04CVT-10
Development of 3-D simulation for power transmitting analysis
of CVT driven by dry hybrid V-belt
Masahide Fujita
Hisayasu Murakami
Power Train Research and Development Division, Daihatsu Motor CO., LTD.
Shigeki Okuno
Mitsuhiko Takahashi
Power Transmission Technical Research Center, Bando Chemical Industries, LTD.
Copyright © 2004 SAE International
ABSTRACT
A new CVT system driven by a dry hybrid V-belt having
a higher transmitting capacity has been investigated for
small cars. This paper reports dynamic FEM analysis of
the CVT, simulating transmitting efficiency and dynamic
strain acting on the belt running at high speeds. We
constructed 3-dimensional models for both belt and
pulley assembly. It is demonstrated that pulley rigidity
and clearance between pulley shaft and movable pulley
significantly affect the power transmitting efficiency of
the system. Dimensional change of the belt due to
permanent deformation of the rubber material also
affects the efficiency at high-speed range at the end of
belt life. The dynamic strain obtained by the FEM
simulated life of the belt in a bench test successfully.
INTRODUCTION
Continuously variable transmissions (CVT) are
increasingly utilized in automobiles aiming reduction in
fuel consumption. In some CVT systems V-belts are
used to transmit engine power. V-belts for CVT are
roughly categorized into two types: metal V-belts in wet
CVT and hybrid V-belts in dry CVT. Advantage of the
metal V-belt is its large transmitting torque capacity.
Recently, metal V-belts are used in large vehicles with
engine volume of 3000 cm3 and more. 1)
On the other hand, the dry hybrid V-belt has an
advantageous feature of its excellent power transmitting
efficiency in small cars, which results reduction of fuel
consumption. Some disadvantages of the dry hybrid Vbelt include smaller torque capacity, reliability of the
component polymeric materials and operating noise. A
new CVT system overcoming the drawbacks has been
investigated. The new system with improved torque
capacity is equipped with a wider hybrid V-belt and an
additional pulley that controlls the belt tension.
This paper concerns simulations of dynamic behaviors
of the new system such as transmitting efficiency, torque
capacity and dynamic strain acting on the components
of the belt running at high speeds. The dynamic strain
can be used to predict belt life. For these purposes 3dimensional dynamic FEM analyses are performed. The
chosen FEM code is PAM-MEDYSA(formerly PAMSHOCK), which has been used in analyses of metal Vbelts. 2) This software has some advanced features in
dynamic CVT system simulation:
1.
The effect of inertia of an object moving at high
speeds can be considered.
2.
Frictional contacts between belts and pulleys can
be analyzed.
3.
Dynamic movements of the belt components and
pulleys can be visualized.
The hybrid structure of the belt is carefully modeled, and
the effect of dimensional change of the belt on its
dynamic behaviors will be examined.
Besides of the model of the hybrid belt, modeling of a
movable pulley is very important. Many articles3)-5) have
reported that pulley rigidity and clearance between a
pulley shaft and a movable pulley greatly affect the
system efficiency, belt strength and the variable-speed
response. In this point of view, the rigidity and the
clearance of the pulley shaft are also taken into account
in the 3-D FE model.
STRUCTURE OF A NEW CVT SYSTEM
EQUIPPED WITH DRY HYBRID V-BELT
. DRY HYBRID V-BELT
Figure 1 shows a dry hybrid V-belt assembly. The belt
has a structure of a pair of tension bands inserted in
many H-shaped blocks. The core of the block is made of
high strength aluminum alloys supporting a high
transverse load from pulleys. The core aluminum alloy is
covered with phenol resin giving the blocks a proper
frictional coefficient with the pulleys. No lubrication is
necessary because of a high frictional coefficient, which
realizes an excellent transmitting efficiency in the CVT
system. Aramid cords supporting belt tension lies at the
center of the tension bands. Heat-resisting rubber
material made of hydrogenated nitrile rubber (HNBR)
surrounds the aramid cords. The surface of the tension
band is reinforced by fabrics preventing crack initiation
and enhancing abrasion resistance of the tension band.
transmission. The contact angle of 172 degree for the
new system at the driving pulley theoretically increases
the torque capacity by 40% in comparison to the
conventional system having the corresponding angle of
154 degree.
The tension pulley, however, imposes additional reverse
bending force to the belt. This reverse bending force
may shorten the belt life.
DYNAMIC SIMULATION OF POWER
TRANSMISSION WITH FEM
Block
Tension band
MODELING OF THE CVT SYSTEM
Modeling of the dry hybrid V-belt
Cord
Rubber
Inserted
Fig.1 Structure of dry hybrid V-belt
The FEM model of the belt is depicted in Figure 3. The
H-shaped block was modeled as an aluminum core
covered with resin. An assignment of the actual elastic
moduli for the aluminum and the resin realizes the actual
rigidities of the upper and the lower beam of the block in
the model. The boundary between blocks and the
tension bands defined as contacting surfaces at a
properly assigned frictional coefficient.
DRY CVT SYSTEM WITH A TENSION PULLEY
Commercially available CVTs with dry hybrid V-belts
include two V-pulleys, a driving and a driven pulley.
Pulley thrust is applied by spring and torque cam at
driven pulley. The applied thrust is converted to a belt
tension ensuring power transmission without slippage.
On the other hand, our new CVT system is equipped
with an additional tension pulley at the slack side of the
belt span, as shown in Figure2.
Driving Pulley
Tension Pulley
Driven Pulley
Pulley movement at shift
down
Fig. 2 Structure of a new CVT with a dry hybrid
belt System
The tension pulley control the belt tension directly to a
desirable level, which improves transmitting efficiency of
the system. Moreover, the rate of shifting ratio becomes
faster since the axial movements of the driving and the
driven pulley are linked by gears, accomplishing a better
driving response. Another feature of the new system is
a higher torque capacity over the commercially available
systems. This is a simple consequence of larger contact
angles of the belt with the driving and the driven pulley
at the presence of the tension pulley. The increase in
torque capacity with increasing contact angle can be
theoretically calculated with Euler’s equation for power
Block
Resin
Aluminum
Rubber
Upper
beam
Lower
beam
Cord
(Membrane)
Tension band
Fig.3 FEM model of the belt
The tension bands are modeled as a cord material
surrounded by a rubber material. Both the cord and the
rubber have anisotropic elastic moduli; the cord has a
higher modulus in the longitudinal direction, whereas the
short-fiber-reinforced rubber has a higher modulus in the
direction of belt width.
In order to express this
anisotropy, membrane elements with proper elastic
moduli were used for the cord and the rubber material.
Modeling of pulleys
Modeling of pulley as an elastic body
As mentioned earlier, pulley rigidity greatly affects the
dynamic behavior of the system. The pulley rigidity is
related to the deformation of the pulley by an action of
pulley thrust. The deformation of the pulley results in
variation of V-angle and contact conditions with the belt.
Such variation in the contact condition greatly affects the
dynamic behavior of the belt. Experimental results
confirmed that a difference in V-angle of the belt and
pulley greatly influences the torque capacity and heat
build-up in the belt. Therefore, as a first approximation,
the pulleys are modeled as elastic bodies to simulate the
deformation of the pulleys and the resultant variation in
the contact condition with the belt.
COMPARISON OF CALCULATIONS WITH
EXPERIMENTAL RESULTS
Modeling of the elastic pulley with shaft clearance
In a bench test of the belt, a torque capacity was
determined as a torque at the onset of belt slip. The
onset of the belt slip was detected as a sudden variation
in pulley ratio. Under the same condition, the calculation
was performed, and the belt slip was detected in the
same way. In Table 1 the results of calculations at
different radii of driving pulleys at ratio of 1 were
compared with the experimental results. Since the
bench tests were performed with fixed V-pulley, the
calculations were also performed with an elastic-body
model of the pulleys without any shaft clearance. The
calculation was consistent with the experimental results,
confirming the calculated force balance in the
transmission state successfully simulates the actual
situation.
In a scope that the clearance between the pulley shaft
and movable pulley predominantly affects the rigidity of
the pulley, the clearance was also taken into account in
a modified model shown in Figure 4.
Fixed pulley
Movable pulley
Movable pulley
boss and sliding key
TORQUE CAPACITY
Table 1. Comparison of calculated torque capacities with
experimental results (Shaft load: 3900N, Pulley ratio:1)
Resin bush
Fixed pulley shaft
Fig. 4 Modified model for elastic pulley
with shaft clearance
Driving pulley
radius(mm)
59.2
32.75
Experimental
(Nm)
190
89
Calculated
(Nm)
192
92
CALCULATION PROCEDURES
TRANSMITTING EFFICIENCY
The procedures of dynamic simulations with software
“PAM-MEDYSA” were schematically shown in Figure 5.
Calculations using elastic-body models of pulleys
③driven torque
①move the pulley center
initial state
belt set
②driving pulley
rotation
power transmitting
Fig. 5 Calculation procedures
First, the belt has a circular shape without any applied
stress. Then, it was placed on the pulleys by moving the
driving pulley outwards. Subsequently, a rotation of the
driving pulley is applied, which was followed by imposing
driven torque at the driven pulley. Balance of kinetic
forces in power-transmission state was calculated with
respect of time. Therefore, the calculations take a long
time with the elastic-body models. In order to shorten
the simulation time, the calculations were performed by
using rigid-body models under transient conditions. The
models were replaced with elastic models after reaching
steady-state power transmission.
Furthermore, calculations at different operating
conditions were performed using “restart” function of the
software for models with identical dimensions.
In Figure 6, the calculated transmitting efficiencies at
different belt velocities were compared with
corresponding experimental results in the CVT unit
having actual movable pulleys. The test conditions were
a high ratio (0.407) and input torque of 80 Nm. In Figure
6 the belt velocity of 30 m/s corresponds to vehicle
speed of 148 Km/h at the high ratio. The tendency of
decreasing transmitting efficiency with increasing belt
velocity was successfully simulated. The effect of inertia
of running belt at high speed was examined correctly
with the FEM model. However, the absolute values of
calculated efficiencies were higher than those of
experimental results by 2 percentage points. The
calculated results seem to agree with the experimental
results using fixed V-pulleys.
movable pulley. It also causes tilting of the movable
pulley.
Transmitting
efficiency(%)
Ratio：High(0.407) Input torque：80Nm
100
99
98
97
96
95
94
2.
Tilting of the movable pulley due to the clearance
shifts the belt to a deeper position on the pulley.
3.
The motion to the deeper position increases friction
loss between the belt and the pulleys.
4.
Transmitting efficiency decreases due to the friction
loss at the pulley shafts and the increase in the
friction loss between the belt and the pulleys.
Calculated
Measured
0
10
20
30
40
Belt velocity (m/s)
Fig. 6 Transmitting efficiency for the model
of elastic pulleys
Calculations using models of elastic pulleys with
clearance at pulley shaft
In order to explain the reason for the 2 percentage point
discrepancy in the previous section, the clearance
between pulley shaft and movable pulley was
considered in the elastic models of the pulleys. In
Figure 7, the calculated transmitting efficiencies at
different belt velocities were compared with
corresponding experimental results in the CVT unit
having actual movable pulleys. By the modification of
the pulley model the calculated values agreed with the
experimental results in magnitude.
Ratio：High(0.407) Input torque：80Nm
EFFECT OF PERMANENT DEFORMATION OF
THE TENSION BAND ON TRANSMITTING
EFFICIENCY
PERMANENT DEFORMATION OF THE TENSION
BAND DUE TO HEAT AGING
Heat build-up of the belt upon running at high speeds
causes permanent deformation of the rubber material.
Consequently, the clearance between blocks and
tension bands increases as heat aging of the rubber
material proceeds as shown in Figure 8. In this section
the influence of the dimensional change due to
permanent deformation of the tension band on the
transmitting efficiency is examined with the FEM model.
Clearance
between tension
block
band and block
Clearance between
Transmitting
efficiency(%)
tension band and
100
99
98
97
96
95
94
Calculated
Measured
0
10
20
30
40
Belt velocity (m/s)
Fig. 7 Transmitting efficiency for the pulley model with
shaft clearance
The decrease in the efficiency with pulleys having shaft
clearance was elucidated by visualizing the movements
of the belt running at high speed in the simulation. The
observation demonstrated the following mechanism for
the efficiency decrease:
1.
The clearance at the pulley shafts results in
increasing frictional loss between the shaft and the
Fig. 8 Clearance between tension band and block
for a fatigued belt
First, the change in the clearance between the tension
band and blocks was quantified in an accelerated bench
test at belt temperature of 130deg C.
The test
conditions were shown in Figure 9. The measured
clearance was plotted against duration of the test in
Figure 10
Tensioner pitch diamater:81.4mm
Drive pulley
diamater:151.5mm
Driven pulley
diamater:61.5mm
Fig. 11 Storage modulus of rubber material as a
function of aging time at different temperatures.
Test conditions
Running speed of belt:30m/s
Input torque:115Nm
Belt temperature:130deg C
Tensionner load:419N
The aging time at a given aging temperature can be
converted to the time at arbitrary reference temperature
by using Arrehenius equation (Equation 1) with properly
assigned activation energy.
Fig. 9 Belt endurance test layout and conditions
250
0 .3
200
0 .1
150
0 .0 5
0
100
- 0 .0 5
- 0 .1
50
- 0 .1 5
- 0 .2
- 0 .2 5
3
0 .2
0 .1 5
d ista n c e (10 Km )
0 .2 5
E stim a te d ve h ic le r u n n in g
C le a ra n c e b e tw e en ten sion
b a n d a n d b loc ks (m m )
t
t
0
0
100
200
300
400
500
0
1
 E  1
1 
= exp 
−

 R  T 0 T 1 
(Equation 1)
Where t0 is the aging time at a chosen reference
temperature T0, t1 is the aging time at arbitrary aging
temperature of T1, E is the activation energy in J/mol,
and R is gas constant (8.31 Jmol-1K-1). The chosen
reference temperature, T0 is 403.15 K (130deg C).
Figure 12 shows a master curve at the reference
temperature of 130deg C using an activation energy E of
96.3 kJ/mol.
R u n n in g T im e a t 130 C (h )
Fig. 10 Clearance between tension band and blocks,
and estimated vehicle running distance as a function
of duration of accelerated test at 130℃.
As mentioned above, this bench test is an accelerated
test at the belt surface temperature of 130deg C.
Therefore, the duration of the test needs to be converted
to actual vehicle running distance. For this purpose, the
aging test of rubber material was performed by
measuring dynamic storage modulus of the rubber
material at different aging temperatures. Figure 11
shows the results of the modulus as a function of aging
time at different temperatures.
Fig. 12 Master curve of elastic modulus as a function of
aging time. (Reference temperature: 130℃)
End of belt life
100
Transimitting efficiency (%)
Next, Figure 13 shows a model distribution of vehicle
speed and the relation between the vehicle speed and
belt temperature. The distribution was expressed in
percentage of time at each vehicle speed.
The
calculated averaged vehicle speed based on the
distribution is 74 Km/h, meaning that the model
distribution is the high-speed type, and it represents the
severest condition for the belt.
99
98
97
96
95
Vehicle speed 60Km/h
94
123Km/h
93
92
148Km/h
91
173Km/h
90
0
0.05
0.1
0.15
0.2
0.25
0.3
Clearance between tension band and blocks (mm)
Fig. 14 Transmitting efficiency as a function of tension
band-block clearance at different vehicle speed
Fig. 13 A model distribution of vehicle speed and
relation between vehicle speed and belt
temperature. (High-speed model)
The distribution and belt temperature in Figure 13 and
Equation 1 gives cumulative aging time at the reference
temperature per unit vehicle running distance. The
estimated vehicle distance was plotted against aging
time at 130 deg C in Figure 10. Figure 10 indicated that
the tension band-block clearance of 0.2 mm after 400
hour’s running at 130 deg C corresponds to vehicle
distance of 200,000 km, assuming the severest
condition of Figure 13.
CALCULATION OF TRANSMITTING EFFICIENCY AT
DIFFERENT CLEARANCE BETWEEN TENSION BAND
AND BLOCKS
The dimensional change of Figure 10 was considered in
the calculation by creating the clearance in the belt
model.
Figure 14 shows the results, transmitting
efficiency as a function of tension band-block clearance
at different vehicle speeds (belt speeds).
This result clearly demonstrates that the transmitting
efficiency is lowered as the permanent deformation of
the tension band proceeds.
This deterioration is
significant at high vehicle speed range at the end of belt
life. Stated differently, the transmitting efficiency is kept
at a high level if the vehicle speed is below 120 km/h.
An accelerated endurance test of the belt reveals larger
heat generation of the belt at the end of the belt life,
supporting the calculated results of lower efficiency
(higher loss). This energy loss attributes to an increase
in frictional loss between the tension band and blocks at
high speed range.
ANALYSIS OF STRAIN ACTING ON THE
TENSION BAND AND PREDICTION OF CRACK
LIFE OF THE BELT
CALCULATION OF THE DYNAMIC STRAIN ACTING
ON THE TENSION BAND
Figure 15 shows a crack at the tension band at the end
of its life. The crack growth at the lower side of the band
leads to rupture of the belt. Dynamic strain at the
tension band is the cause of the crack initiation.
Therefore, the dynamic strain at high belt speeds must
be evaluated quantitatively using the dynamic FEM
analyses.
Fig. 15 Crack of tension band at the end of belt life
In Figure 16 the calculated dynamic strain acting on the
tension band was shown at belt speeds of 35 and 9.7
m/s. At the higher belt speed of 35 m/s a clear
maximum of the strain was observed at the entrance of
the tension pulley. In Figure 17 the maximum dynamic
strain was plotted against belt speed. The result
revealed that the dynamic strain increases by 20 % at
the belt speed of 30 m/s. The increase in the dynamic
strain was proportional to the square of the belt speed.
position. In case of high belt speeds, the inertia pushes
a block forwards, which causes collision of the blocks.
This interference of neighboring blocks leads to
momentum around the collision point. This momentum
increases spacing between the blocks, which, in turn,
leads to the maximum dynamic strain of the tension
band.
Velocity of
belt
Strain Peak in dynamic behavior
Rotational direction
①Collision
Ratio: High(0.407)　Low(2.449)
①
②Reaction of
Velocity of tension
pulley surface
the collision
Strain
Bending
Strain
②
Increase in
the gap
between
block
0
Period of contact with tension
Pulley
Belt speed:　35m/s　9.7m/s
Calculated strain
S=0.00177V2+7.96
2
1
0
5
10
15
20
25
ﾍﾞﾙﾄ速度(m/s)
30
35
PREDICTION OF CRACK LIFE OF THE BELT
The maximum strain in Figure 17 is used for the
prediction of the crack life of the belt. For this purpose,
S-N curves for the crack life of the tension band were
drawn at different belt temperature in Figure 19. Each
point is the result of experimental data evaluated by a
flex test of a tension band itself (without blocks) with
varying flex diameter and belt temperature.
Bending strain
下ｺｸﾞ表面歪み（%)
Tension band strain(%)
Strain in dynamic behavior
0
Pitch circle of belt
Fig. 18 Dynamic behavior of blocks at the entrance to
the tension pulley
Fig. 16 Strain in the tension band with respect to the
relative position with tension pulley
12
11
10
9
8
7
6
5
4
3
Tension pulley
40
Belt speed(m/s)
Fig. 17 Maximum strain in the tension band as a
function of belt speed.
The visualization of the belt demonstrated a larger pitch
distance between the adjacent blocks at the entrance of
tension pulley at the belt speed of 30 m/s. Additional
static analysis confirmed that the larger pitch distance
corresponds to the 20% increase of the strain due to
tearing force imposed to the tension band by the blocks.
Such dynamic behavior of the blocks at high speeds was
depicted in Figure 18 schematically. The translation
motion of the block changes to a rotational motion upon
contacting with the tension pulley. The contact with the
pulley decelerates the motion of the block at the contact
Fig. 19 S-N curves of crack life of tension band at
different belt temperature
. The S-N curves in Figure 19 and the calculated
maximum strain in Figure 17 predict crack life of the
hybrid belt in bench tests at different belt velocity.
Figure 20 shows the results of predicted crack life at belt
temperature of 130deg C at high ratio. The predicted
(calculated) life is consistent with the experimental
results of bench tests. This consistency proves the
accuracy of the maximum dynamic strains calculated in
the proposed FEM analyses.
30m/s
velocity of belt (m/s)
Calculated
Experiment
value
Calculated
Experiment value
Belt life (hr)
Belt temperature： 130deg C
35m/s
Fig. 20 Comparison of the prediction of crack life
with the results of bench tests.
IMPROVEMENT OF THE CVT SYSTEM
The results in the previous section demonstrated that
belt life is predominantly determined by operating belt
temperature and dynamic strain acting on the belt.
Therefore, lowering the belt temperature and strain is
very important to achieve longer life of the belt. For this
reason the following improvements was undertaken for
the new CVT system:
Cooling of the belt by increasing a ventilation flux
rate by 100%.
the movable. The transmitting efficiency decreases with
increasing belt velocity.
Permanent deformation of the tension bands creates
clearance between the tension bands and the blocks in
a fatigued belt. Dynamic simulations demonstrated that
the clearance causes the decrease in the transmitting
efficiency in a high belt speed range at the end of belt
life. The decreased efficiency leads to heat build-up of
the belt. The increase in belt temperature was confirmed
in a bench test of the belt at the end of the belt life.
Calculations of the dynamic strains acting on the
tension band revealed a peak of the strain at the
entrance to the tension pulley. The reason for the
maximum strain was elucidated by the visualization of
the dynamic motions of the blocks in the simulation.
Crack life of the belt was predicted with the peak strain
and S-N curves. The predicted life was consistent with
the results of some bench tests at different belt speeds.
Finally, improvements in the CVT system were
explained. The improvements leading to the decrease
of belt temperature and dynamic strain are the keys to
achieve a longer belt life and higher transmitting
efficiency of the system.
REFERENCES
1.
Yoshiaki Kato, Hiroshi Yamashita, Yoshihiro
Kono :A Study on the Torque Capacity of Belt CVTs
for 2.0-liter and 3.5-liter Front-Drive Cars JSAE
Autumn Convention, No.65-03, p.1-4 (2003)
2.
Hirofumi Shimizu, Daisuke Kobayashi, Junichi
Kawashima, Yoshiaki Kato: Development of 3-D
Simulation for Analyzing the Dynamic Behavior of a
Metal Pushing V-Belt for CVTs JSAE Spring
Congvention, No.8-99,p.9-12 (1999)
3.
Masanori Shimizu, Kazuya Okubo, Toru Fujii,
Mituhiko Takahashi, Ryuuichi Kido：Variations of
Belt Pitch Line and Power Transmitting Efficiency
due to Pulley-flange Tilting of CVT using a Dry
Hybrid V-belt JSAE Autumn Convention, No.108-2,
p.7-12 (2002).
4.
Toshihiro Saito :Development of Metal Pushing Vbelt Stress Simulation for CVT JSAE Annual
Congress, No.21-03,p.9-12 (2003)
5.
[4]Hirokazu Uchiyama, Takeshi Yoneda, Hironaga
Itou, Yasukazu Nobumoto :On the Shifting
Mechanism of a Metal V-belt CVT JSAE Spring
Congvention, No.8-99，p.21-24 (1999)
Reduction of the maximum belt speed from 35m/s
to 30m/s, which decreases heat generation due to
higher transmitting efficiency, and decreasing
dynamic strain acting on the belt.
Optimization of the belt tension by controlling the
tension pulley with a spring assisted by a hydraulic
cylinder, which improve the transmitting efficiency,
reducing the heat generation of the belt.
CONCLUSION
Dynamic FEM analyses of a new CVT system driven by
a dry hybrid V-belt was performed with 3-dimensional
models of the belt and pulley assembly. The calculated
transmitting efficiency of the CVT as a function of belt
speed was consistent with the experimental results with
a modified elastic-body model of the driving and driven
pulleys having actual clearance between pulley axis and