Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Explain the working principle of epicyclic gear trains.
• Calculate the velocity ratio using two different methods.
• Identify practical applications in South African industries.
• Compare epicyclic gear trains to simple gear trains.
• Understand how to construct a simple model.

Lesson summary

This week, we delve deeper into the fascinating world of advanced mechanisms and gear systems. Building upon your existing knowledge of basic gear systems, we will explore more complex configurations, including epicyclic (planetary) gear trains and their applications in modern technology. Understanding these advanced mechanisms is crucial because they are the backbone of many technologies we use daily, from the automatic gearboxes in taxis and bakkies to the high-precision instruments used in mining and manufacturing industries across South Africa.

Lesson notes

2.1 Epicyclic (Planetary) Gear Trains Epicyclic gear trains, also known as planetary gear trains, are gear systems where one or more gears (the planet gears) revolve around a central gear (the sun gear). A carrier, also known as an arm, connects the planet gears and rotates, causing them to orbit the sun gear. A ring gear, with internal teeth, is often used to constrain the movement of the planet gears.

Components: Sun Gear: The central gear in the system.

Planet Gears: Gears that revolve around the sun gear.

Ring Gear: An internal gear that meshes with the planet gears.

Carrier (Arm): Connects the planet gears and rotates about the sun gear's axis.

Working Principle: The unique feature of epicyclic gear trains is that the axes of some of the gears (planet gears) move relative to the frame. This allows for high gear ratios in a compact space and different input/output configurations. By fixing different components (sun, planet carrier, or ring gear), and applying input to another, we can achieve different speed and torque characteristics. Why Epicyclic Gear Trains?

High Gear Ratios: Epicyclic gear trains can achieve very high or very low gear ratios in a relatively small volume, making them ideal for applications where space is limited, such as automatic transmissions in vehicles.

Compact Size: Compared to traditional gear trains, epicyclic gear trains can achieve the same gear ratio in a smaller package.

Load Sharing: Multiple planet gears distribute the load, resulting in higher torque capacity and improved durability.

Coaxial Input and Output: Some configurations allow for the input and output shafts to be coaxial (aligned), simplifying design in many applications. 2.2 Velocity Ratio (Gear Ratio) Calculation Calculating the velocity ratio (VR) of an epicyclic gear train is more complex than for simple gear trains because of the relative motion involved.

Two common methods are: a)

Tabular Method: This method involves constructing a table to track the rotations of each component. The key is to first imagine locking the entire system and giving it one rotation. Then, unlock the system and compensate for the initial rotation by adding an appropriate number of rotations to one of the components.

Steps: Condition 1: Arm Fixed (Simple Gear Train): Determine the gear ratios when the carrier (arm) is held stationary. This simplifies the system into a simple gear train.

Condition 2: Arm Rotates +1 Revolution: Assume the entire system is locked together and rotate the arm by +1 revolution.

Condition 3: Compensate for Arm Rotation: Unlock the system and rotate one gear (usually the ring gear if it's fixed) by an amount that cancels out its rotation from step

2. Total Rotations: Sum the rotations for each component across all conditions.

Velocity Ratio: Calculate the VR using the total rotations of the input and output components.

Example: Consider an epicyclic gear train with a sun gear (S) of 20 teeth, a planet gear (P) of 10 teeth, and a ring gear (R) of 40 teeth. The ring gear is fixed. Calculate the velocity ratio if the sun gear is the input and the carrier (arm) is the output. | Component | Arm Fixed (Condition 1) | Arm +1 (Condition 2) | Ring Compensation (Condition 3) | Total Rotation | | --------- | ------------------------ | ------------------- | ------------------------------- | -------------- | | Arm | 0 | +1 | 0 | 1 | | Sun | -(40/20) = -2 | +1 | 0 | -1 | | Ring | 0 | +1 | -1 | 0 | VR = Output / Input = Arm / Sun = 1 / -1 = -1 This means that for every one rotation of the sun gear in one direction, the arm rotates once in the opposite direction. b)

Formulaic Method: A general formula can be used, but it's crucial to understand its derivation.

A common form is: VR = (N R + (VR fixed arm * N S )) / (N S ), where: VR is the overall velocity ratio N R is the number of teeth on the ring gear N S is the number of teeth on the sun gear VR fixed arm is the velocity ratio when the arm is fixed (ring/sun if ring fixed)

For our example above: VR = (40 + (-40/20) * 20) / 20 = (40 - 40) / 20 = 0/20 =

0. This result needs adjusting because the basic formula needs careful application based on the fixed part. A more appropriate general formula for the specific case where the ring gear is fixed is VR = (N S / (N S +N R )) Therefore, VR = 20 / (20 + 40) = 20/60 = 1/3 (Sun input, Arm output). Note that in this formula, the input (sun) rotates three times for every rotation of the output (arm). This is the inverse of the previously (and incorrectly) calculated ratio using the table. The negative from the tabular method reflects the direction of rotation. The formula above is simplified for this scenario and requires careful adjustment depending on which component is fixed and which are input/output. The tabular method, while initially complex, is far more robust and less error-prone in most situations.