MDAU_SHC

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Modelling of dynamic characteristics of an automatictransmission during shift changes

N Zhang1, D K Liu1*, J M Jeyakumaran2 and L Villanueva21Faculty of Engineering, University of Technology, Sydney, New South Wales, Australia2Research and Development Department, BTR Automotive, Fair� eld, New South Wales, Australia

Abstract: This paper describes modelling of the transient dynamics of an automatic transmissionduring gear changes. A brief introduction to the automatic transmission system and the dynamiccharacteristics of the transmission components during the gear changes are presented. Then, detailedmathematical models of a four-speed automatic transmission manufactured by BTR Automotive,Australia, are developed. A mode description method is used to describe the transient shifting processand a modular structure of the transmission system, which consists of a torque converter module,geartrain module, hydraulic system module and modules of clutches and bands, is presented. As anapplication, the developed simulation system is applied to investigate the transient performance ofthe automatic transmission during the 1–2 shift process. The output torque pro� les predicted by themodel simulation correlate very well with the experimental data measured from vehicle tests.

Keywords: dynamic modelling, transient characteristics, automatic transmissions, gear shifting

NOTATION r radiusR

ddrum radius of the band

Ri

inner radius of the clutch friction platesA(t) � ow areaR

oouter radius of the clutch friction platesc ori� ce � owrate coeYcient

t timeCR

radial clearance of the valveT torqued diameter of the valvesV(t) oil volume as a function of timeF forceX stroke of the valve/pistonH Heaviside step function

i speed ratio bB

band wrap angleI mass moment of inertia b( p) bulk modulus as a function of pressurek0, k

1clutch/band stiVness b

0bulk modulus at p=p

0l � ow length between ends 1 and 2 ca

air ratio at pressure pL

ilength vectors c

0air ratio at p=p

0Mi

modes of operation e, k oil property constantsn number of contact surfaces in the multidisc hÇ angular velocityclutch pack h angular acceleration

p pressure q dynamic viscosity of the oilpÇ pressure rate m

Cstatic or dynamic friction coeYcient of the

p0

atmospheric pressureclutch

p1 p

2pressure drop across the ends 1 and 2 m

Bstatic or dynamic friction coeYcient of theQÇ � owrateband

Qi

generalized coordinates

Subscripts

B bandThe MS was received on 8 October 2001 and was accepted after revision B1 band B1for publication on 26 June 2002. C clutch* Corresponding author: Faculty of Engineering, University of

C2 clutch 2Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia.

I07201 © IMechE 2002 Proc Instn Mech Engrs Vol 216 Part I: J Systems and Control Engineering

332 N ZHANG, D K LIU, J M JEYAKUMARAN AND L VILLANUEVA

CR carrier eters, the coeYcients were obtained through regression� ts of the known torque converter characteristics. ThisE engine

FS forward sun model was also used in the automatic transmissionmodel developed by Pan and Moskwa [2 ] to study theLP long pinion

LU lock-up clutch transient characteristics of a Ford automatic trans-mission. Simpli� ed dynamic models of an engine, torqueOWC one-way clutch

P pump converter and vehicle are often used in the investigationof the transient characteristics of a powertrain equippedR ring gear

RS reverse sun gear with an automatic transmission (see, for example, refer-ences [3 ] to [6 ]). In particular, Jo et al. [7 ] applied vari-SP short pinion

T turbine ous simulation techniques in order to analyse the shiftcharacteristics of the vehicle powertrain with automaticV vehicletransmissions.

This paper focuses on the development of a simulation1 INTRODUCTION system to investigate the transient characteristics during

gear changes in a BTR four-speed automatic trans-mission. The outcomes of this study will then be used inVehicles equipped with automatic transmissions provide

advantages such as easy operation, smooth acceleration the design and optimization of the shift control elements.A state mode description method is used for describingand safety. Demands for improved performance of the

vehicle, such as driveability, passenger comfort, early the shifting process, and the governing equations ofmotion of the integrated powertrain system are derived.malfunction detection and fuel economy, have been

increasing dramatically in recent years. In automatic A modular structure of the automatic transmission,including the torque converter module, hydraulic systemtransmissions, frequent stops and starts and rapid

acceleration/deceleration require frequent gear ratio module, geartrain module and modules of clutches andbands, is developed. Experimental data are also used inchanges, which perturb vehicle acceleration and engine

speed. These conditions are the main causes of poor shift the simulation processes to allow the model to be tunedand validated with reasonable accuracy. The transientquality, poor fuel economy and unwanted emissions.

Thus, an in-depth knowledge of the transient character- characteristics of a 1–2 shift are presented, and themodel is then used to determine optimum pressureistics during a gear ratio change is very important for

the design and control of an automatic transmission pro� les for the shifting elements.system.

Computer simulation is a powerful tool for investi-gating complex transmission systems. This can lead to 2 AUTOMATIC TRANSMISSION SYSTEM ANDshorter product design cycles, reduce development cost THE 1–2 SHIFT PROCESSand allow engineers to explore many options early in thedesign phase. Simulating the transient characteristics of

The BTR four-speed automatic transmission system con-an automatic transmission is, however, complicatedsists of a torque converter with a single-face lock-upbecause many factors aVect the shift quality during gearclutch, four multiplate clutch assemblies, two brakechanges. These factors include the magnitude of the ratiobands, two one-way clutches and a planetary geartrainchange, clutch and band con� gurations, hydraulic con-as shown in Fig. 1. In � rst gear, the C2 clutch is appliedtrol system, operating temperatures, shift scheduling, etc.and the engine torque, T

E, and consequently the turbineSystem-level dynamic models of the transmission com-

torque, TT

, are transmitted to the forward sun (FS) gearponents, together with a complete understanding of thethrough the C2 clutch. The clockwise rotation of the FShydraulic system instabilities, and system-level inter-gear causes counterclockwise rotation of the shortference forces are necessary for a model-based designpinions (SP). The carrier (CR) is held by the 1–2 one-and control of the automatic transmission systems.way clutch (1–2 OWC ), which makes the ring (R) gearThe powertrain elements that describe an automaticrotate around its axis in a clockwise direction. Torquetransmission system consist of an engine, a hydro-is transmitted from the FS gear to the output shaftdynamic torque converter, a planetary gearset, frictionsplined to the ring gear. The � rst gear ratio is the ratioelements and driveline elements. Most studies carriedbetween the FS gear speed, hÇ

FS, and the R gear speed,out to date on modelling have been the result of individ-

hÇR, i.e.ual transmission manufacturers looking at their speci� c

powertrain requirements and applications. As aniFS1

=hÇ

FShÇ

R=

rR

rFS

(1)example, Kotwicki [1 ] developed a dynamic model of anautomatic transmission system based on the steady stateoperations of the torque converter. Instead of deriving The speed ratios of other gear elements, such as the short

pinions, long pinion (LP), reverse sun (RS) gear andmodel coeYcients from the converter physical param-

I07201 © IMechE 2002Proc Instn Mech Engrs Vol 216 Part I: J Systems and Control Engineering

333MODELLING OF DYNAMIC CHARACTERISTICS OF AN AUTOMATIC TRANSMISSION

Fig. 1 The schematic diagram of the BTR four-speed automatic transmission

carrier, to the ring gear are

iSP1

=hÇ

SPhÇ

R=

rR

rSP

(2)

iLP1

=hÇ

LPhÇ

R=

rR

rLP

(3)

iRS1

=hÇ

RShÇ

R=

rR

rRS

(4)

Fig. 2 Typical output torque pro� le with torque and inertiaThe 1–2 shift is accomplished by applying the B1 bandphasesto hold the RS gear and by overrunning the 1–2 OWC.

Similarly, the speed ratios in the second gear are listedas follows:

3 MODE DESCRIPTION METHOD

iFS2

=hÇ

FShÇ

R=

rR(r

FS+r

RS)

2rFS

(rRS

+rLP

)(5)

An automatic transmission can be described as a para-metric system due to the discrete speed changes that

iSP2

=hÇ

SPhÇ

R=

rR(r

RS r

SP)

2rSP

(rRS

+rLP

)(6) occur during the shifting process. In each gear, the

system dynamics is determined from various combi-nations of active elements involved with the power � ow,

iLP2

=hÇ

LPhÇ

R=

rR

2rLP

(7) such as clutches, bands and gear elements. Each combi-nation is called a ‘mode’ of operation [8 ] and each modehas it own distinct set of dynamic equations of motion.

iCR2

=hÇ

CRhÇ

R=

rR

2(rRS

+rLP

)(8) As a result, the number of governing equations also

changes during each phase of the shift simulation. Aparticular shift process may be divided into severalwhere the subscripts 1 and 2 represent the � rst and

second gears respectively. modes of operation, {M1, M

2, . .. , M

n}, with one combi-

nation of transmission elements involved in each mode.The 1–2 shift sequence can be divided into two phases,a torque phase where the speed ratios do not change but A set of vectors of generalized coordinates,

{Q1, Q

2, .. . , Q

n}, and a set of length vectors,the output torque is reduced by the application of B1

band and an inertia phase where the speed ratios begin {L1, L

2, .. . , L

n}, are de� ned for each mode of operation.

The modes are arranged in increasing time steps, withto change. In the torque phase, the output torquechanges according to the frictional characteristic of the M

nrepresenting the last mode during the shifting pro-

cess. The eYciency of the numerical simulation dependsB1 band and the inertial properties of the geartrainsystem. A typical torque and speed pro� le during the on the choices made on modes during the various phases

of the shifting process. As an example, a poorly chosen1–2 shift is shown in Fig. 2. In general, two parametersmay be used to assess the quality of the shift, time width, mode with too many sets of dynamic equations will slow

down the simulation. In addition, switching betweeni.e. duration of the torque phase, and torque hole, i.e.torque drop during the torque phase. In general, the modes will increase the complexity of the shift

simulation.shorter the duration of the torque phase, the less thetorque hole is during the torque phase. According to the mode description method, the 1–2



shift process can be divided into four modes, i.e.

M={M1, M

2, M

3, M

4} (9)

where modes 1 and 4 represent the steady state con-ditions of the � rst and second gears respectively andmodes 2 and 3 de� ne the torque and inertia phasesduring the 1–2 shift respectively. It is noted that inmode 4, there is slip in the torque converter before thesecond gear steady state condition is achieved. Referringto equation (9), the corresponding set of vectors ofgeneralized coordinates is

Q={Q1, Q

2, Q

3, Q

4} (10)

where

Q1={h

E, h

P, h

T, h

FS, h

SP, h

LP, h

RS, h

R,

FSP

, FLP

, FRS

, FR}

Q2={h

E, h

P, h

T, h

FS, h

SP, h

LP, h

RS, h

R,

FSP

, FLP

, FRS

, FR}

Q3={h

E, h

P, h

T, h

FS, h

SP, h

LP, h

RS, h

CR, h

R,

FSP

, FLP

, FRS

, FR}

Q4={h

E, h

P, h

T, h

FS, h

SP, h

LP, h

CR, h

R,

Fig. 3 Modular models of the automatic transmissionFSP

, FLP

, FRS

, FR}

Hence, the set of length vectors used in the 1–2 shiftsimulation is the scope of this paper. Instead, static torque converter

models are used in this study; i.e. the torque converterL={12, 12, 13, 12} (11)characteristics are approximated from regression � ts ofactual experimental steady state performance curves.

4 MODEL STRUCTURE

4.3 Hydraulic control system modelFigure 3 shows the modular models of an automatictransmission consisting of an engine module, torque con- The hydraulic circuit of an electronically controlledverter module, geartrain module, hydraulic control automatic transmission is complex and consists of manymodule and the modules for clutches and bands. A brief elements such as a pump, electromagnetic actuators,description of these modules follows. regulator and control valves, etc., as part of a complex

system. It controls hydraulic pressures in clutches andbands according to the signals from the electronic con-

4.1 Engine model trol unit of the automatic transmission. It is essential forthe hydraulic pressure at friction elements to be preciselyIn this study, a simpli� ed engine model based on thecontrolled in relation to the amount of torque trans-numerical tabulation of the measured engine perform-mitted during the shift process in order to achieveance curve is used. A cubic interpolation method is usedoptimum shift quality. The pressure � uctuations into obtain the engine torque during intermediate throttlethe hydraulic system often create fatigue problems inpositions and engine speeds.the system components and lead to unacceptable shiftqualities and low-frequency airborne hydraulic noise.Understanding the dynamic performance of each4.2 Torque converter modelhydraulic element is therefore essential in predicting thetransient characteristics of the frictional torque. A com-The torque converter provides torque multiplication

during vehicle launch and smooth acceleration during prehensive mathematical model of the hydraulic controlsystem is developed to calculate the pressure applied togear changes. The derivation of detailed dynamic models

of the torque converter that captures the accelerations the frictional elements during shift changes.Because of the computational complexity of theof the � uid � ow as well as time lags associated with the

establishment of steady state � ow conditions is beyond hydraulic circuit, a pragmatic approach to the math-



ematical description of the hydraulic elements is required The � owrate passing through the valves and ori� cescan be de� ned in terms of Reynolds number; i.e. thein order to execute the simulation. As an example, the

� ow characteristic (pressure drop/� ow resistance) of the standard ori� ce � ow equations of turbulent � ow arehydraulic circuit at low temperature is dominated by

QÇ (t)=cA(t) ã p1(t ) p

2(t) (13)the viscosity eVect rather than the turbulent ori� ce � ow;

thus the pressure drop due to viscosity eVects should be and viscous � ow conditions for laminar � ow areincorporated at low operating temperatures. Math-ematical representation of each hydraulic element can QÇ (t)=

p1(t ) p

2(t)

l

ðdc3R12q C

1+3

2 Ae

CRB

2

D(14)

be formulated from � rst principles. The pressure changesin terms of various � ow quantities are described by the

The stiVness characteristics of the clutch can be divideddiVerential equations for � uid dynamics, and the � ow-

into two regions; one is a linear region where the stiVnessrate passing through valves and ori� ces is described by

is governed by the clutch return spring and the secondlaminar and turbulent ori� ce � ow equations.

region is dominated by the compressibility of the facingIn particular, Newton’s second law is used to derive

material of the clutch. Based on the measured exper-the equation of motion of the regulator and control

imental data, the clutch non-liinear characteristics canvalves, which is primarily dictated by pressure diVer-

be written asences, spring forces and jet forces. Fluid dynamics

FC(t)=k

0X+H(X X

0)k

1(X X

0)2 (15)models are used to de� ne the pressure changes in the

hydraulic system. As an example, sudden pressureAs an example, simulation results of the hydraulic

changes in a valve signi� cantly aVect the dynamics ofsystem are compared with measured pressure pro� les in

the valve. The diVerential equations that describe theFig. 4. In particular, Fig. 4 compares the band pressure

pressure changes in terms of various system � ow resulting from a typical current pro� le of the trans-quantities can be written in the following generalized mission control unit (TCU). The pressure pro� les showform:

very good correlation, with the exception of the periodbetween 4.8 and 5.3 s. It is noted that the inertia phase

pÇ(t)=QÇ (t)V(t )

b( p) (12) begins at t=4.8 s and the system enters the non-lineartransient gear state.

where the bulk modulus can be de� ned as a function ofpressure, i.e.

4.4 Clutch and band modelb( p)=

b0pk

(1 ca)pk+c

ab

0 Frictional torques of clutches and bands are obtainedfrom empirical formulae [9 ]. The torque transmitted byandclutches is proportional to the axial force across theclutch plates and the coeYcient of friction on the contactc

a=

c0( p

0/p)0.7

1 c0+c

0( p

0/p)0.7 surfaces. The torque transmitted through the multidisc

Fig. 4 Comparison of the simulated band pressure with measured data



clutch can be expressed as follows: 5 MATHEMATICAL MODEL

TC=m

CnR

mF

C(16)

Figure 6 shows the free-body diagram of the BTR four-where speed automatic transmission system. The equations of

motion of the transmission elements are developed usingR

m=

2(R3o R3i

)

3(R2o R2i

)the following assumptions:

1. All links and rotating elements of the transmissionIt is noted that the dynamic coeYcient of friction is gen- are rigid.erally a function of the slipping speed of the frictional 2. All links have only one rotational degree of freedom.couple. 3. Gears exhibit no backlash and bearings have no play.

The torque transmitted by the band depends on 4. Friction eVects of rotating elements except bands andwhether the band is in the energized mode or in the clutches are neglected.de-energized mode. In the energized mode, the band

The dynamic equations are expressed as follows:torque is given by [9 ]

Engine:TB=R

dF

B(eí

Bâ

B 1) (17)

IEh

E=T

E T

LU T

P(23)and in the de-energized mode the band torque is

Torque converter shaft:TB=R

dF

B(1 e Õ í

Bâ

B) (18)

I1h

1=T

LU+T

T T

C2 T

C4(24)Accurate values of F

Bare determined from the model

of the hydraulic control system described in the pre- Forward sun gear:vious section. A typical band force pro� le used in the

IFS

hFS

=TC2

+TC4

3FSP

rFS

(25)simulation of the 1–2 shift process is shown in Fig. 5.

Short pinion:

3ISP

hSP

=3rSP

(FSP

FLP

) (26)4.5 Planetary geartrain model

Long pinion:The energy method and free-body analysis method areused to derive the governing equations of the geartrain 3I

LPh

LP=3r

LP(F

LP F

R) 3r

LPF

RS(27)

during shift changes. Assuming that there is no backlashRing:between gear meshings and the gear elements are in� -

nitely stiV, the speed and torque relationships of the BTR (IR+I

V)h

R=3r

RF

R T

V(28)

Ravigneaux planetary gearset areReverse sun gear:

hÇFS

iFS1

hÇR (1 i

FS1)hÇ

CR=0 (19)

IRS

hRS

=3rRS

FRS

TB

(29)hÇ

RS+ i

RS1hÇ

R (1+ i

RS1)hÇ

CR=0 (20)

Carrier:T

R= i

RS1T

RS i

FS1T

FS(21)

AhCR

= TOWC1

+3FSP

(rFS

+rSP

)+3FRS

(rRS

+rLP

)T

CR= (i

RS1+1)T

RS+(i

FS1 1)T

FS(22)

3FR(r

RS+r

LP) 3F

LP(r

SP+r

LP) (30)

where equations (19) and (20) describe the speedThe � ow chart of the 1–2 shift simulation is shown in

relationship and equations (21) and (22) describe theFig. 7. Referring to the mode description method

torque relationship.described in Section 3, modes 1, 2 and 3 represent thesteady state in the � rst gear, torque phase and inertiaphase respectively. The last mode, mode 4, is a phaseafter the gear ratio change. The reverse sun gear is heldin the second gear, and the angular velocities and thetorque are � xed by the second gear speed ratios.

6 RESULTS AND DISCUSSION

Figure 8 shows the typical speed ratio pro� le during a1–2 shift for the B1 band force pro� les shown in Fig. 5.The simulation is performed at a 100 per cent throttlecondition at an engine speed of 4800 r/min. The 1–2 shiftFig. 5 Typical force applied to the B1 band during the 1–2

shift is initiated at t=0.25 s and the speed ratio remains in



Fig. 6 Free-body diagram of the BTR four-speed automatic transmission

Fig. 7 Flow chart of the 1–2 shift simulation

Fig. 8 Change in the speed ratio during the 1–2 shift

the � rst gear ratio (iFS1

=2.39) during the torque phase torque pro� le with the torque hole (rapid decrease inoutput torque) during the torque phase and gradualbetween t # 0.25 s and t # 0.49 s. The speed ratio then

continuously decreases to the second gear ratio (iFS2

= torque increase during the inertia phase. To examine thesensitivity of the time width and torque hole, the 1–21.45) during the inertia phase between t # 0.49 s and

t # 1.47 s. Consequently, the engine speed also decreases shift simulation is performed with various combinationsof band pressures and frictional coeYcients.during the inertia phase. Figure 9 shows that the corre-

sponding output speed pro� le of the ring gear increases Variations in band forces are common in high-volumeproduction transmissions because of the dimensionalslightly during the 1–2 shift. Figure 10 shows the output



Fig. 9 Output speed pro� le of the ring gear

Fig. 10 Output shaft torque pro� le of the ring gear

Fig. 11 Four diVerent forces applied to the B1 band



tolerances associated with manufacturing transmission begins at t=0.25 s and ends at t=0.55 s, which is longerthan the shift event presented in Fig. 10 (i.e. t=0.25 scomponents, in particular, high-precision hydraulic

elements. In addition, variations in oil properties such and t=0.49 s respectively). Similarly, the inertia phasebegins at t=0.55 s and ends at 1.525 s. The inertia phaseas frictional characteristics, oil viscosity, oil ageing, etc.,

alter the performance of the hydraulic system and con- duration of 0.975 s is similar to the one shown in Fig. 10(duration of 0.98 s). Results con� rm that a small changetribute to variations in the band supply pressure. To

simulate the variability of a typical production trans- in the dynamic friction coeYcient has a signi� cant eVecton the time width and torque hole in the torque phasemission, simulations are performed using the four

slightly diVerent band force pro� les shown in Fig. 11. of the shift process. The static and dynamic frictioncoeYcients vary during the service of automatic trans-These band force pro� les are obtained by changing the

regulator valve (an important hydraulic system compo- missions as a result of changes in oil temperature, oilviscosity, and components and oil ageing.nent) dimensions within the production tolerances. The

output shaft torque pro� les corresponding to the four The developed simulation model was then rearrangedto investigate what B1 band pressure and torque pro� lesinput forces are compared in Fig. 12. Cases 1 and 3 show

a sharp decrease in output torque at the end of the torque were required to achieve a desired output torque pro� le.In the rearranged model, the desired shaft torque is usedphase and case 2 shows the longest duration of the shift

and sharp changes in output torque at the end of the as one of the input parameters and the correspondingB1 torque is calculated as part of the simulation. As aninertia phase. Alternatively, case 4 shows the best torque

pro� le; i.e. it exhibits the minimum torque hole and example, a desired output torque pro� le, shown inFig. 14, which is smoother than the torque pro� le pre-output torque disturbance. Results presented in Fig. 12

show that variations in the band force have a signi� cant sented in Fig. 10, is used as an input to the simulationmodel and then the 1–2 shift simulation is executed. TheeVect on the output shaft torque variations.

To investigate the eVect of the dynamic friction resulting B1 band torque required to produce the desiredoutput torque pro� le is obtained and shown in Fig. 15.coeYcient of the band material, the 1–2 shift simulation

is performed with a 2 per cent decrease in the friction Here again the results show that a smooth B1 bandtorque pro� le is required to achieve smooth outputcoeYcient. The results corresponding to the output

torque pro� le are presented in Fig. 13. The torque phase torque variations during the 1–2 shift. This model can

Fig. 12 Output torque pro� les corresponding to the band forces shown in Fig. 10



Fig. 13 Output torque pro� le with a 2 per cent decrease in the friction coeYcient of Band B1

Fig. 14 Desirable output torque pro� le

Fig. 15 Band torque required for the desirable output torque pro� le shown in Fig. 14



now be used to explore diVerent torque pro� les, which REFERENCEScan drive an optimal shift strategy used in a vehicle.

1 Kotwicki, A. J. Dynamic models for torque converterequipped vehicles. SAE technical paper 820393, 1982.

2 Pan, C.-H. and Moskwa, J. J. Dynamic modelling and simu-lation of the Ford automobile transmission. SAE technical7 CONCLUSIONSpaper 950899, 1995.

3 Hojo, Y., Iwatsuki, K., Oba, H. and Ishikawa, K. ToyotoA mathematical model and simulation system of the � ve-speed automatic transmission with application ofBTR four-speed automatic transmission have been modern control theory. SAE technical paper 920610, 1992.developed in order to investigate the transient character- 4 Hwang, S.-J., Chen, J.-S., Liu, L. and Ling, C.-C. Modelling

and simulation of a powertrain–vehicle system with auto-istics during gear changes. A mode description methodmatic transmission. Int. J. Veh. Des., 2000, 23(1/2), 145–160.of the shift process, a modular structure of the power-

5 Megli, T. W., Haghgooie, M. and Colvin, D. S. Shift charac-train system and the � owchart of the simulation are pre-teristics of a 4-speed automatic transmission. SAE technicalsented. Simulation results show that the dynamic modelpaper 1999-01-1060, 1999.accurately predicts dynamic transient performance of an

6 Kim, Y. H., Yang, J. and Lee, J. M. A study on the transientautomatic transmission during the 1–2 shift process. characteristics of automatic transmission with detailedThis is a very useful tool for determining optimum dynamic modelling. SAE technical paper 941014, 1994.pressure pro� les for the shifting elements. 7 Jo, H.-S., Park, Y.-I., Lee, J.-M., Jang, W.-J., Park, J.-H.

and Lim, W.-S. A study on the improvement of the shiftcharacteristics for the passenger car automatic transmission.Int. J. Veh. Des., 2000, 23(3/4), 307–328.

8 Munns, S. A. Computer simulation of powertrain compo-ACKNOWLEDGEMENTSnents with methodologies for generalized system modelling.MS thesis, University of Wisconsin-Madison, 1996.

The � nancial support for this work is provided jointly 9 Design Practices: Passenger Car Automatic Transmissions,by BTR Automotive, Australia, and the Australian 3rd edition, 1994, AE-18 (SAE Inc., Warrendale,

Pennsylvania).Research Council SPIRT Grant C00107787.


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