cvt dynamics

12th IFToMM World Congress, Besançon (France), June18-21, 2007 CK-xxx

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Dynamics of CVTs: A comparison between theory and experiments

G. Carbone* L. Mangialardi† P.A.Veenhuizen‡ Dept. Mech. Eng. Dept. Mech. Eng. Dept. Mech. Eng. Politecnico di Bari, Italy Politecnico di Bari, Italy Eindhoven Univ. of Technology

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. Keywords: Continuously variable transmissions, automatic transmissions, CVT, V-belt CVT, pushing-belt CVT, metal-chain CVT, shifting dynamics.

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

*E-mail: [email protected] †

E-mail: [email protected] ‡

E-mail: [email protected]

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 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.

ψωψββ

tanrr

costantan

s

s

==

(1)

In Eq. (1) [see also Fig. 1] r is the local radial position of the one-dimensional belt, β is the pulley half-opening

angle, sβ is the half-opening angle in the sliding plane, ψ is the sliding angle, and sω is the local sliding angular velocity of the belt, defined as ωω −Ω=s , with ω being the pulley rotating velocity, and Ω the local angular velocity of the belt.


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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 half-opening angle βs 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:

( )0

0

222

ββ

πθθββ

−=

+−∆+=

tanRu

sin c (2)

0β is the groove angle of the undeformed pulley,

110 3 <<≈∆ − is the amplitude of the sinusoid, cθ 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

20u

tanRtanr −= ββ (3)

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

1<<ϕ on most part of the contact arc, and to assume the

radius of curvature R≈ρ everywhere but at the edges of the contact arc. With these assumptions, and neglecting second order terms, the continuity equation can be written as

0=∂∂+

θθv

vr (4)

where rv is the radial sliding velocity of the belt and θv is its tangential sliding speed. Taking the time-derivative of Eq. (3) and using the Sattler’s relations (2) gives

( )cr sinRadtdR

v θθω −∆+= (5)

where ( ) ( )002 21 ββ sin/cosa += . 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. 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

( )

( )ψβµβσω

ψβµβψβµ

θσω

σω

coscossinRRF

p

coscossinsincosRF

RF

s

s

s

−−=

−=

∂−∂

−

0

220

22

22

2

1

(6)

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 cθ as

( ) ( ) θθθθθθθ αα dcosp/dsinptan c = 00 (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

( )( )RFFT

pRdsincosS s

21

0

−=

+= θβµβα

(8)

The theory [1] shows that the governing parameter which significantly affect the CVT shifting behaviour is

( )0

20

1

21β

βω cos

sinR

RA

DRDR

DR

+∆=

(9)

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 DNDR S/S , as described by the following equation


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( )

−

=

eqDN

DR

DN

DR

SS

lnSS

lncA τ (10)

where ( )eqDNDR S/S is the clamping force ratio at

equilibrium, i.e. in steady-state conditions and ( )τc can be calculated by the model. Eq. (10) can be rephrased in terms of the geometric speed ratio τ as

( ) ( )

−

+∆=eqDN

DR

DN

DRDR S

Sln

SS

lngsin

cos τβ

βωτ0

02

21

(11)

where ( )τg can be calculated by the model. Eq. (11) shows that the shifting speed τ is proportional to the primary pulley angular velocity DRω , to the parameter ∆ , and that it depends linearly on ( )DNDR S/Sln .

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 mm 155=d , mm 649=L , and kg/m 21.=σ . As an example, in steady state conditions (i.e. 0=τ ) with

02.=τ , °=100β , RPM2000=DRω , mm370.RDR = , mm135.RDN = , N2670=minF , N6228=maxF , and

090.=µ , we get ( ) kN846CMM .SDR = and

( ) kN525CMM .SDN = , whereas Tenberge's model gives

( ) kN 664T .SDR = and ( ) kN027T .SDN = . 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

0β . The velocity field and the friction forces at the belt-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 one-dimensional 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].

Fig. 3. A comparison with the Tenberg’s model (adapted from Ref. [8])

and the CMM model for steady-state running conditions.

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.

Fig. 4. The test rig utilized for the experiments.

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 DNS and

the primary angular velocity DRω have been fixed. The geometrical quantities of the pushing-belt CVT utilized for the experimental activity, are: belt length

mm 703=L , center-to-center distance of the pulleys mm 168=d , groove angle °= 110β . The friction

coefficient has been estimated equal to 090.=µ .


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V. Steady-state Measurements In steady-state conditions, the clamping force DRS , acting on the primary pulley, has been measured as a function of the geometrical speed ratio DNDR R/R=τ , for a fixed

value of the driven pulley clamping force DNS . 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, ( )eqDNDR S/Sln , as a function of τln , 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 ( RPM 300020001000 ,,DR =ω ) and two different values of the secondary clamping force ( kN3020,SDN = ). 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 DNS , nor the angular velocity of the primary pulley

DRω have a significant influence on the clamping force ratio in steady-state. All the experimental data, instead,

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.

Fig. 7. The shifting speed ratio as a function of clamping force ratio.


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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 ( )DNDNDN SR/T for kN30=DNS ,

RPM1000=ω and different values of the speed ratio and torque load ( Nm 100804020 ,,,TDN = ). 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 ( )DNDNDN SR/T is about µ2 . 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 DNS and the

primary angular velocity DRω . The primary clamping

force DRS 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 ( )DNDR S/Sln for each value of τ , DNS and DRω .

Fig. 8. The shifting speed as of clamping force ratio for different primary

pulley velocities. In Fig. 7 the theoretical results are compared with the experimental ones, for kN 20=DNS , RPM 1000=DRω

and for different values of the speed ratio 81 41 01 60 .,.,.,.=τ . The agreement with the theoretical

calculations (thick lines) looks very good. In particular, for fixed values of τ and DNS , all the measured data fall on a straight line. This proves the linear dependence of τ on ( )DNDR S/Sln , 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 DNS . Furthermore, because of the linear elastic response of the system, a linear relation is expected between ∆ and DNS [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 ( )eqDNDR S/S , rather than to a different slope of the

curves. Fig. 8 shows the effect of the primary angular velocity on the shifting behaviour of the system.

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 1=τ and the other one for

21.=τ . In both cases, kN 20=DNS , whereas the angular

velocity is respectively RPM2000 1000,DR =ω . 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 DRω . Similar results have been also obtained in all the other cases, i.e. for different values of τ and of the secondary clamping force

DNS . Fig. 9 shows the rate of change of the speed ratio τ as a function of the force ratio DNDR S/S , instead of

( )DNDR S/Sln . The figure clearly shows that in the linear-linear diagram the curve deviates significantly from a straight line, especially when τ is small, thus showing


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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

DNDR S/S . The linear relation between τ and

( )DNDR S/Sln has also been compared with the Ide's formula, which is, instead, a linear relation between τ and DNDR S/S . 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

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

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cvt dynamics