Innovative tooth contact analysis with non-uniform rational b-spline surfaces

Abstract

More and more simulation tools are being used in the development of gears in order to save development time and costs while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip, root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g. micro-pitting can only be considered insufficiently.
Hence, a new method for surface approximation with non-uniform rational b‑spline surfaces (NURBS) is presented. Its application can significantly improve the surface representation compared to the target geometry, leading to more realistic results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction of lifetime and acoustics is demonstrated by application to different gear geometries.

Full text

ORIGINALARBEITEN/ORIGINALS
https://doi.org/10.1007/s10010-021-00521-7
Forsch Ingenieurwes (2022) 86:273–281
Innovative tooth contact analysis with non-uniform rational b-spline
surfaces
Felix Müller1·StefanSchumann
1· Berthold Schlecht1
Received: 26 March 2021 / Accepted: 12 July2021 / Published online: 13 September 2021
© The Author(s) 2021
Abstract
More and more simulation tools are being used in the development of gears in order to save development time and costs
while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid,
beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine
the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable
surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface
representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip,
root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g.
micro-pitting can only be considered insufficiently.
Hence, a new method for surface approximation with non-uniform rational b-spline surfaces (NURBS) is presented. Its
application can significantly improve the surface representation compared to the target geometry, leading to more realistic
results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces
are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction
of lifetime and acoustics is demonstrated by application to different gear geometries.
Innovative Zahnkontaktsimulation mit nicht-uniformen rationalen B-Spline-Flächen
Zusammenfassung
Bei der Entwicklung von Verzahnungen werden zunehmend Simulationswerkzeuge eingesetzt, um die Entwicklungszeit
sowie -kosten einzusparen und gleichzeitig die Verzahnungen zu verbessern. Hierfür bietet BECAL ein umfassendes Soft-
warepaket für die Zahnkontaktanalyse (ZKA) an Kegel-, Hypoid-, und Stirnradgetrieben. Die Verzahnungsgeometrie wird
durch eine Fertigungssimulation oder einen Geometrieimport bereitgestellt. Zur Ermittlung der exakten Kontaktverhältnisse
in der ZKA werden die diskreten Flankenpunkte in eine kontinuierliche und differenzierbare Flächendarstellung überführt.
Derzeit handelt es sich um eine Approximation mittels Bézier-Tensorproduktflächen. Bei dieser Flächendarstellung kön-
nen in Abhängigkeit von der Zahngeometrie erhebliche Abweichungen zu den Vorgabepunkten auftreten. Insbesondere
Kopf-, Fuß- und Endrücknahmen, stark gekrümmte Zahnfußgeometrien oder diskontinuierliche topologische Messdaten
z. B. durch Grauflecken können damit nur unzureichend berücksichtigt werden.
Hierzu wird ein neues Verfahren zur Flächenapproximation mit nicht-uniformen rationalen B-Spline-Flächen (NURBS)
vorgestellt. Mit dieser Methode kann die Flächenabbildung im Vergleich zu den Vorgabepunkten deutlich verbessert werden,
was zu realistischeren Ergebnissen hinsichtlich der Flankenpressung, der Zahnfußspannung sowie der Drehwegabweichung
führt. Um die Vorteile zu verdeutlichen, werden NURBS-basierte Flächen mit den Bézier-Tensorproduktflächen verglichen.
Schließlich wird das Potenzial des neuen Ansatzes in Bezug auf die Vorhersage von Tragfähigkeit und Akustik durch die
Anwendung auf verschiedene Verzahnungsgeometrien demonstriert.
Code availability The program BECAL, which is a basis
of this article, is a program of the Forschungsvereinigung
Antriebstechnik (Drive Technology Association) FVA (http://fva-
net.de). It was developed by the Institute of Machine Elements
and Machine Design in cooperation with the Institute of Geometry
of the TU Dresden on behalf of the FVA.
Felix Müller
felix.mueller1@tu-dresden.de
1Institute of Machine Elements and Machine Design (IMM),
TU Dresden, Dresden, Germany
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274 Forsch Ingenieurwes (2022) 86:273–281
Fig. 1 BECAL calculation process
1Introduction
A major focus in the development of gear boxes is the opti-
mization of the gears. The aim is to increase the transferable
power as well as the efficiency and load capacity of those
components. At the same time, the demands on the design
process are growing in terms of development time and fi-
nancial effort. The use of simulation tools can shorten the
design process and significantly reduce the need for costly
gear tests.
BECAL is a comprehensive simulation tool for the tooth
contact analysis (TCA) of bevel, hypoid, beveloid and cylin-
drical gears, developed at the IMM (TU Dresden) in coop-
eration with the Institute of Geometry at the Faculty of
Mathematics (TU Dresden). It allows the calculation of lo-
cal stress on the tooth flank as well as in the tooth root.
Derived from this, the local load capacity regarding flank
and root can be determined. In addition, this simulation
tool enables the consideration of the efficiency as well as
an analysis of the acoustic excitation in the tooth mesh. The
BECAL calculation process is shown in Fig. 1[1].
The flank and root geometry of the gear is provided by
a manufacturing simulation, a geometry import or measure-
ment data. The geometry is thus represented by a discrete
3D point cloud. However, in order to perform the stress
calculation, it is necessary to determine the coordinates,
normals and curvatures for each of these points. In addi-
tion, for tooth stiffness determination and tooth root stress
calculation (BEM method [1,2]orFEAmethod[3]), it is
necessary to identify arbitrary points in the tooth root.
To do this, the discrete data points are transformed into
a mathematically closed surface representation. At present,
this is an approximation with Bézier tensor product sur-
faces. Depending on the tooth geometry, this surface rep-
resentation can result in significant deviations between the
approximated surface and the given points. Especially lo-
cal modifications, such as tip and root relief or local flank
damage (e.g. micro pitting, wear) can be insufficiently rep-
resented with this approach.
Thus, surface deviations can have a direct impact on
the calculation of flank and root stresses. For example, the
curvature on the flank has a significant influence on the
Hertzian pressure, which in turn is the basis for the load
capacity against pitting and micro pitting. The same occurs
with the representation of the root geometry, which can
have an effect on the root stress and, accordingly, on the
tooth root capacity. Finally, such deviations are compen-
sated for by an increased safety factor in the gear design,
which means that the potential of the gears cannot be com-
pletely utilized.
In this context, this paper focuses on the surface rep-
resentation with non-uniform rational B-spline surfaces
(NURBS) with the aim of significantly improving the
approximation of the flank and root geometry. For this
purpose, both the Bézier and NURBS approach are com-
pared based on their mathematical properties. Furthermore,
the results of the representation accuracy, the local stress
for flank and root, and the local load capacity are used
to demonstrate the potential of the alternative NURBS
approach for an improved gear design process.
2 Gear geometry representation in the
tooth contact analysis
There are several simulation tools for the tooth contact anal-
ysis (TCA), which differ in the range of available gear types
and in the degree of complexity and thus in the computa-
tional effort and time required. In the following section, two
additional tools are presented to give a classification of the
BECAL TCA.
RIKOR, developed at the FZG (TU Munich), is a TCA
program for the calculation of spur, helical and double he-
lical gears. It is able to consider tooth flank modifications
but also to determine shaft deformations and bearing de-
flections of the surrounding system and to take these into
account in the tooth contact simulation. The geometry of the
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Forsch Ingenieurwes (2022) 86:273–281 275
cylindrical gears is described purely analytically from the
involute profile according to DIN 3960. Hence, the deter-
mination of the contact distances in the tooth mesh, which
is necessary for the further stiffness and stress calculation,
is also conducted analytically and therefore requires less
computing time. However, by using the theoretical geome-
try from DIN 3960, the actual geometry resulting from the
manufacturing process cannot be taken into account. Fur-
thermore, the geometry of bevel and hypoid gears cannot be
calculated with this analytical approach due to the multiply
curved flank and root geometry [4].
ZaKo3D, developed at the WZL (RWTH Aachen Univer-
sity), is another TCA tool that is comparable to BECAL in
terms of manufacturing simulation and calculation of stiff-
ness and stress, but has a significant difference in the repre-
sentation of flank and root geometry. Since in BECAL the
flank surface is approximated in a manner that the contact
distance between the flanks in the mesh can be determined
analytically, the contact analysis is accurate and very fast
[1,5]. In contrast, ZaKo3D uses an interpolation algorithm
based on Bézier surfaces or a triangulated mesh to achieve
the highest accuracy with respect to the output data from the
manufacturing simulation. Both approaches make it neces-
sary to determine the contact distances with numerical al-
gorithms, such as the Newton scheme. The advantage of
the interpolation lies in the consideration of detailed manu-
facturing-related micro geometric flank deviations, such as
feed marks and profile section deviations. However, this de-
tailed flank description requires a high computational effort
[6,7].
In summary, BECAL offers a compromise between a suf-
ficiently accurate description of the flank geometry (includ-
ing the manufacturing process) and a high computational
speed. For this reason, the surface approximation concept
will be pursued. In order to increase the surface accuracy,
the use of an alternative surface approach with NURBS is
further investigated.
Fig. 2 Parameterization of the
flank geometry (a) and root
geometry (b)
3 Approximation of ank and root geometry
For the approximation of the tooth geometry in BECAL,
both, the flank and the root geometry are represented with
separate surface descriptions by Bézier tensor product sur-
faces. For the purpose of surface approximation, the target
points, e.g. as provided by the manufacturing simulation,
must first be parameterized. In BECAL, the parameteriza-
tion is based on spherical coordinates, as shown in Fig. 2.
Furthermore, the parameters are limited to a range R2Œ0,1
to fit the polynomial surface representation. The approxi-
mation, which determines the Bézier surface coefficients, is
done by solving a linear minimization problem (least square
fit) [1,5].
To describe the flank surface X, the spherical coordi-
nates R and θare used as parameters for the representation
of φaccording to
'=X.R;/ (1)
showninFig.2. This form of surface mapping has a major
advantage: In order to determine the contact distances in
the load-free tooth contact simulation, the tooth is divided
into spherical sections along the face width. Based on this,
the contact line and the contact distances along this line are
calculated. Thus, the representation as defined in Eq. 1has
the benefit that the spherical sections as well as the contact
distances can be determined directly and precisely. This
avoids the use of complex contact determination algorithms
and therefore enables high processing speeds [1,5,8].
The description of the root surface is also based on spher-
ical coordinates. In contrast to the flank, an additional pa-
rameter uis introduced. This parameter is needed to de-
scribe the potentially strongly curved root geometry. The
representation of the root surface is defined as
'
=X.R;u/ (2)
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276 Forsch Ingenieurwes (2022) 86:273–281
Fig. 3 Bézier surface with polynomial degree n=m=4
Following the flank description, spherical sections can
also be calculated with a constant Ralong the uparameter,
which is direct and fast [1].
4 Comparison of the Bézier and NURBS
surface approach
To understand the differences between Bézier and NURBS
surfaces and the resulting advantages of NURBS, their
mathematical description is explained and contrasted, and
possibilities for approximation with these approaches are
shown.
4.1 Bézier surfaces
Bézier surfaces result from the tensor product of two Bézier
curves and are following the equation
X.t;s/=
n
X
i
m
X
j
bij Bin .t/ Bjm.s/: (3)
The Bernstein polynomials Bin(t), Bjm(s) are used as basis
functions. Here n and m define the polynomial degree of
the Bernstein polynomials along the parameters tand s.
This results in Œn +1m+1Bézier points bij, which define
a bidirectional net of control points [9–11]. Fig. 3shows the
Fig. 4 Basis system of Bernstein
polynomials for degree n=4
control net and the resulting Bézier surface for a polynomial
degree n=m=4.
The Bernstein polynomials Bin(t) form a basis system
of linearly independent functions, which are calculated ac-
cording to Eq. 4.
Bin .t/ =n
i.1−t/n−iti;i =0,1; :::; n (4)
Corresponding to the polynomial degree n, the basis sys-
tem is defined with Œn +1functions on the range tin [0,
1]. Fig. 4shows the system of Bernstein polynomials for
degree n=4.
The function values of the Bernstein polynomials Bin(t),
Bjm(s) determine the influence of the corresponding Bézier
point bij along the parameters t and s. Due to the nature
of the Bernstein polynomials, which have a function value
greater than 0 on the range t, s in (0,1), the location of a sin-
gle Bézier point influences the shape of the entire surface.
In addition, the number of Bézier points is directly coupled
to the polynomial degree [9–11].
4.2 Non uniform rational b-spline surfaces (NURBS)
NURBS surfaces are a general form of the rational B-spline
surfaces as described in Eq. 5
X.t;s/=Pn
iPm
jNik .t / Njl .s/ wij dij
Pn
iPm
jNik .t / Njl .s/ wij
(5)
Here the function X(t,s) describes a surface of order k
and lalong the parameters tand s.TheŒn m De Boor
points dij represent the control net over the surface. In ad-
dition, a rational denominator is added to the function of
the surface. Here, the weights wij .> 0/represent addi-
tional design parameters. The basis functions Nik(t)form
a linearly independent basis system of functions analogous
to Bernstein polynomials (acc. Eq. 4). These are piecewise
defined (segmented) polynomials of order k(polynomial de-
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Forsch Ingenieurwes (2022) 86:273–281 277
Fig. 5 Basis functions,
k=4,n=6,=
f0,0;0,0;0.25,0.5,1;1,1;1g
gree k− 1) and their properties are similar to the Bernstein
polynomials’ properties. The basis functions are described
as follows:
k=1 Ni1.t/ =1where tit<t
i+1
0otherwise ;(6)
k>1Nik .t/ =t−ti
ti+k−1 −ti
Ni;k−1
.t/ +ti+k−t
ti+k−ti+1
Ni+1;k−1 .t / :
(7)
Equation 7represents a recursive function. In contrast to
Bernstein polynomials, basis functions are based on a knot
vector τ, with the knots ti. These must be provided in as-
cending order. Its defined as:
=ft0;t
1;t
2; :::; tn+kg;t
iti+1:(8)
Depending on the distribution of the knots in the knot
vector, uniform rational b-splines have equidistantly dis-
tributed knots, while NURBS contain arbitrarily distributed
knots. The knots may occur multiple times in the knot vec-
tor. If the first knot t0= 0 and the last knot t.n+k/ =1
occur k-times in the knot vector, the basis functions are
N0k.0/=Nn−1;k.1/=1(seeFig.5). This is especially
relevant for the description of surfaces, since, in this case
Fig. 6 Basis functions,
k=4,n=6,=
f0,0;0,0;0.25,0.25,1;1,1;1g
the boundary points of the control net thus lie on the sur-
face. If the interior knots occur l-times, the differentiability
of the surface at the point along parameter treduces to
Ck−1−l. Basis functions of order k= 4 with single interior
knots are therefore continuously differentiable at most three
times, which is, for instance relevant for the computation
of curvatures [9,10].
Examples for basis functions are shown in Figs. 5and 6.
The bars shown above the functions indicate the range
Nik.t/ > 0 for the corresponding basis function. The basis
functions in both Figures differ regarding the knot vector.
In summary, compared to the Bernstein polynomials,
the basis functions are defined in segments, which can
be specifically influenced via the knot vector. Transferred
to a rational B-spline surface, this means that the control
points (De Boor points) have only a local influence on the
surface. Furthermore, the surface order or the polynomial
degree is independent of the number of De Boor points, as
long as nk− 1. Additionally, the influence of single con-
trol points can be controlled by the weights wij. Thus, com-
pared to the Bézier representation, NURBS surfaces have
a much larger degree of freedom for modeling surfaces,
which satisfies the requirement of an accurate representa-
tion of complex shapes.
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278 Forsch Ingenieurwes (2022) 86:273–281
Fig. 7 Tooth geometry exam-
ples; tip and root relief (a);
strongly curved root geome-
try (b)
Fig. 8 Comparison of local
deviation and stress results;
hypoid gear with tip and root
relief
4.3 Approximation algorithms
To represent the surfaces with the mentioned approaches,
the corresponding control points and weights have to be
determined. By means of the surface approximation, these
parameters are calculated on the basis of a given set of
points. Hence, it is required to represent the given points
accurately, i.e. to minimize the deviations. There are var-
ious mathematical methods for solving this minimization
problem. The most common method is the least square fit
[10], which is used in BECAL for the surface approxima-
tion with Bézier surfaces [1,5].
When approximating with NURBS surfaces, the weights
wij (acc. Eq. 5) must be determined in addition to the De
Boor points. When using the least square fit method, there
is a possibility that the weights become negative values.
This would mean that the denominator of the approxima-
tion function becomes X.ti;s
i/= 0 for certain parame-
ters and consequently the surface has a singularity at that
point. Therefore, negative weights must be unconditionally
avoided. For this purpose, Elsässer [12] developed a method
with the advantage that no nonlinear optimization problems
occur. The minimization problem follows
F=F+G !min (9)
Here, Fdescribes the distance function in Eq. 10,G
represents the control term as defined in Eq. 11.
F=
N
X
i
X2.ti;s
i/−X.t
i;s
i/Pi2;where X=
PPNik .t / Njl .s/ wij
PPNik .t / Njl .s/ wij dij 
(10)
G=
N
X
i
0
@
n
X
j
m
X
l
wjNik .ti/N
jl .si/−1
1
A
2
(11)
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Forsch Ingenieurwes (2022) 86:273–281 279
Tab le 1 Comparison of stress and load capacity results; hypoid gear with tip and root relief
Description Symbol Unit Bézier NURBS
Pinion Gear Pinion Gear
Max. Hertzian Stress HŒMP a 1722.0 1938.4
Min. Pitting Safety SH;min Œ−1.03 0.87 0.79 0.67
Min. Micro Pitting Safety S;min Œ−2.68 3.05 1.69 2.17
Min. Scuffing Safety SS;min Œ−0.80 0.74
Tab le 2 Comparison of stress and load capacity results; forged
differential gear with a strongly curved root geometry
Description Symbol Unit Bézier NURBS
Max. Root Tensile
Stress
FŒMP a 256.7 252.8
Min. Safety (Tooth
Root Fracture)
SF;min Œ−3.67 3.73
The factor λcontrols the influence of the control term.
It is first estimated in order to determine the weights wij
and to check whether these are negative. If there are nega-
tive weights, λis increased and the weights are determined
again. This process is carried out until positive weights are
obtained.
5 Potential of the NURBS surface approach
in the TCA
Finally, both surface approaches are used in the BECAL
TCA and compared with respect to their influence on the re-
sults of stress, load capacity and acoustic excitation. There-
fore, two different gear types (acc. Fig. 7) are analyzed. The
first example shows a hypoid gear set with a linear tip relief
and a spherical root relief on the drive side of the pinion.
The second example describes a forged differential gear
characterized by a strongly curved tooth root geometry.
The flank deviation and stress results of the first ex-
ample are summarized in Fig. 8. As shown in the upper
left figure, the flank deviations of the Bézier surface range
from –5.45 µm to 3.3 µm with a root mean square value of
1.42 µm. The difference in curvature, especially at the tooth
tip at the transition from the involute profile to the linear tip
relief, causes the Bézier surface to oscillate. This results in
two local maxima of Hertzian stress along the tooth profile.
In contrast, the NURBS surface has an overall deviation
of 2.35 µm, which is strongly concentrated on the transi-
tion line at the tooth tip. In addition, the surface is less
oscillating, resulting in flank deviations of less than 0.1 µm
for the most of the flank. The root mean square value is
0.21 µm. Thus, a more realistic pressure distribution can be
achieved with the NURBS surface, while the maximum of
the Hertzian stress is located at the tip. Table 1lists the
minima and maxima of stress and load capacity.
It is apparent that the maximum Hertzian stress is under-
estimated with the Bézier surface. Accordingly, the safeties
calculated for the flank, especially for pitting and micro
pitting, are too high compared to the results obtained with
the NURBS surface. Contrary to expectations, the tip relief
does not result in lower stress at the tooth tip in this exam-
ple. This can be explained by the high total contact ratio
of ">2.5, which leads to a smooth pressure distribution
along the path of contact and thus to a low transmission
error. Hence, the common effect of tip relief, to reduce the
transmission error, is not given. Instead, it leads to a limi-
tation of the contact pattern and therefore to an increase of
the contact stress. However, the higher accuracy of the ap-
proximation surface allows for more detailed design of the
micro geometry, such as applying higher profile crowning
or using a spherical relief to reduce the stress at the tip.
Another important criterion for the gear design, is the
acoustic excitation. In Fig. 9the excitation spectrum for
the transmission error under load is shown. By comparing
Bézier and NURBS, two different spectra can be obtained.
In particular, the first three harmonics differ significantly in
the level of excitation, resulting in calculated volume levels
of 94.3 dB (Bézier) and 89.5 dB (NURBS).
The results for the second example are shown in Fig. 10
and Table 2. The calculation of the root tensile stress is
performed using FEM influence vectors according to [13].
Similar to the results from the first example, the deviations
at the tooth root can be reduced by a factor of 3.7 by means
of root mean square. Accordingly, Bézier gives deviations
in the range of –78.9 µm to 114.24 µm, while the NURBS
surface only shows deviations in the range of –43.08µm to
24.52 µm. It can be seen that both surfaces are primarily
oscillating at the transition from the toe to the straight mid-
section and from there to the heel. However, unlike for the
flank surface, these deviations have no significant influence
on the tensile stress in the tooth root and thus on the safety
against tooth root fracture.
6 Summary
For the tooth contact simulation in BECAL, it is neces-
sary to transfer the point cloud-based tooth geometry into
a mathematical description. This is performed by an ap-
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280 Forsch Ingenieurwes (2022) 86:273–281
Fig. 9 Comparison of acoustic
results; hypoid gear with tip and
root relief
Fig. 10 Comparison of local
deviation and stress results;
forged differential gear with
a strongly curved root geometry
proximation of the target points, which, in contrast to an
interpolation, enables an efficient and fast TCA. In order to
improve the mapping accuracy of the flank and root geom-
etry, only the use of a higher-order approach compared to
the presently used Bézier tensor product surfaces must be
considered.
For this purpose, NURBS and Bézier surfaces are ana-
lyzed and compared with each other. The comparison con-
cludes that NURBS provide a significantly higher degree
of freedom to describe complex surfaces. In particular, this
results from the independence of polynomial degree and
number of control points as well as the additional weights.
Finally, two different gear geometries are used to demon-
strate that NURBS surfaces can improve the mapping ac-
curacy by a factor greater than 3. In particular, the flank
representation shows a significant influence on the pressure
distribution as well as the results of the flank load capacity.
Since the correct curvature is primarily relevant here, devi-
ations in the range of a few microns already lead to a false
statement regarding the flank load capacity and acoustic ex-
citation. In comparison, deviations of the geometry in the
tooth root are found to be less relevant for the calculation
of the tooth root load capacity.
Funding The author would like to thank the Forschungsvereinigung
Antriebstechnik e. V. (FVA) for funding the research project on which
this paper is based on. (FVA 223 XIX BECAL – Auslgeichsflächen-
Studie)
Funding Open Access funding enabled and organized by Projekt
DEAL.
Open Access This article is licensed under a Creative Commons At-
tribution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, pro-
vide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included
in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
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