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Mechanism and Machine Theory 129 (2018) 261–278
Contents lists available at ScienceDirect
Mechanism and Machine Theory
journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
A revised time-varying mesh stiffness model of spur gear
pairs with tooth modifications
Yanning Sun a, Hui Ma a,b,∗, Yifan Huangfu a, Kangkang Chen a, LinYang Che a,
Bangchun Wen a
a
School of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 110819, PR China
Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, Shenyang,
Liaoning 110819, PR China
b
a r t i c l e
i n f o
Article history:
Received 28 January 2018
Revised 3 April 2018
Accepted 2 August 2018
Keywords:
Tooth modifications
Spur gear pairs
Time-varying mesh stiffness
Revised analytical model
Thin slice method
a b s t r a c t
Based on the thin slice assumption, a revised time-varying mesh stiffness (TVMS) model of
spur gear pairs with tooth modifications is developed. The spur gear is divided into many
individual slices along tooth width, and considering the revised fillet-foundation stiffness,
the nonlinear contact stiffness, the extended tooth contact and the tooth profile errors, the
stiffness of each slice gear pair is figured out. According to the relationship between the
deformation and the total stiffness in mesh period, the TVMS of spur gear pairs can be
worked out. Meanwhile, relative to the finite element (FE) method, the errors of the proposed method under different modification quantities are discussed. The proposed method
is more accurate than those previous methods but there are still some errors. Taking the
FE model as a benchmark, the TVMS is further revised based on a simple model updating
technique. Based on the revised model, the effects of the tooth width and torque on mesh
stiffness are also studied. The result shows that based on the proposed method, the TVMS
under any given modification quantities in a suitable range can be calculated accurately.
© 2018 Elsevier Ltd. All rights reserved.
1. Introduction
Tooth modification is an important method to reduce vibration and noise, which can be carried out by tooth profile
and lead crowning modifications. In recent years, many scholars have studied the vibration and noise behaviors of gears
with tooth modifications [1–9]. Lin and He [1] developed a finite element (FE) method to calculate the transmission error
of gear transmission systems with machining errors, assembly errors and modifications. Wang et al. [2] and Maatar and
Velex [3] analyzed the effects of gear modifications on the contact and dynamic characteristics of a gear pair. Bruyère
et al. [4] investigated the transmission errors of modified gears and obtained the optimal profile modifications which can
minimize the transmission error fluctuations. Velex et al. [5] proposed a design criterion for tooth modifications minimizing
dynamic tooth loads. They also pointed out that the transmission error fluctuations are related to the tooth modifications
[6]. Bahk and Parker [7] studied the effects of tooth profile modifications on the vibration of spur planetary gears. Ma et al.
Abbreviations: 3D, Three-dimensional; CW1∼CW9, Nine modification quantities for the gear pair 1; CL1∼CL8, Eight modification quantities for the gear
pair 2; FE, Finite element; FEM, Finite element model; PAM, Proposed analytical model; TVMS, Time-varying mesh stiffness.
∗
Corresponding author at: School of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 110819, PR China.
E-mail address: huima@me.neu.edu.cn (H. Ma).
https://doi.org/10.1016/j.mechmachtheory.2018.08.003
0094-114X/© 2018 Elsevier Ltd. All rights reserved.
262
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Nomenclature
Cq
C
Ca
dz
E
Epc
Lead crowning quantity
Tooth profile error in mesh position caused by tip relief along horizontal direction (x-direction in Fig. 2b)
Amount of profile modification
Width of each piece spur gear pair
Young’s modulus
Tooth errors of each slice gear caused by lead crowning relief along horizontal direction (x-direction in
Fig. 2b)
Epi
Total tooth profile error of the ith tooth pair of each slice gear pair along line of action
Eri
Static transmission error of the ith tooth pair of each slice gear pair at meshing point
F
Total meshing force
Fi (i = 1, 2)
Meshing force of ith meshing tooth pair
Fn (n = 1∼N) Total meshing force of the nth piece tooth
j
Number of meshing position
ka , kb , ks
Axial compressive stiffness, bending stiffness and shear stiffness
kn
TVMS of the nth piece tooth
ktooth
Total mesh stiffness of meshing teeth pairs
kitooth
Stiffness of the ith tooth pair
k∗h
Hertzian contact stiffness
kih
Nonlinear Hertzian contact stiffness of the ith tooth pair
kmean
Mean mesh stiffness of gear pairs
kit1 , kit2
Tooth stiffness of the ith tooth pair and subscripts 1 and 2 denote the driving and driven gears, resepectively.
kf
Stiffness of fillet-foundation
K
Total mesh stiffness of spur gear pairs with tooth modifications
L
Width of the tooth
La
Length of profile modification
lsfi
Load-sharing ratio of the ith tooth pair
q
Number of tooth pair
N
Number of slice gear pair
R
Radius of the lead crowning circular curve
Tl
Torque applied to the driving gear
Z
Number of tooth
zn
Coordinate of each sliced spur gear pair along the axial direction (z-direction in Fig. 3c)
Greek symbols
λ1 , λ2 (i = 1, 2) Coefficients of the fillet-foundation stiffness, subscripts 1 and 2 denote the driving gear and driven
gear, respectively
λk
TVMS correction coefficient
ν
Poisson’s ratio
[8] proposed a mesh stiffness model for profile shifted gears with addendum modifications and tooth profile modifications,
and determined the optimum profile modification curve under different amounts of tooth profile modifications.
The vibration characteristics of gear transmission systems will be significantly affected by the time-varying mesh stiffness
(TVMS) [10–13], which will change with tooth modifications. In the earlier study, the TVMS of healthy spur gear pairs can
be evaluated by analytical method on the basis of the potential energy method in elastic mechanics [14–18]. However, the
stiffness of the spur gear with tooth modifications or tooth faults is difficult to be determined accurately by this analytical
method, and the effects of gear errors are usually ignored. Subsequently, based on the analytical method, Chen and Shao
[19] took the gear errors into account and developed a TVMS calculation model, in which the relationship between the mesh
deformation and the total mesh stiffness in mesh period is determined. Furthermore, they also proposed a more general
calculation method for both healthy and tooth root crack cases, and developed an analytical model to calculate the mesh
stiffness of spur gear pairs with non-uniformly distributed tooth root crack, and the dynamic simulation of spur gears with
tooth root crack propagating along tooth width and crack depth is also carried out [20–22]. The effects of extended tooth
contact (ETC) on TVMS should not be ignored in TVMS calculation [23,24] and Ma et al. [25–28] established an improved
TVMS model for healthy gear pairs, cracked spur gears and gear pairs with tip relief, in which the transition curve, revised
fillet-foundation stiffness, nonlinear contact stiffness and ETC were considered. Liu et al. [29] studied the mesh stiffness of
a gear pair with tooth profile modification by analytical method and determined the optimal modification amount.
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
263
With the development of computer technique, the application of the FE method is becoming more and more widely.
And the flexibility of gears, tooth errors and modifications can be easily considered in the finite element model (FEM). So
the TVMS can be acquired by applying a meshing force on the theoretical engagement position of gear pairs [30–32] or
employing contact elements between the two contact faces of gears [33–35]. Ma et al. [36,37] developed an FEM of a spur
gear pair to calculate TVMS, and they also studied the influences of tip reliefs on the vibration responses. Based on the
FE method, Li [38,39] developed a model to calculate tooth surface contact stress and tooth root bending stress of spur
gear pairs, in which manufacturing errors, setting errors and tooth modifications were all considered. And the influences
of misalignment error, tooth modifications especially lead crowning relief and transmission torque on mesh characteristics
were further studied [40]. Compared with the analytical method on the basis of potential energy method, the FE method is
more accurate, but it is also relatively time-consuming. In order to combine the accuracy of FE method and the efficiency
of analytical method, an analytical-FE method is put forward which adopts a linear FE method to calculate the deformation
(shearing and bending) of the tooth and the whole gear body and employs a non-linear analytical formulation to compute
the local contact deformation near the contact point [41–44]. Fernandez et al. [41,42] researched the tooth profile modifications of spur gears and planetary gears by the analytical-FE method. Taking the profile errors into account, a non-linear
model of spur gears is developed based on an analytical-FE approach and the influences of the tip relief lengths on mesh
stiffness were studied [43].
In order to calculate TVMS of gear pairs with lead crowning relief, tooth shape deviations or alignment errors, the thin
slice method was developed [44–51]. That is, the tooth is divided into many individual slices along tooth width and each
slice can be regarded as a spur gear pair with rather small thickness. Supposing that the meshing deformation of the sliced
spur gear pair which has the minimum lead crowning quantity is the total mesh deformation of gear pairs, Wang et al.
[46] evaluated the TVMS of spur gear pairs with lead crowning relief and analyzed the vibration responses by the thin slice
method. Based on this method, an analytical model was also developed to calculate TVMS and contact stress of helical gear
pairs with tooth profile errors [47]. Considering tooth shape deviations and alignment errors, Ajmi and Velex [48] presented
a novel method to calculate deflections and load distributions on solid spur gears and helical gears, in which the elastic
couplings (also known as elastic convective effects) were also considered.
Generally, modifications include tooth profile modifications and lead crowning relief, moreover, tip relief is the most
common in tooth profile modifications and lead crowning relief with circular curve is mostly used. The analytical method
for mesh stiffness calculation of spur gears with tip relief has a higher accuracy because it can consider the influences of
nonlinear contact stiffness, ETC and revised fillet-foundation stiffness [27]. However, for lead crowning relief, the analytical
method can be challenging due to the different amounts of modification along the tooth width. Based on the slice method,
this paper recommends a revised TVMS model of spur gear pairs with tooth modifications including simultaneously tip relief
and lead crowning relief, which upgrades the overall calculation accuracy by improving the TVMS calculation precision of
each slice gear. The proposed method is also verified by the FE method. A further revision for the TVMS is carried out
by comparing TVMS obtained from the proposed analytical method (PAM) with that from the FE method. In addition, the
effects of tooth widths and transmission torques on mesh stiffness are also discussed.
2. Mesh stiffness model for spur gear pairs with tooth modifications
2.1. Analytical model
2.1.1. TVMS calculation for heathy spur gear pairs
The total mesh stiffness of meshing teeth pairs can be expressed as:
ktooth =
q
kitooth =
q
i=1
1
i=1 kih
1
+
1
kit1
+
1
kit2
,
(1)
where q denotes the number of tooth pairs in engagement; kitooth is the stiffness of the ith tooth pair; kih denotes the
nonlinear Hertzian contact stiffness which can be elaborated in Fig. 1; k∗h denotes the Hertzian contact stiffness without
nonlinearity, which can refer to Ref. [46]; kit1 and kit2 denote the stiffness of teeth. Subscripts 1, 2 denote the driving and the
driven gears, and superscript i denotes the ith tooth pair (see Fig. 2a); E represents Young’s modulus; ν denotes Poisson’s
ratio; L is the tooth width and F is the total meshing force; Fi is the meshing force of the ith meshing tooth pair; lsfi is the
load sharing ratio of the ith meshing tooth pair.
The tooth stiffness kit1 and kit2 can be expressed as:
kit1 =
1
1
kib1
+
1
kis1
+
1
kia1
, kit2 =
1
1
kib2
+
1
kis2
+
1
kia2
,
(2)
where the bending stiffness kb , the shear stiffness ks and the axial compressive stiffness ka can be written as:
1
=
kb
α
β
2
[cosβ (yβ − y1 ) − xβ sinβ ] dy1
(cosβ (yβ − y2 ) − xβ sinβ )2 dy2
dγ +
dτ ,
π
E Iy1
dγ
E Iy2
dτ
αC
2
(3)
264
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. 1. Calculation flowchart of nonlinear Hertzian contact stiffness of healthy gear pair.
Fig. 2. Schematic of tooth modifications for each slice spur gear: (a) schematic of meshing gear pair, (b) schematic of tooth with modifications, (c) enlarged
view of tooth with modifications.
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
265
Fig. 3. Schematic of the spur gear with tooth modifications. (For interpretation of the references to color in this figure, the reader is referred to the web
version of this article.)
α
β
1.2cos2 β dy1
1.2cos2 β dy2
dγ +
dτ ,
π
GAy1 d γ
GAy2
dτ
αC
2
α
β
1
sin2 β dy1
sin2 β dy2
=
dγ +
dτ ,
π
ka
E
A
d
γ
y1
αC E Ay2 dτ
2
1
=
ks
(4)
(5)
E
where α C is the pressure angle of the involute starting point; G = 2(1+
v ) is the shear modulus; β is the operating pressure
dy1 dy2
angle; and α is the pressure angle. xβ , yβ , y1 , y2 , dγ , dτ , Iy1 , Iy2 , Ay1 andAy2 can be found in Appendix A and Ref. [25].
The stiffness of fillet-foundation kf can be expressed as:
2
1
cos2 β ∗ uf
=
L
kf
EL
Sf
+M
∗
u f
Sf
+ P (1 + Q tan β ) ,
∗
∗
2
(6)
where the detailed expressions of uf , Sf , L∗ , M∗ , P∗ andQ∗ are elaborated in Ref. [17].
Based on the above equations, the total mesh stiffness kn of two meshing teeth can be redefined and kn of each slice
spur pair can be expressed as:
kn = 1/ (
1
λ1 kf1
+
1
+
ktooth
1
λ2 kf2
),
(7)
where λ1 and λ2 are the correction coefficients of the fillet-foundation stiffness which can refer to Ref. [27].
2.1.2. TVMS calculation for spur gear pairs with tooth modifications
The schematic of spur gears with tooth modifications is shown in Fig. 3. Following assumptions are made to calculate
mesh stiffness:
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Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
(1) The tooth in Fig. 3a can be divided into N individual slices (see Fig. 3b) along tooth width where N is the total slice
number. The top view is shown in Fig. 3c, and the convective (or coupling) effects among individual thin tooth pieces
are ignored.
(2) The quantity of modification is so small that teeth contact at the theoretical action plane.
(3) Each slice gear pair can be equivalent to a spur gear pair with tooth errors, and its TVMS can be calculated by Eq. (7).
As is shown in Fig. 3, the parts of the green dashed and dot line and the red dot line represent the tip relief and the lead
crowning relief, respectively. The linear tip relief (described by the length of profile modification La , the amount of profile
modification Ca ) and the arc lead crowning relief (described by lead crowning quantity Cq ) are adopted. Tooth profile error
in mesh position along horizontal direction (x-direction in Fig. 2b) in tip relief part can be defined as:
C = Ca (
u s
) , u ∈ [0, La ],
La
(8)
where u is the vertical distance between ideal mesh position and starting point of tip relief (see Fig. 2b); s is the index of
tip relief curve and s = 1 in this paper.
As for lead crowning relief part, the tooth errors of each slice gear along horizontal direction (x-direction in Fig. 2b) can
be expressed as:
Epc = R −
L L
R2 − zn 2 , zn ∈ [− , ],
2 2
(9)
where zn is the coordinate of the each slice gear along the axial direction (z-direction in Fig. 3c); R is the radius of the lead
crowning circular curve of the gear (see Fig. 3), which can be calculated as follows:
R=
(L/2 )2 + Cq 2
2Cq
.
(10)
The total tooth profile errors of the ith tooth pair of each slice spur gear along line of action can be expressed as:
Epi = (C + Epc ) cos β .
(11)
Considering the tooth modifications, the mesh stiffness due to teeth deformation can be expressed as:
ktooth =
F
Er2
− Epi
, where Epi = min(Ep1 , Ep2 ),
(12)
where Er2 is the static transmission error ignoring the fillet-foundation deformation and it can refer to Ref. [26]; superscript
1 or 2 denotes the 1st or 2nd tooth pair. Based on Refs. [19,26], load sharing ratio lsfi (i = 1 or 2) of the gear with tooth
modification can be calculated by:
ls f1 = 1 − ls f2 , ls f2 =
k2tooth (Er2 − Ep2 )
F
,
(13)
Substituting Eq. (12) into Eq. (7), the stiffness of each slice gear pair with tooth modifications can be obtained. Based on
the above analysis, the total mesh stiffness can be obtained as follows [19,46]:
F·
K=
F+
N
n=1
N
n=1
kn
,
(14)
kn Epc
Considering the nonlinear contact stiffness, the revised fillet-foundation stiffness and the ETC, the calculation procedure
of the TVMS is shown in Fig. 4. BC (see Fig. 2a) denotes the single-tooth contact region; Fn is the force of the nth slice gear
pair; rb1 is the radius of base circle of driving gear; Tl is the torque and detailed parameters about ETC can be found in Ref.
[26].
2.2. Model verification
Based on the basic parameters of a spur gear pair (defined as gear pair 1, see Table 1), a three-dimensional FE model
considering tooth modifications is established in ANSYS software (see Fig. 5). In the model, the gears are discretized using
Solid185 element, the contact between teeth are simulated using contact pairs (Targe170 and Conta174 elements), as shown
in Fig. 6. In the figure, the master node of the driven gear (O2 ) is completely constrained. Only the rotational degree of
freedom around the gear axis direction of the master node of driving gear (O1 ) is retained. Based on the rotation direction
of the gears, a constant loaded torque Tl in the right-handwise direction is applied to the master nodes of the driving gear.
According to the deformation of gears, the mesh stiffness can be acquired by the rotational angular displacement of the
master node of the driving gear.
Nine cases (CW1-CW9) for different lead crowning relief parameters are listed in Table 2. It should be noted that La
and Ca are all zero under these nine cases. The TVMS obtained from the proposed analytical method (PAM) is compared
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. 4. Calculation procedure of TVMS of a spur gear pair with tooth modifications.
Table 1
Parameters of gear pair 1.
Parameters
Driving gear/ driven gear
Parameters
Driving gear/ driven gear
Number of teeth Z
Module m (mm)
Tooth width L (mm)
Hub radius rint (mm)
Torque load Tl (Nm)
30, 25
2
20
5.5, 4.5
100
Pressure angle α (°)
Addendum coefficient ha ∗
Tip clearance coefficient c∗
Young’s modulus E (GPa)
Poisson’s ratio ν
20
1
0.25
210
0.3
267
268
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. 5. Schematic of the spur gear with tooth modification.
Fig. 6. Schematic of the FEM.
Table 2
Nine cases for different modification quantities of the gear pair 1.
Modification quantities
Cq (μm)
Modification quantities
Cq (μm)
Modification quantities
Cq (μm)
CW1
CW2
CW3
0
2.5
5
CW4
CW5
CW6
7.5
10
12.5
CW7
CW8
CW9
15
17.5
20
with those from the FEM and Wang’s model [46]. Wang’s model belongs to the traditional analytical model which neglects
the effects of some influence factors (such as extended tooth contact and nonlinear Hertz contact) so that the model is
not accuracy enough. The TVMS obtained from the three models (PAM, FEM and Wang’s model) under CW3 and CW5 are
shown in Fig. 7. Some errors between the PAM and the FEM, Wang’s model and the FEM under double- and single-tooth
contact regions (positions A and B) are listed in Table 3. It is clear that taking the revised fillet-foundation stiffness, the
nonlinear contact stiffness and the ETC into account, PAM is more accurate to calculate the TVMS but there are still some
errors. The error between the PAM and the FE method is relatively small and the maximum is about 9.8% but the maximum
error between Wang’s method and the FE method is as high as 30.8%.
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
269
Fig.7. TVMS using different methods: (a) CW3, (b) CW5.
Table 3
Errors between the PAM and FEM, Wang’s model and the FEM under double- and single-tooth contact regions.
Modification quantities
TVMS (× 108 N/m)
FEM
PAM
Error (%)
Wang’s model
Error (%)
CW3
kA
kB
kA
kB
1.782
1.466
1.633
1.327
1.954
1.564
1.786
1.457
9.7
6.7
9.4
9.8
2.330
1.477
1.909
1.273
30.8
0.8
16.9
4.1
CW5
Fig. 8. TVMS considering the tooth modifications: (a) PAM, (b) FEM.
3. A further revision for improving the calculation accuracy
3.1. Revision of PAM under lead crowning relief
The TVMS obtained from PAM and FEM is shown in Fig. 8, which shows the TVMS error between two methods. To
further improve the accuracy of the PAM, a revision method is proposed. The mean square error function between the PAM
and the FEM can be defined as:
φ (λ ) =
(λkA−PAM − kA−FEM )2 + (λkB−PAM − kB−FEM )2 ,
(15)
where λ is the independent variable of this function; kA-PAM and kB-PAM denote the TVMS from the PAM at meshing positions
A and B, respectively; kA-FEM and kB-FEM denote the TVMS from the FEM at meshing positions A and B, respectively.
Based on the minimum mean square error, the TVMS correction coefficient λk under CW1, CW3, CW5, CW7 and CW9
are plotted in Fig. 9 (red circles). The best fitting curve of λk and Cq is figured out by polynomial interpolation method and
270
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. 9. The best fitting curve for correction coefficient λk. . (For interpretation of the references to color in this figure, the reader is referred to the web
version of this article.)
Table 4
TVMS errors compared with the FEM.
Modification quantities
Mesh stiffness (× 108 N/m)
FEM
PAM before revision
Error (%)
PAM after revision
Error (%)
CW1
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
2.096
1.682
2.015
1.922
1.560
1.843
1.782
1.466
1.717
1.701
1.385
1.635
1.633
1.327
1.572
1.588
1.285
1.525
1.543
1.247
1.483
1.514
1.212
1.451
1.479
1.181
1.420
2.157
1.688
2.076
2.050
1.624
1.982
1.954
1.564
1.895
1.866
1.509
1.815
1.786
1.457
1.740
1.713
1.409
1.673
1.646
1.364
1.608
1.584
1.321
1.549
1.526
1.282
1.494
2.9
0.4
3.0
6.7
4.1
7.5
9.7
6.7
10.4
9.7
8.9
11.0
9.4
9.8
10.7
7.9
9.6
9.7
6.7
9.4
8.4
4.6
8.9
6.8
3.2
8.5
5.2
2.116
1.650
2.037
1.940
1.537
1.872
1.801
1.442
1.748
1.706
1.379
1.656
1.630
1.330
1.587
1.578
1.298
1.536
1.527
1.266
1.493
1.489
1.241
1.455
1.449
1.217
1.417
1.0
1.9
1.1
1.0
1.5
1.6
1.1
1.6
1.8
0.3
0.4
1.3
0.9
0.2
1.0
0.6
1.0
0.7
1.0
1.5
0.7
1.7
2.4
0.3
2.0
3.0
0.2
CW2
CW3
CW4
CW5
CW6
CW7
CW8
CW9
J
Note: kmean is the mean stiffness which can be calculated by: kmean =
j=1
J
kj
, where J is the number of mesh positions.
it can be written as Eq. (16):
λk = p1Cq3 + p2Cq2 + p3Cq + p4 ,
(16)
where p1 = − 2.931 × 10−5 , p2 = 1.408 × 10−3 , p3 = − 1.803 × 10−2 and p4 = 0.981 are the coefficients of this polynomial function. Compared with the FEM, the errors before and after revision are shown in Table 4.
Calculation procedure of correction coefficient λk is shown in Fig. 10. The correction coefficient λk is calculated based on
the calculation procedure under CW2, CW4, CW6 and CW8 (see Fig. 9, black triangles), which agree well with best fitting
curve. The TVMS under those modification quantities is shown in Fig. 11. Those figures show that TVMS obtained from the
PAM and the FEM are similar. So it is reasonable to improve the calculation accuracy by λk .
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. 10. Calculation procedure of correction coefficient λk .
Fig. 11. TVMS obtained from different methods: (a) FEM, (b) PAM before revision, (c) PAM after revision.
271
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Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Table 5
Parameters of gear pair 2.
Parameters
The driving/driven gear
Parameters
The driving/driven gear
Tooth number Z
Module m (mm)
Tooth width L (mm)
Hub radius rint (mm)
Torque load Tl (Nm)
20, 30
4
40
11.7, 38.3
98
Pressure angle α (°)
Addendum coefficient ha ∗
Tip clearance coefficient c∗
Young’s modulus E (GPa)
Poisson’s ratio ν
20
1
0.25
212
0.3
Table 6
Eight modification quantities for driving gear.
Modification quantities
Ca (μm)
La (mm)
Cq (μm)
CL1
CL2
CL3
CL4
CL5
CL6
CL7
CL8
0
0
0
0
0
0
8
8
0
0
0
0
0
0
0.6
0.6
0
2
4
6
8
10
0
10
Fig. 12. TVMS under different modification quantities: (a) PAM, (b) FEM.
3.2. Revision of PAM under lead crowning relief, tip relief and combination of both
To further verify the proposed method, the gear pair in Ref. [40] is adopted, in which only the tooth of the driving
gear is modified. And the detailed gear parameters are listed in Table 5 (defined as gear pair 2). The TVMS obtained from
the PAM and the FEM under different modification quantities (CL1–CL6, see Table 6) is shown in Fig. 12a and Fig. 12b,
respectively. The correction coefficient λk under CL1-CL6 and the best fitting curve are plotted in Fig. 13. The correction
coefficient increases with the increasing Cq . When Cq = 10 μm, the maximum error is about 13.1% before revision while
about 4.9% after revision (see Table 7). The formula of λk can be expressed as:
λk = 1.130 × 10−3 · Cq2 + 3.115 × 10−3 · Cq + 0.977,
(17)
The TVMS under CL1, CL6, CL7 and CL8 is shown in Fig. 14. It indicates that the PAM can consider the combined effects
of tip relief and lead crowning relief on TVMS and accurately figure out TVMS under any given modification quantities.
4. Effects of different parameters on TVMS
The transmitting load will also affect the contact behavior between a mating toot pair under a certain amount of modification. And tooth widths may influence the validity of slicing principle. In this section, the effects of tooth widths and
torques on TVMS of spur gear pair 1 are discussed.
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
273
Fig. 13. The best fitting curve of correction coefficient λk .
Table 7
Mesh stiffness obtained from PAM and FEM.
Modification quantities
Mesh stiffness (× 108 N/m)
FEM
PAM before revision
Error (%)
PAM after revision
Error (%)
CL1
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
7.357
5.031
6.781
6.277
4.491
5.813
5.525
4.052
5.140
5.033
3.713
4.706
4.739
3.474
4.411
4.464
3.258
4.193
7.770
5.027
7.133
6.445
4.455
6.001
5.517
3.999
5.174
4.830
3.629
4.557
4.301
3.321
4.080
3.881
3.061
3.695
5.6
0.1
5.2
2.6
0.8
3.2
0.1
1.3
0.7
0.1
2.3
3.2
9.2
4.4
7.5
13.1
6.0
11.9
7.608
4.922
6.969
6.348
4.388
5.928
5.550
4.023
5.213
4.999
3.756
4.723
4.649
3.590
4.383
4.335
3.419
4.143
3.4
2.2
2.8
1.1
2.3
2.0
0.5
0.7
1.4
0.7
1.2
0.4
1.9
3.3
0.6
2.9
4.9
1.2
CL2
CL3
CL4
CL5
CL6
Fig. 14. TVMS under different modification quantities: (a) PAM after revision, (b) FEM.
274
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. 15. TVMS of the spur gear pairs with modification under different L: (a) PAM after revision, (b) FEM.
Table 8
Mesh stiffness obtained from FEM and PAM after revision.
L (mm)
Mesh stiffness (× 108 N/m)
PAM after revision
FEM
Error (%)
20
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
1.630
1.330
1.585
2.776
2.328
2.704
3.634
3.109
3.543
4.304
3.738
4.202
1.633
1.331
1.572
2.744
2.232
2.638
3.656
2.970
3.511
4.445
3.619
4.283
0.2
0.1
0.8
1.2
4.1
2.5
0.6
4.5
0.9
3.3
3.2
1.9
40
60
80
4.1. Effects of tooth widths on TVMS
Based on the proposed method and FE method, the TVMS under different tooth widths (L = 20 mm, 40 mm, 60 mm and
80 mm) and the modification quantity CW5 is shown in Fig. 15. Two methods all show that the magnitude of mesh stiffness
increases with the increase of L, the TVMS obtained from the revised PAM agrees well with that from the FEM and the
maximum error is only 4.5%, and kmean is more accurate (maximum error is about 2.5%, see Table 8). This also suggests that
the proposed method is also valid for wide-faced spur gears.
4.2. Effects of torques on TVMS
TVMS under different transmission torques (Tl = 50 Nm, 100 Nm, 150 Nm and 200 Nm) and the modification quantity
CW5 is shown in Fig. 16. The figure shows that TVMS obtained from PAM agrees well with that from the FEM. The maximum
error under double-tooth contact region is about 4.8%, and maximum error of mean stiffness is about 5.5% (see Table 9). The
stiffness increases with the increasing Tl and increases slowly under larger torques. In addition, the region of single-tooth
engagement becomes smaller. That is to say, the increasing torque Tl enhances the effects of the ETC on mesh stiffness.
5. Conclusions
Based on the thin-slice method, a previously developed analytical model for healthy spur gear pairs is further improved
to consider the effects of tooth modifications including tip relief and lead crowning relief. The proposed model is verified
and revised by comparing with the finite element model (FEM), and the TVMS correction coefficient is figured out based on
the minimum mean square error. The revision process is elaborated by two cases with different modification types (case 1:
only lead crowning relief for driving and driven gear; case 2: lead crowning relief, tip relief and combination of both only
for driving gear). The effects of tooth widths and torques on mesh stiffness are also discussed. Some detailed conclusions
are summarized as follows:
(1) Under case 1 before revision, the errors between two models are very small and the maximum error before revision is
about 11% at Cq = 7.5 μm where Cq denotes the lead crowning quantity. However, the maximum error after revision is
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
275
Fig. 16. TVMS of the spur gear pairs with modification under different Tl : (a) PAM after revision, (b) FEM.
Table 9
Mesh stiffness obtained FEM and PAM after revision.
Tl (Nm)
Mesh stiffness (× 108 N/m)
PAM after revision
FEM
Error (%)
50
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
kA
kB
kmean
1.388
1.164
1.316
1.630
1.330
1.546
1.733
1.399
1.650
1.790
1.437
1.710
1.454
1.164
1.393
1.633
1.327
1.572
1.735
1.427
1.676
1.804
1.488
1.747
4.8
0
5.5
0.2
0.2
1.7
0.1
2.0
1.6
0.8
3.5
2.1
100
150
200
about 3.0% at Cq = 20 μm. Under case 2, the maximum error before revision is about 13.1% at Cq = 10 μm. However, the
maximum error after revision is about 4.9% at Cq = 10 μm.
(2) The proposed analytical method (PAM) is also valid for wide-faced spur gears. Under different tooth widths, the TVMS
obtained from the revised PAM is in good agreement with that from the FEM and the maximum error is only 4.5%. Under
different torques, the TVMS obtained from the PAM is also in good agreement with that from the FEM and the maximum
error is 5.5%. With the increasing torque, the region of single-tooth engagement becomes smaller, i.e., the large torque
will enlarge the region of the extended tooth contact.
At the end, it should be noted that the FEM is the prerequisite to determine the correction coefficient in order to make
sure that the proposed analytical method can yield accurate results. Thus it is inevitable that the computation time will
increase using the proposed method. However, once the correction coefficient is determined, the calculation efficiency will
be greatly improved for the studied gear pair in the optimizing process of relief parameters. In our future work, we will
focus on the improvement of calculation accuracy by considering the effects of elastic couplings among thin slices or using
analytical-FE method.
Acknowledgment
This project is supported by the National Natural Science Foundation (Grant no. 11772089), the Fundamental Research
Funds for the Central Universities (Grant nos. N170308028, N160313004 and N160312001).
Appendix A.
The gear tooth is modeled as a nonuniform cantilever beam on root circle in this paper (see Fig. A.1). Radii of the base
circle rb and root circle rf of the gear can be expressed as:
rb =
1
1
mZ cosα , rf = mZ − (h∗a + c∗ )m,
2
2
(A.1)
276
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
Fig. A.1. Geometric model of the gear profile.
where m is the module, Z is the number of teeth, α is the pressure angle of the gear pitch circle, ha ∗ is the addendum
coefficient and c∗ is the tip clearance coefficient. If the gear is a standard spur gear with ha ∗ = 1, c∗ = 0.25 and α = 20°
Tooth profile curve can be divided into four parts: addendum curve AB, involute curve BC, transition curve CD and dedendum curve DE (see Fig. A.1). The transition curve is cut out by the cutter tip. From the perspective of the machining
process of a gear tooth, transition curve is the tooth profile between the involute starting point and the root circle, and no
matter how many teeth the gear has, the transitional part will always exist. The shape of the transition curve is directly
dependent on the shape of the cutter tip. When the shape of the cutter tip is ordinary fillet, the transition curve equations
are [25,52]:
x1 = r × sin(
) − (a1 /sin γ + rρ ) × cos(γ − )
, ( α ≤ γ ≤ π /2 ),
y1 = r × cos(
) − (a1 /sin γ + rρ ) × sin(γ − )
(A.2)
where r is the radius of the pitch circle; = (a1 / tan γ + b1 )/r, a1 = (h∗a + c∗ ) × m − rρ , b1 = π m/4 + h∗a m tan α + rρ cos α ;
rρ = c∗ m/(1 − sin α ).
Equations of involute curve are expressed as follows:
x2 = rb [(τ + θb )cosτ − sinτ ]
, (αC ≤ τ ≤ αa ),
y2 = rb [(τ + θb )sinτ + cosτ ]
(A.3)
where τ (α C ≤ τ ≤ α a ) is the pressure angle of the arbitrary point at the involute curve, in which αC = arccos(rb /rC ),αa =
arccos
(rb /ra ) is the pressure angle of the involute starting point and the addendum circle, respectively; rC =
(rb tan α − h∗a m/sin α )2 + rb2 is the radius of the involute starting point circle. θ b is the half tooth angle on the base circle
of the gear, θb = 2πZ + inv α .
β is the operating pressure angle and the coordinate of contact point (τ = β ) can be expressed as follows:
xβ = rb [(β + θb )cosβ − sinβ ]
,
yβ = rb [(β + θb )sinβ + cosβ ]
(A.4)
Iy 1 , Ay 1 , Iy 2 , Ay 2 represent the area moment of inertia and the cross-sectional area, respectively, which can be calculated
by Eqs. (A.5) and (A.6):
Ay1 = 2x1 L, Ay2 = 2x2 L,
(A.5)
2 3
2
x L, Iy2 = x32 L,
3 1
3
(A.6)
Iy1 =
dy1
dγ
and
dy2
dτ
can be calculated as follows:
dy1
=
dγ
α1 sin
α1 / tan γ +b1 r
tan2 γ
(1 + tan2 γ )
α1 cos γ
α1 / tan γ + b1
+
sin γ −
2
r
sin γ
Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278
α1
α1 / tan γ + b1
−
+ rρ cos γ −
sin γ
r
dy2
= rb (τ + θb ) cos
dτ
τ,
α1 1 + tan2 γ
1+
r tan2 γ
277
(A.7)
(A.8)
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