Hobbing.com's Nano Error Curve Calculation of Gear Tooth Geometry





gear tooth profile drawn with finite number of line segments with uniform curve errors
red: profile curves, yellow: line segments to draw curves, white: curve errors


curve error: maximum distance between curve and line segment
nano error: millimeters per module = meters error per module
                  This is somewhat tricky. And yet, we have the record of the calculation with millimeters error per module as shown in the last picture on this page.
nono error curve calculation: computer calculation of line segments for a specific curve with uniform curve errors which are closest to and less than nano error, minumum number of points to be generated for short curves = 2, equivalently 1 line segment
minumum number of points to be generated for tip or root rounding = 3, equivalently 2 line segments

calculation of circular arc: by equations
calculation of involute: by equations and numerical solution methods
calculation of root fillet: by equations and numerical solution methods

minimum number of points to generate: 2 (or 3 as needed)

limitation of the culculation: currently only for 2-dimensional curves, not 3-dimensional ones

limitation of application to software: up to 1/10000 (or 1/100000 as needed)

monitoring method: recalculation of curve errors, by calculating variances of statistic to confirm uniform errors
A surge in variance occurs when the calculation fails.

verifying method:
Users may have equations to recalculate curve erros of involute and arc using generated coordinates.
It is very difficult for users to recalculate curve errors of root fillet. However, they can do approximately by means of equations of circle.

contributions to gear engineering:
1. It is the best way to generate coordinates to draw teeth.
2. Numbers of points to draw teeth can be selected according to the engineering characteristics, manufacturing accuracy, and capacity of CAD software and computer hardware.
3. Formalization as engineering data (coordinates)

Examples of solid modeling with 1/50, 1/1000, 1/1000000 curve errors (millimeters error per module)


[example 1]


curve errors of computer input and output
input curve error: 1/ 50.0
 
                   output curve errors, variances and numbers of points generated
 root circle:    1/     6309.9263325237,         0.0 by nature,      2
 root fillet:    1/       57.5048628249,         0.0000000000,       4
 involute:       1/       81.3047171389,         0.0000000000,       4
 outside circle: 1/      729.0076491572,         0.0 by nature,      2


[example 2]


curve errors of computer input and output
input curve error: 1/ 1000.0
 
                   output curve errors, variances and numbers of points generated
 root circle:    1/     6309.9263325237,         0.0 by nature,      2
 root fillet:    1/     1078.7659396816,         0.0000000000,      14
 involute:       1/     1065.0553196748,         0.0000000000,      12
 outside circle: 1/     2915.9851413732,         0.0 by nature,      3


[example 3]


curve errors of computer input and output
input curve error: 1/ 1000000.0
 
                   output curve errors, variances and numbers of points generated
 root circle:    1/  1066374.3504170855,         0.0 by nature,     14
 root fillet:    1/  1000939.5128114824,         0.0000000000,     397
 involute:       1/  1000312.8587131152,         0.0000000000,     339
 outside circle: 1/  1052665.1809664315,         0.0 by nature,     39


Challenge for 1/10000000, 1/100000000, 1/1000000000 ( = millimeters error per module) to have the record for the calculation

[example 4]
curve errors of computer input and output
input curve error: 1/10000000.0
 
                   output curve errors, variances and numbers of points generated
 root circle:    1/ 10095851.6748497560,         0.0 by nature,     41
 root fillet:    1/ 10005207.4119485900,         0.0000000000,    1253
 involute:       1/ 10005858.1710100970,         0.0000000000,    1070
 outside circle: 1/ 10150491.5497110710,         0.0 by nature,    119


[example 5]
curve errors of computer input and output
input curve error: 1/100000000.0
 
                   output curve errors, variances and numbers of points generated
 root circle:    1/100176088.0754979700,         0.0 by nature,    127
 root fillet:    1/100043395.4854906400,         0.0000000000,    3960
 involute:       1/100030389.8627611700,         0.0000000000,    3381
 outside circle: 1/100339256.3559257100,         0.0 by nature,    372




[example 6]


curve errors of computer input and output
input curve error: 1/1000000000.0
                   output curve errors, variances and numbers of points generated
 root circle:    1/1004543549.6043290000,         0.0 by nature,    400
 root fillet:    1/1000040048.3040519000,         0.0000000000,   12518
 involute:       1/1000023106.6602092000,         0.0000000000,   10688
 outside circle: 1/1001332430.6911339000,         0.0 by nature,   1173
Download a full data sheet of this example.


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