Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear
The approach provides simulation of the gear design with high-precision profile configuration parameters to accurately predict its load capacity and stress state, which requires incessant efforts, as gear failure modes depend largely on its profile representation. The finite element analysis (FEA) i...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2015
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| Цитувати: | Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear / O.T. Yusuf, Z. Gang, W. Wei, S.O. Onuh // Проблемы прочности. — 2015. — № 1. — С. 26-36. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-1732602020-11-29T01:26:43Z Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear Yusuf, O.T. Gang, Z. Wei, W. Onuh, S.O. Научно-технический раздел The approach provides simulation of the gear design with high-precision profile configuration parameters to accurately predict its load capacity and stress state, which requires incessant efforts, as gear failure modes depend largely on its profile representation. The finite element analysis (FEA) is a well-known effective computational method, which many authors used to estimate the gear tooth strength against the primary failure modes of gears. This method is not only known to be time-consuming but often expensive, it employs approximate geometry that resulted in a coarse mesh. Данный подход позволяет моделировать конструкцию зубчатой передачи с высокоточными параметрами геометрии профиля для достоверного прогнозирования допускаемой нагрузки и напряженного состояния, что требует постоянных исследований, поскольку вид разрушения зубчатой передачи в значительной мере зависит от способа задания ее профиля. Конечноэлементный анализ хорошо известный эффективный вычислительный метод, который многие авторы использовали для оценки прочности зубьев зубчатого колеса в условиях независимых отказов зубчатых передач. Известно также, что этот метод не только трудоемкий, но и зачастую дорогостоящий, использует приближенную геометрию, что дает в результате неточную сеть. 2015 Article Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear / O.T. Yusuf, Z. Gang, W. Wei, S.O. Onuh // Проблемы прочности. — 2015. — № 1. — С. 26-36. — Бібліогр.: 27 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173260 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Yusuf, O.T. Gang, Z. Wei, W. Onuh, S.O. Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear Проблемы прочности |
| description |
The approach provides simulation of the gear design with high-precision profile configuration parameters to accurately predict its load capacity and stress state, which requires incessant efforts, as gear failure modes depend largely on its profile representation. The finite element analysis (FEA) is a well-known effective computational method, which many authors used to estimate the gear tooth strength against the primary failure modes of gears. This method is not only known to be time-consuming but often expensive, it employs approximate geometry that resulted in a coarse mesh. |
| format |
Article |
| author |
Yusuf, O.T. Gang, Z. Wei, W. Onuh, S.O. |
| author_facet |
Yusuf, O.T. Gang, Z. Wei, W. Onuh, S.O. |
| author_sort |
Yusuf, O.T. |
| title |
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear |
| title_short |
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear |
| title_full |
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear |
| title_fullStr |
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear |
| title_full_unstemmed |
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear |
| title_sort |
simulation based on trivariate nurbs and isogeometric analysis of a spur gear |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2015 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173260 |
| citation_txt |
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur Gear / O.T. Yusuf, Z. Gang, W. Wei, S.O. Onuh // Проблемы прочности. — 2015. — № 1. — С. 26-36. — Бібліогр.: 27 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT yusufot simulationbasedontrivariatenurbsandisogeometricanalysisofaspurgear AT gangz simulationbasedontrivariatenurbsandisogeometricanalysisofaspurgear AT weiw simulationbasedontrivariatenurbsandisogeometricanalysisofaspurgear AT onuhso simulationbasedontrivariatenurbsandisogeometricanalysisofaspurgear |
| first_indexed |
2025-07-15T09:51:00Z |
| last_indexed |
2025-07-15T09:51:00Z |
| _version_ |
1837706057648963584 |
| fulltext |
UDC 539.4
Simulation Based on Trivariate NURBS and Isogeometric Analysis of a Spur
Gear
O. T. Yusuf,
a
Z. Gang,
a
W. Wei,
a
and S. O. Onuh
b
a Mechanical Engineering and Automation, BUAA, Beijing, China
b Center for Satellite Technology Development, NASRDA, Abuja, Nigeria
The approach provides simulation of the gear design with high-precision profile configuration
parameters to accurately predict its load capacity and stress state, which requires incessant efforts,
as gear failure modes depend largely on its profile representation. The finite element analysis (FEA)
is a well-known effective computational method, which many authors used to estimate the gear tooth
strength against the primary failure modes of gears. This method is not only known to be
time-consuming but often expensive, it employs approximate geometry that resulted in a coarse mesh.
The tooth strength and deformation modes were obtained from a three-dimensional gear model based
on the trivariate nonuniform rational B-splines (NURBS) under loading and investigated using the
isogeometric analysis (IGA) for simulation. The results produced with the IGA simulation are
compared with the corresponding values derived from the classical estimation method with the Lewis
equation and FEM analysis using ANSYS. Both results are in good agreement and confirm that the
IGA model is more accurate than the FEM one. This study demonstrated that IGA was practically
applicable, providing an alternative means of using high-precision geometry of 3D models to
estimate the behavior of gear bodies under loading and modify their strength calculations.
Keywords: three-dimensional model, carbon steel, spur gear, gear profile, trivariate NURBS,
finite element method, isogeometric analysis.
Introduction. Gears are used in various types of machinery in many engineering
areas, including automobiles, marines, aeronautics and astronautics, where extreme operating
speeds of rotating members increase the load input on machinery, thereby demanding high
strength gearing component to achieve positive power transmission. In operation, the teeth
deform elastically under the induced radial and tangential components of pressure load via
the contact points on the mating teeth profiles (as shown in Fig. 1) in the direction of the
pressure line.
© O. T. YUSUF, Z. GANG, W. WEI, S. O. ONUH, 2015
26 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
Fig. 1. Loaded solid spur gear contact teeth at the pitch point (diamonds) and transfer along the
pressure line (arrows).
Fatigue failure of gear tooth occur, as a result of high stress concentration at the root
of a gear tooth subjected to bending loads. To achieve high-strength gearing with the
highest efficiency power transmission requires a design minimizing this bending stress.
Ristiã [1] investigated the optimal fillet tooth root radius to minimize the root stress
intensity in order to create a valid numerical model for a tooth root geometrical discontinuity
at static loading using finite element method (FEM). As shown in study [2], modification of
root fillet profile using a circular root fillet instead of the standard trochoidal one can
improve the tooth strength in a spur gear. These investigations bordered on accuracy and
geometrical modification of gear profile that can be easily achieved using the volumetric
polynomial functions, such as tensor product NURBS [3], which allows geometries to be
represented exactly with a smooth surface that can be directly used in the IGA [4]. IGA is a
way of invoking the isoparametric concept of representing the field variables with the same
smooth NURBS basis functions for numerical computation. Moreover, IGA allows for an
easy model modification. With NURBS model, one can exclude geometric discontinuities
induced by the conventional geometry discretization (which causes jumps in the contact
surfaces unless finest meshes are used), as well as avoid the time- and memory-consuming
mesh generation operation for the transformation of CAD model into a finite element (FE)
mesh before it can be used in the analysis.
Both theoretical calculation method via the Lewis equation [5], and finite element
method have been widely used in the field of gear strength numerical calculation. The
solution of load distribution in the meshed gear tooth pairs with numerical FEM [6–8]
resulted in many important conclusions on the spur gear stress distrubution.
IGA has been successfully applied in various fields for solving PDEs of domains
formed by smooth basis functions of NURBS. Such fields include structural mechanics and
optimization [9, 10], solid mechanics [11, 12], fluid mechanics [13], fracture mechanics
[14, 15], biomechanics [16], and contact mechanics [17–19]. However, this approach has
not been fully implemented into commercialized sofware packages due to trimless nature of
NURBS for representing complex NURBS domains.
Analysis of gear stress using NURBS-based isogeometric analysis is recently presented
for a two-dimensional case [20], but [21] shows that 2D representation method cannot
embrace the actual stress field in a spur tooth fillet. We generalize this idea to create an
accurate 3D conforming multi-patch gear tooth model based on NURBS and investigate the
stress-strain states and deflection of the gear under the influence of loads at the points of
contact using the isogeometric analysis. This provides more insight into the load distribution
along the tooth face width of gear, as well as stress distribution in the gear tooth root. The
numerical results fobtained by IGA are compared to those derived from the available
empirical and finite element analyses from ANSYS workbench 14.0 and are found to be in
a good agreement.
1. Basic Characteristics of the Analyzed Spur Gear Model.
1.1. Gear Geometrical Data. The gear geometrical parameters used in this study are
shown in Table 1. These parameters are used to develop a pair model of solid spur gears in
contact (as shown in Fig. 1) to illustrate the active gear contact zone, where the maximum
torque is transferred along the line of action between the mating gears.
1.2. Materials of Spur Gear. In the manufacturing of spur gear, wide varieties of
materials are used, including steels, nylon, aluminum, bronze, cast iron, phenolic, bakelite,
and plastics. In this study, a steel gear made from 20% carbon steel with no subsequent
treatment with the properties shown in Table 2 is used.
1.3. Trivariate NURBS. NURBS solids S ( , , )� � � have a topological structural form
with grids of control points P Rijk
w � 4 with associated weights { }wijk ijk
nml
�1 that provide
the basic information on their geometry, such as position, size, and aspect ratios of the
isogeometric mesh. The node vectors in parametric directions �, �, and �, partition the
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 27
Simulation Based on Trivariate NURBS and Isogeometric Analysis ...
parametric domain of the basis functions N i p, ( ),� M j q, ( ),� and Lk r, ( )� into piecewise
rational polynomial functions Rijk
pqr
( , , ).� � �
The result of mapping of the parametric domain �� into the physical domain � is
given as S : �� �� with
S N M L Pw
i p
kj
m
i
n
j q k r ijk( , , ) ( ) ( ) ( ), , ,� � � � � ��
���
1
1
11
w �
�
���
w N M L Pijk
k
i p j q k r ijk
j
m
i
n
1
1
11
, , ,( ) ( ) ( )� � � (1)
in the perspective division of homogenous coordinates:
W N M L wi p j q k r ijk
kj
m
i
n
( , , ) ( ) ( ) ( ) ., , ,� � � � � ��
���
1
1
11
(2)
T a b l e 1
Spur Gear Geometrical Data
Description Symbol Formula Values
Number of teeth z – 12
Pressure angle (deg)
– 14.5
Diametral pitch Pd – 4
Addendum (mm) ha h pa � 0 3183. 6.35
Base circle diameter (mm) Db D db � cos
73.8
Circular pitch (mm) p �m 19.95
Circular tooth thickness (mm) ctt p 2 9.97
Dedendum (mm) hf 125. m 7.94
Module (mm) m d z 6.35
Outside diameter (mm) D0 m z( )� 2 88.9
Pitch circle diameter (mm) d zm 76.2
Face width (mm) F – 50.8
Whole depth (mm) h 2 25. m 14.28
Top land width (mm) t0 0 25. m 1.59
T a b l e 2
Properties of Material
Properties Gear Properties Gear
Material Carbon steel Poisson’s ratio 0.3
Elastic modulus (GPa) 210 Density (kg/m3) 7850
Yield strength (MPa) 758
28 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
O. T. Yusuf, Z. Gang, W. Wei, and S. O. Onuh
NURBS solid � R 3 can be realized using
S
W
N M L w Pi p j q k r ijk ijk
k
( , , )
( , , )
( ) ( ) ( ), , ,� � �
� � �
� � ��
�
1
1
1
11
��
�
j
m
i
n
�
���
R Pijk
pqr
kj
m
i
n
ijk
1
1
11
( , , ) .� � � (3)
Figure 2 shows an example of a quadratic model of trivariate NURBS constructed from
uniform open node vectors �� { , , , , , },0 0 0 1 1 1 �� { , , , , , },0 0 0 1 1 1 and � � { , , , , , }0 0 0 1 1 1.
Its associated control grid, as shown in Fig. 2b, is constructed from piecewise interpolation
of 27 control points. Figure 2c shows the result of the nodal spans for a hexahedron
element achieved by mapping of a parametric domain into a physical one via Eqs. (1)–(3).
2. Modeling of NURBS Based Spur Gear. The specified primary parameters in the
previous section serve as input to define the topological part of solids NURBS S ( , , )� � �
geometry, in which the gear coordinate data (control points) { } ,Pijk ijk
nml
�1 associated with
weights { } ,wijk ijk
nml
�1 are embedded into four-dimensional homogeneous coordinates in the
projective space R 4. To obtain the spur gear generation, each tooth of the gear of twelve
teeth, in view of their symmetry, comprises of four conformal NURBS patches with three
coupling interfaces defined by the product of node vectors of non-decreasing positive real
numbers in �, �, and � parametric directions given as �� { , , , , , },0 0 0 1 1 1 ��
� { , , , , , },0 0 0 1 1 1 and � � { , , , , , },0 0 0 1 1 1 from which the basis functions
{ ( , , }N ijk
pqr
ijk
mnl� � � �1 of degree p� 2, q� 2, and r� 2 are defined. The association of the
Fig. 2. Parametric domain with basis functions (a); physical domain (b); trivariate NURBS model (c).
c
a
b
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 29
Simulation Based on Trivariate NURBS and Isogeometric Analysis ...
control points with the basis functions defines one patch of the tooth solid NURBS using
Eqs. (1)–(3), this procedure is repeated for other three patches to obtain one-twelve of a
single tooth sector spur gear shown in Fig. 3 with the four patches showing in different
colors.
2.1. Theoretical Determination of Gear Load Capacity and Tooth Bending Stress.
The spur gear strength is expressed in terms of the bending strength that can be estimated
with the Lewis equation [5], which is related to the power and forces to be transmitted via
the line of action between the pinion and gear. As illustrated in Fig. 4, the maximum
transmission force F, which is between the contacting teeth at pressure angle of
to pitch
line tangent, can be resolved into a tangential component Ft and a separating component
Fn . We first compute the fundamental equation relating the tangential force Ft at the pitch
circle, power P and torque T to have:
F
P
v
P
d n
T
d
t
w w
� �
�
102 195 10 20006.
, (4)
P
F v
F d n
t
t w� �
�
102
10
195
6
.
, (5)
T
F d P
n
t w� �
2000
974
, (6)
F Fn t� tan ,
(7)
F
Ft�
cos
,
(8)
v
d nw�
19100
, (9)
Fig. 3 Fig. 4
Fig. 3. Geometrical NURBS model of a single-tooth solid gear.
Fig. 4. Direction of forces acting on the mating gear teeth.
30 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
O. T. Yusuf, Z. Gang, W. Wei, and S. O. Onuh
where v is the tangential speed of working pitch circle (ms�1), dw is working pitch
diameter, dw � 76.2 mm, n is rotating speed, n� 1000 rpm,
v
d nw� �
� �
19100
762 1000
19100
399 1.
. ,ms
T
F d P
n
t w� � �
�
�
2000
974 974 5
1000
4 87 4776. . ,kgf m N m
F
P
v
P
d n
T
d
t
w w
� �
� �
�
102 195 10 2000 2000 4 87
762
12782
6. .
.
. kgf N� 12535. ,
F Fn t� �
�tan . . . .
12535 02586 32416 N
The Barth velocity factor K v increases the Lewis stress approximately by
K
V
v �
�61
61
.
.
, (10)
where V is the velocity at the pitch diameter (in m/s),
K v �
�
�
61 3989
61
165
. .
.
. .
The combined expression for tooth bending stress � b is then
� b
v t
a
K F
b mY
� , (11)
where ba is face width (mm), m is module (mm), and Y is the Lewis form factor,
� b �
�
165 12535
508 635 0355
1806
. .
. . .
. MPa, � �theor b� � 1806. MPa.
2.2. Construction of a Gear Tooth Fillet. The gear root fillet radius (rf ) is the critical
cross section of a gear tooth root as it is exposed to bending and combined action of the
radial and tangential components of pressure load, which has a direct influence on the
stress concentration factor and thus affects the gear service life. The fillet does not have a
constant radius, but is shaped as a curve generated by a hobbing tool or cutter [22] with a
tip radius (rt ) for 14 5. � pressure angle which is standardized as
r
P
t
d
�
0235.
, (12)
where Pd is diametral pitch, Pd � 4, r Pt d� � �0235 0235 4. . 0.05875 in � 1.49 mm.
The radius of a gear tooth fillet produced by the tool of radius rt is denoted by
r
h r
Z P h r
rf
f t
d f t
t�
�
� �
�
( )
( ) ( )
,
2
2
(13)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 31
Simulation Based on Trivariate NURBS and Isogeometric Analysis ...
rf �
�
� �
� �
( . . )
. .
. .
794 149
12 8 794 149
149 672
2
mm.
3. Finite Element Analysis Using ANSYS.
3.1. Models and Mesh. Due to the symmetry of the gear structure and loading, only
one-twelfth of the spur gear was modeled using Rhinoceros V5 software as shown in Fig.
5. In its IGES format, it was exported to the commercial finite element package ANSYS
workbench 14.0, which was used for a linear static finite element analysis. Three-dimensional
isoperimetric hexahedron solid elements defined by eight nodes were used for generating
the FEA mesh. For improvement of the calculation accuracy, a fine mesh of hexahedron
solid elements is used consisting of 19,737 elements and 86,722 nodes.
3.2. Boundary and Loading Conditions. The boundary conditions implies the
application of displacement constraints and external loads on the 3D FEM mesh. All nodes
in the symmetric sections are constrained against displacement in the direction perpendicular
to the symmetry plane, i.e., along the surfaces that separate the modeled gear segment from
the rest of the gear body. Displacement constraints of fixed support are also placed at the
nodes on gear tooth of the inner bore border. The FEA models were subjected to the
combined load by the tangential force of 1253.5 N and the normal force of 324.16 N at the
pressure angle to pitch line tangent. Figure 6 illustrates the finite element mesh boundary
and loading conditions for the single-tooth model.
3.3. Material Properties. The material parameters applied in the model include the
Young modulus and Poisson’s ratio for the carbon steel as specified in Table 2.
4. Isogeometric Analysis of Spur Gear. The IGA simulation, which is an alternative
to the traditional finite elements, is based the geometric input model composed by four
NURBS patches as shown in Fig. 7 (level 4) with three coupling interfaces. The model is
discretized using the tricubic functions (for patch 1, patch 2, patch 3, and patch 4), each
consisting of 16 16 16� � elements. This results in the total number of 16,384 elements and
19,652 control points.
4.1. Mesh Definition and Model Refinement. The physical domain � � R 3 of the
spur gear tooth is represented by four single-patch NURBS mappings S i( ) for i� 1 to 4,
each of which maps the parameter domain �� into the patch � �( ) ( ) ( � ).i iS� The
superscript i represents the node vectors, degrees, NURBS basis functions, control points
associated with the mapping S i( ) of the individual patch.
The mesh M i( ) is directly defined by the NURBS mapping process of Eqs. (1)–(3),
which is quasi-uniform for all i� 1, 2, 3, 4. The domain is subdivided into subdomains
Fig. 6. 3D FEM boundary and loading conditions.Fig. 5. 3D gear tooth CAD model.
32 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
O. T. Yusuf, Z. Gang, W. Wei, and S. O. Onuh
� ��e
e by hexahedral elements, such that � � �i j m� � � � for i j m� � . The
elements are determined by the nodal spans: � e
i i j j m m� � �� � �[ , ] [ , ] [ , ],� � � � � �1 1 1
where hi denotes the element size of M i( ) . This yields a family of meshes { }
( )
M h
i
h
i i
, such
that there exists the most coarse mesh M h
i
i0
( )
that does not provide a sufficiently smooth
contour plot, which could be directly used to produce accurate results of the analysis. Four
levels of this mesh refinement are executed (as shown in Fig. 7) to have a higher resolution
mesh. The latter is suitable to obtain accurate results, and the description of the geometry is
fixed at the level of M h
i
i0
( )
. When the mesh is refined, the control points are adjusted in a
such way that the originality of geometrical shape of the model S i( ) remains unchanged.
Thus, the geometry and its parameterization are held fixed in the refinement process [23].
The meshing operation in FEA is thereby replaced with the model refinement by
simple insertion of nodes into IGA, which results in additional degrees of freedom.
4.2. Performing the IGA Analysis. The boundary conditions for the IGA analysis are
similar to those for the FEM analysis (Fig. 6). The loads and material properties for the
numerical simulation using IGA are identical to those defined in Subsections 3.2 and 3.3.
An algorithm in matlab codes is developed, adopting some built-in functions and
structures within the GeoPDEs, IGA, FEM, and Abaqus NURBS libraries [24–26] to
construct the geometry based on the gear model data input, which includes the control
points, node vectors, orders, the boundaries and interfaces of the NURBS patches needed to
perform the analysis. The topological entities in the model were selected, where force and
displacement constraints were set, and performed the analysis using the Abaqus NURBS
plugins [26].
5. Results and Discussion. In accordance with theoretical investigations given in [22]
and FEM [7, 8], the expected maximum concentrated stress occurring at the fillet root at the
base of the gear tooth which is equal to tensile stress that causes crack failure shown in
Fig. 8 illustrates distribution of the Mises stresses in the tooth root. Figure 9 shows the
corresponding tooth deflections calculated for a single-tooth 3D spur gear geometry based
on the NURBS four patches and on the 3D FEM model. The compared values are the
equivalent Mises stresses and the total deformation.
Fig. 7. Four-level refinement of the IGA meshes generated by insertion of nodes.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 33
Simulation Based on Trivariate NURBS and Isogeometric Analysis ...
Table 3 compares the calculated results of the tooth root stresses and deflections
obtained: the bending stress results of the theoretical calculation via the Lewis equation,
finite element analysis, and isogeometric analysis are in good agreement. The difference
between the theoretical calculation results with those obtain via FEA and IGA are 6.47 and
2.16%, respectively. The IGA approximates the theoretical calculation better than FEA.
Moreover, the IGA makes it possible to use the h-refinement method, in order to
refine a coarse mesh of the gear geometric model, obtain additional degrees of freedom in
the model, and produce the result of the required quality without changing the overall
geometrical shape. This allows one to avoid the time- and cost-consuming mesh generation
process in FEA, insofar as FEA requires additional meshing efforts to discretize the 3D gear
tooth CAD model into fine elements for the FE analysis.
Updating the NURBS model for IGA by simulation results, in order to realize the
best fillet radius to optimize the stress concentration, seems to be a quite easy task, since
the geometry representation and the simulation results cover the same spline domain.
However, many researchers have problems with application of the CAD-model for similar
modification due to the discrepancies between CAD and FEM meshes in the FEA approach.
T a b l e 3
Comparison of the Maximum Mises Stresses and the Total Deformed Shape
Obtained from IGA and FEA
�theor ,
MPa
�FEA ,
MPa
�IGA ,
MPa
Deformation
in FEA (mm)
Deformation
in IGA (mm)
Difference (%)
(� �theor FEA� )
Difference (%)
(� �theor IGA� )
18.06 16.89 18.45 0.0018 0.0020 6.47 2.16
Fig. 8 Fig. 9
Fig. 8. The Mises stress distributions in the tooth gear (MPa): (a) FEA model; (b) IGA model.
Fig. 9. Total deformed shape of the tooth gear (mm): (a) FEA model; (b) IGA model.
b b
a a
34 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
O. T. Yusuf, Z. Gang, W. Wei, and S. O. Onuh
Conclusions. The developed application of 3D NURBS-based high-precision model
for isogeometric analysis of the stress-strain state provides not only an excellent insight into
the stress distribution, but also reveals the exact position of the maximum stress value point
in the gear tooth root.
The results obtained by the IGA show a good agreement with the well-known FEA
method, which makes the IGA an alternative numerical method to compute spur gear
stress-strain behavior under various loading conditions.
This study is mainly focused on application of the NURBS basis functions for the 3D
geometric simulation of a spur gear and isogeometric analysis of stresses and deflections in
the gear tooth. Further development of this tudy will include the use of other smooth
parametric functions, such as T-splines [27], for the gear model geometric description and
analysis. Another lucrative direction is the IGA extension to contact stress analysis.
Acknowledgments. The authors acknowledge the support by National Nature Science
Foundation of China (Project Numbers 61170198 and 51305106) and the Fundamental
Research Funds for the Central Universities in China.
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Received 20. 10. 2014
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