Program 60-150—Angular Gear Transmission Error
Introduction
This IGS module calculates the angular position errors (index errors) produced by a
train of gears due to the errors allowed by the AGMA tolerances for the gear quality
class to which the individual gears are made. It is assumed that the maximum
errors allowed are present in all the gears in the system. The statistical effect of the
scatter in the actual errors is accounted for by calculating the root-mean-square
index error.
References
“Angular Errors in Gears” by T.C. Nielsen, Associate Engineer, IBM Federal Systems
Division, Owego, N.Y., in Gear Design and Application, Edited by N.P. Chironis,
published by McGraw-Hill 1967
Data Extracted from ANSI/AGMA 2000-A88, Gear Classification and Inspection,
with the permission of the publisher, American Gear Manufacturers Association,
1500 King Street, Suite 201, Alexandria, Virginia 22314
1
UTS Integrated Gear Software
Example
This example is a control gear system consisting of three stages with a total ratio of
30 to 1. To obtain the correct rotation direction the second stage has an idler gear.
The first stage pinion is connected directly to a stepping motor. It is desired that a
drum driven by the last gear in the third stage repeat its motion as closely as
possible to hold the register on multi-color printed sheets. We wish to find the
probable error in the drum angular position with respect to the stepping motor.
The gear trains have the following specifications:
1st Stage
12 tooth pinion driving a 36 tooth gear
Normal DP = 48
Normal PA = 20 deg
Helix Ang = 23 deg
Face
= 0.375 inches
AGMA Q#
8
2nd Stage
26 tooth pinion driving a 26 tooth idler and a 130 tooth gear
Normal DP = 24
Normal PA = 22.5 deg
Helix Ang = 0 deg
Face
= 0.625 inches
AGMA Q#
8
3rd Stage
22 tooth pinion driving a 44 tooth gear
Normal DP = 10
Normal PA = 20 deg
Helix Ang = 0 deg
Face
= 1.0 inches
AGMA Q#
8
2
60-150—Angular Gear Transmission Error
The model can be calculated with US customary or metric (SI) units: in the screen
image below you can see that there are cells for Normal Diametral Pitch, Module,
and Face Width in inches and in millimeters. Make certain that you blank the cells
for the set of units you are working with: if you are using metric units, blank the
cells for Normal DP and for Face Width in inches; conversely, for US customary,
blank the cells for Module and Face Width in mm.
In this example we are working in US/British units, with lengths in inches.
The first step is to enter the gear data into the interactive form. The form as it
appears when a new analysis is opened is shown below.
3
UTS Integrated Gear Software
There are seven gears in all, so we must add six columns to the form. Click the “Add”
button for the number of columns needed, then enter the numbers of teeth in the
entire train, starting with the first pinion. The form with the numbers of teeth
entered is shown below.
Now enter the normal diametral pitch for all the gears.
4
60-150—Angular Gear Transmission Error
Similarly, enter the pressure angles, helix angles, face widths (remember we’re
working in inches), and AGMA quality numbers. (You can enter a measurement in
millimeters, but if there are values for the face width in inches and in millimeters,
the inch measurement takes precedence.)
5
UTS Integrated Gear Software
Now we need to enter the relative speed of the gears with respect to each other. Any
speed can be entered for any of the gears as long as the speeds of all other gears are
in proportion to their actual speeds. This is necessary to assess the effect of the gear
ratios on the transfer of index error from each gear to the output gear.
6
60-150—Angular Gear Transmission Error
When the last relative speed is entered, solve the model by clicking the “Solve”
button. The results are shown in the Output Table of the interactive form; scroll up
and down to read them. Part of the solution is shown as it appears in the form, on
the next page. The complete solution, as it appears in TK Solver, is in Table 1.
7
UTS Integrated Gear Software
8
60-150—Angular Gear Transmission Error
Table 1
9
UTS Integrated Gear Software
When the model is solved, a dialog box warns that you should switch to the Power
User form to avoid losing data. Do so by clicking on the Power User button on the
Toolbar. The appearance of the interactive form will not change.
The model has produced these results:
1. The reference pitch diameter for all gears.
2. The AGMA tolerances for runout, pitch, profile and lead in accordance with
the specified quality number.
3. The angular errors (in radians) produced by the various errors for each gear
with respect to the axis of rotation of each gear. The total tolerance is used to
calculate the effect of each error except for the lead tolerance. One half of the
lead tolerance is applied to the angular error from lead error and half is
applied to angular error from wobble. The effect from both is usually the
same amount (but not always) except that lead error is usually seen once per
revolution of the gear and wobble is seen twice per revolution of the gear.
4. The maximum total angular error (in radians) produced by the maximum
value of all the individual errors combined for each gear with respect to the
axis of rotation of each gear.
5. The mean total angular error (in radians) produced by one half the maximum
value of all the individual errors combined for each gear with respect to the
axis of rotation of each gear.
6. The RMS angular error (in radians) produced by the maximum value of all
the individual errors combined by taking the square root of the sum of the
squares of the errors for each gear with respect to the axis of rotation of each
gear. More than 95% of systems should have less angular error than the
RMS value.
7. Maximum, mean and RMS total angular error for each gear taken with
respect to the output gear. This is the angular error produced by each gear
measured at the output gear. It is found by dividing the error at each gear
with respect to its axis by the total ratio from the gear to the output gear.
8. Maximum, mean and RMS angular transmission error for the whole system
measured in radians at the output shaft. This is the angular error measured
at the output shaft with a uniform rotation of the input gear.
9. Maximum, mean and RMS angular transmission error for the whole system
measured in degrees at the output shaft.
The 12 tooth pinion, for example, has a RMS total angular error with respect to its
own axis of 0.00558 radians produced by the allowable runout, pitch, profile and lead
tolerances for quality class 8. The total angular error is calculated for each of the
gears.
10
60-150—Angular Gear Transmission Error
When the effect of the 12 tooth pinion angular error is calculated at the output gear
we see that the RMS angular error at the output gear due to the pinion is only
0.00019 radians. The effect of the gear ratio has been taken into account.
The RMS angular transmission error for our example is 0.00336 radians (0.1919
degrees) with a uniform rotation of the input pinion.
We will assume that this is a little too high and see what might be done to reduce it
by changing the quality class of some of the gears.
First we will check the effect of changing the input train to AGMA quality class 10.
Move the cursor to AGMA Quality # for gears 1 and 2 and enter 10 for both, as shown
in Sheet 1-2.
Sheet 1-2
After solving, you should have the results shown in Table 2.
11
UTS Integrated Gear Software
Table 2
12
60-150—Angular Gear Transmission Error
The RMS angular error has been reduced from 0.00336 radians to 0.00319 radians.
This is not much decrease (about 5 percent) for the added expense of the input train.
Let's see what happens if we change the output train to quality class 10 instead of
the input train. See Table 3.
13
UTS Integrated Gear Software
Table 3
14
60-150—Angular Gear Transmission Error
This time the RMS angular error has been reduced from 0.00336 radians to 0.00236
radians, or about 30 percent. For our example it would be much more effective to
change the output train than the input train. Any proposed changes should be
entered in the model and checked before any decision is taken because the effect of
changes for some of the gears is not readily apparent.
Unlike earlier versions of this software, there is no need to rework the model to
calculate metric values; they are included. Remember to blank the cells with
US/British units when you work the model with metric units.
15