The aim of this work is to develop a software that allows the inspection of spur gear manufactured in the sub-millimeter range. The measurements are made using a digital optical machine and using an analysis proprietary software implemented in Matlab®, which is able to handle images, captured using the digital optical machine. The software allows to evaluate the profile and pitch deviations as establish in the ISO/TR 10064-1:1992 standar

Barajas Fernandez, Cintia

Berzal Rubio, Miguel

Caja Garcia, Jesus

Gomez Garcia, Emilio

Maresca, Piera

English

Archivo Digital UPM

10064-1:1992 Cylindrical gears - Code of inspection practice - Part 1: Inspection of corresponding flanks of gear teeth,

101:2008. Evaluation of measurement data - Supplement 1 to the &quot;Guide to the expression of uncertainty in measurement&quot; - Propagation of distributions using a Monte Carlo method,

1328-1:1995 Cylindrical gears - ISO systems of accuracy - Part 1: Definitions and allowable values of deviations relevant to corresponding flanks of gear ,

A multiscale turning angle representation of object shapes for image retrieval&quot;

Gear Geometry and applied theory 2nd Ed,

Alternative method for the metrological characterization of spur gears in the sub-millimeter range using optical equipment

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Alternative Method for the Metrological 
Characterization of Spur Gears in the Sub-
Millimeter Range Using Optical Equipment 
J. Caja, E. Gómez, C. Barajas, P. Maresca, M. Berzal 
Departamento de Mecánica Industrial - Universidad Politécnica de Madrid 
Ronda de Valencia, 3 - 28012 Madrid - España 
jesús. caja@upm. es 
Abstract. The aim of this work is to develop a software that allows the inspection of spur gear 
manufactured in the sub-millimeter range. The measurements are made using a digital optical 
machine and using an analysis proprietary software implemented in Matlab®, which is able to 
handle images, captured using the digital optical machine. The software allows to evaluate the 
profile and pitch deviations as establish in the ISO/TR 10064-1:1992 standard [1], 
Keywords: Metrology, Optical Measuring Machine, Gear Measurement, Spur Ger 
PACS: 06.20.-f, 06.30.Bp, 06.60.Mr, 89.20.Bb, 89.20.Ff, 89.20.Kk 
INTRODUCTION 
Systems for power transmission based on gears are one of the most widely used 
today in fields such as automotive, aerospace, machine tools and other applications 
[2]. Cause of this so extensive use, greater demands in terms of transmitted power, 
durability, life... are required. Also, a reduction of their production costs, noise, etc. 
This situation has led to the tolerances allowed for these elements become increasingly 
narrow, forcing more precise manufacturing processes, being of vital importance the 
dimensional verification. 
The verification of the gears is a costly process, and that is currently carried out by 
using three Coordinates Measuring Machine (CMM) or specific machines for 
measuring gear, equipped with numerical control systems. The use of this equipment 
reduces the measurement time and the associated uncertainties. Measured phases with 
this equipment can be summarized as: alignment of the workpiece, the scanning 
surface, data evaluation and presentation of results. 
The development of micro-manufacturing techniques has allowed the manufacture 
of spur gears in the sub-millimeter range. In this case, the use of traditional system of 
gear check, described above, is unfeasible due to the impossibility of probing on the 
surfaces to be evaluated. In this assumption the special optical equipment and optical 
digital machines are a viable alternative. The use of these equipments has, among 
others, the following advantages: no probing, speed of data acquisition (taking a series 
of images) and no need to align the workpiece to be evaluated. Also, we can also use 
this technique to evaluate this type of gear (straight) in any dimension. 
From an analytical point of view, optical equipment can be considered a three 
measuring coordinates that without Z axis. This assumption implies that the plane on 
which to perform the measurement of the workpiece is parallel to the XY plane of the 
machine, restriction that limits the type of workpiece to characterize. 
Among the parameters to be evaluated according to the technical recommendation 
ISO/TR 10064-1:1992 [1] are: pitch deviations (Figure 1), single and cumulative, and 
the evaluation of the profile deviations of the gear tooth (Figure 4). 
EVALUATION OF THE PITCHES DEVIATIONS 
In gear metrology, the pitch and deviations from it are always arcs of a circle. 
Depending on the considered radius (base radius, reference radius, tip radius, root 
radius,...) different values for the pitch and its deviations are defined. Among these, 
the measured pitch on the reference circle (pi) and the one measured on the base circle 
ipb) are most relevant. The pitch pt is defined as the length of the arc measured on the 
reference circle between two consecutive flanks (right or left) of the tooth profile 
(Figure 1). 
pt =n-m = rp-6 
z 
Also, the base pitch is defined as the length of the arc measured on the base circle 
(right or left) between two consecutive flanks of the tooth profile. 
pb =n-m = rh-6 
z 
Where m is the module of the gear, z is the number of teeth, rp is the reference 
radius of the gear defined as z-m , r¿ is the base radius calculated as rp cosa and a is 
the pressure angle of gear. 
Determination of the pitch deviations is one of the most important checks to realize. 
We can distinguish different types of deviations: single deviations between two 
adjacent flanks (fpt) and deviations pitch accumulated on a specific number of pitches 
(Fpk), among others (Figure 1). Traditional methods, based on instruments that do the 
probing of the surface, present two major problems: the need of a rotary table and 
accurate determination of the axis rotation of gear (made by different methods). 
FIGURE 1. Circles features of a gear and pitch deviations. 
EVALUATION OF THE INVOLUTE PROFILE 
For the evaluation of the involute profile, we have started from the work of 
Shumungan [3]. Figure 2 shows an involute curve generated from point C to P from 
the tangent line at this point, rotated of an angle 6. The family of tangent lines that 
surrounds the base circle can be expressed parametrically as: 
x • cos 0 + y • sin 0 - rb - 0 (3) 
where x and y represent the coordinates of a point belonging to the involute curve 
and rb is the radius of the base circle. 
Likewise, and seeing the Figure 2, the length of the segment PT must be equal to 
the length of the arc TC. This relationship can be expressed analytically as: 
(x - rb • cos Of + (y-rb- sin Of =(rb- Of (4) 
Relating equations 3 and 4, the coordinates x, y of a point belonging to the involute 
profile can be expressed as [4]: 
x-r- cos 6 + r • 0 • sin 0 
(5) y = rb-sm6-rb-6- cos 6 
By varying the rotation angle 6 from 0 to a suitable value, a series of points, that 
describe the involute profile so that r = s]x2 + y1 range between base circle and tip 
circle rc, are obtained. 
FIGURE 2. Involute curve. 
Verification of the involute profile deviations 
According to the standard ISO 1328-1:1995 [5], the involute profile deviations 
correspond to the normal distances corresponding between a point on the gear tooth 
profile and the one corresponding to theoretical profile of the involute curve. 
Moreover, these deviations are measured in planes transverse to the tooth. 
The calculation model used to determine these deviations will be the following: 
According to Figure 3, the x-axis must pass through point C, base of the involute 
curve. Suppose that the point M of coordinates (xm, ym) corresponds to a point on the 
gear tooth profile, the tangent line to the base circle that passes through this point can 
be obtained by the following equation: 
x • cos 6>m + ym- sin 6m 
m m J m m 
•
rb 0 (6) 
where the only unknown value is the rotation angle 9m. Mathematical operator, this 
can be obtained as: 
cos 6L = 
-^b
2
-x
2
m+(x2m+y2m)(y2m-rb2) 
( 
2 2 \ 
xm+ym) 
(V) 
The normal deviation between the measured point and the theoretical point, seeing 
Figure 3, can be expressed as: 
\ =PM=P,M*=T,M,-T,P' 
m 
A =J(x -r,-cos& ) +(v -r,-sine? ) —r,-6 
m \ \ m b m) \s m b m) b r 
FIGURE 3. Deviations from the theoretical involute profile 
These deviations are represented in the typical diagram of the profile (Figure 4). 
Deviations of the curve respect to a straight line represent the deviations of the profile 
about involute curve generated from the base circle of the gear inspected. 
Starting from the tip of the tooth (point A, Figure 4), the theoretical profile 
contrasts with the measured profile in the range LAE (including the evaluation length 
Lac more the tip area of the tooth). The distance between the highest and lowest 
profile, measured by a horizontal line, represents the total profile deviation 
Fa. Midline of the profile (dashed line) is determined and two lines parallel to this, 
containing all points of the profile, are established. Distance measured according to a 
horizontal line between these two lines is the shape deviation of the profile f/a- Fua 
represents the deviation of the profile (distance between a vertical line and the midline 
of the profile measured to the distance Lac)-
FIGURE 4. Diagram of the involute profile deviations of a gear tooth 
IMPLEMENTATION OF COMPUTATIONAL ALGORITHMS 
The developed algorithms allow to (a) get the edge of the gear from the digital image 
obtained by the CCD camera, (b) identifying pixels belonging to it and (c) determine 
the results of the various checks described above. 
The made measurements allow to vary the used magnification of the optical 
equipment and the number of pixels supposed on the profile. The developed 
algorithms have been implemented in MatLab R2009b, together effective library for 
image analysis, Toolbox Image Processing. From the work of Midoh et al [6, 7], the 
edge detection technique based on the Canny filter, which for metrological images 
obtained for workpieces of sub-millimeter range is the most effective in terms of 
probability, uncertainty and precision [8], has been decided to use. Also, to determine 
more accurate edge, techniques subpixel and thresholding have been used [9]. Stages 
of the measurement process for previous checks are indicated and illustrated then. As 
initial image obtained by a digital optical machine Tesa Visio 300, from a gear module 
1 and 32 teeth, has been used. One of the problems of the edge determination is to 
differentiate whether a potential outlier is part of a manufacturing defect or is a foreign 
particle (dust or dirt). For an user is a simple task, but it is certainly difficult to 
implement automatically. 
Case 1: Evaluation of the pitch deviations 
Step 1: The algorithm receives as inputs: (a) the image to be treated (Figure 5), (b) 
the coordinates of pixels in the upper left and lower right corners, and the module and 
the number of gear teeth. 
FIGURE 5. Initial Screen gear, module 1 and 32 teeth. (Area 36.5 x 36.5 mm) 
Step2: To perform optimally edge detection, this is done from circles defined by the 
user on the image captured by the optical machine (in green color showed in Figure 6). 
Then, the algorithm establishes rectangular areas of evaluation (Figure 6) to which the 
Canny filter and the threshold value are applied [10]. Finally, the algorithm selects the 
edge pixels and determines their coordinates (Figure 7). 
FIGURE 6. Edge detection zones. 
Step 3: With these coordinates, and a computation iterative process, the algorithm, 
based on the coordinates of pixels, determines the position of the rotation center of the 
gear. From this center, feature circles of the gear are established (Figure 7). 
FIGURE 7. Detected edge and feature circles 
Step 4: From center established in the previous step, the points, located on the 
flanks of the teeth, that have a distance to the rotation center of the gear equal to 
feature circle one in which we want to evaluate the pitch deviations, are determined. A 
detection interval is stabled due to the difficulty of finding a point on the feature 
circle. 
Step 5: Finally, and from the above points detected, taking into account equations 1 
and/or 2, pitch deviations are calculated. Figure 8 shows some of the results that the 
program provides. 
S 
B 0.04 
32 1 2 3 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 27 28 29 30 31 32 
FIGURE 8. Pitch deviations of the right flank, 7^ [mm] 
Case 2: Evaluation of the profile deviations 
Step 1: The algorithm receives as inputs: (a) the image to be treated (Figure 9), (b) 
the coordinates of pixels in the upper left and lower right corners, and information 
about the rotation center of the gear and feature circles, predetermined. 
FIGURE 9. Initial image of the tooth profile 
Step2: The edge detection is made, similar to the one form explained in paragraph 
4.1-Step 2, except that the study area, defined by the user, is rectangular. The pixels of 
the edge and their coordinates are obtained. 
FIGURE 10. Edge detected on the tooth profile. 
Step 3: The algorithm separates the pixels on the edge belonging to the right and 
left flank. Also, we determine that the pixels of the edge are on the active area of the 
profile, between the base circle and the head, lowered a certain value (Figure 10). 
After realizing a change of coordinates system to use the equations of the involute 
profile (eq. 5), the theoretical profile (blue points) against the measured profile (red 
points) are showed in Figure 11. 
0.7 p 
• Borde Detectado 
• Perfil Teórico 
15 15.2 15.4 15.6 15.8 16 16.2 16.4 16.6 16.8 17 
FIGURE 11. Measured profile versus theoretical profile 
Step 4: Finally, the normal deviations between the theoretical profile and the 
measured profile (eq. 8) are obtained. The Figure 12 shows the typical representation 
of these deviations. 
0 50 100 150 200 250 300 350"\. 400 45C 
FIGURE 12. Diagram of deviations of the involute profile 
ESTIMATING UNCERTAINTY 
The use of the Monte Carlo method to obtain the uncertainty of the parameters 
previously calculated has been used. We have followed the recommendation of the 
Supplement 1 to the 'ISO Guide to the Expression of Uncertainty in Measurement' 
[11]. The considered components of the uncertainty are: 
Associated to the coordinates of the pixels lying at the edge. The digital optical 
machine has been calibrated, so that the traceability of the above coordinates is 
ensured. This coordinates, are sampled from a normal distribution with the mean value 
equal to the coordinate one and the standard deviation equal to the standard 
uncertainty obtained in the calibration of the machine 
Following the recommendations in section 7.2.1 of the Supplement 1 to the 'ISO 
Guide to the Expression of Uncertainty in Measurement' [11], the model is replicated 
N = 10 times, establishing a coverage interval for an approximate probability of 95%. 
The results obtained in different simulations and with different magnifications (high 
values of magnification give lower value of pixel size) show that the expanded 
uncertainty of the measurements takes values between 2 or 3 times the value of the 
pixel size. 
CONCLUSIONS 
The algorithms developed in this study, have allowed to evaluate the geometric 
parameters, in particular the pitch and the involute profile of the gears teeth, the pitch, 
and the involute profile of the teeth, from a digital image obtained with a digital 
optical machine. These algorithms have been proven robust in use during the 
verification of gear of sub-millimeter dimensions. 
The use of optical equipment allows verify gear that otherwise would not be 
possible, with a number of advantages such as: 
• Process automation 
• High speed of the verification process (only is necessary to determine a 
number of images of the gear, function of the used magnification) 
• Measurements with an associated uncertainty of about 2 to 3 times the 
instrument resolution. 
• Elimination of problems such as the need to use rotary table and accurate 
determination of the rotation axis of the gear. 
• The use of the developed algorithms can be extended to larger gears, 
obtaining the same advantages as described above. 
REFERENCES 
1. ISO, ISO/TR 10064-1:1992 Cylindrical gears - Code of inspection practice - Part 1: Inspection of 
corresponding flanks of gear teeth, 1992 
2. G. Goch, Manufacturing Technology 52 (2003). 
3. M.S. Shumungan, S.V.R. Surya and V. Jayaprakash, Mech. Mack Theory 33 (1998). 
4. F. L. Litvin and A. Fuentes, Gear Geometry and applied theory 2nd Ed, Cambridge:Cambridge 
University Press, 2004. 
5. ISO, ISO/TR 1328-1:1995 Cylindrical gears - ISO systems of accuracy - Part 1: Definitions and 
allowable values of deviations relevant to corresponding flanks of gear , 1995 
6. Y. Midoh, K. Nakamae and H. Fujioka, Meas. Sci. Technol 18 (2007). 
7. Y. Midoh, K. Miura, K. Nakamae and H. Fujioka, Meas. Sci. Technol 16 (2005). 
8. G. Iannizzotto and L. Vita, "A multiscale turning angle representation of object shapes for image 
retrieval" in Visual 99, Amsterdam, (The Netherlands), 1999. 
9. R. C. Gonzalez, R. E. Woods and S. L. Eddins, Digital Image Processing Using MATLAB 2nd Ed, 
Knoxville:Gatesmark Publishing, 2009. 
11. P. Maresca, E. Gómez, J. Caja, C. Barajas and M. Berzal, Measurement, 43 (2010). 
12.BIPM, JCGM 101:2008. Evaluation of measurement data - Supplement 1 to the "Guide to the 
expression of uncertainty in measurement" - Propagation of distributions using a Monte Carlo 
method, 2008, URL: http://www.bipm.org/en/publications/guides/gum.html 


Alternative method for the metrological characterization of spur gears in the sub-millimeter range using optical equipment

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