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A comparison of the performances of full and half toroidal traction drives
Mechanism and Machine Theory
Mechanism and Machine Theory 39 (2004) 921–942
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A comparison of the performances of full and half toroidal traction drives G. Carbone *, L. Mangialardi, G. Mantriota Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, V.le Japigia 182, 70126 Bari, Italy Received 5 May 2003; received in revised form 21 January 2004; accepted 10 April 2004
Abstract The efficiency of two different typologies of the toroidal traction drive, the full-toroidal and the halftoroidal, is estimated in order to point out which of them offers the higher mechanical efficiency. A fully flooded isothermal contact model between the discs and rollers, based on the results of EHL theory, is used to evaluate the slip, the spin losses and the mechanical performances of the variators. It is shown that the half-toroidal traction drive offers higher efficiency and higher maximum transmissible torque. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The request for a higher energy efficiency and CO2 reduction has pushed several researchers to find new technical solutions to improve the emission performance of nowadays IC engine vehicles. While waiting for new and renewable forms of energy to become effective and cost reasonable, new solutions have to be found: among different and several technical solutions, new drive train systems are being investigated to accomplish this purpose. The continuously variable transmission (CVT) represents one of the most promising solution since it is able to provide an infinite number of gear ratios between two finite limits, and, thus, to allow the IC engine to operate closer to its optimal efficiency line. Several studies have shown that it is possible to improve the fuel savings and to reduce the vehicle emissions by adopting this kind of transmission [1,2]. It has been shown, for instance, that when the transmission is optimally controlled the mid class vehicles equipped with the CVT and
*
Corresponding author. Tel.: +39-080-596-2746; fax: +39-080-596-2777. E-mail addresses: [email protected] (G. Carbone), [email protected] (L. Mangialardi), [email protected] (G. Mantriota). 0094-114X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.04.003
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IVT variators may achieve less fuel consumption (about 10% for CVT and 6% for IVT) and higher comfort in the urban traffic [3]. Moreover applications of CVTs to wind power systems have been proposed, and some papers [4–6] have shown that a significant increase of the energy production may be attained. The metal pushing V-belt and the metal chain CVTs are able to achieve these results, but they have some drawbacks as the strict dependence of the shifting speed on the clamping forces acting on the moving pulley sheaves [7–9], and the smaller torque capacity when compared to toroidal traction drives. The robotized gearbox may be a different solution, since it combines the economy of a welldriven manual transmission with the easy to use of a conventional automatic transmission, and get the further advantage to retain the simplicity and the economies of scale of an established manual design. But, in this case, some problems regarding the shift quality arise: the torque interruption becomes intrusive because the driver is not able to predict or anticipate the gear shift. For these reasons more promising solutions have to be found, and the toroidal traction drives may be one of these. They are being extensively investigated because of their high torque capacity, that makes them suitable for application in larger engine cars and even trucks. The most attractive typologies are the full-toroidal [10,11] and the half-toroidal traction drives [12–14]. The main components of these transmissions are the input and output discs, designed to create a toroidal cavity (see Fig. 1), coupled with an appropriate number of rollers. The high torque capacity of these transmissions is obtained by coupling together in a series scheme two or more single units [15–17]. Moreover the particular geometry of the toroidal traction drive makes it able to rapidly adjust its speed ratio to the request of the driver, thus improving the driving comfort [18–21]. Between the roller and the discs no metal–metal contact occurs, the torque is transmitted by means of the shearing action of a special oil referred to as traction oil. The lubrication regime of such a system is the hard EHL with pressures up to 3 GPa. The high pressures lead to a much higher oil viscosity (several order of magnitude) than in the normal hydrodynamic regime, thus enabling the transmission of high torque despite the very small area of contact. In the technical literature many works can be found concerned with the design, fatigue life, and thermal stresses of the half-toroidal CVT [22–26], but not many contributions compare the full-
(a)
Fig. 1. The traction drive CVTs: (a) half-toroidal; (b) full toroidal.
(b)
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Table 1 CVT geometric data Half toroidal CVT:
Full toroidal CVT
Cavity radius r0 ¼ 40 mm Roller curvature r22 ¼ 32 mm Half cone-angle h ¼ p=3 Aspect ratio k ¼ e=r0 ¼ 0:625 Speed ratio range s ¼ 0:5–2:0
Cavity radius r0 ¼ 40 mm Roller curvature r22 ¼ 26:4 mm Half cone-angle h ¼ p=2 Aspect ratio k ¼ e=r0 ¼ 0:25 Speed ratio range s ¼ 0:5–2:0
toroidal and half-toroidal traction drives as regards the mechanical efficiency and their traction capabilities. Only few papers provide some experimental results on their efficiency [27–29]. The main scope of this work is to propose a theoretical model of the variators, based on the results of EHL lubrication theory, that is able to estimate the traction capability and the mechanical efficiency of these two different CVT typologies, and to point out which of them offers the higher mechanical efficiency. The analysis is limited to the simple variator as it is, made up of the input discs, the rollers, the support bearings and the output discs. The two variator are supposed to have the same cavity radius and the same radial size, the latter condition requires different values of the aspect ratio k ¼ e=r0 as reported in Table 1. The principal advantage of the proposed method with respect to other similar methodologies [30,31] consists of three points: the model is independent of the specific traction drive under consideration, the formulation presented does not require the determination of the traction curve by experiments performed on the given traction drive with the specific traction oil. The model is also able to take into account the influence of the spin motion on the mechanical efficiency and the traction performance of the variator. The main drawback is related to the large number of equations to deal with, that makes the overall computation time-consuming.
2. Geometric and kinematic quantities of full and half toroidal CVTs 2.1. Geometrical description of CVTs Fig. 1 shows the main geometrical features of the toroidal variators. During the steady state operation of the CVT the swing center of the roller coincides with the cavity center O, and its axis of rotation is tilted of c. The tilting angle c (positive if clockwise directed) controls the distance r1 and r3 of the contact points A and B from the main axis of the variator, and, consequently, controls the ideal speed ratio srID ¼ r3 =r1 . In the same Fig. 1, r12 ¼ r23 ¼ r0 represent radius of the toroidal cavity, that is also one of the two principal radii of curvature of the input and output discs. The quantity r11 is the second principal radius of curvature of the input disc, whereas r33 is the second principal radius of curvature of the output disc. Moreover r2 and r22 are the two principal radii of curvature of the roller with r22 < r12 . The quantity e is the distance of the toroidal cavity from the disc axes, it is related to the aspect ratio k ¼ e=r0 of the toroidal traction drive. Moreover in a half-toroidal CVT, the half cone-angle h of the roller is about 50–70°, whereas in a full-toroidal CVT the cone angle is 90°. All the remaining geometrical parameters are reported in Table 1.
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2.2. Kinematic analysis of CVTs In this subsection the kinematics of the toroidal traction drives is analyzed during the steady state operation of the variator. Consider the roller and the discs as rigid bodies and assume no-slip at points A and B (see Fig. 2). Under these conditions, the motion of the roller relative to the input disc is a spherical rigid motion, of which the instantaneous axis of rotation can be easily determined as the straight-line through the points of null relative velocity A and X (the point X is the intersection of the roller absolute axis of rotation and the input-disc absolute axis of rotation). Similar arguments hold when studying the relative velocity field between the roller and the outputdisc, in this case the instantaneous axis of rotation of the relative motion is the straight-line BX. Let x1 , x2 and x3 be, respectively, the absolute angular velocities of the input disc, of the roller and that of the output disc. The angular velocity of the roller relative to the input disc is, therefore, x21 ¼ x2 x1 , whereas that one relative to the output disc is x23 ¼ x2 x3 . Figs. 2 and 3 show these two relative velocities of rotation x21 and x23 both for the half-toroidal and fulltoroidal CVTs. It is clearly shown that, since the point of intersection H (see Fig. 2) of the two tangents to the toroidal cavity at points A and B does not always coincides with the point X, both the relative angular velocities x21 and x23 have non-zero spin vector components x21spin and x23spin , respectively. Moreover, the ratio ðx21spin Þin =jx21 j ¼ ðx23spin Þout =jx23 j ¼ sin a, assumes its maximum value for srID ¼ r3 =r1 ¼ 1, since the distance between the points H and X is maximum for this value of the ideal speed ratio. Furthermore, Fig. 2 shows that, for the half-toroidal CVT, two points exist at which H and X coincide and the spin motion vanishes. Different considerations have to be done for the full-toroidal traction drive. This time, Fig. 3 shows that the point H goes to infinity, thus the spin motion never vanishes, and, because of the bigger angle a, it is always bigger than in the case of the half-toroidal CVT. Once again, the worst situation occurs for srID ¼ 1, when (see Fig. 3) the modulus of the spin vector components x21spin and x23spin equals the modulus of absolute angular velocity of the output and input discs, respectively. What happens when slip occurs, namely when torque is transmitted, is only slightly different from the above written scenario since the slip is always very small and, hence, the axes of rotation
(a)
(b)
Fig. 2. The half toroidal CVT: spin motion and rolling motion of the roller depicted for no-slip conditions; (a) speed ratio equal to 1; (b) no spin condition (speed ratio differs from the unit value).
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(b)
(a)
Fig. 3. The full-toroidal traction drive: (a) speed ratio equal to 1; (b) same speed ratio as in (Fig. 2(b)) but the spin is always different from zero.
of the roller relative to the discs would result only slightly tilted relatively to those depicted in the Figs. 2 and 3. Now consider the contact area between the roller and the discs. In this region the elastic deformations of the bodies have a large influence on the relative velocity motion, and this one can no more be classified as a rigid motion. The region of contact is an elliptical area centered at the point of contact. The ellipse principal axes, (see Fig. 4) lay on the y-axis (the major) and on the x-axis (rolling direction, the minor). In order to calculate the shear strain of the lubricant we need to find an explicit formulation of the relative velocity field in the contact region. Observe that, over the contact area the bodies cannot penetrate each other, thus the relative velocity, because of the elastic deformations, do not have any component normal to the area of contact. Hence, assuming a negligible tangential deformation of the elastic bodies, the velocity of the roller points relative to the input and output discs, respectively, can be written, in the region of contact, as: v21 ¼ v21A þ x21spin ^ ðPin AÞ
ð1Þ
v23 ¼ v23B þ x23spin ^ ðPout BÞ
ð2Þ
Pin and Pout are points of the roller, while the velocity vectors v21A and v23B stand for the relative velocity between roller and discs at the center points A and B of the contact areas. Since we are considering only steady-state behavior v21A and v23B do not have components along the y-axes, yin and yout (see Fig. 4), but only along the x-axes. The previous Eqs. (1) and (2) show that the relative motion between the roller and the discs in the contact region, can be split into a pure translation, given by the vectors v21A and v23B , and a pure spin motion about the z-axes. 2.3. Practical aspects for CVTs When studying the toroidal traction drives, it is useful to define the input and output slip coefficients, usually referred to as creep coefficients Crin and Crout : Crin ¼
jx1 jr1 jx2 jr2 ; jx1 jr1
Crout ¼
jx2 jr2 jx3 jr3 jx2 jr2
ð3Þ
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Fig. 4. The reference frames used at the input and output points of contact.
A small amount of creep must be always present to allow the transmission of torque. Besides the creep coefficients defined above, it is useful, for the next calculations, to introduce the following dimensionless geometric quantities (remember that r12 ¼ r23 ¼ r0 , see also Fig. 1): r1 ~r1 ¼ ¼ 1 þ k cosðh þ cÞ r0 ð4Þ r3 ~r3 ¼ ¼ 1 þ k cosðh cÞ r0 By means of the creep coefficients and considering that srID ¼ ~r3 =~r1 it is possible to write the actual speed ratio sr ¼ jx3 j=jx1 j as: sr ¼ ð1 Crin Þð1 Crout Þ
1 þ k cosðh þ cÞ ¼ ð1 Crin Þð1 Crout ÞsrID 1 þ k cosðh cÞ
and also define the speed efficiency mspeed of the variator as the ratio sr =srID : sr mspeed ¼ ¼ 1 Cr srID
ð5Þ
ð6Þ
where 1 Cr ¼ ð1 Crin Þð1 Crout Þ stands for the global sliding coefficient between the output disc and input one. In a similar way, it is also possible to write the spin ratios as a function of the creep coefficients: 1 þ k cosðh þ cÞ tan h 1 þ k cosðh cÞ tan h
r21 ¼ sinðh þ cÞ ð1 Crin Þ r23 ¼ sinðh cÞ
1 1 Crout
ð7Þ ð8Þ
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1.1
(σ 21)no-slip
1
0.9 0.8 0.7 0.6 Full Toroidal Half Toroidal
0.5 0.4 0.3 0.2 0.1 0 0.5
0.75
1
1.25
1.5
1.75
s r ID 2
Fig. 5. The spin-ratios as a function of the ideal speed ratio srID , for no-slip conditions (traction drive data in Table 1).
The above written Eqs. (4)–(8) hold true also for the full toroidal traction drive with h ¼ p=2. As already discussed before, it is shown that, for the ideal case of no-slip, the worst condition as regards the magnitude of the spin occurs for c ¼ 0, that is to say for srID ¼ 1. In fact, replacing both Crin and Crout by zero, Eqs. (7) and (8) become: ðr21 Þno-slip ¼
cos c ð1 þ kÞ cos h ¼ ðr23 Þno-slip sin h
ð9Þ
Equation (9) shows that the two spin ratios r21 and r23 assume their maximum value when cos c ¼ 1, i.e. c ¼ 0. Observe that if cos h < ð1 þ kÞ1 two different values of the tilting angle c also exist at which the spin ratios vanish, as pointed out in Section 2.2 (see also Fig. 2). For full toroidal CVT replacing h by p=2 in Eq. (9) we obtain ðr21FT Þno-slip ¼ ðr23FT Þno-slip ¼ cos c which is always bigger than the spin ratios of an half-toroidal CVT. Fig. 5 shows, for no-slip conditions, the spin-ratios as a function of the ideal speed ratio srID . The CVT geometrical characteristics are reported in Table 1, where the radii of curvature r22 have been chosen in order to obtain the same maximum shear stress in both CVTs (see also Section 4.1). As before predicted the spin ratio of the full-toroidal CVT is about five times higher than that of the half-toroidal one.
3. Forces, spin momentum and efficiency The presence of spin motion affects the full-toroidal traction drive more than the half-toroidal variator. But, on the other hand, the latter is affected by the support bearing losses, since the normal forces FN acting on the roller, at the points of contact, do not balance out (see Fig. 6). Therefore, a resulting axial force FR has to be supported by an axial bearing (one for each roller), that, because of its internal losses, causes a reduction of the CVT mechanical efficiency. On the other hand, the full-toroidal variator is not affected by this problem, since h ¼ p=2, and the normal forces balance out. But, on the other hand, the full-toroidal CVT is much more affected by spin losses, thus it is necessary to carry out a comparison in order to single out which typology of CVT offers the higher mechanical efficiency.
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Fig. 6. The free body diagram of the discs and roller; FN is the normal force at the point of contact, FR is the resulting load on the axial bearing, FDin and FDout are the axial clamping forces on the input and output discs, respectively, FTin and FTout are the traction (tangential) at the input and output points of contact, respectively, Tin and Tout are the input and output torques, MSin and MSout are the spin momenta at the input and output points of contact, respectively, TBL is the torque resistance of the axial bearing, n is the number of rollers per each cavity, and m is the number of cavities.
As regards the support bearing losses some studies have been carried out on this aspect of the HT-traction drives. For example in [25] the authors estimate the spinning losses of the power roller bearing by means of an elastic–plastic lubricant model, and obtain results in good agreement with the experiments. For our scope we will make use of the empirical relation Eq. (10), already used in [17], that gives the torque loss as a function of the axial thrust acting on the roller FR , and that results in agreement with the experimental results reported in [25]. In Eq. (10) the axial thrust acting on the roller FR is measured in (N) and the torque bearing loss TBL is measured in (Nm). TBL ¼ 4:6 105 FR1:03
ð10Þ
Fig. 6 shows the free body diagram of the input and output discs, and that one of the roller. The support bearing is modelled as a revolute movable joint, that prevents the translatory motion of the roller along the direction of its axis of rotation. The force balance of the roller gives: FR ¼ 2FN cos h
ð11Þ
FTin r2 FTout r2 TBL þ MSin cos h þ MSout cos h ¼ 0
ð12Þ
and where FTin and FTout stand for the traction forces at the points of contact, and MSin and MSout are the corresponding spin momenta. The equilibrium of the input and output discs, once chosen the number of rollers n and the number of cavity m, gives: FDin ¼ nFN sinðh þ cÞ
ð13Þ
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Tin =m ¼ nFTin r1 þ nðMspin Þin sinðh þ cÞ
ð14Þ
FDout ¼ nFN sinðh cÞ
ð15Þ
Tout =m ¼ nFTout r3 nðMspin Þout sinðh cÞ
ð16Þ
where Tin and Tout stand for the input and output torque, and FDin and FDout stand for the axial load on the input and output discs. Let us now define the traction coefficient l as the ratio between the traction tangential force FT and the normal force FN , at the input and output sides of the variator: lin ¼ FTin =FN
ð17Þ
lout ¼ FTout =FN Also the spin momentum coefficients vin and vout can be defined as: vin ¼ MSin =ðFN r1 Þ
ð18Þ
vout ¼ MSout =ðFN r3 Þ Together with these coefficient it is useful to define the following dimensionless quantities: fDin ¼ FDin =ðnFN Þ;
fDout ¼ FDout =ðnFN Þ
fR ¼ FR =FN tin ¼ Tin =ðmnFN r1 Þ;
tout ¼ Tout =ðmnFN r3 Þ
ð19Þ
tBL ¼ TBL =ðFN r0 Þ With the above mentioned dimensionless quantities, Eqs. (11)–(16) can be rephrased in a dimensionless form as: fR ¼ 2 cos h
ð20Þ
tBL ¼ ðlin lout Þ sin h þ fvin ½1 þ k cosðh þ cÞ þ vout ½1 þ k cosðh cÞg cos h
ð21Þ
fDin ¼ sinðh þ cÞ;
ð22Þ
fDout ¼ sinðh cÞ
tin ¼ lin þ vin sinðh þ cÞ;
tout ¼ lout vout sinðh cÞ
ð23Þ
The above defined quantities, enable us to find a simple expression of the mechanical efficiency of the variators: m¼
Tout x3 r1 Tout r3 x3 r1 Tout sr r1 Tout ¼ ¼ ¼ mspeed ¼ mtorque mspeed Tin x1 r3 Tin r1 x1 r3 Tin srID r3 Tin
where the torque efficiency mtorque has been defined as: mtorque ¼
ðTout =r3 Þ tout lout vout sinðh cÞ ¼ ¼ ðTin =r1 Þ lin þ vin sinðh þ cÞ tin
ð24Þ
It represents the ratio between the actual output torque Tout and the output torque that would be transmitted if the the spin momenta MSin and MSout and the torque loss in the roller bearing TBL were absent. The overall CVT mechanical efficiency can, therefore, be rewritten as:
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
m ¼ mspeed mtorque ¼ ð1 CrÞ
lout vout sinðh cÞ lin þ vin sinðh þ cÞ
ð25Þ
4. Contact model A fully flooded isothermal contact model between disks and rollers, based on the results of EHL theory, is adopted to evaluate the slip and spin losses. Because of the severe fluid contact conditions the Bair and Winer non-Newtonian model is used to describe the rheological behavior of the fluid. The influence of the pressure on the limiting shear stress is also taken into account, and according to Roelands [32], the effect of the pressure on the fluid viscosity is considered too. The film thickness of the traction oil is estimated by means of the Hamrock and Dowson formulas [32,33], whereas the pressure distribution over the contact area is supposed to obey to the Hertz law for the dry contact. This last hypothesis is commonly adopted in hard-EHL contacts because the very high contact pressure results in an almost constant thickness of the oil film, except for a very narrow area near to the outlet region of the contact [34–36]. The model does not account for the influence of the temperature gradients on the fluid properties, since there is not an universally accepted technique to calculate this effect. The preferred methods are based on the fluid flash temperature, but these techniques have been developed for line contacts, and they are very difficult to validate experimentally since there is no direct way to measure the fluid temperature. Therefore, it is expected that, for high values of the creep coefficients, the performances of the traction drives will be worse than those calculated by the proposed model. But, typically, the creep coefficient are limited to 2–3% and the proposed analysis can be still considered accurate. 4.1. Contact pressure distribution The evaluation of the pressure distribution and the calculation of the extension of the contact region need the knowledge of the equivalent radius of curvature of the contacting surfaces. Moreover, also the ellipticity parameter e and the complete elliptic integrals of the first and second kinds I1 and I2 need to be known. The equivalent curvature of the contacting surfaces is easy to calculate since the geometry of the variators is given. Therefore, once defined the dimensionless radii of curvature as the ratio of the dimensional radius of curvature and the cavity radius r0 , i.e. ~j ¼ qj =r0 and ~r22 ¼ r22 =r0 (the subscript j refers to the generic radius of curvature) the following q relations hold true at the input and output points of contact, respectively: 1 ~eqX q
¼ in
1 ~eqY q
¼
in
1 ~eqin q
r0 qeqX
¼ in
r0 qeqY
in
¼
¼
1þk ; 1 þ k cosðh þ cÞ 1 1; ~r22
1 ~eqY q
out
¼
1
¼
~eqX q
out
r0 qeqY
out
r0 cosðh þ cÞ 1 ¼ þ ; qeqin 1 þ k cosðh þ cÞ ~r22
¼
r0
¼
qeqX
out
1þk 1 þ k cosðh cÞ
1 1 ~r22 1
~eqout q
¼
ð26Þ ð27Þ
r0 qeqout
¼
cosðh cÞ 1 þ 1 þ k cosðh cÞ ~r22
ð28Þ
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To evaluate the pressure distribution, the semi-axes aX and aY of the contact ellipse have to be calculated, the simplified approach of Hamrock and Brewe [32] is adopted, thus the ellipticity 1=2 R p=2 parameter e ¼ aY =aX , and the elliptic integrals I1 ¼ 0 1 ð1 1=e2 Þ sin2 / d/ and 1=2 R p=2 2 2 I2 ¼ 0 1 ð1 1=e Þ sin / d/ can be estimated as [32]: e ¼ n2=p p p I1 ¼ þ 1 ln n 2 2 p 1 I2 ¼ 1 þ 1 2 n
ð29Þ ð30Þ ð31Þ
where the dimensionless quantity n stands for the ratio between the principal radii of curvature ~eqY =~ n¼q qeqX . The calculation of the semi-axes aX and aY of the contact ellipse can be done, now, by means of Eq. (34) where the dimensionless semi-axes ~ aX and ~aY , defined in Eq. (33) appear. The contact length parameter is: 1=3 6FN r0 ð32Þ K¼ pE0 and the dimensionless semi-axis of the contact ellipse are: aX aY ~ aX ¼ ; ~ aY ¼ ð33Þ K K where the quantity E0 is the effective elastic modulus defined as E0 ¼ E=ð1 m2 Þ, m is the Poisson’s ratio and E is the modulus of elasticity of both roller and disc. The Hertz formulas give [32]: !1=3 1=3 ~ q I 2 eq ~eq ~ ; ~ aX ¼ ð34Þ aY ¼ e2 I2 q e By introducing the dimensionless pressure p~ ¼ pK2 =FN , the dimensionless maximum half-amplitude of the subsurface orthogonal shear stress ~s0 ¼ s0 K2 =FN and the dimensionless co-ordinates X ¼ x=aX , Y ¼ y=aY the Hertz theory yields: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 ~ pMax ¼ ð35Þ p¼~ pMax 1 X 2 Y 2 ; ~ 2 p~ aX ~ aY ~s0 ¼ ~ pMax
ð21 1Þ1=2 21ð1 þ 1Þ
ð36Þ
where the auxiliary quantity 1 satisfies the following relation: e2 ð12 1Þð21 1Þ 1 ¼ 0
ð37Þ
Fig. 7 shows that, by means of an appropriate choice of the rollers curvature r22 , it is possible to obtain, over the whole ratio range, almost the same value of ~s0 ¼ s0 K2 =FN for both variators investigated in this paper. Observe that this quantity is one of the most important parameter to evaluate the stress severity as regards the fatigue life of the variator. A good choice consists in
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0.36
0.32
0.28 Half-Toroidal Full-Toroidal
0.24
0.2 0.5
0.75
1
1.25
1.5
1.75
2
s r ID
Fig. 7. The dimensionless maximum half-amplitude of the subsurface orthogonal shear stress ~s0 ¼ s0 K2 =FN versus the ideal speed ratio srID for the following dimensionless parameters ~r22HT ¼ r22HT =r0 ¼ 0:8, ~r22FT ¼ r22FT =r0 ¼ 0:66.
making ~s0 exactly the same when the ideal speed ratio is srID ¼ 1. To obtain this result the following values of the dimensionless parameter ~r22 ¼ r22 =r0 has been chosen: ~r22HT ¼ r22HT =r0 ¼ 0:8;
~r22FT ¼ r22FT =r0 ¼ 0:66
ð38Þ
4.2. EH lubrication model In EHL contacts the pressure can rise up to 3 GPa, thus producing very severe lubricant operative conditions. According to the Roelands model [32] and considering the isothermal contact hypothesis the following Eq. (39) enables us to estimate the fluid viscosity over the whole contact region: !Z 1 # " g p~ p g0 logð~ gÞ ¼ log 1 log ¼ 1þ ð39Þ g0 6R~cp g1 where ~cp ¼ cp =E0 , cp ¼ 1:96 108 Pa, g is the absolutely viscosity at the pressure p, g0 is the absolute viscosity at the atmospheric pressure for the given temperature, g1 ¼ 6:31 105 Pa s, the dimensionless constant Z1 is the viscosity-pressure index and the new dimensionless load parameter is: 0 2 1=3 r0 pE r0 ð40Þ R¼ ¼ K 6FN The non linear behaviour of the traction oil is described in Eq. (41) according to the normal rule used in the plasticity theory to split the shear strain along the different directions: ovi ovj sij þ ¼ Cðse Þ oxj oxi se
ð41Þ
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
In the above written Eq. (41) the equivalent stress se has been defined as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi se ¼ ðsij sij Þ=2
933
ð42Þ
The function Cðse Þ is representative of the non-linear behavior of the oil. One of the most commonly used explicit representation of this function is the one by Bair and Winer [32]: sL 1 ln Cðse Þ ¼ ð43Þ 1 se =sL g The quantity sL is the limiting shear stress of the lubricant and is normally evaluated, for a certain value of the oil temperature, by means of the following Eq. (44): sL ¼ sL0 þ ap
ð44Þ
where sL0 is the limiting shear stress at the atmospheric pressure [32]. Observe that, on that part of the contact region where significant shear stresses are involved, the film thickness of the lubricant could be considered almost constant, thus the following relation hold true at the input and output side of the variator, respectively, (h is the oil film thickness): v21X ovx s21X ¼ ¼ Cðse Þ; h oz se v23X ovx s23X Cðse Þ; ¼ ¼ h oz se
v21Y ovy s21Y ¼ ¼ Cðse Þ h oz se v23Y ovy s23Y Cðse Þ ¼ ¼ h oz se
ð45Þ ð46Þ
From the above written Eqs. (45) and (46) and from the Bair and Winer model (see Eq. (43)) it is possible to obtain an explicit formulation for the shear stress acting on the rollers at the input contact area: gjv j gjv j 6 v21X 6 v21Y 21 21 ~s21X ¼ R~sL 1 e hsL ; ~s21Y ¼ R~sL 1 e hsL ð47Þ p p jv21 j jv21 j and at the output contact region: gjv j 6 v23X hs23 L ~s23X ¼ R~sL ; 1e p jv23 j
~s23Y
gjv j 6 v23Y hs23 L ¼ R~sL 1e p jv23 j
ð48Þ
where ~s21X ¼ s21X K2 =FN , ~s21Y ¼ s21Y K2 =FN , ~s23X ¼ s23X K2 =FN , ~s23Y ¼ s23Y K2 =FN and ~sL ¼
sL sL0 p p p~ ¼ 0 þ a 0 ¼ ~sL0 þ a E 6R E0 E
ð49Þ
It is possible to rephrase Eqs. (47) and (48) in terms of the following dimensionless parameters: ~Y v21X r21 a v21Y r21 ~ aX ¼ Crin Y; ¼ X; jx1 jr1 R ~r1 jx1 jr1 R ~r1 0 !2 !2 11=2 jv21 j r21 ~ r21 ~ aY aX ¼ @ Crin Y þ X A jx1 jr1 R ~r1 R ~r1
ð50Þ
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
~Y v23X Crout r23 a v23Y r23 ~aX ¼ Y; ¼ X; jx3 jr3 1 Crout R ~r3 jx3 jr3 R ~r3 0 !2 !2 11=2 jv23 j Crout r23 ~ r23 ~aX aY þ ¼@ Y þ X A jx3 jr3 1 Crout R ~r3 R ~r3
ð51Þ
21 j 23 j and gjv , that appear in Eqs. (47) and (48), the following Whereas, as regards, the quantities gjv hsL hsL relations hold true: 0 !2 !2 11=2 ~ g gjv21 j r21 ~ r21 ~aX aY ~ 1 ð1 þ kÞ@ Crin ¼ Y þ X A ð52Þ x Hin~sL hsL R ~r1 R ~r1
0 !2 ~ ~ g gjv23 j Cr r a out 23 Y ~ 1 ð1 þ kÞð1 Crin Þð1 Crout ÞsrID @ þ ¼ Y x Hout~sL hsL 1 Crout R ~r3
þ
!2 11=2 ~X r23 a X A R ~r3
ð53Þ
where the dimensionless rotating velocity has been defined as: g x1 ~1 ¼ 0 0 x E
ð54Þ
This quantity takes into account the effect of the rotating speed of the input disc on the variator behaviour. Hin ¼ h=qeqX in is the dimensionless thickness of the oil film at the input contact zone, it can be evaluated by means of the hard EH lubrication formulas [32,33], where the above defined dimensionless parameters are used: h i0:67 0:134 ~f0:53 R0:201 q ~1 ~eqX 1 0:61 e0:73ein ð55Þ Hin ¼ 2:81 ð1 þ kÞð1 0:5Crin Þx in
In Eq. (55) the dimensionless pressure–viscosity coefficient ~f of the oil has been defined as (see [32]): ~f ¼ fE0 ¼ Z1 ln g0 ð56Þ ~cp g1 As regards the dimensionless oil film Hout at the output contact, it can be calculated by means of the relation: Hout 2 Crout 1 0:61 e0:73eout ¼ ð1 Crin Þ0:67 s0:536 rID Hin 2 Crin 1 0:61 e0:73ein
ð57Þ
Equation (57) shows that, since the creep coefficients in a normal operative condition are sufficiently small and the last term in Eq. (57), related to the ellipticity parameters ein and eout , is very close to the unity, the ratio Hout =Hin can be evaluated by means of the simpler well approximate relation
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
Hout ¼ s0:536 rID Hin
935
ð58Þ
that Eq. (58) shows that the oil film thickness at the input and output points of contact, respectively, may differ significantly, especially at the extreme values of the speed ratio. 4.3. Calculation of the traction coefficient and spin momentum The contact model described in Section 4, enable us to find an integral relation that allows for the calculation of the traction coefficients lin and lout and the spin momentum coefficients vin and vout , as reported in Eqs. (59) and (60). Z 2p Z 1 ~s21X R dw aXin ~ dR lin ¼ ~ aYin 0 0 ð59Þ Z 2p Z 1 aXout ~aYout lout ¼ ~
dR 0
~s23X R dw 0
Z 2p Z ~ aXin ~ aYin 1 ~ vin ¼ dR aXin ~s21Y cos w ~aYin ~s21X sin w R2 dw R~r1 0 0 Z 2p Z 1 ~ aXout ~ aYout ~ dR vout ¼ aXout ~s23Y cos w ~aYout ~s23X sin w R2 dw R~r3 0 0 where the following co-ordinate transformation has been applied:
X ¼ R cos w ; 0 6 R 6 1; 0 6 w 6 2p Y ¼ R sin w
ð60Þ
ð61Þ
5. Results Since the scope of the paper is to compare the half-toroidal and the full-toroidal traction drives, it is necessary to specify which quantities will be kept constant during the calculations. These are the ideal speed ratio srID , and the normal force FN at the points of contact, whereas the geometrical quantities are reported in Table 1, and the fluid properties in Table 2. Fig. 8 shows the efficiency m of the two variators as a function of the input dimensionless torque tin , for srID ¼ 1, and R ¼ 24. This last parameter corresponds, for the given geometry to a maximum pressure value close to 2.2–2.3 GPa. Moreover, the angular velocity of the input disc has been chosen equal to ~ 1 ¼ 2:95 1012 . As shown in jx1 j ¼ 2000 [RPM], that corresponds to a dimensionless value x Fig. 8 the better efficiency of both variators is reached in the region of high tin . Moreover, the efficiency of the full toroidal traction drive is smaller of about 4–5% points, compared to that of the half-toroidal variator over the whole range of tin values. The reason of this difference is caused, in the region of low tin , mostly by the worse torque efficiency of the full-toroidal variator (see Fig. 9). In fact, Fig. 10 shows that, because of the torque loss in the roller bearing and the larger contact area (the radius of curvature ~r22 is bigger than in full-toroidal variator), the half-toroidal
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Table 2 The fluid properties Fluid properties: T ¼ 99 °C g0 ¼ 3:25 103 Pa s Z1 ¼ 0:85 f ¼ 1:71 108 Pa1 sL0 ¼ 0:02 109 Pa a ¼ 0:085 cp ¼ 1:96 108 Pa g1 ¼ 6:31 105 Pa s
Absolute viscosity at the atmospheric pressure Viscosity–pressure index Pressure–viscosity coefficient Limiting shear stress at atmospheric pressure Limiting shear stress constant Pole pressure constant of Roelands viscosity model Pole viscosity of Roekands viscosity model
ν
1 0.95 0.9
s r ID =1 0.85
ℜ = 24 Half-Toroidal Full-Toroidal
0.8
2000 [RPM]
0.75 0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 8. The efficiency of the variators as a function of the input traction coefficient tin .
ν torque
1
0.95 0.9
s r ID =1 0.85
ℜ = 24 Half-Toroidal Full-Toroidal
0.8
2000 [RPM]
0.75 0.7 0
0.02
0.04
0.06
0.08
t in
0.1
Fig. 9. The torque efficiency of the variotors as a function of the input traction coefficient tin .
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 χ in
937
0.003 0.0025
s r ID =1
ℜ = 24
0.002
2000 [RPM] 0.0015 0.001
Half-Toroidal Full-Toroidal
0.0005 0 0
0.02
0.04
0.06
0.08
0.1
t in
Fig. 10. The spin momentum of the variators as a function of the input traction coefficient tin .
traction drive is more effected by the spin momentum. But, because of its very small spin velocity (see Fig. 5), the energy losses due to spin are less important than in the full-toroidal traction drive. Fig. 9 shows that also for mid and high tin values the efficiency of full-toroidal CVT is smaller than that of the half-toroidal traction drive, this is due mostly to the worse speed efficiency of the full-toroidal variator as shown in Fig. 11. In fact, the large values of the spin motion, that affect the full-toroidal CVT, cause the global sliding coefficient Cr to increase very fast as the requested torque tout increases (see Fig. 12), thus causing larger power losses. Regarding the influence of the dimensionless parameter R, Figs. 13 and 14 show how it affects the efficiency of both CVT variators: the bigger R the higher the mechanical efficiency of the variators. The explanation is simple: the efficiency of the variators is largely affected by the spin losses, that in turn depends on the extension of the elliptical contact area. By reducing the load, i.e. by incrementing R, the area of contact reduces its extension. This, in turn, causes a reduction
ν speed
1
0.95 0.9
s r ID =1 0.85
ℜ = 24 Half-Toroidal Full-Toroidal
0.8
2000 [RPM] 0.75 0.7 0
0.02
0.04
0.06
0.08
0.1
t in
Fig. 11. The speed efficiency of the variators as a function of the input traction coefficient tin .
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 t out
0.1 0.09 0.08 0.07 0.06
s r ID =1
0.05
ℜ = 24
0.04
Half-Toroidal Full-Toroidal
0.03 0.02
2000 [RPM]
0.01 0 0
0.05
0.1
0.15
0.2
Cr
Fig. 12. The traction capabilities of the variators: output traction coefficient tout as a function of the global sliding coefficient Cr.
ν
1
Half-Toroidal
0.95
ℜ = 48
0.9
ℜ = 42 ℜ = 36
0.85
s r ID =1
ℜ = 30
0.8
2000 [RPM]
ℜ = 24
0.75
ℜ = 18
0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 13. The efficiency of the half-toroidal variator as a function of the input traction parameter tin and for different values of the load parameter R.
of the spin momentum, and therefore of the energy dissipated by the spin motion. Morevoer, since the spin motion affects the FT variator more than the HF one, it is expected that the full-toroidal variator be more sensible to the load variations than the half-toroidal CVT, as clearly demonstrated in Figs. 13 and 14. Therefore, it may happen that for higher values of R the efficiency of the full-toroidal variator overcomes that of the half-toroidal traction drive: see, for example, the curves plotted for R ¼ 48 in Figs. 13 and 14. But, on the other hand, too big values of R reduce the traction capability of the variators, i.e the maximum values of tin . ~ 1 does not affect appreciably the efficiency of The simulations have shown that the parameter x both variators. Moreover, it was shown that the FT traction drive, because of its particular
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 ν
1
939
Full-Toroidal
ℜ = 48 ℜ = 42
0.95 0.9
ℜ = 36
0.85
ℜ = 30
0.8
s r ID =1
ℜ = 24
2000 [RPM] 0.75
ℜ = 18 0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 14. The efficiency of the full-toroidal variator as a function of the input traction parameter tin and for different values of the load parameter R.
ν
1
Half-Toroidal
0.95 0.9 2000 [RPM] 0.85
ℜ = 24 s r ID =2/3 s r ID =1
0.8
s r ID =1.5 0.75 0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 15. The efficiency of the half-toroidal variator as a function of the input traction parameter tin and for different values of the speed ratio srID .
symmetry, is almost insensible to the actual value of ideal speed ratio srID , whereas the HT variator results to be largely affected as shown in Fig. 15. In conclusion, the analysis carried out, shows that the full-toroidal traction drive is unable to achieve the same efficiency of the HT variator, despite the torque losses in the support bearing that affect the latter. The main reason of this result is the very high spin motion of the FT variator that produces high values of energy dissipation. It has been also shown that only two parameters influence significantly the mechanical efficiency of the variators, these are the dimensionless load parameter R and the dimensionless input torque tin .
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6. Conclusions The paper deals with the mechanical efficiency of full and half toroidal traction drives and has the aim of comparing the performances of these two CVT typologies. A fully flooded isothermal model of the contact between discs and rollers, based on the results of EHL theory, has been implemented to evaluate the slip and spin losses. The analysis has shown that by optimizing the roller geometry of the full toroidal variator, it is possible to reduce its spin momentum below the value of the half-toroidal variator. But, since the energy dissipation due to the spin losses is the product of the spin momentum and the spin velocity, the full-toroidal variator always results to have a smaller efficiency because of its much bigger spin velocity. Moreover the full-toroidal traction drive needs higher values of global creep (smaller speed efficiency) to transmit the same torque of the HT traction drive. This causes an additional heating of the lubricant and a further reduction of its traction capability and also of the mechanical efficiency. Moreover, the mechanical efficiency of the half-toroidal CVT, in comparison to that of the full-toroidal variator, is less affected by the value of the normal contact forces and often over the threshold of 90% on the most part of the torque range. The full-toroidal CVT, instead, shows a different behavior, its efficiency varies within a bigger range of values, and it is more affected by the normal load at the contact between the rollers and discs.
References [1] 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, in: Proceedings of International Congress CVT’99, Eindhoven, The Netherlands, September 16–17, 1999, pp. 27–33. [2] C. Brace, M. Deacon, N.D. Vaughan, C.R. Burrows, R.W. Horrocks, Integrated passenger car diesel CVT powertrain control for economy and low emissions, in: ImechE International Seminar S540, Advanced Vehicle Transmission and Powertrain Management, September 25–26, 1997. [3] G. Carbone, L. Mangialardi, G. Mantriota, Fuel consumption of a mid class vehicle with infinitely variable transmission, SAE Transaction 2001 Journal of Engines 110 (Section 3) (2002) 2474–2483. [4] L. Mangialardi, G. Mantriota, The advantages of using continuously variable transmission in wind power systems, Renewable Energy 2 (3) (1992) 201–209. [5] L. Mangialardi, G. Mantriota, Continuously variable transmission with torque-sensing regulators in waterpumping windmills, Renewable Energy 7 (2) (1992) 807–823. [6] L. Mangialardi, G. Mantriota, Automatically regulated C.V.T. in wind power systems, Renewable Energy 4 (3) (1994) 299–310. [7] G. Carbone, L. Mangialardi, G. Mantriota, Influence of clearance between plates in metal pushing V-belt dynamics, ASME Journal of Mechanical Design 124 (September) (2002) 543–557. [8] G. Carbone, L. Mangialardi, G. Mantriota, EHL visco-plastic friction model in CVT shifting behaviour, International Journal of Vehicle Design 32 (3–4) (2003) 332–357 (A special Issue on Advancements in the field of vehicle transmission). [9] G. Carbone, L. Mangialardi, G. Mantriota, The influence of pulley deformations on the shifting mechanism of MVB-CVT, ASME Journal of Mechanical Design (in press). [10] M. Patterson, the Full-Toroidal Variator in Theory and in Practice, Torotrak (Development) Ltd, in: Proceedings of CVT ’96 Congress, Paper No. 9636394, SAE of Japan, Yokohama, September 11–12, 1996, pp. 95–100.
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[11] M. Patterson, Traction drive contact optimisation, in: Proceedings of the Leeds-Lyon Symposium, Vehicle Tribology, in: D. Dowson (Ed.), Tribology Series 18, Paper X (iv), Elsevier Science Publisher B.V., Amsterdam, 1991, pp. 295–300. [12] H Machida, S Aihara, State of art of the traction drive CVT applied to automobiles, in: Proceedings of the LeedsLyon Symposium 17, in: D. Dowson (Ed.), Tribology Series 18, Paper X (i), Elsevier Science Publisher B.V., Amsterdam, 1991, pp. 267–275. [13] M Taniguchi, H Machida, S Aihara, Some experimental findings at EHL contacts under traction with a heavy-duty traction drive, in: Proceedings of the Leeds-Lyon Symposium 17, in: D. Dowson (Ed.), Tribology Series 18, Paper X (iii), Elsevier Science Publisher B.V., Amsterdam, 1991, pp. 287–294. [14] H. Machida, Traction drive CVT up to date, in: Proceedings of the CVT’99 Congress, Eindhoven, The Netherlands, September 16–17, 1999, pp. 71–76. [15] H. Kumura, M. Nakano, T. Hibi, J. Sugihara, K. Kobayashi, Performance of a Dual-Cavity Half-Toroidal CVT for Passenger Cars, in: Proceedings of the CVT’96 Congress, 1996. [16] H. Kumura, J. Sugihara, Y. Partita, Y. Arakawa, M. Nakano, N. Maruyama, Development of a dual-cavity halftoroidal CVT, in: Proceedings of the CVT’99 Congress, 1999, Eindhoven, The Netherlands, September 16–17, 1999, pp. 65–70. [17] H. Machida, H. Itoh, T. Imanishi, H. Tanaka, Design principle of high power traction drive CVT, SAE Technical Paper, in: International Congress and Exposition Detroit, No. 950675, Michigan, February 27–March 2, 1995, pp.79–89. [18] P. Tenberge, Ratio stability of toroidal traction drives, in: Proceedings of the CVT’99 Congress, inserire pag., Eindhoven, The Netherlands, September 16–17, 1999. [19] H. Mori, T. Yamazaki, K. Kobayashi, T. Hibi, A study on the layout and ratio change characteristics of a dualcavity half-toroidal CVT, JSAE Review 22 (2001) 299–303. [20] H. Tanaka, J. Yokohama, Aspect of variator control of half-toroidal CVT, in: Proceedings of the CVT 2002 Congress, 2002, pp. 23–33. [21] H. Tanaka, Speed ratio control of a parallel layout double cavity half-toroidal CVT for four-wheel drive, JSAE Review 23 (2002) 213–217. [22] H. Tanaka, H. Machida, H. Hata, M. Nakano, Half-toroidal traction drive continuously variable power transmission for automobiles, JSME Internal Journal, Series C 38 (4) (1995). [23] H. Machida, T. Abe, Fatigue life analysis of a traction drive CVT, in: Proceedings of the CVT’96 Congress, 1996, pp. 101–106. [24] H. Tanaka, H. Machida, Half-toroidal traction-drive continuously variable power transmission, Proceedings of the Institution of Mechanical Engineers 210 (July) (1996) 205–212. [25] T. Yamamoto, K. Matsuda, T. Hibi, Analysis of the efficiency of a half-toroidal CVT, JSAE Review 22 (2001) 565– 570. [26] H. Machida, Y. Murakami, Development of the POWERTOROS Unit half toroidal CVT, Motion and Control No. 9 NSK, Oct. 2000, pp. 15–26. [27] T. Imanishi, H. Machida, H. Tanaka, A geometric study of toroidal CVT, in: Proceedings of CVT ’96 Congress, Paper No. 9636411, SAE of Japan, Yokohama, Sept. 11–12, 1996, pp. 107–111. [28] T. Imanishi, H. Machida, Development of POWERTOROS Unit Half-Toroidal CVT (2), Motion and Control No. 10 NSK, April 2001, pp. 1–8. [29] M. Raghavan, Kinematics of the full-toroidal traction drive variator, ASME Journal of Mechanical Design 124 (Sept.) (2002) 448–455. [30] Y. Zhang, X. Zhang, W. Tobler, A systematic model for the analysis of contact, side slip and traction of toroidal drives, ASME Journal of Mechanical Design 122 (December) (2000) 523–528. [31] Z. Zou, Y. Zhang, X. Zhang, W. Tobler, Modeling and simulation of traction drive dynamics and control, ASME Journal of Mechanical Design 123 (December) (2001) 556–561. [32] B.J. Hamrock, Fundamentals of fluid film lubrication, Series in Mechanical Engineering, Book, McGraw-Hill, 1994 (ISBN 0-07-025956-9). [33] D. Dowson, Modelling of elastohydrodynamic lubrication of real solids by real lubricants, Meccanica, Kluwer Academic Publishers, vol. 33, 1998, pp. 47–58.
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[34] K.L. Johnson, J.L. Tevaarwek, The influence of fluid rheology on the performance of traction drives, Journal of Lubrication Technology, Transaction of the ASME 101 (July) (1979) 266–274. [35] K.L. Johnson, J.L. Tevaarwek, Shear behaviour of elastohydrodynamic oil films, Proceedings of the Royal Society of London A 356 (1977) 215–236. [36] K.L. Johnson, R. Cameron, Shear behaviour of elastohydrodynamic oil films at high rolling contact pressures, in: Proceedings of IMechE Tribology Group 182(Part 1), 1967–68.
Mechanism and Machine Theory 39 (2004) 921–942
www.elsevier.com/locate/mechmt
A comparison of the performances of full and half toroidal traction drives G. Carbone *, L. Mangialardi, G. Mantriota Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, V.le Japigia 182, 70126 Bari, Italy Received 5 May 2003; received in revised form 21 January 2004; accepted 10 April 2004
Abstract The efficiency of two different typologies of the toroidal traction drive, the full-toroidal and the halftoroidal, is estimated in order to point out which of them offers the higher mechanical efficiency. A fully flooded isothermal contact model between the discs and rollers, based on the results of EHL theory, is used to evaluate the slip, the spin losses and the mechanical performances of the variators. It is shown that the half-toroidal traction drive offers higher efficiency and higher maximum transmissible torque. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The request for a higher energy efficiency and CO2 reduction has pushed several researchers to find new technical solutions to improve the emission performance of nowadays IC engine vehicles. While waiting for new and renewable forms of energy to become effective and cost reasonable, new solutions have to be found: among different and several technical solutions, new drive train systems are being investigated to accomplish this purpose. The continuously variable transmission (CVT) represents one of the most promising solution since it is able to provide an infinite number of gear ratios between two finite limits, and, thus, to allow the IC engine to operate closer to its optimal efficiency line. Several studies have shown that it is possible to improve the fuel savings and to reduce the vehicle emissions by adopting this kind of transmission [1,2]. It has been shown, for instance, that when the transmission is optimally controlled the mid class vehicles equipped with the CVT and
*
Corresponding author. Tel.: +39-080-596-2746; fax: +39-080-596-2777. E-mail addresses: [email protected] (G. Carbone), [email protected] (L. Mangialardi), [email protected] (G. Mantriota). 0094-114X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.04.003
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IVT variators may achieve less fuel consumption (about 10% for CVT and 6% for IVT) and higher comfort in the urban traffic [3]. Moreover applications of CVTs to wind power systems have been proposed, and some papers [4–6] have shown that a significant increase of the energy production may be attained. The metal pushing V-belt and the metal chain CVTs are able to achieve these results, but they have some drawbacks as the strict dependence of the shifting speed on the clamping forces acting on the moving pulley sheaves [7–9], and the smaller torque capacity when compared to toroidal traction drives. The robotized gearbox may be a different solution, since it combines the economy of a welldriven manual transmission with the easy to use of a conventional automatic transmission, and get the further advantage to retain the simplicity and the economies of scale of an established manual design. But, in this case, some problems regarding the shift quality arise: the torque interruption becomes intrusive because the driver is not able to predict or anticipate the gear shift. For these reasons more promising solutions have to be found, and the toroidal traction drives may be one of these. They are being extensively investigated because of their high torque capacity, that makes them suitable for application in larger engine cars and even trucks. The most attractive typologies are the full-toroidal [10,11] and the half-toroidal traction drives [12–14]. The main components of these transmissions are the input and output discs, designed to create a toroidal cavity (see Fig. 1), coupled with an appropriate number of rollers. The high torque capacity of these transmissions is obtained by coupling together in a series scheme two or more single units [15–17]. Moreover the particular geometry of the toroidal traction drive makes it able to rapidly adjust its speed ratio to the request of the driver, thus improving the driving comfort [18–21]. Between the roller and the discs no metal–metal contact occurs, the torque is transmitted by means of the shearing action of a special oil referred to as traction oil. The lubrication regime of such a system is the hard EHL with pressures up to 3 GPa. The high pressures lead to a much higher oil viscosity (several order of magnitude) than in the normal hydrodynamic regime, thus enabling the transmission of high torque despite the very small area of contact. In the technical literature many works can be found concerned with the design, fatigue life, and thermal stresses of the half-toroidal CVT [22–26], but not many contributions compare the full-
(a)
Fig. 1. The traction drive CVTs: (a) half-toroidal; (b) full toroidal.
(b)
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
923
Table 1 CVT geometric data Half toroidal CVT:
Full toroidal CVT
Cavity radius r0 ¼ 40 mm Roller curvature r22 ¼ 32 mm Half cone-angle h ¼ p=3 Aspect ratio k ¼ e=r0 ¼ 0:625 Speed ratio range s ¼ 0:5–2:0
Cavity radius r0 ¼ 40 mm Roller curvature r22 ¼ 26:4 mm Half cone-angle h ¼ p=2 Aspect ratio k ¼ e=r0 ¼ 0:25 Speed ratio range s ¼ 0:5–2:0
toroidal and half-toroidal traction drives as regards the mechanical efficiency and their traction capabilities. Only few papers provide some experimental results on their efficiency [27–29]. The main scope of this work is to propose a theoretical model of the variators, based on the results of EHL lubrication theory, that is able to estimate the traction capability and the mechanical efficiency of these two different CVT typologies, and to point out which of them offers the higher mechanical efficiency. The analysis is limited to the simple variator as it is, made up of the input discs, the rollers, the support bearings and the output discs. The two variator are supposed to have the same cavity radius and the same radial size, the latter condition requires different values of the aspect ratio k ¼ e=r0 as reported in Table 1. The principal advantage of the proposed method with respect to other similar methodologies [30,31] consists of three points: the model is independent of the specific traction drive under consideration, the formulation presented does not require the determination of the traction curve by experiments performed on the given traction drive with the specific traction oil. The model is also able to take into account the influence of the spin motion on the mechanical efficiency and the traction performance of the variator. The main drawback is related to the large number of equations to deal with, that makes the overall computation time-consuming.
2. Geometric and kinematic quantities of full and half toroidal CVTs 2.1. Geometrical description of CVTs Fig. 1 shows the main geometrical features of the toroidal variators. During the steady state operation of the CVT the swing center of the roller coincides with the cavity center O, and its axis of rotation is tilted of c. The tilting angle c (positive if clockwise directed) controls the distance r1 and r3 of the contact points A and B from the main axis of the variator, and, consequently, controls the ideal speed ratio srID ¼ r3 =r1 . In the same Fig. 1, r12 ¼ r23 ¼ r0 represent radius of the toroidal cavity, that is also one of the two principal radii of curvature of the input and output discs. The quantity r11 is the second principal radius of curvature of the input disc, whereas r33 is the second principal radius of curvature of the output disc. Moreover r2 and r22 are the two principal radii of curvature of the roller with r22 < r12 . The quantity e is the distance of the toroidal cavity from the disc axes, it is related to the aspect ratio k ¼ e=r0 of the toroidal traction drive. Moreover in a half-toroidal CVT, the half cone-angle h of the roller is about 50–70°, whereas in a full-toroidal CVT the cone angle is 90°. All the remaining geometrical parameters are reported in Table 1.
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
2.2. Kinematic analysis of CVTs In this subsection the kinematics of the toroidal traction drives is analyzed during the steady state operation of the variator. Consider the roller and the discs as rigid bodies and assume no-slip at points A and B (see Fig. 2). Under these conditions, the motion of the roller relative to the input disc is a spherical rigid motion, of which the instantaneous axis of rotation can be easily determined as the straight-line through the points of null relative velocity A and X (the point X is the intersection of the roller absolute axis of rotation and the input-disc absolute axis of rotation). Similar arguments hold when studying the relative velocity field between the roller and the outputdisc, in this case the instantaneous axis of rotation of the relative motion is the straight-line BX. Let x1 , x2 and x3 be, respectively, the absolute angular velocities of the input disc, of the roller and that of the output disc. The angular velocity of the roller relative to the input disc is, therefore, x21 ¼ x2 x1 , whereas that one relative to the output disc is x23 ¼ x2 x3 . Figs. 2 and 3 show these two relative velocities of rotation x21 and x23 both for the half-toroidal and fulltoroidal CVTs. It is clearly shown that, since the point of intersection H (see Fig. 2) of the two tangents to the toroidal cavity at points A and B does not always coincides with the point X, both the relative angular velocities x21 and x23 have non-zero spin vector components x21spin and x23spin , respectively. Moreover, the ratio ðx21spin Þin =jx21 j ¼ ðx23spin Þout =jx23 j ¼ sin a, assumes its maximum value for srID ¼ r3 =r1 ¼ 1, since the distance between the points H and X is maximum for this value of the ideal speed ratio. Furthermore, Fig. 2 shows that, for the half-toroidal CVT, two points exist at which H and X coincide and the spin motion vanishes. Different considerations have to be done for the full-toroidal traction drive. This time, Fig. 3 shows that the point H goes to infinity, thus the spin motion never vanishes, and, because of the bigger angle a, it is always bigger than in the case of the half-toroidal CVT. Once again, the worst situation occurs for srID ¼ 1, when (see Fig. 3) the modulus of the spin vector components x21spin and x23spin equals the modulus of absolute angular velocity of the output and input discs, respectively. What happens when slip occurs, namely when torque is transmitted, is only slightly different from the above written scenario since the slip is always very small and, hence, the axes of rotation
(a)
(b)
Fig. 2. The half toroidal CVT: spin motion and rolling motion of the roller depicted for no-slip conditions; (a) speed ratio equal to 1; (b) no spin condition (speed ratio differs from the unit value).
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
925
(b)
(a)
Fig. 3. The full-toroidal traction drive: (a) speed ratio equal to 1; (b) same speed ratio as in (Fig. 2(b)) but the spin is always different from zero.
of the roller relative to the discs would result only slightly tilted relatively to those depicted in the Figs. 2 and 3. Now consider the contact area between the roller and the discs. In this region the elastic deformations of the bodies have a large influence on the relative velocity motion, and this one can no more be classified as a rigid motion. The region of contact is an elliptical area centered at the point of contact. The ellipse principal axes, (see Fig. 4) lay on the y-axis (the major) and on the x-axis (rolling direction, the minor). In order to calculate the shear strain of the lubricant we need to find an explicit formulation of the relative velocity field in the contact region. Observe that, over the contact area the bodies cannot penetrate each other, thus the relative velocity, because of the elastic deformations, do not have any component normal to the area of contact. Hence, assuming a negligible tangential deformation of the elastic bodies, the velocity of the roller points relative to the input and output discs, respectively, can be written, in the region of contact, as: v21 ¼ v21A þ x21spin ^ ðPin AÞ
ð1Þ
v23 ¼ v23B þ x23spin ^ ðPout BÞ
ð2Þ
Pin and Pout are points of the roller, while the velocity vectors v21A and v23B stand for the relative velocity between roller and discs at the center points A and B of the contact areas. Since we are considering only steady-state behavior v21A and v23B do not have components along the y-axes, yin and yout (see Fig. 4), but only along the x-axes. The previous Eqs. (1) and (2) show that the relative motion between the roller and the discs in the contact region, can be split into a pure translation, given by the vectors v21A and v23B , and a pure spin motion about the z-axes. 2.3. Practical aspects for CVTs When studying the toroidal traction drives, it is useful to define the input and output slip coefficients, usually referred to as creep coefficients Crin and Crout : Crin ¼
jx1 jr1 jx2 jr2 ; jx1 jr1
Crout ¼
jx2 jr2 jx3 jr3 jx2 jr2
ð3Þ
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
Fig. 4. The reference frames used at the input and output points of contact.
A small amount of creep must be always present to allow the transmission of torque. Besides the creep coefficients defined above, it is useful, for the next calculations, to introduce the following dimensionless geometric quantities (remember that r12 ¼ r23 ¼ r0 , see also Fig. 1): r1 ~r1 ¼ ¼ 1 þ k cosðh þ cÞ r0 ð4Þ r3 ~r3 ¼ ¼ 1 þ k cosðh cÞ r0 By means of the creep coefficients and considering that srID ¼ ~r3 =~r1 it is possible to write the actual speed ratio sr ¼ jx3 j=jx1 j as: sr ¼ ð1 Crin Þð1 Crout Þ
1 þ k cosðh þ cÞ ¼ ð1 Crin Þð1 Crout ÞsrID 1 þ k cosðh cÞ
and also define the speed efficiency mspeed of the variator as the ratio sr =srID : sr mspeed ¼ ¼ 1 Cr srID
ð5Þ
ð6Þ
where 1 Cr ¼ ð1 Crin Þð1 Crout Þ stands for the global sliding coefficient between the output disc and input one. In a similar way, it is also possible to write the spin ratios as a function of the creep coefficients: 1 þ k cosðh þ cÞ tan h 1 þ k cosðh cÞ tan h
r21 ¼ sinðh þ cÞ ð1 Crin Þ r23 ¼ sinðh cÞ
1 1 Crout
ð7Þ ð8Þ
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
927
1.1
(σ 21)no-slip
1
0.9 0.8 0.7 0.6 Full Toroidal Half Toroidal
0.5 0.4 0.3 0.2 0.1 0 0.5
0.75
1
1.25
1.5
1.75
s r ID 2
Fig. 5. The spin-ratios as a function of the ideal speed ratio srID , for no-slip conditions (traction drive data in Table 1).
The above written Eqs. (4)–(8) hold true also for the full toroidal traction drive with h ¼ p=2. As already discussed before, it is shown that, for the ideal case of no-slip, the worst condition as regards the magnitude of the spin occurs for c ¼ 0, that is to say for srID ¼ 1. In fact, replacing both Crin and Crout by zero, Eqs. (7) and (8) become: ðr21 Þno-slip ¼
cos c ð1 þ kÞ cos h ¼ ðr23 Þno-slip sin h
ð9Þ
Equation (9) shows that the two spin ratios r21 and r23 assume their maximum value when cos c ¼ 1, i.e. c ¼ 0. Observe that if cos h < ð1 þ kÞ1 two different values of the tilting angle c also exist at which the spin ratios vanish, as pointed out in Section 2.2 (see also Fig. 2). For full toroidal CVT replacing h by p=2 in Eq. (9) we obtain ðr21FT Þno-slip ¼ ðr23FT Þno-slip ¼ cos c which is always bigger than the spin ratios of an half-toroidal CVT. Fig. 5 shows, for no-slip conditions, the spin-ratios as a function of the ideal speed ratio srID . The CVT geometrical characteristics are reported in Table 1, where the radii of curvature r22 have been chosen in order to obtain the same maximum shear stress in both CVTs (see also Section 4.1). As before predicted the spin ratio of the full-toroidal CVT is about five times higher than that of the half-toroidal one.
3. Forces, spin momentum and efficiency The presence of spin motion affects the full-toroidal traction drive more than the half-toroidal variator. But, on the other hand, the latter is affected by the support bearing losses, since the normal forces FN acting on the roller, at the points of contact, do not balance out (see Fig. 6). Therefore, a resulting axial force FR has to be supported by an axial bearing (one for each roller), that, because of its internal losses, causes a reduction of the CVT mechanical efficiency. On the other hand, the full-toroidal variator is not affected by this problem, since h ¼ p=2, and the normal forces balance out. But, on the other hand, the full-toroidal CVT is much more affected by spin losses, thus it is necessary to carry out a comparison in order to single out which typology of CVT offers the higher mechanical efficiency.
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
Fig. 6. The free body diagram of the discs and roller; FN is the normal force at the point of contact, FR is the resulting load on the axial bearing, FDin and FDout are the axial clamping forces on the input and output discs, respectively, FTin and FTout are the traction (tangential) at the input and output points of contact, respectively, Tin and Tout are the input and output torques, MSin and MSout are the spin momenta at the input and output points of contact, respectively, TBL is the torque resistance of the axial bearing, n is the number of rollers per each cavity, and m is the number of cavities.
As regards the support bearing losses some studies have been carried out on this aspect of the HT-traction drives. For example in [25] the authors estimate the spinning losses of the power roller bearing by means of an elastic–plastic lubricant model, and obtain results in good agreement with the experiments. For our scope we will make use of the empirical relation Eq. (10), already used in [17], that gives the torque loss as a function of the axial thrust acting on the roller FR , and that results in agreement with the experimental results reported in [25]. In Eq. (10) the axial thrust acting on the roller FR is measured in (N) and the torque bearing loss TBL is measured in (Nm). TBL ¼ 4:6 105 FR1:03
ð10Þ
Fig. 6 shows the free body diagram of the input and output discs, and that one of the roller. The support bearing is modelled as a revolute movable joint, that prevents the translatory motion of the roller along the direction of its axis of rotation. The force balance of the roller gives: FR ¼ 2FN cos h
ð11Þ
FTin r2 FTout r2 TBL þ MSin cos h þ MSout cos h ¼ 0
ð12Þ
and where FTin and FTout stand for the traction forces at the points of contact, and MSin and MSout are the corresponding spin momenta. The equilibrium of the input and output discs, once chosen the number of rollers n and the number of cavity m, gives: FDin ¼ nFN sinðh þ cÞ
ð13Þ
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
929
Tin =m ¼ nFTin r1 þ nðMspin Þin sinðh þ cÞ
ð14Þ
FDout ¼ nFN sinðh cÞ
ð15Þ
Tout =m ¼ nFTout r3 nðMspin Þout sinðh cÞ
ð16Þ
where Tin and Tout stand for the input and output torque, and FDin and FDout stand for the axial load on the input and output discs. Let us now define the traction coefficient l as the ratio between the traction tangential force FT and the normal force FN , at the input and output sides of the variator: lin ¼ FTin =FN
ð17Þ
lout ¼ FTout =FN Also the spin momentum coefficients vin and vout can be defined as: vin ¼ MSin =ðFN r1 Þ
ð18Þ
vout ¼ MSout =ðFN r3 Þ Together with these coefficient it is useful to define the following dimensionless quantities: fDin ¼ FDin =ðnFN Þ;
fDout ¼ FDout =ðnFN Þ
fR ¼ FR =FN tin ¼ Tin =ðmnFN r1 Þ;
tout ¼ Tout =ðmnFN r3 Þ
ð19Þ
tBL ¼ TBL =ðFN r0 Þ With the above mentioned dimensionless quantities, Eqs. (11)–(16) can be rephrased in a dimensionless form as: fR ¼ 2 cos h
ð20Þ
tBL ¼ ðlin lout Þ sin h þ fvin ½1 þ k cosðh þ cÞ þ vout ½1 þ k cosðh cÞg cos h
ð21Þ
fDin ¼ sinðh þ cÞ;
ð22Þ
fDout ¼ sinðh cÞ
tin ¼ lin þ vin sinðh þ cÞ;
tout ¼ lout vout sinðh cÞ
ð23Þ
The above defined quantities, enable us to find a simple expression of the mechanical efficiency of the variators: m¼
Tout x3 r1 Tout r3 x3 r1 Tout sr r1 Tout ¼ ¼ ¼ mspeed ¼ mtorque mspeed Tin x1 r3 Tin r1 x1 r3 Tin srID r3 Tin
where the torque efficiency mtorque has been defined as: mtorque ¼
ðTout =r3 Þ tout lout vout sinðh cÞ ¼ ¼ ðTin =r1 Þ lin þ vin sinðh þ cÞ tin
ð24Þ
It represents the ratio between the actual output torque Tout and the output torque that would be transmitted if the the spin momenta MSin and MSout and the torque loss in the roller bearing TBL were absent. The overall CVT mechanical efficiency can, therefore, be rewritten as:
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
m ¼ mspeed mtorque ¼ ð1 CrÞ
lout vout sinðh cÞ lin þ vin sinðh þ cÞ
ð25Þ
4. Contact model A fully flooded isothermal contact model between disks and rollers, based on the results of EHL theory, is adopted to evaluate the slip and spin losses. Because of the severe fluid contact conditions the Bair and Winer non-Newtonian model is used to describe the rheological behavior of the fluid. The influence of the pressure on the limiting shear stress is also taken into account, and according to Roelands [32], the effect of the pressure on the fluid viscosity is considered too. The film thickness of the traction oil is estimated by means of the Hamrock and Dowson formulas [32,33], whereas the pressure distribution over the contact area is supposed to obey to the Hertz law for the dry contact. This last hypothesis is commonly adopted in hard-EHL contacts because the very high contact pressure results in an almost constant thickness of the oil film, except for a very narrow area near to the outlet region of the contact [34–36]. The model does not account for the influence of the temperature gradients on the fluid properties, since there is not an universally accepted technique to calculate this effect. The preferred methods are based on the fluid flash temperature, but these techniques have been developed for line contacts, and they are very difficult to validate experimentally since there is no direct way to measure the fluid temperature. Therefore, it is expected that, for high values of the creep coefficients, the performances of the traction drives will be worse than those calculated by the proposed model. But, typically, the creep coefficient are limited to 2–3% and the proposed analysis can be still considered accurate. 4.1. Contact pressure distribution The evaluation of the pressure distribution and the calculation of the extension of the contact region need the knowledge of the equivalent radius of curvature of the contacting surfaces. Moreover, also the ellipticity parameter e and the complete elliptic integrals of the first and second kinds I1 and I2 need to be known. The equivalent curvature of the contacting surfaces is easy to calculate since the geometry of the variators is given. Therefore, once defined the dimensionless radii of curvature as the ratio of the dimensional radius of curvature and the cavity radius r0 , i.e. ~j ¼ qj =r0 and ~r22 ¼ r22 =r0 (the subscript j refers to the generic radius of curvature) the following q relations hold true at the input and output points of contact, respectively: 1 ~eqX q
¼ in
1 ~eqY q
¼
in
1 ~eqin q
r0 qeqX
¼ in
r0 qeqY
in
¼
¼
1þk ; 1 þ k cosðh þ cÞ 1 1; ~r22
1 ~eqY q
out
¼
1
¼
~eqX q
out
r0 qeqY
out
r0 cosðh þ cÞ 1 ¼ þ ; qeqin 1 þ k cosðh þ cÞ ~r22
¼
r0
¼
qeqX
out
1þk 1 þ k cosðh cÞ
1 1 ~r22 1
~eqout q
¼
ð26Þ ð27Þ
r0 qeqout
¼
cosðh cÞ 1 þ 1 þ k cosðh cÞ ~r22
ð28Þ
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
931
To evaluate the pressure distribution, the semi-axes aX and aY of the contact ellipse have to be calculated, the simplified approach of Hamrock and Brewe [32] is adopted, thus the ellipticity 1=2 R p=2 parameter e ¼ aY =aX , and the elliptic integrals I1 ¼ 0 1 ð1 1=e2 Þ sin2 / d/ and 1=2 R p=2 2 2 I2 ¼ 0 1 ð1 1=e Þ sin / d/ can be estimated as [32]: e ¼ n2=p p p I1 ¼ þ 1 ln n 2 2 p 1 I2 ¼ 1 þ 1 2 n
ð29Þ ð30Þ ð31Þ
where the dimensionless quantity n stands for the ratio between the principal radii of curvature ~eqY =~ n¼q qeqX . The calculation of the semi-axes aX and aY of the contact ellipse can be done, now, by means of Eq. (34) where the dimensionless semi-axes ~ aX and ~aY , defined in Eq. (33) appear. The contact length parameter is: 1=3 6FN r0 ð32Þ K¼ pE0 and the dimensionless semi-axis of the contact ellipse are: aX aY ~ aX ¼ ; ~ aY ¼ ð33Þ K K where the quantity E0 is the effective elastic modulus defined as E0 ¼ E=ð1 m2 Þ, m is the Poisson’s ratio and E is the modulus of elasticity of both roller and disc. The Hertz formulas give [32]: !1=3 1=3 ~ q I 2 eq ~eq ~ ; ~ aX ¼ ð34Þ aY ¼ e2 I2 q e By introducing the dimensionless pressure p~ ¼ pK2 =FN , the dimensionless maximum half-amplitude of the subsurface orthogonal shear stress ~s0 ¼ s0 K2 =FN and the dimensionless co-ordinates X ¼ x=aX , Y ¼ y=aY the Hertz theory yields: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 ~ pMax ¼ ð35Þ p¼~ pMax 1 X 2 Y 2 ; ~ 2 p~ aX ~ aY ~s0 ¼ ~ pMax
ð21 1Þ1=2 21ð1 þ 1Þ
ð36Þ
where the auxiliary quantity 1 satisfies the following relation: e2 ð12 1Þð21 1Þ 1 ¼ 0
ð37Þ
Fig. 7 shows that, by means of an appropriate choice of the rollers curvature r22 , it is possible to obtain, over the whole ratio range, almost the same value of ~s0 ¼ s0 K2 =FN for both variators investigated in this paper. Observe that this quantity is one of the most important parameter to evaluate the stress severity as regards the fatigue life of the variator. A good choice consists in
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 (τ 0 Λ2)/F N 0.4
0.36
0.32
0.28 Half-Toroidal Full-Toroidal
0.24
0.2 0.5
0.75
1
1.25
1.5
1.75
2
s r ID
Fig. 7. The dimensionless maximum half-amplitude of the subsurface orthogonal shear stress ~s0 ¼ s0 K2 =FN versus the ideal speed ratio srID for the following dimensionless parameters ~r22HT ¼ r22HT =r0 ¼ 0:8, ~r22FT ¼ r22FT =r0 ¼ 0:66.
making ~s0 exactly the same when the ideal speed ratio is srID ¼ 1. To obtain this result the following values of the dimensionless parameter ~r22 ¼ r22 =r0 has been chosen: ~r22HT ¼ r22HT =r0 ¼ 0:8;
~r22FT ¼ r22FT =r0 ¼ 0:66
ð38Þ
4.2. EH lubrication model In EHL contacts the pressure can rise up to 3 GPa, thus producing very severe lubricant operative conditions. According to the Roelands model [32] and considering the isothermal contact hypothesis the following Eq. (39) enables us to estimate the fluid viscosity over the whole contact region: !Z 1 # " g p~ p g0 logð~ gÞ ¼ log 1 log ¼ 1þ ð39Þ g0 6R~cp g1 where ~cp ¼ cp =E0 , cp ¼ 1:96 108 Pa, g is the absolutely viscosity at the pressure p, g0 is the absolute viscosity at the atmospheric pressure for the given temperature, g1 ¼ 6:31 105 Pa s, the dimensionless constant Z1 is the viscosity-pressure index and the new dimensionless load parameter is: 0 2 1=3 r0 pE r0 ð40Þ R¼ ¼ K 6FN The non linear behaviour of the traction oil is described in Eq. (41) according to the normal rule used in the plasticity theory to split the shear strain along the different directions: ovi ovj sij þ ¼ Cðse Þ oxj oxi se
ð41Þ
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
In the above written Eq. (41) the equivalent stress se has been defined as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi se ¼ ðsij sij Þ=2
933
ð42Þ
The function Cðse Þ is representative of the non-linear behavior of the oil. One of the most commonly used explicit representation of this function is the one by Bair and Winer [32]: sL 1 ln Cðse Þ ¼ ð43Þ 1 se =sL g The quantity sL is the limiting shear stress of the lubricant and is normally evaluated, for a certain value of the oil temperature, by means of the following Eq. (44): sL ¼ sL0 þ ap
ð44Þ
where sL0 is the limiting shear stress at the atmospheric pressure [32]. Observe that, on that part of the contact region where significant shear stresses are involved, the film thickness of the lubricant could be considered almost constant, thus the following relation hold true at the input and output side of the variator, respectively, (h is the oil film thickness): v21X ovx s21X ¼ ¼ Cðse Þ; h oz se v23X ovx s23X Cðse Þ; ¼ ¼ h oz se
v21Y ovy s21Y ¼ ¼ Cðse Þ h oz se v23Y ovy s23Y Cðse Þ ¼ ¼ h oz se
ð45Þ ð46Þ
From the above written Eqs. (45) and (46) and from the Bair and Winer model (see Eq. (43)) it is possible to obtain an explicit formulation for the shear stress acting on the rollers at the input contact area: gjv j gjv j 6 v21X 6 v21Y 21 21 ~s21X ¼ R~sL 1 e hsL ; ~s21Y ¼ R~sL 1 e hsL ð47Þ p p jv21 j jv21 j and at the output contact region: gjv j 6 v23X hs23 L ~s23X ¼ R~sL ; 1e p jv23 j
~s23Y
gjv j 6 v23Y hs23 L ¼ R~sL 1e p jv23 j
ð48Þ
where ~s21X ¼ s21X K2 =FN , ~s21Y ¼ s21Y K2 =FN , ~s23X ¼ s23X K2 =FN , ~s23Y ¼ s23Y K2 =FN and ~sL ¼
sL sL0 p p p~ ¼ 0 þ a 0 ¼ ~sL0 þ a E 6R E0 E
ð49Þ
It is possible to rephrase Eqs. (47) and (48) in terms of the following dimensionless parameters: ~Y v21X r21 a v21Y r21 ~ aX ¼ Crin Y; ¼ X; jx1 jr1 R ~r1 jx1 jr1 R ~r1 0 !2 !2 11=2 jv21 j r21 ~ r21 ~ aY aX ¼ @ Crin Y þ X A jx1 jr1 R ~r1 R ~r1
ð50Þ
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
~Y v23X Crout r23 a v23Y r23 ~aX ¼ Y; ¼ X; jx3 jr3 1 Crout R ~r3 jx3 jr3 R ~r3 0 !2 !2 11=2 jv23 j Crout r23 ~ r23 ~aX aY þ ¼@ Y þ X A jx3 jr3 1 Crout R ~r3 R ~r3
ð51Þ
21 j 23 j and gjv , that appear in Eqs. (47) and (48), the following Whereas, as regards, the quantities gjv hsL hsL relations hold true: 0 !2 !2 11=2 ~ g gjv21 j r21 ~ r21 ~aX aY ~ 1 ð1 þ kÞ@ Crin ¼ Y þ X A ð52Þ x Hin~sL hsL R ~r1 R ~r1
0 !2 ~ ~ g gjv23 j Cr r a out 23 Y ~ 1 ð1 þ kÞð1 Crin Þð1 Crout ÞsrID @ þ ¼ Y x Hout~sL hsL 1 Crout R ~r3
þ
!2 11=2 ~X r23 a X A R ~r3
ð53Þ
where the dimensionless rotating velocity has been defined as: g x1 ~1 ¼ 0 0 x E
ð54Þ
This quantity takes into account the effect of the rotating speed of the input disc on the variator behaviour. Hin ¼ h=qeqX in is the dimensionless thickness of the oil film at the input contact zone, it can be evaluated by means of the hard EH lubrication formulas [32,33], where the above defined dimensionless parameters are used: h i0:67 0:134 ~f0:53 R0:201 q ~1 ~eqX 1 0:61 e0:73ein ð55Þ Hin ¼ 2:81 ð1 þ kÞð1 0:5Crin Þx in
In Eq. (55) the dimensionless pressure–viscosity coefficient ~f of the oil has been defined as (see [32]): ~f ¼ fE0 ¼ Z1 ln g0 ð56Þ ~cp g1 As regards the dimensionless oil film Hout at the output contact, it can be calculated by means of the relation: Hout 2 Crout 1 0:61 e0:73eout ¼ ð1 Crin Þ0:67 s0:536 rID Hin 2 Crin 1 0:61 e0:73ein
ð57Þ
Equation (57) shows that, since the creep coefficients in a normal operative condition are sufficiently small and the last term in Eq. (57), related to the ellipticity parameters ein and eout , is very close to the unity, the ratio Hout =Hin can be evaluated by means of the simpler well approximate relation
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
Hout ¼ s0:536 rID Hin
935
ð58Þ
that Eq. (58) shows that the oil film thickness at the input and output points of contact, respectively, may differ significantly, especially at the extreme values of the speed ratio. 4.3. Calculation of the traction coefficient and spin momentum The contact model described in Section 4, enable us to find an integral relation that allows for the calculation of the traction coefficients lin and lout and the spin momentum coefficients vin and vout , as reported in Eqs. (59) and (60). Z 2p Z 1 ~s21X R dw aXin ~ dR lin ¼ ~ aYin 0 0 ð59Þ Z 2p Z 1 aXout ~aYout lout ¼ ~
dR 0
~s23X R dw 0
Z 2p Z ~ aXin ~ aYin 1 ~ vin ¼ dR aXin ~s21Y cos w ~aYin ~s21X sin w R2 dw R~r1 0 0 Z 2p Z 1 ~ aXout ~ aYout ~ dR vout ¼ aXout ~s23Y cos w ~aYout ~s23X sin w R2 dw R~r3 0 0 where the following co-ordinate transformation has been applied:
X ¼ R cos w ; 0 6 R 6 1; 0 6 w 6 2p Y ¼ R sin w
ð60Þ
ð61Þ
5. Results Since the scope of the paper is to compare the half-toroidal and the full-toroidal traction drives, it is necessary to specify which quantities will be kept constant during the calculations. These are the ideal speed ratio srID , and the normal force FN at the points of contact, whereas the geometrical quantities are reported in Table 1, and the fluid properties in Table 2. Fig. 8 shows the efficiency m of the two variators as a function of the input dimensionless torque tin , for srID ¼ 1, and R ¼ 24. This last parameter corresponds, for the given geometry to a maximum pressure value close to 2.2–2.3 GPa. Moreover, the angular velocity of the input disc has been chosen equal to ~ 1 ¼ 2:95 1012 . As shown in jx1 j ¼ 2000 [RPM], that corresponds to a dimensionless value x Fig. 8 the better efficiency of both variators is reached in the region of high tin . Moreover, the efficiency of the full toroidal traction drive is smaller of about 4–5% points, compared to that of the half-toroidal variator over the whole range of tin values. The reason of this difference is caused, in the region of low tin , mostly by the worse torque efficiency of the full-toroidal variator (see Fig. 9). In fact, Fig. 10 shows that, because of the torque loss in the roller bearing and the larger contact area (the radius of curvature ~r22 is bigger than in full-toroidal variator), the half-toroidal
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
Table 2 The fluid properties Fluid properties: T ¼ 99 °C g0 ¼ 3:25 103 Pa s Z1 ¼ 0:85 f ¼ 1:71 108 Pa1 sL0 ¼ 0:02 109 Pa a ¼ 0:085 cp ¼ 1:96 108 Pa g1 ¼ 6:31 105 Pa s
Absolute viscosity at the atmospheric pressure Viscosity–pressure index Pressure–viscosity coefficient Limiting shear stress at atmospheric pressure Limiting shear stress constant Pole pressure constant of Roelands viscosity model Pole viscosity of Roekands viscosity model
ν
1 0.95 0.9
s r ID =1 0.85
ℜ = 24 Half-Toroidal Full-Toroidal
0.8
2000 [RPM]
0.75 0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 8. The efficiency of the variators as a function of the input traction coefficient tin .
ν torque
1
0.95 0.9
s r ID =1 0.85
ℜ = 24 Half-Toroidal Full-Toroidal
0.8
2000 [RPM]
0.75 0.7 0
0.02
0.04
0.06
0.08
t in
0.1
Fig. 9. The torque efficiency of the variotors as a function of the input traction coefficient tin .
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 χ in
937
0.003 0.0025
s r ID =1
ℜ = 24
0.002
2000 [RPM] 0.0015 0.001
Half-Toroidal Full-Toroidal
0.0005 0 0
0.02
0.04
0.06
0.08
0.1
t in
Fig. 10. The spin momentum of the variators as a function of the input traction coefficient tin .
traction drive is more effected by the spin momentum. But, because of its very small spin velocity (see Fig. 5), the energy losses due to spin are less important than in the full-toroidal traction drive. Fig. 9 shows that also for mid and high tin values the efficiency of full-toroidal CVT is smaller than that of the half-toroidal traction drive, this is due mostly to the worse speed efficiency of the full-toroidal variator as shown in Fig. 11. In fact, the large values of the spin motion, that affect the full-toroidal CVT, cause the global sliding coefficient Cr to increase very fast as the requested torque tout increases (see Fig. 12), thus causing larger power losses. Regarding the influence of the dimensionless parameter R, Figs. 13 and 14 show how it affects the efficiency of both CVT variators: the bigger R the higher the mechanical efficiency of the variators. The explanation is simple: the efficiency of the variators is largely affected by the spin losses, that in turn depends on the extension of the elliptical contact area. By reducing the load, i.e. by incrementing R, the area of contact reduces its extension. This, in turn, causes a reduction
ν speed
1
0.95 0.9
s r ID =1 0.85
ℜ = 24 Half-Toroidal Full-Toroidal
0.8
2000 [RPM] 0.75 0.7 0
0.02
0.04
0.06
0.08
0.1
t in
Fig. 11. The speed efficiency of the variators as a function of the input traction coefficient tin .
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 t out
0.1 0.09 0.08 0.07 0.06
s r ID =1
0.05
ℜ = 24
0.04
Half-Toroidal Full-Toroidal
0.03 0.02
2000 [RPM]
0.01 0 0
0.05
0.1
0.15
0.2
Cr
Fig. 12. The traction capabilities of the variators: output traction coefficient tout as a function of the global sliding coefficient Cr.
ν
1
Half-Toroidal
0.95
ℜ = 48
0.9
ℜ = 42 ℜ = 36
0.85
s r ID =1
ℜ = 30
0.8
2000 [RPM]
ℜ = 24
0.75
ℜ = 18
0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 13. The efficiency of the half-toroidal variator as a function of the input traction parameter tin and for different values of the load parameter R.
of the spin momentum, and therefore of the energy dissipated by the spin motion. Morevoer, since the spin motion affects the FT variator more than the HF one, it is expected that the full-toroidal variator be more sensible to the load variations than the half-toroidal CVT, as clearly demonstrated in Figs. 13 and 14. Therefore, it may happen that for higher values of R the efficiency of the full-toroidal variator overcomes that of the half-toroidal traction drive: see, for example, the curves plotted for R ¼ 48 in Figs. 13 and 14. But, on the other hand, too big values of R reduce the traction capability of the variators, i.e the maximum values of tin . ~ 1 does not affect appreciably the efficiency of The simulations have shown that the parameter x both variators. Moreover, it was shown that the FT traction drive, because of its particular
G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942 ν
1
939
Full-Toroidal
ℜ = 48 ℜ = 42
0.95 0.9
ℜ = 36
0.85
ℜ = 30
0.8
s r ID =1
ℜ = 24
2000 [RPM] 0.75
ℜ = 18 0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 14. The efficiency of the full-toroidal variator as a function of the input traction parameter tin and for different values of the load parameter R.
ν
1
Half-Toroidal
0.95 0.9 2000 [RPM] 0.85
ℜ = 24 s r ID =2/3 s r ID =1
0.8
s r ID =1.5 0.75 0.7 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
t in
0.1
Fig. 15. The efficiency of the half-toroidal variator as a function of the input traction parameter tin and for different values of the speed ratio srID .
symmetry, is almost insensible to the actual value of ideal speed ratio srID , whereas the HT variator results to be largely affected as shown in Fig. 15. In conclusion, the analysis carried out, shows that the full-toroidal traction drive is unable to achieve the same efficiency of the HT variator, despite the torque losses in the support bearing that affect the latter. The main reason of this result is the very high spin motion of the FT variator that produces high values of energy dissipation. It has been also shown that only two parameters influence significantly the mechanical efficiency of the variators, these are the dimensionless load parameter R and the dimensionless input torque tin .
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G. Carbone et al. / Mechanism and Machine Theory 39 (2004) 921–942
6. Conclusions The paper deals with the mechanical efficiency of full and half toroidal traction drives and has the aim of comparing the performances of these two CVT typologies. A fully flooded isothermal model of the contact between discs and rollers, based on the results of EHL theory, has been implemented to evaluate the slip and spin losses. The analysis has shown that by optimizing the roller geometry of the full toroidal variator, it is possible to reduce its spin momentum below the value of the half-toroidal variator. But, since the energy dissipation due to the spin losses is the product of the spin momentum and the spin velocity, the full-toroidal variator always results to have a smaller efficiency because of its much bigger spin velocity. Moreover the full-toroidal traction drive needs higher values of global creep (smaller speed efficiency) to transmit the same torque of the HT traction drive. This causes an additional heating of the lubricant and a further reduction of its traction capability and also of the mechanical efficiency. Moreover, the mechanical efficiency of the half-toroidal CVT, in comparison to that of the full-toroidal variator, is less affected by the value of the normal contact forces and often over the threshold of 90% on the most part of the torque range. The full-toroidal CVT, instead, shows a different behavior, its efficiency varies within a bigger range of values, and it is more affected by the normal load at the contact between the rollers and discs.
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