Week #12: Nov 17 – 21

Lecture #33: Introduction to Gears – terminology and limits for good design.

Book Sections: 9.0 – 9.6

The early parts of chapter 9 focus on gear terminology. Perhaps the quickest way to gain an understanding of the terminology is to examine and understand the figures in these sections. In particular, Figs. 9-5 to 9-23 should be understood. If you don’t understand what is being shown in the figure, read the text associated with the figure to gain an understanding. If that does not work, bring your questions to class! Also spend some time examining and understanding Tables 9-1 to 9-5.

Some of the more important ideas of this section for the designer include: pressure angle, center distance, backlash, and gear sizes to avoid interference and undercutting. Some understanding of the involute profile is also beneficial in understanding the behavior of gears.

Gear Train Class Exercises: Work in groups of 2 or 3 for the following assignment:

1) Design THE SMALLEST POSSIBLE reverted gear train to produce a gear ratio of 43:1. You must solve both of the following options:

(a) Produce that ratio within 2% using equal steps

(b) Produce the exact gear ratio.

2) Complete problem 9-36 from the text.

3) Consider a differential such as the one shown in Fig. P9-3 on page 482. Prove that the sum of the output speeds of the left and right axles is always a constant.

Total assignment = 20 points. Value = 22 points for Nov. 21 submissions, 19 points for Nov 24 submissions; 16 points for Dec. 1 submissions.

Lecture #34: Basic gear trains: Simple, compound and reverted.

Book Sections: 9.7, 9.8

Gear Train Basics

The velocity ratio, m_V, of a gear train relates the output velocity to the input velocity. For example, a gear train ratio of 5:1 means that the output gear velocity is 5 times the input gear velocity.

Simple Gear Trains – A simple gear train is a collection of meshing gears where each gear is on its own axis, as shown in Fig. 9-27. The train ratio for a simple gear train is the ratio of the number of teeth on the input gear to the number of teeth on the output gear (equation 9.7 in text). A simple gear train will typically have 2 or 3 gears and a gear ratio of 10:1 or less. If the train has 3 gears, the intermediate gear has no numerical effect on the train ratio except to change the direction of the output gear.

Compound Gear Trains – A compound gear train is a train where at least one shaft carries more than one gear (see Figs. 9-28 & 9-29 for examples). The train ratio is given by the ratio (eqn 9.8b):

m_V = (product of number of teeth on driver gears)/(product of number of teeth on driven gears)

A common approach to the design of compound gear trains is to first determine the number of gear reduction steps needed (each step is typically smaller than 10:1 for size purposes). Once this is done, determine the desired ratio for each step, select a pinion size, and then calculate the gear size. Example 9-2 in the book demonstrates this process.

Reverted Gear Trains – A reverted gear train is a special case of a compound gear train. A reverted gear train has the input and output shafts in –line with one another, as shown in Fig. 9-30. Assuming no idler gears are used, a reverted gear train can be realized only if the number of teeth on the input side of the train adds up to the same as the number of teeth on the output side of the train (equation 9.9c in text). Example 9-3 gives a couple of approaches to the design of reverted gear trains.

NOTE: Do not worry about the book section that describes “An Algorithm for the Design of Compound Gear Trains” (unless you are really into programming that kind of stuff!)

Lecture #35: Planetary gear trains I; definitions, relative velocity analysis.

Book Sections: 9.9

A planetary or epicyclic gear train is a 2 degree of freedom geared system. A couple of common applications of epicyclic gear trains include transmissions, and differentials. These systems require the control of 2 inputs to produce a single output.

Terminology

The basic components of a planetary gear train include (Fig. 9-33):

Sun Gear, Planet Gears, Arm, Ring Gear (optional)

The sun and ring gears and the arm all have a common axis. The input is typically applied to two of these three and the output is on the third one. (One of the inputs is commonly to fix one of the three so it does not move).

Analysis

Planetary gear trains can be analyzed using relative velocity methods. Two methods presented in the text include the tabular method and the formula method. We will focus on the tabular method as it gives a more complete solution and is (perhaps) easier to remember.

The basis for the tabular method is equation 9.12 in the text. The total motion of any gear can be found as the sum of the motion of the arm plus the motion of the gear with respect to the arm. We focus on the motion relative to the arm because, if we were to hold the arm stationary, we get a simple gear train. Thus, all motion relative to the arm is analyzed as a simple gear train.

Work through example 9-5 in the text to prepare for class.