Loop Simulation Thesis Presented in Parti

Modeling and Performance Analysis of a 10-Speed Automatic Transmission for X-in-the-

Loop Simulation

Thesis

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

the Graduate School of The Ohio State University

Clayton Thomas

Mechanical Engineering

The Ohio State University

2018

Thesis Committee

Dr. Shawn Midlam-Mohler, Advisor

Dr. Krishnaswamy Srinivasan

Dr. Punit Tulpule

Copyrighted by

Clayton Thomas

2018

Abstract

Vehicle integration testing has been increasingly front-loaded in the automotive development cycle to reduce prototyping and testing costs. One method of performing these tests is X-in-the-loop integration, which allows a verification and validation workflow from the design of control algorithms in offline high-fidelity models (MIL) to the online integration verification with prototype control hardware (HIL). A 10-speed automatic transmission is used as an example to traverse the gap between an offline high- fidelity model and a real-time capable online model. The model is split into three subsystems and each is built up from the component level using one-dimensional mechanics and zero-dimensional hydraulic fluid flow. The high-fidelity model parameters are perturbed to judge sensitivity of output performance metrics. The model is reduced by removing higher-order derivatives and faster dynamics at the component level. Multiple reduced models were generated and tested for errors relative to the high- fidelity version and increases in model execution speed. After reduction, a full automatic transmission model with hydraulic actuation circuit and dynamic torque converter has been implemented on a dSpace HIL simulator for real-time testing without control hardware in the loop.

Dedication

First and foremost, I would like to dedicate this work to my family for their support throughout my time here at Ohio State. To Natalie – you have been more amazing and encouraging than I could ever explain. Thank you for being there for me through this process and putting up with me as I slowly drove myself crazy. To Mom, Dad, and Cole

– you have obviously all helped push me throughout my life, so I can attribute where I am today to your love and support. Dad – thank you for driving me around in a borrowed

Corvette for an hour on my 12th birthday.

I would also like to thank Dr. Shawn Midlam-Mohler for working with me throughout my time at SIMCenter and taking me in as an inexperienced intern three years ago. Being a part of SIMCenter has been the most enjoyable and rewarding experience of my college career and I want to thank all of my colleagues for making that happen.

Finally, the importance of mental health in stressful environments cannot be overstated. If you or a loved one is struggling to get through a tough time, the signs may not always be visible to those nearby. Do not be afraid to ask for help or be willing to provide it.

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Acknowledgments

I would like to thank Honda R&D Americas for their support and feedback during this project. I would also like to thank Dr. Punit Tulpule and Dr. Shawn Midlam-Mohler for providing valuable guidance on this project since its inception. Dr. Krishnaswamy

Srinivasan has also provided valuable time and feedback on this work, which is much appreciated.

Vita

May 2011 …………………………………………. Westerville South High School

August 2015 to December 2016 …………………... Undergraduate Research Associate, The Simulation Innovation and Modeling Center, The Ohio State University

December 2016 ……………………………………. B.S. Mechanical Engineering, The Ohio State University

December 2016 to Present ………………………… Graduate Research Associate, The Simulation Innovation and Modeling Center, The Ohio State University

Fields of Study

Major Field: Mechanical Engineering

Table of Contents

Abstract ...... ii Dedication ...... iii Acknowledgments...... iv Vita ...... v List of Tables ...... x List of Figures ...... xiii Chapter 1. Introduction ...... 1 1.1. Background ...... 1 1.2. Motivation ...... 1 1.3. Objectives ...... 3 1.4. Methods...... 3 1.5. Structure of Document ...... 5 Chapter 2. Literature Review ...... 6 2.1. Transmission Modeling ...... 6 2.2. Torque Converter Modeling ...... 10 2.3. Drivability and Metrics ...... 12 2.4. Simulation and Software ...... 14 2.5. Summary of Prior Work ...... 18 Chapter 3. Requirements and Metrics ...... 19 3.1. Functional Requirements ...... 19 3.1.1. Stability ...... 19 3.1.2. Model Boundaries ...... 20 3.1.3. Software Interfacing ...... 20 3.1.4. Fidelity ...... 23 3.1.5. Real-Time HIL Model ...... 23 vi

3.1.6. Replicate Functional Behavior ...... 24 3.2. Performance Metrics ...... 24 3.2.1. Clutch Engagement Hydraulics ...... 25 3.2.2. Torque Converter Metrics ...... 29 3.2.3. Overall Shift Metrics...... 35 3.2.4. Summary of Metrics ...... 42 Chapter 4. High-Fidelity Model Development ...... 43 4.1. High-Fidelity Hydraulic Network ...... 44 4.1.1. Solenoid Valves ...... 47 4.1.2. Regulation Valves ...... 50 4.1.3. Switch Valves ...... 55 4.1.4. Two Way Clutch Valve ...... 57 4.1.5. Hydraulic Modeling Summary ...... 60 4.2. High-Fidelity Gearbox ...... 60 4.2.1. Planetary Gearset ...... 61 4.2.2. Friction Clutch ...... 63 4.2.3. Two-Way Clutch ...... 65 4.2.4. Hydraulic Clutch Actuation Cylinder ...... 66 4.2.5. Gearbox Modeling Summary ...... 69 4.3. Torque Converter ...... 69 4.4. Simulation Settings and Notes ...... 72 4.4.1. Preprocessing Routines ...... 72 4.4.2. BDF and MEBDF Solvers ...... 76 4.4.3. CVODE Solver (from SUNDIALS) ...... 78 4.4.4. Fixed Step Solvers ...... 79 4.4.5. Performance Analyzer ...... 80 4.4.6. Additional Notes ...... 81 4.4.7. Summary of Simulation Environment ...... 83 Chapter 5. Parameter Sensitivity ...... 84 5.1. Design and Setup of Experiments ...... 84 5.1.1. Gearbox Parameters ...... 90 5.1.2. Gearbox Design of Experiments and Simulation Setup ...... 94

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5.1.3. Hydraulic Network Parameters ...... 96 5.1.4. Hydraulic Network Design of Experiments and Simulation Setup ...... 100 5.1.5. Torque Converter Parameters ...... 102 5.1.6. Torque Converter Design of Experiments and Simulation Setup ...... 104 5.2. Performance Metric Sensitivity ...... 105 5.2.1. Gearbox Parameter Sensitivity ...... 111 5.2.2. Hydraulic Network Parameter Sensitivity ...... 116 5.2.3. Torque Converter Parameter Sensitivity ...... 128 5.3. Analysis and Discussion ...... 133 Chapter 6. Reduced Model Development ...... 141 6.1. Hydraulic Network ...... 143 6.1.1. Reduced Solenoid Valves ...... 143 6.1.2. Reduced Regulation Valves ...... 147 6.1.3. Reduced Switch Valves ...... 154 6.1.4. Reduced Two-Way Clutch Valve ...... 156 6.1.5. Integrated Hydraulic Network Levels ...... 157 6.2. Gearbox ...... 161 6.3. Torque Converter ...... 166 6.4. Simulation Settings and Notes ...... 168 Chapter 7. Model Reduction Study ...... 171 7.1. Reduction Simulation Setup and Notes ...... 171 7.2. Variable-Step Simulation Sensitivity ...... 173 7.2.1. Hydraulic Network Parameter Reduction ...... 173 7.2.2. Gearbox Parameter Reduction ...... 177 7.2.3. Torque Converter Parameter Reduction ...... 178 7.3. Fixed-Step Simulation Sensitivity ...... 179 7.3.1. Hydraulic Fixed-Step Reduction ...... 179 7.3.2. Gearbox Fixed-Step Reduction ...... 183 7.3.3. Torque Converter Fixed Step Reduction ...... 184 7.4. Analysis and Discussion ...... 186 Chapter 8: Reduced Model Implementation ...... 190 8.1. Model Portability Options ...... 190

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8.2. Implementation Hardware and Software ...... 192 8.3. Implementation Examples ...... 194 Chapter 9. Conclusions ...... 200 9.1. Key Takeaways and Implications ...... 200 9.2. Project Obstacles and Notes ...... 202 9.3. Future Work ...... 205 Bibliography ...... 209 Appendix A. Fixed-Step Model Reduction Metric Plots ...... 212

List of Tables

Table 1: Solver Order Specifications (ESI ITI GmbH, 2017) ...... 80

Table 2: Shift Schedule for Repeatable Transient Experimentation ...... 86

Table 3: Gearbox Friction Clutch Parameters ...... 90

Table 4: Gearbox Planetary Gearset Parameters ...... 92

Table 5: Gearbox Clutch Cylinder Parameters ...... 93

Table 6: Gearbox Two-Way Clutch Parameters ...... 93

Table 7: Gearbox Parameter Variation Experimental Overview ...... 96

Table 8: Hydraulic Network Linear Solenoid Electrical Parameters ...... 97

Table 9: Hydraulic Network Linear Solenoid Mechanical Parameters ...... 97

Table 10: Hydraulic Network Shift Solenoid Electrical Parameters ...... 98

Table 11: Hydraulic Network Shift Solenoid Mechanical Parameters ...... 99

Table 12: Hydraulic Network Regulation Circuit Mass/Damping Parameters ...... 100

Table 13: Hydraulic Network Parameter Variation Experimental Overview ...... 101

Table 14: Torque Converter Blade Angle Parameters ...... 102

Table 15: Torque Converter Empirical Factors ...... 103

Table 16: Torque Converter Geometrical Parameters ...... 104

Table 17: Torque Converter Parameter Variation Experimental Overview ...... 105

Table 18: Hypothetical Output Table - Input Parameters Augmented with Outputs for Each Shift ...... 107

Table 19: Hypothetical Metric Handling – Metrics Averaged across All Shifts ...... 108

Table 20: Hypothetical Output Table - Mean Metrics Augmented with Parameters ..... 108

Table 21: Hi-Fi Gearbox Clutch Friction Sensitivity Data ...... 111

Table 22: Hi-Fi Gearbox Hydraulic Cylinder Sensitivity Data ...... 112

Table 23: Hi-Fi Gearbox Planetary Gearset Sensitivity Data ...... 113

Table 24: Hi-Fi Gearbox Two-Way Clutch Sensitivity Data ...... 114

Table 25: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Electrical Sensitivity Data ...... 116

Table 26: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Electrical Sensitivity Data ...... 117

Table 27: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Mechanical Sensitivity Data ...... 118

Table 28: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Mechanical Sensitivity Data ...... 119

Table 29: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Electrical Sensitivity Data ...... 120

Table 30: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Electrical Sensitivity Data ...... 121

Table 31: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Mechanical Sensitivity Data ...... 122

Table 32: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Mechanical Sensitivity Data ...... 123

Table 33: Hi-Fi Hydraulic Network (Flow-Sourced) Regulation Valve Sensitivity Data ...... 125

Table 34: Hi-Fi Hydraulic Network (Pressure-Sourced) Regulation Valve Sensitivity Data ...... 127

Table 35: Hi-Fi Torque Converter Blade Angle Sensitivity Data ...... 129

Table 36: Hi-Fi Torque Converter Empirical Factor Sensitivity Data ...... 131

Table 37: Hi-Fi Torque Converter Geometrical Sensitivity Data ...... 132

Table 38: Gearbox Sensitivity Parameter Rankings by Absolute Sensitivity ...... 134

Table 39: Importance Score for Model Parameters (Banerjee, Asl, Azad, & McPhee, 2012) ...... 137

Table 40: Comparison of Torque Converter Sensitivity Rankings to Analytical Method from Literature ...... 138

Table 41: Linear Solenoid Reduction Summary ...... 145

Table 42: Shift Solenoid Reduction Summary ...... 147

Table 43: Main Regulator Valve Reduction Summary ...... 149

Table 44: Lubrication Regulation Valve Reduction Summary ...... 150

Table 45: Torque Converter Regulation Valve Reduction Summary ...... 151

Table 46: Line Pressure Accumulator Valve Reduction Summary ...... 152

Table 47: Lockup Clutch Control Valve Reduction Summary ...... 154

Table 48: Selected Switch Valve Reduction Summary ...... 155

Table 49: Two-Way Clutch Valve Reduction Summary ...... 157

Table 50: Integrated Hydraulic Network Reduction Summary and Combinations ...... 157

Table 51: Integrated Gearbox Reduction Summary and Combinations ...... 163

Table 52: Comparison of Torque Metrics between Online HIL and Offline VLVL 4 Gearbox Only ...... 197

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List of Figures

Figure 1: Shift Torque Response Influences (Deur, Asgari, & Hrovat, 2006) ...... 8

Figure 2: Model Boundaries/Scope ...... 20

Figure 3: Example Delay Time Measurement ...... 25

Figure 4: Example Rise Time Measurement ...... 26

Figure 5: Example Corner Time Measurement ...... 27

Figure 6: Example Corner Pressure Measurement ...... 28

Figure 7: Example Overshoot Percentage Measurement ...... 28

Figure 8: Example Stall Torque Ratio, adapted from (Robinette, Grimmer, & Beikmann, 2011) ...... 30

Figure 9: Example Fluid Coupling Speed Ratio Measurement, adapted from (Robinette, Grimmer, & Beikmann, 2011) ...... 31

Figure 10: Example Peak Efficiency Measurement, adapted from (Banerjee & McPhee, 2012) ...... 32

Figure 11: Example Peak Efficiency Measurement, adapted from (Banerjee & McPhee, 2012) ...... 33

Figure 12: Example RMSKE Measurement, adapted from (Robinette, Grimmer, & Beikmann, 2011) ...... 34

Figure 13: Example RMSTE Measurement, adapted from (Robinette, Grimmer, & Beikmann, 2011) ...... 35

Figure 14: Example Torque Hole Measurement, adapted from (Kulkarni, Shim, & Zhang, 2007) ...... 37

Figure 15: Example Torque Overshoot Measurement, adapted from (Kulkarni, Shim, & Zhang, 2007) ...... 38

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Figure 16: Example Torque Settling Time, adapted from (Kulkarni, Shim, & Zhang, 2007) ...... 39

Figure 17: Example Torque Phase, adapted from (Coric, Ranogajec, Deur, Ivanovic, & Tseng, 2017) ...... 40

Figure 18: Example Torque Phase, adapted from (Coric, Ranogajec, Deur, Ivanovic, & Tseng, 2017) ...... 41

Figure 19: Overall High-Level Hydraulic Network Breakdown ...... 45

Figure 20: Linear Proportional Solenoid Schematic, adapted from (Lee, Sung, & Kim, 2012) ...... 49

Figure 21: High-Fidelity Linear Proportional Solenoid Model ...... 49

Figure 22: Example Shift Solenoid Schematic (Watechagit & Srinivasan, 2003) ...... 50

Figure 23: Example Regulation Valve, adapted from (Marano, Moorman, Whitton, & Williams, 2007)...... 51

Figure 24: High-Fidelity Main Regulation Valve Model ...... 52

Figure 25: High-Fidelity Lubrication Regulation Valve Model ...... 53

Figure 26: High-Fidelity Torque Converter Regulation Valve Model ...... 54

Figure 27: High-Fidelity Line Pressure Accumulator Valve Model ...... 54

Figure 28: High-Fidelity Lockup Clutch Control Valve Model ...... 55

Figure 29: Selected High-Fidelity Switch Valve Model ...... 56

Figure 30: High-Fidelity Cut Valve Model ...... 57

Figure 31: Diagram of Detent Mechanism Definition (ESI ITI GmbH, 2017) ...... 59

Figure 32: High-Fidelity Two-Way Clutch Model ...... 59

Figure 33: High-Fidelity Planetary Gearset Model ...... 62

Figure 34: Clutch Plate Diagram and Parameters (ESI ITI GmbH, 2017) ...... 64

Figure 35: High-Fidelity Disk Clutch Hydraulic Actuation Cylinder Model ...... 67

Figure 36: High-Fidelity Brake Clutch Hydraulic Actuation Cylinder Model ...... 69

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Figure 37: Dynamic Torque Converter Blade Approximation Diagram ...... 71

Figure 38: Solution Workflow for SimulationX Dynamic Simulation (ESI ITI GmbH, 2017) ...... 74

Figure 39: Example Performance Analyzer Output for Hi-Fi Gearbox Simulation ...... 81

Figure 40: Hydraulic Network Subsystem Overview ...... 85

Figure 41: Gearbox and Clutch Cylinder Subsystem Overview ...... 89

Figure 42: Hi-Fi Gearbox Clutch Friction Sensitivity Plot ...... 111

Figure 43: Hi-Fi Gearbox Hydraulic Cylinder Sensitivity Plot ...... 112

Figure 44: Hi-Fi Gearbox Planetary Gearset Sensitivity Plot ...... 113

Figure 45: Hi-Fi Gearbox Two-Way Clutch Sensitivity Plot ...... 114

Figure 46: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Electrical Sensitivity Plot ...... 116

Figure 47: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Electrical Sensitivity Plot ...... 117

Figure 48: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Mechanical Sensitivity Plot ...... 118

Figure 49: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Mechanical Sensitivity Plot ...... 119

Figure 50: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Electrical Sensitivity Plot ...... 120

Figure 51: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Electrical Sensitivity Plot ...... 121

Figure 52: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Mechanical Sensitivity Plot ...... 122

Figure 53: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Mechanical Sensitivity Plot ...... 123

Figure 54: Hi-Fi Hydraulic Network (Flow-Sourced) Regulation Valve Sensitivity Plot ...... 124

Figure 55: Hi-Fi Hydraulic Network (Pressure-Sourced) Regulation Valve Sensitivity Plot ...... 126

Figure 56: Hi-Fi Torque Converter Blade Angle Sensitivity Plot ...... 128

Figure 57: Hi-Fi Torque Converter Empirical Factor Sensitivity Plot ...... 130

Figure 58: Hi-Fi Torque Converter Geometrical Sensitivity Plot ...... 132

Figure 59: Linear Solenoid Reduced Models ...... 145

Figure 60: Shift Solenoid Reduced Models ...... 146

Figure 61: Main Regulator Valve LVL 3 ...... 149

Figure 62: Lubrication Regulation Valve LVL 3 ...... 150

Figure 63: Torque Converter Regulation Valve LVL 3...... 151

Figure 64: Line Pressure Accumulator Valve LVL 2 (left) and LVL 3 (right) ...... 152

Figure 65: Lockup Clutch Control Valve LVL 3...... 153

Figure 66: Selected Switch Valve Reduced Models ...... 155

Figure 67: Two-Way Clutch Valve Reduced Models...... 156

Figure 68: Eigenvalue Overview of Hydraulic Network Reduction ...... 159

Figure 69: Step-by-Step Comparison of Eigenvalues for Reduced Hydraulic Network Models...... 160

Figure 70: Example of Planetary Gearset Torque Bench for Elasticity Testing ...... 162

Figure 71: Eigenvalue Overview of Gearbox Reduction ...... 164

Figure 72: Step-by-Step Comparison of Eigenvalues for Reduced Gearbox Models .... 164

Figure 73: Shift Event Comparison between LVL 2 and LVL 3 Gearbox Models ...... 166

Figure 74: Flow-Sourced Model Accuracy Reduction Plots (Reference: LVL1Q) ...... 173

Figure 75: Pressure-Sourced Model Accuracy Reduction Plots (Reference: LVL1P) ... 175

Figure 76: Pressure-Sourced RMS Error Plots, with (left) and without (right) Overshoot % ...... 176

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Figure 77: Gearbox Model Accuracy Reduction Plots ...... 177

Figure 78: Gearbox RMS Error Plot ...... 178

Figure 79: Torque Converter RMS Error Plot (Sampled Stable Results) ...... 179

Figure 80: Fixed-Step Hydraulic Network Reduction Summary Plots ...... 180

Figure 81: Zoomed-In Plot of Hydraulic Network Real-Time Factor vs. Step Size ...... 182

Figure 82: Fixed-Step Gearbox Reduction Summary Plots ...... 183

Figure 83: Real-Time Tradeoff Plot for Fixed-Step Gearbox Reduction ...... 184

Figure 84: Metric Error vs. Real-Time Factor Distribution Plots for RMSTE (left) and RMSKE (right)...... 184

Figure 85: Fixed-Step Torque Converter Real-Time Factor vs. Step Size ...... 185

Figure 86: Fixed-Step Torque Converter Error vs. Step Size ...... 185

Figure 87: Fixed-Step Torque Converter Error vs. Real-Time Factor ...... 186

Figure 88: dSpace ControlDesk Signal Recording Environment ...... 194

Figure 89: Comparison of HIL Implementation Sweeps with and without TC Lockup 195

Figure 90: Comparison of Online HIL Upshift Sweep with Offline LVL 4 Gearbox Only ...... 197

Figure 91: Visual Example of Instability in Model Leading to Signal Inaccuracies ...... 198

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Chapter 1. Introduction

1.1. Background

Mathematical models of automotive powertrain systems may be constructed using a wide variety of techniques that determine the resulting accuracy of simulation. Offline simulation models may be constructed primarily with simulation accuracy in mind, requesting the highest-fidelity mathematical descriptions of the system. When verifying control functionality via hardware-in-the-loop simulation, real-time capability becomes an additional constraint on both the governing equations and the execution environment.

Migrating control software verification testing from the track to a virtual environment often saves testing resources and time, as well as providing a controlled environment for repeatable experimentation.

1.2. Motivation

A recent increase in demand for virtual control verification in the industry necessitates a dynamically representative model for maneuver-based testing sequences.

Some identified use-cases of hardware-in-the-loop (HIL) vehicle simulation in industry are identified as: confirming supplier specifications, validating software countermeasures, and developing new control strategies (Monsma, 2009). For future testing requirements, a large fraction of planned test cases may involve transient vehicle maneuvers.

The automotive industry currently utilizes transmission models of varying complexities depending on the intended purpose. One potential use case is for HIL simulation, where the model behavior must feed information back to a hardware-based controller to support controller verification testing. One modern implementation of a transmission which is modeled for such applications would be a 10-speed conventional automatic transmission consisting of multiple planetary gearsets, friction clutches, and a hydraulic actuation circuit. To translate testing confidence in the transmission controls from the simulation space to the test track, it is important to understand the effect of model fidelity on accuracy and simulation capability.

Gaps in available literature are present between the low-level component modeling activities and the high-level HIL implementation and integration activities.

Modern sources do reference integrating system-level models of engines, transmissions, and vehicle models under the umbrella of a real-time machine. However, these works do not generally touch on model development or to what extent a model of the chosen fidelity fulfills the requirements of a HIL bench in a software verification role. In contrast, much of the older literature dives into nonlinear actuator dynamics for offline control development, with some of the more recent work looking into linearization techniques for model-based control. For gearbox and torque converter modeling, much of the work involves offline design optimization or performance prediction, without the constraints of compiling for real-time implementations.

There is thus an intersection between first-principles-based physical modeling and computational realization which is left out of the development process. Having different

2 teams build and integrate physical models may naturally lead to a disconnection with translating the modeling goals into a constrained computational environment. Integrating the modeling process with a computational understanding of the limits of implementation requires a bridge between the system dynamics, numerical methods, and computer science.

1.3. Objectives

In this project, a physically-representative dynamic transmission model will be compiled and examined for xIL validation and verification tasks. The construction of this real-time-capable full transmission model – from torque converter impeller shaft to gearbox output shaft – represents the ultimate output of this work. However, the process of developing and reducing the model from the component level is a significant outcome of the modeling activity as well. A goal of this modeling study is to determine the relative impact of varying subsystem construction on the accuracy of simulation results and required computational time.

1.4. Methods

The task of creating a high-fidelity simulation model would involve a large calibration effort and subsequently large testing burden to perform controlled experiments to identify model parameters. To combat this burden and to prepare for this future task, a “loosely-calibrated” high-fidelity transmission model will be constructed and used to evaluate the sensitivity of subsystems and parameters. Ultimately, a

3 transmission model must be calibrated to physically represent the dynamic response, but first a more in-depth understanding of the model capabilities will be performed in this work.

A model with estimations of physical parameters and empirical factors is sufficient for this study to understand the relationships between the components and specified output metrics. The full transmission model will be split into subsystems for the torque converter, hydraulic network, and mechanical gearbox to more-easily analyze the input-output relationships. The highest fidelity presented in this work will remain one- dimensional in the mechanical domain, while fluid flow will only be analyzed as a series of lumped reservoirs and restrictions (no one-dimensional flow).

Using estimated values (or measured values, when known) for nominal levels, parameters of the subsystems will be deviated from nominal conditions to measure impacts of each parameter on specified output metrics and simulation efficiency. This process will be known as “parameter sensitivity analysis”. Furthermore, as component models are reduced in fidelity, either by removing fast dynamic states, lumping elements together, or mapping empirical relationships, the output metrics will be tracked relative to those of the highest-fidelity model. This activity will be known as the “model reduction study” and will include analysis of variable step solving results as well as fixed-step implementations. Finally, the subsystems will be integrated into an application for real- time simulation capable of running in a HIL environment.

1.5. Structure of Document

A literature review is present at the beginning of this work to give examples of prior work in modeling, simulation, and analysis of dynamic automotive systems. The list is not exhaustive, though it represents a subset of the relevant work done in the industry reaching back multiple decades. Chapter 3 outlines project requirements and metrics used for evaluation throughout this project.

Next, Chapter 4 gives an in-depth description of the modeling of each subsystem shown from a systems level. The reader is assumed to be familiar with simple dynamic models such as mass-spring-damper mechanical subsystems and basic sharp-edged orifice flow equations. In Chapter 5, the aforementioned parameter sensitivity study is outlined with specific designs of experiments, analyses, and results.

The model reduction process is outlined in Chapter 6, where component models from each of the subsystems are reduced by eliminating fast dynamic states which would lower the critical time step of the subsystems. Chapter 7 outlines the model reduction study in terms of the performance and accuracy of the reduced models (referring to the set of reduced subsystem models of decreasing fidelity).

Finally, the model implementation of a real-time capable transmission model which has resulted from the model reduction study will be outlined in Chapter 8. This includes simulation software and environment settings, as well as dynamic comparisons to higher-fidelity offline transmission models. This work is concluded with a discussion of overall takeaways, modeling obstacles, and future work.

Chapter 2. Literature Review

2.1. Transmission Modeling

Modeling for real-time simulation and modeling for control fall under many overlapping categories. Many control-based models must simplify actuator dynamics to feed an online model either for feedforward model-based control or state observers for state feedback. For this reason, many sources which fall under the “modeling for control” umbrella are especially suited to be computationally more efficient yet still functionally accurate for online simulation. Additionally, older sources which outline model-based control development many times use models which may be implemented more easily with modern solutions. Simply increasing the clock speed of processors over the past decade has made some modeling techniques more relevant, let alone the increase in commercial packages which lower the barrier to efficient computational tools.

One crossover study which merged modeling and computation came from The

Romanian Academy (Bataus, Maciac, Oprean, & Vasiliu, 2011). The research focused on clutch modeling and the computational impacts of incorporating certain levels of fidelity into a one-dimensional clutch friction model. Discontinuous stick-slip maps were studied along with smoothed empirical curves of from the simple hyperbolic tangent curve to the empirical Stribeck friction curve. One effect was found to increase the simulation speed by 10-17% regardless of the inclusion of Stribeck friction: the Karnopp friction model

6 implements a dead zone of Δ휔 near zero slip speed at which the slip speed is forced to zero. This helps to prevent small oscillations about zero speed due to the frequent sticking and slipping during a lower-energy engagement. In a slight contrast, Coric,

Ranogajec, Deur, Ivanovic, & Tseng (2017) used various friction models for a shift control optimization strategy, in which the Karnopp model was not used due to inaccuracies induced by smoothing for optimization. Instead, a “classical” friction model was used which replaced the sticking dead zone with a linear interpolation for the friction coefficient through the near-zero-speed regime.

In a study about modeling for a dual-clutch transmission (Cavina, Olivi, Corti,

Poggio, & Marcigliano, 2012), a pressure response trace was calibrated with respect to experimental measurements. The key points taken from this model were that trapped air volume percentage contributes a significant portion of the hydraulic compressibility in fluid volumes. The addition of 1.4% trapped air by volume decreased the effective bulk modulus by 230 times, which greatly improved simulation capability. Without this effect, step sizes below 1 휇푠 were required for stable fixed-step simulation, while adding the trapped air increased the step size to 0.5 푚푠.

A look at vehicle transmission and driveline modeling for so-called “garage shifts” (shifts at zero speed) outlines elements of torque response signals and the related influential parameters (Deur, Asgari, & Hrovat, 2006). These types of torque characteristics are key to measuring drivability with respect to shift quality and may help diagnose inconsistencies during future calibration efforts. A figure summarizing the analysis is shown below:

Figure 1: Shift Torque Response Influences (Deur, Asgari, & Hrovat, 2006)

Kulkarni, Shim, and Zhang (2007) modeled a dual-clutch transmission in which thermal effects are ignored, hydraulic delays are neglected, backlash is not included, and all losses are lumped into a road load force at the wheels. With a simplified overall model, the goal was to study gear shift output torque dynamics by controlling clutch pressures. The lumped vehicle model is presented with parameter values for benchmarking and example output torque traces, along with visual breakdowns of shift phases referenced in Chapter 3 with respect to shift metrics.

In (Lucente, Montanari, & Rossi, 2007), Stateflow was used to transition discontinuities such as hard end stops within the hydraulic actuation system. The researchers note that line dynamics are neglected, as the lengths are too short to contain significant 1-D dynamics. In addition to using a dither-signal-based method of frequency response analysis, this paper also includes a comprehensive list of model parameters used in the hydraulic system, gearbox, and other model subsystems.

A more in-depth look at hydraulic line dynamics was performed in another reference (Favennec, Sall, Lebrun, & Alirand, 2003) where each pipe was modeled as multi-mode pipes using a varying number of volume elements. Line lengths varied from

12 to 100 cm, and the study found that only the longest line (100 cm) displayed dynamics relevant to the pressure output signal, thus deserving of a five-element lumped line model.

One study done by Claytex modeled a full vehicle powertrain and driveline in

Dymola and illustrated the motivation of implementing dynamic torque converters

(Roberts, Dempsey, & Picarelli, 2013). The team showed that using a static torque converter model, an engineer was not able to discern between two engine selections which, in reality, would have different firing frequencies. While the goal was to highlight the usefulness of integrated Dymola libraries, the case for dynamic torque converter models makes sense for transient applications where design decisions may be made.

A research team at Ohio State studied the current-to-pressure hydraulic control dynamics with a fully dynamic electrical solenoid model controlling the supply pressure to the clutch chamber (Watechagit & Srinivasan, 2003). In the calibration effort, it was found that neglecting the electromechanical solenoid model made the valve characterization more error-prone and unrealistic. The team used a critical Reynold’s number to transition from turbulent to laminar flow at low fluid velocities. Though computational resources are now much more advanced, the authors had noted the need for model simplification for online simulation.

2.2. Torque Converter Modeling

Though torque converter research stretches back into the 1940’s, modern popular models are often built off of the work done by Allan Kotwicki (1982). In this work, conservation of angular momentum equations were used to derive the relationships between the speeds and torques of the input and output shafts, while the conservation of mechanical power defined the flow rate of the fluid. Using first principles incorporated the geometrical parameters of the torque converter, including blade angles, element radii, and flow areas. However, the equations were derived using the assumption of constant speed for the rotational elements and fluid, making the model ultimately algebraic in nature. In its original form, the Kotwicki model accepts input and output speeds and calculates the corresponding torques, with the flow rate as an intermediate state.

An extension of the Kotwicki steady model (Pritchard, Gould, & Johnson, 2014) uses the moment of momentum derivation scheme from Kotwicki to include a stator torque-speed equation into the power flow loop. The most significant addition for this paper is the derivation of the model equations for the “overrunning” regime, where the turbine rotates faster than the impeller shaft, causing a flow reversal by the fluid.

The other main physical modeling basis for torque converters contains hydrodynamic equations without assuming constant speed or flow (Hrovat & Tobler,

1985). Four ordinary differential equations are presented for the rotational dynamics of the turbine, impeller, stator, and fluid inertias. Reverse flow derivation was done in an appendix of this work, which focused on creating a matrix-based equation structure for a state-space model. Graph theoretic modeling techniques were used in another work

(Banerjee & McPhee, 2012) which derived the same model equations as Hrovat &

Tobler, with the inclusion of all geometrical model parameters for benchmarking. While modeling for control, a method is presented for the integration of 3D CFD flow data into the one-dimensional torque converter equations (Lee J.-H. , 2017).

Additional dynamic studies of the Hrovat & Tobler dynamic model look at the torque converter as a rotational damping element (Robinette, Grimmer, & Beikmann,

2011) and multiple works have studied its frequency response. With respect to parameters in the Hrovat & Tobler model, a qualitative illustration of the effects of varying some geometrical parameters is shown for efficiency and capacity factor curves

(Asl, Azad, & McPhee, 2011). A more analytical approach is taken in another source by many of the same researcher (Banerjee, Asl, Azad, & McPhee, 2012) where an energy loss function is used as the basis for “importance analysis”, measuring the partial derivatives of the objective with respect to the constituent model parameters. Finally, a

Master’s thesis from Chalmers University of Technology (Li & Sunden, 2016) looks at the implementation of static map-based torque converter models alongside the hydrodynamic model and a proprietary Dymola torque converter scheme. The so-called

“Drenth” dynamic model from Dymola combines the dynamic states of the Hrovat &

Tobler equations with the capacity factor and torque ratio curves from the empirical map- based model. This model is not published explicitly due to the proprietary nature of the model used exclusively in Dymola’s modeling software.

2.3. Drivability and Metrics

Drivability in this work will mostly refer to shift quality metrics, though tip-in and

NVH measures are relevant and play important roles in perceived quality. Metrics for drivability can relate to torque delay, rough operation, instability, and longitudinal acceleration or jerk (Wei & Rizzoni, 2004). While acoustics and high-frequency noise contributes to perceived quality, models for the powertrain are mostly concerned with torque-based metrics such as acceleration and jerk.

Aggregate drivability scoring algorithms seek to combine elements from all vehicle operational modes into quality scores, such as those found in AVL DRIVE. Other efforts have sought to correlate features of torque and speed traces with subjective drivability semantics such as “sportiness” or “responsiveness” (Jeon & Kim, 2014). In the study by Jeon & Kim, these subjective factors were correlated with jerk values, delay times, speed gradients, and other metrics to create radar plots for different vehicle configurations.

One study which bears resemblance to this work involves using Dymola to study drivability in progressively-reduced driveline models (Dempsey, Biggs, & Dixon, 2005).

This work splits the vehicle model into four subsystems and defines multiple levels of complexity based on the number of lumped inertial elements in each subsystem. These varying subsystems levels are integrated in four different combinations and simulated for a tip-in transient torque response. Peak torque overshoot percentage and torque settling time were used in conjunction with the simulation speed to show that the second-most- complex model achieved almost the same performance characteristics as the highest

12 fidelity driveline model, while reducing the simulation time by 63%. The lowest-level models simulated much faster than even this model, however a more significant drop in accuracy was observed.

Backlash in the powertrain was studied in a modeling exercise between Ford and

Ohio State where the “clunk” effect of tooth impact was studied with experimental validation (Gurm, et al., 2007). The researchers varied backlash angles from 2-10 degrees in a tooth contact model and measured the peak-to-peak acceleration and the number of impact events. The authors note a general increase in the clunk measure with lash angle, but the amount required to notice a change varies considerably. If no double-sided impacts occur, increasing the lash regime does not affect the response, and sometimes decreasing the lash increases the acceleration RMS because multiple impacts are induced.

Overall metric evaluation and uncertainty was examined for many drivability and fuel economy tools in a study by Geller & Bradley (2012). This work aggregated fuel economy metrics and outlined the uncertainty bands for multiple computational methods, such as AVL DRIVE and a LabView economy measurement, showing that many fuel- economy metrics contain uncertainty levels near 10%. For more acceleration-based metrics, uncertainty comes down to the 2-5% range for output validation. This source also inspects cumulative power measures when using different solving methods, illustrating how errors of up to 0.7% in system electrical energy can result from using a lower-order solver.

2.4. Simulation and Software

High-fidelity modeling software exists to study systems at a component level through computational fluid dynamics or finite element/volume methods. At a systems level, insights on the internal structure of components (stress/temperature distributions) are not as useful as the overall behavior described by stiffness, inertia, and damping elements. One-dimensional rotational dynamics can be modeled by hand with ordinary differential equations and implemented in a generic mathematical modeling environment such as MATLAB/Simulink. This method is applicable in general to model any system of ordinary differential equations, yet in practice, this process does not scale to systems consisting of many components. Simulink is also a signal-based modeling environment, where values flow in directions specified by the modeler during the model creation.

These choices fix causality in one direction and thus the underlying physical relationships are translated into more-abstract mathematical equations.

To allow for a more scalable modeling endeavor, systems modeling software has been developed with applications to many dynamic systems. The automotive field is naturally taking advantage of this by employing commercial modeling software designed to take the mathematical load off the modeling engineers. One advancement that has enabled easier implementation has been the growth of object-oriented modeling languages. The most common example of this in dynamic system modeling has been the

Modelica language. Developed by the architect of Dymola, the Modelica language was made into a collaborative effort among the dynamic modeling community in the late

1990’s. The idea was that equations should be thought of as relationships, and the

14 known/unknown variables could be different depending on the problem. Dymola (and

Modelica) leave the causality of a dynamic system unspecified (“acausal” or “non- causal”) until preprocessing steps which allow the formulation of models without the need to rewrite the equations based on the direction of information flow. As Dymola grew and eventually adopted the unified Modelica language as its base language, the company made many advancements in the simulation of dynamic systems and has now sparked a new growth of object-oriented modeling and integration of Modelica into software packages (Elmqvist, 2014).

Recently, systems-level modeling environments have become more supportive of symbolic algebraic manipulation algorithms, allowing for the rise of acausal modeling as a commercial solution. Dymola has been a leader in this segment, but the Modelica language along with SimScape from The MathWorks have become more accessible in recent years with support from ESI, Wolfram, MapleSoft, Siemens, and the Open Source

Modelica Consortium. While most commercial implementations provide users with block diagram representations, the underlying (and sometimes accessible) code is increasingly written in the Modelica language. Commercial tools vary in their built-in libraries, analysis tools, preprocessing algorithms, and often solver options.

As mentioned above, one of the enabling algorithms of the Modelica language is the symbolic manipulation of equations to automatically arrange the system of dynamic equations into a reduced set (Iwagaya & Yamaguchi, 2013). This process is detailed by the team using MapleSim as an example, where the differential-algebraic equations are

15 substituted and rearranged to display a 96% reduction in simulation time with respect to the standard OpenModelica configuration.

This speed increase was also documented with Dymola (93% simulation time decrease compared to OpenModelica) while using the same type of DASSL solver in each software package (Floros, Bergero, Cellier, & Kofman, 2011). This source, along with a related work (Bergero, Floros, Fernandez, Kofman, & Cellier, 2012), also demonstrates how a new type of quantized state systems (QSS) solver can overcome the performance difference between Dymola and OpenModelica by implementing this algorithm instead of the standard DASSL solver. The work posits that implementation of this solver alongside the preprocessing capabilities of Dymola and other commercial packages would yield the fastest simulation of systems with many events. These works also showed comparable simulation accuracy, if not better, for the QSS implementation versus DASSL.

With respect to implementation, some sources use commercial software to do simulation profiling on the computational tasks and graphically examine task execution times (Anthony, 2013) whereas some older references manually separate multi-rated task scheduling (Hagiwara, Terayama, Takeda, Yoda, & Suzuki, 2002). Regardless of the method, when implementing large system models in which time scales can vary widely, the execution times of subsystems may also have disparate values. Thus, multi-rating models by scheduling fast tasks together and important or longer tasks on dedicated resources helps balance out simulation loads for hardware-constrained simulations. The ease of implementation for multi-rating varies wildly by the software in use. It can be as

16 straightforward as specifying a sample time for a subsystem block within Simulink or as complicated as writing custom task-level code for custom real-time applications. Some software, such as Concurrent Real-time Workbench integrates a task-scheduler to help the user arrange schedules manually and profile execution times.

Finally, a good example of modern HIL integration is displayed by a collaboration between ESI and dSpace (Klein, et al., 2017). In this work, the flexibility of the new dSpace architecture is used to integrate a GT-Power-RT engine model, dSpace

ASM Simulink vehicle model, and dual-clutch transmission from SimulationX into an application run in dSpace’s VEOS virtual environment. This kind of example is much more common in industry as commercial tools have added system analysis tools and model export options like the Functional Mockup Interface (FMI). Functional Mockup

Units can be generated as containers of models with integrated solvers for real-time implementation or porting to other modeling environments such as Simulink. These containers are parameterizable but contain the entire model description in non-editable code-form. Therefore, to alter the model configuration outside of the tagged inputs, outputs, and parameters, the model must be altered in the native modeling environment and re-compiled to FMI. This restricts debugging transparency, so the modeler should take care to verify model functionality in the native environment before export. The

SCALEXIO processing architecture and dSpace ConfigurationDesk allow for applications to be integrated and configured from different sources by adopting support for the FMI model standard.

2.5. Summary of Prior Work

The above sample of representative literature shows a distinct gap in prior work at the intersection of modeling for offline simulation and modeling for real-time testing.

This gap is accentuated in the field of transmission modeling, as engine, driveline, and chassis models have been investigated more-thoroughly to this point with respect to model reduction. Parametric studies have been done to produce mostly qualitative or graphical differences in output metrics to this point, with little work being done to quantify sensitivity values. This work will aim to supplement system-level transmission modeling work with a study of the real-time capability of various levels of fidelity in transmission models.

Studies which investigate applicable drivability metrics for real-world vehicle testing can be carried into the virtual testing realm to predict some of these behaviors.

This would be applicable for control design, but these can also be used to evaluate transient performance of a dynamic system. To accurately replicate experimental testing, drivability evaluation should be done on a fully-calibrated vehicle and powertrain, however, these metrics can also be used to help evaluate models in comparison to one another, without real-world validation.

Chapter 3. Requirements and Metrics

3.1. Functional Requirements

3.1.1. Stability

All final models, regardless of fidelity, must simulate continuously with no hard simulation errors, failures, or other events that prematurely interrupt the simulation. This must be true of models in any software environment where the models are intended to be used. If this requirement is met in native simulation software but fails to carry over the stability criteria to an external environment (say, Simulink), the model or solver settings should be adjusted in the native software to allow for universal stability of the exported model. Other options include an acceptance by the modeler that the model cannot be simulated in the exact same way (e.g. if solver is different) and settings such as the time step or convergence tolerance should be adjusted with appropriate notes. The expectation here is that the modeler would choose to stabilize the simulation at the cost of using different (albeit not necessarily worse) solver specifications. The modeler must make a concerted verification effort to ensure that this stability criterion is met for operating conditions including: steady state in all gears, step loading, gear shift events, virtual component “failure” (e.g. loss of supply pressure, loss of clutch friction, etc.), and any transitions between discrete dynamical states (e.g. stick-slip or hard stops).

3.1.2. Model Boundaries

All final models shall have boundaries drawn such that the torque converter pump shaft and transmission output gearset are both included in the model, as well as the mechanical structures in the torque path between these two components. From the actuation point of view, the model shall take the solenoid currents as inputs to the actuation model. A summary of the scope of the model in this work is shown below:

Figure 2: Model Boundaries/Scope

3.1.3. Software Interfacing

The intended modeling environments may end up being different between the

HIL-level transmission model and the offline MIL-level model. In either case, the final model form must be able to interface with the desired modeling environment successfully, thereby allowing the model to be simulated alongside other dynamic vehicle models. This requirement is twofold: signal ports must match the hooks expected by

20 other dynamically coupled elements and the model must be portable outside of its native software.

First, depending on the overall architecture of the dynamic vehicle model, the signal connections for the torque path may take on a couple of configurations. In acausal modeling software, a shaft element may take torque or speed as inputs, and the solver is responsible for propagating this choice throughout the model during simulation. If the vehicle model is in acausal form, this choice should be left to the software for optimal simulation. However, if the model is to be ported into a causal environment such as

Simulink or dSpace ConfigurationDesk, software hooks should be established to accommodate the most restrictive element of the dynamical system. This will likely not be the transmission in acausal form, meaning the transmission model must have adaptable causality to take inputs of torque or speed to the input and output shafts, depending on the scenario. This is straightforward for the gearbox model, as the

Modelica language allows for flexible definition of causality by simply changing the source type from torque to speed or vice-versa. For the hydraulic system and the torque converter, this may not be possible without some physical alterations. To change the source of the line supply from flow to pressure, the line pressure is thus eliminated as a dynamic state and its level is set as an input. To provide force or voltage to the solenoid models rather than current, modifications to the solenoid side of the solenoid valves will be necessary. In order to switch the causality on the torque converter from torque to speed inputs, the dynamic equations discussed later require re-arrangement or conversion to lower dynamic orders.

Second, the portability of the model from a software perspective is important to ensure maximum utilization in the future. Coupled simulations involving multiple native environments solving their respective pieces of the model is referred to as “co- simulation”. The so-called “master” simulation environment or software is the environment with the highest authority with which it controls the overall simulation. In offline simulation, this may be Simulink, even though the model also talks to GT-Power or SimulationX (“slave” environments) through S-functions. In online simulations for

HIL, it will likely be the dSpace software suite which configures and schedules the tasks.

Based on the choice of the master simulation software, the model should be able to be ported directly to the master software, co-simulated as a slave, or both. Co-simulation can be realized through the Functional Mockup Interface or through proprietary dynamical connection software, while simulation within the master either involves code generation

(if the master is not native transmission software) or no effort (if the model native software is used as the master). The speed benefits of co-simulation can vary depending on computational resources and task scheduling capabilities. If multiple modeling environments are open at the same time working on different parts of the same model, communication between these often involves some simplification and/or latency.

However, the net outcome could be positive if the native simulation environment is especially-suited for that section of the model (e.g. GT-Power solving a 1-D engine air path model and communicating with Simulink, which could not have solved such a system). These tradeoffs must be weighed on a per-project basis.

3.1.4. Fidelity

One goal of the transmission study, as defined in prior documentation, is to generate a high-fidelity model useful for offline simulation. This Model-in-the-Loop

(MIL) model shall be of equal or higher fidelity than the final HIL-level transmission model. Due to the nature of this modeling study, calibration of these models will not be the main focus. Therefore, it is possible that the final HIL-level model presented will have a higher dynamic accuracy than the MIL-level model. However, this requirement is set to ensure that the MIL-level model fidelity is only evaluated based on physical dynamic considerations. Specifically, the MIL-level model shall have a higher dynamic order than the HIL-level model as well as a higher overall bandwidth to characterize higher-frequency dynamics than the HIL-level model.

3.1.5. Real-Time HIL Model

The final HIL-level model should be real-time capable in the intended simulation environment to accommodate the physical components integrated into the simulation loop. Original specifications desired a model which was specifically under real-time on a specific HIL hardware system. This specification was deemed too restrictive, as this may miss out on key combinations of software and hardware that would allow higher-fidelity

HIL simulation. This is an important point for the design of the current system as well as future HIL networks which may not be tied to legacy architecture.

The HIL-level model may overrun during online simulation in some situations. As with other dynamic models, harsh instantaneous inputs or state events can induce numerical inefficiencies which take longer to handle. These types of overruns are

23 allowed, provided that the real-time simulation software can detect these overruns and they do not impact the stability requirement mentioned above. Sustained overruns are only permissible when accuracy of a given model can be verified to the set standards

(defined by metrics, discussed below).

3.1.6. Replicate Functional Behavior

The final versions of both xIL models shall replicate the highest-level functionality at steady state as specified in the input/output truth table specifications for the transmission under investigation. This means that the gear ratio of the virtual transmission should match the specified gear ratio when the appropriate input combinations of input currents are applied to the clutch actuation subsystem. Since the transmission control unit is not under investigation, the inhibits and other software- related events which would be affected by the order of input commands are inherently bypassed, allowing the physical transmission to be commanded at will.

3.2. Performance Metrics

Quantitative metrics have been chosen to characterize the simulated transmission performance. These may be applicable to the validation, calibration, and model reduction processes. They will be used to evaluate subsystem models as they are reduced to monitor changes in accuracy related to the model complexity, not as criteria for model reduction. This section has been separated by subsystem with metric definitions for each.

3.2.1. Clutch Engagement Hydraulics

The following are measures of the hydraulic responses that may be used for validation, calibration, and model reduction. Each of these pressure traces are measured at the output pressures of the hydraulic system where it interfaces the mechanical system.

Delay Time, 풕풅풆풍풂풚

Figure 3: Example Delay Time Measurement

The delay time is affected by all of the upstream connections before the output pressure. Delays in solenoid current, valve edge overlaps and preloads, clutch filling dynamics, or valve plunger motions may contribute to the overall pressure delay. This is defined as the time between the current rise above 2% of its steady state initial value and the rise of the pressure signal past 2% of the steady state operating pressure.

Total Rise Time, 풕풓풊풔풆

Figure 4: Example Rise Time Measurement

Total time from hydraulic pressure rise above 2% initial steady state value to reaching the final steady state value. This, along with the delay time, characterizes how long it takes for the pressure to reach the targeted value. This times the hydraulic system from start to finish, after which the clutch normal force becomes approximately constant.

“Corner” Time, 풕풄풐풓풏풆풓

Figure 5: Example Corner Time Measurement

This time is connected to the time it takes for the clutch piston to reach its hard stop, after which, the pressure rise is purely related to the pressure equalization through the orifice. This time starts at +2% of initial steady pressure and ends when the clutch piston reaches its end stop.

“Corner” Pressure, 푷풄풐풓풏풆풓

Figure 6: Example Corner Pressure Measurement

The pressure difference between initial steady state pressure and pressure at 푡 =

푡푐표푟푛푒푟. This shows how much the pressure has risen when the piston reaches the end of its travel and the clutch disk makes contact.

Percent Overshoot, 푶푺%

Figure 7: Example Overshoot Percentage Measurement

The pressure rise in the clutch chamber has spring elements and fluid compressibility, making overshoot possible. However, this value may be low, as the application is a ramp function as opposed to a pure step. This overshoot amplitude should be characterized such that an accurate trace of normal force is transmitted to the friction plate, which would affect the stick-slip curve behavior. The percent overshoot is defined as in a 2nd-order dynamical system as:

(푃 − 푃 ) %푂푆 = 100% × 푝푒푎푘 푓 (1) 푃푓 − 푃𝑖푛𝑖푡 where 푃𝑖푛𝑖푡is the initial pressure level, 푃푓 is the final settled pressure value, and 푃푝푒푎푘 is the maximum pressure level after 푡 = 푡푟𝑖푠푒.

3.2.2. Torque Converter Metrics

Experimental models of the torque converter have been derived by measuring the torque ratio and capacity factor as a function of the speed ratio across the torque converter. Therefore, by definition, the simple experimental model would be accurate at steady state. In this project, a dynamic torque converter model is developed from first principles, with the goal of matching these steady state maps after calibration. The steady state characteristics of interest are as follows:

Torque Ratio at Stall, 푻푹풔풕풂풍풍

Figure 8: Example Stall Torque Ratio, adapted from (Robinette, Grimmer, & Beikmann, 2011)

The stall ratio is related to the geometry of the torque converter and is inversely related to the efficiency below fluid coupling. High torque ratios (near 2.5) generally have worse efficiencies, but the stall ratio may be as low as 1.5. This stall ratio should be accurate for an empirical model, as this information would be embedded in the 푇푅 푣푠. 푆푅 curve.

Speed Ratio at Fluid Coupling, 푺푹풄

Figure 9: Example Fluid Coupling Speed Ratio Measurement, adapted from (Robinette, Grimmer, &

Beikmann, 2011)

This is the speed ratio at which the torque ratio reaches a value of 1, thereby ending the torque multiplication phase of operation.

Peak Amplification Efficiency, 휼풑풆풂풌

Figure 10: Example Peak Efficiency Measurement, adapted from (Banerjee & McPhee, 2012)

Efficiency curves can come from multiplying the torque and speed ratios, whether from a steady state or dynamic test. This metric, the peak efficiency during the torque amplification phase, comes from taking the maximum value of the efficiency curve while the torque ratio is greater than 1.

Speed Ratio at Peak Efficiency, 푺푹휼

Figure 11: Example Peak Efficiency Measurement, adapted from (Banerjee & McPhee, 2012)

The speed ratio at which the maximum torque amplification efficiency is 푆푅휂.

This helps to characterize the shape of the efficiency curve, to show if the peak is leaning high or low on the speed ratio axis.

Root-Mean-Squared Capacity Factor Error, 푹푴푺푲푬

Figure 12: Example RMSKE Measurement, adapted from (Robinette, Grimmer, & Beikmann, 2011)

As mentioned above, an algebraic model for the torque converter based on torque ratio and capacity factor will be, by default, accurate when compared with experimental data at steady state. The capacity factor is related to damping behavior of the torque converter in an inverse relationship (Robinette, Grimmer, & Beikmann, 2011) and the steady state 퐾 error will be calculated via RMS error. This will be defined as a point-to- point root-mean sum of squares:

푁 2 1 (퐾 − 퐾 ) 푅푀푆퐾퐸 = √∑ 푠𝑖푚,𝑖 푡푟푢푒,𝑖 (2) 푁 퐾2 𝑖=1 푡푟푢푒,𝑖 where 푁 is the number of points in each trace, 퐾푠𝑖푚 and 퐾푡푟푢푒 represent the simulated and experimental capacity factor curves, respectively, and the index variable 푖 iterates through the two vectors.

Root-Mean-Squared Torque Ratio Error, 푹푴푺푻푬

Figure 13: Example RMSTE Measurement, adapted from (Robinette, Grimmer, & Beikmann, 2011)

Note the definition of 푅푀푆퐾퐸, above. The analogous RMS error for the torque ratio curves is given by:

푁 2 1 (푇푅 − 푇푅 ) 푅푀푆푇퐸 = √∑ 푠𝑖푚,𝑖 푡푟푢푒,𝑖 (3) 푁 푇푅2 𝑖=1 푡푟푢푒,𝑖

Here, similar definitions apply as to the RMSKE variables listed above. The “sim” and

“true” torque ratios correspond to the simulated and experimental values of the two vectors at each point, while the index 푖 steps through 푁 points.

3.2.3. Overall Shift Metrics

Shift quality metrics have been the topic of discussion in many resources aiming to predict drivability early in the driveline design process. Drivability has agreed-upon features, but true validation and evaluation has typically been performed by expert drivers in a representative vehicle. Many sources and software packages aim to replicate 35 this inherently subjective evaluation, while others aim to draw out the components of the dynamic response that may contribute to the subjective score.

Below, drivability metrics are proposed for quantification of shift events using physical signals and without aiming to explicitly predict “shift quality”. Future applications of this research may require accurate dynamic characterization of prototype vehicles in a virtual environment. This activity would benefit from having the capability to predict at-limit or safety-critical behavior as well as standard operational test modes such as the EPA FTP75 and US06 driving cycles. For these tests, shift quality as it relates to torque (ignoring auxiliary dissipative effects such as acoustic noise) would directly affect performance in these modes. With a fully integrated control system in the loop, having the correct physical responses to driver inputs would greatly increase the likelihood that aggregate evaluation metrics will be consistent with experimental tests

(e.g. number of shifts or time in each gear for EPA drive cycles with an automatic transmission controller in the loop).

Torque Hole, 푻풉풐풍풆

Figure 14: Example Torque Hole Measurement, adapted from (Kulkarni, Shim, & Zhang, 2007)

The torque hole is the magnitude of the torque signal dip below the final output torque. Percentages were not used here in as they would be misleading at low torque values. The torque trace in question will be measured at the output shaft torque, as opposed to the wheel, to ensure these measurements are on similar scales regardless of downstream vehicle architecture. Where 푇푚𝑖푛 represents the minimum value of torque after the initial drop and 푇푓 is the final settled value, the following defines the torque hole:

푇ℎ표푙푒 = 푇푚𝑖푛 − 푇푓 (4)

Torque Recovery Overshoot, %푶푺푻

Figure 15: Example Torque Overshoot Measurement, adapted from (Kulkarni, Shim, & Zhang, 2007)

After the torque hole, the recovery will contain a compliant response with low damping due to the elastic torque build-up in the system. The torque may bounce back past the final operating torque, leading to an overshoot. This is measured as:

푇푂푆 − 푇푓 %푂푆푇 = 100% × , 푇푓 − 푇푚𝑖푛 (5) where 푇푚𝑖푛 is the minimum torque value before it recovers (as mentioned above), 푇푓 is the final settled torque value, and 푇푂푆 is the maximum torque value after recovery from the torque hole.

Settling Time after Recovery, 풕풔풆풕풕

Figure 16: Example Torque Settling Time, adapted from (Kulkarni, Shim, & Zhang, 2007)

To complete the torque recovery characterization, the settling time will be measured as the time from first recovery (푇 = 푇푓) to when the output shaft torque becomes bounded within 2% of its final value. This settling time is highly dependent on the stiffness and damping of the transmission while completely engaged (purely oscillatory, possibly including backlash). High damping (more internal friction) will lead to a faster settling time.

Torque Phase Duration, 풕풕풓풒

Figure 17: Example Torque Phase, adapted from (Coric, Ranogajec, Deur, Ivanovic, & Tseng, 2017)

The torque phase is characterized by the transition of torque from one clutch

(offgoing) to another (oncoming). During this phase, torque changes path in the transmission, but the gear ratio and output speeds have not yet been affected. The duration of the torque phase is shown to be the time from the start of the shift (first torque decline) to time when the offgoing clutch torque drops to zero.

Inertia Phase Duration, 풕풊풏풆풓

Figure 18: Example Torque Phase, adapted from (Coric, Ranogajec, Deur, Ivanovic, & Tseng, 2017)

Following the torque phase, the torque recovery is achieved once the torque in the off-going clutch has dropped to zero and the speeds are given time to come to their final values in the new gear. This will be defined as the time it takes for the oncoming clutch relative speed to reach zero (engaged) from the start of its drop. This phase is usually longer than the torque phase, as the inertias play a role in allowing the speeds to change to their new values.

Peak Jerk Torque, 푻̇ 풑−풑

Normally taken as longitudinal jerk directly from an accelerometer on the vehicle, jerk can also be calculated from the torque trace, by dividing out the inertia

(normalization) and differentiating with respect to time. This metric is highly correlated with shift feel, as the driver usually experiences hard jerks with a shift event or throttle

41 maneuver. This metric will be determined by the maximum peak-to-peak measurement of the torque derivative across the duration of the shift.

3.2.4. Summary of Metrics

The values of the above metrics change depending on the intended uses of the model under evaluation. For offline control development, it is more important to capture the dynamic response accurately, which may place weight on most of these metrics plus others. For online control verification, the purpose of the model in the control loop is to provide plausible feedback to the control unit, meaning that the intermediate dynamics of some of the states is less important to the test.

When evaluating these metrics through model reduction (Chapter 7), it is important to remember that some of the intermediate dynamics will likely deviate, but the overall behavior is more critical to maintain at the HIL-level model. For the hydraulic system, this means that the delay time and rise time are the metrics which should be important at all model levels, while the corner-feature measures may not be influential on the control verification step. For the gearbox, the gear-shift timing is important to testing software inhibits which prevent actions from occurring until after shifts, for example. For this reason, the HIL-level model will be more concerned with torque and inertia phase durations than the drivability-focused metrics such as the jerk torque and settling time.

Finally, the torque converter model on a HIL verification bench should be able to provide an accurate stall torque ratio and fluid-coupling point to capture the important points of the torque amplification phase. The capacity factor is more related to damping and drivability, and the overall efficiency may not be of concern to control verification.

Chapter 4. High-Fidelity Model Development

The transmission model is best examined when broken down into three subsystems for modeling purposes: the hydraulic circuit, the mechanical gearbox, and the torque converter. The hydraulic system accepts electrical signals for the solenoid valves and provides pressure to the hydraulic clutch pistons. The gearbox model in this work includes the clutch piston models, which are hydraulically actuated, and the rotational inertias within the transmission along with the coupling elements (gears and clutches).

Finally, the torque converter transmits power from the input source to the input shaft of the transmission. The high-fidelity modeling process separates each of these subsystems and works from the component level and through the interface level.

The modeling environment for this project has been chosen as SimulationX, from

ESI ITI GmbH. This software is built upon Modelica as the base acausal, object-oriented language, as discussed previously. SimulationX includes the standard Modelica library which is available as open-source, along with proprietary libraries with additional options and configurations not available through the standard Modelica blocks. ESI writes its own code on top of the base Modelica classes to provide blocksets for automotive, power electronics/generation, thermal fluids, and many more applications. The libraries utilized for this modeling project primarily come from the Mechanics, Power Transmission (1-D),

Hydraulics, and Signal Blocks libraries.

SimulationX contains built-in options for many of their proprietary blocks which allow the modification of the underlying models with simple switches and settings. For example, end-stop blocks may be defined with stiffness and damping elasticity or a coefficient of restitution for impacts. These selectable options help to configure models to various levels of fidelity without exchanging blocks every time a small change is made.

While SimulationX does allow the construction and simulation of 3D multi-body structures and distributed fluid inertial models, this work will focus on 1-D mechanical elements (rotational or translational) and lumped zero-dimensional fluid flow models

(restriction-reservoir orifice flows). While the provided libraries are extensive, there are times when a combination of options are not available in a certain configuration, so custom groupings of components are combined to form a custom class called a

“compound” model. These are saved and kept available alongside the traditional

SimulationX libraries and may be implemented as reusable model pieces with multiple copies containing different parameters within the same model. At the most-extreme with respect to customizability, Modelica classes can be written from the ground up in the text editor via extensions of existing classes or original work.

4.1. High-Fidelity Hydraulic Network

The hydraulic network is built up from valve component models to capture the dynamics of every part of the circuit. Some of these valves are best modeled through custom compounds in SimulationX. These include the solenoid valves, of which there are two types: “linear proportional” solenoids and “shift” solenoids. Most remaining valves

44 in the hydraulic circuit are one-off valves with only one implementation. Such valves are still saved as custom classes in SimulationX and parameterized with the relevant parameters exposed to the subsystem mask (available to edit as custom component properties).

The hydraulic subsystem is composed as a hierarchy of sub-models down to the component level. Components are combined to create valves, valves are combined to create circuits, circuits are combined into the overall hydraulic network, and the hydraulic network subsystem is interfaced with the other subsystems with only necessary input and output ports exposed. This overview is shown below:

Figure 19: Overall High-Level Hydraulic Network Breakdown

Many types of fluids and their properties are tabulated within SimulationX.

Transmission fluids have two options from the list, and Dextron VI was chosen as the closest comparative transmission fluid. The temperature of the fluid was held constant through all simulations at 40℃. Dynamic temperature changes occur in the transmission during operation, but a full heat transfer model was not developed in this work.

The fluid is treated as compressible, with a tabulated nominal bulk modulus, but trapped air is also included in the fluid in one of multiple ways. In the present work, the trapped air is set at a nominal volume percentage, and the amount that is dissolved in the fluid varies with the static pressure of the fluid. This creates a nonlinear relationship with the fluid compressibility (effective spring constant), but since it is algebraically dependent on the static pressure level, this relationship does not add any dynamic states.

The fluid properties which vary as a function of temperature include the viscosity, density, maximum gas solubility, and fluid compressibility. The fraction of undissolved gas a major influence on the fluid compressibility, and is calculated according to the

SimulationX documentation as:

훼푈 = 훼푈,푟푒푓 − (훼퐷,푚푎푥 − 훼푉) (6)

In the above equation, 훼푈,푟푒푓 is the nominal undissolved gas fraction at standard pressure and temperature. The parameter 훼푉 is the Bunsen Coefficient and is tabulated as a part of the SimulationX fluid library. Calculating this fraction online during the simulation gives a more realistic fluid behavior when work is done on or by the fluid. This fraction is used in a weighted mixture equation for the bulk modulus of the oil and gas mixture.

What follows is a breakdown of some of the representative components in the hydraulic network. Example valves are shown from the group of linear solenoids, regulation valves, and switch/cut valves which check certain pressures via thresholds. In general, each valve consists of a spring-mass-damper element with specified preload and/or maximum stroke length. The displacement of the valve plunger is coupled to valve

“edge” elements from the SimulationX Hydraulics Library, which use parameterized geometric relationships to calculate the flow area as a function of plunger displacement.

These “edge” elements provide a coupling between the valve position and the fluid flow and maintain the acausal relationship by only controlling the orifice dimensions. Whether the source/sink connections of the edge are pressure or flow sources, the valve maintains the same internal relationships while adapting the causality direction depending on the supplied variables (given flow, calculates pressure drop and vice versa).

4.1.1. Solenoid Valves

The electronic control unit interfaces with the hydraulic network subsystem through solenoid-operated valves. Linear proportional solenoids are used to modulate the hydraulic pressure delivered to the power clutches and torque converter lockup clutch.

Flow-control switch (“shift”) solenoids operate more like switches and are used to redirect flow in the network. Each of the linear and shift solenoids receives current inputs from the transmission controller, which ranges from zero to one ampere.

Specifications were laid out for the linear solenoid output pressure given input current and supply pressure. These specifications were used to develop target steady-state values using assumptions about spring-mass-damper characteristics. Physical

47 measurements were not available for the internal solenoid geometry, so the target pressures were reached by choosing spool mass, spring stiffness, spring damping, and spring preload for the translational system in the solenoid valve.

For the high-fidelity valve model, two proportional edge components were coupled to a spring-damper element, mass element, and resistor-inductor solenoid model, chosen from the SimulationX Gearbox Actuation library. The control current input is transformed into a voltage which is supplied to the dynamic solenoid model, which outputs a force from a current-force map in the component block. The current-force map was tuned to provide adequate force at each known steady state point from the hydraulic specification.

The linear proportional solenoids function by providing an output pressure feedback pathway for the fluid to push back against the solenoid force. This feedback action allows the variable orifice to open until the feedback pressure becomes high enough to close the orifice again. This holds the output pressure as approximately linearly proportional to the applied force by the solenoid, which is desirable from a clutch pressure control perspective. An example schematic is shown below, followed by the

SimulationX block diagram representation.

Figure 20: Linear Proportional Solenoid Schematic, adapted from (Lee, Sung, & Kim, 2012)

Figure 21: High-Fidelity Linear Proportional Solenoid Model

Both proportional edge models were approximated by a sharp-edged cylindrical orifice. The edge model includes Bernoulli flow-force feedback on the spool, imparting kinetic energy of the fluid as a force in the direction of the flow entering the valve chamber.

So-called “shift” solenoids are similar in nature to simple ball valves, but they are actuated by a solenoid integrated into the valve. These behave like a variable orifice opening without feedback pathways present in the linear solenoid model above. An example schematic is shown for the shift solenoid valves below:

Figure 22: Example Shift Solenoid Schematic (Watechagit & Srinivasan, 2003)

4.1.2. Regulation Valves

The pressure regulation circuit includes the hydraulic fluid flow source, main regulation valve, lubrication regulation valve, and torque converter regulation valve.

Each of these spool valves is connected at multiple fluid ports, and the internal geometry dictates the linear motion resulting from the interacting fluid pressures. The fluid connections can be sources, outputs, or feedback pressures. A generic schematic of an example regulation valve is pictured below. The spool-type valves in the regulation circuit behave similarly to this example valve, however the number of connections and orifices in the feedback pathway vary from valve to valve. 50

Figure 23: Example Regulation Valve, adapted from (Marano, Moorman, Whitton, & Williams, 2007)

For the main regulator valve, fluid flow is provided to the entire hydraulic network through an ideal flow source. In this application, a fluid pump is not modeled explicitly. The fluid supply flow is split among the components connected to it, but the main regulator valve is responsible for keeping line pressure at the target valve, regardless of the opening and closing of other flow paths in the hydraulic system. The supply and feedback paths within the valve open and close three proportional edge models which connect: 1) regulated line pressure to the regulated lubrication pressure, 2) regulated line pressure to the regulated torque converter and lockup clutch supply pressure, and 3) the fluid pump suction level to the torque converter and lockup clutch supply. Each of these has a specified overlap designed into the valve and included in the known geometric specification. The overlap specification also calls out a chamfer on two of the edges, which is available as a geometric option within the edge models in

SimulationX. The chamfer size is used to calculate the variable flow area in terms of the spool displacement. The chamfers effectively crack open the connection earlier than a

51 sharp-edged opening, but at a very small flow area. This valve is shown in block- diagram-form below.

Figure 24: High-Fidelity Main Regulation Valve Model

The lubrication regulation valve regulates the supply pressure to the lubrication circuit in the transmission. This is done by modulating connections between the output lubrication pressure and the “suction” pressure at the inlet of the oil pump. The output lubrication pressure is also fed back through an orifice to self-regulate based on the feedback piston area. A chamfered overlap is specified for one of the edge models, leaving the other as a sharp edge. This regulation valve is shown in block diagram form below.

Figure 25: High-Fidelity Lubrication Regulation Valve Model

The torque converter lockup clutch supply pressure is regulated by the torque converter regulator valve. The torque converter regulation valve has two specified orifices in the feedback pathway, which must be coupled by an intermediate fluid volume. This way, the blocks are connected with so-called “reservoir” elements (volume) between every “restriction” element (orifice). These small volumes do, however, contribute to the total number of dynamic states. The single edge connection in this valve is chamfered and specified as such. The torque converter regulator valve is shown below.

Figure 26: High-Fidelity Torque Converter Regulation Valve Model

A line pressure accumulator valve is incorporated to the regulation circuit to act as a damper to higher frequency line pressure oscillations. This valve oscillates with the line pressure actuating the piston that opens the connection from lubrication pressure to suction. Opening this connection drains the lubrication pressure and causes the other regulation valves to work to restore it.

Figure 27: High-Fidelity Line Pressure Accumulator Valve Model

Finally, the torque converter lockup clutch has its own regulation valve which keeps the supply pressure to the lockup clutch within a range of controllability for the lockup clutch engagement. There are two orifices in the feedback pathway for this valve, which gives it a larger number of states than many other valve components. The “SRC” pressure adds a force that partially pressurizes the output, while the “SOL” pressure source from the corresponding linear solenoid controls the pressure at a finer resolution.

Figure 28: High-Fidelity Lockup Clutch Control Valve Model

4.1.3. Switch Valves

Switch and cut valves operate more as binary pressure switches, as opposed to the more continuous outputs from the linear solenoids and regulation valves. These valves take a pressure signal as an input and use a spring stiffness and preload to oppose the valve opening until the supply pressure reaches a designed threshold pressure. The spring stiffness and preload values were provided in the specification for the hydraulic circuit, as 55 well as the target threshold pressure. Some of the valves were designed to pass through the command pressure once it reaches a threshold, while some use a third command pressure to link a source and sink together.

An example of a simple switch valve is the following, which uses mechanical preload to set a threshold pressure to pass-through the input pressure to the output.

Figure 29: Selected High-Fidelity Switch Valve Model

A significant component in the cut-valve category is shown below, which can cut pressure supply to four different locations (on three connections) with the actuation of one shift solenoid. The lines which may be switched to drain pressure are clutch supply lines for three different power clutches and the output pressure from a linear solenoid which normally controls the main regulator valve. All six of the proportional edge models for the flow connections are treated as sharp-edged orifices with the Bernoulli flow force considered. This valve is modeled using a spring-mass-damper mechanical system, with

56 an elastic end stop. The end stop is to ensure that the valve does not travel too far in any direction and open up a larger orifice area than physically realizable. In this case, no stroke distance was specified, so reasonable assumptions were made based on the range of comparable valves with stroke dimensions.

Figure 30: High-Fidelity Cut Valve Model

4.1.4. Two Way Clutch Valve

The two-way clutch (TWC) valve contains no explicit linear springs, but instead relies upon detent mechanisms to provide a piecewise stiffness profile. Four supply pressures push against each other on two sides of the valve to determine its position.

Pressures from two shift solenoids push the valve in the direction of “drive”, while two more shift solenoids push the valve toward “reverse”. The cross-sectional areas of each of

57 the piston chambers are designed to be approximately equal, so the position of the plunger is effectively dependent on a signed summation of the input pressures.

The dual detent mechanisms provide variable resistance for the spool motion.

Pressure variations in the hydraulic system happen due to the dynamics of opening and closing of orifices, so the detents allow some travel on each end with a threshold required to push it across the separating hump in the middle. The rolling balls provide little translational resistance when the spool profile is flat, at either height. The ramp edge requires that the detent ball springs be compressed gradually to make it onto another flat portion of the spool profile. This acts like a spring stiffness that is variable with the spool displacement, with a backlash behavior on the ends. The detent mechanism model comes from a built-in SimulationX library for actuation. It includes a support profile definition to provide data points along the spool length, as well as ball size. The spring is defined as a spring and damping element with preload pressing the ball into the spool to maintain contact during transients. The profile is modeled by tracing a closed polygon and defining ball geometry, as shown below from the SimulationX documentation.

Figure 31: Diagram of Detent Mechanism Definition (ESI ITI GmbH, 2017)

Figure 32: High-Fidelity Two-Way Clutch Model

For this transmission model, the detent mechanism was slightly simplified by assuming that the ball maintains contact with the support at all times. This may not be entirely accurate in situations with high pressure gradients, but the effect will not be noticed as the missing dynamics would be seen normal to the translational path, not having a high impact on the resistance forces. The simplified model also provided stability benefits to the initialization and simulation of the TWC valve in isolation. There

59 were situations when the model would not initialize due to the ball vertical displacement, so this state was removed to resolve these issues at minimal cost.

In this model, the stroke sensors are not modeled, and instead the output of the

TWC valve is a mechanical position, read and translated by the TWC in the gearbox model.

4.1.5. Hydraulic Modeling Summary

As presented above, physical valve schematics were translated into lumped models for valve plunger motion and fluid flow between orifices. At the highest level of fidelity, each valve in the hydraulic is modeled as a spring-mass-damper system with at least one additional state for pressure (depending on the number of connections). With the number of valves and other components (some of which are not detailed here), the total number of dynamic states for the hydraulic system totals 137. Some of these states are key to supplying pressures to the clutches, while others simply dictate whether fluid pressures have reached a high-enough threshold to pass through. Reduction of the hydraulic system will seek to maintain accuracy in the output pressure traces as various sections of the hydraulic system are simplified.

4.2. High-Fidelity Gearbox

The mechanical gearbox and clutch piston cylinders are combined into the

“gearbox” subsystem. The rotational part of the gearbox consists of eight rotational inertia elements, four planetary gearsets, and six friction clutches (three each of disk and

60 brake varieties). In addition to the standard friction clutches, a special “two-way” locking clutch operates as a selective one-way clutch in certain situations.

When all connections are considered rigid, four of the eight rotational inertia elements are found to be dependent, meaning that their motion is predetermined and not actually a true degree of freedom. This means that three clutches must be locked to fully define an input-output relationship and leave one degree of freedom for one-dimensional rotation. This can mean either three friction clutches are engaged or two friction clutches and a locked two-way clutch.

4.2.1. Planetary Gearset

SimulationX provides a detailed base to allow gear teeth to be defined geometrically through helical gear interaction sub-models. There are explicit helical gear- pair models available, but there is a framework designed for planetary gearsets, which model each helical interaction separately. This provides the freedom to individually define the planets, carrier, sun, and ring gears with inertias, tooth stiffness, backlash, and bearing effects.

A planetary gearset package is defined with the three input rotational connections, and in this case, the elements are lumped masses accounted for outside of the planetary package. The planetary inertias are included in this planetary gearset model both as rotational inertias about their own axes and about the main rotational axis of the planetary package.

Figure 33: High-Fidelity Planetary Gearset Model

The planetary inertias are computed based on an assumption of a solid disk and

1 using 퐼 = 휌푉푟2 to calculate the moments of inertia about their own axes, with 푟 푠푒푙푓 2 being the mean radius of the planet gear. The moments of inertia about the central axis of

2 the planetary package were calculated using 퐼표푟푏𝑖푡 = 휌푉푅 where 푅 is the distance between the sun axis and planet axis (carrier radius). Using the known gear geometry to estimate these values, the following relations are developed:

1 1 1 푍 푚 4 퐼 = 휌푉푟2 = 휌푏 휋푟4 = 휌푏 휋 ( 푝 푛) 푠푒푙푓 2 2 푤 2 푤 2 (7)

푍 푚 2 퐼 = 휌푉푅2 = 휌푏 휋푟2푅2 = 휌푏 휋 ( 푝 푛) 푅2 표푟푏𝑖푡 푤 푤 2 2 푍 푚 2 푚 = 휌푏 휋 ( 푝 푛) ( 푛 (푍 + 푍 )) 푤 2 2 푝 푠 (8)

In the above (final) equations, 휌 is the material density, 푏푤 is the overall width of the planet gears, 푍푝 and 푍푠 are the number of teeth on the planet and sun gears, and 푚푛 represents the module of the gears.

4.2.2. Friction Clutch

The friction clutch model in the mechanical subsystem of the gearbox has multiple options for representation in SimulationX. One of these options is a pre- packaged hydraulic clutch, which connects (acausally) to a hydraulic pressure port and couples the friction disks based on the normal force generated by the hydraulic pressure.

This is a simplification of the clutch actuation system, which does not allow for custom setting of spring/backlash behavior of the clutch piston. A more representative hydromechanical coupling element was thus created in a custom block, described below.

The friction clutch portion of the clutch was thus separated from the actuation method by choosing a SimulationX library block for a disk clutch which takes normal force as the input. This clutch piston force is calculated externally, and then routed to the clutch disk blocks using “signal” connections, which fixes causality in the chosen direction. This breaks the overall system into causal subsystems which exchange information in a predetermined direction, which is an appropriate simplification for the force application on the clutch piston.

Figure 34: Clutch Plate Diagram and Parameters (ESI ITI GmbH, 2017)

The clutch model assumes that the interaction between the friction surfaces happens with a uniform pressure distribution, no asperity contact, no fluid film squeeze effects, and simultaneous contact of all friction surfaces (푁푓). The straightforward one- dimensional friction torque is calculated by using a parameterized friction coefficient slip curve.

1 푑3 − 푑3 표 𝑖 (9) 푇푓푟𝑖푐 = 퐹푎푝푝휇(휔푠푙𝑖푝)푓𝑖푁푓 2 2 3 푑표 − 푑𝑖

In the above equation for transmitted friction torque, the friction reduction factor,

푓𝑖, is a way to reduce the friction coefficient externally (manually) or via an internal

푁 empirical function of the number of friction surfaces: 푓 = 1.01 − 푓 . The number of 𝑖 100 friction surfaces, 푁푓 multiplies the friction torque, while the inside and outside plate diameters (푑𝑖 and 푑표) are used to get an effective radius of the friction force. The applied normal force, 퐹푎푝푝 is the input to the function. The other varying factor in the above torque equation is the friction coefficient, 휇. When the slip speed is zero, this coefficient is fixed at 휇0, the sticking friction coefficient. At nonzero slipping speeds, the coefficient

64 varies with the slip speed and is partially determined by the materials involved in the interaction. SimulationX has tabulated empirical coefficients for continuous functions of

휇(휔푠푙𝑖푝) based on the selected materials. These are of the form:

휇(휔푠푙𝑖푝) = 휇0 − (휇0 − 휇퐾1) tanh(휔푠푙𝑖푝퐾2) + 휔푠푙𝑖푝퐾3 (10)

The coefficients 퐾1−3 are tabulated internally in SimulationX, otherwise the user may define custom coefficients or bypass these functions with simplified stick-slip behavior or custom slip functions. In addition, the nominal values of the slipping and sticking friction coefficients (휇 and 휇0) must be provided to fully define the slipping friction coefficient above. These are estimated from testing data and varied as part of the parameter sensitivity study below. The clutch geometry with respect to the number of friction surfaces and inside and outside disk diameters are known and held constant through any simulations discussed below.

4.2.3. Two-Way Clutch

Inside the gearbox, the two-way clutch (TWC) is a one-way clutch which can lock up on command from a hydraulically-controlled valve. In normal operation, the TWC overruns in one direction but locks when the clutch wants to rotate in reverse. In first gear, the shaft on the hub of the TWC wants to rotate in reverse, but the TWC locks it in this direction. This fixes one of the degrees of freedom of the gearbox and thus requires one less friction clutch to define the gear ratio.

The same is true when the transmission is shifted into reverse. In reverse, the

TWC valve is pushed by a commanded hydraulic pressure and mechanically switches the

TWC from a one-way clutch to a fully locked connection.

There are no options for a selectable one-way clutch in block-diagram form, so the TWC behavior was modeled using custom Modelica code as an elastic end stop. The position of the TWC valve was input to the TWC lockup code to create a conditional equation for the torque applied by the TWC connection.

When the hub of the TWC was rotating in the positive direction, no additional torque was applied to the element. However, when the speed dropped below a threshold

(most obviously, zero speed), an external torque was applied in the positive direction, preventing the hub from gaining negative speed. This is according to the following elastic torque definition:

휏푇푊퐶 = −푏푇푊퐶휔푇푊퐶 − 푘푇푊퐶휃푇푊퐶 (11)

The hub rotational angle (휃푇푊퐶) was incorporated to this equation to potentially smooth the lockup of the TWC by adding an integral effect and simulating stiffness of the connection. This rotational angle had to be reset every time the speed dropped below the locking threshold to avoid build-up of torque during forward freewheeling. Without the stiffness element, the TWC torque may oscillate or go unstable if the values of the damping coefficient are not chosen correctly.

4.2.4. Hydraulic Clutch Actuation Cylinder

The bridge between the hydraulic network and mechanical gearbox is the set of hydraulic clutch cylinders. The hydraulic pressure is applied to a volume element on the backside of a piston, which generates a force. On the opposite end of the piston, a spring- damper element is present. Instead of including it with the spring element, the preload force is separated into a separate external force on the piston, which pushes it back

66 against its seat (away from disk engagement). After overcoming the preload force, the generated piston force must traverse a backlash regime installed to maintain a gap between the friction surfaces when the clutches are not engaged. After the spring has been slightly compressed and traverses this gap, it begins engaging the clutch plates. At this coupling point, the hydraulic pressure force (reduced by the preload) is applied through the spring and onto the friction plates as a normal force. This normal force is used to determine the friction torque within the friction clutch model. This setup is shown below:

Figure 35: High-Fidelity Disk Clutch Hydraulic Actuation Cylinder Model

As shown above, a “force sensor” block is placed between the end of the backlash and the fixed housing reference. This measures the force applied to the clutch plate by the piston after it traverses the lash regime. This signal is sent causally to the corresponding disk, brake, or lockup clutch.

The other two elements present in the above model of the clutch cylinder are forces acting on the piston due to hydraulic interactions in the cylinder. Since the disk clutches are integrated into the mechanical gearbox, the hydraulic pressure is applied to a rotating chamber. The fluid rotating within this chamber exerts a force on the piston by itself which, if unchecked, would start to apply the clutches unintentionally. This centrifugal force by the fluid is modeled as an external force as a function of the rotational speed of the clutch cylinder.

휋휌휔2 퐹 = (푟4 − 푟4 − 2푟2(푟2 − 푟2)) (12) 푐푒푛푡 4 표 𝑖 푝 표 𝑖

In the above equation, the inside and outside radii are of the hydraulic cylinder, while 푟푝 is the fluid entry radius to the cylinder. The density term here (휌) is of the fluid.

To combat this centrifugal effect in the clutch cylinders, an extra fluid chamber was designed into the clutch actuation system which provides a similar effect in the opposite direction, partially cancelling the effect. This cancellation effect is modeled in the system via an opposing external force to the centrifugal fluid force. In the equation below, the subscripts 푐 denote that they are measurements for the cancellation chamber.

휋휌휔2 퐹 = (푟4 − 푟4 − 2푟2 (푟2 − 푟2 )) (13) 푐푎푛푐푒푙 4 표,푐 𝑖,푐 푝,푐 표,푐 𝑖,푐

The brake clutch cylinders are stationary with respect to the transmission housing because they only brake one rotating component rather than coupling two rotational elements. For this reason, there is no centrifugal force exerted by the fluid, as shown below for the brake clutch subsystem.

Figure 36: High-Fidelity Brake Clutch Hydraulic Actuation Cylinder Model

4.2.5. Gearbox Modeling Summary

The highest number of dynamic states in a gearbox component belongs to the planetary gearsets. Each planetary element has four planets which contribute to the number of states via four degrees of freedom each (angle and velocity about itself and about the central axis). Each gear-tooth interaction also incorporates a material-based estimation of tooth stiffness and damping to make impacts elastic after traversing small backlash distances. The clutch piston compounds are represented with spring-mass- damper dynamics and interact with a centrifugal fluid force from the rotating fluid inertia within the clutch cylinder. Finally, the clutch friction characteristics use empirical functions to create a speed-dependent slip curve for the friction coefficient. This avoids the assumption of constant sticking and slipping coefficients used in many simplified transmission models.

4.3. Torque Converter

The three-element dynamic torque converter model was implemented based on the ordinary differential equations for the three rotational elements and an additional 69 equation for the fluid inertial dynamics. These equations, as discussed previously, are taken from the one-dimensional hydrodynamic equations derived by Hrovat and Tobler

(1985). The major pieces of this model are as follows, with detailed parameter definitions available in the provided reference (Hrovat & Tobler, 1985):

푅 푄 푅 푄 퐼 휔̇ + 휌푆 푄̇ = −휌 (휔 푅2 + 푝 tan(훼 ) − 휔 푅2 − 푠 tan(훼 )) 푄 + 휏 (14) 푝 푝 푝 푝 푝 퐴 푝 푠 푠 퐴 푠 푝

푅 푄 푅 푄 퐼 휔̇ + 휌푆 푄̇ = −휌 (휔 푅2 + 푡 tan(훼 ) − 휔 푅2 − 푝 tan(훼 )) 푄 + 휏 (15) 푡 푡 푡 푡 푡 퐴 푡 푝 푝 퐴 푝 푡

푅 푄 푅 푄 퐼 휔̇ + 휌푆 푄̇ = −휌 (휔 푅2 + 푠 tan(훼 ) − 휔 푅2 − 푡 tan(훼 )) 푄 + 휏 (16) 푠 푠 푠 푠 푠 퐴 푠 푡 푡 퐴 푡 푠

휌퐿 휌(푆 휔̇ + 푆 휔̇ + 푆 휔̇ ) + 푓 푄̇ 푝 푝 푡 푡 푠 푠 퐴 2 2 2 2 2 2 2 2 2 = 휌(푅푝휔푝 + 푅푡 휔푡 + 푅푠 휔푠 − 푅푠 휔푝휔푠 − 푅푝휔푡휔푝 − 푅푡 휔푠휔푡) 푄 + 휔 휌(푅 tan 훼 − 푅 tan 훼 ) 푝 퐴 푝 푝 푠 푠 (17) 푄 + 휔 휌(푅 tan 훼 − 푅 tan 훼 ) 푡 퐴 푡 푡 푝 푝 푄 + 휔 휌(푅 tan 훼 − 푅 tan 훼 ) − 푝 푠 퐴 푠 푠 푡 푡 퐿

휌 푝 = 푠푔푛(푄)(퐶 푉2 + 퐶 푉2 + 퐶 푉2 ) 퐿 2 푠ℎ,푝 푠ℎ,푝 푠ℎ,푡 푠ℎ,푡 푠ℎ,푠 푠ℎ,푠 휌푓 (18) + 푠푔푛(푄)(푉∗2 + 푉∗2 + 푉∗2) 2 푝 푡 푠

The four dynamic equations and algebraic pressure loss term are implemented in

SimulationX using the Modelica Text Editor embedded in the modeling environment.

This allows a user to write native Modelica code alongside proprietary SimulationX implementations in text form and create custom blocks to interface with other blocks.

Preprocessing in Modelica environments usually involves automatically rearranging and substituting parameters, algebraic variables, and dynamic states to form an explicit or

70 implicit system of (likely nonlinear) differential equations. The burden of putting equations in a standard form is thus partially taken off of the user, who is able to code the differential equations as shown above. As the preprocessor works through the code, it will check each symbol or variable name for an explicit definition to determine if it is a constant parameter, algebraic variable, or dynamic state. The product of preprocessing is a number of differential equations equal to the order of the dynamic system.

In the standard model, shown above, the assumptions are that the flow is in the forward direction. When the fluid flow (푄) drops below zero, the flow reverses and travels backwards to the designed profile. In the equations, this is represented by exchanging angles, speeds, and radii to re-derive the equations for the new order of connections. This is apparent when viewing the following diagram, which also illustrates that there may be large differences in blade angles between the two rotational directions.

Figure 37: Dynamic Torque Converter Blade Approximation Diagram

The reversed flow path uses entry blade angles instead of exit blade angels, defined with respect to the forward flow direction.

Geometrical parameters were measured from available 3D models for the torque converter. These include blade angles, radii, fluid cross-sectional area, fluid path length, and approximations of the element shape factors (푆𝑖). The shock and friction loss factors were tuned using initial values from literature (Banerjee & McPhee, 2012) and adjusting to reach the target stall torque ratio. At this time, the parameters have not been optimized or otherwise tuned to reduce error in any other performance metrics.

4.4. Simulation Settings and Notes

The following major section outlines many notes about the SimulationX environment and dynamic simulation in general. This is provided to the reader with the intent to brief them on the standard routines which are executed by SimulationX either before, during, or after simulation of a model. These notes include information on available solvers, which is relevant to this work and should be understood by the reader if they wish to replicate these methods in a similar modeling environment for a related project. If the amount of preprocessing required to manipulate an acausal system is familiar to the reader, they may skip Section 4.4.1., while Sections 4.4.2., 4.4.3., and

4.4.4. detail the specifications of the types of solvers considered for this work.

4.4.1. Preprocessing Routines

Pressing the “Start” button in the SimulationX user interface begins by running a

Global Symbolic Analysis (GSA). This procedure starts with the base modeling language

(Modelica) and produces a set of computationally-friendly equations. The SimulationX

User Manual provides a summary of the functions of the GSA algorithm that runs prior to simulation:

 “Conversion of the hierarchical model into a “flat” one

 Resolution of references

 Substitution of simple equations

 Determination of index range of loop variables

 Determination of the dimension of variables

 Conversion of multidimensional equations into one-dimensional equations

 Resolution of language constructs

 Creation of discontinuities (Zero-functions / root functions)

 Introduction of variables for special functions

 Assignment of variables to equations

 Rearrangement of equations with respect to variables

 Determination of the calculation order of the equations

 Determination of states and equations

 Index reduction

 Symbolic calculation of Jacobian matrix”

After this initial step is done, the following workflow is followed to implement the solving algorithm (from the SimulationX User Manual):

Figure 38: Solution Workflow for SimulationX Dynamic Simulation (ESI ITI GmbH, 2017)

The calculation of initial values is a critical step in very large models due to the possibility of unforeseen coupling that may constrict the subset of allowable initial states.

For example, if a planetary gearset model is to be simulated in isolation, initial conditions of all independent inertias should be provided, along with input torques or speeds to define the forcing function. If this sub-model is copied into a full gearbox multiple times and the mechanical connections are made as in the physical system, some of the initial conditions that were necessary in isolation are now possibly redundant due to coupling of some elements. SimulationX will issue warning messages upon finding these types of conflicts, but if there are few conflicts and they are easily remedied, the solver will automatically make some assumptions to follow through with the simulation. If the DAE index is high or the solver runs into a set of inconsistent algebraic equations that cannot be resolved, the program will produce a hard error and stop the initialization procedure.

It is recommended to attempt to define many/most initial conditions as “fixed”

(user sets this value) as opposed to “free” (user suggests, algorithm decides). This will ensure that the computed initial state is the one desired by the user. If the desired initial states are not obtainable for the given set of constant values, the program will produce a hard error, rather than selecting arbitrary values itself and possibly not fulfilling the user’s requests. This process is most critical when dealing with sets of differential- algebraic equations, where normally-independent dynamic systems are coupled by algebraic constituent equations (like two inertias connected by a gearset). This algorithm is also executed during a transient simulation after a state event (discontinuity) is detected. In this case, all variables have a distinct value due to the time evolution of the system, as opposed to the initialization process at 푡 = 0 which must populate any missing values before proceeding. This usually makes the online execution of this algorithm more reliable and efficient as long as the first initialization procedure completed without errors.

The following are examples of states that the algorithm will typically leave “free” for automatic determination following a discontinuous state/logical event (from the

SimulationX User Manual):

 “Purely algebraic state variables

 State variables with a derivative which is not used at any point in the model

 The highest derivatives of the state variables

 The 1st derivative if the 2nd derivative exists but is not used anywhere in the

model”

Other state variables are typically considered to be “fixed” and maintain their values through a discontinuous event. Special exceptions exist for rigid end-stops in the linear and rotational domain (including as part of a larger system), which can allow discontinuous jumps in dynamic state variables only when using BDF and MEBDF solvers (non-code-generation solvers). If initial conditions are under-defined, homotopy techniques may be used to select a distinct set of initial conditions for the model after issuing a warning to the user about a lack of fixed initial states. More on this advanced topic can be found in the SimulationX User Manual (Section 6.1.2.3).

Following the Global Symbolic Analysis, Index Calculation, and the Calculation of Initial Conditions, the selected transient solver steps into the “flattened” set of dynamic equations and begins event iteration or time evolution according to the above flowchart.

4.4.2. BDF and MEBDF Solvers

Two variants of “Backward Differential Formulas” are present in the solver settings. The standard BDF method is described as an extension of the popular DASSL algorithm employed by other object-oriented system modeling packages. This is the default selection for Dymola and OpenModelica, however a performance increase is seen through Dymola based on proprietary equation-tearing methods prior to the DASSL execution (Floros, Bergero, Cellier, & Kofman, 2011).

The BDF method present in SimulationX is related to the standard DASSL solver.

It is a predictor-corrector method of varying order and step size which iterates every time step to achieve a set tolerance of error specified by the user (solver settings). During the preprocessing stage, the entire set of system equations are arranged in implicit nonlinear

76 form (zero on the left-hand side and everything on the right-hand side). During each step, the linearized forms of these equations are solved for the residuals, which are monitored for convergence. If the step size and order are not sufficient to achieve this accuracy requirement, the time step is reduced and the process is repeated until convergence or the minimum step size is reached.

The method for computing the residuals in each iteration can be specified as

“Sparse Matrix Solver” or “Gaussian”. The sparse matrix solver is highly superior when large systems contain components that are coupled only to other nearby components (few direct interactions) as opposed to each component directly affecting every other component (as in vector-field-based forces). The sparse matrix assumption holds for mechanical systems such as this transmission model, because elements can be broken into a few interactions and the rest of the interaction terms in the system of equations are zero, thereby yielding a sparse matrix. Gaussian matrix solutions are only useful when the matrix is dense with interactions among components. This method is highly sensitive to ill-conditioned matrices which arise from disparate eigenvalue magnitudes and large condition numbers. In dynamic systems, this arises when there are wide ranges of time constants within the same set of equations, classifying the model as stiff.

These methods are implicit and the order of the solver specifies how far backward the transient solver will look to compute the next point. A single backward step has order of 1 and corresponds to implicit-Euler.

It is noted in the User Manual that the SimulationX BDF and MEBDF each have code-generation variants which take more time to prepare and compile, yet perform with

77 generally lower simulation times, provided the system does not violate code-export requirements. The main violation encountered for code-generation is the inclusion of rigid end-stop elements. To get around this, some library blocks with internal end-stops allow for an elastic definition of the end-stop contact. If this is not exposed and a contact model must be rigid, a code-generation solver may not be implemented.

4.4.3. CVODE Solver (from SUNDIALS)

The Lawrence Livermore National Laboratory (LLNL) in Livermore, California has been doing research and development in the computational domain for over 60 years.

A division of LLNL, the Center for Applied Scientific Computing, is responsible for developing and publishing open-source software for use in a wide variety of applications.

The Suite of Nonlinear and Differential/Algebraic Solvers (SUNDIALS) was developed by this branch of LLNL and includes an algorithm for solving DAE systems which are compiled prior to execution. The main method of SUNDIALS, CVODE, is included in

SimulationX for code-generation only.

CVODE is a solver which operates on a set of explicit nonlinear equations

(highest derivative on the left-hand side, everything else on the right-hand side). Like the other BDF methods, this is variable-step, variable-order, and backward-facing time- implicit. This solver is coupled with different options for iteration evaluation (solving for the residuals) than the standard BDF method, which include Krylov subspace iterative solvers (namely, generalized minimum residual) on top of the dense/sparse direct Newton iteration schemes. If the model is not stiff, the BDF implicit-time integration can be replaced with a higher-order Adams explicit-time linear multistep integration method.

SimulationX recommends this solver for large systems due to the compilation process making the execution more efficient. Simulation time should be saved, especially for fixed system configurations, at the cost of increased preparation time. If the user is only varying exposed parameters between simulation runs, the model does not need to be re-compiled and the increased solving efficiency can be utilized consistently. For this reason, parameter variation studies are aided by the use of this compiled solver.

4.4.4. Fixed Step Solvers

Various options for fixed-step solvers are integrated into SimulationX, but all are implemented with code-generation meaning that the models must comply with all code- export guidelines. These solvers were included for testing and export with real-time applications. Fixed-step methods may be used in native SimulationX simulations, but they will likely incur a penalty to efficiency over the variable step methods mentioned above. Native fixed-step solutions are, however, valuable in order to select the settings, limits, and expectations for a project intended for export to an external platform. Real- time simulation on dSpace, ETAS, National Instruments, and other HIL-based platforms require fixed resources to reliably synchronize with a real-time operating system.

The selections that the user may make for fixed-step implementations are the step size (set as “minimum calculation step size”) and the integration method. The fixed-step methods are all explicit-time methods for solving systems of explicit ODEs (separate-out the highest derivative in each equation). The integration method is essentially selecting the order of the explicit-time integration method. The orders available for selection are 1,

2, 3, and 5. There are two second-order methods (ITI-Standard and classical Heun) and

79 no fourth-order methods. Looking through these methods, one can notice that these are common options for ODE solvers in causal modeling packages such as

MATLAB/Simulink. They are as follows (from the SimulationX User Manual):

Table 1: Solver Order Specifications (ESI ITI GmbH, 2017)

RHS Method Name Order Evaluations Euler Forward 1 1 ITI Standard 1 2 Heun 2 2 Runge-Kutta 2/3 3 3 Dormand-Prince 5 6 5

4.4.5. Performance Analyzer

SimulationX has built-in a tool called “Performance Analyzer” to judge the computational load of solving each of the dynamic states and tracking discrete events.

This tool runs in parallel to the simulation and collects data on the residuals cumulatively over the transient simulation. The meaning of the “Influence” column shown below is essentially a measure of how much each state is affecting the step size and number of iterations of the solver. For example, if one state’s residuals are taking a long time to converge in the iterative scheme of the solver while all the other states settle quickly, this dynamic state would be classified as computationally cumbersome and its Influence would be higher relative to other states. The absolute value of the “Influence” does not have physical meaning, only the ratios between the values among the states. This is a very useful tool for identifying states which may take a long time to converge.

If states take a long time to converge and they represent a high-order derivative or ultra-fast dynamics, these would be tagged as good candidates for model reduction along the critical path of simulation speed. If the states in question do not include dynamics interesting to the modeler, their removal or reduction should also not significantly affect output behavior of interest to that modeler. If a conflict is present between reduction for speed and loss of dynamics important to the modeler, they must decide whether or not to pursue reduction for their application. Chapter 7 will investigate the accuracy tradeoffs associated with eliminating states during the model reduction process in Chapter 6, which is partially informed by Performance Analyzer judgements of simulation bottlenecks.

Figure 39: Example Performance Analyzer Output for Hi-Fi Gearbox Simulation

4.4.6. Additional Notes

An option that is not included in current Modelica-based simulation packages is a quantized-state system (QSS) solver. For implicit-time BDF-based methods like the

SimulationX BDF, DASSL, and Sundials CVODE solvers, the time axis is divided into steps and the algorithm determines the amount of variation that occurred in the state

81 variables across that time step. To get finer granularity, the time step is reduced, either manually or automatically, to get more samples in the same amount of time.

With a QSS solver, the state-axis (or “y-axis”) is incremented by one “quantum” and the solver is tasked with determining the amount of time that has passed for such a change in the variable to occur. For this to be possible on a large scale, each state variable is actually integrated using a paired time-vector, making the state variables asynchronous.

Since the only options for the state variable are increase, decrease, or remain constant, the handling of discontinuities is superior to time-implicit methods. For QSS methods, execution times increase linearly with increasing number of events, whereas stiff/DAE solving time usually has a quadratic dependency on the number of events.

For ODE systems with comparable time constants, this method would not provide much, if any, advantage because common ODE solvers can handle well-conditioned systems of equations with ease. However, when a model is of very high order (many states), it is likely that the time constants of the system vary significantly and the system is classified as stiff. For increasing model complexity, QSS scales approximately linearly while implicit-time methods like ODE solvers and DASSL-based methods typically increase cubically. This is due to the absence of an iteration scheme at each time step.

QSS methods have been integrated with compiled Modelica code from

OpenModelica and been shown to compare quite well to the DASSL (BDF) execution of

Dymola. This is significant because Modelica code processed by Dymola is much more efficient due to their equation-tearing algorithms, even though Dymola and

OpenModelica use very similar DASSL solving algorithms. This study showed that the

82 difference in preprocessing complexity (one of Dymola’s commercial advantages) can be overcome by upgrading the solver to a quantized-state method. It is possible that these solving methods, along with efficient equation tearing, would produce the most efficient simulation combination.

Note that performance is highly dependent on the type and orientation of the system under test, and it may be the case that complex solvers are more resource- consuming to implement than simply multi-rating a model with classical stiff ODE solvers.

4.4.7. Summary of Simulation Environment

An in-depth discussion has been provided about the operations included in the solution of a model in SimulationX, with much of this information being generalizable to object-oriented modeling language simulation. For offline simulation of the high-fidelity models presented in this chapter, the CVODE variable-step solver has been chosen with a common maximum step size of 1 푚푠. The minimum step size allowed has been chosen as 1 푛푠, while the absolute and relative tolerances for the solver convergence criteria were each set to 1 × 10−6. These settlings were chosen to provide a set of robust solver settings which allowed the simulation to run from start to finish with a range of parameter values (detailed in Chapter 5). Step sizes or tolerance values which were higher than those chosen may have worked for nominal cases, but ran into trouble solving near some state events and would sometimes march along indefinitely at the minimum time step.

Chapter 5. Parameter Sensitivity

The process of finding sensitivity of an output to an input in an analytical dynamic equation with multiple states is performed by taking the partial derivative of one variable with respect to the other. When the full dynamic system is not entirely known, there are many nonlinearities or discontinuities present, or if the outputs are not directly mapped to dynamic states, other means may be necessary. One way to do this is by experimentally disturbing the system in a methodical way and then taking a statistical approach to analyzing the results.

The dynamic system of the automatic transmission contains many discontinuities and is constructed within proprietary modeling software, which partially obscures a clear set of dynamic equations with many conditional statements. In addition, the chosen performance metrics for the subsystems in this model are not simply linear transformations from the states in the dynamic system. For these reasons, this model is an appropriate use case for the deviation-based method of studying sensitivity of the system.

The parameters of the system are chosen and varied according to the outline of experimentation which follows below.

5.1. Design and Setup of Experiments

The gearbox and hydraulic actuation cylinders were grouped into the same experimental setup due to the relatively lower number of states compared to the hydraulic

84 network. There is also a feedback signal required to go from some of the gearbox rotational elements to the hydraulic cylinders of the disk clutches due to the centrifugal force on the fluid in the cylinders.

A simplification was made to cut the hydraulic connections between the output pressures of the hydraulic network and the input pressures of the hydraulic clutch actuation. In place of the connection, equivalent volumes were inserted on the output pressures from the hydraulic network to take the place of the hydraulic volumes removed as part of the hydraulic actuation cylinders. Equivalent hydraulic volumes were inserted on the pressure inputs to the hydraulic actuation cylinders to complete this assumption.

For the hydraulic network parameter studies, the inputs to the network are the commanded current values to all solenoids. The outputs are the hydraulic pressures commanded to the clutch actuation cylinders for the disk clutches, brake clutches, and torque converter lockup clutch. Finally, the TWC valve position is an output of the hydraulic network to be fed into the gearbox to select its locking behavior.

Figure 40: Hydraulic Network Subsystem Overview

The hydraulic network was given a commanded current trajectory that corresponded to a shifting sequence which started in reverse and shifted every two seconds up through 10th gear, and then back down in the opposite order. The total simulation time is 50 seconds, which is common between the gearbox and hydraulic network experiments. This shift schedule is shown below:

Table 2: Shift Schedule for Repeatable Transient Experimentation

Time Stamp [sec] Event Time Stamp [sec] Event Simulation Start Release 푇 0 25 𝑖푛 Engage REV Engage BRAKE 2 Engage PARK 26 Shift 10th  9th 4 Engage NEU 44 Engage NEU 6 Engage 1st GEAR 46 Engage PARK 24 Shift 9th  10th 48 Engage REV 50 Simulation Stop

Input maps are used for each solenoid which take the gear state as an input and output a binary signal to activate or deactivate a solenoid. Further downstream mapping controls the final DC offset of the commanded output current to adjust for different currents being requested by different solenoids. These constitute the binary section of the makeshift transmission controller which selects the correct solenoid activation given an input gear. After the engagement/disengagement signals are passed from this section, trajectory maps are used to apply a 0.5-second ramp signal of current to the oncoming clutch solenoid and a matching down-ramp of the same duration for the offgoing clutch solenoid. The opposing ramp signals are executed at the same time, and the delays in the physical system of actuation end up delaying the application of the oncoming clutch until

86 after the offgoing clutch has reduced its force to some extent. Steps in current would slam clutches together as fast as physically possible, which is not realistic for vehicle operation. Additionally, a smooth, repeatable signal transition is desired as an input for the purposes of studying the input/output relationships.

In a modern automatic transmission, feedback control may be used to vary commanded current in response to speed sensors and state observers and thus ease the mechanical engagement of the clutch. The feedback in this case would render gear shift events non-repeatable and thus less useful for physical parameter sensitivity, due to the controller in the loop. For these reasons, a simple filtered ramp trajectory was used to apply a command current ramp to the solenoid models in the transmission controller. In control verification applications, this controller would have the prototype software in the loop to test the algorithms. However, to test the model itself, the added complexity of a software-in-the-loop feedback controller should not be incorporated.

The pressure output results from the hydraulic network simulation using the baseline nominal parameter values were recorded and used as consistent playback signals throughout all gearbox/actuation simulations. This is to create a realistic (not just ramp signals) trajectory of pressures for the physical gearbox simulation and maintain the same inputs across all levels of fidelity. Regardless of the parameters chosen in the gearbox model, this same master pressure trajectory was used for the seven hydraulic clutch cylinders.

The gearbox and clutch actuation parameter study was set up to take clutch pressures as inputs, as well as overall torque input and the TWC valve position. The

87 outputs of the gearbox setup are those signals which are necessary for the calculation of the selected performance metrics. These include the output torque and speed, but also the internal states in the gearbox like clutch speed differences and clutch torques to determine the clutch engagement metrics.

Resistance was added to the gearbox model by way of a transformation of vehicle drag into a simple resistive torque element and lumped inertia on the output shaft. This shaft was also used to brake the gearbox for its deceleration phase. In practice, the driveline compliance will affect oscillation and damping of the output torque trace, but phase timing and the torque hole should remain a function of the clutch actuation

(application force and friction characteristics). The main shift metrics for control verification are the timing metrics because the controller can judge whether or not the transmission is in gear. Other performance metrics like the peak jerk torque and the settling time will be damped out at a rate dependent on the elastic parameters of the lumped driveline. An elastic driveline model may filter the high-frequency oscillations of this model’s measured torque trace, though the driveline’s own dominant frequency should also show up as the halfshaft elastic annotations in Figure 1 (Section 2.1). The aerodynamic resistance torque and lumped vehicle inertia were resolved according to the following formulas:

3 1 푅푡𝑖푟푒 2 휏푎푒푟표 = − 휌푎𝑖푟퐶퐷퐴 ( ) 휔표푢푡 (19) 2 푍퐹퐷

2 푅푡𝑖푟푒 퐽푒푞 = 푀 ( ) (20) 푍퐹퐷

The above equation includes the tire radius (푅푡𝑖푟푒), the final drive ratio (푍퐹퐷), the vehicle drag coefficient and frontal area (퐶퐷 and 퐴), and the output shaft speed from the transmission (휔표푢푡). The following diagram illustrates the connections between the subsystems on the highest level:

Figure 41: Gearbox and Clutch Cylinder Subsystem Overview

For torque converter sensitivity, the element is isolated with torque sources attached at each end, since the hydrodynamic model uses torques as inputs to solve for speed. Since only static torque converter metrics are studied in this work, a sweep of operating points is conducted by holding the input torque constant and using a PID feedback controller to adjust the turbine torque to achieve ever-increasing speed ratios.

This sweep of speed ratios allows the sampling of torque ratio, capacity factor, and efficiency values at each settled point. Since no transient tests were performed with the torque converter model, damping effects within the torque converter model are only studied through the lens of the steady state capacity factor curves.

Mean values for some of the parameters below are not shown due to confidentiality. For these cases, the absolute deviations used in the sensitivity study are still presented.

5.1.1. Gearbox Parameters

The parameters in the gearbox sensitivity simulations were grouped into parameters relating to the clutches, the planetary gearsets, the hydraulic actuation mechanical system, and the two-way clutch elastic torque parameters.

To investigate the performance metric sensitivity to the clutch parameters, the four parameters chosen were the input torque to the gearbox, the slipping friction coefficient, the sticking friction coefficient, and the disk/plate thickness. The input torque is expected to impact the output performance metrics significantly, as the ultimate driving torque of the system. The friction coefficients define the slip-휇 Stribeck friction curve modeled in the clutch block, and thus should be significant for engagement metrics.

Finally, the clutch stiffness is linearly dependent on the thickness of the clutch plates, so this should affect overall stiffness of the engagement as well as the damping of lockup.

The following is a summary of the parameters varied in the clutch models throughout the sensitivity study:

Table 3: Gearbox Friction Clutch Parameters

Clutch Number of Nominal Low (횫) High (횫) Parameters Levels (concealed) 푇𝑖푛 3 150 푁푚 50 푁푚 250 푁푚 휇푠 3 - −0.02 +0.02 휇0 3 - −0.015 +0.015 푡퐷 3 2 푚푚 1 푚푚 4 푚푚

For the highest-fidelity gearbox model, the planetary gearsets contain many unknown parameters that may have an impact on the performance of the transmission.

First is the gear tooth backlash for each gear-meshing interface. In a planetary gearset, the backlash determines how much space one tooth has to accelerate before impact with its opposite-facing tooth. The larger this gap, generally, the larger the impact energy when the teeth eventually collide. There is also the possibility that decreasing the backlash increases the torque oscillations as the impacts become double-sided, bouncing back and forth between teeth on the opposing gear. For the high-fidelity planetary gearset, backlash is defined as a linear measurement (as opposed to an angular definition). This lash is known to be small but has not been measured and thus is varied in the model simulation experiment. Zero is included as a possibility for backlash to determine its effect on the output performance metrics. For the helical gear interfaces, the model block takes care of the geometric transformations necessary to turn a linear lash value into gaps between helical faces in all three dimensions.

The next parameters for the planetary structures are associated with the bearing connections of the carrier. The bearings within the planetary gearset are detailed in the model development section, and the mechanical stiffness, damping, and backlash for the carrier-planet interactions are varied as parameters in the experiments. These interactions, along with the tooth stiffness, define the overall stiffness behavior for the planetary gearset.

The last set of parameters to be varied are multiplication factors on the planetary inertias. Since the planet inertias are not known, the inertias were estimated using the

91 formulas discussed in the model development section, and overall multiplication factors were used on top of these equations to vary the inertia and judge its effect on the output performance metrics of the shifts.

The following is a summary of the planetary gearset parameter subset:

Table 4: Gearbox Planetary Gearset Parameters

Planetary Gearset Number of Nominal Low High Parameters Levels Input Torque 3 150 푁푚 50 푁푚 250 푁푚 Tooth Lash 3 0.025 푚푚 0 푚푚 0.05 푚푚 Bearing Stiffness 3 10 푘푁/푚푚 7.5 푘푁/푚푚 12.5 푘푁/푚푚 Bearing Damping 3 1000 푁푠/푚 800 푁푠/푚 1200 푁푠/푚 Bearing Lash 3 0.025 푚푚 0 푚푚 0.05 푚푚 Planet Jfactor (self) 3 1 0.75 1.25 Planet Jfactor (orbit) 3 1 0.75 1.25

For each of the seven hydraulic clutch actuation cylinders, multiple unknown parameters are key to test for output sensitivity. First, the input torque is again used to test the other parameters at different operating loads. The clutch piston return spring stiffness and overall linear damping coefficient as well as the piston’s mass are all unknown and should be varied to determine their effects on transient metrics. The installed clutch plate clearance (“piston lash”) was also used as a variable in the mechanical system, as well as the hydraulic dead volume in the cylinder where pressure is applied.

The following is a summary of the factor levels for the hydraulic clutch actuation cylinders:

Table 5: Gearbox Clutch Cylinder Parameters

Hydraulic Clutch Number of Nominal Low High Cylinder Parameters Levels Input Torque 3 150 푁푚 50 푁푚 250 푁푚 Spring Stiffness 3 50 푁/푚푚 25 푁/푚푚 75 푁/푚푚 Overall Damping 3 500 푁푠/푚 250 푁푠/푚 750 푁푠/푚 Piston Mass 3 200 푔 100 푔 300 푔 Piston Lash 3 5 푚푚 2.5 푚푚 7.5 푚푚 Dead Volume 3 100 푐푚3 75 푐푚3 125 푐푚3

Finally, the two-way clutch parameters that were varied (along with the input torque) were the lockup stiffness, damping, and locking speed threshold. It was important to test the option of a two-way clutch with zero lockup stiffness to test the feasibility of using pure damping to lock the clutch (effectively a proportional-gain feedback controller). The following is a summary of the two-way clutch torque parameters:

Table 6: Gearbox Two-Way Clutch Parameters

Two-Way Clutch Number of Nominal Low High Torque Parameters Levels Input Torque 3 150 50 250 Lockup Damping 3 2000 푁푠/푟푎푑 3500 푁푠/푟푎푑 500 푁푠/푟푎푑 Lockup Stiffness 3 50 푁/푟푎푑 0 푁/푟푎푑 100 푁/푟푎푑 Lockup Speed Threshold 3 0.025 푟푎푑/푠 0 푟푎푑/푠 0.05 푟푎푑/푠

Each of these subsets of parameters were split into a separate design of experiments in order to handle the data and simulations more effectively. This means that that parameter interactions can be studied statistically only within these chunks. No inferences can be made about the interactions between parameters from different subsets.

In this work, there will only be a statistical study of the main effects (first-order terms) to

93 develop the sensitivity values, therefore interactions are not included at this time. The data was collected such that future modeling exercises could use the data to develop multiple-linear regression models for the output metrics as a function of the parameter values and detect two-way interaction terms and quadratic effects. Such higher-order terms would only be applicable within the parameter subsets listed here, not between, to reduce the size of the experimental designs. To fill-in this data, one might only run parameter variants which interact between the parameters of interest and may add these runs to the existing data set.

Within each subset of parameters, an experimental design procedure was selected to give a good balance of number of runs and coverage of the design space. Full factorial experiments work for applications with few parameters, but a two-level full factorial increases the number of runs with 2푘, where 푘 is the number of parameters. If nonlinear effects are to be detected, a three-level full factorial design scales the total number of runs with 3푘. For this reason, the following experimental designs have a combination of fractional- and full-factorial design matrices.

5.1.2. Gearbox Design of Experiments and Simulation Setup

The gearbox parameters are split into the four subsets discussed above. For the clutch parameters, only four factors were studied, and thus a four-factor, three-level full factorial design was within reason at 81 total runs. The full factorial design is the most straightforward, but also most thorough design of experiments, as every combination of parameters is tested among the chosen factor levels. The resulting output also yields the

94 most information for modeling purposes, with up to four-way interactions available for study and no aliasing of signals.

The planetary gearset parameters are more plentiful. There are seven parameters including the input torque which is used to test multiple load conditions on the gearbox for each parameter subset. To avoid doing a three-level full factorial with 2187 runs, a central composite design was implemented, which starts with a base two-level design and fills in center points in the design space to test for curvature (nonlinearities) of effects. In this case, a two-level half-fractional factorial design was used as the base of the central composite design of experiments. This creates 64 runs (27−1), or half the number of runs of a two-level full factorial. From there, there are 2푘 axial points added to test one parameter at a time while holding the others at their center points. The 14 axial points and

10 additional overall center points of the design space (every parameter set to its halfway value between minimum and maximum) bring the design total to 88 runs – a reduction of almost 96% from the full-factorial option. The half-fractional factorial has a design resolution of seven, meaning that main effects are aliased with six-way interactions, and two-way interactions are aliased with five-way interactions. The unlikely nature of five- and six-way interactions means that this is an appropriate method to test linear and second-order effects.

For the hydraulic clutch actuation cylinders, six parameters were varied including the input torque. As with the previous subset, each parameter had three levels at which to test, but the number of parameters would come to 729 for a full factorial design, which is prohibitively large. Instead, the same half-fractional central composite design was used as

95 the previous subset, but with six parameters instead of seven, lowering the design resolution of the two-level “cube” part of the design space to a design resolution of six.

This still allows the main effects to be aliased with unlikely five-way interactions, but now the two-way interactions are aliased with four-ways. This is also generally acceptable, as the probability of four-way interactions being statistically significant is much slimmer than that of any second-order effects one would be interested in. The

“cube” portion of this design thus comes out to 26−1 = 32 runs. Adding 12 axial points and 9 overall center points brings the total to 53 experimental runs.

Finally, there are four variables in the two-way clutch torque subset, including the input torque. These four are varied with a three-level full-factorial design which consists of 81 runs, similarly to the clutch parameter design mentioned above.

The following is a summary of the gearbox parameter designs:

Table 7: Gearbox Parameter Variation Experimental Overview

Design Number of Parameter Subset Design Chosen Resolution Number of Runs Parameters (factorial part) Clutch Friction 4 3푘 Full Factorial N/A 81 Planetary Gearsets 7 ½ Fractional CCD VII 88 Hydraulic Cylinders 6 ½ Fractional CCD VI 53 Two-Way Clutch 4 3푘 Full Factorial N/A 81

5.1.3. Hydraulic Network Parameters

The hydraulic network parameters under test were broken into subsets for the linear proportional solenoid valves, the shift solenoid valves, and the regulation valves.

Within each of the solenoid subsets, there were two additional subsets of parameters: the 96 electrical parameters and the spring-mass-damper mechanical parameters. The electrical parameters include resistance of the solenoid, inductance, and solenoid back-EMF. The resistance and inductance values form the time constant for the electrical response of the solenoid. In the physical system, the command current gets filtered through the resistor-

퐿 inductor circuit with time constant ⁄푅. This leads to the prior assumption that raising the inductance of the solenoid will delay the pressure rise time as the plunger’s motion will be delayed by a smaller cutoff frequency from the solenoid. The opposite is true for the resistance, with an inverse relationship with the time constant.

Table 8: Hydraulic Network Linear Solenoid Electrical Parameters

Linear Solenoid Number of Nominal Low (횫) High (횫) Electrical Parameters Levels (concealed) Resistance 3 - −3 Ω +3 Ω Inductance 3 0.01 퐻 0.002 퐻 0.05 퐻 Back-EMF 3 1 0.2 5

The mechanical parameter subset for the linear solenoids included the mass of the solenoid plunger, the spring stiffness, overall damping coefficient, spring preload force, and output hydraulic volume.

Table 9: Hydraulic Network Linear Solenoid Mechanical Parameters

Linear Solenoid Number of Nominal Low High Mechanical Parameters Levels Plunger Mass 3 5 푔 2.5 푔 7.5 푔 Spring Stiffness 3 15 푁/푚푚 12.5 푁/푚푚 17.5 푁/푚푚 Plunger Damping Coefficient 3 10 푁푠/푚 12.5 푁푠/푚 17.5 푁푠/푚 Spring Preload 3 1 푁 0.75 푁 1.25 푁 Output Volume 3 100 푐푚3 75 푐푚3 125 푐푚3

The electrical parameters for the shift solenoids are the same as for the linear solenoids and are also varied for all shift solenoids at the same time. It stands to reason that these electrical parameters will not have a significant effect on the output pressure traces, as they are used to command the shift solenoids. However, it is possible that the changes in eigenvalues by altering the R-L parameters changes the simulation speed capability.

Table 10: Hydraulic Network Shift Solenoid Electrical Parameters

Shift Solenoid Number of Nominal Low (횫) High (횫) Electrical Parameters Levels (concealed) Resistance 3 - −3 Ω +3 Ω Inductance 3 0.01 퐻 0.002 퐻 0.05 퐻 Back-EMF 3 1 0.2 5

The parameter subset for the shift solenoid mechanical system was slightly different due to their different modes of operation. The linear solenoids generally operate within a range of strokes, depending on the command current, and use feedback to modulate the position of the plunger near the opening threshold of the orifice.

Conversely, the shift solenoids are concerned with simply opening and closing flow connections, and thus spends most of their operating time at one of the two ends of their stroke-lengths. For this reason, the end stop parameters are more important to the shift solenoid operation and the eigenvalues associated with the elastic end-stops may play a larger role in the simulation of the hydraulic network. The parameters varied as part of the shift solenoid mechanical parameter subset were the plunger mass, spring stiffness,

98 overall damping coefficient, preload force, maximum plunger stroke, and the elastic end stop stiffness.

Table 11: Hydraulic Network Shift Solenoid Mechanical Parameters

Shift Solenoid Number of Nominal Low High Mechanical Parameters Levels Plunger Mass 3 20 푔 10 푔 30 푔 Spring Stiffness 3 500 푁/푚 350 푁/푚 650 푁/푚 Plunger Damping 3 2.25 푁푠/푚 1.75 푁푠/푚 2.75 푁푠/푚 Spring Preload 3 2 푁 1.5 푁 2.5 푁 Stroke Length 3 3 푚푚 2 푚푚 4 푚푚 End Stop Stiffness 3 10 푘푁/푚푚 7.5 푘푁/푚푚 12.5 푘푁/푚푚

The regulation circuit is important to simulation, though its impact on the clutch engagement and thus output metrics is less likely, as the pressures are supposed to be regulated at near-constant levels for the linear solenoid sources. The response times for the regulation valves need to be fast for the internal feedback to quickly regulate the flow connections. Altering the mechanical properties of these components surely affects the ability to respond to these changes. The main way that varying these parameters would affect the output metrics is if a change in a parameter affects a valve’s ability to quickly regulate the pressure it is tasked with.

Geometric specifications are given for the regulation valves, along with spring stiffness and preload values. Unknown are the masses and damping coefficients, which are both key to defining the maximum frequency of each valve. For this reason, each of the regulation valves’ plunger masses and damping coefficients were used as parameters in the experiment. The regulation circuit includes the main regulation valve, torque converter regulation valve, lubrication regulation valve, lockup clutch control valve, and 99 the line pressure accumulator valve. Each of these five valves had their masses and damping coefficients varied at three levels in a full factorial small design matrix. The naming convention for these parameters is that masses begin with “m”, while the damping multiplication factors begin with “b”. Each is followed by the abbreviation for the valve in question (e.g. “mMRV” would be the name of the main regulator valve plunger mass variable).

Table 12: Hydraulic Network Regulation Circuit Mass/Damping Parameters

Regulation Valve Number of Nominal Low High Parameters Levels Plunger Mass (m___) 3 1 푔 0.75 푔 1.25 푔 Damping Factor (b___) 3 0.05 0.025 0.075

5.1.4. Hydraulic Network Design of Experiments and Simulation Setup

Since there are only three parameters in the linear solenoid electrical subset, a three-level full factorial design was used with 27 total runs to characterize the effects of these parameters on the output metrics. This was the case for both the linear solenoids and the shift solenoids. Each of the three variables were changed across all solenoids when varied, instead of varying each solenoid’s parameters individually (ensuring the design only has 3 variables, rather than treating each solenoid as unique).

With five parameters in the linear solenoid mechanical subset, a three-level full factorial experiment would come out to 243 total runs. For this reason, a full factorial central composite design was used, where the base “cube” portion of the design space had 25 = 32 points and the curvature was tested by incorporating 10 face-centered axial

100 points and 10 overall center points. The total for this design becomes 52 total runs, while maintaining a design resolution of five for the factorial “cube” portion of the design.

The six parameters of the shift solenoid mechanical subset were varied within a half-fractional factorial central composite design with 32 “cube” points, 12 face-centered axial points, and 9 overall center points. This creates an experimental design with 53 runs and a design resolution of six.

For the regulation valve subset of parameters, the two parameters at three levels each produces a small 9-run design performed separately for each valve. While varying one valve’s mass and damping, the eight parameters from the rest of the regulation valves are held constant at their respective midpoints. This was done to test the impact of each valve on the system since each valve is unique, in contrast to the linear or shift solenoids where all solenoids are interchangeable within the same type. With 9 runs per valve, the subset design totals 45 runs.

The following is a summary of the hydraulic system parameter subset designs:

Table 13: Hydraulic Network Parameter Variation Experimental Overview

Design Number of Parameter Subset Design Chosen Resolution Number of Runs Parameters (factorial part) Linear Solenoid 3 3푘 Full Factorial N/A 27 (electrical) Linear Solenoid Full Factorial 5 N/A 52 (mechanical) CCD Shift Solenoid 3 3푘 Full Factorial N/A 27 (electrical) Shift Solenoid 6 ½ Fractional CCD VI 53 (mechanical) 3푘 Full Factorial Regulation Valves 2 (x5) N/A 45 (x5)

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5.1.5. Torque Converter Parameters

The torque converter can be parameterized with three major subsets: the blade angles, empirical factors, and other geometrical parameters. The blade angles within the torque converter model are interesting because while there is a correct geometrical measurement, the model may still work best when the measured values are deviated slightly. For this reason, it may be helpful to interpret the blade angles (along with the radii) as effective values. Both the entry and exit values are used in the dynamic model, though the entry blade angles are only used in the power loss term (shock velocity) in forward flow. Therefore, in reverse flow, the entry angles become more prominent because they are now the exit angles for the fluid. For each of the blade angles, the values were disturbed by two degrees in each direction.

Table 14: Torque Converter Blade Angle Parameters

Torque Converter Number of Nominal Low (횫) High (횫) Blade Parameters Levels (concealed) Input Torque 3 - −20 푁푚 +20 푁푚 Impeller Exit Angle 3 - −2 푑푒푔 +2 푑푒푔 Turbine Exit Angle 3 - −2 푑푒푔 +2 푑푒푔 Stator Exit Angle 3 - −2 푑푒푔 +2 푑푒푔 Impeller Entry Angle 3 - −2 푑푒푔 +2 푑푒푔 Turbine Entry Angle 3 - −2 푑푒푔 +2 푑푒푔 Stator Entry Angle 3 - −2 푑푒푔 +2 푑푒푔

The empirical factors of interest are the blade shape factors and the loss factors.

The blade shape factors are theoretically dependent on the blade radius and angle as an integral along the blade lengths. They represent inertial effects, and are thus only present in dynamic simulations. The shock loss factors can be considered separately for each

102 element, and are started at values which give an experimentally accurate stall torque.

These loss factors amplify or attenuate the speed differences between the elements when the fluid reaches a boundary. Finally, the fluid friction factor accounts for power loss from the fluid dragging along the walls of the mechanical elements.

Table 15: Torque Converter Empirical Factors

Torque Converter Number of Nominal Low (횫) High (횫) Empirical Factors Levels (concealed) Input Torque 3 - −20 푁푚 +20 푁푚 Impeller Shape Factor, 푆_푝 3 - −0.0002 +0.0002 Turbine Shape Factor, 푆_푡 3 - −0.0002 +0.0002 Stator Shape Factor, 푆_푠 3 - −0.0002 +0.0002 Impeller Shock Loss, 퐶_푝 3 - −0.15 +0.15 Turbine Shock Loss, 퐶_푡 3 - −0.1 +0.1 Stator Shock Loss, 퐶_푠 3 - −0.15 +0.15 Fluid Friction Factor, 푓 3 - −0.05 +0.05

As mentioned above with the blade angles, the torus radii may be measured, but the “true” value that is correct for the mean flow path of the fluid may be different. This would be an area improved by the used of CFD fluid flow through the torque converter to track the mean fluid path radius, area, and mean path flow length from 3D simulations.

For the geometrical parameters, the element radii were each varied by 5 푚푚 in each direction. This was done to get a balance between checking large variations and maintaining stability of the torque converter model. It was found in some experimentation that having the turbine exit radius exceed the stator exit radius caused the simulation to become unstable, even at supposed steady state. Therefore, values were chosen such that these parameters would not overlap in such a way. The fluid flow length is related to the total fluid inertia in the dynamic equations, and thus should not 103 significantly impact the steady state metrics evaluated here. The fluid flow area is present in many equations, so this is likely influential on both the transient and steady state characteristics.

Table 16: Torque Converter Geometrical Parameters

Torque Converter Number of Nominal Low (횫) High (횫) Geometrical Parameters Levels (concealed) Input Torque 3 - −20 푁푚 +20 푁푚 Impeller Exit Radius 3 - −5 푚푚 +5 푚푚 Turbine Exit Radius 3 - −5 푚푚 +5 푚푚 Stator Exit Radius 3 - −5 푚푚 +5 푚푚 Total Fluid Length, 퐿푓 3 - −50 푚푚 +50 푚푚 Common Flow Area, 퐴 3 - −0.002 푚2 +0.002 푚2

5.1.6. Torque Converter Design of Experiments and Simulation Setup

To test the significance of the blade angles, each of the six blade parameters were varied at three levels within a half-fractional central composite design and then replicated at three input torque values. The 2푘 factorial part of the CCD used 32 points. Added with

9 overall center points and 12 axial points, this gives a design with 53 runs at each torque level.

The empirical factors were studied using the same technique, but for seven parameters rather than six. 64 cube points result from a 27−1 half-factional central composite design, coupled to 10 overall center points and 14 axial points. This totals to

88 runs per torque level.

Finally, the five geometrical parameters are varied within a full-factorial central composite design, which gives a total of 52 runs at each torque level. This comes from

104 using 32 cube points, 10 center points, and 10 axial points. These are all summarized below:

Table 17: Torque Converter Parameter Variation Experimental Overview

Number of Design Resolution Number of Parameter Subset Design Chosen Parameters (factorial part) Runs Blade Angles 6 (+1) ½ Fractional CCD VI 53 (x3) Empirical Factors 7 (+1) ½ Fractional CCD VII 88 (x3) Geometrical Parameters 5 (+1) Full Factorial CCD N/A 52 (x3)

5.2. Performance Metric Sensitivity

Parametric sensitivity has been demonstrated via design of experiments similar to what has been shown in this work (Van Voorhees & Bahill, 1995). Specifically, this conference paper showed an example of varying parameters via 3푘 factorial designs to estimate sensitivities for main effects, quadratic effects, and higher-order interaction terms via a method called the “Yates Algorithm”. This algorithm is essentially a way to individually estimate the least-squares coefficients by measuring the Δ훼 change in parameter and Δ퐴 effect on an output. It is noted that the Yates Algorithm is a part of least-squares regression on a DOE parameter set and uses the same F-statistic to measure significance of the coefficients (technically, the F-distribution is a squared Student’s t- distribution, but the critical values are transferrable and results are equivalent). This resource also mentions that sensitivity can be measured in absolute terms (units of partial derivative) or relative (percent output versus percent input deviation). Most examples use this type of study to experimentally determine input/state output sensitivity or state output

105 sensitivity with respect to the model parameters. This is opposed to the idea presented here of using the sensitivity on an output metric – that is, not a linear transformation of the output states.

This is also a reminder that this work only includes main effects in the sensitivity study, and there are likely multiple significant quadratic and two-way interactions with respect to the output metrics. Depending on how close a parameter is to a local maximum or minimum and the magnitude of the parameter variation, capturing the main effect could yield the local derivative or the larger overall trend. The degree to which this is useful depends on the expected variation/uncertainty in the parameter values as well as the nominal levels. For this study, reasonable ranges for parameter values were chosen around estimated nominal values which could be physically realizable, however it is not guaranteed that these parameters will remain in these ranges post-calibration. A full investigation of parameter sensitivity including higher-order terms could be done in the future if the need is there for propagation of uncertainty and understanding of nonlinear relationships.

After the system parameters are identified and varied according to the above experimental outline, the collected output signals are post-processed to calculate the performance metrics discussed earlier. The result of the first post-processing step is a design matrix of the variables (푥𝑖) augmented with the output metrics (푌𝑖) for each variant at each of the shift points. An output file like this is generated for each experimental design subset outlined above. An example of such an output table is shown below

(hypothetical values):

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Table 18: Hypothetical Output Table - Input Parameters Augmented with Outputs for Each Shift

풙 풙 풙 풀 풀 풀 ퟏ ퟐ ퟑ ퟏ,ퟏ→ퟐ ퟏ,ퟐ→ퟑ ퟏ,ퟑ→ퟒ … (Mass) (Stiffness) (Damping) (푻풉풐풍풆,ퟏ) (푻풉풐풍풆,ퟐ) (푻풉풐풍풆,ퟑ) 5 10 2 -239 -142 -120 … 5 10 3 -250 -145 -132 … 5 15 2 -235 -141 -117 …

For each shift event in the shift schedule, a shift metric calculation is triggered by the post-processing algorithm and different metrics are calculated depending on the subsystem under test. This means that for a shift schedule accelerating up through 10th gear and decelerating back to 1st contains 18 shift events to be analyzed. Looking at the full processed metric data set in raw aggregate form allows one to see varying magnitudes in the metrics across the different shift events.

However, to parse this data and draw meaningful conclusions about relationships, it is useful to condense these 18 samples of each metric into one mean value for the whole variant run. This is possible because of the way the parameters are varied across all like components in the model at the same time. For example, a chosen value of solenoid inductance is applied to all solenoids for that run, so any shift event uses the same variant of solenoid parameters. If each solenoid were treated independently, it would only be meaningful to study those shift events where the altered solenoid was being utilized. This way, all shift events involve the chosen solenoid parameters and one may study the overall trend in pressure delay times as a result of varying all inductance values, for example.

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By taking the mean of metrics across all shift events, a reduced summary table is produced where the columns of parameter level sets are augmented with columns of single metrics.

Table 19: Hypothetical Metric Handling – Metrics Averaged across All Shifts

풀풊 12 23 34 45 56 67 78 … 풀흁

푻풉풐풍풆 -145 -155 -145 -155 -145 -155 -145 … -150

풕풔풆풕풕 0.55 0.65 0.55 0.65 0.55 0.65 0.55 …  0.60 푶푺% 42 38 42 38 42 38 42 … 40

푻̇ 풑풆풂풌 2.3e5 2.7e5 2.3e5 2.7e5 2.3e5 2.7e5 2.3e5 … 2.5e5

Table 20: Hypothetical Output Table – Mean Metrics Augmented with Parameters

풙ퟏ 풙ퟐ 풙ퟑ 풀ퟏ,흁 풀ퟐ,흁 풀ퟑ,흁 5 10 2   50.6 -142 0.50 5 10 3 47.8 -145 0.35 5 15 2 56.2 -141 0.61

This table may now be used in multiple linear regression to correlate each output column with the variations in factors. To model each output metric, enough information was gathered through the design of experiments to measure the effects of all factors, two- way interactions between factors, and quadratic effects of each factor. This is ensured by using design resolutions of five or higher and using three-level factorial or central composite designs to test nonlinearities. However, to judge sensitivity, only main effects need be calculated. To reduce multicollinearity of inputs, the design matrices discussed in

108 previous sections were formed by deviating the factors equal amounts in each direction when possible. This allows the factor design matrix to be coded, by normalizing each parameter with respect to its range. After coding, the lowest input factor level becomes

−1, while the maximum of the same factor becomes +1. The midpoint naturally is zero on this re-mapped number line. Coding before regression ensures that interaction terms are independent of their constituents.

A coded regression coefficient represents the deviation in the output for a deviation of the input factor by one 푐표푑푒푑 unit (since the maximum minus the midpoint in coded form is always 1). Therefore, the slope of the correlation between the output and selected input factor becomes the coded regression coefficient divided by the deviation of the selected input factor in uncoded units (maximum factor value minus the mean factor level). To remove the effect of relative magnitudes and compare each coefficient on the same level, units were removed by normalizing the coefficient into percentage deviations in the numerator and denominator. This is shown below for the definition of relative sensitivity used in this analysis:

% 표푢푡푝푢푡 푑푒푣푖푎푡푖표푛 Δ푦/푦̅ 푥̅ 퐶 푥̅ 푠푒푛푠푖푡푖푣푖푡푦 = = = 퐶 = 푥,푐표푑푒푑 (21) % 푖푛푝푢푡 푑푒푣푖푎푡푖표푛 Δ푥/푥̅ 푥 푦̅ Δ푥 푦̅

It is important to not only produce the coefficients for each relationship, but also test the correlation for statistical significance. A Student’s t test is used to test the null hypothesis that estimates using a linear model of the form 푦̂ = 훽0 + 훽푛푥푛 is statistically the same as using 푦̂ = 훽0 as an estimation model. The confidence level is chosen to be

95% for each parameter, and p-values are tabulated for each coefficient. The following sections detail the calculated sensitivities (without the raw coefficients) along with the 109 statistical significance of the effect. Importantly, sensitivity values whose corresponding p-values are above the significance level of 훼 = 0.05 should not be considered and are included for completion and visual comparison. In charts, statistically insignificant effects have been greyed-out to reduce emphasis and focus the charts on the bold-colored significant effects.

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5.2.1. Gearbox Parameter Sensitivity

Figure 42: Hi-Fi Gearbox Clutch Friction Sensitivity Plot

Table 21: Hi-Fi Gearbox Clutch Friction Sensitivity Data

Torque Sticking mu Slipping mu Disk Thickness p sens p sens p sens p sens simTime 0.588 -0.014 0.097 0.209 0.054 -0.249 0.944 0.002 stepCount 0.137 0.018 0.000 0.245 0.005 -0.172 0.208 0.015 dTqMax 0.000 0.362 0.000 1.108 0.000 -0.519 0.000 0.054 OSperc 0.000 0.676 0.000 0.370 0.000 -0.839 0.735 -0.005 t_tq 0.000 -0.048 0.000 -0.120 0.008 0.071 0.392 -0.004 t_in 0.000 0.275 0.000 -0.569 0.003 0.235 0.907 0.002 t_sett 0.000 0.296 0.205 0.119 0.048 -0.191 0.738 0.006 T_hole 0.000 0.166 0.000 0.875 0.130 -0.069 0.048 -0.018

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Figure 43: Hi-Fi Gearbox Hydraulic Cylinder Sensitivity Plot

Table 22: Hi-Fi Gearbox Hydraulic Cylinder Sensitivity Data

Trq Piston Stiffness Piston Damping Piston Mass Piston Lash Piston Volume

p sens p sens p sens p sens p sens p sens simTime 0.000 0.024 0.995 0.000 0.064 0.016 0.819 0.002 0.507 0.006 0.104 -0.029 stepCount 0.000 0.022 0.308 -0.008 0.198 0.010 0.963 0.000 0.991 0.000 0.481 -0.011 dTqMax 0.000 0.372 0.130 -0.018 0.270 -0.013 0.043 0.024 0.527 0.007 0.745 -0.008

OSperc 0.000 0.658 0.000 0.069 0.042 -0.027 0.137 -0.020 0.000 0.053 0.762 0.008 t_tq 0.000 -0.031 0.000 -0.078 0.119 -0.009 0.088 0.010 0.000 -0.067 0.730 0.004 t_in 0.000 0.324 0.004 -0.038 0.936 -0.001 0.253 0.015 0.239 -0.015 0.792 0.007 t_sett 0.000 0.256 0.523 -0.012 0.700 -0.007 0.184 -0.024 0.134 -0.028 0.852 -0.007

T_hole 0.000 0.171 0.000 -0.055 0.022 0.017 0.412 0.006 0.000 -0.032 0.986 0.000

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Figure 44: Hi-Fi Gearbox Planetary Gearset Sensitivity Plot

Table 23: Hi-Fi Gearbox Planetary Gearset Sensitivity Data

Bearing Bearing Planet jFact Planet jFact Torque Tooth Lash Bearing Lash Stiffness Damping (self) (orbit) p sens p sens p sens p sens p sens p sens p sens sim 0.364 -0.035 0.000 0.457 0.949 -0.007 0.000 -0.482 0.162 0.036 0.434 0.080 0.043 0.209 Time step 0.808 0.008 0.000 0.361 0.773 0.025 0.000 -0.390 0.000 0.124 0.797 0.022 0.124 0.133 Count dTq 0.000 0.307 0.000 0.036 0.000 0.492 0.000 -0.100 0.000 0.084 0.898 -0.002 0.857 -0.003 Max OS 0.000 0.698 0.010 -0.014 0.063 0.039 0.835 0.005 0.434 -0.004 0.976 -0.001 0.341 -0.020 perc t_tq 0.000 -0.019 0.000 -0.001 0.018 0.002 0.000 0.022 0.000 -0.001 0.673 0.000 0.498 -0.001 t_in 0.000 0.298 0.057 -0.009 0.000 -0.124 0.266 -0.026 0.000 -0.030 0.972 -0.001 0.431 0.015 t_sett 0.000 0.295 0.001 -0.028 0.000 -0.200 0.000 -0.787 0.000 -0.039 0.981 -0.001 0.781 0.010 T_ 0.000 0.134 0.002 0.011 0.020 0.032 0.000 -0.144 0.000 0.012 0.945 -0.001 0.299 0.014 hole

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Figure 45: Hi-Fi Gearbox Two-Way Clutch Sensitivity Plot

Table 24: Hi-Fi Gearbox Two-Way Clutch Sensitivity Data

Torque TWC Damping TWC Stiffness TWC Locking Speed

p sens p sens p sens p sens simTime 0.001 -0.376 0.472 -0.071 0.078 0.131 0.412 0.061 stepCount 0.001 -0.172 0.601 0.023 0.064 0.061 0.483 0.023 dTqMax 0.000 0.342 0.368 0.004 0.251 -0.003 0.965 0.000 OSperc 0.000 0.639 0.809 0.003 0.866 -0.002 0.335 -0.010 t_tq 0.000 -0.045 0.388 -0.004 0.944 0.000 0.421 -0.003 t_in 0.000 0.240 0.883 0.002 0.338 -0.009 0.582 -0.005 t_sett 0.000 0.291 0.809 0.004 0.913 0.001 0.994 0.000 T_hole 0.000 0.176 0.270 0.007 0.846 -0.001 0.022 0.011

It is somewhat surprising that the two-way clutch parameters do not have a significant effect on any of the simulation parameters, let alone the performance outputs.

It was hypothesized that using primarily damping to lockup the TWC may improve the simulation efficiency, but there was no evidence to support this claim. The clutch friction parameters were significant with respect to almost all outputs except the settling time,

114 which was influenced more by the planetary gearset parameters. At this highest fidelity level, it is worth noting that there is a high correlation between the planet tooth backlash gap and bearing damping with simulation speed. The sensitivity of +0.5 for 푗푡 implies that the removal of backlash (-100% change) would reduce the simulation time by 50%.

It is also worth noting that very little correlation was found between the clutch hydraulic parameters and the torque trace outputs, other than the clutch piston stiffness and lash having approximately the same effects. This would make sense because both of these parameters would similarly delay the time it takes for the clutch plates to make contact.

However, it seems that the major timing variables are associated almost exclusively with friction characteristics. If the torque oscillatory settling time needs to be reduced for calibration purposes, the planetary gearset elastic parameters are the place to start, seeing a strong negative correlation with the stiffness and damping values.

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5.2.2. Hydraulic Network Parameter Sensitivity

Figure 46: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Electrical Sensitivity Plot

Table 25: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Electrical Sensitivity Data

resistance inductance backEMF LVL1Q p sens p sens p sens finishTime 0.001 -2.851 0.578 0.066 0.072 -0.217 stepCount 0.001 -3.148 0.630 0.063 0.066 -0.244 Overshoot_perc 0.125 0.598 0.000 -0.423 0.415 0.045 RiseTime 0.005 0.001 0.000 -0.001 0.781 0.000 DelayTime 0.000 -0.009 0.000 0.007 0.118 0.000 CornerTime 0.002 0.022 0.011 -0.003 0.776 0.000 CornerPressure 0.083 0.010 0.001 -0.003 0.238 0.001

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Figure 47: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Electrical Sensitivity Plot

Table 26: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Electrical Sensitivity Data

resistance inductance backEMF LVL1P p sens p sens p sens finishTime 0.041 -2.658 0.358 -0.170 0.542 -0.113 stepCount 0.045 -3.186 0.355 -0.208 0.530 -0.141 Overshoot_perc 0.000 -0.044 0.000 0.035 0.367 -0.001 RiseTime 0.009 0.001 0.000 0.000 0.000 0.000 DelayTime 0.000 -0.008 0.000 0.007 0.520 0.000 CornerTime 0.194 -0.034 0.000 0.014 0.662 -0.002 CornerPressure 0.820 0.002 0.000 0.005 0.271 -0.001

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Figure 48: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Mechanical Sensitivity Plot

Table 27: Hi-Fi Hydraulic Network (Flow-Sourced) Linear Solenoid Mechanical Sensitivity Data

mass stiffness damping preload volume LVL1Q p sens p sens p sens p sens p sens finishTime 0.568 -0.094 0.573 -0.277 0.320 0.327 0.683 -0.134 0.638 0.155 stepCount 0.424 -0.142 0.433 -0.418 0.229 0.428 0.478 -0.252 0.753 0.111 Overshoot 0.000 1.075 0.621 -0.179 0.264 0.270 0.078 -0.428 0.000 -2.281 percent RiseTime 0.000 0.003 0.000 -0.010 0.000 -0.007 0.000 -0.066 0.000 -0.006 DelayTime 0.067 0.001 0.000 0.039 0.011 0.002 0.000 0.130 0.000 0.017 Corner 0.000 -0.042 0.000 0.243 0.003 -0.061 0.018 0.049 0.000 0.078 Time Corner 0.000 0.034 0.327 0.015 0.000 0.061 0.000 0.098 0.017 0.025 Pressure

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Figure 49: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Mechanical Sensitivity Plot

Table 28: Hi-Fi Hydraulic Network (Pressure-Sourced) Linear Solenoid Mechanical Sensitivity Data

mass stiffness damping preload volume LVL1P p sens p sens p sens p sens p sens finishTime 0.665 -0.118 0.913 -0.090 0.024 1.247 0.379 -0.482 0.524 0.348 stepCount 0.553 -0.198 0.998 0.002 0.022 1.545 0.266 -0.743 0.432 0.524 Overshoot 0.738 0.182 0.035 3.495 0.834 0.229 0.000 -6.054 0.005 -3.095 percent RiseTime 0.005 0.016 0.009 -0.046 0.003 -0.034 0.157 -0.016 0.008 -0.031 DelayTime 0.111 -0.008 0.000 0.067 0.042 0.021 0.000 0.111 0.001 0.036 Corner 0.333 0.194 0.430 -0.475 0.033 -0.863 0.171 0.550 0.210 -0.503 Time Corner 0.000 0.041 0.010 -0.039 0.000 0.057 0.000 0.065 0.000 0.061 Pressure

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Figure 50: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Electrical Sensitivity Plot

Table 29: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Electrical Sensitivity Data

resistance inductance backEMF LVL1Q p sens p sens p sens finishTime 0.131 0.999 0.261 0.119 0.255 0.121 stepCount 0.102 1.233 0.303 0.124 0.172 0.165 Overshoot_perc 0.614 0.033 0.000 0.086 0.000 0.044 RiseTime 0.012 -0.001 0.000 0.001 0.502 0.000 DelayTime 0.327 -0.001 0.000 0.001 0.141 0.000 CornerTime 0.003 -0.025 0.020 0.003 0.527 0.001 CornerPressure 0.000 -0.013 0.000 0.009 0.000 0.002

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Figure 51: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Electrical Sensitivity Plot

Table 30: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Electrical Sensitivity Data

resistance inductance backEMF LVL1P p sens p sens p sens finishTime 0.081 -10.252 0.462 -0.684 0.195 -1.214 stepCount 0.081 -6.784 0.451 -0.465 0.168 -0.857 Overshoot_perc 0.062 -3.473 0.035 0.634 0.034 0.637 RiseTime 0.028 -0.001 0.368 0.000 0.017 0.000 DelayTime 0.150 -0.001 0.000 0.001 0.096 0.000 CornerTime 0.493 0.015 0.041 -0.007 0.727 0.001 CornerPressure 0.779 0.001 0.008 -0.002 0.022 0.002

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Figure 52: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Mechanical Sensitivity Plot

Table 31: Hi-Fi Hydraulic Network (Flow-Sourced) Shift Solenoid Mechanical Sensitivity Data

mass stiffness damping preload endStop stopStiffness LVL1Q p sens p sens p sens p sens p sens p sens finish 0.532 -0.135 0.151 -0.519 0.117 -0.765 0.717 0.156 0.140 0.480 0.529 -0.272 Time stepCount 0.579 -0.128 0.145 -0.564 0.121 -0.809 0.536 0.286 0.110 0.557 0.532 -0.289 Overshoot 0.000 0.228 0.222 -0.076 0.304 0.086 0.000 -0.858 0.136 -0.084 0.200 0.096 percent RiseTime 0.000 0.002 0.013 -0.001 0.839 0.000 0.000 -0.010 0.752 0.000 0.727 0.000 Delay 0.000 -0.001 0.052 0.001 0.004 0.002 0.000 -0.061 0.723 0.000 0.419 0.000 Time Corner 0.077 0.006 0.223 0.007 0.000 0.028 0.000 0.083 0.648 0.002 0.537 0.004 Time Corner 0.000 0.036 0.000 -0.027 0.000 0.026 0.000 0.140 0.758 0.001 0.922 -0.001 Pressure

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Figure 53: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Mechanical Sensitivity Plot

Table 32: Hi-Fi Hydraulic Network (Pressure-Sourced) Shift Solenoid Mechanical Sensitivity Data

mass stiffness damping preload endStop stopStiffness LVL1P p sens p sens p sens p sens p sens p sens finish 0.529 -0.080 0.489 -0.147 0.378 -0.252 0.017 0.616 0.461 0.141 0.016 -0.621 Time stepCount 0.739 -0.050 0.314 -0.253 0.224 -0.414 0.055 0.584 0.637 0.107 0.033 -0.651 Overshoot 0.620 -0.101 0.714 -0.124 0.913 -0.050 0.000 -6.214 0.896 -0.040 0.853 0.075 percent RiseTime 0.439 0.000 0.561 0.000 0.191 0.001 0.000 -0.009 0.906 0.000 0.460 0.000 Delay 0.000 -0.001 0.000 0.001 0.000 0.002 0.000 -0.055 0.763 0.000 0.766 0.000 Time Corner 0.486 0.006 0.344 -0.014 0.645 0.009 0.480 0.013 0.013 0.034 0.428 0.014 Time Corner 0.264 -0.002 0.038 -0.005 0.984 0.000 0.000 -0.018 0.027 0.005 0.292 -0.003 Pressure

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Figure 54: Hi-Fi Hydraulic Network (Flow-Sourced) Regulation Valve Sensitivity Plot

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Table 33: Hi-Fi Hydraulic Network (Flow-Sourced) Regulation Valve Sensitivity Data

mMRV bMRV mTCRV bTCRV mLRV bLRV LVL1Q p sens p sens p sens p sens p sens p sens Finish 0.879 0.150 0.568 -0.281 0.007 2.707 0.673 0.208 0.676 -0.412 0.013 -1.245 Time Step 0.847 -0.204 0.681 -0.217 0.006 2.988 0.655 0.236 0.650 -0.480 0.016 -1.295 Count Overshoot 0.349 -0.069 0.002 0.120 0.388 -0.064 0.763 0.011 0.000 0.270 0.092 0.063 Percent Rise Time 0.130 0.001 0.221 0.000 0.042 -0.001 0.065 0.000 0.894 0.000 0.011 0.001 Delay 0.263 -0.001 0.683 0.000 0.578 0.000 0.366 0.000 0.353 0.001 0.770 0.000 Time Corner 0.608 0.006 0.086 -0.011 0.325 0.012 0.040 -0.013 0.112 -0.020 0.133 0.009 Time Corner 0.537 0.004 0.857 0.001 0.887 0.001 0.668 0.001 0.286 -0.007 0.439 -0.002 Pressure

mLCCV bLCCV mPLAV bPLAV LVL1Q (cont.) p sens p sens p sens p sens Finish Time 0.002 -3.080 0.767 -0.146 0.017 2.387 0.014 -1.235 Step Count 0.002 -3.337 0.612 -0.268 0.020 2.497 0.020 -1.244 Overshoot 0.198 -0.096 0.000 -0.140 0.003 -0.228 0.024 0.085 Percent Rise Time 0.366 0.000 0.440 0.000 0.941 0.000 0.332 0.000 Delay Time 0.923 0.000 0.908 0.000 0.440 -0.001 0.069 -0.001 Corner Time 0.324 -0.012 0.009 0.016 0.266 0.014 0.427 0.005 Corner Pressure 0.235 -0.008 0.510 0.002 0.000 0.048 0.000 0.014

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Figure 55: Hi-Fi Hydraulic Network (Pressure-Sourced) Regulation Valve Sensitivity Plot

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Table 34: Hi-Fi Hydraulic Network (Pressure-Sourced) Regulation Valve Sensitivity Data

mMRV bMRV mTCRV bTCRV mLRV bLRV LVL1P p sens p sens p sens p sens p sens p sens

finishTime 0.000 -22.036 0.000 -11.179 0.983 -0.105 0.983 0.051 0.912 -0.537 0.999 -0.002

stepCount 0.000 -10.483 0.000 -5.654 0.947 -0.160 0.949 0.077 0.562 -1.396 0.939 -0.092

Overshoot_perc 0.001 0.051 0.537 -0.005 0.138 -0.022 0.354 0.007 0.452 -0.011 0.342 0.007

RiseTime 0.790 0.000 0.385 0.000 0.334 0.000 0.042 0.000 0.012 0.001 0.230 0.000

DelayTime 0.026 0.001 0.860 0.000 0.908 0.000 0.638 0.000 0.009 -0.002 0.046 -0.001

CornerTime 0.041 -0.065 0.233 -0.019 0.174 -0.043 0.519 -0.010 0.004 0.095 0.012 0.040

CornerPressure 0.005 0.017 0.001 0.010 0.016 -0.014 0.442 -0.002 0.820 -0.001 0.758 -0.001

mLCCV bLCCV mPLAV bPLAV LVL1P (cont.) p sens p sens p sens p sens finishTime 0.999 0.008 0.992 0.024 0.988 0.073 0.837 -0.503 stepCount 0.992 0.025 0.985 0.022 0.876 0.375 0.283 -1.297 Overshoot_perc 0.840 -0.003 0.610 0.004 0.110 0.024 0.944 0.001 RiseTime 0.000 0.002 0.034 0.000 0.287 0.000 0.867 0.000 DelayTime 0.004 -0.002 0.331 0.000 0.586 0.000 0.075 -0.001 CornerTime 0.000 0.148 0.642 0.007 0.788 -0.009 0.609 0.008 CornerPressure 0.764 -0.002 0.073 0.005 0.058 0.011 0.016 -0.007

The hydraulic network parameters show some form of dependence on the choice of line source. That is, the sensitivity of the output metrics changes based on whether the supply pressure is fixed as a source, or sourced by a constant flow rate. The most significant effects observed from the solenoid valves were the volume sizes and preload values and their negative correlations with the overshoot percentage. A higher preload value would cause the valve to delay more before opening, which could reduce the flow rate under transient opening events and lower potential overshoot. The delay time was noted to increase in these scenarios, but the rise time was not, implying that once opened, the pressure trace rose at the same expected rates. Multiple parameters had strong effects on the variable-step simulation time, but they varied somewhat in the regulation circuit as 127 to which elements were the cause. It was surprising to see that the main regulator valve had more of an impact on simulation time when the line pressure was set versus when the constant pressure source was supplied.

5.2.3. Torque Converter Parameter Sensitivity

Figure 56: Hi-Fi Torque Converter Blade Angle Sensitivity Plot

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Table 35: Hi-Fi Torque Converter Blade Angle Sensitivity Data

Trq ap at as LVL1 p sens p sens p sens p sens finishTime 0.058 -0.006 0.000 0.225 0.068 0.141 0.990 -0.001 stepCount 0.000 0.000 0.262 0.000 0.127 0.000 0.000 -0.001 TRstall 0.785 0.000 0.000 -0.128 0.000 0.197 0.000 0.505 Kstall 0.968 0.000 0.000 -0.156 0.000 0.193 0.000 0.790 CouplingPt 1.000 0.000 1.000 0.000 0.009 0.015 0.009 0.015 etaPeak 0.965 0.000 0.000 -0.013 0.000 0.098 0.000 0.128 SRetaPeak 1.000 0.000 1.000 0.000 0.009 0.015 0.009 0.015 RMSTE 0.337 -0.034 0.000 -2.362 0.000 4.116 0.000 11.646 RMSKE 0.517 0.000 0.000 0.065 0.000 -0.194 0.000 -0.552

LVL1 bp bt bs (cont.) p sens p sens p sens finishTime 0.849 -0.013 0.170 0.066 0.739 0.001 stepCount 0.000 0.001 0.000 0.000 1.000 0.000 TRstall 0.000 -0.120 0.000 -0.162 0.000 0.002 Kstall 0.000 0.118 0.000 0.160 0.000 -0.002 CouplingPt 0.009 0.013 1.000 0.000 1.000 0.000 etaPeak 0.000 -0.090 0.000 -0.033 0.000 -0.001 SRetaPeak 0.009 0.013 1.000 0.000 1.000 0.000 RMSTE 0.000 -3.259 0.000 -3.112 0.276 0.052 RMSKE 0.000 -0.238 0.000 -0.101 0.000 -0.002

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Figure 57: Hi-Fi Torque Converter Empirical Factor Sensitivity Plot

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Table 36: Hi-Fi Torque Converter Empirical Factor Sensitivity Data

Trq Sp St Ss LVL1 p sens p sens p sens p sens finishTime 0.000 0.022 0.207 -0.006 0.416 0.002 0.989 0.000 stepCount 0.000 0.000 0.495 0.000 0.670 0.000 0.765 0.000 TRstall 0.957 0.000 1.000 0.000 1.000 0.000 1.000 0.000 Kstall 0.987 0.000 1.000 0.000 1.000 0.000 1.000 0.000 CouplingPt 0.000 0.007 0.931 0.000 0.931 0.000 0.931 0.000 etaPeak 0.000 0.003 0.935 0.000 0.935 0.000 0.936 0.000 SRetaPeak 0.000 0.007 0.931 0.000 0.931 0.000 0.931 0.000 RMSTE 0.859 0.008 0.994 0.001 0.995 0.000 0.996 0.001 RMSKE 0.370 0.002 0.990 0.000 0.991 0.000 0.992 0.000

LVL1 Cp Ct Cs f (cont.) p sens p sens p sens p sens finishTime 0.503 -0.002 0.034 -0.020 0.006 0.013 0.205 -0.007 stepCount 0.002 0.000 0.000 0.000 0.000 0.000 0.017 0.000 TRstall 0.000 -0.054 0.000 -0.431 0.000 -0.030 0.000 -0.041 Kstall 0.000 0.055 0.000 0.426 0.000 0.028 0.000 0.038 CouplingPt 0.000 -0.161 0.000 -0.032 0.000 -0.010 0.000 -0.012 etaPeak 0.000 -0.154 0.000 -0.064 0.000 -0.019 0.000 -0.017 SRetaPeak 0.000 -0.161 0.000 -0.032 0.000 -0.010 0.000 -0.012 RMSTE 0.126 -0.136 0.000 -1.795 0.364 -0.101 0.025 -0.300 RMSKE 0.000 -0.325 0.000 -0.193 0.000 -0.030 0.000 -0.034

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Figure 58: Hi-Fi Torque Converter Geometrical Sensitivity Plot

Table 37: Hi-Fi Torque Converter Geometrical Sensitivity Data

Trq Rp Rt Rs Lf A LVL1 p sens p sens p sens p sens p sens p sens finish 0.002 0.011 0.718 -0.024 0.289 0.037 0.401 -0.040 0.119 0.010 0.006 0.028 Time step 0.000 0.000 0.309 0.000 0.000 0.000 0.111 0.000 0.168 0.000 0.000 0.000 Count TR 0.943 0.000 0.000 -0.205 0.000 0.165 0.000 0.041 0.996 0.000 0.988 0.000 stall Kstall 0.999 0.000 0.000 -2.020 1.000 0.000 0.000 0.508 1.000 0.000 0.000 -0.517 Coup. 0.104 0.006 0.000 0.598 0.000 -0.137 0.000 -0.325 0.266 0.007 0.509 -0.007 Pt eta 0.243 0.002 0.000 0.556 0.000 -0.182 0.000 -0.306 0.240 0.004 0.395 -0.005 Peak SR eta 0.104 0.006 0.000 0.598 0.000 -0.137 0.000 -0.325 0.266 0.007 0.509 -0.007 Peak RMS 0.474 -0.092 0.000 -13.58 0.000 -6.968 0.000 9.610 0.999 0.000 0.620 -0.182 TE RMS 0.678 0.014 0.077 -1.112 0.000 -1.318 0.032 0.994 0.996 0.000 0.001 0.318 KE

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Almost all parameters with strong significant effects impact the RMS torque error curve. As this is an important measure of the steady state accuracy of the model, it makes sense that many of the parameters play a role. However, it should be noted that there are some parameters to note which had a lower relative sensitivity. One of these was the stator entrance blade angle, 훽푠. Upon inspection, this parameter would have benefitted from an absolute sensitivity measurement. The nominal value was close to zero and variations during the DOE process produced much larger relative (%) deviations than for the parameters whose nominal values were much higher. Looking at the data table provided, the stator entry angle has significant correlations with similar metrics as for the turbine entrance angle, except the step count and RMS torque error. It was also somewhat surprising to note that the stator shock loss and friction loss factors were significantly less sensitive than the turbine and pump shock losses. This does make sense as more of the kinetic energy is in these two elements, however the stator shock loss and friction factors do not suffer the same limitations as the stator entrance angle mentioned above.

5.3. Analysis and Discussion

Studying the gearbox sensitivity outputs, obvious extremes are present with the correlations under study. Some parameters, such as the clutch sticking friction coefficient, showed significant correlations for every output metric except the torque settling time (and simulation speed). Others, such as the hydraulic piston mass, showed very small correlations, even when judged as significant.

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Looking at the metrics individually and eliminating insignificant parameters, the friction coefficients are impactful on every physical metric outside of the simulation speed and step count. The sticking friction coefficient is ranked in the top-3 parameters for each output metric except for settling time and is the most impactful parameter for the peak jerk torque, torque phase duration, inertia phase duration, and torque hole. Many of these correlations make intuitive sense, but having a quantitative measure associated with the effects can be enlightening. For example, the torque hole is primarily affected by the sticking friction coefficient with a sensitivity of 0.875, while the second-place influencer

(input torque) has a drastically-lowered impact on the torque hole, at only 0.166. It is also validating to see that the overall torque settling time had the highest sensitivity to the internal planetary damping factor, at −0.787. The top-5 parameters for each physical output metric are shown below:

Table 38: Gearbox Sensitivity Parameter Rankings by Absolute Sensitivity

푑푇푞/푑푡푚푎푥 푂푆 % 푡푡푞 푡𝑖푛 푡푠푒푡푡 푇ℎ표푙푒

휇0 1.11 휇 -0.84 휇0 -0.12 휇0 -0.57 bBt -0.79 휇0 0.88 휇 -0.52 Trq 0.68 pistonK -0.08 Trq 0.28 Trq 0.30 Trq 0.17

kBt 0.49 휇0 0.37 휇 0.07 휇 0.24 kBt -0.20 bBt -0.14 Trq 0.36 pistonK 0.07 pistonL -0.07 kBt -0.12 휇 -0.19 pistonK -0.06 bBt -0.10 pistonL 0.05 Trq -0.05 pistonK -0.04 LBt -0.04 pistonL -0.03

The hydraulic network sensitivity study revealed that there are more significant effects on the solution speed than many of the measured output metrics. While this may seem like the results do not reveal much about the metrics, what one can say is that changing the parameters within the hydraulic network has lower impact on the output 134 metrics. There are the apparent correlations of the spring-mass-damper components with overshoot percentage (especially in the linear solenoids), but many other parameters had little to no effect on the outputs. Preload forces appear to influence the delay time and corner pressure, which makes intuitive sense because increasing the preload force increases the threshold for movement of the plunger. Another obvious note is that the volume size of the linear solenoids influences the overshoot percentage by damping out the overshoot as the volume is increased.

The relatively few significant effects with respect to the output metrics paints a picture of a subsystem which is somewhat resistant to many of the unknown dynamic parameters. While possible, it is also likely that the actuation scheme and measurement metrics used in this work have masked significant dynamic features which do, in fact, vary significantly with the model parameters. Clutch filling dynamics were neglected in this modeling exercise and would thus contribute a missing physical effect on the pressure output response. Another possible cause of this lack of correlation is that the nominal conditions for the model parameters were not entirely representative of the

“true” hydraulic system. In reality, it is most likely that an intersection of these effects is being observed, with some effects being masked by the current-ramp inputs and the constant pressure/flow rate source assumptions, while others are masked due to incomplete dynamic modeling and yet more are simply insignificant on these chosen output metrics.

Under ideal regulation conditions, a clutch supply pressure which is being metered via feedback control of a linear solenoid should not significantly vary with any

135 parameters other than the linear solenoid electrical and mechanical parameters. In practice this would not be true, especially if the timing of upstream dynamics were such that coupled pressures had not sufficiently settled prior to the opening of a solenoid valve. Ultimately, pressure overshoot would likely not have a significant effect on the clutch engagement dynamics and it should be noted that many of these output metrics may not be felt by a clutch plate as significantly in the presence of feedback control.

However, it should be noted that for control verification testing, these types of judgements are made by internal checks in control software, which allow the controller to verify that the hydraulic system is working as intended and operating at correct levels. It is therefore more relevant to control verification that this subsystem behaves with the correct timing and steady state levels, as opposed to the gearbox findings, which are more-tightly tied to performance and drivability.

For the torque converter parameter sensitivity study, it should be noted that many previous works have focused on qualitatively examining the sensitivity of parameters with respect to efficiency and capacity factor. Such works have essentially ended with graphical representations of the torque ratio, capacity factor, and/or efficiency curves.

However, one resource has used an objective function based on energy loss to study the sensitivity of parameters analytically (Banerjee, Asl, Azad, & McPhee, 2012). The resulting ranking of parameters with respect to the loss function is shown below:

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Table 39: Importance Score for Model Parameters (Banerjee, Asl, Azad, & McPhee, 2012)

These results can be compared with the rankings of significant sensitivity to the two aggregate metrics used in this work: the RMS errors of the torque ratio and capacity factor curves. The values of the sensitivity scores are different due to differing methodologies and the rankings should not be taken as a direct comparison due to the differing objective function(s). The following list removed parameters from the above study which were not tested in this work and removed parameters from this work which were found to be statistically insignificant.

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Table 40: Comparison of Torque Converter Sensitivity Rankings to Analytical Method from Literature

Analytical Experimental Experimental Importance Ranking Sensitivity Ranking Sensitivity Ranking (Table 36) (RMS TR Error) (RMS K Error)

푅푝 푅푝 푅푡 푅푠 훼푠 푅푠 훼푠 푅푠 훼푠 퐴 푅푡 퐶푝 푅푡 훼푡 퐴 훼푡 훽푝 훽푝 훼푝 훽푡 훼푡

It can be seen based on the above rankings that many of the same parameters have been found to be important. One of the only major differences between the literature ranking and the RMSTE ranking is the inclusion of the flow area as a significant parameter. The flow area was not found to be significant for RMSTE, but it was significant (ranked fifth) for the RMSKE analysis. The inclusion of the fluid density high on the list from the above study suggests a significant effect can be seen on torque converter losses when the density of the fluid is varied. The density would also affect the inertial terms in the dynamic equations, which were not investigated here.

Overall, it has been shown that the blade angles consistently have a large effect of the torque ratio steady-state curve. The shock loss factors for the pump and turbine also have a large effect on the measured metrics, with the turbine shock loss factor having a large negative correlation with the RMS torque error (albeit still outranked by radii and blade angles). The element radii and blade angles have been shown to have the largest effects on the aggregate RMS error metrics, though there are also significant (lesser) effects on the stall measures of torque ratio and efficiency. Dynamic effects were not 138 studied in this work and should be incorporated with a vehicle model to analyze transient events such as shifts, tip-in, and tip-out with respect to the given parameters. It should be expected that the shape factors, fluid path lengths, inertias, and fluid density should play more of a role in the transient dynamics.

Taking a step back, the parameter sensitivity exercise has provided insight to the strengths and directions of correlations between the high-fidelity model parameters and the simulation performance metrics. The significant parameters from this study should be especially cared for when working through the model calibration process, due to the large impacts on the chosen metrics in this work. Uncertainty in “sensitive” model parameters could propagate with more severe implications to future model accuracy. In addition, controller calibration exercises should take care to study robustness to changes or uncertainty in the highly-sensitive model parameters.

If this work is to be extended, one may also create higher-order sensitivity models which include quadratic parameter effects and potentially 2nd- or 3rd-order interaction terms. As noted in the design of experiments, all data here was collected to allow for the correlation of up to three-way interactions without aliasing of signals, provided the constituent terms belong to the same parameter subset.

The sensitivity of the transmission may vary in magnitude from transmission to transmission based on possible changes in nominal values (equivalent to changing the linearization point). The gearbox parameter sensitivity should be especially transferrable to other clutch-to-clutch-based transmission architectures. However, it would require significantly different deviation scales or linearization points to change the direction of

139 the correlations found in this study. All experimental studies are reliant on the operating/nominal point, and it is usually possible to fall into a local maximum/minimum of the “true” behavior to produce a localized, near-zero correlation (read: derivative).

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Chapter 6. Reduced Model Development

The simulation speed of any dynamic system is going to be related to the frequencies of the eigen-modes. To eliminate dynamic states on the critical path of simulation speed, the dynamic states associated with high-frequency eigenvalues should be reduced with higher priority. The eigenvalues of the system cannot always be calculated using the Natural Frequencies Tool in SimulationX, and they are only applicable to an operating point around where the [linearized] snapshot is taken.

Therefore, the monitoring of state eigenvalues is not a truly repeatable measure of simulation speed, though a steady-state snapshot like the ones provided in this chapter can give insight to obvious states. Some states may hold up the convergence time in earlier segments in the simulation, while others dominate once the system has reached some other operating point. For this reason, the Performance Analyzer tool is used to measure the states based on their cumulative influence on the convergence speed. Higher derivatives will also be targeted, with higher priority given to those states which have the largest simulation speed “influence” as defined by the built-in Performance Analyzer in

SimulationX. These higher derivatives will usually be inertial elements in compounds such as valves and gearsets. This method of working through the bottlenecks in simulation should allow the system to operate with larger time steps as the limiting highest frequencies are progressively eliminated.

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Eigenvalue analysis is applicable to linear or linearized dynamic systems, and thus a fully defined set of first-order linear differential equations is required to determine the 퐴 matrix of a dynamic system (from the state-space form 푥̇ = 퐴푥 + 퐵푢). Solving the eigenvalue equation 퐴푥 = 휆푥 gives the complex eigenvalues for the full system at that point in time. Unfortunately, an automatic transmission and its hydraulic system have many discontinuities which are frequently changing and intricately coupled. After rearrangement and substitution of the hundreds of equations into a full-rank dynamic system, the full-system set of equations is unreadable due to nesting of if-else statements.

The only way these can be adequately analyzed is within the SimulationX software, through a tool which calculated natural frequencies and mode shapes. This tool is highly dependent on the dynamic system being near steady state, where all state derivatives are near zero. This means that the eigen-analysis would only be feasible at incremental points during a dynamic simulation. Once something changes (like clutch lockup, for example), the analysis would need to be re-run with the new connections. This type of analysis is thus best-suited for continuous dynamic systems, even if nonlinearities require linearizing the system at multiple operating points.

A proxy for determining the speed of eigenvalues is another tool in SimulationX called “Performance Analyzer”. This tool runs in parallel to a simulation and tracks the convergence of residuals for all states, along with event triggers for discontinuous functions. Knowing the dynamic states which hold up the solver convergence at each time step is related to finding the fastest eigen-values of the dynamic system, but it is tracked numerically. This bypasses the need to analytically determine the linearized

142 system at each time step, as this is built into the solver’s algorithm. If there are any doubts about the fastest eigenvalues or where to target order reduction, this tool is used to find the simulation bottleneck.

The goal of the model reduction process is to create progressively reduced component models, which are then replaced in their respective subsystems to test the impact of components on the simulation. The impact is measured by the aforementioned performance metrics, depending on the subsystem of interest, and simulation speed using the variable step solver. Finally, once a reduced subsystem model is able to run with a fixed-step solver, the real-time factor is used to correlate the simulation improvement alongside the accuracy degradation with respect to the highest-fidelity model.

Subsystems are reduced by removing complexity from their constituent component models. For a valve model, this may involve removing small masses from the system or lumping volumes previously separated by flow orifices. In a mechanical model, this could be accomplished by removing discontinuities induced by backlash regimes.

This process is detailed at the component level in the following sections, with aggregate subsystem models being defined at different stages as “levels” of fidelity.

6.1. Hydraulic Network

6.1.1. Reduced Solenoid Valves

The linear solenoid model discussed previously was reduced by first removing the electrical dynamics from the component model. The resistance-inductance circuit for the solenoid acted as a first-order filter on the force produced by the solenoid. Removing this

143 filter and replacing the solenoid model with a map of current-to-force was the first level of reduction. The force map was transferred directly from the inside of the solenoid model, and thus has the same steady state performance by default.

The second act of reduction on the linear solenoids was to remove the mass of the plunger from the equation. Removing these made the plungers act like first-order mechanical responses within the operation range.

The third level of reduction (fourth version of the solenoid) was completed by removing the feedback orifice from the flow path, which lumped the feedback volume with the output volume. Finally, the most basic model for the linear solenoid is an algebraic feed-through map. This map turns the input current into a normalized signal (푤) from 0 to 1 which acts as a weighting factor between the two input pressure levels

(“DRAIN” and “SRC”). The output pressure equation thus looks like:

푝표푢푡 = 푤(푖)푝푠푟푐 + (1 − 푤(푖))푝푑푟푎𝑖푛 (22)

These iterations are shown below and tabulated with the number of states present.

a) b)

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c) d) Figure 59: Linear Solenoid Reduced Models

Table 41: Linear Solenoid Reduction Summary

Linear Number of Defining Features Solenoid Level States LVL 1 Full fidelity 5 LVL 2 (a) No electrical (R-L) circuit 4 LVL 3 (b) No plunger mass 3 LVL 4 (c) No feedback orifice, lumped volumes 2 LVL 5 (d) Algebraically mapped pressure output 0

Shift solenoids are reduced in a similar manner for the first two reduction levels.

First the electromagnetic dynamics are removed, followed by the masses of the plungers.

For the third reduction step, the forces are taken out of the plunger model. This means that instead of mapping force to input currents, the input map translates the input current into a plunger displacement, removing the displacement as a dynamic state. For the linear solenoids, this step was not feasible, as the plunger relied on its feedback pressure chamber to self-regulate the output pressure based on the input force. The shift solenoids do not operate in this way, instead functioning as switch valves which open the valve

145 orifices based on the generated solenoid force. There is no feedback path to require forces be calculated dynamically.

Finally, the furthest the shift solenoids could be reduced was an algebraic feedthrough equation of the same form as above in the “LVL 5” linear solenoid model.

These reduction steps are shown below, along with a table showing the corresponding number of dynamic states for each component level.

a) b)

c) d)

Figure 60: Shift Solenoid Reduced Models

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Table 42: Shift Solenoid Reduction Summary

Shift Solenoid Number of Defining Features Level States LVL 1 Full fidelity 4 LVL 2 (a) No electrical (R-L) circuit 3 LVL 3 (b) No plunger mass 2 LVL 4 (c) Linear displacement mapped, edge models retained 1 LVL 5 (d) Algebraically mapped pressure output 0

6.1.2. Reduced Regulation Valves

The regulation circuit was reduced with the intention of preserving mechanical functionality with internal feedback, similarly to the linear solenoid model. The regulation circuit valves must be capable of handling both types of input: flow rate and pressure. Therefore, it is important to maintain causality in the fluid flow path by preserving the hydraulic edge models and attempting to reduce the mechanical system.

The first reduction in complexity for the main regulator valve comes from replacing the piston force blocks with volume elements and pressure-force transformation equations. This breaks the possibility of the plunger pushing back onto the hydraulic volume and increasing the pressure. Now, pressure is determined to flow into the volume element of the former piston model and the pressure is measured without interacting back the other way by displacing a fluid piston. The same number of dynamic states are present, but this action fixes the piston volumes rather than allowing volume as a function of the displacement, 푉(푥).

The third fidelity level (second reduced component) for the main regulator valve removed the mass of the plunger from the system. “LVL 4” for the main regulator valve

147 switched off the internal Bernoulli flow forces that act on the valve plunger through the edge models. This is the final mechanically representative level of fidelity. The most basic regulation valve model simply turns the input into an output, and sets the output (in this case, line pressure) to its target value based on the specification. This also necessitates the spoofing of the other flow connections, as there is no longer a true orifice flow connection between two volumes. The volume elements which were attached to input pressure ports via the piston force blocks were maintained as a “dead-end”, to keep the volumes consistent across the varying fidelity levels. “LVL 3” is shown below, with the piston force blocks replaced and the mass removed.

In the table below, two values for each component level are reported for the number of dynamic states. The first, Q, is the number of dynamic states when a flow source is used as the input. Defining the flow as the input is the more accurate assumption, as the oil pump in the transmission will generate a flow rate based on its rotational speed. However, setting the input as the target pressure (P) eliminates the line pressure as a state, since it is being held constant. This does the job of the main regulator valve with respect to regulation, but it still allows the valve to work normally and feed fluid to the other output ports. One can think of the pressure-sourced regulation valve as a reduced version of the entire network, since the components themselves are still identical.

This type of situation illustrates the power of using acausal modeling techniques, as no component needs to be altered to account for the change in causality.

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Figure 61: Main Regulator Valve LVL 3

Table 43: Main Regulator Valve Reduction Summary

Main Number of Regulation Defining Features States Valve Level LVL 1 Full fidelity 5 (Q), 4 (P) LVL 2 Decoupled hydraulic piston 5 (Q), 4 (P) LVL 3 No plunger mass 4 (Q), 3 (P) LVL 4 No valve flow forces 4 (Q), 3 (P) LVL 5 Pressure output, no dynamics 0 (both)

The lubrication regulation valve was first reduced by “decoupling” the hydraulic piston forces and making them causal. Second, the mass was removed from the plunger model. The next step was to lump the volumes together by removing the feedback orifice, similarly to the linear solenoid model reduction. Finally, the lowest form of the lubrication valve was a direct pressure command, which set the lubrication pressure to the specified target value while spoofing the other port volumes. The decoupled, massless

“LVL 3” lubrication regulation valve is shown below, along with the reduction in states. 149

Figure 62: Lubrication Regulation Valve LVL 3

Table 44: Lubrication Regulation Valve Reduction Summary

Lubrication Number of Regulation Defining Features States Valve Level LVL 1 Full fidelity 4 LVL 2 Decoupled hydraulic piston 4 LVL 3 No plunger mass 3 LVL 4 No feedback orifices, lumped volumes 2 LVL 5 Pressure output, no dynamics 0

Reduction of the torque converter regulation valve begins with breaking the acausal link between the feedback pressure and the force generated by the piston. This does not reduce the number of states, but does reduce the complexity of the system by removing a nonlinear term. Second, the mass of the plunger was removed. Lumping the volumes together by removing the feedback orifices further reduced the number of states, while the least complex model was a constant output pressure with port volumes

150 maintained on all connections. The “LVL 3” model with the mass removed and the decoupled piston are shown below to contrast the high-fidelity model discussed previously.

Figure 63: Torque Converter Regulation Valve LVL 3

Table 45: Torque Converter Regulation Valve Reduction Summary

Torque Converter Number of Defining Features Regulation States Valve Level LVL 1 Full fidelity 5 LVL 2 Decoupled hydraulic piston 5 LVL 3 No plunger mass 4 LVL 4 No feedback orifices, lumped volumes 2 LVL 5 Pressure output, no dynamics 0

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The first reduction step for the line pressure accumulator valve is the same as for the previous few regulation valves: decoupling the hydraulic piston force. Second, the mass was removed, but then there are no additional reductions to be made before the valve is replaced by static volume elements with no flow connections. “LVL 2” and

“LVL 3” are shown below to illustrate the decoupling of the hydraulic piston and removal of the plunger mass.

Figure 64: Line Pressure Accumulator Valve LVL 2 (left) and LVL 3 (right)

Table 46: Line Pressure Accumulator Valve Reduction Summary

Line Pressure Number of Accumulator Defining Features States Valve Level LVL 1 Full fidelity 3 LVL 2 Decoupled hydraulic piston 3 LVL 3 No plunger mass 2 LVL 4 Volume element connections only, no flow 0

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The lockup clutch control valve has more volume elements than other regulation valves due to the multiple orifices in the feedback pathway. These small volume elements are necessary to couple the orifices which are specified in the component design. The first level of reduction is to decouple the hydraulic piston. Second is to remove the plunger mass from the system, while the next level is to cut out the orifice elements and couple three volume elements into one output volume. Finally, like the valves above, the least complex valve model is a simple pressure output with port volumes. The “LVL 3” lockup clutch control valve is shown below.

Figure 65: Lockup Clutch Control Valve LVL 3

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Table 47: Lockup Clutch Control Valve Reduction Summary

Lockup Clutch Number of Control Valve Defining Features States Level LVL 1 Full fidelity 5 LVL 2 Decoupled hydraulic piston 5 LVL 3 No plunger mass 4 LVL 4 No feedback orifices, lumped volumes 2 LVL 5 Pressure output, no dynamics 0

6.1.3. Reduced Switch Valves

The “switch”-type valves in the hydraulic network are similar to the shift solenoids discussed above. Instead of the electrical system, a pressure command is the actuation action to move the plunger. However, most of these switch valves do not include any feedback pathways which cause them to oscillate around an opening threshold. These valves are primarily operating in fully-open or fully-closed positions.

Reducing the switch valves is very similar across all valves, so the generic process is applicable to most valves of this type. The first reduction step (“LVL 2”) removed the hydraulic piston for a fixed-volume element, as discussed in above valves.

Next, masses were removed while maintaining the spring-damper elements and linear motion degree of freedom. “LVL 4” is a pure pressure-to-displacement map, which has the threshold pressure embedded to prevent plunger motion before the threshold is reached. The plunger has no degrees of freedom at this point, so the only state in the component is the output pressure. Finally, the fifth level of fidelity is the algebraic feedthrough, where the output pressure is calculated as a weighted sum of the two input pressures. 154

An example of the reduction in these shift valves is illustrated by the switch valve shown below, which allows the source through to the output port when it reaches a designed threshold pressure.

a) b)

c) d) Figure 66: Selected Switch Valve Reduced Models

Table 48: Selected Switch Valve Reduction Summary

Switch Valve Number of Defining Features Level States LVL 1 Full fidelity 3 LVL 2 Decoupled hydraulic piston 3 LVL 3 No plunger mass 2 LVL 4 No valve forces, mapped displacement 1 LVL 5 Pressure output, no dynamics 0

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6.1.4. Reduced Two-Way Clutch Valve

The two-way clutch valve is somewhat unique relative to the other hydraulic valve components, as the useful output of the valve model is the plunger displacement, rather than a pressure signal. For the reduction process, the hydraulic piston elements are replaced by decoupled forces on the plunger. The detent mechanisms were retained for the first reduced level but were eliminated for the next reduced model while the plunger mass was also removed. For the next two reduced models, the plunger hydraulic forces were combined into a net force on the plunger before the displacement is evaluated.

a) b)

c) d) Figure 67: Two-Way Clutch Valve Reduced Models

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Table 49: Two-Way Clutch Valve Reduction Summary

TWC Valve Number of Defining Features Level States LVL 1 Full fidelity 8 LVL 2 (a) Decoupled hydraulic pistons 8 LVL 3 (b) No plunger mass, detents replace with hysteresis 1 LVL 4 (c) Mass involved, no spring/damper, only one net force 2 LVL 5 (d) Net force mapped algebraically to position 0

6.1.5. Integrated Hydraulic Network Levels

When these varying component levels are available, the combinations of such components begin to multiply. Integrating these component models with each other into a full hydraulic network subsystem provided the opportunity to test combinations of different valve fidelities. The following table shows the eight different hydraulic network configurations that came out of the model reduction process, along with the corresponding number of states at each level. The first five levels were tested using both flow and pressure sources for the line pressure, denoted as 푄 and 푃, respectively.

Table 50: Integrated Hydraulic Network Reduction Summary and Combinations

Hydraulic Number of Details Network Level States LVL 1 (푃/푄) Full fidelity (w/o and w/ flow source) 136 / 137 LVL 2 (푃/푄) Removed electrical dynamics 121 / 122 LVL 3 (푃/푄) LVL3 solenoids, LVL2 switch valves, LVL1 regulation 104 / 105 LVL 4 (푃/푄) LVL4 all valves, including regulation circuit 51 / 52 LVL 5 (푃/푄) Removed network orifices, removed accumulator masses 47 / 48 LVL 6 (푃) LVL5 regulation circuit, LVL4 other valves 40 LVL 7 (푃) LVL5 all switch valves, LVL4 linear solenoids 20 LVL 8 (푃) Volume elements increased to slow pressure oscillations 20

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It is not practical to report eigenvalues for this discontinuous dynamic system at every operating point (say, every gear) across all different parameter variations.

However, for the nominal parameter values, an eigenvalue analysis was done after model initialization to get an estimate about the eigenvalues associated with the hydraulic subsystem. The plots which follow plot all eight levels of the pressure-sourced hydraulic subsystem on the same complex plane. The first contains all eight configuration levels, while the plots that follow compare successive reduction levels (e.g. LVL2 vs. LVL3,

LVL3 vs. LVL4, etc.). The presence of right-hand plane eigenvalues would normally imply instability, however, in this case they illustrate a limitation of measuring eigenvalues at a fixed operating point. At the measured (initialization) point, the valves with internal feedback have not crossed certain displacement thresholds to open feedback pathways. The behavior of such valves presents as unstable until a discontinuity (valve opening) is reached. Please note the changes in axis limits as the magnitudes decrease.

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Figure 68: Eigenvalue Overview of Hydraulic Network Reduction

159

continued

Figure 69: Step-by-Step Comparison of Eigenvalues for Reduced Hydraulic Network Models

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Figure 69 continued

6.2. Gearbox

The gearbox mechanical subsystem is reduced by focusing on the planetary gearset elements. As discussed previously, the planetary gearset components used in the high-fidelity gearbox model were custom compound elements with helical gear tooth interactions and internal planet inertias. When analyzed both via eigenvalues and the

Performance Analyzer in SimulationX, the planet inertia angles and angular velocities were by far the most influential dynamic states with respect to solver convergence. The small inertia values and high stiffness made for a highly oscillatory response of each planetary gearset.

For the first act of model reduction, the planetary gearset was lumped into a simplified planetary model with no geometric elements. The custom compound was replaced by a lumped elastic planetary gearset block, which simply focuses on one- dimensional rotational coupling, as opposed to the many degrees of freedom calculated from the geometrical helical models. The stiffness, backlash, and damping for the 161 planetary element are lumped onto the sun element. No inertias are present within the planetary model, so the only inertias now present in the gearbox are those of the main rotational degrees of freedom.

The stiffness of the overall high-fidelity planetary gearset is different for each of the four gearsets. This is due to the differing radii of the gear interactions within the gearset. To maintain the same approximate stiffness as the high-fidelity components, the high-fidelity planetary gearsets are isolated in a model which fixes the ring and carrier elements while applying torque to the sun element slowly. This simulation testing allowed the calculation of the gearset’s torsional stiffness and backlash, which are different for each of the four gearsets. These values were used for the lumped version of each of the gearsets in the first reduced gearbox model. The idea of this mapping is to create as much parameter consistency as possible during the reduction process. An example test bench for these elastic values is shown below, along with the output curve:

Figure 70: Example of Planetary Gearset Torque Bench for Elasticity Testing

The second reduction step is to simplify the output gear to make it a simple torque amplification element. This eliminated any elasticity and helical geometry. Finally, the 162 lowest level of complexity investigated was for a rigid planetary gearset element. This gave all elasticity to the clutch elements and made the planetary gearsets into simple torque amplification blocks. A summary of the four fidelity levels for the gearbox model is shown in the table below.

Table 51: Integrated Gearbox Reduction Summary and Combinations

Number of Gearbox Level Details States LVL 1 Full fidelity – helical planetary gears with planet inertias 103 LVL 2 Simplified planetary gears – lumped elasticity and backlash 39 LVL 3 Simplified output gear – replaced helical output gear with simple 37 LVL 4 Rigid planetary – eliminated lumped elasticity in planetary sets 29

As with the hydraulic subsystem, the gearbox (with clutch cylinders) was analyzed through the “Natural Frequencies and Mode Shapes” tool within SimulationX.

The four fidelity levels were plotted on the same complex plane for the initialized model state at nominal parameter values. The first plot shows all fidelity levels, while the plots which follow compare adjacent fidelity levels. In Chapter 7, implications of eliminating states associated with the eigenvalues shown below are detailed with respect to errors in the output metrics defined in Chapter 3. Again, note the axis changes when removing higher levels from the plots.

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Figure 71: Eigenvalue Overview of Gearbox Reduction

continued

Figure 72: Step-by-Step Comparison of Eigenvalues for Reduced Gearbox Models

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Figure 72 continued

As examples of some of the effects of model reduction on the output torque trace, the following are looks at a sample shift event, where two different model levels are overlaid on top of one another. The effects of removing the helical output gear during the step down from LVL 2 to LVL 3 are at least evident upon inspection of these curves. One can easily see the high-frequency oscillations superimposed on the torque trace that are present under the LVL 2 model. Removing these should increase simulation speed while generally only removing these slight deviations that come from tooth contact elasticity.

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Figure 73: Shift Event Comparison between LVL 2 and LVL 3 Gearbox Models

6.3. Torque Converter

The torque converter model is inherently different from the other two main subsystems. As only forward flow regimes are investigated here, there is only one discontinuity in the unlocked model: the locking of the stator one-way clutch. This is in contrast to the many discontinuities within the hydraulic network and gearbox models which make those subsystems more complex. Within each phase of operation (e.g. torque amplification phase) of the torque converter, the dynamic equations are continuous.

Only one dynamic model was investigated for this project for multiple reasons.

First, the torque converter is a component which cannot be easily discretized into hierarchal sub-components the way that the hydraulic subsystem is comprised of valves, which are comprised of translational mechanical elements. When moving down in fidelity, the model quickly becomes algebraic in nature.

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The algebraic torque converter model comes from the extended Kotwicki model presented in 2014 by North Carolina State University (Pritchard, Gould, & Johnson,

2014). This work models the steady state torque converter equations developed in 1982 in terms of pressure drops through the flow circuit and extends the original work by incorporating derivation for the overrunning case (푆푅 > 1). A benefit of using the

Modelica modeling language for such equations is the flexibility to conditionally fix different variables to change an input to an output and vice versa. This is done for the stator one-way clutch, which uses a condition on the torque ratio to unlock the stator:

0 = 푖푓 푇푅 > 1 푡ℎ푒푛 휔푠 푒푙푠푒 푇푠 (23)

The model equations for this static model use a lot of the same geometrical parameters, without any of the inertial terms or factors. One difference in this model is the pressure loss term being treated more simply than the hydrodynamic model. As opposed to part of the 푝퐿 term from Hrovat and Tobler, the pressure flow loss in this

1 model is approximated by 휌퐶 푄2. The major difference in this model is the omission of 2 푓 the derivative terms from the hydrodynamic model. This is shown below:

푅 푄 푅 푄 0 = −휌 (휔 푅2 + 푝 tan(훼 ) − 휔 푅2 − 푠 tan(훼 )) 푄 + 휏 (24) 푝 푝 퐴 푝 푠 푠 퐴 푠 푝

푅 푄 푅 푄 0 = −휌 (휔 푅2 + 푡 tan(훼 ) − 휔 푅2 − 푝 tan(훼 )) 푄 + 휏 (25) 푡 푡 퐴 푡 푝 푝 퐴 푝 푡

푅 푄 푅 푄 0 = −휌 (휔 푅2 + 푠 tan(훼 ) − 휔 푅2 − 푡 tan(훼 )) 푄 + 휏 (26) 푠 푠 퐴 푠 푡 푡 퐴 푡 푠

The derivation of the flow equation is similar to the work of Kotwicki to separate all terms of an energy balance such that the resulting equation is quadratic with respect to

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푄. The solution for the flow is thus obtained by using the quadratic formula. This derivation is present in literature for the torque amplification, fluid coupling, and overrunning phases of torque converter operation.

Analyzing the hydrodynamic torque converter has gotten more attention in literature, and the hydrodynamic model has been implemented here without a simulation speed bottleneck. Therefore the simulation sensitivity for the torque converter model is limited to a study of step size with the hydrodynamic (LVL1) model.

6.4. Simulation Settings and Notes

All models were solved using the CVODE variable-step solver discussed in

Chapter 4 with a maximum time step of 1 푚푠, a minimum time step of 1 푛푠, and solver tolerances set to 1 × 10−6 (both relative and absolute). As the models were reduced, there were some subsystems which were capable of running with higher maximum time steps, which greatly increased the simulation speed. However, sometimes a simulation would run faster at the start and then unexpectedly hang on an event, unable to converge to the set tolerance. These issues were avoidable by keeping the maximum time step at a millisecond for all models, as well as the added benefit of ensuring consistency between compared run times. This occasional instability did not appear to affect significantly- reduced models, mostly only occurring in middle-to-upper fidelity levels.

As mentioned above, the eigenvalue analysis tool is not practical for large-scale use cases such as for studying the eigen-modes of every parameter variant. Since these have to be calculated from within the user interface, it stands in contrast to the batch

168 simulation tools which are available for use from the command line on the compiled executable models. It would be desirable to have the full system equations at a certain time and be able to plug in the parameters to study the variations in eigen-modes from the variations in parameters. This is obscured by the generated model code at this time, so this would be left to future work which has easily-accessible model equations for this type of analysis.

The use of the fully algebraic torque converter model was not fully implemented for this work due to instabilities encountered during parameter variation, which were left to be resolved in future work. Two apparent steps to reducing the torque converter model between the full hydrodynamic model and a fully-algebraic model come from assuming constant rotational speed or flow rate. Using a quasi-static approach for both would yield the reduced algebraic model but eliminating only the inertial effects of the rotational elements and keeping the dynamic flow equation would leave one dynamic state to examine. Additionally, one could assume steady fluid flow and maintain the rotational inertia dynamics to have a model with three states. Each of these middle steps was attempted as an implementation solution, but experienced issues in some way.

Making the steady-flow approximation forced an algebraic formulation of the flow rate, which suffered the same inaccuracy issues as the above algebraic model.

Assuming that the rotational elements are at constant speed while keeping the dynamic flow equation maintained model accuracy at nominal conditions, but proved unpredictable and unstable while performing parameter variant simulations. For these reasons, the reduced torque converter model has been filed under future work, along with

169 the reverse flow regime for the hydrodynamic model. At this time, the LVL 1 torque converter was the only stable option for continuation of the model integration step.

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Chapter 7. Model Reduction Study

Having produced multiple levels of fidelity for the hydraulic subsystem and mechanical gearbox, the next step is testing the models for simulation speed and errors relative to the high-fidelity versions. Here, one may use information provided through the model reduction process to assess the accuracy impacts of removing certain eigenvalues from the system, as shown in Chapter 6. To measure speed of execution, the real-world simulation time is normalized relative to the duration of the simulation to create a “real- time factor” which shows how much slower than real-time a simulation has been completed. For example, a simulation of 50 seconds of driving which takes a computer

100 seconds to run would have a real-time factor (RTF) of 2, which means it runs twice and slow as real-time (i.e. the critical value for RTF is 1).

7.1. Reduction Simulation Setup and Notes

To compare parameter variations one-to-one and provide a distribution of tests

(rather than just the nominal parameter values), common parameter sets have been selected for use in each of the reduced models of the same type. This is necessary because using the planetary gearset DOE described in Chapter 5 would work with the highest fidelity level, but the LVL 2 gearbox contains drastically different planetary gearset components, meaning they have a different set of parameters. Therefore, it is

171 necessary to choose components which do not vary significantly during the model reduction.

For the hydraulic subsystem, the best example is the linear solenoid mechanical parameters, as the only change throughout the model reduction is the elimination of the mass element. This is overcome by maintaining the same parameter inputs and keeping the solenoid plunger mass as a “dummy” parameter. The variable set will input a value just like for the higher-fidelity models, but it will not actually be used for any element.

This allows the same parameter set to be used all the way down, with the same number of data points which can be compared directly for real-time capability and output accuracy.

For the gearbox, two parameter sets remain constant through the model reduction: the clutch friction parameter DOE and the hydraulic cylinder DOE. These may both be used for comparison of simulation speed and accuracy, though it should not necessarily matter which is chosen. What is important for this stage of the analysis is that each variant at successively-reduced levels has a paired variant at the highest-fidelity level with which to judge errors directly.

As stated above, the torque converter model was not studied completely in reduced form due to instabilities with the model implementations. Instead, a sample of the stable results from the steady-speed (one dynamic state) reduced model were taken and compared with the hydrodynamic model. The common parameter DOE between the two models was chosen as the geometrical parameters. The visualized sample makes up approximately 60% of the original data set used for parameter sensitivity. The high- fidelity torque converter is also studied at multiple step-sizes in Section 7.3.3.

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7.2. Variable-Step Simulation Sensitivity

For the first round of reduction testing, only the model accuracy will be shown as a function of the number of states in the respective subsystem. This means that all reduced subsystems were solved using the variable-step CVODE solver, prioritizing accuracy over repeatable simulation speed and real-time applicability. On each of the following metric plots, the LVL 1 (high-fidelity) model is shown with the highest number of states on the x-axes and a default error of zero for all variants – this is for visual reference. Subsequent DOE runs are performed at each reduced level and plotted individually such that the mean deviation and variance in each data set is apparent.

7.2.1. Hydraulic Network Parameter Reduction

continued

Figure 74: Flow-Sourced Model Accuracy Reduction Plots (Reference: LVL1Q)

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Figure 74 continued

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Figure 75: Pressure-Sourced Model Accuracy Reduction Plots (Reference: LVL1P)

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The following plot of RMS metric error is shown twice. The second is shown without the overshoot percentage which was two orders of magnitude over the next highest peak. Conceptually, this is possible when overshoot values are sufficiently small to begin with and deviations cause this to increase.

Figure 76: Pressure-Sourced RMS Error Plots, with (left) and without (right) Overshoot %

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7.2.2. Gearbox Parameter Reduction

Figure 77: Gearbox Model Accuracy Reduction Plots

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Figure 78: Gearbox RMS Error Plot

7.2.3. Torque Converter Parameter Reduction

The same sets of plots cannot be provided for a reduction of the torque converter model however, a sample of stable results was obtained with an approximate 60% success rate (unacceptable for use as a model, yet useful to inspect errors incurred by reduction alone). There are only two points for each metric on the plot below.

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Figure 79: Torque Converter RMS Error Plot (Sampled Stable Results)

7.3. Fixed-Step Simulation Sensitivity

The second part of the model reduction study incorporates the error-tracking and adds the complexity of varying step sizes using the fixed-step solver. This shows the real capability of each of these reduced models of running in a fixed-step environment. What follows is a graphical description of the model reduction sensitivity with respect to simulation time and fixed step size for those reduced subsystem models capable of running with fixed step sizes under a real-time factor of 100. These plots show the output parameters for the gearbox and hydraulic subsystems while reducing the number of dynamic states and all three subsystems while varying the fixed step sizes. Detailed plots breaking down each metric individually can be found in Appendix A.

7.3.1. Hydraulic Fixed-Step Reduction

The flow-sourced hydraulic network models were not capable of running on a fixed-step solver at a feasible rate due to the much smaller required time steps for the

179 oscillating regulation circuit. Therefore, the hydraulic network fixed-step reduction which follows only examines the pressure-sourced hydraulic network. This would imply that for real-time integration, the flow-sourced regulation circuit must either be modeled differently or be provided with a set line pressure moving forward. In addition, LVL 3P was not able to be stabilized at a reasonable time step, so the reduction process skips over this level, from LVL 2P to LVL4P. The error plot which shows RMS error for each metric against the number of states has the same issue with scaling for the overshoot percentage as discussed above. Each reduced fixed-step model is compared with the variable-step solution from the LVL 1P model. For this reason, this plot is duplicated with and without the overshoot percentage skewing the vertical axis.

continued

Figure 80: Fixed-Step Hydraulic Network Reduction Summary Plots

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Figure 80 continued

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The figure below shows a zoomed-in look at the RTF vs. step-size figure which visually separates the points more clearly at RTF values below 20.

Figure 81: Zoomed-In Plot of Hydraulic Network Real-Time Factor vs. Step Size

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7.3.2. Gearbox Fixed-Step Reduction

Figure 82: Fixed-Step Gearbox Reduction Summary Plots

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Figure 83: Real-Time Tradeoff Plot for Fixed-Step Gearbox Reduction

7.3.3. Torque Converter Fixed Step Reduction

Figure 84: Metric Error vs. Real-Time Factor Distribution Plots for RMSTE (left) and RMSKE (right)

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Figure 85: Fixed-Step Torque Converter Real-Time Factor vs. Step Size

Figure 86: Fixed-Step Torque Converter Error vs. Step Size

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Figure 87: Fixed-Step Torque Converter Error vs. Real-Time Factor

7.4. Analysis and Discussion

The goal of this work is to quantify and illustrate the performance accuracy and simulation speed of progressively-reduced automatic transmission models. As such, this section has presented charts which illustrate the expected performance metric errors as functions of model complexity (number of states), step-size, and real-time factor.

It has been shown that Pareto-type tradeoffs can be plotted for the reduction of the three subsystems. Within the hydraulic subsystem, there were also two different reduction paths taken: flow-sourced (푄) and pressure-sourced (푃). As discussed in the modeling section, the 푄 models should be most-representative of the pressure dynamics in the hydraulic network, with the 푃 models being more applicable to simulation speed- ups as noted in this section (fixed-steps only possible in 푃-path).

Upon inspection of the variable-step flow-sourced model reduction path, the RMS error increases steadily and monotonically from LVL 1Q to LVL 3Q. The pressure

186 overshoot error approaches -100% at LVL 3Q, implying that all overshoot originally present in the high-fidelity model has been eliminated due to removal of the electrical solenoid dynamics and solenoid plunger masses. Both the rise time and the delay time remain under 5% error down to LVL 3Q, and the corner pressure and corner time remain under 10% error for LVL 2Q before increasing more significantly. This illustrates that the removal of the solenoid plunger masses and the decoupling of the switch valve piston force are more significant effects than simply eliminating electrical dynamics.

In the variable-step reduction sensitivity (Section 7.2), the pressure-sourced hydraulic metric errors suffered from an issue with small scales in the denominator inflating the pressure overshoot percentage as a performance metric. This type of error was plausible and stems from the lack of significant overshoot in the model when using a ramp control input and a non-dynamic line pressure. Using a step input would have generated a more significant overshoot percentage, but step applications are not generally used in clutch actuation control strategies. On the other hand, the LVL 3 hydraulic network displayed a much higher mean overshoot percentage across its 50-second shift sweep.

As shown above, it is not possible to achieve a real-time factor below 1 unless the lowest-fidelity hydraulic network is used (LVL 8P, 20 dynamic states) and a time step above 15 휇푠 is used. Keeping in mind that the hardware-in-the-loop simulator does not have the same processing specifications as the laptop being used for these tests, the “sub- one” real-time factor is still an appropriate target to shoot for offline for compiled fixed- step models. As mentioned at the end of Chapter 5, it is likely that many of the hydraulic

187 pressures will be judged more so on delay/rise timing metrics and steady-state pressures during online HIL testing than the corner features and/or overshoot. However, it should be strongly noted that the corner time has been deviated by about 140% by the time the model is reduced to LVL 8. Conversely, the corner pressure, delay time, and rise time all remain near 5-12% RMS error even after many dynamic states have been removed.

The gearbox reduction process shows that the torque metric error increases approximately monotonically as the number of states decreases, with the exception being the inertia phase duration. The peak jerk torque decreases initially, up to LVL 3 of the gearbox, before jumping up past 200% RMS error when the planetary gearsets were made to be rigid. Very little stayed below 25% RMS error once LVL 4 was reached. The torque phase duration proved to be the most consistent metric across all reduction levels.

The tradeoff from this increased error is that the LVL 4 gearbox model allows for an increase of the maximum stable time step from 1 휇푠 to 80 휇푠, decreasing the offline real-time factor to 0.1. This also leaves room to integrate more into the model (like the torque converter and lockup clutch) and be confident that there is sufficient overhead to remain under real-time. The peak jerk torque is something which can be calibrated if drivability requirements are to be met online, as currently, the rigidity of the LVL 4 gearbox makes the torque gradients approximately three times as severe as the LVL 1 gearbox model.

At this point, the reduction process has produced the LVL 8 hydraulic network,

LVL 4 mechanical gearbox, and a hydrodynamic torque converter as leading candidates for real-time integration. It is important to note that while many physical dynamics have

188 been reduced from the model through this process, the fully-integrated transmission model now still includes:

 hydro-mechanical solenoid valves with a full network of valve connections,

 hydro-mechanical clutch actuation cylinders with a full mass-spring-damper,

 speed-dependent slip friction coefficients within the clutch models,

 dynamic torque converter with lockup clutch and actuation cylinder.

The following model integration process will give insight as to the product of implementing this full automatic transmission on real-time hardware.

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Chapter 8: Reduced Model Implementation

8.1. Model Portability Options

A Functional Mockup Unit (FMU) is a packaged, standardized, container which includes a pre-processed set of system equations, an optional solver, and an open interface to many software options. The Functional Mockup Interface (FMI) is being adopted by physical modeling companies such as SimulationX, Dymola, Amesim, and more. FMI has been developed by the Modelica Association Project as a way to standardize Modelica models into cross-platform packages. Simulation integration software such as dSpace ConfigurationDesk, AVL Model.CONNECT, and Concurrent

Simulation Workbench are also adopting the standard in order to import these models under a unified simulation umbrella and/or execute on a real-time operating system.

SimulationX has the ability to export models directly to FMU for Co-Simulation or Model Exchange. Model Exchange includes only the system configuration and equations, while the Co-Simulation variant is packaged with a solver. SimulationX can export to either version, but the limiting factor is usually related to the target environment. For example, Simulink can import FMU blocks with or without a solver, while dSpace ConfigurationDesk can only receive Co-Simulation FMU blocks. While the dSpace DS1006 architecture required a direct compile from SimulationX, the newer

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SCALEXIO architecture will accept FMUs that are integrated through

ConfigurationDesk.

Another option for porting the model out of the native SimulationX environment is a lower-level compilation of the model and (optionally) a solver into C-code or a system-executable file. The lower-language level of this option makes it more efficient without the packaging or interface layers that other generated containers must work with.

This model is executed at the system-level with the same solver options as an FMU

(CVODE, fixed-step, or none). It runs from the command line and can work out of a few generated text files for configurations. Generally, variable-step packages with the

CVODE solver are larger in size than fixed-step outputs due to the larger libraries incorporated with the CVODE package.

A “parameters” text file is used to specify the values of variables which were explicitly defined as external parameters upon code generation. This means that the user can only modify parameters post-generation that have been tagged as “parameters” during the code-export process. These parameters are put into a table in the

“parameters.txt” file. Each row in this text file corresponds to a variation of the model, so adding rows to the “parameters.txt” adds variants to the simulation. When executing the

.exe, the program will run through all rows in the parameter file back to back, generating results files for each row.

Further gains can be realized through the selection of the parallel processing option during code-export. This allows the local hardware to use as many threads as possible in order to perform parallel model simulations with different values of

191 parameters. This is akin to using the native “Variant Wizard” in SimulationX, but there is less overhead for the software environment since the code is already packaged into the executable format. An advantage of using this method from the command line is the integration of the simulation with a scripting language to set up, run, and process a DOE with more freedom than the SimulationX interface can provide.

While the executable file is useful for fully-contained simulation outside of

SimulationX, the Functional Mockup Interface allows for more straightforward time synchronization and input/output interfacing for HIL implementation. The FMU option is used for hardware-in-the-loop simulation due to the adoption of the FMI standard across the software packages used in this project.

8.2. Implementation Hardware and Software

The dSpace SCALEXIO architecture has been chosen as the processing hardware for the HIL-phase of simulation. SCALEXIO is a modular hardware architecture which has greater customizability and is designed to be integrated with other processing boards or full HIL units. The processing unit contains an Intel Xeon E3-1275v3 with a 3.55GHz base clock speed, four cores, and 4GB of RAM.

Without access to prototype vehicle hardware, a HIL implementation for this project must necessarily be self-contained, with control inputs being simulated through a simplified shift controller. Outputs are also not converted to physical signals on a HIL

I/O board, so the environment is essentially deploying the compiled application within the real-time operating system of the SCALEXIO processing unit. In practice, the

192 powertrain output signals would be routed to voltages at the I/O pins on the front of the

HIL rack unit. The SCALEXIO interfaces with the host PC via Ethernet, while multiple

HIL simulators would be linked together through a faster optical connection.

The software associated with the SCALEXIO hardware is a matching dSpace suite. ConfigurationDesk is used to integrate all models into a real-time application, configure processing tasks and data acquisition, and route all signals which are passed between models. The gear shift, torque, and braking trajectories were all outsourced to a

Simulink block diagram to remove the prescribed trajectories from the model. The solenoid mapping was kept within the hydraulic system model such that the inputs to the hydraulic subsystem were: target gear, shift direction, engine torque, and output shaft friction coefficient. This way, if the Simulink controller were decoupled and replaced with constant input blocks, the hydraulic subsystem could still translate these requests into current signals for the solenoids.

The Simulink-based control model, the hydraulic system model, and the gearbox model make three separate model packages which are integrated into one real-time application within ConfigurationDesk. Each model is given a communication time step of

1 푚푠, as this is the same interval that signals were stored during offline simulation and is representative of the time step of many of the faster real-time processes in control verification. The communication time steps are not required to be the same as the solver step size. The solver step sizes were derived from the work on the reduction study in order to use variations of each subsystem model that could fit under real time on its own.

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The control model, hydraulic model, and gearbox (plus torque converter) used fixed solver time steps of 1 푚푠, 30 휇푠, and 50 휇푠, respectively.

The host PC also ran dSpace ControlDesk in parallel with the running real-time application on the SCALEXIO unit to control and measure signals. When prescribed trajectories were not implemented, ControlDesk numeric input boxes and sliders were used to input the gear, torque, and brake commands to a continuous simulation of the transmission model.

Figure 88: dSpace ControlDesk Signal Recording Environment

8.3. Implementation Examples

To compare the simulation with the offline implementation, gearbox clutch torques and speeds were recorded along with the output torque signal and the command

194 pressures. These signals were recorded at 1 푚푠 intervals and fed into the same analysis algorithms used on the offline parameter DOEs. Three different variants were recorded for visual inspection below: 1) full 50-second shift sweep with the lockup clutch always fully disengaged, 2) the full 50-second shift sweep with the lockup clutch always fully engaged, and 3) the full 50-second shift sweep with normal actuation of the lockup clutch. Since the torque converter was never integrated with the gearbox model for parameter sensitivity studies, the lockup clutch engagement is a new feature for the online simulation and will thus affect the output trace.

The following plot shows a comparison of the first two configurations mentioned above, with the torque converter either permanently unlocked or permanently locked.

Figure 89: Comparison of HIL Implementation Sweeps with and without TC Lockup

It is apparent that the dynamic characteristics of the torque trace during shift events have been dampened by the inclusion of a torque converter model. The initial 195 torque rise predictably shows a delay in torque delivery followed by an overshoot before beginning the first shift event. The torque converter model also slows the torque settling after the shift events and smooths out many of the highest peaks and valleys within the torque trace.

During normal operating conditions, a nominal amount of pressure is consistently applied to the lockup clutch such that it is consistently in contact while still slipping.

Upon actuation of the lockup clutch engagement, the linear solenoid which supplies the lockup clutch opens the path for higher pressure to enter the clutch chamber and fully engage the lockup clutch. Without the use of feedback control signals (for repeatability), the full transmission model has been actuated in such a way that lockup is engaged upon shifting from first to second gear, then remains in fully-locked state until re-entering first gear. The following plot illustrates the third configuration discussed above, with more realistic lockup clutch actuation, alongside a torque output trace of the same gearbox model, only isolated in a parameter study. The offline gearbox model has no torque converter at all, which is different than having a completely locked torque converter.

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Figure 90: Comparison of Online HIL Upshift Sweep with Offline LVL 4 Gearbox Only

It is evident that the addition of the torque converter and lockup clutch in the online model changes the response of the first upshift. The largest visual differences in the shift transients are the reduced torque oscillation amplitudes and the more gradual settling of the output torque. The quantitative shift metrics are shown below for both runs using the same algorithm and averaging across all shifts (up and down).

Table 52: Comparison of Torque Metrics between Online HIL and Offline VLVL 4 Gearbox Only

Mean Metric Value Across All Shifts

푑푇푟푞/푑푡푚푎푥 푂푆 [%] 푡푡푟푞 [푠푒푐] 푡𝑖푛 [푠푒푐] 푡푠푒푡푡[푠푒푐] 푇ℎ표푙푒 [푁푚] Gearbox Only 206038.2 52.3 0.249 0.069 0.255 −169.0 (offline) Full 10AT 136676.6 47.7 0.252 0.065 0.298 −167.4 HIL Diff. −33.7% −8.7% +1.0% −6.5% +17.1% −0.9% Percent

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If there are errors in the model solution, there are excessive task overruns, or if the fixed step size is too large to maintain accuracy, signals may appear jittery in nature.

If these are due to a step size issue, this effect may be observed online or offline and may not cause errors. Below is an example of the output torque trace of the online HIL 10AT model when the torque converter model is set to oscillate around instability while locked.

Figure 91: Visual Example of Instability in Model Leading to Signal Inaccuracies

The simulation instability shown above is explained more in Sections 6.4 and 9.2 but occurs when the torque converter flow rate reaches zero or turns slightly negative.

This causes the model to go unstable and the fixed-step application calculates incorrectly.

The countermeasure was implemented to maintain stability of the model at or near zero- flow such that these step failures do not occur while the lockup clutch is engaged. As noted in the next chapter, future work should include a permanent, physically-based solution to this issue. 198

It is worth noting that the hydraulic network model is found to overrun continuously when implemented in real-time on the dSpace hardware. This is likely due to the increased overhead associated with compiling the model to FMU and then to the real-time application. However, instead of causing jitter or instability as exemplified above, the hydraulic pressures simply lag by 25 푚푠, which also induces a 25 푚푠 lag in the gearbox operation relative to offline models.

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Chapter 9. Conclusions

9.1. Key Takeaways and Implications

The analysis of this 10-speed automatic transmission has provided insight to its response to varying both its parameters and fidelity. Evaluation metrics were chosen which reflect the dynamic performance of the hydraulic network and mechanical gearbox, while also looking at the torque converter from a steady-state-map perspective.

Two different ways of regulating the hydraulic circuit were presented, where the constant-flow source represented the more realistic implementation as opposed to the pressure-sourced circuit.

The gearbox was reduced by focusing almost exclusively on the gear models within. The resulting reduced gearbox displayed a much harsher shift transient jerk due to the rigidity of the gearbox, but the rest of the system was maintained at high-fidelity levels throughout, leaving the actuation system intact, along with the speed-dependent clutch models. The gearbox was integrated with a hydrodynamic torque converter and lockup clutch with acausal port connections within SimulationX for real-time implementation.

The fully-reduced subsystem models presented in this work were integrated into a dSpace SCALEXIO real-time environment using the code-export wizard in SimulationX to configure portable FMUs for the hydraulic network and mechanical system separately.

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These were integrated within dSpace ConfigurationDesk and the result was verified to work with similar torque metric errors as measured from the plain offline LVL 4 gearbox model.

The hydraulic subsystem overruns continuously at the chosen time step, but inaccuracies in the pressure traces have not been observed outside a constant lag. To combat this phenomenon in the future, the simplest remedy would be to implement the model on higher-specification real-time hardware. A higher clock speed running the model would be useful to test the current form of the model, however solutions which simply increase the number of parallel threads would likely not see any increase in simulation speed, as the hydraulic model in its current form would sit on only one thread.

A more-likely solution to the overrunning problem would be to work on a further- reduced hydraulic model. The linear solenoid dynamics can be kept fairly easily, as seen in this work, however the regulation circuit and the many switch valves upstream of the linear solenoids should be reduced further or moved to a separate model. In its most- reduced form, the hydraulic model upstream of the linear solenoids has many fixed- causality components which may benefit from moving into a modeling environment such as Simulink and feeding the SimulationX FMU. To get source pressure dynamics online, further investigation should be done to focus on the regulation circuit and reduce these components for use in fixed-step solutions with sufficiently large time steps (> 10 휇푠).

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9.2. Project Obstacles and Notes

Throughout the work on this project, there have been multiple times when obstacles have presented themselves. A few of these are listed below which range from simulation instability to unforeseen time investments.

The first issue with modeling was the apparent instability with the torque converter reverse flow equations. A large amount of time was spent attempting to diagnose the instability of the dynamic torque converter flow rate in the reverse flow regime. As soon as the flow rate touched zero during simulation, the entire simulation would hang up indefinitely. It was thought to be an issue with the parameters entered for blade angles or radii, however the same issue presented itself when using parameters taken entirely from literature. The model was also implemented in other modeling environments such as Simulink and OpenModelica, only to find the same instability with the reverse flow equations. This helped to verify that the instability is with the model equations for reverse flow and would need to be investigated further to sufficiently model this regime.

The second problem encountered during this work was a solver implementation inconsistency. This was noticed early in the project when running large batches of simulations in parallel using the OpenMP library. It was noted that certain variants of parameters took orders of magnitude longer to simulate than the rest of the variants in the set. This was noticed during the batch simulation of the hydraulic network subsystems, and it is believed to be related to the disparate eigenvalues present in the system, though there is evidence of solver inconsistency nonetheless. When running a batch simulation

202 of 50 copies of the same set of parameters, tasks which were scheduled on the master processing thread executed differently than the rest of the threads. This problem was consistent regardless of the number of cores on the computer. Threads were manually manipulated on multiple Windows platforms and the result remained that one thread executed much differently than the rest. It was found that this difference did not always lead to a slower performance, but the different solution convergence path simply took a different number of steps for all of the same settings. This was found to be extremely sensitive to the solver convergence tolerance, but still did not explain the computational inconsistencies among the exact same variations. The problem only existed when compiled with the OpenMP parallel computing library, thus parallel computing was eliminated from the workflow to ensure consistency.

With respect to the SimulationX software environment, there were times when the graphical user interface became burdensome with respect to responsiveness. Under normal operation with moderately-sized models open, the GUI was responsive and useful as expected. When large models were open, however, it was not uncommon for simple tasks to slow down, sometimes even hanging up the program. It is expected that processing tasks like compiling and simulation would take much longer to execute, but simply opening menus was also slowed by a large model being loaded. This was not a problem with the host PC, as plenty of RAM and CPU overhead were left untouched during normal operation. Additionally, when running the preprocessing algorithm or solver, SimulationX would occasionally crash upon an event that should just trigger a processing error (e.g. “divide by zero” or “inconsistent initial values”). These types of

203 errors with the software were not sufficient to overcome the overall excellent modeling and analysis environment, though there were unforeseen delays associated with these issues. It is expected that future iterations of this software will have these bugs resolved.

One task which proved tedious in model debugging was the determination of a complete set of initial conditions for the entire model. SimulationX provides a lot of help in diagnosing conflicts and undefined values, but many times manipulation of initial states took longer than anticipated. SimulationX includes an initial value calculation algorithm in the preprocessing flow which is very useful for filling in under-defined systems. Problems arise when an initial condition set becomes overdetermined, which was the case sometimes when the model had been reduced, which required the elimination of some initial constraints. Solving would often not even begin due to these conflicts, but the code generation step would not be satisfied if even soft warnings were left unresolved. Therefore, great care was taken to define the initial states at each implementation of a subsystem – a necessary task which sometimes took more time than anticipated to diagnose.

Finally, some items which simply took longer than expected were scattered throughout the duration of the project. One such task was the creation and verification of the response metric algorithms. Many of the features of a torque response trace (for example) are straightforward for a human eye to pick out, yet not always as straightforward for an algorithm to calculate in a robust way. In general, handling the large amount of data output from these simulations took multiple attempts at organization. Through each transient shift sweep, there were multiple metrics to be

204 calculated at each shift event, and these sweeps were performed hundreds of times across different settings. Finally, it was important not to get pulled too deeply into a task that the greater picture of the project became obscured. There were some tasks which were started before deciding that they did not appreciably affect the ultimate outcome of the study

(one such task was a more in-depth monitoring of processor threads at the system level).

9.3. Future Work

There are more extreme fidelity limits on both ends of the spectrum which were not investigated in this work. For example, literature has outlined fluid film squeeze dynamics for clutch modeling, which were not presented in these models. Squeeze dynamics do include empirical factors, yet they are another step further toward a first- principles model of clutch friction as opposed to the empirical slip curves used in this project. On the other hand, the clutch friction models were also not reduced any further during the reduction process, as most of the focus was on the gearsets. The result was a real-time-capable gearbox subsystem, but an iteration with constant friction coefficients was not tested.

The hydraulic system components were modeled with a relatively-high fidelity from the start, but the integration of the components used simple lumped volume elements for lines which may actually contain unexamined dynamics. In addition, temperature effects were not investigated formally, as the fluid models chosen in

SimulationX assume a fluid of constant temperature. Efforts at changing this temperature online were unsuccessful, and the thermal fluid libraries were not utilized. At present

205 time, there is not enough information about heat sources and sinks to attempt to validate a thermal model. Further friction calibration should be done prior to implementing dynamic thermal effects. A thermal model should be implemented for components whose mechanical behavior is significantly altered by the fluid viscosity, namely clutch drag and torque converter drag.

On the other hand, the hydraulic system could stand to be reduced even further.

Though the hydraulic subsystem executed in real time, there were task overruns present which did not appreciably affect the output pressure responses but would induce delays in communicating updated values to the other subsystems. To do verification and validation of some hydraulic operations, it would be useful to playback true transmission controller signals to the model and judge states such as idle-stop and park. Due to the intended use of such a model, it should be apparent that a future step in this process would be to integrate this model into a hardware-in-the-loop environment with physical controller hardware present.

The torque converter model should be extended to work in the reverse flow regime to study engine braking effects, as noted in some prior literature (Pritchard,

Gould, & Johnson, 2014). If possible, the fluid flow path parameters would likely be more accurate if pulled from a 3D CFD analysis, rather than from a static geometrical measurement (Lee J.-H. , 2017).

One part of this work which did not get extended as intended was the inclusion of other simulation hardware platforms. The intent was to use the same compiled executable models on multiple processor platforms to judge the real-time capability with both

206 improved and worsened computational resources. This work was performed on a desktop

PC with an Intel Xeon processor (12 threads, 3.5GHz base clock) and 64GB of RAM, but many industry engineers are tied to their own laptops for modeling work, which have much lower hardware specifications. It was noted that a consumer-grade laptop with an

Intel i7 processor (8 threads, 2.6GHz base clock) and 16GB of RAM executed some variable-step simulation variations approximately four times slower than the desktop tower. Server-grade processors were also tested in multiple configurations, but improvements were marginal if nonexistent for single-thread performance when compared to the Xeon-based PC (parallel batch speed was helped by increasing the number of CPU threads). It is also noted that the desktop Xeon processors are similar in performance to the high-performance variant of the SCALEXIO processing unit, based on PassMark scores (CPU Benchmarks: High-End CPUs - Intel vs AMD, 2018). The

SCALEXIO variant used in this project, however, is a slower Xeon variant with a

3.55GHz clock frequency and 4GB of RAM compared to the current high-end. There are still platforms and implementations (both software environments and hardware specifications) which are applicable to real-time simulation and were left for future work.

The largest planned future step is the calibration of the transmission models. The highest-fidelity gearbox models should be calibrated for friction behavior of the clutches and the stiffness, damping, and backlash of the planetary gearsets. If more fidelity is added to the gearbox such as bearing friction models, these drag models would also be key to understanding the in-gear efficiencies. The hydraulic clutch piston models should be tuned or improved with measured stiffness values. The hydraulic network should be

207 calibrated with respect to the regulation system, with the regulation circuit having a fundamental effect of the supply pressures of the other components. The flow rate is assumed constant even in the flow-rate-based hydraulic network models, so an approximate pump model should be included depending on the engine speed. The torque converter geometry (radii and blade angles) should be tuned or generally improved upon by CFD data to get the steady-state maps calibrated. The lockup clutch should also be treated like the other clutch models, with the piston model and clutch friction improved using test measurements or geometrical data.

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Appendix A. Fixed-Step Model Reduction Metric Plots

Hydraulic Network Fixed-Step Reduction Plots:

The following plots separate each individual output for the hydraulic response metrics from two perspectives: metric error with respect to LVL 1P versus step size, and metric error versus number of states. Colors in the plots on the right show distinctions between model implementations (ex. one color may be LVL 3 at 10 휇푠). These are meant to show the metric error with respect to simulation step size on the left, with the same metric and its error progression with decreasing fidelity (right to left on the x-axis).

Points on the plots on the left represent the mean values for the model metric error across all data points from the linear solenoid mechanical subset of parameters evaluated at different model fidelity levels and step sizes.

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Gearbox Fixed-Step Reduction Plots:

The following plots show gearbox response relative metric errors with respect to metrics evaluated at the highest fidelity levels (LVL1, fixed step). Each output metric’s relative error is shown as functions of the step size (left) and number of states (right) as a proxy for model fidelity.

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Torque Converter Fixed-Step Reduction Plots:

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