Velocity Ratio Calculations
Velocity Ratio
A measure of how fast one part of a mechanical system moves compared to another, helping designers control speed, torque, and efficiency.
Key formula:
$$\text{Velocity Ratio (VR)} = \frac{\text{Input Speed}}{\text{Output Speed}}$$
Velocity ratio is a dimensionless quantity, meaning it has no units. It simply expresses a relationship between two speeds.
Velocity Ratios in Gear Systems
- Gears are used to transmit torque and rotational motion between shafts.
- The velocity ratio in a gear system is determined by the number of teeth on the gears.
- A large output gear = slower speed, more torque
Formula:
$$\text{VR} = \frac{\text{Number of teeth on output gear}}{\text{Number of teeth on input gear}}$$
Input gear = 20 teeth, Output gear = 40 teeth → VR = 40 / 20 = 2
- When calculating velocity ratios, always identify which gear is driving and which is driven.
- This ensures accurate calculations and understanding of the system's behavior.
Velocity Ratios in Pulley Systems
- Pulleys are used to change the direction of force and provide mechanical advantage.
- The velocity ratio in a pulley system depends on the diameter of the pulleys.
Formula:
$$\text{VR} = \frac{\text{Diameter of driven pulley}}{\text{Diameter of driver pulley}}$$
Driver pulley = 50 mm, Driven pulley = 100 mm → VR = 100 / 50 = 2
- Consider a bicycle with different-sized wheels.
- A larger rear wheel (driven) compared to the front wheel (driving) means the rear wheel will rotate more slowly, but with greater force.
Velocity Ratios in Belt-Driven Systems
- Belt-driven systems use belts to transfer motion between pulleys.
- The velocity ratio is calculated similarly to pulley systems, where $$\text{VR} = \frac{\text{Diameter of driven pulley}}{\text{Diameter of driver pulley}}$$
- A smaller driven pulley = higher output speed
Driver = 150 mm, Driven = 75 mm → VR = 75 / 150 = 0.5 (speed increase)
- Belt-driven systems can experience slippage, which may affect the actual velocity ratio.
- This is an important consideration in real-world applications.
- VR > 1 → output is slower, but has more torque
- VR < 1 → output is faster, but has less torque
- Ideal in applications like bikes, conveyors, and lathes.
Practical Applications of Velocity Ratios
- Gears: Used in bicycles, cars, and machinery to adjust speed and torque.
- Pulleys: Found in elevators, cranes, and conveyor systems to lift heavy loads.
- Belt-Driven Systems: Common in engines and industrial machinery for efficient power transmission.
In a car's transmission system, gears with different velocity ratios allow the vehicle to accelerate quickly (low gear) or maintain high speeds efficiently (high gear).
- Try calculating the velocity ratio for a gear system where the driving gear has 15 teeth and the driven gear has 45 teeth.
- What does this ratio tell you about the system's performance?
