12th IFToMM World Congress, Besançon (France), June18-21, 2007
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Dynamics of CVTs: A comparison between theory and experiments
G. Carbone*
L. Mangialardi†
Dept. Mech. Eng.
Dept. Mech. Eng.
Politecnico di Bari, Italy Politecnico di Bari, Italy
Abstract—We present1 an experimental investigation of the
pushing V-belt CVT dynamics with the aim of comparing the
experimental data with the theoretical predictions of the
Carbone, Mangialardi, Mantriota (CMM) model [1-2]. A very
good agreement between theory and experiments is found. In
particular it is shown that, during creep-mode (slow) shifting,
the rate of change of the speed ratio depends linearly on the
logarithm of the ratio between the axial clamping forces acting
on the two movable pulleys. The shifting speed is also shown to
be proportional to the angular velocity of the primary pulley.
The results of this study are of utmost importance for the design
of advanced CVT control systems and the improvement of the
CVT efficiency, cars' drivability and fuel economy.
P.A.Veenhuizen‡
Dept. Mech. Eng.
Eindhoven Univ. of Technology
a reliable model of the CVT mechanical behaviour. This is
necessary in order to appropriately regulate the clamping
forces, the speed ratio and the shifting speed, thus
allowing the engine to operate on its economy line. In a
previous paper [1] Carbone, Mangialardi and Mantriota
(CMM) have developed a model that describes both the
steady-state and the shifting dynamics of the V-belt CVT.
Aim of this study is to investigate the reliability of the
CMM theory by comparing its predictions with the
experimental outcomes. Also a brief comparison with
other models is carried out.
II. The theoretical model
Keywords:
Continuously
variable
transmissions,
automatic transmissions, CVT, V-belt CVT, pushing-belt
CVT, metal-chain CVT, shifting dynamics.
In this section we briefly review the CMM model of CVT
dynamics presented in Ref. [1]. The theory treats the belt
as a one-dimensional continuum body having a locally
rigid motion, the belt is indeed considered as an
inextensible strip with zero radial thickness and infinite
transversal stiffness. Although the model may appear
more suitable for the chain belt, as it does not take into
account the influence of the bands-segments interaction
and that of the varying gap among the steel segments, the
experimental investigations, carried out on a Van Doorne
type pushing-belt, have shown that the main predictions of
the CMM theory do not depend on the actual design of the
belt. The friction forces, at the interface between the
pulley and the belt, are described by means of the simple
Coulomb-Amonton's friction law, i.e. the friction
coefficient µ is taken to be constant. Figure 1 shows the
kinematical and geometric quantities involved into the
problem, which satisfy the following relations.
tan β s = tan β cosψ
(1)
rω s = r tanψ
In Eq. (1) [see also Fig. 1] r is the local radial position of
the one-dimensional belt, β is the pulley half-opening
I. Introduction
In the last decades, a growing attention has been focused
on the environmental question. Governments are
continuously forced to define standards and to adopt
actions in order to reduce the polluting emissions and the
green-house gasses. In order to fulfil these requirements,
car manufacturers have been obligated to dramatically
reduce vehicles' gas emissions in relatively short times.
Thus, a great deal of research has been devoted to find
new technical solutions, which may improve the emission
performances of nowadays internal combustion (IC)
engine vehicles. Among all the proposed solutions, the
hybrid technology is very promising for the short term.
But hybrid vehicles often need a complicated drive train
to handle the power flows between the electric motor, the
IC engine and vehicles' wheels. A very good solution may
be that of using a continuously variable transmission
(CVT), which is able to provide an infinite number of gear
ratios between two finite limits. CVT transmissions are
even potentially able to improve the performances of
classical IC engine vehicles, by maintaining the engine
operation point closer to its optimal efficiency line [3-6].
However, in order to achieve a significant reduction of
fuel consumption, it is fundamental to have a very good
control strategy of the transmission, which in turns needs
angle, β s is the half-opening angle in the sliding plane,
ψ is the sliding angle, and ωs is the local sliding angular
velocity of the belt, defined as ω s = Ω − ω , with ω being
the pulley rotating velocity, and Ω the local angular
velocity of the belt.
*E-mail: [email protected]
†
E-mail: [email protected]
‡
E-mail: [email protected]
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12th IFToMM World Congress, Besançon (France), June18-21, 2007
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vr =
(
dR
+ a∆ωR sin(θ − θ c )
dt
(5)
)
where a = 1 + cos 2 β 0 / sin(2β 0 ) . Besides the above
written equations, we also need to write the equilibrium
equations, where the forces acting on the belt are shown in
Figure 2.
Fig. 1. Geometrical and kinematical quantities. (a) the sliding angle ψ,
the angular co-ordinate θ, the radial co-ordinate r, the radius of curvature
ρ and the slope angle ϕ . (b) the belt’s local sliding velocity vs, its
components r and Rωs, the pulley half-opening angle β and the halfopening angle β s in the sliding plane.
The pulley bending is described on the basis of the
Sattler's model [7], where trigonometric functions are used
to represent the varying groove angle β and the local
elastic axial deformations u of the pulley sheaves:
∆
π
β = β 0 + sin θ − θ c +
(2)
2
2
Fig. 2. Forces acting on the belt.
The equilibrium of the belt involves the tension F of the
belt, the linear pressure p acting on the belt sides, the
inertia force and the friction forces. Neglecting second
order terms, the equilibrium of the belt gives
1
∂ F − σω 2 R 2
µ cos β s sinψ
=
2 2
∂
θ
sin
β
F − σω R
0 − µ cos β s cosψ
(6)
F − σω 2 R 2
p=
2 R(sin β 0 − µ cos β s cosψ )
with σ being the mass per unit length of the belt. The last
equation of the model allows to calculate the center of the
wedge expansion θ c as
u = 2 R tan(β − β 0 )
β 0 is the groove angle of the undeformed pulley,
(
∆ ≈ 10 −3 << 1 is the amplitude of the sinusoid, θ c is the
center of the wedge expansion and R stands for the pitch
radius of the belt, i.e. the distance from the pulley axis that
the belt would have if the pulley sheaves were rigid. The
varying groove makes the radial motion of the belt non
uniform along the contact arc, thus affecting the sliding
angle ψ , the direction of friction forces at the belt-pulley
interface, as well as the pressure and tension distributions.
By using the Sattler's relations (2), the local radial position
of the belt can be easily calculated as
u
r tan β = R tan β 0 −
(3)
2
Though the quantity r is not uniform along the belt [and
therefore the slope angle ϕ differs from zero on the
contact arc (see Fig. 1)], it is always possible to consider
ϕ << 1 on most part of the contact arc, and to assume the
tan θ c =
α
0
)
p(θ ) sin θ dθ /
α
0
p(θ ) cos θ dθ
(7)
where α is the extension of the wrap angle. Once the
pressure and tension distribution have been calculated, it
is possible to easily calculate the axial clamping force and
torque on the pulley as
α
(cos β + µ sin β s ) pRdθ
T = (F1 − F2 ) R
S=
0
(8)
The theory [1] shows that the governing parameter which
significantly affect the CVT shifting behaviour is
sin(2β 0 )
1 RDR
(9)
A=
∆ ω DR RDR 1 + cos 2 β 0
radius of curvature ρ ≈ R everywhere but at the edges of
the contact arc. With these assumptions, and neglecting
second order terms, the continuity equation can be written
as
∂v
vr + θ = 0
(4)
∂θ
where vr is the radial sliding velocity of the belt and vθ
is its tangential sliding speed. Taking the time-derivative
of Eq. (3) and using the Sattler’s relations (2) gives
where DR stand for drive-pulley (the driven-pulley will
be referred to with the subscript DN ). The CMM model
also shows that during creep mode shifting (slow shifting)
A is almost a linear function of the logarithm of the
clamping force ratio S DR / S DN , as described by the
following equation
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12th IFToMM World Congress, Besançon (France), June18-21, 2007
S DR
S
− ln DR
S DN
S DN
A = c(τ ) ln
where
(S DR / S DN )eq
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(10)
eq
is the clamping force ratio at
equilibrium, i.e. in steady-state conditions and c(τ ) can be
calculated by the model.
Eq. (10) can be rephrased in terms of the geometric speed
ratio τ as
τ = ω DR ∆
1 + cos 2 β 0
S
S
g (τ ) ln DR − ln DR
sin(2β 0 )
S DN
S DN
(11)
eq
where g (τ ) can be calculated by the model. Eq. (11)
shows that the shifting speed τ is proportional to the
primary pulley angular velocity ω DR , to the parameter ∆ ,
Fig. 3. A comparison with the Tenberg’s model (adapted from Ref. [8])
and the CMM model for steady-state running conditions.
and that it depends linearly on ln(S DR / S DN ) .
IV. Experimental Validation
In order to validate the CMM model, a detailed
experimental investigation has been carried out. Tests
have been undertaken on a pushing-belt CVT by van
Doorne Transmissie, mounted on the power-loop test rig
available at the Automotive Engineering Science
Laboratory - Eindhoven University of Technology, as
shown in Fig. 4.
III. Comparison with other models
In this section the CMM model predictions will be
compared with those provided by Tenberge [8], who
considered the case of a chain belt CVT and used a FEM
approach to calculate the Green function, i.e. the elastic
response of the pulley. The comparison focuses on both
the sliding velocity field and the friction forces at the
pulley-belt interface, and on the axial clamping forces.
The CVT is a metal chain variator with the following
properties d = 155 mm , L = 649 mm , and σ = 1.2 kg/m .
As an example, in steady state conditions (i.e. τ = 0 ) with
τ = 2.0 , β 0 = 10° , ω DR = 2000 RPM , RDR = 70.3 mm ,
RDN = 35.1 mm , Fmin = 2670 N , Fmax = 6228 N , and
µ = 0.09 ,
we
get
(S DR )CMM = 46.8 kN
and
(S DN )CMM = 25.5 kN , whereas Tenberge's model gives
(S DR )T = 46.6 kN and (S DN )T = 27.0 kN . The agreement
is very good, with a difference of less than 5% on the
driven pulley. Furthermore, observe that this difference
may be due to some uncertainties in the value of µ and
Fig. 4. The test rig utilized for the experiments.
β0 . The velocity field and the friction forces at the belt-
Steady-state experiments under no-load and load
conditions have been carried out, whereas shifting
experiments have been carried out only at zero torque load
because the control of the test rig did not allow safe
shifting experiments under load conditions. In both kinds
of experiments, the secondary clamping force S DN and
pulley interface have been also calculated, and, as shown
in Fig. 3, the agreement between the two models is very
good.
We may conclude that the simpler continuum onedimensional model of the belt, proposed in Ref. [1], gives
very good results despite the continuum approximation
which might be expected not to be suitable for the chain
belt due to the presence of a discrete number of contact
points. The CMM model solves a very small number of
equations and does not need to deal with the very large
number of degrees of freedom of the system. For this
reason, it runs very fast on a PC, mostly in steady- state,
when the magnitude ∆ of the pulley bending does not
affect the pressure and tension distributions along the
contact arc [1].
the primary angular velocity ω DR have been fixed. The
geometrical quantities of the pushing-belt CVT utilized
for the experimental activity, are: belt length
L = 703 mm , center-to-center distance of the pulleys
d = 168 mm , groove angle β 0 = 11° . The friction
coefficient has been estimated equal to µ = 0.09 .
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12th IFToMM World Congress, Besançon (France), June18-21, 2007
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V. Steady-state Measurements
follow a master curve, see continuous thin line in Fig. 5,
which is very close to the theoretical thick line. However,
we have also observed that very new pushing-belts have a
relatively significant different behaviour in comparison
with used belts, mainly at low clamping force values.
This, of course, cannot be predicted by the theory and
needs further investigations. One possible key factor
might be the interaction between the steel rings and the
segments which has not been taken into account by the
CMM model.
In steady-state conditions, the clamping force S DR , acting
on the primary pulley, has been measured as a function of
the geometrical speed ratio τ = RDR / RDN , for a fixed
value of the driven pulley clamping force S DN . The speed
ratio τ has been measured by using axial position sensors
placed on one of the two movable half-pulleys.
Fig. 5. The clamping force ratio as a function of the geometrical speed
ratio (log-log representation) under no torque load running.
A. No-load tests
Fig. 5 shows the logarithm of the clamping forces ratio,
ln(S DR / S DN )eq , as a function of ln τ , in steady-state
conditions. Circles represent the measurements, the thin
line is the fit of the experimental data, while the thick one
represents the theoretical prediction of the CMM model.
Data have been measured for different primary angular
velocities ( ω DR = 1000, 2000, 3000 RPM ) and two
different values of the secondary clamping force
( S DN = 20, 30 kN ). The agreement with the theoretical
calculation is very good.
Fig. 6. The clamping force ratio as a function of the applied torque TDN
for different values of the geometric speed ratio.
Experiments have shown that, as predicted by the CMM
model, neither the magnitude of the secondary clamping
force S DN , nor the angular velocity of the primary pulley
ω DR have a significant influence on the clamping force
ratio in steady-state. All the experimental data, instead,
Fig. 7. The shifting speed ratio as a function of clamping force ratio.
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12th IFToMM World Congress, Besançon (France), June18-21, 2007
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and for different values of the speed ratio
τ = 0.6 , 1.0, 1.4, 1.8 . The agreement with the theoretical
calculations (thick lines) looks very good. In particular,
for fixed values of τ and S DN , all the measured data fall
on a straight line. This proves the linear dependence of τ
on ln(S DR / S DN ) , which was one of the most significant
results of CMM model. Experiments [2] have shown that
the slope of the curves depends slightly, on the secondary
clamping force. This can be interpreted as due to a change
of the magnitude of the pulley deformation and in
particular of the dimensionless parameter ∆ . Indeed, it is
expected that increasing the clamping force increases also
the magnitude of the pulley deformation, i.e. ∆ . Thus,
different values of ∆ have to be used for different values
of the secondary clamping force S DN . Furthermore,
because of the linear elastic response of the system, a
linear relation is expected between ∆ and S DN [2]. Also
observe that the small difference between the theory and
the experiments, sometimes observed in Fig. 7, is mainly
due to a different value of the steady-state clamping force
ratio (S DR / S DN )eq , rather than to a different slope of the
B. Load tests
Steady-state experiments have been also performed under
load conditions. Fig. 6 shows the logarithm of the
clamping forces ratio as a function of the dimensionless
torque load TDN / (RDN S DN )
for
S DN = 30 kN ,
ω = 1000RPM and different values of the speed ratio and
torque load ( TDN = 20 , 40 , 80, 100 Nm ). Fig. 6 shows a
very good agreement between theory and experiments for
all the tested speed ratios, thus confirming the validity of
the CMM model at least within the range of torque values
utilized during the experiments. We observe that the
maximum value of the parameter TDN / (RDN S DN ) is
about 2 µ . Thus, additional experiments are needed to test
the theory over the whole range of the dimensionless
torque load.
VI. Shifting measurements
Shifting tests have been carried out only under no load
conditions, as the test bench control system did not allow
to perform load shifting tests under safe conditions. The
experiments have been carried out by fixing the shifting
speed τ , the secondary clamping force S DN and the
curves.
Fig. 8 shows the effect of the primary angular velocity on
the shifting behaviour of the system.
primary angular velocity ω DR . The primary clamping
force S DR and the speed ratio τ were measured as a
function of time. A very good way to represent the
experimental results is to plot the quantity τ as a function
of ln(S DR / S DN ) for each value of τ , S DN and ω DR .
Fig. 9. The rate of change of speed ratio as a function of the clamping
force ratio in a linear-linear diagram.
Two cases are shown, one for τ = 1 and the other one for
τ = 1.2 . In both cases, S DN = 20 kN , whereas the angular
velocity is respectively ω DR = 1000, 2000 RPM .
A very good agreement with the results predicted by the
CMM model is again clearly shown. This confirms a
direct proportionality between the shifting speed τ and
the primary pulley angular velocity ω DR . Similar results
have been also obtained in all the other cases, i.e. for
different values of τ and of the secondary clamping force
S DN .
Fig. 9 shows the rate of change of the speed ratio τ as a
function of the force ratio S DR / S DN , instead of
Fig. 8. The shifting speed as of clamping force ratio for different primary
pulley velocities.
ln(S DR / S DN ) . The figure clearly shows that in the linearlinear diagram the curve deviates significantly from a
straight line, especially when τ is small, thus showing
In Fig. 7 the theoretical results are compared with the
experimental ones, for S DN = 20 kN , ω DR = 1000 RPM
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12th IFToMM World Congress, Besançon (France), June18-21, 2007
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Management, September 25-26, 1997.
[5] Carbone G., Mangialardi L., Mantriota G., 2002: "Fuel
Consumption of a Mid Class Vehicle with Infinitely Variable
Transmission", SAE J. Engines, 110(3), pp. 2474-2483.
[6] Carbone G., Mangialardi L., Mantriota G., Soria L.: Performance of
a City Bus equipped with a Toroidal Traction Drive. IASME
Transactions, 1 (1), pp. 16-23, January 2004.
[7] Sattler H., 1999: "Efficiency of Metal Chain and V-Belt CVT".
Proc. CVT'99 Congress, Eindhoven, The Netherlands, pp. 99-104.
[8] Tenberge P.: "Efficiency of Chain-CVTs at Constant and Variable
Ratio. A new mathematical model for a very fast calculation of
chain forces, clamping forces, clamping ratio, slip and efficiency",
paper no. 04CVT-35, 2004 International Continuously Variable and
Hybrid Transmission Congress, UC Davis, September 23-25, 2004.
[9] Ide T., Uchiyama H., Kataoka R., 1996: "Experimental
Investigation on Shift Speed Characteristics of a Metal V-Belt
CVT". JSAE paper 9636330.
[10] Ide T., Udagawa A., Kataoka R., 1995: "Simulation Approach to
the Effect of the Ratio Changing Speed of a Metal V-Belt CVT on
the Vehicle Response", Veh. Syst. Dyn., 24, pp. 377-388.
[11] Bonsen B. , Carbone G. , Simons S.W.H. , Steinbuch M. and
Veenhuizen P.A., "Shift dynamics modeling for optimizing slip
control in a continuously variable transmission", 31st FISITA
Congress, Yokohama 22 - 27 October 2006
[12] Simons S.W.H., Klaassen T.W.G.L., Steinbuch M., Veenhuizen
P.A. and Carbone G. " Shift dynamics modeling and optimized
CVT slip control ", in preparation
again that the logarithmic relation proposed in the CMM
model is more suitable than a linear relation as proposed
instead by Ide [9-10].
VII. Conclusions
In this work a detailed experimental investigation
concerning the V-belt CVT dynamics has been carried
out, in order to compare the theoretical predictions of the
so-called CMM theoretical model by Carbone,
Mangialardi and Mantriota [1] with the experimental
results. A very good agreement between theory and
experiments has been found, both in steady-state and
during shifting manoeuvres. This confirms all the most
important predictions of the model. In particular, it has
been shown that during relatively slow shifting maneuvers
(creep-mode) the rate of change of the speed ratio τ is a
linear function of the logarithm of the clamping force ratio
S DR / S DN . The linear relation between τ and
ln(S DR / S DN ) has also been compared with the Ide's
formula, which is, instead, a linear relation between τ
and S DR / S DN . The experiments have shown that Ide's
relation may well approximate the real CVT shifting
behaviour only for speed ratio values greater than 1,
whereas in all other cases the approximation is less good.
Experiments have also confirmed that the shifting speed is
also proportional to the angular velocity of the primary
pulley. The CMM predictions have been also compared
with those by Tenberge [8] for the chain belt case. Also in
this case, the agreement was really very good, showing
that the continuum belt approximation, which is the basis
of the CMM model, works very well, not depending on
the typology of the considered belt, i.e. both for the
pushing and chain belts. On the basis of the very good
obtained results, the authors also propose a relatively
simple differential equation to describe the creep-mode
evolution of the variator. Very few parameters appear in
the formula, which may be calculated either
experimentally or theoretically. The results of this study
are already being used to design advanced CVT control
systems which may improve the CVT efficiency, cars'
drivability and fuel economy [11-12]
References
[1] Carbone G., Mangialardi L., Mantriota G.: "The influence of pulley
deformations on the shifting mechanisms of MVB-CVT". ASME
Journal of Mechanical Design, 127, 103-113 (2005).
[2] Carbone G., Mangialardi L.; Bonsen B.; Tursi C., Veenhuizen P.A.,
CVT Dynamics: Theory and Experiments, Mechanism and
Machine Theory, in press (2006)
[3] Brace C., Deacon M., Vaughan N. D., Horrocks R. W., Burrows C.
R., 1999: "The Compromise in Reducing Exhaust Emissions and
Fuel Consumption from a Diesel CVT Powertrain over Typical
Usage Cycles", Proc. CVT'99 Congress, Eindhoven, The
Netherlands, pp. 27-33.
[4] Brace C., Deacon M., Vaughan N. D., Burrows C. R., Horrocks R.
W., 1997: "Integrated passenger cat diesel CVT powertrain control
for economy and low emissions". ImechE International Seminar
S540, Advanced Vehicle Transmission and Powertrain
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