Effect of Pinion Profile Modification On Rack and Pinion Steering Gear
Effect of Pinion Profile Modification On Rack and Pinion Steering Gear
Effect of Pinion Profile Modification On Rack and Pinion Steering Gear
2005-01-1273
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2005-01-1273
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Considering the steering gear alone, the torque misalignments [7],[8]. Also the rack and pinion are spring-
required to rotate the pinion provides the means for loaded and, hence, have zero backlash. Zero backlash
predicting the performance of the steering system. For a gives dual flank contact. Hence, each tooth is in contact
stand alone steering gear the torque required to rotate the with two adjacent teeth of mating gears. But when two
pinion is termed as Free Pinion Torque (FPT). The Free teeth are in contact, theoretically dual contact will give four
Pinion Torque (FPT) results from three frictional interfaces, contact points. Since, no two teeth are identical in practice
namely, 1) Rack and pinion, 2) Rack and plunger, and 3) so the contact occurs at any two points. But these two
Rack and bush. Friction between rack and plunger can be contact point out of four theoretically possible contact
reduced by replacing sliding contact with by rolling contact. points are difficult to determine due to randomness of tooth
But due to manufacturing difficulties and cost, use of better thickness error [13]. The number of contact points at any
material combination like metal and plastic is commonly instant determines the number of frictional interfaces at
adopted. The friction between rack and pinion is the major that instant. Increase and decrease in number of frictional
source of the torque variation of the steering gear. Baxter interfaces while the teeth go through mesh leads to tooth
et al. [3] used a vectorial approach to develop a friction excitations. Tooth friction excitations [10] ,[11] are
mathematical model of mesh friction and mechanical mainly controlled by the transverse contact ratio. In RPS,
efficiency of rack and pinion steering. This mesh friction once per tooth excitations are enhanced due to larger tooth
model is limited to single tooth contact for a roll angle of depth. As the number of teeth in contact changes, the
one involute flank. Steering gear provides a gear ratio of number of frictional interfaces and, hence, the mesh
15-20:1, which means that 4-5 turns of steering wheel are stiffness varies causing vibrations in the gear system. To
required to steer the wheels through an angle of 60o [4]. obtain lower turning effort and less variation in the torque, it
The rack travel being limited due to track-width of vehicle is necessary to analyze the gear mechanics of the RPS,
and tie rod linkages, the gear ratio is decided by the pinion which is the contribution of this paper. The paper is
diameter. This puts limitation of pinion diameter. The tooth organized as follows: Next section explains the manual
load capacity of the pinion is increased by increasing the Rack and Pinion Steering (RPS) system under study,
tooth thickness and reducing the number of teeth [5]. followed by the concept of profile modification. Sliding
Reduced number of teeth gives poor contact ratio. velocities and mesh friction are calculated next, while the
Similarly the circular cross section of rack is prone to roll torque characteristics are presented in the “Effective radius
while translating leading to slight misalignments in the gear for torque transmission.” Finally conclusions are given.
mesh. In helical gears overall contact ratio can be
represented as combination of profile (transverse) and face MANUAL RACK AND PINION STEERING (RPS)
contact ratio. This combination in helical gears gives SYSTEM
averaging of once per tooth effect with smoother load
transfer between teeth [6]. In RPS, the rack and pinion has In the Manual Rack and Pinion Steering (RPS) system,
non-parallel, non-intersecting shafts, which improves the the steering gear box essentially consists of a rack, a
contact ratio as well as accommodates slight pinion, the yokenut assembly, a bush and a rack tube.
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The manual RPS behaves as constant ratio steering should be sufficient to accommodate the composite
system as it gives same gear ratio over the entire span error, which is unavoidable due to manufacturing
of the rack movement. The rack and pinion gear process. This will avoid frequent jamming of teeth and
assembly is designed to fulfill the following higher torque requirements. Similarly, the yoke
requirements: clearance should not be large so as to produce noise
due to vehicle vibration. This suggests that the gear
1) Backlash Elimination: Backlash resulting due to gear errors need to be closely controlled in order to keep the
errors or wear is eliminated by meshing the rack and the center distance variation less than the of yoke
pinion under the spring force, as shown in Fig. 3. This clearance.
leads to dual flank contact in the gear mesh. The rack is
forced upwards to mesh with the pinion due to the spring 2) Stiffness of Steering System: The rack and pinion in
compression. Any upward motion of rack to take up the an RPS is forced to mesh under the spring force. The
backlash is supported by the spring-loaded plunger. spring preload acts as normal load at gear mesh,
Similarly, downward motion of rack acts against the leading to higher frictional torque. The nominal torque
spring preload. The downward motion of rack is limited value required to rotate the pinion depends upon this
due to small clearance available for plunger to move preload. The preload is so adjusted that the steering is
downwards. This small clearance is known as yoke neither too responsive during the straight ahead
clearance, as denoted in Fig. 3. Manufacturing and motions, nor too stiff during turning. Considerably larger
assembly errors often lead to center distance variation value of spring preload keeps the RPS backlash free.
between the gears. If this variation is constrained by
fixing the axis of gears then it leads of jamming of teeth PROFILE MODIFICATION
during mesh. This situation can be predetermined by
meshing the gear with a master gear under a spring With the reduction in number of teeth, the
force. Then the total center distance variation gives the tendency for undercutting of the mating gear increases.
composite error. If this composite error [13] is less than So the pinion with less number of teeth needs to be
intentional backlash provided in the gear mesh, then modified so as to avoid the interference. The pinion is
gears will operate smoothly. cut by withdrawing the hob by certain amount, which
results in shift of the whole involute profile radially
outwards.
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The above formula gives 20 teeth considering the tool The pinion in the RPS under study has 6 Number
clearances. For best profile precision, the teeth number of teeth. To avoid undercutting, the pinion profile is shifted
should be 22. So when it is necessary to design gears radially outwards. The pinion tooth thickness is increased
with few teeth, undercutting is avoided by resorting to at the expense of tooth space. As a result of this, the
enlargement. A modified gear makes the use of involute datum line or axis of symmetry of the rack does not roll
profile further out from the base circle, making it along the pitch circle of the pinion but displaced laterally
stronger. from it by an amount equal to the profile shift. For the rack
and pinion gear mesh, extended center distance Ce, is
As discussed earlier, the pinion diameter is given by following formula [5], i.e.,
limited due to limited rack gain. To avoid undercutting,
pinion is cut with hob withdrawn by an amount equal to
the profile shift. The resulting pinion has increased tooth
Ce= ∑ (k p + k w )m (2)
width with corresponding decrease in tooth space. True
kinematic pitch line of the rack does not roll with the Where, kp : Profile modification coefficient on pinion;
pitch circle of the pinion but is displaced by an amount kw : Profile modification coefficient on rack; m : Module
equal to the profile shift. Distance between the line of
symmetry of rack and pinion pitch circle is the extended Pinion behaves as all addendum gear with
center distance as shown in Fig. 4. Also there is an larger addendum exposed to the rack. Contact occurs
increase in pinion addendum and corresponding on one side of flank only. Approach part being negligible,
decrease in rack addendum. This system is commonly larger recess part of the line of action contributes to
known as ‘Long and short addendum system’ and the motion transmission. The effect of this modified
gears operate at extended center distance [5]. The geometry is to increase the sliding velocity as well as
profile-shifted pinion behaves as recess-action gear with effective radius for torque transmission.
a long addendum and short dedendum. The gear is call
recess-action gear, because most of the contact occurs
SLIDING VELOCITY AND MESH FRICTION
on one side of the pitch point. Recess action gears are
preferable because of their better contact properties and
The nature of sliding in invloute gear teeth
less wear along the recess portion of flank [12].
consists of sliding in one direction during approach,
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[ ]
active profile constitutes of addendum of the pinion. Due
V 2
Where Vs = r − r 2 b − rp sin φ (3) to increased tooth depth and larger addendum, leverage
r offered by the point of contact varies considerably.
p Theoretically, the pitch circle radius and normal tooth
load determines average torque requirement. But it
In which, V: Pitch line velocity; rp: Pitch circle radius of cannot produce realistic results, because effective radius
pinion; rb: Base circle radius of pinion; r: Radius of point for torque transmission varies through one third of the
of contact; φ :Pressure angle. pitch circle radius and active part of the involute flank
lies almost on one side of the pitch circle radius. Thus,
Relation between the sliding velocity and the major portion of active flank is which actually contributes
friction coefficient based on the above formulae is to the torque transmission is considerably larger than the
plotted Fig. 5 for the following specifications: pitch circle radius. So torque determined by pitch circle
radius is always less than the actual torque. The torque
Pinion Base Radius (rb): 5.58mm required by the pinion depends upon the instantaneous
radius of point of contact, which varies considerably due
Pitch circle Radius (rp): 6.06mm to enlarged pinion tooth depth keeping the pitch circle
diameter same. So there is considerable variation in
Pinion Helix Angle (ψ) : 30 o torque from engagement to disengagement of the tooth.
A similar effect was reported in Baxter et. al. [3], where
Installation Angle (γ) : 15o friction coefficient is considered constant. The recess
torque (Tr) considering the friction given by eq. 3, is
Pressure Angle (φ) : 20o calculated as follows: [14]:
Number of Teeth (n) : 6 Where θx: Angle of action (roll angle); f : Coefficient of
friction; Wn: Tooth load.
Bush Friction Preload (Wb) : 0.18 N
5
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ACKNOWLEDGMENTS
REFERENCES
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9. Michalec, G.W., 1966, Inherent Gear Errors, 12. Mabie, H. H., Ocvirk, F. W., 1978, Mechanisms
Precision Gearing – Theory and Practices, John and Dynamics of Machinery, John Wiley and
Wiley and Sons, Inc., NY, 56-80. Sons.
10. Velex, P., Sainsot, P., 2002, An Analytical Study 13. Smith J. D., 1983, Gears and Their Vibrations,
of Tooth Friction Excitations in Errorless Spur Gear-box Modelling, Marcel Dekker, Inc. pp,
and Helical Gears, Journal of Machines and 103-116.
Mechanism Theory, 37, 641-658. 14. Buckingham, E., 1963, Analytical Mechanics of
11. Lin, J., Parker, R., 2002, Mesh Stiffness Gears, Efficiencies of gears, Dover Publications,
Variation Instabilities in Two Stage Gear NY 1949, pp. 395-425.
Systems, Transactions of ASME, vol 124, 68-76.