DYNAMIC MODELING, FRICTION PARAMETER ESTIMATION ...

DYNAMIC MODELING, FRICTION PARAMETER ESTIMATION, AND

CONTROL OF A DUAL CLUTCH TRANSMISSION

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in

the Graduate School of The Ohio State University

By

Matthew Phillip Barr, B.S.M.E

Graduate Program in Mechanical Engineering

The Ohio State University

2014

Master's Examination Committee: Approved by

Professor Krishnaswamy Srinivasan, Advisor

Professor Shawn Midlam-Mohler

Advisor

Department of Mechanical and Aerospace Engineering

Copyright by

Matthew Phillip Barr

2014

ii

ABSTRACT

In this thesis, a mathematical model of an automotive powertrain featuring a wet

dual clutch transmission is developed. The overall model is comprised of models that

describe the dynamic behavior of the engine, the transmission mechanical components,

the hydraulic actuation components, and the vehicle and driveline. A lumped-parameter

model, that incorporates fluid film dynamics and a simplified thermal model, is used to

describe wet clutch friction. The model of the hydraulic actuation system includes

detailed models of the clutch and synchronizer actuation subsystems. A simulation of the

dynamic powertrain model is built using AMEsim and MATLAB/Simulink.

The powertrain simulator is used to demonstrate how changes in transmission

parameters affect the quality of clutch-to-clutch shifts and the overall dynamic response

of the powertrain. Based on this model, measurements of clutch pressure and the

rotational speeds and estimated accelerations at the input and output sides of the clutch

are used in the design of a friction parameter estimation scheme that can be implemented

offline using past simulation data or online using current simulation signals. For both

offline and online cases, simulation results demonstrate that friction parameters are

estimated with reasonable accuracy.

iii

An integrated powertrain controller is developed with a model-based feedforward

controller and multiple feedback loops. The feedforward controller, which generates a

pressure command to either clutch, is developed by inverting a simplified model of the

powertrain, and using a static friction model to relate clutch pressure to friction torque.

The inputs to the feedforward controller are speeds and estimated accelerations of the

engine and clutches. The feedforward controller adapts to changes in friction

characteristics by updating the friction parameters used in the static friction model using

the values generated by the estimation scheme. The feedback controller contains loops

that control clutch slip and engine speed by manipulating clutch pressure, throttle angle,

and spark advance. Simulation results for the proposed controller demonstrate that for

upshifts, the adaptation of the feedforward controller to varying friction parameters

results in improved shift quality relative to the non-adaptive case where the friction

parameters input to the feedforward controller are not varied along with the simulated

friction characteristics.

iv

DEDICATION

This one’s for you, Pop-pop.

v

ACKNOWLEDGMENTS

I wish to thank my advisor, Professor Krishnaswamy Srinivasan, for his

encouragement, patience and support, and technical expertise. I have learned a great deal

from him over the last two and a half years, and it is due to his guidance that I will leave

The Ohio State University a much more confident engineer. I also wish to thank my

committee member, Professor Shawn Midlam-Mohler for his time, as well as his helpful

suggestions and comments.

I want to express my thanks to Professor Ahmet Selamet for giving me the

opportunity to be a teaching assistant, and the Department of Mechanical and Aerospace

Engineering for providing financial support throughout my graduate studies. Through this

experience, I found that I truly enjoy teaching and I hope to return to the classroom

someday.

Last, but not least, I wish to thank my parents, Gilbert and Gale Barr, my brother,

Jason Barr, and my friends for their constant encouragement, love, and support. Without

all of you, I could not have made it to this point.

vi

VITA

June 2007 .......................................................Solon High School

December 2011 ..............................................B.S. Mechanical Engineering, The Ohio

State University

January 2012 to present ................................Graduate Teaching Associate, Department

of Mechanical and Aerospace Engineering,

The Ohio State University

FIELDS OF STUDY

Major Field: Mechanical Engineering

System Dynamics, Hydraulic Systems, Modeling and Control of Dual Clutch

Transmissions

vii

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ ii

DEDICATION ................................................................................................................... iv

ACKNOWLEDGMENTS .................................................................................................. v

VITA .................................................................................................................................. vi

FIELDS OF STUDY.......................................................................................................... vi

LIST OF TABLES .............................................................................................................. x

LIST OF FIGURES .......................................................................................................... xii

NOMENCLATURE ...................................................................................................... xviii

CHAPTER 1: INTRODUCTION ....................................................................................... 1

1.1 Background and motivation .............................................................................. 1

1.2 Objectives of the research ................................................................................. 4 1.3 Organization of the thesis ................................................................................. 5

CHAPTER 2: LITERATURE REVIEW ............................................................................ 7

2.1 Modeling of a dual clutch transmission ............................................................ 7 2.1.1 Overview of dual clutch transmissions .............................................. 7

2.1.2 Gear synchronization and the clutch pressure control valve ............. 9 2.1.3 Compensation for lost torque converter benefits ............................. 11

2.2 Control of clutch and gear engagement in a dual clutch transmission ........... 13 2.2.1 Phases of a clutch-to-clutch shift during upshifting and downshifting

....................................................................................................... 13 2.2.2 Clutch slip control and friction parameter estimation using pressure

and speed measurements ............................................................... 15 2.2.3 Gear preselection and gear skipping ................................................ 23

2.3 Conclusion ...................................................................................................... 26

CHAPTER 3: POWERTRAIN MODEL .......................................................................... 28

3.1 Top level structure of powertrain model ......................................................... 28

viii

3.2 Mean-value engine model ............................................................................... 30 3.2.1 Intake manifold air dynamics ........................................................... 32 3.2.2 Intake manifold fueling dynamics ................................................... 33

3.3 Dual clutch transmission mechanical system ................................................. 35

3.3.1 Rotational dynamics of the dual clutch transmission ...................... 38 3.3.2 Effect of gear selection on the torque acting on the input shafts ..... 41 3.3.3 Implementation of Karnopp friction model .................................... 44 3.3.4 Dynamic clutch friction model ........................................................ 48

3.4 Longitudinal vehicle dynamics ....................................................................... 54

3.4.1 Vehicle dynamics with tire-road interaction .................................... 55 3.4.2 Simplified vehicle dynamics for feedforward control ..................... 57

3.5 Hydraulic component actuation ...................................................................... 58 3.5.1 Pressure regulation system ............................................................... 63

3.5.1.1 Pressure regulation valve .................................................. 65 3.5.1.2 Pressure regulation control solenoid ................................. 70

3.5.2 Clutch actuation system ................................................................... 75 3.5.2.1 Clutch pressure control valve, N215 ................................. 77

3.5.2.2 Clutch piston, K1 .............................................................. 84 3.5.3 Synchronizer actuation system ........................................................ 85

3.5.3.1 Multiplexer valve .............................................................. 87

3.5.3.2 Multiplexer control solenoid ............................................. 89 3.5.3.3 Shift fork, SF13 ................................................................. 93

3.5.3.4 Synchronizer solenoid, N88 .............................................. 95 3.6 Model limitations ............................................................................................ 99

3.7 Conclusion .................................................................................................... 101

CHAPTER 4: MODEL SIMULATION ......................................................................... 102

4.1 Modeling in AMEsim ................................................................................... 103 4.2 Solver (or integrator) options in AMEsim .................................................... 105 4.3 Co-simulation with MATLAB/Simulink ...................................................... 107

4.4 Simulation results for vehicle launch ............................................................ 109 4.5 Conclusion .................................................................................................... 115

CHAPTER 5: FRICTION PARAMETER ESTIMATION ............................................ 116

5.1 Clutch friction characteristics as the material/fluid ages .............................. 116 5.2 Friction parameter estimation ....................................................................... 120

5.2.1 Sensitivity of coefficient of friction to variation in individual

parameters ................................................................................... 120 5.2.2 Estimation scheme and results ....................................................... 123

5.3 Conclusion .................................................................................................... 135

CHAPTER 6: INTEGRATED POWERTRAIN CONTROL OF CLUTCH SLIP ........ 136

ix

6.1 Transmission shift control strategies from literature .................................... 137 6.2 Proposed integrated powertrain controller .................................................... 145 6.3 Simulation results.......................................................................................... 162 6.4 Conclusion .................................................................................................... 177

CHAPTER 7: CONCLUSIONS AND FUTURE WORK .............................................. 179

7.1 Summary ....................................................................................................... 179 7.2 Contributions................................................................................................. 180 7.3 Recommendations for future work ............................................................... 181

REFERENCES ............................................................................................................... 183

APPENDIX A: AUXILIARY HYDRAULIC SUBSYSTEMS ..................................... 188

A.1 Clutch cooling system ............................................................................. 188 A.2 Safety systems ......................................................................................... 194

APPENDIX B: EVEN GEAR COMPONENT ACTUATION SYSTEM ..................... 196

APPENDIX C: SIMULATION PARAMETERS ........................................................... 208

x

LIST OF TABLES

Table 3.1: Model parameters and sources......................................................................... 29

Table 3.2: Clutch and output gear engagement schedule for VW02E .............................. 38

Table 3.3: Solenoid operation schedule for output gear engagement ............................... 87

Table 4.1: Powertrain simulator solver parameters ........................................................ 109

Table 5.1: Ingredients in paper-based friction materials [42] ......................................... 116

Table 5.2: Controlling factors of wet friction materials [42] .......................................... 118

Table 5.3: BW-6100 coefficient of friction parameters .................................................. 119

Table 5.4: Coefficient of friction data subsets ................................................................ 130

Table 5.5: Offline estimation of coefficient of friction parameters ................................ 133

Table 5.6: Simulated coefficient of friction parameters for parameter estimation ......... 133

Table 5.7: Online estimation of varying sets of coefficient of friction parameters ........ 134

Table 6.1: Simulated coefficient of friction parameters for controller comparison ....... 163

Table 6.2: Peak-to-peak and root-mean-square driveshaft accelerations for the ideal case,

case 2 and case 5: Hebbale and Kao’s strategy, launch-1st-2

nd-1

st .................................. 166

Table 6.3: Peak-to-peak vehicle acceleration and jerk for the ideal case, case 2 and case

5: Hebbale and Kao’s strategy, launch-1st-2

nd-1

st ........................................................... 166


case 2 and case 5: strategy proposed by Bai et al., launch-1st-2

nd-1

st ............................. 167

xi


5: strategy proposed by Bai et al., launch-1st-2

nd-1

st ....................................................... 167


and cases 1-5: proposed strategy, launch-1st-2

nd-1

st ........................................................ 171


and cases 1-5 with/without feedforward adaptation: proposed strategy, upshift from 1st-

2nd

.................................................................................................................................... 172


and cases 1-5 with/without feedforward adaptation: proposed strategy, downshift from

2nd

– 1st ............................................................................................................................ 173

Table 6.9: Peak-to-peak vehicle acceleration and jerk for cases 1-5 with/without

feedforward adaptation: proposed strategy, upshift from 1st – 2

nd .................................. 175


feedforward adaptation: proposed strategy, downshift from 2nd

– 1st ............................ 176

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LIST OF FIGURES

Figure 1.1: Example stick diagram of a dual clutch transmission [7] ................................ 3

Figure 2.1: Input shaft torques during gear synchronization [14] .................................... 10

Figure 2.2: Typical direct acting solenoid valve [15] ....................................................... 11

Figure 2.3: Normalized clutch pressure profiles at launch [7] .......................................... 12

Figure 2.4: Transmission output torque at launch as a function of launch times [7] ........ 13

Figure 2.5: “First generation” clutch-to-clutch power-on shift: upshift, clutch 1 to 2 (left)

downshift, clutch 2 to 1 (right) [12] .................................................................................. 15

Figure 2.6: Closed loop pressure control block diagram [21] .......................................... 20

Figure 2.7: Tracking of commanded pressure – open loop and closed loop [21] ............. 20

Figure 2.8: “Second generation” clutch-to-clutch power-on shift: upshift, clutch 1 to 2

(left) downshift, clutch 2 to 1 (right) [12] ......................................................................... 22

Figure 2.9: “Third generation” clutch-to-clutch power-on upshift [25] ........................... 23

Figure 2.10: Multiple-shifts from clutch 1-2-1: double-upshift (left) double-downshift

(right) [12] ......................................................................................................................... 25

Figure 2.11: Output torque and gearbox component speeds for an upshift from first gear

to third gear [12] ............................................................................................................... 26

Figure 2.12: Output torque and gearbox component speeds for a downshift from third

gear to first gear [12]......................................................................................................... 26

xiii

Figure 3.1: Block diagram of overall powertrain model................................................... 31

Figure 3.2: VW02E DCT stick diagram ........................................................................... 36

Figure 3.3: Synchronizer components [31] ....................................................................... 37

Figure 3.4: Rotational dynamics of VW02E engine, flywheel and gearbox .................... 39

Figure 3.5: Generalized Stribeck friction curve [44] ........................................................ 45

Figure 3.6: Generalized Stribeck friction curve: a) Steep-line approximation b) Karnopp

model [44] ......................................................................................................................... 45

Figure 3.7: Simplified rotational dynamics for implementation of Karnopp friction

model: a) K1 sticking, K2 disengaged/slipping b) K2 sticking, K1 disengaged/slipping 46

Figure 3.8: Free body diagram for vehicle dynamics model ............................................ 54

Figure 3.9: Free body diagram for simplified model of vehicle dynamics ....................... 58

Figure 3.10: Pressure regulation system ........................................................................... 60

Figure 3.11: Simplified clutch actuation system............................................................... 60

Figure 3.12: Synchronizer actuation system - solenoids and multiplexer valves ............. 61

Figure 3.13: Synchronizer actuation system - shift forks and synchronizers ................... 62

Figure 3.14: Fixed displacement internal gear pump [40] ................................................ 63

Figure 3.15: Pressure regulation valve .............................................................................. 64

Figure 3.16: Pressure regulation control solenoid, N217 ................................................. 65

Figure 3.17: Ball poppet geometry ................................................................................... 71

Figure 3.18: PWM and VFS voltage pulse trains ............................................................. 73

Figure 3.19: Steady-state pressure versus solenoid current: N217 ................................... 74

Figure 3.20: Odd gear clutch actuation system ................................................................. 76

xiv

Figure 3.21: Steady-state pressure versus solenoid current: N215 ................................... 80

Figure 3.22: Multiplexer valve, MPV ............................................................................... 86

Figure 3.23: Multiplexer control solenoid, N92 ............................................................... 89

Figure 3.24: 1-3 shift fork and synchronizers ................................................................... 92

Figure 3.25: Synchronizer solenoid, N88 ......................................................................... 95

Figure 4.1: Example of component causality ................................................................. 104

Figure 4.2: Line pressure during launch ......................................................................... 110

Figure 4.3: Clutch #1 measured and commanded pressures during launch .................... 110

Figure 4.4: Normalized shift fork position and synchronizer state during engagement of

output gear #2 ................................................................................................................. 111

Figure 4.5: Clutch (hub and gearbox side) and engine angular speeds during launch ... 113

Figure 4.6: Clutch #1 state during launch ....................................................................... 113

Figure 4.7: Engine and clutch #1 friction torque during launch ..................................... 114

Figure 4.8: Differential torque during launch ................................................................. 114

Figure 4.9: Longitudinal vehicle velocity during launch ................................................ 115

Figure 5.1: curves with positive and negative slopes [11] ................................... 117

Figure 5.2: curve for BW-6100 friction plate [41] ............................................... 118

Figure 5.3: as is varied from baseline .................................................... 121

Figure 5.4: as is varied from baseline ..................................................... 121



Figure 5.7: as is varied from baseline ..................................................... 123

xv

Figure 5.8: Simulink implementation of discrete Kalman estimator .............................. 126

Figure 5.9: Derivative of clutch #1 pressure during launch: calculated using Kalman

estimator and in AMEsim ............................................................................................... 127

Figure 5.10: Derivative of clutch #1 speed during launch: calculated using Kalman

estimator and in AMEsim ............................................................................................... 127

Figure 5.11: Offline estimation and simulation of friction torques during a 2-3 upshift 128

Figure 5.12: Online estimation and simulation of friction torques during a 2-3 upshift 129

Figure 5.13: Estimated coefficient of friction as a function of slip speed and viscosity 132

Figure 6.1: Flow chart of Hebbale and Kao’s shift control strategy ............................... 137

Figure 6.2: Feedback control of clutch slip using Simulink’s built-in PID controller with

anti-windup ..................................................................................................................... 138

Figure 6.3: PID controller with integrator clamping ...................................................... 141

Figure 6.4: Flow chart of shift control strategy presented by Bai et al. ......................... 144

Figure 6.5: Goetz’s control strategy for an upshift [12] ................................................. 145

Figure 6.6: Goetz’s control strategy for a downshift [12] .............................................. 147

Figure 6.7: Flow chart of proposed integrated powertrain control strategy ................... 149

Figure 6.8: Top level schematic of the proposed integrated powertrain controller ........ 150

Figure 6.9: Pressure control solenoid, duty cycle versus clutch pressure data points used

in lookup table................................................................................................................. 152

Figure 6.10: Feedback control of clutch pressure using Simulink’s built-in PID controller

......................................................................................................................................... 152

xvi

Figure 6.11: Feedback control of engine speed using Simulink’s built-in PID controller

with anti-windup (manipulation of clutch pressure) ....................................................... 154


with anti-windup (manipulation of throttle angle and spark advance) ........................... 154

Figure 6.13: Simplified rotational dynamics for feedforward controller development .. 156

Figure 6.14: Generalized model inversion for calculation of feedforward clutch pressure

......................................................................................................................................... 156

Figure 6.15: Driveshaft accelerations for the ideal case, case 2 and case 5: Hebbale and

Kao’s strategy, launch-1st-2

nd-1

st .................................................................................... 164

Figure 6.16: Driveshaft accelerations for the ideal case, case 2 and case 5: strategy

proposed by Bai et al., launch-1st-2

nd-1

st ........................................................................ 164

Figure 6.17: Driveshaft accelerations for the ideal case, and case 1 with/without

feedforward adaptation: proposed strategy, launch-1st-2

nd-1

st ........................................ 168



nd-1

st ........................................ 169



nd-1

st ........................................ 169



nd-1

st ........................................ 170



nd-1

st ........................................ 170

xvii

Figure 6.22: Simulation of ideal friction parameters, actual and feedforward pressures at

clutch #2: proposed controller, inertia phase of 2-1 downshift ...................................... 174

Figure 6.23: Simulation of ideal friction parameters, clutch #2 slip speed: proposed

controller, inertia phase of 2-1 downshift ....................................................................... 174

Figure A.1: Schematic of pressure regulation and clutch cooling systems .................... 188

Figure A.2: Clutch cooling system ................................................................................. 189

Figure A.3: Schematic of safety and clutch actuation systems ....................................... 194

xviii

NOMENCLATURE

Variables

Cross-sectional area [m2]

Equivalent flow area [m2]

Normalized air fuel influence

Frontal area of vehicle [m2]

Pressure acting area of clutch piston [m2]

Orifice area, j=1-41 [m2]

- Dimensionless parameters used to calculate

Viscous damping coefficient [Nm/rad/s]

Longitudinal stiffness factor

Mechanical damping coefficient [Nm/rad/s]

Discharge coefficient

Heat capacity [J/K]

Shape, peak factors

Engine torque constant [Nm/kg/s]

Longitudinal drag coefficient

Diameter [m]

(If used with dynamic friction model) Thickness of friction

material [m]

Hydraulic diameter [m]

Pump displacement [m3/rad]

Duty cycle of PWM signal to solenoid

Force [N]

Force applied to synchronizer, j-k=1-3,2-4,6-R,5-N [N]

Frequency of PWM signal to solenoid [Hz]

Rolling resistance

Surface roughness factor

Young’s modulus for the friction material [Pa]

Film thickness [m]

, , Dimensionless values relating film thickness to surface roughness

Gear ratio

IG1/R, IG3, IG5 Input gears #1,#3,#5,R on input shaft #1

IG2, IG4/6 Input gears #2,#4,#6 on input shaft #1

Solenoid current [A]

xix

Inertia [kgm2]

Stiffness [Nm/rad]

K1, K2 Clutch #1, clutch #2

Proportional gain

Integral gain

Derivative gain

Linear valve gain [m3/s/Pa]

Inductance [H]

Mass [kg]

Mass flow rate [kg/s]

MBT Spark timing for maximum brake torque

N Number of synchronizer cones, clutch plates or grooves (in

friction material)

- Inductance polynomial coefficients for N88 [H/m2],[H/m],[H]




- Inductance polynomial coefficients for N92 [H/m

3], [H/m

2],

[H/m], [H]


3], [H/m

2],

[H/m], [H]


3], [H/m

2],

[H/m], [H]



Asperity density of the friction material [1/m2]

Filter coefficient for derivative control [rad/s]

Number of flow restrictions in series

OG1-OG6, OGR Output gears #1-#6, reverse output gear

OSG1, OSG2 Output shaft pinions gears

Pressure [Pa]

Normalized pressure influence at the throttle body inlet

(If used with dynamic friction model) Geometric scaling factor

Flow rate [m3/s]

(If used in electromagnetic circuit) Resistance [Ω]

Ideal gas constant for air [J/kg/K]

Inner radius [m]

Outer radius [m]

Mean radius [m]

S13,S24,S6R,S5N 1-3,2-4,6-R,5-N synchronizers

Spark advance [deg BTDC]

Normalized spark influence

Synchronizer state

xx

Torque [Nm]

Throttle angle [deg]

Normalized throttle opening

Durations of pulses for a PWM solenoid[s]

Rise time [s]

Duration of clutch slip control [s]

, Air intake, spark to torque production delay [s]

Volume [m3]

Solenoid voltage [V]

Displacement [m]

Overlap displacement for specified port [m]

Underlap displacement for specified port [m]

Angle [deg]

Acceleration tolerance [rad/s2]

Bulk modulus of ATF [Pa]

Air-to-fuel ratio

Asperity tip radius [m]

Viscous friction coefficient [s/rad]

Viscosity-independent, speed dependent friction coefficient

[1/(Pa rad)]

Longitudinal slip value

Heat conductivity [N/s/K]

(If used in electromagnetic circuit) Flux linkage [H A]

Flow number

Shape factor scaling coefficient

Friction coefficient scaling factor

Stribeck factors [unitless], [rad/s]

Beavars and Joseph factor

Engine volumetric efficiency

Temperature [K]

Angular displacement between grooves [rad]

Coefficient of friction

Viscosity [Pa s]

Density of ATF fluid [kg/m3]

Density of the air entering the intake manifold [kg/m3]

RMS roughness of mating surfaces (steel and friction material)

[m]

Effective fueling time constant [s]

Friction material permeability [m2]

Patir and Cheng’s flow factors

Beavars and Joseph slip coefficient

xxi

Maximum engine speed [rpm]

Angular speed [rad/s]

Angular acceleration [rad/s2]

Static-dynamic threshold speed [rad/s]

Variable subscripts

Net quantity of air in intake manifold

Quantity related to accumulator j=1-5

Quantity of air entering intake manifold

Quantity of air exiting intake manifold

Atmospheric parameter

Quantity related to the geometry of the ball in a ball poppet valve

Torque applied by brake

(If used with feedback control) Commanded variable

Quantity related to dry contact between surfaces

Clutch cooling valve

- Quantities for various chambers in clutch cooling valve

Manipulated variable in a feedback control loop

Flow to the clutch cooling path

Quantity related to synchronizer cones

Quantity at output of a pressure control valve

Pressure at which valve cracks

Differential

Gear ratio between output shaft #1 and the differential shaft

Gear ratio between output shaft #2 and the differential shaft

Desired quantity

Error signal to a controller

Engine

Indicated engine parameter

Mechanical connection between engine and flywheel

Quantity related to the closed loop control of engine speed

Quantity that defines the engagement of a component

Quantity corresponding to an equilibrium value

EST Quantity that refers to an estimated parameter

ET Quantities lumped at the engine crankshaft

Quantity related to valve exhaust

Quantity related to acting at clutch #1 friction

Quantity related to acting at clutch #2 friction

Sliding tire friction

Actual quantity of fuel entering the combustion chamber

Commanded quantity of fuel entering the combustion chamber

xxii

Quantity that is output from a feedforward controller

Flywheel

Number of output gears, and gear ratio between gearbox input and

output shafts (1-6)

Quantity related to the geometry and/or engagement of clutch #1

and #2

(If used with torque) Quantity introduced on gearbox side of clutch

to algebraically solve for friction torque as a function of external

torques

Clutch hub

Quantity introduced on clutch hub side of clutch to algebraically

solve for friction torque as a function of external torques

Quantity related to the high speed subset of coefficient of friction

data used to estimate

Transmission housing

Quantity related to a valve inlet

Initial value of a variable

Inertia phase

Quantity common to both clutch #1 and clutch #2

Clutch #1 half that is fixed to input shaft #1

Clutch #2 half that is fixed to input shaft #2

Quantity related to the fluid supply

Quantity related to the friction lining material

Quantity related to the low speed subset of coefficient of friction

data used to estimate

Quantity defined for both clutch #1 or clutch #2

Intake manifold

ag Magnetic force

ax Maximum quantity of a variable

Quantity related to multiplexer valve

Quantity for chamber A in the multiplexer valve

Quantity related to synchronizer solenoid N88




Quantity related to multiplexer control solenoid N92

Quantity related to clutch pressure control valve N215

- Quantities for various chambers in N215

Quantity related to clutch pressure control valve N216

- Quantities for various chambers in N216

Quantity related to pressure regulation control solenoid N217

Quantity related to clutch cooling control solenoid N218

Quantity related to safety valve N233

xxiii

Quantity related to safety valve N371

Net quantity of a variable

Quantity related to the near zero speed subset of coefficient of

friction data used to estimate

Quantity related to the ATF oil

Flow out of a valve

Quantity related to pressure relief valve

Pressure regulation valve

- Quantities for various chambers in pressure regulation valve

Quantity related to pressure sensor

Quantity related to fixed displacement hydraulic pump

Reference input to a control loop

Flow to return line

Road load

Quantity related to road incline

Quantity related to static friction

Static tire friction

Quantity related to ball poppet valve seat

Quantity related to the shift fork ( ) chamber that pressurizes to

engage gear #1 (OG1 or G1)










disengage gear #5 (N)




engage gear #R (OGR or GR)

Input shaft #1

Input shaft #2

Quantity related to clutch slip

Output shaft #1

Output shaft #2

Slip speed at which the viscous damping coefficient term

becomes non-negligible [rad/s]

Quantity upstream of the odd gear clutch and synchronizer actuation

systems

xxiv

Quantity upstream of the even gear clutch and synchronizer

actuation systems

Quantity related to a synchronizer

Quantity related to the throttle body

Torque phase

Quantity related to viscous effects on a moving surface

(If used to describe vehicle dynamics) Vehicle

Wheel

Front wheels/tires

Quantity related to wind

Rear wheels/tires

Longitudinal force acting on front axle

Net flow force through a valve

Flow force acting on a spool, caused by flow through the exhaust

Flow force acting on a spool, caused by flow through the inlet

Longitudinal force acting on rear axle

Vertical force

Abbreviations

ATF Automatic transmission fluid

DCT Dual clutch transmission

DMF Dual-mass-flywheel

DSG Direct shift gearbox

OWC One way clutch

PT Planetary automatic transmissions

1

CHAPTER 1: INTRODUCTION

1.1 Background and motivation

Over the last forty years, the planetary automatic transmission (PT) has been the

most widely used transmission in passenger vehicles driven in the United States; PTs can

be found in 70%-90% of said vehicles produced from 1975-2012 [1]. Planetary

automatics transmit power from the engine to the gearbox through a torque converter.

The most common torque converter is the fluid-filled three-element device that consists

of a pump (or impeller), stator, and turbine. The engine turns the pump, which circulates

hydraulic fluid to the turbine. Fluid flow through the turbine causes it, as well as the

gearbox input shaft, to rotate. Upon its exit from the turbine, the fluid is directed to the

pump inlet by the stator.

The biggest advantages gained by using a torque converter are vibrational

damping and torque amplification during vehicle launch. Internal combustion engines

produce torque pulsations because the individual combustion events for each cylinder

occur at different points during the engine cycle. The torque converter’s fluid coupling of

the engine to the gear train damps the vibrations caused by the engine torque pulsations,

as well as lurch associated with poor clutch-to-clutch shifts. Additionally, at low vehicle

speeds the vehicle inertia forces the turbine speed to be much lower than the pump speed.

This causes torque amplification at the turbine, and in turn, improves the vehicle

acceleration at launch [2].

2

To meet the ever increasing fuel efficiency demands – the corporate fuel economy

average (CAFE) standard, which applies to passenger vehicles and light trucks (pickup

trucks, vans and sports utility vehicles), was raised from 27.5 miles per gallon (mpg) in

2011 to 35.5 mpg in 2016 and 54.5 mpg in 2025 [3] – automotive engineers must

improve powertrain efficiency. For planetary transmissions, which are typically 85%-

87% efficient [4], the majority of the power losses are attributed to the torque converter.

Losses in the torque converter are caused by leakage flow, viscous flow, and flow

discontinuities between elements [5].

Manual transmissions, which operate with efficiencies of 96% or better [4], are an

alternative to planetary automatic transmissions. These transmissions incorporate a

layshaft design; the gear train consists of the input shaft, to which input gears are fixed,

and a parallel output shaft which is splined to synchronizers and supports, via bearings,

the output gears [6]. The engine is normally coupled to the gearbox input shaft. When the

driver compresses the clutch pedal, the engine is decoupled from gearbox so that the

desired output gear can be manually engaged. During the shift the transmission of torque

from the engine to the wheels is interrupted resulting in sharp changes in vehicle

acceleration during the shift. Also, the engine continues to consume fuel while providing

no power to the wheels.

The dual clutch transmission (DCT) was designed to overcome the issue of torque

interruption by combining two parallel layshaft transmissions. Each of the two branches

of the transmission is independently coupled to the engine via a clutch, and similar to

planetary automatic transmissions, power flow is switched between branches by a clutch-

to-clutch shift. The input shafts for each branch are concentric: a solid shaft is positioned

3

within a hollow shaft. The odd input gears (1, 3, 5, etc.) and the even input gears (2, 4, 6,

etc.) are fixed to input shafts #1 and #2, respectively. Typically, the output gears

corresponding to the low gears (1-3 or 1-4) are held by output shaft #1, and the remaining

gears are held by output shaft #2. Both output shafts are permanently meshed with the

differential gear. An example of a six-speed dual clutch transmission is presented in

Figure 1.1.

Figure 1.1: Example stick diagram of a dual clutch transmission [7]

The primary deficiencies of the dual clutch transmission in comparison with the

planetary automatic stem from the lack of a torque converter and shifting between just

two clutches rather than a combination of four or more clutches and brakes. Without the

torque amplification provided by a torque converter, DCTs can be perceived as sluggish

during vehicle launch. Also, DCTs do not have the benefit of the driveline damping

provided by a torque converter operating in the fluid coupling mode. Compared to a

4

planetary automatic, both clutches in a DCT are involved in every shift and are more

susceptible to wear. If the shifts are not well controlled, the clutches wear rapidly and the

overall life of the transmission is reduced.

1.2 Objectives of the research

There are three primary objectives of this work. The first objective is to develop a

dynamic model of a powertrain with a dual clutch transmission that can be used to

demonstrate how changes in transmission parameters and the control of clutch-to-clutch

shifts and output gear engagement affect the overall dynamic response of the powertrain.

Note that without the means for experimentally validating the powertrain model, the

model developed in this work is not meant to represent with any accuracy the

performance of a specific powertrain. Instead, simulations of subsystems of the

powertrain model have been compared to similar systems found in literature; by

demonstrating that simulations of each subsystem produce reasonable results, the

combined response of the whole model should also produce reasonable results. The

resulting powertrain model is thus a good test bed for evaluating the clutch friction

estimation and powertrain controller methods proposed here.

The second objective of this research is to use measurements of clutch pressure

and the rotational speeds on the hub and gearbox sides of the clutch to estimate changes

in clutch friction characteristics over the life of the transmission. As discussed in Chapter

3, the model of clutch friction used in this work incorporates fluid film dynamics and a

simplified thermal model. For most engagement conditions, the static friction model

commonly used in literature does not accurately describe clutch friction; however, for

5

high slip speed and high pressure clutch engagements, the static friction model is

reasonably accurate [8]. In this work, the measured signals described above and the static

clutch friction model are used to calculate the clutch coefficient of friction as a function

of slip speed. The estimated coefficient of friction data is then fit to a known structure so

that individual parameters such as the sliding and static coefficients of friction and the

viscous damping coefficient can be determined.

The third objective of this work is to design an integrated powertrain controller

that uses measurements of engine speed, clutch slip speed, and clutch pressure, and the

estimated friction parameters to smoothly control clutch engagement during both the

torque and inertia phases. In particular, a feedforward controller based on a simplified

version of the powertrain described in Chapter 3 is designed for both phases of the shift.

The resulting feedforward controller can be updated as clutch friction characteristics

change over the life of the transmission, thus reducing the degradation of the shift quality

with change in clutch friction characteristics.

1.3 Organization of the thesis

This thesis is organized as follows. In Chapter 2, a review of technical literature

pertaining to the modeling and control of dual clutch transmissions is presented. First, a

general explanation of DCT types – dry or wet – is given. Next, the modeling of

synchronizer engagement and pressure control solenoids, and the means by which DCTs

compensate for not utilizing a torque converter, are discussed. The control of a dual

clutch transmission includes two main operations: power transfer between clutches

(clutch-to-clutch shift) and the preselection of the next output gear. Clutch-to-clutch shift

6

control strategies that are common to both planetary automatics and dual clutch

transmissions are discussed. Finally, the benefits of having access to clutch pressure

measurements are presented.

In Chapter 3, a detailed model of the powertrain of interest is presented. The

simulation of said powertrain is discussed in Chapter 4. The powertrain simulator is built

in AMEsim and MATLAB/Simulink. Simulation results for vehicle launch are provided

in this chapter; results pertaining to a clutch-to-clutch shift are deferred until Chapter 6,

where the shift controller is presented.

Chapter 5 presents the method for estimating clutch friction characteristics. The

results show that the estimation algorithm is accurate enough to show significant

deviations in friction characteristics, such as a change in sign of the friction gradient with

slip speed, over the life of the transmission. Chapter 6 presents the development of an

integrated powertrain shift controller. Both feedforward and feedback control are

employed in this work. The feedforward controller is based on inversion of the simplified

powertrain model described in the same chapter. The feedback control uses

measurements of engine speed, clutch slip speeds, and clutch pressures in the control of

clutch engagement. Finally, conclusions of the research and recommendations for future

work are given in Chapter 7.

7

CHAPTER 2: LITERATURE REVIEW

This chapter provides a review of literature pertaining to the modeling of dual

clutch transmissions, the control of a clutch-to-clutch shift, and the preselection of the

next output gear. The review is divided into three parts. Section 2.1 describes the

modeling of a dual clutch transmission. Section 2.2 focuses on the control of clutch and

output gear engagement in a dual clutch transmission. Note that current clutch-to-clutch

shift control strategies that are common to planetary automatic and dual clutch

transmissions are also discussed here. Concluding remarks are provided in Section 2.3 .

2.1 Modeling of a dual clutch transmission

2.1.1 Overview of dual clutch transmissions

Dual clutch transmissions can be designed with either single-plate dry clutches or

multi-plate wet clutches; the clutch type affects shift quality, efficiency and clutch torque

capacity. Dry DCTs are more efficient than wet DCTs because of the reduced drag torque

acting on the clutches and the decreased weight of the transmission. The drawback to dry

DCTs is that there is purely mechanical contact between clutch plates; when single-plate

clutches are slipping, heat is generated due to kinetic friction, and without any

lubrication, the cooling capabilities are limited. To reduce the heat transferred during a

shift, and in turn clutch wear, the clutch torque capacity should be limited and the

8

engagement time must be very short [9]. For these reasons, dry DCTs are primarily used

in small passenger vehicles such as the Ford Focus and Dodge Dart. Since these

transmissions are not a viable replacement for planetary transmissions in vehicles that

require high torque capacity, dry DCTs will not be discussed further in this work.

For wet DCTs, transmission fluid coats the clutch plate surfaces. The fluid film

provides additional cooling of the clutch plates, which offers the following advantages:

(1) multiple plates can be used, providing increased clutch torque capacity and (2) heat

transfer due to clutch slip is reduced, allowing smoother, less aggressive clutch

engagement. The friction torque developed in a wet-clutch is also affected by the fluid

film. When the clutch plates are coated with transmission fluid, the clutch friction is due

to viscous effects; as fluid is squeezed from the clutch plates, the cause of clutch friction

transitions from viscous effects to mechanical asperity contact. Ivanović et al. [10]

experimentally characterized the wet clutch coefficient of friction as it is influenced by

the following parameters: (1) the coefficient of friction of the lining material, (2) the

separator plate surface roughness, (3) the transmission fluid viscosity, (4) the friction

interface temperature, (5) and the engagement conditions (clutch slip speed and applied

force). The authors demonstrated that, with relatively new lining material and

transmission fluid, the coefficient of friction decreases significantly as the interface

temperature increases, and that at very high applied forces, the coefficient of friction

decreases as slip speed increases. Berglund et al. [11] showed experimentally that lining

and lubricant age causes a negative gradient of coefficient of friction with slip speed,

which leads to system instability and clutch judder [12]. With knowledge of how clutch

9

friction changes with operating conditions and age (i.e., if friction characteristics can be

estimated), a more robust shift control system can be developed.

2.1.2 Gear synchronization and the clutch pressure control valve

Most of the DCT simulations in the literature focus on predicting clutch-to-clutch

shift characteristics. The dynamics associated with gear synchronization are often ignored

because the gear is engaged prior to the clutch-to-clutch shift. In the works presented by

Zhang et al. [13] and Kulkarni et al. [7], gear synchronization is modeled as a switch: the

gear is either engaged or disengaged. These research groups believe that any dynamic

effects associated with gear selection do not affect the quality of the clutch-to-clutch

shift, because the next gear is fully engaged prior to the start of the shift. Galvagno et al.

[14] completed two different simulations to test whether the gear synchronization

dynamics can be ignored; the first and second simulations modeled the synchronizers as

switches and conic clutches, respectively. Figure 2.1 presents the input (or primary) shaft

torques plotted versus time for both simulations. The authors concluded that neglecting

synchronizer dynamics during clutch-to-clutch shifts caused oscillations in torque

transmitted through the gearbox that were not observed when the synchronizers were

modeled as conic clutches. Since said oscillations in torque attenuated in less than 0.1s,

synchronizer dynamics may be ignored if there is a sufficient delay between gear

synchronization and the initiation of the clutch-to-clutch shift. Thus to minimize the

effect of gear engagement on shift quality, synchronizers will be modeled as conic

clutches in this work.

10

Similar to planetary automatics, wet DCTs actuate, and control, clutches

hydraulically. The main difference between the hydraulic systems for both transmissions

is the mechanism for clutch pressure control. PTs use pilot solenoids to operate pressure

control valves; hydraulic fluid flows from the output of the pressure control valve to the

clutch cavity. The pilot solenoid controls the pressure in a command chamber of the

pressure control valve. Thus, the response time for pilot-operated valves, and in turn the

clutch pressure, is dependent on the pressure dynamics of the pressure control valve’s

command chamber. For this reason, Walker et al. [15] stated that pilot-operated solenoids

do not respond quickly enough to meet the short shift times demanded by dual clutch

transmissions. Instead, high flow, direct acting solenoids, which do not require a pressure

difference generated by an additional valve to operate, are used to directly control the

clutch pressure. An example of a typical direct acting solenoid valve is shown in Figure

2.2.

Figure 2.1: Input shaft torques during gear synchronization [14]

11

Figure 2.2: Typical direct acting solenoid valve [15]

2.1.3 Compensation for lost torque converter benefits

As stated in Section 1.1 , the torque converter (with open torque converter clutch)

utilized in planetary automatics provides torque amplification and vibrational damping

during launch (i.e., shifting from parked or neutral into 1st gear while the vehicle is

stopped). To maximize the torque available at the wheels during launch, dual clutch

transmissions utilize gear trains with wider and non-uniform gear ratio spreads; the drop

in gear ratio from 1st to 2

nd gear can easily be twice as large as the drop experienced

during higher gear shifts [16].

To improve driveline damping, a torsional damper [17] or dual mass flywheel

(DMF) [18] is inserted between the engine crankshaft and the transmission input shafts.

These elements reduce the transmittance of vibrations throughout the powertrain.

Additionally, Kulkarni et al. [7] illustrated that the oncoming clutch pressure profile

12

should be optimized so that overshoot of transmission output torque, felt by the driver as

lurch (or jerkiness), is limited without degrading the vehicle acceleration. Figure 2.3

displays clutch pressure profiles (normalized by maximum clutch pressure), and Figure

2.4 presents simulated output torques corresponding to slow, fast and optimized launch

times. Steady-state output torque is achieved in nearly the same time for all three cases;

however, the optimized pressure profile results in a reasonably quick launch with

minimal torque overshoot. We note here that the clutch pressure profiles are determined

during transmission calibration, so the effectiveness of any optimization during said

calibration diminishes with time.

Figure 2.3: Normalized clutch pressure profiles at launch [7]

13

Figure 2.4: Transmission output torque at launch as a function of launch times [7]

2.2 Control of clutch and gear engagement in a dual clutch transmission

2.2.1 Phases of a clutch-to-clutch shift during upshifting and downshifting

Both planetary automatics and dual clutch transmissions avoid torque interruption

between the engine and the wheels during gear shifts by carefully coordinating clutch-to-

clutch shifts. An upshift refers to a shift that causes a decrease in gear (or speed) ratio

between the transmission input and output shafts, while a downshift refers to a shift that

causes an increase in gear ratio. Shifts can occur when the engine provides power to the

wheels (power-on), or when the vehicle’s inertia causes the wheels to drive the engine

while coasting or braking (power-off). Compared to power-off shifts, power-on shifts

have a much greater effect on shift quality and will be discussed in more detail

throughout the remainder of this section [12].

14

Since both transmission designs rely upon friction elements for transmitting

power, a clutch-to-clutch shift consists of a torque (or load transfer) phase and an inertia

(or speed adjustment) phase. Note that the following description of these phases during

an upshift or downshift assumes that the engine is controlled independently of the

transmission: this is referred to as a “first generation” shift control scheme. Descriptions

of “second generation” and “third generation” shift control schemes, which are

accomplished with integrated powertrain control, are deferred until Section 2.2.2 . For

power-on upshifts, the torque phase precedes the inertia phase. During the torque phase,

the currently engaged (or offcoming or offgoing) clutch is disengaged by decreasing

clutch pressure. As the clutch pressure is reduced, the offcoming clutch begins to slip.

Once said clutch slips, the offgoing clutch pressure is reduced to zero. Simultaneously,

the oncoming (or target) clutch pressure is ramped up so that the slip speed between the

clutch surfaces is reduced. The torque phase concludes when the offcoming clutch is

completely disengaged and the full engine torque is transmitted through the oncoming

clutch. During the inertia phase, the engine must be decelerated to match the reduced

speed ratio of the oncoming gear. This deceleration is accomplished by increasing the

oncoming clutch torque above the required level; the increased clutch torque also causes

an increase in transmission output torque. Upon completion of the inertia phase, the

oncoming clutch slip speed and torque are reduced to zero and the level required to

transfer engine torque, respectively.

In the case of power-on downshifts, the inertia phase precedes the torque phase.

During the inertia phase, the engine must be accelerated to match the increased speed

ratio of the oncoming gear. This acceleration is accomplished by decreasing the offgoing

15

clutch torque until said clutch slips. Once the engine speed matches the target speed, the

offgoing clutch torque is increased to its original value; the inertia phase is concluded at

this point. During the torque phase, the offgoing clutch pressure is reduced to zero while

the oncoming clutch pressure is increased to the level required to transfer engine torque;

the torque phase is concluded at this point [12]. Figure 2.5 displays clutch torque and

gear ratio profiles, indicating the torque and inertia phases, for a “first generation” power-

on upshift (1-2) and downshift (2-1).

Figure 2.5: “First generation” clutch-to-clutch power-on shift: upshift, clutch 1 to 2 (left)

downshift, clutch 2 to 1 (right) [12]

2.2.2 Clutch slip control and friction parameter estimation using pressure and

speed measurements

Clutch slip control is critical for smooth gearshifts. The complexity of the control

scheme depends on the types of components involved in the shift. When a shift involves a

clutch and a one-way clutch (OWC), which is a coupling element that transmits torque in

16

one direction and spins freely in the other, clutch control is simple; only one variable, the

oncoming clutch pressure, is controlled and the OWC smoothly releases the offgoing

clutch just when the oncoming clutch is able to transmit the full engine torque [5]. Since

dual clutch transmissions do not use one-way clutches, the pressures in the offgoing and

oncoming clutches need to be controlled in a coordinated manner; said coordination may

be accomplished by controlling the clutch slip in both clutches in a predetermined

manner. Clutch-to-clutch slip control methods will be discussed in the remainder of this

section.

For all transmissions with hydraulically activated clutches and without OWCs, the

quality of the gearshift is influenced by the coordination of the engagement of the

oncoming clutch with the disengagement of the offgoing clutch. If both clutches slip at

the same time due to inadequate clutch capacity to transmit torques, the total torque

transferred through the clutches decreases and engine flare occurs; the net torque acting

on the engine crankshaft increases, so the engine accelerates. Clutch fighting occurs when

the oncoming clutch torque capacity becomes excessive before the offgoing clutch slips;

engine power is transmitted through two power branches rather than one, again causing a

decrease in output torque.

To minimize the disturbance to output torque caused by engine flare and clutch

fighting, offgoing and oncoming clutch pressures should follow coordinated profiles so

that the engine torque is smoothly transferred between clutches. Planetary automatic

transmissions can have four or more friction elements (i.e., clutches and brakes) and

multiple solenoid valves to actuate each clutch and brake; due to the cost, size, and

required number of pressure sensors, production transmissions typically have not

17

included pressure sensors for each clutch or brake to enable, among other enhancements,

closed loop clutch pressure control. Thus, clutch coordination for planetary automatic

transmissions has been typically achieved using open loop, rather than closed-loop,

pressure control strategies. The clutch maximum pressure and fill-time are calibrated for

a wide range of engine speeds and torques, driver inputs (throttle positions), transmission

fluid properties, transmission input and output speeds, clutch friction material properties,

and “current” transmission gear. Open loop control can be very good when the

transmission parameters match those used for calibration. However, age and wear of

transmission components, namely the transmission fluid and the friction material, as well

as build-to-build variations affect the clutch friction properties, and in turn, the shift

quality. Therefore, the quality of clutch-to-clutch shifts has remained an area needing

improvement.

Adaptive strategies have been designed to make clutch slip control schemes less

sensitive to deviation in calibrated parameters. Hebbale and Kao [19] presented a method

that uses sensed transmission output shaft acceleration to refine the clutch pressure

profile (fill-rate and fill-time). Note that shaft acceleration is found by real-time

differentiation of shaft speed measurements using a Kalman filtering technique. When the

oncoming clutch pressure increases too rapidly (clutch fighting), the transmission output

torque drops. This drop corresponds to a drop in output shaft acceleration. The clutch

pressure fill-rate is decreased for successive shifts so that the error between the nominal

and previous shift shaft accelerations is reduced. When the oncoming clutch pressure

increases too slowly (engine flare) the transmission input speed overshoots the required

value. This speed overshoot is reduced by increasing the clutch pressure fill-rate over

18

successive shifts. Hebbale and Kao concluded that the performance of the adaptive

control is limited by the consistency of clutch solenoid valves over a wide range of

operating conditions and the effectiveness of the Kalman filter at reducing the noise

present in the acceleration estimates.

Bai et al. [20] avoided the issue of accurately determining fill-time by linking the

operation of the two pressure control solenoid valves involved in the shift. This is

accomplished by using the oncoming clutch pressure as a “washout” signal to the

offgoing clutch valve; when the oncoming clutch chamber is full and the pressure in the

chamber builds, the offgoing pressure is proportionally reduced. Once the oncoming

clutch reaches critical pressure capacity, the offgoing clutch pressure is reduced to zero.

Despite the independence of the shift to clutch fill-time, the washout gain must be tuned.

Thus, the algorithm is still sensitive to build-to-build variations in the hydraulic system.

In addition, the special purpose nature of the valve hardware adds to the cost of the

transmission.

Unlike planetary automatics, production dual clutch transmissions typically

include pressure sensors, thus enabling a number of improvements in clutch-to-clutch

shifts. With pressure feedback, clutch slip control schemes can be improved in the

following ways: previously calibrated parameters such as clutch pressure fill-time and

fill-rate can be identified on-line and adjusted adaptively; speed measurements can be

used to estimate shaft accelerations, and in turn, clutch friction torque; along with clutch

pressure measurements, estimated clutch friction torque can be used to recognize broad

changes in clutch friction characteristics as the transmission ages; and, finally, pressure

19

measurements can be used in closed-loop clutch pressure control to improve how the

actual clutch pressures track the commanded clutch pressure trajectories.

Zheng et al. [21] presented a closed loop pressure control system for a planetary

automatic that uses solenoid pressure feedback to control the solenoid current. Both

feedforward and feedback controllers are implemented. The feedforward controller maps

the commanded solenoid pressure on a solenoid pressure-current curve to determine the

open loop solenoid current. In addition, the controller monitors the total commanded

solenoid current (sum of open loop and closed loop commanded current) and the

measured solenoid pressure; when these values are steady for a given amount of time, the

solenoid pressure-current curve is adjusted and is set as the new baseline curve. The

feedback controller makes the final adjustments to the commanded current to achieve the

correct solenoid output pressure. Figure 2.6 displays the pressure control block diagram

used in their work. Robust optimization techniques are used to select the optimal

controller and system design parameters; the sets of parameters corresponding to the best

and worst optimization results are labeled as optimal and worst case, respectively. As

displayed in Figure 2.7, the authors proved that the optimal and worst case closed loop

control system tracks a commanded solenoid pressure more accurately then a purely open

loop control system. A modified version of this pressure control system can be

implemented in dual clutch transmissions; clutch pressure rather than solenoid pressure

would be the feedback state, and the feedforward controller would be a function of

measured clutch pressure. Such a control scheme is implemented in this work and

described in Chapter 6.

20

Figure 2.6: Closed loop pressure control block diagram [21]

Figure 2.7: Tracking of commanded pressure – open loop and closed loop [21]

Output torque has also been investigated as a control variable for clutch slip

control. Due to high cost and low durability, torque sensors are not utilized in production

transmissions; however, output torque may be estimated using speed sensors. Li et al.

developed a virtual torque sensor (VTS) that could accurately estimate relative changes in

21

transmission output torque. Existing magnetic rotary encoders located at the transmission

output and wheels were used to measure the torsion of the driveshaft, which was then

related to relative changes in torque [22]. In subsequent work, Li et al. demonstrated that

by adding a unique feature to the encoder wheel so that a full revolution could be

detected and by adding an identical sensor at the driveshaft output, the absolute

transmission output torque could be accurately estimated [23].

Minowa et al. [24] proposed a clutch-to-clutch shift controller based on speed and

estimated transmission output torque feedback. They determined that a sharp rise in

output torque signifies the end of the torque phase. Additionally, the estimated output

torque is used by the engine control unit (ECU) to control engine speed; a smooth change

in transmission gear ratio during the inertia phase is provided by smoothly decelerating

(in the event of an upshift) or accelerating (in the event of a downshift) the engine.

Although this method uses some feedback control, the method still relies on the

calibration of clutch pressure fill-rate and fill-time. Further, the estimation of

transmission output torque is based upon torque converter static characteristics and the

engine map, and is susceptible to significant estimation error.

There are three generations of clutch-to-clutch shift control schemes. For the “first

generation” shift control scheme described in Section 2.1.1 , the transmission output

torque does not smoothly decrease during a shift. Instead, a torque hole and hump is

observed during the respective torque and inertia phases. Goetz [12] proposed an

integrated powertrain controller that allows a small controlled amount of slip in the

offgoing clutch during the torque phase and controls engine torque during the inertia

phase to eliminate the torque hole and hump. Goetz’s control strategy results in one

22

example of a “second generation” shift control scheme, where the torque smoothly and

monotonically decreases during an upshift, and increases during a downshift. For Goetz’s

control strategy, ideal traces of clutch and engine torque, as well as normalized gear ratio

and torque ratio, are displayed in Figure 2.8. Bai et al. [25] proposed an integrated

powertrain controller that controls engine torque during the torque and inertia phases.

The result of this control strategy is a “third generation” shift, where a constant

transmission output torque is maintained throughout the duration of the shift. For the

control strategy proposed by Bai et al., ideal traces of clutch, engine, and transmission

output torque, as well as engine speed, are displayed in Figure 2.8.

Figure 2.8: “Second generation” clutch-to-clutch power-on shift: upshift, clutch 1 to 2

(left) downshift, clutch 2 to 1 (right) [12]

23

Figure 2.9: “Third generation” clutch-to-clutch power-on upshift [25]

2.2.3 Gear preselection and gear skipping

Gear preselection is a necessary component of the dual clutch transmission gear

shift; prior to shifting the engine load from the current clutch to the target clutch, the next

output gear must be selected. Consider an upshift from first to second gear for the DCT

shown in Figure 1.1. The target gear, output gear 2, is meshed with a gear that is fixed to

the shaft splined to clutch 2. Output gear 2 is normally disengaged from the branch’s

output shaft (or intermediate shaft 1) and thus does not rotate at the same speed as said

shaft. Before the shift from first to second gear occurs, output gear 2 should be locked to

intermediate shaft 1. Hence, the input shaft is forced to decelerate so that the target gear

ratio is achieved once the clutch-to-clutch shift is completed [17].

When the driver requires a significant change in gear ratio, consecutive and

rapidly executed upshift or downshifts may be required. Multiple quick shifts should be

avoided for the following reasons: (1) the total shift time between the desired and target

gears is too slow, and (2) too many consecutive shifts may be uncomfortable for the

24

driver. To avoid these issues, multiple gearshifts or skip-shifting can be built into the shift

control strategy.

Goetz et al. [17] described a multiple shift method for dual clutch transmissions.

Since one branch holds the odd gears and the other holds the evens gears, a multiple

upshift (or double-upshift) or multiple downshift (or double-downshift) requires

transferring power flow between two odd or two even gears on the same branch. For

upshifts and downshifts, the torque-transmitting clutch must be disengaged to shift from

the initial gear to the target gear. Consider a double downshift from third to first gear for

the DCT shown in Figure 1.1. As the offgoing clutch, clutch 1, begins to slip, the “fill-in

clutch”, clutch 2, is partially engaged to prevent power loss at the wheels; power

temporarily flows through second gear, i.e. the gear that is being skipped. Further, the

fill-in clutch is partially engaged to help with engine acceleration during a downshift and

engine deceleration during an upshift. While clutch 2 is transmitting torque, the initial

gear, output gear 3, is released and the target gear, output gear 1, is synchronized to

intermediate shaft 1. Once output gear 1 is synchronized, engine torque is transferred

from clutch 2 to clutch 1; at this stage of the shift, clutch 1 is the oncoming clutch.

Ideal traces of clutch torques, gearbox component speeds, and engine torque for

the control of a double-upshift and double-downshift (clutch 1-2-1) are displayed in

Figure 2.10; the torque and inertia phases are indicated on each plot. Here, the torque

phase includes the initial transfer of engine torque from the offgoing clutch to the fill-in

clutch and the inertia phase includes the synchronization of the next output gear and the

transfer of engine torque from the fill-in clutch to the oncoming clutch. Simulated traces

of transmission output torque and gearbox component speeds for an upshift from first

25

gear to third gear and a downshift from third gear to first gear are displayed in Figure

2.11 and Figure 2.12, respectively. For both types of double-shifts, the driver may

perceive the double-shift as one continuous shift if the output torque smoothly decreases,

in the case of an upshift, or increases, in the case of a downshift, throughout the entire

shift.

Figure 2.10: Multiple-shifts from clutch 1-2-1: double-upshift (left) double-downshift

(right) [12]

26

Figure 2.11: Output torque and gearbox component speeds for an upshift from first gear

to third gear [12]

Figure 2.12: Output torque and gearbox component speeds for a downshift from third

gear to first gear [12]

2.3 Conclusion

This chapter provides a survey of the modeling of DCTs and clutch-to-clutch shift

control strategies common to planetary automatic and dual clutch transmissions. The

most significant differences between the two transmission types are the lack of a torque

converter, and the presence of clutch pressure sensors, in production dual clutch

transmissions. The torque converter damps engine and driveline vibrations and provides

torque amplification during vehicle launch. Damping in a DCT is improved by inserting a

27

torsional damper or dual-mass flywheel between the engine and the gearbox input shaft.

To maximize the torque available at the wheels during launch, DCTs are designed with

wider gear ratio spreads.

Planetary automatics, which typically do not have clutch pressure sensors, rely on

open loop control of clutch pressures during a clutch-to-clutch shift. Clutch pressure

profiles are calibrated for a wide range of operating conditions; open loop control can be

very good when the transmission parameters match those used for calibration. However,

as the transmission parameters deviate from calibrated values over the life of the

transmission, the performance of the open loop controller degrades. The inclusion of

clutch pressure sensors in production dual clutch transmissions makes closed loop

pressure control possible. Closed loop pressure control reduces the calibration effort and

improves the accuracy with which the actual clutch pressure tracks the commanded

clutch pressure. In addition, pressure and speed measurements provide the sensed

information required to estimate transmission output torque and to capture changes in

clutch friction characteristics as the clutch friction material and transmission fluid age.

Chapter 5 describes an estimation scheme and the extent of its effectiveness.

28

CHAPTER 3: POWERTRAIN MODEL

The following chapter presents a time-domain powertrain model with emphasis

on the dual clutch transmission; a brief description of the overall model is provided in

Section 3.1 , and a block diagram of said model is shown in Figure 3.1. Next, the model

subsystems are presented in the following order: mean-value engine model (engine

torque production and engine mechanical dynamics blocks), transmission mechanical

system (clutch dynamic models, friction models, and gearbox dynamics blocks),

longitudinal vehicle dynamics with strictly forward motion, and hydraulic component

actuation system (clutch pressure command and regulator systems, synchronizer

command and actuation systems blocks). Model limitations are discussed in Section 3.6

and concluding remarks pertaining to model development are provided in Section 3.7 .

Validation of the powertrain model through simulation is discussed in Chapter 4.

3.1 Top level structure of powertrain model

The transmission modeled in this work is based on the Volkswagen VW02E (or

DQ250) direct-shift gearbox (DSG). Direct shift gearboxes are more commonly referred

to as dual clutch transmissions (DCT), or to a lesser extent, twin-clutch transmissions.

Developed by Volkswagen and BorgWarner, the VW02E has been used in multiple

Volkswagen and Audi vehicles since 2003. A remanufactured VW02E from 2011,

29

including a complete mechatronic unit (valve body, solenoids, and transmission control

module), has been obtained; when available, geometric parameters have been identified

through direct measurement or determined experimentally.

Submodel Parameter Source

Mean-Value

Engine

Model

Engine volume Literature [26]

Air-to-fuel ratio Literature [26]

Manifold volume Literature [26]

Spark timing Literature [26]

Torque map Literature [27]

Engine torque constant Estimated

Transmission

Mechanical

System

Clutch plate mass, geometry Measurement

Clutch spring stiffnesses Measurement

Shaft, clutch, gear, synchronizer, differential

inertias Measurement

Shaft and clutch stiffnesses, mechanical damping

coefficients Estimated

Flywheel stiffness, inertia, damping coefficient Literature [18]

Clutch coefficient of friction Literature [28, 29]

Synchronizer friction Literature [30, 31]

Friction material properties used in dynamic

model Literature [32-34]

Vehicle and

Tire Model

Vehicle mass, frontal area Literature [27]

Tire radius, rolling resistance Literature [27]

Drag coefficient Literature [27]

Tire and wheel inertia Literature [27]

Constants used in simplified Pacejka model Literature [35]

Hydraulic

Component

Actuation

System

Hydraulic circuit Measurement

Valve, solenoid geometry and spring stiffnesses Measurement

ATF fluid properties Literature [32]

Solenoid electromagnetic properties Literature [36, 37]

Solenoid control pressures Literature [36, 37]

Pump displacement, flow rate Literature [38]

Line pressure, clutch cooling flow rate Literature [38]

Table 3.1: Model parameters and sources

30

Model parameters pertaining to the hydraulic component actuation system, clutch friction

material and solenoid specifications have been obtained from Volkswagen and

BorgWarner literature and Volkswagen vehicle data published informally on the internet

[27-29, 32, 36-41]. Parameters needed to model synchronizer friction and to describe the

properties of automatic transmission fluid have been estimated from literature [30, 31, 38,

42]. Since a complete vehicle could not be obtained, engine and vehicle parameters have

been selected from data published on the 2005 Volkswagen Passat [27]. The sources of

model parameters are summarized in Table 3.1.

3.2 Mean-value engine model

The mean-value engine model described here was originally developed by Cho

and Hedrick [43], and later modified by Zheng [26]. Zheng modeled a General Motors

powertrain with a 3.8L V6 spark-ignition engine, sequential port fuel-injection, and

natural aspiration (i.e., operates without a turbocharger or supercharger). Volkswagen has

typically paired a dual clutch transmission with a compression ignition engine. However,

since the primary focus of this research is the modeling and control of DCTs, the spark-

ignition engine model used by Zheng, with minor modification, is incorporated here.

31

Figure 3.1: Block diagram of overall powertrain model

31

32

3.2.1 Intake manifold air dynamics

For a mean-value engine model, the air intake process can be modeled by

applying the concept of conservation of mass to the intake manifold. Following this

approach, the net mass flow rate of air ( ) through the manifold is defined as:

(3.1)

, an empirical representation of compressible flow through the throttle body and into

the intake manifold, is modeled as:

(3.2)

Defined as the maximum mass flow rate of air into the intake manifold, is

described by:

(3.3)

where is the total engine displacement, is the maximum engine speed

(5000rpm), and is the density of the air entering the intake manifold. is the

normalized throttle opening, which can be described as a function of throttle angle

( ). In this case,

{

(3.4)

, defined as the normalized pressure influence, is a function of the ratio of the intake

manifold pressure to ambient pressure at the throttle body inlet. The expression for

is:

( (

))

(3.5)

33

Assuming the air in the intake manifold behaves as an ideal gas, the manifold

pressure can be described by the following expression:

(3.6)

where is the ideal gas constant, and and are the respective temperature and

volume of the intake manifold air.

The mass flow rate of air exiting the intake manifold and entering the combustion

chamber, , is defined as:

(3.7)

where is the engine speed in rad/s and is the engine volumetric efficiency; an

empirical expression for is provided by equation (3.8).

(3.8)

Combining equations (3.1)-(3.8), the intake manifold dynamics is modeled by the first

order differential equation given by (3.9).

(3.9)

3.2.2 Intake manifold fueling dynamics

For fueling systems, including sequential port fuel-injection, the fueling dynamics

are affected by lag and transport delay. The transport delay is a function of injector firing

time and duration (i.e., fuel that is sprayed while the intake valve is closed cannot enter

the combustion chamber until the intake valve opens again), the injection solenoid

34

response time and accuracy, and engine speed. If lag and transport delay are combined

into an effective fueling time constant ( ), the fueling dynamics are described by:

(3.10)

where and are the actual and command mass flow rates of fuel into the

combustion chamber. can be modeled by the following first order lag:

(3.11)

where is the desired air-to-fuel ratio. The first term in equation (3.11)

represents the lag, while the second term represents the transport delay.

The engine torque production, which is discrete in nature, can be modeled in the

time domain using the following expression:

(3.12)

Here, is the engine torque constant, which represents the maximum torque produced

by the engine for a given charge of air and engine speed. For the work presented here,

has been adjusted so that the maximum torque produced by the engine more closely

approximates that of the compression ignition engine used in the 2005 Volkswagen

Passat. , defined as the normalized air fuel influence, is a scaling factor that reduces

engine torque production as the air-to-fuel ratio deviates from an optimal value. An

empirical expression for is given by equation (3.13).

( ) (3.13)

35

The normalized spark influence, , is a scaling factor that reduces engine torque

production as the spark timing deviates from MBT (spark timing for maximum brake

torque). An empirical expression for is provided by equation (3.14).

( )

(3.14)

To capture the discrete, and cyclical, behavior of a four-stroke engine, two delays,

and , are used. , defined as the air intake to torque production delay, is modeled

as:

(3.15)

, defined as the spark to torque production delay, is modeled as:

(3.16)

3.3 Dual clutch transmission mechanical system

In the VW02E dual clutch transmission, a stick diagram of which is shown in

Figure 3.2, engine torque is transmitted through a dual-mass-flywheel (DMF) to the

clutch hub. The clutch hub holds both the odd gear (K1) and even gear (K2) clutch packs.

Each clutch pack consists of a hydraulically actuated piston and steel separator plates. K1

friction plates are mechanically connected to input shaft #1 (or the odd gear input shaft or

si1), and K2 plates are connected to input shaft #2 (or the even gear input shaft or si2).

The separator and friction plates, which together comprise the clutch pack, are immersed

in automatic transmission fluid (ATF). To reduce the size of the gearbox, the input shafts

are arranged in a layshaft configuration, where input shaft #1 is installed inside the

36

hollow input shaft #2. The pump shaft, which is also splined to the clutch hub, rotates in a

through hole in input shaft #1.

Figure 3.2: VW02E DCT stick diagram

Each input shaft is permanently connected to a set of gears; the odd and even

shafts carry input gears IG1/R, IG3, and IG5, and IG2 and IG4/6, respectively. Each

input gear is continuously meshed with the output gear(s) of the same number (i.e., IG3 is

meshed with OG3, and IG4/6 is meshed with OG4 and OG6). Gears OG1, OG2, OG3,

and OG4 are supported by output shaft #1 (or so1), while gears OG5, OG6, and OGR are

held by output shaft #2 (or so2). The synchronizers, which are comprised of a hub,

sleeve, friction cone(s), engagement and blocker rings, and detents, are splined to the

output shafts and follow a similar numbering system to the gears; synchronizer S13

controls the engagement of OG1 and OG3, S24 controls the engagement of OG2 and

OG4, etc. Figure 3.3 shows the exploded view of a typical synchronizer [31]. Two pinion

37

gears, OSG1 and OSG2, which are permanently connected to output shafts #1 and #2, are

continuously meshed with the differential. The differential is connected to the front

wheels of the vehicle via two half shafts. However, since it is assumed that torque is

transferred evenly to both front wheels, the differential output is modeled as one shaft

connected to both front wheels.

Figure 3.3: Synchronizer components [31]

When either K1 or K2 is engaged, torque is transmitted from the engine to either

input shafts #1 and #2, respectively. Whether each output gear can spin freely or is

mechanically locked to its respective output shaft is dependent on the state of the gear’s

synchronizer. When both the clutch and an output gear from the same branch (odd gear or

even gear) are engaged, torque is transmitted to the wheels. Thus, the engagement of the

38

clutches and output gears determine the overall transmission gear ratio; Table 3.2 shows

the clutch and output gear engagement schedule, as well as the active synchronizer, and

the gearbox and final drive gear ratios. As discussed in Section 2.1.3 , the VW02E

utilizes a non-uniform gear ratio spread with larger changes in gear ratio for the low

gears. For example, the gear ratio drops by a factor of 0.592 from 1st to 2

nd gear but only

by a factor of 0.827 from 5th

to 6th

gear. Also, the spread in gear ratios from the lowest to

the highest gear is roughly 6.2:1.

Transmission

Gear

Clutch

Status Active

Synchronizer/

Gear

Engaged

Gearbox Gear

Ratio

(Input to output

shaft)

Final Drive Ratio

(Output shaft

pinion to

differential) K1 K2

1st X - S13/OG1 3.462

4.118 2

nd - X S24/OG2 2.050

3rd

X - S13/OG3 1.300

4th

- X S24/OG4 0.902

5th

X - S5N/OG5 0.914

3.043 6th

- X S6R/OG6 0.756

Reverse X - S6R/OGR 3.989

X = Engaged, - = Disengaged

Table 3.2: Clutch and output gear engagement schedule for VW02E

Note that the output gear is preselected prior to a clutch-to-clutch shift. Also, only

forward driving conditions are considered in this research, so the reverse operation will

not be modeled.

3.3.1 Rotational dynamics of the dual clutch transmission

To capture low frequency dynamic effects due to events such gear shifting and

output gear selection, a powertrain model with lumped inertias and shaft compliances is

39

considered. A schematic of said powertrain (through the differential) is shown in Figure

3.4; the vehicle dynamics will be described separately in Section 3.4 .

Figure 3.4: Rotational dynamics of VW02E engine, flywheel and gearbox

By applying Newton’s second law for rotation at each of the inertias, the

following set of equations describing the rotational dynamics of the powertrain (through

the differential) is obtained:

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

40

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

where

is inertia

is angular velocity

is the shaft stiffness

is the mechanical damping coefficient

is the viscous damping coefficient

is the friction torque acting at clutches K1, K2

is the gear ratio between the gearbox input and output shafts, with

subscript j denoting the transmission gear (1st —6

th )

is the gear ratio between output shafts #1 and #2, and the differential

output shaft

41

For the variables in equations (3.17)-(3.33), the subscripts E,F,H,L1,si1,L2,si2,so1,so2,

and D denote the engine, flywheel, clutch hub, gearbox side of the odd gear clutch, input

shaft #1, gearbox side of the even gear clutch, input shaft #2, output shaft #1, output shaft

#2, and differential output shaft, respectively. We note here that all gear and synchronizer

inertias are lumped at their respective shafts, and that the subscript L designates a

variable that applies to both the odd and even gear clutches.

is the net torque produced by the engine . ,

defined as the torque required to drive the fixed displacement internal gear pump, is

modeled as:

(3.34)

where is the pump displacement, and and are the supply line and sump

pressures. The sump pressure is equal to atmospheric pressure; for the sake of simplicity,

the line pressure, as well as any other pressure in the hydraulic system, is represented as a

gauge pressure. Thus, and equation (3.34) reduces to:

(3.35)

3.3.2 Effect of gear selection on the torque acting on the input shafts

and , defined as the load torques acting on input shafts #1 and #2, are

direct functions of the gearbox configuration; when all of the output gears are free-

spinning (i.e., the transmission is in neutral) the gearbox input and output shafts are

decoupled, and and are both equal to zero. When an output gear is engaged, the

torque acting on the corresponding input shaft is equal to the synchronizer torque

42

reflected from the output shaft to the input shaft. Expressions for and are given by

equations (3.36) and (3.37).

{

(3.36)

{

(3.37)

Here, refers to synchronizer torque; the numerical subscripts denote the output gear

to be synchronized.

The synchronizer experiences three states (denoted by SS) during the engagement

process: disengaged (SS=0), when the synchronizer engagement ring is not in direct

contact with the output gear; synchronizing (SS=1), when the slip speed between the

synchronizer rings and output gear is reduced to zero, but the synchronizer sleeve hasn’t

moved far enough axially to engage the dog teeth of the output gear and the engagement

ring with the detents; and mechanically locked (SS=2), when the synchronizer sleeve has

reached its maximum axial displacement and the dog teeth of the output gear and the

engagement ring are engaged with the detents. The following expressions describe the

synchronizer torque, during each state, for all of the output gears:

{

( )

(3.38)

43

(

) (

) (3.39)

{

( )

(3.40)

(

) (

) (3.41)

{

( )

(3.42)

(

) (

) (3.43)

{

( )

(3.44)

(

) (

) (3.45)

where

is the number of synchronizer cones (or rings)

is the coefficient of friction for the synchronizer cone-gear interface

is normal force applied to the synchronizer

is the mean cone radius

is cone angle

is the synchronizer stiffness, when mechanically locked

is the synchronizer damping coefficient, when mechanically locked

44

Again, the numerical subscripts denote variables corresponding to the output gear to be

synchronized.

During the synchronizing phase, the synchronizer is modeled as a friction

element. The torque transmitted through the synchronizer is proportional to the dynamic

coefficient of friction of the synchronizer friction cone, the number of friction cones, the

normal force applied to the synchronizer sleeve, and geometric properties of the cones.

Note that the coefficient of friction is considered constant with slip speed. The VW02E

transmission utilizes three types of synchronizers: single-cone, for synchronization of

OG4, OG5, and OG6; double-cone, for synchronization of OGR; and triple-cone, for

synchronization of OG1, OG2, and OG3 [38]. Due to higher slips speeds in 1st-3

rd gear,

triple-cone synchronizers provide larger heat transfer surfaces for improved cooling. The

synchronizer rings are made of molybdenum coated brass (single and triple-cone) or steel

(double-cone). When in the mechanically locked phase, the synchronizer is modeled as a

compliant element. It is assumed the output gear is perfectly fixed to the output shaft

during this phase, allowing for the synchronizer stiffness and damping coefficient to be

set equal to the same values for the output shaft.

3.3.3 Implementation of Karnopp friction model

The generalized Stribeck curve shown in Figure 3.5 can be described by equation

(3.46). Note that the notation reflects what is used in reference [44].

[

|

|

] (3.46)

45

and are the Coulomb and static torques, and are Stribeck factors, and is the

viscous friction coefficient. As the slip speed ( ) is reduced to zero, torque can

instantaneously jump to the positive or negative static friction torque; i.e., the friction

model is not uniquely defined at zero slip speed, so it should not be used in computer

simulations. As shown in Figure 3.6a), this issue can be overcome by simply

approximating the zero slip speed region of the torque curve as a very steep line.

Unfortunately, this method can result in longer simulation times and incorrect prediction

of stick-slip behavior.

Figure 3.5: Generalized Stribeck friction curve [44]

Figure 3.6: Generalized Stribeck friction curve: a) Steep-line approximation b) Karnopp

model [44]

46

To address these simulation issues at zero slip speeds, Karnopp introduced a

model that, depending on the slip speed, describes friction torque in two ways [45]. In a

narrow stiction region defined for and is the static-dynamic threshold

speed, friction torque is calculated as a function of external torques applied to the friction

element. In the slipping region ( ), friction torque is modeled as a function of

viscous and/or mechanical contact effects. The Karnopp model applied to the generalized

Stribeck curve is shown in Figure 3.6b). See Section 3.3.4 for a detailed description of

the friction model used in this research.

Figure 3.7: Simplified rotational dynamics for implementation of Karnopp friction model:

a) K1 sticking, K2 disengaged/slipping b) K2 sticking, K1 disengaged/slipping

To implement the Karnopp model in this work, additional torques are introduced

– on the sticking clutch branch – between sections of the clutch hub inertia ( or )

and the load inertias ( or ). Figure 3.7 shows the powertrain in the following two

47

configurations: Figure 3.7 a), when the odd gear clutch is sticking and the even gear

clutch is disengaged or slipping; and Figure 3.7b), when the even gear clutch is sticking

and the odd gear clutch is disengaged or slipping. For the configuration shown in Figure

3.7a), equations (3.21) and (3.22) can be rewritten as:

(3.47)

(3.48)

(3.49)

(3.50)

When the odd gear clutch is engaged, the hub and gearbox sides of the clutch rotate with

the same angular velocity (i.e., the slip speed is zero). Both sides of the clutch must have

the same angular accelerations to maintain a zero slip speed. Setting and

combining equations (3.48) and (3.50), the sticking friction torque is described by:

(

)

(3.51)

where and are calculated using equations (3.47) and (3.49), respectively.

Applying the same procedure to the configuration where the even gear clutch sticks, the

expression for sticking friction torque is obtained using the following set of

equations:

(3.52)

(3.53)

48

(3.54)

(3.55)

(

)

(3.56)

3.3.4 Dynamic clutch friction model

The dynamic wet-clutch friction model presented here, which is adapted from

Deur et al. [8, 10], is a lumped-parameter model that incorporates fluid film dynamics

and a simplified thermal model in an effort to improve the accuracy in which the clutch

engagement process is described. The accuracy of the commonly used static model is

dependent on the energy level of the engagement; the static model most accurately

represents clutch friction during a high-energy engagement (high clutch pressure and

high slip speed), and least accurately during a low-energy engagement (low clutch

pressure and small slip speeds). The static model is inaccurate in large part because it

does not describe the fluid film dynamics during engagement.

Wet-clutch engagement can be divided into three phases: hydrodynamic

lubrication, partial lubrication and mechanical contact. In the hydrodynamic lubrication

phase, the clutch separator and friction plates are completely separated by a fluid film. As

the plates are pushed together, fluid is forced out of the clutch pack- as the film thickness

decreases, the viscous friction increases. Partial lubrication occurs when the film

thickness is reduced past the asperity height of the friction material, and the friction due

to asperity contact increases from zero. The final phase occurs when the remaining fluid

49

is squeezed out of the friction material asperities. At this point the viscous friction is

reduced to zero, and all of the clutch friction is due to mechanical asperity contact.

Starting with the Reynolds equation in polar coordinates, Deur et al. developed a

lumped-parameter clutch model that includes fluid film dynamics. This model can be

applied to clutches with ungrooved plates, and with less accuracy, to clutches with

friction plates that have simple groove patterns. Even though both VW02E clutches use

double-sided grooved friction plates, the irregularity of the groove patterns would further

limit the accuracy of the grooved model. Thus, double-sided ungrooved plates are

modeled in this work.

The equations describing the lumped-parameter model for ungrooved plates are

given as follows. The clutch friction, , is described by:

(3.57)

where and are the viscous and mechanical contact components of the total

friction. The subscript m refers to a variable that is specific to clutch K1 or K2. As

mentioned in the previous section, equation (3.57) models friction torque when the clutch

is slipping. When the clutch is sticking, is the maximum static friction

torque. If the static friction torque calculated by equations (3.51) or (3.56) is greater than

the maximum static friction, that clutch will begin to slip.

The viscous and mechanical contact frictions are modeled as:

(

)

( ) ( ) (3.58)

50

{

(

)

( )

(

)

( )

(3.59)

where

are the outer and inner radii of the clutch friction plates

is the number of friction interfaces

( = 2 x (# of separator plates – # of friction plates))

is the number of grooves in the friction material ( = 1 for ungrooved)

is the clutch slip speed ( )

is the angular displacement between grooves ( = 2π for ungrooved)

is the fluid film thickness

Fluid viscosity is calculated as a function of oil temperature, , and is described by:

(3.60)

where and are coefficients found using the viscosity versus temperature curve

presented by Lam et al. [42]. Note that refers to the oil temperature immediately after

it makes contact with the clutch mating surfaces. At this location, we expect the oil

temperature to vary noticeably; however, the oil temperature throughout the remainder of

the hydraulic system is assumed to be constant. The thermal model used in this research

is a single lumped parameter model given by:

(∑ ( ) )

(3.61)

51

where is the housing temperature, and is the heat conductivity of the steel

separator plates. CHC , defined as the total heat capacity of the odd and even gear

clutch packs, is described by the following polynomial:

(3.62)

and , referred to as Patir and Cheng’s flow factors, describe how

fluid flow across surface asperities affects friction torque during the hydrodynamic and

partial lubrication phases [33, 34]. and , defined as the pressure flow

factor and shear stress factor for a smooth surface, respectively, are described by the

following equations:

{

(3.63)

{

( )

(3.64)

H, z, and , all defined in terms of fluid film thickness and the RMS roughness of the

mating surfaces (steel and friction lining), σ, are described by:

;

;

⁄ (3.65)

The mating surfaces in this work are assumed to have an isotropic surface roughness

pattern. For this case, Patir and Cheng determined that the shear stress factor coefficients

are , , , and . According to the

52

Greenwood and Williamson model [46], which assumes a Gaussian distribution of the

friction material asperities, the pressure due to mechanical asperity contact, , is

calculated as:

√

[

(

√ )

√

(

√ )] (3.66)

where

is the Young’s modulus of the friction material

is the asperity density of the friction material

is the asperity tip radius of the friction material

The fluid film dynamics are described by the following equation:

[

(

)]

(

)

(3.67)

where

is the friction material permeability

is the friction material thickness

is the pressure applied to the clutch piston

is the pressure-acting area of the clutch piston

is the friction material area ( (

))

and

(

) (3.68)

53

The surface roughness factor, , is modeled as:

[ (

√ )] (3.69)

, defined as the Beavars and Joseph factor, is described by:

√

(3.70)

Here, is the Beavars and Joseph slip coefficient. , which is a geometric scaling

factor that is a function of the friction material geometry and groove pattern, is defined

by:

(

(

)

(

)

(

)

(

)

)

(3.71)

The clutch coefficient of friction, , is described by the Stribeck model given

by equation (3.72).

( (

| |

)

)

| | (3.72)

Here, and are the Coulomb and static coefficients of friction, and are the

Stribeck coefficients, is the viscous friction coefficient, and is the static-dynamic

threshold speed. Setting and

, the remaining parameters are

identified by fitting the structure described above to experimental clutch friction data

provided by Tersigni et al. of Afton Chemical [28, 41].

54

3.4 Longitudinal vehicle dynamics

Two models of the longitudinal vehicle and driveline dynamics are provided in

the following section. The first one utilizes the simplified Pacejka 89 tire model [35], and

considers the vehicle body dynamics separately from the driving (front) and driven (rear)

shaft dynamics. It should be noted that Pacejka has updated his model multiple times

since its publication in the late 1980s. However, the simplified Pacejka 89 model is still

available in many commercial simulation packages, so it will be used in this research.

The first model is simplified into the second one by neglecting the tire-road interaction,

and lumping the vehicle and rear wheel inertias at the driving front wheels. The

differential and road load torques are inputs to both models.

Figure 3.8: Free body diagram for vehicle dynamics model

55

3.4.1 Vehicle dynamics with tire-road interaction

The longitudinal vehicle model utilizing the simplified Pacejka 89 [35] tire model

is described here. A free body diagram for the vehicle model is shown in Figure 3.8. The

driving front wheel and driven rear wheel dynamics are described by:

( ) (3.73)

and

( ) (3.74)

The longitudinal dynamics of the vehicle body are described by:

(3.75)

where

is the combined inertia of the front wheels/tires

is the combined inertia of the rear wheels/tires

is the rolling resistance

is the longitudinal vehicle velocity

is the vehicle mass

is the mean tire radius

is the brake torque applied to one wheel

, defined as the combined vertical force acting on either the front or rear axle, is

modeled as:

(3.76)

where is the acceleration due to gravity and is the angle of the road incline. For

to be equal at the front and rear axles, the vehicle center of gravity is assumed to be

56

equidistant, in the longitudinal direction, from both axles. and are the combined

longitudinal forces acting at the front and rear wheels, respectively. The expressions for

these forces are given by equations (3.77) and (3.78).

( (

)) (3.77)

( (

))

(3.78)

and are the longitudinal slips values at the respective front and rear wheels. The

expressions for these slip values are given by equations (3.79) and (3.80).

(3.79)

(3.80)

is defined as the longitudinal stiffness factor. , defined as the shape factor,

is described by:

( (

)

)

(3.81)

where and are the static and dynamic coefficients of friction for the tire and

road interface, and is the shape factor scaling coefficient. , defined as

the peak factor, is modeled as:

(3.82)

where is the road grip factor and is the friction coefficient scaling factor.

57

The road load force, , which includes the force due to aerodynamic drag and road

incline, is described by:

(3.83)

Here, is the longitudinal drag coefficient and is the frontal area of the vehicle.

, defined as the net velocity of air flowing around the vehicle, is equal to the sum of

the vehicle and wind velocities ( ).

3.4.2 Simplified vehicle dynamics for feedforward control

For the purposes of feedforward clutch slip control, which is described in detail in

Chapter 6, the vehicle model from the previous section is simplified by neglecting the

tire-road interaction. The resulting model is expected to be accurate for low tire slip

conditions. A free body diagram for the simplified vehicle model is shown in Figure 3.9.

By ignoring tire slip, the vehicle velocity and wheel angular velocity are related in the

following way:

(3.84)

Using the relationship defined in equation (3.84), the inertias of the vehicle and the rear

wheels are lumped at the driving front wheels. The resulting vehicle dynamics are

described by:

(3.85)

where

is the lumped inertia of one wheel/tire

58

The road load torque, , which includes the torque due to rolling friction, aerodynamic

drag and road incline, is described by:

(

) (3.86)

Figure 3.9: Free body diagram for simplified model of vehicle dynamics

3.5 Hydraulic component actuation

The hydraulic actuation system significantly affects the operation of the dual

clutch transmission. Output gear selection is accomplished by applying a differential

pressure to a shift fork. The shift fork forces the synchronizer into contact with the target

gear so that the synchronization process described in the previous section can be

completed. During the gear shifting operation, torque is transferred between the odd and

even gear branches by way of a clutch-to-clutch shift; the pressures of the oncoming and

offgoing clutches must be precisely controlled to ensure a fast and smooth shift. A

mathematical model of the hydraulic actuation system is therefore helpful when

analyzing the transmission operation, as well as designing a controller that improves gear

shift quality [2]. This is particularly true for the work presented here; since a vehicle is

unavailable for controller testing, the fidelity of the model of the hydraulic actuation

system is relied upon to demonstrate relative improvements between controllers.

59

Consequently, considerable attention has been devoted to the development of the model

of the hydraulic actuation system.

The VW02E hydraulic system is comprised of five subsystems. These subsystems

can be broken into two categories: essential and auxiliary. The essential subsystems,

collectively the hydraulic actuation system, are those that affect the output gear selection

and clutch-to-clutch shift processes. They are: the pressure regulation system shown in

Figure 3.10; the clutch actuation system presented in Figure 3.11; and the synchronizer

actuation system displayed in Figure 3.12 -Figure 3.13. The auxiliary subsystems, which

have a negligible effect on the actuation of the synchronizers and clutches, are: the clutch

cooling system and safety system. The model presented in this chapter is a highly

nonlinear, lumped-parameter, and dynamic model of the hydraulic actuation system.

Complete schematics of the hydraulic system, as well as the mathematical model of the

clutch cooling subsystems, are presented in Appendix A.

60

Figure 3.10: Pressure regulation system

Figure 3.11: Simplified clutch actuation system

61

Figure 3.12: Synchronizer actuation system - solenoids and multiplexer valves

61

62

Figure 3.13: Synchronizer actuation system - shift forks and synchronizers

63

3.5.1 Pressure regulation system

The pressure regulation system includes the pump, the pressure relief valve (PR),

the pressure regulation valve (PRV), and the pressure regulation control solenoid (N217).

Figure 3.14 shows a fixed displacement internal gear pump [40]. The red port indicates

flow at line pressure, while the blue ports represent flow at sump pressure. Powered by

the engine, the pump supplys flow to the hydraulic system at the volumetric flow

rate, , described by:

(3.87)

Figure 3.14: Fixed displacement internal gear pump [40]

The pressure relief valve opens a direct path from the supply line to the sump when the

line pressure is greater than or equal to the cracking pressure. Thus, the volumetric flow

rate through the pressure relief valve is modeled as:

{

( )

(3.88)

where is the pressure relief valve gain and is the cracking pressure.

64

The line pressure is controlled by the position of the pressure regulation valve.

Figure 3.15 shows the PRV in the closed position; in said position, the flow paths that

connect the supply line to the pump return and the clutch cooling system are blocked. For

the valve to open, the pressure force at chamber A must overcome the pressure force at

chamber D and the spring force.

Figure 3.15: Pressure regulation valve

The pressure in chamber D is controlled by the pressure regulation control solenoid

shown in the closed position in Figure 3.16. N217 is a 2-way, normally-closed, and pulse-

width-modulated solenoid valve. When the solenoid is de-energized, the plunger is

pushed to the left hand side by the spring located inside the coil. In this position, the

plunger closes the inlet port by pushing a steel ball into its seat. When closed, flow

cannot be exhausted through N217, so the pressure in chamber D is at its maximum.

When the solenoid is energized, the magnetic force generated by the coil pulls the

plunger away from the closed position. As the magnetic force is increased, the steel ball

65

is moved further away from its seat, resulting in increased flow to the exhaust port and a

reduction of the pressure in chamber D. As the chamber D pressure is decreased, the net

force acting on the PRV spool becomes increasingly positive and the spool moves to the

right. As the inlet port to chamber B is uncovered, the flow rate between the supply and

return lines is increased, which results in a reduction in line pressure.

Figure 3.16: Pressure regulation control solenoid, N217

3.5.1.1 Pressure regulation valve

The mathematical model of the pressure regulation valve consists of two

subsystems. They are: the spool mechanical system and the fluid flow system.

Spool mechanical system

The PRV spool motion depends on the pressure forces at chambers A and D, the

spring force, the inertia force, and the viscous damping force. Note that the pressure

forces in chambers B and D are not included because the difference in land cross-

66

sectional area is zero for both chambers. The static friction force and the flow forces

acting on the spool are assumed to be negligible. Thus, the PRV spool mechanical

dynamics are described by:

(3.89)

where

is the mass of the spool


is the spring constant

, are the land cross-sectional areas at chambers A and D

, are the pressures in chambers A and D

is the spring preload

, defined as the spool displacement and measured from the closed position, is

described by:

{

(3.90)

Fluid flow system

The net volumetric flow rate to the pressure regulation valve, , is

determined by applying the continuity equation at the pump outlet. The resulting

expression is given by equation (3.91).

(3.91)

67

Here, is the volumetric flow rate produced by the pump, while , ,

, , and are the volumetric flow rates to the pressure

relief valve, the clutch cooling branch, the odd gear clutch and synchronizer actuation

systems, the even gear clutch and synchronizer actuation systems, and the multiplexer

control solenoid, respectively. , which must equal the sum of the volumetric flow

rates into or out of the various chambers of the pressure regulation valve and into the

pressure regulation control solenoid, can also be described by:

(3.92)

where and are the flow rates into/out of chambers A and D,

and are the metered flow rates into chambers B and C, and is the exhaust

flow rate through N217.

In this work, volumetric flow rates through an orifice of a constant or variable

area are modeled as a nonlinear function of pressure drop. The following set of equations

describes the flow rates into or out of chambers A, B, C and D:

√

| | ( ) (3.93)

√

| | ( ) (3.94)

√

| | ( ) (3.95)

√

| | ( ) (3.96)

68

Here, and are the pressures in chambers B and C, is the density of the

transmission fluid, and and are the constant areas of sharp-edged orifices #1 and

#2. Throughout the rest of this thesis, constant area orifices will be defined in this

manner, where denotes flow area and the numerical subscript refers to the orifice

number. and , which are defined as flow areas that vary with

spool position (or area gradient [47]), are described by:

{

(3.97)

{

(3.98)

where and are the diameters of the lands at the inlet to chambers B and C,

and and are the overlap lengths for the port opening to chambers B and

C. In general, the overlap length is defined as the axial distance that the spool or plunger

must move before the closed port opens.

The discharge coefficient, , is defined as a function of the flow number, .

The flow number is a non-dimensional term that is used to indicate the transition from

laminar to turbulent flow. For a sharp-edged orifice, the transition (or critical) flow

number is . As becomes greater than the critical flow number, the discharge

coefficient approaches its maximum values (0.61 for sharp-edged orifices) [48]. The

expression used to calculate flow number is given by:

√

(3.99)

69

where is the pressure drop across the orifice. In general, the hydraulic diameter, , is

defined as:

cross sectional area

wet perimeter (3.100)

For a circular orifice, the hydraulic diameter is equal to the diameter of the orifice.

However, for spool and ball poppet valves, the hydraulic diameter is modeled as:

{

2 valve opening for spool valves

2 valve opening cos( ) for ball poppet valves (3.101)

where is the central angle corresponding to the pressure acting diameter of the ball.

By applying the continuity equation at chambers A, B, C and D, the following set

of equations describing the flow through the pressure regulation valve is obtained:

(3.102)

(3.103)

(3.104)

(3.105)

where , , , and , are the volumes in chambers A, B, C and D,

respectively, is the transmission fluid bulk modulus, is the volumetric flow

rate from chamber B to the pump return, and is the volumetric flow rate from

chamber C to the clutch cooling system inlet. Note that the leakage flow from chamber A

to the exhaust is assumed to be negligible. The volumes in chambers A and D, which

vary with spool position, are given by:

70

(3.106)

(3.107)

Here, and are the chamber volumes of A and D at zero spool

displacement.

3.5.1.2 Pressure regulation control solenoid

The mathematical model of the pressure regulation control solenoid consists of

three subsystems. They are: the plunger mechanical system, the electromagnetic circuit,

and the fluid flow system.

Plunger mechanical system

The N217 plunger motion depends on the pressure force acting on the steel ball at

the inlet port, the magnetic force, the spring force, the inertia force, and the viscous

damping force. As with the pressure regulation valve, the static friction force and the

flow forces acting on the plunger are assumed to be negligible. Thus, the mechanical

dynamics for the N217 solenoid valve are described by:

(3.108)

where

is the mass of the plunger



is the magnetic force acting on the plunger

71

is the pressure acting on the steel ball


, defined as the plunger displacement and measured from the closed position, is

described by:

{

( )

( )

(3.109)

Figure 3.17 shows a general ball poppet valve with a conical seat; the angle of the conical

seat, , and the ball diameter, , are labeled on the sketch. Note that all valves of

this type will share the same geometry. As previously defined, is the central

angle corresponding to the pressure-acting diameter of ball. The expression for

is given by the following equation.

(3.110)

The pressure-acting area of the steel ball, , is described by:

( ( ))

(3.111)

Figure 3.17: Ball poppet geometry

72

Electromagnetic circuit

The model of the solenoid electromagnetic circuit used here is adapted from

Watechagit [2, 49]. Watechagit assumed that all magnetic flux is uniform across, and

contained within, the core, and that leakage flux is negligible. The solenoid is modeled as

an inductor in series with a resistor. From Faraday’s law, the flu linkage, , inside the

system is given by:

(3.112)

where is the induced voltage in the solenoid coil, and is described by:

(3.113)

Here, is the coil inductance as a polynomial function of plunger position , and

is the solenoid current. By substituting equation (3.113) into (3.112), and applying

Kirchhoff’s voltage law to the solenoid electrical circuit, the following e pression

describing the electrical dynamics is obtained:

( (

)) (3.114)

Here, is the voltage applied to the solenoid and R is the coil resistance. It is assumed

that there is no loading effect on the voltage input, and there is no self-heating of the coil

resistance [2, 49]. The magnetic force, , generated by the solenoid is modeled as a

function of inductance and current. can be calculated from the following equation.

(3.115)

There are three types of solenoids used in this work: ON/OFF, PWM, and

variable force solenoid (VFS). ON/OFF solenoids have two possible voltage inputs,

73

which are: maximum voltage and zero. When the solenoid is de-energized (zero voltage

and current), its plunger is in the OFF position. At maximum voltage (and solenoid

current), the magnetic force is large enough to move the plunger to its ON position.

PWM and VFS solenoids use more complex circuitry to generate and send voltage pulse

train signals to the solenoid. By varying the duty cycle (dc) and frequency (f) of the pulse

train, the solenoid current, and in turn, magnetic force is varied, allowing for multiple

plunger positions to be achieved. Figure 3.18 shows the pulse train signal used in this

work.

Figure 3.18: PWM and VFS voltage pulse trains

and are the solenoid voltages during the ON and OFF pulses. The pulse

durations, and , and the off duration, , are all set as a function of

duty cycle and frequency, where:

74

(3.116)

Note that in this form, the duty cycle should be between 0 and 1.

Up to this point, the magnetic model has been developed in general terms. To

model a specific solenoid, , , and should be defined. For N217,

(3.117)

where , , and are the inductance polynomial coefficients.

Figure 3.19: Steady-state pressure versus solenoid current: N217

These coefficients are varied until simulated responses reasonably match steady-state

pressure versus solenoid current data provided by BorgWarner [36] – Figure 3.19 shows

simulated and supplier data on the same plot.

75

Fluid flow system

, which is previously defined as the exhaust flow rate through N217, is

described by:

√

(3.118)

Here, is the equivalent flow area for two restrictions- O3 and N217- connected

in series. is calculated using equation (3.119).

√

∑

(3.119)

Here, is the number of restrictions in series and is the flow area of each restriction;

in this case, . , defined as the ball poppet valve flow area, is

modeled as:

( ) ( )

(3.120)

where is the diameter of the seat.

3.5.2 Clutch actuation system

The clutch actuation system consists of two subsystems, which are: the odd gear

clutch actuation system, and the even gear clutch actuation system. Each system includes

a clutch pressure control valves (N215 or N216), a clutch pistons (K1 or K2), and a

pressure sensor. The system also includes a check valve that allows a malfunctioning

clutch pressure control valve to be bypassed, however, this flow path will not be modeled

here. The model structure for both the odd and even gear clutch actuation systems is

76

identical. Thus, the odd gear system shown in Figure 3.20 will be described in this

section and the model equations for the even gear system will be given in Appendix B.

Figure 3.20: Odd gear clutch actuation system

The clutch pressure control valves modeled here are direct-acting,variable force,

and normally closed solenoid valves. For direct-acting valves, the magnetic force

generated by the solenoid is directly applied to the spool valve that controls fluid flow to

the clutch piston. Compared to a pilot controlled solenoid valve, which affects the

position of a second valve by increasing or decreasing the pressure in one of its

chambers, the direct-acting valve allows for more responsive and accurate pressure

control [15]. When N215 is de-energized, the spool valve inlet is closed and the exhaust

is open; i.e. the clutch piston cavity can exhaust through N215. The spool displacement

increases with increasing solenoid current; the exhaust port is closed when the spool

displacement is greater than the underlap length (the axial distance that the spool or

plunger must move before the open port closes). There is a dwell length where neither the

77

exhaust or inlet ports are open. After the dwell length is recovered, the inlet port opens

and the clutch piston cavity can be pressurized.

Unlike most conventional automatic transmissions that, due to cost, have too

many friction elements to instrument with pressure sensors, each clutch piston cavity in

the VW02E is instrumented with pressure sensors. Real-time clutch pressure

measurements allow for on-line friction parameter estimation, and the additional

feedback variable can be used to improve shift control. Both of these topics will be

covered in more detail in later chapters of this thesis.

3.5.2.1 Clutch pressure control valve, N215

The mathematical model of the clutch pressure control valve consists of three

subsystems. They are: the spool and accumulator mechanical systems, the

electromagnetic circuit, and the fluid flow system.

Spool and accumulator mechanical systems

The N215 spool motion depends on the difference in pressure forces acting on the

spool, the magnetic force, the net spring force, the inertia force, the viscous damping

force, and the steady-state flow forces due to flow through the exhaust and inlet ports.

The static friction force and the transient flow forces are assumed to be negligible. Thus,

the mechanical dynamics for the N215 valve are described by:

And (3.121)

78

where




is the net force applied to the spool

is the magnetic force acting on the spool

are the pressures in chambers A and D

is the difference in land cross-sectional areas at chamber A

is the land cross-sectional areas at chamber D

is the net spring preload at static equilibrium


described by:

{

(3.122)

where is the unpressurized and de-energized static equilibrium

position. , which is defined as the net steady flow force acting on the spool, is

given by:

(3.123)

Here, and are the forces acting on the spool that are caused

by flow through the exhaust and inlet ports. The sign of the flow forces is determined by

the flow direction. Flow exits the valve through the exhaust port on the right hand side of

79

the land; the flow force tends to close the exhaust port, and accordingly, move the spool

valve in the positive direction. On the other hand, flow exits the valve through the inlet

port on the left hand side of the land, so the flow force tends to close the inlet port and the

spool valve [50]. The flow forces are modeled as:

{

(3.124)

{

And ( )

(3.125)

Here, and are the land diameters at chambers B and C. and

are the pressures in chambers B and C, while is the controlled

pressure at the output of N215. and are the underlap and overlap

lengths for the exhaust and inlet ports, respectively.

The N215 accumulator (or accumulator #5) mechanical dynamics are a function

of spring force and applied pressure to the accumulator piston. Friction and inertial forces

are ignored. The accumulator mechanical dynamics are described by:

(3.126)

where

is the spring displacement measured from static equilibrium


is the piston area

is the pressure acting on the piston


80


Again, the general model of the solenoid magnetic circuit described in section

3.5.1.2 can be applied to a specific solenoid by defining , , and for said

solenoid. For N215,

(3.127)

where , , , and are the inductance polynomial coefficients.

Figure 3.21: Steady-state pressure versus solenoid current: N215

81

As with N217, these coefficients are varied until simulated responses reasonably match

steady-state pressure versus solenoid current data provided by BorgWarner [37]; Figure

3.21 shows simulated and supplier data on the same plot.

Fluid flow system

For the simplified hydraulic system, the net volumetric flow rate to the odd gear

actuators is equal to the flow rate exiting the safety valve ( ). By applying

the continuity equation at the safety valve exit, the flow rate at the inlet of N215, ,

is described by:

(3.128)

Here, and are the flow rates to the synchronizer control solenoids N88 and

N89, respectively.

The flow through chambers B and C of the clutch pressure control valve is

defined by applying the continuity equation to both chambers while the clutch is either

filling or exhausting. For the filling phase:

(3.129)

(3.130)

For the exhausting phase:

(3.131)

(3.132)

82

Here, is flow rate into chamber B, and are the volumes of

chambers B and C, and and are the pressures in chambers B and C.

and , which are the metered flow rates at the output of N215 and

the exhaust, respectively, are modeled as:

√

And

(3.133)

√

(3.134)

and , which are defined as the flow areas at the exhaust

and inlet ports, are described by:

{

(3.135)

{

(3.136)

where and are the diameters of the lands at the inlet to chambers B and

C, and and are the underlap and overlap lengths for the respective port

opening at chambers B and C.

The flow into and out of chambers A and D is described by applying the

continuity equation at those chambers. The resulting equations are given by:

(3.137)

(3.138)

83

and

√

And

(3.139)

√

And

(3.140)

Here, and are the flow rates into or out of chambers A and D. The

volumes in chambers A and D, which vary with spool position, are given by:

(3.141)

(3.142)

Here, and are the chamber volumes of A and D at zero spool

displacement.

Finally, the continuity equation is applied at the N215 accumulator cavity.

(3.143)

is the volume in the accumulator, and the flow rate across orifice

#18, , is given by:

√

And

(3.144)

84

3.5.2.2 Clutch piston, K1

The mathematical model of the clutch piston consists of two subsystems. They

are: the clutch piston mechanical dynamics and the fluid flow system.

Clutch piston mechanical dynamics

The K1 piston motion depends on the pressure force acting on the piston, the

spring force, the inertia force, and the viscous damping force. The static friction force and

the flow forces are assumed to be negligible. Thus, the mechanical dynamics for K1 is

described by:

(3.145)

where

is the mass of the clutch piston



is the pressure acting area of the clutch piston

is the pressure applied to the clutch piston


, defined as piston displacement and measured from the unloaded static equilibrium,

is described by:

{

(3.146)

85

Fluid flow system

The net volumetric flow rate to or from the clutch piston cavity, , is

determined by applying the continuity equation at the outlet of the clutch pressure

regulation valve. The resulting expression is given by equation (3.147).

(3.147)

, which is also equal to the flow rate across orifice #21, is modeled as:

√

And

(3.148)

The pressure dynamics of the clutch piston cavity are determined by applying the

continuity equation at the clutch cavity.

(3.149)

, which is the position dependent clutch volume, is given by:

(3.150)

is the piston volume at zero displacement.

3.5.3 Synchronizer actuation system

The synchronizer actuation system includes the multiplexer control solenoid

(N92), the multiplexer valve (MPV), the synchronizer control solenoids (N88, N89, N90

and N91), and four shift forks (SF13, SF24, SF6R, SF5N). The model structure is

identical for all synchronizer control solenoids and shift forks. Thus, the model equations

86

for N88 and SF13 will be described in this section; equations for the remaining

components will be given in Appendix B.

The multiplexer valve shown in Figure 3.22 is implemented to allow four

ON/OFF solenoids (N88-N91) to control four shift forks into eight positions. The

multiplexer valve has two position: off (or home) and on (or work). When N92 is de-

energized, MPV is in the off position. By energizing N92, MPV is moved to the on

position. In the off position, the control ports of N88 and N89 are connected to opposite

sides of the 1-3 shift fork (SF13). When N88 is energized and N89 is de-energized, the

resulting pressure gradient pushes the shift fork, and the synchronizer sleeve, toward

output gear #1. On the contrary, when N88 is de-energized and N89 is energized, SF13 is

pushed toward output gear #3. If the multiplexer valve is in the on position, the control

ports of N88 and N89 are connected to opposite sides of the 5-N shift fork (SF5N). When

N88 is energized and N89 is de-energized, SF5N is moved toward output gear #5. If N89

is energized and N88 is de-energized, SF5N is moved toward the neutral position; if all

other synchronizer sleeves are in their center positions, then the gearbox is in neutral.

Figure 3.22: Multiplexer valve, MPV

87

To center a shift fork, there must be a brief pressure gradient in the desired

direction of travel. The method for accomplishing this task is best explained with the

following example. To move the 1-3 shift fork from output gear #3 to its center position,

N88 should be energized. After a brief delay, during which the shift fork should be near

its desired position, N89 should be energized. Table 3.3 provides a summary of the

solenoid operation schedule for ouput gear engagement.

Solenoid (Status) Target Ouput Gear

N92(OFF)

N88(OFF) N89(ON) OG3

N88(ON) N89(OFF) OG1

N90(OFF) N91 (ON) OGR


N92(ON)

N88(OFF) N89(ON) Neutral


N90(OFF) N91 (ON) OG4


Table 3.3: Solenoid operation schedule for output gear engagement

3.5.3.1 Multiplexer valve

The mathematical model of the multiplexer valve consists of two subsystems.

They are: the spool mechanical dynamics and the fluid flow system.

Spool mechanical dynamics

The MPV spool motion depends on the pressure force acting on the spool, the

spring force, the inertia force, and the viscous damping force. Again, the static friction

force and the flow forces are assumed to be negligible. The spool dynamics are described

by:

88

(3.151)

where




is the pressure in chamber A

is the land cross-sectional areas at chamber A


, defined as the spool displacement and measured from off position, is described by:

{

(3.152)

Fluid flow system

The multiplexer valve position is controlled by the pressure in chamber A. The

chamber A pressure dynamics are described:

(3.15

3)

where is the flow rate from the N92 valve to chamber A. , which is the

position dependent volume of chamber A, is given by:

(3.154)

is the chamber A volume at zero spool displacement.

89

3.5.3.2 Multiplexer control solenoid

The model of the multiplexer control solenoid shown in Figure 3.23 consists of

three subsystems, which are: the plunger mechanical dynamics, the electromagnetic

system, and the fluid flow system.

Plunger mechanical dynamics

The multiplexer control solenoid plunger motion depends on the pressure force

acting on the steel ball at the inlet port, the magnetic force, the spring force, the inertia

force, and the viscous damping force. As with the pressure regulation control valve, the

static friction force and the flow forces acting on the plunger are assumed to be

negligible. Thus, the mechanical dynamics for the N92 valve are described by:

Figure 3.23: Multiplexer control solenoid, N92

90

(3.155)

where






is defined as the plunger displacement. There are two possible positions for the

plunger: closed, when de-energized; and fully open, when energized. The plunger

displacement, which is measured from the closed position, is given by:

{

(3.156)


( ( ))

(3.157)

where and are the ball diameter and the central angle corresponding to

the pressure acting diameter of the ball.


The general model of the solenoid magnetic circuit described in Section 3.5.1.2

is applied here. For N92,

(3.158)

91

where , , , and are the inductance polynomial coefficients. The

voltage input to N92 is set to its minimum (0V), or when energized, its maximum ( ).

The inductance polynomial coefficients are selected so that when the solenoid is

energized, the generated magnetic force is sufficiently large enough to move the plunger

to its fully open position.

Fluid flow system

, which is previously defined as the inlet flow rate to N92, is modeled

separately if the exhaust port is closed ( ) or open ( ). Here,

refers to the underlap length for the N92 exhaust port. When the exhaust port is

closed, the flow rate through N92 is equal to the flow rate into chamber A of the

multiplexer valve ( ). In this case, is described by:

√

| | ( ) (3.159)

, defined as the equivalent flow area for three orifices - O24, O25, and N92 -

connected in series, is calculated using equation (3.119). The ball poppet valve flow area,

, is modeled as:

( ) ( )

(3.160)


When the exhaust port is open, the continuity equation is applied at the node

downstream of the solenoid seat. Thus, the flow rate to chamber A of the MPV is

modeled as:

92

(3.161)

where,

√

| | ( ) (3.162)

√

| | ( ) (3.163)

√

(3.164)

Here, is the pressure downstream of the N92 seat. , defined as the

equivalent flow area for orifices O24 and N92, is calculated using equation (3.119). The

exhaust flow area, , is described by:

{

where is the plunger diameter.

Figure 3.24: 1-3 shift fork and synchronizers

93

3.5.3.3 Shift fork, SF13

The mathematical model of the 1-3 shift fork displayed in Figure 3.24 consists of

two subsystems: the shift fork mechanical dynamics, and the fluid flow system.

Shift fork mechanical dynamics

The shift fork motion depends on the net pressure force acting on the shift fork,

the Coulomb and static friction forces, the inertia force, and the viscous damping force.

Thus, the shift fork mechanical dynamics are described by:

( ) ( ) (3.165)

where

is the combined mass of the shift fork and the movable half of

the synchronizer


is the pressure-acting area of either shift fork side

, is the pressures applied to move synchronizer toward gear 1, 3

is the Coulomb friction force

The shift fork position, , is calculated using equation (3.165) when the applied

force, ( ), is greater than the static friction force, , or the

shift fork velocity is nonzero. If these conditions are not met, the applied force should be

less than or equal the static friction force, and the shift fork velocity is set to zero.

is measured from the neutral position (or center) of the synchronizer. As defined by

94

equation (3.166), is limited to the synchronizer displacements required to engage

output gears #1 and #3 ( and ).

(3.166)

The normal force applied to the shift fork, , is defined by:

( ) (3.167)

Fluid flow system

The synchronizer engagement process relies on creating a pressure gradient across

the shift fork. To model this process, we must consider the pressure dynamics in the

chambers on either side of the shift fork. Note that the chamber number corresponds to

the gear that is engaged when said chamber is pressurized. For the 1-3 shift fork, if the

pressure in chamber #1 is sufficiently greater than the pressure in chamber #3, the shift

fork will move toward gear #1. The pressure dynamics for chambers #1 and #3 are

described by:

(3.168)

(3.169)

where and are the flow rates into chambers #1 and #3. The volumes in

chambers #1 and #3, which vary with shift fork position, are given by:

(3.170)

(3.171)

95

and are the volumes of chambers #1 and #3 when the shift

fork is in the neutral position.

3.5.3.4 Synchronizer solenoid, N88

The matematical model of the synchronizer solenoid shown in Figure 3.25

consists of three subsystems, which are: the plunger mechanical dynamics, the

electromagnetic system, and the fluid flow system.

Figure 3.25: Synchronizer solenoid, N88


The synchronizer solenoid plunger motion depends on the pressure force acting

on the steel ball at the inlet port, the magnetic force, the spring force, the inertia force,

and the viscous damping force. The static friction force and the flow forces acting on the

plunger are assumed to be negligible. Thus, the mechanical dynamics for the N88 valve

are described by:

96

(3.172)

where






is defined as the plunger displacement. There are two possible positions for the

plunger: closed, when de-energized; and fully open, when energized. The plunger

displacement, which is measured from the closed position, is given by:

{

(3.173)


( ( ))

(3.174)

where and are the ball diameter and the central angle corresponding to

the pressure acting diameter of the ball.


The general model of the solenoid magnetic circuit described in Section 3.5.1.2

is applied here. For N88,

(3.175)

97

where , , and are the inductance polynomial coefficients. As with the

multiplexer control solenoid, the inductance polynomial coefficients are selected so that

when the solenoid is energized, the generated magnetic force is sufficiently large enough

to move the plunger to its fully open position. The voltage applied to N88 is either 0V, or

when energized, .

Fluid flow system

The model describing , which is previously defined as the inlet flow rate to

N88, is a function of the multiplexer valve position (off or on), and whether the exhaust

port is closed ( ) or open ( ). Here, refers to the

underlap length for the N88 exhaust port. When the exhaust port is closed, is equal

to the flow rate into either chamber #1 of the 1-3 shift fork or chamber #5 of the 5-N shift

fork.

:

{

√

√

(3.176)

Here, is the equivalent flow area of restrictions O32, O33, N88, and O41

connected in series. is the equivalent flow area of restrictions O32, O33,

98

N88, and O40 connected in series. These areas are calculated using equation (3.119). The

ball poppet valve flow area, , is modeled as:

( ) ( )

(3.177)


When the exhaust port is open, the continuity equation is applied at the node

downstream of the N88 seat. In this case, the difference between and the exhaust

flow rate, , is equal to the net flow rate to either the 1-3 or 5-N shift forks.

:

{

√

√

(3.178)

In this case, is the equivalent flow area of restrictions O33 and O41

connected in series. is the equivalent flow area of restrictions O33 and

O40 connected in series. Both areas are calculated using equation (3.119). is the

pressure downsteam of the N88 seat. The inlet and exhaust flow rates are described by:

99

√

(3.179)

√

(3.180)

is the equivalent flow area of restrictions O32 and N88 connected in series. The

exhaust flow area, , is described by:

{

(3.181)

where is the plunger diameter.

By describing the flow rate through N88 and into the shift fork chambers as we

have in this section, two key assumptions are made. First, we assume that the volumes of

the multiplexer valve chambers B-I are small enough that the pressure dynamics in each

chamber is insignificant. Thus, the net flow rate through N88 is approximately equal to

the flow rate into or out of the shift fork chamber. The second assumption is that the shift

fork chamber pressures must be at steady-state prior to the repositioning of the

multiplexer valve. This assumption is used in the synchronizer control logic, and it

ensures that the synchronizers are at the desired position prior to moving the multiplexer

valve.

3.6 Model limitations

The powertrain model presented in the previous sections describes, with a low

fidelity thermal model, the heat transfer through the clutch, models the vehicle dynamics

100

just in the longitudinal direction, and estimates driveline damping based on shaft

stiffnesses and an assumed damping ratio for all components in the driveline. The thermal

model describes the oil temperature at the clutch interface as a function of heat generated

by sliding friction and heat lost by convection (oil to housing); the oil cooling system,

which regulates the temperature of the oil in the hydraulic system, is not modeled. Thus,

the powertrain simulation should not be used to describe hundreds or thousands of

consecutive shifts, as the temperature of the oil at the clutch interface would not reach

steady state at any point during the simulation.

Since the engine mounts, vehicle suspension, and the lateral forces acting on the

tires are not modeled, the vehicle dynamics are greatly simplified to describe purely

longitudinal, and translational, motion. This low fidelity vehicle and tire model can only

be used to correlate changes in longitudinal vehicle acceleration to shift quality; however,

passengers in a vehicle also perceive excessive vibrations in the vertical direction, and

varying pitching and roll moments as uncomfortable motions that should be avoided.

Thus to provide a more complete illustration of shift quality, a higher fidelity vehicle

model should be incorporated with the rest of the powertrain model.

The structural damping coefficients for compliant torsional elements in the

driveline are estimated by assuming that each element represents a second order

rotational system. Given the mechanical stiffness and inertia of each element, and

assuming that each component is lightly damped (damping ratio on the order of 0.05), the

structural damping coefficient is calculated. Thus without experimentally validating the

powertrain model, there is an element of uncertainty in the actual level of damping

present in the driveline.

101

3.7 Conclusion

This chapter presents the development of a powertrain model with an emphasis on

the dual clutch transmission. Simulation of the overall model, which is comprised of

models of the engine, the transmission mechanical system, the longitudinal vehicle

dynamics, and the hydraulic actuation system, is described in Chapter 4. Particular

importance is placed on the implementation of a dynamic clutch friction model and the

development of a representative hydraulic model, as the powertrain model provides the

basis for the estimation of clutch friction parameters and the controller design; these

topics will be discussed in Chapters 5 and 6, respectively.

102

CHAPTER 4: MODEL SIMULATION

The following chapter presents the powertrain simulator developed using the

dynamic equations derived in Chapter 3 and the control scheme developed in Chapter 6.

The simulator is built with two software packages: AMEsim 11.2 and MATLAB

R2011b/Simulink 7.8. AMEsim, which is an acronym for Advance Modeling

Environment for performing Simulations, is a multi-physics, one-dimensional, and

component based simulation tool produced by LMS. It belongs to a class of simulation

software referred to as physical simulation software and it provides high level support for

simulation of complex physical systems. When a dynamic system can be modeled

entirely out of components available in AMEsim, the overall model can be constructed

much faster than the mathematical model can be programmed from scratch with a

software package such as MATLAB/Simulink. However, the AMEsim modeling

environment is limited by the extent of its libraries; when a physical system cannot be

adequately described in AMEsim, MATLAB/Simulink can be used instead. In addition,

Simulink is the preferred software for controller design. Thus, co-simulation with

AMEsim and Simulink allows for reduced modeling time without losing the

programming capabilities required for high-fidelity modeling and controller design.

This chapter is organized as follows. Section 4.1 introduces modeling in

AMEsim, while Section 4.2 describes the solver options. The powertrain simulator,

103

which is co-simulated with AMEsim and MATLAB/Simulink, is discussed in Section 4.3

. Finally, simulation results and concluding remarks are provided in Sections 4.4 and 4.5

, respectively.

4.1 Modeling in AMEsim

The AMEsim modeling environment consists of extensive libraries of

preprogrammed, “stock”, components that represent the mathematical models of physical

objects. There are 40 libraries included with the educational/research license of AMEsim

11.2; libraries are formed based on the physical domain or engineering function that can

be described using components from said library. For example, a component representing

a spur gear, spool valve, inductor and PID controller may be found in the mechanical,

hydraulic, electrical, and control libraries, respectively. For a component to be included

in AMEsim, AMEsim developers first create the component to meet the modeling needs

of a specific LMS customer. Only after the customer experimentally validates the

component, will it be made available to the general AMEsim user. Since a vehicle with a

VW02E transmission is unavailable for experimentation, confidence in the simulation is

built by comparing simulation results to literature and is bolstered by the fact that

AMEsim library components are validated experimentally.

Components are associated with icons that can be inserted into an AMEsim

model. Each icon is a graphical representation of the physical object. Typically, icons are

based on standard symbols used in industry (ISO symbols for hydraulic components,

block diagram symbols for control components, etc.) If no industry standard exists, then a

picture or sketch of the physical object is used [51]. The ports of an icon represent the

104

input(s) and output(s) of the mathematical model that describes the component.

Components that have opposite input(s) and output(s) may be connected to each other.

For instance, the rotational dynamics of an inertia element are modeled by Newton’s

second law for rotation, where torque (T) is the input and angular displacement (θ) and

angular speed (ω) are the outputs. For a torsional damper, which can be used to model

shaft damping, the causality is reversed – angular speed is the input to the damper, while

torque is the output. Figure 4.1 shows the inputs and outputs of both components; since

the inertia and damper inputs and outputs are opposite, they may be connected to each

other.

Figure 4.1: Example of component causality

Connections between components are made via a line that represents the physical

connection regardless of the domain; i.e., a line is used to represent the shaft connecting

an inertia element to a damper, or the wiring between a resistor and capacitor. A model is

constructed by connecting together multiple components. Once inserted into the overall

model, the user can adjust a fixed set of parameters specific to the mathematical model of

each component; however, the structure of the mathematical model cannot be changed.

105

If the components available in the AMEsim library are insufficient, AMEsim

users can create a new component, or modify the structure of an existing one, using the

submodel editing tool (or AMEset). Here, submodel refers to the mathematical model of

a component. In AMEset, users can write the code for a component in C or Fortran 77.

Users may also determine the set of parameters that can be adjusted once the component

is inserted into the AMEsim environment [52].

4.2 Solver (or integrator) options in AMEsim

The solvers options in AMEsim can be broken into two categories: (1) fixed step,

fixed order and (2) variable step, variable order. There are three explicit fixed step

integration methods - Euler, Adams-Bashforth, and Runge-Kutta – that are available. For

the latter two methods, the user can select the order (2nd

-4th

), while the order is fixed for

Euler’s method (1st) [51]. The step size and solver order are fixed throughout a

simulation.

AMEsim’s variable step, variable order solver is referred to as the “standard

integrator”. The standard integrator dynamically switches integration methods based on

whether the model contains implicit variables and the overall model stiffness. If the

model includes implicit variables, AMEsim selects the differential algebraic system

solver (DASSL) [51]. The DASSL algorithm is a collection of backward differentiation

formulae (BDF) of the order of 1st-5

th that are used to solve differential algebraic

equations (DAEs). Hindmarsh and Petzold [53] define the BDFs in the following form:

(

∑

) (4.1)

106

Here, and are the time and state variable(s) at the nth

step, is the state

variable(s) at the (n-1)th

step, is the step size, is the solver order, and and are

coefficients that are functions of previous step sizes and solver order.

If the model does not contain implicit variables, AMEsim selects the Livermore

solver for ordinary differential equations (LSODA). The LSODA solver switches

between non-stiff and stiff integration methods depending on the stiffness of the overall

model. The non-stiff solver is based on the collection of 1st-12

th order Adams methods.

Hindmarsh and Petzold [53] define Adams method as:

∑

(4.2)

where is a coefficient that is a function of solver order and previous step sizes, and

is the derivative of the state variable(s) evaluated at the (n-j)th

step. The stiff solver

utilizes the same DASSL algorithm described by equation (4.1).

A system is considered stiff if a small perturbation to a state variable causes the

system to rapidly restore itself to equilibrium. The faster the system responds to said

perturbation, the shorter the step size required to accurately describe the behavior of the

system. Given multiple perturbations to the system, the longest and shortest time scales

are defined as the largest and smallest step sizes - over the total time for which a solution

is found - required to achieve an accurate solution. The more stiff the system, the larger

the ratio of the longest to shortest time scales-this is called the stiffness ratio [54].

Although AMEsim has not published this information, the stiffness ratio is likely used

internally by the LSODA solver to determine when to switch between the non-stiff and

stiff solvers.

107

4.3 Co-simulation with MATLAB/Simulink

The powertrain model developed in Chapter 3 includes models of the engine, the

transmission mechanical system, the hydraulic component actuation system, and the

longitudinal vehicle dynamics. The gearbox sans the clutches, the hydraulic actuation

system, and the vehicle dynamics are modeled entirely within AMEsim. This portion of

the powertrain model contains over 200 explicit states; given the larger number of states

and the high numerical stiffness of the model due to the hydraulic system, AMEsim’s

LSODA solver is an effective option.

The dynamic clutch friction model presented in Chapter 3 is not available in the

AMEsim library. Instead, AMEsim clutch components primarily use static friction

models, and the Karnopp method is not used for simulating clutch friction around zero

slip speed. As an alternative to the Karnopp method, the clutch friction torque is

described using the tanh model [55] defined as:

(

) (4.3)

Here, is the clutch friction torque, is the maximum static torque, is

the static-dynamic threshold speed, and is the slip speed. Using this method,

phenomena such as stick-slip can be misrepresented because clutch friction torque is

calculated as function of slip speed and proportional to the maximum static torque rather

than a function of the external torques acting on the clutch. Thus, the dynamic clutch

friction model, as well as the mean-value engine model and the powertrain controller, is

programmed in Simulink.

108

A combination of AMEsim and Simulink models can be interfaced in the

following ways: (1) the AMEsim model can be imported as an S-function into Simulink,

and (2) C-code generated from the Simulink model can be imported into AMEsim.

Generally, it is best to import the AMEsim model into Simulink if the goal for the co-

simulation is to test and develop a controller. However, if the goal is to develop an

extensive plant for which the AMEsim standard integrator would be the most effective

solver, then it is best to import the Simulink model into AMEsim. For the powertrain

simulator developed in this work, the Simulink model is imported into AMEsim.

Mathworks Simulink Coder (formerly Real-Time Workshop) is used to create the

C-code of the Simulink model. For successful C-code generation, the Simulink model

must use a continuous fixed-step solver and it cannot contain any algebraic loops. The C-

code is used by AMEsim to create a component representing the Simulink model that can

be directly inserted into the AMEsim model. During a simulation, the AMEsim integrator

– fixed or standard – solves the AMEsim model and Simulink’s fi ed step solver solves

the Simulink model. AMEsim and Simulink do not need to have the same step size;

however, the AMEsim print interval should match the Simulink step size. For each time

step taken by the Simulink solver, AMEsim and Simulink exchange data. AMEsim

registers this data exchange as a discontinuity; the more discontinuities registered by

AMEsim, the longer the overall simulation time. To reduce number of discontinuities, the

user may specify a number steps (or undersample) taken by the Simulink solver between

data exchanges. The powertrain simulator solver parameters are summarized in Table 4.1.

109

Software

Package

Integrator

Type

Relative

Tolerance

Step Size

(s) Undersample

Print

Interval (s)

AMEsim Standard 1e-6 X X 1e-4

Simulink

Fixed step,

4th

Order

Runge-Kutta

X 1e-5 10 X

Table 4.1: Powertrain simulator solver parameters

4.4 Simulation results for vehicle launch

Simulation results for vehicle launch are presented by Figure 4.2-Figure 4.9. The

simulation is completed under the following conditions: the engine speed is stepped up to

1000rpm at the simulation start time (t0), the vehicle starts from rest at t0, the engine is

operated at wide open throttle and spark timing for maximum brake torque until the shift

into 1st gear is completed, the odd gear clutch pressure is ramped up from 0bar to line

pressure (17-18bar) in an open loop fashion, and output gears #1 and #2 are engaged

before and after the shift into 1st gear, respectively. Note that simulation results for

power-on upshifts and downshifts will be discussed in Chapter 6, when the integrated

powertrain control of clutch-to-clutch shifts is presented.

Figure 4.2 shows the line pressure during launch. The initial pressure transients

are caused by introducing a step input in pump flow rate. The first peak is clipped – due

to the cracking of the pressure relief valve – at 32bar. After the transients die out, the line

pressure continues to oscillate about a mean pressure; this behavior can be attributed to

the line pressure feedback acting on the pressure regulation spool valve. Figure 4.3

110

displays the measured and commanded pressure at clutch #1; a PID controller is used to

improve tracking of the commanded pressure by the measured pressure.

Figure 4.2: Line pressure during launch

Figure 4.3: Clutch #1 measured and commanded pressures during launch

111

Figure 4.4 shows the normalized shift fork displacement and the synchronizer

state associated with the engagement of output #2. Note that the synchronization of

output gear #1 is excluded, since the synchronization of said gear is completed prior to

the engagement of clutch #1. With no power transfer from the engine to the wheels, the

difference between the synchronizer and gear speeds is zero. Thus, the engagement of

output gear #1 does not significantly affect differential torque and vehicle acceleration.

After clutch #1 is engaged, the synchronization of output gear #2 is completed in 26.5ms,

which is on the order of the gear synchronization times – 0.1s or less – described by

Galvagno et al. [14]. Razzacki and Hottenstein state that the typical synchronization time

for layshaft transmissions is 0.2s [4]; however, the synchronization time can be less than

0.1s if the actuation force is significant ( 1000N). For the transmission described here,

the typical actuation force is between 800N and 2500N.

Figure 4.4: Normalized shift fork position and synchronizer state during engagement of

output gear #2

112

The engagement of clutch #1 begins at 147.5ms and ends at 286.1ms. Per the

Audi self-service manual, a typical clutch-to-clutch shift time for the VW02E

transmission is approximately 200ms [38]; however for fast launch, the engagement time

can be between 100 and 200ms [7]. Thus, 140ms is a reasonable duration for clutch

engagement at launch. Figure 4.5 displays the angular speeds for the clutch hub, the

gearbox sides of clutch #1 and #2, and the engine during launch. Figure 4.6 shows the

state of clutch #1; when the state changes from 0 to 1, clutch #1 is engaged and the slip

speed between the clutch hub and the gearbox side of clutch #1 is reduced to zero.

Figure 4.7 shows the engine torque and the clutch #1 friction torque; over the shift

duration, the clutch friction torque is calculated using the dynamic friction model

described in Section 3.3.4 . Once locked, the clutch friction torque is calculated as a

function of external torques acting on the clutch. Due to the light structural damping

throughout the gearbox, transient torque and acceleration responses of gearbox

components are observed in the clutch friction torque. Figure 4.8 and Figure 4.9 show the

differential torque and the longitudinal vehicle speed, respectively, during launch. Once

clutch #1 is engaged, the engine power is transmitted to the wheels, causing the vehicle to

accelerate. At the end of the 0.6s simulation, the vehicle speed is 1.7m/s.

113

Figure 4.5: Clutch (hub and gearbox side) and engine angular speeds during launch

Figure 4.6: Clutch #1 state during launch

114

Figure 4.7: Engine and clutch #1 friction torque during launch

Figure 4.8: Differential torque during launch

115

Figure 4.9: Longitudinal vehicle velocity during launch

4.5 Conclusion

In this chapter, a powertrain simulator is developed using the model presented in

Chapter 3. The simulator is built using two software packages: AMEsim and

MATLAB/Simulink. Most of the powertrain model is built using components found in

AMEsim; however, MATLAB/Simulink must be used to describe physical systems or

processes that are not included in AMEsim. Since a vehicle with a comparable powertrain

is unavailable for experimentation, confidence in the structure of the simulator is

strengthened since AMEsim library components are validated experimentally. Simulation

results of the powertrain at launch are presented and compared to results found in

literature. With confidence in the powertrain simulator, it can be useful in demonstrating

how changes in the integrated powertrain controller described in Chapter 6 affect the

dynamic response of the powertrain, particularly during a clutch-to-clutch shift.

116

CHAPTER 5: FRICTION PARAMETER ESTIMATION

5.1 Clutch friction characteristics as the material/fluid ages

For wet multi-plate clutches in automatic transmissions used in passenger

vehicles, the friction material is typically paper-based. Additional materials such as

organic and inorganic compounds, cellulose, and fibers may be bonded to the paper layer

[5]. Typical ingredients used in paper-based friction materials are given in Table 5.1.

Table 5.1: Ingredients in paper-based friction materials [42]

To achieve smooth and consistent torque transfer through a clutch, the coefficient of

friction should vary consistently with slip speed, applied pressure, and temperature, and

the overall coefficient of friction versus slip speed (or or ) curve should

have a positive slope [56]. Figure 5.1 displays curves with both positive and

negative slopes. A positive slope is desirable for stable operation and control of the clutch

117

slip [5]. As the slope becomes negative, the clutch may experience: stick-slip,

when static friction is higher than sliding friction; and/or shudder, when driveline

damping cannot sufficiently suppress vibrations caused by an increase in friction

associated with a decrease in slip speed [11].

Figure 5.1: curves with positive and negative slopes [11]

For wet clutches, the interactions of the automatic transmission fluid (ATF) and

the friction material affect the quality of the clutch engagement. Lam et al. [42] explain

that the type of oil additives, the chemical adsorption affinities, and the oil film

characteristics affect the clutch torque capacity and the slope at low speeds (i.e.,

transmission fluid with the proper additives allows static friction to be less than sliding

friction). As the ATF ages due to oxidation, thermal degradation, shearing, vaporization,

or hydrolysis, the performance of the clutch degrades [11]. Table 5.2 summarizes the

factors that affect the wet clutch performance.

118

Table 5.2: Controlling factors of wet friction materials [42]

Figure 5.2: curve for BW-6100 friction plate [41]

119

The VW02E transmission uses BW-6100 friction plates produced by Borg

Warner. Using the low-speed SAE#2 friction test on the BW-6100 friction plate, Tersigni

et al. [41] of Afton Chemical generated the curves shown in Figure 5.2. The

friction tests were conducted using new, aged, and end-of-test (EOT) DCTF-1 fluid– the

transmission fluid used in the VW02E model released in 2002 – at an interface

temperature of 120°C. Here, “aged” refers to transmission fluid that was continuously

stirred at 150°C for 200 hours. “EOT” refers to transmission fluid from a Volkswagen

GTI that was tested for over 60,000 miles on a chassis dynamometer. By fitting the

Stribeck friction model described by equation (3.72) to the data corresponding to

new oil, the parameters given in Table 5.3 are identified.

Parameter Value [Unit]

0.1558

0.1192

-1.960e-4 [s/rad]

0.8669 [s/rad]

0.5030

Table 5.3: BW-6100 coefficient of friction parameters

Again, and are the Coulomb and static coefficients of friction, and

are the Stribeck coefficients, and is the viscous friction coefficient. Note that is set

to zero in the dynamic friction model presented in Chapter 3, because the contact and

viscous friction torques are calculated separately (i.e., the function given by equation

(3.72) is purely used to describe contact friction). For the static friction model used in the

estimation scheme described in the next section, the combination of contact and viscous

friction is represented by including a nonzero value for in the Stribeck model. To

120

include thermal effects in the static friction model, the viscous damping coefficient can

be described as a function of temperature dependent fluid viscosity ( ) and the

viscosity-independent and speed dependent friction coefficient . Thus, equation (3.72)

can be rewritten as:

( (

| |

)

)

| | (5.1)

5.2 Friction parameter estimation

5.2.1 Sensitivity of coefficient of friction to variation in individual parameters

To demonstrate how individual parameters affect the overall clutch coefficient of

friction, the baseline curve is compared to curves with all but one parameter

held constant at its baseline value. Figure 5.3-Figure 5.5 display sets of curves

as a function of , , , , and , respectively, where each variable is set to 50%,

75%, 115%, 130% and 150% of its baseline value denoted by the subscript B. We note

here that ( ), where (

) is the ATF viscosity at the nominal operating

temperature (120°C), and that any percent change in from its baseline value is

proportional to the change in .

As expected, Figure 5.3-Figure 5.4 show that relative variations in the Coulomb

and static coefficients of friction can cause a change in the slope of the curve,

and that a large variation in the Coulomb term causes a vertical shift in the overall

coefficient of friction. Figure 5.5 clearly indicates that, for a constant ATF temperature,

the effect of variations in the viscous damping coefficient depends on slip speed; the

121

overall coefficient of friction changes marginally for slip speeds less than 10 rad/s and

becomes progressively more sensitive to variations in as slip speed increases.

Figure 5.3: as is varied from baseline


0 5 10 15 20 25 30

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

Slip Speed, slip

[rad/s]

Coeff

icie

nt

of

Friction,

K(

slip

)

Baseline

C = 0.5*

C, B

C = 0.75*

C, B

C = 1.15*

C, B

C = 1.3*

C, B

C = 1.5*

C, B

0 5 10 15 20 25 300.11

0.12

0.13

0.14

0.15

0.16

0.17

Slip Speed, slip

[rad/s]

Coeff

icie

nt

of

Friction,

K(

slip

)

Baseline

S = 0.5*

S, B

S = 0.75*

S, B

S = 1.15*

S, B

S = 1.3*

S, B

S = 1.5*

S, B

122



0 5 10 15 20 25 300.11

0.12

0.13

0.14

0.15

0.16

0.17

Slip Speed, slip

[rad/s]

Coeff

icie

nt

of

Friction,

K(

slip

)

Baseline

V

= 0.5*V,

B

V

= 0.75*V,

B

V

= 1.15*V,

B

V

= 1.3*V,

B

V

= 1.5*V,

B

0 5 10 15 20 25 300.11

0.12

0.13

0.14

0.15

0.16

0.17

Slip Speed, slip

[rad/s]

Coeff

icie

nt

of

Friction,

K(

slip

)

Baseline

S = 0.5*

S, B

S = 0.75*

S, B

S = 1.15*

S, B

S = 1.3*

S, B

S = 1.5*

S, B

123


Figure 5.6-Figure 5.7 demonstrate that the Stribeck factors, and , have a marginal

effect on the shape of the curve at low slip speeds and have a negligible effect

for mid to high slip speeds. Since the overall coefficient of friction is relatively

insensitive to variations in and , both parameters will be considered fixed at their

respective baseline values – only , , and will be estimated.

5.2.2 Estimation scheme and results

Friction torque at either clutch #1 or #2 is estimated by applying linear least

squares to the following system of equations:

(5.2)

(5.3)

(5.4)

0 5 10 15 20 25 300.11

0.12

0.13

0.14

0.15

0.16

0.17

Slip Speed, slip

[rad/s]

Coeff

icie

nt

of

Friction,

K(

slip

)

Baseline

S = 0.5*

S, B

S = 0.75*

S, B

S = 1.15*

S, B

S = 1.3*

S, B

S = 1.5*

S, B

124

The resulting expressions for and are given by equation (5.5).

[

]

And,

[

] , [

]

(5.5)

The flywheel torque ( ), the torque transmitted through input shafts #1 and #2 (

and ), and the speeds of the clutch hub, clutch #1, and clutch #2 ( , , and )

are obtained directly from the simulation. We note here that for implementation in a

vehicle, , , and may be calculated using the additional measurements of

engine and wheel speed.

The accelerations of the clutch hub, clutch #1, and clutch #2 ( , , and )

may be estimated by numerical differentiation or by applying the Kalman estimation

technique. The 4th

order central difference method shown in equation (5.6) is used for

offline differentiation, where the speed signals saved from previous shifts are expected to

be less noisy.

{

}

where,

( )

( )

(5.6)

125

( )

( )

Here, is the independent variable at the kth

step, h is the time step, is the output

function evaluated at , and n is the total number of data points.

For online (or real-time) differentiation, a modified version of the discrete

Kalman estimator presented by Bai et al. [57] is implemented. The Kalman estimator is

designed as follows. The discrete plant is modeled as:

(5.7)

where is the vector of state variables at the nth

point, is the state matrix, is the

measurement matrix, and is the state noise matrix. For acceleration estimation using

speed measurements, , , , and are given by equations (5.8)-(5.11).

[

] (5.8)

[

] (5.9)

(5.10)

[

] (5.11)

Here, is the sample period. and are process weights for each state variable. The

state estimator is described by:

( )

[

]

(5.12)

126

is the vector of estimated state variables at the nth

point and is the Kalman

estimator gain vector. may be calculated using the built-in MATLAB function shown in

(5.13).

(5.13)

and , which are the Kalman estimator system equations and the covariance matrix,

respectively, are not used in this work. and are the process noise covariance matrix

and measurement noise covariance matrix, respectively. The discrete Kalman estimator

as implemented in Simulink is shown in Figure 5.8.

Figure 5.8: Simulink implementation of discrete Kalman estimator

To demonstrate the effectiveness of the discrete Kalman estimator, the estimated

derivatives of the pressure and angular speed at clutch #1 are compared to the same terms

calculated in AMEsim. Figure 5.9 and Figure 5.10 display the derivatives of pressure and

speed at clutch #1, respectively, during launch; clearly the derivatives calculated using

the Kalman estimator and in AMEsim match well.

127

Figure 5.9: Derivative of clutch #1 pressure during launch: calculated using Kalman

estimator and in AMEsim

Figure 5.10: Derivative of clutch #1 speed during launch: calculated using Kalman

estimator and in AMEsim

128

Estimated and simulated friction torques for a 2-3 upshift are shown in Figure

5.11, where the estimation is completed offline using data from past shifts, and Figure

5.12, where the estimation is completed online using data from current shifts. In both

cases, clutch #2 is engaged from 1.5s to when the shift begins to slip at 2.08s. The shift

ends when clutch #1 sticks at 2.62s (offline estimation) and 2.51s (online estimation). For

both slipping and sticking conditions, the estimated – online and offline – and simulated

friction torques match reasonably well for both clutches. We note here that the online

estimated torques are noisiest when the clutch is completely disengaged, however, data

during the completely disengaged state is not used to estimate coefficient of friction

parameters.

Figure 5.11: Offline estimation and simulation of friction torques during a 2-3 upshift

1.5 2 2.5 3-100

0

100

200

300

400

Time [s]

Clu

tch F

riction T

orq

ue [

Nm

]

Estimated (offine) TF1

Simulated TF1

Estimated (offine) of TF2

Simulated TF2

129

Figure 5.12: Online estimation and simulation of friction torques during a 2-3 upshift

The total coefficient of friction can be calculated using the following static

friction model:

( ) (5.14)

As in Chapter 3, the subscript m refers to a variable that is specific to clutch #1 or #2.

is the mean radius of the clutch friction material. The friction torque is

calculated using equation (5.5) and the clutch pressure is obtained from the

simulation. The coefficient of friction parameters are estimated separately for each clutch

and averaged together to get an overall estimate for , , and .

To estimate the coefficient of friction parameters, only data generated with

equation (5.14) that corresponds to a positive coefficient of friction ( ) and

approximately steady slip speeds (| | ) is stored. Here, is defined as

the acceleration tolerance. The data is further partitioned into three subsets: near-zero slip

130

speed ( ), low slip speed ( ), and high slip

speed ( ). Criteria for each subset, as well as the parameter

estimated using said subset of data, are summarized in Table 5.4.

Data

Subset Criteria

Estimated

Parameter

Near-Zero

Slip Speed

( )

1.

2. | |

1.

2. || | |

Low Slip

Speed

( )

1.

2. | |

High Slip

Speed

( )

1.

2. | |

Table 5.4: Coefficient of friction data subsets

The subset corresponds to coefficient of friction data when the clutch piston

is at its maximum displacement, and the difference between slip speed and the static-

dynamic threshold speed is greater than zero but less than or equal to the stick

tolerance . The static coefficient of friction is estimated by taking the

average of the subset.

The subset corresponds to coefficient of friction data when the clutch

piston is at its maximum displacement, and the slip speed is greater than but less

than or equal to

. The low to high speed threshold is

131

defined as the slip speed at which the viscous damping coefficient term

becomes non-negligible. The Coulomb coefficient of friction is estimated using

the linear least squares method described by equation (5.15). It’s assumed that

| | , and is the most recent estimate of the static coefficient of

friction.

( )

And,

( (

| |

)

)

( (

| |

)

)

(5.15)

The subset corresponds to coefficient of friction data when the clutch

piston is at 85% of its maximum displacement (i.e., a fluid film is present in the gap

between clutch plates), and the slip speed is greater than . For such

conditions, the viscous damping term is not negligible, and the constant is estimated

using the linear least squares method described by equation (5.16). The oil temperature

is obtained from the simulation, and and are the most recent estimates

of the static and Coulomb coefficients of friction.

( )

And,

| |

( ) ( (

| |

)

)

(5.16)

132

Figure 5.13: Estimated coefficient of friction as a function of slip speed and viscosity

Data from six sequential upshifts (launch through 6th

gear) is used for offline

parameter estimation. The estimation tolerances and are set at

and

, respectively. The estimated parameters summarized in Table 5.5 are used to

generate Figure 5.13; in said figure the coefficient of friction is plotted as a function of

slip speed and temperature-dependent viscosity. The percent error in the estimates of the

static and Coulomb coefficients of friction and the constant are 0.03%, 0.46% and

-0.68%, respectively. Clearly , , and are estimated with enough accuracy to

recognize changes in the slope; given data from thousands of shifts rather

than six, the estimation accuracy may be improved by setting the acceleration tolerance

much closer to zero.

-40 -20 0 20 40

0

0.005

0.010.1

0.11

0.12

0.13

0.14

0.15

0.16

Slip Speed, slip

[rad/s]Viscosity, [Pa*s]

Coeff

icie

nt

of

Friction,

K(

slip

)

133

%Error

%Error

[1/(Pa*rad)]

%Error

Simulated 0.1558 N/A 0.1192 N/A -0.0444 N/A

Estimated 0.1559 0.03 0.1198 0.46 -0.0441 -0.68

Table 5.5: Offline estimation of coefficient of friction parameters

For online estimation, computer memory is allotted for vectors , ,

, , , and of the respective lengths , , and

. The coefficient of friction data is continuously sampled, so each vector element is

populated sequentially. Once the last element is populated, the next data point overwrites

the first element in the vector. For example, is populated with the first points.

Datas points and replaces the values stored in elements 1 and 2,

respectively.

Ideal 0.1192

-0.0444

Case 1 ⁄

Case 2 ⁄

Case 3 ⁄

Case 4 ⁄

Case 5 ⁄

Table 5.6: Simulated coefficient of friction parameters for parameter estimation

To demonstrate that the friction parameter estimation scheme performs well for

ideal and poor friction characteristics, simulations of six sequential upshifts and five

134

downshifts (launch to 6th

gear, 6th

gear to 1st gear) are completed for each set of

parameters listed in Table 5.6. Note that the ideal parameters are identical to those found

in Table 5.3. For cases 1-5, the static coefficient of friction is increased while the

Coulomb coefficient of friction is decreased. Both parameters are scaled relative to their

ideal values, while the constant is fixed at its ideal value. Note that the slope of the

curve is positive for the ideal case and case 1, but becomes increasingly

negative for cases 2-5. The number of points stored in the near-zero slip speed, low slip

speed, and high slip speed vectors are 5, 50000, and 5000, respectively. The stick

tolerance ( ) is set at

and the acceleration tolerance ( ) is set at

. For all six parameter sets, the percent error in the estimate of each friction

parameter is provided in Table 5.7.

%Error

%Error

[1/(Pa*rad)]

%Error

Ideal Simulated 0.1558 N/A 0.1192 N/A -0.0444 N/A

Estimated 0.1588 1.93 0.1168 -2.01 -0.0432 -2.70

Case 1 Simulated 0.1416 N/A 0.1311 N/A -0.0444 N/A

Estimated 0.1444 1.98 0.1323 0.92 -0.0452 1.80


Estimated 0.1318 1.54 0.1407 -1.61 -0.0426 -4.05


Estimated 0.1222 2.00 0.1483 -4.32 -0.0433 -2.48


Estimated 0.1127 1.26 0.1647 -1.32 -0.0456 2.70


Estimated 0.1074 3.36 0.1764 -1.34 -0.0428 -3.60

Table 5.7: Online estimation of varying sets of coefficient of friction parameters

135

The online estimation scheme performs well regardless of the simulated friction

parameters – the percent error is less than 4.4% for all parameters. We note here that the

estimation criteria, primarily the length of the vectors described above, are relaxed so that

the friction parameters may be estimated over the course of 11 shifts. Since in reality

these parameters vary over thousands of shifts rather than 11, the data should be sampled

at a lower frequency and the vector lengths should be orders of magnitude larger for this

scheme to be implemented in a vehicle. Further, the inclusion of additional data points in

the friction parameter estimation would serve to reduce the effect of noise due to the

estimation of acceleration signals using current speeds.

5.3 Conclusion

In this chapter, the friction characteristics of a wet clutch are discussed for new

friction material and automatic transmission fluid. As the material and/or fluid ages, the

clutch may experience stick-slip or shudder when static friction ( ) becomes greater

than Coulomb friction ( ). In addition, the sign of the slope changes from

positive, which is desirable for stable operation and clutch slip control, to negative when

. To recognize changes in friction characteristics, a friction parameter estimation

scheme is developed. The parameter estimation can be completed using stored signals

(offline estimation) or real-time signals (online estimation). Results are provided for

offline and online estimation – for both cases, the friction parameters are estimated with

reasonable accuracy. As discussed in Chapter 6, the friction parameters used in

feedforward clutch slip control are updated based on the estimation results.

136

CHAPTER 6: INTEGRATED POWERTRAIN CONTROL OF CLUTCH SLIP

In Section 6.1 , transmission shift control strategies developed by Hebbale and

Kao [19] and Bai et al. [20], which are reviewed in Chapter 2, are revisited. Other than

their implementation of closed loop clutch slip control in the inertia phase, both methods

rely heavily on open loop control of clutch pressures. An extensive calibration effort is

required for successful implementation of open loop control, and significant deviations in

system and/or control parameters from their respective calibrated values reduce the

effectiveness of the control scheme; to demonstrate that the type of variations in friction

characteristics discussed in Chapter 5 reduce the effectiveness of primarily open loop

control strategies, the approaches presented by Hebbale and Kao and Bai et al. are

implemented with the powertrain model developed in this work. As discussed in Chapter

2, the scheme presented by Hebbale and Kao is adaptive; however, the adaptive portion

of Hebbale and Kao’s strategy is not implemented here.

The development of an integrated powertrain control strategy designed to improve

the smoothness and robustness of clutch engagement is presented in Section 6.2 . During

the torque and inertia phases, the proposed controller utilizes a model-based feedforward

component and closed loop control of clutch slip and engine speed. The simulation

results presented in Section 6.3 demonstrate that the adaptation of the feedforward

controller to changes in friction parameters can lessen the degradation of shift quality.

137

6.1 Transmission shift control strategies from literature

A flow chart of the shift control strategy –for both upshifts and downshifts –

presented by Hebbale and Kao [19] is shown in Figure 6.1.

Figure 6.1: Flow chart of Hebbale and Kao’s shift control strategy

138

Step labels that include an ‘a’ or ‘b’ refer to control actions at the oncoming and offgoing

clutches, respectively.

Once an upshift is commanded, the torque phase of the shift begins with the

filling of the oncoming clutch (step 1a). Note that the fill time and fill pressure are

calibrated as a function of clutch speed and transmission fluid temperature.

Simultaneously, the offgoing pressure is exponentially reduced at a predetermined rate so

that a target pressure is reached by the end of the fill phase (step 1b). During step 2a the

oncoming pressure is reduced to a level slightly above the pressure required to compress

the clutch spring after which it is ramped up in an open loop fashion. The pressure ramp

time and target pressure are calibrated as function of the input torque and the oncoming

gear ratio; by the end of step 2a, the oncoming pressure should be sufficient for

transmission of the full input torque. As the oncoming pressure is ramped up, the

offgoing pressure is stepped down once a dip in input or output acceleration is detected

(step 2b). The output acceleration is continuously monitored; when the acceleration falls

to a target value predetermined as a function of oncoming and offgoing gear ratios, the

offgoing pressure is exponentially reduced to zero (step 3b). The torque phase is

completed once a change in gear ratio is detected.

Figure 6.2: Feedback control of clutch slip using Simulink’s built-in PID controller with

anti-windup

139

Closed loop control of the oncoming clutch slip is initiated at the beginning of the

inertia phase (4a). Using Simulink’s built-in PID controller shown in Figure 6.2 and

described by equation (6.1), the oncoming clutch pressure is manipulated to control

clutch slip.

( )(

) ( )

And,

(6.1)

Here, is the clutch pressure command generated by the PID controller and

is the slip speed error. , , and are the proportional, integral, and derivative gains,

respectively. Note that the controller gains are defined generically to illustrate the

structure of the PID controller; in practice each gain is tuned specifically for a given

control loop. For each loop, the following steps are taken to manually tune the controller

gains:

1. is selected so that an acceptable rise time is achieved – and remain

zero.

2. is selected so that steady state error is eliminated. is fixed at the value

determined in step 1 and remains zero.

3. is selected to achieve an appropriate overshoot and settling time. and

are fixed at their values determined in steps 1 and 2, respectively.

140

4. Each gain is varied – within +/- 10% of the values determined in steps 1-3 – until

an acceptable rise time, settling time, overshoot, and steady state error is

achieved.

To eliminate large variations in the control signal due to high frequency components of

the error signal, the derivative control action is filtered with a low-pass filter. The

derivative control action approximates ideal derivative control (

) for low

frequencies, and acts a constant gain (

) for high frequencies. The

break frequency ( ) is selected to be ~10% greater than the highest acceptable

frequency observed in the error derivative. The desired slip trajectory, , is

described by:

(

) (6.2)

is the clutch slip speed when the slip control is initiated. , defined as the

duration of the clutch slip control, is limited by the rise time ( ). The rise time is

selected based on shift durations found in literature. Walker et al. [15] and Goetz [12]

present simulations of clutch-to-clutch shifts with shift durations in the range of 300ms-

450ms and 500ms-700ms, respectively. In both cases, the shift durations are much longer

than the ~200ms given in the Audi self-service manual [38]. Recognizing that the shift

time given in the Audi manual is averaged over many shifts, the rise time is chosen so

that the duration of the inertia phase is between 200ms and 300ms – the longer inertia

phases occur when the transmission is in the lower gears. Thus with torque phase

durations between 50ms and 100ms, the overall shift is completed within the range of

141

250ms-400ms. The oncoming gear ratio is achieved once the slip trajectory is reduced to

zero, after which the closed loop clutch slip control is terminated and the oncoming

pressure is ramped up to maximum pressure (5a). The inertia phase, and in turn the

upshift, is completed when the clutch pressure is maximum.

The anti-windup method based on integrator clamping is implemented in the

closed loop slip control described above and the Simulink block diagram of said method

is displayed in Figure 6.3.

Figure 6.3: PID controller with integrator clamping

Integrator clamping prevents the integral of the error from building to extremely high

levels, which leads to increased controller response time, by setting the integral term to

zero when the calculated controller output exceeds the capabilities of the actuator. For the

clutch slip controller described here, the integrator clamps if the pressure commanded by

142

the clutch slip controller is greater than the absolute value of the maximum clutch

pressure.

After a downshift is commanded, the inertia phase of the shift begins with the

filling of the oncoming clutch (step 1a). The calibration of the fill parameters is similar to

that of the upshift case. During the fill phase, the offgoing pressure is dropped to a target

value then ramped down until a change in gear ratio is detected (step 1b). Again, the

slope of the offgoing pressure trajectory is predetermined so that a target pressure is

achieved by the end of the fill phase. Using the PID controller described above, the

offgoing clutch pressure is manipulated to control oncoming clutch slip (step 2b). The

inertia phase is completed once the oncoming gear ratio is achieved. During the torque

phase (steps 3a and 3b), the oncoming pressure is ramped up to maximum and the

offgoing pressure is reduced to zero. The torque phase, and in turn the downshift, is

completed when the clutch pressure is maximum.

Figure 6.4 displays a flow chart of the shift control strategy –for both upshifts and

downshifts –presented by Bai et al. [20]. Again, step labels that include an ‘a’ or ‘b’

refer to control actions at the oncoming and offgoing clutches, respectively. The torque

phase of an upshift begins with the oncoming clutch fill phase (step 1a). As with Hebbale

and Kao’s control strategy, the fill time and fill pressure are calibrated parameters. The

offgoing pressure is stepped down to just above the holding pressure, which is the

minimum pressure required to prevent the clutch from slipping (step 1b). During step 2a

the oncoming pressure is reduced to a level slightly above the pressure required to

compress the clutch spring after which the pressure is ramped up until the clutch can

carry the full input torque. As the oncoming pressure is ramped up, the offgoing pressure

143

command is stepped down to the holding pressure (step 2b) and reduced at a rate

proportional to the rate of increase of the oncoming pressure. This approach

electronically represents the “washout” function of the modified pressure control

solenoids used in the work presented by Bai et al. The washout gain, defined as the

desired ratio of the rates of change of the offgoing clutch pressure to the oncoming clutch

pressure [20], is a calibrated parameter. The torque phase is completed once a change in

gear ratio is detected.

Closed loop control of the oncoming clutch slip is initiated at the beginning of the

inertia phase (step 3a). Using the PID controller described above, the oncoming clutch

pressure is manipulated to control oncoming clutch slip: the slip trajectory is given by

equation (6.2). Once the oncoming gear ratio is achieved, the clutch slip control is

terminated, the oncoming pressure is stepped up to maximum pressure, and the offgoing

pressure is reduced to zero (steps 4a and 4b). The inertia phase, and accordingly the

upshift, is completed when the clutch pressure is at its maximum.

The inertia phase of a downshift begins with the filling of the oncoming clutch

(step 1a); concurrently, the offgoing pressure is ramped down until a change in gear ratio

is detected (step 1b). The calibration of the fill parameters is similar to that of the upshift

case. During step 2a, the oncoming pressure is reduced to, and held at, a level slightly

above the pressure required to compress the clutch spring, and closed loop control of the

offgoing clutch slip is initiated during step 2b. Again, the PID controller described above

is utilized in the slip control. The inertia phase is completed once the oncoming gear ratio

is achieved. During the torque phase (steps 3a and 3b), the oncoming pressure is stepped

144

up to maximum and the offgoing pressure is reduced to zero. The torque phase, and in

turn the downshift, is completed when the clutch pressure is maximum.

Figure 6.4: Flow chart of shift control strategy presented by Bai et al.

145

6.2 Proposed integrated powertrain controller

The control strategy proposed in this work utilizes concepts from Goetz [12] and

Bai et al [25]. Figure 6.5 and Figure 6.6 display flow charts of Goetz’s control strategy

for upshifts and downshifts, respectively. For both the upshift and downshift case, Goetz

labels the offgoing and oncoming clutches as C1 and C2, and a standard PID controller

with anti-windup is utilized in all feedback control loops.

Figure 6.5: Goetz’s control strategy for an upshift [12]

The torque phase of the upshift begins by reducing the pressure at C1 to a

pressure level that barely maintains clutch engagement (step 1). The target pressure in

step 1 is calibrated as a function of transmission output torque and the current

transmission gear. We note here that Goetz assumes transmission output torque is a

146

measurable signal; however, due to the additional cost and limited durability, torque

sensors have not been included in production transmissions. By applying the method

described in Chapter 5, the suggested calibration may be completed by using measured

speeds and estimated accelerations to calculate transmission output torque. Concurrent

with step 1, C2 is filled (step 2). To allow the pressures at C1 and C2 to steady, a time

delay of 0.05 seconds is imposed. During step 3, the clutch slip control of C1 is activated;

the offgoing clutch pressure is manipulated to maintain a slip speed of 5 rad/s at the

offgoing clutch. Simultaneously, the pressure at C2 is ramped up (step 4). To maintain

the 5 rad/s slip speed, the clutch slip controller reduces the pressure at C1. The torque

phase is completed when a change in the sign of clutch slip speed is detected. At this

point, the clutch slip control of C1 is deactivated (step 5).

The inertia phase of the upshift begins by activating the engine speed controller

(step 6), where both throttle angle and spark advance are manipulated so that engine

speed tracks the semi-cosine profile defined by equation (6.3).

(

) (6.3)

Here, is the reference engine speed, is the engine speed when the control

is initiated, and and are the prescribed and actual durations of the engine speed

control, respectively. Goetz proposes that the engine deceleration can be completed

solely by the engine speed controller, so the oncoming pressure is held constant until the

end of the inertia phase. Once C2 engages and the oncoming gear ratio is achieved, the

engine speed controller is deactivated (step 8) and the pressure at C2 is stepped up to line

pressure (step 9). At this point, the inertia phase, as well as the upshift, is completed.

147

Figure 6.6: Goetz’s control strategy for a downshift [12]

The inertia phase of a downshift begins by reducing the pressure at C1 in a similar

manner to step 1 of the torque phase of an upshift. During step 2, the throttle angle is

increased after which a 0.05 second delay is imposed. The engine speed controller is

activated during step 3. Assisted by the increased throttle angle, the pressure at C1 is

manipulated so that the engine speed follows the following reference trajectory:

(6.4)

where is the desired engine acceleration. Prior to end of the inertia phase, the

oncoming clutch is filled and the throttle angle is decreased to 10 degrees higher than the

initial angle at the start of the inertia phase (steps 4 and 5). The inertia phase is

148

completed once the oncoming gear ratio is achieved; at this time, the engine speed

controller is deactivated (step 6).

Similar to the upshift case, the torque phase of a downshift begins with the

activation of the clutch slip controller at C1 (step 7); the offgoing clutch pressure is

manipulated so that the offgoing clutch slip speed is maintained at 5 rad/s. During step 8,

the pressure at C2 is ramped up, forcing a reduction in the pressure at C1. The torque

phase is completed when the pressure at C1 is reduced to zero and the offgoing clutch

slip changes signs; at this point, the clutch slip controller is deactivated and the pressure

at C2 is stepped up to line pressure (steps 9 and 10).

A flow chart of the integrated powertrain control strategy – for both upshifts and

downshifts – proposed in this work is shown in Figure 6.7. Step labels that include an ‘a’,

‘b’, or ‘c’ refer to control actions at the oncoming clutch, offgoing clutches, and engine,

respectively. There are many control actions common to both Goetz’s and the proposed

strategy, and these are not repeated here. The modified and/or additional actions taken in

the proposed strategy are given as follows: the end of the clutch fill phase is identified by

comparing the measured clutch pressure to the pressure required to compress the clutch

spring and by registering when the derivative of clutch pressure increases beyond the

clutch spring rate; in all phases of an upshift (steps 3b and 4a) and downshift (steps 2b

and 3b), a portion of either the oncoming or offgoing clutch pressure command is

generated using model-based feedforward control; during the torque phase of an upshift

(step 2c), the throttle angle and spark advance are manipulated to increase engine torque

with the goal of eliminating the torque hole [25]; during the inertia phase of an upshift

(step 4a), the oncoming clutch pressure is manipulated to assist in engine deceleration;

149

and during the torque phase of a downshift (step 3b), the offgoing clutch is smoothly

released rather than controlling the slip speed to maintain a constant value.

Figure 6.7: Flow chart of proposed integrated powertrain control strategy

150

Figure 6.8: Top level schematic of the proposed integrated powertrain controller

151

The top level schematic of the proposed integrated powertrain controller is shown

in Figure 6.8. The clutch pressure and engine speed controller contains and executes the

control strategy for upshifts and downshifts shown in Figure 6.7. This supervisory

controller determines when feedforward and/or feedback control should be active,

provides open loop commands when neither feedforward or feedback control is required,

and generates the desired reference inputs for all components of the control. The inputs to

this controller are: the commanded gear, the states of both clutches, the pressure at each

clutch, the speeds of the engine, the clutch hub, both clutches, and the driveshaft, the

estimated derivative of pressure at each clutch, and the pressure commands for both

clutches generated by the feedforward and feedback controllers. The outputs from this

controller are: the reference inputs (speeds at the engine, the clutch hub, and both

clutches, and the oncoming, offgoing, and differential gear ratios), the measured gear

ratio (

), the open loop throttle angle and spark advance commands, and the total

pressure commands for both clutches.

The pressure commands generated by the clutch pressure and engine speed

controller are converted via a lookup table to duty cycle commands to the clutch pressure

control solenoids. The data points included in said lookup table are displayed in Figure

6.9. For pressures within the range of 0-18.2 bar, duty cycles are calculated by linear

interpolation. If the pressures are outside this range, the duty cycle is clipped at 0 and 1.

152

Figure 6.9: Pressure control solenoid, duty cycle versus clutch pressure data points used

in lookup table

Figure 6.10: Feedback control of clutch pressure using Simulink’s built-in PID controller

A minor pressure control loop using Simulink’s built-in PID controller is shown

in Figure 6.10. This loop is implemented to ensure that the actual clutch pressure tracks

the commanded clutch pressure. The duty cycle command generated by the control loop

( ) is calculated using the following expression:

153

( ) (

) (6.5)

where is the commanded pressure at each clutch.

The proposed controller contains five feedback control loops in addition to the

two minor pressure control loops. They are: a clutch slip control loop and engine speed

control loop for each clutch, where clutch pressure is the manipulated variable (four

separate loops); and an engine speed control loop, where spark advance and throttle angle

are the manipulated variables.

The clutch slip control loops used in the proposed controller are of the same

structure shown in Figure 6.2 and Figure 6.3, and the PID controllers are represented by

equation (6.1). The reference slip speed varies by shift phase. During the torque and

inertia phases of an upshift, the reference slip speed is set to a constant 5 rad/s and the

trajectory described by equation (6.2), respectively. During the inertia phase of a

downshift, the reference slip speed increases from zero with a slope equal to that of the

engine reference speed given by equation (6.4). The engine speed control loops are

shown in Figure 6.11, where clutch pressure is manipulated, and Figure 6.12, where

throttle angle and spark advance are both manipulated. The engine reference speeds are

given by: equation (6.4), for the torque phase of an upshift and the inertia phase of a

downshift; and equation (6.3), for the inertia phase of an upshift. The expressions for all

three PID controllers are given by the following equations:

( )(

) (6.6)

154

( )(

) (6.7)

( )(

) (6.8)

where , , are the manipulated clutch pressure, spark advance, and

throttle angle. We note here that the anti-windup method based on integrator clamping is

used with each of the engine speed PID controllers.


with anti-windup (manipulation of clutch pressure)


with anti-windup (manipulation of throttle angle and spark advance)

155

The feedforward controller is developed using a simplified version of the

powertrain described in Chapter 3. To simplify said powertrain model, the following

assumptions are made: the input, output, and differential shafts, as well as the connection

between the engine crankshaft, flywheel and clutch hub, are considered stiff; the road is

nearly flat ( ); and tire slip is considered to be negligible. As a result of these

assumptions, inertias and damping coefficients for multiple elements are lumped at the

engine crankshaft, both input shafts, and the differential. A schematic of the simplified

powertrain is shown in Figure 6.13 and the simplified rotational and vehicle dynamics are

described by the following set of equations:

(6.9)

(6.10)

(6.11)

(6.12)

(6.13)

(6.14)

(

) (6.15)

156

Figure 6.13: Simplified rotational dynamics for feedforward controller development

Figure 6.14: Generalized model inversion for calculation of feedforward clutch pressure

157

An expression for model-based feedforward pressure is obtained by inverting

equations (6.9)-(6.15) and using the static friction model described by equation (5.14).

The inputs to the inverted model are engine torque, engine and clutch slip speeds, and

engine and clutch slip accelerations. Here, engine torque is estimated using a two-

dimensional lookup table, where engine torque is mapped as a function of engine speed

and open loop throttle command. Recalling from Figure 6.7, the feedforward controller is

active at the offgoing clutch and oncoming clutch during the respective torque and inertia

phases of an upshift, and is active at the offgoing clutch during both phases of a

downshift; thus, the expressions used in the model inversion vary with the phase of the

shift (torque phase or inertia phase) and the clutch order in the shift (clutch #1 – clutch #2

or clutch #2 – clutch #1), but the inversion process is the same for all phases and shifts. A

flow chart of the general inversion process is given by Figure 6.14.

To demonstrate the model inversion process, the expressions defining

feedforward pressure at clutch #1 (offgoing clutch during the torque phase) and clutch #2

(oncoming clutch during the inertia phase) are developed for an upshift from 1st – 2

nd

gear. For an upshift from 1st – 2

nd gear, equations (6.9)-(6.15) are further simplified by

recognizing that only two output gears (#1 and #2), and output shaft #1, are active during

said shift. Thus, equation (6.12) reduces to:

(6.16)

In addition, the dynamics of output shaft #2, which is described by equation (6.13), and

the torque transmitted through said shaft ( ) may be ignored. Of course, these terms

would not be ignored for a shift between 4th

and 5th

gear.

158

Feedforward pressure at clutch #1 during the torque phase of a 1-2 upshift

The inversion process to find the feedforward pressure at clutch #1 ( )

during the torque phase of an upshift from 1st-2

nd gear begins by reflecting equations

(6.11) and (6.14) to output shaft #1. The resulting equation is given by equation (6.17).

(

)

(

)

(6.17)

To replace , equation (6.17) is reflected to equation (6.10):

(6.18)

where

(

)

(6.19)

(

)

(6.20)

Here, and are the inertia and damping coefficient, respectively, lumped at the

offgoing input shaft (input shaft #1). The subscript TP12 refers to shift phase (torque

phase of a 1-2 upshift). Assuming, for the sake of simplicity, that ,

and subtracting equation (6.9) from equation (6.18) yields:

(

) (

)

(

) (

)

(6.21)

159

where

(6.22)

(6.23)

and are the inertia and damping coefficient, respectively, lumped at the engine

crankshaft. From equation (6.21), the following expression for is obtained:

(

) [(

) (

)

(

)

]

(6.24)

And,

(

( )

) (6.25)

For a static friction model, the feedforward pressure at clutch #1 is expressed as:

( )

(6.26)

where is calculated using equation (6.24).

Feedforward pressure at clutch #2 during the inertia phase of a 1-2 upshift

The inversion process to find the feedforward pressure at clutch #2 ( )

during the inertia phase of an upshift from 1st-2

nd gear starts by reflecting equations

(6.10) and (6.14) to output shaft #1. The resulting equation is described by equation

(6.27).

160

(

)

(

)

(6.27)

To replace , equation (6.27) is reflected to equation (6.11):

(6.28)

where

(

)

(6.29)

(

)

(6.30)

and are the inertia and damping coefficient, respectively, lumped at the

oncoming input shaft (input shaft #2). The subscript IP12 refers to the shift phase (inertia

phase of a 1-2 upshift). Subtracting equation (6.9) from equation (6.28), using the

expression to replace , and solving for yields:

(

) [(

) (

)

(

)

]

(6.31)

And,

(

( )

) (6.32)

Using a static friction model, the feedforward pressure at clutch #2 is expressed as:

161

( )

(6.33)

where is calculated using equation (6.31).

Limitations of the feedforward controller

The effectiveness of the feedforward controller is limited by the accuracy of the

engine torque estimation and the static friction model, as well as the simplifications made

to the rotational dynamic model. The engine torque used in the feedforward model, which

is calculated using a torque map with inputs of engine speed and open loop throttle

command, instantaneously changes with varying inputs; however, the simulated mean-

value engine model accounts for delays in engine torque production due to the air intake

and spark ignition processes. Thus, the engine torque used in the feedforward model

leads the simulated engine torque during transient conditions, and accordingly, causes

feedforward pressure, which is proportional to engine torque, to lead simulated clutch

pressure.

Recalling from Section 3.3.4 , the static clutch friction model, compared to the

dynamic model of wet clutch friction, is least accurate for low-energy and mid-energy

clutch engagements. The static model’s inaccuracy during said types of engagement is

attributed to the model’s inability to describe the delay and overshoot in clutch friction

torque caused by viscous effects [8]. It is reasonable to conclude that viscous effects

cause a similar effect during the low-energy and mid-energy (high clutch pressure and

low slip speed) disengagement of a wet clutch. Since the majority of the pressure

162

command during the inertia phase of a downshift comes from the feedforward pressure,

the commanded pressure tends to lead the simulated pressure during this phase.

The feedforward controller is further limited by the assumption that all compliant

elements in the driveline are stiff. Without the damping provided by the compliant shafts

and flywheel, the feedforward pressure is highly sensitive to oscillations in the speed and

acceleration inputs to the feedforward model. To attenuate these oscillations, both types

of inputs must be filtered using a low-pass filter. The filtering process can lead to an

offset between the filtered and unfiltered signals, and in turn, the feedforward and

simulated clutch pressures. Overall, the method of calculation of engine and clutch

friction torques, as well as the assumptions made in the development of the feedforward

model, cause an offset and lag between the feedforward and simulated clutch pressures.

6.3 Simulation results

The transmission shift controllers developed by Hebbale and Kao and Bai et al.,

as well as the integrated powertrain shift controller described in Section 6.2 , are

implemented with the powertrain model described in Chapter 3. To test the robustness of

each controller to variations in friction characteristics, and in the extreme case the change

in the sign of the slope of the curve, simulations are completed with the

Coulomb and static coefficients of friction presented in Table 6.1.

Recalling from Chapter 5, Tersigni et al. of Afton Chemical produced coefficient

of friction versus slip speed data for new and aged friction material and transmission

fluid. As before, the friction parameters corresponding to new material and fluid are

referred to as ideal. For case 1, the Coulomb coefficient of friction is reduced from its

163

ideal value by a factor of 1.1 and the static friction term is increased by a factor of 1.1.

The resulting curve, which still has a positive slope, resembles the

curves for the aged and end-of-test cases shown in Figure 5.2. Cases 2-5, where the

slope becomes more negative with each case, are extreme examples of clutch

wear and fluid aging that will be used to test the robustness of each controller.

Ideal 0.1192

-0.0444

Case 1 ⁄

Case 2 ⁄

Case 3 ⁄

Case 4 ⁄

Case 5 ⁄

Table 6.1: Simulated coefficient of friction parameters for controller comparison

Figure 6.15 and Figure 6.16 show driveshaft accelerations for the control

strategies proposed by Hebbale and Kao and Bai et al., respectively. The three traces on

each plot are results of simulations using ideal coefficients of friction, and the values

corresponding to cases 2 and 5. On each plot, the driveshaft acceleration transient

responses due to vehicle launch, upshifting from 1st – 2

nd gear, and downshifting from 2

nd

– 1st gear are labeled with green, blue, and black boxes. For the data contained in each

box, maximum peak-to-peak and root-mean-square accelerations are calculated; the peak-

to-peak value indicates the jerkiness of the shift and the root-mean-square value provides

insight to how well the driveline attenuates acceleration transients. To demonstrate how

164

acceleration transients at the driveshaft correlate to quantities that vehicle passengers

directly feel, peak-to-peak values of vehicle acceleration and jerk in the longitudinal

direction are calculated for an upshift from 1st-2

nd gear and a downshift from 2

nd-1

st gear.

Figure 6.15: Driveshaft accelerations for the ideal case, case 2 and case 5: Hebbale and

Kao’s strategy, launch-1st-2

nd-1

st

Figure 6.16: Driveshaft accelerations for the ideal case, case 2 and case 5: strategy

proposed by Bai et al., launch-1st-2

nd-1

st

0 0.5 1 1.5 2 2.5 3-2000

-1500

-1000

-500

0

500

1000

1500

2000

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]

Launch transients

Upshift 1-2 transients

Downshift 2-1 transients

Ideal

Case 2: Minor Negative Slope

Case 5: Severe Negative Slope

0 0.5 1 1.5 2 2.5 3-2000

-1500

-1000

-500

0

500

1000

1500

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]

Launch transients Upshift 1-2 transients


Ideal

Case 2: Minor Negative Slope

Case 5: Severe Negative Slope

165

Table 6.2 and Table 6.4 list the peak-to-peak and root-mean-square driveshaft

accelerations observed during launch, a 1-2 upshift, and a 2-1 downshift, as well as the

percent difference of said quantities from their ideal values, for the controllers developed

by Hebbale and Kao and Bai et al., respectively. For the same controllers, Table 6.3 and

Table 6.5 list the peak-to-peak values of the vehicle acceleration and jerk observed during

a 1-2 upshift and 2-1 downshift, as well as the percent difference of said quantities from

their ideal values. As expected, the driveshaft acceleration transients are harshest for case

5. For Hebbale and Kao’s controller, the ma imum percent difference in peak-to-peak

and root-mean-square driveshaft accelerations is 206.61% and 97.64%, respectively. For

the controller developed by Bai et al., the maximum percent differences are even higher;

the peak-to-peak and root-mean-square driveshaft accelerations are 306.35% and

226.58% larger than ideal values. The vehicle acceleration transients are also harshest for

case 5; for this case, the maximum percent difference in peak-to-peak vehicle

acceleration and jerk are 3.93% and 120.21% (Hebbale and Kao) and 10.62% and

133.66% (Bai et al.). As anticipated, shifts become more uncomfortable for passengers in

the vehicle as the slope becomes more negative.

Both controllers are unable to maintain the shift quality demonstrated by the ideal

case because of their reliance on calibrated clutch pressure commands. As the slope of

the curve becomes more negative, and the calibrated portion of the pressure

command remains unchanged, small variations in error input to the closed loop clutch

slip controller cause the clutch to experience stick-slip and shudder. Rapid clutch

engagement and disengagement leads to large oscillations in the driveshaft acceleration,

166

and given the lightly damped driveline modeled in this work, these oscillations are not

suppressed very effectively for cases 2 and 5.

Peak-to-

Peak (PP)

[

]

% Difference

from Ideal

PP Value

Root-Mean-Square

(RMS) [

]

% Difference from

Ideal RMS Value

Launch - - - -

Ideal 1116.99 0.00 94.47 0.00

Case 2 1194.15 6.91 94.83 0.38

Case 5 1195.55 7.03 97.37 3.07

1-2

Upshift - - - -

Ideal 885.22 0.00 80.83 0.00

Case 2 1215.04 37.26 94.93 17.45

Case 5 2714.23 206.61 159.74 97.64

2-1

Downshift - - - -

Ideal 2457.81 0.00 271.62 0.00

Case 2 1732.08 -29.53 202.91 -25.30

Case 5 2785.87 13.35 281.63 3.68


case 2 and case 5: Hebbale and Kao’s strategy, launch-1st-2

nd-1

st

PP of Vehicle

Acceleration

[

]

% Difference

from Ideal PP

Value

PP of Vehicle

Jerk [

]

% Difference

from Ideal PP

Value

1-2 Upshift - - - -

Ideal 6.46 0.00 220.65 0.00

Case 2 6.49 0.47 222.72 0.94

Case 5 6.71 3.93 485.90 120.21

2-1 Downshift - - - -

Ideal 11.63 0.00 370.96 0.00

Case 2 11.63 0.00 369.59 -0.37

Case 5 11.63 0.00 390.72 5.33


5: Hebbale and Kao’s strategy, launch-1st-2

nd-1

st

167

Peak-to-

Peak (PP)

[

]

% Difference

from Ideal PP

Value

Root-Mean-

Square (RMS)

[

]

% Difference from

Ideal RMS Value

Launch - - - -

Ideal 1305.24 0.00 109.49 0.00

Case 2 1122.26 -14.02 100.08 -8.59

Case 5 1184.95 -9.22 97.76 -10.71

1-2

Upshift - - - -

Ideal 1017.14 0.00 78.61 0.00

Case 2 861.95 -15.26 77.59 -1.30

Case 5 1667.74 63.96 114.66 45.86

2-1

Downshift - - - -

Ideal 745.38 0.00 61.45 0.00

Case 2 1615.43 116.73 121.10 97.06

Case 5 3028.85 306.35 200.70 226.58


case 2 and case 5: strategy proposed by Bai et al., launch-1st-2

nd-1

st

PP of Vehicle

Acceleration

[

]

% Difference

from Ideal PP

Value

PP of Vehicle

Jerk [

]

% Difference

from Ideal PP

Value

1-2 Upshift - - - -

Ideal 6.37 0.00 157.79 0.00

Case 2 6.44 1.16 152.62 -3.28

Case 5 6.44 1.17 224.05 41.99

2-1 Downshift - - - -

Ideal 8.38 0.00 169.35 0.00

Case 2 7.05 -15.86 258.42 52.60

Case 5 9.27 10.62 395.69 133.66


5: strategy proposed by Bai et al., launch-1st-2

nd-1

st

168

Figure 6.17-Figure 6.21 show driveshaft accelerations for the proposed integrated

powertrain control strategy. The first trace is a result of the simulation of ideal

coefficients of friction. The remaining two traces are results of simulations of non-ideal

friction parameters; the second trace corresponds to simulations using the feedforward

controller in its non-adaptive mode (ideal coefficients of friction are still input to the

feedforward model), and the third trace corresponds to simulations using the non-ideal

friction parameters as inputs to the feedforward model. In this mode, the feedforward

controller is considered adaptive. Again, the driveshaft acceleration transient responses

due to vehicle launch, upshifting from 1st – 2

nd gear, and downshifting from 2

nd – 1

st gear

are labeled with green, blue, and black boxes.



nd-1

st

0 0.5 1 1.5 2 2.5 3-3000

-2000

-1000

0

1000

2000

3000

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]

Launch transientsUpshift 1-2 transients


Ideal

Case 1: No FF Adaptation

Case 1: FF Adaptation

169



nd-1

st



nd-1

st

0 0.5 1 1.5 2 2.5 3-3000

-2000

-1000

0

1000

2000

3000

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]



Ideal



0 0.5 1 1.5 2 2.5 3-3000

-2000

-1000

0

1000

2000

3000

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]



Ideal



170



nd-1

st



nd-1

st

0 0.5 1 1.5 2 2.5 3

-3000

-2000

-1000

0

1000

2000

3000

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]

Launch transients

Upshift 1-2 transients


Ideal



0 0.5 1 1.5 2 2.5 3-3000

-2000

-1000

0

1000

2000

3000

Time [s]

Dri

ves

haft

Acc

eler

ati

on,

_!

D[r

ad/s]

Launch transientsUpshift 1-2 transients


Ideal



171

Peak-to-

Peak (PP)

[

]

% Difference from

Ideal PP Value

Root-Mean-

Square (RMS)

[

]

% Difference from

Ideal RMS Value

Launch - - - -

Ideal 1664.00 0.00 129.34 0.00

Case 1 1538.38 -7.55 120.31 -6.98

Case 2 1453.23 -12.67 114.56 -11.43

Case 3 1396.57 -16.07 114.60 -11.40

Case 4 1371.04 -17.61 110.84 -14.30

Case 5 1302.55 -21.72 107.97 -16.52


and cases 1-5: proposed strategy, launch-1st-2

nd-1

st

Table 6.6-Table 6.8 list the peak-to-peak and root-mean-square accelerations

observed during launch, a 1-2 upshift, and a 2-1 downshift, respectively, as well as the

percent difference of said quantities from their ideal values. For cases 1-3 and case 5, the

adaptation of the feedforward controller to changes in friction parameters during the

upshift operation resulted in lower percent differences, when compared to the non-

adaptive cases, in peak-to-peak and root-mean-square driveshaft accelerations. This trend

is not as clearly seen for downshifts, as the adaptation of the feedforward controller

improved the attenuation of driveshaft acceleration transients in just two out of five cases

(cases 1 and 4). For the remaining cases, adaptation of the feedforward controller resulted

in approximately the same or slightly larger peak-to-peak and root-mean-square

driveshaft accelerations. This inconsistency is due to a 1-4 bar offset and slight lag

between the actual pressure at which the clutch begins to slip and the feedforward

pressure that is observed during the inertia phase of the downshift.

As shown by Figure 6.22, this offset in pressure is observed even for simulation

of ideal friction. The PID gains for the clutch slip feedback loop are tuned to compensate

172

for this pressure error. As the slope of the curve became more negative, the

PID gains for the clutch slip feedback loop are no longer appropriate and the PID

controller begins to overcompensate for small slip speed errors. The feedback controller

generates much larger pressure commands than required, which as shown in Figure 6.23,

results in the clutch engaging and disengaging multiple times before fully releasing, and

in turn, harsher acceleration transients.

Peak-to-

Peak (PP)

[

]

% Difference

from Ideal PP

Value

Root-Mean-

Square

(RMS) [

]

% Difference

from Ideal RMS

Value

Ideal 365.20 0.00 38.42 0.00

Case 1 - No

FF Adaptation 552.25 51.22 48.95 27.40

Case 1 - FF

Adaptation 304.42 -16.64 35.18 -8.42

Case 2 - No

FF Adaptation 1014.25 177.72 75.97 97.74

Case 2 - FF

Adaptation 463.88 27.02 50.73 32.05

Case 3 - No

FF Adaptation 1333.75 265.21 82.49 114.71

Case 3 - FF

Adaptation 459.02 25.69 39.55 2.96

Case 4 - No

FF Adaptation 2100.80 475.25 159.86 316.10

Case 4 - FF

Adaptation 3571.32 877.91 292.56 661.49

Case 5 - No

FF Adaptation 1865.56 410.83 107.36 179.45

Case 5 - FF

Adaptation 761.44 108.50 62.08 61.60


and cases 1-5 with/without feedforward adaptation: proposed strategy, upshift from 1st-

2nd

173

Peak-to-

Peak (PP)

[

]

% Difference

from Ideal PP

Value

Root-Mean-

Square

(RMS) [

]

% Difference

from Ideal RMS

Value

Ideal 2698.86 0.00 269.34 0.00

Case 1 - No

FF Adaptation 2570.40 -4.76 259.36 -3.71

Case 1 - FF

Adaptation 2473.51 -8.35 223.47 -17.03

Case 2 - No

FF Adaptation 2509.13 -7.03 249.62 -7.32

Case 2 - FF

Adaptation 2699.87 0.04 228.37 -15.21

Case 3 - No

FF Adaptation 3585.97 32.87 231.97 -13.87

Case 3 - FF

Adaptation 3625.60 34.34 240.17 -10.83

Case 4 - No

FF Adaptation 4808.18 78.16 299.15 11.07

Case 4 - FF

Adaptation 3748.77 38.90 306.00 13.61

Case 5 - No

FF Adaptation 3809.43 41.15 281.14 4.38

Case 5 - FF

Adaptation 3826.70 41.79 330.44 22.69


and cases 1-5 with/without feedforward adaptation: proposed strategy, downshift from

2nd

– 1st

174

Figure 6.22: Simulation of ideal friction parameters, actual and feedforward pressures at

clutch #2: proposed controller, inertia phase of 2-1 downshift

Figure 6.23: Simulation of ideal friction parameters, clutch #2 slip speed: proposed

controller, inertia phase of 2-1 downshift

175

PP of Vehicle

Acceleration

[

]

% Difference

from Ideal

PP Value

PP of Vehicle

Jerk [

]

% Difference

from Ideal

PP Value

Ideal 4.23 0.00 139.04 0.00

Case 1 - No FF

Adaptation 3.86 -8.86 145.94 4.96

Case 1 - FF

Adaptation 4.09 -3.37 149.15 7.27

Case 2 - No FF

Adaptation 4.33 2.39 139.23 0.14

Case 2 - FF

Adaptation 4.22 -0.35 150.55 8.28

Case 3 - No FF

Adaptation 4.57 8.05 167.97 20.81

Case 3 - FF

Adaptation 4.22 -0.25 141.40 1.70

Case 4 - No FF

Adaptation 4.74 11.96 307.14 120.90

Case 4 - FF

Adaptation 4.95 17.06 422.93 204.17

Case 5 - No FF

Adaptation 5.00 18.17 219.83 58.10

Case 5 - FF

Adaptation 4.20 -0.59 170.11 22.35


feedforward adaptation: proposed strategy, upshift from 1st – 2

nd

Peak-to-peak values of vehicle acceleration and jerk, which are used to quantify

how harsh shift events feel to a passenger in a vehicle, are listed in Table 6.9-Table 6.10.

Both peak-to-peak values, as well as the differences of said quantities from their ideal

values, are provided for a 1-2 upshift and 2-1 downshift. Unfortunately, the same

conclusions about feedforward controller performance that are drawn from the

attenuation of driveshaft acceleration transients are not as evident when comparing

vehicle acceleration and jerk because the vehicle body and the compliant axles, which

176

can be thought of as mass-spring-damper system, filter the driveshaft acceleration

transients. Even though the PID gains for the clutch slip feedback loop are tuned to

achieve smooth driveshaft accelerations, the vehicle body may resonate at different

frequencies. Thus to improve the quality of the shift as perceived by passengers in the

vehicle, the PID gains should be tuned to achieve smooth vehicle acceleration and jerk.

PP of Vehicle

Acceleration

[

]

% Difference

from Ideal PP

Value

PP of Vehicle

Jerk [

]

% Difference

from Ideal PP

Value

Ideal 9.02 0.00 2.20E-4 0.00

Case 1 - No FF

Adaptation 8.49 -5.86 2.31E-4 4.91

Case 1 - FF

Adaptation 8.33 -7.65 2.25E-4 2.37

Case 2 - No FF

Adaptation 8.01 -11.18 2.35E-4 6.63

Case 2 - FF

Adaptation 8.37 -7.17 2.26E-4 2.60

Case 3 - No FF

Adaptation 8.23 -8.74 2.21E-4 0.34

Case 3 - FF

Adaptation 7.89 -12.49 2.24E-4 1.94

Case 4 - No FF

Adaptation 8.59 -4.77 2.33E-4 5.71

Case 4 - FF

Adaptation 8.35 -7.37 2.24E-4 1.74

Case 5 - No FF

Adaptation 7.25 -19.63 2.28E-4 3.73

Case 5 - FF

Adaptation 7.78 -13.71 2.21E-4 0.22


feedforward adaptation: proposed strategy, downshift from 2nd

– 1st

177

6.4 Conclusion

In this chapter, transmission shift control strategies developed by Hebbale and

Kao and Bai et al. are implemented with the powertrain model developed in Chapter 3 to

demonstrate that primarily open loop control strategies perform poorly when clutch

friction characteristics vary significantly from when the controller is calibrated. For cases

where the sign of the slope becomes strongly negative, simulations of both

controllers result in peak-to-peak and root-mean-square driveshaft accelerations

significantly greater than if ideal friction parameters are simulated.

An integrated powertrain control strategy is then developed with the goal of

improving the robustness of the controller to changes in friction characteristics. As

demonstrated in Chapter 5, the friction parameters that govern the slope of the

curve may be estimated. The ability to estimate friction parameters provided the

motivation to implement – along with multiple feedback control loops – a model-based

feedforward controller that uses inputs of speed and estimated acceleration to generate a

clutch pressure command. The pressure calculated from the feedforward model is a

function of coefficient of friction; by estimating friction parameters such as the Coulomb

and static coefficients of friction, the feedforward controller may be updated as the

friction characteristics vary from their ideal values. During an upshift, adaptation of the

feedforward controller to varying friction parameters results in improved shift quality

relative to the non-adaptive case; however, the same trend is not as clear for downshifts.

Due to an offset between the feedforward pressure and the actual pressure at which the

clutch begins to slip, the PID controller begins to overcompensate for small slip speed

errors as the friction parameters deviate further from their ideal values. The feedback

178

controller generates much larger pressure commands than required, which results in

clutch stick-slip, and in turn, harsher acceleration transients.

179

CHAPTER 7: CONCLUSIONS AND FUTURE WORK

7.1 Summary

This thesis focuses on the modeling and control of a powertrain utilizing a wet

dual clutch transmission. Particular emphasis is placed on the modeling of the clutch and

synchronizer hydraulic actuation systems and the dynamics associated with wet clutch

friction. A simulation of the dynamic powertrain model is built using AMEsim and

MATLAB/Simulink. The powertrain simulator is used to demonstrate how changes in

transmission parameters affect the quality of clutch-to-clutch shifts and the overall

dynamic response of the powertrain. Based on this model, measurements of clutch

pressure and the rotational speeds and estimated accelerations at different gearbox shafts

are used in the design of a friction parameter estimation scheme. Friction parameters are

artificially changed in the simulation to represent how clutch friction characteristics may

change as clutch friction material and transmission fluid ages over the life of the

transmission; simulation results show that the friction parameters are estimated with

reasonable accuracy.

Two transmission shift control strategies found in literature are implemented with

the powertrain model to demonstrate that primarily open loop controls are not robust to

large changes in friction characteristics from their ideal values. When the sign of the

slope of the coefficient of friction versus slip speed curve becomes negative, the quality

180

of the clutch-to-clutch shift degrades for both controllers. An integrated powertrain

controller is developed with a model-based feedforward controller and multiple feedback

loops. The feedforward controller uses speed and estimated accelerations of the engine

and clutches, and a static friction model that may be updated as friction parameters vary,

to generate a pressure command to either clutch. The feedback controller contains the

following control loops: a clutch slip control loop for each clutch, where clutch pressure

is manipulated; a minor pressure control loop for each clutch, where the duty cycle

command to the pressure control solenoid is manipulated; and an engine speed control

loop, where spark advance, throttle angle, and the pressure applied to the torque-carrying

clutch during the inertia phase of either an upshift or a downshift are manipulated.

Simulation results for the proposed controller demonstrate that for upshifts, the

adaptation of the feedforward controller to varying friction parameters results in

improved shift quality relative to the non-adaptive case (ideal friction parameters are

inputs to the feedforward controller). The same conclusion cannot be drawn as clearly for

downshifts. Due to an offset between the feedforward pressure and the actual pressure at

which the clutch begins to slip, the PID controller begins to overcompensate for small

slip speed errors as the friction parameters deviate further from their ideal values. The

feedback controller generates much larger pressure commands than required, which

results in the clutch alternating between engaged and disengaged states multiple times

before fully releasing, and in turn, harsher acceleration transients.

7.2 Contributions

The contributions of this thesis are summarized as follows:

181

I. A detailed nonlinear dynamic model of the clutch and synchronizer hydraulic

actuation systems of a wet dual clutch transmission is developed.

II. Clutch pressure measurements are used to identify different phases of a shift.

The end of the clutch fill phase is recognized by comparing clutch pressure and

the estimated derivative of clutch pressure to the pressure required to compress

the clutch spring and the spring stiffness, respectively. The end of the torque

phase of an upshift and downshift is identified in part by recognizing when the

offgoing clutch pressure is zero.

III. Measurements of clutch pressure, and the rotational speeds and estimated

accelerations at different gearbox shafts are used in the design of a friction

parameter estimation scheme. For each clutch, the Coulomb and static

coefficients of friction and the viscous damping coefficient are estimated with

reasonable accuracy.

IV. An integrated powertrain controller with a model-based feedforward controller

and multiple feedback loops is developed. For upshifts, the adaptation of the

feedforward controller to varying friction parameters results in improved shift

quality relative to the non-adaptive case.

7.3 Recommendations for future work

The work presented here is purely simulation based, so the contributions

described in the previous section are limited. Thus the first recommendation for future

work is that the friction parameter estimation scheme and powertrain controller are

validated experimentally. To do so, the estimation and control logic described in this

182

work must be integrated with the engine and transmission control modules of a test

vehicle.

It was found that the adaptation of the feedforward controller to changes in

friction characteristics did not improve downshift quality. This is due to the large

difference between the feedforward pressure and the actual pressure at which the clutch

began to slip that was observed during the inertia phase of the downshift. Thus, the

quality of the downshift was strongly dependent on the feedback control. As the slope of

the curve became more negative, the PID gains for the clutch slip feedback

loop were no longer appropriate; large slip speed errors caused by the discrepancy

between the feedforward and actual clutch pressures caused the feedback controller to

generate much larger pressure commands than required. This resulted in the clutch

engaging and disengaging multiple times before fully releasing. Hence, the second

recommendation for future work is to identify and correct the error in the development of

the feedforward model that causes the mismatch between the feedforward pressure and

the actual pressure observed during the inertia phase of the downshift.

183

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Hydraulic System for a Stepped Automatic Transmission," SAE International.

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188

APPENDIX A: AUXILIARY HYDRAULIC SUBSYSTEMS

A.1 Clutch cooling system

Figure A.1 shows a complete schematic for the pressure regulation and clutch

cooling systems. The clutch cooling system, shown in Figure A.2, consists of the clutch

cooling valve (CCV) and the clutch cooling control solenoid (N218).

Figure A.1: Schematic of pressure regulation and clutch cooling systems

189

Figure A.2: Clutch cooling system

Clutch cooling valve

The mathematical model of the clutch cooling valve consists of two subsystems.

They are: the spool and accumulator mechanical system and the fluid flow system.

Spool and accumulator mechanical system

The CCV spool mechanical dynamics are described by:

(A.1)

where




is the land cross-sectional area at chambers A and C

, is the pressure in chambers A and C

190



described by:

( )

( )

(A.2)

The CCV accumulator (or accumulator #1) mechanical dynamics are described by:

(A.3)

where

is the spring displacement measured from static equilibrium


is the piston area

is the pressure acting on the piston

Fluid flow system

The net flow from the supply line to the clutch cooling valve, , is

described by:

(A.4)

Here, is the flow rate to or from the accumulator, is the flow rate into or out

of chamber A of the CCV, and is the N218 exhaust flow rate.

The accumulator pressure dynamics are modeled as:

(A.5)

and,

191

√

| | ( ) (A.6)

where is the accumulator volume.

The chamber A pressure dynamics are given by:

(A.7)

and,

√

| | ( ) (A.8)

Here, the chamber A volume, , is defined as:

(A.9)

is the chamber A volume at zero spool displacement.

The chamber B pressure dynamics are given by:

(A.10)

and,

√

And

(A.11)

√

And

(A.12)

192

√

(A.13)

and are the volume and pressure in chamber B. and

are the pressure and flow rate at the clutch cooling valve output (or the cooling passages).

, which is the flow rate into chamber B, is equal to the flow rate from the pressure

regulating valve ( ) minus the flow rate to the fluid filter/cooler ( ).

is the flow rate into or out chamber C, and is the exhaust flow rate. The flow

areas and are given by:

{

(A.14)

{

(A.15)

where and are the land diameters at chamber B and C, and and

are the overlap and underlap lengths for the inlet and exhaust ports,

respectively.

The chamber C pressure dynamics are given by:

(A.16)

Here, the chamber C volume, , is defined as:

(A.17)

is the chamber C volume at zero spool displacement.

193

Clutch cooling control solenoid, N218

The structure of the mathematical model of the clutch cooling control solenoid is

identical to the model of the pressure regulation control solenoid, N218. The N218 model

equations are determined by modifying the subscripts in equations (3.108)-(3.120).

Plunger mechanical system

(A.18)

{

( )

( )

(A.19)

(A.20)

( ( ))

(A.21)


(A.22)

Fluid flow system

√

(A.23)

( ) ( )

(A.24)

194

A.2 Safety systems

Figure A.3: Schematic of safety and clutch actuation systems

194

195

The full schematic of the safety and clutch actuation systems is shown in Figure

A.3. The safety valves protect the downstream components from overpressurization;

during normal operation, the safety valves are fully open and have a negligible effect on

component actuation.

196

APPENDIX B: EVEN GEAR COMPONENT ACTUATION SYSTEM

The model structures of the following systems are identical to the model of a

similar component described in the thesis body; the model equations are generated by

modifying variable subscripts.

B.1 Clutch pressure control valve, N216

Spool and accumulator mechanical systems

And

(A.25)

{

(A.26)

(A.27)

{

(A.28)

{

And ( )

(A.29)

(A.30)

197


(A.31)

Fluid flow system

(A.32)

For the filling phase:

(A.33)

(A.34)

For the exhausting phase:

(A.35)

(A.36)

√

And

(A.37)

√

(A.38)

{

(A.39)

198

{

(A.40)

(A.41)

(A.42)

√

And

(A.43)

√

And

(A.44)

(A.45)

(A.46)

(A.47)

√

And

(A.48)

199

B.2 Clutch piston, K2

Clutch piston mechanical dynamics

(A.49)

{

(A.50)

Fluid flow system

(A.51)

√

And

(A.52)

(A.53)

(A.54)

200

B.3 Shift forks, SF24, SF5N, SF6R

SF24


( ) ( ) (A.55)

(A.56)

Fluid flow system

(A.57)

(A.58)

(A.59)

(A.60)

SF5N


( ) ( ) (A.61)

(A.62)

201

Fluid flow system

(A.63)

(A.64)

(A.65)

(A.66)

SF6R


( ) ( ) (A.67)

(A.68)

Fluid flow system

(A.69)

(A.70)

(A.71)

(A.72)

202

B.4 Synchronizer solenoids, N89, N90, N91

N89


(A.73)

{

(A.74)

( ( ))

(A.75)


(A.76)

Fluid flow system

:

{

√

√

(A.77)

203

is the equivalent flow area of restrictions O30, O31, N89, and O37


N89, and O36 connected in series.

( ) ( )

(A.78)

:

{

√

√

(A.79)

is the equivalent flow area of restrictions O31 and O37 connected in

series. is the equivalent flow area of restrictions O31 and O36 connected in

series.

√

(A.80)

√

(A.81)

{

(A.82)

204

is the equivalent flow area of restrictions O30 and N89 connected in series.

N90


(A.83)

{

(A.84)

( ( ))

(A.85)


(A.86)

Fluid flow system

:

{

√

√

(A.87)

205




( ) ( )

(A.88)

:

{

√

√

(A.89)


series. is the equivalent flow area of restrictions O27 and O34 connected

in series.

√

(A.90)

√

(A.91)

{

(A.92)

206


N91


(A.93)

{

(A.94)

( ( ))

(A.95)


(A.96)

Fluid flow system

:

{

√

√

(A.97)

207




( ) ( )

(A.98)

:

{

√

√

(A.99)


series. is the equivalent flow area of restrictions O29 and O38 connected

in series.

√

(A.100)

√

(A.101)

{

(A.102)


208

APPENDIX C: SIMULATION PARAMETERS

Engine Model


9.97e5 [Nm/kg/s]

30 [CAD]

101.3 [kPa]

287 [J/kg/K]

293 [K]

, 0.0038, 0.0027 [m3]

13.5

1.185 [kg/ m3]

5000 [rpm]

Transmission Mechanical System and Vehicle Dynamics


, , 2.26, 4.5e-3, 6e-3 [m2]

, , , , ,

, ,

0.2, 0.05, 0.05, 0.05, 0.05,

0.005, 0.005, 4 [Nm/rad/s]

, , , , , ,

, 2, 1.8, 2.28, 0.318, 0.66, 25,

0.318, 0.66 [Nm/rad/s]

0.29

0.56e-3 [m]

60e6 [Pa]

0.05

, 75, 200

, , , , , ,

, , , , ,

0.4, 0.2, 0.08, 0.0234, 0.0057, 8.1e-4,

8.7e-4, 3e-4, 4.5e-4, 0.5, 1.8, 1.8 [kg∙m2]

, , , ,

, , , , 9.5e3, 2e4, 7.2e3, 4.2e4,

1.8e4, 5.2e4, 8.3e3, 1.8e4, 5.2e4, [Nm/rad]

1479 [kg]

2.5e7 [m-2

]

, 3, 1 [cones]

, 8, 10 [clutch friction interfaces]

209

10.34 [Pa∙s/°C]

0.3

0.3

, , , ,

37, 0.0965, 0.0815, 0.0685, 0.0525 [mm]

0.316 [m]

6.5 [degree]

0.15e-3 [m]

-1.960e-4 [s/rad]

2.54 [Nm/s/K]

1.594

0.5030

0.15

0.1558

0.1192

8.41e-6 [m]

4.93e-12 [m2]

0.0513 [rad/s]

0.8669 [s/rad]

Hydraulic Component Actuation System


, , , , ,

, ,

6.32, 9.05, 74.8, 2275,

535, 10, 13.54, 3.41, 700 [N/m/s]

0.61

3.183e-6 [m3/rad]

, ,

, ,

, ,

8.4, 3.8,

0.406, 2700,

1330, 20.2, 100 [N]

, 64, 1000 [Hz]

, , , , ,

, ,

1.2, 3.8, 0.454, 1800, 190,

2.3, 0.54, 8.5 [N/mm]

, , , ,

, , , ,

0.017, 0.011, 0.035, 1.467,

0.771, 0.021, 0.011, 0.011, 0.6 [kg]

- -0.0261 [H/m2], 0.0607 [H/m], 0.0409 [H]

- 0.227 [H/m3], -0.430 [H/m

2], 0.213 [H/m], 0.174 [H]

- 0.0074 [H/m3], 0.032 [H/m

2], 0.06 [H/m], 0 [H]

- 0.0055 [H/m2], 0.0052 [H/m], 0.021 [H]

0 [bar]

, ,

,

6.9, 16.5,

4.8, 4.5 [Ω]

210

, , ,

,

2.9, 2.1,

0.15,

0.85, 1.2 [mm]

, , ,

, , ,

4.5, 2.5, 1.78,

0.1, 6.7, 2,

2 [mm]

, ,

, ,

,

3, 11,

4, 8,

6, 6 [mm]

7000 [bar]

761 [kg/m3]

PID Parameters for Controllers Proposed by Hebbale and Kao and Bai et al.


, ,

, (Slip Speed Control via Clutch

Pressure Manipulation)

7e-2 [bar/rad/s], 8e-3 [bar/rad],

1e-5 [bar/rad/s2], 150 [rad/s]

PID Parameters for Proposed Controller


, ,

, (Clutch Pressure Control via

Solenoid Duty Cycle Manipulation)

1.3e-2 [1/bar], 0 [1/(bar∙s)],

0 [s/bar], N/A [rad/s]

, ,


Pressure Manipulation, Torque Phase)

4.1e-1 [bar/rad/s], 5e-2 [bar/rad],


, ,


Pressure Manipulation, Inertia Phase)

6e-2 [bar/rad/s], 5e-3 [bar/rad],


, ,

, (Engine Speed Control via

Throttle Angle Manipulation)

4 [deg/rad/s], 3e-1 [deg/rad],

1e-5 [deg/rad/s2], 200 [rad/s]

, ,


Spark Advance Manipulation)

3 [CAD/rad/s], 2e-1 [CAD/rad],

1e-5 [CAD/rad/s2], 200 [rad/s]

, ,


Clutch Pressure Manipulation)

5e-2 [bar/rad/s], 0 [bar/rad],


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