Modeling and Simulation of Toroidal Traction Drives

Copyright @ IFAC Mechatronic Systems, Darmstadt, Germany, 2000

MODELING AND SIMULATION OF TOROIDAL TRACTION DRIVES Dipl.-Ing. H. Bark· Prof. Dr.-log. Dr. h.c. F. Pfeiffer·

• Lehrst1.l.hl B for Mechanik, TU-Miinchen, D - 85748 Germany

Abstract: An overview over the modeling issues of a toroidal traction drive is given. A traction drive consists of a loading cam, a hydraulic feedback control circuit and two toroidal cavities, each ofthem including two discs and two power rollers. The modeling of both the mechanical and the hydraulic components is discussed. An existing rheology model is used to describe the tangential forces generated in the contacts between the discs and the rollers. The transmission ratio is feedback controlled by a hydraulic circuit which is modeled according to mechanical principles. Finally, simulation results of transmission ratio changes are compared with measurements. Copyright @2000 IFAC

Keywords: Simulation, hydraulic, traction, toroidal, rheology

1. INTRODUCTION

The automotive industry is seeking new concepts for a continuously variable transmission (CVT) in the driveline. One possible solution for a CVT design are toroidal traction drives, providing a high torque capacity with quick transmission ratio response and a coaxial in- and output. Many companies are currently developing new traction drives designs, where two different applications from Nissan and Torotrak already exist. A traction drive unit consists of two toroidal discs that face each other and rotate around a common main axis. Between the two discs several power rollers are arranged, where each power roller is supported by a trunnion to freely rotate. The input and output toroidal discs are pressed together such that high normal forces in the contact points are generated. In the contacts tangential traction forces are generated by differential surface velocities. Thus, power is transmitted by tangential forces from the input to the power rollers, and from the power rollers to the output disc. In Figure 1 a halftoroidal traction drive according to Nissan (Nakano,M. and al., 1996) is depicted. It utilizes two cavity units to compensate high axial forces, a hydraulic circuit for transmission ratio feedback

Fig. 1. Halftoroidal traction drive control, a loading cam for moment proportional axial forces, and a conveyor for startup.

2. MECHANICAL MODELING The mechanical model of a traction drive cavity consists of four rigid bodies. Each body is symmetrical to its rotational axis. The movements of the rigid bodies are described in a body fixed 109

coordinate frame Ri that does not follow the revolving motion of the bodies. The coordinate system is attached to the body for all movements except the revolving motion around the main axis z of the body. Figure 2 depicts the mechanical model of the traction drive cavity. The angle er defines the transmission ratio of the cavity. The

Fig. 4. Parametrisation of power roller

z

Together with the normal vector n they build the coordinate system C in the contact point. Introducing the distance vector r d between two points of the bodies, the parameters that define the contact points can be found according to (Meitinger, 1998) by solving a set of nonlinear equations

Fig. 2. Mechanical model of a traction drive cavity equation of motion is derived from the moment and the moment of momentum using the principle of Newton-Euler written according to the formulation of Jourdain (Bork,H. and Pfeiffer,F., 1999)

0=

JT

T

((p) - Fe)

+J R

T

(t - Me)

z

(1)

O nT , T n l V3 -- 0 l U3 =

(4)

rbvl = 0

(5)

rbul =0,

Since high tangential forces require high normal forces elastic deformations of the toroidal disc have to be considered as well. The deformed surface of each toroidal disc is described by the toroidal radius R . The radius is then a function of the parameter E, the contact point Ej as well as the normal force Fn

In the most general case, where movements and rotations of the bodies are considered an analytic determination of the contact point is impossible and the contact point has to be determined by a surface parametrisation. The parametrisation of a symmetrical body is given by two independent variables. In Figure 3 the parametrisation of a toroidal disc is shown and in Figure 4 the power roller is depicted analogously. In the coordinate

(6)

R(E)=Ro+dr(E,Ej)Fn dr(E,Ej) =a(ej}(E - 71")

+ b(ej)(E - 71")2

(7)

The influence function dr( E, ej) is found by an least squares approximation of finite element calculations for different contact points Ej. The coefficients a( ej) and b( ej) are a polynomial of second order. In Figure 5 the results of a finite element calculation are shown. As the disc is deformed the Ea

1.6791-06

~i -6.1t....-Ob

Fig. 3. Parametrisation of toroidal disc -2 . ':I6<)P-05

system R a surface point re may be described by the two parameters E and -y -6.107..-0:;

(2) By differentiation of this vector with respect to the parameters the tangential vectors are obtained. Fig. 5. Deformation of toroidal disc

arc U=

arc

a-y ,v=a;-

(3)

normal vector of the contact plane changes. Con110

elusively the normal forces yield a turning torque on the power roller, that has to be compensated for stationary operation.

viscous term (Johnson,K.L. and Tevaarwerk,J.L., 1979), yielding a relation between shear velocity "r and shear stress T

The loading cam shown in Figure 6 generates axial pressing force proportional to the transmitted torque. Between two discs with curved

"r = "rvis + "reI = ~T + G at

1

1 aT

(12)

with viscosity f} and shear modulus G. The viscous term is considered to be nonlinear, Le. a weighting function f (Teq) is introduced that depends on the equivalent shear stress Teq = JT~ + T; in the element. Thus, for each direction i = 'U, V a differential equation is obtained . 1 aTi 'Yi=--

G at

Ti -TO • h (Te q ) +sm -

Teq q f(Teq) = ~ sinh (Te ) Teq To

surfaces facing each other several loading rollers are arranged. By applying input torque to the loading cam, the two discs are rotated relative to each other. Through the cam surface of the discs, relative axial elongation of the cam mechanism results. This elongation produces axial forces in the main shaft. The generalised coordinates Sd of the loading cam are the rotations and the axial positions of the discs. The equation of motion for the loading cam is given by

(8)

dSd

The contact parameter numerically.

+ Wd(€, €, Sd) €

(14)

... ': ' . X

where the constraint force.>.. of the loading rollers is determined by a bilateral constraint = W

(13)

TO

that describes the increase of the shear stresses by shear velocity and viscosity. Figure 7 shows a calculation result for the shear stresses in an example contact with tangential velocities and spinning. The fluid reacts sluggish to the change in viscosity with pressure and the shear velocities, thus that the increase of the shear stresses is limited by the elastic part. The viscosity drops elose to the exit of the contact. Therfore, the largest values of the shear stresses are obtained in the second half of the contact ellipse. The spinning motion introduces negative tangential velocities in the lower part of the ellipse, reducing the resulting tangential force.

Fig. 6. Model for loading cam

!in

f}

10'

1.5

-

.' .

N

a z

(9)

..

..

';:' o.s

o

and € can be found

..{).S

-1 .. ' . 1

3. FLUID MODELING

X

In the contact a traction fluid is subjected to severe pressures and different surface velocities. As a matter of fact the contact kinematics yield differential tangential and transverse velocities coupled with a spinning motion around the normal axis. Therfore, the differential velocities in the contact ellipse, that is calculated according to Hertzian formulae, are C~VCl,3

=

ACI (IVC3 - IVCr)

10-3

Fig. 7. Shear stress caused by tangential velocity and spinning motion

4. HYDRAULIC FEEDBACK CONTROL CmCUIT

(10) (11)

4.1 Feedback control

The increase of the shear stresses in the contact is described by a Maxwell body with a nonlinear

The transmission ratio in a traction drive is changed by offsetting the trunnion and the power III

where the pressures of the adjacent knots are projected by the matrix Wi towards the velocity Vi of the line. The model of the piston with two compressible chambers and two connected lines is depicted in Figure 9. The equation of motion for

roller from the main axis of rotation. As the contact point moves on the discs, transverse velocities are generated by the discs revolution speed. As both discs rotate in opposite direction the transverse velocities cause transverse forces that turn the power roller, i.e. accelerate the tilting angle a. The offset motion of each trunnion is realised by pistons subjected to hydraulic pressure. Pressure changes are obtained by the relative position between a feedforward sleeve and a feedback spool. The feedforward sleeve is moved according to the desired tilting angle ao. The feedback spool is moved according to the actual tilting angle, piston position of one power roller and the realized control parameters b and a. In Figure 8 the hydraulic feedback control circuit and its model is depicted. The relative opening ~z between sleeve and spool

.Y..2.

Pia

Ak

~ Pol

fXk Pkl

Fig. 9. Model of the Piston the piston is given by

The two hydraulic chambers are modeled compressible, such that the pressures Pkl and Pk2 depend on the velocity of the piston Vk as well as on the velocity of the connected line Vi (18) The equation of motion for the two lines connected to the hydraulic chambers is given analogously to equation 16 and the hydraulic subsystem consisting of the piston, the two compressible chambers and the two connected lines is characterised by the generalised coordinates v z and the resulting equation of motion is given by

Ps

Fig. 8. Hydraulic feedback control circuit

Vz

is given by ~Z =

=

Mzv z

z/ -

Zb

= a(ao - ad -

bXI

(15)

+ Wzpz

= fAt,v z )

(20)

The subsystem four-way valve consists of the 4 valves and 4 lines and is depicted in Figure 10. The generalised coordinates of the subsystem are

The offset position x is used as additional feedback information to create damping in the system.

p.

POI

4.2 Hydraulic modeling

V3

~

The hydraulic circuit is modeled according to (Borchsenius,F. and Pfeiffer,F., 1998). Stiff differential equations for the hydraulic circuit are avoided, because the knots are modeled incompressible. Analogously to the mechanical principle, stiff couplings are introduced as kinematic constraints. Thus the volume flow at a knot is considered as an algebraic constraint. The elements of the control circuit are lines, a four-way valve and the two pistons. They are modeled independently and are coupled afterwards. A line is subject to two pressures, and the equation of motion for the incompressible line is given by mivi + Wi [;;] = hi

(19)

[Vk,VI,V2,Pkl,Pk2]T

Al

~ PI

A2

i

V4

V2

~

~

A3

~ P3

A4

P4

i

VI

Po

Fig. 10. Model for four-way Valve the fluid velocities of the lines VI and the valves V V • The lines are connected with the boundary pressures Po and Pr· Considering the pressures PI through P4 of the internal knots gathered in Plc the equation of motion for the 4 lines is given by

(16)

112

5. RESULTS

Furthermore, in each valve pressure loss fdp occurs which is a function of the opened valve area A". yielding the equation of motion for the valves fdp,.

M"

P = 2 (a"A"i(~z»2

V" +

WvlcPIc

IVil ViAi

(22)

= -fvr + fdp

(23)

In the following, simulation results are compared with measurements for a transmission ratio change under a no-load condition. The measurements are taken from (Kobayashi, 1994). In 0.2,...-....,..,.........--.....-----.----.---,

From Figure 10 it is obvious that the generalised velocities depend on the valve velocities. Therefore, the valve velocities are a representation of the minimal coordinates for the subsystem and the generalised coordinates are expressed by the Jacobian matrix

(24)

[::] =Jv"

~ .051...---"---..i...----'---.......- - - '

o

05

o ]J

M"

2

1.5 I

yielding the mass matrix of the subsystem

2.5

rsl

Fig. 11. Simulation-Measurement: power roller offset

(25)

and the equation of motion in minimal coordinates

" ....... ; ........ .

20 ......... :

~

"

.

10

]"

0

."

1:5

-10 .

-20

The pressure losses in the valves and in the lines are included in the h vectors. The boundary pressures are projected by the matrices W r and Wo · The internal pressures Plc are constraint forces in the mechanical sense. By projection of the equation of motion towards the minimal coordinates the constraint forces are eliminated.

(28)

the corresponding constraint matrix W for the hydraulic system is found

= WVhyd = 0

I

Figures 11 and 12 the simulation results are compared with measurements for the offset position of the piston and the tilting angle. At the beginning of the transmission ratio change the piston is moved fast. The tilting angle is accelerated towards the desired transmission ratio change speed. The tilting angle increases the input speed and effects the transverse velocity components, thus that the offset during the ratio change is reduced. Damped oscillations in the offset and the tilting angle can be seen at the beginning and at the end of the ratio change. They are effected by the choice of the control parameters b and a. In Figure 13 the piston pressures are shown. To accelerate the input speed tangential slip is increased during the ratio change. The resulting tangential forces are in equilibrium with the differential hydraulic pressure.

By collecting the generalised coordinates of the complete hydraulic circuit, for example for n lines, two pistons and one four-way valve

[WI, W2, ••. , WIc)T Vhyd

measUTemc:n~

Fig. 12. Simulation-Measurement: power roller tilting angle

(27)

= [VI, V2, ..• , V n , Vd, V z 2, V,,]T

simulation

-~I...---"---~-~--~-~ 2 2.5 1.5 o 0.5 I rsl

The hydraulic subsystems are connected by knots. For each incompressible knot i a kinematic constraint on the fluid velocities is given

Vhyd

1-

...• .- .

(29)

Using a Singular value decomposition the Jacobian matrix J hyd is found, which is orthogonal to the constraint matrix

The following measurements were published in (Nakano,M. and al., 1996). They show a simultaneous jump in input torque and a transmission ratio change command with constant output speed. The transmission ratio should be changed

(30)

113

decrease the output torque. Therefore, the drop in the output torque is obtained.

1 2 r - -........- - - - - - - . - - - - - - - ,

300 ...... ; . . . ... ; . . . . . .. .. . .. .. ..

\

95 " •. .. _

.

simp2

.. 2OO.1~ =~~tl. ... ~ .

.

!

9 . .. . .. . -

simpl meas p2 . . .. . . . . .... . . . .... . . ... meas I

85


L --===::::::L--,---,,---1

o

05

2.5

2

1.5

t

r.1 _IOOL---'_--'-_........._....i...._~_'-----J

o

Fig. 13. Simulation-Measurement: Piston pressures


-30'--~-----"""""-""""'---'------J

0.4

0.5

0.5

0.6

0.7

The modeling and the simulation of toroidal traction drives is discussed in this paper. The model starts from a general description of the rigid body movements and uses a continuous surface parametrisation to determine the contact points. In the contact the traction fluid is severely subjected to normal pressure and differential surface velocities. The generated tangential forces are calculated with an existing rheology model. The model for the CVT-cavity is extended to account for elastic deformations of the discs and is coupled with the model of the loading cam and the hydraulic feedback control circuit. Conclusively the dynamic behaviour of such a toroidal transmission may be examined. The simulation results are compared with measurements and the transmission ratio change is explained with the results.

~ -1O

03

0.4

6. CONCLUSION

3"

0.2

0.3

Fig. 16. Simulation-Measurement: output torque

o ...

0.1

0.2

t rsl

in about 0.5 s from a overdrive position to a 1:1 ratio. The simulation result for the power roller titling angle is compared with the measurement in Figure 14.

o

0.1

0.6

Usl

Fig. 14. Simulation-Measurement: power roller tilting angle

3300 . .... . ... ... . . ..... .

,.-., 2900 ...... \ , . .. . .. . . E: ~ ~ 2500 c.. .S a 2100 .

7. REFERENCES

0.6

Borchsenius,F. and Pfeiffer,F. (1998). Alternative verfahren zur modellierung hydraulischer systeme. VD1-Berichte Nr. 1406 pp. 567-580. Bork,H. and Pfeiffer,F. (1999). Simulation des transienten verstellvorganges in toroid reibradgetrieben. Antriebstechnisches Kolloquium Aachen, 1ME pp. 291-302. Johnson,K.L. and Tevaarwerk,J.L. (1979). The influence of fluid rheology on the performance of traction drives. Transactions of the ASME pp. 266-274. Kobayashi, K. (1994). Shift control system for continously variable traction roller transmission. U.S. Patent 5.286.240. Meitinger, Th. (1998). Dynamik automatisierter Montageprozesse. number 476 In: 2. VDI Fortschritt-Berichte. VD 1-Verlag, Diisseldorf. Nakano,M. and al. (1996). Performance of a dualcavity half-toroidal cvt for passenger cars. SAE 9636466.

0.7

Fig. 15. Simulation-Measurement: input speed The titling angle increases the input speed, that is depicted in Figure 15. As the input torque is available a short time after the ratio change command, the output torque depicted in Figure 16 does not show the desired behavior. At the beginning the output torque increases with the input torque. As the power roller starts turning tangential forces are generated, because the tangential velocities at the toroidal discs change with the contact radii and the tilting angle. These tangential velocities effect the power transmitted by the rollers and 114