Impact Predictions of Face Gear Rim Thicknesses on Mesh Stiffness of Face Gear Drives
H.S. Chen, Z.M.Q. Li, X.S. Liu, H. Wu, J. Wang College of Mechanical and Electrical Engineering
Nanjing University of Aeronautics and Astronautics China
Abstract-Mesh stiffness is one of fundamental parameters of face gear dynamics. Thus, a calculation solution of mesh stiffness of face gear drives was constructed by finite element method. The impact of face gear rim thicknesses on mesh stiffness was predicted. The analytical results indicate that a face gear rim is thicker, average mesh stiffness is larger, but changes of average mesh stiffness would slow versus changes of face gear rim thickness. The contributions are benefit to improve the design of face gear drives associated with low vibrations.
Keywords- face gear drives; tooth contact analysis; finite element method; gear mesh stiffness
I. INTRODUCTION Face gear drives are addressed by many scholars due to its
insensitive manufacture and alignment errors versus bevel gear drives [1~5]. Face gear dynamics is a part of studies of face gear drives, and mesh stiffness is one of the fundamental parameters of face gear dynamics. Mesh stiffness calculation solutions of spur and bevel gear drives have been constructed [6~9]. However, mesh stiffness of face gear drives with a spur gear is yet to be further researched. Thus, a constructed solution of face gear drives mesh stiffness was provided, and the impact of face gear rim thicknesses on mesh stiffness is predicted in this study. It is based on the contact finite element method (FEM) for face gear drives. The geometric modelling of a face gear drive is constructed. The mesh stiffness solution and data processing were analysed in FEM software. The deformations of different rims are also computed. And then mesh stiffness of face gear drives of different rim thicknesses are obtained. The conclusions of this study indicate that a thicker face gear rim will lead to larger average mesh stiffness, but changes of average mesh stiffness would slow versus changes of face gear rim thickness. Therefore, the paper provides an approach for calculating mesh stiffness of face gear drives, and the impact prediction is meaningful for its design process.
II. CONSTRUCTION OF A GEOMETRIC MODELLING SOLUTION
A. Equations of Face Gear Tooth Surface Face gears are manufactured by involute generating cutters.
The coordinate system of a cutter [10] is given by Fig. 1. φs is the rotational angle of the cutter. The tooth surface vector equations [10] of the cutter are expressed as:
[ ]0 0
0 0
[sin( ) cos( )][cos( ) cos( )]
( , )
1
bS S S S S S
T bS S S S S SS S S S S S
S
rr
r u x y z tu
θ θ θ θ θθ θ θ θ θ
θ
± + − +⎡ ⎤⎢ ⎥− + + +⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦
r
(1)
Where rbs is the base circle radius of the cutter; uS and θS are the axial parameter and angle parameter of a point on cutter surface, respectively; φS is the rotational angle of the gear cutter. θS0 is an angle parameter from the symmetric plane of the tooth space to the initial point of the involute; “±” corresponds to the involutes at the two sides of the tooth space, namely γ-γ and β-β, respectively.
Unit normal of the cutter tooth surface [10] is expressed as:
0
0
cos( )/ /
sin( )/ /
0
S ST S S S S
S Sx Sy Sz S SS S S S
r r un n n n
r r u
θ θθ
θ θθ
+⎡ ⎤∂ ∂ ×∂ ∂ ⎢ ⎥⎡ ⎤= = = − +⎣ ⎦ ⎢ ⎥∂ ∂ ×∂ ∂
⎢ ⎥⎣ ⎦
mr rr
r r
(2)
The processing coordinate systems of an orthogonal face gear are presented in Figure 2. There are four coordinate systems, which are two fixed ones for the cutter S and the face gear 2, named OS0xS0yS0zS0 and O2x20y20z20, respectively; two rotating ones for the cutter S and the face gear 2, named OSxSySzS and O2x2y2z2, respectively. The motion relationship of the gear shaper and a face gear is shown in Figure 2 (a).
FIGURE I. THE COORDINATE SYSTEM OF THE CUTTER TOOTH
SURFACE [10].
FIGURE II. THECOORDINATE SYSTEMOF FACE GEARS MACHINE
[10]
ys0
xs0
ys
xsφs
γ
β
γβ
θs
θsθs0
International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015)
© 2015. The authors - Published by Atlantis Press 180
Tooth surface equations of a face gear are deduced by the conversion relationship from coordinate system SS to coordinate system S2. They are expressed [10] as:
2 2,( , , ) ( , )
( , , ) 0S S S S S S S
S S S
r u M r u
f u
θ ϕ θ
θ ϕ
⎫⎡ ⎤= ⎪⎣ ⎦ ⎬= ⎪⎭
r r
(3)
Where [M2, S] represents the transformation matrix of the two coordinate systems, and the second equation can be achieved by the correct meshing conditions. Therefore, the tooth surface equations can be expressed[10] as below:
22
2
22 2 2 2 2
2
sin[cos (sin cos ) ]
coscos
( , ) [ ] [sin (sin cos ) ]cos
(sin sin )
bS SS
TS S bS S
S
bS S
rq
r u x y z rq
r
θ θθ
θ θθ
θ θ
ϕϕ ϕ θ ϕ
ϕϕ
θ ϕ ϕ θ ϕϕ
ϕ θ ϕ
⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥
= = − +⎢ ⎥⎢ ⎥⎢ ⎥− ±⎢ ⎥⎣ ⎦
m
rm
(4)
Where q2S is the gear ratio of the face gear and the gear cutter; φ2 =q2Sφ2; φθ =φS± (θS0+θS).
B. Tooth Contact Analysis The point contact face gear drives are studied because of
unbalanced loading of line contact. A medium gear is a gear that can meet the line contact with the face gear and the pinion at same time. It is an imaginary gear. Its geometry parameters are always consistent with the cutter, and the tooth profile coordinate is shown in Figure3 [11]. In Figure 3, rb is the base circle radius of the medium gear; rk is the radius vector of a point on the tooth profile, named k. θk is the angle between rk and y.
Hence, the tooth profiles equations [11] of the medium gear are expressed as:
sincos
m k k
m k k
m k
x ry rz u
θθ
= ±⎧⎪ =⎨⎪ =⎩ (5)
Where uk is a variable of the tooth profile equations of the medium gear in the tooth width direction. “+” represents the left tooth profile; “-” represents the right tooth profile in xm. Hence the homogeneous equation of the tooth profile is
[ 1]Tm m mx y z=mR , , , .
The coordinate systems of the point contact face gear drives are shown in Figure4 [11]. In the coordinate systems, OgXgYgZg and Og'Xg'Yg'Zg' are the static coordinate and mobile coordinate of the pinion, respectively; OmXmYmZm and Om'Xm'Ym'Zm' are the static coordinate and mobile coordinate of the medium gear, respectively; OfXfYfZf and Of'Xf'Yf'Zf' are the static coordinate and mobile coordinate of the face gear, respectively. The plane OgXgYg and OmXmYm are in the same plane. The distance between Og and Om is a. The distance between the plane OmXmZm and OfXfYf is ra, and it is equal to the radius of the addendum circle of the medium gear. The plane OmXmYm and OfXfZf are parallel, and the distance between them is d.
FIGURE III. TOOTH PROFILES COORDINATE OF THE MEDIUM GEAR.
FIGURE IV. THE DRIVE COORDINATE SYSTEM OF POINT CONTACT
FACE GEARS [11].
The surfaces equations and the envelope condition can be expressed[11] as eqn. (6) and eqn. (7). These equations are under the envelope principle.
' 'm f m m=zR M R (6)
0mz mz mz
k k muα ϕ∂ ∂ ∂
× ⋅ =∂ ∂ ∂R R R
(7|)
Where Mf'm' is the homogeneous matrix that the medium gear generates the face gear with envelope method. The surfaces equations of the medium gear can be expressed as eqn (8)The surfaces are enveloped by the spur gear. The envelope condition is expressed as eqn(9) [11].
' 'g m g g=zR M R (8)
0gz g gz
k k muα ϕ∂ ∂ ∂
× ⋅ =∂ ∂ ∂
R R R
(9)
Where Rg=Rm, Mm'g' is the homogeneous matrix when the pinion envelopes the medium gear. The four equations eqn(6), (7), (8) and (9) are contact equations of a face gear drive.
C. Geometry Model of a Face Gear Discrete data points of the tooth surface can be solved
from the equations of the face gear tooth surface by numerical calculation. The parameters of the face gear drive are shown in Table 1. The gear tooth surface can therefore be simulated as presented in Figure5. In the picture, the top land is sharpened at the inner radius and the dedendum is undercut at the outside radius. The contact points can be resolved as shown in Figure5 according to the above contact equations. To optimize the transmission, an appropriate face gear width like in Figure6 is chosen so that the contact points are on the middle of face width. Then a three-dimensional solid model can be simulated with the face gear surface in Figure6in software. The model of
xm Om
ym
rb
rkθk
OmOg
X f '
X f
Y f 'Y f
Zf '
Zf
Of
ωf
φf
ra d
X m
X g X m'Xg'
YgYg'Ym
Y m'φm
φg
ZmZm'
ZgZg'ωm
ωg
aFace gear
PinionMedium gear
181
theset
A.
mochgesho
thepindoAf
e face gear is t as 0.5mm to
FIGU
FIGU
TABLE
Descript
Module (Number PressureTooth wiModifica
FIGURE V
FIGUR
III. THE L
The FEM MReduced inte
odel of the fachosen to reducar rim thicknown in Figure
Rigid couplie gear axis annion profile. T
o not have rotafterwards, rota
shown in Figapproximate
URE V. TOOTH
URE VI. TOOTH
E I. THE FA
ion
(mm) of teeth
e angle (°) idth (mm) ation coefficient
VII. THE
RE VIII. F
LTCA OF A FA
Model egration elemce gear drive. ce the amounness into cone8.
ing is created nd the faces, This is becausational degreeational veloci
gure7. Here ththe situation o
SURFACE SIMU
H CONTACT AN
ACE GEAR DRIVE P
VaSpur gear
4 24 20 30 0
E FACE GEAR S
EM MODEL CO
CE GEAR DRIV
ment C3D8R isA partial face
nt of computansideration. T
between the as well as these three-dimene of freedom iity and torque
e length of itsof no rim.
ULATION.
NALYSIS.
PARAMETERS.
alues Face gear
4 96 20 30 0
SOLID MODEL
ONSTRAINT
VE WITHOUT A
s applied to the gear drive mations and takThe FEM mo
references poe inner ring ansion solid elin the FEA soe can be appl
s rim is
A RIM
he FEM model is ke face odel is
oints on and the ements ftware. lied on
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B.
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rare
stetw
asco
stefixitsonbesesafreonth
C.
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anThloapextimcovaitedede
he reference pmodels of spur ace gear driven
PreprocessoThe preproc
ow to select thads, boundarhether the reschnologies. T
(1) Definitioatio of the twoespectively.
(2) Analysieps to ensure
wo general ana
(3) Definitios contact pairontact property
(4) Boundaep, the face gxed with the es rotational axn the spur geetween two gecond analysisame as the laseedom of degn the referenche maximum li
. Results of thIn the pape
ivided into fiftevery positio
an be calculate
The middlend the tooth che picture shong ellipse an
pplied. The txported as a fme varying coontact forces oariable. The erative resulteflection of theformation, re
FIGUR
points. The logear and face
n by the spur g
ors of the FEMcessors of a fahe grid elemenry conditionssults are accu
The main steps
on of materialo gears mater
s Type: Analyconvergence
alysis steps.
on of contact:rs, and set thy options.
ary conditionsgear is compexception of thxis. At the samear around thgears and accs step, the cost step. At thegree of the spuce point of theinearly in the
he Finite Elemr, the meshinteen positions
on. The loads aed by FEM so
e tooth surfacecontact force ows that the cd it leans on total elastic field variable ontact forces of middle geaimage is giv
t of the cohe tooth surfacespectively. Th
RE IX. CONTAC
oads will be tre gear. Therebgear can be sim
M Analysis Moace gear drivent, the settingss, etc. All urate. Therefos are:
l: The elastic mrial are all set
ytical processof tooth conta
: Define the che friction c
s and loads: pletely fixed ahe rotational d
me time, applyhe axis to getcelerate the c
onstraints of te same time, rur gear and ape spur gear. Tmode of analy
ment Analysis ng process of s. The face geaand deformati
oftware.
e of the gear distribution icontact area othe tooth surdeflection ofand presentecan be picke
ar surface areven in Figurntact force
ce are labelledhey are shown
CT FORCE DIST
ransferred to by, the motionmulated.
odel eFEM model s of the analyof these det
ore, they are t
modulus and Pas 206GPa a
s is divided inact analysis; t
ontact tooth scoefficient as
In the first aand the spur degree of free
y a little displat the perfect convergence; the face gear release the ro
pply a 200N·mThe load increysis step time
a single gearar rotates 0.5 dion of these po
sectors is anis shown in Fof the face gerface when a f the gear toed in Figure 1ed up when the regarded as re 11. The uand the ma
d as the force n in Table 2.
TRIBUTION.
the FE n of the
include sis type, termine the key
Poisson and 0.3,
nto two they are
surfaces 0.1 in
analysis gear is
edom of acement contact In the are the tational
m torque eases to .
r pair is degrees ositions
nalyzed, Figure9. ear is a load is
ooth is 10. The he total history
ultimate aximum and the
182
byTa
whfactotnatheap
addebacuploshomeslo
FIGURE X. T
FI
TABLE II.
Rotational
angle (°)
Cont
forcN
0 947.40.5 13111 1750
1.5 24762 2787
2.5 31703 3343
3.5 3395
IV. THE INFL
The contact y referring to able 1.
Face gear rihich are illustrce gears to cotal deformatio
amely the defoe inner ringsplied on the o
Then, the todded. The mulrived by ove
ased on the curves and averotted in Figurow that a largesh stiffness, ow versus cha
Tota
l con
tact
forc
e:F n/
kN
TOOTH SURFAC
IGURE XI. CONT
CONTACT FORCEPO
ntac
ces()
Deformaton (mm)
437 5.32×10-3
1.23 4.62×10-3
0.94 5.35×10-3
6.49 5.92×10-3
7.14 5.40×10-3
0.46 5.52×10-3
3.93 5.66×10-3
5.48 5.77×10-3
LUENCE OF RIMSTI
ratio of the fliterature [10
ims are takenrated in Figurompute their don of one facormation of gs of these rimouter rings. Th
otal deformatilti-tooth mesherlapping the ontact ratio.
rage mesh stifre 13 and Figuger rim thicknbut changes
anges of face g
0.2 0.4 0.60
1
2
3
4
n
CE DEFLECTION
TACT FORCE C
E AND DEFORMATOSITION.
ti)
Rotational
angle (°)
3 4 3 4.5 3 5 3 5.5 3 6 3 6.5 3 7 3
M THICKNESS OIFFNESS face gear driv0]. The param
n to be 2.5mmre 12. The rimdeformation ace gear is divear teeth and ms are fixed,he results are p
on of the threh stiffness cur
results of siThe time-var
ffness of threeure 14, respecness will leadof average m
gear rim thickn
6 0.8 1 1.Time:t/s
N DISTRIBUTIO
CURVE.
TION AT EVERY A
Contact
forces(N)
Defoon (
3304.62 5.692926.75 5.652730.44 5.682071.68 4.881667.10 4.811352.99 5.151023.60 6.04
ON FACE GEAR
ve is solved asmeters are sho
m, 5mm and 7ms are separatealone. Therefovided into two
its rim. In thi, and the torpresented in T
ee face gears rve can therefingle mesh strying mesh ste face gear drictively. The pd to larger fac
mesh stiffness ness.
2 1.4
ON.
ANGLE
rmatimm)
×10-3
×10-3
×10-3
×10-3
×10-3
×10-3
×10-3
R MESH
s 1.791 own in
7.5mm, ed from ore, the o parts, is step, rque is
Table 3.
can be fore be tiffness tiffness ves are
pictures ce gear would
FI
FI
widedrlincoanstirimavrimfeca
TAB
Rim thicknes
Deformation
IGURE XII.
FIGURE XIII.
GURE XIV.
Three-dimenith face gearerived by genrive was consnear brick reontact force analysis. The iiffness was anm is thicker, average mesh sm thickness. asible and
alculation of fa
aver
age
mes
hsti
ffne
ss/N
·mm
-1
BLE III. THE
ss/mm 2.
n/mm 1.74×
THE FACE G
GEAR METHIC
AVERAGE MTHIC
V. Cnsional solid r tooth surfacnerating methostructed by thduced integraand total deflinfluence of dnalyzed, and average meshtiffness wouldThe solutioneffective me
face gear drive
0 51
2
3
4
5x 105
Rotation
Tota
l mes
h sti
ffne
ss/N
·mm
-1
2.5mm
2.51.5
2
2.5
3
3.5
4
x 105
Ri
aver
age
mes
h sti
ffne
ss/N
·mm
DEFORMATION O
.5 5
×10-2 8.59×
GEAR MODELS
ESH STIFFNESSCKNESSES.
MESH STIFFNECKNESSES
CONCLUSION model of a face equations.od. The FE mhe element oation elementflection was cdifferent rimsthe results sh
h stiffness is lad slow versus n constructedethod for ges.
10 15 20nal angle of face gear/°
5mm 7
5im thickness/mm
F THE RIMS.
7.5
×10-3 5.70×1
S OF DIFFEREN
AT DIFFERENT
SS AT DIFFERE
ace gear was . The equatiomodel of a faf 8-node hext. The tooth computed by s on face geahow that a faarger, but chachanges of fa
d in this studgear mesh s
25
7.5mm
7.5m
10-3
T RIMS.
T RIM
ENT RIM
created ons are ce gear
xahedral surface LTCA
ar mesh ce gear
anges of ace gear dy is a stiffness
183
ACKNOWLEDGMENTS The authors are grateful for the financial support provided
by the National Natural Science Foundation of China under no.51105194 and no.51375226, and the Fundamental Research Funds for the Central Universities under no. NS2015049.
In addition, the authors declare that there is no conflict of interests regarding the publication of this article.
REFERENCES [1] Zachary S. Henry. Bell helicopter advanced rotorcraft transmission
(ART) program. Bell Helicopter Textron, Inc., Texas. NASA, CR 195497, 1995.6.
[2] Litvin F. L., Zhang Y., Wang J. C., et al. Design and geometry of face-gear drives. Transactions of the ASME, Journal of Mechanical Design, 114(4), pp. 642-647,1992.
[3] Litvin F. L., Wang J. C., Bossler R. B. Application of face gear drives in helicopter transmissions. Journal of Mechanical Design, Transactions of the ASME, 116(3), pp. 672-676, 1994.
[4] M Guingand, JP De Vaujany, CY Jacquin. Quasi-static analysis of a face gear under torque, Comput. Methods Appl. Mech. Engrg, 194, pp. 4301-4318.2005.
[5] Guanghu JIN, Rupeng ZHU, Zibing ZHU, et al. Analysis of the coupling vibration of a face gear transmission system, Mechanical Science and Technology for Aerospace Engineering, 28(1), pp. 124-128, 2009.
[6] Fakher C., Tahar F., Mohamed H. Analytical modelling of spur gear tooth crack and influence on gearmesh stiffness, European Journal of Mechanics A/Solids,28, pp. 461-468,2009.
[7] Zhaoyao SHI, Yan KANG, Jiachun LIN. Comprehensive dynamics model and dynamic response analysis of a spur gear pair based on gear pair integrated error, Journal of Mechanical Engineering, 46(17), pp. 55-61,2010.
[8] Yanzhong WANG, Shu WANG, Zhimin TIAN, et al. Research on loading tooth contact analysis of face gear transmission, Journal of Mechanical Transmission, 36(10), pp.4-7,2012.
[9] Xiaozhen LI, Rupeng ZHU, Zhengminqing LI, et al. Influence of Setting Error of Tool on Tooth Profile and Contact Point of Face Gear Drive. Transactions of Nanjing University of Aeronautics &Astronautics.31(4), pp. 370~376, 2014.
[10] Rupeng ZHU. Reserch of meshing characteristic of face gear drives, Nanjing University of Aeronautics and Astronautics, Nanjing, 2000.
[11] Li Zhengminqing, Zhu Rupeng. Load Tooth Contact Analysis on Face Gear Driver, Journal of Nanjing University of Aeronautics & Astronautics, 42(2), pp.219-223, 2010.
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