1. Introduction

* Corresponding author. Address: Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, v.le Japigia 182, 70126 Bari,Italy. Tel.: +39 80 596 2746; fax: +39 80 596 2777.

E-mail address: carbone@poliba.it (G. Carbone).

Mechanism and Machine Theory 42 (2007) 409428

www.elsevier.com/locate/mechmt

MechanismandMachine Theory0094-114X/$ - see front matter 2006 Elsevier Ltd. All rights reserved.In the last decades, a growing attention has been focused on the environmental question. Governments arecontinuously forced to dene standards and to adopt actions in order to reduce the polluting emissions and thegreen-house gasses. In order to fulll these requirements, car manufacturers have been obligated to dramati-cally reduce vehicles gas emissions in relatively short times. Thus, a great deal of research has been devotedto nd new technical solutions, which may improve the emission performances of nowadays internal combus-tion (IC) engine vehicles. Among all the proposed solutions, the hybrid technology is very promising for theshort term. But hybrid vehicles often need a complicated drive train to handle the power ows between the elec-tric motor, the IC engine and vehicles wheels. A very good solution may be that of using a continuouslyAbstract

We present a detailed experimental study of the pushing V-belt CVT dynamics and compare the experimental data withthe theoretical predictions of the Carbone, Mangialardi, Mantriota (CMM) model [G. Carbone, L. Mangialardi, G.Mantriota, The inuence of pulley deformations on the shifting mechanisms of MVB-CVT, ASME Journal of MechanicalDesign 127 (2005) 103113]. A very good agreement between theory and experiments is found. In particular it is shownthat, during creep-mode (slow) shifting, the rate of change of the speed ratio is a linear function of the logarithm ofthe ratio between the axial clamping forces acting on the two movable pulleys. The shifting speed is also shown to be pro-portional to the angular velocity of the primary pulley, and to increase as the clamping force on the secondary pulley isincreased. Indeed, a growth of the clamping force increases the pulley bending and, therefore, in agreement with the CMMmodel, increases the shifting speed. The authors also propose a relatively simple dierential equation to describe the creep-mode evolution of the variator. Few parameters appear in the formula, which may be calculated either experimentally ortheoretically. The results of this study are of utmost importance for the design of advanced CVT control systems and theimprovement of the CVT eciency, cars drivability and fuel economy. 2006 Elsevier Ltd. All rights reserved.CVT dynamics: Theory and experiments

G. Carbone a,b,*, L. Mangialardi a, B. Bonsen b, C. Tursi a, P.A. Veenhuizen b

a Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, v.le Japigia 182, 70126 Bari, Italyb Department of Mechanical Engineering, Eindhoven University of Technology, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands

Received 21 February 2006; accepted 11 April 2006Available online 9 June 2006doi:10.1016/j.mechmachtheory.2006.04.012

CVT car may achieve fuel savings of about 10% in comparison to the traditional manual stepped transmissions[25].dameing o

shifting speed being proportional both to the magnitude of the pulley bending and to the angular velocity

410 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428of the primary pulley. The main purpose of this paper is to carry out a detailed experimental investigationof the CVT steady-state and shifting dynamics, in order to compare the theoretical predictions with the exper-imental outcomes. The experimental analysis shows a very good agreement with the CMM model. This allowsthe authors to propose a relatively simple dierential equation to describe the CVT creep-mode shifting. Thisequation may constitute the basis of optimized CVT control strategies [8,9].

2. Mechanical model

In this section we briey review the CMM model of CVT dynamics presented in Ref. [1]. The theory treatsthe belt as a one-dimensional continuum body having a locally rigid motion, i.e. the belt is considered as aninextensible strip with zero radial thickness and innite transversal stiness. Although the model may appearmore suitable for the chain belt (see Fig. 1(a)), as it does not take into account the inuence of the bandsbeltinteraction (Fig. 1(b)), the experimental investigations, carried out on the Van Doorne type pushing-belt, haveshown that the main predictions of the CMM theory do not depend on the actual design of the variator. Thepulley deformation is described on the basis of the Sattlers model [10], where trigonometric functions are usedto represent the varying groove angle and the local elastic axial deformations of the pulley sheaves. The fric-tion forces, at the interface between the pulley and the belt, are described by means of the simple CoulombAmontons friction law, i.e. by means of a constant friction coecient l. Fig. 2 shows the kinematical andgeometric quantities involved in the model. These quantities satisfy the following relations:

tan bs tan b cosw 1rxs _r tanw 2

where r is the local radial coordinate of the one-dimensional belt, b is the pulley half-opening angle, bs is the half-opening angle in the sliding plane, w is the sliding angle, and xs is the local sliding angular velocity of the belt,dened as xs = X x, with x being the pulley rotating velocity, and X the local angular velocity of the belt.

The varying groove makes the radial motion of the belt non-uniform along the contact arc, thus aectingshifting speed and allow the engine to operate on its economy line, without aecting the CVT mechanical e-ciency. In the case of V-belt CVTs, which are the main focus of interest of our investigation, the clamping forcesshould not be too high, to avoid very high pressures at the pulleybelt interface. At the same time, they shouldnot be too small, in order to avoid very high slip factors. However, without an accurate and reliable CVT the-oretical model, the above aim could hardly be fullled. In a previous paper [1] Carbone, Mangialardi and Man-triota (CMM) have developed a model that describes both the steady-state and the shifting behavior of the V-belt CVT. The CMMmodel has been shown being able to explain why, during creep-mode shifting, the rate ofchange of the speed ratio is strictly related to the actual value of the axial clamping forces acting on the movablepulleys. The theory also shows why increasing the rate of change of the speed ratio leads to a transition from acreep-mode to a slip-mode behavior, which, in turn, is characterized by a complete independence of the clamp-ing force ratio from the actual value of the rate of change of the speed ratio [6,7].

In Ref. [1] it has been also pointed out that, during creep-mode shifting, the pulley bending has a crucialrole in determining the actual shifting response of the variator. The theory also predicts a linear relationbetween the rate of change of the speed ratio and the logarithm of the clamping force ratio, and shows thethe slHowever, Refs. [4,5] show that, in order to achieve a signicant reduction of fuel consumption, it is fun-ntal to have a very good control strategy of the transmission, able to adjust the axial clamping forces act-n the movable pulley with great precision. This is necessary in order to regulate the speed ratio and thevariable transmission (CVT), which is able to provide an innite number of gear ratios between two nite limits.CVT transmissions are even potentially able to improve the performances of classical IC engine vehicles, bymaintaining the engine operation point closer to its optimal eciency line. Several studies have shown, indeed,that CVTs may improve the fuel savings and reduce the vehicle polluting emissions. For instance, a mid classiding angle w, the direction of friction forces at the beltpulley interface, as well as the pressure and ten-

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 411sion distributions. It is worth to notice that the belt transversal deformation has a negligible inuence in deter-mining the actual path of the belt. In fact Fig. 3 (adapted from Refs. [11,12]) shows that for a rigid pulley

Fig. 1. The chain belt CVT (a), and the pushing V-belt CVT (b).

(a) (b)

Fig. 2. Kinematical and geometric quantities involved: (a) planar view; (b) 3D view.

412 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428(contversathe bcan bbendangleAppe

b0 iscenteaxis tsinusearlychangSattle

Thoucontasumeand n

Fig. 3.eect oinuous line) the radial position of the belt is almost uniform along the contact arc despite the belt trans-l deformation, whereas when the pulley bending is taken into account (dashed line) the radial position ofelt may vary of about 0.51 mm along the wrap angle. This shows that the belt transversal deformatione neglected when calculating the actual sliding path of the belt. For this reason, in Ref. [1] only the pulleying has been taken into account by using the Sattlers formula [10], which describes the varying grooveb and the axial displacement u of the pulley groove by means of simple trigonometric functions (seendix A for additional details)

b b0 D2sin h hc p

2

3

u 2R tanb b0 4the groove angle of the undeformed pulley, D 1 103 1 is the amplitude of the sinusoid, hc is ther of the wedge expansion and R stands for the pitch radius of the belt, i.e. the distance from the pulleyhat the belt would have if the pulley sheaves were rigid. It is worth to notice that the amplitude of theoid strictly depends on the actual value of the clamping forces, since increasing the clamping forces lin-increases the elastic deformation of the pulley. Thus, D cannot be considered constant during speed ratioing. Nonetheless, its value cannot vary too much being always in the range (1 0.5) 103. By using thers relations (3) and (4), the local radial position of the belt can be easily calculated as

r tan b R tan b0 u2

5

gh the quantity r is not uniform along the belt [and therefore the slope angle u diers from zero on thect arc (see Fig. 2)], it is always possible to consider juj 1 on most part of the contact arc, and to as-the radius of curvature q R everywhere but at the edges of the contact arc. With these assumptions,eglecting second order terms, the continuity equation can be written as

Radial displacement of the belt, both on the driven and driving side. Dashed line: the eect of pulley bending. Continuous line: thef belt transversal deformation only. Adapted from Refs. [11,12].

where

and t

whereequat

Ththe inbe wr

withof the

whereis pos

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 413vr oh 0 6

v is the radial sliding velocity of the belt and v is its tangential sliding speed. Moreover, Eq. (2) yieldsovhFig. 4. The forces acting on the belt.r h

tanw vhvr

7

aking the time-derivative of Eq. (5) leads to

vr dRdt

aDxR sinh hc 8

a = (1 + cos2b0)/sin(2b0). Besides the above written equations, we also need to write the equilibriumions, where the forces acting on the belt are shown in Fig. 4.e equilibrium of the belt involves the tension F of the belt, the linear pressure p acting on the belt sides,ertia force of the belt element and the friction forces. Neglecting second order terms, two equations canitten, which describe the equilibrium of the belt along the tangential and radial directions [11]

1

F rx2R2oF rx2R2

oh l cos bs sinwsin b0 l cos bs cosw

9

p F rx2R2

2Rsin b0 l cos bs cosw10

r being the mass per unit length of the belt. The last equation of the model allows to calculate the centerwedge expansion hc as

tan hc R a0ph sin hdhR a

0 ph cos hdh11

a is the extension of the wrap angle. Once the pressure and tension distribution have been calculated, itsible to easily calculate the axial clamping force and torque on the pulley respectively as

wher_

wherEq. (

whichthe marc, athe radistrithe m

Hoaccou

wherauthoclampto de

414 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428vh vr dh DaxR cosh hc k 14

2 2 2 2steady-state, i.e. when R 0, integrating Eq. (6) givesZ

e F1 and F2 are the tensile belt forces at the edge of the beltpulley contact (see Fig. 5). Observe that inand

T F 1 F 2R 13S 0

cos b l sin bspRdh 12

Z aFig. 5. A schematic view of the V-belt variator, with the tensile forces F1 and F2 acting on the branches of the belt.e both hc and k only depend (see Ref. [1]) on the tensile force ratio (F1 rx R )/(F2 rx R ). Therefore7) yields

tanw cosh hc ksinh hc 15

shows the sliding angle distribution not depending on D during steady running. Thus, in steady-state,agnitude D of the pulley deformation cannot aect the tension and pressure distributions on the contacts these quantities only depend on w. Therefore, we have to conclude that in stationary conditions, besidestio (F1 rx2R2)/(F2 rx2R2), only the shape of the deformed pulley aects the tension and pressurebutions, and, hence, the clamping forces, whereas the actual magnitude of deformations only inuencesechanical eciency of the variator.wever, the main focus of this work is on CVT shifting dynamics, which may be simply taken intont by means of the following dimensionless parameter

A 1D

_RDRxDRRDR

sin2b01 cos2 b0

16

e DR stand for drive pulley (the driven pulley will be referred to with the subscript DN). In Ref. [1] thers have shown that during creep-mode phases A is almost a linear function of the logarithm of theing force ratio SDR/SDN, and they have been able to propose the following relatively simple equationscribe the variator shifting behavior

the fo

l pq 2q arcsin q 2 1 q 20

time-

(16) a

angul

4. Di

Thfunda

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 415_s f RDR;RDN; SDR; SDN; TDR; TDN; v; L; d 24e Buckinghams pi-theorem [1315] allows us to simplify the above written relation Eq. (24) by using, asIn this section we will show, by using dimensional analysis, that the symmetry of the system under no-loadconditions (we neglect the torque losses in the variator and the slip between the belt and the pulleys) leads to asimpler expression for the quantity g(s). First, consider that the rate of change of the speed ratio _s can beexpressed as a function of the clamping forces SDR and SDN, the torques TDR and TDN, the belt velocity v,the belt length L, the pulleys center-to-center distance d, and the pitch radii of the belt RDR and RDN. There-fore, we can writear velocity xDR, to the parameter D, and that it depends linearly on ln(SDR/SDN).

mensional analysiswhere g(s) = [1 + sh(s)]sc(s). Eq. (23) shows that the shifting speed _s is proportional to the primary pulley_s xDRD 1 cos2 b0

sin2b0gs ln SDR

SDN

ln SDR

SDN

eq

23D xDR 1 cos b0 s1 shsand using Eq. (17) we nally have" #s

A 1 1 sin2b02

_s 22_RDR

p 2 arcsinq1 s=s hs 21

which shows that also _RDN= _RDR hs only depends on s. Thus, using Eqs. (18) and (21), we can rewrite Eq.derivative of Eq. (19), we get

_RDN p 2 arcsinq1 s=ss s s s

Eq. (20) shows that q is a function of s only. Now, neglecting the belt longitudinal deformation, and taking thellowing dimensionless quantities have been dened: q(s) = RDR/d and l = L/d

1 s 1 s 1 s 1 s 2s

where d is the center-to-center distance of the pulleys. Eq. (19) can be rewritten in a dimensionless form, onceL pRDN RDR 2RDN RDR arcsin RDN RDRd

2d2 RDN RDR2

q19where (SDR/SDN)eq is the clamping force ratio at equilibrium, i.e. in steady-state conditions.

3. Geometric relations

Eq. (17) can be rephrased in terms of the geometric speed ratio s. Thus, taking the time-derivative ofs = RDR/RDN gives:

_s s_RDRRDR

1_RDN_RDR

s

18

In Eq. (18) we need to express the quantity _RDN= _RDR as a function of s. Observe that the length of the belt L isA cs lnSDN

lnSDN eq

17SDR

SDR " #mental units, the quantities RDN, SDN and v. Thus Eq. (24) takes the form

that,stantmetri

righttual s

4.1. C

416 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428lnSDRSDN

Hln s 33eq

U ln s; ln SDRSDN

eq

" # 0 32

Eqs. (31) and (32) show that under no-load conditions ln(SDR/SDN)eq is an odd function of lns, i.e.U ln s; lnSDRSDN

" # 0 31and since di/dt = s2ds/dt, substituting this result in Eq. (28) gives

_s vdsU ln s; ln SDR

SDN

29

Thus a comparison between Eqs. (27) and (29) shows that the symmetry of the system under no load condi-tions requires that

U ln s; ln SDRSDN

U ln s; ln SDR

SDN

30

The symmetric condition Eq. (30) allows to nd an approximate relation for U, at least under no-load condi-tions. First consider that in steady-state conditions, i.e. when _s 0, Eqs. (27), (29) yielddidt v

diU ln i; ln

SDNSDR

28where v = xDRRDR, and q(s) = RDR/d. Now consider the symmetric condition s! 1/s = i and SDR/SDN ! SDN/SDR, where i = xDR/xDN is the reduction ratio. In this case the symmetry of the system requiresthe reduction ratio i to fulll the same relation Eq. (27), i.e._s vdsU ln s; ln

SDRSDN

xDRsqsU ln s; ln SDRSDN

27Now, suppose that the CVT is running under no-load conditions, i.e. TDN = 0. In this case there is no wayto distinguish between the drive and the driven pulley, i.e. the system is physically symmetric. Eq. (26) becomesVT symmetry under no-load conditionsThus, the dimensional analysis allows us to simplify the design of the experimental activity, since only threequantities need to be varied in order to map the whole dynamical response of the variator.hand side of Eq. (26). Eq. (26) states that the dynamical response of the system depends only on the ac-peed ratio, on the clamping force ratio and on the dimensionless torque coecient TDN/(RDNSDN).Because of symmetry and considering that s = RDR/RDN > 0, we have used lns instead of s as the argument ofthe unknown function U. Moreover, for convenience, we have also introduced a multiplying factor s at theonce the geometry of the system has been xed and in particular the quantity l = L/d is taken to be con-, the implicit relation Eq. (20) allows to write d/RDN = s/q(s) which, of course, is a function of the geo-c speed ratio s only. Therefore, Eq. (25) can be rephrased without any loss of generality as

_s vdsU ln s; ln

SDRSDN

;TDN

RDNSDN

26Now, observe that s = RDR/RDN = TDR/TDN, and that TDR/(RDNSDN) = sTDN/(RDNSDN). Also notice_s vdG

RDRRDN

;SDRSDN

;TDR

RDNSDN;

TDNRDNSDN

;L

RDN;

dRDN

25eq

with

Using

good

0 DN DN

S

wherecondi

Eqdition

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 417SDN eq RDNSDNwherelnSDR Hln s K TDN 44(41) should still hold true. The CMM model shows indeed that under load conditions the eect of torque loadmay be included in the model simply by modifying the quantity ln(SDR/SDN)eq as . (41) has been obtained in the case of zero torque load. However, we may expect that under load con-s the basic dependence from the speed ratio and the clamping forces will not change signicantly, i.e. Eq.Eqs. (42) and (43) show that the shifting response of the variator is determined only by the quantities a01 and a02

and m, that can be easily calculated by means of the CMM model.

4.2. Load conditionsai aiD1 cos b0= sin2b0 with i = 1,2. Comparing Eq. (41) and Eq. (23) gives under no loadtions

gs sqsa01 a02ln s2 43DN eq

0 2lnSDR Hln s mf 42eq

with

_s xDRD 1 cos b0

sin2b sqsa01 a02ln s2 ln

SDRS

ln SDRS

41As a consequence, Eq. (27), in case of zero torque load, nally becomes

2 " #veq Hf mf Of 40

Now, notice that Eq. (34) gives

3choice. Hence, Eq. (38) becomes

Uf; v a1 a2f2v veq 39In Section 6 both theory and experiments show that a rst order approximation in (v veq) is already a veryEqs. (36) and (37), Eq. (35) becomes

Uf; v a1 a2f2v veq b1fv veq2 38Uvvf;Hf 2b1f Of3 37

Uvf;Hf a1 a2f2 Of4 36For convenience, let us dene the quantities f = lns, and v = ln(SDR/SDN). We can expand the function Uabout the steady-state point v = veq as

Uf; v Uvf;Hfv veq 1

2Uvvf;Hfv veq2 Of3 35

where U(f,veq) = 0. Recalling Eq. (33), we can write veq = H(f). The symmetry condition Eq. (30) impliesUv(f,v) = Uv(f,v) and Uvv(f,v) = Uvv(f,v), thus in steady-state we have Uv(f,H(f)) = Uv(f,H(f))and Uvv(f,H(f)) = Uvv(f,H(f)). On the basis of the these considerations, the MacLaurin series ofUv(f,H(f)) about f = 0 (i.e. s = 1) must contain only even terms, whereas the MacLaurin expansion ofUvv(f,H(f)) contains odd terms only, i.e.H ln s Hln s 34the K function can be calculated by the theory.

It is very important to notice that Eq. (41) has been obtained by means of only dimensional analysis andsymmetry considerations. Therefore, we may expect the formula (41) to be of general validity, i.e. to hold true,with dierent values of a01 and a

02 and m, also in the case of dry-hybrid belts and rubber belts, even though, in

some cases, additional terms of the Taylor expansion might be needed.

5. Comparison with other models

In this section the CMM model predictions will be compared with those provided by Tenberge [16], whoconsidered the case of a chain belt CVT and used a FEM approach to calculate the Greens function, i.e.the elastic response of the pulley.

The comparison has focused on both the sliding velocity eld and the friction forces at the pulleybelt inter-face, and on the axial clamping forces. In this case the CVT is a metal chain variator with the following prop-erties d = 155 mm, L = 649 mm, and r = 1.2 kg/m. As an example, in steady-state conditions (i.e. _s 0) withs = 2.0, xDR = 2000 rpm, RDR = 70.3 mm, RDN = 35.1 mm, Fmin = 2670 N, Fmax = 6228 N, b0 = 10, andl = 0.09, we get (SDR)CMM = 46.8 kN and (SDN)CMM = 25.5 kN, whereas Tenberges model gives(SDR)T = 46.6 kN and (SDN)T = 27.0 kN. The agreement is very good, with a dierence of less than 6% onthe driven pulley. However, observe that this dierence may be due to some uncertainties in the value of land b0. The velocity eld and the friction forces at the beltpulley interface have been also calculated, and,as shown in Fig. 6, the agreement between the two models is still very good.

A further comparison has been carried out for a dierent running condition, with _s 0, s = 0.5,xDR = 2000 rpm, RDR = 35.1 mm, RDN = 70.2 mm, Fmin = 709 N, Fmax = 723 N, b0 = 10 and l = 0.09.The calculated clamping forces are (SDR)CMM = 5.2 kN and (SDN)CMM = 6.0 kN, whereas the Tenbergesresults are (SDR)T = 5.4 kN and (SDN)T = 6.0 kN, showing again a very good agreement with the CMM

418 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428model. The corresponding velocity eld and the friction forces are also drawn in Fig. 7, which conrms theagreement between the two theories. Therefore, we may conclude that the simpler continuum one-dimensionalmodel of the belt, proposed in Ref. [1], gives very good results, despite the discrete number of contact pointsbetween the chain and the pulley (due to the presence of chain pins). Furthermore, the CMM model solves avery small number of equations and does not need to deal with the very large number of degrees of freedom of

Fig. 6. A comparison between the Tenberges model (adapted from Ref. [16]) and the CMMmodel for steady-state conditions. (a) Slidingvelocity eld and (b) friction forces. The following data have been used: s = 2.0, xDR = 2000 rpm, RDR = 70.3 mm, RDN = 35.1 mm,

Fmin = 2670 N, Fmax = 6228 N, b0 = 10, and l = 0.09.

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 419the system. For this reason, it runs very fast on a PC, mostly in steady-state, when the magnitude D of thepulley bending does not aect the pressure and tension distributions along the contact arc.

6. Experimental validation of the CMM model

Fig. 7. A comparison between the Tenberges model (adapted from Ref. [16]) and the CMMmodel for steady-state conditions. (a) Slidingvelocity eld and (b) friction forces. The following data have been used: s = 0.5, xDR = 2000 rpm, RDR = 35.1 mm, RDN = 70.2 mm,Fmin = 709 N, Fmax = 723 N, b0 = 10 and l = 0.09.In order to validate the CMM model, a detailed experimental investigation has been carried out. Tests havebeen undertaken on a pushing-belt CVT by van Doorne Transmissie, mounted on the power-loop test rigavailable at the automotive Engineering Science Laboratory Eindhoven University of Technology, as shownin Fig. 8. Steady-state experiments under no-load and load conditions have been carried out, whereas shiftingexperiments have been carried out only at zero torque load, because the control of the test rig does notyet allow safe shifting experiments under load conditions. In both kinds of experiments, the secondary clamp-ing force SDN and the primary angular velocity xDR have been xed. The geometrical quantities of the push-ing-belt CVT utilized for the experimental activity, are: belt length L = 703 mm, center-to-center distance ofthe pulleys d = 168 mm, b0 = 11. Moreover the friction coecient has been estimated equal to l = 0.09.

6.1. Power-loop test rig layout

In Fig. 9 the layout of the power-loop test rig is shown. It consists of a drive motor and two variators cou-pled in parallel. The drive motor shaft is the primary side, the other one is the secondary side. Variator A ismounted between the drive motor and variator B. Subscripts (1, 2, a, b) indicate the integrated manifold forthe hydraulic system. The couplings connecting the beltboxes can be released without changing the position ofthe beltboxes. This enables quick (dis)-assembly of the test rig, without the need to realign the complete setup.The bearings and belt are lubricated by a separate hydraulic circuit, which is fed by the lubrication pump(La,Lb). These circuits also feed the pressure pumps (Pa,Pb), which are used to control the pressure(p2a,p2b) in the secondary pulley cylinders of the variators. The primary pulley cylinders are pressurized bythe ratio pumps (Ra,Rb), which control the ow between the primary and secondary pulley cylinders. Bidirec-tional external gear pumps are used, with a displacement of 1.0 [cc/rev]. PWM controlled brushless 42 [V] DCservomotors are used to drive the pressure and ratio pumps. The hydraulic feed of the pulley cylinders is real-ized by an axial connection, which uses a sealed close clearance bushing to prevent excessive leakage. For the

420 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428shaft connected to the motor, the axial connection is not available and therefore a radial oil feed has beendesigned. It consists of a chamber, sealed with two rings in a groove on the shaft. For control and measuringpurposes the test rig is equipped with sensors for pressure (p1a,p2a,p1b,p2b), rotational speed (x1,x2), move-able pulley sheave position (x1b,x2a) and torque (T1,T2).

6.2. Steady-state measurements

In steady-state conditions, the clamping force SDR, acting on the primary pulley, has been measured as afunction of the geometrical speed ratio s = RDR/RDN, for a xed value of the driven pulley clamping forceSDR. The speed ratio s has been measured by using axial position sensors, which allow the calculation ofthe running radius RDR of the belt.

Fig. 8. The power-loop test rig at the automotive Engineering Science Laboratory Eindhoven University of Technology.

Fig. 9. Power-loop test rig layout. Pressure circuit in solid lines, lubrication circuit dashed.

6.2.1.

Fig. 10. The logarithm of the clamping force ratio ln(SDR/SDN)eq as a function of the logarithm of the geometrical speed ratio ln(s).Circles represent the experimental data, the thick line represents the theoretical prediction, whereas the thin line is the cubic t of theexperimental data. The friction coecient is l = 0.09, and the pulley groove angle is b = 11.

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 421Fig. 10 shows the logarithm of the clamping forces ratio, ln(SDR/SDN)eq, as a function of lns, in steady-state conditions. Circles represent the measurements, the thin line is the t of the experimental data, whilethe thick one represents the theoretical prediction of the CMM model. Data have been measured for dierentprimary angular velocities (xDR = 1000, 2000, 3000 rpm) and two dierent values of the secondary clampingforce (SDN = 20, 30 kN). The agreement with the theoretical calculation is very good. Experiments haveshown that, as predicted by the CMM model, neither the magnitude of the secondary clamping force SDN,nor the angular velocity of the primary pulley xDR have a signicant inuence on the ratio (SDR/SDN)eq insteady-state. This also conrms that the parameter D does not inuence the steady-state CVT behavior, sinceotherwise we should have noticed a strict dependence of (SDR/SDN)eq on SDN. All the experimental data,instead, follow a master curve, see continuous thin line in Fig. 10, which is very close to the theoretical thickline. The CMM model allows also the calculation of the friction coecient inuence on the ln(SDR/SDN) vslns curve in steady-state conditions. Fig. 11 shows indeed that this inuence is very signicant. The theoret-ically calculated steady-state curves in Fig. 11 have been obtained for dierent values of l (l = 0, 0.06, 0.09,0.12, 0.15), and the diagram clearly shows that increasing the friction coecient strongly reduces the slope ofthe curves: observe that a friction coecient l = 0.15 can already reduce the slope almost to zero. Of course, inorder to avoid the belt self-locking, the friction coecient must not exceed the limiting value llim = tanb0,which, in our case, is llim = tan11 0.19. Also observe that at zero friction the steady-state clamping forcesratio (SDR/SDN)l=0 can be easily obtained by means of energy considerations only. In fact, under no load con-ditions and with no-friction at the pulleybelt interface, the principle of virtual works requires that

SDRdxDR SDNdxDN 0 45Fig. 11ratio lnThe puNo-load tests. The logarithm of the clamping force ratio ln(SDR/SDN)eq at steady-state as a function of the logarithm of the geometrical speed(s). Curves have been calculated using the CMM model for dierent values of the friction coecient l = 0, 0.06, 0.09, 0.12, 0.15.lley groove angle is b = 11.

where dxDR and dxDN are the virtual axial displacements of the primary and secondary pulleys. Observe thatdxDR = 2dRDRtanb0 and dxDN = 2dRDNtanb0 where dRDR and dRDN are the virtual displacements of therunning radius of the belt. Thus, Eq. (45) gives

SDRSDN

l0

dRDNdRDR

46

and using Eq. (21) we get

dRDNdRDR

hs p 2 arcsinRDN RDR=dp 2 arcsinRDN RDR=d

aDRaDN

47

where aDR and aDN are the wrapped angles on the driver and driven pulley respectively. Therefore, Eq. (46)leads to the very simple relation

SDRSDN

l0

a1a2

48

that, of course, satises the symmetry condition given by Eq. (33).

422 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428Fig. 12. The logarithm of the clamping force ratio ln(SDR/SDN)eq as a function of the TDN/(RDNSDN) ratio, dierent values of thegeometrical speed ratio (s = 0.50, 0.66, 0.80, 1.00, 1.25, 1.50 and 2.00) and torque load (TDN = 20, 40, 80, 100 N m). Circles represent theexperimental data, the thick line represents the theoretical prediction, whereas the thin line is the t of the experimental data. The frictioncoecient is l = 0.09, and the pulley groove angle is b = 11. The secondary clamping force is SDN = 30 kN and the primary pulley

rotatin6.2.2. Load tests

Steady-state experiments have been also performed under load conditions. Fig. 12 shows the logarithm ofthe clamping forces ratio as a function of the dimensionless torque load TDN/(RDNSDN) for SDN = 30 kN,x = 1000 rpm and dierent values of the speed ratio (s = 0.50, 0.66, 0.80, 1.00, 1.25, 1.50, 2.00) and torqueload (TDN = 20, 40, 80, 100 N m).

Fig. 12 shows a very good agreement between theory and experiments for all the tested speed ratios, thusconrming the validity of the CMM model. It is worth to notice that the experimental curves show slightlydierent slopes, if compared to the theoretical ones. This dierence may be due to some uncertainties inthe estimation of the friction coecient that has been used in the theoretical calculations.

However, the experiments have shown that changing the secondary clamping force SDN and the rotatingvelocity xDR causes actually a small modication of the ln (SDR/SDN)eq vs TDN/(RDNSDN) curve in steady-state. This behavior is predicted neither by the CMM model nor by the dimensional analysis, and a possibleexplanation may be the following one. First of all, as already mentioned before, it is important to remark thatthe theoretical model does not consider the inuence of bandssegments interaction, that could not always benegligible, especially in case of too low values of the clamping forces SDR and SDN. The second aspect is thatthe inuence of the lubrication conditions at the pulleybelt interface has not been considered as a relevantparameter in the theoretical investigation (we simply used a constant friction coecient l = 0.09). Actually,g speed is x = 1000 rpm.

Fig. 13. The rate of change of speed ratio as a function of ln(SDR/SDN) for xDR = 1000 rpm, for dierent values of s and for two values ofthe secondary clamping force SDN = 20, 30 kN. The friction coecient is l = 0.09, and the pulley groove angle is b = 11. Thick lines arethe theoretical calculations, thin lines connect the experimental data.

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 423

A very good way to represent the experimental results is to plot the quantity ln(S /S ) as a function ofthe shexper

3

424 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428where SDN is expressed in kN. If SDN = 0, the only quantity that aects the value of D is the clearance betweenthe pulley and its shaft. Placing SDN = 0 in Eq. (49), we get D = 0.6 103.

Fig. 13 shows that D is not, or at least is only slightly aected by the actual value of the speed ratio s.Observe also that the small dierence between the theory and the experiments, sometimes observed inFig. 13, is mainly due to a dierent value of the steady-state clamping force ratio (SDR/SDN)eq, rather thanto a dierent slope of the curves.

However, results obtained at s = 1 require some considerations. In this case, a step-type variation of theexperimental curves is shown as they intersect the origin of the diagram. In order to understand this unex-pected behavior, rst observe that, when the speed ratio s is equal to 1, the system is again in a situationof complete symmetry between the _s < 0 case and the _s > 0 one. Symmetry requires that

SDRSDN

_s

SDNSDR

_s

50

which, in terms of ln(SDR/SDN), means that the experimental curves must be antisymmetric with respect to theorigin of the axes. Nonetheless, this is shown not to happen. The deviation from symmetry may again becaused by a not strictly uniform value of the friction coecient along the radial direction, which breaks thesymmetry. But more likely this deviation may be caused by the bandsegments interaction, which has not beentaken into account in the CMM theory. In all the other cases, the dierence between theory and experiments isnegligible.

Fig. 14 shows the eect of the primary angular velocity on the shifting behavior of the system. Two casesare shown, one for s = 1 and the other one for s = 1.2. In both cases, SDN = 20 kN, whereas the angular veloc-ity is respectively xDR = 1000, 2000 rpm. A very good agreement with the results predicted by the CMMmodel is again clearly shown. This conrms that a direct proportionality between the shifting speed _s andthe primary pulley angular velocity xDR actually holds true. Similar results have been also obtained in all

the oD 1 0:02SDN 20 10 490.8, 1.0, 1.2, 1.4, 1.6, 1.8. Observe the very good agreement with the theoretical calculations (thick lines). Inparticular, for xed values of s and SDN, all the measured data fall on a straight line. This proves the lineardependence of _s on ln(SDR/SDN), which was one of the most signicant results of the CMM model. Observealso that the slope of the curves depends, at least slightly, on the secondary clamping force. This can be inter-preted as due to a change of the magnitude of the pulley deformation and in particular of the dimensionlessparameter D. Indeed, it is expected that increasing the clamping force makes the magnitude of the pulley defor-mation, i.e. D, grow. Thus, dierent values of D have been used for dierent values of the secondary clampingforce SDN; in particular, D = 0.0012 has been used for SDN = 30 kN, and D = 0.001 for SDN = 20 kN. Fur-thermore, because of the linear elastic response of the system, we also expect a linear relation between Dand SDN to hold true, that isDR DN

ifting speed _s for each value of s, SDN and xDR. In Fig. 13 the theoretical results are compared with theimental ones, for SDN = 20, 30 kN, xDR = 1000 rpm and for dierent values of the speed ratio s = 0.6,lubrication conditions, as for instance the oil lm thickness, may depend signicantly on the pulley clampingforce and on the rotating speed. Thus, changing the clamping forces and/or the angular velocity of the beltmay modify the friction at the beltpulley interface, thus leading to dierent behaviors of the system. Theauthors will report on these aspects of the problem in a next publication.

6.3. Shifting measurements

Shifting tests have been carried out only under no load conditions, since the test bench control system didnot allow to perform load shifting tests under safe conditions. The experiments have been carried out by xingthe shifting speed _s, the secondary clamping force SDN and the primary angular velocity xDR. The primaryclamping force SDR and the speed ratio s were measured as a function of time t.ther cases, i.e. for dierent values of s and of the secondary clamping force SDN.

(a)

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 425(b)Fig. 14. The rate of change of speed ratio as a function of ln(SDR/SDN). The primary pulley speed is x1 = 1000, 2000 rpm, the speed ratiois s = 1.0, 1.2 and the secondary clamping force is S2 = 20 kN. The friction coecient is l = 0.09 and the pulley groove angle is b = 11.Filn(SDstraigappro

Howeeasilystate

Fig. 15x1 = 1b = 11

Thickg. 15 shows the rate of change of the speed ratio _s as a function of the force ratio SDR/SDN, instead ofR/SDN). The gure clearly shows that in the linearlinear diagram the curve deviates signicantly from aht line, especially for small values of s, thus showing again that the logarithmic relation is much morepriate than the Ides formula [6,7]

_sIde 1SDR=SDNeqSDRSDN

SDR

SDN

eq

" #51

ver, it is worth to observe that at higher speed ratios the deviation becomes less signicant. This might beexplained considering that the Taylor expansion of ln(SDR/SDN) ln(SDR/SDN)eq about the steady-point (SDR/SDN)eq is

. The rate of change of speed ratio as a function of (SDR/SDN), for dierent values of the speed ratio s. The primary pulley speed is000 rpm and the secondary clamping force is SDN = 20 kN. The friction coecient is l = 0.09 and the pulley groove angle is. The curve shows a signicant deviation from a straight line, especially when s < 1.

lines are the theoretical calculations, thin lines connect the experimental data.

52

(SDR/SDN)eq is decreased below 1, that is to say when the speed ratio of the system is s < 1 (see Fig. 10).

creep-mode evolution of the variator. Very few parameters appear in the formula, which may be calculated

G.mont

Appe

Inmode

426 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428ndix A. Groove angle and axial displacement of the pulley sheaves

this appendix we provide a brief clarication about Eqs. (3)(5) which constitute the basis of the Sattlersmost of this research project has been performed. G. Carbone also thanks Ir. J. van Rooij and Ir. G. Com-missaris by Gear Chain Industrial B.V. (Nuenen NL) for their support during the experimental activity.Carbone would like to thank prof. M. Steinbuch and Dr. P.A. Veenhuizen for the support during threehs visit at the Department of Mechanical Engineering Eindhoven University of Technology, whereeither experimentally or theoretically. This equation is of utmost importance to design advanced CVT controlsystems, which aim at improving the CVT eciency, cars drivability and fuel economy.

AcknowledgementsIncreasing s makes the term (SDR/SDN)eq increase, and when s > 1, being (SDR/SDN)eq > 1, the correction be-comes less important.

7. Conclusions

In this work a detailed experimental investigation concerning the V-belt CVT dynamics has been carriedout, in order to compare the theoretical predictions of the so-called CMM theoretical model by Carboneet al. [1] with the experimental results. A very good agreement between theory and experiments has beenfound, both in steady-state and during shifting maneuvers. This conrms all the most important predictionsof the model. In particular, it has been shown that during relatively slow shifting maneuvers (creep-mode) therate of change of the speed ratio _s is a linear function of the logarithm of the clamping force ratio SDR/SDN.The authors have also shown, by means of dimensional analysis and using the physical symmetry of the CVTunder no-load conditions, that the linear relation between _s and ln(SDR/SDN) is a relatively robust property ofV-belt CVTs, not depending on whether the belt is a chain belt or a pushing belt. The linear relation between _sand ln(SDR/SDN) has also been compared with Ides formula, which is, instead, a linear relation between _s andSDR/SDN. The experiments have shown that Ides relation may well approximate the real CVT shifting behav-ior only for speed ratio values greater than 1, whereas in all other cases the approximation is less good. Exper-iments have also conrmed that, as predicted by the CMM model, the shifting speed is also proportional tothe angular velocity of the primary pulley, and that it increases as the magnitude of pulley deformation isincreased, i.e. as the clamping forces on the pulleys are increased. The CMM predictions have been also com-pared with those by Tenberge [16] for the chain belt. Also in this case, the agreement is really very good, show-ing that the continuum belt approximation, which is the basis of the CMM model, works very well, notdepending on the typology of the considered belt, i.e. both for the pushing and chain belts. On the basis ofthese very good results, the authors also propose a relatively simple dierential equation to describe theTherefore the dierence between the Ides relation Eq. (51) and the CMM Eq. (23) can be rewritten as

_sIde _s 1SDR=SDN2eqSDRSDN

SDR

SDN

eq

" #2 53

Eq. (53) shows that the dierence between the Ides linear relation and the CMM one rapidly increases aslnSDRSDN

ln SDR

SDN

eq

1SDR=SDNeqSDRSDN

SDR

SDN

eq

" #12

1

SDR=SDN2eqSDRSDN

SDR

SDN

eq

" #2 l [10]. Fig. 16 shows the pulley bending being a consequence of two contributions. The former is related

to the

of theReferpulley

G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428 427tions of the belt the following equations hold true

b 2R tan b0 db 2r tan b d u A1

where b is the constant transversal width of the belt. Eq. (A1) yields

r tan b R tan b0 u A2whichsheavand (

Refer

[1] GM

[2] C.coN

[3] C.ecSe

[4] GEn

[5] GTr

[6] T.96

[7] T.repulley sheaves, determined by the pressure distribution at the interface between the belt and the pulleys.ring to Fig. 16 (where b0 is the grooves angle of the undeformed pulley, and R is the distance from theaxis that the belt would have if the pulley sheaves were rigid), and neglecting the transversal deforma-modication b0 ! b of the half-opening angle of the groove, the latter is related to the elastic displacement u/2

pulley tilting which may be caused by clearance between the moving pulley and the shaft and produce aFig. 16. Varying groove angle and pulley bending. r is the local radial coordinate of the belt, b is the actual pulley half-opening angle, b0 isthe groove angle of the undeformed pulley, and R is the pitch radius of the belt.2

is the same as Eq. (5). By using FEM calculations, Sattler has shown in Ref. [10] that in case of highe stiness the varying groove angle b and the varying axial groove width u can be described by Eqs. (3)4).

ences

. Carbone, L. Mangialardi, G. Mantriota, The inuence of pulley deformations on the shifting mechanisms of MVB-CVT, ASME J.ech. Des. 127 (2005) 103113.Brace, M. Deacon, N.D. Vaughan, R.W. Horrocks, C.R. Burrows, The compromise in reducing exhaust emissions and fuelnsumption from a diesel CVT powertrain over typical usage cycles, in: Proceedings of the CVT99 Congress, Eindhoven, Theetherlands, 1999, pp. 2733.Brace, M. Deacon, N.D. Vaughan, C.R. Burrows, R.W. Horrocks, Integrated passenger cat diesel CVT powertrain control foronomy and low emissions, in: ImechE International Seminar S540, Advanced Vehicle Transmission and Powertrain Management,ptember 2526, 1997.. Carbone, L. Mangialardi, G. Mantriota, Fuel consumption of a mid class vehicle with innitely variable transmission, SAE J.gines 110 (3) (2002) 24742483.. Carbone, L. Mangialardi, G. Mantriota, L. Soria, performance of a city bus equipped with a toroidal traction drive, IASMEans. 1 (1) (2004) 1623.Ide, H. Uchiyama, R. Kataoka, Experimental investigation on shift speed characteristics of a metal V-belt CVT, JSAE paper36330, 1996.Ide, A. Udagawa, R. Kataoka, Simulation approach to the eect of the ratio changing speed of a metal V-belt CVT on the vehiclesponse, Veh. Syst. Dyn. 24 (1995) 377388.

[8] B. Bonsen, G. Carbone, S.W.H. Simons, M. Steinbuch, P.A. Veenhuizen, Shift dynamics modeling for optimizing slip control in acontinuously variable transmission, submitted to 31st FISITA World Automotive Congress in Yokohama from 22 to 27 October2006.

[9] B. Bonsen, G. Carbone, S.W.H. Simons, M. Steinbuch, P.A. Veenhuizen, Shift dynamics modeling and optimized CVT slip control, inpreparation.

[10] H. Sattler, Eciency of metal chain and V-belt CVT, in: Proceedings of CVT99 Congress, Eindhoven, The Netherlands, 1999, pp.99104.

[11] J. Srnik, F. Pfeier, Dynamics of CVT chain drives: mechanical model and verication, in: Proceedings of the 1997 ASME DesignEngineering Technical Conferences, DETC97/VIB-4127, 1997.

[12] J. Srnik, F. Pfeier, Dynamics of CVT chain drives, Int. J. Veh. Des. 22 (1999) 5472.[13] E. Buckingham, On physically similar systems: illustrations of the use of dimensional equations, Phys. Rev. 4 (1914) 345376.[14] E. Buckingham, The principle of similitude, Nature 96 (1915) 396397.[15] E. Buckingham, Model experiments and the form of empirical equations, Trans. ASME 37 (1915) 263.[16] P. Tenberge, Eciency 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 eciency, Paper no. 04CVT-35, 2004 International Continuously Variable andHybrid Transmission Congress, UC Davis, September 2325, 2004.

428 G. Carbone et al. / Mechanism and Machine Theory 42 (2007) 409428

CVT dynamics: Theory and experimentsIntroductionMechanical modelGeometric relationsDimensional analysisCVT symmetry under no-load conditionsLoad conditions

Comparison with other modelsExperimental validation of the CMM modelPower-loop test rig layoutSteady-state measurementsNo-load testsLoad tests

Shifting measurements

ConclusionsAcknowledgementsGroove angle and axial displacement of the pulley sheavesReferences