Tolerance verification permits to check the product conformity and to verify assumptions made by the designer. For conformity assessment, the uncertainty associated with the values of the measurands must be known. In the ISO TS 17450 part 2, the notion of the uncertainty is generalized to the specification and the verification. The uncertainty is divided into correlation uncertainty, specification uncertainty and measurement uncertainty. Correlation uncertainty characterizes the fact that the intended functionality and the controlled characteristics may not be perfectly correlated. Therefore, we propose a new specified characteristics based on the statistical tolerancing approach which is directly in relationship with the design intent: the probability distribution of maximum range of the transmission error (the transmission error is the main source of vibratory and acoustic nuisances), and the evaluation of this characteristic based on 3D acquisition by Monte Carlo simulation and Tooth Contact Analysis. Moreover, the measurement uncertainty of the evaluation of this characteristic is estimated by Monte Carlo Simulation

BAUDOUIN, Cyrille

BIGOT, Régis

BRUYERE, Jérôme

DANTAN, Jean-Yves

VINCENT, Jean-Paul

English

SAM;  the Arts et Métiers ParisTech open access repository

Science Arts & Métiers (SAM)
is an open access repository that collects the work of Arts et Métiers ParisTech
researchers and makes it freely available over the web where possible.
This is an author-deposited version published in: https://sam.ensam.eu
Handle ID: .http://hdl.handle.net/10985/11193
To cite this version :
Jean-Paul VINCENT, Cyrille BAUDOUIN, Jérôme BRUYERE, Jean-Yves DANTAN, Régis BIGOT
- Gear metrology of statistical tolerancing by numerical simulation - In: International  Conference
on Advanced Mathematical and Computational Tools in Metrology and Testing/ AMCTM VIII,
France, 2008-06 - Advanced mathematical&computational tools in metrology&testing VIII - 2009
Any correspondence concerning this service should be sent to the repository
Administrator : archiveouverte@ensam.eu
GEAR METROLOGY OF STATISTICAL TOLERANCING BY 
NUMERICAL SIMULATION 
JEAN-PAUL VINCENT, CYRILLE BAUDOUIN, JÉRÔME BRUYERE, 
JEAN-YVES DANTAN, REGIS BIGOT  
Laboratoire de Génie Industriel et de Production Mécanique. 
E.N.S.A.M. de Metz, 4 rue A. Fresnel, 57070 METZ Cedex, France. 
 E-mail: jean-yves.dantan@metz.ensam.fr 
Tolerance verification permits to check the product conformity and to verify assumptions 
made by the designer. For conformity assessment, the uncertainty associated with the 
values of the measurands must be known. In the ISO TS 17450 part 2, the notion of the 
uncertainty is generalized to the specification and the verification. The uncertainty is 
divided into correlation uncertainty, specification uncertainty and measurement 
uncertainty. Correlation uncertainty characterizes the fact that the intended functionality 
and the controlled characteristics may not be perfectly correlated. Therefore, we propose 
a new specified characteristics based on the statistical tolerancing approach which is 
directly in relationship with the design intent: the probability distribution of maximum 
range of the transmission error (the transmission error is the main source of vibratory and 
acoustic nuisances), and the evaluation of this characteristic based on 3D acquisition by 
Monte Carlo simulation and Tooth Contact Analysis. Moreover, the measurement 
uncertainty of the evaluation of this characteristic is estimated by Monte Carlo 
Simulation.  
1. Introduction
As technology increases and performance requirements continually tighten, the 
cost and required precision of assemblies increase as well. There is a strong need 
for increased attention to tolerance design to enable high-precision assemblies to 
be manufactured at lower costs. Tolerance verification permits to check the 
product conformity and to verify assumptions made by the designer. Tolerance 
verification defines inspection planning and metrological procedures for 
functional requirements, functional specifications and manufacturing 
specifications.  
For gears, two kinds of metrology exist [1], [2]: Geometrical metrology and 
kinematic metrology. Geometrical metrology is performed to ensure that parts 
meet tolerance specified on its geometric definition. Kinematic metrology 
allows determining transmission error of the product. Concerning the gears 
standards (ANSI/AGMA 2009-B01; DIN 3962), three type of geometrical 
deviation are identified: the flank deviation (profile or form deviation…), the 
situation deviations between flanks (cumulative and single pitch deviations…) 
and the situation deviations between teeth and hole (runout, …) As geometric 
deviations impact the kinematic behaviour of the product, it is possible to use a 
computer program, able to predict the total composite deviation F’I and the 
tooth-to-tooth composite deviation f’I knowing the geometric deviation of the 
product. This program evaluates the virtual kinematic error based on the CMM’s 
data points from the geometrical measurement of gears. 
The notion of uncertainty is by now well entrenched in metrology. In the 
ISO TS17450-2, the notion of the uncertainty is generalized to the specification 
and the verification. In fact, the uncertainties through the product life cycle 
begin from the design intent to the inspection activity. The uncertainty is divided 
into correlation uncertainty, specification uncertainty and measurement 
uncertainty. Correlation uncertainty characterises the fact that the intended 
functionality and the controlled characteristics may not be perfectly correlated. 
The specification uncertainty characterizes the ambiguity in the specification 
expression. And the measurement uncertainty is considered by the metrologists 
and well described in GUM. The measurement uncertainty includes all the 
causes of variation from the geometric specification to the result of inspection. 
To decrease the correlation uncertainty, a new specified characteristic based 
on a statistical tolerancing approach is proposed: the probability distribution of 
the transmission error of the studied gear. This characteristic is in direct 
relationship with the design intent: the requirement on the transmission error 
(this approach could be apply to others requirements – backlash, ...) In fact, it is 
interesting to predict the kinematic behaviour the studied gear with a gear 
randomly chosen from production i.e. to know the probability distribution of the 
transmission error of the studied gear. 
2. Global Approach
The objective of this paragraph is to present the global approach followed to 
first estimate the total composite deviation F’I and the tooth-to-tooth composite 
deviation f’I for a virtual master gear meshing with a studied gear based on its 
acquisition CMM’s data points, and secondly to estimate the probability 
distribution of the transmission error of the studied gear. 
The estimation of F’I and f’I (Approach 1) is detailed in Figure 1a): 
1. To obtain an image of a manufactured gear, the acquisition equipment
chosen, has been a CMM. 
2. The result of the CMM acquisition is a set of points
3.  A set of points of the nominal model of the master gear geometry is
generated
4.  Based on the result of the CMM acquisition [3], substitute tooth surfaces in
the global coordinate system are reconstructed using Bezier surfaces. The
geometrical modelling of parts is detailed in the third chapter.
5.  Tooth Contact Analysis (TCA) [4], [6], program computes a simulated
transmission error then F’I and f’I are calculated.
Secondly, the estimation of the probability distribution of the transmission error 
of a studied gear (Approach 2) is detailed in Figure 1 b): 
1 2
4 5
3
-200
-100
0
100
200
300
400
500
600
-0,5 0 0,5 1 1,5 2 2,5
Set of points
Nominal model
of master gear
4 5 6
3’ 4 5
7
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40 50 60
Random Deviations
1 2
3’
Set of points
Nominal model
of master gear
a)
b)
Figure 1: a) Detail of the estimation of F’I and f’I. b) Detail of the estimation of the probability 
distribution of the transmission error of a studied gear. 
3'. The first and the second steps of this approach are the same as before. The
 geometrical deviations of the wheel and the assembly are modelized by
 random variables. These geometries provide an image of the manufactured   
 gears. 
The fourth and fifth steps of this approach are the same as before. 
6.  A Monte Carlo simulation computes a set of meshing simulations. Each
 meshing simulation is computed by a computer program based on Tooth 
Contact Analysis. The result of this metrology and numerical simulation. 
7.  The probability distribution of the maximum range of the transmission error
of the measured pinion is then calculated. 
To validate the program of virtual meshing simulation, a simulated 
kinematic error, based on the set of points of the CMM’s measurement of a pair 
of gears, has been compared to an experimental kinematic error of this same pair 
of gear. Several measurements have been performed in order to assess the 
measurement uncertainty of the kinematic metrology. For each test, the tooth-to-
tooth composite deviation has been calculated. The simulation meshing program 
allows evaluating the transmission error of a manufactured gears meshing with 
an other gear, master, or manufactured one. It provides virtual kinematic 
metrology estimation. In order to test the robustness of this approach, the 
confidence interval must be evaluated. 
3. Confidence interval of virtual kinematic metrology
In general, the result of a measurement is only an approximation or estimate of 
the value of the specific quantity subject to measurement. In this case, the 
measurands are the total composite deviation F’I and the tooth-to-tooth 
composite deviation f’I. The uncertainty of the result of this measurement must 
be evaluated by statistical methods. Furthermore, the TCA computer program is 
not base on an explicit model, that’s why the evaluation of uncertainty by the 
statistical analysis is made by Monte Carlo simulation [5]. 
The uncertainty and confidence interval of the result given by the meshing 
simulation program depend on several factors like the approximation error of the 
fitting operation, the point’s density used to reconstruct surfaces, the numerical 
precision of the TCA algorithm and the acquisition uncertainty due to the CMM. 
In order to evaluate the effects of acquisition uncertainty of the step 1 on the 
result of the simulation, the following method has been followed:  
• The evaluation of the acquisition uncertainty is based on CMM’s
manufacturer’s specifications, 
• The measurand Y (the total composite deviation F’I or the tooth-to-tooth
composite deviation f’I) is not measured directly, but is determined from N 
other quantities X1, X2. . . XN (the result of the acquisition: the set of points) 
through a functional relation: Y = f(X1, X2, …, Xn). The input quantity Xi is 
estimated from an assumed probability distribution based on the evaluation 
of the acquisition uncertainty. N independent observations Xi,k of Xi are 
obtained by a random generator like a Monte Carlo simulation, and the 
probability distribution and the combined standard uncertainty of the 
measurand Y is estimated (Figure 2a)) 
2'. Points of the first geometry are perturbed around there initial position. The 
perturbation is taken around the value of the uncertainty measurement of a 
CMM. For example 4 µm. Steps 3’ 4, 5, and 6 are the same as shown in 
section 2.  
Figure 2 b) shows the set of simulations for a set of geometries (perturbed 
points from 4 µm to their initial position). The confidence interval of the tooth-
to-tooth composite deviations for this set of simulation is 2 µm. 
2’
4 5
3’
6
2’ 4 5
Set of points
Master gear 
with deviations
0.06 0.08 0.10 0.12 0.14 0.16
Mesh cyle
_10
_5
5
10
Transmison eror in_m
Perturbations
0.06 0.08 0.10 0.12 0.14 0.16
Mesh cycle
_10
_5
5
10
Transmission error in _m
a)
b)
T
ra
n
s
m
is
s
io
n
 e
rr
o
r
Meshing cycle
Figure 2: a) Uncertainties evaluation approach due to acquisition uncertainties, b) A set of 
simulations based on perturbed geometries 
4. Gear Metrology of Statistical Tolerancing on Transmission Error
In this part, a statistical tolerancing approach is proposed to provide a new 
specified characteristic: the probability distribution of the transmission error of a 
studied gear. This characteristic, in direct relationship with the design intent, 
could decrease the correlation uncertainty (ISO TS 17450 part 2). 
Statistical tolerancing is a more practical and economical way of looking at 
tolerances and works on setting the tolerances so as to assure a desired yield [6]. 
By permitting a small fraction of assemblies to not assemble or function as 
required, an increase in tolerances for individual dimensions may be obtained, 
and in turn, manufacturing costs may be reduced significantly. Statistical 
tolerance analysis computes the probability that the product can be assembled 
and will function under given individual tolerance. 
Gear roll testing allows knowing the kinematic error of a gear from 
production meshing with a master gear, but to know how a manufactured gear 
behaves with others, randomly chosen from production i.e. to know the 
probability distribution of the kinematic error of the studied gear, a large number 
of meshing must be performed. To do so, geometrical deviations are introduced 
in the model of gears. Different models can be use to introduce these deviations 
i.e. vectorial tolerancing or zone tolerancing. 
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40 50 60
f’i (micrometers )
N
u
m
b
e
r
0 10 20 30 40 50 60
0
400
800
1200
Figure 3: Probability distribution of the maximum range of the transmission error. 
Usually, statistical analysis uses a relationship of the form Y = f(X1, X2, …, 
Xn) where Y is the response (characteristic such as gap or functional 
characteristics) of the assembly and X ={X1,X2, …, Xn} are the values of some 
characteristics (such as situation deviations or/and intrinsic deviations) of the 
individual parts. f can be an explicit function or implicit expression. As it is 
implicit in or case, statistical analysis methods like Linear Propagation (RSS) 
and Taylor series cannot be employed. Quadrature technique can be use if the 
function is approximable. Monte Carlo Simulation is a powerful tool to know 
the probability distribution of a functional characteristic of a product for 
example. The following figure shows the result of a Monte Carlo simulation 
with 4000 samples: probability distribution of the maximum range of the 
transmission error: a measured pinion meshing with randomly generated 
geometries of wheel (equivalent to ISO Quality 6) (Figure. 3). 
5. Conclusion
The important point of the proposed specification and metrology for a gear 
based on dimensional metrology and numerical simulation is to provide a 
solution which is in adequacy with the design intent. There is no more 
interpretation for the designer, the manufacturer and the metrologist. This 
approach could be decrease the correlation uncertainty [ISO TS 17450 – part 2]. 
Further more, in order to test the robustness of the proposed approach (to 
evaluate the impact of the measurement uncertainties on the simulated 
transmission error), the confidence interval of the proposed characteristic has 
been evaluated using Monte Carlo simulation. 
Reference 
1. G. Goch, Gear metrology, keynote papers, Annals of the CIRP, 52(2), 1–37.
(2003)
2. S. Boukebbab, H. Bouchenitfa, H. Boughouas, J. M. Linares. Applied
iterative closest point algorithm to automated inspection of gear box tooth,
Computers & Industrial Engineering, 52, 162-173. (2007)
3. C. Baudouin, R. Bigot, S. Leleu, P. Martin,  Gear geometric control
software: approach by entities, The International Journal of Advanced
Manufacturing Technology.(2007)
4. F.L. Litvin, A. Funetes and K. Hayasaka. Design, manufacture, stress
analysis, and experimental tests of low-noise high endurance spiral bevel
gears, . Mechanism and Machine Theory, 41, 83-118. (2006).
5. K.D. Sommer, O. Kuehn, A. Weckenmann, Use of MCM for Uncertainty
Evaluation of a Non-Linear and Multivariate Dimensional Measurement
Task.  CIRP Seminar on Computer Aided Tolerancing, Germany. (2007).
6. J. Bruyère, J.Y. Dantan, R. Bigot, P. Martin, Statistical tolerance analysis of
bevel gears by Tooth Contact Analysis and Monte Carlo simulation,
Mechanism and Machine Theory,  42( 10), 1326-135.(2007).


Gear metrology of statistical tolerancing by numerical simulation

https://sam.ensam.eu/bitstream/handle/10985/11193/LCFC_AMCTM_2009_BAUDOUIN.pdf?sequence=1&isAllowed=y

Gear metrology of statistical tolerancing by numerical simulation

Abstract

Similar works

Full text

Available Versions

SAM; the Arts et Métiers ParisTech open access repository