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

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Rigid couplie gear axis annion profile. T

o not have rotafterwards, rota

shown in Figapproximate

URE V. TOOTH

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E I. THE FA

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(mm) of teeth

e angle (°) idth (mm) ation coefficient

VII. THE

RE VIII. F

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SURFACE SIMU

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ACE GEAR DRIVE P

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4 24 20 30 0

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FIGUR

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RE IX. CONTAC

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TRIBUTION.

the FE n of the

include sis type, termine the key

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182

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whfactotnatheap

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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

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l con

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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

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results of siThe time-var

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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.

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[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.

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