Design of a Pair of Noncircular Gears Meeting Ackermann Steering Principle

Sheng Zhao, Jing-Shan Zhao and Zhi-Jing Feng Department of Precision Instruments and Mechanology

Tsinghua University Beijing 100084, P. R. China

zhaoc08@mails.tsinghua.edu.cn

Abstract—This paper focuses on the design of a pair of noncircular gears, which is used in a concept automobile steering system to make the steering meet Ackermann steering principle. Based upon Ackermann steering principle, the pitch curves of the noncircular gears are formulated. The tooth profiles, addendum and dedendum curves, and transition curves of the noncircular gears are designed. Moreover, the phenomenon of the tooth profile reflection is discussed. Three-dimensional model of the noncircular gears is developed in Pro/ENGINEER, where a motion simulation is also carried out to prove the design.

Keywords-Noncircular Gears; Ackermann Steering Principle; Pitch Curves; Tooth Profiles; Transition Curves

I. INTRODUCTION Ackermann steering principle [1] can be simply stated that

the automotive wheels should all move about a common centre point when turning. Meeting Ackermann steering principle can make the wheels do pure rolling without sideslip, so that can reduce the tire wear. The most commonly used four-bar linkage steering trapezoid cannot always meet the Ackermann steering principle exactly. So many researchers have done a series of work trying to improve the steering system. Fahey [2] introduced a Fahey eight-member mechanism (FEMM), which could control the angle error of the inside steering wheel in ±0.03° even in a large steering range of 61°. Pramanik [3] synthesized a Pramanik six-member mechanism (PSMM), which has 5 exact points meeting the Ackermann steering principle. The overall accuracy of PSMM is higher than that of the traditional four-bar linkage steering trapezoid, but lower than that of FEMM. Gautam and Awadhiya [4] also adopted a six-bar linkage to make the steering have 7 exact points meeting the Ackermann steering principle. Although its overall accuracy is higher than that of the traditional four-bar linkage steering trapezoid, it is lower than that of either FEMM or PSMM. Carcaterra [5] added a differential mechanism in the steering system, which provides substantial improvements for the large steering angles. Emura and Arakawa [6] investigated noncircular gears in the design of the steering system.

This paper also focuses on an automobile steering system meeting the Ackermann steering principle. It consists of a pair of noncircular gears that we use to reach this goal. Noncircular gears [7] are a kind of gears that have noncircular pitch curves, and hence can be used to transfer non-uniform speed ratio

motions. The main process of designing a pair of noncircular gears consists of two steps [7]. First, determine the pitch curves of the noncircular gears. Second, determine other parameters of the noncircular gears, especially the tooth profiles, to make sure that the two noncircular gears can mesh correctly.

II. ACKERMANN STEERING PRINCIPLE AND A CONCEPT AUTOMOBILE STEERING SYSTEM

This section first introduces the Ackermann steering principle formulation and then introduces a concept steering system, in which the pair of noncircular gears are used.

As shown in Fig. 1, by Ackermann steering principle, the angles turned through by the two front wheels should satisfy the following relation [9]:

cot cot ,rw lwl l bϕ ϕ− = (1)

where l denotes the wheel base, b denotes the wheel tread, lwϕ and rwϕ represent the angles turned through from the

forward direction by the front left wheel and the front right wheel, respectively. lwϕ and rwϕ are positive when the vehicle turns left.

A concept automobile steering system with pure rotation transfers is proposed [10] to meet the Ackermann steering principle, as shown in Fig. 2. In the concept automobile steering system, steering rotational input is transferred to the right steering shafts by a middle pair of bevel gears. The rotation of the right steering shafts is then, on the one hand, transferred to the front right wheel by a right pair of bevel gears; and on the other hand, transferred to the left steering shafts through a pair of noncircular gears. Then the rotation of the left steering shafts is transferred to the front left wheel by a

lw rw

Figure 1. Ackermann Steering Principle

217978-1-61284-459-6/11/$26.00 ©2011 IEEE

left pair of bevel gears. The left and the right steering shafts are both made up of a pair of constant velocity joints.

To further describe the concept steering system, the noncircular gear which is connected with the left steering shafts is named the left noncircular gear, and the other noncircular gear is named the right noncircular gear. The positive directions of the angular displacements of the gears that have horizontal shafts are defined to point rightwards. Therefore, we have

/,

/lb ln lw

rb rn rw

i

i

ω ωω ω

== −

(2)

where lnω ( rnω ) is the angular velocity of the left(right) noncircular gear, lwω ( rwω ) is the angular velocity of the front left(right) wheel, lbi ( rbi ) is the transmission ratio of the left(right) pair of bevel gears from the steering shaft to the wheel.

Considering the symmetry design between the left pair of bevel gears and the right pair of bevel gears, we let

.lb rbi i= (3)

In this paper, we assume (3) always holds. Combining (2) with (3), there will be

/ / .ln rn lw rwω ω ω ω= − (4)

III. DESIGN OF THE PAIR OF NONCIRCULAR GEARS From (4), we can find that the angular velocities of the two

front wheels are determined by the angular velocities of the two noncircular gears. We will have the angles turned through by the two front wheels meet the Ackermann steering principle when we design the noncircular gears properly. So next, we discuss the design of the pair of noncircular gears, consisting of the design of pitch curves, addendum curves, dedendum curves, tooth profiles and transition curves.

A. Design of Pitch Curves According to (1), we have

( ) ( )2 2

1 ./ sin / sin 2 1

lw

rw rw rw

dd b l b l

ϕϕ ϕ ϕ

=− +

(5)

Since

/ / ,lw rw lw rwd dω ω ϕ ϕ= (6)

combining (4), (5), with (6), there is

( ) ( )2 2

1 ./ sin / sin 2 1

ln

rn rw rwb l b l

ωω ϕ ϕ

= −− +

(7)

The meshing noncircular gears can be treated as pure rolling along their pitch curves, which means

/ / ,ln rn rn lnr rω ω = − (8)

where rnr and lnr are the polar radii of the pitch curves of the right noncircular gear and the left noncircular gear with the polar origins located at each shaft centre, respectively. Hence, substituting (7) into (8), we obtain

( ) ( )2 2

1 ./ sin / sin 2 1

rn

ln rw rw

rr b l b lϕ ϕ

=− +

(9)

Assume the center distance of the two noncircular gears is d , then there is

.ln rnr r d+ = (10)

Combining (9) and (10), we can obtain

( ) ( )2 2.

/ sin / sin 2 2rn

rw rw

dr

b l b lϕ ϕ=

− + (11)

We now try to replace rwϕ with rnθ , where rnθ denotes the polar angle of the right noncircular gear. For this purpose, we introduce a variable rnϕ , which represents the angular displacement of the shaft of the right noncircular gear. Specify that 0rnϕ = when 0rwϕ = ; 0rnθ = when 0rnϕ = ; rnθ has the same positive direction as rnϕ . Assume the front right wheel turns through an angle rwϕ from the zero position, therefore,

/ / .rn rw rn rwtime time

dt dtϕ ϕ ω ω= (12)

Applying (2) to (12), we will have

( )/ / .rn rw rb rw rwtime time

i dt dtϕ ϕ ω ω= − (13)

Since rbi is a constant, it can be drawn out from the integral sign, therefore,

/ .rn rw rbiϕ ϕ = − (14)

Note that when the shaft of the right noncircular gear rotates by an angle of rnϕ , it is at the polar radius corresponding to the polar angle rnθ− that the pure rotation takes place. Hence, we have

.rn rnθ ϕ= − (15)

Combining (14) with (15), one can obtain

/ .rw rn rbiϕ θ= (16)

Substituting (16) into (11), we finally get the polar equation of the right noncircular gear as following:

Figure 2. A Concept Automobile Steering System

218

( ) ( ) ( )2 2.

/ sin / / sin 2 / 2rn

rn rb rn rb

dr

b l i b l iθ θ=

− + (17)

Similarly, we can get the polar equation of the left noncircular gear as following:

( ) ( ) ( )2 2.

/ sin / / sin 2 / 2ln

ln lb ln lb

dr

b l i b l iθ θ=

− + (18)

Comparing (17) with (18), it can be found that ln rnr r= when ln rnθ θ= . However, since lnθ and rnθ have the same positive direction and they all start from the center connection line of the two noncircular gears, it means the polar curves of the two noncircular gears are centrosymmetric about the midpoint of their center connection line. Therefore, the two noncircular gears can be designed centrosymmetrically. As a result, we will just investigate the left noncircular gear later in this paper, and the right noncircular gear is obtained using the centrosymmetric characteristics.

We establish a plotting Cartesian coordinate system, p p po x y , with the origin, po , located at the center of the left

noncircular gear. The px -axis is along the center connection line of the two noncircular gears, pointing from the center of the left noncircular gear to the center of the right one. The py -axis is located where 90lnθ = . From (18), we have

cos

.sin

ln ln ln

ln ln ln

x r

y r

θθ

==

(19)

Thus we can plot the pitch curve of the left noncircular gear using (18) and (19) with b , l , d , and lbi all known.

B. Design of addendum curves and dedendum curves Having known the pitch curves of the two noncircular

gears, the addendum and dedendum curves can be obtained in line with the equidistance line formula [11]. In the plotting coordinate system, p p po x y , the addendum curve equation of the left noncircular gear can be represented as following:

2 2

2 2

/,

/

aln ln a ln ln ln

aln ln a ln ln ln

X x h my x y

Y y h mx x y

∗

∗

= + +

= − + (20)

where lnx and lny are given as (19), ah∗ is the addendum factor, m is the gear module, lnx is the derivative of lnx with respect to lnθ , and lny is the derivative of lny with respect to

lnθ . The dedendum curve equation of the left noncircular gear can also be obtained that:

2 2

2 2

/,

/

dln ln f ln ln ln

dln ln f ln ln ln

X x h my x y

Y y h mx x y

∗

∗

= − +

= + + (21)

where fh∗ is the dedendum factor.

C. Design of tooth profiles The profiles of the noncircular gears are designed based

upon the pitch curves using the analytical method [8].

For any profile point, P , located outside/inside the pitch curve, suppose the inward/exterior normal of the profile drawn through P and the pitch curve intersect at a point Q , as shown in Fig. 3. We use QR to denote the tangent of the pitch curve through point Q , ( )0,μ π∈ to denote the angle turned through from OQ to QR anticlockwise about the point Q . Then we will have [11]

( )( )

tan ,ln Q

ln Q

ln ln

ln ln

r

rθ θ

θ θ

θμ

θ=

=

=′

(22)

where ( )0,μ π∈ , ( )ln lnr θ is shown as (18), ( )ln lnr θ′ is the derivative of ( )ln lnr θ with respect to lnθ , Qθ is the polar angle of OQ .

According to the analytical method [8], the coordinate of the profile is:

( ) ( )( ) ( )( ) ( )( ) ( )

cos cos cos

sin cos sin,

cos cos cos

sin cos sin

Q

Qn

Q

Qn

Q

Qn

Q

Qn

lp Q Q ln ln ln n Q n

lp Q Q ln ln ln n Q n

rp Q Q ln ln ln n Q n

rp Q Q ln ln ln n Q n

x r r d

y r r d

x r r d

y r r d

θ

θ

θ

θ

θ

θ

θ

θ

θ θ θ α θ μ α

θ θ θ α θ μ α

θ θ θ α θ μ α

θ θ θ α θ μ α

= ± + −

= ± + −

= + +

= + +

(23)

where subscript lp denotes the profile through nQ is a left profile, subscript rp denotes the profile through nQ is a right profile, Qr is the polar radius of point Q , Qθ is the polar angle of point Q , the upper sign in ± or represents the profiles outside the pitch curve, while the lower sign represents the profiles inside the pitch curve, ( )nQ n ∈ is the intersection points of the profiles and the pitch curve, represents the integer set,

nQθ is the polar angle of point

nQ , ( )0,μ π∈ and can be worked out using (22), nα is the pressure angle. Each point Q will result in a profile point P , and can be appointed from the point nQ by increasing

nQθ piece by piece as you wish.

Now we discuss how ( )nQ n ∈ come about. On the pitch curve, the pitch p and the tooth thickness s are both / 2mπ ,

μ

Qθ

Figure 3. The Profile of a Noncircular Gear

219

where m is the gear module. Therefore, the arc length determined by the two neighboring intersection points of the profiles and the pitch curve is / 2mπ . To keep the centrosymmetric characteristics on profiles of the left and right noncircular gears, we set

0Qθ to be 0°. Then the polar

angles of other intersection points, nQθ , can be determined by

solving the integral equation

( )0

/ 2.Qn

Qln ln lnr d n m

θ

θθ θ π= (24)

When plotting profiles, a reflection of the tooth profile will occur once the profile get intersected with the base curve, as shown in Fig. 4. The reason for the reflection is that the involutes are expanded outsides the base curve. So there could not be involutes insides a base curve. The analytical method we use here is derived based upon the existence of the involutes. In fact, if we investigate the reflection of the tooth profile as Fig. 3 and choose the inward normal drawn through the point P , we can find (23) is also available.

D. Transition curves, undercutting and tooth interference For rack cutters, the transition curves are the envelope

curves of the rack tooth tip fillet and are formed naturally by the rack cutters when machining [8]. The transition curves can be formulated as following.

As shown in Fig. 5, the rack cutter has a tip fillet with a radius of ρ . The points S and E represent the endpoints of the tip fillet. fO is the center of the fillet. The point T is any point on the tip fillet. The line fTO and the pitch line intersect at the point U . Since fO is the center of the fillet, the line

fTO , in fact, is the inward normal of the tip fillet. Tα is the acute angle determined by TU and the pitch line, so

,90T nα α∈ . nα is the pressure angle. Thus, we have

( )*0 sin / sin ,f a n TUO h m ρ α α= − (25)

where fUO is the length of the line segment fUO , *0ah is the

addendum factor of the rack and is generally equal to 1, and m is the gear module. Therefore,

,fUT UO ρ= + (26)

and

*0 tan cos cot ,n a n n f TQ U h m UOα ρ α α′ = + + (27)

where UT is the length of the line segment UT , and nQ U′ is the length of the line segment nQ U′ .

When the rack pitch line and the pitch curve of the noncircular gear are tangent at the point U , since the line TU is the inward normal of the tip fillet, the rack cutter tip fillet and the transition curve of the noncircular gear will get contacted at the point T [8]. Therefore, if we work out the position of the point T at the time, we can get the corresponding point on the transition curve.

Suppose the rack pitch line and the pitch curve of the noncircular gear are tangent at the point ( )n nQ Q′ firstly. Next the rack cutter moves rightwards, making the rack cutter and the noncircular gear contact at the point U , as shown in Fig. 6. We use Uθ to denote the polar angle of OU , ( )0,Uμ π∈ to denote the angle μ at the point U . Hence, Uθ can be worked out using

( ) ,U

Qnln ln ln nr d Q U

θ

θθ θ ′= (28)

where nQθ is computed by (24), and nQ U′ is shown as (27).

Then Uμ can be worked out using (22) where ln Uθ θ= .

Thus, after specifying the angle Tα , we can obtain the polar angle of UT as

2 .UT U U Tθ θ μ α π= + + − (29)

Therefore, the coordinate of the point T is

( )( )

cos cos,

sin sin

T ln U U UT

T ln U U UT

x r UT

y r UT

θ θ θ

θ θ θ

= +

= + (30)

where Uθ can be worked out using (28), UT is equal to UT

102 103 104 105 106 107 108 109 110 111 112

65

66

67

68

69

70

71

72

Figure 4. The Reflection of the Tooth Profile

0cm∗

0cm∗

0ahm

∗0ahm

∗/ 2mπ

ρ

Figure 5. The Rack cutter and Tip Fillet

ρ

Figure 6. A Point, T, on the Transition Curve

220

as shown in (26), and UTθ can be obtained by (29).

Change Tα from nα to 90 , then the position of the point

T will change on the tip fillet, and we can get a number of points on the transition curve. Connect these points smoothly, and the transition curve can be obtained. Since the pitch curve of the noncircular gear is noncircular, each transition curve is not the same as others. So, we need to design the transition curves one after one.

For rack cutters, to avoid undercutting, there should be [8]

2max min 0sin / ,n am hρ α ∗≤ (31)

where maxm is the maximal gear module allowed, minρ is the minimal radius of curvature of the pitch curve, nα is the pressure angle, 0ah∗ is the addendum factor of the rack and is generally equal to 1. The noncircular gears machined by rack cutters have no tooth interference [8].

For wire cutting, we can design the noncircular gears as processed by the rack cutter. Moreover, if the tooth profiles can reach the addendum curve, then, for the convenience of design, we can use a simplified transition curve that a fillet with the radius of [12]

0.2 ,cr m= (32)

where m is the gear module.

IV. A SAMPLE DESIGN Here, we give a sample design of the noncircular gears.

The quantities known are shown in Table I.

TABLE I. QUANTITIES KNOWN IN THE SAMPLE

m /mm

d /mm

b /mm

l /mm

nα

/ lbi rbi ah∗ fh∗

3 200 1380 2300 25 1 1 1 1.25

Considering the angle turned through by the front wheel

rwϕ is generally less than 60°, we let 60 ,60rnθ ∈ − in line

with (16). Similarly, we let 60 ,60lnθ ∈ − . Specify the profile of the left noncircular gear which passes through point

0Q is a left profile, then we can obtain the noncircular gear assembly in Pro/ENGINEER as Fig. 7. If we set the angular velocities of the left and right noncircular gears in accordance with (7), and set the initial meshing position at 0ln rnθ θ= = , then a simulation can be run to prove the meshing motion.

CONCLUSIONS A pair of noncircular gears is designed, which is used in a

concept automobile steering system to make the steering meet Ackermann steering principle. The pitch curves, addendum curves, dedendum curves, tooth profiles and transition curves of the noncircular gears are formulated. Moreover, the phenomenon of the tooth profile reflection is discussed. A sample is given in Pro/ENGINEER, where a motion simulation is carried out to prove the meshing motion.

ACKNOWLEDGMENT This research was supported by the Beijing Municipal

Natural Science Foundation under Grant 3112014, the National Natural Science Foundation of China under Grant 50805083, a Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 200741 and the Program for New Century Excellent Talents in University. The authors gratefully acknowledge these support agencies.

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Figure 7. The Assembly of the Pair of Noncircular Gears

221