”NE SITY LIBRARIES

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This is to certify that the

thesis entitled

USING TOP DOWN MULTIPORT MODELING FOR
AUTOMOTIVE APPLICATIONS

presented by
Mark Andrew Minor

has been accepted towards fulfillment
of the requirements for

M.S. degree in Mechanical Engineering

KIM

Major professoa

Date Mé/%

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MSU ieAn Affinnetive Action/Emmi Opportunly inetttuion
WW1

USING TOP DOWN MULTIPORT MODELING FOR
AUTOMOTIVE APPLICATIONS
By

Mark Andrew Minor

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1996

ABSTRACT

USING TOP DOWN MULTIPORT MODELING FOR
AUTOMOTIVE APPLICATIONS

By

Mark Andrew Minor

The objective of this research is to explore the development and use of a model
library by simulating the powertrain and rigid body dynamics of a typical automobile. A
top down hierarchical structure of an automobile system is developed and used as a
foundation for a model library. The multiport macro node is used as the building block
that allows the iconification of components or effects. Component models are developed,
verified, then stored in the library. A vehicle model is constructed from the iconified
component models. Issues raised by this exercise include multiport connectivity, valid
operating range, parameter management, initial condition management, and standard
model verification tests. As a result of the connectivity issues, causality and power flow

must also be considered.

MD.

I would like to dedicate this work to my Grandparents and Parents. Their support

and guidance have made this possible.

iii

ACKNOWLEDGMENTS

Special thanks to the guys is in the Computational Design Laboratory at Michigan
State University. Dr. Rosenberg, Giuseppe DeRose, Michael Hales, and Wei-Wen Deng
have given me tremendous support throughout the process of completing this thesis.

Without their assistance, this would not be possible.

iv

Table of Contents

Table of Contents ............................................................................................................ v
List of Tables ................................................................................................................ viii
List of Figures ................................................................................................................. x
1. Introduction ................................................................................................................. 1
1.1 Problem Definition ................................................................................................. 1
1.2 Statement of Objective ........................................................................................... 2
1.3 Thesis Structure ..................................................................................................... 3

2. Multiport Modeling ..................................................................................................... 4
2.1 Nodes .................................................................................................................... 4
2.2 Ports and Connectors ............................................................................................. 5
2.3 Hierarchy ............................................................................................................... 6

3. Design of a Model Library ........................................................................................... 8
3.1 Reusability and Libraries ........................................................................................ 8

3 .2 Background ........................................................................................................... 9
3.3 A Proposed Library Structure .............................................................................. 10

3 .4 A Library Example: A Propelled Two Dimensional Rigid Ring Tire Model .......... 13
3.4.1 Model Description and Assumptions ............................................................. 13
3.4.2 Ports and Parameters ..................................................................................... 19
3.4.3 Model Verification ........................................................................................ 20

4. Top Down Model Structure ....................................................................................... 25

V

4.1 Top Level ............................................................................................................ 26

4.2 Vehicle System .................................................................................................... 26
4.3 Power Train ......................................................................................................... 29
4.4 Vehicle Dynamics ................................................................................................ 29

5. Case Study: An Automobile Powertrain and Rigid Body Kinematics Model .............. 31
5.1 Model Introduction .............................................................................................. 32
5.2 Objectives ............................................................................................................ 37
5.3 Results ................................................................................................................. 39
5.3.1 Model Verification ........................................................................................ 39
5.3.2 Additional Model Capabilities ........................................................................ 42
5.3.3 Exploring Design Alternatives ....................................................................... 45

5.4 Discussion ............................................................................................................ 46

6. Conclusions ............................................................................................................... 49
Appendix A Model Library ......................................................................................... 51
Al Two Dirnensiona] Rigid Body Vehicle Dynamics Model ...................................... 52
A. 1.1 Ports and Parameters .................................................................................... 54

A. 1.2 Verification Results ...................................................................................... 56

A2 An Unpropelled Two Dimensional Rigid Ring Tire Model ................................... 62
A21 Model Description and Assumptions ............................................................. 62
A22 Ports and Parameters .................................................................................... 65
A23 Verification Results ...................................................................................... 66

A3 A Simple Open Difi‘erential .................................................................................. 67
A31 Model Description and Assumptions ............................................................. 67
A32 Ports and Parameters .................................................................................... 69
A33 Verification Results ...................................................................................... 70

AA A Lumped Mass Manual Transmission with Clutch .............................................. 73
A41 Model Description and Assumptions ............................................................. 73

vi

A.4.2 Ports and Parameters .................................................................................... 76

A43 Verification Results ...................................................................................... 77
A5 Hollow Shafi Models .......................................................................................... 81
A51 Model Description and Assumptions ............................................................. 81
A52 Ports and Parameters .................................................................................... 84
A6 Mean Torque Full Throttle Engine Model ............................................................ 85
A61 Model Description and Assumptions ............................................................. 85
A62 Ports and Parameters .................................................................................... 87
A63 Verification .................................................................................................. 87
A7 Longitudinal Wind Effects ................................................................................... 89
A7 .1 Model Presentation and Assumptions ........................................................... 89
A72 Ports and Parameters .................................................................................... 90
A73 Verification .................................................................................................. 91
Appendix B FORTRAN Subroutines ........................................................................... 92
B.1 ZZSU01 - Parameter Multiplication ..................................................................... 92
B.2 ZZSU02 - Ground Contact Force Function .......................................................... 93
B.3 ZZSU03 - Multi Parameter and Variable Multiplication ....................................... 94
B.4 ZZSU04 - Coulomb Damping Function ............................................................... 95
B.5 ZZSUOS - Parameter Selection Based Upon Signal .............................................. 96
B.6 ZZSU08 - Engine Torque-Speed Curve ............................................................... 98
B.7 ZZSU23 - Tire Slip Function ............................................................................... 99
B.8 ZZSU50 - Automatic Tranmission Torque Converter ........................................ 101
B9 ZZSU51 - Determination of CVT Ratio ............................................................. 102
7. Literature Cited ....................................................................................................... 104

vii

Table 1:
Table 2:
Table 3:
Table 4:
Table 5:
Table 6:
Table 7:
Table 8:
Table 9:
Table 10:
Table 11:
Table 12:
Table 13:
Table 14:
Table 15:
Table 16:
Table 17:
Table 18:
Table 19:
Table 20:

List of Tables

Rigid Rim Tire Model node descriptions .......................................................... 18
Port description for a Rigid Rim tire model ...................................................... 18
Parameter description for a Rigid Rim tire model ............................................. 19
Top Level Variables ........................................................................................ 27
Vehicle System Level Connectors .................................................................... 28
Powertrain Level Connectors. ......................................................................... 28
Nodes associated with the Rigid Body Vehicle Model ..................................... 54
Ports associated with the 2D Rigid Body Vehicle Model ................................. 55
Parameters associated with the 2D Rigid Body Vehicle Model ........................ 55
Unpropelled Rigid Rim Tire Model node descriptions .................................... 65
Port description for an Unpropelled Rigid Ring Tire Model ........................... 65
Parameter description for an Unpropelled Rigid Rim Tire Model ................... 66
Port description for a Simple Open Differential ............................................. 69
Parameter Description for a Simple Open Differential .................................... 69
Lumped mass manual transmission model node description ............................ 76
Port description for a lumped mass manual transmission. ............................... 77
Parameter description for a lumped mass manual transmission. ...................... 77
Ports for the hollow shaft model. ................................................................... 84
Parameters for the hollow shaft model. ......................................................... 84
Node description for a mean torque full throttle engine model. ...................... 86

Table 21: Ports for a mean torque full throttle engine model. ........................................ 87
Table 22: Parameters for a mean torque full throttle model. .......................................... 87

Table 23: Ports for the longitudinal wind effects model. ................................................ 9]

ix

List of Figures

Figure 1: Node Examples. (a) Typical Atom node. (b) Special Atom node. (c) Macro
node ......................................................................................................................... 4

Figure 2: Macro Node Illustration. (a) Macro node appearance. (b) Macro node

contents. .................................................................................................................. 5
Figure 3: Sample Top Down Model Hierarchy ................................................................ 7
Figure 4: Diagram of Rigid Ring Tire Model ................................................................. 14
Figure 5: Diagram of modified Rigid Ring Tire Model .................................................. 14
Figure 6: Fraction of transmitted Normal Force as a function of longitudinal slip, x. ...... 15
Figure 7: Non-linear Contact Force Function ................................................................ 17
Figure 8: Bond graph submodel of a Rigid Ring tire model .......................................... 17

Figure 9: Test 1 - Damped and undamped response of vertical tire deflection to initial

load application. ..................................................................................................... 21
Figure 10: Test 2 - Effective Tire Radius after a 2000N load is applied. ....................... 22
Figure 11: Test 3 - Torque as a result of longitudinal deflection and vertical forces. ..... 22
Figure 12: Test 4 - Longitudinal and angular velocity of the tire. ................................... 24
Figure 13: Top Level System Graph .............................................................................. 25
Figure 14: Vehicle System, Level 1. .............................................................................. 27
Figure 15: Powertrain, Level 1.1 ................................................................................... 28
Figure 16: Vehicle Dynamics ........................................................................................ 30
Figure 17: Vehicle Coordinate System and Tire Numbering .......................................... 30

X

Figure 18:

Vehicle System Diagram .............................................................................. 34

Figure 19: Hierarchical Automobile Model .................................................................... 35
Figure 20: Flat Vehicle Model ....................................................................................... 36
Figure 21: Axle deflection response to a 4lN-m step in engine torque. ......................... 41
Figure 22: Test Case: Hrovat's (1991) Clutch/Axle response to a step in engine torque.41
Figure 23: Wide Open Throttle Vehicle Driveaway Response ........................................ 43
Figure 24: Wide Open Throttle Driveaway with CVT ................................................... 46
Figure 25: Diagram of a Two Dimensional Rigid Body Vehicle Model .......................... 53
Figure 26: Bond graph submodel of a Two Dimensional Rigid Body Vehicle Model ..... 53
Figure 27: Test 1 - Initial Settling of the 2D Rigid Body Model .................................... 57
Figure 28: Test 2 - 2D Rigid body model subjected to a 1000N step force at rear
suspension. ............................................................................................................ 58
Figure 29: Test 2 - 2D Rigid Body model subjected to a 1000N step force at front
suspension. ............................................................................................................ 58
Figure 30: Test 3 - Vehicle Response to longitudinal wind
drag. ...................................................................................................................... 60
Figure 31: Test 2 - Vertical forces due to a longitudinal step force with and without Wind
Lift Considered ...................................................................................................... 60
Figure 32: Test 4 - Vehicle Response due to application of load in tow ......................... 61
Figure 33: Non-Linear Contact Force Function ............................................................. 63
Figure 34: Diagram of modified Unpropelled Rigid Ring Tire Model ............................. 64
Figure 35: Bond graph submodel of an Unpropelled Rigid Ring Tire Model .................. 64
Figure 36: Verification Test - Damped response of vertical tire deflection to initial load
application. ............................................................................................................ 67
Figure 37: Diagram of a typical open face differential .................................................... 68
Figure 38: Bond graph submodel of a Simple Open Difl‘erential .................................... 68

Figure 39: Simple Open Differential Verification Results: Test 1, the comparison of
input and output shaft speeds. ................................................................................ 71

Figure 40: Simple Open Differential Verification Results: Test 2, the loss of traction to
W2 ......................................................................................................................... 71

Figure 41: Simple Open Differential Verification Results: Tests 2 and 3, torque applied

to each axle. ........................................................................................................... 72
Figure 42: Clutch stick-slip properties. .......................................................................... 74
Figure 43: Diagram of a Lumped Mass Manual Transmission with Clutch ..................... 75
Figure 44: Bond graph of a Lumped Mass Manual Transmission ................................... 75

Figure 45: Test 1 - Steady state input torque and output velocity for a lumped mass
manual transmission. .............................................................................................. 79

Figure 46: Test 2 - Lumped mass manual transmission response to clutch engagement
and disengagement. ................................................................................................ 79

Figure 47: Test 3 - Lumped mass manual transmission response during shifting and clutch

operation. .............................................................................................................. 81
Figure 48: Bond graph of a hollow shaft model with flow-flow external causality. ......... 82
Figure 49: Bond graph of a hollow shaft model with effort-efl‘ort causality. ................... 83
Figure 50: Bond graph of a hollow shaft model with effort-flow causality. .................... 83
Figure 51: Diagram of a mean torque full throttle engine model. ................................... 86
Figure 52: Bond graph of a mean torque fiill throttle engine model ................................ 86

Figure 53: Torque as a function of engine speed for a mean torque engine model based

upon a GM 4.3L V6 engine .................................................................................... 88
Figure 54: Engine speed during idle mode operation. .................................................... 89
Figure 55: Bond graph of wind effects model. ............................................................... 90
Figure 56: Wind force as a firnction of relative wind speed. ........................................... 91

xii

1. Introduction

1.1 Problem Definition

Mathematical modeling of systems and components is now an accepted strategy
for improving the efficiency of product design. A recognized tool for constructing
mathematical models of complicated systems is the bond graph language. Bond graphs
are typically constructed from a fundamental set of elements which are used to describe

mixed domain systems and components. (Karnopp, 1990)

In the past few years an element known as Macro node has come into use. Macro
nodes are used to describe complex behavior associated with a device or component, such
as a pump or motor, using a single node to represent the device. A Macro node is
displayed as an icon with connection ports which allow communication. The icon
represents an underlying structure which typically is constructed of basic elements and
possibly macro nodes. An intricate machine can be broken down by its subsystems and
Macro nodes can be used to represent each subsystem. This compact representation
simplifies system modeling and naturally leads to the reuse of models. A library of

component/system models naturally follows.

Software applications, such as CAMAS (Broenink, 1995) and OLMECO (Top,
1995), have adopted the Macro node style of modeling. The problem is that the ENPORT
(Hales, 1995) environment to date does not fully incorporate the hierarchical mentality

associated with multiport modeling. In order to move forward the ENPORT group must

2

firrther explore issues related to the use of Macro node modeling and the creation of a

model library.

1.2 Statement of Objective

The objective of this research is to develop a general powertrain and rigid body
kinematic model library for an automobile and investigate its use. The structure of the
library will be based upon a top-down perspective of an automobile. The top down
perspective dissects the automobile until all the systems can be modeled in terms of
components such as the transmission, engine, or tire. The framework of the model library
will be based upon the classifications which the top down structure creates. For each
component an appropriate Multiport Macro node is created and it holds a place within the

model library structure.

The model library will serve fixture researchers with a selection of models to use
and build upon. The top-down structure guides the implementation and connection of the
multiport nodes. The user is not to be burdened with the task of developing each
individual model from scratch and deciding how they should communicate. Yet, if a user
needs to customize a component model it is only a matter of editing the underlying
structure of the multiport node. Further, the modified model can be saved as a new model

in the library and reused at a later date.

In an industrial setting, the library and model structure also grant greater flexibility.
Given that a model structure has been defined separate teams can work simultaneously on
modeling different components or they can develop variations of the same component. In

either case the multiport model allows faster development with more options for design

3

alternatives. Vehicle models can be produced faster and component model

interchangability is guaranteed by the clearly defined interface.

1.3 Thesis Structure

The structure of this thesis begins with a brief introduction to the problem
definition and research objectives. Section 2 is dedicated to discussing the fundamentals
of Hierarchical Multiport modeling. Section 3 discusses reusability and provides
requirements for a model library. A sample library entry is given. Section 4 presents the
top down model structure that the library will be based upon. A case study applied to an
automobile will be examined in section 5 which explores the use of Macro nodes and a
model library. Finally, in section 6 conclusions will be made based upon the case study
results. Appendix A contains a more complete model library and Appendix B contains the

subroutines used in this work.

2. Multiport Modeling

In this chapter we explore the issues associated with multiport modeling. There
are three basic elements associated with multiport modeling which we will present and

discuss. Then we examine how these elements fit together in a hierarchical structure.

2.1 Nodes

A node is a general term used to describe an object which graphically represents
some mathematical model of a component, system, or phenomenon. There are two types
of nodes: the Atom node and the Macro node. The Atom node is the most basic type and
is used as a building block. They are used to describe physical or mathematical
phenomenon and are the only node capable of directly stating a mathematical relationship.

Figure 1 illustrates two cases of Atom nodes. Figure 1a represents a typical atom node

 

 

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Figure 1: Node Examples. (a) Typical Atom node. (b) Special Atom node. (c)
Macro node.

4

5
from the bond graph language (Karnopp, 1990) and Figure 1b represents a specialized
atom node for a transfer fimction. The Macro node, shown in Figure 1c, is used to iconify
an underlying graphical node structure. This graphical structure may consist of both Atom

and Macro nodes in conjunction. Figure 2b illustrates this point.

Power

Ports Signal
Y Ports O 1 __>. R<l

 

 

 

 

 

 

 

 

.................................................

(a) (b)

Figure 2: Macro Node Illustration. (a) Macro node appearance. (b) Macro node
contents.

2.2 Ports and Connectors

A port is an object associated with a node that allows the node to communicate
with its environment. There are two primary types of ports and a node may posses
multiples of each. The first type of port is the Power port and it allows other nodes to
communicate with the node. This implies that the port exchanges effort and flow data, but
causality may also be a concern. Causality is determined by the underlying model which
may have indifferent, constrained, or restricted causality amongst its power ports. The

Signal port is the other type of port. It allows one variable to flow through the port and

6

the direction of that flow is indicated by the direction it points. Figure 2 illustrates the use
of both Power and Signal ports. If a node has multiple ports it is referred to as a
Multiport node. There are Multiport Macro nodes and Multiport Atom nodes. Figures 1

and 2 both illustrate Multiport nodes.

Connectors permit ports, and ultimately nodes, to communicate. Connector types
include bonds and signals. Bonds carry effort and flow information between ports.
Causality must be assigned to a bond to facilitate computability. In contrast, Signals only
carry data and direction of data flow is indicated by the Signals’ arrow. Observing Figure
2, note that a port symbol is only used when a port is not connected. It is implied that a

port is present where a connector contacts a node.

2.3 Hierarchy

Top down modeling hierarchy views a model as consisting of multiple levels of
information which vary in detail. At the highest level the model is viewed from its most
simple perspective and as we move into lower levels of the model we zoom in on its
structure. At the very highest level of this hierarchy, the model appears simply as a Macro
node with Ports that allow the model to interface with its environment. The simplest
perspective of an automobile is as a complete system which interfaces with the driver,

road, and environmental conditions.

A convenient database has been developed by Hales (1995) which represents this
structure. Rather than referring to parent and child levels, Figure 3 illustrates a
hierarchical structure which numbers the various levels. Here the simplest perspective of
the system is the System Graph, which is viewed at level 1. This is how the system would

appear if used as a Multiport in a parent level. The system graph can be expanded to view

7

the children at the next level in 1.1. Level 1.1 contains other Macro nodes which act as
parents to their own subsequent underlying structure. Accordingly the child levels of
these other macro nodes are referred to as 1.1.1 and 1.1.2. This same numbering scheme
can be extended to the individual atoms which are used to build a sub level, but that aspect

is not explored here.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

r1 .......... b .......... ,
Top Level \erw E a M c E
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: ........ d ........ . E
11.11 ........................ R ............. i
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ExpandedVIew ; a MélsgglgM c 3
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11.1.1 . . 1'3172' """""""""" .
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Figure 3: Sample Top Down Model Hierarchy

3. Design of a Model Library

3.1 Reusability and Libraries

Further observation of Figure 3 reveals that child levels 1.1 and 1.2.1 contain
essentially the same sub-model. This is not an uncommon situation in modeling given that
systems often contain multiple occurrences of similar components. For example, an
automobile model requires four tire models that are similar. On a broader scale a
component may be used in other models as well. That same tire model may be applicable
over a wide range of applications. Much time and energy could be saved if a standard
model of that tire was created and available for usage. The next issue which must be

considered is the accessibility of those models.

Reusability of component models is only feasible if the models are stored and
organized in structured manner. A library of component models would facilitate storage
and organization of those models. The next question that has to be addressed is how to
organize the library. There are many different schemes that are possible, but one scheme
has to be chosen. The scheme selected in this research is based upon a top down
perspective of the automobile. The top down hierarchy is presented in the next chapter,
but its key features will be presented here to give the reader a better impression of the
library structure. From the highest level the vehicle is viewed as a Multiport macro node
which interfaces with its environment. We expand this macro node and discover that the
automobile is considered in terms of its powertrain and kinematics. Each of these sub-

systems can be further expanded until we are viewing the component models which make

9

them up. The library developed here will be structured based upon this hierarchical

viewpoint.

The refinement of the hierarchical breakdown is determined by the requirements of
the users. If the modeler is interested in the general performance of a vehicle then the
breakdown should consider major components such as the engine and transmission.
Different combinations of engine and transmission could be examined for affect on vehicle
performance. If the user is interested in building transmission models then the library
should have smaller scale models for pumps and valves. In either case the refinement of
the library is determined by the requirements of the user. This research is directed towards
exploring the development of a model library for the kinematics and powertrain of a
vehicle. Development of larger component models, such as the engine, which fit within
this hierarchy are suflicient for exploring the issues involved with developing and

implementing such a library.

3.2 Background

This section of the paper discusses the fundamental concepts and information
required to build a library of component models. Currently, there are two primary schools
of thought regarding model libraries. Stein (1995) supports the development of a two
level modeling approach which utilizes an expandable component model library. The first
level of Stein’s approach is equivalent to viewing a component model as a node. The
second level views the physical model which describes the component. Expandability
refers to the model’s ability to capture varying levels of complexity. For example, a drive
shaft could be viewed as a simple lumped mass torsional member or several lumped mass

members. In his current work, the physical model which Stein refers to is a traditional

10

bond graph model. Stein does not focus on the development of a multilevel library which

can contain nested sub-models.

Similar to Stein, Top (1995) suggests that a model library needs to contain a
variety of models varying in levels of complexity and validation status. However, Top
further suggests that the models can be based upon generic building blocks and hence
contain nested sub-models. This type of modeling structure is much more conducive to
hierarchical modeling which should support the birth of new component models through
the combination of existing models. For example, if a user is building a model of an
automatic transmission (s)he should be able to select valves from a set of validated valve
models and not have to build five identical valve models. Top also points out that if a
model is to be reusable it has to be described in terms of what type of component it is,
how the model is conceptually intended to be used, and what mathematical relations apply.
Top’s work is much more focused on developing a fimdamental library structure which is

adapted to reuse and growth.

3.3 A Proposed Library Structure

The author of this thesis agrees with Top that a library structure must support the
reuse of component models in system models and in the development of hierarchical
component models. Further, a component model should be able to be modified fi'om its
original form to support model refinements and modifications when an implementation of
the model requires it. The library structure should also provide sufficient information to
its users such that they can easily understand the intended use of the stored multiport
models. Subjects such as energy domain, port type, range of valid usage, and units must

not be neglected.

11

Clearly the component view point indicates that the library be divided by type of
component. However, it is insufficient to state that a library of component models is
available without fiirther qualifying what energy domain(s) the models apply to and
whether the models are mean value or dynamic in nature. Therefore the library structure
must clearly indicate what the component is and what physical characteristics of the
component are under consideration. A library should contain sufficient information
regarding a model such that the user can evaluate the intended usage and applicability of a
model. This means that a model’s underlying assumptions and dynamic qualities must be
clearly described. To assure proper implementation of a model library the port type,
domain, and units must be indicated for every port; parameter usage, units, and range must
be described as well. The developer of a component model must be aware that the model
(s)he is developing is intended for reuse and take proper precautions in documenting

assumptions, parameters, and intended qualitative behavior.

A summary of the relevant information which a library should contain is as follows:

what component the model applies to

what aspect of the component the model captures
what assumptions were made in constructing the model
port connection and domain data

parameter usage, range, and units

validation results

In the structure of the library developed here all of the above are addressed in three
separate sections for each component. The first section is “Model Description and
Assumptions” which describes what component the model applies to, what aspect of the
component is considered, and what assumptions were made in the construction of the
model. The second section is “Ports and Parameters." For each port this section indicates

what information the port transfers, port type (signal versus power flow), and what units

12

apply. This same section also indicates what parameters are used in the component model
as well as what their corresponding units and relevant range of applicability are. The third
section is “Verification Results.” This section describes what verification tests a model
had to pass in order to be deemed valid. A description of the tests and results is included

as well as a discussion.

13

3.4 A Library Example: A Propelled Two Dimensional Rigid Ring
Tire Model

A component model is presented as an example of how a typical component model
should be described in a model library. Information associated with the component model
is presented in the fashion prescribed above. A more usefiil model library is included in
Appendix A which contains a set of component models sufficient to illustrate the use of a

model library applied to an automobile’s powertrain and kinematic dynamics.

3.4.1 Model Description and Assumptions

The tire model presented here is a combination of work by Pacejka (1991) and
Zegelaar (1993). Work by Zegelaar is used to construct the interface between the tire
carcass and the rim. (see Figure 4) His model is capable of capturing torsional, vertical,
and longitudinal dynamics. The interface between the tire carcass and the ground is
described by Pacejka's "Magic Tyre Formula" applied solely to longitudinal slip in

conjunction with non-linear springs which consider tire contact with the road.

In the Zegelaar model it is assumed that the behavior of the tire acts as a linear
spring about its equilibrium point and that damping is negligible. A rigid ring is used to
represent the tire carcass and is connected to the rim via translational and rotational
springs. The result is a model which is capable of capturing the zeroth and first modes of
the tire and rim interaction in both translation and rotation. Since the model approximates
the lower modes only, the model is only applicable to lower frequency usage up to 90 Hz.
A small amount of damping has been introduced as shown in Figure 5 which represents
the tire-rim interactions more realistically and provides for a numerically better

conditioned model.

l4

 

Rigid Ring TIre

Figure 4: Diagram of Rigid Ring Tire Model

Rigid Ring Tire

Fa.

I'm ken

T

1'" e 4193
+

nun

Rim

 

Figure 5: Diagram of modified Rigid Ring Tire Model

15

The Zegelaar model also suggests the use of a first order transfer function to
approximate the tire carcass and longitudinal tractive interactions. However, since very
small phase and amplitude effects are noted at lower frequencies, a steady state model
presented by Pacejka is used which more accurately describes tractive force as a function

of longitudinal slip. Here longitudinal slip, x, is described as,

rQ—V

V (1)

 

x:

where r is the rolling radius, 0 is the angular velocity of the tire, and V is the longitudinal
velocity of the rigid rim. Figure 6 shows the fraction of normal force transferred to
longitudinal force as a firnction of longitudinal slip, x, for a set of parameters
characteristic of dry pavement. A more complete description of these parameters follows

in the Ports and Parameters section.

 

 

 

 

 

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s 0.20 1 9 )1
1* i t t t 333 I v v I ¢
-1 .00 -000 0.00 0.40 0 20 o 20 0.40 0 00 0.00 1.00
en
0 1 '
-0 I
w 4»
0.00 ‘1

 

4.00 A

Figure 6: Fraction of transmitted Normal Force as a fimction of longitudinal slip, x.

16

Ground and tire contact is described by a non-linear Spring which has a stiffness
profile as shown in Figure 7. Note that the stiffness of the ground tire interaction is
recommended to be at least ten times greater than the stiffness of tire itself. While this
minimizes displacements as a result of road surface deformation, it creates a much stiffer
model. The result is that the model is much more computationally intensive to solve but is
more accurate regarding road contact. A slight amount of damping has also been included

in the ground interface.

Figure 8 is the bond graph of the Rigid Ring tire model. The structure of the
graph will not be examined but the meaning of the nodes, except 0's and 1's, will be
explained in Table 1. There the node name, primary coordinate, type, relevant
subroutines, and a brief description of the node are included. Several subroutines are

referenced by the nodes and are as follows:

ZZSUOI - Multiplies several model parameters and a model variable.
ZZSU02 - Non-linear contact stiffness. See Figure 7.
ZZSU23 - Pacejka's longitudinal traction model as a fiinction of slip. See Figure 6.

These user subroutines can be found in Appendix B.

0.80 <

0.60 u

0.40 '

0.20 *

Gonna Force

0.00 <

020 ..

~0.40 ‘

060 A»

-0.80 4*

-1.00

17

Ground Contact Force

 

 

Kgr

 

-1.00 -0.80 -O.60 -0.40 020 0.00 0.20 0.40 0.60 0.80 1.00

Relative Position of Rigid Ring to Ground

Figure 7: Non-linear Contact Force Function

1 1 30
IFIX CBX T- 1 TX RBX IBX CROTB RBFIOT

13

3
29 sFtX \fl
10 8103

12 7
BDXO IYRw-OTXV—imr—OGXTRGFlv-TFR IROTB 1TROT IROTR

SER

25

20

/
iRZ

 

I

31 \5 a. 1
‘IR
11

 

 

 

 

see 0113 ‘oRoT‘I ROTFt OSHFTIN
2s 35 22
l /
302 0 izR—LQOTZ £128 ——-15——\ooI=I —-2-1——>rer
0R2
23 13 23
I 1 r
1T2 lBZ 10R
18 24 14 27
I I
082 1:32 con R02

 

Figure 8: Bond graph submodel of a Rigid Ring tire model

18

Table 1: Rigid Rim Tire Model node descriptions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nay DPS PM Associated Description
Coordinate Subroutine
lRO'I'B 1 0 Sums the torques applied to the tire (mm.
BDX port x Rim connection to axle
BDZ port 2 Rim connection to axle
CBX C x Tire stiffness
CBZ C z Tire stiffness
CGR C 2 ZZSU02 Ground contact stiffness
CROTB C 0 Tire torsional stiffness
IBX I x Tire Intertia
IBZ I 2 Tire Inertia
IROTB I 0 Rotational Inertia of the tire
IROTR I 0 Rotational Inertia of the rim
IRX I x Rim Inertia
IRZ I z Rim Inertia
NEG fen 2 Conditions normal force signal such that it is
positive when in compression
RBROT R 0 Tire/Numerical torsional damgng
RBX R x Tire/Numerical Damping
RBZ R z Tire/Numerical Damping
RDEFL src z Deflection of tire from free radius
RGR R x ZZSU23 Pacejka hiagic Tyre model slip function
RGZ R z Ground contact damme
RTIRE src 2 Free radius of tire
SEB SE z ZZSU01 Gravitational force of tire
SER SE 2 ZZSU01 Gravitational force of rim
SF GRND port 2 Road profile
SHF'I‘IN port 0 Rim connection to axle
SUM sum 2 Sums RDEFL and RTIRE tpgive actual tire radius
TFR TF x <—> 0 Effect of longitudinal force on torque applied to
nm
TFZ TF z <—> 0 Efl'ect of normal force on torque applied to rim
Table 2: Port description for a Rigid Rim tire model
Port Tm Units Description
BDX Bond N Longitudinal force to rim
m/s Lonfludinal velocity of rim
BDZ Bond N Vertical force to rim
m/s Vertical velocity of rim
SHFI‘IN Bond N-m Torque to rim
rad/sec Aggy velocity of rim
RX Bond N Traction Force transmitted to ground
m/s Velocity of ground profile
RZ Bond N Vertical Force transmitted to ground
m/s Velocity of ground profile

 

 

 

19

Table 3: Parameter description for a Rigid Rim tire model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 .4.2 Ports and Parameters

 

 

Parameter Node Units Typical Value Description
YSS RGR -none- 0.84 Asymptotic F-traction/F-normal for tire at
high slip
YPK RGR -none- 0.95 Peak value of F-traction/F-normal
XPK RGR -none- 0.22 Slip value where YPK occurs
THETA RGR -none- 80° - 87° Initial slope of traction curve.
KBO CROTB N-m/rad 70100 Torsional stiffness of tire.
KB CBX, N/m 1.47 x 10r Translational stiffness of tire
CBZ
MRIM IRX, IRZ kg 5.0 Mass of rim
MTIRE IBX, IBZ kg 4.52 Mass of tire
JRIM IROTR kg-m2 .100 Rotational Inertia of rim
JTIRE IROTB k -m2 .375 Rotational Inertia of tire
RTIRE RTIRE m .288 Undeforrned tire radius
RNUM RBZ, 100 Tire/Numerical Damping
RGZ,
RBROT
G SER, rn/s’2 9.81 Acceleration due to gravity
SEB
KGR CGR N/m 7.35 x 10" Contact stiffness with ground
RG RGZ l x 10‘ Contact damping with ground

The external ports of the Rigid Ring tire model are indicated in Figure 8 as BDX,
BDZ, SHFTIN, RX, and RZ. Since each port is connected to a bond there are two pieces
of information associated: efl‘ort and flow. For each port Table 2 indicates the name, port

type, a brief description, and units associated with that port.

Parameters associated with the Rigid Rim tire model are described in Table 3. The
parameter name, associated node, units, a brief description, and a typical range have been
indicated for each parameter. The parameters ranges listed for the node RGR are typical
for the longitudinal slip-traction characteristics of a tire on dry pavement. (Wagner, 1995)
The remaining parameters describe the linear characteristics of a smaller tire with a 70%

aspect ratio. (Zegelaar, 1993)

20

3 .4.3 Model Verification

Verification of the Rigid Ring tire model must be a multistep process since the
model is relatively complicated and certain aspects of the model's performance must be
verified methodically to insure proper performance of the model as a whole. The

following tests must be completed:

Test 1 - Tire deflection in the z-direction.

Test 2 - Effective tire radius passed to TFR

Test 3 - Torque resulting from longitudinal deflection and normal force
Test 4 - Steady state longitudinal velocity of tire.

For Test 1, tire deflection in the z-direction can be tested by applying a load to
port BDZ with all other inputs and initial condition set to zero. The anticipated result is
an initial transient, which with the added tire/numerical damping will damp out to a steady
state equilibrium without oscillations. If tire/numerical damping was not present the
oscillations would result in steady harmonic equilibrium. Figure 9 demonstrates the effect
of added damping on tire deflection with a 2000N load which is appropriate for most cars,
applied to port BDZ. Special care must be taken when simulating this model because of
stiff ground contact. For example, since the stiflhess of the ground contact is 50 times
greater that the tire stiffness the simulation time-step must be set appropriately to capture
the affects of 460 Hz rather than 90 Hz, which is the natural frequency of the tire.
Further, the benefit of added damping can be observed by viewing the Undamped
response in Figure 9 as compared to the Damped response and noting the steady state
behavior of the two. Clearly, the Damped response is much more realistic and both have a
mean steady state deflection of 0.139 cm. Ifwe compute the expected deflection of the
tire given the net 2050N load and 1.47 MN/m tire stimiess, we also get 0.139 cm. While
this deflection seems a bit small for an initial load application, we have to keep in mind

that the stifl‘ness we are using is intended to be implemented about an equilibrium point.

21
Given that the steady state deflection is accurate and that the numerical damping seems to

be of benefit, we can assume that the damped vertical deflection of the model is valid.

Test #2 of the model is intended to assure that the effective radius of the tire is
accurate after a load is applied. The tire radius is calculated by a summing node, SUM,
and two signal source nodes, RDEFL and RTIRE. This test is to assure that the
calculated radius is of the expected value. Given that the same 2000N load is applied as in
Test 1, we know that the tire radius should be 0.139 cm smaller resulting in a new radius
of 0.2866m. Figure 10 illustrates the actual effect of the applied load on tire radius. It can
be observed that the tire radius does decrease as expected and that the response has a

transient dynamic nature. We can therefore assume that the results of this test are valid.

Tire Response to InitIel Load Application
0000

-0.001

IAIIAA
vvvvvvvw

Damped
Deflection (m)

-0. 002

—0.003
0.000

-0.001

-0.002

Undemped
Deflection (m)

-o.003 I . I . I . I . I . I . I I I I I . I . I
0.0 0.1 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (sec)

 

Figure 9: Test 1 - Damped and undamped response of vertical tire
deflection to initial load application.

LongItudInaI

Bond 22
Torque (N-m)

Deflection (m)

 

22

Effective Tire Radius after Load is Applied

 

 

0w—
1
0m-
? Initial Radius
‘0'; Min“... ..................
”-3 “nun" Actual Radius
‘6 1
c:
9
.1:
0m-
0-27 Irifi'Iri*rvirvi'i
0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.8 0.9 1.0
Time (sec)

Figure 10: Test 2 - Effective Tire Radius after a 2000N load is applied.

0.000

43.004

 

 

Normal Force Reaction Torque and Moment Arm

 

..... ll. _-_-___,_,___-_-_,___-___-,
_II._. _ ...................................
i ----- uv - --------------------------
I I T I Y I I T I l
0.0 01 02 0.3 04 05

Time (sec)

Figure 11: Test 3 - Torque as a result of longitudinal deflection

and vertical forces.

23

Test #3 is intended to verify the torque applied to the rim and axle as a result of
vertical forces and longitudinal deflection. It is anticipated that this torque will counteract
the positive input torque at SHFTIN being applied to the rim. We can confirm this by
inputting a step torque to SHFTIN and observing the longitudinal deflection and the effort
associated with bond 22. Figure 11 illustrates that as the tire pulls the inertia of the
vehicle there is a negative reaction torque resulting from the vehicle lagging behind the tire
and generating a torque proportional to the longitudinal deflection. Based upon these

results it is confirmed that the normal force resultant torques are valid.

Test #4 verifies the longitudinal velocity of the tire to a given input shaft speed. In
this test the input shaft speed is 20 radians/second and it is anticipated that the final
longitudinal velocity of the wheel will be 5.73 m/s given the steady state tire radius of
0.2866 m. Figure 12 shows the angular and longitudinal velocities of the tire when the
angular velocity is ramped up from zero to 20 rad/sec in a one second time period. Note
that the longitudinal velocity goes through a transient and then reaches its steady state
velocity of 5.73 m/s in around 1.2 seconds. The final speed is as expected and it is

concluded that the model is verified.

24

Angular and Longitudinal TIre Velocity

 

Longitudinal
Velocity (mls)

 

 

Angular
Velocity (radlsec)

 

 

TIme (sec)

Figure 12: Test 4 - Longitudinal and angular velocity of the tire.

4. Top Down Model Structure

The structure of a mathematical model describing the propulsion/braking
characteristics of a typical automobile is presented here. The model focuses on the
powertrain and rigid body kinematics of the automobile. The structure of the model is
presented from a top down perspective where the major systems under consideration are
subdivided into sub-systems. This top down perspective of the automobile will serve as a
hierarchical structure for the model library created by this research. Since our goal is to
explore the development and use of a model library we will only explore the hierarchical
structure of the major systems and components in the vehicle. We will not be examining
the structure and details of a particular engine or transmission. This structure will be more

general and applicable to a wide range of configurations.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Driver
6
s
asasio can
I: S
1.
Vehicle System ®®®
A _ A
If.” 9 tr. 5
3. 4. 5.
Rood Surface @flg Load In Tow

 

 

 

 

 

 

 

Figure 13: Top Level System Graph
25

26

4.1 Top Level

The top level model structure is shown in Figure 13. This level describes the
interface between the vehicle system and its external environment; namely the driver, road
surface, wind effects, and load in tow. Note that at this level, the vehicle system is shown
as a macro node and is labeled "1. Vehicle System. " The ports are not drawn here, but it
is understood that where a connector is adjacent to a macro node a port is present.
Further note that "6. Ambient Conditions" shows connectors draw to symbols such as G)
which indicate that this information may be used elsewhere in the model rather than
drawing a signal to the necessary ports. The symbol can then be inserted where necessary
and connected appropriately. Connections between many of these sub-systems are made
using multiport vector bonds and vector signals. (Margolis, 1989) Such a connector type
is a convenient means of simplification and generalization. Table 4 describes the

connectors from Figure 13, and their type and units.
4.2 Vehicle System

The vehicle system is shown in Figure 14 below. Note that the Macro node "1.
Vehicle System" has been opened to show its child level. Further note that it has two
children; macro nodes "1.1 Power Train" and "1.2 Vehicle Dynamics." The child levels of
these macros will be shown later. While it is not necessary to show sub-systems external
to the macro node, the Driver, Road Surface, Air/Wind Effects, and Load in Tow are
shown here to assist the reader in visualizing the Vehicle System's external connections.
Further use has also been made of vector bonds and signals here to exploit their
compactness and generality. Table 5 below details the connector internal to this macro

node at this level.

...........

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E 2.
Driver
<5 ridge: 5 (‘3 _
t: 8
0
L111 8°
f]...véhi.c.l.e.-_.—- l-- --I -- - ........................... 1- —-.--._ ..........E
: Warn 2
E I .1 1 .2 f
3 tax Vehicle :
' ' === '
3 ®®® ®®® g
I j I ;
’ r: S
3
E3. 24. Air/\Mnd 5 """"""""""""
5 i ' LOOd In Tow
,5____R°Od ””0021; Effects ______________________________________________
Figure 14: Vehicle System, Level 1.
Table 4: Top Level Variables
Symbol Decemtion Connector Type Units
togt Engine Speed Signal rad/sec
r. Steering torque Bond N-m
to, Steering Angular velocity rad/sec
0t Throttle angle flgnal radians
CLUTCH Clutch engagement Signal -
Ft, Applied brake force Bond N
Vb Brake velocity m/s
F8 Force of ground Vector Bond N
\A Velocity of ground m/s
F. Force of load Vector Bond N
VI Velocity of load m/s
F. Force of wind Vector Bond N
V. Velocij of wind m/s
GEAR Gear selection Signal -
Pans Engine oil pressure SW kPa
T3 Engine Temperature Sign“ °C2
a Driver perceived acceleration Vector Signal m/s

 

 

 

28

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a 65‘ a .1
i— o. a: w 0
11...]. POO-Wei Tiéii‘f - - .1 .............. - .. ................................ I
i 1.1.1 ' 1.1.2 1.1.3 : 1'2
I E ' [cf , _ fdr DlflefenflOI ; 1 VehIC|e
2 ng ne “’0’ Tr E 1 "’3' and Axles 1% E Dynamics
: ®®® ®®® ®®® ;
Figure 15: Powertrain, Level 1.1
Table 5: Vehicle System Level Connectors.
gsymbol Description Connector Type Units
1., Axle/body torques Vector Bond N-m
cog Axle/body relative angular velocity rad/sec
Table 6: Powertrain Level Connectors.
Symbol Description Connector Type Units
1,, Crankshaft/Transmission Torque Vector Bond N-m
toIt Crank/Trans relative angular velocity rad/sec
rd, Transmission/Differential Torque Vector Bond N-m
21, Trans/Diff relative angular velocity rad/sec

 

 

 

 

29

4.3 Power Train

Figure 15 shows the child level of the "1.1 Powertrain" macro node from Figure 14
above. This level details the structure of the Powertrain macro node. Note that the
children of this level are "1.1.1 Engine", "1.1.2 Transmission", and "1.1.3 Differential and
Axles. " This division was chosen because each of these sub-systems interacts as a set of
distinct components with well defined modes of interaction. This correlates well with the
preferred library structure discussed in Chapter 2. Table 6 below describes the connectors

in this level.

4.4 Vehicle Dynamics

The internal structure of the "1.2 Vehicle Dynamics" macro node is shown in
Figure 16. The children of this macro node include "1.2.1 Vehicle Body/Frame", "1.2.2
Front Suspension", "1.2.3 Brake System", "1.2.4 Rear Suspension", "1.2.5 Tire 1", "1.2.6
Tire 2", "1.2.7 Tire 3", and "1.2.8 Tire 4". Unique to this level is the connector
distribution box, shown as WI which signifies that a vector bond or signal is split into
components. In this case those components include the connectors between the ground
and the tires. The tires in this case are numbered one through four. Figure 17 illustrates
the numbering of the tires with reference to a typical automobile as well as the SAE
standard vehicle coordinate system (Gillespie, 1992). This coordinate system will be used

in the simulations and sub-models following.

 

----------------------------------------------------------------------------------------

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{rig'msi 1.2.2 ‘ 1.23 1.24 i
1 wer " tax 2
1 1101111 WA 1:1th + 81619559311 —%=i RechuspercsiCeénQ) 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ES

 

 

14- Air/Wind
' Effects

altood Surface iTsLoodln Tow

 

 

Figure 16: Vehicle Dynamics

 

Front of Vehicle
Ix
l 2
Y 1
Front of Vehicle
3 4

 

 

 

 

Figure 17: Vehicle Coordinate System and Tire Numbering

5. Case Study: An Automobile Powertrain and Rigid
Body Kinematics Model

To this point we have shown that a model library aimed towards modeling the
kinematics and dynamics of an automobile can be created. The structure of this library is
based upon the hierarchical structure of an automobile’s powertrain and kinematics. Now
we will use the model library to construct a powertrain and rigid body kinematics model of
an automobile. The model constructed here is based upon the model library developed by
this research which is shown in Appendix A. Enport/PC 5.4 Professional was used to
build, simulate, and debug all of the models shown in Appendix A as well as the vehicle

model constructed here.

The version of Enport implemented here does not fully support hierarchical
modeling. Enport facilitates traditional “flat” system models which consist of the basic
bond graph elements and user defined subroutines. It does not support the usage of
multiport macro nodes nor does it support hierarchical structuring. However, many
aspects of multiport modeling can be explored through the construction of a large scale
flat model which imitates the structure of a hierarchical model. This technique lacks the
benefits of multiport modeling software, but it does force the researcher to thoroughly
consider issues related to the construction of a hierarchical model. The researcher is

forced to consider issues which must be anticipated in the design of library software.

31

32

5.1 Model Introduction

The vehicle model constructed here is based upon one published by Hrovat (1991).
Hrovat’s model was intended to show that simple bond graphs are capable of modeling
basic vibration characteristics of an automobile. For example, in his paper Hrovat points
out that his model predicts that there will be a mode of vibration where the engine, clutch,
and longitudinal vehicle kinematics oscillate together. This is the shuffle mode of vibration
and Hrovat points out that the frequency and amplitude of oscillation are dependent on
transmission gear and vehicle speed. In this research we will be examining these modes as
well as demonstrating that the model can be easily extended and applied to a much

broader range of applications.

The model developed by Hrovat and the one developed here are similar in nature
but different in structure and applicability. Hrovat’s model is generally linearized about
specific operating ranges, such as shift points. The models used in this research are
generally non—linear and applicable over a broader range of problems. Another difi‘erence
is that Hrovat’s makes many simplifications in the form of model reductions which cannot
be duplicated easily using a multiport model. Hrovat’s model reductions result in a much
simpler model, but no attempt is made to imitate them. While the underlying structure of
the multiport vehicle model is very complex, the outward appearance of the model is much
clearer than Hrovat’s own simplified model. Hrovat’s model is much less computationally
intense though. His work serves us as a source of data, a general outline, and a baseline

reference.

The vehicle model will now be presented. Since the major component models are
developed and tested in the appendix A, we will not have a thorough discussion of the

vehicle model. Rather, we will briefly present the major features of the vehicle. The

33

system diagram of the vehicle is shown in Figure 18. The vehicle model is a propelled
rigged body bicycle model of a rear wheel driven vehicle. Essentially rigged body means
that the kinematic characteristics of the vehicle are summarized by the vehicle mass and
inertia. Further, bicycle implies that the model has been simplified to being supported by
two tires; the fiont and back. This assumes that any road effects, including traction, are
felt equally by the passenger and driver side tires. Since the vehicle is rear wheel driven,
the front tire is described by a simple tire model which only captures the vertical vibration
of the tire, rim, and suspension. As the rear wheel is propelled by the powertrain, the
vertical, longitudinal, and torsional vibrations of the tire and rim are modeled. The same
applies for the rear suspension except that there are no torsional dynamics to consider.
The engine is treated as a source of effort with inertia and rolling resistance. The
transmission is treated similarly as a lumped mass inertia with damping and compliance.
The clutch is firlly considered as having torsional dynamics associated with the pressure
and fiiction plate contact as well as the torsional compliance associated with the friction
plate isolator springs. The drive shaft and axle shaft are each treated as single lumped
masses with associated stifl‘ness and inertia. The differential is considered as an ideal

transformer with no energy loss.

The hierarchical multiport model of the vehicle is shown in Figure 19. The major
components are labeled clearly in both Figures 18 and 19 and can be easily identified. As
the reader may recall, the software implemented does not have the capability to construct
a hierarchical model such as that shown in Figure 19. The hierarchical model shown is
simply an illustration of what the model should appear as. The actual model is flat and is
shown in Figure20. The components are not clearly labeled in the flat model, but their
location in the figure corresponds relatively to the location of the components in the

hierarchical model.

34

    

 

 

 

  

 

 

 

 

 

 

 

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Figure 18: Vehicle System Diagram

35

 

 

 

 

 

 

 

 

 

 

 

MJ M M

V h' l ' .
Wind e Ice Dynamics Load 111 Tow

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M M

Front Suspension Back Suspension

Lu

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M M
Front Tire Back Tire
M M
Ground Axle Shaft
M
Driver
M M M M
Engine Transmission Drive Shaft Differential

Figure 19: Hierarchical Automobile Model

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Figure 20: Flat Vehicle Model

37
5.2 Objectives

Typically a vehicle model like this is developed for the purpose of exploring a
design alternative such as transmission gear ratios for optimal acceleration performance.
In this case, the model has been developed for the purpose of exploring the usage of a
model library and multiport macro nodes. So, rather than optimizing the vehicle
performance through iterative solutions, we will only be solving the model a few times so

that we can demonstrate the vehicle model’s functionality.

What is intended by the fimctionality of the model is rather vague, so, we will
begin by clarifying what we mean. First of all, by functionality we do not limit ourselves
to just this model. We are interested in the applicability of multiport modeling to real
problems. The baseline paper by Hrovat was chosen as a real problem because it presents
a validated model that is accurate yet straight forward. His work presents a model that
possesses the absolute minimum complexity to achieve a goal. Hrovat’s goal was to show
that bond graphs could easily be applied to the modeling of automotive powertrains. Our
goal is similar in that we wish to show that multiport modeling and the use of a model
library is applicable to real problems. Hrovat’s paper gives us a baseline to aim for where
we can achieve this goal and explore issues associated with this type of modeling without

being muddled by a horribly complex system.

The vehicle model developed here is capable of more than imitating Hrovat’s
results. Hrovat’s model is linear and as a result is only applicable over a narrow range of
operating points. The model developed here is non-linear and applicable over a much
broader operating range. For example, Hrovat’s tire model required that he linearize the
traction resistance at each shift point. The tire model implemented by this research is non-

linear and is valid (Pacejka, 1991) over the entire operating range of the tire from full tire

38

spin to steady-state propulsion. Simple parameter modifications can allow for varying
road surfaces and environmental conditions. We will show that the vehicle model is
capable of capturing the affects of a broader range of problems such as vehicle
acceleration from a stand still. However, there are still inherent limitations to the model
developed here. The largest limitation is the engine model, which as Hrovat points out,
does not include any intake manifold or fiiel delays. The engine model is most accurate at

high speeds, which is where the model will spend the majority of its operating time.

We will firrther demonstrate and explore the ability of the vehicle model to
undergo major design changes. In true multiport modeling this means that we are going to
delete the icon representing a major component, such as the transmission, and insert a new
component in its place. This is a true test of the applicability of the model library to a real
problem. To facilitate this test we will delete the manual transmission in the vehicle and
replace it with an experimental Constantly Varying Transmission (CVT). We will then
perform several simulations to compare the vehicle performance with the manual
transmission and the CVT. Discussion of the issues involved with this design change will

follow.

In summary, we can state that the objectives that we have for this case study are

to:

1. Show that the model library is capable of repeating previous work and explore the
issues involved in developing the model.

2. Demonstrate that the model can easily capture a broader range of vehicle dynamics.

3. Demonstrate that major design changes are easily facilitated and explore issues
which must be considered when modifying the model.

39
5.3 Results

5 .3.1 Model Verification

First we must show that the model developed here is capable of repeating results
accomplished by other authors. To facilitate this a vehicle model has been developed
which is similar to one developed by Hrovat (Hrovat, 1991). As discussed above,
Hrovat’s model is quite fundamental and is intended to demonstrate the applicability of
bond graphs to vehicle body and powertrain kinematics. One test implemented by Hrovat
strives to predict the clutch torsional vibrations which result from a 41N—m engine toque
step. Hrovat’s model showed that the resulting oscillations would have a fi'equency of
approximately 3 Hz with a damping ratio of 0.16 superimposed upon a slower first order
decay. The first order response results fi'om the vehicle acceleration counteracted by wind
and rolling resistances. The harmonic component is a result of what is known as shuffle
mode vibrations and is expected to be in the range of 2-10 Hz with a damping ratio below
0.20. Shuffle mode vibrations are largely associated with the engine inertia and

powertrain compliance oscillations.

The vehicle model developed here is based upon the same vehicle configuration
used by Hrovat. As we have discussed previously, our model is constructed fi'om
Multiport Macro nodes extracted from the developed model library. Hrovat’s model has
been reduced to an absolute minimum complexity and our’s is in its full complex form
represented by the multiport icons. To repeat Hrovat’s results we have applied a 41 N-m
engine torque step and we are observing the clutch spring deflections. The resulting
deflections are shown in Figure 21. Interpretation of the figure reveals that the results
posses a slow first order component and a faster harmonic component. The frequency of

the harmonic component is about 3 Hz with a corresponding damping ratio of about 0.14.

40

The slower first order component does not decay as rapidly as the one in Hrovat’s
response. This can be explained by additional powertrain losses that are considered by
our model. These losses require effort to overcome them and as a result do not allow the
clutch deflection to decay as rapidly as Hrovat’s. Further observation of Figure 21 also
reveals that there is an additional high frequency component of about 50 Hz that becomes
apparent after about 0.50 seconds. This is a result of additional torsional and inertial
effects, such as the transmission and suspension, taken into consideration by our model.
Hrovat’s model neglects components which do not have a dominant effect on the
powertrain dynamics. Comparison with Figure 22, which shows Hrovat's (1991) results,

illustrates that our results match quite closely.

41

010———-—-— —————— ____-__-___._____.

009-1— ————,-——_-_-_-T-___-____-.

0.08mi-—- '-—-—-—--——-————__.-__-_-___-

0.07;~—-- - ————— I_-_-i__-_| _____ l
0.06~1—- - —— _|__-_-_I_-._-._| _____ I
0.05:1 - -—1— ———|—-———--T—_—__| _____ I

0.04—~ -— —-—-————-__-___._________-

CWAdWDdWKfiGflOEd)

0.03—— ——--——' ————— '____-_-___

0.02‘ _ — " ——1 ‘ "'— — — "l —— - — - -—1— — — - — .— —— .. __ - 1

0.01— -—-—1 ————— 1———-—__-_._1_____-

0.00 fl 1L I i 1 . J1
0.0 0.4 0.8 1.2 1.6
Time (sec)

 

Figure 21: Axle deflection response to a 41N-m step in engine torque.

SOOLE SIDE-1
1.00

 

 

 

 

 

0.20

AXLE/CLUTCH DISPLACEHENT (RAD)
\

 

 

 

 

 

 

 

 

 

 

 

 

0.00
0.00 0.40 0.80 1.80 1.6. a.ee
TIRE SCRLE SIQE 0
(SECONDS)

Figure 22: Test Case: Hrovat's (1991) Clutch/Axle response to a step in engine torque.

42

5.3.2 Additional Model Capabilities

Hrovat further verifies his model by estimating shuffle mode characteristics during
a Wide Open Throttle (WOT) driveaway for a light duty manual transmission truck. The
driveaway starts from a dead stop and the driver shifts aggressively through the first four
gears of the transmission. Due to the linearized nature of Hrovat’s model, he is limited to
estimating powertrain behavior near the shift points. For his purposes that is sufficient.
He is solely interested in demonstrating that bond graphs can be used to model dominant
drivetrain oscillations. However, we are interested in demonstrating that our model can
exceed the capabilities of a typical linearized model. We will actually predict the behavior

of the vehicle during the WOT driveaway.

As just discussed, we will simulate the WOT driveaway behavior of a vehicle. The
objective of this test is to demonstrate that the vehicle model developed is capable of
functioning in a richer class of applications while still functioning well in basic
environments. A basic environment requires that only the minimum functionality of a
device is modeled. A basic transmission model is one where only the inertia and
compliance of the transmission is considered. A richer transmission model would also
allow the driver to operate the clutch and shift gears. The WOT driveaway provides a
richer environment in that much more is expected of the model. To accurately capture a
driveaway with shifting it is necessary that the driver is capable of operating the clutch,
throttle, and shifting gears. The component models must be sufficiently general to
accurately capture a device’s dynamics while still being specific enough to imitate the

operational characteristics of the component.

43

 

 

l
10
Time (sec)

 

 

 

é 1.,

 

 

 

lrl'lfil'

 

1
8 81115381 .9 °§ § § ° 3 ° 'nresupo
(radians/sec)
Vehicle Speed Engine 5 I (N—m)

Axle Shaft Torque

Figure 23: Wide Open Throttle Vehicle Driveaway Response

44

The model validation above demonstrates that the vehicle model is capable of
basic implementations. Now we will replace the engine model used above with one that
represents the WOT behavior of a six cylinder engine. In conjunction with a more
elaborate driver model, the new engine will enhance the vehicle model sufliciently so that
we can simulate the WOT driveaway. Modification of the other component models, such
as the transmission, tires, and wind resistance, will not be necessary. Their sophistication

is sufficient for our purposes.

Figure 23 represents our results of a WOT driveaway. In that figure we find the
vehicle velocity, engine speed, axle shaft torque, and tire slip. These variables have been
chosen since they characterize the behavior of the vehicle quite compactly. During a
driveaway the prime concerns are whether we have sufficient traction, if the engine is over
revving, and what our vehicle speed is. As a possible design issue, we are also interested

in the dynamic torque applied to the axle shafts.

Our results indicate that during the first two seconds of driveaway there is a large
amount of tire slip. This slip is associated with the initial acceleration of the vehicle from
a stand still and the rapid engagement of the clutch. During this period the engine speed
increases and the axle torque oscillates slightly about a maximum value. After 1.5 seconds
the tire slip stabilizes and the axle torque and engine speed equilibrate. At each shift point
(4, 9, and 15 seconds corresponding to 1“ to 2"“ gear, 2"" to 3", and 3rd to 4th gears) the
engine and clutch are disengaged and the transmission gear is shifted. During this period,
the engine speed decreases to idle and the vehicle speed decreases slightly. There is a
transient in axle shaft torque and tire slip that occurs when the clutch is disengaged. This
happens because the torque applied to the transmission by the engine is instantly
discontinued when the clutch pedal force is applied. From a qualitative standpoint the

transients seem too large. For the purpose of demonstration this transient is acceptable,

45

but it is recommended that the shift timing involved in the driver model be examined.
Approximately half a second after disengagement the engine and transmission are
reengaged. There is a slight transient in slip, axle torque, and engine speed associated
with this engagement. These transients that occur after reengaging the clutch and engine
are associated with the shuffle mode of vibration. As Hrovat points out, the fi'equency of

these oscillations should increase in higher gears, and our model indicates this.

5.3.3 Exploring Design Alternatives

So far we have shown that the vehicle model is capable of duplicating previous
results. With slight modifications to the engine and driver models we have also shown
that the model is capable of predicting a much richer class of vehicle performance. Now
we will show that by replacing a component model, such as the transmission, we are
capable of exploring design alternatives. In this case we will replace the manual
transmission with a preliminary model of a Constantly Varying Transmission (CVT)
coupled to the engine via a torque converter. We will simulate the driveaway performance
of the vehicle with the new transmission and briefly compare the performance of the two

vehicle configurations.

Figure 24 represents the performance of the vehicle during a WOT driveaway with
a CVT installed. Figure 24 illustrates the vehicle and engine speeds. The figure represents
the investigation of a design alternative to the manual transmission used above. In
comparison with Figure 23, it can be seen that the manual transmission vehicle reaches a
slightly higher speed after ten seconds (22 m/s) than the CVT vehicle (20m/s). Note
however that the engine with the manual transmission reaches a maximum speed of nearly
500 rad/sec whereas the CVT vehicle reaches a maximum engine speed of 270 rad/sec.

Further note that the acceleration of the CVT vehicle is not discontinuous like that of the

46

manual transmission vehicle. We have explored a design alternative which has great
potential for passenger comfort. Thanks to the multiport macro node it was simply a

matter of replacing one node in our system diagram of Figure 19.

 

 

ao—I
1
to?
es ‘
"' 10—
s
0_i
280
240
11 200
13% 160
O
fig 120
80
T 1 1 1m 1 r I ff 1 l T r l 1
o 1 2 3 4 5 6 7 8 9 1o
Time(sec)

Figure 24: Wide Open Throttle Driveaway with CVT

5.4 Discussion

The objectives discussed above are aimed at exploring the use of a model library in
a hierarchical multiport modeling environment. The first set of results proves that the
automobile model we have developed yields reasonably accurate results. The second set
of results demonstrates that our automobile model is capable of simulating more complex

phenomena with minimal effort. The third set of results illustrates that multiport

47

modeling simplifies exploring design alternatives. While our model managed to “jump
through the hoops” provided by these tests, the real issue at hand was exploring what

issues arose during these tests. We can list these issues as:

1. Connection of multiports

2. Model misuse

3. Parameter management

4. Initial condition management

One of the greatest difficulties encountered during the initial assembly of the
vehicle model was the connection of component models. During the construction of the
model library, each component model was built individually and tested. During this phase,
power flow direction and causality at each port were determined by what seemed the most
functionally logical form for each component and port. However, this logic rapidly
decayed when it came time to assemble the component models. Issues of incompatible
causality became evident and incompatible power flow soon resulted in sign errors.
Essentially, even though each component model fimctioned very well individually there
were no guarantees as to how they would function as a whole. Smart assembly
procedures managed to remedy many of these problems, but the issues must be considered

in software implementations.

Model misuse was an issue encountered during the use of the automobile model.
It seems to occur in the form of using the model outside of its valid range. This can refer
to excessive amplitude of effort, flow, power, or frequency at a port. The engine model
implemented here is most subject to this misuse. The engine’s torque source is derived
fi'om an eighth order polynomial fit through torque-speed data and is quite accurate in the
range of 80 to 550 radians/second. However, outside of this range the engine model is
very unreliable. A safety for this effect was provided by limiting the valid operating range

of the model. This is an important issue in larger scale models. As hierarchical modeling

48

tools develop larger models will become common place. It will be very difficult for the
user to fully understand the underlying complexities of each multiport model. Large
amounts of time will be saved if a safety feature is built into each component models

which monitors its operating status.

Parameter management is another issue which needs to be addressed. As
hierarchical tools develop model complexity will increase exponentially. With increased
model sophistication comes greater parameterization. The issue is whether parameters
should be local to each component model, or, whether parameters should be able to be
passed through the hierarchical structure. In essence, the issue is whether models should
be able to use global or local parameters. For greatest flexibility it is recommended that
parameters originate at the local and global levels and that they can be linked to each

other.

Initial conditions also require further attention. Much like parameters, as the
model complexity increases, so does the number of initial conditions. Management and
control of all these initial conditions can rapidly become very tedious. It is recommended
that some sort of management tool be developed that streamlines the management of

initial conditions.

6. Conclusions

A library of models supporting a powertrain and rigid body kinematics model of an
automobile has been developed. The vehicle model has been assembled from component
models stored in the library. Issues related to the construction of the library and
hierarchical vehicle model have been explored. Following is a brief summary of the

modeling and library environment design to which this exercise has led.

Software which manages model libraries of the type described here is not readily
available at this time. If the hierarchical modeling and model libraries are going to be
useful tools, software must be developed which manages the model libraries and makes
them readily accessible. The user must be capable of selecting a model fiom a structured
list where key characteristics of each model are available. Key characteristics include the
contents of the model, port configuration, parameters, characteristics captured, and valid

range of operation.

Intelligent hierarchical modeling tools must accompany good library management
software for effective modeling. Good library management software provides the
information discussed above to both the user and modeling software. The modeling
software must be sufficiently intelligent to handle issues which arise during the structuring
and assembly of the model. The issues faced by modeling software includes the
connection of component models, assurance that a component model is operating within a

valid range, parameter management, and initial condition management.

49

50

Connection of component models requires that causality and power flow be
considered at every part. It is essential that these issues be handled appropriately or the
resulting model will be useless. It is recommended that future library management
software be capable of indicating a port's preferred or required causality. In some cases it
may be necessary for software to manipulate the system equations into an appropriate
form. In some cases the connection of component models may simply not be compatible.
The issue of power flow direction is equally important at a port, but this issue can be

handled much easier than causality.

It is also important that each component model function properly for a given
operating range. It is easy to unknowingly allow a component model to operate outside of
its valid range. It is recommended that the modeling software be made capable of
monitoring the amplitude of effort, flow, power, and fi'equency range across any given

component.

Parameter and initial condition management is also a point of key interest to the
user. A local and global parameter management scheme is recommended where
parameters can exist solely at the local level or they can be linked to global system

parameters.

The reliability of reused models would benefit from a standard set of verification
test conditions. A standard set of verification tests would improve the reliability of reused
models by providing general operating specifications for a component. Significant
deviation from specification would indicate that modifications made to the component

model may be erroneous.

Appendix

Appendix A

Model Library

This section of the paper presents a brief library of component models critical to
modeling and simulating vehicle kinematic and powertrain dynamics. While the library
presented here does not provide a plentifirl selection of models to choose from, it does
provide a selection of models sufliciently large to illustrate hierarchical modeling and the

use of a component model library. The following models are presented:

A. 1 Two Dimensional Rigid Body Vehicle Dynamics Model
A.2 An Unpropelled Two Dimensional Rigid Ring Tire Model
A3 A Simple Open Differential

A4 A Lumped Mass Manual Transmission with Clutch

A.5 Hollow Shaft Model

A.6 Mean Torque Full Throttle Engine Model

A.7 Longitudinal Wind Effects

51

52

A.1 Two Dimensional Rigid Body Vehicle Dynamics Model

Following is a two dimensional rigid body vehicle dynamics model designed to
capture longitudinal, vertical, and pitch dynamics. The model considers the mass and
inertia of the vehicle body to be concentrated at the center of gravity as a single lumped
mass with inertial properties. It is firrther considered that there are two suspension
connection points (see Figure 25) at the front and rear of the vehicle which can transmit
both vertical and longitudinal forces. Wind drag forces are modeled as a point load
applied to the vehicle in the longitudinal direction and wind lift forces are modeled as
forces applied to each suspension point. A load in tow is treated as a pair of point loads
applied to the rear of the vehicle in the longitudinal and vertical directions. The user of
this model is required to know the vehicle mass, moment of inertia of the vehicle about the
y-axis, and the location where all external loads are applied. External loads include wind

drag, wind lift, suspension loads, and the load in tow.

The bond graph representing the system is shown below in Figure 26. Since this
model is largely non-linear a series of sources and sinks have been used to calculate the
module of the transformers. The sources in concern are SUM] through SUM6, SINO, and
C080. SUMl through SUM6 actually calculate the corresponding moduli of TF1
through TF6. SINO and C080 are used as a basis in the previously mentioned
calculations. The sinks M1 through M7 simply act as destinations for the signals from the
sources. Further explanation of the nodes associated with the model can be found in

Table 7.

 

 

 

 

 

 

 

 

 

 

 

Front Suspension Mount Rear Suspension Mount

Figure 25: Diagram of a Two Dimensional Rigid Body Vehicle Model

20 s
-é———NTV OLX -——>|SINI<

  

BDIX——A1031x OSZXI;—92—BDZX TF5 m——)181NK

4
FW1 gown 117:3TF2/19 ESU -

M2

INKU m am E

x \m Mm

HE

 

HE

IZ

 

Has
Figure 26: Bond graph submodel of a Two Dimensional Rigid Body Vehicle Model

 

54

Table 7: Nodes associated with the Rigid Body Vehicle Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ninp m Bri_r_nap_rv_1 Associated Description
Coordinate Subroutine
BDlX port x Suspension connection point 1
BDlZ port 2 Suspension connection point 1
BD2X port x Suspension connection point 2
BDZZ port 2 Suspension connection point 2
FLX port x Load in tow
FLZ gm 2 Load in tow
FWl port x Longitudinal wind effects
FW2 port z Vertical wind effects at connection point 1
FW3 port z Vertical wind effects at connection point 2
II I 0 Vehicle in-plane rotational intertia.
INTVW int 0 Integrates angular velocity and fies anglp.
INTVX int x Integrates velocity and gives position
INTVZ int 2 Integrates velocity and gives position
IX I x Vehicle translational inertia
ll 1 z Vehicle translational inertia
SEZ SE 2 Weight of vehicle
TF1-TF7 TF Transmits effects of forces to the vehicle's
rotational intertia

 

 

A. 1.1 Ports and Parameters

 

 

The ports and parameters associated with this model are listed in Tables 8 and 9.

Ports associated with this sub-model allow an interface with suspension forces, wind

forces, and forces resulting from a load in tow. Parameters associated with this model

determine the location of the forces interfacing with sub-model, vehicle mass, and vehicle

inertia.

55

Table 8: Ports associated with the 2D Rigid Body Vehicle Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

jig; Location Units Description
BDlZ Bond 91 N Vertical force at suspension point 1
m/s Vertical velocity of suspension point 1
BDZZ Bond 96 N Vertical force at suspension point 2
m/s Vertical velocity of suspension point 2
BDlX Bond 90 N Longitudinal force at suspension point 1
m/s Longitudinal velocity of susp. point I
BD2X Bond 92 N Longitudinal force at suspension point 2
m/s Longitudinal velocity of susp. point 2
PW] Bond 95 N Longitudinal Wind Forces
m/s Velocig of wind relative to vehicle
FW2 Bond 93 N Vertical wind force (lift) at susp. point 1.
m/s Velocity of lift force
FW3 Bond 94 N Vertical wind force (lift) at susp. point 2.
m/s Velocity of lift force
FLX Bond 20 N Longitudinal load in tow force
m/s Relative velocity of load in tow
FLZ Bond 22 N Vertical load in tow force
m/s Relative velocity of load in tow
Table 9: Parameters associated with the 2D Rigid Body Vehicle Model
Parameter Node Units T ical Ran e Description
MBODY IX, IZ, kg 1400 Vehicle mass
SEZ
L1 TF3, TF4 m 1.26 Longitudinal distance from CC. to front
suspension mount
L2 TF1, TF2 m 1.45 Longitudinal distance from CC. to rear
suspension mount
hl TF3, TF4 m .25 Vertical distance fi'orn C.G. to fiont
suspension mount
h2 TF1, TF2 m .25 Vertical distance from C.G. to rear
suspension mount
j SEZ m/s2 -9.8l Acceleration due WWW
JBODY IJ N-rn-s2 1862 In plane inertia
h3 TF5, TF6 m .25 Vertical distance from C.G. to point of
load in tow connection
h4 TF7 m .25 Vertical distance from CC. to location of
wind drag point lead
L3 TF5, TF6 m 2.00 Longitudinal Distance from CC. to point
of load in tow connection

56

A. 1.2 Verification Results

Verification of the Two Dimensional Rigid Body Vehicle Model must be a
multistep process since there are multiple ports communicating with the model. Logically,
the model must first be analyzed for settling performance with only the vehicle's weight
and vertical suspension inputs. After this test has been completed then the performance
of each of the other ports can be evaluated. The evaluation procedure will be

implemented as follows:

Test 1 - Model settles properly under its own weight
Test 2 - Model response to a longitudinal impulse
Test 3 - Model response to wind affects

Test 4 - Effect of a load in tow on the model

Evaluation of Test 1 requires that suspension components are added to the ports
BDlZ and BD2Z. Characteristic suspension parameters were used from a mid-size sedan.
Namely, the spring stiffness was 13.8 kN/m and damping was 3.22thsec/m. The
remaining parameters used are listed in Table 9. The evaluation consisted of starting with
the vehicle at rest and level and the suspension undeforrned. The vehicle was then allowed
to settle upon the suspension and reach equilibrium. The results are shown in Figure 16.
Upon analytical evaluation of the suspension deflection we first assume that the moment
arms of the suspension are unafl‘ected by angular displacement. While slight error is
introduced, this serves as a good approximation for model verification. Inspection of
Figure 16 indicates that the deflection of front suspension is 0.267m, the deflection of the
rear suspension is 0.23 1m, and that the angular displacement of the center of gravity is
0.0133 radians. Our theoretical results are 0.266m, 0231111, and 0.129 radians
correspondingly. Given that nonlinearities were neglected we can assume that the results

match and that the model has passed this test.

57

Initial ZD Rigid Body Settling Characteristics

0.012 /

Angular
Displacement (rad)

 

 

 

 

 

 

E —‘9— 051
_‘:c—’ —e— 053
:2
Ea, z
00
>ro
a
g c e e e c o
c c c c c <>
j I I r I l
o 1 2 3 4 5

Time (sec)

Figure 27: Test 1 - Initial Settling of the 2D Rigid Body Model

Test #2 requires that we now add a longitudinal step force to the vehicle and
observe the vehicle's response. Two separate scenarios will be observed where a step
force is first applied to the rear suspension point and then where the step is applied to the
front suspension point. Here the qualitative behavior of the vehicle's pitch (in plane
rotation), and corresponding deflection of the suspension will be observed. In either
scenario of applying a step force we expect that the angular displacement will decrease,
the rear suspension will compress further, and the fiont suspension will be extended.
Further, we expect that since the force is a step force the vehicle will reach a new steady
state orientation. Figure 28 illustrates the vehicle's response to a step force applied to the
rear suspension. The vehicle responded as we expected and can assume that this portion
of Test #2 is satisfied. Further examination of Figure 29 where a step is applied to front
suspension reveals similar results to those in Figure 28. This is expected since both forces

were applied at an equal distance below the center of gravity.

Angular

Vertical

58

Response to a Longitudinal Step Force

 

 

 

 

[1 012 :h at the Rear Suspension
6‘ A < --— Theta
9 q
S 0003— \ ___________________________________
E a
8 4
9 .1
3 0.004—
C) ..

0.000—

-020—
A l —A— 051 —e— 054 —— 2
é .
5-0.24—\9 0 e e o o e 0 e o
E
8 d g '9' 3 3, £- 9. a. ._. 4;,
El “/4:—
O -0 28—

.

 

1 2

T
3 4 5
Time (sec)

Figure 28: Test 2 - 2D Rigid body model subjected to a 1000N step force
at rear suspension.

Angular
Displacement (rad)

Vertical
Displacement (m)

 

 

 

 

0.014 e Response to a Longitudinal Step Force

1 at the Front Suspension

\ - — - Theta
0.012 ~ \
1

i
0.010 a
0 .000 — \ _ J , .. ——————————————————————————————
0 .005 -
-020 —

. —A— 051 —e— 054 — z
-0 24 :\e o o o o 0 e 0 e o

1H”-
029 —

i

l l

—1
--4
—‘

Time (sec)

Figure 29: Test 2 - 2D Rigid Body model subjected to a 1000N step force
at front suspension.

59

Test #3 evaluates the model's response to applied wind loads. Since there are
multiple sources of wind load, longitudinal wind load will be evaluated first. For the sake
of simplicity the longitudinal wind load will be modeled simply as linear viscous damping.
A step load will be applied to the vehicle as in Test #2 and the model will reach a steady
state velocity proportional to the damping coefiicient and applied load. In this case a
375N step load will be applied to the vehicle while the wind drag coefficient of viscous
damping will be 15N-sec/m. The resulting final speed is expected to be 25m/sec. Figure
30 illustrates that the vehicle speed exponentially approaches 25m/sec and we can

conclude the model has passed this test.

Test #3 also requires that the effects of wind lift be tested. Using a similar test to
the previous one, the vehicle's orientation will be monitored with and without wind effects
included. If the wind afl‘ects are considered correctly the vehicle should lifi slightly when
wind afl‘ects are included and the force transmitted to the ground should decrease. For
simplicity the lift affects will be considered to be linearly proportional to vehicle velocity.
The lift coefficient for the front suspension will be considered to be SN-sec/m and the rear
suspension lifi coefficient will be 13N-sec/m. The same 375N step load will be applied
and the longitudinal viscous drag coefficient will again be 15N-sec/m. As it can be seen
from Figure 31 the vertical forces at the front and rear suspensions both decrease due to
wind lift. This is as expected and it is concluded that the wind lift interface is functioning

correctly.

60

Vehicle Velocity with Longitudinal Wind Drag

 

 

30 —
20 d
i=1
g .............................................................................
(:5
2
10 —
0 r” 1
U 100 200
Time (sec)
Figure 30: Test 3 - Vehicle Response to longitudinal wind drag.
Vertical Forces due to a
c 2 3680 Longitudinal Step Force
0
'2 o
as
3 [L
w a
'5 e
a g

 

Rear Suspension
Verfical Force (N)

 

 

 

 

l
0 100 200
Time (see)

Figure 31: Test 2 - Vertical forces due to a longitudinal step force with and without Wind
Lifi Considered

61

Test #4 requires that the efi‘ect of a load in tow be examined. In order to evaluate
this affect the vertical and longitudinal forces due to the load must be evaluated separately.
The vertical forces will be investigated first by applying the load to the vehicle and
observing the settling performance. Figure 32 shows the settling of the vehicle with and
without the load in tow. As expected, the load in tow results in greater compression of

the suspension and the vehicle pitch decreases.

Vehlcle Settllng Response with 3 Load In Tow

0.016

0.01 2

 

0.000

0.004

(radians)
Vehicle Pitch

0.000

 

Front Suspension
Vertical Force (N)

— With load
— - - Without Load

 

Rear Suspensnon
Vertical Force (N)

 

 

 

Y I I I l U U I f r Y I U V I V I I
0.0 0.5 1.0 1.5 2.0
Time (sec)

Figure 32: Test 4 - Vehicle Response due to application of load in tow

62

A.2 An Unpropelled Two Dimensional Rigid Ring Tire Model

A.2.1 Model Description and Assumptions

The tire model presented here is a much simpler version of the Propelled Two
Dimensional Rigid Ring Tire Model based upon work by Zegelaar (Zegelaar, 1993). In
this case though the tire model has been limited to only capture vertical translational
dynamics of the rim and tire. The model does not include the ability to transmit torque
from an input shaft to the ground and hence does not need to consider longitudinal forces.
The model is purely concerned with normal forces between the tire and ground and

between the rim and axle.

Much like the Propelled Rigid Ring Tire Model, this unpropelled tire model
captures the zeroth and first modes of the tire and rim. The resulting limitation is that the
model is only accurate up to 90 Hz. Further, since the tire-ground interface is modeled as
a non-linear stiff spring there are computational limitations which require that solution
strategies are capable of capturing frequencies up to SOOHz. A small amount of damping
has also been added to this model which damps out otherwise present harmonic steady
state motion and results in a steady state equilibrium. As mentioned previously, this is

both more realistic and computationally desirable.

63

Ground Contact Force

 

1‘00 ,_ a _ _ _. ._
0.80 l
0.60 i»
0.40 4»

0.20 «r

 

Contact Form
D
8

040 «» Kgr

 

 

-1.00 -080 -0.60 -0 40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00
Relative Position of Rigger! Ring to Ground

Figure 33: Non-Linear Contact Force Function

A diagram of the unpropelled tire model is presented in Figure 34. The tire model
consists of a rigid ring which represents the tire carcass, the rim, and a linear spring in
parallel with a damper which represents the tire side wall. As mentioned above, the
damper is not an original part of Zegelaar's model, but has been added to damp out
oscillations and make the model more numerically appealing. The bond graph of the
model is shown in Figure 35. Table 10 describes the pertinent nodes of the model and
their type, units, and associated subroutines. Note that the following subroutines are

used in this model and can be found in Appendix A:

ZZSU01 - Multiplies several model parameters and a model variable
ZZSU02 - Non-linear contact stiffness. See Figure 33.

64

 

Rigid Ring Tire

Figure 34: Diagram of modified Unpropelled Rigid Ring Tire Model

SER SEB

25 28

92 J 19 17 J 15 9
BDZ—AiZRF—QOTZ—Aizal—QOEBRASFGRND

 

 

 

 

20 23 16 28
-l / -l I
IRZ T IBZ 10R
24 27
18 14
1’
(X32 FUBZ CXBFR FK3Z

Figure 35: Bond graph submodel of an Unpropelled Rigid Ring Tire Model

65
Table 10: Unpropelled Rigid Rim Tire Model node descriptions

 

 

 

 

 

 

 

 

 

 

 

Egg 1mg Pym Associated Description

Coordinate Subroutine
BDZ port 2 Rim connection to axle
CBZ C 2 Tire stiffness
CGR C 2 ZZSU02 Ground contact stifl'ness
IBZ I 2 Tire Inertia
IRZ I z Rim Inertia
RBZ R z Tire/Numerical Damflg
RGZ R 2 Ground contact datum
SEB SE 2 ZZSU01 Gravitational force of tire
SER SE 2 ZZSU01 Gravitational force of rim
SF GRND port 2 Road profile

 

 

 

 

A.2.2 Ports and Parameters

There are two ports associated with this model. As indicated in Figure 35 and
Table 11 the two ports are BDZ and SFGRND. Table 11 describes the ports, their type,

and associated units.

The parameters associated with this model are described in Table 12. There the
parameter name, a description, associated node, units, and a typical range of usage are
listed. The parameters describe the linear characteristics of a smaller 70% tire. (Zegelaar,

1933).

Table 11: Port description for an Unpropelled Rigid Ring Tire Model

 

 

 

 

Pg; Location Units Description
BDZ Bond 92 N Vertical force to rim
m/s Vertical velocity of rim
SFGRND Bond 93 N Force transmitted to ground
m/s Velocity of ground profile

 

 

 

66
Table 12: Parameter description for an Unpropelled Rigid Rim Tire Model

 

 

 

 

 

 

 

 

 

Parameter Node Units m m
__ ..___ ,___

KB CBX, N/m 1.47 x 10 Translational stiflhess of tire
CBZ

MRIM IRZ kg 5.0 Mass of rim

MTIRE IBZ kg 4.52 Mass of tire

RTIRE RTIRE m .288 Undeforrned tire radius

RNUM RGZ 100 Tire/Numerical Damping

G SER, m/s2 9.81 Acceleration due to gravity
SEB

KGR CGR N/m 7.35 x 107 Contact stiffness with ground

RG RGZ 1 x 104 Contact damping with ground

 

 

 

 

A.2.3 Verification Results

The verification of this model is rather simple since only the vertical steady state
behavior must be checked. This will be accomplished by applying a 2000N load to the
port BDZ with the model at initially at rest in a relaxed position. The net load applied to
the tire will 2050N and since the tire stiffness is 1.47 MN/m it is anticipated that the net
deflection of the tire will be 0. 139cm. Figure 36 illustrates the damped dynamic response

of the load application and verifies the steady state deflection of 0.139cm.

67

Tire Response to lnitid Load Application

 

 

0.000 - ---------------------------
5 -0.001 -------------------------
C .
lllll . . ......
"3 ill
., 0.002 1 — .........................
D
-00003 I I I I T I —l I I I I
0.0 0.2 0.4 0.6 0.8 1 .0

Time (see)

Figure 36: Verification Test - Damped response of vertical tire deflection to initial load
application.

A.3 A Simple Open Differential

A.3.1 Model Description and Assumptions

A difl‘erential is used in automotive applications to distribute power from the
transmission or drive shaft between the driven axles. Further, a difl‘erential compensates
for speed variations between the driven wheels when a vehicle follows a curve. In this
case, the difi‘erential is described as open because the torque transmitted to either axle is
always the same while the speed of each axle may vary. The result is that if one axle loses

traction the other axle can only provide as much traction as the axle which is slipping.

This is a simple model of a differential typically found in automotive applications.
Gear backlash, viscous damping, contact stifi‘ness, and inertial efl‘ects have been neglected
and only the essential characteristics of the gear ratios are considered. Figure 37 below
indicates that this model captures two separate stages of gear reduction between the Drive

Pinion and Ring Gear and between the Differential Pinion gears and the Side Gears. In

68

Figure 38 this is represented as two separate transformers TFDIFF and TFPIN

correspondingly. These transformers are also indicated in Figure 37.

 

 

 

 

 

 

 

 

 

 

 

Drive a)
. . s
P'"'°" Np) TFDIFF
/ Ring Gear and
,.'-~ , Differential Case
Differential '
Pinions
031:
(02 (01
—\ \
l l R l l
L/ Wr— 1!
Side Gears '

TFPIN

Figure 37: Diagram of a typical open face differential

 

 

 

 

 

 

1W1 \{W1
7.
4
1 2 3
SFSHFTlr \TFDIFF: STFPIN} 3 OPiN
5
I 7
1W2} \w2

Figure 38: Bond graph submodel of a Simple Open Differential

69

A32 Ports and Parameters

The ports of the Simple Open Face Differential are indicated in Figure 38 by the
nodes SHFTIN, W1, and W2. Each of these nodes are connected to bonds and therefore
each carry two pieces of information: effort and flow. Since the ports are all connected to
rotating shafts effort corresponds to torque and flow corresponds to rotational velocity.

Table 13 below describes the port information.

Table 13: Port description for a Simple Open Differential

 

 

 

 

 

 

Port Location W Description
SHFTIN Bond 1 N-m Input shaft torque
rad/sec Input shafi angular velocity
W1 Bond 6 N-m Output shalt #1 torque
rad/sec Output shaft #1 anjular velocity
W2 Bond 7 N-m Output shalt #2 torque
rad/sec Output shafl #2 angular velocity

 

 

Parameters associated with this model correspond to the gear ratios between the
Drive Pinion and Ring Gear (node TFDIFF) and between the Differential Pinion gears and
the Side Gears (node TFPIN). Table 14 below describes the parameters, their associated

nodes, units, and recommended range of usage.

Table 14: Parameter Description for a Simple Open Differential

 

 

 

 

Parameter Node Units mall—ans: m
———u ——_ .
NDIFF TFDIFF -none- 2.73 to 4.11 Drive Pinion and Ring Gear reduction
ratio.
NPIN TFPIN -n0ne- 2 Two times reduction ratio of differential
pinion gear teeth to side gear teeth.

 

 

 

 

70

A33 Verification Results

Since this is a relatively simple model its verification is also correspondingly
simple. That is to say that only a few tests are required and those tests are largely

qualitative in nature. Several aspects of the Simple Open Differential need to be tested:

Test #1 - Proper input-output shaft speed reduction
Test #2 - Appropriate behavior during traction loss to a driven axle
Test #3 - Equal torque provided to each output axle

In all of the validate tests the following parameters were used: NDIFF=4,
NPIN=2. The objective of Test #1 was to demonstrate that the speed reduction of the
difl‘erential was appropriate. In the case of the given parameters, it was expected that the
reduction of input shafi speed to output shaft speed was 4:1. The results of Test #1 are

shown in Figure 39 below and indicate that this test is satisfied.

The objective of Test #2 was to evaluate the difl‘erential's performance when
traction was lost to one axle. In this case the traction of W2 was ramped down from fiill
traction to zero traction. It was anticipated that the effect of the traction loss would be
two-fold. First, it was expected that as the traction of the axle W2 was decreased the
torque applied to W1 would also decrease equivalently. Since the angular velocity of W1
has been set to be proportional to its supplied torque it was expected that as its torque
decreased so would its speed. Thus, as the traction of W2 decreased it was expected that
the speed of W2 would increase. Figure 40 confirms the anticipated variations in output
shaft velocity and Figure 41 confirms the anticipated variations in torque. Further, Figure
41 also resolves the verification of Test 3 because it indicates that the torque applied to

each output shafi is equal.

Angular Velocity (radiate)

71

 

 

 

 

 

input and oinput sit-n Velocities
5‘00 ......................... . .......... . ..... . ..............................................................................
4.50 l s
4.00 e 4f 4 e e e e e e e,
3.50 «i

g 3.00 1*-

E

g 2.50

>

5

a zoo ..

2"
1.50 ..
1.00 I
0.50 ..
0'00 i i i i t t 1 ¢ ¢ 1‘

0.00 0.10 0.20 0 30 040 0.50 0.60 0.70 0.80 0.90 1.00
The (one)

 

[+ SHFTIN +W1 +w2|

Figure 39: Simple Open Differential Verification Results: Test 1, the
comparison of input and output shafi speeds.

 

hputmeutpmMVdocfllu

 

 

4.50 «»

3.50 ‘-

9’
8
1

N
8
1

N
8
1

ii

1.00!

0.50 «-

 

 

0.00 i . 1 i 1 1 . 1 4'
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.00 0.90 1.00
Tine (one)
|+SHFTIN +w1 +wfl

Figure 40: Simple Open Differential Verification Results: Test 2,
the loss of traction to W2.

 

 

Angular Velocity (rodlooc)

72

Input and Output Shit Torque:

 

jm I ...................................................... . ................................................................. l
0.90 4

0.80 <>

.0
\l
o

p
8

'8

.o
8

o
8

0.20 «>

0.10 ->

 

0.00 1 1 1 1 1 1 . 1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Tlrno(ooc)
[+SHFTIN +W1 +W2]

Figure 41: Simple Open Differential Verification Results: Tests 2 and 3,
torque applied to each axle.

 

 

73

A4 A Lumped Mass Manual Transmission with Clutch

A.4.1 Model Description and Assumptions

A generic manual transmission model is being presented which assumes that the
stiffness, damping, and inertial effects of a manual transmission can be treated in a lumped
fashion. Equivalent stiffness, damping, and inertia are used to capture the rotational
dynamics of the transmission. The clutch is treated as having stribeck damping and

torsional compliance.

The model is a combination of work completed by Hrovat (Hrovat, 1991) and
Runde (Runde, 1986) complemented with work by the author of this thesis. Hrovat
contributed the framework for a lumped mass transmission with torsional compliance due
to the clutch. Runde's work details the characteristics of clutch during engagement where
slip may or may not be present. The author of this thesis extended Hrovat's assumption of
lumped mass to equivalent damping and torsional rigidity. Runde's work states that the
causality of a clutch switches during slip and stick conditions. His work was modified
slightly to include viscous damping such that the causality of the clutch is unchanged
during slip and stick conditions. Figure 42 illustrates the viscous effects near zero clutch
velocity and the stribeck damping which dominates as the magnitude of slip velocity is
increased. Note that the coefficient of viscous damping, be], is sufficiently large to

minimize slip velocity and simulate stick conditions.

A diagram of the model is shown in Figure 43 and the system bond graph is shown
in Figure 44. Note that contact stifiiiess springs are shown on the pressure plate of the
clutch. These springs allow the clutch pad and pressure plate to establish a normal force

which balances the forces generated by the clutch springs. The force Fe] is the clutch

74

disengaging force which the driver applies to disengage the clutch. The reduction ratio of
the transmission is facilitated via the gear set and the node RATIO in Figure 44 dictates
what the actual reduction ratio is based upon driver input. Throughout the model several
subroutines have been used to model efl‘ects where the ENPORT fiinction library was

inadequate. The following subroutines have been used in this model:

ZZSU02 - Establishes the normal force in the clutch as a result of the contact stiffness.

ZZSU04 - Determines the torque transmitted by the clutch.
ZZSUOS - Selects a transmission reduction ratio based upon driver input.

Table 15 highlights the nodes in the bond graph model (Figure 44) and describes their

type, associated subroutine, and firnction.

Clutch Torque Transmission

1.0—
1 K

« bcl

9:)
m
l

P
01
l

 

Transmitted Torque
P
a.
. r

5::
M
l

 

CD
CD

I I I v . v I V
.2 _1 1 2
.112 Clutch Slip Veloclty

-0.4
-0.6

-0.8

Figure 42: Clutch stick-slip properties.

75
Torsional Clutch Spring

 

 

 

 

 

 

 

 

 

 

Clutch Pressure Springs Equivalent Damping, Inertia,
. . and Stiffness
Contact Stiffness Springs
Pressure Plate
\_ _ Gear Set
Fe!
11% L 1W . . -
Tin Kcl * --
(Din Fc' H I 03%
711 a Jeq M
Kcont de ‘1 __J Keq Tout

 

 

 

 

 

\ beq
Backing Plate

Clutch Pad

Clutch Disengagement Forces

Figure 43: Diagram of a Lumped Mass Manual Transmission with Clutch

 

 

 

, SFN 11 x, l 92
RCL \ och 1 01.x“ FOL
2 10 1 3 12
TF COONT CPRES ICLX

TRINIflOCLPD A‘l1cr.r=Dl—1§001_44|'rr=<hsii=ii|1SHi--r|i OSHFrF—E—ATRNOUT
I

 

 

7 9
16 3 RATIO 8
ICL COL [RATIO] REG EC 0
A
81
R SE

 

Figure 44: Bond graph of a Lumped Mass Manual Transmission

76

Table 15: Lumped mass manual transmission model node description.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Em 111E Associated Description

Subroutine
CCL C Clutch torsional rigidity
CCONT C zzsu02 Clutch contact stiffness
CEQ C Equivalent stiffness
CPRES C Clutch pressure springs
FCL port Clutch disengagement force
GR_SEL port Driver gear selection
ICL 1 Clutch rotational inertia
ICLX I Clutch translational inertia
IEQ I Equivalent inertia
RATIO fcn zzsuOS Determines gear ratio from driver input
RCL R zzsu04 Clutch force transmission
REQ R Equivalent damping
TFCL TF Relates clutch forces to torques
TFGR TF Modulus of the transmission
TRIN port Input from engine
TRNOUT port Output to drive shaft

A.4.2,Ports and Parameters

In the following tables the ports and parameters used in the lumped mass

transmission model are described. Table 27 describes the ports and Table 17 describes the

parameters. The source of parameters values used in this model are Hrovat (Hrovat,

1991), Runde (Runde, 1986), and derived by the author of this thesis.

 

 

77

 

 

 

 

 

Table 16: Port description for a lumped mass manual transmission.
Pflt Location Units Description
FCL BOND 92 N Force applied to clutch
m/s Velociy of clutch
GR SEL signal 81 none Driver gear selection
TRIN BOND 90 N-m Motor torque at input
rad/sec Motor angular velocity
TROUT BOND 91 N-m Output shalt torque
rad/sec Output shalt angular velocity

 

 

 

Table 17: Parameter description for a lumped mass manual transmission.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Node Units Typical Range Description

JCL ICL kg-m2 0.0296 Rotational inertia of the clutch pad/backing
plate

MCL ICLX kg 5.93 Mass of clutch pad/backing plate

JEQ IEQ kg-m2 0.0105 Equivalent inertia of the transmission

KEQ CEQ N-m/rad 7000 Equivalent Stifl‘ness of the transmission

REQ REQ N-s 0.0027 Equivalent viscous damping of the
transmission

KCONT CCONT N/m 1x109 Contact stiffness

KCL CCL N-m/rad 678 Torsional stiffness of clutch springs

KCLX CPRES N/m 2.95x10S Stiffness of clutch pressure springg

RCL TFCL m 0.1016 Clutch radius

MUSTAT RCL -n0ne- 0.4 Static coefficient of friction

MUDYN RCL -n0ne- 0.3 Dynamic coefficient of friction

GEAR] RATIO -none- 3 1st gear ratio

GEAR2 RATIO -none- 2 2nd gear ratio

GEAR3 RATIO -none- 1.5 3rd gear ratio

GEAR4 RATIO -none- 1 4th gear ratio

GEARS RATIO -none- 0 5th gear ratio or neutral

GEARRV RATIO -n0ne- -3.5 Reverse gear ratio

BCL RCL N-s/m 1 x 10‘ Viscous clutch damping

 

 

A.4.3 Verification Results

 

 

The lumped mass manual transmission model must pass several tests before it can

be treated as a valid model. The major qualities of the transmission operation which must

78

be tested include the proper speed and torque reduction, clutch, and shifting. These

qualities will be verified in the following tests:

Test 1 - Steady state input/output torque and velocity.
Test 2 - Affect of clutch engagement/disengagement on transmission input/output.
Test 3 - Influence of shifting gears on transmission input-output.

Test #1 requires that the steady state response of the transmission is evaluated. To
facilitate this, an angular velocity step of lrad/sec is applied to the input of the
transmission and the response is observed. Figure 45 illustrates that after an initial
transient the steady state output velocity is 0.33 rad/sec. This is expected since the
reduction ratio during this test is 1/3. It can be concluded that the steady state throughput

characteristics of the transmission are appropriate.

Test # 2 requires that the operation of the clutch is verified. To accomplish this
the transmission will operate at a steady initial condition and then the clutch will be
disengaged and re-engaged. A small amount of viscous damping has be added to the
output of the transmission such that when the clutch is disengaged the output velocity will
decrease. Figure 46 contains the results of the trial. The output velocity does decrease
when the clutch is disengaged and then there is a slight transient as the output velocity
returns to its initial state. Further note that the clutch normal forces also experience a
slight transient alter re-engagement. This is expected since the contact of the clutch and

clutch plate is represented by a stifl‘ spring.

Output Velocity

Input Torque

79
Transmission Response to a Velocity Step Input

 

rad/sec

 

Nm

 

 

'20 1 I I I I I I I I I
0 l 2 3 4
Time (sec)

Figure 45: Test 1 - Steady state input torque and output velocity for a
lumped mass manual transmission.

Transmission Response to Clutch

 

 

1’ ~ Disengagement and Engagement
‘1, input Torque
3 1 a
gA
o E
'- 2
15 a.
o. 0 4"“
E
-1 -d
0.36 -
gg —— Output Velocity
034 -
o 3
—S E
a» g 0 32 a
E n
> 4
0 30 ~
8000.0 —
E I
3 1
“E: 4000.0 -« — - - Applied Clutch Force
: —— Clutch Pad Normal Force
4
0.0 1 , 1 1

 

 

 

3 4 5
Time (sec)

Figure 46: Test 2 - Lumped mass manual transmission response to
clutch engagement and disengagement.

80

Test #3 requires that the shifting characteristics of the transmission are also
evaluated. In this test the transmission is initially operating at steady state conditions. A

brief description of the test cycle follows:

1. Initial steady state operation

2. At 1.0 seconds the input torque is ramped up.

3. At 3.0 seconds the clutch is disengaged, the motor torque is discontinued, and the
transmission shifts gears.

4. From 4.0 to 8.0 seconds the clutch is engaged and the motor torque is ramped up.

5. At 8.0 seconds the clutch is disengaged, the motor torque is discontinued, and the
transmission shifts gears.

6. From 8.0 to 12.0 seconds the clutch is engaged and the motor torque is held at
constant value.

Figure 47 illustrates the response of the transmission during the above test cycle.
There are several key characteristics which determine the proper performance of the
transmission. The first quality is that the output velocity decreases when the clutch is
disengaged. Figure 47 illustrates this during both shift points. The second quality is the
ratio of input speed to output speed. As Figure 47 illustrates, the ratio of input speed to
output speed is directly related to the transmission gear ratio. Initially the input speed is
105.2rad/sec and the output speed is 35.0rad/sec which results in a ratio of 3.00 as
expected. After the transmission shifts to second gear the output speed is 48.6rad/sec
and the input speed is 80.1rad/sec which results in a ratio of 1.97. This ratio is slightly
lower than expected but since the vehicle is accelerating the input and output velocities
will have a slight phase angle. This confirms that as the driver selects new gears, the

transmission changes to the appropriate ratio and the clutch engages and disengages

properly.

81
Transmission Response with Shifting and Clutch Operation

      
  

 

 

 

 

 

 

 

 

 

 

2‘ 12] --- Outpu Speed
:3 3 Input Speed
0.) Q m
> n
a s
3,: 40 __________________________
E
0
some i
8 A
B 5 — Clutch Normal Force
tL 4110.0 -1
. —-- Applied Clutch Force
L.__. _____
r __________
“3’? — Gear Ratio
.9 :‘t I - - - Selected Gear
1
I
i
1 ' I ' I ' I ' I I I I I ' I ' I ' I I I ' I 7—I
0 1 2 3 4 S 6 7 8 9 10 11 12
Time (sec)
Figure 47 : Test 3 - Lumped mass manual transmission response
during shifting and clutch operation.
A.5 Hollow Shaft Models

A.5.l Model Description and Assumptions

The hollow shalt model is used to transmit rotational power between various
submodels in the vehicle model. Since there are difl‘erent causal restrictions between
submodels several shaft models must be developed which are capable of linking these
components. Three different types of shaft models are described here. Each shaft model
is capable of capturing the the zeroth and first modes of a hollow shalt and have identical

characteristics. In all cases the user is required to specify shalt geometry in metric units.

82

Three different types of shaft models will be presented. They will be described by
the causal restrictions on their input and output ports. The first model will be described
as a flow-flow model since it dictates what flows the ports will have based upon the efforts
applied by the outside models. The bond graph of the model can be seen in Figure 48.
The second model will be described as an effort-efi‘ort model since it enforces efl‘ort on its
external ports. The bond graph of this model is shown in Figure 49. The third model is
described as an effort-flow model since it enforces effort on one port and flow on the
other. The bond graph of this model is shown in Figure 50. While each of these shaft

models appears to be different, they are each qualitatively identical.

 

 

 

 

 

 

11_ 12
7 ll"
1 5
90 2 4 91
sew \11A: \0 >118; \SEOUT
3
1’
c

Figure 48: Bond graph of a hollow shaft model with flow-flow external causality.

83
11

 

 

 

T
1
90 4 2 m
SFIN,L \0A \{1A1 \ 0 \{SFOUT
5 3
I i’
01

 

c
Figure 49: Bond graph of a hollow shaft model with effort-effort causality.

 

90 2 91
SEIN 31A}

_ \ SFOUT

 

I
0

Figure 50: Bond graph of a hollow shaft model with effort-flow causality.

A.5.2 Ports and Parameters

84

Table 18 describes the ports and Table 19 describes the parameters for the hollow

shaft models. All three of the models use the same ports and parameters.

 

Table 18: Ports for the hollow shaft model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Port T Units description
INPUT Bond N-m Torque input to shalt
rad/sec Velocity of input to shalt
OUTPUT Bond N Torque input to shaft
m/s Velocity of input to shaft
Table 19: Parameters for the hollow shaft model.
Parameter Node Units Typical Range Description
L C, I 111 Total shalt lepgth
G C Pa 77.9 x 109 Modulus of Rigidity
DI C,I m Inner Diameter
DO C,I in Outer Diameter
rho I win2 76.5 x 103 Material Density

85

A.6 Mean Torque Full Throttle Engine Model

A.6.1 Model Description and Assumptions

The engine model presented is a quasi-static semi-empirical model capable of
predicting the mean value engine torque over a range of operating speeds. The range of
speeds is dependent upon user supplied engine data. The user can choose between two
modes of operation which can be described as being an idle mode and a operating mode.
The idle mode gives the user the opportunity to specify an idle speed and then the engine
will supply sufficient torque to maintain the steady state speed given engine damping.
When the model is in "operating mode" the engine supplies torque as a function of engine
speed. The engine torque is determined by a polynomial equation of up to order 9 and is
determined by the subroutine ZZSU08 shown in appendix B. The model assumes that the
engine inertia and damping can be lumped and treated linearly where the engine torque,
output, and damping are applied to the engine inertia as shown in Figure 51. The bond

graph of the model is shown in Figure 52 and the nodes are explained in Table 20.

Engine data used in this study is based upon published data for a GM 4.3L V6
internal combustion engine (Graham, 1992). A seventh order polynomial was fit through
the torque data and the coefficients were entered as parameters for the model. The
parameter values are shown in the next section and the torque curve is shown with the

verification results.

"9

(1)

7'1

86

/ Damping

 

E T
Engine Inertia ( @———°‘"

7 XI

 

Figure 5]: Diagram of a mean torque firll throttle engine model.

81

 

7" SEENG

RENO

— 90
\{1 ENG I—ATRNOUT

 

 

dz
iENG

Figure 52: Bond graph of a mean torque fiill throttle engine model

 

 

 

 

 

 

Table 20: Node description for a mean torque full throttle engine model.
Ea_m_e Tm Associated Description
Subroutine
IENG I Lumped engine inertia
RENG Lumlied engine dampirg
SEENG SE ZZSU08 Detemiines the applied engine torque as a function
of erggine speed and operatingmode
THRL port Determines engipe mode of operation
TRNOUT port Engine output.

 

 

 

87

A62 Ports and Parameters

The ports and parameters used in this model are shown in Tables 21 and 22

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

respectively.
Table 21: Ports for a mean torque fiill throttle engine model.
Port Tm Units Description
THRL signal Determines operation mode of engine.
0 :> idle mode, 1 :> operation mode.
TRNOUT bond N-m Output torque
rad/sec Output speed.
Table 22: Parameters for a mean torque full throttle model.
Parameter Node Units Typical Range Description

jeng [ENG km 0.1354 Equivalent inertia
reng RENG N-m-s 0.2714 Equivalent damping
X0 SEENG -none- -570.6l65 0th order polynomial term.
X1 SEENG -none- 24.11288 1st order polynomial term.
X2 SEENG -none- -2.749l87e—01 20d order polynomial term.
X3 SEENG -none- 1.6363470-03 3rd order polynomial term.
X4 SEENG ~none- -5.470921e-06 4th order polynomial term.
X5 SEENG -none- 1.03 9598e-08 5th order polynomial term.
X6 SEENG -none- -l.052086e-ll 6th order polynomial term.
X7 SEENG -none- 4.4085870-15 7th order polynomial term.
X8 SEENG -none- 0 8th order polynomial term.
X9 SEENG -none- 0 9th order polynomial term.

 

 

 

 

A.6.3 Verification

The mean torque engine model will be validated for proper torque output and
operating mode performance. Figure 53 shows the mean torque output as a function of
engine speed when the model is in operating mode. The curve shown is a seventh order

polynomial fit through data publish by General Motors (Graham, 1992). The coefficients

88

of the curve are shown in Table 22. The idle mode performance is shown in Figure 54
where the system settles to 100 rad/sec (idle speed) from 200rad/sec. The model satisfies

both of these tests successfiilly.

 

350 _ Engine Torque
- GM 4.3L LT-35
340 —
320 —
’E‘
a -l
0) 300 —
3
g -1
O
'—
200 —
200 a
240 ' r ' T ' 1 ' T C r ' 1
0 100 200 300 400 500 800

 

Engine Speed (radlsec)
Figure 53: Torque as a function of engine speed for a mean torque engine model based
upon a GM 4.3L V6 engine.

89

Idle Mode Performance

200—
180-

160—

Engine Speed
(rad/sec)

140-

120-

 

100-

 

 

Time (sec)

Figure 54: Engine speed during idle mode operation.

A.7 Longitudinal Wind Effects

A.7.1 Model Presentation and Assumptions

Drag on the vehicle resulting from wind is calculated based upon information
presented by Gillespie (Gillespie, 1992). This model only considers wind effects in the
longitudinal direction of the vehicle and is based upon the dynamic pressure of the wind in
conjunction with a drag coefficient. The net forces which the vehicle feels are determined
by the RWIND node in Figure 55 which calculates force as a fimction of relative wind
speed. Varying wind speed and gusts are ported to the model through the SFWIND node
and vehicle speed and forces are ported to the model through the BODY node. The

RWIND node is based upon the following equation,

90
F = 12,,V2 (2)

where R... is the equivalent wind damping and V is the wind speed relative to the vehicle.

 

1 90
RWIND {—4 OWIND\———lBODY
I

2
SFWIND

Figure 55: Bond graph of wind effects model.

A.7.2 Ports and Parameters

The ports associated with the longitudinal wind effects model are shown in Table
23 along with their associated information. The only parameter associated with the wind
effects model is R“, which is based upon work presented by Gillespie (Gillespie, 1992).

The equation for Req is,

R... =1p-A-C. (3)
where p is the density of the air, A is the frontal area of the vehicle, and Cd is the drag
coefficient. In this study R...l = 0.3983 N-s/m. Often the drag coefficient is based upon
experimental results, but typical values for a particular vehicle body style can be used as an

initial value. In this study Cd=0.35 was used as a typical value for an aerodynamic sedan.

 

 

91

Table 23: Ports for the longitudinal wind effects model.

 

 

 

 

fig; Location Units Description
BODY bond 90 N Force exerted on body
m/s Velocity of body
SFWIND bond 2 N Force exerted by additional wind
m/s Absolute velocity of wind

A.7.3 Verification

 

 

The proper functioning of the model will be evaluated by studying the forces

exerted as a function of relative wind speed. Figure 56 shows the wind forces where Req

was set equal to 0.3983 N's/m. It can be seen that at 0 m/s the force is zero Newtons and

that at 50 m/s the force is 996 N which is valid for this model.

1000 -
900 —.:
800 —
700 ;

600 :1

500 -l

iMnd Force (N)

400 —
300 —-1
200 —

100—

 

Wind Force as a Function of Relative Wind Speed.

 

1 ' 1 ' ‘
20 30 40 50
Relative \Nind Speed (rn/s)

Figure 56: Wind force as a function of relative wind speed.

92

Appendix B

FORTRAN Subroutines

The FORTRAN subroutines used by ENPORT are listed below. References to

these subroutines are made throughout Appendix A.

8.1 ZZSU01 - Parameter Multiplication

c...__....
C.._.__
C_._.__

0000000

0

0000

SUBROUTINE ZZSU01(TIME,X,P,Y,STAT)

PROGRAMMING: Mark Minor 3/5/96
DESCRIPTION: Multiplies two parameters
INPUTS: TIME, current time
X, input_variable values
P, parameter values

STATll), =1 iff lST time subrtn is called, =0 else
STAT(2), =1 iff last time subrtn is called, =0 else
STAT(3), =1 iff last time subrtn is called

for this storage interval, =0 else

OUTPUTS: Y, output_variable values
STAT(4) =1, stop the integration, =0 continue
DECLARATIONS:
CHARACTER STR*80
DOUBLE PRECISION TIME, X(20), P(20), Y(20)
INTEGER STAT(10)
EXTERNAL ZZWRIT
P(l) - Pla
P(2) - Plb

C***ZZSU01***************************************~k**********************

C
C____

Description section

IF (STATllO).EQ.1) THEN
STR= ' A function to multiply multiple parameters and vars and'
CALL ZZWRIT(STR)
STR= ' sum it all up.’

93
CALL ZZWRIT (STR)

RETURN

ENDIF
C
C---- Multipling two parameters....
C

Y(1) = P(l)*P(2)

RETURN

END

8.2 ZZSU02 - Ground Contact Force Function

SUBROUTINE ZZSU02(TIME,X,P,Y,STAT)

C---— PROGRAMMING: Mark Minor 2/26/96
C——-- DESCRIPTION: A contact function which only establishes a
tranmitted force when contact is made
C
C--—— INPUTS: TIME, current time
C X, input_variable values
C P, parameter values
C STAT(1), =1 iff lST time subrtn is called, =0 else
C STATlZ), =1 iff last time subrtn is called, =0 else
C STATI3), =1 iff last time subrtn is called
C for this storage interval, =0 else
C
C———— OUTPUTS: Y, output_variable values
C STAT(4) =1, stop the integration, =0 continue
C
C---- DECLARATIONS:
CHARACTER STR*8O
DOUBLE PRECISION TIME, X(20), P(20), Y(20)
INTEGER STATllO)
C
EXTERNAL ZZWRIT
c P(l) - Contact Stiffness
C P(2) - P(2)>O : Compression, P(2)<0 : Tension forces only
C X(l) - Displacement of spring from Free Length
C Y(l) - Contact force
C

C***ZZSU02**************************************************************

C
C—--— Description section
IF (STAT(10).EQ.1) THEN
STR= ' A function which determines whether forces are'
CALL ZZWRIT(STR)
STR= ' transmitted. P(l) is the contact stiffness'
CALL ZZWRIT(STR)
STR= ' P(2) >< 0 => compr/tension contact forces only.‘

8.3 ZZSU03 - Multi Parameter and Variable Multiplication

C____
C____

C
C__-_

0000000

C____
C
C
C_-__

94

CALL ZZWRIT(STR)
RETURN
ENDIF

Contact stiffness subroutine

if (p(2) .gt. 0) then
if (X(l) .ge. 0) then
Y(l) = abs(p(l))*x(l)
else
yll) = 0
endif
else
if (X(l) .le. 0) then
y(1) = abs(p(l))*x(l)
else
y(1) = 0
endif
endif
RETURN
END

SUBROUTINE ZZSU03(TIME,X,P,Y,STAT)

PROGRAMMING: Mark Minor 3/5/96

DESCRIPTION: Multiplies two parameters and a variable for 2
channels and sums it all to give the output result

INPUTS: TIME, current time
X, input_variable values
P, parameter values

STAT(l), =1 iff lST time subrtn is called,

STAT(Z), =1 iff last time subrtn is called,

STAT(3), =1 iff last time subrtn is called
for this storage interval,

OUTPUTS: Y, output_variable values
STAT(4) =1, stop the integration,

DECLARATIONS:

CHARACTER STR*80

DOUBLE PRECISION TIME, X(20), P(20),

INTEGER STAT(lO)

EXTERNAL ZZWRIT

P(l) - Pla

=0 continue

000000

95

P(2) - Plb
X(l) - X1
P(3) - P2a
P(4) - P2b
X(2) - X2

C***ZZSU01***********~k***************iv**i'it******************************

C
C____

C
C__.__.
C

Description section
IF (STAT(lO).EQ.1) THEN
STR= ' A function to multiply multiple parameters and vars and'
CALL ZZWRIT(STR)
STR= ' sum it all up.‘
CALL ZZWRIT(STR)
RETURN
ENDIF

Multipling two parameters....
Y(l) = P(l)*P(2)*X(1)+P(3)*P(4)*X(2)

RETURN
END

B.4 ZZSU04 - Coulomb Damping Function

C____

SUBROUTINE ZZSUO4(TIME,X,P,Y,STAT)
PROGRAMMING: Mark Minor 3/17/96
DESCRIPTION: Coulomb Damping Function

INPUTS: TIME, current time
X, input_variable values
P, parameter values

STAT(l), =1 iff lST time subrtn is called, =0 else
STAT(Z), =1 iff last time subrtn is called, =0 else
STAT(3), =1 iff last time subrtn is called

for this storage interval, =0 else

OUTPUTS: Y, output_variable values

STAT(4) =1, stop the integration, =0 continue
DECLARATIONS:
CHARACTER STR*80

DOUBLE PRECISION TIME, X(20), P(20), Y(20),fd,fd_break,vr,fd_max
DOUBLE PRECISION fsp, fds, fdf

INTEGER STAT(lO)
EXTERNAL ZZWRIT
P(l) Static Coefficient of Frictin

P(2) Dynamic Coefficient of Friction

96

C P(3) Viscous damping which allows evaluation of torque when
vr<>0
C X(l) Normal Force
C X(2) Velocity of slip
C Y(l) Damping Force
C
C***ZZSUO4***************************************~k**********************
C
C—--- Description section
IF (STAT(lO).EQ.1) THEN
STR= ' Coulomb Damping Function'
CALL ZZWRIT(STR)
RETURN
ENDIF
C
C---- Stribeck Coulomb Damping Subroutine
C
vr = X(2)
fd_break = P(1)*x(1)
fd_max = P(2)*x(l)
fsp = X(l)
fds = p(3) * vr ! Forces due to slip

c Forces due to stick friction
fdf=dsign(l,vr)*(fd_max+(fd_break-fd_max)*exp(—abs(vr)))
if (abs(fds) .le. abs(fdf)) then

fd = fds
else

fd = fdf
end if
Y(l) = fd
RETURN
END

B.5 ZZSU05 - Parameter Selection Based Upon Signal

SUBROUTINE ZZSU05(TIME,X,P,Y,STAT)

C-—-- PROGRAMMING: Mark Minor

C—--- DESCRIPTION: Gear selection subroutine....

C—--- INPUTS: TIME, current time

C X, input_variable values

C P, parameter values

C STAT(l), =1 iff lST time subrtn is called, =0 else
C STAT(Z), =1 iff last time subrtn is called, =0 else
C STAT(3), =1 iff last time subrtn is called

C for this storage interval, =0 else

C

C-—-- OUTPUTS: Y, output_variable values

97

C STAT(4) =1, stop the integration, =0 continue
C
C———— DECLARATIONS:
CHARACTER STR*80
DOUBLE PRECISION TIME, X(20), P(20), Y(20)
INTEGER STAT(lO)
C
EXTERNAL ZZWRIT
C P(l) First Gear Ratio
C P(2) Second Gear Ratio
C P(3) Third Gear ratio
C P(4) Fourth Gear Ratio
C P(5) Fifth Gear Ratio
C P(6) Reverse Gear Ratio (should be less than zero)
C X(l) Driver Gear Selection
C Y(l) Output gear ratio
C

C***ZZSU05**************************************************************

C

C—--- Description section
IF (STAT(lO).EQ.1) THEN
STR= ' Subroutine to select transmission gear ratio.‘
CALL ZZWRIT(STR)
RETURN
ENDIF
C
C-——— The Subroutine...
C
if (X(l) .le. 1) then
Y(l) = P(l)
else if (X(l) .le. 2) then
Y(l) = P(2)
else if (X(l) .le. 3) then
Y(1) = P(3)
else if (X(l) .le. 4) then
Y(l) = P(4)
else if (X(l) .le. 5) then
Y(l) = P(5)
else if (X(l) .le. 6) then
Y(l) = P(6)
end if
RETURN

END

98
8.6 ZZSU08 - Engine Torque-Speed Curve

SUBROUTINE ZZSUO8(TIME,X,P,Y,STAT)

C
C---— PROGRAMMING: Mark Minor, 3/24/96
C
C---- DESCRIPTION: Determines engine torque as a function of speed
C
C—--— INPUTS: TIME, current time
C X, input_variable values
C P, parameter values
C STAT(l), =1 iff lST time subrtn is called, =0 else
C STAT(Z), =1 iff last time subrtn is called, =0 else
C STAT(3), =1 iff last time subrtn is called
C for this storage interval, =0 else
C
C---- OUTPUTS: Y, output_variable values
C STAT(4) =1, stop the integration, =0 continue
C
C---- DECLARATIONS:
CHARACTER STR*80
DOUBLE PRECISION TIME, X(20), P(20), Y(20), W
INTEGER STAT(10)
C
EXTERNAL ZZWRIT
c X(l) - Engine speed, rad/sec
C X(2) - 1 => engaged, 0=>disengaged
c Y(l) — Engine torque, N-m
c P(1) - X“0 coeff
c P(2) - X“2 coeff
c P(3) - X“3 coeff
c P(4) - X“4 coeff
c P(5) - X‘s coeff
c P(6) — XA6 coeff
c P(7) — X“? coeff
c P(8) - X“8 coeff
c P(9) - X“9 coeff
c P(10) - offset coeff
C P(ll) - idle speed
C P(12) - equivalent motor damping
C

C***ZZSU08**************************************************************

C
C-——— Description section
IF (STAT(lO).EQ.1) THEN
STR= ' Engine subroutine'
CALL ZZWRIT(STR)
RETURN
ENDIF
C
C-—-- Engine subroutine

99

if (X(l) .lt. 85) then

W = 85

elseIF

(X(l) .GT. 550) THEN

W = 550

ELSE

W = X(l)

endIf

if (X(2) .gt. 0) then

Y(l)

= W*P(1)+P(2)*W**2+P(3)*W**3+P(4)*W**4+P(10)+

/ P(5)*W**5+P(6)*W**6+P(7)*W**7+P(8)*W**8+P(9)*W**9

Y(l)
else

Y(l)
end if

RETURN
END

= Y(l) * 0.70

= P(ll)*P(12)

8.7 ZZSU23 - Tire Slip Function

SUBROUTINE ZZSU23(TIME,X,P,Y,STAT)

C ---- PROGRAMMING:

C ver date who what

c 1.0 9/25/95 Minor Initial creation for testing.

c 1.1 10/9/95 Minor Modify variable notation.

C 2.1 10/14/95 Minor Adapted original single power port
model

C to two-power port version.

C 2.2 10/14/95 Minor Added ability to assign Yss and
ka

c Removed assignment of shape
function.

C

C---- DESCRIPTION: Two power port tire traction function.

C

C--—— INPUTS: X(l) Traction velocity

000000000000

X(2) Wheel velocity
X(3) Normal Force (FNORM)

P(l) Asymptotic FTR/FNORM typical for high slip.
P(2) Peak value of FTR/FNORM

P(3) Slip where peak occurs.

P(4) SLOPE AT ORIGIN (DEGREES)

STAT(l), =1 iff lST time subrtn is called, =0 else
STAT(Z), =1 iff last time subrtn is called, =0 else
STAT(3), =1 iff last time subrtn is called

for this storage interval, =0 else

100

C STAT(4) =1, stop the integration, =0 continue
C
C—--— OUTPUTS: Y(l) Traction Force (FTR)
C Y(2) % SLIP
C
C---- DECLARATIONS:
CHARACTER STR*8O
DOUBLE PRECISION TIME, X(20), P(20), Y(20), E, SLIP, YSS, YPK

DOUBLE PRECISION

THETA, VTR, FNORM, FTR,PI

INTEGER STAT(lO)
C
EXTERNAL ZZWRIT
YSS = P(l)
YPK = P(2)
XPK = P(3)
THETA = P(4)
VTR = X(l)
VWHL = X(2)
FNORM = X(3)
PI = 3.1415927
C
C***ZZSU23**************************************************************
C
C-——- Description section
IF (STAT(lO).EQ.l) THEN
STR= ' Tire Traction-Handling Model'
CALL ZZWRIT(STR)
STR= ' Based upon the Magic Formula Traction model'
CALL ZZWRIT(STR)
STR= ' developed by Pacejka, Delft University'
CALL ZZWRIT(STR)
RETURN
ENDIF
C
C-—-- Subroutine...
C

C CHECKING FOR APPLIED NORMAL FORCE...

IF (X(3)
Y(l) = 0
Y(2) = 0
GOTO 999
END IF

C CALUCLATING COEFFICIENTS...

.LT. 0) THEN

IF (ABS(VTR) .LT. .1) THEN
VTR = 0

ENDIF

IF (ABS(VHWL) .LT. .1) THEN

VHWL=O

101
ENDIF

IF (VTR .EQ. 0) THEN
IF (VWHL .EQ. 0) THEN

SLIP = o

ELSEIF (VWHL .GT. 0) THEN

SLIP =-10

ELSE

SLIP = 10

ENDIF

ELSE

SLIP = (VTR-VWHL)/VTR

ENDIF

SHAPE =((PI-ASIN(YSS/YPK))*2/PI)
B = TANlTHETA/180*PI)/(SHAPE*YPK)
E = (B*XPK-TAN(PI/2/SHAPE))/(B*XPK-ATAN(B*XPK))

C CALCULATING THE OUTPUT

999

FTR = FNORM*YPK*SIN(SHAPE*ATAN(B*SLIP-E*(B*SLIP-ATAN(B*SLIP))))
Y(l) = FTR

Y(2) = FTR

RETURN

END

8.8 ZZSU50 - Automatic Tranmission Torque Converter

This torque converter model is based upon one used by Runde [Runde, 1986].

SUBROUTINE ZZSU50(TIME,X,P,Y,STAT)

C_.__._
C-___
C___....

000000

C_.___.

C__.__.

PROGRAMMING: Mark Minor 4/18/96
DESCRIPTION: Torque converter subroutine

INPUTS: TIME, current time
X, input_variable values
P, parameter values

STAT(l), =1 iff lST time subrtn is called, =0 else
STAT(Z), =1 iff last time subrtn is called, =0 else
STAT(3), =1 iff last time subrtn is called

for this storage interval, =0 else

OUTPUTS: Y, output_variable values

STAT(4) =1, stop the integration, =0 continue
DECLARATIONS:
CHARACTER STR*80

DOUBLE PRECISION TIME, X(20), P(20), Y(20), wp,wt,Tp,Tt
INTEGER STAT(lO)

000000

102
EXTERNAL ZZWRIT

x(1) = wp => pump speed

X(2) = wt => turbine speed
y(1) = Tp => pump torque
y(2) = Tt => turbine torque

C***ZZSU50**************************************************~k***********

C
C____

Description section

IF (STAT(lO).EQ.1) THEN
STR= ' Torque converter subroutine.‘
CALL ZZWRIT(STR)
RETURN

ENDIF

The subroutine

WP=X(1)
wt=X(2)

Tp=3.4325e-3*wp**2+2.2210e-3*Wp*wt-4.604le-3*wt**2
Tt=5.7656e—3*wp**2+3.107e-4*wp*wt-5.4323e—3*wt**2

Y(l)=Tp
Y(2)=Tt
RETURN
END

8.9 ZZSU51 - Determination of CVT Ratio

SUBROUTINE ZZSU51(TIME,X,P,Y,STAT)

C
C____
C
c-_-_
C
C__.__

0000000

C____
C

PROGRAMMING: Mark Minor 4/18/96

DESCRIPTION: CVT Model

INPUTS: TIME, current time
X, input_variable values
P, parameter values

STAT(l), =1 iff lST time subrtn is called,
STAT(Z), =1 iff last time subrtn is called,
STAT(3), =1 iff last time subrtn is called

=0 else
=0 else

for this storage interval, =0 else

OUTPUTS: Y, output_variable values

STAT(4) =1, stop the integration, =0 continue

103

C
C—--- DECLARATIONS:
CHARACTER STR*80
DOUBLE PRECISION TIME, X(20), P(20), Y(20)
INTEGER STAT(lO)
C
EXTERNAL ZZWRIT
c X(1) - Engine Speed
c X(2) - Transmission Speed
C Y(1) — Gear ratio
C P(l) - Proportionality to engine speed
C P(2) - Proportionality to transmission speed
C P(3) — Offset
C ie: Y(1)=X(1)P(1)+X(2)P(2)+P(3)
C

*iri' *************************************************************ir
C ZZSU

C
C---— Description section
IF (STAT(lO).EQ.1) THEN
STR= ' CVT Ratio model'
CALL ZZWRIT(STR)

RETURN
ENDIF
C
C---— CVT Ratio Subroutine
C
Y(1) = X(l)*P(l)+X(2)*P(2)+P(3)
RETURN

END

Literature Cited

7. Literature Cited

Broenink, 1995. PC Version of the Bond Graph Modeling, Analysis, and Simulation Tool CAMAS.
1995 International Conference on Bond Graph Modeling and Simulation. 203-8. San Diego:
Society for Computer Simulation

Gillespie, T. D. 1992. Fundamentals of Vehicle Dynamics. 4th Printing. Warrendale, PA: Society for
Automotive Engineers, Inc.

Graham, C. R 1992. General Motors High Performance 4.3L V6 Engine. SAE Transactions. Vol. 101.
Sect 3. 1305-1320. Warrendale, Pa: Society of Automotive Engineers, Inc.

Hales, MK. 1995. A Design Environment for the Graphical Representation of Hierarchical Engineering
Models. Master of Science Thesis. Michigan State University.

Hrovat, D. 1991. Bond Graph Modeling of Automotive Power Trains. Journal of the Franklin Institute.
Elmsford, NY: Pergamon Press.

Karnopp, D.C., D.L. Margolis, and RC. Rosenberg. 1990. System Dynamics: A Unified Approach. 2nd
ed. Canada: John Wiley and Sons, Inc.

Margolis, D.L. and J. Asgari. 1989. Sophisticated Yet Insightful Models of Vehicle Dynamics Using
Bond Graphs. Advanced Automotive Technologies, Automotive Society of Mechanical Engineers.
DSC Volume 13. 155-168. New York, N.Y.: American Society of Mechanical Engineers.

Pacejka, H.B. and E. Bakker. 1991. The Magic Formula Tyre Model. Tyre Models for Vehicle Dynamic
Analysis: proceeding of the 1 st International Colloquium on Tyre Models for Vehicle Dynamics
Analysis held in Delft, The Netherlands. Supplement to Vehicle System Dynamics, Volume 21 .
1-18. Amsterdam/Lisse, The Netherlands: Swets and Zeitlinger B.V.

Runde, J. K. 1986. Modeling and Control of an Automatic Transmission. Master's Thesis.
Massachusetts Institute of Technology, Department of Mechanical Engineering.

Top, J.L., A.P.J. Breuneuse, J.F.Broenink, J.M. Akkermans. 1995. Structure and Use of a Library for
Physical System Models. 1995 International Conference on Bond Graph Modeling and
Simulation. (ICBGM '95). 97-102. San Diego: The Society for Computer Simulation.

Wagner, J .R. 1995. Optimization of a Tire Traction Model for Antilock Brake System Simulations.
Transactions of the ASA/IE, Journal of Dynamic Systems, Measurement, and Control. Vol. 117.
199-204. New York, N.Y.: American Society of Mechanical Engineers.

Zegelaar, P.W.A., S. Gong, and H.B. Pacejka. 1993. Tyre Model for the Study of In-plane Dynamics.
The Dynamics of Vehicles on Roads and on Tracks. Supplement to Vehicle System Dynamics,
Volume 23. 578-590. Alblasserdam, The Netherlands: Ofl‘setdrukkerij Kanters B.V.

104