?
What does an idler gear change, and what does it not change?

B3.3B Gear and belt-driven system applications (HL only)

B3.3B Gear and belt-driven system applications (HL only)

Subtopics and learning objectives

B3.3.4 Gear and belt-driven system applications

Gear and Belt Systems: Working Out the Output Speed

Understanding the System

Gear and Belt-Driven System Applications

Direction Change Example

Calculating Speed of Rotation in Gear Systems

Speed at Several Points in a Compound Train

Gear and Belt Systems: Working Out the Output Speed

Understanding the System

Gear and Belt-Driven System Applications

Direction Change Example

Calculating Speed of Rotation in Gear Systems

Speed at Several Points in a Compound Train

A. Design in theory

B. Design in practice

C. Design in context

A. Design in theory

B. Design in practice

C. Design in context

B3.3B Gear and belt-driven system applications (HL only)

B3.3B Gear and belt-driven system applications (HL only)

Subtopics and learning objectives

B3.3.4 Gear and belt-driven system applications

B3.3.4 Gear and belt-driven system applications

Gear and Belt Systems: Working Out the Output Speed

Understanding the System

Gear and Belt-Driven System Applications

Direction Change Example

Calculating Speed of Rotation in Gear Systems

Speed at Several Points in a Compound Train

Gear and Belt Systems: Working Out the Output Speed

Understanding the System

Gear and Belt-Driven System Applications

Direction Change Example

Calculating Speed of Rotation in Gear Systems

Speed at Several Points in a Compound Train