Modeling of Pulley Based CVT Systems: Extension of the CMM Model with Bands-Segment Interaction Dct Nr: 2007.024

Traineeship Polytecnico di Bari, Italy

Modeling of pulley based CVT systems: Extension of the CMM model with bands-segment interaction Dct nr: 2007.024

J.F.P.B. Diepstraten

Coach: Dr. Ing. G. Carbone Dr. P.A. Veenhuizen

January 2007 Contents

Contents...... 2 Introduction ...... 3 1. Introduction to CVT systems...... 5 1.1 Pulley Based Transmission...... 5 1.1.1 Metal Pushing V-belt...... 6 1.1.2 Metal V-chain...... 6 2. Existing Models...... 8 2.1 Assumptions ...... 8 2.2 Mechanical Model ...... 8 2.3 Pulley Deformation...... 9 2.4 Momentum Equation ...... 10 2.5 Comparison with other models...... 11 2.6 Simplified and Dimensionless equations...... 13 3. Band Segment Interaction ...... 15 3.1 Assumptions ...... 15 3.2 Geometrical model ...... 15 3.3 Continuity Equation...... 17 3.4 Forces equilibrium...... 17 3.5 Dimensionless equations ...... 19 3.6 Influence of clearance between the segments...... 19 3.7 Parameters ...... 21 3.8 Driving Pulley ...... 22 3.9 Driven Pulley...... 28 3.10 Verify distribution of the kinematical strain...... 35 4. Future Work ...... 40 Conclusions ...... 41 Literature list ...... 42

2 Introduction

Nowadays the car has become the most used transportation application in the world. From some estimation there are nowadays almost 1000 millions registered vehicles over our entire planet [Ref: 1], and this number is still rising. This enormous number of cars has great influence on our environment. The exhaust gases are filled with toxic gases and particles, like nitrogen oxides and sulphur oxides, and also not directly poisonous gases like carbon dioxide and vapour. The toxic gases contribute to smog problems in big cities, and other air pollutions, this leads to all kinds of health problems. Carbon dioxide has a significant contribution to the greenhouse effect. This extended greenhouse effect leads to global warming and climate change we are dealing with. Different solutions are thought of to solve this problem. In the late seventies vehicles became equipped with catalytic converters, these converters reduce the toxic gases in the vehicle exhaust. Also the internal combustion engines have been further developed to produce less exhaust gases. These applications contributed to a decrease of the harmful exhaust gases. In the last decade also other solutions have been thought of. Some of them exist of replacing the combustion engine by a fuel cell, or a combination of the combustion engine with a battery, called hybridization. Another solution is topology change of the drive train. For a few years a sixth gear has been added to the conventional gear box, this to reduce the rotating speed of the engine at high vehicle speed to reduce the exhaust gases. One more solution is the use of a continuously variable transmission, CVT. This transmission is able to provide infinite gear ratios between two constraint limits, without the use of any clutch to disengage the engine from the drive line. By this property the combustion engine can be driven in its optimal working point, the engine speed does no longer depend on the drive line, because this can be freely chosen due the presence of the CVT. This means the combustion engine always can be used in its working point at which it delivers the most power. A well chosen engine speed also leads to a minimum of exhaust gases. This last solution is an interesting one. A CVT can be easily used in place of classical transmission in a normal car. The drive train must be replaced, but the combustion engine and other energy storage devices can keep the “old” configuration. So the already used lay out of the car does not have to be fully changed, as instead is the case of fuel cells or hybrid engines. Further more it has been estimated that fuel reduction of 10% could be obtained using a CVT in comparison with a manual shitted gear box [Ref: 2]. This may significantly reduce the above mentioned environmental problems concerning fossil fuels. Also the driveability of a vehicle equipped with a CVT is very good. The driver does not have to change gear, and by doing this lose his attention on the road. Moreover the comfort rises, because the sudden (de)acceleration during shifting disappears. Torque disengagement will also disappear, because no clutch is used during normal driving. The CVT seems a good transmission to place in modern vehicles. But there are still some problems. A CVT is a complicated transmission and must be controlled by a hydraulic system managed by an electronic control unit (ECU). This control still must be further optimized and investigated, this is necessary to obtain the lowest fuel consumption and emission, and highest driveability and comfort. At this time the control and control strategy are not to be called optimal. The hydraulic clamping forces determine the transmission’s efficiency and maximum torque that is transmitted. The timing and size of these hydraulic forces must be further investigated to reach the optimal desired working point. Another problem is the question wetter the consumer is prepared to buy this new transmission, or not. Especially in Europe the manual shifted gear box has a big market share, about 80% of all new produced cars are equipped with manual gear boxes. In Japan the CVT has already a marking share of 20% and the American market is very promising for the CVT [Ref: 3]. But then the question still remains, will there be enough demand for CVT vehicles?

To be able to determine the behaviour of the CVT transmission different approaches exists, one mathematical models, multi-body models and FEM models. In this paper the mathematical model, especially the CMM model, is investigated. The CMM model is derived by Carbone, Mangialardi and Mantriota from the Polytecnico di Bari, Italy. This mathematical model has been compared with experiments done at the Technical University of Eindhoven, the Netherlands. This comparison showed some inequalities between model and experiments. This can be seen in figure 1. In this figure the geometric speed ratio τ (x-axis) is compared with the clamping force ratio SDR / S DN (y-axis). The fat line represents the CMM model, the thin lines the results of the experiments.

Figure 1: Comparison CMM model and experiments

In this paper a start is made to derive a solution that can cancel these inequalities. The CMM model will be extended with some equations, in such a way that a better approximations can be made with respect to the experiments. Because of time limits the exact new model is not derived, but sufficient equations will be derived to complete the new model.

In the first chapter of this paper a closer look will be taken to different kinds of CVT transmission like belt CVTs and chain CVTs. In the second chapter the CMM model concerning belt and chain CVTs is presented and after this in the third chapter the extended version of this model is derived and some results of this extended version are presented. The last chapter deals with some recommendations about continuation of the research in such a way that the new model can be derived.

4 1. Introduction to CVT systems

In the first chapter is explained why to use a Continuously Variable Transmission (CVT) as drive train in a vehicle. In this chapter a brief history will be given, an overview of the commercial applied CVTs and how they work and its advantages and disadvantages.

Leonardo da Vinci was said to be the first one who thought about a CVT system around 1490. But when the invention of the car was a fact at the end of the 19 th century, CVT became a true issue. The first CVT patent dates from 1886. The first commercial CVT equipped vehicle was produced in 1958 by DAF (van Doorne Automobiel Fabriek), in the Netherlands, this factory was set up by Hub van Doorne in 1928. He invented the Variomatic CVT, this is based on a double V-belt system. The transmission was placed on the rear wheels and the drive train did not contain a differential gear. The system and the produced car can be seen in figure 1.1

Figure 1.1: Variomatic; layout and produced car

This transmission first could only be used for low-powered cars, 600cc, after some improvements this raised to 1400cc. About 1,2 million vehicles were equipped with the Variomatic [Ref: 4]. After the launch of the Variomatic different types of CVTs are produced by different companies.

Two different commercial applied CVTs exist, the pulley based CVT and the toroidal CVT. Some other CVTs exists, but they are only used for research, prototype or at very low scale.

1.1 Pulley Based Transmission The pulley based transmission is the transmission in the above mentioned Variomatic created by DAF. It consists of two pulleys connected with a V-shaped belt, this kind of CVT is therefore also called the V-belt CVT. The pulley that is connected to the engine is called the driving or primary pulley, the other pulley is connected to the wheels and is called the driven or secondary pulley. By changing the axial position of the moveable sheave of each pulley the pitch radius of the belt is changed and in turn the transmission ratio is modified, this is shown in figure 1.2.

Figure 1.2: Concept of V-belt CVT’s

5 In the left figure the transmission is in its lowest gear and in the right figure in its highest gear. One sheave of each pulley is connected with a hydraulic circuit, these controlled sheaves are on the opposite side of the belt. With the hydraulic circuit the clamping force on each pulley can be varied, by modifying the clamping force the radius of each pulley can be changed, and so the transmission ratio. This kind of transmission can provide a speed ratio from 0.4 up to 2 or even higher [Ref: 5]. Two different types of V-belts are used in CVT’s; the metal pushing V-belt and the metal V-chain.

1.1.1 Metal Pushing V-belt The push belt is an enhanced version of the DAF Variomatic. It was invented again by Hub van Doorne, and it consists of two series of thin metal bands, which hold together a number of wedge- shaped steel blocks. This belt is manufactured by VDT, Van Doorne Transmission. When the blocks are compressed, by pushing, they act as one single rigid column, by this it is possible to transmit torque from one pulley to the other one. In figure 1.3 the construction of a metal pushing V-belt is shown.

Figure 1.3: Construction of metal pushing V-belt

A belt consists of two band-sets with 9 to 12 metal bands. These bands give to the belt its flexibility and provide the necessary tensile strength. The number of blocks or segments, depending on the length of the belt, is approximately 400. A segment can have different measures and sizes. A larger block, which means a larger contact area with the sheave, decreases the contact pressure at higher torque load. Size and measures of the segments are optimized for its application. The main failure mechanisms are related to over speeding, misalignment and insufficient lubrication [Ref: 6]. The metal push V-belt is used as drive train by companies like Fiat, Ford, Subaru and Nissan. It was also applied in the Formula 1, developed by Williams and VDT. It was a prototype, but the FIA, Fédération Internationale de l’Automobile, banned the CVT from single-seat racing.

1.1.2 Metal V-chain The metal V-chain CVT transmits the torque through a tension difference between two chain strands. Links and rocker joint pins are available to transfer the chain tension. In figure 1.4 the metal V-chain and the layout of this transmission are shown. The chain is a Luk type chain belt, and is applied in the Audi A6, 2.8L Multitronic.

Figure 1.4: Metal V-chain and layout

Instead is the push belt, the chain can only transmit torque by tension in the chain, not with pressure. The chain consists of a number of chain elements depending on the size of the chain variotor. The main failure mechanism is fatigue of the link plates, and the elements that contact the sheaves, pin or strut. The fatigue is mostly due to tension in the belt and a little less to articulation [Ref: 6].

7 2. Existing Models

In chapter 1 it was made clear that for obtaining the right variation of speed ratio, understanding the behavior of the metal belt CVT is crucial to predict the needed clamping force. In this chapter the CMM model of the metal belt CVT is presented. Experiments made clear that two different shifting behaviors of the metal belt CVT exist. The first mode is the creep mode, this mode is characterized by the fact that the ratio of the clamping force acting on the primary pulley to that acting on the secondary pulley is mainly influenced by the rate of change of the speed ratio and by the tangential velocity of the belt. The slip mode takes place during fast shifting maneuvers and is characterized by the fact that the above mentioned ratio of the clamping forces is neither influenced by the magnitude of the rate of change of the speed ratio nor by the tangential velocity of the belt. These characteristics can be shown by using the CMM model. This model [Ref: 7], made up by Carbone, Mangialardi and Mantriota, uses kinematical and geometrical relations and also takes into account the pulley deformation. The model will be compared with another model, a Multi-Body model.

2.1 Assumptions The model that will be presented is derived by making some assumptions and simplifications. The metal belt is considered as a continuous body, with locally rigid motion. This means there is no longitudinal and transversal deformation, i.e. the belt is considered to be an inextensible strip with zero radial thickness and infinite axial stiffness. Furthermore the bending stiffness of the belt is neglected. The considered Coulomb friction, with the friction coefficient , acting between the segments and the pulleys, has a constant value. The deformation of the pulley will be described on the basis of Sattler’s model. In the derivation of the model second order terms will be neglected.

2.2 Mechanical Model In figure 2.1 the entire CVT variator is shown, the driving (primary) and driven (secondary) pulley are presented. In figure 2.2 the kinematical and geometrical quantities involved are shown.

Fig 2.1: CVT scheme Fig 2.2: Kinematical en geometrical quantities involved. (a) planar view, (b) 3D view Explanation of the parameters in these figures: ψ: sliding angle γ: complementary angle of ψ; − γ = π 2/ −ψ θ: angular coordinate

8 r: radial coordinate ρ: radius curvature φ: slope angle τ: tangent unit vector n: corresponding normal unit vector of τ er, e θ: radial and circumferential unit vector & vs: sliding velocity and its components r and rωs β: pulley half-opening angle βs: pulley half-opening angle in the sliding plane From figure 2.2 the following geometrical equations can be derived. 1 ∂r tan ()ϕ = [2.1] r ∂θ r δ l = δθ [2.2] cos ()ϕ 1 cos (ϕ) ∂ϕ  = 1−  [2.3] ρ r  ∂θ 

tan (β s ) = tan (β )cos (ψ ) [2.4] & r ⋅ωs = r tan ()ψ [2.5] With:

ωs: local sliding angular velocity of the belt; ωs = Ω −ω : local angular velocity of the belt ω: pulley rotating velocity δl: length of a material element of the belt δθ : angular extension of the same material element

After taking the material time derivative of equation [2.2], this becomes: 1 D r& 1 D ∂l = +ϕ& tan ()ϕ + ∂θ [2.6] ∂l Dt r ∂θ Dt Neglecting elongation of the belt, variation of the pulley rotating velocity in tangential direction and the slope angle ϕ , the continuity equation of the belt can be written into: r& ∂ω + s = 0 [2.7] r ∂θ

2.3 Pulley Deformation To calculate the actually path of the belt (the radial position of the belt), transversal deformation and pulley deformation must be known. Experiments made clear that belt transversal deformation does not contribute to radial differentiation in the position of the belt. On the contrary, pulley deformation can vary the radial position of the belt between 0.1 to 1 mm. So belt transversal deformation is neglected and only pulley deformation is taken into account [Ref: 7]. In figure 2.3 the pulley deformation is shown.

Figure 2.3: Pulley deformation

The actually pulley deformation is described by the using the Sattler’s Formulas. They describe the varying groove angle β and the axial displacement u of the pulley. ∆  π  β = β + sin θ −θ +  [2.8] 0 2  c 2 

u = 2 ⋅ R ⋅ tan (β − β0 ) [2.9] With:

β0 : groove angle in undeformed situation ∆ ≈ 10 -3: amplitude of the sinusoid

θc : center of the wedge expansion R: pitch radius of the belt, the distance from the pulley axis that the belt would have if the pulley sheaves were rigid. The local radial position can be calculated by: u r ⋅ tan β = R ⋅ tan β − [2.10] 0 2 Taking the time derivative of the above equation the radial velocity can be calculated. dR v = + a∆ωRsin ()θ −θ [2.11] r dt c 2 where a = (1+ cos β0 )/sin (2β0 )

2.4 Momentum Equation The equilibrium of the belt involves the tension of the belt (F), the linear pressure acting on the belt 2 sides ( p ), the friction force ( f a ), and inertia, centrifugal, force of the belt element ( σ (ω ⋅ R) ). This is visualized in figure 2.4. The friction force f a = µ ⋅ p

Figure 2.4: Forces acting on belt

σ = mass density F = T – P ; the net tension of the belt T: tension of the band P: compressive forces between metal segments A number of assumptions are made to calculate the equilibrium. At first it is possible to calculate the local angular acceleration θ& of the considered belt’s material element with the pulley’s angular velocity ω. Secondly from equation [2.3] follows ρ ≈ r ≈ R , therefore all these three parameters are R&& written as R. Thirdly the term R&& is neglected with respect to ω 2 R , i.e. << 1, furthermore also ω 2 R ϕ << 1. The last assumption is that the belt’s axial and tangential acceleration can be neglected [Ref: 8]. With these assumptions, the two involved equations are:

2 2 1 ∂(F −σ ⋅ω ⋅ R ) µ ⋅cos βs ⋅sin ψ 2 2 = [2.12] F −σ ⋅ω ⋅ R ∂θ sin β0 − µ ⋅cos β s ⋅cos ψ F −σ ⋅ω 2 ⋅ R 2 p = [2.13] 2R()sin β0 − cos β s ⋅cos ψ

With this last equation it is possible to calculate the center of the wedge expansion θc as: α ∫ p()()θ sin θ dθ o tan θc = α [2.14] ∫ p()()θ cos θ dθ o

2.5 Comparison with other models The determined CMM model will be compared with another model, this is a Multi-Body model derived by Srnik and Pfeiffer [Ref: 9]. The results are obtained during steady-state behaviour, in the CMM model this means the parameter A from equation [2.25] has the value zero. Furthermore the wrap angle is 180° and the groove angle β0= 10°. In the figures below, figure 2.5 and 2.6, the results are shown for the driving and driven pulley. The driving pulley has a ξ value of 0.35 and the driven pulley a value of 1/0.35. ξ represents a ratio between the force at the exit and at the entrance of the pulley, see equation [2.33]. In the figures the

11 relation between sliding angle γ and angular coordinate α is presented; here is γ =ψ −π 2/ . In the driving pulley the angular coordinate α is given as α I ,Drive =θ DR +π 2/ and in the driven pulley as

α I ,Driven = θ DN −π 2/ .

Figure 2.5: Result driving pulley Figure 2.6: Result driven pulley a) Multi-Body b) CMM ---) flexible pulley ) rigid pulley

From these figures can be concluded that the CMM model and the Multi-body produce almost similar behaviour, this is off course good. Also must be concluded that pulley deformation has a large influence on the sliding angle.

In figure 2.7 the tensile forces on one chain link are shown during one revolution, and in figure 2.8 the normal forces are presented. The calculation is done under the above mentioned condition. The friction coefficient has a value 0.1, and ξ has a value 0.35. Furthermore the minimal tension force in the belt is 1.6 kN.

Figure 2.7: Tensile forces Figure 2.8: Normal forces a) Multi-Body b) CMM

The agreement is quite good between both models. Some differences exist in the normal forces at the driving pulley, but it is only a small error.

12 So the CMM model is a good model to describe the behaviour of a CVT, and can be used to understand its performance.

2.6 Simplified and Dimensionless equations To be able to calculate the force acting on the belt and the pulley pressure the equations have to be rewritten into dimensionless equations. These equations will be needed in the next chapter. To simplify equations [2.1 and 2.7], consider ϕ << 1. The equations become: 1 ∂r ϕ = [2.15] r ∂θ ∂v v + θ = 0 [2.16] r ∂θ

All the previously derived relations can be rephrased in dimensionless form, using the following dimensionless quantities: R& w = [2.17] ωR

w sin (2β0 ) A = 2 [2.18] ∆ 1+ cos β0 & ~ r 1 sin (2β0 ) vr = 2 [2.19] ωr ∆ 1+ cos β0

~ ωs 1 sin (2β0 ) vθ = 2 [2.20] ω ∆ 1+ cos β0 F −σω 2 R 2 κ = 2 2 [2.21] F0 −σω R ~ p ⋅ R p = 2 2 [2.22] F0 −σω R

In equation [2.28] F 0 is the tension force of the belt at the entry point of the pulley, θ = 0 . Equation [2.5] can be written into: ~ vθ tan ()ψ = ~ [2.23] vr And [2.23] means: δv~ v~ + θ = 0 [2.24] r δθ From equation [2.11] can be concluded that:  π  v~ = A − cos θ −θ +  [2.25] r  c 2  Equation [2.12, 2.13 and 2.14] can be written into: 1 ∂κ µ ⋅sin ψ = [2.26] κ ∂θ 2 2 sin β0 1+ tan β0 cos ψ − µ cos ψ 1+ tan 2 β cos 2 ψ κ ~p = 0 [2.27] 2 2 2 sin β0 1+ tan β0 cos ψ − µ cos ψ

13 α ∫ ~p ⋅ sin ()θ ∂θ 0 tan θc = α [2.28] ∫ ~p ⋅ cos ()θ ∂θ 0 Combining [2.24 and 2.25] result in: θ  θ  v~ = v~ − Aθ − 2sin  sin  −θ  [2.29] θ θ 0  2   2 c  Equation [2.23] can be written into: θ  θ  v~ − Aθ − 2sin  sin  −θ  θ 0  2   2 c  tan ψ = [2.30]  π  A − cos θ −θ +   c 2  To solve these set of equation the following boundary conditions exist: v~ = v~ [2.31] θ θ =0 θ 0 κ =1 [2.32] θ =0 2 2 F2 −σω R ξ = 2 2 [2.33] F1 −σω R The last parameter ξ is the ratio between the belt tension force at the exit and the entry point of the pulley. With a given parameters A and ξ the other parameters can be calculated.

14 3. Band Segment Interaction

As stated in the introduction the presented CMM model from chapter 2 showed some inequalities with performed experiments. In this chapter a possible solution will be given to deal with these inequalities. The idea is that band segments interaction can be the solution, this kind of interaction is not taken into account in the standard CMM model. In this chapter an approximation will be presented to handle this phenomenon. As stated before the belt consists of two types of components, the bands and the segments. On the contact face the difference between the velocity of the bands and the segments will play a significant role, so this velocity should be derived. After this the tension of the bands will be further analyzed, the tension is influenced by the above mentioned velocity. The obtained equations will be added to the CMM model from chapter 2, and some results will be presented.

3.1 Assumptions Besides the assumptions in the last chapter, other assumptions are considered to derive the equations. The segments and the bands can slide with respect to each other, i.e. they can have different velocities. At the contact face between the segments and the bands a Coulomb friction is considered with a constant friction coefficient. Also we will follow the commonly adopted theory in of elastic beams that the cross section of the bands and the segments remains plane during motion. The bands have a tension force T and the segments have pressure force P working on it, in such a way that the total force F acting on the belt is equal to F = T − P . The last assumption is that the segments can be separated from each other, this phenomenon will be further explained in this chapter.

3.2 Geometrical model In figure 3.1 the considered situation is shown, concerning bands and segments. Looking in the right figure, h 1 stands for the height of the bands and h 2 for the height of the segments, and a total height of the belt h.

Figure 3.1: geometrical model

The speed difference between the segments and bands can be written as:

v12 = v(P1 ) − v(P2 ) [3.1] This is the difference in velocity in the contact P1 for the bands and P 2 for the segments. Using the general approach of multi-body dynamics, this equation can be rewritten into:

15 r r v12 = v(O1 ) − v(O2 ) − ω1Z h1eθ − ω2Z h2eθ [3.2]

In this equation v(Oi ) is the velocity of the bands or the segments in the point Oi . The angular velocity ωiz is the angular velocity measured from point P i, where i stands for 1 representing the bands and for 2 in case of representing the segments.

The velocity can also be phrased as follow, v(O ) = r& [3.3] i i r Remembering that ri = ier r . Using this, the derivation can be made, which leads to:  ∂r ∂r r r r v = Ω 1 − Ω 2 e + u e − ()ω h + ω h e [3.4] 12  1 ∂θ 2 ∂θ  r θ θ 1Z 1 2Z 2 θ

In this equation holds uθ = Ω1r1 − Ω2r2 .

& Now taking a more specific look to the term ωiz . In figure 3.2 this parameter is visualized, ω1Z = α .

Figure 3.2: Explanation of angles and ωiZ From the figure can be made up that: α = θ − ϕ [3.5] & & & This means ω1Z = θ −ϕ = Ω1 −ϕ . ∂r The terms i can be taken together. From figure 3.1 follows: ∂θ

r1 = (rint + h1 ) [3.5]

r2 = (rint − h2 ) [3.6] ∂r ∂r ∂r This means that 1 = 2 = . ∂θ ∂θ ∂θ When the above obtained results are substituted in equation [3.4], it gives: ∂r r r r v = ()Ω − Ω e + we + hϕ& e [3.7] 12 1 2 ∂θ r θ θ

With w = rint (Ω1 − Ω2 ).

Ω −ω The first term on the right hand side of equation [3.7] is a second order term, i << 1, and is ω therefore neglected.

16 ∂ϕ To calculate ϕ& equation [3.22] is used, remembering that ϕ& = Ω . Second order terms are again ∂θ neglected. Substituting this result, leads to: r h⋅Ω ∂ 2r r v = we + e [3.8] 12 θ R ∂θ 2 θ 2 2 ∂ r 1+ cos β0 The term 2 can be calculated by using equation [2.12]. Using the parameter a = ∂θ sin( 2β0 ) results in the following equation: r v12 = [w + a ⋅h⋅Ω ∆⋅ ⋅cos (θ −θc )] eθ [3.9]

3.3 Continuity Equation The continuity equation for both the bands and segments becomes: ∂Ω r& + r i = 0 [3.10] i i ∂θ Now the continuity equation of the segments is subtracted from the continuity equation of the bands. ∂r ∂r ∂r ∂r Consider that r& = i + Ω i and that 1 − 2 = 0 . i ∂t i ∂θ ∂t ∂t ∂(r Ω − r Ω ) 1 1 2 2 = 0 [3.11] ∂θ

From this equation it can be concluded that (r1Ω1 − r2Ω2 ) is constant along the arc.

Parameter w can be rephrased into:

w = (r1Ω1 − r2Ω2 ) − Ω(h1 + h2 ) [3.12] Equation [3.12] can also be rewritten into: ∂w ∂(r Ω − r Ω ) ∂Ω = 1 1 2 2 − ()h + h [3.13] ∂θ ∂θ 1 2 ∂θ ∂(r Ω − r Ω ) ∂Ω h Because 1 1 2 2 = 0 and the second order term ()h + h can be neglected, << 1 , ∂θ 1 2 ∂θ R the conclusion can be drawn that w is constant along the arc.

3.4 Forces equilibrium A more specific look will be given to the Coulomb friction forces between the belt and the segments. In figure 3.3 a visualization of the situation is shown. Five forces acting on the belt are considered: two tension forces on both ends of the section, a normal force and a friction force between the band and the segments, and a mass inertia force (centrifugal force).

Figure 3.3: Visualization of forces on one infinitesimal section d θ of the CVT belt

An assumption is that the beltr tension T, is directed in the tangential direction, this means there is no angle between tension and eθ direction. R: radius of the belt T: belt tension

dF N : normal force between band and segments = Pb⋅ R ⋅δθ

with P b: pressure between band and segments

dF µ : friction force between band and segments = f ⋅dF N = f ⋅ Pb ⋅ R ⋅δθ ( f = friction coefficient ) 2 dF in : inertia force of the considered band section = σ b ⋅ω ⋅ R ⋅ R ⋅δθ When the equilibriumr of ther five existingr forces isr written downr the following equation is obtained: (Teθ )θ +dθ − (Teθ )θ + dF N er − dF µ eθ + dF in ⋅ er = 0 [3.14] The first two terms can be combined intor one term: r r ∂(Te ) ()()Te − Te = θ ∂θ [3.15] θ θ +dθ θ θ ∂θ r r Equation [3.14] can be written in the tangential direction, eθ , and in the radial direction, er , doing this gives: ∂T − f ⋅ P ⋅ R = 0 [3.16] ∂θ b 2 2 − T + Pb ⋅ R + σ b ⋅ω ⋅ R = 0 [3.17] 2 2 From equation [3.17] follows Pb ⋅ R = T − σ b ⋅ω ⋅ R . This is substituted in equation [3.16]. ∂T = f ⋅(T −σ ⋅ω 2 ⋅ R 2 ) [3.18] ∂θ b Using this differential equation the tension of the band can be calculated.

The derivative of tension in the belt is determined by the direction of v12 . When v12 is positive, the ∂T ∂T is positive, and the other way around, when v is negative, then is negative. In figure 3.4 a ∂θ 12 ∂θ possible situation that can occur is visualized.

Figure 3.4: Possible situation for values of v12

In the situation above, at two positions the velocity difference between segments and bands, v12 , is ∂T zero. This means on two positions the will change of direction. The two other possible situations ∂θ are when only one point or no point of zero velocity difference exists. Also in these situations the change of tension is determined by the direction of v12 .

3.5 Dimensionless equations To make the equations more understandable the following dimensionless quantities are used. v v~ = 12 [3.19] 12 ah ω∆ w w~ = [3.20] ah ω∆ 2 2 Tb −σ bωb Rb κ b = 2 2 [3.21] F1 −σω R With these quantities equation [3.9] is written as: ~ ~ v12 = w + cos (θ −θc ) [3.22] The quotient between the tension in the band at the input and at the output of the pulley is defined as: 2 2 T2 −σ bωb Rb ξb = 2 2 [3.23] T1 −σ bωb Rb And equations [3.18] can be rewritten into: ∂κ b = f ⋅κ [3.24] ∂θ b The segments pressure becomes: ~ P P = 2 2 [3.25] F1 −σω R

3.6 Influence of clearance between the segments To calculate the tension at the belt, and pressure on the segments, the model must be expanded with the influence of clearance between the segments. This subject is investigated in (Ref: 10). The contact arc exists of two parts, a section where the segments are separated from each other, and one where the segments are pushed together. Considering the driving pulley, the above first mentioned section is the first section on the pulley, see figure 3.5, so here the segments are separated. In this part the only force acting on the belt is the tension in the bands, caused by the band segment interaction. To calculate this tension equation [3.24] is used. In the second section the segments are packed together. Two forces exists, the tension in the bands and

19 because the segments are pushed on each other, the pressure on the segments. The total force becomes F = T − P . Because the total force can be calculated by equation [2.26] and the tension is known by equation [3.24], also the segments pressure can be calculated.

Figure 3.5: Influence of clearance on driving pulley

At the shock section, at angular coordinate θ * , the segments will collide and come in contact, after this, a pressure occurs between the segments. The shock section separates the two mentioned phases. The sliding velocity at the shock section consists of two quantities, one calculated from the kinematical strain and the other the tangential component of the sliding velocity at θ * . The first component is calculated by (Ref: 10): * * − ε DR sDR = * [3.26] 1+ ε DR With the kinematical strain of the belt, db , db is the distance between two segments and ε ε = da ~ * da is the thickness of one segment. The tangential sliding velocity is calculated with vr (θ )⋅ tan (ψ ) . So the sliding velocity at the shock section becomes:

~ * 1 sin (2β0 ) ~ * vθ 0 = sDR ⋅ ⋅ 2 + vr (θ )⋅ tan ()ψ [3.27] ∆ 1+ cos β0 The pressure at the shock section P* , can be calculated by the following equation (Ref: 10): * * * 2 2 PDR = −sDR (1+ sDR )⋅σω R [3.28]

When considering the driven pulley, the sections have changed place, as is shown in figure 3.6. In the first section the segments are packed on each other and in the second section the segments are separated. The equations used on the driving pulley on each section hold true for the corresponding section on the driven pulley.

20 Figure 3.6: Influence of clearance on driven pulley

For both pulley holds that in the section where the segments are separated, the total force is equal to the tension in the bands, therefore can be written from equation [2.26 and 3.24] that in this section holds true that:

µ SP ⋅sin ψ µ BS = [3.29] 2 2 sin β0 1+ tan β0 cos ψ − µ SP cos ψ

With µBS the friction coefficient between the bands and the segments and µSP the friction coefficient between the segments and the pulley. Using the following values the sliding angle ψ can be calculated. µBS = 05.0 , µSP = 09.0 , β0 = 10 deg . The result can be seen in figure 3.7. Equilibrium k = kb 0.8 k 0.6 kb

0.4

0.2

-0.2

-0.4

-0.6

-0.8 0 1 2 3 4 5 6 7 psi

Figure 3.7: Calculation of sliding angle ψψψ

As can be seen in figure 3.7 two solutions exist for the sliding angle. ψ 1 = .0 0481 and ψ 2 = .2 9937 . ~ vθ Remembering that with tan ψ = ~ the right solution can be picked: the distinguishing must be made vr between the driving pulley and the driven pulley. At the driving pulley the section of separated segments is placed at the entrance of the pulley, in this part the sliding velocity is pointed inwards, so the sliding angle must be around π. This is the case for ψ 2 . At the driven pulley the considered section is placed at the exit of the pulley, the sliding velocity is here pointed outwards, so the sliding angle must be around 0. This is the case for ψ 1 . So now the sliding angle is known in the section where the segments are separated for both pulleys.

3.7 Parameters The calculations are done with the following values of the parameters: α = π rad,

A = 0 , µBS = 05.0 , µSP = 09.0 , β0 = 10 deg , ω = 2000 rpm, ∆ = .0 001 , σ = 2.1 kg/m,

σ b = .0 432 kg/m, R = .0 0540 m, h =12 mm. Because the driving pulley needs different input parameters to do the calculation than the driven pulley, the distinguishing is now made between the two pulleys.

21 3.8 Driving Pulley To do the calculation on the driving pulley three input parameters are needed, these are w~ , the dimensionless representation of the difference in angular velocity between bands and segments, θ * the angular coordinate on which the segments come in contact and ε * the kinematical strain in this point. With given values of these parameters the calculation can be done. The assumption is, as stated before, that at the entrance the segments are separated and at position θ * the segments are pushed together. By the use of equation [3.22 and 3.24] the dimensionless tension in the bands can be computed. Because * * * the kinematical strain at θ is given, ε , the other parameters, P and vθ 0 can be calculated. With the help from equation [3.28] P* , the pressure between the segments in this point, is known, and from equation [3.27] vθ 0 can be calculated. From these parameters the dimensionless total force on the belt, the sliding angle, the pulley pressure and θc can be calculated in the same way as presented in chapter 2.

To study in which way the different input parameters influence the results of the calculation the following is done. Two input parameters are fixed at one value, the third parameter is varied in five different values with equal increasing step size. In this was the influence of this parameter is studied.

The only input parameter that significantly influences the behaviour of the tension in the bands is w~ . ~ Suitable values for w can be approximate from equation [3.22] with setting v12 to zero and an ~ estimation of θc . In figure 3.7 the behaviour of κ b is shown for different values of w , variation from w~ = −1 up to w~ = .0 6293 . Dimensionless Tension 1.25 wmin = -1 1.2 w2 w3 w4 1.15 wmax = 0.6293

1.1

[-] 1.05 b κ

0.95

0.9

0.85 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.7: Behaviour for dimensionless tension a varying w~

~ ~ In case of w = −1the tension is decreasing constantly along the contact arc. In the second case, w2 , two point exists where v12 is zero, the tension is first decreasing, then increasing and at the end for a second time decreasing. In the last three cases, one point at which v12 is zero exists. First there is a decreasing tension, and after this an increasing tension. Variation of w~ can therefore cause a rise or a drop of tension at the exit point of the pulley with respect to the tension at the entrance point.

22 To calculate the other quantities the other two input parameters are fixed, ε * = .0 0025 and θ * = 70 o . The results are given in figure 3.8, 3.9 and 3.10. Dimensionless forces on the belt 1.4

1.2

0.8

0.6 Forces [-] Forces w = -1 min 0.4 w 2 w 3 0.2 w 4 w = 0.6293 max 0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.8: Dimensionless forces with varying w~ Sliding angle 6 w = -1 min w 5.5 2 w 3 w 5 4 w = 0.6293 max 4.5 [rad]

Ψ 4

3.5

2.5 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.9: Sliding angle with varying w~

23 Dimensionless pulley pressure 4 w = -1 min w 2 3.5 w 3 w 4 w = 0.6293 max 3

2.5 Pulley[-] pressure

1.5 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.10: Pulley pressure with varying w~

In figure 3.8 the fat solid lines represent the total force, the thin solid lines the tension and the fat dotted lines the segments pressure. As can be seen in this figure a segments pressure occurs at θ * . From this point on the sliding angles rises, causing the total force to decrease, and the segments pressure to increases. The sliding angle has expected behaviour; until θ * a constant sliding angle calculated from equation

[3.29], and after this point a big increase until θc . The small difference is due a different θc , caused by a different tension at θ * . The pulley pressure has also expected behaviour, a more or less constant value until θ * , and after this a great enhancement until θc . The difference is the consequence of different θc and a different values of the total force, caused by different w~ .

Variation of ε * does not significantly change the behaviour of the tension, the small difference is * caused by a minor difference in θc . The variation of ε has the largest influence on vθ 0 , which determines an increase or decrease of the total force, ε * is varied from 0.0018 up to 0.0030. The other two input parameters are fixed at w~ = − .0 4905 and θ * = 70 o . The results are shown in figure 3.11, 3.12 and 3.13.

24 Dimensionless forces on the belt 1.4

1.2

0.8

0.6

0.4 Forces [-] Forces ε* = 0.0018 0.2 min * ε 2 0 * ε 3 ε* -0.2 4 * = 0.0030 ε max -0.4 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

* Figure 3.11: Dimensionless forces with varying ε Sliding angle 7 * = 0.0018 ε min * 6 ε 2 * ε 3 5 * ε 4 * = 0.0030 ε max 4 [rad]

Ψ 3

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

* Figure 3.12: Sliding angle with varying ε

25 Dimensionless pulley pressure 9 * = 0.0018 ε min 8 * ε 2 ε* 7 3 * ε 4 * = 0.0030 6 ε max

4 Pulley[-] pressure

1 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

* Figure 3.13: Pulley pressure with varying ε

In figure 3.11 the same lines represent the same forces as in figure 3.8. In the first case ε * has a very * small value. This causes small values of P and vθ 0 . This causes a decrease of the sliding angle, and therefore a rise of the total force, because of this the segments pressure decreases below zero. This means the pressure between the segments is first positive, but then becomes negative, with other words, the segments are after θ * first in contact, but further along the arc the segments are separated again. This means the pulley is no longer a driving pulley, but a driven pulley. Remember that the segments pressure cannot be negative, and therefore the presented solution is not correct. The second case is a remarkable situation. In the same way as mentioned above the segments pressure drops below zero, but further along the arc the total force decreases and the segments pressure is again positive. This would mean that the segments first are in contact, become separated, and than again come in contact. This would mean the pulley could be a driving pulley. Because the presented solution is not exactly correct, an additional research would be needed to examine this situation, to investigate if this kind of behaviour can occur. The last three cases have logically, expected solutions. The sliding velocity at θ * is far enough below zero to cause a rise of the sliding angle after θ * , wherefore the total force decreases and the segments pressure increases, the solutions are also affect by a different θc . The solutions for the pulley pressure concerning these last three cases are correct and expected. Almost constant up to θ * , and after this an increase, and then becomes smaller because the total force becomes smaller. The other two solutions are not correct, because the total force distribution is not correct since the pressure cannot be below zero.

The third parameter θ * also does not significantly influence the behaviour of the tension distribution. * The difference is again caused by a different θc . The parameter θ determines the size of the arc where the segments are in contact and by this influences whether the total force increases or decreases. The variation of θ * is from 60° up to 80°, chosen on the hand of experimental data. The other input parameters are set to w~ = − .0 4905 and ε * = .0 0025 . The results are presented in figure 3.14, 3.15 and 3.16.

26 Dimensionless forces on the belt 1.5

0.5 Forces [-] Forces

θ* = 60 ° 0 θ* = 65 ° θ* = 70 ° θ* = 85 ° θ* = 80 ° -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.14: Dimensionless forces with varying θ * Sliding angle 7

4 [rad]

Ψ 3

2 θ* = 60 ° θ* = 65 ° 1 θ* = 70 ° θ* = 85 ° θ* = 80 ° 0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.15: Sliding angle with varying θ *

27 Dimensionless pulley pressure 9 θ* = 60 ° 8 θ* = 65 ° θ* = 70 ° 7 θ* = 85 ° θ* = 80 °

4 Pulley[-] pressure

1 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.16: Pulley pressure with varying θ *

In the first case the sliding angle decreases, causing a rise of the total force. After θ * the segments pressure is positive, but when the total force increases, the pressure between the segments becomes negative, which is physically not acceptable as in agreement with the case mentioned at varying ε * . Therefore this presented solution is not correct In the second case, in the first stadium the same occurs as in the first case. The segments pressure is below zero, but the sliding angle rises, and the total force decreases even this far that the segments pressure is positive again. This would mean that the segment are first in contact, then separated and at the end once more come in contact. This is in agreement with the situation mentioned above at varying ε * . Because the presented solution is not correctly determined, this situation must be further studied to draw correct conclusions. The last tree cases are correct and show expected behaviour. An increase of the sliding angle causes a drop of the total force and a rise of the segments pressure. In these three cases the pulley pressure is as expected, constant up to θ * , and after this an increase followed by a decrease, which is caused by a decreasing total force.

3.9 Driven Pulley The calculation on the driven pulley also needs three input parameters, these are w~ , the dimensionless ~ representation of the difference in angular velocity between bands and segments, P1 , the pressure between the segments at the entrance of the pulley and vθ 0 at the entrance point. With given values of these parameters the calculation can be done. As mentioned before, at the entrance of the pulley the segments are packed together, on a certain position, θ * , they will be separated. First the tension and the total force distribution along the contact arc are computed, equations [2.26, 3.22 and 3.24] At the angular position when these two quantities are equal, the segments are separated because the segments pressure equals zero, P = T − F , this position is θ * . Hence, before θ * there is a segments pressure and a tension, after θ * only a tension exists.

To verify the influence of the tree input parameters on the driven pulley the same approach is done as with the driving pulley. So two input parameters are fixed on a certain value and the third input

28 parameter is varied with equal increasing step size. In this way the influence of this input parameter on the results of the calculation is studied.

Also for the driven pulley holds that w~ is the only parameters which significantly influence the tension ~ distribution. The values of w are estimated by using equation [3.22], setting v12 to zero and assume a certain θc . In figure 3.17 the dimensionless tension distribution on the bands is shown. Dimensionless Tension 2.2 wmin = -1 w2 2.1 w3 w4 wmax = 0.1736 2

[-] 1.9 b κ

1.8

1.7

1.6 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad] ~ Figure 3.17: Behaviour of tension with varying w

As can be seen in the figure above the parameter w~ can cause a higher or a lower tension at the exit with respect to the tension at the entrance point. In case of w~ = −1a constant decrease occurs, no point exists where v12 = 0 , so a constant decrease takes place. In the second case two points exist where v12 = 0 , this means that on two positions the derivative of the tension changes, thus first a decrease takes place, then an increase and at the exit for a second time a decrease in tension occurs.

The last three cases have one point where v12 is zero, therefore first a decrease takes place followed by an increase of the tension. The other results, total force, segments pressure, sliding angle and pulley pressure are presented in figure 3.18, 3.19 and 3.20. To be able to do the computation the other input parameters are fixed on ~ ~ P1 = 0,1 and vθ 0 = − .0 5870 and the variation of w is from -1 up to 0.1736.

29 Dimensionless forces on the belt 2.5

1.5

1 Forces [-] Forces w = -1 0.5 min w 2 w 3 0 w 4 w = 0.1736 max -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad] ~ Figure 3.18: Dimensionless forces with varying w Sliding angle 4 w = -1 min 3.5 w 2 w 3 3 w 4 w = 0.1736 2.5 max

2 [rad] Ψ 1.5

0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad] ~ Figure 3.19: Sliding angle with varying w

30 Dimensionless pulley pressure 14 w = -1 min w 12 2 w 3 w 10 4 w = 0.1736 max 8

6 Pulley[-] pressure 4

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad] ~ Figure 3.20: Pulley pressure with varying w

In figure 3.18 the fat solid lines represent the total force, the thin solid lines the tension and the fat dotted lines the pressure between the segments. In this figure can be seen that θ * is positioned at the end of the contact arc. On this position the segments pressure becomes zero, and the total force is equal to the tension in the bands. The sliding angle is decreasing until θ * , and the total force is rising, wherefore the segments pressure is decreasing. After θ * the sliding angle is constant, equal to the value calculated from equation [3.28]. The pulley pressure has expected behaviour, rising in agreement with the total force, and more or less constant after θ * .

~ Variation of P1 does not significantly influence the distribution of the tension, only the level is influenced, with this it determines wetter the segments will be separated or not. The variation is done ~ with values of P1 from 0.40 up to 1.59, and the other input parameters are fixed ~ on w = − .0 1865 and vθ 0 = − .0 5870 . The results are given in figure 3.21, 3.21 and 3.22.

31 Dimensionless forces on the belt 3

2.5

1.5

1 Forces [-] Forces P = 0.40 min 0.5 P 2 P 3 0 P 4 P = 1.59 max -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

~ Figure 3.21: Dimensionless forces with varying P1 Sliding angle 4 P = 0.40 min 3.5 P 2 P 3 3 P 4 P = 1.59 2.5 max

2 [rad] Ψ 1.5

0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

~ Figure 3.22: Sliding angle with varying P1

32 Dimensionless pulley pressure 14 P = 0.40 min P 12 2 P 3 P 10 4 P = 1.59 max 8

6 Pulley[-] pressure 4

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

~ Figure 3.23: Pulley pressure with varying P1

In figure 3.21 the lines represents the same forces as in figure 3.18 As can bee seen in the first three cases, at a certain θ * the segments pressure becomes zero, from this point on the total force is equal to the tension in the bands. The sliding angle is decreasing and after separation of the segments the constant value. The pulley pressure decreases and is almost constant at the end of the contact arc. All expected results. For the last two cases the pressure between the segments never becomes zero, this means the segments are always packed together and never separate, therefore θ * does not exists. The total force is increasing along the complete arc, the sliding angle is decreasing along the arc and the pulley pressure is always increasing.

The variation of vθ 0 does not significantly influence the distribution of the tension, variation of vθ 0 determines wetter a separation of the segments takes place or not. vθ 0 is varied from -1.2 up to 0, the ~ ~ other input parameters are set to w = − .0 1865 and P1 = 0,1 . The results are shown in figure 3.24, 3.25 and 3.26.

33 Dimensionless forces on the belt 2.5

1.5

1 Forces [-] Forces v = -1.2 0.5 θ0 min v θ0 2 v 0 3 0 θ v θ0 4 v = 0 θ0 max -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.24: Dimensionless forces with varying vθ 0 Sliding angle 4.5 v = -1.2 θ0 min 4 v θ0 2 v 3.5 θ0 3 v θ0 4 3 v = 0 θ0 max

2.5 [rad]

Ψ 2

1.5

0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.25: Sliding angle with varying vθ 0

34 Dimensionless pulley pressure 14 v = -1.2 θ0 min v 12 θ0 2 v θ0 3 v 10 θ0 4 v = 0 θ0 max 8

6 Pulley[-] pressure 4

0 0 0.5 1 1.5 2 2.5 3 3.5 Angular coordinate θ [rad]

Figure 3.26: Pulley pressure with varying vθ 0

In figure 3.24 the lines represents the same forces as before. In the first two cases, when vθ 0 has the largest value below zero, the sliding angle is decreasing the least. As a result the total force is increasing the least. This leads to a segments pressure which never reaches zero, as a result the segments are never separated. The sliding angle has no constant value and also the pulley pressure does not reach a constant value. In the other three cases the results are as expected, the rise of total force is larger and the pressure between the segments drops below zero and the segments are separated. In the part after θ * the total force is equal to the tension, the sliding angle has the constant value and the pulley pressure reaches an almost constant value.

I final note must be made about the step in the value of the sliding angle and the pulley pressure at the coordinate were a pressure between the segment occurs. From the continuity equation can made up that this is not in agreement. The curves should be continuous, this can be done by adjusting ψ (θ = 0) by changing vθ 0 .

3.10 Verify distribution of the kinematical strain The assuming of a θ * , which is previously done, can be verified. It must be ensured that in the contact area where the segments assumed to be separated, the segments do not come in contact. In other words for the driving pulley in the part before θ * the value of ε , kinematical strain, must not reach zero. The expecting behaviour is that the kinematical strain is decreasing from the entrance point up to θ * , this would imply that the gap between the segments becomes smaller, moving from the entrance to θ * . For the driven pulley the kinematical strain must not reach a zero value in the part after θ * , the expected distribution is an increasing value of ε from θ * up to the exit point of the pulley. This would mean that the gap between the segments becomes larger from θ * to the exit.

To calculate the distribution of the kinematical strain equation [2.6] can be used. This equation must be rewritten, and becomes the following continuity equation:

35 1 ∂ε r& ∂ω ω = + s [3.29] 1+ ε δθ r ∂θ Because the sliding angle ψ in this part of the contact arc is known by equation [3.28], the parameters & r and ωs can be computed with equation [2.23 and 2.25]. Two parameters influence the distribution * of the kinematical strain, these are θ and θc . In applying a variation of those two parameters the distribution is examined.

* For the driving pulley the kinematical strain at θ * is known as ε . With this initial condition the above * equation can be solved. The results are shown in figure 3.27 and 3.28.When varying θ the value of θc * is fixed on 120°, and when varying θc the value of θ is fixed on 90°.

-3 x 10 Kinematical strain ε for different θ* 12 θ* = 70 ° 11 θ* = 80 ° * = 90 10 θ ° θ* = 100 ° 9 θ* = 110 ° [-] ε 8

5 Kinematical strain

2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Angular coordinate θ [rad]

Figure 3.27: Kinematical strain with varying θ *

36 -3 Kinematical strain ε for different θ x 10 c 11 = 90 θc ° 10 = 100 θc ° = 110 9 θc ° = 120 θc ° [-] 8 ε = 130 θc ° 7

5 Kinematical strain 4

2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Angular coordinate θ [rad]

Figure 3.28: Kinematical strain with varying θc

* As can be seen in the above figures the value of ε at coordinate θ * is equal to ε = .0 0025 . The value of ε is decreasing from the entrance point up toθ * . Variation in θ * does not influence the slope of ε , but it does influence the level of the distribution. A variation in θc manipulates the slope of distribution, and by this also in some way the level ε at the entrance. With these results it can be sure that there is not a second point in this part of the contact arc where the segments are packed together.

* The driven pulley has an initial condition of ε = 0 at θ * . The distribution of the kinematical strain * on the driven pulley is given in figure 3.29 and 3.30. When varying θ the value of θc is fixed on * 120°, and when varying θc the value of θ is fixed on 150°.

37 Kinematical strain for different ε θc 0.014 = 100 θc ° = 110 0.012 θc ° = 120 θc ° 0.01 = 130 θc ° [-] ε = 140 θc ° 0.008

0.006

Kinematical strain 0.004

0.002

0 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 Angular coordinate θ [rad]

Figure 3.29: Kinematical strain with varying θ * Kinematical strain ε for different θ* 0.015 θ* = 130 ° θ* = 140 ° θ* = 150 ° θ* = 160 ° θ* = 170 ° [-] 0.01 ε

0.005 Kinematical strain

0 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 Angular coordinate θ [rad]

Figure 3.30: Kinematical strain with varying θc

As is shown in these figures the kinematical strain is zero at θ * , the value of ε is increasing from θ * up to the exit of the pulley. * Variation of θ manipulates the slope of the distribution and the end level of ε . The variation of θc also influences the slope of the distribution and by this the kinematical strain at the exit point of the driven pulley.

38 This results show that after θ * no point exists where the segments make contact again, the assumption of θ * is correct.

39 4. Future Work

The CMM model is expanded with the band segment interaction, but it is not completely finished yet. The tree input parameters needed to do the calculation on each pulley should be eliminated, this can be done by coupling the two pulleys. Mathematically this must be done using six equations, which can eliminate the six unknowns; three times two input parameters. For remembering the input parameters ~ * * on the driving pulley are: w , ε DR and θ DR , on the driven pulley the input parameters are: ~ ~ w , vθ 0 and P1 . Looking at figure 4.1, five equations can be derived.

Figure 4.1: Equilibrium parameters

In the above figure the following parameters play an important role, F total force, v , the velocity of the belt, P pressure between the segments. The number one represents the entry point of the pulley and number two the exit point of the pulley. The character DR represents the driving pulley and the character DN the driven pulley.

The following equilibrium equations can be derived:

F1DR = F2DN [4.1]

v1DR = v2DN [4.2]

F2DR = F1DN [4.3]

P2DR = P1DN [4.4]

v2DR = v1DN [4.5] Still one equation is needed two solve the problem. The following equation can be used. ε ∫ ∂l = ∆ L [4.6] L ε +1 On the left hand of this equation the integral of the kinematical strain on each infinitesimal piece of the belt, ∂l , is taken, this equals the total clearance of the belt ∆ L . This total clearance is the difference of the length of all segments placed one after each other and the length of the bands.

With the help of these six equations it should be possible to eliminate the six input parameters. In this way results from this new model can be calculated in the same way as done for the model from chapter 2. With given conditions parameters, physically important parameters like clamping force and torque on each pulley can be calculated.

40 Conclusions

The continuously variable transmission is a promising transmission for all kinds of drive trains, good results can be obtained in the field of emissions, efficiency and driveability. The pulley based CVT can be divided in two categories, the metal push belt and the metal chain. The working principle of those two CVTs is more or less the same. Other kinds of CVTs exist, but they are not investigated in this paper. To be able to reach the optimal in the categories of emissions, efficiency and driveability, more understanding is needed of the behaviour of the CVT. This can be done by doing experiments and modelling. In this paper a closer look was taken to a mathematical model of the CVT, the CMM model. The CMM model is derived by Carbone, Mangialardi and Mantriota from the Polytecnico di Bari, Italy. After validation of this model, by doing several experiments, some differences seem to appear. A probable solution seems to be the band segment interaction, which was not taken into account in the CMM model. The first step that is taken is the derivation of the equations, concerning band segment interaction. These equations are related to friction between the bands and the segments, this friction is assumed to be a Coulomb friction. Because the friction force is determined by the velocity difference between the bands and the segments, this velocity is derived. To be able to add the band segment interaction to the model also clearance between the segments has to be taken into account. With this approach it was possible to modify the CMM model to take into account the band-segments interaction. The new model has been utilized for a first estimation of the influence of band-segments interaction on the CVT behaviour. Because of the increased number of degrees of freedom, some extra input parameters are needed to be able to do the calculation. The first results are promising. Forces on the belt, tension in the bands and pressure between the segments have been calculated, together with the pulley pressure and the sliding angle. Also the influence of the input parameters is investigated. However, the new model is not yet completed. The need of the input parameters should be eliminated by the use of some extra equations, which make it possible to combine the equations for the two pulleys and so couple the two pulley into one CVT. These equations are also presented, and so the next step that has to be taken is to eliminate the need of the extra input parameters by using the extra equations, and in this way the new CVT model must be completed.

41 Literature list

[1] www.autoblog.nl, Wall Street Journal, April 2006 [2] C. Brace, M. Deacon, N.D. Vaughan, R.W. Horrocks, C.R. Burrows, The compromise in reducing exhaust emissions and fuel consumption from a diesel CVT powertrain over typical usage cycles , Proceedings of the CVT’99 Congress, Eindhoven, The Netherlands [3] B. Vroemen, Continuously Variable Transmission, Course Vehicle Drive Trains, Eindhoven, The Netherlands [4] Van der Wal S., De opkomst van het automobilisme in Nederland , University of Maastricht, April 2003 [5] Harris W., How CVT works , http://auto.howstuffworks.com/cvt.htm [6] Carbone G., Shifting dynamics in continuously variable transmission , Bari, January 2002 [7] Carbone G., Mangialardi L. and Mantriota G., The influence of pulley deformations on the shifting mechanism of metal belt CVT , Journal of mechanical design, January 2005 [8] Carbone G., Mangialardi L. and Mantriota G., Theoretical model of metal v-belt driver during apid ratio changing , Journal of mechanical design, March 2001 [9] Srnik J. and Pfeiffer E., Dynamics of CVT chain drives , Int. J. of vehicle design, vol. 22, 1999 München [10] Carbone G., Mangialardi L. and Mantriota G., Influence of clearance between plates in metal pushing v-belt dynamics, Journal of Mechanical Design, September 2002