IRJET- Sensitivity Analysis Study of CVT Parameters using Mathematical Model

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 12 | Dec 2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1012
Sensitivity Analysis Study of CVT Parameters using
Mathematical Model
NIKHIL SINGH1, SURAJ S1, BHARANI S ANAND1, SUPREETH A GOWDA1
1Students, Dept. of Mechanical Engineering, JSS Academy of Technical Education, Bengaluru, Karnataka, India
---------------------------------------------------------------------***----------------------------------------------------------------------
Abstract – The paper presented here discusses one of the
most valuable invention in thefieldofautomobileengineering,
the Continuously Variable Transmission or the CVT.
Although the invention of CVT dates back to 1490s there is a
lot more to learn about this technology. This paper will
present the currentproblemofmanufacturingandassemblyof
the CVT setup of a two wheeler, and also discuss how a small
variation in the CVT parameter such as length of V-belt, mass
of the rollers, spacer length, angle between the pulley, center
distance between the pulley, even within the tolerance range
can cause drastic changes in the performance of the vehicle.
This paper will also study how and what kind of different
combinations of above mentioned CVT parameters( keeping
one or the other parameters constants) will cause different
effects on vehicle performance parameters such as fuel
efficiency, acceleration time , reliability etc. All of this will be
studied using a mathematical model, suchthatvariousoutputs
can be generated for various inputs at least possible time, and
plotting these results as a graphical representation.
Key Words: CVT performance, Mathematical Model,
Microsoft Excel, Roller Mass, Spacer Length, Pulley, V-
Belts.
1. INTRODUCTION
Continuously Variable Transmission or simply CVT is the
most common power transmission system in the field of
automobile. In general, there aretwopulleys thataredivided
perpendicular to their axis of rotation, and with a V-belt
running in between them. The gear ratio can be
automatically changed by moving the two sections of one
pulley closer together and the two sections of the other
pulley little bit apart. Due to the V-shaped cross section of
the belt, this causes the belt to ride higher on one pulley and
lower on the other. Because of this there are changes in the
effective diameters of the pulleys, which changes the overall
gear ratio. The distance between the two pulleys cannot
change, and neither does the length of the V- belt changes ,
so changing the gear ratio means both pulleys must undergo
adjustment (one can bigger, the othersmaller,or bothcan be
of same size) simultaneously to maintain theproperamount
of tension on the belt. The V-belt needs has to be very stiff in
the pulley's axial direction in order to makeonly small radial
movements while moving in and out of the pulleys.
Fig -1: General Layout of the CVT.
COMPONENTS OF A CVT SYSTEM
 A V type rubber belt
 A variable-input "driving" pulley
 An output "driven" pulley
Fig -2: Components of CVT
The V- belt and the variable-diameter pulleys are the main
part of the CVT system. Each pulleyismadeoftwo23-degree
cones facing each other. A belt drives in the seams between
of the two cones of the pulleys. V-belts are preferred if the
belt is made of rubber. When the two cones of the pulley are
far apart (i.e, the diameter increases), the belt drives lower
in the seams, and the radius of the belt loopgoingaround the
pulley gets smaller. When the cones are close together (i.e.,
the diameter decreases), the belt drives higher in the seams,
and the radius of the belt loop going around the pulley gets
larger. CVTs may use hydraulic pressure,centrifugal force or
spring tension to create the force necessary to adjust the
pulley halves. Variable-diameter pulleys must always come
in pairs. One of the pulleys, known as the drive pulley (or
driving pulley), is connected to the crankshaft of the engine.
The driving pulley is also called the input pulley because it's
where the energy from the engine

Fig -3: Working Principle of CVT
Enters the transmission. The second pulley is called the
driven pulley because the first pulley is turning it. As an
output pulley, the driven pulley transfers energy to the
driveshaft. When one pulley increases its radius, the other
decreases its radius to keep the belt tight. As the two pulleys
change their radii relative to one another, they create an
infinite number of gear ratios -- from low to high and
everything in between. When the pitch radius is small onthe
driving pulley and large on the driven pulley, the rotational
speed of the driven pulley decreases resulting in a lower
gear ratio. When the pitch radius is large on the driving
pulley and small on the driven pulley, then the rotational
speed of the driven pulley increases, resulting in a higher
gear ratio. Thus, in theory, a CVT has an infinite number of
"gears" that it can run through at any time, at any engine or
vehicle speed. The simplicity and steeples nature of CVTs
make them an ideal transmission for a variety of machines
and devices, not just automobiles. CVTs have been used for
years in power tools and drill presses. They've also been
used in a variety of vehicles, includingtractors,snowmobiles
and motor scooters. In all of these applications, the
transmissions have relied on high-density rubber belts,
which can slip and stretch, thereby reducing their efficiency.
The distance between the centers of the pulleystowherethe
belt makes contact in the seams is known as thepitchradius.
When the pulleys are far apart, the belt rides lower and the
pitch radius decreases. When the pulleys are close together,
the belt rides higher and the pitch radius increases.
1.1 LITERATURE REVIEW
1.1.1 The Kim Kwangwon & Hyunsoo Kim Model
According to this method of modelling, the meandering
radius on both the actuators must undergo the changes
because of the belt force differentiation. The experimental
research results, describedinthework [4],indicatethatsuch
belt behaviour occurs only for the driven wheel. For the
driving wheel, the meandering radius remains same on the
complete wrap angle. This behaviour can be explained with
the help of self-locking phenomenon occurrence.
As a result, it is presumed that the force exerted by the belt
is constant. Thus, the circumferential friction component is
neglected. For the primary (driving) actuator,onlytheradial
static friction is assumed. Thus, single contact area is
considered. The total axial force for the primary actuator is
expressed by equation(3)andforthesecondaryactuatorthe
two contact areas are assumed.
In the passive contact area (at the entrance), the belttension
is constant. In this area the belt elementforcesdistributionis
familiar with the belt element forces distribution on the
primary actuator. The kinetic force in radial and
circumferential direction is considered in the active contact
area. The total axial force of the secondary actuator is
expressed by equation (4).
1.1.2 The Cammalleri model
The main difference between the Kim-Kim described
models and the Cammalleri model [1] lies in the
consideration of the belt flexibility and the changes of the
slip angle along the pulley wrap angle.
1.2 OBJECTIVES
1. Create a Mathematical Model of the sensitivity analysis
of the CVT parameter.
2. Verification of the Mathematical Model with the practical
values.
3. Plot graphs for different parameters and analysis of how
the changes in the CVT parameters effects the vehicle
performance.
2. NOMENCLATURES
1. FR Axial Force Produced by Rollers
2. M Mass of Rollers
3. yg Distance of Roller Mass center G from pulley axis
4. β Angle Formed by the Pulley Axis and the Tangent
To the Curved Profile at Contact Point
5. ϒ Angle of the Back Plate with Roller
6. 𝜑sx Angle of Sliding Friction between Roller and Ramp
7. 𝜑dx Angle of Friction of the Guide
8. FO Pre Load on the Secondary Actuator Spring
9. K Stiffness of the spring
10. D Mean Diameter of Spring
11. d Mean Diameter of the Helical Guide
12. 𝜁 Angle of Helical Guide Slope
13. V Poisson’s Ratio
14. Ca Torque at Primary Actuator
13. Cb Torque at Secondary Actuator
14. Fr Axial Force of the Driver (Primary) Actuator
15. Fn Axial Force of the Driven (Secondary) Actuator
16. T1 Tension on Tight Side of the Belt
17. T2 Tension on Slack Side of the Belt
18. θa Active Wrap Angle
19. θ Length of Arc of Contact
20. α V-Belt Wedge Angle
21. µ Co-efficient of Friction between Belt and Actuator
22. Ra Active Radius of Driver (Primary) Actuator
23. Rb Active Radius of Driven (Secondary) Actuator

3. METHODOLOGY
The tests was carried out on a Dynamometer test rig.
The test gives us the following outputs
i. The engine RPM with respect to the vehicle speeds at
different loading conditions.
ii. The wheel force.
Using these values as input we determine the various
parameters that affects the CVT performance.
I. The Primary Actuator.
The primary actuator or the primary pulley is directly
connected to the crank shaft of the engine and thereby it is
assumed that the speed of the primary actuator (na) is equal
to the engine speed in rotations per minute (RPM).
Fig -4: Primary Actuator
This actuator consists of number of centrifugal mass rollers
which are placed on a curved profile called ramp, so as the
speed of the engine increases which indeed increases the
speed of the primary actuators and vice versa, there is a
centrifugal force which is acting on these rollers which
creates an axial force and acts on the back plate, thereby
pushing the back plate forward or backward depending on
the speed of the actuators and thereby varying the radius of
the belt. The free body diagram of the roller is given in the
figure below.
Fig -5: Free Body Diagram
The figure 2 shows FBD for the case upshift i.e, increase in
the speed ratio. Where Fcen is the centrifugal force acting on
the rollers. Imposing radial,axial,androtational equilibrium,
it is possible to write:-
____(1)
____(2)
____(3)
As Fcen=Mn2
ayg and FR= Fdx sin(β− 𝜑sx), the equations 1
and 2 gives the axial forced produced by the centrifugal
rollers and is given by the below equation
____(4)
From equations 2 and 3 the roller motion is given by
____(5)
It is possible to come to a conclusion that,
for ,
[𝜑dx from equation (5)] and that the roller rolls on the back
plate and slips on the left ramp.
While, for

[𝜑sx from equation (5)] and the roller rolls on the left ramp
and slips on the back plate.
(Where 𝜑a is the angle of sliding friction between the
roller and the ramp).
And is given by the equation:-
In the case of downshift, the frictional forces reverse their
angles 𝜑dx and 𝜑sx and the directions change their sign in the
above equations, but all the results on the roller motion
remain the same.
Yg is determined with the helpofCADdrawingoftheprimary
actuator by finding 3 different roller position from the
rotational axis to the centre of the roller and also the angle
tangent to the ramp (α). Then using polynomial equation the
roller position along with tangent angle can be found at any
instant.
Chart -1: Determining the position and angle of roller
II. V-Belt
The V-belt is the linking member between the primary and
secondary actuator. It is known that the belt is elastic in
nature and due to this property of the belt, it penetrates into
the side walls of the actuators and due to which there is a
radial component of frictional force that comes into action
and this plays an important role in the V-belt mechanics.
Fig -6: V-Belt Geometry
Using the Speed Ratio-Torque and Thrust Relationship
The axial force that is developed by the belt on the primary
actuator can be given by the equation below
____(6)
Also the axial force that is developed on the secondary
actuator due to the V-belt is given by the equation
___(7)
Where,
____(8)
Using the value of Fr obtained in equation 4 and substituting
it in the equation 6 to obtain the value of T1. Then by using
the belt equation given below
___(9)
Finding the value of T2 from the value obtained by equation
6. Then by substuting the value of T1 and T2 obtained from
equations 6 and 8 respectively into equation 7 to find the
value of the axial force developed by the V-belt on the
secondary actuator.
III. Secondary Actuator
The axial force that is needed to assure the connection
between the V-belt and the secondary actuator is generally
produced by a helical spring compressed in between the
half-secondary movable pulley and a fixed contrast wall. In
addition , in motorcycle actuators , an extra force which is
proportional to the output torque is generated by the slope
of the sliding guides of a pre-defined constant angle 𝜁 as
shown in the figure below.

Fig -7: Secondary Actuator
The equation for axial force equilibrium of the movable half
actuator is given by
____(10)
Depending on the constraintsonthespring,thehalf-actuator
movement may cause some torsional effect on the spring. In
such situation the reaction torque produces an extra axial
thrust, which is the function of speed ratio. All together, the
equation of the actuator axial force remains the same
provided that the spring stiffness is given by
____(11)
The torque that is generated atthesecondaryactuator(Cb)is
obtained by substuting the value of Fn from equation 7 in
equation 10. As the output of the dynamometer test is
measured in terms of the wheel force, the torque obtained
from the equation 10 need to be converted into wheel force,
for which torque is multiplied by the reduction gear ratio
and divided by the dynamic rollingradiusofthe wheel. Using
all the formulas above an excel table is formulated in order
to find various parameters at different speeds of the vehicle.
4. RESULTS
At first it is important to verify that the mathematical model
hold true or not, for that a dyno test was conducted on the
standard vehicle and results were compared with the
mathematical model and a graph representing this
comparison is shown below:-
Chart -2: Verification of the Mathematical Model
As it can be seen the blue line represents the mathematical
model values and orange line represents experimental
values and both of them are comparable, for speeds upto
15Kmph the values divert a lot due to slippageofthebeltson
the actuators. Now we can compare variousCVTparameters
using this mathematical model.
Chart -3: Varying the mass of the rollers
Similarly all other parameters such as Length of the V-belt,
Pulley wedge angle, center distance between the pulley,
spacer length and spring stiffness can varied in the
mathematical model and can be studied for each and every
combination of the parameter to obtain the best results in
terms of power or torque transfer depending on the users
need without conducting the actual test by varying each
parameters.

5. CONCLUSIONS
After carrying out various test by varyingthecombinationof
the CVT parameters the following conclusion are drawn:-
1. Mass of the rollers: - The wheel force increases by
decreasing the mass of the rollers and vice versa.
2. Length of Belt: - The wheel force decreases with increase
in the length of V-Belt, in other words wheel force is
inversely proportional to the length of belt.
3. The Wheel Force increases with increase in the wedge
angle of the pulley, center distance of the pulley and
spring stiffness.
ACKNOWLEDGEMENT
We would like to thank TVS Motor Company, Hosur for
letting us carry out our research at their state of the art
Research and Development center, and guiding us in using
advanced equipments.
REFERENCES
[1] Cammalleri, M. A. R. C. O. "A new approach to the design
of a speed-torque-controlled rubber V-belt
variator." Proceedings of the Institution of Mechanical
Engineers, Part D: Journal of Automobile
Engineering 219.12 (2005): 1413-1427.
[2] Cammalleri, M., A. Conti, and F. Sorge. "Experimental
results for a belt variator in transient conditions." XVIII
Congresso AIMETA. 2007.
[3] Grzegożek, Witold, Krzysztof Dobaj, and Adam Kot.
"Experimental verification and comparison of the
rubber V-belt continuously variable transmission
models." IOP Conference Series: Materials Science and
Engineering. Vol. 148. No. 1. IOP Publishing, 2016.
[4] Kim, Kwangwon, and Hyunsoo Kim. "Axial forces of a V-
belt CVT." KSME Journal 3.1 (1989): 56-61.
[5] Kim, Hyunsuk, et al. "Steady state and transient
characteristics of a rubber belt CVT with mechanical
actuatiors." KSME international journal 16.5 (2002):
639-646.
[6] Oliver, Larry R., et al. Design equations for a speed and
torque controlled variable ratio V-belttransmission.No.
730003. SAE Technical Paper, 1973.
[7] Zhu, C. C., et al. "Experimental research on the effect of
structural parameters on the governing characteristics
of a pulley-drive, continuously variable
transmission." Proceedings of the Institution of
Mechanical Engineers, Part D: Journal of Automobile
Engineering 224.6 (2010): 775-784.
[8] Dolan, J. P., and W. S. Worley. "Closed-form
approximations to the solution of V-belt force and slip
equations." Journal of Mechanisms, Transmissions, and
Automation in Design 107.2 (1985): 292-300.
[9] Kong, Lingyuan, and Robert G. Parker. "Mechanics and
sliding friction in belt drives with pulley
grooves." Journal of Mechanical Design 128.2 (2006):
494-502.
[10] Julio, G., and J-S. Plante. "An experimentally-validated
model of rubber-belt CVT mechanics." Mechanism and
Machine Theory 46.8 (2011): 1037-1053.
BIOGRAPHIES
NIKHIL SINGH, Graduated in B.E
Mechanical Engineering, JSS
Academy of Technical Education
affiliated to VTU Belagavi, 2018
Batch.
SURAJ S, Mechanical Engineering
student at JSS Academy of
Technical Education affiliated to
VTU Belagavi.
BHARANI S ANAND, Graduated in
B.E Mechanical Engineering, JSS
affiliated to VTU Belagavi, 2018
Batch.
SUPREETH A GOWDA,Mechanical
Engineering student at JSS
affiliated to VTU Belagavi.

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IRJET- Sensitivity Analysis Study of CVT Parameters using Mathematical Model