Design and parametric analysis of a half toroidal [623866]

Design and parametric analysis of a half toroidal
continuously variable transmission (CVT)
Ionuț Răzvan NECHITA , Trai an CICONE*
*Department of Machine Elements and Tribology , University “Politehnica” of
Bucharest; E -mail: [anonimizat]
Abstract: A continuously variable transmission (CVT) is a transmission that can change permanently
through a continuous spectrum of gear ratios , allow ing an infinitely number of transmission ratio in a
finite rang e. In contrast with other types of mechanical transmissions, than only offer a fixed number of
gear ratios, a continuously variable transmission, through its flexibility, allows the input shaft to keep
a constant angular velocity. The objective of this research was to analyze the existing types of CVTs,
verification of the elasto -hydrodynamic regime and to design and perform a parametric analysis of a
half toroidal CVT to see if can be found an opt imized version of a CVT, both in terms of dimensions and
performance. At first, there were presented existing mechanical CVTs, together with their specific
characteristics and operating principles. The analyzed configuration is half toroidal (HT) CVT, for
which it was presented the geometry, and the calculation model. Then, the elaso -hydrodynamic
mechanism was presented, together with the verification of the EHD regime . The parametric analysis
was performed for two different situations with their specific parameters; for each situation being
realized a 3D model of the CVT, to have a better overview of the differences between each situation.
Keywords: CVT (Continuously Va riable Transmission), Half -Toroidal, Extroid CVT, Torotrak CVT,
Parametric analysis, 3D model, EHD mechanism , EHD regime.
1. INTRODUC TION
One hardly needs to be an automotive engineer to understand that the less fuel an engine uses,
the fewer pollutants produces, and the cleaner is the air we breathe. Unfortunately, improving
the variables in this equation is becoming more and more difficult . In order to achieve
additional fuel economy improvements, we began to focus on increasing efficiency in areas
where improvements are much more difficult and more expensive to achieve, mainly on
powertrain components such as transmissions.
The base for th is is the fact that transmissions operate over a variety of power conditions,
such as low speed – high torque or high speed – low torque regimes, as well as through a wide
range of gear ratios. In order to achieve something in this area I have challenged t he
engineering thinking associated with existing designs and the knowledge of my coordinator
and all the persons that helped me throughout the period on which I realized this research. It
is known that the use of toroidal CVT as powertrain increases the to rque capacity of the
transmission, in comparison with a normal CVT. Also, the engine operates continuously in its
most efficient or highest performance range and the fuel consumption reduces together with
its emissions.

2. ANALYSIS OF THE EXISTING CONSTRUCTIVE SOLUTIONS
The finite number of shifting steps, present in traditional gearboxes, cannot entirely exploit
the power produced by an internal combustion engine (ICE). With the use of a continuously
variable transmission, the engine can operate at the ideal operating point from economy or
performance point of view.
There are various kinds of continuously variable transmissions, but the most common
used in automotive industry are push belt and variable pulleys CVT. Also, due to their higher
torque c apacity, toroidal transmissions have a growing interest in them.
2.1. Variable pulleys CVT
A variable diameter CVT is a pulley based CVT, that has two pulleys connected by a
flexible element that can be a composite V -belt or a steel belt. The first pulley (the driving
pulley) is connected to the crankshaft of the engine, whilst the second pulley (the driven
pulley) is connected to the rear axle. [1] Each of the two pulleys forming this type of CVT is
made of two 20 -degrees cones that are facing each other and in the groove between the two
cones, a belt ride. The movable side of the pulley is operated by a hydraulic actuator that can
increase or decrease t he amount of space between the two sides of the pulley.

Figure 2.1 a. Low gear pulley system b. High
gear pulley system [2]
As the radii relative to the two pulleys
are changed, an infinite number of gear ratios are created between the two pulleys,
from low to high. As in the case of figure
2.1a, a lower gear occurs when the pitch
radius is small on the driving pulley and
large on the driven pulley and the
rotational speed of the output pulley
decreases. An opposite case is presented in
figure 2.1b where the pitch radius is larger
on the input pulley and smaller on the
output pulley and the rotational speed of
the driven pulley increases, producing a
higher gear. [2]
As it was previously mentioned, the flexible element that connects the two pulleys can be
a composite V -belt o r a steel belt. With the introduction of the new materials, CVTs became
more reliable and efficient. The design and development of the metal belts used to connect the
pulleys was one of the most important advances that CVT got. This type of belts are flexible
and are composed of numerous (generally 9 or 12) thin bands of steel that hold together high –
strength, bow -tie shaped pieces of metal. As opposed to rubber belts, metal belts are quieter,
do not slip and they are highly durable, thing that en ables the continuously variable
transmissions to handle more engine torque.
When using a chain -based CVT, a film of lubricant must be applied to the pulleys. The
film of fluid must be thick enough to prevent the pulley and the chain from touching each othe r
and it must be thin in order not to waste power when each element of the chain submerge into
the lubrication film. Additionally, around 12 steel bands are stabilized by the chain elements.
Each band is thin enough so that it bends conveniently. In the as semblage of bands, one by
one band correspond to a slightly different gear ratio, hence they shift over each other and

need lubrication. The outer bands slide through the stabilizing chain, while the center band has
the role to link the chain. [3]

Figure 2.2 Metal belt design layout [4]
2.2. Toroidal CVT
The main components of this type of CVT are the input and output discs, designed to
create a toroidal cavity, coupled with a number of rollers. Also, t he principle of operation of
toroidal CVT is similar to that of the variable diameter pulley . The power from the engine is
transmitted to the input disc, the motion of rotation being transferred to the power rollers, and
afterwards from the rollers to the output disc. The toroidal CVT performs smooth and
continuous gear ratio development because the power rollers are having the tilt changed
continuously.

Figure 2.3 a. Low gear toroidal CVT [5] Figure 2.3 b. High gear toroidal CVT [5]

In the moment when the edges of the power rollers are in contact close to the center of the
driving disc, they will hit the driven disc close to the rim. This results in an increase in torque
and a reduction in speed and it is correlated to the low gear range of a conventional
transmission (figure 2.3a). Opposite, when the edges of the power rollers are in contact near
the center of the driven disc and the rim of the driving disc occurs a decrease in torque and an
increase in speed and it is correlated to the high gear range of a transmission (figure 2.3b).
The transmission also includes a self -contained, closed -loop hydraulic and lubrication
system utilizing a special traction fluid , elasto -hydrodynamic lubricant. The input end of the
main shaft is driving a gear driven lube pump through a gear arrangeme nt formed of a drive
and an idler. The lubrication for the hydraulic roller control and the transmission is supplied
by the pump. A series of oil jets linked together by an oil system delivers the lubrication to all

rolling elements, the gears being lubric ated by splash lubrication, with the oil flowing to the
transmission sump by gravity return. [6]

Figure 2.4 NSK lubricating oil film [7]
3. GEOMETRICAL DESIGN
During this chapter was analyzed in detail the toroidal CVT for which there w as presented the
geometry of the toroidal CVT and the calculation model to obtain the maximum Hertzian
stress .
3.1. Geometry
The toroidal CVTs are analyzed to a large extent due to their high torque capacity, which
makes them convenient for use with large engines in automotive industry . The configurations
that are most commonly encountered and investigated are full -toroidal and semi -toroidal.
In figure 3.1a are presented the components of the toroidal CVT, whilst figure 3.1b shows
the main geometrical characteristics of the half-toroida l variator. When the CVT operates in
steady state, the oscillating center of the roller corresponds with the tor oidal cavity center O
and its axis of rotation is inclined with an angle .
The tilting angle mentioned above, , has positive values if it is clockwise oriented and
thus it controls the distances r1 and r3 of the contact points A and B with respect to the main
axis of the variator, are therefor , it is manipulating the ideal speed ratio srID = r3 / r1. In the
same figure, r12 = r23 = r0 illustrate s the radius of the toroidal cavity, which happens to be one
of the two main radii of curvature of the input and output discs. The radius r11 represents the
second principal radius of curvature of the input disk, at the same time r33, being the s econd
principal radius of curvature of the output disk (figure 3.2).

Figure 3.1 Half -toroidal CVT a. Components b. Geometrical characteristics

Figure 3.2 Half -toroidal CVT. Geometrical characteristics
Furthermore, r2 and r22 express the
roller radius and the curvature of the roller,
also they are the two principal radii of
curvature of the roller, with the mention
that r22< r 0 (figure 3.3) . The distance from
the disc axis, of the toroidal cavity is
denoted with e and it is linked to the aspect
ratio k = e/r 0 of the toroidal traction. What
is more, in a half -toroidal CVT, the half
cone angle  of the roller is between 50 –
70.
Figure 3.3 I llustration for r22 <r0
3.2. Calculation model
Table 3.1 CVT geometrical parameters
Parameter Value
Cavity radius, r0 40 mm
Roller curvature, r22 32 mm
Half cone angle,  60
Aspect ratio, k 0.625
Speed ratio change,  0.52
Tilting angle,  20
The rest of the parameters that were mentioned in the paragraph above were calculated
based on an existing model, presented by G. Carbone [8], are the ir respective calculation
formulas and values obtained are presented below:
𝑟2=𝑟0×sin()=34.64 𝑚𝑚 (1)

𝑟1=𝑟0×(1+𝑘−cos(+))=58.05 𝑚𝑚 (2)
𝑟3=𝑟0×(1+𝑘−cos(−))=34.36 𝑚𝑚 (3)
𝑟11=𝑟1
cos (+)=334 .32 𝑚𝑚 (4)
𝑟33=𝑟3
cos (−)=44.85 𝑚𝑚 (5)
Radii of curvature chosen are based also on the varying parameter
, and are denoted
further on as follows:
➢ Principal radius of curvature of the input and output discs, rax= -r0.
➢ The second principal radius of curvature of the input disc, ray= r 11.
➢ Principal radius of curvature of the roller, rbx= r 22.
➢ The second principal radius of curvature of the roller, rby = r 2.
The following part will describe the deformation due to concentrated loading, but also it
will bring into discussion the associated surface and subsurface stresses. The attention will be
fully on the elliptical contacts. The theory on which a ll the following calculations are made is
based on the work of Hertz in 1881, and therefore will be further referred as Hertzian contact.
The condition of the elliptical contact will be explained in the next paragraph which is an
extract from Steven R. Sch mid, Bernard J. Hamrock and Bo O. Jacobson [9]:
“The undeformed geometry of nonconformal contacting solids can be represented in
general terms by two ellipsoids. Two solids with different radii of curvature in a pair of
principal planes (x and y) passing through the conjunction make contact at a single point
under the condition of zero applied load. Such a condition is called point or elliptical contact
because of the shape of the contact patch. It is assumed throughout this book that conve x
surfaces like those in Figure 2.2 exhibit positive curvature; concave surfaces exhibit negative
curvature. The importance of the sign of the radius of curvature will be shown to be extremely
important and allows extension of Hertzian contact equations to a wide variety of machine
elements.”
Curvature sum, is important in analyzing contact stresses and deformation, and is
calculated based on the values of the curvature radii, presented before (eq s. 1 – 5).
However, to obtain the value of the curvature sum, it was required to perform a
verification beforehand, as it is mentioned by B. Horovitz [10].
1) If:
1
𝑟−1
𝑅1≤1
𝑅1+1
𝑅2 (6)
then:
1
𝑅𝑒𝑐ℎ=1
𝑟−1
𝑅1 or,
2) If: 1
𝑟−1
𝑅1≥1
𝑅1+1
𝑅2 (7)
then:
1
𝑅𝑒𝑐ℎ=1
𝑅1+1
𝑅2
In our case: r=r 22, R1=r0 and r2=r11 (or r33 in
case of the output disc) (see figure 3.4)

Figure 3.4 Curvature sum [10]
Using numbers, the result of the check performed is:
1
32−1
40≤1
40+1
334 ,32 (𝑠𝑎𝑢 44,85)
Analyzing the result and replacing the numbers, the value of the curvature sum:
1
𝑅𝑒𝑐ℎ=1
𝑟−1
𝑅1 (8)
𝑅𝑒𝑐ℎ=𝑟22×𝑟0
𝑟0−𝑟22 (9)
Following e quations define the problem of two ellipsoidal solids in contact in terms of an
equivalent solid of radii r0 and r22 in contact with a plane.
The radius ratio, r, is:
𝑟=𝑟0
𝑟22=1,25 >1 (10)
Further on, the ellipticity parameter, ke, which is defined as the radius ratio r power 2/:
𝑘𝑒=(𝑟)2
=1,153 (11)
Note the fact that the ellipticity parameter is function only of the radii of curvature and
not of load. The only explanation for this is that if as the load increases, the semi -axes in the
x and y directions of the ellipsoidal contact increase proportionally to each other so the
ellipticity parameter remains unchanged.
After weighting all the possibilities of materials that can be u sed for this type of CVTs ,
and based on the research made on the internet, the most suitable material for toroidal CVT is
medium and high-speed alloys steel, with the following characteristics:
➢ Modulus of elasticity, E: 𝐸 = 200 𝐺𝑃𝑎
➢ Poisson’s ratio,  :  = 0.3
Based on the values of modulus of elasticity and Poisson’s ratio, one can compute the
effective modulus of elasticity denoted with Eeff, using the following formula:
𝐸𝑒𝑓𝑓=2
2×(1−2
𝐸)=219 ,78 𝐺𝑃𝑎

The normal force, Fn, acting on the
contact point (see figure 3.5) has a
proposed value of 10kN , representing the
average force at which the Hertzian stress
has acceptable values .
Traction moment is given by :
𝑀𝑡=𝐹𝑛×𝑟1(𝑠𝑎𝑢 𝑟3) [𝑁𝑚] (12)
Figure 3.5 Normal force
When r11 is used, Mt determined is valid only for input disc and when r3 is used, Mt
determined is valid for output disc.
Next, it was calculated the print radius, a, and the maximum deformation at the contact
center , max:
𝑎=√3×𝐹𝑛×𝑅𝑒𝑐ℎ
2×𝐸𝑒𝑓𝑓3≈1,14√𝐹𝑛×𝑅𝑒𝑐ℎ
𝐸𝑒𝑓𝑓3 [𝑚𝑚 ] (13)
𝑚𝑎𝑥 =𝑎2
𝑅𝑒𝑐ℎ=√9×𝐹𝑛2
4×𝐸𝑒𝑓𝑓2×𝑅𝑒𝑐ℎ3
[𝑚𝑚 ] (14)
Last, but not least, the maximum Hertzian stress (or maximum pressure) σ Hmax, was
computed based on the values obtained for Eeff, Fn and Rech.
𝜎𝐻𝑚𝑎𝑥 =1
𝜋√3×𝐹𝑛×𝐸𝑒𝑓𝑓2
2×𝑅𝑒𝑐ℎ23
[𝐺𝑃𝑎 ] (15)
4. ELASTO -HYDRODYNAMIC MECHANISM (EHD)
Between the two components of the CVT, there must be, mandatory, a film of lubricant, in
order for them not to burn out of fail quickly. This lubricant does not have only the capability
to lubricate, but it also serves as a medium to transfer power, since the two elements are not in
close contact. In use for CVTs are available three classes of lubricants: boundary lubrica nts,
hydrodynamic lubricants and elas to-hydrodynamic lubricants/fluids, or EHLs.
In the case of the boundary lubricants, the film is sheared under high pressures, and the
hydrodynamic fluids tend to keep a film of oil on the surfaces where the two metal element s
are in contact. When low forces are applied, that film remains in its place, but when the forces
increase, the fluids have a tendency to shear. Surface damage to the metal parts is most likely
to occur and the opposing parts undergo gross slippage.
The ability of the lubricant to transfer power is very important due to the fact that the area
of contact between the two elements is very small (the two parts of the CVT are curved rather
than flat). As for the hydrodynamic lubricants, they have a relatively low traction coefficient,
being used in applications of less robust CVTs. As a fact, a CVT build with hydrodynamic

lubrication will have great problems related to the size, being quite large to be installed on a
vehicle and uneconomical to manufacture.
The following paragraph is an extract from a paper of Manuel E. Joaquim [11]: ”EHLs
have higher traction coefficients and behave very differently. Under the high normal and
tangential forces needed for transmissions in heavier vehicles, the film of EHL on a metal
surface is actually much thinner than the film of a hydrodynamic fluid would be. But where
the two metal elements are in rolling contact, the EHL briefly ceases to be a fluid and turns
into a glassy solid ( figure 3.6). The thickness of the solid is about0.254 micro -meters and the
solid state lasts for only a few microseconds. As rotation carries the EHL out of the contact
area, it immediately becom es a fluid again and regains its normal properties. But during the
brief period as a solid, the EHL actually stops being a lubricant and becomes part of the
machinery. By doing so, it achieves a much higher traction coefficient than is possible for any
hydrodynamic fluid.”

Figure 3.6 EHD lubricants – instantaneous solid state [11]
As it was obtained experimentally,
EHLs have the highest traction coefficient
of any known lubricant. As an order of
magnitude, the traction coefficients of
hydrodynamic fluids are somewhere
between 0.05 and 0.06, while the traction
coefficients of EHLs go up to a value of
0.1, the efficiency at transferring power of
EHLs being around 50 to 100% better than
that of the hydrodynamic fluids.
Figure 3.7 Behavior of petroleum -based
lubricants and synthetic EHLs under increasing
pressure [11]
Verification of the EHD regime
It is mandatory, that during the operation of the toroidal variator, to avoid metal to metal
contact between the discs and the rollers. In order for this to happen the following condition
must be fulfilled:
𝜆= ℎ𝑚𝑖𝑛
√𝑅𝑑2+𝑅𝑟2≥(2.5÷3) (16)
where: hmin represents the minimum thickness of the EHD film, Rd and Rr are the roughness
of the surfaces in contact (the disc and the roller) and λ is the EHD lubrication coefficient.
According to N. Popinceanu et. al. [12], the dimensionless value of the minimum film
thickness of the EHD lubricant can be computed as follows:

𝐻𝑚𝑖𝑛 =3.63×𝑈0.68×𝐺0.49×𝑊−0.073×(1−𝑒−0.68×𝑘𝑓) (17)
with U, G, W and kf being dimensionless parameters that can be calculated.
𝑈=0×𝑉
𝐸𝑒𝑓𝑓×𝑅𝑥=10−11(estimated) (18)
𝐺=𝐸𝑒𝑓𝑓
𝑝𝑖𝑣,𝑎𝑠=5.582 ×103 (19)
𝑊=𝐹𝑛
𝐸𝑒𝑓𝑓×𝑅𝑥2=1.955 ×10−6 (20)
𝑘𝑓=1.03×(𝑅𝑦
𝑅𝑥)0.64=0.363 (21)
where: 0 – dynamic viscosity of the oil at normal conditions ( 0 = 0.027 Pa·s );
V – velocity of the contact bodies;
Eeff – effective modulus of elasticity ( Eeff = 219.78 GPa );
Rx – curvature radius on x direction;
Ry – curvature radius on y direction;
where:
𝑅𝑥=𝑟𝑎𝑥×𝑟𝑏𝑥
𝑟𝑎𝑥+𝑟𝑏𝑥=−𝑟0×𝑟22
−𝑟0+𝑟22=160 𝑚𝑚 (22)
𝑅𝑦=𝑟𝑎𝑦×𝑟𝑏𝑦
𝑟𝑎𝑦+𝑟𝑏𝑦=𝑟11×𝑟2
𝑟11+𝑟2=31.39𝑚𝑚 (23)

piv,as – asymptotic isoviscous pressure obtained by Reynolds ( piv,as = -1,  being the
viscosity – pressure coefficient of the oil,  = 25.4 GPa-1);
Fn – normal load ( Fn = 10 kN).
The roughness of the two components, the discs and the rollers, are considered to be
equal, having a value of : R d = R r = 0.05m.
Further on, after all these parameters were computed, the dimensionless value for the
minimum film thickness was obtained:
𝐻𝑚𝑖𝑛 =3.63×(10−11)0.68×(5.582 ×103)0.49×(1.955 ×10−6)−0.073×(1
−𝑒−0.68×0.363) (24)
𝐻𝑚𝑖𝑛 =4.704 ×10−6
In order to obtain the real value for the minimum film thickness, one used the formula
below:
𝐻𝑚𝑖𝑛 =ℎ𝑚𝑖𝑛
𝑅𝑥 (25)

obtaining a value for hmin = 0.752 m.
Last, but not least, after obtaining the real value of the minimum film thickness,
relation (35) can be verified.
𝜆= 0.752
√0.052+0.052=10.645 >2.5÷3 (26)
Also, the dimensionless value of the film thickness in the center of the contact can be
calculated using the following formula:
𝐻𝑐=2.69×𝑈0.67×𝐺0.53×𝑊−0.067×[1−0.61×(𝑒−0.68×𝑘𝑓)] (27)
𝐻𝑐=1.403 ×10−5
And the real value of central film thickness is :
𝐻𝑐=ℎ𝑐
𝑅𝑥 (28)
with hc = 2.244 m.
5. PARAMETRIC ANALYSIS OF A HAL F TOROIDAL CVT
The aim of this study was to make an optimization of the geometrical parameters of a toroidal
CVT. In order to perform such study, it was necessary to establish a number of parameters.
Two of these parameters have already been presented in table 3.1 (see c hapter 3) , namely the
aspect ratio of the toroidal cavity k, which will henceforth be referred to as k1 and the cavity
radius , r0.
For the third parameter, it was
necessary to find a relation that would link
the roller to the dis k. The only geometrical
feature of the CVT that links the roller to
the disk is the roller curvature . As a result,
the third parameter defined for study is
denoted k2 and represents the ratio between
the roller curvature and cavity radius.
(r22/r0).
To observe different variations of the
geometry of the CVT forced us to perform
this study taking into account two
situations. For the first one the cavity
radius, r0, and the normal force, Fn, were
kept constant, whilst for the second
situation the normal force varied .
5.1. Case I – Fn constant Table 5.1 Case I – parameters
Nr. Crt. Parameters
r0 [mm] k2
1
30

40

60 0,51
2 0,59
3 0,642
4 0,675
5 0,712
6 0,763
7 0,8
8 0,85

For each value of r0, k2 varies between 0.51 and 0.85. After establishing this data, there
were calculated the curvature sum, Rech, the maximum deformation, δmax, and the maximum
Hertzian stress , σHmax, then the respective graphs were plotted.
As can be se en in figure 5.1 , in all 3 situations analyzed in case I, the curvature sum
has a similar behavior. We can s tate the fact that Rech increases more as the values of the cavity
radius , r0, are higher.
In the case of the maximum deformation (figure 5.2) , δmax, things are similar to
curvature sum . The values of the maximum deformation are inversely proportional to the
values of the cavity radius , r0. For small values of r0 we have a large deformation, and as these
value s increases, the deformation decreas es.

Figure 5.1 Curvature sum , Rech

Figure 5.2 Maximum deformation , δmax
Lastly, picture 5.3 represents the maximum Hertzian stress, σHmax, which was plotted
with respect to k2. It can be observed that this does not have a linear behavior, which is an
important aspect of th e study and, like the maximum deformation, the values of the maximum
stress are inversely proportional to the values of the cavity radius . 050100150200250300350
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85Rech [mm]
k2r0=ct
Fn=ct
Mt≠ct
0.0200.0250.0300.0350.0400.0450.0500.055
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85δmax [mm]
k2r0=ct
Fn=ct
Mt≠ct

Figure 5.3 Maximum Hertzian s tress, σHmax
5.2. Case II – Fn variable
For each of the 3 proposed configurations (see table 5.2) , the normal force, the maximum
deformation and of course, the maximum Hertzian stress will be calculated. As can be seen in
table 5. 2 the variation of the force depends on the tilt ing angle, γ , (parameter k2 is present) .
Table 5.2 Case II – parameters
Nr. Crt. Parameters
r0 [mm] k1 k2
1 30 0,83 0,59
2 40 0,625 0,8
3 60 0,42 0,712
The fact that the normal force is
variable implied the traction moment
would have to have a fixed value . The
value chosen for this case is: Mt = 550Nm ,
representing a value close to the maximum
obtained value using the initial geometric
parameters. ( r0 = 40mm and r22 = 32mm ,
Mtmax = 580.54Nm ).
For each of the 3 proposed
configurations seen in table 5.2. (1 – 3)
will now be presented the corresponding
3D model ( figure 5.4. – 5.6), realized in
CATIA v5.
Figure 5.4 3D model for the 1st configuration:
r0=30mm, k 1=0,83, k 2=0,59 0.40.91.41.92.42.93.4
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85σHmax
[GPa]
k2r0=ct
Fn=ct
Mt≠ct

Figure 5.5 3D model for the 2nd configuration:
r0=40mm, k 1=0,625, k 2=0,8
Figure 5.6 3D model for the 3rd configuration:
r0=60mm, k 1=0,42, k 2=0,712
In figure 5.7 is presented t he variation of the maximum stress which is directly
proportional with respect to the normal force. The "chaotic" distribution and the discrepancy
between the 3 stress curves is due to the 3 parameters that were used for this study : the valu es
chosen for k1 decrease progressively (0.83 -> 0.625 -> 0.42), while the values chosen for k2
are randomly chosen (0.59; 0.8; 0.712.

Figure 5.7 Maximum Hertzian stress, σHmax
6. CONCLU SIONS
The objective of this research was to analyze the existing types of CVTs, verification of the
elasto -hydrodynamic regime and to design and perform a parametric analysis of specific type
of CVT to see if can be found an optimized version of it, both in term s of dimensions and
performance.
The solution that was of main interest for the present paper is the toroidal CVT which is
less known and explored, being considered ”the next generation CVT”. Toroidal transmissions
have a growing interest in them, due to t heir higher torque capacity than that of the other types
of existing CVTs . Considering that the traditional transmissions have been in continuous
development and improvement in the last century, we can say that the development of the
variators is still in an incipient state. A consequence of this fact is reflected in the online
environment, which does not offer a wide range of information, most of the information found
being the same.
0.700.951.201.451.701.952.202.452.702.95
7.00 9.00 11.00 13.00 15.00 17.00σHmax [GPa]
Fn [kN]r0=ct
Fn≠ct
Mt=ct

Analyzing different models CVTs encountered during the period on which I studied this
topic, it was noticed that the principle of operation is similar from model to model, the
differences being from point of view of the components.
Based on the available data regarding the geometry of this types of transmissions and the
results obtained using the calculation model, we were able to design a compact transmission
having a maximum Hertzian stress of 0.74GPa and a film thickness of 0. 752m, which it is
believe d to be a real solution for a vehicle.
Analyzing the data obtained for the parametric study , the existing geometric model is
indeed an optimized one, both in terms of dimensions and performance, but it is encouraging
that there are al ternative solutions, that can become real options, depending on the applications
where it will be implemented such transmission.
Even though CVTs are no so used around the world, its applications and benefits can
increase based on research and development that is being done. As automotive market
continues to develop CVTs, more and more car manufacturers will implement them, which
will lead to an increasing desire to improve them more and more.
What is more, an increase in the development of CVTs will coincide with an increase in
the competition among manufacturers which then will lower the manufacturing costs of such
transmissions. Any technology that has any connection with the produc tion of CVTs will
experience a period of prosperity, because CVTs have only just begun to blossom.
7. REFERENCES
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Metal Pushing Belt and Variable Pulleys , International Journal of Automotive Engineering,
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[2] William Harris, Howstuffworks Pulley -based CVTs, 27 April 2005. Retrieved March 11,
2017, from website: http://auto. howstuffworks.com/cvt2.htm .
[3] http://en.wikipedia.org/wiki/Continuously_variable_transmission#Variablediameter_pull
ey_.28VDP.29_or_Reeves_drive
[4] http://www.crzforum.com/forum/general -discussion/57946 -possible -replacecvt -belt.html
[5] William Harris, Howstuffworks Toroidal CVTs , 27 April 2005. Retrieved Ma rch 11, 2017,
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[8] Carbone G., Mangialardi L., Mantriota G., A comparison of the perfo rmances of full and
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[11] Manuel E. Joaquim, EHL’s: The Secret Behind CVTs , Findett Corporation, pag. 2 -7.
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