TECHNICAL REPORT

Analysis of Ball Rotation Behavior of Rzeppa Type Constant
Velocity Joint (CVJ)
T. NAKAMURA K. ICHIKAWA H. KOBAYASHI S. SUZUKI
Y. SHINODA A. MORI Y. KIMURA Y. SUGIYAMAThe Rzeppa type constant velocity joint is widely used in the tire side of driveshafts. Analyzing joint ballrotational behavior could contribute to durability and functional improvement. To analyze ball behavior, we applieda multibody dynamics approach including a stick slip friction force model. This model allows the calculation ofball angular velocity. We have established an experiment method which quantifies ball angular velocity throughmultipoint measuring and graphical analysis, enabling the accurate prediction of ball behavior.
Key Words: rzeppa constant velocity joint, ball behavior, multibody dynamics, graphical analysis

1. Introduction
The driveshaft (Fig. 1) is a part which transfers power
to the tires from the engine, etc. while responding tochanges in the direction the vehicle is travelling andfluctuation due to the unevenness in the road surface.
Rzeppa type constant velocity joints (CVJ) are amechanical element of the power transfer system locatedon the tire side of driveshafts which take the axial
intersecting angle between the input axis and output axis
(joint angle) and constantly transfer rotary motion. Dueto the improvement in vehicle fuel efficiency and weight
reduction of recent years, a demand has emerged for
CVJs to also be lightweight and compact. Predicting the
behavior of CVJ balls could lead to further lightening andhigher functionality.
To lighten CVJs, there is a need to find a way toachieve longevity under high surface pressure but thereis the difficulty of handling tribology, which deals withthe dynamic contact, friction and so on of multiple parts.
Moreover, due to the fact that the entire CVJ rotates, it
was difficult to quantify and analyze CVJ ball behavior,as well as verify the results through experiment. As
an experimental example, Nagatani and company, theauthors of “Study of the BJ Ball Rotational Motion,
Transactions of the JSME”, have measured the rotation
1)
movement using a magnetized ball .
In the experiment of this study, we established amethod to quantify ball behavior through arithmetic
processing using the Euler Parameters concept on therotation matrix obtained from multipoint simultaneous
measurement using a high-speed camera.

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

Moreover, in the analysis stage, we used a model able
to reproduce static friction force and dynamic friction
force at a high level and built an analysis model which
simulates the contact load and friction force createdbetween parts.
This made it possible to verify the analysis results in anexperiment and predict ball behavior with good accuracy.
We report on the details below.

Driveshaft

Fig. 1 Location of driveshaft on vehicle

2. Analysis Model
2. 1 Configuration of CVJ
Figure 2 shows the configuration of a CVJ. Generally
speaking, CVJs are configured from an outer race, inner
race, cage and six balls. The inner race is held in the
inner spherical surface of the cage, while the cage is heldin the inner spherical surface of the outer race. Both theouter race and inner race have six torus-shaped grooves

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−Analysis of Ball Rotation Behavior of Rzeppa Type Constant Velocity Joint (CVJ)−

2. 3 Contact model

Cage
Inner race

Outer race

When individual surface shapes overlap, the contactload found from the penetration amount works on bothcontact portions. Below is the defining equation. Contactload f n is established as the exponent of penetration
amount d expressed in the following equation and index
m , coefficient C are established to suit the contact profile.
mBallFig. 2 Configuration of CVJ

at even intervals with each of the six balls being held ineach groove and these are relatively displaced with theinner race and outer race in line with the joint angle andaxial rotatory motion. The cross-sections of the grooves
are a Gothic arch shape and the face which makes contact
with the balls differs depending on the torque transfer
direction.

fn ＝C d⑴

2. 4 Friction model
2. 4. 1 Friction categories
For the friction generated within a CVJ, only the
relative sliding of the contact faces of two physicalobjects was considered and the differential sliding, spin
sliding, etc. which occurs due to the difference in the ballrotation radius of the center and edge of the ball contact
ellipse, is not taken into consideration. Moreover, friction
due to shearing of the grease which passes between thecontact faces is also not considered.

2. 2 Overview of the analysis techniqueFigure 3 shows the arrangement of CVJ parts at a joint
angle.
For the analysis, we used general-purpose mechanism
analysis software to establish an intermediate shaft,individual CVJ parts, the differential-side CVJ modeledusing the contraposition which can transfer constant speed
rotation on any angle and the sliding contraposition, andthe testing apparatus as the rigid bodies, and arranged
these on three-dimensional global coordinates.
Part of the intermediate shaft uses a beam element.
Next, with contact elements alone, we correlated theouter race ball grooves, inner race ball grooves, cagewindow and balls, outer race inner spherical surface andcage outer spherical surface, and the cage inner sphericalsurface and outer spherical surface of the inner race.
We connected the outer race to the outer race axis andthe inner race to the intermediate shaft, and connected
the outer race axis and intermediate shaft to the testing
apparatus. We applied rotation, torque and joint angle
from the testing apparatus and analyzed CVJ behavior.

2. 4. 2 Friction coefficient and friction force of thecontact portion
Sliding friction force ft is the sliding friction coefficient
multiplied by the contact load, as expressed in thefollowing formula.

ft ＝l fn⑵

As shown by the bottom line of Fig. 4, for theconventional mechanism analysis, the friction force ofthe contact portion was smooth when there was relative
speed from a speed of zero, however in reality, as Fig. 5
shows, there is both static friction and dynamic friction,
resulting in large error with the preconditions of analysis
particularly where sliding speed is slow. Moreover, thefriction coefficient is affected by surface pressure andsliding speed. “SN” in Fig. 5 standards for “standard
pressure”.
Therefore, the preconditions derived from the static
friction coefficient and the dynamic friction coefficient
are introduced. Each of the friction parameters were
determined from a friction coefficient measurement
performed in a ball-on-plate experiment.

Fig. 3 Arrangement of CVJ parts at a joint angle

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−Analysis of Ball Rotation Behavior of Rzeppa Type Constant Velocity Joint (CVJ)−

Stick slip friction
Frictioncoefficient µls

3. Analysis Results

ls：Stick friction
ld：Slip friction

3. 1 Analysis conditions
For analysis conditions, the same joint angle, rotating
speed and torque were adopted that were used as theverification conditions in the experiment describedhereinafter.

ld
Conventional friction
Transition velocity

3. 2 Analysis results of ball behavior
Figure 8 shows the CVJ internal force. Moreover, eachof Fig. 9, 10 and 11 have the CVJ rotational phase as the
X axis, and respectively show the ball groove contactload, the panel angle defined as having a common contact
plane for the inner race groove and ball and the outer
race groove and ball, and the size of the angular velocity
x with a ball datum which only separates the rotating
element.

Fig. 4 Function of friction coefficient

Friction coefficient

ls

0

0.9SN
SN
1.4SN
1.8SN
2SN
2.5SN

ld

SN : Standard Pressure0

Velocity

Fig. 5 Measurement results of friction coefficient fromthe experiment

Outer Race Contact
Load
Cage Window
Panel Angle

2. 4. 3 Simulation examples of static friction anddynamic friction
Friction force

Ball Moving Direction
Inner Race Contact
Load
Tangent Plane between
Inner race and Ball

Fig. 8 CVJ internal force
Contact load

Figure 7 shows an example of simulating friction force
by incorporating static friction and dynamic friction ona simple ball-on-plate model (Fig. 6). In this way it waspossible to confirm the occurrence of friction force inareas where sliding speed is slow, which conventionallycould not be reproduced.

0
Compulsory displacement

0

60

120

180

240

300

360

Rotational phase angle, °

Load

Ball

Tangent Plane between
Outer race and Ball

Friction force

Fig. 9 Calculation results of ball contact load

Friction force
Velocity
Panel angle

Plate

Fig. 6 Simple ball on plate model0

Stick slip friction force

0

60

120

180

240

300

360

Rotational phase angle, °0Fig. 10 Calculation results of panel angle
Conventional friction force
0

0.2

0.4

0.6

0.8

1

time, sFig. 7 Calculation results of ball on plate model

Angular velocity

Friction force
Velocity

Tangent velocity

0

0

60

120

180

240

300

360

Rotational phase angle, °Fig. 11 Calculation results of angular velocity

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−Analysis of Ball Rotation Behavior of Rzeppa Type Constant Velocity Joint (CVJ)−

4. Verification of analysis results throughexperiment

B ＝ GA

4. 1 Measurement and analysis method of ballrotating angular velocity

Here，A＝

For the measurement of ball rotating motion, we
devised a way to stop the ball that would normally berotating and observe coordinates. Moreover, until now it
had been impossible to observe behavior at the back of
the joint however this became possible by opening a holein the outer race.
Three points carved into the ball surface at random
were simultaneously measured using a high-speed camera
fixed to the outer race and graphical analysis was used to
calculate the coordinates of these three points on the ballsurface with the outer race as the datum.

⑷

x 1A x 2A x 3A

y 1A y 2A y 3A , B＝ y 1B y 2B y 3B ,z 1A z 2A z 3A
g 11 g 12

G＝ g 21

g 23

g 31 g 32

g 33
TMeanwhile, if u and k＝［l m n ］ are made thedirectional vectors of the rotating angle and the rotating
axis respectively and the Euler Parameter of EAB＝［e0Te1 e2 e3］ as defined in Formula ⑸ is used to express2)
the rotating matrix G, Formula ⑹ will be the case .

4. 2 Calculation method of ball angular velocity

dr→i
dt

→
→
＝ x ×r i （ i ＝ 1, 2, 3 ）⑶Angular velocity vectors can be obtained by solving
the above simultaneous equation.
Rotational→
Axis xr→（
1 t + Dt ）

r→（
3 t）
r→（
3 t + Dt ）
r→（
2 t）

r→（
1 t）

r→（
2 t + Dt ）Marker Position of T = t
Marker Position of T = t + Dt

Fig. 12 Definition of ball angular velocity

Specifically, the relationship shown in Formula ⑸ isestablished between the three-dimensional coordinates
which have the three ball centers as the datum in T＝
TA and T＝TB, A and B, which are expressed by a 3×3
matrix and the rotating matrix G between TA and TB.

70

z 1B z 2B z 3B

g 13

g 22

cos

If the three arbitrary coordinates on the ball surface
which use the outer race as the datum are expressed withpositional vectors of r→1，r→2，r→3 with the ball center as
the starting point, the result will be as shown in Fig. 12.
The following relationship is established between these
positional vectors and angular velocity vectors.

x 1B x 2B x 3B

e0
EAB＝

e1
e2

l sin
＝

m sin

e3

n sin

e 12 − e 22 − e 32 ＋ e 02

u
2
u
2
u
2
u
2

u
2
u
2

cos
＝
k sin

⑸

2（ e 1e 2 − e 3 e 0 ） 2（ e 3 e 1 ＋ e 2 e 0 ）G＝ 2（ e 1e 2 ＋ e 3 e 0 ） e 22 − e 32 − e 12 ＋ e 02 2（ e 2 e 3 − e 1 e 0 ） ⑹2（ e 3 e 1 − e 2 e 0 ）

2

2

2

2

2（ e 2e 3 ＋ e 1 e 0 ） e 3 − e 1 − e 2 ＋ e 0

The angular velocity vector x was obtained from therotating angle u which was obtained by comparing therotating matrix G obtained from experimentation with
Formula ⑷ with the rotating matrix G of Formula ⑹, thedirectional vector k of the rotating axis and the time Dt
between TA and TB.

4. 3 Results of a verification of experimental andanalysis accuracy
Figures 13 and 14 show comparisons of the analysis
results and experimental results. We made the rotational
phase for comparison 120 to 270 degrees, as this range
can be observed. As shown in Fig. 15, the target analysis
accuracy was achieved within the rotational phaseactually measured when the range from 240 degrees was
excluded and if we focus on direction, the phase at which
direction changes abruptly is also matching, therefore we
judged there was good consistency.

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

Angular velocity−Analysis of Ball Rotation Behavior of Rzeppa Type Constant Velocity Joint (CVJ)−

Experimental result

shearing friction and rolling friction. It is believed thesefactors have a significant impact at the phases where loaddissipates.

Calculation result

Target of accuracy
0
120
150
180

210

240

270

Rotational phase angle, °Directional
vector：z

Directional
vector：y

Directional
vector：x

Fig. 13 Ball angular velocity comparison between calculation
and actual behavior
1.0

Calculation

0.0
−1.0
120

Actualbehavior
150
180
210
240
Rotational phase angle, °

1.0

270

Actual behavior

0.0
−1.0
120

1.0

Calculation
150
180
210
240
Rotational phase angle, °270

Calculation

0.0
−1.0
120

Actual behavior
150
180
210
240
Rotational phase angle, °270

Fig. 14 Directional vector comparison between calculation
and actual behavior

Error ratiowith target

2.0
1.0
0
120

150

180

210

240

270

Rotational phase Angle, °5. Conclusion
For CVJs (Rzeppa type constant velocity joints), wequantified the ball angular velocity vectors within the
CVJ on the preconditions of static friction and dynamic
friction. Moreover, we performed multipoint simultaneous
measurement using a high-speed camera and judged that
there was accuracy in the analysis.
The establishment of this analysis method made thefollowing items possible.
1) With conventional analysis, it was not possible togenerate friction force in the portions with slow
sliding speed however this has become possible byconsidering static friction and dynamic friction basedon measurement results of friction coefficients in aball-on-plate experiment.
2) With the preconditions of a static friction and dynamic
friction model for the contact portion, it has become
possible to measure CVJ ball behavior with angularspeed vectors.
3) As the entire CVJ rotates, it was previously difficult
to observe and quantify ball behavior, however
quantification was made possible through innovative
photography methods and multipoint simultaneous
measurement. When compared with the analysis
results, the target analysis accuracy was achieved
excluding the phase range where the ball loaddissipated, making it possible to predict ball behavior
in theory through analysis.
This means there is a possibility of being able to predict
joint durability at high accuracy and propose the optimal
friction coefficient in grease development, therefore
future design techniques can be expected to improve.

Fig. 15 Error between calculation and actual behavior

References
4. 4 Observations of ball behavior analysis results
During CVJ rotation, one rotation has two load peaks,
as shown in Fig. 9, and from a rotating phase angle of 240
degrees onwards, there are phases where load dissipates.
Moreover, the ball panel angle is the rotational primary
period, as shown in Fig. 10, and determines the directionof the contact load and friction force applied to the balls.
It is believed that the ball behavior is determined by thesum of the friction force moment applied to the balls, andwe predict that the analysis results reproduce contact load
and its direction with good accuracy.
Meanwhile, possible factors causing error may bethose other than sliding friction which are not taken into
consideration as analysis preconditions, namely grease

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

1)

2)

H. Nagatani, G. Saito, M. Imoto, S. Ishiguro: Study of
the BJ Ball Rotational Motion, Transactions of the Japan
Society of Mechanical Engineers, Sries C, Vol. 72, No.
716 (2006) 1249-1257
H. Tajima: Fundamentals of Multibody Dynamics, Tokyo
Denki University Press, (2006) 77

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−Analysis of Ball Rotation Behavior of Rzeppa Type Constant Velocity Joint (CVJ)−

T. NAKAMURA* K. ICHIKAWA** H. KOBAYASHI*

S. SUZUKI*** Y. SHINODA**** A. MORI****

Y. KIMURA**** Y. SUGIYAMA******

Experiment & Analysis Dept., Driveline Systems
Operations Headquarters**
Experiment & Analysis Dept., Driveline Systems
Operations Headquarters, Doctor of Engineering*** Driveline System Engineering Dept. 1, Driveline
Systems Operations Headquarters**** Drivetrain System Development & Engineering
Management Div., TOYOTA MOTER CORPORATION***** Material Development Div., TOYOTA MOTERCORPORATION72

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