In robotics epicyclic gears is an important subject and are used a lot in manufacturing robots. Lets have a look to the definition and calculation of the simple epicyclic gear system.
An epicyclic gear (also called a planetary gear) is a gear system consisting of:
Sun Gear: The central gear.
Planet Gears: Gears that rotate around the sun gear and mesh with it.
Ring Gear: An outer gear with internal teeth that mesh with the planet gears.
Carrier: The arm that holds the planet gears and rotates around the sun gear.
Key Features:
Compact and efficient design.
Multiple gear ratios by fixing different components.
Used in applications like automatic transmissions, bicycles, and industrial machinery.
Advantages:
High power density.
Smooth operation.
Versatile speed and torque configurations.
The fundamental equation for epicyclic gear systems is derived from the relative motion of the components:
Where:
- ωsun, ωring, ωcarrier = angular velocities of the sun gear, ring gear, and carrier, respectively.
- Nsun, Nring = number of teeth on the sun gear and ring gear, respectively.
The negative sign indicates opposite rotation directions.
3. Gear Ratio Calculations
The gear ratio depends on which component is fixed, which is the input, and which is the output. Here are common configurations:
- Case 1: Ring Gear Fixed
- Input: Sun Gear
- Output: Carrier
- Gear Ratio:
- Case 2: Sun Gear Fixed
- Input: Ring Gear
- Output: Carrier
- Gear Ratio:
- Case 3: Carrier Fixed
- Input: Sun Gear
- Output: Ring Gear
- Gear Ratio:
Torque Distribution
Torque distribution depends on the gear ratio and the power flow. The torque on each component can be calculated using:
Where T is torque.
Worked Example
Problem:
A planetary gear system has:
- Sun gear: 20 teeth
- Ring gear: 60 teeth
- Carrier: Holds 3 planet gears
If the ring gear is fixed and the sun gear rotates at 100 rpm, what is the carrier's speed?
Solution:
Case 1: Ring Gear Fixed
- Fixed Component: Ring Gear (ωring=0)
- Input: Sun Gear (ωsun=100 rpm)
- Output: Carrier (ωcarrier)
Case 2: Sun Gear Fixed
- Fixed Component: Sun Gear (ωsun=0)
- Input: Ring Gear (ωring=100 rpm)
- Output: Carrier (ωcarrier)
Case 3: Carrier Fixed
- Fixed Component: Carrier (ωcarrier=0)
- Input: Sun Gear (ωsun=100 rpm)
- Output: Ring Gear (ωring)
Summary of Speeds
Assume the input torque is (100 Nm) the torque calculation will be:
Case 1: Ring Gear Fixed
- Fixed Component: Ring Gear (0ωring=0)
- Input: Sun Gear (ωsun=100 rpm)
- Output: Carrier (ωcarrier=25 rpm) (from earlier calculation)
Torque Relationships:
The torque on the sun gear (Tsun) and carrier (Tcarrier) are related by:
Torque on Ring Gear:
Since the ring gear is fixed, it reacts to the torques on the sun gear and carrier. Using equilibrium:
Case 2: Sun Gear Fixed
- Fixed Component: Sun Gear (ωsun=0)
- Input: Ring Gear (ωring=100 rpm)
- Output: Carrier (ωcarrier=75 rpm) (from earlier calculation)
Torque Relationships:
Torque on Sun Gear:
Since the sun gear is fixed, it reacts to the torques on the ring gear and carrier. Using equilibrium:
Case 3: Carrier Fixed
- Fixed Component: Carrier (ωcarrier=0)
- Input: Sun Gear (ωsun=100 rpm)
- Output: Ring Gear (ωring=−33.33 rpm) (from earlier calculation)
Torque Relationships:
Torque on Carrier:
Since the carrier is fixed, it reacts to the torques on the sun gear and ring gear. Using equilibrium:
Summary of Torque Results
