The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.PS6.007

Contact Pattern Simulation and Stress Analysis of Intersected Concave Conical Involute Gear Pairs Generated by Shaper Cutters

Chia-Chang Liu1 Yi-Cheng Chen2 Yu-Lin Peng3 Chien Hsin University of National Central University Chien Hsin University of Science and Technology Taoyuan, Taiwan Science and Technology

Taoyuan, Taiwan Taoyuan, Taiwan

Abstract: This paper gives an overview of the most important model collections in Europe at that time and the current status of preservation and use. This study investigates the contact patterns and contact stresses of intersected concave conical gear pairs generated by shaper cutters. The mathematical models of gear pair have been derived based on the theory of gearing and the generation mechanism. Then, contact pattern simulation have been performed by SolidWorks. Meanwhile, the commercial software, ABAQUS, capable of contact analysis of two 3-D deformable bodies is applied to evaluate the stress distribution on the tooth surfaces. Several numerical examples are presented to demonstrate the results of contact pattern Simulations and stress analyses of the gear pairs with various design parameters. The results show that concave conical gear pairs relieve the high contact stress problem which is inherent in conventional conical involute gear pairs.

Keywords: Conical involute gears, Contact Pattern, Contact Stress.

I. Introduction Conical involute gears, also known as beveloid gears [1],

are involute gears with tapered tooth thicknesses, roots and outside diameters. Conical involute gear represent the most general type of the involute gear, which can mesh conjugately with most involute type gears of spur gears, helical gears, conical involute gears, worms and racks to serve the motion transmission between parallel, intersected and crossed axes in any relative position. Owing to its tapered tooth thickness, the backlash of a conical involute gear pair can be easily eliminated by axial adjustments without affecting its center distance. Nowadays, progresses in the design and production method of conical involute gears enable their use in an increasingly wide range of applications. The most familiar application of conical involute gears is the reduction gear in marine transmissions. A conical involute gear is meshed with a spur or helical pinion to provide the down angle for the output shaft which enables an optimum placement for the engine. Conical involute gears are also frequently applied to backlash control gears used in machine tools, timing gears for compressors, miter gears in gear grinding machines, differential gears for robots and sector gears in the power steering device of vehicles. The use of conical involute gears is particularly ideal for small shaft angles, and the latest development has been realized for the AWD(All-Wheel Drive) version of a automatic transmission [2].

1 ccliu@uch.edu.tw 2 ethan@ncu.edu.tw 3 m10021022@uch.edu.tw

Theoretically, the bearing contacts of conical involute gear pairs under non-parallel axes meshing are point contacts. The contact ellipses are relatively small [3], and the tooth surface durability is generally low owing to its high contact stress. Recently, Ohmachi et al. [4-5] investigated the fatigue strength of an intersected-axes conical involute gear pair experimentally. The tooth surface life has been evaluated by the pitting area rate, while the critical value of the circulating torque has been obtained as well. The low-load capacity thus limits the application of this kind of gear pairs to power transmission. To overcome this drawback, Mitome et al. [6-8] proposed the idea of infeed grinding for the concave conical involute gear generation. According to the concept of infeed grinding, the author [9-10] derived the mathematical models of concave conical involute gear pairs. Investigations including tooth contact analysis, contact pattern simulation and stress analysis are performed. The results indicated that the concave conical involute gear pairs solve the problems associated with low-load capacity by enlarging the contact area. On the other hand, Mitome et al. [11-12] also introduced another cutting method of concave conical gears by applying shaper cutters. In this study, the mathematical models of concave conical involute gears generated by shaper cutters are derived based on the theory of gearing and the generating mechanism. Investigations including contact pattern simulations and finite element analysis (FEA) will be performed to examine the contact characteristics of concave conical involute gears with intersected axes. The effects of design parameters on the dimensions of contact patterns will be studied as well. The results of the FEA and contact ellipse simulations can be verified mutually.

II. Mathematical Model of Concave Conical GearGenerated By Shaper CutterThe concave conical involute gear studied in this study

is cut by a spur shaper cutter with an involute tooth profile. Figure 1 shows the cutting mechanism and the relationship between the shaper cutter and the gear blank. The axis of the shaper and the gear blank rotation axis form an angle that equal to the gear’s conical angle. The shaper and the gear perform rotation between intersected axes with angular velocities

c and g that are related as follows:

c

g

g

c

N

N

,(1)

where Nc and Ng are the tooth numbers of the shaper

and the generated gear.

Fig. 1. Schematic view of the cutting mechanism of concave conical gear

The shaper cutter used for the generation can be considered to have a two-dimensional tooth profile which is identical to the cross-section of the spur gear. Figure 2 illustrates the normal section of the involute-shape shaper cutter

c expressed on the cc YX plane in coordinate

system ),,( cccc ZYXS .

The shaper cutter resembles a spur gear in appearance. Designating by u the surface parameter in the direction of zc, the shaper surface and its unit normal can be represented in coordinate system Sc by

u

qrqr

qrqr

z

y

x

bb

bb

c

c

c

c )cos()sin(

)sin()cos(

R

,

(2)

and

0

)cos(

)sin(

q

q

c

n

.

(3)

where tcb rr cos .

Herein, br and

cr are the radii of the base circle and the

pitch circle of the shaper cutter, respectively, represents

the involute profile parameter(i.e. the involute polar angle), t is the pressure angle of the shaper cutter. The

parameter q determines the width of the space of the shaper on the base circle and is represented for a standard shaper by the equation

)inv2

( tcN

q . (4)

The coordinate systems between the shaper cutter and the generated gear during the generation process can be depicted in Fig.3. Coordinate systems Sc and Sg are attached to the shaper cutter c and the generated gear g, respectively. The plane Zc - axis, which represents the rotational axis of the shaper, is set to form an inclination angle with respect to Zg - axis, which represents rotational axis of the generated gear. Furthermore, c and g are rotational angles of the shaper and the generated gear, respectively, during gear generation.

q

2N

c

invt

rc

t

M

rb Yc

cX

Fig. 2. Normal section of the shaper cutter

Based on the theory of gearing [13], the generated gear surface g can be obtained by simultaneously considering the locus of the shaper surface c, represented in gear coordinate system Sg, together with the equation of meshing between the shaper and the generated gear. The locus of the shaper cutter and its unit normal in coordinate system Sg can be obtained as follows:

gggcg

ccgb

ccgbg

rru

qqr

qqrx

coscoscossincos

)cos()sin(sin

)sin()cos(coscos

gggcg

ccgb

ccgbg

rru

qqr

qqry

sincossinsinsin

)cos()sin(cos

)sin()cos(coscos

sincos

)sin()cos(sin

c

ccbg

ru

qqrz

(5)

)sin(sin

)cos(cos)sin(cossin

)cos(sin)sin(coscos

c

cgcg

cgcg

g

q

qq

qq

n

,

(6)

where g

ccg N

N .

The equation of meshing can be obtained according to the orthogonality of the relative velocity and surface common normal of the shaper cutter surface c and the generated gear surface g. The following equation can thus be observed:

0)( )()()()()( gf

cf

cf

cgf

cf VVnVn (7)

Equation (7) is known as the equation of meshing in the theory of gearing, where )(c

fn is the common normal

vector, and )(cgfV is the relative velocity of the two

mating surfaces at their instantaneous contact point, represented in coordinate system

fS , respectively.

Z ,Zf

gO ,Of

rgY f

g

gY

P

Xg

0

g

cY

cXrc

c

hX

hY

cZ ,Zh

Oh

cO ,

fX

Fig. 3. Coordinate relationship between the shaper cutter and generated gear

The equation of meshing can be arranged and simplified as

follows:

gcc

gcb rrq

mru

cos

)cos(

cos

sin

1 , (8)

where c

ggc r

rm .

Therefore, the mathematical model of the concave conical gear surface g generated by a shaper cutter c can be obtained by considering Eqs. (5)and (8) simultaneously.

III. Contact Pattern SimulationWhen tooth surfaces are meshed with each other, their

instantaneous contact point is spread over an elliptical area owing to elastic deformation. Figure 4(a) shows that two mating surfaces 1 and 2 are tangential to each other at their instantaneous contact point OT. Plane T denotes the common tangent plane of the two mating surfaces. The separation distance of two mating surfaces can be defined by d1+d2, which equals to )2()1(

TT zz in coordinate system

TS .The equal distance-separation line for two mating

surfaces can be represented by the following system of nonlinear equations:

mmZZ TT 00632.0)2()1( (9)

(a)

(b)

Fig. 4. Measurements on tooth surface separation

Herein, the value of 0.00632 mm in Eq.(9) represents the thickness of the coating paint used for contact pattern tests. In this study, the “Interference detection” function of SolidWorks is used to simulate the contact pattern of the mating tooth surfaces. First, the 3-D mating tooth models are established in SolidWorks with their tooth surfaces, 1 and 2, tangent to each other at their instantaneous contact point OT. Then, tooth surface 1 is forced to make a displacement of -0.00632 mm along N-axis, as shown in fig.4(b). The interference highlights in red in the graphics area, as illustrated in Fig.5.

Fig. 5. Contact pattern simulation by “Interference detection” function of SolidWorks

IV. Finite Element Contact Stress AnalysisDue to the progress of computer technology and the

computational techniques, the FEA becomes a popular and powerful analysis tool to determine the formation of

bearing contacts and stress distributions of gear drives. The author [10] have studied the contact stress of concave conical involute gear pairs generated by infeed grinding method with the FEA software, ABAQUS. Similar analysis procedures are adopted to evaluate the contact stress of the proposed concave conical involute gear pairs in this study.

V. Simulation Results and Discussions In this study, the gear pair is composed of a straight

conical involute pinion and gear mounted with an intersected angle of 10 . Figure 6 shows three different models of this gear pair with different assembly of 1 and

2 . The gear 2 is statically fixed and a torque of 200 N-m

is applied directly to the rotational axis of the pinion 1. The major design parameters for the gear pairs in the following examples are listed in Table 1. Meanwhile, the medium carbon steel AISI 1045 with the material properties listed in Table 2 has been chosen for the gear material for the FEA.

Table 1 Major design parameters of the gear pairs

Pinion 1 Gear 2

Number of teeth N1= N2=25

Face width mmWW 4021

Normal pressure angle

20n

Normal module teethmmmn /4

Intersected angle 10

Table 2 Material properties of AISI 1045

Young’s Modulus 205 GPa

Poisson’s Ratio 0.292 Allowable Contact Stress 980 MPa

Figure 7 illustrated the maximum contact stresses of

conical gear pairs with different Nc. Among the three

models, the reduction of the maximum contact stresses in

model A is the most significant. The maximum contact

stress is about 37 percent when Nc decreases from 80 to 25.

Figures 8 to 10 illustrate the results of contact pattern

simulations together with the distributions of von-Mises

stress by FEA. Shaper cutters with different tooth numbers

(Nc) are also chosen to investigate the effects of Nc on the

contact pattern and contact stress in each case. With the

decrease of Nc, the contact area enlarges and the contact

stress reduces significantly. The results of FEA agree with

results of contact pattern simulations. Observing Fig.8, the

contact of the gear pair approaches line contact as Nc

decreases from 80 to 25. However, unfavorable edge

contacts and phenomenon of stress concentrations occurs

when Nc = 20. Therefore, choosing an appropriate tooth

number of the shaper cutter is very important.

(a) Model A （ 01 , 102 ）

(b) Model B ( 5.21 , 5.72 )

(c) Model C ( 51 , 52 )

Fig.6 Three models of conical involute gear pairs with different

assembly of 1 and 2 .

Fig.7 Maximum contact stresses of conical gear pairs with different Nc

Pinion

1

Gear 2

Pinion

1

Pinion

1

Gear 2

Gear 2

Model A（ 01 , 102 ）

cN Contact Pattern Stress distribution by FEA

20

25

40

60

80

Fig.8 Contact pattern and stress distribution of Model A

Model B ( 5.21 , 5.72 )

cN Contact Pattern Stress distribution by FEA

20

25

40

60

80

Fig.9 Contact pattern and stress distribution of Model B

Model C ( 51 , 52 )

cN Contact Pattern Stress distribution by FEA

20

25

40

60

80

Fig.10 Contact pattern and stress distribution of Model C

VI. Conclusions This study investigated the contact pattern and the

contact stress of generated by shaper cutters. Several illustrative numerical examples are presented as well. The simulation results indicate that the dimensions of the contact pattern can be enlarge and contact stress can be reduced significantly by choosing a smaller tooth numbers of the shaper cutter Nc for the generation of concave conical involute gear pairs. The simulated results of contact pattern simulation are consistent with those of contact stress analysis.

Acknowledgment The authors would like to thank the Ministry of

Science and Technology of Taiwan for financially supporting this research under Contract No. MOST 103-2221-E-231-005

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[5] Ohmachi T., Sato J., Mitome K., “Allowable contact strength of normalized steel conical involute gears,” Proceedings of the JSME International Conference on Motion and Power Transmission (MPT2001-Fukuoka), Fukuoka, Japan, 15–17 November 2001.

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