Original Article
A numerical investigation on the hybrid
spur gears: Stress and dynamic analysis
Proc IMechE Part C:
J Mechanical Engineering Science
0(0) 1–16
! IMechE 2020
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DOI: 10.1177/0954406220982007
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Tufan G Yılmaz1 , O
guz Do
gan2 and Fatih Karpat1
Abstract
In this study, the effect of rim thickness of hybrid gears on the root stress, joint stress, tooth stiffness, natural frequency,
and dynamic behavior are examined numerically. Hybrid gears consist of two materials, which are steel for the teeth-rim
and hub regions of gear, carbon fiber reinforced plastic (CFRP) for the web region. Adhesive bonding is assumed for the
joining of steel and composite materials. FE method is used to evaluate tooth root stress, joint stress, tooth deformation, and the natural frequency of hybrid gears. The adhesive is defined by cohesive zone modeling (CZM). Moreover,
2-DOF dynamic analyses are implemented to obtain dynamic factors and static transmission error. According to results,
hybrid gears have substantial potential to reduce the mass of gear transmission systems with no adverse effect on root
stress and dynamic factor if the design parameters are appropriately selected. Besides, rim thickness is found as a critical
parameter for the hybrid gears since when its value changes from 0.5xm to 3xm, the root stress decreases 10% while the
tooth stiffness and torque capacity increase 20% and 65%, respectively.
Keywords
Hybrid spur gears, finite element stress analysis, cohesive zone modeling, dynamic analysis
Date received: 18 August 2020; accepted: 25 November 2020
Introduction
According to an international report, the energy
demand will grow by 56% up to 2040 due to the
increasing population and, consequently, the
number of vehicles.1 Based on this critical point,
researchers have focused on different ways to
decrease fuel usage and CO2 emission rates. Among
these ways, reducing the structural mass is a highly
preferred way since the weight of the vehicle is
responsible for 75% of the total fuel consumption.2
The studies reported that 6-8% of fuel savings could
be ensured a 10% reduction from the weight of the
vehicle.3 Replacing the materials of components with
a higher specific strength (have a high strength to
density ratio) is the most convenient way to reduce
total weight. This material shifting can be totally or
partially for a component. As a total change instance,
the aluminum alloys that a type of low-density materials can be used in the manufacturing of body panels
or gearbox casings instead of steel since these components are subjected to low external forces/stresses.4
Regarding power transmission machine elements,
the high strength steel alloys are still the best option
with their mechanical, physical, and manufacturability properties. However, partial material change can
be possible for this type of component provided that a
joining method that ensures the structural integrity of
the gear under the loading conditions is found.5
Involute spur gears are decent instances for this
type of machine elements. During the running time,
involute spur gears are subjected to Hertzian stress on
the contact region while the tensile and compressive
stresses on the root region of the tooth.6 Except for
these areas, the stress values are quite low, as illustrated in Figure 1.
Based on this situation, different materials can be
used for high and low-stress regions in the designing
of the spur gear. Carbon fiber reinforced plastics
(CFRP) and aluminum alloys are the candidate lightweight materials for the low-stress region with their
adequate mechanical properties. Stiffness is another
significant parameter for spur gears as it affects the
static transmission errors, which are the primary
source of gear whine and noise. CFRP materials are
1
Department of Mechanical Engineering, Bursa Uludag University,
Bursa, Turkey
2
Department of Mechanical Engineering, Kahramanmaraş Sütçü _Imam
University, Kahramanmaraş, Turkey
Corresponding author:
Fatih Karpat, Department of Mechanical Engineering, Bursa Uludag
University, Bursa, Turkey.
Email: karpat@uludag.edu.tr
2
Figure 1. Stress regions of a spur gear.
superior to aluminum alloys with their higher stiffness
as well as lower density and well damping characteristics. CFRP materials have been used in aerospace,
automotive, and several industries as the material of
structural components with its tailorable and customizable mechanical properties for decades.7,8 They are
preferred to reduce noise and vibration in machine
elements as well.9,10 Recently, CFRP materials have
found application areas in involute gears as a hybrid
design material with steel alloys. However, there are
still very few studies on hybrid gears. Initial important studies were initiated within NASA Glenn
Research Center. Handschuh et al. generated and
tested a prototype lightweight hybrid gear. Adhesive
bonding was used to join steel and composite. They
conducted numerical and experimental studies to
obtain free vibration modes of full steel and hybrid
gear for comparison aim. Besides, dynamics tests
were conducted for several torque and rpm values
to measure vibration and sound level for different
driven-driving gear configurations. An endurance
test was carried out at the level of 10,000 rpm and
300 Nm torque for 300000 cycles. According to the
results, there was no visible fatigue damage. The
hybrid gear has lower natural frequency values than
the full steel gear. In addition to a 20% weight reduction, low vibration values were obtained at only high
load and speed values for hybrid driving hybrid configuration.11,12 Based on these studies, It is thought
that if the rim thickness is selected properly, the stress
and fatigue damage will not be an important problem
for hybrid gears. LaBerge et al. investigated the
hybrid gear performance under no lubrication condition. They stated that there is a potential risk factor
due to the excessive temperature increase, which
results in damage interface of steel and composite.
With this increase, plastic flow occurred, and black
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
debris found on the gear teeth.13 Catera et al. proposed a modeling method for composite material to
obtain natural frequencies of hybrid gear by finite
element analysis (FEA). The method was validated
with previous experimental values from the literature.14 Catera et al. conducted a non-linear FEA
study to obtain stress values on gear teeth and adhesive for a special designed hybrid gear under the
heavy loading condition. Besides, mesh stiffness and
STE are determined for ply to ply and homogeneous
composite representations. According to the results,
the obtained values of these two approaches are very
close. The proposed hybrid gear design ensures a
lower peak to the peak value of the STE when comparing lightweight steel gear for the same mass.15 In
this study differently from Catera’s research, the rim
thickness effect on root and adhesive stress is taken
into consideration. Contartese et al. proposed a novel
quick method to specify the effect of steel teeth and
composite body on the mesh stiffness of hybrid gears
by FEA. The spring elements were used instead of
gear teeth and body. The results show a good agreement with the full gear model.16 Catera et al. conducted a comparative numerical and experimental
study to observe the effect of two different joining
methods on the frequency response of hybrid gears.
According to results, adhesive bonding has better
damping properties, while tooth stiffness is higher in
the interference fitting method. The numerical results
were verified with experimental ones.17 Karpat et al.
compared the bimetallic gear and hybrid gears in
terms of root stress, mesh stiffness, and weight for
same ring rim thicknesses.18 Gauntt and Campell
investigated the effect of composite type and lay up
sequence on the natural frequency and mode shapes
of hybrid gear. According to results, the natural frequency is highly dependent on elasticity modulus.19
Gauntt et al. conducted an optimization study to
decrease weight and increase the joint strength of
hybrid spur gears with a special algorithm. They proposed a sinusoidal geometry for the composite part as
it ensures the highest shear strength for adhesive
bonding.20 Kim et al. tried to find the optimum
radial distance of the composite part between steel
hub and steel teeth in terms of vibration and sound.
FEM and BEM are used to force reaction and modal
analysis. According to results, the vibration levels
reduce 11.9-33.1% while noise levels reduce 4.643.2% when it is compared to full steel gear between
range 2000 to 3000 rpm.21 These studies are mostly
focused on natural frequency, vibration, and sound
situation of hybrid gears. There are no studies about
the effect of gear parameters on the performance of
the hybrid gear. Among these gear parameters, the
rim thickness stands out as the most significant
parameter for weight reduction. For this reason, the
root stress, joint stress, tooth deflection, and natural
frequency are determined with FEA for different rim
thicknesses in this study. In addition, torque capacity
Yılmaz et al.
3
Figure 2. 3D model of hybrid gear for FEA.
of hybrid gears is specified for different rim thicknesses. Tooth stiffnesses and weights are obtained to
calculate mesh stiffness and rotational inertia. Based
on these calculated values, the 2-DOF dynamic analyses are conducted to find dynamic factors and STE.
Evaluation of tooth root and joint region
stress of hybrid gears
During the torque transmission, involute spur gears
are subjected to tooth root stress and tooth deformation. Tooth root stress in the tensile side of the tooth
is one of the primary and most significant reasons for
the tooth root crack.22 A 30% reduction in gear
stresses can lead to a 46% increase in the crack initiation cycle.23 For this reason, tensile root stress
should be examined in hybrid gears, as in standard
gears with a single material. There are a few analytical
methods to calculate the root stress of gear tooth,
such as AGMA, DIN3990.24 However, adequate,
accurate results are not obtained when they applied
on non-standard gears such as asymmetric, asymmetric trochoid, or bimetallic gears. In literature, the
finite element method (FEM) is generally preferred
as a robust tool for stress analysis of this kind of
spur gears.25–28 Based on these points, FEM is used
with ANSYS 16.0 to obtain root tensile stress and
joint stress of hybrid spur gears. 3 D Finite element
hybrid gear model is designed in CATIA V5 with the
points of gear tooth, which are imported from a generated MATLAB code based on Litvin’s
approach.29,30 FE model used in the study is presented in Figure 2. The design parameters of the FE
model are illustrated in Table 1.
Stress analyses (for root and joint regions) are conducted in the same FE model. Steel teeth and hub
parts are defined as isotropic material. Its elasticity
modulus and Poisson’s ratio is selected as
E ¼ 210 GPa and t ¼ 0.3, respectively. As to CFRP
for the web region, it is assigned as a homogenous
orthotropic material. It is assumed that the CFRP
material consists of the joining of 12 uni-directional
(UD) laminas with 2 mm thicknesses. The angle (Ø)
Table 1. Gear properties.
Design parameters
Values
Module m (mm)
Number of teeth z
Pressure angle an
Addendum ha (mm)
Dedendum hf (mm)
Tip radius of cutter pfp (mm)
Profile shifting x
Facewidth b (mm)
Rim thickness (xm)
Hub thickness (xm)
Shaft hole diameter (mm)
Gear ratio i
Teeth-rim and hub material
Web material
3
20
20
1m
1.25 m
0.47 m
0
24
0.5-1-1.5-2-2.5-3-Solid
1
10
1
16MnCr5
CFRP
between consecutive lamina is determined as 30 in
the following symmetric lay up: [0/30/60/90/-60/30]s. The mechanical constants of UD laminas15 are
presented in Table 2.
Analytical equations are used to obtain the
homogenous orthotropic material properties (as demonstrated in Appendix A). Based on this analytical
procedure,31 which is programmed in MATLAB,
the CFRP material constants, which are illustrated
in Table 3, are calculated.
The detail views of composite laminas with the
fiber angles are presented in Figure 3.
Adhesive bonding is assumed for the joining
method of steel parts and CFRP. To evaluate the
joint stress of hybrid gear, the contact region between
steel teeth-rim and CFRP is defined with cohesive
zone modeling (CZM). CZM can predict interface
damage for adhesive bonding with different materials
without an initial defect.32,33 It is characterized by a
cohesive law based on tractions-displacements on the
normal (Mode I) or tangential (Mode II) directions in
the contact region. If these tractions-displacements
occur in both directions, then it is called mixed
mode. Detail view of cohesive law for mixed-mode
presented in Figure 4.33
4
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Table 2. Material constants of UD lamina.
E2
E3
E1(GPa) (GPa) (GPa) t1
230.6
6.23
6.23
t2
t3
G12
G13
G23
(GPa) (GPa) (GPa)
0.38 0.31 0.38 3.29
3.29
2.38
Table 3. Material constants of CFRP laminate.
E1
E2
E3
(GPa) (GPa) (GPa) t1
81.65 81.65 6.81
t2
t3
G12
G13
G23
(GPa) (GPa) (GPa)
0.32 0.27 0.27 30.76 2.76
2.76
Figure 4. Cohesive law for mixed-mode.
Table 4. Properties of cohesive zone material.
Figure 3. Detail views of composite laminas: (a) Left view.
(b) Front view.
According to Figure 3, the stress (rn) has a linear
relationship with displacement until dns for the adhesive. Until this value, this region is accepted as safe,
and there is no debonding. After this value, damage
begins, and full separation occurs when displacement
reaches the value of dnc. The same is true in Mode II.
As to mixed-mode, it is a combination of Mode I and
II. Details of the debonding equations can be found in
a previous study.33 It is well known that the transmitted load during the meshing is composed of two components, which are tangential and radial load in the
spur gears. The tangential load is responsible for
tensile-compression and shear stress in the tooth
root and joint region, while radial force is the
reason for only the compressive stress. Based on
these facts, the mixed debonding mode, which is a
combination of Mode I (Tensile-Compressive) and
Mode II (Shear), is selected to define contact region
behavior under the loading for strength analyses. In
the analyses, a cohesive zone material32 with zero
thickness is defined based on Fracture-Energies
debonding in ANSYS. In Table 4, the properties of
cohesive zone material that are used in the analyses
are presented.
Property
XNR6823
Young Modulus E (MPa)
Tensile failure strength rn (MPa)
Shear Modulus G (MPa)
Shear Failure Strength sn (MPa)
Toughness in tension GIC (J/m2)
Toughness in shear GIIC (J/m2)
2600
57
1000
32.9
1180
1500
20 node SOLID 186 hexahedral solid elements are
used with an element size 0.3 for meshing steel and
CFRP parts, while TARGE170 and CONTA174
mesh elements are applied to the joint region for
CZM. The 3 D full gear model is used for analyses.
The shaft hole is fixed that no rotation or displacement is allowed. Mesh and boundary conditions are
presented for root stress and joint stress analyses in
Figure 5.
To obtain the torque capacity of hybrid gear as
well as root stress, first, starting from 100 Nm, 50
Nm increments up to 400 Nm, torque equivalent
normal force is calculated, then this normal force
divided into the facewidth (b). In the analyses, this
unit force is applied to HPSTC. As the spur gears
are subjected to planar loading, facewidth has a
linear effect on the root stress. Based on this fact,
Facewidth is taken as 1 mm to decrease CPU and
analysis time in FEA. The relation between normal
force and torque is presented in Appendix B.
The root stress results of hybrid gear with different
rim thicknesses for 100 Nm torque value are illustrated in Figure 6.
According to the results of the analyses, root tensile stress decreases as the rim thickness increases. The
root tensile stress of the hybrid gear with a rim thickness of 0.5xm is 11% higher than that of full steel
gear. This ratio decreases to 4.6% and 1.9% in
hybrid gears with 1xm rim thickness and 1.5xm rim
thickness, respectively. After 1.5xm rim thickness,
there is almost no difference in root stress between
Yılmaz et al.
5
Figure 5. Mesh and boundary conditions for root stress and joint stress analyses.
steel gear and hybrid gear. The root stress of full steel
gear is 117.39 MPa. To validate of FEA, DIN3990
Method B (details given in Appendix C) is used.
The root stress is calculated as 122.62 MPa with the
analytical standard. The difference between the FEA
and DIN (4.2%) is found very little that FEA has
adequate accuracy. The other stress results are presented in graphical data (Figure 7) as the relation
between torque and stress is linear.
The normal stress values of hybrid gears on the
contact region for 100 Nm torque are illustrated in
Figure 8.
In Figure 8, the stress values on the tensile side
(minimum values) are taken into consideration as
these stresses cause separation in the normal direction
for joint. According to results, as the distance
between the root and joint regions increases, the
normal stresses in joint decrease. Rim thickness is a
highly influenced parameter on normal stress. When
the rim thickness is increased from 0.5xm to 3xm, the
normal stress decreases 97% in the joint region. This
ratio is nearly 46%, 72%, 82% and 87% in 1xm,
1.5xm, 2xm and 2.5xm rim thicknesses, respectively.
The shear stresses on the contact region for 100 Nm
torque are illustrated in Figure 9.
The maximum values (compressive side) in Figure
9 are the shear stresses that occur in the joint.
According to the results, as the distance between the
root stress region and the joint region increases, shear
stresses on the joint decrease. Shear stress for 0.5xm
rim thickness is 16.404 MPa. When the rim thickness
is increased 0.5xm to 3xm, the shear stress decreases
by approximately 65%. After 2xm rim thickness, the
shear stress is extremely low. For other torque values,
the stress results are illustrated in Figures 10 and 11.
When Stress results in Figures 10 and 11 are
observed, it can be concluded that shear stress is
more significant than normal stress in terms of
damage initiation. In these figures, the torque value
for damage initiation indicates that the stress value
reaches the failure strength pointed out in Table 4.
The limit torque for 0.5xm rim thickness is
199 MPa. When the rim thickness is increased from
0.5xm to 1xm, the torque limit value is increased by
50%, and when it is increased to 1.5xm, it becomes
double. For other rim thickness values, the torque
limit exceeds 400 Nm. The hybrid gear with 0.5xm
rim thickness is found as the weakest option in
terms of joint stress. When Figures 10 and 11 are
examined, a linear relationship between torque and
joint stresses can be understood. Based on this, the
torque capacity of the hybrid gears for different rim
thicknesses is presented in Figure 12.
On the other hand, According to the previous
experimental and numerical study, the temperature
of the contact region does not exceed 100 C under
normal torque and rotational speeds. In addition, the
temperature value is lower than the contact region on
the internal regions.34 So if the higher rim thickness is
selected, the temperature effect on the joint decreases.
For these reasons, it is assumed that joint stress due
to the temperature remains low that it could be
neglected. Nevertheless, for higher torque and rotational speeds, the temperature effect on joint stress
could be a future study. A paper from NASA
researchers also proved that usage of hybrid gears
does not increase the gearbox temperature when comparing steel gears for under six different speeds and
five different levels of applied torque.35
Evaluation of tooth stiffness of hybrid
gears
As the involute spur gears are subjected to vibration
due to the higher rotational speeds during the operation, they have to be investigated in terms of dynamic
performance. Tooth stiffness is a highly significant
parameter for dynamic analysis. It can be calculated
6
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Figure 6. Root stress results for 100 Nm torque: rim thicknesses of (a) 0.5xm, (b) 1xm, (c) 1.5xm, (d) 2xm, (e) 2.5xm, (f) 3xm,
(g) full steel.
as the proportion of normal force to the total deflection of the gear tooth.36–38 Total deflection consists of
five components, which are bending, compressive,
shear, rim, and Hertzian deformation. It can be calculated by an analytical method for standard gears.39
However, for gears with non-standard geometry such
as asymmetric or gears with multi-material, the finite
element method is more convenient. Finite element
analysis was used to obtain total deflection, consequently tooth and mesh stiffness for bimetallic spur
gears,40 asymmetric gear,41 or hybrid gear.15 Based on
these, FEA is preferred to obtain tooth deformations.
Yılmaz et al.
7
Figure 7. Root stress results of hybrid spur gears for different torques.
The mesh and boundary conditions of deformation
analysis are presented in Figure 13.
In deformation analyses, the normal force (100 N)
is implemented from five lines of active flank, which
are on radii of 33, 31.875, 30.7266, 29.625, 28.5 mm,
respectively. The shaft hole is fixed as in stress analyses. Mesh size is reduced to 0.15 mm to include
Hertzian deformation, as proposed by Coy and
Chao.42 All contacts between components of hybrid
gear are defined as purely bonded. The adhesive bond
is modeled as a physical layer to include its elasticity.
The material of adhesive is assigned as isotropic with
the Poisson’s ratio of 0.3 and 2.6 GPa Young modulus. Adhesive young modulus is relatively low when
comparing it to CFRP and steel Young modulus. For
this reason, if the cohesive zone is enhanced, then the
tooth stiffness decreases because of the deformation
of the tooth increases. The thickness of the adhesive
layer for steel/CFRP joints generally varies from
0.25 mm to 1.5 mm in literature. In analyses,
0.25 mm is selected as thickness of adhesive bond,
as used in a previous study.15 Since stiffness is one
of the most significant parameters for especially
transmission error and noise, the bond thickness
should be kept as minimum as possible with taken
into consideration bond/slip condition and fracture
energy in practice.
Steel and CFRP properties are identical to stress
analyses. Total deformation values are recorded for
each rim thickness, then the normal force (100 N) is
divided into the total deformation to calculate tooth
stiffness for five radii. The single tooth stiffness values
for each point are calculated by equations (1) to (4).
kp;1 ¼
F
xp;1
(1)
kg;1 ¼
F
xg;1
(2)
kp;2 ¼
F
xp;2
(3)
kg;2 ¼
F
xg;2
(4)
Where F is the applied load on the tooth profile, x
is the total deformation of the tooth geometry, (mm),
and kp and kg are the single tooth stiffnesses of the
pinion and gear, respectively (N/mm). 1 and 2 indicate that the first and second tooth pair. Finally, high
precision (eave<0.002) curves are fitted to obtain the
tooth stiffness curves. The tooth stiffness equations
are used to calculate mesh stiffnesses, which are
inputs for dynamic analyses. In Figure 14, the tooth
stiffness curves are illustrated for hybrid gear with
different rim thickness and full steel gear.
According to results, the tooth stiffness increases
from tip to the bottom of the tooth for all gears. The
stiffness values of hybrid gears are lower than full
steel gear since the elasticity modulus of composite
material is a third of the steel approximately. When
the rim thickness is decreased, the tooth stiffness
decreases. The average stiffness value decreases by
nearly 48% for hybrid gear with 0.5xm rim thickness
compared to steel gear. This decrease rate is found
37%, 31%, 28%, 27% and 26% for 1xm, 1.5xm,
2xm, 2.5xm, 3xm, rim thicknesses respectively. The
effect of the rim thickness on stiffness decreases considerably after 2xm. Weights of hybrid gear are also
obtained for different rim thicknesses to compare full
steel gear and to calculate rotational inertia for
dynamic analyses. The density of steel is taken as
8
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Figure 8. Normal stresses on joint region for 100 Nm torque: rim thicknesses of (a) 0.5xm, (b) 1xm, (c) 1.5xm, (d) 2xm, (e) 2.5xm,
(f) 3xm.
7.86 g/cm3 while it is 1.53 g/cm3 for CFRP. The weight
of the adhesive is neglected. The weight status of
gears is presented in Table 5.
While the rim thickness is decreased, the weight
decreases as well, since the portion of the steel
part reduces. It turns out that the weight changes
inversely with the stress while it is in direct proportion
with stiffness. For this reason, a weight benefit
factor should be defined to obtain optimum rim
thickness. The weight benefit factor is calculated by
weight reduction percentage divided by the stress
increase percentage for stress and stiffness decrease
percentage for stiffness. In Table 6, the relation
between weight benefit factor and rim thickness is
illustrated.
Hybrid gear with 3xm rim thickness is found as the
best option in terms of weight benefit factor for stress
while 1.5xm rim thickness in terms of weight benefit
factor for stiffness. Indeed, the difference between
hybrid gear with 1.5xm rim thickness and solid gear
Yılmaz et al.
9
Figure 9. Shear stresses on joint region for 100 Nm torque: rim thicknesses of (a) 0.5xm, (b) 1xm, (c) 1.5xm, (d) 2xm, (e) 2.5xm,
(f) 3xm.
is only 1.9%. For this reason, 1.5xm rim thickness can
be preferred to gain more weight reduction.
Modal analysis of hybrid gears
Finite element analyses are also conducted to understand the effect of rim thickness on the natural
frequencies and mode shapes of hybrid gears under
free-free boundary conditions. These analyses are
conducted for only comparison aim between Solid
(Steel) and hybrid gears. The first six modes are related to rigid body movements, so these modes are not
taken into consideration. Natural frequencies of steel
gear and hybrid gears for Mode 7, 8, and 9 are presented in Table 7.
According to results, the frequency variation
between hybrid gears with different rim thickness is
rather low. When the rim thickness increases from
0.5xm to 3xm, the natural frequency increases by
nearly 5% for Mode 7 and 8 while it is 13% for
10
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Evaluation of dynamic behavior of hybrid
gears
Calculation of time-varying mesh stiffness
Figure 10. Normal stress results of hybrid spur gears on the
joint for different torques.
Figure 11. Shear stress results of hybrid spur gears on the
joint for different torques.
Dynamic analyses are performed for the determination of the dynamic loads and resonance regions of
the steel and hybrid gear mechanisms. The timevarying mesh stiffness (TVMS) is one of the most
critical parameters for the dynamic models. Thus
the (TVMS) directly affects the dynamic loads and
the dynamic response of the gear mechanisms.
Hence, the TVMS should be calculated carefully. In
the literature, there are several methods proposed for
the calculation of the TVMS. These methods can be
classified as analytical, numerical, and experimental
methods. Liang et al. calculated the TVMS for planetary gears by using the potential energy method.43
Munro et al. calculated TVMS by using back to back
test rig experimentally for lower speeds.44 In this
study, the TVMS is calculated by using the finite element method, which was developed in the literature.34–38 During the power transmission of the spur
gears, one and two tooth pairs contact and leave the
meshing, respectively. In a gear mechanism with a
normal contact ratio, at least one and at most two
tooth pairs transmit the torque. The contact starts at
the point of A for tooth pair p,1/g,1. At this time,
another tooth pair (p,2/g,2) is in contact at the
point of D (Figure 16). When the first tooth pair
comes to the point B, and the second tooth pair
leaves the contact. Thus, the regions A–B and D–E
are defined as double tooth contact region. Between
B–D, only the first tooth pair is in contact, so the B–
D region is defined as the single tooth region. For
calculation of the TVMS, the equivalent stiffness of
gear pairs is calculated as a serial-connected spring in
itself. On the other side, in the double tooth pair
region, the equivalent mesh stiffness is calculated by
the sum (assumed as parallel-connected spring) of
each equivalent stiffness of the pairs.
The equivalent stiffness of the first pair of teeth is:
K1 ¼
Figure 12. Torque capacity of hybrid gears with different rim
thicknesses.
kp;1 kg;1
kp;1 þ kg;1
(5)
The equivalent stiffness of the second pair of teeth
is:
Mode 9. Solid (Steel) gear has the highest natural
frequencies for all modes as expected. The reason
why steel has the highest elasticity modulus. As
the volume of steel increases in hybrid gears with
increasing rim thickness, natural frequency values
increase. Mode shapes are similar for all gears. For
this reason, mode shapes of hybrid gear with only
0.5xm rim thickness are also presented in Figure 15
as a sample.
K2 ¼
kp;2 kg;2
kp;2 þ kg;2
(6)
If there is contact between B–D in the single tooth
region:
K1 6¼ 0 and K2 ¼ 0
If there is contact between A–B or D–E in the
double tooth contact region:
K16¼0 and K26¼0
Yılmaz et al.
11
Figure 13. Mesh and boundary conditions for deformation analyses.
Table 5. Weight of solid gear and hybrid gears with different
rim thicknesses.
Figure 14. Tooth stiffness curves for hybrid gears and steel
gear (solid).
The TVMS results for the steel and hybrid gears
with different rim thicknesses are given in Figure 17.
It is seen that the TVMS takes the highest value when
it is made of all steel. The TVMS values are decreased
with the increase in the volume of composite material
in the gear structure. The TVMS nearly two-times
decreases for the minimum rim thickness case,
which is 0.5xm. Moreover, after the 2xm rim thickness
value, there is no remarkable difference in terms of
mesh stiffness.
Rim thickness (xm)
Weight (kg/b)
Reduction (%)
0.5
1
1.5
2
2.5
3
Solid (Steel)
0.009221
0.010655
0.012000
0.013228
0.014400
0.015483
0.019442
52
45
38
32
26
20
–
Where Jp and Jg are polar mass moments of inertia
of pinion and gear, hp and hg are angular displacements of pinion and gear, T is transmitted torque, rbp
and rbg are the radii of base circles, F is dynamic load,
qp and qg radius of curvatures, m is coefficient of friction. The sign of friction force depends on the linear
velocity of the pinion and gear throughout the meshing. If the linear velocity of the pinion is higher than
the velocity of gear, the sign of the friction force is
positive; otherwise, it is negative. In the analyses, the
contact points on the line of action corresponding
angular rotation angle of pinion and gear are used
based on equations (9) and (10).
xp ¼ rbp hp
(9)
xg ¼ rbg hg
(10)
The dynamic model of hybrid spur gear
In this study, to calculate dynamic loads and the
static transmission error of the steel and hybrid gear
mechanisms, a 2-DOF dynamic model is developed.
The equations of motion can be defined by using the
free body diagram seen in Figure 18.
h p ¼ Tp rbp ðF1 þ F2 Þqp;1 l1 F1 qp;2 l2 F2
Jp €
(7)
h g ¼ rbg ðF1 þ F2 Þ TG qg;1 l1 F1 qg;2 l2 F2
Jp €
(8)
Equivalent masses of pinion and gear can be written as;
mp ¼
Jp
r2bp
(11)
mg ¼
Jg
r2bg
(12)
12
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Table 6. Weight benefit factors in terms of stress and stiffness.
Rim thickness (xm)
Stress
Increase (%)
Average Tooth
Stiffness Decrease (%)
Weight benefit factor
for stress
Weight benefit factor
for tooth stiffness
0.5
1
1.5
2
2.5
3
Solid
11
4.6
1.9
1.2
0.7
0.5
–
47
38
32
28
27.2
26.9
–
4.72
9.78
20
26.66
37.14
40
–
1.1
1.18
1.19
1.14
0.95
0.74
–
Table 7. Natural frequencies of hybrid gears and steel gear.
Modes (Hz)
Rim thickness (xm)
0.5
1
1.5
2
2.5
3
Solid
7
8
9
15086
15387
18213
15282
15587
18878
15326
15632
19870
15434
15742
20626
15664
15977
20653
16006
16326
20984
17536
17888
22198
The static load (FD) that applied on the gears can be defined as;
FD ¼
Tg Tp
¼
rbg rbp
(13)
The above expressions are added to equations (7) and (8). If the difference of two equations is taken and
required arrangements are conducted on the equations, the static transmission error and general equation of
motion are obtained as in equations (14) and (15), respectively. Derivation of equations is given in Appendix D
section in detail.
The static transmission error can be described as;
ðmg mp ÞFD þ K1 e1 fp;1 mg þ fg;1 mp þ K2 e2 fp;2 mg þ fg;2 mg
xs ¼
K1 fp;1 mg þ fg;1 mp þ K2 fp;2 mg þ fg;2 mp
(14)
Finally, the equation of motion of the system can be written as;
"
#12
K1 fp;1 mg þ fg;1 mp þ K2 fp;2 mg þ fg;2 mp
K1 fp;1 mg þ fg;1 mp þ K2 fg;2 mg þ fg;2 mp
nx r þ
xr
xr þ 2
mg mp
mg mp
ðmg mp ÞFD þ K1 e1 fp;1 mg þ fg;1 mp þ K2 e2 fp;2 mg þ fg;2 mg
¼
mg mp
(15)
Where n is the damping of the gear pair, and m is the mass of the pinion and gear. The equation of motion is
solved by using the fourth-order Runge – Kutta method and static transmission error results, and maximum
dynamic forces are taken into consideration, and the dynamic factor (DF) is calculated between 1000 and
40,000 rpm pinion speed. The dynamic factor is calculated according to equation (16).
DF ¼
Maximum Dynamic Load
Static Load
(16)
STE and dynamic factor results
Static transmission error is one of the most critical factors for power transmission and gear noise and vibration.
Thus, the static transmission error should be predicted by the gear designers. In this study, static transmission
Yılmaz et al.
13
Figure 18. Free body diagram for 2 – DOF dynamic model.40
Figure 15. Mode shapes of gears (a) Mode 7, (b) Mode 8,
(c) Mode 9.
errors are investigated for steel gear and hybrid gears
for comparison aim. The static transmission errors of
the gear mechanisms are calculated by using equation
(14) in the dynamic model. The damping of meshing
gears and the coefficient of friction varies throughout
the line of action. These expressions can be calculated
with equations (17) and (18), respectively.41
2
n ¼ 2nt 4
l¼
Figure 16. The meshing of spur gears.18
Figure 17. Time-varying mesh stiffness results for steel and
hybrid gears.
K1 þ K2
r2bp
JP
0:05
e0:125Vs
312
5
(17)
pffiffiffiffiffi
þ 0:002 Vs
(18)
þ
r2bg
Jg
Where nt the damping ratio of meshing gears, J is
the rotational inertia of gears and Vs is the sliding
velocity between pinion and gear. Damping ratio of
meshing gears are taken as 0.1.
The effect of rim thickness on the static transmission error is illustrated in Figure 19. Static transmission error can be defined as a reverse function of the
time-varying mesh stiffness. Thus the results of the
TVMS and static transmission error should be compatible. Actually, a small fluctuation can be noticed at
the middle of the single tooth contact region as differently from TVMS. The reason for this situation is
the change in the coefficient of friction throughout
the line of action. The values of static transmission
error are obtained for 1 rpm pinion angular velocity.
When Figure 19 is examined, it is seen that the minimum static transmission error is obtained for solid
gear because it has the highest time-varying mesh
stiffness. When the rim thickness of the hybrid gear
increases, the static transmission error decreases as
the time-varying mesh stiffness increases with the
rim thickness. The effect of rim thickness on STE
reduces drastically after 2xm. The dynamic factor of
the steel and hybrid gears between “1000 and
40,000 rpm” is given in Figure 20. It is seen that the
differences in the dynamic response for the different
14
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
modeled as a physical layer to include its elasticity
on deformation. Single tooth stiffness of hybrid gears
was calculated with load divided by deformation value
obtained from analyses results. Finally, 2-DOF
dynamic analyses were performed to understand the
effect of rim thickness on the dynamic response for
hybrid gears. According to results,
Figure 19. Static transmission error results for steel and
hybrid gears.
Figure 20. Dynamic response of the steel and hybrid gears
between 1000 and 40,000 rpm values.
cases are minimum. The first resonance region is
nearly 9000 rpm for all cases, and the value of a
dynamic factor is almost 1.43. Similarly, the second
resonance regions are also close, and the value of the
dynamic factor is approximately 2.23 around nearly
30,000 rpm. The backlash effect is not taken into consideration in this study. According to the results, the
situation does not worsen prominently in terms of the
dynamic factor when using lightweight hybrid gears.
Conclusion
In this study, the static and dynamic behavior of
hybrid gears with different rim thicknesses were investigated and compared with the solid steel gear, numerically. 3 D full gear models for steel and hybrid gears
were created in CATIA. To obtain root tensile and
joint stress, finite element analyses were conducted.
In these analyses, cohesive zone material (CZM) was
used to define joint region behavior for hybrid gears.
For deformation analyses, the adhesive bonding is
• In hybrid gears maximum shear stress on joint
region is higher than maximum normal stress so,
adhesive bonds with high shear strength should be
used in hybrid gear design.
• Rim thickness is the most important parameter for
weight reduction so, it should be defined properly.
According to results 1.5xm rim thickness is found
best option in terms of stress and stiffness. After
this value stress and stiffness values change very
little. For this value, root tensile stress is nearly
equal with root stress of steel gear.
• Rim thickness is more effective on joint normal
stress than shear stress. For hybrid gear with 3xm
rim thickness, the shear stress decreases nearly by
65% while normal stress decreases %87 with
regard to 0.5xm rim thickness.
• When the rim thickness is increased from 0.5xm to
3xm, the torque transmit capacity of hybrid gear
increases 20% nearly.
• Static transmission error decreases approximately
25% when the rim thickness becomes 3xm with
regard to 0.5xm rim thickness.
• For dynamic factor there is no noticeable difference
between hybrid gears for different rim thicknesses.
Although static transmission errors increase at a
specific rate, it can be reduced by changing gear
parameters. It can be concluded that using hybrid
design in spur gears can ensure 30–50% weight reduction with regard to rim thickness when comparing to
steel gear without a higher performance drop. Of
course, it is not possible to replace steel for every
application area, however, it offers a very good alternative. Hybrid spur gears can be used in several
industries where the fuel conception is an important
issue, such as the automotive and aerospace industries. The fact that it has the potential to dampen
noise and vibration due to its structure, and provides
design-dependent flexibility in terms of dynamic
properties, can enable hybrid gears to find application
areas in these sectors.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) disclosed receipt of the following financial
support for the research, authorship, and/or publication
of this article: This work was partially supported by the
Yılmaz et al.
Scientific and Technical Research Council of Turkey
(TUBITAK) under Grant No. 118 M 810.
ORCID iDs
Tufan G Yılmaz
https://orcid.org/0000-0003-3772-7871
Fatih Karpat
https://orcid.org/0000-0001-8474-7328
Supplemental material
Supplemental material for this article is available online.
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