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This is to certify that the

dissertation entitled

MODULAR MODELING OF ENGINEERING SYSTEMS
USING FIXED INPUT-OUTPUT STRUCTURE

presented by

Brooks Philip Byam

has been accepted towards fulfillment
of the requirements for

 

Ph . D . degree in Mechanical
Engineering

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

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PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

11:3 a a 2304

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MODULAR MODELING OF ENGINEERING SYSTEMS
USING FIXED INPUT-OUTPUT STRUCTURE

By

Brooks Philip Byam

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1999

 

 

 

 

ABSTRACT

MODULAR MODELING OF ENGINEERING SYSTEMS
USING FIXED INPUT-OUTPUT STRUCTURE

By

Brooks Philip Byam

Computer modeling is common in the design and development of complex
engineering systems. A system model is built up by connecting the inputs and outputs of
several subsystem models. This process requires flexible modeling tools. Models with
arbitrary input-output structure have this flexibility but must have their internal equations
reformulated to agree with the inputs and outputs used. The flexibility achieved with
arbitrary input-output structure occurs at the cost of globally reformulating the equations
of each subsystem and component model with every change. Each model equation
formulation requires performance verification because every formulation does not have
the same guaranteed performance. This can be particularly cumbersome in large models.
A fixed input-output structure allows elements to be used without modification of their
internal equations and eliminates equation reformulation. Modular modeling models
have the property that their elements have fixed input-output structure. The cost is a
connector to assemble elements adding complexity to the global model. Modular
modeling is a systematic realization of compatible standardized modular elements and
connectors that maintain modularity with the flexible assembly required in today’s large

complex modeling environments. Structural and automotive examples are given.

Copyfight by
Brooks Philip Byam
1 999

In memory of a dear fi'iend Martha Somers

iv

ACKNOWLEDGMENTS

I would like to thank my major professor Dr. Clark Radcliffe for his patience and

guidance. Dr. Radcliffe is my role model, colleague, and friend.

Dr. Rosenberg, Dr Haddow, and Dr MacCluer deserve my thanks for serving as

members of my Ph.D. committee and for providing useful suggestions and insight.

My thanks to Dr. Tom Kullgren and the faculty of the Mechanical Engineering

Department at Saginaw Valley State University for giving me an opportunity.

Charles Birdsong, Mark Minor, Joga Setiawan, Joe DeRose, Brennan Sicks, Gary
Gosciak, April Bryan, Roe] Maes, Nancy Albright, and Carol Bishop also provided

valuable assistance via their fi'iendship.

My thanks go to my parents and my family. They provided the love and support

for me to realize this goal.

Finally, Sarah Brandt, deserves great appreciation for the support, sacrifice, and

encouragement she has given me.

TABLE OF CONTENTS

List of Tables ..................................................................................... viii
List of Figures ....................................................................................... ix
Introduction ........................................................................................ 1
Chapter 1 Modular Modeling: The Impact of Causality ................................... 4
1.1 The Impact of Causality: Model Internal Formulation ............................ 6
1.2 Modular Modeling Elements ......................................................... 8
1.3 Connector for Element Interconnections .......................................... l 1

Chapter 2 Assembly and Solution of Linear Modular Models Using Fixed Input-

Output Structure .................................................................. 18
2.1 Modular Modeling .................................................................... 18
2.2 Linear Modular Modeling Analysis ................................................ 21
2.2.1 Linear Algebraic Differential Equations ................................ 22
2.2.2 Linear Algebraic Equations ............................................... 29
2.2.3 Linear Algebraic Boundary Value Problem Equations ............... 31
2.3 Guaranteed Computational Sparseness ............................................. 34
2.4 Examples ............................................................................... 37

Chapter 3 Assembly and Solution of Nonlinear Modular Models Using Fixed Input-

Output Structure ................................................................... 53
3.1 Nonlinear Modular Modeling Analysis ............................................ 53
3.1.1 Nonlinear Algebraic Differential Equations ............................ 53
3.1.2 Nonlinear Algebraic Equations .......................................... 62
3 .2 Some Modular Computational Advantages ........................................ 64

vi

3.3 Examples ............................................................................... 65

Chapter 4 Conclusions ........................................................................ 77
4.1 Contributions ........................................................................... 77
4.2 Future Work ........................................................................... 79
List of References ................................................................................... 82

vii

LIST OF TABLES

Table 1.1 Measurement Perspective Modular Modeling Element Fixed Port

Causality Across Multiple Energy Domains ................................... 9

viii

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 1.6

Figure 1.7

Figure 1.8

Figure 2.1

LIST OF FIGURES

Two Possible Different Input-Output Causal Definitions At A Multi DOF
Element Port i ....................................................................... 6
2 Possible Different Input-Output Causal Definitions At n Power Ports
Yields 2" Possible Different Useful Multi DOF Element Formulations —
Three Port Hydraulic Pump Example ............................................ 7

1 Possible Input-Output Causal Definition At n Power Ports Yields A

Single Multi DOF Element Formulation ........................................ 8
Modular Modeling Element Graphical Notation .............................. 11
Modular Modeling Connector Graphical Notation ............................ 12

Structural Models Of 3 Bars Connected With A Shear Pin: a) Physical
System b) Negligible Pin Deflection c) Large Pin Deflection .............. 13
A Conceptual Rear-Wheel Drive Power Train Model: a) Physical System
b) Modular Model ................................................................ 16
Simplified Modular Modeling Notation of the Conceptual Power Train
Model in Figure 1.7 ............................................................... l7
Modular Modeling Element Graphical Notation. Power flows into port 1
if ul and y1 are both positive. The fixed causality input variable, u,, is
always shown on the top or to the left of the port. The fixed causality

output variable, y, , is always shown on the bottom or to the right of the

port. ................................................................................. 19

ix

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 3.1

Figure 3.2

Figure 3.3

Modular Connector Graphical Notation. Power flows into the connected
elements if yc and u‘ are both positive. The input variable, u, is always
shown on the top or to the left of the port. The output variable, y , is
always shown on the bottom or to the right of the port. ..................... 21
Electric Automotive Vehicle Power Train Modular Modeling Elements a)
Electric Motor b) Transmission c) Clutch ...................................... 38
Electric Automotive Vehicle Power Train Modular Model With An
Electric Motor And A Transmission Modular Modeling Elements ........ 39
Electric Automotive Vehicle Power Train Modular Model With An
Electric Motor, A Compliant Clutch, And A Transmission .................. 41
Three Structural Linear Algebraic BVP Modular Modeling Elements: a)
Axial Force Bar b) Bending Pin c) Fixed Point ............................... 46
Three Axial Force Bar And A Fixed Point Modular Modeling Elements In
A Bracket-Pin-Clevis Configuration With No Pin Deflection .............. 46
Three Axial Force Bar, Two Fixed Point, And A Bending Pin Modular
Modeling Elements In A Bracket-Pin—Clevis Configuration ................. 49
Nonlinear Modular Modeling Internal Constraint Stabilizer Block
Diagram ............................................................................. 59
Electric Automotive Vehicle Power Train Modular Modeling Elements a)
Electric Motor b) Transmission c) Fluid Clutch .............................. 66
Electric Automotive Vehicle Power Train Modular Model With An
Electric Motor, A Fluid Clutch, And A Transmission Modular Modeling

Elements ............................................................................ 67

Figure 3.4 Three Structural Nonlinear Algebraic Modular Modeling Elements: a)
Axial Force Bar b) Fixed Point .................................................. 71
Figure 3.5 Three Axial Force Bar And A Fixed Point Modular Modeling Elements In

A Bracket-Pin-Clevis Configuration With No Pin Deflection .............. 71

INTRODUCTION

Increasingly large and complex computer models are becoming standard
practice in design and development of engineering systems. Models should have
sufficient complexity to predict actual behavior of complex engineering systems
(Ferris etal, 1994). Chrysler used a large computer model to completely design
and analyze the geometry of their 1998 Chrysler Concorde and 1998 Dodge
Intrepid. The Chrysler large car models were limited to mechanical geometry
studies and still contained representations of over 5500 interconnected physical
subsystems (”Computers In Engineering: Chrysler designs paperless cars”, 1998).
Chrysler engineers resolved design issues by developing this computer model
instead of physical prototypes reducing the cycle time from 39 to 31 months
saving the company more than $75 million (Jost, 1998). Efficient design,
development, and refinement of large computer models of complex engineering
systems are critical to engineers and their companies. A systematic approach to
design and development is the most efficient (Shigley and Mischke, 1989).

Several existing modeling methodologies use systematic techniques for
the design and development of engineering system computer models. Kinematic
and dynamic models of mechanical systems are developed using the systematic
method of generalized Cartesian coordinates (Haug, 1989 and Nikravesh, 1988).
Electrical system models are developed using the systematic method of applying
Kirchhoff Laws from a network topology (Chua and Lin, 1975, Vlach and
Singhal, 1983, and Calahan, 1972). These methods are useful but are limited to
their respective energy domains.

A systematic method that includes different energy domains is Finite

Element Analysis (FEA). FEA methods systematically construct grids of similar

elements to model engineering components and systems in mechanical and
thermal energy domains (Zienkiewiez, 1977). These methods are useful for
systematic generation of model equations and models grow to be quite large and
complex. However, FEA models are not generated with an input-output
structure that allows them to be easily interconnected. Therefore, assembling
several of these independent models requires refonnulating an entirely new
model or using special purpose software like PDESolve (PDESolve, BEAM
Technologies) to ”connect” them, which is both cumbersome and expensive.

Another systematic inter-energy domain modeling method is bond
graphs. Bond graphs have a systematic approach of using graphical multi-port
elements and junctions to develop component and system models in mechanical,
electrical, hydraulic, and thermal energy domains (Kamopp etal, 1990). System
model equations are systematically generated with some hierarchical design but
the assembly reformulation issue still exists due to arbitrary input-output
definitions (Karnopp etal, 1990). Recent research has further enhanced the
hierarchical design of bond graph models such that some reformulation can be
avoided (Hales, 1995). Reformulation of model equations is the practice that
prevents efficient development of large, complex, models.

A new approach to systematic modeling across multiple energy domains
provides a modular fixed input-output structure modeling method. Modular in
the sense that the physics that describes each subsystem model remains the same
whether the subsystems are separate or assembled into a system (Hogan, 1987).
Fixed input-output structure in the sense that the inputs and outputs are
standardized so the internal equations of mathematical subsystem models of
engineering systems have the same modularity as the engineering system.
Modular modeling with fixed input-output structure is a power-based physically
intuitive top-down methodology systematic modeling method that eliminates

equation reformulation from large model development across multiple energy
domains (Byam and Radcliffe, 1999).

Modular modeling is a top-down systematic equation assembly scheme
well suited to multi DOF components. Modeling efficiency is diminished for
development and assembly of idealized single DOF model components. This
method has a standardized input-output definition for all multi port multi
degree of freedom modular modeling elements resulting in a single standardized
element formulation. A single standardized formulation allows modelers to gain
performance verification experience enhancing the verification process. The
single standardized modular modeling element combined with the compatible
modular connector makes modular model realization a simple systematic
process. Modular modeling makes the large model design, development, and

refinement process systematic, functional, and physically intuitive.

Chapter 1

MODULAR MODELING: THE IMPACT OF CAUSALITY

Modular modeling is defined by a fixed input-output structure. This strict input-
output approach standardizes the internal equation formulation of multi degree of
freedom (DOF) subsystem models. The objective is mathematical models of engineering
systems with equations that have the same modularity as the engineering system. This
method is not intended for single DOF model elements or single component modeling. It
is intended for large system models with many multi DOF subsystems. Modular
modeling is particularly advantageous for large system models because the multi DOF
subsystem models fixed formulation simplifies large model design, development, and
refinement.

Large complex models of engineering systems contain a large number of multi
DOF subsystem models (5500 in the Chrysler large car models). Each physical multi
DOF subsystem model has one or more connections through which it is attached to other
subsystem models. For example, the transmission subsystem model of a pick-up truck
has connections to the engine model, the flame model, the driver controls model, and the
drive shaft model. Each physical subsystem model connection has an input-output
definition, conveys input-output variables to interconnected subsystem models, and hence
defines the internal formulations of the subsystem models. Controlling the number of
internal formulations of interconnected multi DOF subsystem models is key for design,
development, and refinement of large models.

Subsystem model internal formulations result from input-output connections
defined here in two general forms: signal-type and natural-type. Signal-type model
connections contain single variables and can only be defined as input or an output. For

example, the driver controls-to-transmission model connection is a signal-type model

input connection. The only reasonable model is a selected-gear control signal input to the
transmission model. Indicator lights are an example of a signal-type model connection
only reasonably defined as a model output. Once defined, signal-type connections have
only one possible definition and their effects on the model’s internal equations is fixed.

Natural-type model connections may have many variables and many valid input-
output definitions resulting in many useful subsystem model equation formulations. For
example, a transmission-to-drive shaft model connection may have mechanical rotation
and mechanical translation variables. Each model must have an input-output definition
and hence internal formulation that provides the appropriate input and output variables at
the connection. An output of one subsystem must be an input to the other. There are
many possible usefirl input-output definitions, which could be used to assemble these
elements. Each definition requires a different, useful, well-posed internal formulation of
the connected multi DOF subsystems’ internal equations.

Power-based models represent natural-type physical model connections with
power ports. A power port is a place where physical systems are connected and exchange
power. Power is commonly modeled as the product of two variables such as force and
velocity, pressure and volume flow rate, or voltage and current. The variable pairs are
often referred to as the effort-flow variable pair, erfi, at each power port (Karnopp etal,
1990). Power-based simulations pair these power variables at each port. Each port has
causality, an input-output definition (Karnopp etal, 1990). Causality manages connected
power port’s physical cause and effect between the variable pairs by defining one
variable as a port input and the other variable as a port output. Connected ports must
have reciprocal causality (Rosenberg and Karnopp, 1983). In other words, the input of
one port is the output of its connected port, and vice versa. Each port affects the model’s

internal formulation through its input-output causality definition.

1.1 The Impact of Causality: Model Internal Formulation

Causality has a great impact on the number of possible different, useful, internal
equation formulations of a computer model. Power port causality allows two possible,
different, reciprocal, input-output definitions. Let the two variables, e,- and j}; be the
input-output variables at the multi DOF element model port i. There are two possible
different input-output definitions at a multi DOF element port 1' (Fig. 1.1). For example,
the mechanical shaft of a pump can be modeled with a torque input and an angular

velocity output or with the reciprocal causality. Useful pump models can be formulated
with either causality.

~.

    
  

e
—' .

a.

t
23-: .

 

Flow Out ‘_
f.-

e r

<-

Effort Out ”” ..<

Flow In _>

I?
Figure 1.1: Two Possible Different Input-Output Causal Definitions At A
Multi DOF Element Port 1'

A multi DOF element model with n power ports with arbitrary causality
permitting 2 possible reciprocal input-output definitions at each port will have Na
possible different internal equation formulations (Fig. 1.2).

N, = 2" (1.1)
Typical multi DOF subsystem models have 1 or more power ports. For example, a
simple hydraulic pump model may have 3 power ports (1 for the mechanical shaft, and 2
for the hydraulic high and low pressure connections). This leads to
23 = 8 (1.2)

possible specific causal formulations.

”‘8“ High High mg},
P233?”

J

   

 

Pressure Shaft
Port “*2
Hydraulic Pump Hydraulic Pump Hydraulic Pump Hydraulic Pump
Formulation l Formulation 2 Formulation 3 Formulation 4
High High Highm Highm
Pleasure
Pay-{.215 Pun-est“: .3 Pmpr: 1;:
«fix;
Shaft g “'0st Shah if? :rwure SW 5.6,, PM“ 3““ 1i
Hydraulic Pump Hydraulic Pump Hydraulic Pump Hydraulic Pump
Formulation 5 Formulation 6 Formulation 7 Formulation 8

Figure 1.2: 2 Possible Different Input-Output Causal Definitions At n Power

Ports Yields 2" Possible Different Useful Multi DOF Element
Formulations - Three Port Hydraulic Pump Example

2" possible useful different element formulations requires 2" different model

verifications. Verifying 2" possible correct element formulations is a staggering task.

Consider interconnecting 5500 multi DOF model elements (e.g. 1998 Chrysler large car

models) with reciprocal causality. In the most simplistic interconnected form, 2 power

ports per element, there are 5500x2 = 11,000 power ports. There is a 50% probability of

a perfect input-output causal match but the number of possible useful system model

formulations to verify is an impossible task.
50% of 2"°°° = 1.06885 x103310 (1.3)

For this reason users of models with arbitrary causality have made the choice to verify at
the component level and reformulate the system equations after every change.

Fixing power port causality at every multi DOF element port selects a single
element internal equation formulation (Fig. 1.3). This standardized functionality
simplifies the design, deve10pment, and refinement of large models. There is only one
system configuration, which enables verification at the subsystem level. The key concept
of modular modeling is a physically intuitive fixed causality at every port of every

modular modeling element.

 

Single Element Formulation

Figure 1.3: 1 Possible Input-Output Causal Definition At n Power Ports Yields
A Single Multi DOF Element Formulation

1.2 Modular Modeling Elements

Modular modeling elements are multi-port multi DOF subsystem models with a
fixed causality at every power port and a single fixed formulation. The element internal
equation formulation fits the fixed causality and the number of ports. Since the causality
is fixed, the formulation of element’s equations does not change. Once formulated
modelers can gain validation experience with each specific element. The fixed causality
at every port leads to a standardized element functional form.

y = element(u ) (1.4)

~nXI ”"X|

n is the number of element power ports, u is a vector of inputs and y is a vector of

”"XI ~nXl

outputs. The multi port multi DOF element calculates a vector of outputs, y , given the

~nxl

vector of inputs, u , and some internal element parameters. This functional form can be

~nxl

generated for most engineering elements with any number of power ports.

Modular modeling standardizes the choice of causality for every port per energy
domain to realize the objective of an internal equation formulation with the same
modularity as physical systems. The choice of the modular modeling element fixed port
causality is motivated by physical measurements. Ideal physical measurements occur at a
natural power port with a specific physical location and zero power flow such that there
is no effect on the response of the system. In other words, physical measurements have a

specific physical location, an externally sensed output at that location, and zero input at

that location to zero the power flow. This measurement perspective defines the fixed port
causality of modular modeling elements. The port output is the variable related to the
externally sensed physical quantity. The port input is the variable related to the internal
physical quantity typically assumed zero to attain zero power flow.

The measurement perspective modular modeling element fixed port causality for
engineering systems across multiple energy domains are shown in Table l. Externally
sensed physical quantities are electrical potential, curvilinear mechanical motion, angular
mechanical motion, hydraulic pressure, acoustic sound pressure, and temperature.
Internal physical quantities typically assumed to be zero are electrical current, mechanical
force, mechanical torque, hydraulic volume flow rate, acoustic volume velocity, and
thermal heat flux. The resulting measurement perspective modular modeling element
fixed port causality of electrical, mechanical translation, mechanical rotation, hydraulic,
acoustic, and heat transfer systems are current input-potential output, force input-velocity
output, torque input-angular velocity output, flow input-pressure output, flow input-

pressure output, and heat flux input-temperature output respectively.

 

 

 

 

 

 

 

 

 

 

MEASUREMENT PERSPECTIVE
ENGINEERING MODULAR MODELING ELEMENT
SYSTEM FIXED PORT CAUSALITY
Electrical Current Input — u = Current,
Potential Output y = Potential
Mechanical Force Input — u = Force,
Translation Velocity Output y = Velocity
Mechanical Torque Input - u = Torque ,
Rotation Angular Velocity Output y = Angular Velocity
Hydraulic Volume Flow Rate Input - u = Volume Flow Rate,
Pressure Output y = Pressure
Acoustic Volume Velocity Input - u = Volume Velocity,
Sound Pressure Output y = Sound Pressure
Heat Transfer Heat Flux Input — u = Heat Flux,
Temperature Output y = Temperature

 

Table 1.1:

Measurement Perspective Modular Modeling Element Fixed Port

Causality Across Multiple Energy Domains

 

Measurement perspective causality is equivalent to implementing nonessential or
natural boundary conditions at every port (Meirovitch, 1967). This ensures an internal
formulation with a mathematical eigen-structure that does not change whether the model
is separate or assembled into a system model. Essential boundary conditions change the
model’s differential operators and impose specific geometric constraints. Modular
modeling implements essential boundary conditions with modeling elements. For
example, a mechanical fixed-point element would output a zero velocity regardless of the
force input. Measurement perspective fixed port causality enables modular modeling
elements with physical system modularity

A benefit of measurement perspective fixed port causality is the flexibility to
maintain any number of physically “open” power ports at discretionary physical locations
without affecting the element internal formulation. A physically “open” power port i is
physically disconnected with zero input.

u, = 0 (1.5)

The input, up is zero to achieve no power flow but the output, y, , is open to be defined
by the element. For example, a modular (multi DOF) mechanical beam model element
whose ports are defined can maintain any number of “open” zero force-input ports
without changing the model’s mathematical formulation. A modular modeling beam
element will have the same internal formulation and the same performance regardless of
the modeling task. The response of the system can be “measured” at any “open” port

with the power port’s output variable. An element formulated from velocity input-force
output “fixed” causality does not have this flexibility. This flexibility enables the single
formulation of modular modeling elements to maintain a large number of power ports
with no reformulation or revalidation cost.

The modular modeling elements graphical notation represents a multi port multi
DOF physical subsystem model with a rectangle (Fig. 1.4). The bold lines represent the n
power ports with fixed measurement perspective causality (Table 1.1). Arrows on the

10

port lines define the standardized direction of positive power for modular modeling
elements, when u] and y1 are both positive, power flows into port 1. Port power
orientation has a similar 2" effect on model formulation but does not affect the eigen-
structure of the internal formulation. The fixed causality input variable, 14,, is always

shown on the top or to the lefi of the port. The fixed causality output variable, y,, is

always shown on the bottom or to the right of the port.
Port 2 Port n

 

Figure 1.4: Modular Modeling Element Graphical Notation

1.3 Connector for Element Interconnections

Modular modeling requires a connector to join incompatible modular modeling
element power ports. The connector provides the compatible port causality. Modular
modeling connectors provide the proper physical connection constraints for connecting
the ports of engineering subsystems. The connector is not a model of a physical
connection subsystem. It is a power transmission mechanism that enforces constraints
between subsystems.

The connector graphical notation represents the connector with a circle (Fig. 1.5).
The bold lines represent the connected modular modeling element ports, 1' and j. Arrows
on the port lines define the standardized direction of positive power for connectors, when
u, and y, are both positive, power flows out of port i. The input variable, u, is always
shown on the top or to the lefi of the port. The output variable, y, is always shown on

the bottom or to the right of the port.

11

u,- u-
Porti ‘710 11:» Port j
Figure 1.5: Modular Modeling Connector Graphical Notation
Modular modeling connectors provide a power constraint Power is conserved
across modular connectors because modular connectors are power transfer mechanisms.
The power at the connected modular modeling element ports sum to zero.

2P, :0 (1.6)

k=l.2
Recall that the product of port variables, u, and y, , is power.

P, = uiy, (1.7)
An additional constraint is required for the connected modular modeling element ports.
Connectors are defined to constrain connected modular modeling element ports’ outputs

to be the same implementing connections of elements.

y. =y,- (1.8)
From (1.6) - (1.8) the inputs at the connected ports must be equal and opposite.
u, = —uj (1.9)

The functional definition of the connector is a 2-port power constraint with port causality
compatible with modular modeling element port causality (1.4). In modular modeling,
(1.8) and (1.9) are the defining equations for all modular connectors.

There are two important characteristics of modular modeling elements and
connectors to aid in the assembly of a modular modeling system models. First, properly
connecting any number of power ports of modular modeling elements with the 2-port
connector requires a junction structure inside modular modeling elements for each port.
The internal junction structure has the affect of summing port inputs and constraining the
port outputs to have the same output. This is consistent with the measurement
perspective that defines the fixed modular modeling causality. For example, measuring a
voltage at a point on an electrical circuit will have one voltage but there may be several

currents input to that point. Second, the explicit difference between modular modeling

12

elements and connectors is that modeler defined equations are in modular modeling
elements. Modular connector equations are fixed (1.8) and (1.9). All modeling is done
in modular modeling elements, none in modular connectors. Modeling a physical
connection requires modular modeling elements.

Structural models of 3 bars connected with a shear pin will provide a
demonstration (Fig. 1.6). The simplest model (Fig. 1.6 b) neglects deflection of the pin.
In this simple case, no pin deflection model is required and the bars are connected with
displacement and force constraints. When the pin deflection is large (Fig. 1.6 c), a model

for the physical deflection of the pin is required and a pin modeling element is used.
BAR 2

      

 

Figure 1.6: Structural Models Of 3 Bars Connected With A Shear Pin: a)
Physical System b) Negligible Pin Deflection 0) Large Pin
Deflection
The first rigid pin modular model is a demonstration of the choice to not model
the pin. The pin is ignored and 2 modular connectors join the 3 bars. The modeler
defines the multi-port multi-DOF modular modeling elements BAR 1, BAR 2, and BAR 3.
The modular modeling element BAR 1 has a port 45 with two input-output pairs 4 and 5

where power is transferred from BAR 2 and BAR 3 respectively. Each modular connector

13

is associated with a power transfer mechanism between subsystem models. More than
one power transfer mechanism can occur at a single port of a model element because of
the internal junction structure. Port 45 power pair variables 4 and 5 are input and output
from the same apparent geometric point on BAR 1. The internal junction structure has the

affect of summing the 4 and 5 inputs and constraining the 4 and 5 to have the same output

at that port. The right end of BAR 1 has the input force as and the output velocity v45.
£15 = £1 + F;

V45 = V4 = Vs

(1.10)

Modular connectors 4,6 and 5,7 are the power transfer mechanisms that join port

45 of BAR 1 to port 6 of BAR 2 and port 7 of BAR 3 respectively. The connectors enforce
the fixed power constraints (1 .8) and (1.9) to the power transfer such that the respective
outputs are the same and the respective inputs are equal and opposite. Two 2-port
modular connectors properly constrain the power transfer to join 3 modular modeling
element ports at the apparent same geometric point. This can be extended to any number
of connected modular modeling element ports. Multiple ports at one-point increases the
number of ports in the model but modular modeling elements can maintain any number
of “open” ports with no internal formulation change.

The second flexible pin modular model is a demonstration of the choice to model
the pin. The modeler defines the multi-port multi-DOF modular modeling elements PIN,
BAR 1, BAR 2 and BAR 3. PIN has three power ports 5, 6, and 8 at the connection or
power transfer points with BAR 1, BAR 2 and BAR 3 respectively. Modular connectors
4,5 , 6,7 , and 8,9 are the power transfer mechanisms that join the power ports 4, 7, and
9 of BAR 1, BAR 2 and BAR 3 to the PIN power ports 5, 6, and 8 respectively. The

connectors facilitate the power transfer between power ports implementing (1.8) and

(1.9). For example, connector 4,5 enforces power conservation constraints on the power

transfer between power ports 4 and 5 such that v4 and v5 are equal and E, and F; are

14

equal and opposite. Connectors 6,7 and 8,9 facilitate power transfer on their respective

ports enforcing the same constraints.

An automotive model of a conceptual rear-wheel drive power train (Fig. 1.7) is
another modular modeling example. The physical system (Fig. 1.7a) consists of an
engine, a transmission, three mounts, a flex plate, and a frame. The engine has physical
connections to the transmission through a flex plate and to the frame through mounts on
the either side of the engine. The transmission has physical connections to the engine
through a flex plate and to the frame through a transmission mount. The frame has
physical connections to the engine and the transmission through the mounts. The
modular model of the drive train (Fig. 1.7 b) shows the modeling choice not to model the
flex plate deflection. The deflection of the flex plate is small relative to the deflection of
three mounts. The rigid flex plate is not considered an element, so the engine and
transmission transfer power to one another through modular connectors. The flexible
mounts require modeler-defmed equations to describe their deflections and are
considered modular modeling elements. Modular connectors are power transfer
mechanisms composed of standard constraints to conserve power. This is quite different

from multi DOF modular subsystem elements, which implement all modeling analysis.

15

Engine

  

Transmission
' l_‘l
a
) 3 Frame
Left& Flex Trans
Right Plate Mount
b)

 

Figure 1.7: A Conceptual Rear-Wheel Drive Power Train Model: a) Physical
System b) Modular Model

The modular connector can not implement the constraint (1.8) and (1.9) on ports

of different energy domains. For example, from Table 1.1 and constraint (1.8), an

electrical potential output and a mechanical velocity output can not be equal. The

solution is to define an appropriate modular transducer model element with fixed

measurement perspective port causality, then use modular connectors to join the modular

transducer element ports and the element ports in different energy domains. Modeler-

defined multi-port modular transducer elements have the same form and function as any

modular modeling elements and are simply another element in the modular model.

The standardized port causality and sign conventions of modular modeling give

modular connectors the exact same appearance (Fig. 1.7). The modular connector

16

notation (Fig. 1.5) can be replaced with a single line. This simplified modular modeling
notation for the conceptual power train example has a traditional block diagram
appearance (Fig. 1.8). This simpler notation aids only in graphically communicating the
model. The power-based fixed measurement perspective port causality and sign
convention input-output structure implicit in every port line is critically important. For
example, implicit in the port line connecting the Engine and Transmission modeling
elements are two power variables E2“ and v1,” following the input-output causality in
Table 1.1 constrained by (1.8) and (1.9).

Input ={ F' = F‘; Output =>{ v, = v, = v,,‘ (1.11)

F: = ‘Fizc

The dogmatic input-output structure of modeling elements is the key concept of modular
modeling.

     

C C C C C C C C C C
F73 V78 F 910 V910 F 1112 V1112 F1314 V1314 F2324 V2324 F 2526 V2526

C C
C C C C C C C C C C C
V1516 Fl718 V1713 F1920 V1920 F2122 V2122 F2728 ”2728 F 2930 V2930

F1516

Figure 1.8: Simplified Modular Modeling Notation of the Conceptual Power
Train Model in Figure 1.7

17

Chapter 2

ASSENIBLY AND SOLUTION OF LINEAR MODULAR MODELS USING FIXED
INPUT-OUTPUT STRUCTURE

2.1 Modular Modeling

Modular modeling is a modeling method designed to eliminate model equation
reformulation and enhance model performance verification. Modular models are
constrained assemblies of power-based multi degree of freedom modular modeling
elements of physical systems with a single standardized equation formulation.
Standardized modular modeling elements enhance performance verification because their
formulation does not change. Modular modeling elements have incompatible power
ports by definition, which requires a compatible modular modeling connector for
assembly. Modular modeling connectors do NOT describe physical connection elements
like welds, shear pins, and bolts. Modular modeling elements implement all modeling
activity in modular models. Modular modeling connectors are two-port power transfer
mechanisms that implement standardized power constraints between connected modular
element ports (Byam and Radcliffe, 1999).

The key concept of modular modeling is the fixed measurement perspective -
input-output causality at every port of every multi degree of freedom modular modeling
element per energy domain. This causality is based on physical measurements. The port
output variables are defined as the typically sensed physical system response.
Development of the modular modeling method reveals that the port input variable can be
assumed zero for zero power transfer and zero effect on system performance. This
causality choice (Table 1.1) standardizes modular element formulations and allows

modular elements to maintain any number of “open” or physically disconnected power

18

ports. Modular modeling elements with fixed measurement perspective causality have
physical system modularity.

Modular modeling elements have a standardized user-defined functional form that
does not change.
y = element(u ) (2.1)
_ pxl ~ pxl
Where p is the number of element power ports, u 1 is a vector of inputs at those ports

- px

and y is a vector of outputs at those ports defined by the fixed measurement
- pxl

perspective input-output power port causality. This ftmctional form can be generated for
any energy domain in Table 1.1 with any number of power ports.

Modular modeling element graphical notation represents user-defined multi-port
multi-DOF subsystem models with a rectangle (Fig 2.1). The bold lines represent the n
power ports with implicit standardized direction of positive power into the element and
standardized input-output port causality. Standardization of positive power direction and
input-output causality standardizes the modular modeling elements internal formulation,
which is the essence of modular modeling. All modeling equations in modular models
are in modular modeling elements by definition. Modeling the physical behavior of a

connection like an engine mount or a shear pin requires a modeling element.
Port 2 Port n

Port 1

 

Figure 2.1: Modular Modeling Element Graphical Notation. Power flows into
port 1 if uI and yI are both positive. The fixed causality input
variable, u,, is always shown on the top or to the left of the port.

The fixed causality output variable, y,, is always shown on the

bottom or to the right of the port.

19

Modular modeling connectors implement standard output and power constraints.
There are no physical model equations in connectors. They simply connect modular
element ports by constraining their outputs to match and conserve power flow.
Connectors are used exclusively to implement output and power constraints between
ports of modular modeling elements.

The connector forces two connected modular element ports’, i and j, outputs to be
equal to connect the ports and their power flow to sum to zero to conserve power. The
power flow constraint is translated to equal and opposite inputs at connected modular
element ports since the product of power port variables is power. Connected modular

element ports, i and j, have the same output and equal and opposite inputs.
y, = y]. = ye (2.23)

_ C

(2.2b)

u j = —u‘
In modular modeling, the defining relationship for all connectors (2.2) does not change.
The modular modeling connector graphical notation represents connectors with a
bold port line between modular elements (Fig 2.2). By definition, connectors have the
compatible input-output structure to modular elements. The bold lines have implicit

standardized direction of positive power into the connected modular element ports, i and j

_..~__.~.__h
_ _..-.. .-._....

and modular connector constraints (2.2). The modular connecto: 1133196 flexibility to-
assembl’eub‘yppairs any number of modulaerer portsflbecause modular
modeling elements have an‘iritgrnale ungfion structure at each port (Byam andmRadcliffe,
1999). The only function of modular modeling connectors is to constrain connected

modular element ports.

20

1 ' = 1 u‘ f ' '
1e lement—relement

J 53.23”}. ..

Figure 2.2: Modular Connector Graphical Notation. Power flows into the

connected elements if y‘ and u‘ are both positive. The input

variable, 11 , is always shown on the top or to the left of the port.
The output variable, y, is always shown on the bottom or to the
right of the port.

The objective of modular modeling is single standardized modular formulations
for all user-defined multi-port multi-degree of freedom modeling elements. This requires
separate connector constraints, which adds complexity to models. However, each
standardized modular modeling element has a single formulation. A single fixed
formulation allows modelers to gain experience verifying their formulation’s
performance. Modular models, which are constrained assemblies of n modular modeling
elements, are also used with no reformulation and no reverification. Modular models in

pre-assembled form are simply a concatenation of n modular modeling elements.
‘ (y, - (element1 (ul ).

y2 element2 (1:2)
“ = : " (2.3)

 

 

 

 

y, element, (u,)

The standardized input-output structure enables a systematic direct-insertion realization

of modular models fiom modular elements and connectors.

2.2 Linear Modular Modeling Analysis

Linear modular modeling analysis is a systematic direct-insertion realization of
linear modular models of the form (2.3). Using a known input-output topology
constrained by the output and power constraints of the modular modeling connector (2.2)
the constrained modular model is realized. Linear modular modeling elements are

independently user-formulated power-based multi-port multi-degree of fieedom linear

21

modeling equations with a standardized input-output causality and sign convention.
Connectors are two-port output and conservative power constraints between modular
modeling element ports. Modular connector constraints are known. Modular elements
have a known form but their equations are user-defined.

The possible user-defined linear modular element equations are linear algebraic
differential equations and algebraic equations. The linear algebraic differential equations
can be represented in a form convenient for the application of the modular modeling
fixed input-output functional form. A state-space form is the best fit because inputs and
outputs are explicit. Any algebraic differential equation model can be written in state-
space form. Algebraic equation models are typically expressed in an explicit input-

output form.

2.2.] Linear Algebraic Differential Equations
Modular modeling elements with user-defined linear algebraic differential equations

have a traditional state-space form where the inputs and outputs are explicit.
x = A x+ Bu

" “ "' (2.4)
y = Q x+ l_)u

The fixed input-output functional form (2.1) of modular elements is seen in (4), where u

is a vector of port inputs and y is a vector of port outputs ordered in port pairs. x is a

vector of states. A , B , g, and Q are matrices with time invariant coefficients
independent of the x-variables and u-variables.

Consider a modular model in unconnected form as a concatenation of n
independently formulated user-defined linear modular elements of the form (2.4). Let

this model have s total states and p total input-output power ports.

22

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{1 ”fl Q Q‘ ’35:- ‘5 Q 9‘ ”If
.-. = 9.5:» :2 .924” “.2
‘ ' " 9 ' ° 9 ' (2.5a)
' 9 9 A. x. 9 9. B. ,.
x" - _-sxs - “ -sxl _ —..sxp L. ~ _pxl
-"'—.rxl
R=¢AX+3U
{’1 ”g Q .. Q' F351 '13, Q .. 9‘ 1:;
Y2 9 C2 x, 9 D2 i “2
‘I = . I_. " + . .—‘. 1'
. . .. Q . . .. Q . (2.5b)
yr: L9 Q Ca xn Q 9 D" u"
L. " -pxl —-sz- " Jsxl - —..po - " -pxl
Y=CX+aU

For example, the modular model vector X is the concatenation of the n modular element
state vectors with a total size 3 X1 and U is the concatenation of the n modular element
input vectors with a total size p X]. The fixed input-output functional form (2.3) of a
modular model is seen in (2.5). The modular model (2.5) has an equal number of inputs
and outputs because all modular elements power ports always has one input and one
output. Each modular element of (2.5) is completely independent and uncoupled fi'orn
the other modular elements. Given the element’s input-output topology the modular
connector constraints (2.2a-2.2b) provide the coupling between the modular element
equations.

The key concept of modular model analysis is isolating the internal element input-
output power ports from the external element input-output power ports in a known input-
output topology. External input-output power ports have known inputs. Internal input-
output power ports are ports joined to other element ports through connectors (Fig. 2.2).
The input-output topology of (2.5) has a total of p input-output power ports from the n
modular elements. Let m be the number of connectors, hence there are 2m internal

element input-output power ports, which leaves q = p — 2m external element input-

output power ports. The standardized form of modular elements (2.1) and modular

23

connectors (2.2) makes isolating external and internal element ports in the unconnected
modular model (2.5) a simple reordering of the systems’ concatenated input and output
vectors U and Y.

A transformation matrix reorders the concatenated input and output vectors of
(2.5). The vectors are reordered so all external ports’ variables appear first in the vectors
followed by all the internal ports’ variables. The internal ports’ variables are ordered
such that connected port pairs’ appear together. For example, if port i and port j are
connected u, should be followed immediately by u j , similarly for the outputs y, and y j.
The reordered input and output vectors are the input and output vectors of (2.5) pre-
multiplied by the transformation matrix T

-Um

IY=

 

w] (2.6b)

_ w
The transformation matrix I does not add, remove, or combine variables of the original
vectors; it only changes the order in which variables appear. This makes _7__' a linear and
nonsingular reordered p x p identity matrix 1.

11’ = l (2.7)
The external input Um is the q x1 vector of external port inputs. The internal input U,"
is the 2m x1 vector of internal port inputs. The external output Y”, is the q x1 vector of
external port outputs. The internal output Yb, is the 2m x1 vector of internal port

outputs. The mechanism for reorganizing the input-output topology of (2.5) to isolate the

external and internal ports is the matrix (12),”.

The reordered modular model equations are written in terms of the external and

internal inputs and outputs where U = TT[U,,,U,,,]T.

 

. U...-

X = AX + 31’[ U. (2,8a)
TY- Y‘“ -T X T :f-U‘” 28b
"-Y... -_C +_@__UW (-)

24

Using the transformation matrix I the matrices in (2.8) are partitioned to further isolate
internal and external equations. The 3 matrix is partitioned by IT into coefficients of
the external and internal inputs. The C matrix is partitioned by 1‘ into the states’

coefficients of the external and internal outputs. The ,9 matrix is partitioned by IT and

1‘ into external and internal inputs’ coefficients of the external and internal outputs.

31" = [3. .15.] (2.9a)
c...
T = 2.9b
_c [cw] ( )
,_ a... .5...
L91 -[am‘ .7. . ] (2.9c)

The external partition of the 3 matrix 3,, is a s x q matrix. The internal partition of

the 3 matrix 3,, is a s x 2m matrix. The external partition of the C matrix C", is a
q x 3 matrix. The internal partition of the C matrix Q," is a 2m x 3 matrix. The sizes

ofthe partitions ofthe ,9 matrix @ww, am, an”, and and," are qu, qx2m,
2m x q , and 2m X 2m respectively. The reordered unconnected modular model

equations are rewritten in terms of the external and internal inputs and outputs and the

partitioned matrices.
it: Gamay, +3,U,,, (2.10a)
Y“, = Q,X + @waw + gmUw (2.10b)
Y," = QNX + 5,,wa + QMMUW (2.10c)

The internal outputs Y," of (2.10c) are ordered in connected internal port pairs.

At each connected pair, the two output values are constrained by (2.2a) to have equal
values. In order to apply the outputs of (2.10c) to the constraint (2.2a) the connected

output pairs need to be selected from the matrix equation (2. 10c). Define two m x 2m

equation selector matrices .5, and .5, to select the odd and even equations of (2.10c)

respectively.

25

5:2202...52 s=iiiomis (2.11)
2155 05 5355 50
_000010j _0000 01

 

 

 

 

MXZm .me2m

Pre-multiplying (2.10c) by ,3, selects the odd (1”, 3rd, 5th, , m — 1") internal port
output equations. Pre-multiplying (2. 10c) by §, selects the even (2“, 4th, 6th, , m'h)

internal port output equations. Substitute selected equations in the output constraint

(2.2a).
gm=r an»
gym = Y‘ (2.12b)

The constrained internal ports’ output Y ‘ is a m x1 vector of the m constrained inputs at
the 2m connected internal ports (Fig. 2.2) of the system (2.5a-2.5b). Rewrite the output
constraint (2.12a-2.12b) in terms of a difference to eliminate the constrained internal
port’s output.

(a. - m... = 9,, (2.13)

Substituting the internal output equation (2.10c) into the internal output constraint (13)

results in the output constraints of the modular model.
(a.-§.)c...X+(§.-§.).a...U..+(§.-§.).a....U.-.. =9. (2.14)

The three output constrained modular system equations are written in terms of the states,

X , the external inputs, U

ext ’

the external outputs, Ya, , and the internal inputs, U m.
it = 0er + 3,0,, + 3M" (2.15a)
Y“, = CmX + @mew + awn, (2.15b)
(a. - r.)c...X + (a - small... + (a. - swam... = 9 (2.159
The three output constrained modular model equations (2.15a-2.15c) have three

unknowns, the states, X , the external outputs, Ya, , and the internal inputs, Um . The state

equation (2. 15a) and the output constraint equation (2.15c) are independent of the
external outputs, Ya, , and can be solved for the states, X, and the internal inputs, Um, ,

26

then substituted in the external output equation to find the external outputs, Ya“. Rewrite

(2.15a) and (2.150) as a system of two equations with the two unknowns the states, X ,

and the internal inputs, U W.
Sr— 021x — (gym = 3,,U,, (2.16a)

-(§. — r.)c.-..X - (a. - steamy... = (.5. — steamy... (2.161))

The objective is to find the internal input, U.

rnt ’

in terms of the states, X , and external
inputs, Um, that satisfies (2.16b).
The internal input, U m , of the output constrained modular system model is

dependent on the state response of the system.
X(t) = <D(t,to)Xo + j <I>(t,0')(3,,U,-,..(O') + 3,,Uw(o))do (2.17)

The state response of the output constrained modular system (2.17) at some time t

involves the state transition matrix d>(t,to) from some initial time to and an initial state

X0. Substitute the state response of the system (2.17) into the output constraint equation

(2.16b).

‘0 ] (2.1 8)
= (£0 — g: )anturrUint + (go — £2 )anran
Group the internal inputs, U," , and the external inputs, Um together using the screening

-(§. - §. )c..{<1>(t. t.)Xo + j <D(t,0')(3,,U,,,(0') + 3,,Uw(a))do'

property of the Dirac delta.
i (g, - £:)(C..¢(t,0)3.. + .21....60 — a))U,,,(a)da =

‘0

1 (2.19)
-(_S.. - é. )[C-flwvxo + j(C-..<I>(t.o).z, + amao — o))U,,,(o)do]
The internal input, Um , that satisfies the constrained output equation (2.19) requires the

definition of two matrices.

Q(O)=(§. -§..)(c..¢(t.o)oz..+.a....8(t-o)) (ma)
:0) = Imogen: (2.201»)

27

Rewrite 10‘) using the screening property of the Dirac delta and the identity
(
(119‘ = Elf.
l 1 i l C
:0) = (r. - s.)(.e...-..e>..... 5; + .21....03. c... + $3.22.... )(5... — a)’

+ (£0 _ §e )C‘m Kim (‘09 t)cnt.(§o - E: )T
(2.21a)
Lung» = j <I>(t,o)3,,3,,'<l>‘(t,a)do. (2.21b)

The * represents the complex conjugate transpose of a matrix, 8 is arbitrarily small
positive constant, 3.1.100”) is the system’s “internal controllability grammian”. The

internal input is defined by the construction of (2.19).
U...(a) = -§‘(a)r‘ (r)(§. - .5.)-

' (2.22)
[ammo + j(c..,<1>(r. axe... + .23....60 — 0))Uw(o)do]

The internal input, Um, exists if the matrix ___Y__(t) is nonsingular. The matrix _Y_(t) is
nonsingular if it is positive definite. This derivation is similar to derivation of output
controllability in Skelton, 1988, which defines a positive definite matrix made up from
the system matrices and state transition matrix similar to (2.21a). The derivation here is
concerned with the controllability the internal inputs have over the internal outputs or its

internal output controllability. If a modular system model (2.5a-2.5b) has a positive
definite matrix 10), it is internally output controllable and the internal input U," exists.

The condition of existence of UW is less restrictive using the output
controllability approach of modular modeling then previous methods. Hogan found that
assembling linear modular component models with simple “nonenergic” connections
required either the invertability of a matrix involving only the D matrix or that the D

matrix be zero. Modular modeling clearly is less restrictive because the invertability of
X(t) is dependent on the A, B, C, and D matrices where all can be nonzero.

28

The internal port input pairs, U . are ordered the same way as the internal port

mt ’
output pairs, so the m x 2m equations selector matrices _.S_‘, and _S, can be used to define
the m x1 constrained input U c. The constrained input U c is the result of the modular

connector power constraint (2.2b) constraining internal port input pairs to be equal and

opposite.
U... = (a. — MW (223)

Isolate the constrained input U c by pre-multiplying (2.23) by (§, — §,).
(§. - SW... = 21.0" (2.24)
The matrix 1, is an m x m identity matrix. Substitute the internal input (2.22) into (2.24)

and solve for the constrained internal input, U c .
U‘(o) = for. - anion-Rod. - s.)-

. (2.25)
[C490, t(0X0 + I(Gm¢(t,0)& + awdo - 0))U,,,(a)da]

The fully connected modular system equations are realized by substituting (2.23)
into (2.15a-2.15c).
X = aqX+3nUw +3,(§, -§,)TUC (2.26a)
Y... =c..X+2>.....U..+.e.....(§.—§.)’U‘ (2.261»
(a. — s.)c.-..X + (r. — shaman. + (a. - EJQWU. - aft!“ = 9 (2.26.)
Solving the three equations (2.26a-2.26c) for the three unknowns, X , U C, and Y“,
requires the modular system (2.5a-2.5b) to be internally output controllable where U c is
given by (2.25). The connected modular system (2.26a-2.26c) has maintained its
modularity using “nonenergic” connectors with less restrictive conditions for solution

then previous methods designed to maintain modularity (Hogan, 1987).

2.2.2 Linear Algebraic Equations
Modular modeling elements with user-defined linear algebraic equations have the

same form as (2.4) but the matrices A , B, and g, are zero.

29

y = Qt} (2.27)

Following the same procedure as in (2.5a—2.5b), consider a modular model in

unconnected form as a concatenation of n independently formulated user-defined

modular modeling elements in the algebraic form (2.27) with p total input-output power

 

 

 

 

 

 

ports. _ _ _ _
{1 "g Q 9‘ 'f'
y. g _D_, s “2
' — 5 " . Q 5 (2.28)
y. 9 9 D. ..
_"..1pxl '- —-PXPL"_pxl
Y=@U

The same input-output topology, reordering, and output constraint analysis applies to the

algebraic output constrained modular system equations. These equations are written in

terms of the external inputs, Um, the external outputs, Ya, , and the internal inputs, U W.
(§.—§.).a...v..+(§.-.S..)a....U.-.. =_.,.. (2.29a)
Y”, = @mUw + @WUW (2.29b)

Solving (2.29a) for internal inputs, U," , in terms of the Um, the external outputs

using the same techniques as the linear algebraic differential case (2.22).
-1

U...(o> = «am-.50. - a)’((§. — shaman-.50., - sf) (5. - immune)
(2.30)

The internal input, Um, exists if the matrix (5', - §,)ammam‘(§, - §,)T is

nonsingular. This matrix is nonsingular if it is positive definite or the modular system

(2.28) is internally output controllable. The constrained internal input, U ‘, for the

algebraic modular system is found following the same analysis in (2.23)»( 2.24).

U‘(a) = —%(§. — §.)~flm'(§. — s.)’((§. - anaemia. - r.)’)"(§. - a)
$11.91!... (a)

(2.31)

30

The fully input and output constrained algebraic modular system equations are
written in terms of the two unknowns constrained internal input, U C, and the external
outputs, Y“, , where U C is given by (2.31).

(§.—§.).a.....v..+(§. -§.).a....(§.-§.)’Uc =9 (232a)
Y, = amt/w + 5mg, - §,)TU‘ (2.32b)

2.2.3 Linear Algebraic Boundary Value Problem Equations

The linear algebraic equations that result from a boundary value problem (BVP)
has the form of the generalized responses, x, pre-multiplied by a stiffness matrix, Li ,

equal to the generalized excitations, w (Segerlind, 1984).
K x = w (2.33)

Without boundary conditions the stiffness matrix, K, is singular. These equations occur
in structural, solid mechanics, heat transfer, and irrotational flow models typically
developed by Finite Element Analysis (FEA). Consider a modular model in unconnected

form as a concatenation of n of these modeling elements without boundary conditions

 

 

 

 

 

 

applied. _ _ _
”_Igl 9 9 1 {51“ '81
Q E, E J‘2 W2
5 '° . Q ‘ — ‘ (2.34)
Q Q K, x, w,
- —-:xs- " -.rxl _ " dsxl
fl = W

The generalized responses, X , and the generalized excitations, W , are defined by
element dependent geometry’s which differ for each of the n modeling elements. In
order to maintain modularity, the n modeling elements in (34) will be connected through
inputs, U, and outputs, Y, that are linear interpolations of the generalized responses, X ,

and the generalized excitations, W.

31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PE]- 9 ‘0- " _f3— FPYI- —C1T Q Q 1 ”1:!“
Q {:2 x2 W2 9 C2T : “2
= '- 0 s " ' = '- 9 5 (2.35a)
Q 9 K. x, Wu 9 9 Cf n
_ —_sx:_ " _rxl — " asxl L _ 'SXPL - _pxl
.zX= W= c’U
P” 'C. 9 0 ‘ ‘1
Y2 Q 2% x2
3 ' 5 ' . Q - (2.35b)
y" 9 Q C" x"
.m‘jpxl - —_pxs—"-sxl
Y=CX

Notice the input-output matrix C is the same in both (2.35a) and (2.35b). This is
because the co-located inputs and outputs are defined as ports. BVP modeling elements
in this form (2.35a-2.35b) can be connected together maintaining modularity.

The fixed input-output functional form (2.3) of a modular model is seen in (2.35a-
2.35b). The modular BVP model (2.35a-2.35b) has an equal number of inputs and
outputs because all modular elements power ports always has one input and one output.
Each modular BVP element of (2.35a-2.35b) is completely independent and uncoupled
from the other modular elements. Given the element’s input-output topology the modular
connector constraints (2.2a-2.2b) provide the coupling between the modular element
equations.

The same input-output topology, reordering scheme, and input-output constraint
analysis applied to the algebraic differential and algebraic equations applies to the

modular BVP system equations (2.35a-2.35b). These fully connected equations are
written in terms of the external inputs, Um, the external outputs, Ya, , and the constrained

internal inputs, U ‘, using the equation selector matrices, S, and _S_',.

.zX=c...’U.,.+c..’(§.-§.)’U‘ (2.36a)
Y... = CwX (2.36b)
(so - é: )CintX = me1 (2.360)

32

The input constraint is applied immediately to the BVP equation (2.36a) where
U,“ = (§, — §,)T U C. This realization is different then the state equations (2.26a-2.26c)

and (2.32a-2.32b) because the stiffness matrix 3 is singular prior to application of
boundary conditions.

Solving (2.36a-2.36c) for the generalized responses, X , the constrained inputs,
U c, and the external outputs, Ya, , in terms of the external inputs, U ,, requires finding a
set of generalized responses that satisfy the output constraints (2.36c). Any basis of the
nullspace of (.5, - s, )Q, will define a set of output constrained generalized responses
that satisfy (2.36c). The nullspace of (§, - §,)C.,, has dimension 3 - m because it has
rank of m (Leon, 1986). It has rank of m because it represents m connectors at m
different physical locations resulting in m independent rows seen in the structure of
(s, - §,). Define a s X s — m matrix, ’9', that is a basis ofthe nullspace of (S, - _.S_',)C,,
to transform the output constrained generalized responses, X ‘, to the generalized
responses, X .

X = 12X " (2.37)
This procedure is similar to the procedure Meirovitch uses to eliminate rigid body modes
from differential equations (Meirovitch, 1967). In Meirovitch a basis of the nullspace of
the rigid body modes eigenvectors defines the transformed responses. Substitute (2.37)
into (2.36a) and pre-multiply by 12’ with the knowledge that VTCWTQ, - §, )7 = 9
because of the orthogonality of nullspaces (Leon, 1986).
v’xyX‘ = VTQTU... (2-38)

The matrix 117ng has dimension 3 - m x s - m. The matrix V combines rows and
columns of the system stiffness matrix .3 connecting the otherwise independent
element stiffness matrices g. This pre and post multiplication has the same efl‘ect as
applying the direct stiffness method. The direct stiffness method is used to build up

stiffness matrices in BVP (Segerlind, 1984). Solving for X ‘ requires the matrix
1,731) to be nonsingular. This requires the elimination of all rigid body modes,

33

which can be done by applying more output constraints or a further coordinate

transformation. The solution of (2.38) X c can be substituted in (2.37) and then used in
(2.36b) to find the external outputs Ym. Solving for the constrained internal inputs U "

requires pre-multiplying (2.36a) by (,3, — §,)Q,, .

(a. — snafu. - .S..)"U‘ = (.51. — annex - (a. - anew... (2.39)
The matrix (5', - §,)Q,,Q,,T(§, - §,)T is nonsingular because (§, — §,)C,, has rank of
m. This solves the modular linear BVP (2.36a-2.36c).

The linear algebraic differential, linear algebraic, and linear BVP constrained
modular system equations (2.26a-2.26c), (2.32a-2.32b), and (2.36a-2.36c) respectively
are written in terms of independently formulated modular modeling elements. The
reordering and selector matrices, _7_‘ and (3‘, — §,), are comprised of ones and zeros and
defined by the size and input-output topology of the modular system model. The
modular modeling element matrices all, 3, C, a, and ‘3 can be directly inserted
into the constrained modular system equations and solved like any other system of
equations. The advantage of modular modeling is that each multi DOF modeling element
is independent and can be connected together with simple connectors and assembled with

a simple physically intuitive top—down direct insertion method.

2.3 Guaranteed Computational Sparseness

Linear modular models have a guaranteed diagonal modular structure because of
their standardized input-output formulation. This guaranteed modular structure defines
an advantageous guaranteed Sparseness for computation at the cost of additional
constraint equations not found in an arbitrary input-output structure or non-modular
formulated models. These extra equations add computations to modular models but
enable an advantageous modular construction for computation.

A number of computations (NOC) analysis shows the computational advantages

of the modular construction. A NOC analysis is a count of the number of multiplications

34

and additions in a linear system.. The modular system NOC is determined by analyzing
the constrained modular system equations (2.26a-2.26c). Each of the n modular
elements has states, internal port inputs, external port inputs, and external port outputs.
Let a modular element i have 3, states, 2m, internal ports, and q, external ports. The
contribution from (2.26a) to the modular system NOC comes from the state matrix times
the states plus the internal input matrix times the constrained internal inputs plus the
external input matrix times the external inputs (Redish, 1961). Because of modular
construction the state matrix times the states and the external input matrix times the
external inputs can be computed on an element basis which is the modular advantage.
The internal input matrix times the constrained internal inputs must be computed on a

system basis where m is the number of connectors in the modular system.

(2.26a) NOC = 23,2 + ZS. .q, + In: 3, (2.403)

i=1 i=1
Modular Advantage NOC Per Elem Constraints NOE Per System

 

Similarly, the contributions from equations (2. 26b-2. 26c) add to the system NOC.

 

 

(2.26b) NOC: 2.1-q, +23, + miq, (2.40b)
Modulaijdvantage Ni: Per Element Common :1 Per System
(2.26c) NOC: mzs, +m +m2q, (2.40c)
k 3.1.1... uneasy; ‘

Summing the NOC contributions (2.40a-2.40c) together and then simplifying results in an

expression for the NOC for a modular system model.
NOCMm,sy,,m = 2(3 +q,.)2 + m 2+2m2:( (s. +q,.) (2.41)

i=1
W
ModulaAdvantageNOCPerElemeu WNiScrasm

 

The NOC for a modular system (2.41) is guaranteed by the construction of modular
models.

The guaranteed computational Sparseness of a modular system results in
comparable computational efficiency when there are a large number of interconnected

multi DOF subsystems. A non-modular system model of the form (2.4) has a NOC of

35

(s + p)2 where s is the number of states and p is the number of input-output power

ports. In this case, there are no guarantees about the system’s Sparseness. So, s is the
number of states for the entire system and p is the number of power ports for the entire
system. In other words, the NOC is determined by summing the number of all subsystem
states and ports and squaring the sum. Notice in modular advantage portion of (2.41) that
the number of multi-DOF elements’ states and ports are first squared and then summed.
Modular modeling is not, by any means, the minimal realization of a system. Other non-
modular methods can realize models for the same complex multi-port multi DOF systems
with fewer states and fewer ports with some effort. Modular models have a guaranteed
Sparseness, which results in a comparable and in some cases better computational
efficiency.

For example, consider a system model with n =10 multi DOF elements with

s, = 5 states each. A modular model requires at least m = 9 connectors to connect the 10

10
elements and, say, 2 external ports for a total of 2 p, = 20 ports. Each element will have

i=1
p, = 2 ports. Two elements will have 1 external port and 1 internal port leaving 8
elements with 2 internal ports. The modular model NOC has 3, = 5 , q, = qlo =1, and

q2-9 = 2'
10

10
NOCM,,,,,,, = 2(5 + (1,.)2 + 92 + 2(9)Z(5 + 4,.) = 1289 (2.42)
i=1 i=1

A non-modular model of the same system may require only, say 75% of the states in the
modular model s = 0.75 * 50 = 37.5 and only the external ports p = 2 . The non-modular

model NOC, with 75% the modular model states and a tenth of the modular model ports

is larger.

NOCN _,,,,,,,, = (37.5+2)2 = 1560.25 (2.43)

Consider the same system but with n = 2 elements, m = 1 connector, and external ports
q, = q2 = 1 for the modular model. The NOC for modular and non-modular models

become closer in magnitude with the non-modular being less.

36

2 2
N0C,,,,,,, = 2(5+1)2 +12 +2(1)2(5+ 1) = 97 (2.44a)
i=1 i=l

NOCN,,_M,,,,,, = (7.5+2)2 = 90.25 (2.44b)

The above NOC examples show cases where the computational efficiency for modular
models is better and comparable then non-modular methods. Guaranteed computational
Sparseness makes modular modeling one of the modeling methods of choice for large

systems.

2.4 Examples

An electric automotive vehicle power train example model is used to illustrate
modular modeling of linear algebraic differential modular models and linear algebraic
modular models. A structural three bar and connecting shear pin fixed to a point example
model is used to illustrate modular modeling of linear algebraic BVP modular model. The
electric automotive vehicle power train example model is a combination of three simple
dynamic models found in Phillips and Harbor, 1996 and Minor, 1996. The three bar and
connecting shear pin example model is a combination of four simple BVP models fi'om
Segerlind, 1984.

The electric automotive vehicle power train example model contains three simple
modular modeling elements, an electric motor model, a clutch model, and a transmission
model. The three modular modeling elements are all defined with standardized modular
modeling causality (I able 1.1). The electric motor modular modeling element has one
electrical power port with current input i voltage output e causality and one rotational

mechanical power port with torque input I angular velocity output a) causality (Fig.

:5, = [0])., + [514 51:] (2.45a)

[211313313

2.3a).

37

The motor model has a state, Jr”, a lumped motor-shaft rotational inertia, J , a motor coil
resistance, R , a motor back emf constant, K 1,: and a motor torque constant, K,. The

transmission modular modeling element has two rotational mechanical power ports with

torque 1', and 1,, input angular velocity to, and to, output causality (Fig. 2.3b).

O T
x,=—ix,+ g i “ (2.46a)
J J J, r,

0) ”GR v 0 0 t
[wzHllxvio 011:] W
The transmission model a state, x,,, a lumped transmission rotational inertia, J”, an
equivalent transmission damping, 0,, and a transmission gear ratio, GR . The clutch
modular modeling element has two rotational mechanical power ports with torque 1",, and

1, input angular velocity co,I and to, output causality (Fig. 2.3c).

 

 

. 0 o —Kc' i 0
xcl KJcl cl Jcl 1 1'
x,2 = 0 0 6' x,2 + 0 — "' (2.47a)
' Jcl Jcl T!
x., 1 -1 0 x,, 0 0
co, 100x"+00¢, 247,,
62,—010x‘2001, (')
x

03

The clutch model has three states, [x,l,x,2,x,3]7 , a lumped clutch rotational inertia, J,,,

and an equivalent torsional stiffness, K,, .

 

a) b) c)

38

Figure 2.3: Electric Automotive Vehicle Power Train Modular Modeling
Elements a) Electric Motor b) Transmission c) Clutch
These three modular modeling elements are assembled in two different
configurations to illustrate the flexibility of modular modeling. The first configuration
considers a power train with the modular motor modeling element and modular

transmission modeling element (Fig. 2.4).
1 2 3 4

fig]. 1):". -_>._ .1 , '1'." ii§.-‘.fi.'- .- .Ijra' “ 1.3:: c Rfil‘n: x :1 (h a" -,... .faj‘fi
—, electric motor 9

‘6. ‘ : : C , , '
wt war untrue-W1! wry-Load nirvana! 3 1

   

Figure 2.4: Electric Automotive Vehicle Power Train Modular Model With An
Electric Motor And A Transmission Modular Modeling Elements
The unconnected two element modular system is built up'from directly inserting the

motor and transmission modular element matrices into (2.5a-2.5b).

 

 

 

 

 

 

 

,-,_[_2_.]= .sz 3U:[9.5.__9__.I§g]{Min-.1111"? ..... 9-1 (2.43.)
; 0:-C.,/J.. x. 0 0 :GR/J. 1/J._ T.
. _r._
fe' '19,! 0' "R 050 01’1“
Y: 59- =CX+@U= -1--i--9-- ”51+ 3342-9 5- (2.48b)
a), OgGR_x, 0 0:001,
Law,“ _0: 1_ _0 0:0 O__r',_

 

 

The input-output topology of the two element power train modular model defines
the size and structure of the modular modeling reordering and equation selector matrix 1‘

and (§, - §,) respectively. This modular model has four power ports and one connector.

This results in a 4 x 4 reordering matrix 1' and a 1x 2 equation selector matrix

(ta-5.).

1"]
ll

---------- . (3.3.141 —11 (2.49)

 

 

39

Ports 1 and 4 are the external power ports so I reorders the input U = [i, 1', I“, TAT and

output Y = [e, a), a)“, (0dr so i, rd, e, and 0’4 appear first in the vectors. Ports 2 and 3

are connected internal power ports reordered to appear in connected pairs afier the

external ports. Apply the reordering matrix I to the modular model (2.48a—2.48b) to

partition the inputs and outputs internally and externally.

[£1351] = T’U = 3‘1.

1

T
T

-61

 

a":

(0
La)

 

 

o :—1/J
1/J. E 0

0T

 

amen

T TT =
‘0‘ [22...

am]:
a...

 

 

0
GR/ 1,,

 

l

(2.50a)

(2.50b)

(2.50c)

(2.50d)

(2.50e)

The two element electric automotive vehicle power train modular model (2.48a-2.48b) is

now in the form to apply modular modeling constraints (2.23-2.2b).

Apply the internal input and output constraints to the two element electric power

train modular model (Fig. 2.4) and rewrite in the fully connected modular system form

(2.26a-2.26c).

5c: 04X+£nUw +3..(§. —§.)’UC

x'm_o o x, K,/J o i -1/J
[Jio -c,,/J,,Ix,]+[ o l/JIrd]+[ o

40

O

GR/J"

(2.51a)
lun‘

=C¢er + anarU ext + $mim(§a -' £8 )T UC

53+: m M :r]

(S0 _S e)c7mX+(So -5 e)'ameerw+(So— —S e)'9mrinr(So— —S er) Uc =9

[1 it (Eel: ] v «[3 EL] [1 «[3 31:11.2

The internal input uzf exists if (2.51a-2.5 1c) is internally output controllable or the
matrix (2.21) is positive definite. Since am." is zero, in this example, showing C213":
has rank equal to the number of internal outputs shows positive definiteness of (2.21).

This is typical practice to show output controllability (Skelton, 1988).

_ l 0 -1/1 0 _ —1/J 0 252
““73" ’ 0 GR 0 GR/J, ’ o GRz/J, ( ' )
Clearly CW3" has a rank of 2, which is the number of internal outputs. The internal

input u23‘ exists and the modular two element electric vehicle model (2.51a-2.51c) can be

solved like any other state system model.
The second configuration considers a power train with the modular motor
modeling element, a modular compliant clutch modeling element, and modular

transmission modeling element (Fig. 2.5).

 

1 2 3 4 5 . . p v 6
u] EM»? WWW?“ 2 f'\\ 2'hfrnfég “2 c rim-{F -Fs‘; 534g! u45c ‘ ; «maxim yrfifigflhnlew ”6
# eIeCIric motor _°clutch— gransmzsszonb
. ' 3;" . I;
y] fit“? ‘3: w: Lava fry: Tit-92$ y23c bar 22; "3:975; yljc in: i r -;~ .27 re «and y6

Figure 2.5: Electric Automotive Vehicle Power Train Modular Model With An
Electric Motor, A Compliant Clutch, And A Transmission
The unconnected three element modular system is built up from directly inserting the

motor, clutch, and transmission modular element matrices into (2.5a—2.5b).

41

 

 

 

 

 

 

 

 

 

 

 

 

 

] (2.5411)

0
0

=1

 

\
J

‘
C
“4:

l

42

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o w m a w
o... a a m. m.
121 a m 0. O
]
-l T— M 95— Uiflu y ‘3...“
_ _T 7.1 T. l u u ]
1 i- C C
q . . f. .l T- M Tor-Tu d m ][U u” a“
O _O 0 Orl_Jl— _T . T. e 0 0 1 _ .Ill[|l.l_
“ u . . m 0 0 l l
. . 0 Ono Ono 0 m 1 4 0 0 .
. _R or. o _ .
0.0 00 JOOHOOHOO w . R.1.1_.00
_ . IIIII TIIIIL IIIII o 0 0 0 06—]! fll]|__
11111 1111111111 11111
. 1. 0 Ono Duo 0 m 0 0
0.0 1_Je0.0 . . m 0 O 1_ doc 00
u u 0 0“0 0."0 0 d J
_ I _ IIIII _ ''''' . lllll 0 O
O "1_Je0 Ono 00.00.00 m, 01_J.e0 00
Illl1 llllllllll TIIII . . m 0 O
1_J_ _ R Ono Ono 0 1_J +
. nu “U n! n. . . . _ n. no nvnu 4|:
- u u + p b .1- fed
91 _ _ 1 . l 2 3. J 1%- fl
K_J" O O 0" 0 xn.x.o xa xa_x1 w 1 11
_ L F. _ . t . r L . R 0
+. n . . 1J u .U nu no.0 nYl_I. rIIIL
1. . _ o 0.0 cum 1 m. ..
m l..r In. In. t l . r
XHXc Xc Xcux IIIII "11111..“ IIIII O . pun—JO 0 0 0 0 . x. xd X... x
__ ..00.00.00.m® + ...
_ m" .r. . _ 6 . . O 1
n3 no AU nvc_rJ . . .t n2 . 4 AW 4 1
I” u _ O Ono luo 0 m1 fil-m PX x x x x. 0 0
1. 1111111111 +1111 00.10.00 .m a. 1.. 00
_1. 1. . 11111 r11111_ 11111 6 0 0 0 Dee—1
OKCJCu no.0 . _ 4... 2. . oo
_ K J _ Kbluo Duo 0 m 2
n _ u _ . .m ,m. (firm A 1 rm.u
. _ __ . 0 KJ..0 0 _ .
O" 0 0 1:0 _ i m. cm _ ._
m r u d
_ _ e w. . l . J
n: no no .l_nU .11 _ m. w 1w “w. .m m .0 no no . .0 e MW
. . y .m =
o ano XI...” 0 1mm 0 Inner". x11 I” 1m .o n d. o .m. o d. r.
_ _ x x. _ A m Fx x X x x.

 

x4

-Fx.
1 0 () 0 O
() l O 0 0
O () l 0 O

f
K

--100‘
001—11

1

The internal input [112,2 11,; ]’ exists if (2.54a-2.54c) is internally output controllable or

the matrix (2.21) is positive definite. Since am is zero, in this example, showing
C1113» has rank equal to the number of internal outputs shows positive definiteness of

(2.21).

l

C1113," =

OOOH

 

OOHO

Or-‘OO

COCO

 

-— o 0

J
0 q 0 i o
0 Id 1
0 0 0 —
Jcl
GR. 0 o 0
0 o 0

(2.55)

 

 

 

 

Clearly Q3,” has a rank of 4, which is the number of internal outputs. The internal

input [u23‘, 1145‘]? exists and the modular 3 element electric vehicle model (2.54a-2.54c)

can be solved like any other state system model.

The linear algebraic modular model example equations are defined by taking the

Laplace transform of the linear algebraic differential model equations. The two element

electric automotive vehicle power train modular model equations (2.48a-2.48b) after a

Laplace transformation have the form of the unconnected linear algebraic modular model

(2.28).

Y

@U

 

 

 

 

'K,K,/J+R Kb/J
S S
K,/J —l/J
--..--.5' ........ S --_
o o
o o

 

 

 

I

1 0 0

I

l

5 o o

.- ———————————————
1 01171,, GR/J,

E s+c,,/J,, s+c,,/J,,
: 0122/], IN,

1 s+c,,/J,, s+c,,/J,,J

 

 

(2.56)

The two element linear algebraic electric vehicle power train modular model (2.56) has

the same input-output topology (Fig. 4) as the linear algebraic differential system (2 .48a-
2.48b). So, (2.56) can be reordered and constrained with the I and (£0 — S.) in (2.49) to

obtain the fully connected linear algebraic modular model form (2.32a-2.32b).
(.s... — gnaw... + (r. — emu-Ar. - i.)’U° = 9

K,/J

[1 -1

o
GR/J,

T11 +[1 -1

s+c,,/J,,

43

:u

0

018/1,
5' + c” / 1,, _

q

 

(2.5711)

T

Yer! = exrexIUexr + $extint (£0 _ ée) UC
K,K,/J+R Kb/J
e _ ——S——— 0 ,- + S 0 1 6 (2.5711)
60.1 - o _1/_Je_ 1' 0 _QML -1 “23

CH

s+c./J. s+c.,/J.

 

The internal input uzf exists if (2.57a-2.5 7b) is internally output controllable or the
matrix (S, - §,).a,m&m‘(§o - §,)T is positive definite.

_U] 0 —UJ 0
a T S S 1
_ . . - = 1 -1
(£0 §.)$m&m1 (~31 251) l 0 GRz/J" 0 GRz/J, -1]
s+c,,/J,, s+c,,/J,,

_ l/J2 + 0114/1;

> 0
S2 (s + c;,/J,,)2

 

(2.58)
The internal input uu" exists and the modular linear algebraic two element electric
vehicle power train model (2.57a-2.57b) can be solved like any other linear algebraic
system model. A similar analysis can be done for the modular three element electric
vehicle power train model.

The linear algebraic BVP modular model example is a structural BVP model with
three bars and connecting shear pin fixed to a point. This structural modular BVP model
example contains three simple modular modeling elements, an axial force bar model, a
bending pin model, and a fixed point model. The three modular modeling elements are
all defined with standardized modular modeling causality (Table 1.1). Structural BVP
models do not have traditional power ports because the output variable is a displacement
instead of velocity. They are representations of physical points with two variables that
require a causal definition, which fits the application of modular modeling and the
measurement perspective standardized causality (Byam and Radcliffe, 1999).

The axial force bar modular modeling element is a two node Finite Element

Model (F BA) from Segerlind, 1984. This model has two mechanical ports with force

input F axial displacement output 5 causality (Fig. 2.6a), which is written in linear

algebraic BVP modular modeling form (2.35a-2.35b).
Ex = Q u

AE 1 —l x, _ 1 o F, (2.59.11)
71—1 tlxi-[o in]

y=§x

6, _ 1 o x, (25%)
[62]_[0 11%]

The modular bar BVP (2.59a-2.59b) has two states [xv xb]T, a cross-sectional area A , a

modulus of elasticity E, and a length L.

The bending pin modular modeling element is a three node FEA from Segerlind,
1984. This model has three mechanical ports with force input F transverse displacement
output 5 causality and three mechanical rotational ports with torque input 7: angular
displacement output 0 causality (Fig. 2.6b), which is written in linear algebraic BVP
modular modeling form (2.35a—2.35b).

 

 

 

 

 

 

_{c=_C_‘th
r12 6L —12 6L 0 on; '1 o o o 0 0‘7;
6L4L2—6L2L20 011,01000013
E]—12—6L240—126Lx,001000F4
76L 2L2 0 8L2 —6L 2L2 x,=0001001,
00-12—6L12—6Lx8000010F,
_o 0 6L 2L2 —6L 4L2J_x,,_ _o o o o o l__'t,‘
(2.60a)
y=£§
”63‘ '1 o o o o 0121,]
9, 01000011,
6, o 010 o o x, (2.6%)
0,=000100x,
5, OOOOleg
-95. _o o o o 01__x,,_

 

 

 

 

 

45

The modular pin BVP (2.60a-2.60b) has 6 states [xv xd, x,, xi, x8, xhr, a modulus of

elasticity E, a length L, and a 2“d moment of area I.

The fixed point modular modeling element is unique to modular modeling
because of the standardized measurement perspective causality. The one port fixed point
modular modeling element outputs a zero displacement regardless of the input and
applies to any energy domain (Fig. 2.6c). The fixed point modular modeling element is
written in linear algebraic BVP modular modeling form (2.35a—2.35b).

Ex = 9’ u
' " (2.61 a)
[11x1 = [011%
y = 9’ x ’
~ " (2.61b)
66 = [Oixi

The modular fixed point modular modeling element has one state x,

 

Figure 2.6: Three Structural Linear Algebraic BVP Modular Modeling
Elements: a) Axial Force Bar b) Bending Pin c) Fixed Point
Two example linear algebraic BVP modular models will use the three modular
modeling elements (Fig. 2.6) in two configurations. First, three axial force bars and a
fixed point will be connected together in a bracket-pin-clevis configuration with no pin
bending deflection (Fig. 2.7). Second, the three axial force bars will be connected with

the bending pin (Fig. 2.8).

  
  
  

 

“3562‘“ ‘ “6
_. . . .1 1 2. 3 1111,:—
"i 1 i" y c-.. y
fixed porn ,1; bar, .5 35c 6
e13 ban; i5—
y47 @‘uf’tk121‘éfl‘. ya
7 8

Figure 2.7: Three Axial Force Bar And A Fixed Point Modular Modeling
Elements In A Bracket-Pin-Clevis Configuration With No Pin
Deflection
The input-output topology of the three bar no pin modular model shows ports 6
and 8 are external ports and three connectors. The internal ports are connected in pairs 1

to 2, 3 to 5, and 4 to 7. This simple topology analysis defines the reordering matrix I
and the equation selector matrix (g, — §,) .

to 0 0 0 0 1 0 0‘

0 0 0 0 0 0 0 1

i"6"6'0"0"6"0"0 ,
01000000 140000

1:6...6 ..... 100000, (£0-§¢)= O 0 1 _1 0 0 (2.62)

00001000 00001—1
0""0W0 ..... 1""0W0W0W0

_0 0 0 0 0 0 l 0_

 

 

The first two rows of I identify the external ports and the remaining six rows order the
internal ports in connected pairs.

Build up the unconnected modular model from the three axial force bar and fixed
point modular modeling elements and write in linear algebraic BVP modular form

(2.35a-2.35b).

47

 

 

 

 

 

 

 

 

 

 

 

”0:0 0 0:0 0:0 0'“1
_ Ix ,' 07'1'"0’0'F0"070"0 “2
rfi...:0:0:0_f‘f_ : : : u,
----- 1.----.1_-__—1.—_—- x 0'0 1 1'0 0'0 0
0 :Lm: 9 : 9 1..., --: -------- u. (2.63a)
..... 1.----1—_-__e_-—- ——-- = 010 0 011 010 0
9 i 9 15m: 9 11..., l l l “5
“o“r‘o“"."6“rk"‘ 9-1.9-.<1-9_19--1.19.-9
~ ‘ ' ‘ ' " '_b“"-x"_“’3 0:0 0 0:0 0:10 "‘5
" "' I I I
_0:0 0 0:0 0:01_“’
_“11_
Y=cx
"u,' ”0:0 0:0 0:0 0“
"I-"'-T"""I """
u2 0:10:0 0:0 0- -
I I I xfixpt
u3 0:01:00:00__.__.
u, 0:01:0 0:0 0 x1... (2.63b)
= --L----L----4 ..... -;;_
115 0:0 011050 0 x11,
11, 0:0 0:01:0 o 33‘
--P-"-f'--'fl ----- 3
117 0:0 0:0 0:10- " .
_u,d _oio oio 010 1_
The K matrices and x vectors in (2.63a-2.63b) are given in the modular modeling

element equations (2.59) and (2.61). Note that the Q matrix for bar: has an extra row to
accommodate the input output ports 3 and 4 on the right hand side of the bar. These two
ports occur at the same geometric point so they have the same output and the inputs sum
in the modular elements internal junction structure (Byam and Radcliffe, 1999). Apply
reordering and input output constraints and rewrite (2.63a-2.63b) in constrained BVP

modular form (2.36a-2.36c)
xx = cm’U... + cits. — s.)’U‘

 

 

 

0 0 '0 0 0
-'xfi, - 0 0 -1 0 0
HQ” 9 Q 9 ~11 0 0 0 l 1 " c
0 r... 0 9 ‘56 'u. “'2, (2.6410
0 O K 0 x = O 0 + O -l 0 1435
" """" ‘ b” 1 0 3“ 0 0 0 c
o o o r... x" 3‘”
_ - — — 231‘ 0 0 0 0 -1
_0 l_ _0 0 0‘

 

 

 

 

48

xfixpt-1
[y6]_[0 0 0 0 1 0 0' ‘er (2.64b)

y8 0000001_xbar2

~

xbar,

~
- d

 

 

(£0 _ g: )Cian = Qua/2x1

xfixpt
0—10 0 0 0 o'xb; (2640)
001—1000~=0 '

xbarz
00100-10_~ 0
xbur,

~
- _

 

 

Find the basis for the nullspace of (go — ,S_‘

J

(2.65)

ooooot—g,
oer—coco?
o~o-oo
~oooooo

 

 

0

At this point the linear algebraic BVP modular system (2.63a-2.63b) can be solved using
(2.37), (2.38), and (2.39). Linear algebraic BVP modular modeling elements were
directly inserted into the modular model with given input output topology. The input
output topology determines the reordering and equation selector matrices. The modular
system model is rewritten in constrained form and the basis of the nullspace of the output
constraint is found with a computer software package. The model is solved like any other
linear algebraic BVP model.

49

Figure 2.8:

 

Configuration

Three Axial Force Bar, Two Fixed Point, And A Bending Pin
Modular Modeling Elements In A Bracket-Pin-Clevis

The three bar bending pin modular BVP model (Fig. 2.8) is solved similarly to the

previous model. There are two fixed point modular modeling elements in the modular

model because the bending pin modular modeling element requires a rotational constraint

to eliminate the rotational rigid body mode. The modular modeling element fixed point2

eliminates the rotation.

The matrices in this example model are quite large and will be present separately.

The input-output topology of the three bar bending pin modular model shows ports 10

and 12 are external ports and five connectors. The internal ports are connected in pairs 1

to 2, 3 to 5, 4 to 9, 8 to 6, and 7 to 11. This simple topology analysis defines the

reordering matrix I and the equation selector matrix (so — § ).

0

fl
0

0

0

0

0

0

1

0

 

 

 

 

 

OOOOOOOOOF‘O
OOOOOOOOWOO

 

OOOOOOOHOCO

COOOOt‘OOOOO

oooooo~ooooo

COG—00900060

CHOOOOOOOOO

COHOOOOOOOO

COCO—‘OOOOCO

OOOOOOOOOOO

HOOOCOOOOOO

OOOOOOOOOC"

0

CO

coo—o
o
oo—oo
.L
cuooo
o
~oooo

.(é. — s.) (2.66)

II
OOOOH
COCO
O
OOOO

 

50

fy the external ports and the remaining ten rows order the

The first two rows of T identi

internal ports in connected pairs. The unconnected ‘3 matrix is built up from the

modular modeling elements _Ig matrices in (2.59a-2.59b), (2.60a-2.60b), and (2.61a-

2.61b).

 

)
7:
,D
04
(
- - — o - n d
_ _ . . . 3
.U__ . .. 2U: W
_ _ . _ .vfi
IIFIIFIIFIIFILII.
. . _ _ _
_ _ _ u m;
_ . .
_ _ . . __nx
_ . _ _ .
1141111111111411.
. _ . m. _
_ _ _ P. .
. _ _ _. .

_ _ . . .
1141111111111411.
. _ h. . .

_ . P.." n.
. . . .
_ . fl: _ _

_ _ T _ .
11p11r11r11F1L11
. _ _ . .

. n. . . _

n2. w. u u .

_ _ .
_VA. _ _ _
1141111111111411.
n. . _ _ .
P. . . _ _

£0: _. . _ .0.
. . . .
vA. _ _ . .
— L
_

 

(2.68)

 

001

“T“-"T“I"'"""-"""""-"""'T"'"-I----'

I) O

I) 0 I) O I) 0

O

00

O I)
9.12.2-9.
0 0
0 0
0 0

O I) O I) O I) I) O
O I) O I) O I)

_-_-------___-_L

O
L

l 0

O

 

The C matrix and the 1} matrix complete the model.
'0

0 l

51

0

O
A

(2.69)

OOOOOOOOOOOOOr—
OOOOOOOOOOv—OOO
OOOOOOOt—OOOOO
OOOOOOHOOOHOO
OOCOr—OOOOOOOOO
OO—‘OOOOOHOOOO
OOi-‘OOOOOOOOOOO
CHOOOHOOOOOOOO
HOOOOOOOOOOOO

O
O
O

 

 

— d

Note that in the C matrix the _Q matrix for the bending pin is missing the 2ml and 6‘”

rows because those inputs are zero. Using (2.37), (2.38), and (2.39) this example model
can be solved like any other linear algebraic BVP model.

52

Chapter 3

ASSEMBLY AND SOLUTION OF NONLINEAR MODULAR MODELS USING
FIXED INPUT-OUTPUT STRUCTURE

3.1 Nonlinear Modular Modeling Analysis

Nonlinear modular modeling analysis is a systematic direct-insertion realization
of nonlinear modular models of the form (2.3). Using a known input-output topology
constrained by the output and power constraints of the connector (2.2) the constrained
modular model is realized. Nonlinear modular modeling elements are independently
user-formulated power-based multi-port multi-degree of freedom nonlinear modeling
equations with a standardized input-output causality and sign convention. Connectors are
two-port output and conservative power constraints between modular modeling element
ports. Modular connector constraints are known. Modular elements have a known form
but their equations are user-defined.

The possible user-defined nonlinear modular element equations are nonlinear
algebraic differential equations and algebraic equations. The nonlinear algebraic
differential equations can be represented in a form convenient for the application of the
modular modeling fixed input-output functional form. A state-space form is the best fit
because inputs and outputs are explicit. Any algebraic differential equation model can be
written in state-space 'for'm. Algebraic equation models are typically expressed in an

explicit input-output form.

3.1.1 Nonlinear Algebraic Differential Equations
Modular modeling elements with user-defined nonlinear algebraic differential

equations have a traditional state-space form where the inputs and outputs are explicit.

53

5c = f (x, u)
” " ' ” (3.1)
y = 80}. If)

The fixed input-output functional form (2.1) of modular elements is seen in (3.1), where
u is a vector of port inputs and y is a vector of port outputs. x is a vector of states, f

and g are vectors of functions of the x-variables and u-variables.

Consider a modular model in unconnected form as a concatenation of n
independently formulated user-defined nonlinear modular elements in state space form.

Let this model have 3 total states and p total input-output power ports.
F o '

 

 

 

 

 

 

 

 

{I F.6(xjr 1:1)-

£2 = (202,1?) '1‘

'5 ‘ (3.2a)
1;" $19.16.)
.. " usxl h ‘8)“

_ x=_r(x, U) _
'12 €109,111)
>:2 5.2%?)

' _ 3 (3.2b)
ya gn(xn’un)
_~-pxr -~ ~ - -pxr

Y= 9(X, U)

For example, the modular model vector X is the concatenation of the n modular element
state vectors with a total size s x1 and U is the concatenation of the n modular element
input vectors with a total size p x 1. The fixed input-output functional form (2.3) of a
modular model is seen in (3.2). The modular model (3.2) has an equal number of inputs
and outputs because all modular elements power ports always has one input and one
output. Each modular element of (3.2) is completely independent and uncoupled from
the other modular elements. Given the element’s input-output topology the modular

connector constraints (2.2a-2.2b) provide the coupling between the modular element

equations.

54

The key concept of modular model analysis is isolating the internal element input-
output power ports from the external element input-output power ports in a known input-
output topology. External input-output power ports have known inputs. Internal input-
output power ports are ports joined to other element ports through connectors (Fig. 2.2).
The input-output topology of (3.2) has a total of p input-output power ports from the n
modular elements. Let m be the number of connectors, hence there are 2m internal
element input-output power ports, which leaves q = p — 2m external element input-
output power ports. The standardized form of modular elements (1) and modular
connectors (2.2) makes isolating external and internal element ports in the unconnected
modular model (3.2) a simple reordering of the systems’ concatenated input and output
vectors U and Y.

A transformation matrix reorders the vectors of (3.2). The vectors are reordered
so all external ports’ variables appear first in the vectors followed by all the internal

ports’ variables. The internal ports’ variables are ordered such that connected port pairs’
appear together. For example, if port i and port j are connected u, should be followed

immediately by u I. , similarly for the outputs y, and y j. The reordered input and output

vectors are the input and output vectors of (3.2) pre-multiplied by the transformation

matrix I. _
Uar
[U = Um] (3.3a)
FY“
IY= .Yw] (3.3b)

 

The transformation matrix I does not add, remove, or combine variables of the original
vectors; it only changes the order in which variables appear. This makes 1 a linear and
nonsingular reordered p x p identity matrix.

LIT = l (3 .4) ‘
The external input Um is the q x1 vector of external port inputs. The internal input U,"

is the 2m x1 vector of internal port inputs. The external output Y“, is the q x1 vector of

55

external port outputs. The internal output Yin, is the 2m x1 vector of internal port

outputs. The mechanism for reorganizing the input-output topology of (3.2) to isolate the

external and internal ports is the matrix (1),,”-

The reordered modular model equations are written in terms of the external and

internal inputs and outputs where U = IT[Um,UW]T.

 

. in“:
X=f(X, 1: U ) (3.5a)
TY- Y‘” -T X :T-FU‘“ 35b

Using the transformation matrix I the function vector 9 in (3.5) is partitioned to firrther

isolate internal and external equations. The function vector 9 is partitioned by 1‘ into

Io=[°“‘] / (3.6)

an
The external partition of the 9 vector 9w is a q x1 vector. The internal partition of the

external and internal outputs.

9 vector 9," is a 2m x1 vector. The reordered modular model equations are rewritten in

terms of the external and internal inputs and outputs and the partitioned output function

VCCtOI'.

22 = ear, 1’[ “']) (3.7a)
Y... = 9....(X. IT ) (3-7b)

Y2»: = 96.,(X. .7." ) (3-70)

 

 

The internal outputs Yin, of (3.7c) are ordered in connected internal port pairs. At

each connected pair, the two output values are constrained by (2.2a) to have equal values.
In order to apply the outputs of (3.7c) to the constraint (2.2a) the connected output pairs
need to be selected from the matrix equation (3.7c). Define two m x 2m equation

selector matrices §0 and .3, to select the odd and even equations of (3.7c) respectively.

56

"1000 00' ”0100 00:
0 . 1 . . . E O . 1 . .
so: 5 E 0 E E E §¢= E E E 0 S E (3.8)
E E E I O E E E E E E 0
L0 0 0 0 1 0.....2... _0 0 0 0 0 1-....6...

 

 

 

 

Pre—multiplying (3.70) by Q, selects the odd (1", 3", 5m, , m - 1“) internal port output
equations. Pre-multiplying (3.7c) by Q”, selects the even (2“, 4'”, 6‘“, , m‘“) internal
port output equations. Substitute selected equations in the output constraint (2.2a).
sax," = Y‘ (3.9a)
,5er = Y‘ (3.9b)
The constrained internal ports’ output Y c is a m x1 vector of the m constrained inputs at
the 2m connected internal ports (Fig. 2.2) of the system (3.2a-3.2b). Rewrite the output
constraint (3.9a-3.9b) in terms of a difference to eliminate the constrained internal port’s
output.
(.5. - £.)Y.-... = 9,, (3.10)
Substituting the internal output equation (3.7c) into the internal output constraint (3.10)
results in the output constraints of the modular model.
(§.—§.)a..<X.z’[g“]>=0 (3.11)

The three output constrained modular system equations are written in terms of the states,
X , the external inputs, Um, the external outputs, Ya, , and the internal inputs, Uh, .

1:101, zT[Zf”]) (3.12a)
TUm
Yw=9w(X,1_‘ U. ) (3.126)
(s —s) 1”“ -
_0 —¢ ”(X91 U )‘Q (3.120)

The internal port input pairs, U m , are ordered the same way as the internal port
output pairs, so the m x 2m equations selector matrices _.So and .5, can be used to define

the m x1 constrained input U ‘. The constrained input U ‘ is the result of the modular

57

connector power constraint (2.2b) constraining internal port input pairs to be equal and
opposite. The constrained internal port’s input U c is a m x1 vector of the m constrained

inputs at the 2m connected internal ports (Fig. 2.2) of the system (3.2a-3.2b).

U... =(r. —§.)’U‘ (3.13)
Substitute the internal input (3.13) into (3.12a-3.12c).
. U
._ r an
X - f(X, I [(550 -§¢)TUC]) (3'14a)
Y — (x T’ U‘” 3 146
ext-ax! ’— (§0_§e)TUc ) ( ' )
(s s) (X TT U“ )—0 3140
_0 —¢ 3n: ’— (§O_§‘)TUC —— ( ' )

The three output constrained modular model equations (3.14a-3.l4c) have three
unknowns, the states, X , the external outputs, Ya, , and the constrained internal inputs,

U ‘. The objective is to find the constrained internal input, U C, in terms of the states, X ,
and external inputs, Um, that satisfies (3.14c).

The nonlinear modular system constraint equation (3.14c) is implicit in terms of
the constrained internal input, U ‘ . Rewrite (3. 14c) in terms of an error subtracting the
right-hand side from the lefi-hand side.

— S S X Tr Um
O-(—o-_¢)am( 9_ (§0-§¢)TUC:|) (3.15)

The internal output constraint error, a, is a m x1 vector of internal output constraint
errors. These errors are a measure of the amount the constraint is violated at each
modular connector. The internal output constraint error, a, is driven to zero by the
constrained internal inputs, U ‘. There are many approaches for solving (3.15). One
Option is using an iterative approach like Newton-Raphson (Burden and Faires, 1985).
Another option is a physical approach defining compliant functions to hold the error at
zero. Modular modeling chooses a control logic approach.

A control logic approach to driving an error to a zero set point is a control

stabilization problem (Khalil, 1996). Define a new m x1 constraint state vector, X ‘, as

58

the integral of the internal output constraint error, e. Let the constrained internal inputs,
U ‘, be the results of multiplying constraint state vector X c by a m x m matrix of constant
gains, _Ig',, (Fig. 3.1).
2% =. (3.16a)

U‘ = 5,)? (3.16b)
The internal constraint stabilizer (3.16a-3. 16b) defines the constrained internal inputs that
zero the internal output constraint error, e. Rewrite the internal constraint stabilizer by
substituting the internal constraint stabilizer output (3. 16b) into the internal output
constraint error (3.15) then substitute into internal constraint stabilizer state (3.16a)
eliminating the unknown, 0.

Um
XI: = (£0 '§e)5m(X’ ZT[(S _S )TKIXC]) (3'17)

The internal constraint stabilizer (3.17) may be solved for the constrained states, X ‘, in
terms of the states, X , and the external inputs, Um. The constrained internal inputs, U C,
that satisfy the nonlinear modular system constraint equation (3.14c) are found from
(3.16b).

e XC Uc
K, V

 

 

 

U“, X
T
(§°-'S")"""(X"T' [(2.—r.)’1_¢,X‘]) U...

Figure 3.1: Nonlinear Modular Modeling Internal Constraint Stabilizer Block
Diagram
The internal output constraint error depends on the states, X , which depends on
initial conditions. In this environment where the error is dependent on joining outputs of
models presetting the initial conditions would not be unusual because knowledge of the
modular elements initial conditions exists. Presetting the initial conditions requires the

knowledge of all the states in of all the modular elements in the modular system model
(3.2a-3.2b).

59

The internal constraint stabilizer (3.16a-3.l6b) is defined as dynamic function
with integral control logic. Using the integral control logic does three things. First,
integral control provides a robust stabilization if there are no perturbations in the
system’s parameters (Khalil, 1996). Since the internal constraint stabilizer is regulating
the outputs of models, there are no parameter perturbations in system parameters unless it
is part of the model. Second, it avoids having to define a physical compliant function
with “fast” eigenvalues to quickly reduce the error resulting in stiff systems (Aiken,
1985). Third, no iteration is required so the constraint equations are combined and solved
with the system equations. Combining the internal constraint stabilizer (3.17) with the
constrained modular system equations (3.14a-3.14b) results in the fully connected

modular system equations, which can be solved for the unknown states, X , the unknown
constrained states, X ‘, and the unknown external outputs, Y“, , in terms of the known

external inputs, Um.

. U
_ T at
Um
o Um
X" = (r. —§.)a..<x. r’[( S _ S )1 K, A) (3.180

The stability analysis of the feedback system in the nonlinear algebraic
differential modular system equations (3.18a-3.18c) is done with the passivity approach.
A passive system does not generate its own energy. The system absorbed energy must be
greater than or equal to the system stored energy. If the absorbed energy is greater than
the stored energy the difference must be the dissipated energy by the conservation of
energy and the system is strictly passive. A passive system is globally uniformly stable
and a strictly passive system is globally uniformly asymptotically stable (Khalil, 1996 &
Krstic etal, 1995). A feedback system is strictly passive if both systems are strictly

6O

passive and passive if at least one of the systems is passive (Khalil, 1996 & Krstic etal,
1995)

Stability proofs for feedback systems from Khalil and Krstic etal can be easily
applied to the nonlinear algebraic differential modular system (3.18a-3.18b)
nontraditional feedback connection with some simple relationships. These proofs use the
product of input and output and an instantaneous power inequality to show passivity

(Khalil, 1996 & Krstic etal, 1995). The nonlinear algebraic differential modular system
has a product of input and output with two inputs, the external inputs, U m, and

constrained internal inputs, U ‘, and two outputs, the external outputs, Yw , and internal

output constraint error, 0.
WU: TT 1"” 1" U“ =[Y T .7]T1‘T U“ =Y TU +cTU‘ (319)
Uc a! UC at an '

The integrator feedback system’s input is the constraint error and outputs the constrained
internal input to the nonlinear algebraic differential modular system with a sign change
due to the standardized sign convention of modular modeling (Byam & Radcliffe, 1999)

Wu =-U". (3.20)
The input and output products for the algebraic differential modular system (3.19) and the ’
integrator feedback system (3.20) are in a form applicable to the passivity stability proofs
of Khalil or Krstic etal. If the algebraic differential modular system is passive and the
integrator feedback system is passive the feedback system (3.18a-3.18b) is globally
uniformly stable.

The assembly and solution of nonlinear modular models using the control logic
approach of modular modeling is less restrictive and avoids the stiff system problem of
previous methods. Hogan found that assembling nonlinear modular component models
with simple “nonenergic” connections required the output to bejtotalfily state dependent,

which rs restrictive. Modular modeling clearly rs less restrictive because the outputs can

be dependent on the inputs. Further Hogan stated that “energic” connections must be

61

used to maintain modularity resulting in stiff systems. Modular modeling uses simple

“nonenergic” connections and a control logic approach to avoid stiff systems.

3.1.2 Nonlinear Algebraic Equations

Modular modeling elements with user-defined nonlinear algebraic equations have

the same form as (3.1) but .E: Q.

(3.21)

Following the same procedure as in (3.2a-3.2b), consider a modular model in

unconnected form as a concatenation of n independently formulated user—defined

modular modeling elements in the algebraic form (3 .21) with p total input-output power

ports. -
'f1(x..u.) -

1362.6)

.. IO IO

(3.22a)

 

 

n.-- 3X1

 

 

f,.(x,..u,.)
_ .. f<sz>=9 -
X! €105"?
1.2 8.2926)

dSXI

(3.22b)

 

 

 

 

ya gn(xn’ an)

_- - dPXI

Y=9(X. U)

.1le

The same input-output topology, reordering, and input-output constraint analysis applies

to the algebraic output constrained modular system equations. These equations are
written in terms of the states, X , the external inputs, Um, the external outputs, Ya, , and

the constrained internal inputs, U c .
U...

U
_ 1- ext
Yen - 53(X9 I [(§0 _ §¢)TUC]) (323b)

62

U...
(a. ‘5‘)’"’(X’IT[(§.- r.)’UC])=9 (3.23c)

Solve (3.23c) for constrained internal inputs, U c, in terms of external inputs Um

using the same techniques as the nonlinear algebraic differential case (3.17 & Fig. 3.1).

U...
[(3.—r.)’r,X‘]) (3'24)

The internal constraint stabilizer is defined in the same way as in the nonlinear algebraic

X’ = (E. -§.)a,.,( IT

differential equation case. The fully connected nonlinear algebraic modular system
equations can be solved for the unknown constrained states, X ‘, the unknown states, X ,
and the unknown external outputs, Yer! , in terms of the known external inputs, U m.

U
1- w _
f(X9 I [(S _ é, )TEIXC]) _ 9 (3.258)

U
_ 1- ex:
1:1! _ ”(X’ I [(éo _ §¢)T£[Xc]) (3'25b)

X‘ = (a. - .s_.)a...<x. 1’ (3.250

U...
[(3. - afar}

The stability analysis of the feedback system of the nonlinear algebraic modular
system equations (3.25a—3.25c) is similar to the stability analysis done in the nonlinear
algebraic differential case (3.18a-3.18c, 3.19, 3.20). If the algebraic modular system is
passive and the integrator feedback system is passive the feedback system (3 .25a-3.25c)
is globally uniformly stable.

The nonlinear algebraic differential and algebraic constrained modular system
equations (3.18a-3.18c) and (3 .25a-3.25c) respectively are written in terms of
independently formulated nonlinear modular modeling elements. The reordering and

constraint matrices, I and (_.So - §,) , are comprised of ones and zeros and defined by the

size and input-output topology of the modular system model. The modular modeling
element functions, f and 9, can be directly inserted into the constrained modular system

equations and solved like any other nonlinear system of equations. The advantage of

modular modeling is that each multi DOF modeling element is independent and can be

63

connected together with simple connectors and assembled with a simple physically

intuitive top-down direct insertion method.

3.2 Some Modular Computational Advantages

Nonlinear modular models have a guaranteed modular structure because of their
standardized input-output formulation. Modular models never condense the model
topology relative to the input-output structure. This uncondensed guaranteed modular
structure has an advantage of ease of computation at a cost of additional computations.
The additional computations come from the constraint equations not found in an arbitrary
input-output structure or non-modular formulated models.

The ease of computation advantage arises from the modular construction.
Modular construction enables evaluations to be performed for each uncondensed element
independently. Independent evaluation is set up to avoid complex multi-step solutions
and separate convergence problems. Modular modeling does not completely eliminate
multi-step solutions and convergence problems but it does ease the computation by
maintaining model element modularity.

Another computational advantage of modular modeling is parallel computation.
Parallel computation is a perfect fit for modular modeling because the elements can be
evaluated independently. Modular element evaluation can be run on separate networked
processors. Evaluation of modular models on networked processors will be a subject of a
future paper. Each multi-port multi-DOF element in a modular model can be distributed
to separate processors. The constraint equations (3.18c) or (3 .23b) must reside on a
single processor.

The constraint equations add computations to modular models and another

computational advantage. These extra dynamic equations have stability defined by the
constraint control gains, Er Manipulating the constraint control gains in the constraint

equations, modularity allows for convergence control and adaptation. Convergence

control and adaptation will be a subject of a future paper.

3.3 Examples

An electric automotive vehicle power train example model is used to illustrate
modular modeling of nonlinear algebraic differential modular models. A structural three
bar configuration fixed to a point example model is used to illustrate modular modeling
of nonlinear algebraic modular models. The electric automotive vehicle power train
example model is a combination of three simple dynamic models found in Phillips and
Harbor, 1996 and Minor, 1996. The three bar structural example model is a combination
of a simple nonlinear model from Ross, 1990.

The electric automotive vehicle power train example model contains three simple
modular modeling elements, an electric motor model, a fluid clutch model, and a
transmission model. The three modular modeling elements are all defined with
standardized modular modeling causality (Table 1.1). The electric motor modular
modeling element has one electrical power port with current input 1' voltage output e
causality and one rotational mechanical power port with torque input 1' angular velocity

output a) causality (Fig. 3.2a).
K 1

x; = [0]x,,, + [-f 71:] (3.26a)

e - K" + R 0 i 3 266
a) ' 1 "' 0 0 r ( ' )
The motor model has a state, xm, a lumped motor-shaft rotational inertia, J , a motor coil

resistance, R, a motor back emf constant, K b, and a motor torque constant, K, . The

transmission modular modeling element has two rotational mechanical power ports with
torque 1'“ and “rd input angular velocity (1), and 60‘, output causality (Fig. 3.2b).

65

J Jer

Ir tr Ir

a)“ _ GR 0 0 r“
1.1-1.161.111

The transmission model a state, x", a lumped transmission rotational inertia, J an

tr’

0 T
x” = —ix,, + [Q i][ "] (3.27a)

equivalent transmission damping, c", and a transmission gear ratio, GR. The fluid

clutch modular modeling element has two rotational mechanical power ports with torque

rm and 1', input angular velocity (1)," and 60, output causality (Fig. 3.2c).

xcl = —&(xcl — xc2)2 + it!!!
Jcl Jcl
(3.28a)
I l
xcz = (xcl — xc2) + —Tt
J 1

‘11:! c
(1),,l __ l 0 xcl 3 28b
60, _ O 1 xc2 ( ° )

The clutch model has three states, [xcl,xc2]T, a lumped clutch rotational inertia, 10,, and

an equivalent resistance, Rd.

 

a) b) c)
Figure 3.2: Electric Automotive Vehicle Power Train Modular Modeling
Elements a) Electric Motor b) Transmission c) Fluid Clutch
These three modular modeling elements are assembled in a motor—clutch-
transmission configuration to illustrate the flexibility of modular modeling. The modular
motor modeling element is connected to the modular fluid clutch modeling element,

which is connected to the modular transmission modeling element (Fig. 3.3).

66

l 2 3 4 5 6

..... ..- - .. '.. an. .- I >. .I.,.ro..5 - .
“a

.. q _, . . . _ . ._ C S’
“I 3: “23M ' =' “45 “6
—:;i electric motor —fluzd clutch— transmzssron_
’1; C 3: ‘ ' , :1
y] Err-1:1. :vrn. win: y23 75‘? " ‘1 “‘1 -‘°‘ y45c 55" “-‘T‘r- “3“”‘33'4 y6

Figure 3.3: Electric Automotive Vehicle Power Train Modular Model With An
Electric Motor, A Fluid Clutch, And A Transmission Modular
Modeling Elements
The unconnected three element modular system is built up from directly inserting the

motor, clutch, and transmission modular element matrices into (3 .2a-3.2b).

 

 

 

 

 

 

' W
_ j %i-%r
5'! - f" (xcl - xc2)2 +—Tm
=f(X.U)= x.“ = Rj’ 2 1" (3.29a)
EC? JO (ch-xCZ) +—Tr
-_£l __________ d __.
_ tr_ C”
-—x +—r +—r
J. " J. " . "
' e ' 'Kbxm+Rr
a) x,"
Y (XU)- 0”” — x" - 3296
-9 . - 91:, — “-133"- (. )
a), GRx"
Lwd_ _ xtr d

 

 

 

 

The input-output topology of the three element modular power train model shows
ports 1 and 6 are external ports and two connectors (Fig. 3.3). The internal ports are

connected in pairs 2 to 3 and 4 to 5. This simple topology analysis defines the reordering

matrix T and the equation selector matrix _0(_S_‘- -S ).
Fl 0 I) 0

O l -l 0 0

0’ (§o_§¢)=[0 O 1 _1] (3.30)
0

O

I")
II

oooooooooooooooooooooooooooooooo

 

 

67

Ports 1 and 6 are the external power ports so I reorders the input U = [i, 1', twtntu, rd]?
and output Y = [e, a), wwwnwu, (DAT so i, rd, e, and and appear first in the vectors.
Ports 2 and 3 and ports 4 and 5 are connected internal power ports reordered to appear in
connected pairs after the external ports. Apply the reordering matrix 1’ to the modular

model (3.29a-3.29b) to partition the inputs and outputs internally and externally.
i -

 

 

7.:
Um '1'
[DZ] -_- l‘U = 1'... (3.3la)
TI
TI
' e ' - _ Kbxm + R:
a), x,
[323.] = 17 = w = [939% 9)] = x," (3.3 lb)
You 0),, QNIX, U ) xcl
20.: ....... .xc; ......
_a), .. .. GRx, .

 

 

 

 

The two element electric automotive vehicle power train modular model (3.29a-3.29b) is
now in the form to apply modular modeling constraints (2.2a-2.2b).

The constraints are applied to the internal inputs and outputs. The internal inputs

are constrained to be equal and opposite (3.13).
r 1. -

 

 

 

 

 

 

 

' 1
1,, T -1 0 ”a;
U. = = S —S U‘= 3.32
W 1.: (—0 -—¢) 0 1 Lu‘sc] ( )
_r,_ _0 -1_
The internal outputs are constrained to be equal using integral control logic (3.16a-3.16b)
xfll
- °. 1 -1 0 0‘ x x -x
c= x23 = = _S X = cl = m cl
X L" . (9. ...)a...( .U) [0 0 1 _1_ x.. [x,,—6Rx,](3'33’)
45
LGRx,_
c k 0 c
Uc =l:u23¢]=_K_,Xc =[ II 11230] (3.331))
u” 0 kn x45

68

Apply the internal input and output constraints to the three element electric power train

modular model (Fig. 3.3) and rewrite in the fully connected modular system form (3.18a-

 

 

 

 

3.18c).
_ K, , l c -
_ . .. 7‘ " 'jknxzs
. U 3:? ““IlicLhcr x02 )2 + ( klrxzac)
X = f( X, IT[ “’7 {I} = fl = é’ 2 ‘1’ (3.34a)
(£0 _ §,) K’X x 2 4(xcr " ch) + Tklzxttsc
' _--£l _________ CL _______
_ rr_ C" GR f c
”7"” +J—Tu + ( knx45 )
_ tr tr tr -
Y = 9,,(X :rT U"; )= e = K’x'" + Ri (3.346)
en , — (£0 —§e) KIXC m4 xrr

U x - x
c = _ 1‘ ex! = x23 = m c]
X (£0 g: )ant(Xr I [(S _ §¢)TKIXC]) x. c [x62 _ GRX’] (3.340)

—0 4s
The fully connected 3 element electric vehicle power train modular model (3.34a-
3. 34c) can be solved like any other nonlinear algebraic differential model. The three

WW4;

.J

modular modeling elements 1n this modular model are all strictly passive because they all .

__ww- _._u

have positive damping. Any model with positive damping rs strictly passive (Khalil,
B 4m+ Do 0101‘ (1111601 pom

1996 & Krstic etal, 1995). The integrator of the output constraint (36a-36b), as are all

integrators, is passive (Khalil, 1996). The modular model is then passive and globally

uniformly stable (Khalil, 1996).

The nonlinear algebraic modular model example is a structural model with three
bars fixed to a point. This structural modular model example contains two simple
modular modeling elements, an axial force bar model and a fixed point model. The two
modular modeling elements are all defined with standardized modular modeling causality
(Table 1.1). These structural models do not have traditional power ports because the

output variable is a displacement instead of velocity. They are representations of

physical points with two variables that require a causal definition, which fits the

69

application of modular modeling and the measurement perspective standardized causality
(Byam and Radcliffe, 1999).

The axial force bar modular modeling element is a four node Finite Element
Model (F BA) from Ross, 1990. This model has four mechanical ports with force input F
displacement output 5 causality (Fig. 3.4a), which is written in nonlinear algebraic
modular modeling form (3.22a-3.22b)L

 

 

 

 

xa-xc-F; T ’0-
xc-xa x -x -F 0
{<§,g)=0=if- ( _xX+; _2 " 1= 0 (3.35a)
_(xc-an—xb+xd)—E,d _O_
'51- ”x“-
62 xb
y=8(f.l_¢)= 5 = x (3.35b)
" "' 3 c
-54- -xd-

 

 

 

 

The modular bar (3.35a—3.35b) has four states [x,, x,, xc, xd]T, a cross-sectional area A, a

modulus of elasticity E, and a length L.

The fixed point modular modeling element is unique to modular modeling
because of the standardized measurement perspective causality. The one port fixed point
modular modeling element outputs a zero displacement regardless of the input and
applies to any energy domain (Fig. 3.46). The fixed point modular modeling element is

written in nonlinear algebraic modular modeling form (3.22a-3.22b).
f (in?) = Q = x. = 0 (3368)

y = 8(frlf) = 55 = O (3.361))

The modular fixed point modular modeling element has one state x, .

70

a) b)
Figure 3.4: Three Structural Nonlinear Algebraic Modular Modeling
Elements: a) Axial Force Bar b) Fixed Point
The example nonlinear algebraic modular model will use the two modular
modeling elements (Fig. 3.4) in a bracket—pin-clevis configuration. Three axial force bars
and a fixed point will be connected together in configuration with no pin bending
deflection (Fig. 3.5).

 

Figure 3.5: Three Axial Force Bar And A Fixed Point Modular Modeling
Elements In A Bracket-Pin-Clevis Configuration With No Pin
Deflection
The unconnected five element modular system is built up from directly inserting the bar
and fixed point modular elements into (3 .22a-3.22b) where the f, g, if, y, and if of

each element are found in the element equations (3.35a—3.35b) and (3.36a-3.36b).

71

Y=9(X.U)=

f(X.U)=Q=

 

-yfi! pt.
yfir pr:
3....
y...

yum,

 

'ffi. m, («349'
f.§,..,(§.z3)
f;(§,t_{) _
feigbfq)
f;,,(x.u)
- -_ " " 1

 

I
IO IO IO IO IO

 

 

 

"an. m (3950'
gl'it I": (’3’ L!)

3;... (5.15)

 

 

81..., (f. 1})

(3.37a)

(3.376)

The input-output topology of the three bar no pin modular model shows ports 1 1,

12, 15, and 16 are external ports and six connectors. The internal ports are connected in

pairs 1 to 3, 2 to 4, 5 to 9, 6 to 13, 7 to 10, and 8 to 14. This simple topology analysis

defines the reordering matrix I and the equation selector matrix (S, — §.)-

To

COO

 

OO

0

COO

OO

0

COO

O

COO

COOO

0

COO

00

OOO
COO

72

OOOO

l O

COCO
OOC
OOH
OOOO

0

COO

CHOO

O
J

HOO

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

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

 

(3.38a)

 

'1—10000000000‘
001—100000000
00001-1000000

(§°—§‘)=0000001—10000 (33%)
000000001-100
_00000000001—1j

 

Ports 11, 12, 15, and 16 are the external power ports so I reorders the input
1' T
Uler’Fi’FivmrI‘id and output Y=[61’62’53’°”’616] 309 Fir, Fiz’ E5: E6’ 6119 512:

5,5, and 5,6 appear first in the vectors. Ports 1 and 3, 2 and 4, 5 and 9, 6 and 13, 7 and

10, and 8 and 14 are connected internal power ports reordered to appear in connected
pairs after the external ports. Apply the reordering matrix I to the modular model
(3 .37 a-3.37b) to partition the inputs and outputs internally and externally.

[if -] = 111 = ..... _ (3.39a)

a?“ o7"

5“ ~71

.31

 

 

g:

I
l

73

I
I
I
I

_90 _90 __00 _90
O M N --
”>1 c-R e-R e-R
a“ a“ at" a“

.93 .93:
a
5;

52
Y X.
[-24.] = y = .64. = [935-32] = {’49. (3.3%)

nnnnnnnnnnnn

093
21R

_00
U
.3“

“00
EH
‘

__o.
C
.24
:4

"01
SH
‘

 

 

 

 

.90
‘
L

b -

The five element structural three bar modular model (3.37a-3.37b) is now in the form to
apply modular modeling constraints (2.2a-2.2b).
The constraints are applied to the internal inputs and outputs. The internal inputs

_ -

are constrained to be equal and opposite (3.13).

 

 

F; 1 0 0 0 0 0“
F; —1 0 0 0 0 0
F, 0 1 0 0 0 0
F; 0 -1 0 0 0 0 ”a,"
F, 0 0 1 0 0 0 u,;
F, ,. . 0 0 -1 0 0 0 a;
U...=-,;- =(§.—§.)U= 0 0 O 1 0 0 u...‘ (3.40)
17,, 0 0 0 -1 0 0 a“;
F; 0 0 0 0 1 0 (um:
E, 0 0 0 0 —1 0
F; 0 0 0 0 0 1
_F;,, L0 0 0 0 0 -1,

 

 

 

The internal outputs are constrained to be equal using integral control logic (3.24)

74

 

 

 

 

- . .;
x13c
x246
Xc = x5? =.=(§0 —§e)&m(X’U)=
x61;
x7106
_x814cd
u]; T k“ 0 0 0
uu‘ 0 km 0 0
U‘ = u”: =E,X‘ = O O k,3| 0
14613 0 0 0 ’6,4
“710C 0 0 0 0
_usucd _ O 0 0 O

 

 

x fp. xbla

foz xb'b

xbl "xbz,

xbl —xb3,

xbn, ‘ 11:2,

xbl, " xbs, .

0 0 - .. x]; -
0 0 x2:
0 0 x59”
0 0 x61;
le 0 x'noc
0 klé- _xsch

 

 

 

(3.41a)

(3.41b)

Apply the internal input and output constraints to the five element electric power train

modular model (Fig. 3.5) and rewrite in the fully connected modular system form (3.25a-

2.25c).

. U
— T — -
X — “X, I [(§ - §¢)T£1Xc:l) Q —

 

_ T _
Ya: — ”(X, I [(éo -§¢)TK1XC:|) __.

U
1- an
ff?" ‘1": _(s. -§.)’z,x=_
- U -
ffixpg (LIT a; c
.. 2 ~ L(§o-§e) KIX _
- U -
fbar (x9IT a; c
-I ~ _(lSo-é'e) XIX _
F- U -
T w 1
@1151 _(_S.. -§.)’_Ig.X‘_
1- U -r
1- a:
_ 11:45.2: _(s. -§.)’_Ig,X‘_
-5ll-l pxbzc .1
w 512 ___ xbz.
515 "b3.
L616- be3d -

 

 

 

75

 

V

v

 

V

v

 

 

 

 

 

v

B IO 1c lo

-0,

J(3.42a)

 

(3.42b)

. U
c — — T at =
X "" (£0 .5: )GM(X’ I [(550 - .‘Se )TKIXC])

 

.
C
9‘13
C
C
x24
.
C
3‘59
C
C
x613
.

C
x710
.

C
_xsu _

q

 

 

q

foI - xb‘

 

foz — xblb
xbl, — xbz,
xbt, _ xba,
xbt, - xbz,
xbn, - xba, _

(3.42c)

The fully connected five element three bar structural modular model (3.42a-3.42c)

can be solved like any other nonlinear algebraic differential model. The two modular

modeling elements in this modular model are all strictly passive because they all

memoryless functions that do not generate energy (Khalil, 1996). The integrator of the

output constraint (3.41a—3.41b), as are all integrators, is passive (Khalil, 1996). The

modular model is then passive and globally uniformly stable (Khalil, 1996).

76

Chapter 4

CONCLUSIONS

4.1 Contributions

We have successfully developed mechanisms for modular modeling of
engineering systems with fixed input-output structure. Modular modeling elements with
a power-based fixed input-output structure have fixed power port input-output causality,
fixed sign convention, and a fixed internal equation formulation. Modular modeling
connectors join modular element ports with fixed power constraints. The standardized
internal formulation of modular modeling elements enables top-down modeling and
enhances model verification via modularity. Because a modular modeling element’s
formulation does not change, modular modeling yields more easily verified models.

Unverified models are of no use in today’s industrial environment. A model with
n power ports has 2" possible formulations. The performance of all 2" formulations
simply cannot be simultaneously verified. Reducing the number of verifications flour 2"
different verifications to a single verification is an enhancement in modeling technology.
Modular modeling elements with a known physically validated performance are an asset
to modelers of engineering systems.

Model complexity increases with the modular modeling method because of the
connector requirement. Modular connectors have a fixed definition, which is compatible
with the modular modeling element’s fixed functional definition. The compatible fixed
functional definitions enable a systematic assembly method for modular models with a
fixed mathematical formulation and a fixed computational sparseness.

The measurement perspective fixed port causality used in modular modeling has
the ability to avoid using energic junctions to maintain modularity upon assembly.

Hogan concluded that use of energic junctions guaranteed modularity at the cost of “stiff”

77

system equations with widely spread eigenvalues. Modular modeling can assemble
modular modeling elements with fixed measurement perspective causality and maintain
modularity with the simple power constraint of the modular modeling connector.

Modular modeling reduces the verification task of large model design,
development, and refinement by standardizing the functional form of all multi degree of
freedom modeling elements. The fixed power-based measurement perspective port
causality results in a standardized multi degree of freedom modular modeling element
with a single mathematical model formulation. This formulation has the flexibility to
support any number of physically disconnected ports without reformulation. Assembling
incompatible modular modeling elements requires a causally compatible 2-port modular
connector that facilitates a standardized constrained power transfer between modular
modeling element ports. The separate modular modeling elements and connectors with
explicitly different functions enable subsystem level modeling with no reformulation.
Fixed formulation enhances model verification. Modular modeling has the flexibility to
model any engineering system across multiple energy domains with the benefits of a
fixed input-output structure.

The solution to linear modular models is a systematic direct insertion solution.
Modular modeling elements with nonlinear equations are the subject of a future paper.
Given n modular modeling elements and an input-output topology of their
interconnections a systematic reordering and constraint analysis realizes a built up
modular model that can be solved like any other linear system of equations. Modular
modeling elements maintain their modularity in the assembled modular model. This
method’s maintenance of element modularity while assembling with simple connectors
with a less restrictive condition on the modular elements is previously unavailable in
other methods. The reordering and constraint processes add complexity to the solution

process but the modular model has a fixed computational Sparseness. This formulation

78

results in cases where the number of computations for modular models is less or
comparable to non-modular models.

The solution to nonlinear modular models is a systematic direct insertion solution.
Given n modular modeling elements and an input-output topology of their
interconnections a systematic reordering and constraint analysis realizes a built up
modular model that can be solved like any other nonlinear system of equations. Modular
modeling elements maintain their modularity in the assembled modular model. This
method’s maintenance of element modularity while assembling with simple connectors
with a less restrictive condition on the modular elements and no stiff equations is
previously unavailable in other methods. The reordering and constraint processes add
complexity to the solution process but the modular model has some computational
advantages. This formulation results in cases where the parts of the computation can be
done on separate processors.

4.2 Future Work
Future work in this new modeling technology has several exciting possibilities.

Application of adaptivecoritrol strategies to the control logic solution of nonlinear
modular models. Using modulflgrkmodeling with fixed input-output structure to connect
complex Finite Element Analysis (FEA) models with incompatible meshes. Modular
modeling using Statistical Energy Analysis (SEA) techniques. Utilization of modular
modeling technology in intemet-based or web-based modeling.

Adaptive control strategies such, gain scheduling, could be applied to the
assembly and solution of nonlinear modular models. Each modular modeling element

may have a unique response requiring a unique feedback gain to reduce the assembly

error.

79

Connecting complex FEA models typically requires remeshing for connection.
Modular modeling with fixed input-output structure is formulated to avoid this problem.
The Q matrix in the boundary value problem formulation provides for interpolation
between mesh node points and the inputs and outputs. Interpolated outputs of different
FEA models from number of mesh node points could be connected through a modular
modeling connector.

Statistical Energy Analysis (SEA) models, which are an equal sharing of high
fiequency vibration and sound energy between modes, are applicable to modular
modeling with fixed input-output structure (Lyon, 1975). SEA models have elements and
connectors the store and distribute energy. SEA models have grown to be quite large and
complex with numerous elements and interconnections. Modular SEA models would be
an advancement in SEA modeling technology.

Internet-based or web-based modeling is an emerging modeling technology well
suited for modular modeling with fixed input-output structure. A key design
specification of web-based modeling in security of proprietary model information. A
fixed input-output structure enables such security by restricting the type of input and
output. Reverse engineering is not a trivial exercise when restricted to input and output

information only.

80

LIST OF REFERENCES

List of References

Ferris, J. B., Stein, J. L., and Bernitsas, M. M., 1994, “Development of Proper Models of
Hybrid Systems.”, 1994 ASME Winter Annual Meeting, Proceedings of The
Symposium Automated Modeling Design, ASME, New York, November

“Computers in Engineering: Chrysler designs paperless cars”, 1998, Automotive
Engineering International, Volume 106, Number 6, (June): Page 48

Jost, Kevin, 1998, “Chrysler redesigns its large cars for 1998”, Automotive Engineegn' g
Internatignfl, Volume 106, Number 1, (January): Page 10-15

Shigley, J. E. and Mischke, C. R., 1989, Mechanical Engineering Design, McGraw-Hill,
New York

Haug, E., J ., 1989, Computer-Aided Kinematics And Dynamics of Mechanical Systems:
Volume I: Basic Methods, Allyn and Bacon, Massachusetts

Nikravesh, P., E., 1988, Computer-A ided Analysis of Mechanical Systems, Prentice Hall,
Englewood Clifi‘s, New Jersey

Chua, L. O. and Lin, P., 1975, Computer-Aided Analysis of Electrical Circuits:
Algorithms And Computational Techniques, Prentice Hall, New Jersey

Vlach, J. and Singhal, K., 1983, Computer Methods For Circuit Analysis And Design,
Van Nostrand Reinhold Company, New York

Calahan, D., A., 1972, Computer-Aided Network Design, McGraw-I-Iill Inc., New York
Zienkiewicz, O. C., 1977, The Finite Element Method, McGraw-Hill Inc., New York
PDESolve, BEAM Technologies, 687 Highland Avenue, Needham, Massachusetts

Kamopp, D. C., Margolis, D. L., and Rosenberg, R C., 1990, System Dynamics: A
Unified Approach, John Wiley and Sons, Inc., New York

82

Hales, M. K., 1995, “A Design Environment For The Graphical Representation 0f
Hierarchical Engineering Models”, Masters Thesis, Michigan State University,
East Lansing, Michigan

Rosenberg, R. C. and Kamopp, D. C., 1983, Introduction to Physical System Dynamics,
McGraw-Hill, Inc, New York

Hogan, N., 1987, “Modularity and Causality in Physical System Modelling”, Journal of

Dynamic Systems, Measurement, and Control, Vol. 109, Pages 384-391, ASME,
New York, December

Meirovitch, L., 1967, Analytical Methods In Vibrations, MacMillan Publishing Co., New
York

Byam, B. P. and Radcliffe, C. J ., 1999, “Modular Modeling of Engineering systems
Using Fixed Input-Output Structure”, 1999 IMECE, Proceedings of The
Symposium of Systematic Modeling, ASME, New York, November

Redish, K. A., 1961, An Introduction To Computational Methods, John Wiley & Sons,
New York

Minor, M. A., 1996, “Using T op Down Multiport Modeling For Automotive
Applications”, Masters Thesis, Michigan State University, East Lansing,
Michigan

Segerlind, L. J ., 1984, Applied Finite Element Analysis, 2"“ Edition, John Wiley & Sons,
New York

Skelton, R. E., 1988, Dynamic Systems Control: Linear systems Analysis And Synthesis,
John Wiley & Sons, New York

Leon, S. J ., 1986, Linear Algebra With Applications, 2"“ Edition, MacMillan Publishing
Co., New York

Phillips, C. L. and Harbor, R. D., 1996, Feedback Control systems, 3’“ Edition, Prentice
Hall, New Jersey

Khalil, H. K., 1996, Nonlinear Systems, 2"4 Edition, Prentice Hall, New Jersey

83

Krstic, M., Kanellakopoulos, I., and Kokotovic, P., 1995, Nonlinear and Adaptive
Control Design, John Wiley & Sons, New York

Ross, C. T. F., 1990, Finite Element Methods in Engineering Science, Ellis Horwood,
New York

Lyon, R. H., 1975, Statistical Energy Analysis of Dynamical Systems: Theory and
Applications, MIT Press, Cambridge, Mass.

84

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