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OPEN
Analysis of gear transmission error
in helical gear using enhanced
tooth contact analysis model
considering measured tooth profile
errors
Dongu Im1,2, Woo-Jin Chung1, Hyunggon Lee3, You-Jin Lee1, Hyunjoon Sung3, Minwook Lee3
& Young-Jun Park1,2,4
A loaded tooth contact analysis model was developed to assess transmission error by incorporating
profile errors in a helical gear pair. Profile errors for all teeth of both the pinion and wheel were
measured, and the mean profile error was computed and integrated into the model. To validate
the model, static transmission error (STE) was measured experimentally. The experimental results,
particularly the peak-to-peak static transmission error (PPSTE) in relation to torque and tooth-meshing
harmonics, demonstrated consistency with the simulation outcomes. Subsequently, the model was
employed to investigate the impact of surface waviness (SW) on the STE, focusing on SW amplitude,
spatial frequency, and initial phase. It was observed that both the amplitude and spatial frequency
of the SW influenced the PPSTE and the harmonic characteristics of the transmission error, while the
initial phase of the waviness had no significant effect.
Keywords Helical gear, Analytical model, Loaded tooth contact analysis, Transmission error, Profile error
Gear whine noise is a significant issue in power transmission systems, drawing considerable attention from
the aviation, automotive, and maritime industries. Transmission error in gear pairs has been identified as the
primary cause of gear whine noise, and extensive research has addressed this issue1–4. Transmission error refers
to the angular displacement error during gear meshing and can be considered a form of displacement excitation
along the line of action. Ideally, no transmission errors occur with perfectly rigid gear pairs featuring involute
tooth profiles. However, transmission errors are prevalent in practical gears due to manufacturing imperfections
and tooth elastic deformation, with manufacturing errors being particularly impactful, especially under lowload conditions5.
During the design phase, predicting gear transmission error necessitates calculating the time-varying mesh
stiffness (TVMS) as a function of the gear meshing position, considering the effects of manufacturing errors. The
TVMS of a gear pair can be determined through experimental methods6–8, the finite element method (FEM)9–11,
or analytical models12–18. Experimental methods involve producing prototypes to measure gear transmission
errors, which can be costly and impractical during the design phase. Furthermore, controlling manufacturing
errors and assessing the effects of individual design factors pose challenges. The FEM for contact analysis
requires significant time investment and restricts the inclusion of real manufacturing errors into the simulation
model. Consequently, analytical models are regarded as the most efficient means for performing loaded tooth
contact analysis (LTCA) and calculating the TVMS while accounting for manufacturing errors in a timely and
cost-effective manner.
Li19–21 conducted LTCA for spur gears by incorporating additional mathematical programming into the FEM
model to account for misalignment, tooth modifications, assembly errors, and machining errors. In this context,
1Department of Biosystems Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826,
Republic of Korea. 2Convergence Major in Global Smart Farm, Seoul National University, 1 Gwanak-ro, Gwanak-gu,
Seoul 08826, Republic of Korea. 3Hyundai Motor Company R&D Center, Hyundai Motor Company, Hwaseong-si,
Gyeonggi-do 18280, Republic of Korea. 4Research Institute of Agriculture and Life Sciences, College of Agriculture
and Life Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea. email:
yjpark95@snu.ac.kr
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a simulation model was developed to evaluate the effects of manufacturing errors on transmission error19–21.
Nevertheless, this simulation model was validated by comparing test results of contact patterns and tooth root
deformations without directly comparing transmission errors, indicating a limitation. Furthermore, since the
model was specific to spur gears, validation for helical gears, which involve more complex meshing kinematics,
remains necessary.
Several studies have presented analytical models that address gear manufacturing errors in LTCA17,22–25.
Velex and Maatar developed a dynamic model for gear systems that considered shape deviations and mounting
errors22. Their model compared static load distribution and transmission error with experimental results.
Although the model emphasized dynamic behavior, it did not thoroughly analyze transmission error as a
function of roll angle, limiting its ability to accurately characterize the frequency of gear excitation forces. Ma
et al.17 incorporated extended tooth contact at high loads beyond the line of action in their LTCA model for
spur gears with tip relief. Their analytical model was validated by comparing the TVMS with that obtained via
the FEM; however, the FEM model used for validation did not reflect actual system characteristics and was
confined to spur gears17. Wang and Zhang (2017) proposed an LTCA model for helical gears that accounted for
manufacturing errors24. They constructed their model using a slice approach with helical gear and validated it
by comparing the contact stress with that obtained via the FEM. Similar to Ma et al.17, Wang and Zhang (2017)
validated their model using an FEM model that did not incorporate the actual tooth profile into the analytical
model24.
A more recent study by Sánchez et al.25 integrated the Archard wear model into the LTCA framework to
consider wear characteristics of spur gears. This analytical model predicted the TVMS as a function of the gear
operating cycle and identified optimal parameters for modifying the tooth profile in response to wear25.
In the present study, an LTCA model was developed to address the limitations of previous research by
incorporating real tooth profile errors for both the pinion and wheel. This model applied the TVMS model
proposed by Chung et al.18 and the profile error model proposed by Wang and Zhang24. The mean profile error,
as proposed by Mark26, was calculated to integrate the entire tooth profile error into the analytical model. Gears
with manufacturing errors were produced, and the LTCA model was validated through static transmission error
(STE) testing. Ultimately, the validated analytical model was employed to analyze the effects of helical gear
waviness on STE. The contributions of this study are summarized as follows:
1.The 3D tooth profile error of helical gears was measured, and the mean profile error was calculated and incorporated into the LTCA model.
2.An STE test was conducted to validate the LTCA model with actual profile errors, deviating from the conventional use of the FEM in previous studies.
3.Mark’s theory, which posits that transmission error in tooth-meshing harmonics is influenced by the mean
profile error, was validated through experimentation and analysis. Mark26 mathematically expressed transmission error as a function of the mean profile error but did not validate this theory through LTCA26.
4.The influence of waviness on STE was analyzed, considering the amplitude, spatial frequency, and initial
phase of waviness in helical gears.
5.Comparisons were made between the second-order derivatives of the load-sharing ratio (LSR) and STE to
confirm the similarity of inflection patterns between the two results.
The proposed analytical methodology facilitates accurate prediction of transmission error and can be used for
robust design considerations related to helical gear waviness.
Methods
Background
This section provides a brief overview of the time-varying mesh stiffness (TVMS) model developed by Chung et
al.18. This model facilitates the calculation of TVMS by accounting for the trochoidal root profile, and its accuracy
has been validated through comparisons with finite element models18. Comprehensive details regarding the
TVMS analytical model are documented in the referenced paper18.
In this study, helical gears were analyzed using the sliced theory, which represents helical gears as an assembly
of multiple small spur gear slices with reduced face width. The sliced theory has been extensively utilized in
various studies due to its computational simplicity18,24,27–32.
The meshing forces in a helical gear pair can be decomposed into transverse and axial components, as
described by Eqs. (1) and (2). The sliced theory was employed to partition the helical gear into several small spur
gear slices, and the transverse stiffness of each slice was determined based on the total potential energy of the
slice18. Where Ft , Fn andFa represent the transverse component of the meshing force, total meshing force, and
axial component of the meshing force, respectively.
Ft = Fn cosβ (1)
Fa = Fn sinβ (2)
The transverse mesh stiffness for a slice of a helical gear can be divided into tooth stiffness, contact stiffness, and
foundation stiffness. To determine the tooth stiffness, the gear tooth is commonly modeled as a non-uniform
cantilever. The total potential energy of the tooth is expressed by Eq. (3) as follows:
Utooth = Ub + Us + Ua (3)
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where Ub , Us , and Ua represent the energies generated by bending, shear, and axial compressive deformations,
respectively. The values were calculated as follows:
F2
Ub = t =
2kb
Us =
Ft2
=
2ks
Ft2
Ua =
2ka
∫ d
Mx2
dx(4)
2EIx
0
∫ d
0
=
1.2 Fb2
dx(5)
2GAx
∫ d
0
2
Fac
dx(6)
2EIx
where Ft represents the transverse mesh force, and kb , ks , and ka correspond to the transverse bending stiffness,
the shear stiffness, and the axial compression stiffness, respectively. Mx denotes the moment for the variable
cross-section at a distance x from the mesh point, calculated as Fb x − Fac h. Here, Fb and Fac represent the
bending and axial compression force, respectively. E is Young’s modulus, and h is the distance between the
center axis and the mesh point. Ix , Ax , G were computed as follows:
Ix =
2h3x ∆l
(7)
3
Ax = 2hx ∆l(8)
E
(9)
2(1 + υ)
G=
where Ix , hx , and Ax represent the moment of inertia, the half-tooth thickness, and the section area at a distance
x from the mesh point, respectively. ∆l is the face width of the sliced helical gear. G and υ are the shear modulus
and Poisson’s ratio, respectively. If the mesh position and pressure angle are known, the bending, shear, and axial
compressive stiffness of a sliced helical tooth can be calculated as follows; herein, ξ1 is the roll angle parameter
shown in Fig. 1.
1
=
kb
∫ d
0
1
=
ks
(xcosξ1 − hsinξ1 )2
dx(10)
EIx
∫ d
1
=
ka
0
1.2cos2 ξ1
dx(11)
GAx
∫ d
0
sin2 ξ1
dx(12)
EAx
Fig. 1. Instantaneous pressure angle and roll angle of a spur gear.
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For the contact stiffness, the nonlinear Hertzian contact stiffness formula proposed by Weber (Eq. 13) is used
in which bcon represents the half-width of the contact region on the tooth, which can be calculated using
Eq. (14)18,33.
1
2
=
kh
πb
[(
1 − νp2
Ep
bcon =
){
[
ln
4F
πb
νp
2hxp
−
bcon
2(1 − νp )
{(
1 − νp2
Ep
)
+
(
}
+
(
1 − νg2
Eg
){
1 − νg2
Eg
)} (
/
ln
νg
2hxg
−
bcon
2(1 − νg )
1
1
+
rcp
rcg
)]1/2
}]
(13)
(14)
where rcp , rcg represents the radius of curvature at the point of contact between the pinion and wheel and Ep
and Eg denote Young’s modulus; νp , νg are Poisson’s ratios of the pinion and wheel, respectively.
The calculation of the foundation stiffness, which has been predicted in numerous analytical investigations
through the application of Muskhelishvili’s theory18,24,30,34–36, is as follows:
cos2 ξ1
1
=
kf
E∆l
{
L
∗
(
uf
Sf
)2
+M
∗
(
uf
Sf
)
+P
∗
(
2
1 + Q tan ξ1
∗
}
)
(15)
where uf, Sf represents tooth height at the line of action, length of arc in the tooth root. The coefficients L* , M *
, P * , and Q* were calculated using an approximate polynomial, as follows:
X ∗ (hf , θf ) =
Ci hf
Ai
Di
+ Bi h2f +
+
+ Ei hf + Fi (16)
θf
hf
θf2
where rint , rf , θf represents inner radius, root radius, and tooth centerline-to-root circle angle. hf is the ratio
of rf to rint , and the constants Ai , Bi , Ci , Di , Ei , Fi are listed in Table 1.
After calculating the tooth, foundation, and contact stiffnesses of the sliced helical gear pair, the total
equivalent mesh stiffness was determined, as shown in Eq. (17).
ki,j =
1
1
1
1
1
1
1
1
+
+
+
+
+ ks,g
+ ka,g
+ kf,g
+ k1h (17)
kb,p
ks,p
ka,p
kf,p
kb,g
1
Profile error considerations
The approach proposed by Wang and Zhang was applied to account for profile errors in the helical gear tooth24.
In a manner analogous to the gear stiffness model that was previously described, the profile error associated
with every meshing position was established using the slice method. Figure 2 illustrates the contact between the
individual slices in the helical gear with profile errors. Egm , Egi , and Egj represent the profile errors for the
sliced pairs “m,” “i,” and “j” on the gear, while Epm , Epi , and Epj denote the profile errors for the sliced pairs
“m,” “i,” and “j” on the pinion, respectively. δm , δi , and δj represent the elastic deformations for the sliced pairs
“m,” “i,” and “j,” respectively. The first row in the vertical direction illustrates the contact for an ideal gear without
profile errors. The second row shows the state before contact for sliced pairs “m,” “i,” and “j” with different profile
errors. The third row represents the elastic deformations caused by the contact for each sliced pair “m,” “i,” and
“j.” According to Wang and Zhang (2017), the relationship between the deformation and the profile error for
each slice can be expressed as Eq. (18)24. Assuming that the sliced pair “m” exhibits the smallest profile error
among the slices in contact, it will be the first to engage and will experience the greatest deformation, with the
slice coupling stiffness neglected. Consequently, the sum of the elastic deformation and profile error at each
mesh position remains consistent across all sliced pairs.
δm + Epm + Egm = δi + Epi + Egi = δj + Epj + Egj (18)
If the meshing occurs at slice “i”, the meshing force can be calculated as follows.
Fts = ki δi (19)
In addition, the stiffness for a slice “i” can be calculated according to Eq. (20). When Ei − Em is greater than δm
, there is no contact, and there is no stiffness, where Ei is the sum of Epi and Egi (Ei = Epi + Egi ).
L∗ (hf , θf )
M ∗ (hf , θf )
P ∗ (hf , θf )
Q∗ (hf , θf )
Ai
Bi
Ci
Di
Ei
Fi
−5.574 × 10−5
−1.9986 × 10−3
−2.3015 × 10−4
4.7702 × 10−3
−50.952 × 10−5
185.50 × 10−3
0.0538 × 10−4
−53.300 × 10−3
60.111 × 10−5
−6.2042 × 10−5
28.100 × 10−3
9.0889 × 10−3
−83.431 × 10−4
−4.0964 × 10−4
0.0271
6.8045
−9.9256 × 10−3
0.1624
0.9086
0.2895
0.9236
7.8297 × 10−3
−0.1472
0.6904
Table 1. Constants of the approximate polynomial function.
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Fig. 2. Contact theory for sliced teeth with profile errors.
ki =
{
ki
0
δm > Ei − Em
δm ≤ Ei − Em (20)
The meshing force acting on the entire gear can be calculated by summing all the forces acting on N slices, as
expressed in Eq. (21). The mesh stiffness can be computed using Eq. (22): The relationship between Fn and Ft
is described in Eq. (1).
Fn =
N
∑
(Fts )
i
i=1
K=
1+
cosβb
∑N
∑j=1
N
j=1
(21)
kj
kj Emj
(22)
Fn cosβb
Finally, LSR and STE can be calculated using Eq. (23) and Eq. (24), respectively.
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Parameters
Pinion
Wheel
Tooth number z1/z2
19
43
Normal module mn (mm)
2.45
2.45
Reference circle pressure angle α (°)
20
20
Helical angle β (°)
32
32
Face width b (mm)
30
28
Center distance a (mm)
90
90
Young’s modulus E (Pa)
2.1 × 1011
2.1 × 1011
Poisson’s ratio ν
0.3
0.3
ISO gear class
5
5
Table 2. Main parameters of the gear pairs.
Fig. 3. Helical gear profile measurement.
ki
LSRi = ∑N
k
j=1 j
ST Ei =
(
1+
∑N
j=1
kj Emj
Fn cosβb
)
−
ki Emi
(23)
Fn cosβb
Fn cosβb
+ Epm + Egm (24)
K
Using the minimum elastic potential energy theory allows for LSR through iterative calculations, thus ensuring
the convergence of the STE calculation37,38.
Tooth profile error measurement
The profile errors of both gear pairs were measured to incorporate their influence into the loaded tooth contact
analysis (LTCA) model. The gear parameters are presented in Table 2. The measurements were conducted using
the Gleason 300GMS® nano Gear Metrology System, which ensures repeatability and reliability, surpassing VDI/
VDE2612 standards. All teeth of the pinion and wheel were measured, with eight measurements taken in the
involute direction for each tooth. These measurement positions were evenly spaced along the face width, as
depicted in Fig. 3.
During the gear measurement process, the direction in which the probe moves is defined as the major
direction, while the perpendicular direction is defined as the minor direction26. Since the minor direction was
interpolated using relatively fewer data points, it typically exhibits lower accuracy. According to Mark26, it is
advisable to select the direction where gear micro-geometry modifications are most pronounced as the major
direction for measurement. In this study, the involute direction was chosen as the major direction due to the
more significant modifications observed in that direction26. For the involute direction (major direction), 300
data points were recorded with respect to the roll angle, while for the helix direction (minor direction), eight
data points were recorded along the face width and then linearly interpolated. Figure 3 illustrates a 3D gear
profile example for a single pinion tooth.
Mean tooth profile error
Mark assumed that the attenuation effect had a negligible influence and expressed the transmission error in the
tooth-meshing harmonic region as a mathematical function of gear tooth deformation and mean profile error26.
In this study, only the tooth-meshing harmonics were considered, excluding the shaft rotational harmonics,
sidebands, and ghost noise components of the transmission error. The 3D mean profile errors of both the pinion
and the wheel were derived and integrated into the LTCA model. The mean tooth profile error can be expressed
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Fig. 4. Mean profile error: (a) pinion #1; (b) pinion #2; (c) wheel #1; (d) wheel #2.
Operating condition
Gear pair information
Condition
Description
Input rotational speed
30 rpm
Output load
10, 20, 30, 40, 50, 60, 70, 80, 90, 100 Nm
Gear pair #1
Pinion #1 – Wheel #1
Gear pair #2
Pinion #2 – Wheel #2
Table 3. STE test operating conditions.
as shown in Eq. (25), where N is the number of teeth, j is the gear tooth number, and ηCj (y, z) represents the
profile error corresponding to the y-z plane Cartesian coordinates. In this study, the mean profile error was
determined based on profile errors measured across the entire tooth surface along the face width (y) and the
roll angle (z). Figure 4 plots the mean profile error for each gear sample; the waviness occurred in the involute
direction for all tooth surfaces.
η̄C (y, z) ≜
1 ∑
ηCj (y, z)(25)
N
N −1
j=0
Validation test
Test rig
A test rig for STE was implemented to validate the developed LTCA model. Two gear pairs with identical
macro- and micro-design parameters were manufactured, as listed in Table 2. Despite sharing the same design
parameters, the two gear pairs exhibited distinct tooth surfaces due to profile errors, as shown in Fig. 4. The STE
test conducted in this study was specifically designed to validate the LTCA model, which does not account for
dynamic behavior. Consequently, the input rotational speed was maintained at 30 rpm to minimize dynamic
effects. The STE measurements were systematically performed under varying torque conditions, ranging from
10 to 100 Nm, as detailed in Table 3. This approach was intended to evaluate the influence of applied torque on
the STE. The two gear pairs involved in the test are referred to as pair #1 (pinion #1 and wheel #1) and pair #2
(pinion #2 and wheel #2), as listed in Table 3.
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Fig. 5. Schematic layout of the STE test rig.
Fourier transform
Model coefficient
1024
2048
5000
a1
−8.274
−5.784
−6.093
a2
2.517
2.903
1.972
a3
0.311
0.614
0.288
a4
3.553
1.540
1.493
a5
0.003
0.050
0.020
Table 4. Best-fit model coefficient for encoder error amplitudes in the order domain40.
Figure 5 presents a schematic layout of the test rig. Two AC induction motors were utilized, with the input
motor controlling rotational speed and the output motor controlling torque. To accurately measure the STE, it
was essential to eliminate all extraneous forces. An elastomer with excellent vibration-isolation properties was
used for the shaft connection to mitigate the influence of torque ripple from the motor. A HEIDENHAIN ERN
120 sensor was employed to measure rotational speed, while torque was measured using a Brüel & Kjær HBM
T22 sensor. The data acquisition system (DAS) employed a DEWESoft DEWE-43 A for torque measurement,
and a Rotec RASdeltaFE08 was used for STE measurement. In this study, the primary focus was on the STE in
the tooth-meshing harmonic region. The average transmission error was determined through time-synchronous
averaging at base pitch intervals39. A total of 129 revolutions were performed, using the pinion as the reference,
which corresponds to an average of 2,451 cycles when converted to the base pitch.
Encoder accuracy validation
Palermo et al.40 compared two methods for measuring gear transmission errors using an encoder: the direct
method (DM) and the elapsed time method (ETM). They reported that the ETM offers higher accuracy
and reproducibility40. In this study, an STE measurement system based on the ETM was developed using a
HEIDENHAIN ERN 120 encoder. To verify the accuracy of the STE measurement system, the encoder grating
error best-fit model, represented by Eq. (26), as proposed by Palermo et al. (2018), was employed40. The
coefficients required for the calculations in the best-fit model are provided in Table 4, where O represents the
order number.
log10 [e∆θ (O)] = a1 + a2 exp(−a3 O) + a4 exp (−a5 O)(26)
The results of the error calculation for the HEIDENHAIN ERN 120 used in this study, based on the formula
proposed by Palermo et al. (2018), are shown in Fig. 640. It was confirmed that a measurement error of 8.63 × 10−6
degrees occurred for the 19th order, which is the first order of the gear meshing frequency (GMF) for the test
gear pair. Assuming a target measurement error of 10% or less for the STE, the linear TE on the line of action that
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Fig. 6. Measurement error of the HEIDENHAIN ERN 120 depending on the order of shaft rotation.
Fig. 7. Comparison of the PPSTE between two gear pairs.
the encoder can accurately measure was calculated using Eq. (27). When verifying the measurement precision of
the HEIDENHAIN ERN 120, T ELOA > 0.1275 μm can be measured with an error rate of 10% or less.
eθ <
0.1 × T ELOA
(27)
rb1 + rb2
Results and discussion
Tooth elastic deformation and manufacturing errors contribute to transmission errors. Therefore, accurate
prediction of transmission error requires not only a gear mesh stiffness model but also an analytical model that
accounts for manufacturing errors. In this study, the TVMS model proposed by Chung et al. was employed
to calculate mesh stiffness, while the model by Wang and Zhang was utilized to incorporate profile error into
the LTCA model18,24. The STE rig tests were conducted to validate the LTCA model. The test results were then
used to confirm the validity of the LTCA model, which was subsequently applied to analyze the effects of gear
waviness on STE.
Effect of manufacturing error on the PPSTE
Figure 7 presents the PPSTE test results for the two gear pairs, which were designed with identical macroand micro-parameters. The results demonstrate that manufacturing errors can cause significant variation in
PPSTE, even when the gear parameters are identical. This finding underscores the necessity of accounting for
tooth profile errors when predicting gear pair performance using the LTCA model. A notable difference in
PPSTE performance due to manufacturing errors was observed in the torque range below 60 Nm for the tested
gear pairs. However, at 70 Nm and above, the impact of manufacturing errors was negligible. This suggests
that the influence of manufacturing errors on gear transmission error is particularly significant under lowload conditions. Conversely, as the load increases, the gear teeth experience greater elastic deformation, which
reduces the sensitivity to manufacturing errors.
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LTCA model validation
To validate the accuracy of the LTCA model, a comprehensive comparison was conducted between the simulated
STE and the STE test results across the operational range of the target gears, which extended from 10 to 100 Nm.
As illustrated in Fig. 8, the PPSTE results for gear pairs #1 and #2 exhibited a high degree of concordance between
the experimental outcomes and the LTCA predictions across the entire spectrum of applied loads. Additionally,
an in-depth analysis was performed on the load conditions where the discrepancy in PPSTE between gear pairs
#1 and #2 was most pronounced, by comparing the STE across mesh cycles and tooth-meshing harmonics
with the test results. This analysis, the results of which are presented in Fig. 9 for a specific load condition
of 40 Nm, revealed PPSTE errors of 10.2% and 4.7% for gear pairs #1 and #2, respectively. Notably, the firstand second-order patterns of the tooth-meshing harmonics between the two sets of results displayed a high
degree of similarity. Given that the actual gear tooth profiles cannot be fully measured at each position using
gear inspection equipment, the accuracy of the LTCA model was deemed reasonable. Furthermore, the method
proposed by Mark for predicting transmission errors in the tooth-meshing harmonic region was successfully
validated using the mean profile26. Figure 9 highlights that for gear pairs with identical macro- and micro-design
specifications, not only do the PPSTE characteristics differ significantly, but the frequency attributes of the gear
excitation source also exhibit considerable variations under identical loading conditions. This underscores the
critical importance of accurately accounting for manufacturing errors during the design phase to precisely
predict the vibration behavior of the system.
Effects of surface waviness on gear transmission error
This section examines the influence of surface waviness generated by the manufacturing process on the STE of
helical gears. To assess the impact of amplitude, spatial frequency, and initial phase of the waviness, the waviness
was defined as presented in Eq. (28) and subsequently integrated into the design tooth profile to yield the final
tooth profile, as expressed in Eq. (29). The tooth profile fabrication process is illustrated in Fig. 10. The definition
of the y-z plane is reported in Section “Mean tooth profile error.” Ellen et al. conducted an analysis of surface
topology using the gear manufacturing method and confirmed that grinding induces waviness in the involute
direction41. Consequently, the waviness in the involute direction was also defined in this study, and similar
waviness was observed in the actual measured tooth profiles, as depicted in Fig. 4. The parametric study matrix
employed to analyze the effects of the factors defining waviness is presented in Table 5. The LTCA was performed
at 30 rpm and 60 Nm.
ηC,waviness (z) =
a
× sin (2πnz + φ)(28)
2
ηC (yi , z) = ηC,design (yi , z) + ηC,waviness (z)(29)
Effect of the waviness amplitude on the gear transmission error
Using the same method as Kissling, the amplitude of waviness was determined based on the form deviation
(ff α ) defined in ISO 1328-142. The reference form deviation was obtained from the measurements of the target
gear sample, and the values corresponding to 50%, 100%, and 150% of this reference were used to define the
criteria, as listed in Table 5. Figure 11 illustrates that significant differences in STE occur depending on the
amplitude for the same spatial frequency and initial phase. An increase in the waviness amplitude corresponded
to a corresponding increase in the magnitude of the higher-order STE components within the tooth-meshing
harmonics. In particular, the first order of the tooth-meshing harmonic exhibited the greatest STE magnitude at
0.5 ff α ; as the number of harmonics increased, the amplitude progressively diminished. In contrast, the largest
STE was observed at 1.5 ff α in the third order of the tooth-meshing harmonic, and the amplitude decreased as
the number of harmonics decrease.
Fig. 8. Comparison of the PPSTE between test and simulation results for (a) gear pair #1 and (b) gear pair #2.
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Fig. 9. Tooth-meshing harmonics comparison @ 40 Nm: (a) STE according to mesh cycle of gear pair #1; (b)
tooth-meshing harmonics of gear pair #1; (c) STE according to mesh cycle of gear pair #2; (d) tooth-meshing
harmonics of gear pair #2.
Fig. 10. Profile error fabrication process: (a) design profile; (b) waviness; (c) fabricated profile with waviness;
(d) measured mean profile error of wheel #1.
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Values
Study target
a
0.5ff α , ff α , 1.5ff α
n
φ
Amplitude
6
0◦
Spatial frequency
ff α
3, 6, 9
Initial phase
ff α
6
0◦
π
0, π
4, 2
Table 5. Study matrix for the effect of waviness on STE.
Fig. 11. STE results for different amplitudes of waviness: (a) STE in the mesh cycle; (b) tooth-meshing
harmonics.
Fig. 12. Comparison of LTCA results: (a) LSR and STE 2nd order derivative of 0.5 ffα; (b) LSR and STE 2nd
order derivative of 1.0 ffα.
The PPSTE values were 0.191 μm when no error occurred, 0.236 μm at 0.5 ff α , 0.198 μm at 1.0 ff α , and
0.231 μm at 1.5 ff α . Notably, PPSTE at 1.0 ff α , where the amplitude of the waviness was larger, was 16%
smaller than at 0.5 ff α . This suggests that as the amplitude of the waviness increases, the periodicity of the LSR,
which represents the load distribution between the teeth, becomes more pronounced. Moreover, the energy is
distributed into higher-order components during the load-transfer process, decreasing the PPSTE.
For a more detailed analysis, LSR interpretations were performed for the 0.5 ff α and 1.0 ff α conditions.
Figure 12 depicts the second derivatives of STE and LSR for these conditions. As evident from Fig. 12, the
waviness of the surface causes multiple inflection points in both the LSR and the STE at the same meshing point,
indicating the dispersion of energy into higher-order components during the load-transfer process. The secondderivative function illustrates the inflection patterns of the original functions, indicating that fluctuations in the
LSR owing to waviness during the load-transfer process directly affect the harmonic characteristics of the STE.
Although 0.5 ff α and 1.0 ff α have similar patterns in the second derivative of the LSR, 1.0 ff α , with a larger
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Fig. 13. STE results for different spatial frequencies of waviness: (a) STE in the mesh cycle; (b) tooth-meshing
harmonics.
Fig. 14. Comparison of the LTCA results: (a) LSR and STE 2nd order derivative of n = 3; (b) LSR and STE 2nd
order derivative of n = 9.
waviness amplitude, has a higher value in Fig. 12. This indicates a larger energy dispersion in the higher-order
components, resulting in a lower PPSTE.
Effects of waviness spatial frequency on the gear transmission error
The STE results as a function of the spatial frequency of the waviness are depicted in Fig. 13. For waviness with
the same amplitude and initial phase, the STE characteristics vary significantly with the spatial frequency. When
n is equal to 3, the PPSTE value reaches its maximum at 0.324 μm. In contrast, when n is equal to 9, the PPSTE
decreases by 37% compared to the no-error condition, reaching 0.119 μm. This reduction in PPSTE, as described
in Section “Effect of the waviness amplitude on the gear transmission error,” is attributed to multiple inflections
in the load-transfer process, which depend on the spatial frequency of the waviness, leading to energy dispersion
into higher-order components.
Figure 14 presents the second derivatives of STE and LSR for n values of 3 and 9. As illustrated in Fig. 14,
similar inflection patterns are observed between STE and LSR, consistent with previous results. For n equal to
9, the second derivative reveals larger and more rapid inflections, indicating that fluctuations in LSR and STE
become more abrupt and rapid as the spatial frequency of the waviness increases.
Effects of the initial phase of the waviness on the gear transmission error
The impact of the initial phase of waviness on the STE was examined. Figure 15 displays the STE as a function
of the initial phase of waviness. The initial phase of waviness affects only the spatial arrangement of peaks and
valleys along the involute direction without altering the harmonic energy distribution. Consequently, changes in
the initial phase lead to minimal variations in PPSTE and tooth-meshing harmonics. However, when the spatial
frequency is extremely low—a scenario that does not occur in practical applications—the initial phase may
influence transmission error by determining whether peaks or valleys of waviness align with critical meshing
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Fig. 15. STE results for different initial phases of waviness: (a) STE in the mesh cycle; (b) tooth-meshing
harmonics.
points, such as the start or end of a tooth contact. These localized effects, which arise due to the alignment of
peaks or valleys with engagement points, are suppressed at higher spatial frequencies, where amplitude and
frequency dominate the harmonic characteristics. This observation aligns with the findings of Sundar et al.43.
Conclusion
This study developed an LTCA model to account for the profile errors of helical gears and validated the model
through STE tests. The LTCA model incorporated the previously established TVMS models18 and the profile
error model proposed by Wang and Zhang24. The total tooth profile errors of the pinion and wheel were measured
using CNC inspection equipment. The mean profile error was subsequently integrated into the LTCA model
following the principles outlined by Mark26. The accuracy of the analysis model was verified by comparing the
STE results with experimental outcomes, with a particular focus on the PPSTE and tooth-meshing harmonics.
Finally, using the validated LTCA model, the effects of waviness on STE were analyzed by varying the amplitude,
spatial frequency, and initial phase of the waviness. The conclusions of this study are summarized as follows:
1.An LTCA analysis model was developed by combining the TVMS model of Chung et al.18 with the profile
error model of Wang and Zhang (and the accuracy of this model was confirmed through comparison of the
result with STE test results18,24.
2.The mean profile error derived for the entire tooth profile enabled accurate prediction of the tooth-meshing
harmonic region of the STE. Mark’s26 theory was validated by the LTCA model26.
3.The effects of waviness on STE were analyzed by varying the amplitude, spatial frequency, and initial phase of
the waviness. The results indicated that amplitude and spatial frequency significantly influenced the PPSTE
and harmonic characteristics, whereas the initial phase of the waviness had a negligible effect.
4.Comparison of the second-order derivatives of LSR and STE confirmed the presence of inflection points at
the same mesh positions, demonstrating that fluctuations in LSR directly affect the harmonics of STE.
By applying the proposed analytical model and methodology, LTCA can be performed to account for the profile
error of helical gears. The developed analytical model can be used as a solver for a robust design that accounts
for waviness.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding
author on reasonable request.
Received: 26 November 2024; Accepted: 10 February 2025
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Acknowledgements
Dongu Im and Woo-Jin Chung equally share lead authorship.
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Author contributions
Conceptualization: D.I., Y.J.P.; Methodology: D.I., Y.J.P.; Code development: D.I., W.J.C.; Investigation: D.I., H.L.,
Y.J.L.; Writing—original draft preparation: D.I.; Visualization: H.S., M.L.; Supervision: Y.J.P. All authors have
read and agreed to the published version of the manuscript.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit
sectors. This work was supported by Mid-Career Bridging Program through Seoul National University.
Declarations
Competing interests
The authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to Y.-J.P.
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