TECHNICAL REPORT

Development of Meshing Variation Analysis ofWorm Gear Used in Electric Power Steering
A. KIKUCHI

S. MACHIDA

It was difficult to theoretically analyze meshing variation of the worm gear used in electric power steering by
numerical formula, because gear meshing positions are largely displaced due to the structure and material. Due tothis, we repeated vehicle and bench experiments and conducted countermeasures through trial and error. In orderto provide a more efficient and accurate analysis, which was a challenge during product development, we worked
to create a meshing variation design tool using a finite element method to allow prior estimation of gear meshingvariation. As a result, we have succeeded in creating an analysis technology with correlation to actual results,
which makes prior estimation in the design stage possible. In addition, we have succeeded in visualizing the sharedload on each tooth surface and contact position. This has facilitated the study on reduction of meshing variationand made development more efficient.
Key Words: electric power steering, worm gear, gear vibration, finite element method

1. Introduction
The rise in environmental awareness of recent years has
driven the market in vehicles with high quietness, suchas hybrid vehicles (HVs) and electric vehicles (EVs), andcreated a demand for further quietness of electric power
steering (EPS). The majority of EPS adopt worm gears
and an important issue is how to prevent the vibration andnoise created due to meshing variation when the steering
wheel is steered.
Furthermore, there is also a need for the gear to preventthe rattling noise which occurs when driving vehicles
on rough roads. As such, resin is used as the material of
the worm wheel and backlash is kept at zero constantly
by pressing the worm against the worm wheel with abacklash adjustment mechanism. Therefore, when theassist torque from the motor is inputted into the gear
during steering wheel operation, elastic deformation
occurs in the sections of the worm wheel made of resin
and worm displacement is caused due to the backlash
adjustment mechanism, thus making it difficult to analyze
the gear tooth face’s state of contact. Hence, we were
faced with the challenge during product development of
how to achieve a more efficient and accurate analysis,
and in reality, were forced to repeat vehicle and bench
experiments and establish countermeasures through aprocess of trial and error.

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Meanwhile, the finite element method (FEM), a
representative technique of CAE, is able to support
analysis conditions including contact of a complex shapeand as such has been used as a method to evaluate thestress and contact face pressure of the meshing portion
when investigating the strength and reliability of gears.
This report introduces initiatives towards building adesign investigation tool which applies FEM, is able
to predict meshing variation at the design stage andefficiently investigate a reduction technique. We will
also introduce an example of actually using this analysis
technique to investigate reduction of meshing variation.

2. The role and overview of gears
Figure 1 shows the structure of a steering system.
Assist torque responding to the steering torque by thedriver is generated in the motor, amplified by passing
through the gear and transferred to the output axis.
Figure 2 shows the structure of a gear with a backlash
adjustment mechanism.

JTEKT ENGINEERING JOURNAL English Edition No. 1013E (2016)

−Development of Meshing Variation Analysis of Worm Gear Used in Electric Power Steering−

Here, we investigated calculating the variation in the
output torque generated by meshing variation in the casethat a constant motor assist torque was inputted. Figure 3
provides an outline of the analysis procedure.
The worm and worm wheel models were made by
preparing a processing tool model, creating a tool
trajectory envelope that is generated in actual processing
and removing that from the pre-processing profile model.
Consequently, it is possible to prepare models from anarbitrary tool profile. Regarding the worm wheel model,
we focused on the low-rigidity resin portions and createdan angle scope where the worm wheel meshes with theworm with consideration to elastic deformation.

Gear

Motor

Fig. 1 Structure of steering system

SpringWorm(steel)Housing

3. 2 Load and boundary conditions
Motor

Non-motor side
support bearing

Worm wheel(resin)Support
Motor side
support bearing point Q

Fig. 2 Worm gear with backlash adjustment mechanism

The worm is able to become displaced centered on the
support point Q, which is the center of the motor side
support bearing, due to clearance secured between theworm’s non-motor side support bearing and housing,and a special spring embedded for backlash adjustment.
During motor operation, the inner diameter of the housing
on the non-motor side acts as a stopper and restricts worm
displacement.

3. Analysis technique
3. 1 Flow of analysis
It is a known fact that much of the vibration andnoise generated when the steering wheel is steered hascorrelation with the variation in output torque of the gear.

2. FEM analysis (static analysis)

3. Meshing variation calculation

Output torque1. ModelingAfter placing the worm and worm wheel models at thepredetermined center distance, we meshed them at anarbitrary angle, fixed the inner diameter face of the worm
wheel resin and performed static analysis while applying
constant torque on the worm. For the analysis model,
we combined the worm support bearing and housing in
addition to the worm and worm wheel. Figure 4 shows
the applied load and boundary conditions defined in eachelement model.
Torque equivalent to motor torque was inputted in theworm and the worm was subjected to a reaction force dueto the contact boundary with the worm wheel tooth face
and the contact boundary between the non-motor side
bearing and housing. Moreover, the motor side bearing
and worm press-fit face is bound by a spring coupling
which considers the axial rigidity, radial rigidity and angle
rigidity of the bearing, and expresses the bearing support
rigidity. The spring load has not been considered as itis sufficiently small compared with the housing stopper
contact load at the presumed input torque.
After executing the calculation, we confirmed therotation moment component of the output axis peripheryfrom the reaction force generated on the worm wheel
fixed face. This is equivalent to the output torque.

Meshing angle

Meshing variation

Max.

Min.
Worm wheel rotation angle

Fig. 3 Outline of analysis procedure

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−Development of Meshing Variation Analysis of Worm Gear Used in Electric Power Steering−

Fixed

Output torquePrimary meshing

Spring coupling
Torque input

Contact
boundary
Contact boundary
Fixed

Fig. 6 Experimental data of output torque3. 3 Calculating meshing variation
The abovementioned analysis was repeated whilechanging the meshing angle of the worm and worm
wheel. For the meshing angle, the angle pitch was equallydivided, and the change in output torque within one pitchrotation was discretely calculated. The difference between
the maximum and minimum values of the calculated
output torque was made the meshing variation.

4. Experiment verification results
In order to confirm the validity of this analysis
technique, a comparison was made with the results of an®
actual test using a column for C-EPS . Figure 5 shows anoutline of the examination procedure.

Load

Column-fixing jig

Servo motor

Steering
wheel shaft

Next, we measured the output torque variationon column assemblies with differing gear tooth
specifications. At the same time, we prepared a tooth
profile model of the same specifications and analyzed themeshing variation using FEM. Figure 7 is a comparisonof the measurement results and the calculation results.
2
The correlation coefficient R exceeds 0.7, and it canbe deemed there is no problem in making the meshingvariation calculated using the analysis technique in theprevious chapter a performance index.

Calculated meshing variation, N･mFig. 4 Applied load and boundary conditions

Torque meter

Time

R2＝0.7266

Measured output torque variation, N･mFig. 7 Comparison of measurement results and calculation results
Fig. 5 Outline of examination procedure

The steering wheel shaft was turned at a constantrotation speed using the servo motor, the time axis
waveform for column output torque when load wasapplied to the output axis was measured and the torquevariation was calculated.
Figure 6 shows the measured time axis waveform of
the output torque. We confirmed that the primary meshingcycle calculated from the steering wheel rotation speed
and worm gear reduction ratio was consistent with thecycle of the measured output torque variation.

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5. Investigation of a technique to reducemeshing variation
5. 1 Visualization of meshing variation
An important point of investigating the design
specifications enabling the reduction of meshing variation
is ascertaining the relationship between variation in load
sharing of the contact teeth, variation in the meshingcontact point, and variation in the number of teeth
meshing, as well as the variation in output torque that
corresponds with these. Therefore, at the same time
as using the analysis technique shown in the previous
chapter to calculate the meshing variation of the worm
wheel axis, we used FEM analysis to find the shared loadin the direction of worm wheel rotation (F i ) that occurs
in the corresponding meshing of the worm and worm
wheel tooth (tooth pair), as well as the contact radius (ri ).
Figure 8 provides an overview of that.

JTEKT ENGINEERING JOURNAL English Edition No. 1013E (2016)

−Development of Meshing Variation Analysis of Worm Gear Used in Electric Power Steering−

F1

Output torqueWormTooth pair 1 Tooth pair 2

F2

r1

r2

F3

r3

Tooth pair 3

Worm wheel

0

0.2

0.4

0.6

0.8

1

Angle pitch(a) Relationship between output torque and angle pitch
Direction of rotation

Fig. 8 Shared load and contact radius of meshing tooth
Shared load

Meshing of two pairs of teeth

From the torque balance, output torque T would be as
follows if the m tooth pair was meshing at the same time.
⑴

Regarding the tooth contact portion, we had madethe contact radius the position where the maximum facepressure was generated due to the fact that a constantarea is maintained as a result of worm wheel resin elastic
deformation. We deemed there was no problem with thistechnique as the output torque calculated from Formula
1 and the output torque from the analysis results werein close agreement. Figure 9 shows the transition in theshared load and contact radius of each tooth pair, as wellas the output torque, when the meshing angle is changed
for a gear of certain specifications.
There are changes in shared load and contact radius forthe respective corresponding teeth, therefore the figure
differentiates between tooth pairs 1, 2 and 3. In otherwords, for these particular gear specifications, a state ofmeshing with two teeth alternates with a state of meshingwith three teeth, and the actual meshing ratio is between
two and three. From Fig. 9 (b), the shared load of “tooth
pair 1” decreases as meshing progresses from the angle
pitch of 0, and the output torque of Fig. 9 (a) decreases.
After this, meshing of “tooth pair 3” begins from theangle pitch of 0.6 and it can be observed that the output
torque increases as the shared load increases.

Tooth pair 1

Tooth pair 3
Tooth pair 2

0

0.2

0.4

0.6

0.8

1

Angle pitch(b) Relationship between shared load of each tooth pair and angle pitch
Tooth tip

Tooth pair 3
Tooth pair 2

Contact radius

T =F1 r1 ＋F2 r2 ＋・・・＋Fm rm

Meshing of three Meshing of two
pairs of teeth pairs of teeth

Tooth root

Tooth pair 1

0

0.2

0.4

0.6

0.8

1

Angle pitch(c) Relationship between contact radius of each tooth pair and angle pitch

Fig. 9 Calculation results

5. 2 Example of a technique for reducing meshingvariation
Here, as an example of a countermeasure for meshingvariation, we investigated optimizing the setting angle ofthe hob, which is used to machine the worm wheel. If thehob setting angle when machining teeth is made b, theworm wheel lead angle is made c and the hob lead angle
is made a, the principle will be β＝γ−α as shown in
Fig. 10.
c
HobaWorm wheel lead angle: c
Hob lead angle: aHobsetting angle: b

Worm wheel

Fig. 10 Relationship between worm wheel and hob

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−Development of Meshing Variation Analysis of Worm Gear Used in Electric Power Steering−

Output torqueHob setting angle
▲：b0，
□：b1，
●：b2

0

0.2

0.4

0.6

0.8

1

Angle pitch(a) Relationship between output torque and angle pitch

Tooth pair 1
Shared load

In the case of the gear specifications in Fig. 9, it canbe seen from Fig. 9 (b) that the reason for the decreasein output torque is a decrease in the shared load of “tooth
pair 1”. “Tooth pair 1” meshes with the worm wheel
tooth root therefore if the shared load at the tooth root
can be maintained, the decrease in output torque canbe minimized, thus reducing output torque variation.
Meanwhile, the contact point with the worm in the worm
wheel, as shown in Fig. 11, does not only progress in theradial direction but also the tooth flank direction therefore
by increasing the hob setting angle b it would be possible
to increase contact with the worm in the worm wheel
tooth root area and it can be predicted that this would
maintain shared load.

b1 load increases
Tooth pair 3

Tooth pair 2b: Large
Start ofmeshing

Increased
contactwith worm

End ofmeshing

Tooth root

0.6

0.8

Tooth tip

1

Tooth pair 3

Tooth pair 2

Delay in startof b2 meshing
Tooth pair 1

0

0.2

0.4

0.6

0.8

1

Angle pitch(c) Relationship between contact radius of each tooth pair and angle pitch

Fig. 12 Hob setting angle and meshing variation

In this way, it is possible to find the optimal hob setting
angle in relation to meshing variation and even in an
actual verification, as shown in Fig. 13, a 30% reductionin output torque variation compared with conventional
specifications was observed.
Measured output
torque variation, N･m

In light of this, we conducted further analysis using
gear specifications with two different hob setting angles.
The results of this are shown in Fig. 12. The relationshipbetween the hob setting angles of b0, b1, and b2 in the
figure is b0＜b1＜b2.
Looking at the calculation results of output torquevariation, b1 from Fig. 12 (a) is the lowest and the causeof this is considered from the relationship between sharedload and contact radius. Looking at the analysis results ofb1 from Fig. 12 (b), it can be seen that the shared load of
tooth pair 1”, in comparison with b0, has increased in the
0 to 0.6 pitch range as expected, and reduction of output
torque has been minimized. However, if the hob setting
angle is made as large as b2, the output torque variationfrom Fig. 12 (a) instead becomes larger than b0. Whenshared load and contact radius are compared with otherspecifications, the reduction in output torque for b2 isminimized for the first half of the angle pitch however
the angle at which meshing of “tooth pair 3” starts is later
than other specifications and this is due to a delay of arise in output torque. This is because, if the hob setting
angle increases, on the worm wheel tooth tip side, inother words at the beginning of meshing, the tooth profilechanges in the direction whereby the clearance with theworm increases. This is expressed in Fig. 11.

0.4(b) Relationship between shared load of each tooth pair and angle pitch

Tooth root

Fig. 11 Tooth surface of worm wheel and meshing linewith worm

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0.2

Angle pitch

Contact radiusb: Large0
Tooth tip

30% reduction

Conventional
specifications

Optimalspecifications

Hob setting angle

Fig. 13 Effects of optimization of hob setting angle

JTEKT ENGINEERING JOURNAL English Edition No. 1013E (2016)

−Development of Meshing Variation Analysis of Worm Gear Used in Electric Power Steering−

6. Conclusion
This paper has introduced technology for analyzing
worm gear meshing variation using FEM and an
examination case for reducing such variation. As
explained at the outset, the meshing of worm gears
for EPS is a complex phenomenon and experimental
verification was difficult. However, by using the analysis
technology introduced in this paper, it was possible tovisualize meshing variation and easily understand andexplain the phenomenon. Moreover, because it is now
possible to predict meshing variation, the efficiency ofdevelopment tasks can be increased by reducing manhours in the prototype and evaluation phases. JTEKT
wishes to continue development so that we may promptly
respond to high-level requirements from our customers by
utilizing a variety of analysis technologies.

References
1)

2)

T. Sugiura: Backlash Adjustment Mechanism for
Reduction Gear of Electric Power Steering System,
JTEKT ENGINEERING JOURNAL, No. 1001 (2006)
73.
S. Nishimura: Development of Rattle Noise Analysis
Technology for Column Type Electric Power Steering
Systems, JTEKT ENGINEERING JOURNAL, No. 1009
(2001) 66.

A. KIKUCHI****JTEKT ENGINEERING JOURNAL English Edition No. 1013E (2016)

S. MACHIDA**

Component Development Dept., Steering Systems
Operations Headquarters
Experiment & Analysis Dept. 1, Steering Systems
Operations Headquarters

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