Design and Manufacture of Spiral Bevel Gears With … · The truly conjugated spiral bevel gears...

Proceedings of IMECE08 2008 ASME International Mechanical Engineering Congress and Exposition

October 31 - November 6, 2008, Boston, Massachusetts USA

IMECE2008-66017

DESIGN AND MANUFACTURE OF SPIRAL BEVEL GEARS WITH REDUCED TRANSMISSION ERRORS

V. Simon Budapest University of Technology and Economics

Faculty of Mechanical Engineering Department of Machine and Product Design

H-1111 Budapest, Műegyetem rkp. 3, Hungary [email protected]

Proceedings of IMECE2008 2008 ASME International Mechanical Engineering Congress and Exposition

October 31-November 6, 2008, Boston, Massachusetts, USA

IMECE2008-66017

ABSTRACT

A method for the determination of the optimal polynomial functions for the conduction of machine-tool setting variations in pinion teeth finishing in order to reduce the transmission errors in spiral bevel gears is presented. Polynomial functions of order up to five are applied to conduct the variation of the cradle radial setting and of the cutting ratio in the process for pinion teeth generation. Two cases were investigated: in the first case the coefficients of the polynomial functions are con-stant throughout the whole generation process of one pinion tooth-surface, in the second case the coefficients are different for the generation of the pinion tooth-surface on the two sides of the initial contact point. The obtained results have shown that by the use of two different fifth-order polynomial functions for the variation of the cradle radial setting for the generation of the pinion tooth-surface on the two sides of the initial contact point, the maximum transmission error can be reduced by 81%. By the use of the optimal modified roll, this reduction is 61%. The obtained results have also shown that by the optimal varia-tion of the cradle radial setting, the influence of misalignments inherent in the spiral bevel gear pair and of the transmitted torque on the increase of transmission errors can be considera-bly reduced.

INTRODUCTION

The traditional cradle-type hypoid generators are evolved

into computer numerical control (CNC) hypoid generating machines, as are the Gleason Phoenix series and the Klingelnberg universal spiral bevel gear generating machine. These new CNC hypoid generators have made it possible to perform nonlinear correction motions for the cutting of the face-milled spiral bevel and hypoid gears. The following refer-ences summarize the proposed free-form cutting methods.

Local synthesis of spiral bevel gears with localized bearing contact and predesigned parabolic function of a controlled level for transmission errors was proposed by Litvin and Zhang [1]. The theory of modified roll, the variation of cutting ratio in the process for pinion teeth generation was introduced. In the paper published by Argyris et al. [2], a computerized method of local synthesis and simulation of meshing of spiral bevel gears with pinion tooth-surface generated by applying a third-order func-tion for modified roll was presented. Fuentes et al. [3] and Lit-vin and Fuentes [4] have developed an integrated computerized approach for the design and stress analysis of low-noise spiral bevel gear drives with adjusted bearing contact. The predes-igned parabolic function of transmission errors was achieved by the application of modified roll for pinion tooth generation. Ref. [5] covered the design, manufacturing, stress analysis and results of experimental tests of prototypes of spiral bevel gears with low levels of noise and vibration and increased endurance. Three shapes of blade profile and a modified roll for pinion tooth generation were applied.

Based on the grinding mechanism and machine-tool settings for the Gleason modified roll hypoid grinder, a mathematical model for the tooth geometry of spiral bevel and hypoid gears was developed by Lin and Tsay [6]. Chang et al. [7] proposed a general gear mathematical model simulating the generation process of a 6-axis CNC hobbing machine. The so-called Uni-versal Motion Concept (UMC) was developed by Stadtfeld [8]. In the UMC eight correction mechanisms were introduced into the calculation of machine settings for gear cutting. In order to develop a gear geometry that reduces gear noise and increases the strength of gears, the Universal Motion Concept was ap-plied for hypoid gear design by Stadtfeld and Gaiser [9]. Higher order kinematic freedoms up to the 4th order were applied to achieve the best possible result in noise, sensitivity, and adjust-ability.

Linke et al. [10] presented a method for taking any addi-tional motions mapped in the process-independent mathemati-

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cal model of the generating process into account in the simula-tion of the manufacturing process of bevel gears. It was demon-strated, how a specific influencing of the meshing and stress conditions can be achieved by such additional motions. A mathematical model of universal hypoid generator with sup-plemental kinematic flank correction motions was proposed by Fong [11] to simulate virtually all primary spiral bevel and hypoid cutting methods. The supplemental kinematic flank correction motions, such as modified generating roll ratio, heli-cal motion, and cutter tilt were included into the proposed mathematical model. Wang and Fong in Ref. [12] proposed a methodology to improve the adjustability of the spiral bevel gear assembly by modifying the radial motion of the head cut-ter in the machine plane of the hypoid generator. A method to synthesize the mating tooth-surfaces of a face-milled spiral bevel gear set transmitting rotation with a predetermined fourth-order motion curve and contact path was presented by Wang and Fong [13].

By Achtmann and Bär [14], modified helical motion and modified roll were applied to produce optimally fitted bearing ellipses. The paper published by Fan [15], presented the theory of the Gleason face hobbing process. In Ref. [16], a generic model of tooth-surface generation for spiral bevel and hypoid gears produced by face-milling and face-hobbing processes conducted on free-form computer numerical control (CNC) hypoid gear generators was presented. Fan et al. [17] described a new method of tooth flank form error correction, utilizing the universal motions and the universal generation model for spiral bevel and hypoid gears. Shih and Fong [18] proposed a flank modification methodology for face-hobbing spiral bevel gear and hypoid gears, based on the ease-off topography of the gear drive. Gear generation with supplemental spatial motions (heli-cal motion, tilt motion), particularly interesting for gear genera-tion with modern free-form cutting machines, was presented by Di Puccio et al [19]. The application of modified roll and basic machine root angle variation in spiral bevel pinion finishing was introduced. In Ref. [20] published by Cao et al., third order function for the modified roll was applied in order to achieve a predesigned parabolic function of transmission errors and to improve the contact pattern with the desired shape of contact path in spiral bevel gears. Liu and Wang [21] presented a method for realizing and improving the conventional gear cut-ting, associated with a traditional machine-tool upon a CNC free-form gear cutting machine.

The truly conjugated spiral bevel gears are theoretically with line contact. In order to decrease the sensitivity of the gear pair to errors in tooth-surfaces and to the mutual position of the mating members, carefully chosen modifications are usually introduced into the teeth of one or both members. As a result of these modifications, the spiral bevel gear pair becomes “mis-matched”, and a point contact of the meshed teeth-surfaces appears instead of line contact. In practice, these modifications are usually introduced by applying the appropriate machine-tool setting for pinion and gear manufacture. The generation of tooth-surfaces of the pinion and the gear in mismatched spiral bevel gears was described in Ref. [22]. In this paper a method is presented for the determination of the optimal polynomial function for the conduction of machine-tool setting variation in pinion teeth finishing in order to reduce the transmission errors in mismatched spiral bevel gears.

NOMENCLATURE c = sliding base setting for pinion finishing, mm

ne = composite manufacture and alignment error, mm

pe = basic radial for pinion finishing, mm f = machine center to back, mm g = blank offset for pinion finishing, mm

gpi = velocity ratio in the kinematic scheme of the machine- tool for the generation of pinion tooth-surface

21 N,N = numbers of pinion and gear teeth p = distance of the initial contact point from pinion apex, mm

maxp = maximum tooth contact pressure, Pa

1Tr = pinion finishing cutter radius, mm s = geometrical separation of tooth-surfaces, mm T = transmitted torque, mN ⋅

1α = pinion finishing blade angle, deg. a∆ = pinion offset, mm b∆ = displacement of the pinion along its axis, mm c∆ = displacement of the pinion along the gear axis, mm F∆ = concentrated load, N

ny∆ = composite displacement of contacting surfaces, mm

2φ∆ = angular displacement of the driven gear, deg.

hε = horizontal angular misalignment of pinion axis, deg.

vε = vertical angular misalignment of pinion axis, deg.

21 ,φφ = rotational angles of the pinion and the gear, deg.

2010 ,φφ = initial rotational angles of the pinion and the gear, deg.

1γ = machine root angle for pinion finishing, deg. ( )cω = angular cradle velocity, 1s− 1ψ = angle of pinion rotation during its generation, deg.

cpψ = cradle angle rotation for pinion finishing, deg.

0cpψ = initial cradle angle setting for pinion finishing, deg. THEORETICAL BACKGROUND

The Spiral Bevel Gear Pair. A Gleason type spiral bevel gear pair with generated pinion and gear teeth is treated (see Fig. 1). The pinion is the driving member. The convex side of the gear tooth and the mating concave side of the pinion tooth are the drive sides. The modifications are introduced into the pinion tooth-surface by applying machine-tool setting varia-tions in pinion tooth generation. As a result of these modifica-tions the spiral bevel gear pair becomes mismatched and a point contact of the meshed teeth-surfaces appears instead of line contact.

The relative position of the pinion and the gear is defined by the following equation (based on Fig. 1):

01vhvhv

hh

vhvhv

011202

1000acoscossinsinsin

c0sincosbpsincoscossincos

rrMrrrr

⋅

∆−εεεε⋅ε∆εε−

∆−ε−ε⋅εε⋅ε

=⋅=

(1)


' ,

2x

'

'

z01z1

'z01

ω (2)

'

O01 , ,'

z02 z2z01z02

zz2z1

x1

01x01x

02x

01x 01x 1x

01y

01y ' , 1y

'

δ02

δ01

hε

'

p

φ2

εv

1φ

02y 2y

02y2y

01y 1y

01y

ω (1)

ω (1)

φ 2

φ 1

ω (2)

εh

εv

O01 O1 O01

O2 O02

O01

O1

O02

O01

O02 O2,O01 O01

∆b

∆a

x x,02 2

,

,

01

∆c

O2

Fig. 1: Relative position of the pinion and the gear in mesh

where matrix 12M defines the relation between the stationary coordinate systems 01K and 02K , in which the pinion and the gear are rotating through mesh. This matrix includes the possi-ble misalignments of the pinion: the pinion offset ( a∆ ), the displacements of the pinion along the pinion axis ( b∆ ) and along the gear axis ( c∆ ), and the angular misalignments of the pinion axis in the horizontal ( hε ) and in the vertical ( vε ) plane.

Machine-Tool Setting for Pinion Finishing. The machine-

tool setting used for pinion teeth finishing is given in Fig. 2. The concave side of pinion teeth is in the coordinate system

1K (attached to the pinion, Fig. 2) defined by the following system of equations

)T(

Tp1p2p3)1(

11

1rMMMrrr

⋅⋅⋅= (2a)

0)T(1m

)1,T(1m

11 =⋅ evrr

(2b) where ( )1

1

TTrr

is the radius vector of tool-surface points, matrices

p1M , p2M , and p3M provide the coordinate transformations from system 1TK (rigidly connected to the cradle and head-cutter 1T ) to system 1K (rigidly connected to the being gener-ated pinion). Equation (2b) describes mathematically the gen-

eration of pinion tooth-surface by the head-cutter [22]. The matrices and vectors of system of Eqs. (2a) and (2b) are defined as it follows.

The surface of the head-cutter used for finishing the con-cave side of pinion teeth is in the coordinate system 1TK (at-tached to the tool) defined by the following equation (based on Fig. 2):

θ⋅α⋅+θ⋅α⋅+

−

=θ

1sin)tgur(cos)tgur(

u

),u(1T

1T)T(T

1

11

1rr

(3)

On the basis of Fig. 2 and Eq. (3), for the relative velocity

vector )1,T(1m1v

r of tool 1T to the pinion, and for the unit normal

vector of the tool-surface )T(1m1e

r, it follows

( )( )( )[ ]

−γ⋅−−γ⋅⋅γ⋅+⋅−

γ⋅+⋅⋅ω=

)T(1m1

1T(1m1

)T(1mgp

1)T(

1mgp)T(

1m

1)T(

1mgp)c()1,T(

1m111

11

1

1

ycoscxsinyisingziz

cosgzivr

(4)


y10

ym1

1δ

1x

x m1

xT1

01

f1δ

δk1

O1

α1

OT1

m1y y

10

yT1

zT1

r t

m1z

M M

ω (c)

b f

z10

γ

g

ψcp

T1O

m1O

10O

10O

1Or T1

10O

c

ω (1)

z10

z1

x10

x10

x1

ψ 1

u

f

ψ1

T1z

e

θ

p

m1O

Cutter T1

p

Fig. 2: Machine-tool setting for pinion tooth-surface finishing

θ⋅αθ⋅α

α

⋅=⋅=

0sincoscoscos

sin

1

1

1

p1)T(

1Tp1)T(

1m11 MeMe

rr (5)

where ( ) ( )1

1

1 TTp1

T1m rMr

rr⋅=

For matrices p1M , p2M , and p3M , providing the coordi-nate transformations, on the basis of Fig. 2, it follows

1Tcppcpcp

cppcpcp1Tp11m

1000sinesincos0

cosecossin00001

rrMrrrr

⋅

ψ⋅−ψψψ⋅ψ−ψ

=⋅=

(6)

1m111

111

1mp201

1000g100

sincf0cossincosc0sincos

rrMrrrr

⋅

γ⋅−−γγ

γ⋅−γ−γ

=⋅= (7)

0111

11

01p31

10000cos0sinp010

0sin0cos

rrMrrrr

⋅

ψψ−−

ψψ

=⋅= (8)

while, for the traditional cradle-type hypoid generators

( )0cpcpgp101 i ψ−ψ⋅+ψ=ψ . Angles 10ψ and 0cpψ correspond to the generation of the “initial” contact point on the pinion tooth-surface. Because of the mismatch of the gear pair, only in one point of the path of contact, called as the “initial” contact point, the basic mating equation of the contacting pinion and gear tooth-surfaces is satisfied, producing the correct velocity ratio based on the numbers of teeth. Usually, the middle point of the gear tooth flank is chosen for the initial contact point and the machine-tool settings for pinion and gear manufacture are calculated due to this initial contact point.

In this paper the variations of the cradle radial setting pe and of the modified roll for pinion generation are conducted by polynomial functions of order up to five:

( ) ( ) ( ) ( ) ( )5

0cpcp5e4

0cpcp4e3

0cpcp3e2

0cpcp2e0cpcp1e0pp cccccee ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅+= (9)

( ) ( ) ( ) ( ) ( )50cpcp4i

40cpcp3i

30cpcp2i

20cpcp1i0cpcpgp1 cccci ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅=ψ (10)


where 1ψ is the angle of pinion rotation during its generation; angle cpψ is the angle of rotation of the cradle of the cutting machine.

Two solutions are investigated: in the first case the coeffi-cients in Eqs. (9) and (10) are constant throughout the whole generation process of one pinion tooth-surface, in the second case the coefficients are different for the generation of the pin-ion tooth-surface on the two sides of the initial contact point. The results have shown that the second solution gives much better results.

Potential Contact Lines Under Load. As it was mentioned before, as a result of the modifications introduced into the teeth of one or both members, theoretically point contact of the meshed tooth-surfaces appears instead of the line contact. However, as the tooth-surface modifications are relatively small, even in the case of light loading, the theoretical point contact spreads over a narrow surface along the whole or part of the “potential” contact line. These potential contact lines can be obtained by determining the line of minimal separations of the mating tooth-surfaces along the face width of teeth. The separations are defined as the distances of the corresponding surface points that are the intersection points of the straight line parallel to the common surface normal in the instantaneous contact point, with the pinion and gear tooth-surfaces. Mathe-matically, it means the minimization of the function

( ) ( )( ) ( ) ( )( ) ( ) ( )( )2102

202

2102

202

2102

202 zzyyxxs −+−+−= (11)

where ( )1

02rr

and ( )202rr

are the position vectors of the correspond-ing points on the pinion and gear tooth-surfaces.

It should be mentioned that in the case of edge contact the computer program developed and applied automatically adjusts the values of separations in other points of the potential contact line accordingly to the zero separation of the edge contact point.

The method for the determination of minimal separations and “potential” contact lines is fully described in Ref. [22].

Transmission Errors. The total transmission error consists of the kinematical transmission error due to the mismatch of the gear pair and eventual tooth errors and misalignments of the meshing members, and of the transmission error caused by the deflections of teeth.

It is assumed that the pinion is the driving member and that is rotating at a constant velocity. As the result of the mismatch of gears, a varying angular velocity ratio of the gear pair and an angular displacement of the driven gear member from the theo-retically exact position based on the ratio of the numbers of teeth occurs. This angular displacement of the gear can be ex-pressed as

( ) ( ) s221011202k

2 N/N φ∆+φ−φ⋅−φ−φ=φ∆ (12) where φ10 and φ20 are the initial angular positions of the pinion and the gear corresponding to the initial contact point, φ2 is the instantaneous angular position of the gear for a particular angu-lar position of the pinion, φ1 [22]; N1 and N2 are the numbers of pinion and gear teeth, respectively, and s2φ∆ is the angular displacement of the gear due to edge contact in the case of

misalignments of the mating members when a “negative” sepa-ration occurs on a tooth pair different from the tooth pair for which the instantaneous angular position is calculated [22].

The angular displacement of the gear, ( )d2φ∆ , caused by the

variation of the compliance of contacting pinion and gear teeth rolling through mesh, will be determined in the load distribu-tion calculation.

Therefore, the total angular position error of the gear is de-fined by the equation

( ) ( )d

2k

22 φ∆+φ∆=φ∆ (13)

Load Distribution. The load distribution calculation is based on the conditions that the total angular position errors of the gear teeth being instantaneously in contact under load must be the same, and along the contact line (contact area) of each tooth pair instantaneously in contact, the composite displace-ments of tooth-surface points - as the sums of tooth deforma-tions, tooth surface separations, misalignments, and composite tooth errors - should correspond to the angular position of the gear member. Therefore, in all the points of the instantaneous contact lines, the following displacement compatibility equa-tion should be satisfied:

( ) ( ) ( ) ( )k2

0

D

nk2

d22 r

yφ∆+

⋅×⋅

∆=φ∆+φ∆=φ∆

rear

r

rrr

(14)

where ny∆ is the composite displacement of contacting sur-faces in the direction of the unit tooth surface normal e

r, rr

is the position vector of the contact point, Dr is the distance of the contact point to the gear axis, and 0a

r is the unit vector of the

gear axis. The composite displacement of the contacting surfaces in

contact point D, in the direction of the tooth-surface normal, can be expressed as

( ) ( ) ( )DnDDn zezszwy ++=∆ (15)

where Dz is the coordinate of Point D along the contact line,

( )Dzw is the total deflection in Point D, ( )Dzs is the relative geometrical separation of teeth-surfaces in Point D, and ( )Dn ze is the composite error in Point D, which is the sum of manufac-turing and alignment errors of pinion and gear.

The total deflection in Point D is defined by the following equation: ([23, 24])

( ) ( ) ( ) ( ) ( )∫ ⋅+⋅⋅=

itLDDcFFDdD zpzKdzzpz,zKzw (16)

where itL is the geometrical length of the line of contact on tooth pair ti , ( )FDd z,zK is the influence factor of tooth load acting in tooth-surface Point F on total composite deflection of pinion and gear teeth in contact Point D. dK includes the bend-ing and shearing deflections of pinion and gear teeth, pinion and gear body bending and torsion, and deflections of support-


ing shafts. A finite element computer program is developed for the calculation of bending and shearing deflections in the pin-ion and in the gear [25]. ( )Dc zK is the influence factor for the contact approach between contacting pinion and gear teeth, i.e., the composite contact deformation in Point D under load acting in the same point. ( ) ( )DF zp,zp are the tooth loads acting in Positions F and D, respectively.

As the contact points are at different distances from the pin-ion/gear axis, the transmitted torque is defined by the equation

( ) dzzrTtt

t it

Ni

1i LF0FF ⋅⋅⋅= ∑ ∫

=

=

tprr

(17)

where Fr is the distance of the loaded Point F to the gear axis,

F0tr

is the tangent unit vector to the circle of radius Fr , passing through the loaded Point F in the transverse plane of the gear, and tN is the number of gear tooth pairs instantaneously in contact.

The load distribution on each line of contact can be calcu-

lated by solving the nonlinear system of Eqs. (14)-(17). An approximate and iterative technique is used to attain the solu-tion. The contact lines are discretized into a suitable number of small segments, and the tooth contact pressure, acting along a segment, is approximated by a concentrated load, F

r∆ , acting in

the midpoint of the segment. The actual load distribution, de-fined by the values of loads F

r∆ is obtained by using the suc-

cessive-over-relaxation method. In every iteration cycle, a search for the points of the "potential" contact lines that could be in instantaneous contact is performed. For these points, the following condition should be satisfied:

( )( )

( )

( )

( )zt

t

zt

i,iD

0

ki22

i,in

r

y

⋅

⋅×

φ∆−φ∆≤∆

rear

r

rrr (18)

where ti is the identification number of the contacting tooth pair, zi of the segment.

The details of the method for load distribution calculation in spiral bevel gears are described in Ref. [23]. RESULTS

A computer program, based on the theoretical background presented, has been developed. By using this program, the influence of the character and order of polynomial functions (9) and (10) conducting the variations of the cradle radial setting

pe and the modified roll for pinion generation on transmission errors, motion graphs and maximum tooth contact pressure was investigated.

The calculation was carried out for the spiral bevel gear pair of design data given in Table 1. The basic machine-tool setting parameters for finishing the pinion and the gear teeth blanks were calculated by the method used in Gleason Works and by the method presented by Argyris et al. [2], and are given in Tables 2 and 3.

Table 1 Pinion and gear design data

Pinion Gear Number of teeth 13 50 Module, mm 5 Outside diameter, mm 76.746 251.224 Face width, mm 30 Pitch apex to crown, mm 123.473 30.146 Mean spiral angle, deg 35 Pitch angle, deg 14.5742 75.4258 Face angle of blank, deg 17.7067 76.9478 Root angle, deg 13.0522 72.2933 Addendum, mm 6.068 2.432 Dedendum, mm 3.432 7.068 Working depth, mm 8.500 Whole depth, mm 9.500

Table 2 Pinion machine-tool settings

Concave Convex Point radius of the cutter, mm 86.117 93.433 Cutter blade angle, deg 18.5 21.5 Machine root angle, deg 13.0522 13.0522 Basic cradle angle, deg 51.4895 47.2527 Sliding base setting, mm 0.5602 -0.3882 Machine center to back, mm -2.4805 1.7189 Basic radial, mm 92.8531 99.3590 Blank offset, mm 0.2001 -0.7347 Ratio of roll 3.8470 4.0560

Table 3 Gear machine-tool settings Cutter diameter, mm 180 Cutter point width, mm 2.79 Cutter blade angle, deg 20 Machine root angle, deg 72.2933 Basic cradle angle, deg -49.7814 Basic radial, mm 96.5017 Ratio of roll 1.0317

Two cases were investigated: in the first case, the coeffi-

cients in Eqs. (9) and (10) were constant throughout the whole generation process of one pinion tooth-surface, in the second case, the coefficients were different for the generation of the pinion tooth-surface on the two sides of the initial contact point.

In Fig. 3 the motion graphs are presented for the case, when the cradle radial setting pe variation is conducted by the same polynomial functions up to third order throughout the whole generation process of a pinion tooth flank:
( ) ( ) ( )3
0cpcp2

0cpcp0cpcp0pp 04.0013.0094.0ee ψ−ψ⋅+ψ−ψ⋅−ψ−ψ⋅+= (19)

The maximum angular displacements of the driven gear member for different orders of the polynomial function are: in

the case of 1st , 2nd and 3rdorders sec,arc231.2max2 =φ∆ 2.206


arcsec, 2.206 arcsec, respectively. For the basic cradle radial setting (given in Table 3), the maximum transmission error is 7.039 arc sec. Therefore, a second order polynomial function is satisfactory, while the maximum transmission error is reduced by 68 %.

- for the generation of the part of pinion tooth flank between the toe of the tooth and the initial contact point:

( ) ( ) ( ) ( ) ( ) ( )50cpcp

40cpcp

30cpcp

20cpcp0cpcp0p

1p 6925.64.6025.0ee ψ−ψ⋅−ψ−ψ⋅−ψ−ψ⋅+ψ−ψ⋅−ψ−ψ⋅+= (20)

- for the generation of the part of pinion tooth flank between the initial contact point and the heel of the tooth:

( ) ( ) ( ) ( ) ( ) ( )50cpcp

40cpcp

30cpcp

20cpcp0cpcp0p

2p 2475.8275.0125.0133.0ee ψ−ψ⋅+ψ−ψ⋅+ψ−ψ⋅−ψ−ψ⋅−ψ−ψ⋅+= (21)

0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

BasicLinearSecond-orderThird-order

Fig. 3: Motion graphs for the case when the variation of the radial setting is conducted by the same polynomial functions

up to third order throughout the whole generation process

Table 4 The transmission errors for different orders of the

polynomial function applied for radial setting variation Order of the polynomial

function

Maximum transmission

error [arc sec] Basic 7.039 1st order 3.458 2nd order 1.868 3rd order 1.460 4th order 1.396 5th order 1.358

The maximum angular displacements of the driven gear

member, for different orders of the polynomial function are given in Table 4. It can be concluded that by the 5th order poly-nomial functions (20) and (21), the maximum transmission error can be reduced by 81 %. The variation of the cradle radial setting pe for the generation of pinion tooth, conducted by the two 5th order polynomial functions, is presented in Fig. 5.

In the next trial, the cradle radial setting variation was con-ducted with different polynomial functions up to 5th order for the generation of the pinion tooth-surface on the two sides of the initial contact point. The obtained motion graphs are shown in Fig. 4. The optimized polynomial functions were:

0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

BasicLinearSecond-orderThird-orderFourth-orderFif th-order

Fig. 4: Motion graphs for the case when the variation of the

radial setting is conducted by two different polynomial functions on the two sides of the initial contact point

Table 5 Coefficients of the polynomial functions and the transmission errors

N1 = 9 N1 = 19 Between the toe and the initial

contact point

Between the

initial contact point

and the heel

Between the toe and the initial

contact point

Between the initial contact

point and the heel

ce1 0.088 0.197 0.003 0.075 ce2 -0.40 -0.64 -4.6 -0.006 ce3 0 0.50 -19 -4.48 ce4 33 33.2 -2 12 ce5 -2 18 0 -1

max2φ∆ : ep=const. 8.742 arcsec 3.672 arcsec

max2φ∆ : ep varried 2.130 arcsec 1.332 arcsec Reduction 76 % 64 %


In order to determine the influence of the gear ratio on transmission errors and motion graphs, the polynomial func-tions for pinion tooth numbers of N1 = 9 and N1 = 19, with corresponding gear ratios of 5.5556 and 2.6316, are optimized. The motion graphs for different orders of the polynomial func-tion are shown in Figs. 6 and 7. The coefficients of the opti-mized 5th order polynomial function (9) for the two parts of the pinion tooth flank (on the two sides of the initial contact point) and the corresponding transmission error reductions are given in Table 5.

92,7

92,75

92,8

92,85

92,9

92,95

93

40 45 50 55 60

ψ [deg.]

e p [m

m] Basic

Fifth-order

Fig. 5: Variation of the cradle radial setting for the generation

of one pinion tooth conducted by the two 5th order polynomial functions

0

1

2

3

4

5

6

7

8

9

10

0 0,2 0,4 0,6 0,8 1

N1˙φ1/2˙π

∆φ 2

[arc

sec

]

BasicLinearSecond-orderThird-orderFourth-orderFifth-order

Fig. 6: Motion graphs for pinion tooth number N1 = 9 when

the variation of the radial setting is conducted by two different polynomial functions on the two sides of the


0

0,5

1

1,5

2

2,5

3

3,5

4

0 0,2 0,4 0,6 0,8 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

BasicLinearSecond-orderThird-orderFourth-orderFif th-order

Fig. 7: Motion graphs for pinion tooth number N1 = 19 when

the variation of the radial setting is conducted by two different polynomial functions on the two sides of the


0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N1˙φ1/2˙π

∆φ 2

[arc

sec

]

Basic

Second-order

Third-order

Fig. 8: Motion graphs for the case when the variation of the

modified roll for pinion tooth generation is conducted by the same polynomial functions up to third order

throughout the whole generation process The further investigations were performed for the spiral

bevel gear pair with data given in Table 1 (the pinion tooth number is N1=13).

The motion graphs for the two types of modifications, in-troduced by the variation of the modified roll for pinion tooth flank generation, are shown in Figs. 8 and 9. In the first case, the variation of the modified roll for pinion tooth-surface gen-


eration is conducted by the same polynomial functions up to 3rd order throughout the whole generating process:

( ) ( ) ( )3

0cpcp2

0cpcp0cpcpgp1 0002.00276.0i ψ−ψ⋅+ψ−ψ⋅−ψ−ψ⋅=ψ

(22)

( ) ( ) ( ) ( ) ( )4

0cpcp3

0cpcp2

0cpcp0cpcpgp1

1 0075.0065.02425.0i ψ−ψ⋅−ψ−ψ⋅−ψ−ψ⋅+ψ−ψ⋅=ψ (23)

( ) ( ) ( ) ( ) ( )40cpcp

30cpcp

20cpcp0cpcpgp

21 0075.00205.001925.0i ψ−ψ⋅−ψ−ψ⋅−ψ−ψ⋅−ψ−ψ⋅=ψ (24)

0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

BasicSecond-orderThird-orderFourth-order

Fig. 9: Motion graphs for the case when the variation of the modified roll for pinion tooth flank generation is conducted

by two different polynomial functions on the two sides of the initial contact point

Table 6 The transmission errors for different orders of the polynomial function applied for modified roll variation

Maximum transmission error max2φ∆ [arcsec]

Order of the polynomial

function I. Case II. Case

1st order 7.039 7.039 2nd order 5.580 3.059 3rd order 5.542 2.139 4th order 2.138 Reduction 21% 61%

The corresponding maximum angular displacements of the

driven gear member are given in Table 6. Again, the use of different polynomial functions on the two sides of the initial contact point performs better results: the reduction of the maxi-

In the second case, the optimal variation of the modified roll for pinion tooth flank generation is conducted by two different polynomial functions up to 4th order on the two sides of the initial contact point:

maximum transmission error is 61% against 21% when the same polynomial function is used throughout the whole genera-tion process. The variation of the modified roll for pinion tooth- surface generation, conducted by the two 4th order polynomial functions, is presented in Fig. 10.

3,81

3,82

3,83

3,84

3,85

3,86

40 45 50 55 60

ψ [deg.]

i gp

Basic

Fourth-order

Fig. 10: Variation of the modified roll for the generation of one pinion tooth flank conducted by two different 4th order poly-nomial functions on the two sides of the initial contact point.

The influence of misalignments, as are (Fig. 1): the pinion

offset ( a∆ ), the displacements of the pinion along the pinion axis ( b∆ ) and along the gear axis ( c∆ ), and the angular mis-alignments of the pinion axis in the horizontal ( hε ) and in the vertical plane ( vε ), on transmission errors of spiral bevel gears with pinion whose teeth are processed by the variation of cradle radial setting ( pe ) and of modified roll ( gpi ) is investigated and the obtained results are presented in Figs. 11-15.

It can be noted that in most cases, the modifications intro-duced into the pinion tooth-surface by the variation of the cra-dle radial setting, yield to higher transmission error reductions. In Fig. 15 it can be seen that the transmission errors are insensi-tive to angular misalignments of the pinion axis in the vertical plane.


0

1

2

3

4

5

6

7

8

9

10

-0,1 -0,05 0 0,05 0,1

∆a [mm]

∆φ 2

max

[arc

sec

]

No correctionsCorrections by "ep"Corrections by "igp"

Fig. 11: Influence of pinion offset on transmission errors

0

5

10

15

20

-0,1 -0,05 0 0,05 0,1

∆b [mm]

∆φ 2

max

[arc

sec

]

No corrections

Corrections by "ep"

Corrections by "igp"

Fig. 12: Influence of displacements of the pinion along its axis

on transmission errors The influence of the transmitted torque on transmission er-

rors is investigated, also (Figs. 16–19). The modifications, introduced by the variation of the cradle radial setting, have a stronger effect on the reduction of transmission errors, espe-cially in the case of smaller transmitted torques (Fig. 16). In Fig. 17, the motion graphs for the spiral bevel gear pair with a pinion finished by the basic machine-tool setting, loaded with different torques, is presented. It can be noted that the angular displacements of the driven gear are much bigger for moderate torque values. By the use of tooth modifications, introduced by

the variation of cradle radial setting conducted by different optimal polynomial functions on the two sides of the initial contact point, the transmission errors can be considerably re-duced, especially in the case of moderate torque values (Fig. 18). When the modified roll variation is applied, the angular displacements of the driven gear significantly increase in the case of light loads (Fig. 19).

0

2

4

6

8

10

12

-0,1 -0,05 0 0,05 0,1

∆c [mm]

∆φ 2

max

[arc

sec

]

No corrections

Corrections by "ep"


Fig. 13: Influence of displacements of the pinion along the gear axis on transmission errors

0

1

2

3

4

5

6

7

8

9

10

11

-0,1 -0,05 0 0,05 0,1

εh [deg.]

∆φ 2

max

[arc

sec

]

No corrections

Corrections by "ep"


Fig. 14: Influence of angular misalignment of the pinion axis in

the horizontal plane on transmission errors


1

2

3

4

5

6

7

8

-1 -0,5 0 0,5 1

εv [deg.]

∆φ 2

max

[arc

sec

]

No corrections

Corrections by "ep"


Fig. 15: Influence of angular misalignment of the pinion axis in

the vertical plane on transmission errors

0

1

2

3

4

5

6

7

8

0 50 100 150 200

T [Nm]

∆φ 2

max

[arc

sec

] No corrections

Corrections by "ep"


Fig. 16: Influence of transmitted torque on transmission errors

The maximum tooth contact pressure can be reduced only

moderately by the use of machine-tool setting variations. In comparison with the spiral bevel gear pair, whose pinion is generated by the use of the basic machine-tool setting (Fig. 20), the maximum tooth contact pressure can be reduced by 4% in the case of optimal cradle radial setting variation based on the reduced transmission errors (Fig. 21), and by 7% when the velocity ratio in the kinematic scheme of the machine-tool for the generation of pinion tooth-surface is optimally varied (Fig. 22).

0

1

2

3

4

5

6

7

8

0 0,25 0,5 0,75 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

T= 5 NmT= 10 NmT= 25 NmT= 50 NmT= 80 NmT= 120 NmT= 200 Nm

Fig. 17: Influence of transmitted torque on motion graphs for

the case when no modifications are introduced into the pinion tooth-surface

0

1

2

3

4

5

0 0,25 0,5 0,75 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

T=5 Nm

T=10 Nm

T=25 Nm

T=50 Nm

T=80 Nm

T=120 Nm

T=200 Nm

Fig. 18: Influence of transmitted torque on motion graphs when

the optimal cradle radial setting variation is applied for pinion teeth finishing


0

2

4

6

8

10

12

0 0,25 0,5 0,75 1

N1 ˙φ1/2˙π

∆φ 2

[arc

sec

]

T=5 Nm

T=10 Nm

T=25 Nm

T=50 Nm

T=80 Nm

T=120 Nm

T=200 Nm

Fig. 19: Influence of transmitted torque on motion graphs when optimal modified roll variation is applied for pinion teeth fin-

ishing

Fig. 20: Tooth contact pressure distribution when the pinion teeth are generated with the basic machine-tool setting

CONCLUSIONS

A method for the determination of the optimal polynomial

functions for the conduction of machine-tool setting variations in pinion teeth finishing in order to reduce the transmission errors in spiral bevel gears is presented. Polynomial functions of order up to five are applied to conduct the variation of the cradle radial setting and of the cutting ratio in the process for pinion teeth generation. On the basis of the obtained results the following conclusions can be made.

1. The transmission errors up to 81% can be reduced by the use of the optimal variation of the cradle radial setting in pinion tooth processing, conducted by two different polynomial func-tions of 5th order on the two sides of the initial contact point.

2. The transmission errors can be also considerably reduced by the use of the optimal variation of the cradle radial setting in the case of misalignments inherent in the spiral bevel gear pair, and for different load levels.

3. The maximum tooth contact pressure can be reduced only moderately: in the case of the optimal variation of the modified roll, the reduction is 7%, and in the case of the varia-tion of the cradle radial setting 4%.

4. The investigations have shown that by the combined variation of the cradle radial setting and of the modified roll, no further reduction of transmission errors can be achieved.

Fig. 21: Tooth contact pressure distribution when the pinion teeth are generated with optimal cradle radial setting variation

Fig. 22: Tooth contact pressure distribution when the pinion teeth are generated with optimal modified roll variation

ACKNOWLEDGMENTS The author would like to thank the Hungarian Scientific Re-

search Fund (OTKA) for their financial support of the research under Contract No. T 035207.


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Design and Manufacture of Spiral Bevel Gears With … · The truly conjugated spiral bevel gears...

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