copyright by thomas joseph connolly 2000 - department of
TRANSCRIPT
Copyright
by
Thomas Joseph Connolly
2000
Synthesis of Multiple Energy Active Elements
for Mechanical Systems
by
Thomas Joseph Connolly, B.E., M.S.E.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
May, 2000
Synthesis of Multiple Energy Active Elements
for Mechanical Systems
Approved byDissertation Committee:
Raul G. Longoria, Supervisor
Ronald O. Stearman
Steven P. Nichols
Richard H. Crawford
S.V. Sreenivasan
Dedication
This work is dedicated to the loving memory of my grandparents,
Lawrence and Philomena Barbara Lang
v
Acknowledgements
I would like to thank my family and friends for all of their love and
support through the years, without which, completing this work would not have
been possible. I would especially like to thank my parents Leona and Richard
Connolly for their encouragement, support, and love they have shown to me as I
have journeyed through life.
I would like to extend my gratitude to those in the academic and research
communities who have provided me with instruction, guidance, and support over
the past seven years. First, I would like to thank Dr. Raul Longoria, who
introduced me to the interesting field of dynamic systems analysis and also
suggested the topic for this dissertation. His support, direction, and insight have
been invaluable in the completion of this work. Special thanks to Dr. Ron
Stearman, who gave me my first opportunity in graduate school to work as a
teaching and research assistant. I am fortunate to have worked with him and learn
about the fields of structural dynamics and non-linear systems identification. His
willingness to be of help and his warm personality certainly provided me
inspiration and the encouragement to continue my course of study. I would also
like to thank Dr. Tom Garza, for the time, enthusiasm, and creativity that he has
put into Russian language instruction - it has been an enjoyable and valuable
addition to my graduate education.
vi
Funding for the research associated with this study was provided by th
National Science Foundation funded Industry/University Cooperative Research
Center for Virtual Proving Ground Simulation. In addition, the assistance of Mr.
Damon Weeks and Dr. Joe Beno of the University of Texas Center for
Electromechanics, who furnished technical information and experimental data
that was used in this study, is much appreciated.
My friends are an extension of my family, and I have had the fortune of
meeting some very special people since moving to Austin: Eloy Gonzalez-Ortega,
Andy Berner, Sebnem Ozupek, Mayte Tormo, Judy Ray, Dixie Houston, Robert
and Maryann Notzon, and especially Eugene Podnos. In the transient nature of
life as a graduate student, where people come and go much too frequently,
Eugene has been there since the beginning and I am greatly indebted to him for
his unwavering friendship, support, and especially his encouragement and advice
during both difficult and happy times – thank you ? ?? ?.
I have been especially blessed with the fellowship and support from those
whom I have met through the University Catholic Center. I would like to extend
my gratitude to Fr. Pat Hensy, Michelle Goodwin and Fr. Dave Farnum for their
spiritual direction, support, and inspiration. My involvement with the Longhorn
Awakening program has literally changed my life and has more fully integrated
my faith into all of my words and actions. Special thanks to my friends Penny
Rivero, Becky Odle, and Regina Praetorius who were instrumental in bringing
this experience to me through their selfless work and continual spirit of giving. I
am very happy for the friendship of John Benner, with whom I have had the
pleasure of faith sharing and exploring together our individual callings in life.
vii
Synthesis of Multiple-Energy Active Elements
for Mechanical Systems
Publication No._____________
Thomas Joseph Connolly, Ph.D.
The University of Texas at Austin, 2000
Supervisor: Raul G. Longoria
This work introduces a synthesis methodology for active elements within
mechanical systems that uses frequency response functions as a basis for
describing required behavior. Classical approaches developed by Foster (1924)
and Brune (1931) put forth a general procedure for the synthesis of electrical
systems. The general concept behind these synthesis procedures is to decompose
a driving point impedance function into terms that represent the impedances of
simple electrical elements: resistors, capacitors, and inductors. This is achieved
through a process of repeated partial fraction expansion and polynomial division.
This concept is extended into the mechanical energy domain, and is facilitated by
the use of impedance-based bond graphs.
This work focuses on expanding the existing methodology such that active
elements are permitted in the system synthesis. The restriction that impedances of
system elements be positive definite, which guarantees a passive element, is
removed. Thus, bond graph elements that have negative impedances, generally a
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non-physically realizable phenomenon, are used as a mathematical modeling tool
to represent the behavior of an active element. Negative impedance elements
have been used in active electrical system synthesis (e.g.,Bruton, 1980), and their
mechanical energy domain equivalents are formulated and used in this work. By
using such elements in a computer-based simulation, the time histories of
pertinent variables associated with the active element, such as velocity and force,
can be obtained for a given input to the system.
These histories are used in a time-based simulation environment to
facilitate the physical realization of the active element. In addition, enhanced
bond graphs that contain signal flow information for pertinent state variables are
constructed using the differential equations that define the states of the system.
Such enhanced bond graphs give insight into control schemes for the synthesized
active element. All of this information is used in the selection of critical design
parameters of the active element, such that an appropriate design is obtained. The
method is applicable in the design of a new system or in the retrofit of an existing
system in which an active element is required or desired.
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Table of Contents
LIST OF TABLES .............................................................................................. XIV
LIST OF FIGURES .............................................................................................. XV
CHAPTER 1: INTRODUCTION .........................................................................1
1.1 SYNTHESIS OF ENGINEERING SYSTEMS.................................................1
1.2 IMPORTANCE OF ACTIVE ELEMENT SYNTHESIS.....................................1
1.3 OBJECTIVES AND CONTRIBUTIONS .......................................................2
1.3.1 Objectives of this Work .............................................................2
1.3.2 Contributions of this Work.........................................................3
1.4 OVERVIEW OF SYNTHESIS TECHNIQUES ...............................................4
1.4.1 Impedance-Based Approaches: Passive and Active ElectricalSystems .....................................................................................4
1.4.2 Functional vs. Parametric Approaches to Synthesis ....................5
1.4.3 Extension of Impedance-Based Methods: Passive MechanicalSystems .....................................................................................6
1.4.4 Active Element Synthesis in this Work ......................................6
1.4.5 Other Frequency Domain Approaches to Active ElementSynthesis....................................................................................7
1.5 DISSERTATION ORGANIZATION ............................................................8
CHAPTER 2: ACTIVE ELEMENT SYNTHESIS METHOD..................................11
2.1 INTRODUCTION..................................................................................11
2.2 DEFINITIONS AND CONCEPTS .............................................................11
2.2.1 Passive and Active Elements....................................................11
2.2.2 Classification of Impedances....................................................14
2.2.2 Frequency Domain Behavior of Systems..................................15
2.3 EXAMPLE PROBLEM – SYNTHESIS OF AN UNKNOWN IMPEDANCE ........17
2.3.1 Problem Description ................................................................17
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2.3.2 Problem Solution .....................................................................23
2.4 ACTIVE DEVICE CONTROL SYSTEM ISSUES.........................................37
2.5 ONE VS. TWO PORT SYNTHESIS OF ACTIVE ELEMENT IMPEDANCES ....43
2.6 SUMMARY.........................................................................................45
CHAPTER 3 PHYSICAL REALIZATION OF ACTIVE ELEMENTS.....................46
3.1 INTRODUCTION..................................................................................46
3.2 MECHANICAL SYSTEM ACTIVE ELEMENT REALIZATIONS ...................46
3.2.1 Introduction .............................................................................46
3.2.3 Hydraulic Actuator ..................................................................48
3.2.4 PMDC Motor...........................................................................49
3.3 EXAMPLE PROBLEM - ACTIVE ELEMENT PHYSICAL REALIZATION.......52
3.3.1 Classification of Impedances and Generalized FlowVariables..................................................................................52
3.3.2 Separable Passive Elements .....................................................54
3.3.3 Physical realization of Non-Separable Bond Graph Elements...59
3.4 SUMMARY OF ACTIVE ELEMENT SYNTHESIS AND REALIZATION..........73
CHAPTER 4: PRACTICAL APPLICATIONS OF THE SYNTHESIS METHOD .......76
4.1 INTRODUCTION..................................................................................76
4.2 TWO-STORY BUILDING MODEL VIBRATION ISOLATION SYSTEM .........76
4.2.1 Introduction .............................................................................76
4.2.2 System Description ..................................................................76
4.2.3 Objectives................................................................................78
4.2.4 Synthesis Procedure .................................................................80
4.2.5 Physical Realization of the Vibration Suppression ActiveElement ...................................................................................87
4.2.6 Summary .................................................................................93
4.3 ELECTROMECHANICAL VEHICLE SUSPENSION SYSTEM .......................94
4.3.1 Introduction .............................................................................94
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4.3.2 Description of System..............................................................95
4.3.3 Objectives................................................................................96
4.3.4 Specifics of the Application of the Methodology to thisProblem ...................................................................................97
4.3.5 Results and Comparisons to Existing Test Data...................... 103
4.3.6 Summary ............................................................................... 112
4.4 PHYSICAL REALIZATION OF THE ACTIVE VEHICLE SUSPENSION ........ 112
4.4.1 Introduction ........................................................................... 112
4.4.2 Realization Case 1: Hydraulic Actuator ................................. 113
4.4.3 Realization Case 2: PMDC Motor with Rack and Pinion........ 126
4.4.3 Comparison of Realizations Involving Different EnergyDomains ................................................................................ 141
4.4.4 Summary and Conclusions..................................................... 143
4.5 ANALYSIS AND REFINEMENT OF SYNTHESIZED CONTROL SYSTEMS .. 144
4.5.1 Introduction ........................................................................... 144
4.5.2 Active Device Control for the EM Suspension Example......... 144
4.6 POST-SYNTHESIS MODEL AUGMENTATION: NONLINEAR EFFECTS .... 152
4.6.1 Extension of Actuator Model and Linearization ofConstitutive Laws .................................................................. 152
4.6.2 Inclusion of Nonlinear System Properties............................... 156
4.7 PRACTICAL LIMITATIONS OF THE SYNTHESIS METHOD ..................... 159
4.7.1 Introduction ........................................................................... 159
4.7.2 Transfer Function Selection and Synthesis Results................. 159
4.7.3 Practical Solution: Bond Graph and Physical SystemInterpretations........................................................................ 165
4.7.4 Conclusions ........................................................................... 168
CHAPTER 5: SUMMARY, FUTURE WORK, AND CONCLUSIONS................... 169
5.1 SUMMARY....................................................................................... 169
5.2 EXTENSION TO SYSTEMS WITH MULTIPLE ACTIVE ELEMENTS........... 170
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5.3 CONCLUSIONS ................................................................................. 174
APPENDIX A: BOND GRAPHS IN SYSTEM MODELING ...................................... 175
A.1 INTRODUCTION................................................................................ 175
A.2 POWER FLOW WITHIN SYSTEMS ...................................................... 175
A.3 ENERGY STORAGE AND DISSIPATION ............................................... 176
A.4 POWER SOURCES............................................................................. 177
A.5 CONSTITUTIVE LAWS....................................................................... 178
A.5.1Introduction ........................................................................... 178
A.5.2Capacitive Elements............................................................... 178
A.5.3Inertive Elements ................................................................... 179
A.5.4Resistive Elements ................................................................. 181
A.5.5Nonlinear Elements................................................................ 182
A.6 CONNECTION OF BOND GRAPH ELEMENTS ....................................... 183
A.7 POWER TRANSFORMATION .............................................................. 187
A.8 CAUSALITY ..................................................................................... 190
A.8.1Introduction ........................................................................... 190
A.8.2Causality for Source Elements ............................................... 191
A.8.3Causality for Junction Elements ............................................. 192
A.8.4Causal structure for C elements.............................................. 194
A.8.5Causal structure for I elements ............................................... 195
A.8.6Causal structure for R elements.............................................. 196
A.8.7Causality for Elements with Nonlinear Constitutive Laws...... 197
A.9 SYSTEM ORDER AND STATE VARIABLES .......................................... 197
A.10 SYSTEM ANALYSIS AND SIMULATION USING BOND GRAPHS............. 199
A.11 IMPEDANCE BASED BOND GRAPHS .................................................. 199
A.11.1 Basic Impedances............................................................... 199
A.11.2 Combining Impedances to form Composite Impedances..... 201
A.11.3 Transmission Matrices ....................................................... 203
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A.12 SUMMARY....................................................................................... 206
APPENDIX B: COMPUTATION OF SPECTRAL DENSITY AND FREQUENCYRESPONSE FUNCTIONS FROM EXPERIMENTAL DATA ............................. 208
APPENDIX C: FREQUENCY RESPONSE FUNCTION CURVE FITTING ................ 217
C.1 THEORY .......................................................................................... 217
C.2 CURVE FIT COMPUTATIONAL PROCEDURES...................................... 223
APPENDIX D: MATHEMATICA™ OUTPUT ...................................................... 238
REFERENCES ................................................................................................... 246
VITA ................................................................................................................ 253
xiv
List of Tables
Table 2.1: Basic impedance elements (Generalized and Mechanical) ................14Table A.1: Bond graph transduction elements: transformer and gyrator............189Table A.2: Generalized variables and equivalents in various energy domains...206Table A.3: Constitutive laws for primitive elements .........................................207
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List of Figures
Figure 2.1: Negative bond graph element ...........................................................12Figure 2.2: Constitutive law for negative spring .................................................12Figure 2.3: Negative spring with constant flow source .......................................13Figure 2.4: Stored energy for passive and active elements ..................................13Figure 2.5: Simple mechanical oscillator............................................................17Figure 2.6: Bond graph of simple mechanical oscillator .....................................18Figure 2.7: Impedance bond graph of simple mechanical oscillator ....................18Figure 2.8: Actual and desired frequency response behavior ..............................20Figure 2.9: Desired frequency response and response with simple damper .........21Figure 2.10: System with unknown vibration filter.............................................22Figure 2.11: Bond graph of system with unknown filter impedance, Z(s) ...........22Figure 2.12: Equivalent impedance representation as a series circuit ..................24Figure 2.13: Equivalent impedance representation in bond graph form...............24Figure 2.14: Word bond graph of filter reticulation ............................................25Figure 2.15: Bond graph representation of system reticulation ...........................28Figure 2.16: Schematic representation of system reticulation .............................29Figure 2.17: Passive/active configuration of vibration filter................................30Figure 2.18: Bond graph representation of passive/active filter...........................31Figure 2.19: Fully active configuration of vibration filter ...................................32Figure 2.20: Bond graph representation of fully active filter...............................33Figure 2.21: Force-velocity profile for fully active filter.....................................34Figure 2.22: Swept frequency response comparisons and error calculations........35Figure 2.23: Comparison of step response for passive and active systems ..........37Figure 2.24: Example vibration isolation system ................................................38Figure 2.25: Example vibration isolation system bond graph..............................39Figure 2.26: Active element block......................................................................39Figure 2.27: Active element block replaced with feedback control .....................40Figure 2.28: Corresponding bond graph fragment for feedback control ..............41Figure 2.29: Generalized feedback control system..............................................41Figure 2.30: Overall system bond graph showing control signal flows ...............42Figure 2.31: Bond graph of example system with active device realized.............43Figure 2.32: Two port impedance and transmission matrix equation...................44Figure 3.1: Bond graph indicating active element force and velocity ..................47Figure 3.2: Schematic of hydraulic actuator .......................................................48Figure 3.3: Actuator bond graph model transduction element.............................48Figure 3.4: Bond graph model of actuator ..........................................................49Figure 3.5: Schematic of electric motor and rack and pinion ..............................50Figure 3.6: Bond graph model of a PMDC motor...............................................51
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Figure 3.7: Example system with unknown suspension impedance, Z ................52Figure 3.8: Bond graph showing active element representation ..........................53Figure 3.9: Sample filter configuration...............................................................54Figure 3.10: Corresponding bond graph of sample filter.....................................55Figure 3.11: Bond graph indicating separable portions of active representation ..56Figure 3.12: Passive suspension elements and active element representation ......57Figure 3.13: Schematic of extracted and active suspension elements ..................58Figure 3.14: Bond graph fragment associated with active element......................59Figure 3.15: Bond graph model of actuator ........................................................60Figure 3.16: Bond graph with control system and modulated effort source.........61Figure 3.17: Force-velocity profile for sinusoidal forcing function .....................62Figure 3.18: Force-velocity profile for random input..........................................62Figure 3.19: Bond graph with actuator piston effects added................................63Figure 3.20: Bond graph with new elements lumped with existing ones .............64Figure 3.21: Comparison of force-velocity profiles for ideal and real actuator....65Figure 3.22: Comparison of individual force and velocity plots..........................66Figure 3.23: Swept frequency results and original FRF ......................................67Figure 3.24: Bond graph of new system with non-ideal actuator effects .............68Figure 3.25: Bond graph of system with reticulated suspension..........................70Figure 3.26: Schematic of the reticulated suspension..........................................71Figure 3.27: Bond graph with realized actuator and effort source control ...........72Figure 3.28: Swept frequency simulation results and original FRF .....................73Figure 3.29: Active element synthesis procedure flowchart ................................75Figure 4.1: Two-story building laboratory model photograph and schematic ......77Figure 4.2: Lumped parameter model of two story building ...............................78Figure 4.3: Building model with active vibration suppression element ...............79Figure 4.4: Desired frequency response between base and top floor ...................80Figure 4.5: Bond graph representation of two-story building and filter ...............81Figure 4.6: Model with reticulated vibration suppression active element ............83Figure 4.7: Input velocity and response velocities ..............................................86Figure 4.8: Bond graph of system with active element representation.................87Figure 4.9: Force-velocity profile for actuator....................................................88Figure 4.10: Bond graph model with signal flow for controlled effort source .....90Figure 4.11: Time history of regulated pressure source ......................................91Figure 4.12: Pressure Velocity profiles for different values of piston area ..........93Figure 4.13: Photograph of quarter vehicle experimental setup...........................95Figure 4.14: Quarter vehicle with unknown active suspension element ..............96Figure 4.15: Actuator replaced by an unknown suspension system.....................97Figure 4.16: Impedance-based bond graph of system with unknown suspension.98Figure 4.17: Bond graph fragment for determination of transfer function ...........99
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Figure 4.18: Curve fit of experimental frequency response data .......................100Figure 4.19: Bond graph representation of synthesized suspension system .......102Figure 4.20: Swept frequency comparison results.............................................104Figure 4.21: Simulated and experimental values of sprung mass velocity .........105Figure 4.22: Fully active bond graph representation .........................................107Figure 4.23: Hybrid configuration for synthesized suspension system ..............108Figure 4.24: Hybrid suspension actuator representation....................................109Figure 4.25: Force-velocity profile for active and hybrid suspension actuators .110Figure 4.26: Existing actuator profile that fits desired force-velocity profile.....111Figure 4.27: System bond graph with signal flows and controlled effort source114Figure 4.28: Bond graph illustrating real actuator effects..................................115Figure 4.29: Swept frequency results and the original transfer function ............117Figure 4.30: Bond graph and signal flow (actuator effects compensated)..........119Figure 4.31: Swept frequency results after iteration..........................................120Figure 4.32: Input excitation and response velocities........................................121Figure 4.33: Experimental and simulated sprung mass velocities......................122Figure 4.34: Close up of overlaid sprung mass velocity plots ...........................123Figure 4.35: Sprung mass velocity data in the frequency domain......................124Figure 4.36: Pressure source time histories for different values of dp ................125Figure 4.37: Pressure-velocity profiles for different values of dp ......................126Figure 4.38: Permanent magnet DC motor .......................................................127Figure 4.39: PMDC/rack and pinion actuator layout.........................................128Figure 4.40: Bond graph with unknown impedance..........................................128Figure 4.41: Hybrid suspension bond graph .....................................................130Figure 4.42: Bond graph with controlled effort source and actuator effects.......132Figure 4.43: Bond graph model with lumped real actuator effects ....................134Figure 4.44: Comparison plot of actual vs. required frequency response...........137Figure 4.45: Force velocity profile for PMDC/rack & pinion actuator ..............138Figure 4.46: Simulated and experimental sprung mass velocities......................139Figure 4.47: Voltage time histories for different values of km ...........................140Figure 4.48: Motor power time histories for different values of km ...................141Figure 4.49: Comparison of transients between energy domains.......................142Figure 4.50: Bond graph and signal flow for hydraulic actuator synthesis.........145Figure 4.51: Force components of the overall control signal Fz.........................148Figure 4.52: Comparison of portion of sprung mass velocity time histories ......149Figure 4.53: Comparison of sprung mass velocity auto-power spectra..............150Figure 4.54: Simplified control system for hydraulic actuator ..........................151Figure 4.55: Schematic of control system.........................................................152Figure 4.56: Actuator schematic illustrating pressure sources...........................153Figure 4.57: Original model fragment of active device and controlled source...154
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Figure 4.58: Augmented bond graph model with source and return pressures ...154Figure 4.59: Portion of bond graph illustrating flow across orifice 1.................155Figure 4.60: Bond graph of vehicle system with linear spring effects only........157Figure 4.61: Bond graph of vehicle system with nonlinear spring effects .........158Figure 4.62: Impedance bond graph showing choice of the transfer function ....159Figure 4.63: Bond graph illustrating selection of alternate transfer function .....166Figure 5.1: Space station model with multiple active elements .........................171Figure 5.2: Impedance based bond graph of the space station model ................172Figure A.1: Power flow into a system element .................................................176Figure A.2: Bond graph representation of a linear spring..................................179Figure A.3: Bond graph representation of a capacitor.......................................179Figure A.4: Bond graph representation of a capacitor.......................................180Figure A.5: Bond graph representation of an inductor ......................................181Figure A.6: Bond graph representation of a viscous damper .............................181Figure A.7: Bond graph representation of an electrical resistor.........................182Figure A.8: Bond graph representations of a nonlinear element........................183Figure A.9: Parallel spring-mass-damper system..............................................184Figure A.10: Series LRC circuit .......................................................................184Figure A.11: Series spring-mass-damper system and bond graph model...........186Figure A.12: Parallel LRC circuit and bond graph model .................................186Figure A.13: Hydraulic piston and bond graph transduction element ................187Figure A.14: Electric motor and bond graph transduction element....................188Figure A.15: Connection of two bond graph elements ......................................190Figure A.16: Bidirectional signal flow between two elements ..........................190Figure A.17: Causal stroke on a power bond between two elements .................191Figure A.18: Allowable causal strokes for effort and flow sources ...................192Figure A.19: Causal stroke rule at a 0-junction.................................................193Figure A.20: Causal stroke rule at a 1-junction.................................................193Figure A.21: Causal structures for a C element ................................................195Figure A.22: Causal structures for an I element................................................196Figure A.23: C element with independent causality..........................................198Figure A.24: I element with independent causality ...........................................198Figure A.25: Time-domain representation of an inertive element .....................200Figure A.26: Impedance-based representation of inertive element ....................200Figure A.27: Impedance-based representation of a capacitive element .............201Figure A.28: Impedance-based representation of a resistive element ................201Figure A.29: Impedance-based representation of spring-mass-damper system..202Figure A.30: Composite impedance for spring-mass-damper system................202Figure A.31: Impedance-based representation of parallel LRC circuit ..............203Figure A.32: 1-junction with inertive element ..................................................204
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Figure A.33: Bond graph illustrating the combining of transmission matrices ..205
1
Chapter 1: Introduction
1.1 SYNTHESIS OF ENGINEERING SYSTEMS
Synthesis of engineering systems can be defined as the determination of
the design characteristics and physical structure of a system such that it meets
specified performance criteria. Well formulated synthesis methodologies can be
valuable tools in system design in that they can provide the designer with possible
feasible design strategies, while ruling out others. Of particular interest in this
work is the synthesis of systems whose critical design criteria are based on their
frequency response behavior.
Traditional methods of synthesizing active elements for engineering
systems using desired frequency response characteristics have been restricted to
the electrical energy domain. Classical approaches exist for the synthesis of
electrical systems with active elements. These are impedance-based and use
frequency domain behavior as a basis for the synthesis (e.g., Bruton, 1980).
However, a review of the literature has revealed that this type of approach has not
been developed for the synthesis of active elements within mechanical systems.
1.2 IMPORTANCE OF ACTIVE ELEMENT SYNTHESIS
Since the presence of active devices in an engineering system generally
provides a wider range of operation, controllability, and added adaptability, the
development of a synthesis technique for such systems is desirable. In many
areas, the presence of an active device can provide considerable weight savings,
2
simplify a design that contains only passive elements, or provide considerable
cost savings.
The methodology developed in this work can also be applied to systems in
which it is desired to replace existing passive components with an active device.
Scenarios in which this would occur are the upgrading or redesign of a system and
technology insertion.
This methodology is intended to serve as a guide in the design of linear
lumped-parameter systems with active components, such that desired frequency
response specifications are met. It is not intended to yield exact design
parameters. Rather, it will provide the system designer with an operating
envelope of an active device, such that an appropriate physical realization of the
active device can be selected or designed.
1.3 OBJECTIVES AND CONTRIBUTIONS
1.3.1 Objectives of this Work
There are three main objectives of this work that will lead to pertinent
contributions to scientific knowledge within the design and simulation research
community.
The first objective is to develop an impedance-based methodology for the
synthesis of multiple energy active elements in mechanical systems. This is an
extension of existing electrical system active synthesis methodologies and the
electrical and mechanical system passive synthesis methodologies.
The second objective is to develop a methodology for state equation
development and simulation of systems, by which active devices can be
physically realized and their approximate design parameters and control schemes
3
can be determined. The design parameters of these active devices are based on
simulation results. Active elements in various mixed energy domains will be
investigated, such as hydraulic-mechanical and electromechanical.
The third objective is to demonstrate the validity of this methodology by
applying it to design problems that involve: (1) newly designed systems with
active elements, and (2) redesign or retrofit of existing systems in which an active
element is desired.
1.3.2 Contributions of this Work
The main original contributions of this work are briefly summarized
below.
(1) This work develops a methodology in which the synthesis of multiple-
energy active elements in mechanical systems and their physical realizations are
carried out, using an impedance-based approach, such that desired frequency
domain behavior of a system is achieved.
(2) This methodology aids in the determination of control schemes that are
necessary to control the active element, such that required frequency domain
behavior are met.
(3) This work has significant pedagogical value in that the methodology
can be taught within the framework of a design course or a systems modeling
course, where it can be used as a design tool for systems in which active devices
are required or desired.
4
1.4 OVERVIEW OF SYNTHESIS TECHNIQUES
1.4.1 Impedance-Based Approaches: Passive and Active Electrical Systems
The earliest work that opened up the topic of synthesizing a system based
on specified frequency-dependent behavior was put forth by Foster (1924). His
work involved the synthesis of LC electrical circuits, via the analysis of driving-
point impedances, that exhibited specified resonant behaviors.
Later work by Brune (1931) more closely resembles the problem of
synthesis in the framework in which it is cast within the research community
today. His work involved the synthesis of a finite two-terminal electrical
networks whose driving point impedances were given as prescribed functions of
frequency. Brune’s work also put forth many important definitions and theorems
regarding the physical realization of synthesized systems. He presented the
necessary conditions for an impedance function of a passive system, specifically,
that impedance has to be a positive real function of frequency.
The general concept behind the synthesis procedure is to decompose the
driving point impedance function into terms that represent the impedances of
simple electrical elements: resistors, capacitors, and inductors. This is achieved
through a process of repeated partial fraction expansion and polynomial division.
From the individual impedance terms, the actual topology of the system can then
be determined.
Since the time of Brune’s initial work, numerous authors have presented
further developments in the area of passive electrical synthesis. Baher (1984)
provides a well-connected survey of various electrical system synthesis
techniques that involve passive lumped networks. He then extends existing
synthesis theory and procedures into the areas of distributed parameter networks
and filter design.
5
Electrical network synthesis techniques for active systems were first
developed in the 1960s, particularly in the design of RC active circuits and
operational amplifiers (op amps.) Bruton (1980) provides a detailed treatment of
these techniques and the theory behind them. Of particular interest to the work
addressed in this dissertation, are the concepts of active electrical system
elements. These elements include negative resistors, capacitors, and inductors, as
well as elements that have no passive system counterparts, such as frequency
dependent negative resistors.
1.4.2 Functional vs. Parametric Approaches to Synthesis
The works of Ulrich and Seering (1988,1989) and Finger and Rinderle
(1989) employ a functional and behavioral-based approach to the synthesis and
design of engineering systems. Their works are based on the identification of the
desired function of a system that is represented by a single input-single output
lumped parameter model.
Their approach is significant to the work presented in this dissertation in
that they employ the bond graph methodology as an integral part of the synthesis
process. Their synthesis method includes the use of two-port power
transformation elements to represent active devices such as electric motors,
however the dynamic frequency response and control of such devices is not
addressed in their work.
As a basis of comparison, the approach taken by the above authors is
considered to be a functional approach to synthesis, whereas the approach taken
in this work is classified as parametric.
6
1.4.3 Extension of Impedance-Based Methods: Passive Mechanical Systems
The work of Redfield and Krishnan (1990) extends the classical
impedance-based approaches of passive electrical system synthesis into the
mechanical energy domain. They successfully employed block diagrams and the
bond graph methodology to provide an analogy between basic electrical and
mechanical system elements. Through this analogy, they were able to synthesize
passive mechanical systems based on desired frequency response behavior.
As in the passive electrical network synthesis techniques described above,
the desired system frequency response function is reduced to basic impedance
terms using partial fraction expansion and polynomial division. Individual terms
are compared to a library of bond graph primitives to determine if their individual
forms match the impedance/admittance definition of one of the primitive bond
graph elements. If a particular impedance term does not match, then a further
decomposition of the term is necessary. This process is repeated until all of the
impedances and admittances terms are of basic form. They formulated a
technique in which the synthesis procedure would yield passive elements only,
requiring that each impedance or admittance term be positive real. They
suggested as future work that this procedure be extended to include active
elements in the system synthesis.
1.4.4 Active Element Synthesis in this Work
While other impedance-based methods, such as electric circuit equivalent
models, could be used to model system behavior, bond graphs were selected as
the modeling method to be used in this work for a number of reasons: (1) the bond
graph methodology yields a clear mapping of the topology of a system, (2) it
easily lends itself to impedance-based modeling, (3) there exists a straightforward
method by which state equations can be extracted from the bond graph, (4) it
7
allows for causality information to be contained in the system model, (5) it easily
handles the modeling of systems that involve multiple energy domains, and (6) it
easily allows for the modeling of system elements that exhibit nonlinear behavior.
The proposed methodology follows the same basic principles of classical
impedance-based synthesis techniques, in that the active element impedance
function is decomposed into basic impedances. The restriction that basic
impedances be positive definite is removed, thus allowing both passive and active
elements in the system synthesis.
Thus, it is required to formulate a technique for the decomposition of
systems into basic impedances that are neither positive definite nor represent
traditional primitive bond graph elements. Thus, bond graph elements that
represent system elements with negative impedances, generally a non-physically
realizable phenomenon, will be used as a mathematical modeling tool to represent
the behavior of an active device.
Simulations are carried out in which the time histories of the dynamic
variables associated with the active element are determined. These time histories
are then used to determine the corresponding dynamic variable time histories that
are required for an appropriate physical realization of the active device.
1.4.5 Other Frequency Domain Approaches to Active Element Synthesis
Review of the literature in the field of actuator synthesis and design,
particularly in the area of vehicle suspension systems (Balas, 1998) and seismic
structures, has revealed that other synthesis approaches exist which differ from
the approach taken in this work.
For example, Van de Wal, et. al (1998) approach actuator and sensor
selection for an active vehicle suspension by eliminating actuator and sensor
8
combinations for which robust controllers do not exist. They note in their work,
for particular applications, that this procedure can be computationally intensive.
In their study of the hybrid control of seismic structures, Cheng, et. al.
(1998), begin with a classical structural dynamics approach, in which the control
system and actuator dynamics are incorporated into the matrix equations of
motion used for modal analysis.
While many of these approaches are based on frequency domain behavior
in some way, they differ from the approach taken in this work, in which the active
element synthesis commences by directly identifying the impedance
representation of the active element, and then using this as the criterion by which
the physical realization of the active device is obtained.
1.5 DISSERTATION ORGANIZATION
In Chapter 2, the proposed active element synthesis method is introduced.
The chapter begins with a section that covers concepts and definitions that are
central to this work, particularly the concept of passive and active elements,
impedances, and frequency domain behavior of systems. Next, a simple example
problem is worked through to familiarize the reader with the process of obtaining
a bond graph representation of active element behavior.
In Chapter 3, the process of active element physical realization is
presented. This involves the process of determining the design of and modeling
the real actuating device that is represented by the bond graph obtained in the
previous chapter. Lastly, a flow chart that summarizes the steps of the entire
process is presented.
In Chapter 4, the validity of this methodology is further demonstrated by
its application to two design problems. The first application involves the design of
9
an active vibration suppression system for a laboratory model of a two-story
building. The second application involves the redesign of a electromechanical
vehicle suspension system that contains an active device. Actual test data are
used to determine an active system design, so that comparisons between simulated
behavior and experimental data can be made. Two physical realizations of the
active element from different energy domains are explored and comparisons are
made between the two. For each problem, specific issues related to the
application of the methodology are discussed, and conclusions and
recommendations regarding the synthesized designs are presented.
In Chapter 5, direction for future work on the development of this
methodology is presented, particularly the extension of this methodology to
include systems with multiple active devices. A sample problem involving a
system that requires multiple actuators is presented in order to provide a starting
point from which to develop a synthesis technique for systems with multiple
active elements. Finally, a summary of the research work presented in this
dissertation is provided, along with conclusions regarding the results of the
implementation of this methodology and its applicability as a design tool.
The appendices provide detailed treatments of various techniques and
methods used to implement the proposed active element synthesis method.
Appendix A provides the reader with a tutorial of bond graphs and their
application in system modeling. It is strongly recommended to the reader whom
is not familiar with bond graphs to read Appendix A before reading the main body
of this work. Appendix B is an overview of various spectral density functions and
their use in determining system frequency response behavior. Appendix C details
the implementation of the technique used for frequency response function curve
fitting. Appendix D contains sample Mathematica™ code, which illustrates the
10
computation of transfer functions and the decomposition of an active element
impedance representation, which leads to the synthesis of an active device.
11
Chapter 2: Active Element Synthesis Method
2.1 INTRODUCTION
This chapter introduces the active element synthesis method. It begins
with a presentation of the definitions, concepts, and background theory that is
relevant to the development of the method. The steps are presented via the use of
an example problem in which an active vibration filter for a simple mechanical
oscillator is synthesized. This problem was chosen due to its simplicity, its
previous study in the literature, and its ability to clearly illustrate each step of the
active synthesis methodology.
The chapter concludes with two sections that address various aspects of
the active synthesis methodology with respect to the effects of active control
systems on frequency response calculations and the use of two-port bond graph
representations in active element modeling.
2.2 DEFINITIONS AND CONCEPTS
2.2.1 Passive and Active Elements
Active elements are defined as elements that have a positive net energy or
power output. Beaman and Paynter (1993) use the following simple example to
illustrate this concept. Consider a negative spring, with a stiffness of –k (where
k>0), which can be modeled in a bond graph by a negative C element, as shown
in Figure 2.1.
12
-C: -kFk
xv &=
Figure 2.1: Negative bond graph element
The constitutive law for this element is kxFk −= . This can be represented in the
effort vs. displacement plot of Figure 2.2
Fk
x
Figure 2.2: Constitutive law for negative spring
The energy stored in the negative spring is given by the following
equation.2
21)( kxx −=Ε (2.1)
If we attach a flow source to this element (Figure 2.3), representing a constant
positive compression rate, v, of the spring, we see that the energy stored in the
13
spring decreases in time, which means that the flow source is being provided with
an increasingly positive energy gain.
-C: -kFk
xv &=:Fv(t)=v
Figure 2.3: Negative spring with constant flow source
In the limit as time approaches infinity, the stored energy approaches negative
infinity, which implies that the negative spring provides an infinite amount of
energy back to the flow source. Physically, this implies that the negative spring is
producing positive net energy, which indicates that it is an active element.
Formally stated, the stored energy function of active elements is not
bounded from below, while that of a passive element is bounded from below.
This is illustrated in Figure 2.4.
E
x
E
x
passive element active element
Figure 2.4: Stored energy for passive and active elements
14
2.2.2 Classification of Impedances
An impedance is defined as the ratio of an effort variable to a flow
variable
floweffortimpedance =
Efforts are force-like quantities, such as force, torque, pressure, or voltage.
Flows are velocity-like quantities, such as velocity, angular velocity, volumetric
flow rate, or current.
The proposed synthesis method requires the analyst to classify impedances
according to two criteria. The first criterion is to determine whether an
impedance is basic or composite. In linear systems, there are three basic
impedance types, as shown in the table below.
elementtype
generalizedimpedance
mechanical energydomain coefficient
mechanicalimpedance
inertive sIsZ =)( I = m = mass smsZ =)(
capacitivesC
sZ 1)( = 1/C = k = spring stiffnesssksZ =)(
dissipative RsZ =)( R = b = viscous damping bsZ =)(
Table 2.1: Basic impedance elements (Generalized and Mechanical)
15
An impedance is said to be composite if it can be further broken down,
such that it is represented in terms of the basic impedance forms presented above.
The second criterion is to determine whether the impedance represents a passive
or an active element. Baher (1984) demonstrates that if an impedance of a
passive, lumped, linear, and time invariant impedance is positive real, then it
represents a passive element. He further shows that the positive real condition is
both necessary and sufficient for passivity. He concludes with the statement that
if the impedance of an element is not positive real, it represents an active
element.
2.2.2 Frequency Domain Behavior of Systems
The proposed synthesis method uses the frequency response function
(FRF), which is also sometimes referred to as the transfer function, as its basis.
Strictly speaking, the transfer function and frequency response function are two
different functions in that the transfer function is defined in the s-domain, whereas
the FRF is a special case of the transfer function, where s=jω (i.e., the real part of
s=0.) The FRF is a complex-valued function that describes the response of a
system to a sinusoidal input throughout a specified range of frequencies. Since it
is not feasible to excite a system at a large number of discrete frequencies,
especially over a wide frequency band, a random excitation input can be used. It
can be shown that random data can be represented by an infinite number of
signals at different frequencies. Thus all possible frequencies are input to the
system, and the response consists of a superposition of all possible responses.
The response can be considered in two parts: magnitude and phase. The
magnitude is the ratio of the amplitude of the response of the system to the
16
amplitude of the input. The phase is the phase angle, or lag, between the input
sinusoid and the output.
The FRF can be analytically expressed as a ratio of two polynomials,
where the denominator polynomial is of equal or higher order than the numerator
polynomial. In some cases, the FRF of a system may not be readily available in
analytical form. When this occurs, the FRF can be determined from experimental
system response data.
An FRF can be computed by evaluating certain combinations of spectral
density functions over a set of discrete frequencies. The magnitude is formed by
taking the ratio of the magnitude of cross power spectrum of the input and
response and the auto power spectrum of the input at discrete frequency points.
The phase angle is computed by taking the four-quadrant inverse tangent of the
ratio of the real and imaginary parts of the cross power spectrum at discrete
frequency points. See Appendix C for a more detailed treatment of these
calculations.
After both portions of the FRF are computed, we have two sets of discrete
data points that define two curves: the magnitude ratio and the phase angle. To
determine the polynomial ratio equation that best fits the data, we employ a
complex-plane curve fitting technique based on the works of Levy (1959) and
Sanathanan, et al (1963, 1974). Upon arriving at a satisfactory polynomial ratio
equation that falls within a user-defined error tolerance, the synthesis may then
proceed based on this analytical expression.
17
2.3 EXAMPLE PROBLEM – SYNTHESIS OF AN UNKNOWN IMPEDANCE
2.3.1 Problem Description
The problem of designing a passive vibration filter for a simple
mechanical oscillator, as studied by Redfield and Krishnan (1990), will be used as
a demonstration problem. The system under study is shown in Figure 2.5.
k
m
F(t)
Figure 2.5: Simple mechanical oscillator
The system consists of a mass (m=1kg) attached to a grounded spring
(k=1000N/m), and is subjected to a sinusoidal forcing function
ttf ωsin)( = . The bond graph of this system is shown in Figure 2.6.
18
f(t) = sin ω t :E
C: k
I: m1
Fs
p&x&
x&
Figure 2.6: Bond graph of simple mechanical oscillator
The impedance bond graph is shown in Figure 2.7. Here, the impedance
of the spring is represented as sk
sC=1 and the impedance of the mass is sm,
where s is the Laplace domain variable.
E
C: 1/sC
I: sm1
Figure 2.7: Impedance bond graph of simple mechanical oscillator
In an impedance bond graph, the impedances at a 1-junction add. Thus
the equivalent impedance is
19
skms
sCmsZ +=+= 1 (2.2)
If we are interested in the transfer function between the force input and the
velocity of the mass, we can equate the admittances on both sides of the 1-
junction, since these are the only bonds connected at that junction. The transfer
function is
kmsssTF+
= 2)( (2.3)
A frequency response plot that corresponds to this transfer function is
shown in Figure 2.8. Also shown in this figure is the desired frequency response
behavior.
20
comparison - without filter and desired response
-80
-60
-40
-20
0ph
ase
(deg
)
mag
nitu
de (d
B)
100 101 102 103-100
-50
0
50
100
frequency (rad/sec)
desired response
no vibration filter
Figure 2.8: Actual and desired frequency response behavior
As seen in this plot, there is an undesirable resonance peak. The results of
adding a simple viscous damper (with b=1 N-s/m) to this system is shown in
Figure 2.9, does not adequately modify the system behavior.
21
comparison - response with damper and desired response
-100
-50
0
50
100
150
200
100 101 102 103-100
-50
0
50
100
frequency (rad/sec)
phas
e (d
eg)
m
agni
tude
(dB
)
with simple damper
desired response
Figure 2.9: Desired frequency response and response with simple damper
The transfer function of the system with the viscous damper is
1000)( 2 ++=
ssssTF (2.4)
Whereas the corresponding transfer function for the desired response is
10002201.1422.0001.0086.0001.0)( 234
23
++++++=
ssssssssTF (2.5)
22
The task is to determine the configuration of an unknown vibration filter that
meets the desired frequency response behavior shown in Figure 2.8. This is
represented schematically in Figure 2.10.
k
m
filter
F(t)
Figure 2.10: System with unknown vibration filter
E
C:
Z(s)
I: sm1
sk
sC=1
Figure 2.11: Bond graph of system with unknown filter impedance, Z(s)
23
2.3.2 Problem Solution
If we represent the impedance of this unknown filter as Z(s), and recalling
that impedances add at a 1-junction, the desired transfer function can be
represented as
ssZsssTF
⋅++=
)(1000)( 2 (2.6)
Equating the previous two equations, we can solve for the impedance of
the filter. The resulting impedance is
10008613400012100134
)( 2
2
++++=
ssss
sZ (2.7)
This impedance function can be reduced to separate impedances using a
partial fraction decomposition
sssZ
++
+−=
138.7202.713
86.1302.137134)( (2.8)
In order to provide an analogy with which most readers would be familiar,
this decomposed impedance can be represented as three electrical impedances in
series, each representing a term in Equation 2.8, as shown in Figure 2.12.
24
Z1
Z2
Z3
Figure 2.12: Equivalent impedance representation as a series circuit
Impedances in series add, which is represented in bond graph format using
a 1-junction, as shown in Figure 2.13.
1
Z1= 134
Z2= s+−
86.1302.137
Z3= s+138.7202.713
Figure 2.13: Equivalent impedance representation in bond graph form
Returning our attention to the nature of each impedance term in Equation
2.8, we see that the first impedance term is basic, which means that its impedance
corresponds to a primitive bond graph element, namely a resistive element. It is
also positive definite, thus it is a physically realizable element. The second term
is composite, which means that its impedance does not correspond to a primitive
25
bond graph element and must be further decomposed. It is also not positive
definite, thus its resulting terms will not correspond to physically realizable
elements. The third term is composite, however no conclusion can be made about
positive definiteness of its resulting terms until a further decomposition is carried
out.
Up to this point, we can draw a word bond graph that illustrates the
reticulation of the system. This is shown in Figure 2.14.
E
C: k=1000
I: m=1
1 1
R: b=134
unknown
active
filter
Figure 2.14: Word bond graph of filter reticulation
The two composite impedance terms cannot be broken down any further.
However, we can invert them to form admittances. Knowing that admittances add
at a zero junction, we can further decompose the composite terms. First consider
the second term of Equation 2.8, whose corresponding admittance is
s+−
86.1302.137 (2.9)
26
Performing a polynomial division, we obtain
s0073.01012.0 −− (2.10)
The first admittance term corresponds to a “negative damper” with a viscous
damping coefficient of
884.91012.01 −=− (2.11)
The second admittance terms corresponds to a “negative spring” with a stiffness
of
02.1370073.01 −=− (2.12)
Next, considering the third term of Equation 2.8 (which is composite); the
corresponding admittance is
ss 0014.01012.002.713
138.72 +=+ (2.13)
The first admittance term corresponds to a damper with a resistance of
8841.91012.01 = (2.14)
The second admittance term corresponds to a spring with a stiffness of
27
02.7130014.01 = (2.15)
In summary, the total filter impedance, expressed as a sum of basic
impedances is
s
sZ
springnegativedamper
negative
02.7138841.9
1
s137.02
-884.9-
1134)(+
++=
434 21321
(2.16)
The complete system reticulation is shown in Figure 2.15 in bond graph from and
in schematic form in Figure 2.16.
28
E
C: k=1000
I: m=1
10
R: b=9.884
C: k=713.02
0
-R: b= -9.884
-C: k=-137.02
active element
filter
R: b=134
Figure 2.15: Bond graph representation of system reticulation
29
k=1000
F(t) m
k=713.02
b=9.884
activeelement
k= -137.02
b= -9.884
b=134
Figure 2.16: Schematic representation of system reticulation
The system is determined to be of fourth order; the state equations are
given below
ss m
pvx ==& displacement of mass (2.17)
sssss
f xkxkxkm
pbtfp −−−−= 2211)(& momentum of mass (2.18)
1
111 b
xkmpx
sS −=& deflection of real spring (2.19)
30
2
222 b
xkmpx
SS −=& deflection of negative spring (2.20)
A simulation was performed in MATLAB and the total force associated
with the 0-junction that joins the negative spring and negative damper was
calculated. This was plotted against the velocity of the mass, which is the flow
variable that feeds into the 0-junction. This force/velocity profile can be used to
characterize the required behavior for an active device that would take the place
of the negative spring and damper, since they are not physically realizable.
Fact
vact
k=1000
F(t) m
k=713.02
b=9.884
b=134active
element
Figure 2.17: Passive/active configuration of vibration filter
Note that the above configuration represents a “passive/active”
configuration, which means that the active device replaces only those filter
elements with non-positive real impedances. In this case, the active element
31
replaces the negative spring and negative damper only. This is illustrated in the
annotated bond graph of Figure 2.18.
E
C: k=1000
I: m=1
10
R: b=9.884
C: k=713.02
0
-R: b= -9.884
-C: k=-137.02
active element
actuator
R: b=134
Fact
vact
Figure 2.18: Bond graph representation of passive/active filter
The force and velocity profile of the actuator is determined by the time
history of the respective power conjugate variables (Fact and vact) on the bond that
connects into the box labeled “actuator”.
Another option is to choose a “fully active” configuration, in which the
active element replaces all of the bond graph elements that represent the filter.
32
The physical realization and the annotated bond graph model are shown in
Figures 2.19 and 2.20, respectively.
Fact
vact
k=1000
F(t) m
activeelement
Figure 2.19: Fully active configuration of vibration filter
33
E
C: k=1000
I: m=1
10
R: b=9.884
C: k=713.02
0
-R: b= -9.884
-C: k=-137.02
active element
actuator
R: b=134F0
v1
F1
v1
F2
v1
Figure 2.20: Bond graph representation of fully active filter
In this case, the force-velocity profile of the actuator is determined by the
sum of the time histories of the respective power conjugate variables (Fact and
vact) on the two bonds that connect into the box labeled “actuator”. The equations
for the actuator force and velocity are
1
210
vvFFFF
act
act
=++=
(2.21)
34
A graph of the required force-velocity characteristics for the active device
(due to a sinusoidal forcing function) is shown in Figure 2.21.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 10-4
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025Actuator force-velocity profile
velocity (m/s)
forc
e(N
)
Figure 2.21: Force-velocity profile for fully active filter
A simulation was run to determine if the reticulated system matched the
frequency response of the original system. A swept frequency simulation was
performed and a frequency response plot was constructed. This was compared to
the frequency response plot generated by MATLAB , given the overall system
transfer function. The superimposed plots are shown in Figure 2.22. There is
35
good correspondence between the two plots, in that the relative error between the
two is less than two percent. This indicates that the reticulation scheme that uses
negative bond graph elements produces physically meaningful results.
101
102
30
35
40
45
50magnitude ratios vs. frequency
frequency (rad/s)
mag
nitu
de ra
tio (d
B)
synthesized system simulation data original FRF data used for curve fit
101
102
0
2
4
6
8
10
frequency (rad/s)
perc
ent r
elat
ive
erro
r
Figure 2.22: Swept frequency response comparisons and error calculations
Additional verification tests were performed using the active synthesis
technique on this system. All yielded results similar to those above, with errors
on the order of two percent.
36
For example, to verify the transient response, a step input to the
synthesized active system was compared to the step response of the original
passive system. The results are shown in Figure 2.23.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.5
1
1.5
2input force, mass velocity, and filter force: PASSIVE SYSTEM
inpu
t for
ce (N
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.5
1
filte
r for
ce (N
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8x 10
-3
time (sec)
mas
s ve
loci
ty (m
/s)
37
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.5
1
1.5
2
Simulation of synthesized system with active elements
inpu
t for
ce (N
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.5
1
filte
r for
ce (N
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8x 10
-3
time (sec)
mas
s ve
loci
ty (m
/s)
Figure 2.23: Comparison of step response for passive and active systems
2.4 ACTIVE DEVICE CONTROL SYSTEM ISSUES
In the initial stages of this work, there was a concern that a control system
associated the active device in a system would affect the synthesis procedure.
This would be of particular concern in the case where experimental data was
being used to generate frequency response requirements for use in the system
synthesis. This will be the case in the next section, where the synthesis of an
38
alternate vehicle suspension system will be carried out using experimental data in
which an actuator and control system is already present.
It is the author’s argument that the synthesis of an active element is not
affected by the system that is employed in the control of the active device. To
illustrate this, a simple vibration isolation system, similar to that of example
system in this chapter, will be used. A schematic of the system is shown in Figure
2.24 and its corresponding bond graph is shown in Figure 2.25.
vin(t)
vm
k Z(s)
m
Figure 2.24: Example vibration isolation system
39
F
C: k/sZ(s)
0 I: ms1
1
Fsv
Fsvm
p&vm
Fs vs
FZ vsFk vs
Figure 2.25: Example vibration isolation system bond graph
As a preliminary argument, consider the power bond associated with the
unknown active element impedance, Z(s). The operational regime of the active
device is defined by the effort and flow variables associated with that power bond.
Regardless of the controller that is used in association with the physical
realization of the active device, the actual device must be able to operate within
the envelope defined by the plot of the effort variable vs. flow variable, since this
represents the actual values of the dynamic variables that will be encountered in
the operation of the system.
Consider Figure 2.26, that shows the corresponding block diagram portion
of the active element, where the input and output signals correspond to the effort
and flow variables on the power bond and are consistent with the causality
illustrated in the bond graph.
Z(s)Vs FZ
Figure 2.26: Active element block
40
If we make the assumption that the control system has an effect on the
active element synthesis, then it would be necessary to include the control system
effect in the bond graph. If we replace the Z(s) block with the “actual” active
element impedance Zf(s) and a simple feedback law, such that the overall
impedance is still Z(s), and making sure that the input variables Vs and Fz remain
intact, the resulting block diagram fragment is shown in Figure 2.27.
viVs
+-ve
Zf (s)FZ
FZ
FZVs
ks
Figure 2.27: Active element block replaced with feedback control
Applying the laws of block diagram algebra and considering the
input/output relationships illustrated in the above figure, the corresponding bond
graph fragment would be
41
1 FZVs
0
FZ
vi
FZve
Zf (s)
ks
Figure 2.28: Corresponding bond graph fragment for feedback control
Note that according to this bond graph fragment, the active element now
“sees” vi as the flow variable, not vs, which is inconsistent with the original block
diagram and bond graph. Thus the introduction of the control system into the
bond graph corrupts the original power conjugate variables of the overall system
bond graph.
As another argument, consider the generalized feedback control system
presented by Nise (1995), as shown in Figure 2.29.
+
-
inputtransducer
G2(s) G3(s)
H2(s) H1(s)
G1(s)
R(s)
inputsignal
referencesignal
ir(s) ie(s)
actuationsignal
feedbacksignal
if(s)
controllerplant
(system)
feedbacklaw
outputtransducer
C(s)
outputsignal
Figure 2.29: Generalized feedback control system
42
If we cast the example system into this control system framework, we see
that the controller, in this case the active device (actuator) is actually part of the
plant/system, since it provides an impedance to the system. That is, depending on
the state of the system at any given time, the actuator acts like a combined spring-
damper, negative spring-damper, a spring-negative damper, or a negative spring-
negative damper. Thus, its presence as a controller has an effect on the overall
impedance of the system, since it is introducing power into the system. This
statement supports the fact that it should be modeled as an impedance in the bond
graph of the overall system, as was shown in Figure 2.25.
Turning our attention to the remainder of the control system, it is noted
that the signals associated with the transducer, feedback law, and the actuating
signal, are all operating at low power, and thus can be represented by signal flows
in the overall system bond graph, as shown in Figure 2.30.
F
C
Z(s)
I1)( x&
Fin p&
Fk
FZ
-
1sH1(s)H2(s)
x&xi
if
ir ie
transducerfeedbacklaw (gain)
...x&
x&
x&
x&
Figure 2.30: Overall system bond graph showing control signal flows
43
Since the impedances associated with all of the above parts of the control system
are associated with low power, they are not included in the bond graph as
elements that affect the overall impedance of the system. Thus they do not affect
the impedance of the active element, and play no part in the active device
synthesis procedure.
2.5 ONE VS. TWO PORT SYNTHESIS OF ACTIVE ELEMENT IMPEDANCES
Another issue is whether the decision to model the active element
impedance as a one-port bond graph element is valid. Consider the bond graph of
the example system from section 2.3.1, shown in Figure 2.31, with the active
device realized as an electric motor with rack and pinion gearing, such that it
provides linear actuation.
T
:
km
:E 1 1
I: Ls
R: Rs
I: Jr
R: Br
rvin
iτω
Fact
vact
vm
iΤωvin=Φ (ie)
ie
I)( x&
x&p&
C
Fin
Fk
E
1
active element realization
x&
x&
G
:
Figure 2.31: Bond graph of example system with active device realized
44
Since the realization of the active element involves two energy domains, we could
chose to model the motor as a two-port impedance, as illustrated in Figure 2.32.
In this case, one could argue that a two-port model of the impedance of the active
element is a more valid choice.
Zvin
iFZx&
=
in
m
m
z
vx
dcba
iF &
Figure 2.32: Two port impedance and transmission matrix equation
However, note that in the overall synthesis procedure, we are seeking a transfer
function that relates the output of the system (i.e., the velocity and force
associated with the mass) with the base excitation (input). We are not seeking a
transfer function that relates the input to the active device (i.e., motor input
voltage) with the output of the system, thus there is no need for a two port
element, since we are not seeking a transfer function that involves more than one
energy domain. Although the synthesis procedure involves multiple energy
domains in general, we must note that the methodology calls for us to work in the
s-domain (frequency domain) in order to synthesize the elements that model the
mathematical behavior of the active device. For the process of physically
realizing the active device and its design parameters, we are employing a
simulation-based approach in the time domain.
45
2.6 SUMMARY
This chapter introduced the active synthesis method and background
theory regarding the classification of impedances, the definitions of active and
passive elements, and the analysis of frequency response behavior for linear
systems.
The steps of the active element synthesis method were presented via the
use of a simple example problem in which an active element for the vibration
isolation of a simple mechanical oscillator was synthesized. The example
illustrated the procedure for solving for the unknown impedance of the active
element in the system, and the process in which the composite impedance is
decomposed into basic impedance terms. Next, the procedure for modeling the
system using bond graphs and selecting the impedance terms that represent the
active element behavior was explained. Simulation results were presented, which
showed that the synthesis method yields active element models that produce the
required frequency response behavior.
Lastly, issues regarding the effect of control systems on experimental data
collection and issues related to active element bond graph modeling were
addressed. It was argued that control systems have no effect on experimental data
used to determine frequency response functions for use in the synthesis process.
It was also shown that one-port bond graph models of active element impedances
are sufficient for modeling the behavior of a multiple energy active element
within a system.
46
Chapter 3 Physical Realization of Active Elements
3.1 INTRODUCTION
The next task is to select the type of actual physical element that will be
used to replace the active bond graph elements in the model. The first step is to
determine which energy domains a physical realization may involve. Physical
factors such as weight, and volume, as well as dynamic factors such as transient
response will play a major role in the selection of the actuator type. The use of
the synthesis methodology within a simulation-based design environment will aid
the system designer in finalizing component selection. The comparison of
different physical realizations for a single application will be more fully
illustrated in Chapter 4.
3.2 MECHANICAL SYSTEM ACTIVE ELEMENT REALIZATIONS
3.2.1 Introduction
There is a distinction between the term active element, which refers to the
model representation (including negative bond graph elements, etc.), and the term
actuator, which refers to the actual physical realization of the active element.
In a mechanical system, there can be many classes of active element
physical realizations. In this work, two types of active elements will be
considered: (1) a hydraulic piston actuator, and (2) an electric motor combined
with a rack and pinion system that converts the angular motion of the motor shaft
to linear motion.
47
As indicated by the causality in the bond graph of Figure 3.1, the active
element velocity, vact, is prescribed by the mass, and the active element responds
with a force Fact. Thus, using the terminology of transfer functions, vact is the
“input” and Fact is the “output”.
E
C: k=1000
I: m=1
1
0
R: b=9.887
C: k=713.02
0
-R: b= -9.887
-C: k=-137.02
Factvact
1
R: b=134
Figure 3.1: Bond graph indicating active element force and velocity
48
3.2.3 Hydraulic Actuator
There are many different types of hydraulic actuator bond graph models,
involving transformers or gyrators. Consider a simple piston-type hydraulic
actuator, as shown in Figure 3.2.
F
Q1 Q2
x&
Figure 3.2: Schematic of hydraulic actuator
The bond graph element that represents transduction in this actuator is a
transformer with a constant transformer modulus, A1 , the cross sectional area of
the piston, as shown in Figure 3.3.
T
:
FPvx =&Q∆
A1
Figure 3.3: Actuator bond graph model transduction element
Since we know the time histories of Fact and vact from a simulation
performed using the bond graph that contains the active element representation
49
(negative elements), we can convert these to the time histories of P and V
(pressure and volume, respectively) using different values of the piston area, A.
Upon inspection of these plots, a designer can pick the value of A that best fits
available (e.g., off-the-shelf) actuation devices or falls within design restrictions
of the system.
The bond graph of a more detailed actuator model is shown in Figure 3.4.
The model is more detailed in that it captures real effects, such as viscous
damping and the mass of the piston
TPin
Pin(t) :E 1
R:bp
I:mp
:
Q∆Fv
A1
Figure 3.4: Bond graph model of actuator
3.2.4 PMDC Motor
There are be several different types of motors that could be used for
mechanical system actuation. We will consider a permanent magnet direct
current (PMDC) motor that is coupled with a rack and pinion gearing system to
convert the angular motion of the motor to the linear motion required for
50
actuation. This motor was chosen because it is the simplest type of
electromechanical actuator. Such a setup is shown in Figure 3.5.
Figure 3.5: Schematic of electric motor and rack and pinion
A bond graph model of a PMDC motor is shown in Figure 3.6. The bond
graph element that represents transduction in this actuator is a gyrator with a
linear constitutive law. In this case, the gyrator modulus is the machine constant,
km.
51
T:
G
:
km
:E 1 1
I: Ls
R: Rs
I: Jr
R: Br
rvin
iτω
Fact
vact
vm
iΤωvin(t)
Figure 3.6: Bond graph model of a PMDC motor
As with the hydraulic actuator case, knowing the Fact and vact time
histories from a simulation using the bond graph that contains the active element
representation (negative elements), we can use the bond graph above to aid in the
simulation-based design of an appropriate motor. The following motor design
parameters, represented in the bond graph above, can be varied such that the
required force and velocity profile (Fact and vact) is obtained:
stator coil inductance, Ls
stator coil resistance, Rs
rotor inertia, J
rotor bearing viscous damping, Br
machine constant, km
52
Using this information, the designer can either design a motor, or select an off-
the-shelf motor that falls within prescribed voltage and current profiles.
3.3 EXAMPLE PROBLEM - ACTIVE ELEMENT PHYSICAL REALIZATION
3.3.1 Classification of Impedances and Generalized Flow Variables
Returning to the example system shown in Figure 3.7, and its
corresponding bond graph, shown in Figure 3.8, we now proceed with the active
element synthesis.
k
munknown suspension
impedance=Z
F(t)
Figure 3.7: Example system with unknown suspension impedance, Z
53
F(t):E
I:m
1Fin
0
R:b1F1
F3
C:kFkp&
R:b31
C:k3232v
0-R:b21
-C:k2222v
F2
sv
synthesized active elementrepresentation
negative bond graph elements
sv
svsv
sv
sv
Figure 3.8: Bond graph showing active element representation
In Figure 3.8, the bond graph elements that represent the synthesis of the
unknown active element are enclosed in a parallelogram. As a subset of these
elements, the bond graph elements with negative impedances are enclosed in a
rectangle. As a by-product of the synthesis, there are now four new flow
variables, v21, v22, v31, and v32. Recall that, at this stage, the portion of the bond
graph that represents the synthesized active element is strictly a representation of
the behavior of the active element, and no topological or physical representation
is yet implied. Thus, these four new flow variables, which in the mechanical
energy domain would be associated with velocities, will be referred to as virtual
velocities. They are not yet associated with any velocity of the actual physical
system velocity.
54
3.3.2 Separable Passive Elements
The first step in the physical realization of the active element is to identify
whether any passive elements can be separated from the active element
representation. This can be done by examining the bond graph and/or a schematic
of the synthesized elements. A separable bond graph element must meet two
criteria:
(1) Its impedance must be positive real.
(2) It cannot be in any type of series configuration with a negative
impedance element. From a bond graph perspective, this means that it
cannot be connected in any way to a 0-junction this is connected in any
way to a negative impedance element.
These criteria can be further illustrated with a simple example. Suppose
that the synthesis for the example problem yielded the filter configuration of
Figure 3.9, which is represented in bond graph form in Figure 3.10.
m
original systemspring
C0
C1
R2
C3
-R4 C5
R7
-C6
C8
m
Figure 3.9: Sample filter configuration
55
1
C0
0
C1
R2
0
C3
1
C5
-R4
0
-C6
1
C8
R7
Figure 3.10: Corresponding bond graph of sample filter
In this case, spring 1 and damper 2 are separable elements. Neither spring 3 nor
spring 5 are separable, since they are involved in a “composite” series
combination with a negative element, namely, damper 4. In the bond graph, we
see that the C2 and C5 elements that represent these springs share a common 0-
junction with the negative R4 element, as shown by the grayed-out power bonds.
Similarly, neither the R7 (damper) nor the C8 (spring) are separable, since they
56
are involved in a composite series combination with a negative element, namely,
spring 6.
Returning to the example problem, we see from the bond graph of Figure
3.8, that the virtual velocities v31 and v32 can be physically realized, thus they are
considered to be real velocities. The resistive element, b1, and the combination of
resistive element b31 and capacitive element k32, are separable, as illustrated in
Figure 3.11. These elements can be associated with real system velocities,
namely vs, v31, and v32. In this particular case, it turns out that all passive bond
graph elements are separable. In general, however, this may not be true. An
example of such a case will be encountered in the next chapter.
F(t):E
I:m
1Fin
0
R:b1F1
F3
C:kFkp&
R:b31
C:k3232v
0-R:b21
-C:k2222v
F2
sv
synthesized active elementrepresentation
sv
sv
sv
sv
sv
Figure 3.11: Bond graph indicating separable portions of active representation
57
The next step is to decide whether these separable elements should be
extracted from the active element representation, thus forming a hybrid
passive/active realization, or if all synthesized elements should be used to model
the active element. This decision is made by the designer on the basis of various
design constraints, such as space, weight, power requirements, etc. In general, the
designer has three options: (1) extract all separable elements from the synthesized
representation, (2) extract none of the separable elements, or (3) extract some of
the separable elements, deciding on an element-by-element basis.
For this example, we will choose the option in which all separable
elements are extracted from the synthesized representation. This is illustrated in
the bond graph of Figure 3.12 and schematically in Figure 3.13.
F(t):E
I:m
1Fin
0
R:b1F1
F3
C:kFkp&
R:b31
C:k3232v
0-R:b21
-C:k2222v
F2
sv
synthesized active elementrepresentation
extracted passivesuspension elements
active elementrepresentation
sv
sv
sv
sv
sv
Figure 3.12: Passive suspension elements and active element representation
58
m
Fin
original systemspring
passive suspensionelements
active element(hydraulic actuator)
Figure 3.13: Schematic of extracted and active suspension elements
The state equations for the system are
31
323232
21
222222
323222221
bxk
mp
x
bxk
mp
x
mp
x
xkxkm
pbkxFp
s
sin
−=
−=
=
−−−−=
&
&
&
&
(3.1)
59
3.3.3 Physical realization of Non-Separable Bond Graph Elements
The next task is to realize a physical representation for the negative bond
graph elements that are associated with the virtual velocities v21 and v22. Consider
the power bond that leads into the zero junction that connects to the negative bond
graph elements, whose causality indicates that the active element must provide a
force F2 at the prescribed velocity vs. The bond graph fragment is shown in
Figure 3.14.
0
-R:b21
-C:k2222v
F2
sv
21vF2
F2
Figure 3.14: Bond graph fragment associated with active element
From the bond graph fragment above, the equation for F2 is
22222 xkF = (3.2)
where x22 is determined by the differential equation
21
2222
21
222222 b
xkmp
bxk
vx s −=−=& (3.3)
60
Here we choose the active element to be a hydraulic actuator, with a piston area
A, where a pump is assumed to supply the input pressure to the working fluid.
The bond graph representation of this actuator is shown in Figure 3.15.
P(t) :E T
:P(t)
Q
F2
vs
1A
controlsignal
...
Figure 3.15: Bond graph model of actuator
The next issue to address is the control of the effort source. The input
pressure P(t) must follow a trajectory such that the output force of the actuator is
equal to F2. Using the above equations, we can represent them as a basis of a
control scheme in which the input signal is vs, represented by a signal flow in the
bond graph, and the output signal is pressure signal, P(t) which is sent to a
modulated effort source. The signal flow structure shown below is a graphical
representation of equations 3.2 and 3.3. This is illustrated in Figure 3.16. Note
that the new bond graph elements and signal flow structure in the gray box
replace the bond graph fragment that represents the active element (shown
previously in Figure 3.14.)
61
F(t):E
T: 1/A
I:m
1Fin
sv
Fs
0
R:b1F1
F3
C:k
V&Pin
Fkp&
E:P(t)
R:b31
C:k3232v
sv + 22v 22x 2F
-
signal flow
svsv
sv
sv
s1 k22
21
1b
A1
Figure 3.16: Bond graph with control system and modulated effort source
Force-velocity profiles obtained from the time simulation results are
shown below. Figure 3.17 shows the force-velocity profile for a sinusoidal
forcing function. Figure 3.18 shows the case in which the input is a uniformly
distributed random function.
62
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 10-4
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025Actuator force-velocity profile
velocity (m/s)
forc
e(N
)
Figure 3.17: Force-velocity profile for sinusoidal forcing function
-4 -2 0 2 4
x 10-5
-3
-2
-1
0
1
2
3
4x 10
-3 Actuator force-velocity profile
velocity (m/s)
forc
e (N
)
Figure 3.18: Force-velocity profile for random input
63
In the above case, the actuator is modeled as an ideal hydraulic piston
actuator, which has no losses or energy storage. Since this cannot be physically
realized, we must incorporate effects such as the mass of the piston and the
viscous damping coefficient into the actuator model to obtain a more realistic
model of the system behavior. This is illustrated in the bond graph of figure 3.19.
F(t):E
T: 1/A
I:m
1Fin
sv
Fs
0
R:b1F1
F3
C:k
V&Pin
Fkp&
E:P(t)
R:b31
C:k3232v
sv + 22v 22x 2F
-
signal flow
svsv
sv
sv
s1 k22
21
1b
A1
R:bp
I:mp
real actuatoreffects
sv
Fbp
actp&
sv1
Fact sv
Figure 3.19: Bond graph with actuator piston effects added
64
Since both of the new bond graph elements have dependent causality, it is
helpful to lump them with existing system bond graph elements to determine the
state equations for this augmented model. This is illustrated in the bond graph of
Figure 3.20.
R:b1 + bp
F(t):E
T: 1/A
I:m + mp
1Fin
sv
Fs
0
F1
F3
C:k
V&Pin
Fkp&
E:P(t)
R:b31
C:k3232v
sv + 22v 22x 2F
-
signal flow
svsv
sv
sv
s1 k22
21
1b
A1
Figure 3.20: Bond graph with new elements lumped with existing ones
The addition of the resistive and inertive elements causes a discrepancy
between the desired behavior, which was modeled using an lossless, non-energic
actuator, and the actual behavior of a real actuator whose model contains energic
and dissipative elements. This is reflected in the force-velocity profile
65
comparison plots of Figure 3.21 and in the comparisons of the individual force
and velocity plots, as shown in Figure 3.22.
-4 -3 -2 -1 0 1 2 3 4
x 10-3
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02Actuator force-velocity profile
velocity (m/s)
forc
e (N
)
ideal case losses/energy storage
Figure 3.21: Comparison of force-velocity profiles for ideal and real actuator
66
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02actuator force vs. time
forc
e (N
)
time
ideal case losses/energy storage
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4
-3
-2
-1
0
1
2
3
4x 10
-3 actuator velocity vs. time
velo
city
(m/s
)
time
ideal case losses/energy storage
Figure 3.22: Comparison of individual force and velocity plots
67
Additionally, a swept frequency simulation was performed to determine this
effect at different excitation frequencies. A plot of the actual system response and
the original FRF is shown in Figure 3.23.
100
101
102
-62
-60
-58
-56
-54
-52
-50
-48
-46
-44
-42
Comparison of data-based FRF and actual system response
frequency (rad/s)
mag
nitu
de ra
tio: v
mas
s/in
put f
orce
(dB
)
actual system responseoriginal FRF
Figure 3.23: Swept frequency results and original FRF
To compensate for the addition of the bond graph elements that model the
energic and dissipative properties of the actuator, it is necessary to repeat the
synthesis process such that the original frequency response behavior requirements
are satisfied. The new system is represented by the bond graph in Figure 3.24.
68
F(t):E
I: m + mp
1Fin
svR:bp
C:kFkp&
Z(s)
svsv
sv
Fb
suspension system impedance(ideal active + passive)
Figure 3.24: Bond graph of new system with non-ideal actuator effects
The expression for the theoretical transfer function of the system that
includes real actuator effects is
1000)()()( 2 ++++=
ZbsmmsstTF
pp
(3.4)
Where values of the piston mass, mp=0.1 kg and viscous damping of the piston,
bp=10 N-s/m were chosen. The data-based transfer function remains the same,
and is repeated below.
10002001.1422.0001.0086.0001.0
)( 234
23
++++++=
sssssss
sdTF (3.5)
69
Following the active synthesis procedure outlined earlier, the impedance
of the unknown active element, Z(s), was found to be
1000086010
124000011140011541.0)( 2
23
+++++=
sssss
sZ (3.6)
Decomposition of the impedance into basic elements yields the bond
graph of Figure 3.25. The negative bond graph elements represent the actuator
behavior, while the bond graph elements that represent passive elements were
extracted as shown.
70
F(t):E
I: m + m p
1Fin
sv
R: bpFbp
C:kFkp& sv
sv
sv
0
F3R:b31
C:k3232v
sv 31v
0F4
-R:b41
-C:k4242vsv
41v
-I: m2
F2 2p&=
sv
svF1 R: b1
extracted passiveelements
negative bond graphelements (active)
Figure 3.25: Bond graph of system with reticulated suspension
A schematic of the corresponding system is shown in Figure 3.26.
71
m
Fin
original systemspring
passive suspensionelements
active element(hydraulic actuator)
Figure 3.26: Schematic of the reticulated suspension
The schematic is the same as that of the original synthesis, but the coefficients of
the passive elements are different.
Following the procedure outlined earlier, where the state equation for the
virtual velocity associated with the negative bond graph elements is used to
determine the control of the effort source into the actuator, the bond graph of
Figure 3.27 was obtained. Note here that the bond graph elements that explicitly
represent the actuator, i.e., the transformer element and its controlled effort
source, represent an ideal actuator (lossless and energic), whereas the real effects
of the actuator are topologically separated within the bond graph structure.
72
F(t):E
T: 1/A
I: m + mact
1Fin
sv 0
R:b1+bactF1
F4
C:k
V&Pin
Fkp&
E:P(t)
R:b41
C:k4242v
+ 32v 32x 3F
-
Fs
s1
signal flowsv
41v
s sv& 2F+
+
32 FF +
sv
sv
svsv
sv
sv
m2
k32
31
1b
A1
Figure 3.27: Bond graph with realized actuator and effort source control
Performing a swept frequency simulation of the system reveals that the
revisions made to the controlled effort source in the bond graph above
compensate for the discrepancies in the earlier model. A comparison plot of the
actual system frequency response, as calculated from the time simulations, and
the original data-based transfer function is shown in Figure 3.28.
73
100
101
102
-62
-60
-58
-56
-54
-52
-50
-48
-46
-44
-42
Comparison of data-based FRF and actual system response
frequency (rad/s)
mag
nitu
de ra
tio: v
mas
s/in
put f
orce
(dB
)
actual system responseoriginal FRF
Figure 3.28: Swept frequency simulation results and original FRF
3.4 SUMMARY OF ACTIVE ELEMENT SYNTHESIS AND REALIZATION
This chapter introduced the active element physical realization process.
The two mechanical system active device types used in this work, a hydraulic
actuator and a PMDC motor, along with their corresponding bond graph models,
were presented.
The example problem used to demonstrate the synthesis procedure in
Chapter 2 was used to illustrate the procedure of active element physical
realization. New concepts such as virtual state variables and separable bond
74
graph elements were introduced and their use in the process of active device
physical realization was explained. A hydraulic actuator was selected as the
active device, and a bond graph model of the simple mechanical oscillator and
actuator, including real actuator effects, was developed. The bond graph model
was augmented with a signal flow diagram that provided information regarding
the control system for the regulated pressure source for the actuator.
Lastly, simulation data that was obtained using the bond graph model of
the system and synthesized active element were presented. The results of the
simulation revealed that the synthesis and realization process yielded an active
device that satisfied the required frequency response behavior of the system.
A detailed summary of the steps of the active element synthesis and
realization procedure as presented in Chapters 2 and 3 is given in the flow chart of
Figure 3.29. Each row of blocks corresponds to a major stage in the process, with
the first and last stages being optional, depending on the application.
75
obta
in d
esir
ed tr
ansf
erfu
nctio
n, d
TF
-giv
en in
ana
lytic
al fo
rm
O
R- d
eter
min
e fro
m c
urve
fit o
f exp
erim
enta
l dat
a
dete
rmin
e th
eore
tical
tran
sfer
func
tion,
tTF
- use
impe
danc
e-ba
sed
bond
gra
ph
solv
e fo
r un
kow
nim
peda
nce,
Z(s
)
- equ
ate
dTF
and
tTF
deco
mpo
se Z
(s) i
nto
basi
cim
peda
nces
- pol
ynom
ial d
ivis
ion
and
parti
al fr
actio
nex
pans
ion
iden
tify
sepa
rabl
e bo
ndgr
aph
elem
ents
- sel
ect h
ybrid
or f
ully
act
ive
syst
em
dete
rmin
e st
ate
equa
tions
from
bon
d gr
aph
mod
el
- ide
ntify
virt
ual s
tate
varia
bles
inco
rpor
ate
into
sys
tem
bond
gra
ph m
odel
- rep
lace
unk
now
nim
peda
nce
elem
ent,
Z(s)
cast
into
bon
d gr
aph
fram
ewor
k
- usi
ng p
ositi
ve a
nd n
egat
ive
basi
c im
peda
nces
sele
ct a
ctiv
e de
vice
phys
ical
rea
lizat
ion
- add
idea
lized
bon
d gr
aph
repr
esen
tatio
n to
mod
el
dete
rmin
e co
ntro
l sig
nal
diag
ram
- gra
phic
al re
pres
enta
tion
ofth
e di
ffere
ntia
l equ
atio
ns o
fvi
rtual
sta
te v
aria
bles
perf
orm
sim
ulat
ions
- obt
ain
forc
e-ve
loci
ty p
rofil
ean
d co
ntro
lled
sour
ce h
isto
ry
add
real
(par
asiti
c)ac
tuat
or e
ffec
ts
- rep
eat s
ynth
esis
pro
cess
Figure 3.29: Active element synthesis procedure flowchart
76
Chapter 4: Practical Applications of the Synthesis Method
4.1 INTRODUCTION
In this section, two practical applications of the active element synthesis
methodology will be presented. The first application involves the synthesis of an
active element in a newly designed system. The second application involves the
application of this methodology as a tool for the retrofit or redesign of an existing
system. Actual experimental test data for the system is used in the synthesis of a
new active element. Both of these applications will demonstrate the utility of the
active synthesis methodology at different stages in the useful life of a system.
4.2 TWO-STORY BUILDING MODEL VIBRATION ISOLATION SYSTEM
4.2.1 Introduction
In this section, the synthesis of an active vibration isolation system for a
two-story building will be investigated. Such a system would be useful in the
isolation of the building in the event of an earthquake. Similar applications could
include any type of vibration isolation equipment, such as a platform upon which
unwanted vibrations could disrupt a sensitive experiment or manufacturing
process.
4.2.2 System Description
A photograph and corresponding schematic representation of an
experimental model of a two-story building are shown in Figure 4.1. The model
77
consists of two aluminum blocks and two sets of four slender aluminum beams
that connect each mass to the one above it. The structure is connected to a
moveable base, where the input excitation, v(t), is applied.
m1
m2
v(t)
Figure 4.1: Two-story building laboratory model photograph and schematic
As a first approximation, the two story building is represented by a
lumped parameter model, as shown in Figure 4.2. The geometry of each beam is
such that its stiffness can be determined by considering them to be flex glides
(Burr, 1982). Using this approach, the equivalent stiffness of each beam is found
to be k=100 N/m. The aluminum blocks that represent each floor of the building
were weighed using a digital scale and their masses were determined to be
m1=m2=0.68 kg. Additionally, the structure is assumed to be very lightly damped,
with a viscous damping coefficient of b1=b2=10 N-s/m.
78
m1
m2
b2
k2
b1k1
v(t)
Figure 4.2: Lumped parameter model of two story building
4.2.3 Objectives
The objectives of this section is to determine the active element model of
an active vibration suppression system for the building model, given specific
frequency response requirements, such that the design parameters for such a
device can be determined. Consider the case in which the vibration suppression
device will be attached between the base of the building and the first floor, as
shown in Figure 4.3.
79
m1
m2
b2
k2
b1k1
Z(s)v(t)
Figure 4.3: Building model with active vibration suppression element
The specified frequency response requirement is given in terms of the base
excitation velocity and the velocity of the top floor. The magnitude ratio between
the input and the response was chosen to reflect a reported magnitude ratio
between the base and top floor of an actual building with a passive earthquake
vibration isolation system (Kelly, 1998). A simple frequency response function
that satisfies this requirement is
1000100)(+
=s
sdTF (4.1)
The desired frequency response plot is shown in figure 4.4.
80
-45
-40
-35
-30
-25
-20
-15
-10M
agni
tude
(dB
)
101 102 103 104-100
-80
-60
-40
-20
0
20
Pha
se A
ngle
(deg
)
Desired Frequency Response
frequency (Hz)
Figure 4.4: Desired frequency response between base and top floor
4.2.4 Synthesis Procedure
Using impedance-based bond graph methods, along with a transmission
matrix approach (see Appendix A), we can determine the required force-velocity
behavior of the active element. The bond graph of the system is shown in Figure
4.5, where the impedance of the vibration suppression filter is Z(s). For this case,
we assume that the vibration suppression element will be a hydraulic actuator,
thus the mass of the piston, mp, and its corresponding viscous damping
coefficient, bp, are included in the model.
81
0
1
k1 :C
I: m2
Fs1
Fs1
Vq
Vs2
Fz
Fk1
Vq :F
R: (b1+bp)
V11 Fs2
1
k2 :C
Fk2
Fs2V2
2p&=0
R: b2
(m1+mp) :IFm1Vs1
I: m1
Fs1V1
V11p&=
Fs2 Vs2
Vs2
Fb2Vs2
Fs1
Vs1Fb1 Vs1
Vs1
Z(s)
Figure 4.5: Bond graph representation of two-story building and filter
The transfer function between the velocities of the base and the top floor is
found from the bond graph. We refer to this as the “theoretical” transfer function,
tTF(s).
10000)1003000()10406()68.04.27(476.010000)1003000()10202(2.0)(
234
23
++++++++++++=
sZsZsZssZsZsstTF (4.2)
82
Following the procedure described in the previous chapter, we can
determine the impedance of the unknown vibration filter by equating the desired
transfer function with the theoretical transfer function and solving for Z, as shown
below.
10000)1003000()10406()68.04.27(476.010000)1003000()10202(2.0
1000100
234
23
++++++++++++=
+ sZsZsZssZsZs
s (4.3)
sssssss
sZ90000910058
9000000271000016440023384.47)( 23
234
++−−−−+= (4.4)
Proceeding with the process of partial fraction expansions and polynomial
divisions, a representation of the impedance Z(s) in terms of basic impedances is
shown below. Recall that negative impedance terms indicate the need for an
active element.
ss
ss
sZ
7.440340966124775
07.1491
181724.010053.168)(
++−
+−−−= (4.5)
The bond graph representation of the system with the active element
model is shown in Figure 4.6.
83
0
1
k1 :C
I: m2
Fs1
Fs1
Vq
Vs2
Fz
Fk1
Vq :F
1
0-C: k22
-C: k12
R: b21
1
I: m24
R: b23
F12
F2
24p&
22x&
R: (b1+bp)
actuator
V11
Fs2
1
k2 :C
Fk2
Fs2
V2
2p&=0
R: b2
(m1+mp) :IFm1
Vs1
-I: m13
F13
-R: b11F11
I: m1
Fs1
V1
V11p&=
Fs2 Vs2
Vs2
Fb2Vs2
Fs1
Vs1Fb1 Vs1
Vs1
Figure 4.6: Model with reticulated vibration suppression active element
84
Where:
k1=k2= 100 N/m (equivalent stiffness of the beams)
b1=b2= 10 N-s/m (equivalent viscous damping)
m1=m2= 0.68 kg (mass of the floors of the building)
bp= 10 N-s/m (viscous damping coefficient of the actuator piston)
mp= 0.02 kg (mass of the actuator piston)
b11= -168.53 N-s/m
k12= -100 N/m
m13= -0.81724 kg
b21= 149.07 N-s/m
k22= -24775 N/m
b23= 40966 N-s/m
m24= 4403.7 kg
The system state variables are:
momentum of mass 1, p1
momentum of mass 2, p2
compression of suspension elements between base and floor 1, xs1
compression of suspension elements between floors 1 and 2, xs2
The virtual state variables, associated with the filter elements, are:
compression of filter spring k22, x22
momentum of filter mass m24, p24
85
The state equations are:
24
2423222224
21
2222
24
24
1
122
2
2
1
12
1
11
222
2
1
122
1312
2122121121211222122213112
1312
12211211112121212121
)(
)()()(
mpb
xkp
bxk
mp
mp
vx
mp
mp
x
mp
vx
xkmp
mp
bp
mmmm
xmmkxmmkxmmkxmmkvmmmpmb
mmmmpmbpvmmbpvmmbvmmbpmbpmb
p
q
s
qs
s
p
sssqp
p
qqqpp
−=
−−−=
−=
−=
+
−=
++−++++−
+++
−−+−+++=
&
&
&
&
&
&
L&
(4.6)
Simulations were run in which actual earthquake ground velocity data
(Chopra, 1981) was used for the input velocity of the base, vq. Plots of the ground
velocity input, the velocity of floor 1, and the velocity of floor 2 are shown in
Figure 4.7. The end of the published data was padded with zeros to verify that the
synthesized system response indeed went to zero after the base excitation was
terminated.
86
0 5 10 15 20 25 30 35-0.4
-0.2
0
0.2
0.4in
put v
eloc
ity
0 5 10 15 20 25 30 35-0.04
-0.02
0
0.02
0.04
floor
1 v
eloc
ity
0 5 10 15 20 25 30 35-0.04
-0.02
0
0.02
0.04
floor
2 v
eloc
ity
time (sec)
Figure 4.7: Input velocity and response velocities
The plots clearly show a magnitude reduction factor of 0.1 (-20dB)
between the input velocity and the velocity of floor 2, which corresponds to the
requirement defined by the given frequency response function, dTF(s). Thus, it
can be concluded that the bond graph model of the reticulated filter produces the
desired response characteristics.
87
4.2.5 Physical Realization of the Vibration Suppression Active Element
The next step is to use the simulations in the determination of pertinent
physical characteristics of the active element. Consider again the bond graph
model of the building with the negative impedance representation of the active
element, as shown in Figure 4.8.
0
1
k1 :C
I: m2
Fs1
Fs1
Vq
Vs2
Fz
Fk1
Vq :F
1
0-C: k22
-C: k12
R: b21
1I: m24
R: b23
F12
F2
24p&
22x&
R: (b1+bp)
actuator
V11
Fs2
1
k2 :C
Fk2
Fs2
V2
2p&=0
R: b2
(m1+mp) :IFm1
Vs1
-I: m13
F13
-R: b11F11
I: m1
Fs1
V1
V11p&=
Fs2 Vs2
Vs2
Fb2Vs2
Fs1
Vs1Fb1 Vs1
Vs1
Figure 4.8: Bond graph of system with active element representation
88
By tracking the time history of the effort and flow variables associated
with the power bond that is connected to the active element representation, we
determine the required force-velocity profile of the actuator, as illustrated in the
above figure. A plot of the force-velocity profile is shown in Figure 4.9.
-10 -5 0 5 10 15 20 25 30 35-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
force (N)
velo
city
(m/s
)
operating envelope of actuator
Figure 4.9: Force-velocity profile for actuator
Following the same procedure presented in Chapter 3, we replace the bond
graph representation of the behavior of the active element with a bond graph
model of a hydraulic actuator, along with a controlled effort source, which in this
case is a hydraulic pressure source, P(t). A control system that regulates this
89
source is determined by the procedure illustrated in section 3.3.3, using the
differential equations associated with the virtual state variables, x22 and p24, and
the system state variable xs1.
From the bond graph, we determine that the actuator force, Fz, is given by
the equation
2131211 FFFFFz +++= (4.7)
where:
22222
11313
11212
11111
xkFamFxkFvbF
s
s
s
====
The bond graph that contains the actuator model and the signal flow and
control system for the controlled effort source is shown in Figure 4.10. The
transformer modulus is dependent on the area of the hydraulic piston, Ap.
90
0
1
k1 :C
I: m2
Fs1
Fs1
Vq
Vs2
Fk1
Vq :F
(b1+bp) :R
V11
Fs2
1
k2 :C
Fk2
Fs2
V2
2p&=0
R: b2
(m1+mp) :IFm1
Vs1
I: m1
Fs1
V1
V11p&=
Fs2 Vs2
Vs2
Fb2Vs2
Fs1
Vs1
Fb1
Vs1
T: 1/Ap
QV =&Pin
E:P(t)
+ 22v22x
-
Fz
s1
1sx
2F+
+
signalflow
1sv&
s1
s 13F
+
12F
+ 24p
-s124p&
-
zF
Vp
1sv
+
1sv
1sv
1sv
b11
k12
m13
k22
24
1m
21
22b
k
24
23m
b
A1
Figure 4.10: Bond graph model with signal flow for controlled effort source
91
The required pressure is dependent on the design of the actuator,
particularly on the piston area. A time history plot of the required pressure source
for a piston with Ap=0.01 is given in Figure 4.11.
0 5 10 15 20 25 30 35-1000
-500
0
500
1000
1500
2000
2500
3000
3500Pressure vs. time: Ap=0.01
Pre
ssur
e (P
a)
time (sec)
Figure 4.11: Time history of regulated pressure source
A nominal piston area can be selected by performing simulations with
varying values of Ap and plotting the various pressure-velocity profiles, as
illustrated in Figure 4.12. From the corresponding operating envelopes, a
designer may select an off-the-shelf actuator or design a new actuator based on
this information.
92
-200 -100 0 100 200 300 400 500 600 700-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3Ap=0.05
Pressure (Pa)
velo
city
(m/s
)
-200 -100 0 100 200 300 400 500 600 700-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3Ap=0.1
Pressure (Pa)
velo
city
(m/s
)
93
-200 -100 0 100 200 300 400 500 600 700-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3Ap=0.2
Pressure (Pa)
velo
city
(m/s
)
Figure 4.12: Pressure Velocity profiles for different values of piston area
Additionally, other actuator parameters, such as piston mass and viscous
damping coefficient can be changed to determine their effects on the actuator
pressure-velocity profiles. As illustrated in the previous chapter, this requires the
system designer to repeat the impedance-based reticulation process.
4.2.6 Summary
This section demonstrated the use of the active element synthesis method
in determining the design parameters of an active device for a newly designed
system, namely the two-story building vibration isolation system. The active
element impedance was determined using impedance-based bond graph
94
techniques and the desired frequency response function. This impedance was
then decomposed into basic impedance terms in order to develop a bond graph
model of the active element behavior. Preliminary simulations using this model
showed that the synthesized active element met the desired frequency response
behavior.
As an example, the active element chosen was a hydraulic actuator. After
identifying the separable bond graph elements of the system, a bond graph model
of the hydraulic actuator with a controlled pressure source was developed. A
signal flow diagram was also developed, which provided information about the
nature of the control system needed such that the system exhibited the desired
frequency response behavior.
Simulations were run using this model to determine the effects of
changing selected design parameters of the hydraulic actuator. As an example,
various values for the actuator piston area were simulated. The required force-
velocity profiles and regulated pressure source time histories were displayed for
each case and comparisons were made. This demonstrated the utility of the
synthesis method within a simulation-based design environment.
4.3 ELECTROMECHANICAL VEHICLE SUSPENSION SYSTEM
4.3.1 Introduction
In this section, the synthesis of an alternate active suspension for an
experimental quarter vehicle model will be demonstrated. An active
electromechanical suspension device already exists for this model and
experimental performance data of the active device has been collected. We will
use this data in the synthesis of a new active device for the suspension that
95
involves different combinations of energy domains. Thus, this section effectively
illustrates the use of the synthesis methodology in the context of active system
redesign and/or retrofit.
4.3.2 Description of System
The system under study is a quarter vehicle experimental model that
consists of a tire, an unsprung mass and a sprung mass, between which there is an
electromechanical suspension system, consisting of an electric motor coupled
with a rack and pinion gearing system, that controls the position of the sprung
mass. The suspension system consists of an electromechanical actuator and an air
spring. A double exposure photograph of the experimental setup that shows the
tire, suspension system, and vehicle mass is shown in Figure 4.13.
Figure 4.13: Photograph of quarter vehicle experimental setup
The corresponding schematic of this system is shown in Figure 4.14.
96
k
msm
existingelectromechanical
actuatormum
sprung mass
unsprung mass
suspension spring
Figure 4.14: Quarter vehicle with unknown active suspension element
The parameters of the system are:
sprung mass, msm = 613 kg
unsprung mass, mum = 81 kg
suspension spring stiffness, kp = 35,550 N/m
effective tire stiffness, kt = 250,000 N/m
effective tire damping coefficient, bt = 4000 N-s/m
4.3.3 Objectives
In this section, there are two main objectives: the first is to determine an
alternate design for the suspension system. That is, it is desired to replace the
existing electromechanical actuator with an actuator that involves another
combination of energy domains, such as hydraulic-mechanical.
97
The second objective is to provide a validation of the methodology. Based
strictly on the experimental data, we wish to determine if the synthesis method
yields an active element whose behavior matches that of the existing active
element in the system.
4.3.4 Specifics of the Application of the Methodology to this Problem
We begin by casting the problem into the form presented in the previous
chapter, in that we represent the suspension system as an unknown configuration
with an impedance Z(s), as illustrated in Figure 4.15.
k
msm
mum
sprung mass
unsprung mass
suspension spring
unknown suspensionsystem, Z(s)
Figure 4.15: Actuator replaced by an unknown suspension system
The impedance-based bond graph for this system is shown in Figure 4.16.
98
I: mum
0F: 1 0 1
1
kt :C R: bt
1
kp :C Z(s)
I: mum
Figure 4.16: Impedance-based bond graph of system with unknown suspension
The first step is to consider the transfer function that will be used in the
synthesis process. Given that we have experimental data for the velocity of the
sprung mass and unsprung mass, vsm and vum respectively, we use the following
transfer function
)()(
)(sVsV
sTFum
sm= (4.8)
Using bond graph impedance methods (see Appendix A), we form a
symbolic expression for this transfer function. For this transfer function, we need
only to examine the bond graph fragment shown below.
99
Z(s)
Fs
Vp
0
1
kp :C
I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=
Fkp
Vp
Vum :F
Figure 4.17: Bond graph fragment for determination of transfer function
The theoretical transfer function is
35550)(61335550)(
)( 2 +++==
sZsssZs
VV
stTFum
sm (4.9)
where Z(s) is the impedance of the unknown suspension system.
Next, using the experimental data available, we form the corresponding
frequency response function by computing and manipulating auto and cross-
spectral density functions of vsm and vum (see Appendix B.) After obtaining the
phase and magnitude plots, we perform a complex-plane curve-fitting procedure
in order to obtain the corresponding transfer function in analytical form, namely
as a ratio of two polynomials (see Appendix C.) The results of the curve fitting
procedure yields what we will refer to in this case as the “data-based” transfer
function, dTF(s)
0.100918.000982.0000518.007409.000026.0000744.0
)( 23
2
+++++==
sssss
VV
sdTFum
sm (4.10)
100
As noted in Appendix C, the order of the polynomials is specified by the designer
during the curve fitting process. An accurate as possible curve fit is desired,
however the designer must exercise caution, since selecting too high a degree of
polynomials can cause the curve fitting algorithm to yield unstable transfer
functions. An illustration of the curve fit to the experimental data is shown in
Figure 4.18, where the original data points are represented by asterisks and the
curve fit is represented by a solid line. The mean error over the range of
frequencies is on the order of six percent.
101
102
-30
-28
-26
-24
-22
-20Comparison of magnitude ratios
mag
nitu
de ra
tio (d
B)
original datacurve fit
101
102
-50
0
50
Comparison of phase angles
phas
e an
gle
(deg
rees
)
frequency (rad/sec)
Figure 4.18: Curve fit of experimental frequency response data
101
To solve for the unknown impedance, Z(s), we equate the theoretical and
data-based transfer functions and solve for Z(s).
0.100918.000982.0000518.007409.000026.0000744.0
35550)(61335550)(
23
2
2 +++++=
+++
sssss
sZsssZs (4.11)
sssssssssZ
925908929085102916.3101711.310773.216181245362)(
234
972734
+++×−×−×−−= (4.12)
Application of the impedance decomposition procedure detailed in
Chapter 3 yields the following expression, whose terms consist of basic
impedances only
ssss
sZ
93526.064997.21
00960.003122.0
11008178.600011.0
1355504.9072)( 7
−+−−
+×−−
+−= −
(4.13)
Note that some of the impedance terms are not positive real, thus indicating the
need for an active element.
The bond graph structure that represents the synthesis of the suspension
system impedance is shown in Figure 4.19. Note the representation of the
negative bond graph elements.
102
0
1
kp :C
I: msmFs Fs
Fs
Vum VsmVp
smp&=
Fz
Vp
Fkp
Vp
Vum :F
1
0-C: k32
-C: k12
-R: b31
1
-I: m34
R: b33
F12
F3
34p&
32x&
R: b11
synthesized suspension
0-C: k22
-R: b21
22x&
F2
Figure 4.19: Bond graph representation of synthesized suspension system
At this point, we recognize that the passive damper, represented by the
R:b11 element, is a separable passive component of the synthesized active
element. Thus, we have two options: we can choose to represent the suspension
system solely as an active element that is a combination of the passive and active
bond graph elements above, or choose a “hybrid” configuration, in which the
103
suspension consists of a passive element, namely the damper and an active
element that is represented by the combination of the remaining non-separable
bond graph elements.
4.3.5 Results and Comparisons to Existing Test Data
Consider first the case in which the suspension system consists of an
active element only. We can extract the state equations from the above bond
graph and perform a time-based simulation in order to determine the required
force-velocity profile of the active element. As the input to the system, we use
actual experimental data for the velocity of the unsprung mass, vum(t). The state
equations are
34
3433323234
34
34
31
323232
21
222222
123232222211
mp
bxkp
mp
bxk
mpvx
bxk
mp
vx
mp
vx
xkxkxkxkmp
vbp
smum
smum
smump
pppsm
smumsm
−=
−−−=
−−=
−=
++++
−=
&
&
&
&
&
(4.14)
To verify that the synthesized filter matches the frequency response
behavior of the function Z(s) that is represents, a swept frequency simulation was
performed in which magnitude ratios at various frequencies were calculated.
These values were compared to the frequency response plot of the original
104
function, Z(s) as indicated above. The results of this comparison are given in
Figure 4.20. There is excellent agreement between the two values, which
indicates good correspondence in the frequency domain.
100
101
102
-40
-38
-36
-34
-32
-30
-28
-26
-24
-22
-20
Comparison of data-based FRF (fit) and actual system response
frequency (rad/s)
v un
spru
ng m
ass
/ v s
prun
g m
ass
actual system responseoriginal FRF
Figure 4.20: Swept frequency comparison results
A simulation was run where the input was the unsprung mass velocity,
vum(t), taken from experimental data, and the output was the sprung mass velocity,
vsm(t). A portion of the comparison plot of the actual experimental values of the
sprung mass velocity overlaid with corresponding simulated values is shown in
105
Figure 4.21. This indicates that there is good correspondence between behavior
of the synthesized suspension system and the existing suspension system in the
time domain.
21 22 23 24 25 26
-0.15
-0.1
-0.05
0
0.05
Comparison of simulated vs. actual sprung mass velocities
time (sec)
spru
ng m
ass
velo
city
simulated data experimental data
Figure 4.21: Simulated and experimental values of sprung mass velocity
The error between the simulated and actual data can be attributed to: (1)
error in the curve fitting procedure used to determine the analytical expression for
the desired frequency response function, and (2) nonlinearites that are present in
the actual system (e.g., nonlinear compliance effects due to the air spring in the
106
experimental setup) that cannot be included in the impedance-based bond graph
model of the system, since the methodology only allows for linear, lumped-
parameter models.
Using time simulation data, the data for the force-velocity profile of the
active element can be computed. First, we consider the case where all of the
reticulated elements of the suspension system are considered to contribute to the
active element behavior (i.e., no passive separable elements are removed.) This is
illustrated in the bond graph of Figure 4.22.
107
0
1
kp :C
I: msmFs Fs
Fs
Vum VsmVp
smp&=
Fz
Vp
Fkp
Vp
Vum :F
1
0-C: k32
-C: k12
-R: b31
1
-I: m34
R: b33
F12
F3
34p&
32x&
R: b11
synthesized suspension
0-C: k22
-R: b21
22x&
F2
Figure 4.22: Fully active bond graph representation
Next, consider the hybrid configuration, in which the suspension consists
of a passive element, namely the damper (represented by the R:b11 element) and
an active element that is represented by the combination of the remaining active
and passive bond graph elements. This is shown schematically in Figure 4.23.
108
k
msm
mum
original suspensionspring
actuator
passivedamper
Figure 4.23: Hybrid configuration for synthesized suspension system
The bond graph of the system and the resulting simulation remains the
same, except that now the portion of the bond graph that represents the actuator
behavior changes. The force and velocity profile of the actuator is determined by
the time history of the respective power conjugate variables on the bond that
connects into the dotted box labeled “actuator”. This is illustrated in Figure 4.24.
109
0
1
kp :C
I: msmFs Fs
Fs
Vum VsmVp
smp&=
Fz
Vp
Fkp
Vp
Vum :F
1
0
-C: k32
-C: k12
-R: b31
1
-I: m34
R: b33
F12
F3
34p&
32x&
R: b11
actuator
0
-C: k22
-R: b21
22x&
separable passiveelement F2
Figure 4.24: Hybrid suspension actuator representation
The resulting force-velocity profiles due to random excitation for the fully
active and hybrid cases are shown in Figure 4.25. Note that the profile for the
hybrid case is notably flattened, which indicates that the power requirements for
the actuator in this case are considerably lower. The range of the actuator force
for the fully active case is much wider than that of the hybrid case.
110
In most cases, an actuator requiring lower power would be more desirable.
However it is important to investigate both cases, since there may be a situation in
which a design constraint, such as available space, may rule out a hybrid system.
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-1500
-1000
-500
0
500
1000
1500
2000Force-velocity profile of actuator
velocity (m/s)
forc
e (N
)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-1500
-1000
-500
0
500
1000
1500
2000Force-velocity profile of actuator
velocity (m/s)
forc
e (N
)
Figure 4.25: Force-velocity profile for active and hybrid suspension actuators
111
At this point of the process, the designer has two options: either select an
existing actuator “off-the-shelf” or design an actuator that fits the required force
velocity profile, as illustrated in Figure 4.26.
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-1500
-1000
-500
0
500
1000
1500
2000
velocity (m/s)
forc
e (N
)
force-velocity envelope ofan off-the-shelf actuator
Figure 4.26: Existing actuator profile that fits desired force-velocity profile
There may be cases in which no such actuator is available, or it is desired
to have an actuator that is specifically designed for the prescribed application. In
this case, the process of physical realization, which was outlined in Chapter 3,
must be carried out.
112
4.3.6 Summary
In this section, an active element model for the vehicle suspension system
was obtained. Since this case involved the redesign of an existing system,
experimental data for vehicle dynamic variables was available. This data was
used in determining the required frequency response function for use in the
synthesis procedure, and for validating the synthesis results by making
comparisons between simulated and actual active element behavior. Simulations
were performed and it was shown that the active element model obtained from the
synthesis procedure accurately modeled the desired system behavior in both the
frequency and time domains.
4.4 PHYSICAL REALIZATION OF THE ACTIVE VEHICLE SUSPENSION
4.4.1 Introduction
Following the procedure outlined in Chapter 3, we can begin the process
of physical realization of the active element that was obtained in the previous
section. We will consider the case where it is desired to extract any separable
passive elements such that a hybrid suspension system is obtained.
The first step is to identify separable passive portions of the active
representation. This was illustrated in the previous section, as shown in the bond
graph of Figure 4.23.
There are three virtual state variables for this configuration: the
displacement associated with the -C:k22 element, x22, the displacement associated
with the -C:k32 element, x32, and the momentum associated with the -I:m34
element, p34. The state equations for these variables are
113
34
3433323234
34
34
31
323232
21
222222
mpb
xkp
mp
bxk
vx
bxk
vx
p
p
−=
−−=
−=
&
&&
&
(4.15)
4.4.2 Realization Case 1: Hydraulic Actuator
Choosing a hydraulic actuator as the active element, the bond graph of the
system, along with signal flow for the controlled effort source of the actuator, is
shown in Figure 4.27.
114
1
kp :C
Fkp
b11 :RFbp
T: 1/A
QV =&Pin
E:P(t)
+ 32v 32x
-
Fz
s1
px
3F +
signalflow
s1
+
12F
+ 34p
-s134p&
-
zF
Vp
pvk12
k32
34
1m
31
32b
k
34
33m
b
A1
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
Vp pv
+ 22v22x
-s1
2F
+
k22
21
22b
k
pv
Figure 4.27: System bond graph with signal flows and controlled effort source
Figure 4.28 illustrates the addition of real effects associated with the
actuator piston, such as piston mass and viscous damping, to the bond graph. A
piston mass, mp=1kg, and a piston viscous damping coefficient, bp=100 N-s/m
was selected. It is important to note there the difference between two pertinent
effort variables that are represented in the bond graph below, Fs and Fz. The
115
variable Fs, which represents the force generated by the entire suspension system,
is independent of the energy domains involved in the physical realization of the
active element (i.e., the type of actuator used.) However, the variable Fz, which
represents the force generated by the active device alone, is dependent on the
selection of energy domains, since it will have to compensate for any real actuator
effects, which are dependent on the type of actuation device used.
1
kp :C
Fkp
bp+b11 :RFbp
T: 1/A
QV =&Pin
E:P(t)
+ 32v 32x
-
Fz
s1
px
3F +
signalflow
s1
+
12F
+ 34p
-s134p&
-
zF
Vp
pvk12
k32
34
1m
31
32b
k
34
33m
b
A1
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
Vp pv
+ 22v22x
-s1
2F
+
k22
21
22b
k
pv
mp :I
FkpVp
Figure 4.28: Bond graph illustrating real actuator effects
116
The constitutive parameters for the synthesized system are:b11= 9072.4 N-s/m (passive damper element)
k12= -35550. N/mb21= -9075.3 N-s/m
k22= -1644260 N/m
b31= -32.0272 N-s/m
k32 = -104.214 N/m b33 = 2.6499 N-s/m
m34= -0.93526 kg
Adding these effects changes the behavior of the original system model,
since they are lumped together with system elements, such as the sprung mass
and the extracted passive damping element. This is revealed by the swept
frequency simulation results, shown in Figure 4.29.
117
100
101
102
-40
-35
-30
-25
-20
-15
-10Comparison of data-based FRF (fit) and actual system response
frequency (rad/s)
v u
nspr
ung
mas
s / v
spr
ung
mas
sactual system responseoriginal FRF
Figure 4.29: Swept frequency results and the original transfer function
This figure illustrates the difference between the desired transfer function (the
ratio of the velocity of the sprung mass, Vsm, to the velocity of the unsprung mass,
Vum) and the actual ratio of those values, as obtained from time-based simulations.
There is more of a discrepancy in the lower frequency range. To compensate for
these effects, the synthesis procedure must be repeated in order to obtain the
suspension system impedance such that the original frequency response behavior
requirements are statisfied.
The new expression for the theoretical transfer function is
118
35550)(10061435550)(100
)( 2
2
++++++=sZsss
sZsssstTF (4.16)
Where the values of the piston mass, mp=1 kg and viscous damping of the piston,
bp=100 N-s/m were chosen. The data-based transfer function that represents the
desired response remains the same, and is repeated below
0.100918.000982.000005.00741.000026.000074.0
)( 23
2
+++++=
ssss
sdTF (4.17)
After equating of the data-based and theoretical transfer functions, the impedance
of the suspension system was found to be
ssssssssssZ
185184.1786.181105832.6101932.8105838.5507008970)(
234
8626345
+++×−×−×−−+−= (4.18)
Decomposition of this impedance yields the bond graph structure shown in Figure
4.30. The constitutive parameters for the newly synthesized system are given
below.
b11= 8972.4 N-s/m (passive damper element)
k12= -35550. N/m
m13= -1 kg
b21= -9075.3 N-s/m
k22= -1644260 N/m
b31= -32.0272 N-s/m
k32 = -104.214 N/m
b33 = 2.6499 N-s/m
m34= -0.93526 kg
119
1
kp :C
Fkp
bp+b11 :RFbp
T: 1/A
QV =&Pin
E:P(t)
+ 32v 32x
-
Fz
s1
px
3F +
signalflow
s1
+
12F
+ 34p
-s134p&
-
zF
Vp
pvk12
k32
34
1m
31
32b
k
34
33m
b
A1
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
Vp
pv
+ 22v22x
-s1
2F
+
k22
21
22b
k
pv
mp :I
FkpVp
pv&spv
m13
+
13F
Figure 4.30: Bond graph and signal flow (actuator effects compensated)
A swept frequency simulation was performed again and the results indicate that
the revisions made to the constitutive parameters associated with the controlled
effort source compensate for the discrepancies in the earlier model. A
120
comparison plot showing the desired frequency response and the actual response
from the time-based simulations is given in Figure 4.31.
100
101
102
-40
-35
-30
-25
-20
-15
-10
Comparison of data-based FRF (fit) and actual system response
frequency (rad/s)
m v
uns
prun
g m
ass
/ v s
prun
g m
ass
actual system responseoriginal FRF
Figure 4.31: Swept frequency results after iteration
As seen in the figure, the compensation for the real actuator effects yields
excellent correspondence between desired and simulated frequency response
behavior.
Additionally, a comparison of time-domain results between the
experimental data and the simulated data was made. In the simulation, the input
was the unsprung mass velocity, vum(t), taken from experimental data, and the
121
output was the sprung mass velocity, vsm(t). The input and output are shown in
Figure 4.32.
0 5 10 15 20 25 30-2
-1
0
1
2Input excitation (unsprung mass velocity due to terrain input)
velo
city
(m/s
)
0 5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2Response: sprung mass velocity
time (sec)
vel
ocity
(m/s
)
Figure 4.32: Input excitation and response velocities
Figure 4.33 shows a comparison plot between experimental and
simulated values of the sprung mass velocity. For clarity, Figure 4.34 shows a
portion of an overlay of these comparison plots. There is good correspondence
between behavior of the synthesized suspension system and the actual suspension
system in the time domain, thus further supporting the validity of the active
synthesis procedure.
122
0 5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2Comparison of simulated vs. actual sprung mass velocities
velo
city
(m/s
)
0 5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2
velo
city
(m/s
)
time (sec)
Figure 4.33: Experimental and simulated sprung mass velocities
123
20.5 21 21.5 22 22.5 23 23.5 24 24.5
-0.15
-0.1
-0.05
0
0.05
Comparison of simulated vs. actual sprung mass velocities
time (sec)
spru
ng m
ass
velo
city
(m/s
)
simulated data experimental data
Figure 4.34: Close up of overlaid sprung mass velocity plots
As an additional basis of comparison between simulated and experimental
time histories of the sprung mass velocity, an auto-power (frequency) spectral
analysis was performed on both data sets. The plot reveals satisfactory
correspondence in the frequency domain within the frequency range of interest
(i.e., 1 – 5 Hz), as indicated in Figure 4.35.
124
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510
-2
10-1
100
101
102
Comparison of auto power spectra
frequency (Hz)
mag
nitu
de
simulated experimental
Figure 4.35: Sprung mass velocity data in the frequency domain
To illustrate the utility of the synthesis procedure as a design tool,
simulations were run for different values of the actuator piston area, Ap. Figure
4.36 shows the required regulated pressure source time histories for different
values of the actuator piston diameter, dp.
125
0 5 10 15 20 25 30-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500Comparison of required actuator pressures
time (seconds)
Pre
ssur
e (p
si)
dp= 1.5indp= 2.5indp= 3.5in
Figure 4.36: Pressure source time histories for different values of dp
Figure 4.37 shows the pressure-velocity profiles for different values of the
actuator piston diameter. From both figures, decisions about the design
parameters of the actuator can be made.
126
-5 -4 -3 -2 -1 0 1 2 3 4 5-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500Comparison of pressure/velocity profiles
velocity (ft/s)
Pre
ssur
e (p
si)
dp= 1.5indp= 2.5indp= 3.5in
Figure 4.37: Pressure-velocity profiles for different values of dp
The system designer can use information from the above plot to determine
an optimal combination of piston diameter and regulated pressure source input for
the active device.
4.4.3 Realization Case 2: PMDC Motor with Rack and Pinion
It may be desirable or necessary, based on design considerations, to select
an active element realization that involves another combination of energy
domains. To further demonstrate the validity of the synthesis methodology and
127
physical realization procedure, the case in which the active element for the system
under study is realized as a permanent magnet direct current (PMDC) motor,
coupled with a rack and pinion to provide linear actuation, is now considered.
A schematic of such a setup is shown in Figure 4.38. The motor is
supplied with a voltage, V(t) at a current i, where the current is determined by the
stator coil resistance, Rm. The output of the motor is a torque, τ, at an angular
velocity, ω.
V(t)i
τ ,ωF ,v
Figure 4.38: Permanent magnet DC motor
This assembly is fastened between the sprung mass and the unsprung
mass, such that the rack displacement corresponds to their relative displacement,
as illustrated in Figure 4.39.
128
k
msm
mum
original suspensionspring
rack
motor andmounting
Figure 4.39: PMDC/rack and pinion actuator layout
We begin with the bond graph representation of the system with an
unknown impedance, Z(s), which will eventually become the model of the
motor/rack and pinion assembly, as shown in Figure 4.40.
0
1
I: smsm
Fs Fs
Fs
Vum Vsm
Vp
Vum :F
kp /s :C
Fkp
Vp
Z(s)
Fz
Vp
Figure 4.40: Bond graph with unknown impedance
129
The same frequency domain behavior as in the previous case applies to
this system and the theoretical and data-based transfer functions are repeated
below
35550)(61335550)(
)( 2 +++==
sZsssZs
VV
stTFum
sm (4.19)
0.100918.000982.000005.00741.000026.000074.0
)( 23
2
+++++==
sssss
VV
sdTFum
sm (4.20)
Thus, the same synthesized filter, whose representation consists of a
combination of negative and positive bond graph elements is obtained. If we
select the option in which separable passive elements are removed from the active
element representation (hybrid suspension), the bond graph of Figure 4.41 results.
130
0
1
kp :C
I: msmFs Fs
Fs
Vum VsmVp
smp&=
Fz
Vp
Fkp
Vp
Vum :F
1
0
-C: k32
-C: k12
-R: b31
1
-I: m34
R: b33
F12
F3
34p&
32x&
R: b11
actuator
0
-C: k22
-R: b21
22x&
separable passiveelement F2
Figure 4.41: Hybrid suspension bond graph
We replace the bond graph elements within the box labeled “actuator”
with a model of the motor/rack and pinion assembly (including a gearbox, which
131
is modeled here by using an effective pinion radius), along with the signal diagram
that provides the necessary signals for the controlled effort source. The bond
graph includes real effects due to the motor, such as rotor inertia, Jr, viscous
damping, BR, and electrical resistance, Rm, as well as effects due to the mass of the
rack, ma. The transformer modulus is the effective radius of the pinion, r, and the
gyrator modulus is the torque constant of the motor, km. Lumped in this
parameter is motor box gear reduction ratio of the motor, rg. Thus the gyrator
modulus, represented by km, is actually the motor torque constant multiplied by
the gear reduction ratio. The model is shown in Figure 4.42.
132
T: r
τm
E:V(t)
Fz
0
1
kp :C
I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=
Fkp
Vp
Vum :F
b11:RFbp
Vp
Vp
G: km
VG
1
1
BT :R
JR :I
Rm :R
ω
ωω
ω
τJ
τB
τG
i
i
i
VR
+
+pv
VG
VR
+ 32v 32x
-s1
px
+
signal flow
s1
+
+ 34p
-s134p&
-
pvk12
k32
34
1m
31
32b
k
34
33m
b
pv
+ 22v22x
-s1
+
k22
21
22b
k
pv
pv&spv
m13
+
13F
12F
2F
3F
zF
V(t)
rmk
1 Rm
rkm
Figure 4.42: Bond graph with controlled effort source and actuator effects
133
The system parameters are:
ma = 2 kg (mass of the rack)
Jr = 0.00000162 kg/m2 (moment of inertia of the rotor)
Br = 0.00011 N-s (torsional viscous damping of rotor bearing)
r = 0.3 m (effective pinion radius)
Rm = 19.1 Ω (resistance of stator coil)
Recognizing that these real effects affect the overall impedance of the
suspension system, the effects associated with the rotor inertia and viscous
damping are reflected across the transformer element to the 1-junction that
represents the compression rate of the suspension system, vp. The stator coil
resistance remains in its original place within the bond graph, such that
appropriate causality is maintained with the effort source, V(t).
Again, as in the previous section, the synthesis procedure is repeated such
that the real effects are captured in the overall system impedance. The final bond
graph and control signal representation are shown in Figure 4.43.
134
T: r
E:V(t)
Fz
0
1
kp :C
I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=
FK
Vp
Vum :F
:RFR
Vp
Vp
G: km
VG
1
:I
Rm :R
ωτ
i
i
i
VR
+
+pv
VG
VR
+ 32v 32x
-s1
px
+
signal flow
s1
+
+ 34p
-s134p&
-
pvk12
k32
34
1m
31
32b
k
34
33m
b
pv
+ 22v22x
-s1
+
k22
21
22b
k
pv
pv&spv
m13
+
13F
12F
2F
3F
zF
V(t)
rmk
1 Rm
rkm
Vp
FJ
2rJ
211 rBb r+
Figure 4.43: Bond graph model with lumped real actuator effects
135
The constitutive parameters are:
b11 = 9072.39 N-s/m
k12 = -35550 N/m
m13 = -0.000162 kg
b21 = -9075.29 N-s/m
k22 = -1644260 N/m
b31 = -32.0272 N-s/m
k32 = -104.214 N/m
b33 = 2.64997 N-s/m
m34 = -0.93526 kg
Note in the above figure the shaded portion of the signal flow diagram that
leads to the controlled effort source, V(t). This portion of the diagram converts
the signal for the required actuator force, Fz, into the corresponding voltage signal
V(t). This signal is computed such that, after being introduced as an effort source
and passing through the 1-junction and gyrator and transformer, the effort variable
on the power bond that leads into the 1-junction that represents the suspension
compression, vp, is indeed equal to Fz, the required actuator force.
136
The state equations of the system are
34
3433323234
34
34
31
323232
21
222222
213
221232
23222
222
213
213
211
)()(
)()()(
mpbxkp
mp
bxk
mpvx
bxk
mpvx
mpvx
rmmJxrkxrkxrkxrkvrmvJm
rmmJpvmrbpvmBp
sm
smum
sm
smum
sm
smump
sm
pppumumsm
sm
smumsmsmumsmrsm
−=
−−−=
−−=
−=
+++++++
+
+++
−+−=
&
&
&
&
&&
L&
(4.21)
A swept frequency simulation was performed to ensure that the
synthesized suspension system matched the specified frequency response
behavior. A comparison plot of the actual response and the required response,
provided in Figure 4.44, shows excellent correspondence.
137
100
101
102
-40
-35
-30
-25
-20
-15
-10Data-based FRF and actual response: motor w/inductor modeled
frequency (rad/s)
v un
spru
ng m
ass
/ v s
prun
g m
ass
actual system responseoriginal FRF
Figure 4.44: Comparison plot of actual vs. required frequency response
Time domain simulations were performed in which the velocity of the
unsprung mass was the flow source. The voltage input to the motor, as calculated
by the signal flow diagram, was the effort source. The required force-velocity
profile for the actuator is shown in Figure 4.45.
138
-2 -1.5 -1 -0.5 0 0.5 1 1.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
4 Force-velocity profile of actuator
velocity (m/s)
forc
e (N
)
Figure 4.45: Force velocity profile for PMDC/rack & pinion actuator
A comparison between the simulated time history of the sprung mass
velocity and the experimental values are shown in Figure 4.46. As with the
hydraulic actuator case, there is good correspondence between the two.
139
7 7.5 8 8.5 9 9.5 10 10.5 11
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Comparison of simulated vs. actual sprung mass velocities
time (sec)
spru
ng m
ass
velo
city
(m/s
)simulated data experimental data
Figure 4.46: Simulated and experimental sprung mass velocities
Figure 4.47 contains plots of the required motor input voltage time
histories that would be necessary to provide the needed actuation. Voltage vs.
time is plotted for different values of the motor torque constant, km. Figure 4.48
contains plots of the corresponding motor power for each case. These plots will
be useful to the designer when either selecting an appropriate motor or designing
a motor for this particular application.
140
0 5 10 15 20 25 30-1500
-1000
-500
0
500
1000
1500Voltage time history comparisons
time (sec)
volta
ge (V
)km*rg=72 km*rg=90 km*rg=126
Figure 4.47: Voltage time histories for different values of km
141
0 5 10 15 20 25 30-1
0
1
2
3
4
5
6
7
8x 10
4 Motor power time history comparisons
time (sec)
pow
er (W
)km*rg=72 km*rg=90 km*rg=126
Figure 4.48: Motor power time histories for different values of km
4.4.3 Comparison of Realizations Involving Different Energy Domains
Transient behavior can be one of several factors in making a decision
regarding the energy domains for active device realization. As an example, the
step input response for the vehicle suspension with the hydraulic actuator and that
with the PMDC motor were simulated. The sprung mass velocity transient
behavior for both systems is shown in figure 4.49.
142
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Comparison of Sprung Mass Velocity Transients (Step Input)
time (sec)
velo
city
(m/s
)hydraulic actuatorPMDC motor
Figure 4.49: Comparison of transients between energy domains
The hydraulic actuator suspension system has a larger overshoot than that
of the PMDC motor system. Also, the hydraulic actuator exhibits weak
underdamped behavior, whereas the PMDC motor exhibits an overdamped
response, in that it settles to zero velocity strictly from above. In the case where
such a larger overshoot and oscillatory behavior would not be acceptable, the
PMDC motor realization would be a better choice.
In general, the choice of the physical realization of the actuator depends
on the application, which in turn will affect design requirements, such as transient
143
response (e.g., settling time), space limitation, weight limitations, and power input
limitations.
4.4.4 Summary and Conclusions
In this section, the process of active element physical realization was
further demonstrated, using the vehicle suspension system as an example. Two
separate physical realizations, involving different energy domain combinations,
were carried out: a hydraulic actuator and a PMDC motor. For both cases, a
combined bond graph model and signal flow diagram was developed. In the
hydraulic actuator case, the model contained a controlled pressure source,
whereas the PMDC motor model contained a controlled voltage source.
For the hydraulic actuator case, simulations were run to determine the
effects of changing selected design parameters and adding real actuator effects,
such as piston mass and viscous damping. As an example, various values for the
piston area were used in the simulations, and the required force-velocity profiles
and regulated pressure source time histories were generated and compared.
For the PMDC motor case, comparative simulations were also run, which
included real motor effects such as stator coil resistance, rotor inertia and rotor
bearing friction. Various values of the machine constant were used in
simulations, and the required force-velocity profiles and regulated voltage source
time histories were generated.
As with the previous application, these applications further demonstrated
the applicability of the synthesis methodology as a simulated-based design tool.
144
4.5 ANALYSIS AND REFINEMENT OF SYNTHESIZED CONTROL SYSTEMS
4.5.1 Introduction
We now turn our attention to the control system diagrams that result from
the active element synthesis. These are represented by the signal flow portions of
the bond graph models presented in the previous sections. A detailed examination
of these signal flow diagrams will assist the system analyst in identifying the type
of control system necessary for the active device that arises from the synthesis
procedure.
4.5.2 Active Device Control for the EM Suspension Example
In Section 4.4.2, a hydraulic actuator was synthesized as a retrofit for an
electromechanical vehicle suspension system. Its combined bond graph and
signal flow diagram is repeated in Figure 4.50 below.
145
1
kp :C
Fkp
bp+b11 :RFbp
T: 1/A
QV =&Pin
E:P(t)
+ 32v 32x
-
Fz
s1
px
3F +
signalflow
s1
+
12F
+ 34p
- s134p&
-
zF
Vp
pvk12
k32
34
1m
31
32b
k
34
33m
b
A1
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
Vp
pv
+ 22v22x
-s1
2F
+
k22
21
22b
k
pv
mp :I
FkpVp
pv&spv
m13
+
13F
Figure 4.50: Bond graph and signal flow for hydraulic actuator synthesis
The controlled effort source, P(t), consists of four force components that
are summed prior to the gain A1 , as shown in the summer at the far right portion
146
of the signal flow diagram. The four components are labeled F12, F13, F2 and F3.
Investigation of the nature of each of these signals will give further insight into
the control scheme necessary for the hydraulic actuator.
The component F12 results from the integration of the compression rate of
the suspension system, vp and its multiplication by a constant gain, k12. Thus, this
portion of the control system can be considered to have the characteristics of a
proportional integral (PI) controller. Following similar reasoning, the component
F13 has the characteristics of an output from a proportional differential (PD)
controller. The component F2 can be classified as an output from a proportional
integral controller with additional feedback.
The component F3 results from a more detailed control scheme. If we look
closely at the input and output, namely vp and F3, we see that the velocity vp is
decremented by two velocity feedback signals and is then integrated and
multiplied by a constant gain, k22. Thus, we can consider this to be an equivalent
PI controller with additional feedback, since the input is integrated.
Figure 4.51 shows the time history of each of the four force components.
From the plots it is clear that F13 and F3 each make a very small contribution to
the overall control signal Fz, with respect to the remaining components.
147
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5x 104 F12
time (sec)
forc
e (N
)
0 5 10 15 20 25 30-50
-40
-30
-20
-10
0
10
20
30
40F13
time (sec)
forc
e (N
)
148
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5x 104 F2
time (sec)
forc
e (N
)
0 5 10 15 20 25 30-40
-30
-20
-10
0
10
20
30
40
50F3
time (sec)
forc
e (N
)
Figure 4.51: Force components of the overall control signal Fz
149
Simulations were run in which these components were eliminated from the
control system. Eliminating these components significantly simplifies the control
system without significantly changing the desired performance of the hydraulic
actuator, as shown in the comparison plot of time history of the sprung mass
velocity of Figure 4.52.
12.5 13 13.5 14 14.5 15 15.5-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Comparison of original time series data for sprung mass velocity
time (sec)
velo
city
(m/s
)
original modified control system
Figure 4.52: Comparison of portion of sprung mass velocity time histories
150
As further evidence that the elimination of these control signals does not
affect the desired frequency response behavior of the system, an auto-power
(frequency) spectrum was generated for both cases. A comparison plot between
the cases where F13 and F3 are included and eliminated from the control signal is
shown in Figure 4.53. The auto-power spectral density plots are virtually
identical.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
7
Comparison of auto power spectra
frequency (Hz)
mag
nitu
de
original modified control system
Figure 4.53: Comparison of sprung mass velocity auto-power spectra
The simplified control system is shown in Figure 4.54.
151
1
kp :C
Fkp
bp+b11 :RFbp
T: 1/A
QV =&Pin
E:P(t)
Fz
pxsignalflow
s1
+
12F
zF
Vp
pvk12
A1
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
Vp
+ 22v22x
-s1 2F +k22
21
22b
k
pv
mp :I
FkpVp
Figure 4.54: Simplified control system for hydraulic actuator
The remaining portions indicate that the control system has the
characteristics of a PI control system. This is illustrated schematically in Figure
4.55.
152
controller(hydraulicactuator)
plant/system(vehicle)
actuating signal, P(t)
feedback signal, vp(t)
pxs1
+
12F
zF
pvk12
A1
+ 22v22x
-s1 2F +k22
21
22b
k
pv
P zF
Figure 4.55: Schematic of control system
4.6 POST-SYNTHESIS MODEL AUGMENTATION: NONLINEAR EFFECTS
4.6.1 Extension of Actuator Model and Linearization of Constitutive Laws
In section 4.4.2, a hydraulic actuator was physically realized as a retrofit
for the electromechanical vehicle suspension system. In this section, we will
extend this model to include the supply and return pressure sources for the
hydraulic actuator. The previous model contained a controlled effort source, but
did not specify the physical workings of the pressure source. A schematic of the
actuator, along with the supply and return pressures is given in Figure 4.56.
153
Fact
Q1Q2
actv
x
P0 Ps P0
P1 P2
Figure 4.56: Actuator schematic illustrating pressure sources
The displacement of the control rod, x, allows the pressure chambers of
the actuator to be exposed to either the return pressure or source pressure, as
shown above. This causes a pressure differential between the two chambers,
causing the piston to move in the desired direction, with a velocity vact. The
inclusion of this part of the actuator into the model introduces nonlinear effects
due to the constitutive laws that relate the flow across the orifice and the pressure
differential between the source/return and chamber pressures.
For comparison purposes, the bond graph portion that corresponds to the
controlled effort source, as modeled in section 4.4.2 is shown in Figure 4.57 and
the bond graph of the extended actuator model, including the source and return
pressures, is shown in Figure 4.58.
154
1
T:
E:
A1
Fact vact
V&Pact
P(t)
...
signal flow
vehicle model
Figure 4.57: Original model fragment of active device and controlled source
1
1
Cf :C
R1= Φ 1(x) :R 1
0
0 C: Cf
1
0
0
Ps :E E: P0
1
R2a= Φ 2a(x) :R
1
R: R1a= Φ 1a(x)
T:A1
Fact vact
V&
P1
V&P2
V&P1
1cV&
P2
2cV&
P1
Q1a
P2Q2a1
P∆Q1
2P∆
Q2
Ps P0
aP
2∆ Q2a a
P1
∆ Q1a
Pact
...
signal flow
x
signal flow
x
signal flow
x
vehicle model
R: R2= Φ 2(x)
Figure 4.58: Augmented bond graph model with source and return pressures
155
Taking the fluid capacitance to have a linear constitutive relationship, we
turn our attention to the resistance elements that model the losses due to orifice
flow. Since the system is symmetric, consider the portion of the bond graph
model that corresponds to orifice 1, as shown in Figure 4.59.
Cf :C
R1= Φ 1(x) :R 1
0
0
0
Ps :E E: P0
1
R: R1a= Φ 1a(x)
P1
V&P1
1cV&
P1
Q1a1P∆
Q1
P0
aP
1∆ Q1a
Figure 4.59: Portion of bond graph illustrating flow across orifice 1
Two R elements are needed to model the orifice flow, since the actuator
chamber can be exposed to either the source or return pressure, depending on the
position of the control rod. The constitutive laws of both resistive elements have
the same form, i.e.,
156
xPPg
ckxPQ s )(2
),( 111 −=γ
(4.22)
which is nonlinear in P1 and linear in x. In this equation, γ is the specific weight
of the hydraulic fluid, c is the discharge coefficient (which is experimentally
determined), g is gravitational acceleration, and k is the constant proportionality
between the orifice area and the displacement of the control rod, x.
Linearizing this equation about an operating point and selecting this
operating point to be the origin, the following linear constitutive relation is
obtained (Ogata, 1978)
xPg
ckxQ s
22
)(1 γ= (4.23)
This linearization can be applied to the remaining three resistive elements. This
results in a linearized model of the actuator source and return pressure system, in
which the four resistive elements are modulated by the control rod displacement,
x.
4.6.2 Inclusion of Nonlinear System Properties
As mentioned in Chapter 1, the bond graph methodology easily allows a
system modeler to model components that have nonlinear constitutive laws. The
synthesis method presented in this work only applies to systems with linear
behavior. However, in the case where systems that exhibit weak nonlinearities
are involved, these nonlinearities can be included in the post-synthesis bond graph
model such that nonlinear effects are captured. This will provide a more accurate
system model to be used in simulations that facilitate the selection of active
device parameters.
157
As an example, consider the synthesized vehicle suspension bond graph
model, with the control system omitted for simplicity, as shown in Figure 4.60.
1
kp :C
Fkp
bp+b11 :RFbp
T: 1/A
QV =&Pin
E:P(t)
Fz Vp
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
Vp
mp :I
FkpVp
Figure 4.60: Bond graph of vehicle system with linear spring effects only
Consider the case where the suspension spring is an air spring which has
the nonlinear consistutive law3)( pnppps xkxkxF += (4.24)
The nonlinear part of this expression can be expressed as
158
3)( pnp xkx =Φ (4.25)
where kn is a constant, whose units are N/m3. This nonlinear effect can be easily
added to the bond graph, as shown in Figure 4.61.
1
kp :C
Fkp
T: 1/A
QV =&Pin
E:P(t)
Fz Vp
0 I: msm
Fs Fs
Fs
Vum Vsm
Vp
smp&=Vum :F
Vp
mp :I
FkpVp
bp+b11 :RFbp
VpC:Φ (xp)
Fkn
Vp
Figure 4.61: Bond graph of vehicle system with nonlinear spring effects
After addition of this bond graph element, the system of state equations
can be easily updated and used in a simulation based design environment to
facilitate the determination of active device parameters.
159
4.7 PRACTICAL LIMITATIONS OF THE SYNTHESIS METHOD
4.7.1 Introduction
While investigating various combinations of experimental data sets for the
synthesis of the active element for the vehicle suspension problem, a relevant
finding was made regarding the choice of the transfer function from which the
synthesis procedure should commence.
4.7.2 Transfer Function Selection and Synthesis Results
Consider again the impedance-based bond graph of the quarter-vehicle
model with the unknown suspension system impedance, Z(s), shown in Figure
4.62.
I: mum
0F: 1 0
1
kt :C R: bt
Z(s)
I: mum
Ft Ft
Ft
vterr vum
vt
Fkt Fbtvt
vt
up&
vum
vum
Fz Fz
Fz
=s
p&
vsm
vp
Figure 4.62: Impedance bond graph showing choice of the transfer function
We wish to begin with the synthesis process from the transfer function
160
)()(
sVsV
terr
sm (4.26)
where the terrain velocity is the input and the velocity of the sprung mass is the
output. This choice of variables is associated with the power bonds that are
boxed, as shown in Figure 4.61. To obtain this transfer function, we form a
transmission matrix equation using the transmission matrices of the bond graph
elements between the two boxed power bonds. This equation has the form
=
terr
t
sm
out
VF
DCBA
VF
(4.27)
The 2x2 matrix is the product of the associated individual transmission matrices,
obtained using impedance bond graph methods (see Appendix A), is shown
below.
−
−
=
+−− 1
01
101
101
101
/11
tt bsk
um
Z
sm smsmDCBA
(4.28)
We can then solve for the ratio that is represented by the transfer function given
above. The solution, in terms of the unknown suspension system impedance,
Z(s), is
ZksZbZsmZsmsmksmbsmmZksZb
VV
tTFttsmumsmtsmtumsm
tt
terr
sm
+++++++== 2223
(4.29)
161
Note that this is the “theoretical” transfer function, tTF, that will be equated to the
“data-based” transfer function, dTF, which is determined from experimental time
histories of the terrain velocity, vterr(t), and sprung mass velocity, vsm(t), to be
100333.000108.0076824.011251.01087.8
2
25
++++×=
−
ssss
dTF (4.30)
Equating tTF and dTF and solving for the filter impedance Z, results in the
following expression for Z
2307923603446.242120.7406156.0101773.1107431.12932805084405.4
)( 234
727345
−+++×−×−−−−=
sssssssss
sZ (4.31)
If we substitute this back into the expression for the theoretical transfer function,
tTF(s), and call the resulting expression valTF(s), the value of the transfer
function, we have
12052.0107876.1103152.631562097.1141205.01037.110055.2667420122445.1130822.0
)( 826356
7273456
−×+×+++−×+×++++=
sssssssssss
svalTF (4.32)
Note that the order of the denominator of this transfer function does not match the
order of the denominator of the original data-based transfer function, dTF(s).
While they can be shown to be equal numerically, they are not the same transfer
functions in that the number of poles are different between the two. The
expression for valTF(s) has six poles, two of which match the poles of the data-
based transfer function, but some of the remaining poles have positive real parts.
This precludes this transfer function from being used in a simulation, since it
would cause the numerical results to go unstable. This occurrence has been
162
consistent, despite various approaches used to arrive at a suitable transfer function
that can be used for simulations.
To illustrate the underlying reason for this, let us consider the most
general form of the equations involved in these calculations. Upon forming the
transmission matrix equation for a bond graph in which there is an unknown
impedance, Z, the general form of the theoretical transfer function is
011
1011
1
011
1011
1)(dsdsdsdZcsZcZscZscbsbsbsbZasZaZsaZsa
stTF kk
kk
jj
jj
nn
nn
mm
mm
++++++++++++++++++= −
−−
−
−−
−−
LLLL (4.33)
Expressing the unknown impedance as
D
N
ZZ
sZ =)( (4.34)
and substituting it into the above equation, we have
DDDk
kDk
kNNNj
jNj
j
DDDn
nDn
nNNNm
mNm
m
ZdsZdZsdZsdZcsZcZscZscZbsZbZsbZsbZasZaZsaZsa
stTF01
1101
11
011
1011
1)(++++++++++++++++++= −
−−
−
−−
−−
LLLL (4.35)
Our goal is for tTF(s) to have the same number of poles as dTF(s), which means
that the degree of the denominator polynomial, tTF, must be the same as the
degree of the denominator polynomial, dTF. Additionally, we also want to
preserve the difference between the orders of these two polynomial functions, so
we require that the order of tTF equal the order of dTF,
( ) ( ))()( sdTFOstTFO = (4.36)
163
where the “order of” a polynomial ratio is defined as the degree of the numerator
polynomial minus the degree of the denominator polynomial. This implies the
additional restriction that the degrees of the numerator polynomials of both tTF
and dTF be the same, given that we have added the constraint that the degree of
the both denominator polynomials be the same.
Returning to the equation for valTF above, we note that we are only
concerned with the highest possible power of s that results from the individual
product terms in both the numerator and denominator. There are four product
terms that must be investigated: two from the numerator, and two from the
denominator. The two numerator product terms are Nm
m Zsa and Dn
n Zsb , and
the two denominator product terms are Nj
j Zsc and Dk
k Zsd . Before we can
evaluate these terms, it is necessary to express ZN and ZD as polynomials
themselves, since the unknown impedance Z(s) is assumed to be a ratio of
polynomials
011
1
011
1)(ffsfsf
eseseseZZ
sZ rr
rr
D
N
++++++++
== −−
−−
L
L (4.37)
Thus, the order of the numerator polynomial of valTF will be the higher of the
following two terms: (m+q) or (n+r). One of these sums must be equal to the
degree of the numerator of dTF, and the other must be less than or equal to that
degree. Similarly, the order of the denominator polynomial of valTF will be the
larger of the following two terms: (j+q) or (k+r). Either of these values must be
equal to the degree of the denominator of dTF, and the remaining value must be
less than or equal to that degree.
Expressing dTF(s) in general form
164
01
1
01
1)(hshshgsgsg
sdTF yy
yy
xx
xx
++++++= −
−
−−
LL
(4.38)
we define the degree of the numerator as x and the degree of the denominator as y.
The constraints such that valTF and dTF have the same number of poles can be
summarized in the following way
[ ] [ ]yrnANDyqmORyrnANDyqm ≤+<+<+≤+ )()()()(
AND (4.39)
[ ] [ ]xrkANDxqjORxrkANDxqj ≤+<+<+≤+ )()()()(
To test the validity of the above statement, let’s return to the original
problem presented at the beginning of this section. The pertinent equations are
repeated below
ZksZbZsmZsmsmksmbsmmZksZb
VV
tTFttsmumsmtsmtumsm
tt
terr
sm
+++++++== 2223
(4.40)
Referring back to the general equation for tTF, we see that m=1, n=0, j=2, and
k=3.
100333.000108.0076824.011251.01087.8
2
25
++++×=
−
ssss
dTF (4.41)
Referring back to the general equation for dTF, we see that x=2 and y=2. Thus,
the constraints on q and r, which determine the degrees of the polynomials that
form the numerator and denominator of Z(s) can be expressed by the following
simultaneous equations
2322221 ≤+≤+≤≤+ rqrq (4.42)
165
While the first and third inequalities can be satisfied with q=0, there is no solution
for the second and fourth inequalities, since, by definition, r must be a positive
number. This means that there does not exist a polynomial ratio expression for
Z(s) such that the denominator polynomials of valTF and dTF are of the same
degree, and thus have the same number of poles. This is consistent with the
obtained polynomial ratio expression for Z(s) and the resultant expression for
valTF in the above example.
4.7.3 Practical Solution: Bond Graph and Physical System Interpretations
The author proposes that a resolution of this problem lies in the selection
of the experimental data quantities used to form the transfer function with which
to begin the synthesis procedure. Note that in the previous example that the
choice of the transfer function required a matrix product of four transmission
matrices in order to obtain the theoretical expression of the transfer function.
Consider the following case, where the selected transfer function is
p
sm
VV
(4.43)
This is illustrated in the bond graph of Figure 4.63, where the power bonds of the
associated variables are boxed.
166
I: mum
0F: 1 0
1
kt :C R: bt
Z(s)
I: mum
Ft Ft
Ft
vterr vum
vt
Fkt Fbtvt
vt
up&
vum
vum
Fz Fz
Fz
=s
p&
vsm
vp
Figure 4.63: Bond graph illustrating selection of alternate transfer function
The transmission matrix equation is
=
um
z
sm
out
VF
DCBA
VF
(4.44)
where
−
=
− 101
101
1Z
smsmDCBA
(4.45)
From the above matrix equation, the theoretical transfer function is found to be
ZsZ
VV
stTFum
sm
+==
613)( (4.46)
167
Note that since there are fewer transmission matrices, and thus fewer system
parameters involved, the form of the resulting transfer function is much simpler.
Since there are fewer matrix products involved, there is less of an opportunity for
the unknown impedance Z to be distributed throughout the calculations and form
terms with higher degrees of s. The impact of this outcome will be apparent when
we apply the previously formulated constraint equations on the polynomials of the
transfer function and the unknown impedance.
Expressing Z(s) as the ratio
D
N
ZZ
sZ =)( (4.47)
the expression for tTF becomes
ND
N
ZsZZ
stTF+
=613
)( (4.48)
Based on a curve fit of the transfer function obtained from experimental
data (i.e., vsm(t) and vum(t) ) the data-based transfer function is
108801.00172.086678.088786.002675.0
)( 2
2
++++−=ss
sssdTF (4.49)
We can now apply the constraint equations to check if the unknown
impedance that will result from equating dTF and tTF will produce a transfer
function that has the same number of poles as dTF. Referring back to the general
equations for tTF and dTF, we find that m=0, n=0, j=0, k=1, x=2, and y=2. This
results in the following inequalities
168
21222 ≤+≤≤≤ rqrq (4.50)
which can be satisfied by q=2 and r=1.
The unknown suspension impedance is found to be
199.1812090123841.373
)(2
−++−=
sss
sZ (4.51)
whose polynomial degrees match the prescribed values of q and r above. Upon
substituting this into the expression for the theoretical transfer function, tTF(s),
we obtain
110154.001984.0102432.103086.0
)( 2
2
++++−=
ssss
svalTF (4.52)
4.7.4 Conclusions
There are two practical recommendations to be made such that the
problem presented in this section is minimized or avoided. In the case of
redesign, experimental data on system variables that physically represent elements
that are directly connected to the active element should be collected specifically
for the purpose of forming the data-based transfer function. In the case of a newly
designed system, the desired system transfer function should be expressed in
terms of a system variable for a physical element of the system that is either
directly connected to (or separated by as few elements as possible from) the
element that is associated with the active element that is to be synthesized
169
Chapter 5: Summary, Future Work, and Conclusions
5.1 SUMMARY
This work focused on the development of a methodology for the synthesis
of multiple energy domain active elements within mechanical systems. The
methodology applies to systems whose design requirements are defined in terms
of desired frequency response behavior and that can be modeled as linear,
lumped-parameter systems. An impedance-based modeling approach, using the
bond graph methodology, was used to model the system and the active element to
be synthesized. The impedance of the active element to be synthesized was
obtained from the model and decomposed into positive and negative basic
impedances, which were used to create a mathematical model of the active
element behavior.
This mathematical model, in the form of a bond graph, was used to
develop a system of first-order differential equations that served as a model of the
overall system. Simulations using this mathematical model were run, such that
force and velocity time histories associated with the active element were
determined. Operational profiles of the active element were developed and used
in a simulation-based design approach to physically realize the active device to be
used in the system under study.
The mathematical model of the system also provided insight into the
control schemes for the active device that was yielded by the synthesis and
physical realization procedures. These control schemes were interpreted and
simplified, such that the nature of the control system needed for the control of the
active device became evident.
170
The validity of this methodology was proven by its application to two
design problems. The first application involved the synthesis of an active element
for a newly designed system, whereas the second application involved the
synthesis of an active element for an existing system that was to be retrofitted
with a new type of active device. In the second application, actual test data for
the system under study was used to develop the frequency response behavior
requirements. The test data was also used to further validate the synthesis
methodology by comparing the time-domain behavior of the synthesized active
device with the existing active device. Comparisons of simulated and
experimental data revealed that the active synthesis methodology yielded physical
realizations of active devices that satisfactorily matched required frequency
response behavior.
Physical realization of active devices was carried out for two different
combinations of energy domains: hydraulic-mechanical and electromechanical.
Comparisons between the two cases were carried out and conclusions were made
about how these comparisons could serve as a useful design tool in the selection
of energy domains associated with an active device for a specific application.
5.2 EXTENSION TO SYSTEMS WITH MULTIPLE ACTIVE ELEMENTS
A logical next stage of developing the synthesis methodology presented in
this work is the ability to synthesize systems that require more than one active
element. Such an extension to this methodology would have applicability in the
area of vehicle suspension design (Nolte, Stearman, et al, 1974) and other classes
of systems in which multiple actuation points are required.
171
An example of such an application will be presented in this section in
order to illustrate the starting point from which the synthesis methodology can be
extended.
Active control of flexible space structures with multiple actuators is a
problem that is of considerable interest (Vaillon, et al, 1999) and to which a
multiple active synthesis methodology can be applied. Consider the model of a
partially constructed space station structure shown in Figure 5.1.
m2
m1
unknown activeelement
k1
v1(t)
Z1(s) Z2(s)
k2
v4(t)
unknown activeelement
m3
k3
F(t)
v3(t)
m4
Figure 5.1: Space station model with multiple active elements
A forcing function is applied at mass m4, which can be caused by the
firing of attitude control jets or the docking of a spacecraft to the structure. It is
desired to control the vibrations of a solar array panel, represented by mass m1,
and those of a sensitive experimental palette, represented by mass m3. Two
172
actuators, represented by the impedances Z1 and Z2, are installed to control the
vibrations of these masses such that they meet specified frequency response
behavior. This is defined by two individual transfer functions
011
1
011
1
4
11 )(
)()(
bsbsbsbasasasa
svsv
sdTF nn
nn
nn
nn
++++++++== −
−
−−
LL
(5.1)
011
1
011
1
4
32 )(
)()(
dsdsdsdcscscsc
svsv
sdTF nn
nn
nn
nn
++++++++== −
−
−−
LL
(5.2)
Which are assumed to be given in polynomial ratio form – these are analogous to
the “data-based” transfer functions defined in the development of the synthesis
methodology in this work. The transfer function of equation 5.1 defines the
desired response behavior between velocities of masses m1 and m4, and equation
5.2 defines the desired frequency response behavior between the velocities of
masses m3 and m4.
The impedance-based bond graph model of the system is shown in Figure
5.2.
E 1
I: sm4
C:
0 1
I: sm3
0
1
C: Z2(s)
1
I: sm2
0
1
C: Z1(s)
I: sm1
sk1
sk2
sk3
Figure 5.2: Impedance based bond graph of the space station model
173
Expressions for the transfer functions defined above can be extracted from this
bond graph via the use of transmission matrices (see Appendix A.) The resulting
transfer functions are analogous to the “theoretical” transfer functions defined in
the development of the synthesis methodology in this work. They will be in terms
of the system parameters m1, m2, m3, m4, k1, k2, and k3, and will contain the
unknown impedance terms of the active elements Z1 and Z2, as illustrated in
equations 5.3 and 5.4. The transmission matrix approach will yield these
equations as ratios of two polynomials.
),,,,,,,,,,(),,,,,,,,,,(
)(,2143213212
,21432132111 ssZZmmmmkkk
ssZZmmmmkkkstTF n
n
K
K
ΦΦ
= (5.3)
),,,,,,,,,,(),,,,,,,,,,(
)(,2143213214
,21432132132 ssZZmmmmkkk
ssZZmmmmkkkstTF n
n
K
K
ΦΦ
= (5.4)
Equations 5.3 and 5.4 are both functions of Z1 and Z2. Upon equation of
each set of transfer functions (equations 5.5 and 5.6), a coupled set of equations in
Z1 and Z2 is obtained.
)()( 11 sdTFstTF = (5.5)
)()( 22 sdTFstTF = (5.6)
This requires that these equations be solved simultaneously. Once polynomial
expressions for these two unknown impedances are obtained, the synthesis
procedure as presented in this work can proceed.
A synthesis methodology in which multiple active elements are treated
would be applicable to design problems where multiple actuation points are
needed, as in the control of seismic structures (Cheng, Jiang, 1998.) An obstacle
174
that might be encountered in the development of such a methodology is the
solution of the simultaneous polynomial ratio equations of equations 5.5 and 5.6.
Another issue in the synthesis of systems that require multiple active
elements is the question of optimal active element location within the system such
that frequency response requirements are met. A multiple active element
synthesis methodology can be applied to various configurations to help a system
designer optimize both active element realizations and design characteristics.
5.3 CONCLUSIONS
The multiple energy active element synthesis methodology developed in
this work was shown to be a viable tool in the design of active devices for
mechanical systems. Its impedance-based approach is well suited to systems
whose required behavior is expressed in terms of frequency response. Combined
with the use of bond graphs, this methodology easily allows for simulations to be
run for various active device physical realizations and design parameters, such
that active device operating regimes are obtained. The results of these
simulations allow a system designer to perform comparisons and make decisions
regarding the preliminary design of an active device and its control system.
175
Appendix A: Bond Graphs in System Modeling
A.1 INTRODUCTION
Bond graphs are a powerful method of modeling physical systems in that
they facilitate the modeling of systems that involve multiple energy domains. For
example, an electromechanical system such as a motor, which involves both
mechanical and electrical elements, can be easily modeled using a bond graph
approach. Systems of state equations that contain variables from multiple energy
domains can then be extracted from the bond graph. Bond graphs can be used to
model systems in many energy domains: mechanical, electrical, hydraulic,
magnetic, thermodynamic, and chemical. The following overview focuses on the
energy domains with which the reader is most likely to be familiar: mechanical
and electrical.
This overview is intended to provide the reader with a basic practical
understanding of bond graphs as they relate to this work. For a more theoretical
and in-depth treatment of the bond graph methodology, see the works of Karnopp,
Margolis, and Rosenburg (1990) and Beaman and Paynter (1993).
A.2 POWER FLOW WITHIN SYSTEMS
Bond graphs map the power flow between the elements of a physical
system. Power is represented by the product of two power conjugate variables:
effort and flow. The term power conjugate is used because the product of the
effort and flow yields the power. Efforts generally represent force-like physical
quantities, such as force, torque, voltage, or pressure. Flows generally represent
176
flow-like quantities, such as velocity, angular velocity, current, or volumetric flow
rate.
The general representation of power flow into a system element is shown
in Figure A.1.
Se
f
Figure A.1: Power flow into a system element
Here, the symbol S can represent a whole system or an element of a particular
system. The horizontal line is called a power bond, which represents a physical
port through which power can flow into or out of the system. The effort variable
associated with this power bond, represented by the symbol e, is indicated above
the power bond. The flow variable, represented by the symbol f, is indicated
below the power bond.
A.3 ENERGY STORAGE AND DISSIPATION
The bond graph methodology uses three basic elements or primitives to
represent lumped effects in a system. The three basic lumped parameter modeling
elements are: the capacitive element, the inertive element, and the resistive
element.
The capacitive element, represented by the symbol C, generally represents
potential energy storage in a system. Examples of physical elements that can be
modeled by capacitive elements are mechanical springs, electrical capacitors, and
fluid holding tanks.
177
The inertive element, represented by the symbol I, represents kinetic
energy storage in a system. Examples of physical elements that can be modeled
by inertive elements are masses, electrical inductors, and fluid flowing in a long
pipe.
The resistive element, represented by the symbol R, represents dissipative
effects in a system. Energy is transformed into an unavailable form, usually in the
form of heat in mechanical and electrical systems, or in the form of head losses in
fluid systems. Examples of physical elements that can be modeled with resistive
elements are mechanical dashpots, electrical resistors, and orifices in pipes.
A.4 POWER SOURCES
Sources are used in bond graphs to represent power inputs and power
sinks into or out of a system. Sources are considered to be ideal, which means
that they are unlimited sources of power, without any losses. There are two types
of sources used in the bond graph methodology: effort sources and flow sources.
An effort source, represented by the symbol E, represents a force-like
input to a system. Examples of effort sources are: a forcing function in a
mechanical system, a pressure source in a fluid system (e.g., a pump), and a
voltage source in an electrical system.
A flow source, represented by the symbol F, represents a velocity-like or
flow-like input to a system. Examples of flow sources are: a velocity driver in a
mechanical system, an input volume flow in a fluid system, and a current source
in an electrical system.
178
A.5 CONSTITUTIVE LAWS
A.5.1 Introduction
For each of the energy storing elements (I and C) and the dissipative
element (R) there must exist a relationship between the effort and flow variables
associated with them. This relationship is called a constitutive law, which
depends on the physical nature of the system element.
A.5.2 Capacitive Elements
Consider the equation that relates the force and displacement
(compression) for a linear spring (Hooke’s Law)
CxkxFs == (A.1)
Here, the effort (spring force, Fs) is linearly related to the time integral of the flow
(the flow is the compression rate of the spring, x&.) The constant of
proportionality, the spring stiffness k, can also be written as the compliance of the
spring C, where C=1/k. The bond graph representation of a linear spring is
shown in Figure A.2. Although the capactive parameter, C, is usually indicated
with the C element, in the case of a linear spring, the stiffness k is often indicated
instead, since springs are commonly characterized by their stiffness, not their
compliance.
179
C: kFs
xv &=
Figure A.2: Bond graph representation of a linear spring
The electrical analog of the linear spring is the capacitor. The constitutive
law is
CqV = (A.2)
Similarly, the effort (voltage, V) is linearly related to the time integral of
the current (charge, q.) The capacitance, C, is the constant of proportionality.
The bond graph representation of this element is shown in Figure A.3.
C:CVC
qi &=
Figure A.3: Bond graph representation of a capacitor
A.5.3 Inertive Elements
As an example of an inertive element, consider a mass m with a velocity
xv &= . The momentum of the mass is given by the equation
xmmvp &== (A.3)
180
Recalling that force is equal to the time derivative of the momentum (Newton’s
second law), we have
xmpF &&& == (A.4)
Here, the effort (force, F) is related to the time derivative of the flow ( xv &= ) by
the constant of proportionality, m. The bond graph of this element is shown in
Figure A.4.
I: mxv &=
p&
Figure A.4: Bond graph representation of a capacitor
The electrical analog of a mass is an inductor, whose inductance is L. The
flux linkage, λ, of the inductor is given by the equation
Li=λ (A.5)
Taking the time derivative of both sides, and noting that the time derivative of the
flux linkage is voltage, we have
dtdiLVL = (A.6)
Here the effort (voltage, V) is related to the time derivative of the flow by the
constant of proportionality, L. The bond graph for the inductor is shown in Figure
A.5.
181
I: Li
λ&
Figure A.5: Bond graph representation of an inductor
A.5.4 Resistive Elements
As an example of a dissipative element, consider a mechanical dashpot.
The relation between the effort (force of the dashpot, Fd) and the flow (velocity,
x&) is given by the equation
xbFd &= (A.7)
The constant of proportionality is the viscous damping coefficient, b. The
bond graph for the dashpot is shown in Figure A.6.
R:bFd
x&
Figure A.6: Bond graph representation of a viscous damper
The electrical analog to the dashpot is an electrical resistor R. Here the
effort (voltage, V) is related to the flow (current, i) by the constant of
proportionality, R, the electrical resistance.
iRV = (A.8)
182
The bond graph for the electrical resistor is shown in Figure A.7.
R:Ri
VR
Figure A.7: Bond graph representation of an electrical resistor
A.5.5 Nonlinear Elements
All of the above examples illustrate bond graph representations of system
elements that have linear constitutive laws. The bond graph methodology easily
handles the modeling of system elements whose constitutive laws are nonlinear.
As an example of a non-linear element, consider a spring, whose force-
displacement behavior is given by the cubic equation
3)( kxxF =Φ= (A.9)
The bond graph for this system element can be constructed in either of the
two ways shown in Figure A.8
183
C:Fs
x&3kxF
s=
C:Fs
x&)(xF
sΦ=
Figure A.8: Bond graph representations of a nonlinear element
While the appearance of the bond graph for such elements changes little, it is
important for the modeler to recognize the correct constitutive relationships for
each element when extracting state equations from the bond graph.
A.6 CONNECTION OF BOND GRAPH ELEMENTS
Bond graph elements are connected by two different types of junction
elements that reflect the physical topology of the system: 0-junctions and 1-
junctions. The common feature of junction elements is that they are power
conserving, which means that power is neither dissipated, stored, nor created by a
junction element. Thus, the sum of the power out of a junction element must be
equal to the power going into it.
The 1-junction is used to link elements that share common flow variables.
As an example, consider the parallel spring-mass-damper system and its
corresponding bond graph, shown in Figure A.9.
184
C: k
R: b
I: m1
x&Fb
Fs
pF &=
k
b
m
x&
x&
x&
Figure A.9: Parallel spring-mass-damper system
Here, the topology of the system dictates that all of the system elements
are subjected to the same velocity (i.e., the velocity of the mass, x&.) Consider the
electrical analog of this system, shown in Figure A.10. The analogous system
elements (i.e., capacitor, inductor, and resistor) are in series, and are thus
subjected to the same current (i.e., the same flow variable, i.)
C: C
R: R
I: L1
VR
VC
i
i
i
VL
i
L
R
C
Figure A.10: Series LRC circuit
185
Note that the bond graph structure is identical to that of the mechanical system; the only
differences are that the effort and flow variables labeled on the power bonds are electrical.
Referring back to the mechanical system, and recalling that junction elements are power
conserving, we can write a power balance equation
0)()()( =⋅−⋅−⋅− vFvFvF dashpotspringmass (A.10)
where: Fmass = force on the mass (time rate of change of momentum)Fspring = spring forceFdashpot = damping force
Canceling the common factor, v, and rearranging this equation yields
0=++ massdashpotspring FFF (A.11)
This illustrates an important feature of the 1-junction: the sum of the effort
variables at a 1-junction is equal to zero.
The 0-junction is used to link elements that share common effort variables.
Consider the free vibration of the series spring-mass-damper system and its
corresponding bond graph, shown in Figure A.11. As a result of the topology of
the system, each element experiences the same force, F.
186
C: k
R: b
I: m0
F
F
k b
m
mx& 1x&
Fmx&
1xxm && −
1x&
Figure A.11: Series spring-mass-damper system and bond graph model
The electrical analog to this system and its bond graph are shown in
Figure A.12. The elements are in parallel, and thus all have the same voltage (i.e.,
the same effort variable.)
C: C
R: R
I: L0
V
V
iR
iLiC
VL RC
Figure A.12: Parallel LRC circuit and bond graph model
187
A.7 POWER TRANSFORMATION
There are two elements in the bond graph methodology that are used to
represent the transformation of power from one energy domain into another. The
transformer element, represented by the symbol T, relates an effort and a flow
variable in one energy domain to an effort and a flow variable in another domain,
respectively. The multiplicative factor that relates the effort and flow variable
pairs is called the transformer modulus, which is usually represented by the letter
m. An example of a simple transformer is a hydraulic piston, shown in Figure
A.13, along with its bond graph representation.
T:
F V
x&
A F P
A
x& V&
Figure A.13: Hydraulic piston and bond graph transduction element
Here, the input effort and flow variables are taken to be the force on the
plunger, F, and the velocity of the piston, x&. The output effort and flow variables
are the pressure in the chamber, P, and the time rate of change of the chamber
volume, V& . Here, the transformer modulus is the surface area of the piston, A, as
confirmed by the equations
VxA
APF&&=
= (A.12)
188
The gyrator element, represented by the symbol G, relates an effort and a
flow variable in one domain into a flow and an effort variable in another domain,
respectively. Similarly, the multiplicative factor that relates the effort and flow
variable pairs is called the gyrator modulus.
An example of a physical system that can be represented by a gyrator is an
electric motor. As indicated by the following equations, the ouput torque, T, and
the input current, i, are related by the machine constant, km. A similar relation
exists between the input voltage, v, and the speed of the motor, ω.
ωm
m
kvikT
==
(A.13)
A schematic of an ideal motor and its corresponding bond graph representation
are shown in Figure A.14.
i
v
T ω
km
G
:vi
Tω
km
Figure A.14: Electric motor and bond graph transduction element
189
Essentially, transformers and gyrators perform the same function, in that
they are lossless energy transformation elements. The difference between them is
that the transformer relates one effort variable to another and one flow variable to
another, whereas a gyrator relates a flow variable to an effort variable, and vice-
versa, as shown in the general equations for the transformer and gyrator given
below in Table A.1.
gyrator
transformer
equations bond graph representation
12
12
mffme
e
=
=
re
f
rfe
12
12
=
=
T
:e1
f1
e2
f2
m
G:e1
f1
e2
f2
r
Table A.1: Bond graph transduction elements: transformer and gyrator
190
A.8 CAUSALITY
A.8.1 IntroductionA powerful feature of the bond graph methodology is its ability to display
causal relations between system elements. Consider an arbitrary pair of bond
graph elements, A and B, that are joined by a power bond, as shown in Figure
A.15.
A Bef
Figure A.15: Connection of two bond graph elements
When any two bond graph elements are bonded, one of the elements
defines the flow as a response to the effort imposed upon it by the other element
and, conversely, the other element defines the effort as a response to the flow
imposed upon it by the first element.
This can be illustrated by redrawing the power bond as two directed
signals, as shown in Figure A.16. In this case, element A defines the flow and
element B defines the effort.
A Be
f
Figure A.16: Bidirectional signal flow between two elements
191
In order to avoid drawing two lines for the bond between the elements, the
following standard notation for indicating causality is used: a vertical line, called
the causal stroke, is drawn at the end of the bond where arrowhead for the effort
signal would be located if bi-directional signal lines had been drawn. This is
illustrated in Figure A.17.
A Bef
Figure A.17: Causal stroke on a power bond between two elements
Comparing Figures A.15 and A.17, it is important to note that power flow
direction, indicated by the half arrow, is independent of the direction of the
causality relationship between elements.
A.8.2 Causality for Source Elements
Causality assignments for the source elements, E and F, are
straightforward, as there can only be one possible causal stroke for each of the
source elements. As an example, an effort source by definition prescribes the
effort to a particular bond graph element, regardless of the flow imposed upon it.
Therefore, the only possible causal stroke for an effort source is indicated in
Figure A.18. The same reasoning also applies for a flow source; the only possible
causal stroke is also shown in Figure A.18.
192
E
FFigure A.18: Allowable causal strokes for effort and flow sources
A.8.3 Causality for Junction Elements
Causality assignments for junction elements can also be easily illustrated.
As an example, consider a 0-junction. Recalling that all power bonds joined into a
0-junction must have the same effort. This leads to the conclusion that only one
power bond can direct an effort to the junction. If, for example, two power bonds
prescribed an effort to a zero junction, there is a chance that two different effort
quantities could be prescribed to the 0-junction at a given time. This situation
would violate the rule for a 0-junction and cause conflicting information to be
propagated within the bond graph model. The acceptable causal structure for a 0-
junction, in which all but one of the causal strokes is directed away from the
junction, is shown in Figure A.19.
193
0
This bond definesthe effort
Figure A.19: Causal stroke rule at a 0-junction
A similar argument can be made for a 1-junction. The conclusion is that
only one power bond can define a flow for a 1-junction: all other bonds must
direct their efforts toward the junction. The acceptable causal structure for a 1-
junction, in which all but one of the causal strokes is directed toward the junction,
is shown in Figure A.20.
1This bond defines
the flow
Figure A.20: Causal stroke rule at a 1-junction
194
A.8.4 Causal structure for C elements
To illustrate the implications of causal structure for a capacitive element,
consider the case of a linear spring, whose constitutive law is
kxF = (A.14)
The effort variable, force (F), is a function of the displacement (compression) of
the spring, x, which is the integral of the flow variable. First consider the case
where the effort is directed into the C element (e.g., a specified force is applied to
the spring.) In order to determine the flow variable, x&, we take the time
derivative of both sides of the constitutive law and solve for the flow, x& (the
compression rate of the spring.) This means that the power flow into the element
is known for any time, t. This causal structure is referred to as derivative or
dependent causality, since no other information is needed in order to determine
the power flow into the element.
The second case is where the flow is directed into the C element (i.e., a
specified compression rate is imposed on the spring.) In order to determine the
effort variable F, we must integrate the flow variable with respect to time. This
results in an indefinite integral, which renders a constant of integration in the
solution
0cxdtx +=∫& (A.15)
It can be shown that this constant of integration is the initial condition for
the spring compression, x. Since additional information is required to determine
the power flow, this causal structure is referred to as independent or integral
causality.
195
The two possible causal structures for capacitive elements are shown in
Figure A.21.
C Cindependent or integral
causalitydependent or derivative
causality
Figure A.21: Causal structures for a C element
A.8.5 Causal structure for I elements
A similar line of reasoning can be used to determine the nature of causal
structures for inertive elements. Consider the case of a mass in motion. The
constitutive law is
xmmvp &== (A.16)
Where p is the momentum of the mass. Recalling Newton’s second law,
we also have
xmpF &&& == (A.17)
The effort variable, the force on the mass, F, is a function of the time
derivative of the flow, the velocity of the mass, xv &= . If we first consider the
case where the flow variable is directed into the I element (e.g., the mass is
subjected to a specified velocity), we can use the above constitutive law to
determine the momentum, p, and then take its time derivative to determine the
196
force, F. Therefore, the power can be calculated for any time, t, which implies
derivative, or dependent, causality.
For the case where the effort is directed into the I element (e.g., the mass
is subjected to a specified force), we can integrate Newton’s second law shown
above with respect to time to determine the velocity of the mass. As in the case of
the capactive element, this equation yields a constant of integration
0cxdtx +=∫ &&& (A.18)
This constant of integration is the initial condition of the velocity of the
mass. Again, since additional information is required to determine the power
flow, this causal structure is integral or independent. The two possible causal
structures for inertive elements are shown in Figure A.22.
I Iindependent or integral
causalitydependent or derivative
causality
Figure A.22: Causal structures for an I element
A.8.6 Causal structure for R elements
Consider a viscous dashpot with the following linear constitutive law
xbFd &= (A.19)
197
Where Fd is the damping force, and b is the viscous damping coefficient.
In this type of constitutive law, the effort variable is directly related to the flow
variable (i.e., there are no derivative or integral relations involved.) Therefore, if
either of the effort or flow variables are specified, the other can be calculated
without having to rely on the knowledge of an initial condition. Since the power
flow can be determined for any time in either case, this means that R elements
always have dependent causality, regardless of the orientation of the causal stroke
of the power bond.
A.8.7 Causality for Elements with Nonlinear Constitutive Laws
In the case of a nonlinear constitutive law, the functional relation between
the effort and flow variables may not be invertible, thus allowing only one
possible causal structure.
A.9 SYSTEM ORDER AND STATE VARIABLES
The general rule for determining system order is that the order of the
system is equal to the number of independent elements (i.e., the number of bond
graph elements that have integral causality.) From the above discussion, it can be
concluded that only I or C elements play a role in determining system order.
Each independent element in the system has a state variable associated
with it. For independent C elements, the state variable is the integral of the flow
variable, which is referred to as the generalized displacement. As an example,
consider the bond graph element of Figure A.23 that represents a linear spring,
which happens to have independent causality in a particular system.
198
C:kFs
vx =&
Figure A.23: C element with independent causality
Here, the state variable is the compression of the spring, x.
For independent I elements, the state variable is the integral of the effort
variable, which is referred to as the generalized momentum. As an example,
consider the bond graph element in Figure A.24 that represents a mass, which
happens to have independent causality in a particular system.
I: mv
p&
Figure A.24: I element with independent causality
Here the state variable is the momentum of the mass, p.
Note also, that since each independent element in a system contributes a
state variable, the number of state variables is also equal the order of the system,
thus
system ordernumber of
independentelements
number ofstate variables= =
199
A.10 SYSTEM ANALYSIS AND SIMULATION USING BOND GRAPHS
There are various systematic approaches for extracting a system of first-
order differential equations from a bond graph model of a system. The number of
differential equations is equal to the order of the system. The numerical solution
of the system of differential equations fully describes the system behavior, given a
set of initial conditions. Numerical solutions of these systems of equations can be
performed using a software package such as MATLAB™ .
A.11 IMPEDANCE BASED BOND GRAPHS
A.11.1 Basic Impedances
Another powerful feature of the bond graph methodology is that it easily
lends itself to the expression of system elements in terms of their impedances.
Recall that the impedance of an element, Z, is defined as the ratio of its effort
variable, e, to its flow variable, f.
feZ = (A.20)
In general, efforts and flows are expressed functions of the Laplace
variable, s. To illustrate how an impedance-based bond graph representation of
an element is derived from its time-domain representation, consider the case of a
mass, which is a inertive element, shown in Figure A.25.
200
I: mxv &=
p&
Figure A.25: Time-domain representation of an inertive element
The effort variable, the time rate of change of momentum, can be expressed in the
following manner, where s is the derivative operator.
smvsppe === & (A.21)
The flow variable is equal to the velocity of the mass
vf = (A.22)
Thus, the impedance is
smv
smvfesZ ===)( (A.23)
The impedance-based bond graph representation is shown in Figure A.26.
I: sm
Figure A.26: Impedance-based representation of inertive element
201
Following the same steps, we find that for a capacitive element
sCsk
sxkx
vkx
fesZ 1)( ===== (A.24)
The bond graph representation is
C: sCks 1=
Figure A.27: Impedance-based representation of a capacitive element
The impedance of a resistive element is simply the resistance coefficient,
thus the impedance-based representation for a mechanical dashpot is
R: b
Figure A.28: Impedance-based representation of a resistive element
A.11.2 Combining Impedances to form Composite Impedances
For interconnected bond graph elements, we can form overall terms of a
system impedance via the use of two basic rules for junction structures. Consider
the simple spring-mass-damper system and its corresponding impedance
representation, as shown in Figure A.29.
202
C: k/s
R: b
I:sm1
k
b
m
x&
Figure A.29: Impedance-based representation of spring-mass-damper system
At 1-junctions, impedances add, thus the composite impedance of the
system above is
sksbms
sk
smbsZ++=++=
2
)( (A.25)
The equivalent bond graph representation is
Z(s)= sksbms ++2
Figure A.30: Composite impedance for spring-mass-damper system
Next, consider the parallel LRC circuit and its impedance-based bond
graph, shown in Figure A.31
203
C:1/sC
R: R
I: sL0L RC
Figure A.31: Impedance-based representation of parallel LRC circuit
At 0-junctions, admittances add. Since admittance is the inverse of
impedance, the composite admittance of the system above is
sCsLR
sY ++= 11)( (A.26)
The equivalent impedance is
RsLLRCssRL
sCsLR
sYsZ
++=
++== 211
1)(
1)( (A.27)
A.11.3 Transmission Matrices
Consider the following impedance-based bond graph structure
204
I: I
1e1
f1
e2
f2
e f
Figure A.32: 1-junction with inertive element
Applying the known rules for a 1-junction, we have
fffeee
==−=
12
12 (A.28)
The impedance associated with the I element is
sIfe
sIfe
sZ
=∴
==)( (A.29)
We can combine these results and express it in matrix form
−
=
1
1
2
2
101
fesI
fe
(A.30)
Thus we can view e1 and f1 being the “input” and e2 and f2 as the “output”, where
the impedance of the element is an operator. The 2x2 matrix in the above
equation is called the transmission matrix. Using the same steps as illustrated
205
above, we can form the transmission matrices for the basic bond graph elements
for both 1-junctions and 0-junctions.
Consider the bond graph structure of Figure A.33.
C: C1
0e1
f1
I: I
1 0e2
f2
C: C2
1 2 3
Figure A.33: Bond graph illustrating the combining of transmission matrices
Each numbered boxed structure is associated with a transmission matrix. By
linking these transmission matrices and applying bond graph junction rules, we
form the following matrix equation that relates the input effort and flow variables
to the output variables.
[][ ][]
=
1
1
2
2 123fe
fe
(A.31)
Note the reversed order of the product of the individual transmission matrices.
206
A.12 SUMMARY
Table A.2 provides a summary of generalized variables used in the bond
graph methodology, along with their counterparts in selected energy domains.
GeneralizedVariables
MechanicalTranslation
MechanicalRotation Hydraulic Electrical
EFFORT (e) force (F) torque (τ ) pressure (P) voltage (V)
FLOW (f) velocity (v) angularvelocity (ω ) flow rate (Q) current (i)
DISPLACEMENT (q) displacement(x)
angulardisplacement (η) volume (V) charge (q)
MOMENTUM (p) linearmomentum (p)
angularmomentum (h)
pressuremomentum (Γ ) flux linkage (λ)
Table A.2: Generalized variables and equivalents in various energy domains
207
Table A.3 summarizes constitutive laws in selected energy domains for
the three primitive elements used in the bond graph methodology.
generalized
mechanicaltranslation
mechanicalrotation
hydraulic
electrical
CAPACITIVE ELEMENT Cgeneral linear
)(eqC
Φ=
)(FxC
Φ=
)(τθC
Φ=
)(PVC
Φ=
)(VqC
Φ=
vxrate =&:
ωθ =&:rate
QVrate =&:
iqrate =&:
Ceq =
CFx =)( kxF =
τθ C=)( κθτ =
kC 1=
κ1=C
CPV =
CVQ =
generalized
mechanicaltranslation
mechanicalrotation
hydraulic
electrical
INERTIVE ELEMENT Igeneral linear
)( fpI
Φ=
)(vpI
Φ=
)(ωI
h Φ=
)(QI
Φ=Γ
)(iI
Φ=λ
Fprate =&:
τ=hrate &:
Prate =Γ&:
Vrate =λ&:
Ifp =mvp =
ωJh =
QIf
=Γ
Li=λ
generalized
mechanicaltranslation
mechanicalrotation
hydraulic
electrical
RESISTIVE ELEMENT Rgeneral linear
)( feR
Φ=
)(vFR
Φ=
)(ωτR
Φ=
)(QPR
Φ=
)(iVR
Φ=
Rfe =
bvF =
ωτ B=
RQP =
iRV =
Table A.3: Constitutive laws for primitive elements
208
Appendix B: Computation of Spectral Density and Frequency
Response Functions from Experimental Data
The following definitions will be used in this section:
Transfer Function: Laplace transform of the unit impulse response
function h(τ) of the system. On the imaginary axis, i.e., real part is equal
to zero, the transfer function is called the frequency response function.
Frequency Response Function: H(f) is the Fourier transform of the unit
impulse response function of the system.
ττ τπ dehfH fi∫∞
∞−−= 2)()( (B.1)
The equation that gives the magnitude of the frequency response function
estimate, based on actual experimental data, is given by
[ ])(ˆ
)(ˆ)(ˆ)(ˆ
2/122
kxx
kxykxyk
fG
fQfCfH
+= (B.2)
The phase angle estimate is
= −
)(ˆ)(ˆ
tan)(ˆ 1
kxy
kxykxy
fC
fQfφ (B.3)
209
xxG is the auto-power spectrum estimate, given by the equation
2,,2,1,0)(2)(ˆ
1
2 NkfXtNn
fGdn
iki
dkxx K=
∆= ∑
= (B.4)
xyG is the cross-power spectrum estimate, given by the equation
[ ]2
,,2,1,0)()(2)(ˆ1
* NkfYfXtNn
fGdn
ikk
dkxy K=
∆= ∑
= (B.5)
xyC and xyQ are the real and imaginary parts of the cross-power spectrum
estimate, respectively, where
nd is the number of ensemble averages
N∆t is equal to the time length of each record
If no ensemble averaging is performed, i.e., there is only one time record,
and the equations for the auto and cross-power spectra become
2,,2,1,0)(2)(ˆ 2 NkfX
TfG kikxx K== (B.6)
[ ]2
,,2,1,0)()(2)(ˆ * NkfYfXT
fG kkkxy K== (B.7)
210
MATLAB™ Script for Computing Spectral Density Functions(filename: frf.m)
clearclose all
% This script calculates the estimates of the one-sided% auto and cross-power spectra for one record of two% channels of data, x and y.% The FRF is then calculated using these quantities.
% This script allows for overlapping ensemble averages, it% reads in filtered data from LabVIEW, performs windowing% on each ensemble and includes a window correction factor.
% Tom Connolly% Last Updated 1.03.2000
%**********************************************************% NOTE: frequency bin width (df) depends on T, whereas the% number of frequency bins depends on N. Frequency range% depends solely on dt (sampling rate)%**********************************************************
% Specify filename for loading (filtered output data from% LabVIEW.) Change variable names in this section to match% the filename from which the data is being extracted!
% Enter the number of ensembles (records)nd=30;
% Enter the number of data points per ensemble% (if overlapping needed)N=500
load -ascii mod_7h3.txtdata=mod_7h3;
%load -ascii sim_out.txt%data=sim_out;
% Extract time vector and calculate delta t (dt)t= data(:,1);dt= abs(t(2)-t(1));
211
% Enter MATLAB variable designation for "input": terrain% velocity (m/s)x= data(:,3);
% Enter MATLAB variable designation for "ouput": sprung% mass velocity (m/s)y= data(:,5);
% Subtract the mean value of the x and y data
x= x - mean(x);y= y - mean(y);
% Calculate L= total number of data pointsL=length(x);
if (N>=L) disp('ERROR!!! Ensemble length is greater than numberof data points available!')end
% Check to see if overlapping is needed. If not, assign% the max number of data points to the ensemble.
if (nd*N <= L)disp('*** Overlapping ensembles not required ***')
% Override user-defined value of N and% automatically calculate the number of data points per%ensemble
N=floor(L/nd)
% break data up into ensembles
for q=1:nd % q=ensemble number for r=1:N % r=data point number xx(q,r)=x((q-1)*N+r); yy(q,r)=y((q-1)*N+r); end end
else
212
disp('*** Overlapping required ***')
% Calculate the spacing needed between the beginning ofeach ensemble spc= floor((L-N)/(nd+1))
for q=1:nd for r=1:N if q==1 % start ensemble at data point #1 xx(q,r)=x(r); yy(q,r)=y(r);
elseif q==nd % start ensemble at data point L-N xx(q,r)=x(L-N-1+r); yy(q,r)=y(L-N-1+r);
else xx(q,r)= x(L-N-1 - (nd-q)*spc +r); yy(q,r)= y(L-N-1 - (nd-q)*spc +r);
end end endend
% Generate the vector that corresponds to the window% function (Hanning.)% The dimension of the window vector is the same as the% length of each ensemble, i.e., N.win=hanning(N);win=win'; % convert to a row vector
% Compute the window correction factor (Powers, p.63),which is needed later.WCF= inv (1/N * sum(win.^2));
% Apply the window to each ensemble (both input and% output.)for q=1:nd xx(q,:)=xx(q,:).*win; yy(q,:)=yy(q,:).*win;end
213
% Take the Fourier transform of each recordfor q=1:nd xh=xx(q,:); % pull out each row and store in
% temporary holding vector xh XH=fft(xh); % compute the one-sided Fast Fourier
% Transform XX(q,:)=XH; % stored transformed data into holding
% matrix XX
yh=yy(q,:); YH=fft(yh); YY(q,:)=YH;end
% The first components of X and Y are the sums of the data% (DC component),so remove them from the arrayXX(:,1)=[];YY(:,1)=[];
% N= record lengths of both XX and YY (force them to be the% same)N=length(XX)
% N must be an even number. If it isn't, then truncate% last element of X and Y.if (rem(N,2)~=0) XX(:,length(XX))=[]; YY(:,length(YY))=[];
N=length(XX)end
% Compute the auto-power spectra
for q=1:nd% pull out each row and store it in temporary holding
% vector XH XH=XX(q,:);
% compute APS for X data Gxxh= (2/(N*dt)) * (abs(XH(1:N/2))).^2;
214
% store APS data for the current ensemble in the holding% matrix GXXGXX(q,:)=Gxxh;
% similar APS calculations for Y data YH=YY(q,:); Gyyh= (2/(N*dt)) * (abs(YH(1:N/2))).^2; GYY(q,:)=Gyyh;
end
% Perform ensemble averaging & multiply by the window% correction factor
if (nd==1) Gxx=GXX * WCF; Gyy=GYY * WCF;else Gxx= sum(GXX) * WCF/nd; Gyy= sum(GYY) * WCF/nd;end
% Compute the cross-power spectrum
for q=1:nd
XH=XX(q,:); YH=YY(q,:);
for k=1:N/2 Gxyh(k)=(2/(N*dt)) * (conj(XH(k)) * YH(k)); end
GXY(q,:)=Gxyh;end
% Perform ensemble averaging and apply the window% correction factorif (nd==1) Gxy=GXY * WCF;else Gxy=sum(GXY) * WCF/nd;
215
end
% Compute the magnitude of the cross-power spectrummag_Gxy=abs(Gxy);
% Compute the real and imaginary parts of the cross-power% spectrumCxy=real(Gxy);Qxy=imag(Gxy);
% Compute the magnitude ratio estimatefor k=1:N/2 H(k)=mag_Gxy(k)/Gxx(k);end
% Compute the phase angle estimatefor k=1:N/2 %phi(k)=atan(Qxy(k)/Cxy(k)); phi(k)=atan2(Qxy(k),Cxy(k)); % four quadrant arc
% tangent corrects for% sign change
end
% transform radians into degrees for FRF plotting purposesphi_deg=phi*360/(2*pi);
% Create the frequency vector (bins)% df = frequency bin width = 1/T = 1/(dt*N)df=1/(dt*N);freq=(1:N/2)*df;
% Save magnitude and phase of FRF in file (transpose into% column vectors first)% freq units in Hz% phi is degreesFRF=[freq;H;phi_deg]';save frf.out FRF -ascii -double
% Convert H into decibels and freq into radians/second and% save in another output file.
rad_s=2*pi*freq;dB=20*log10(H);
FRF_raddB=[rad_s;dB;phi]';
216
save frf_raddB.out FRF_raddB -ascii -double
% Plotting...
subplot(2,1,1),plot(t,x)title('Terrain Velocity (input) Time History')
subplot(2,1,2),plot(t,y)title('Sprun Mass Velocity (output) Time History')
figure(2)subplot(3,1,1),plot(freq,Gxx),grid ontitle('Input Auto Power Spectrum')ylabel('magnitude')
subplot(3,1,2),plot(freq,Gyy),grid ontitle('Output Auto Power Spectrum')ylabel('magnitude')
subplot(3,1,3),plot(freq,mag_Gxy),grid ontitle('Magnitude of Cross Power Spectrum')xlabel('frequency (Hz)')ylabel('magnitude')
figure(3)subplot(4,1,1), semilogx(rad_s,dB)title('Magnitude and Phase of FRF')ylabel('magnitude (dB)')
subplot(4,1,2), semilogx(rad_s,phi_deg)xlabel('frequency (rad/s)')ylabel('phase angle (degrees)')
subplot(4,1,3), plot(freq,H)ylabel('magnitude (dimensionless)')
subplot(4,1,4), plot(freq,phi_deg)xlabel('frequency (Hz)')ylabel('phase angle (degrees)')
217
Appendix C: Frequency Response Function Curve Fitting
C.1 THEORY
Given experimental system response data, it is desired to obtain an
analytical expression of the transfer function of the system, which is often
expressed as a ratio of two polynomials. The method of Levy (1959) with
modifications by Sanathanan, et al (1963, 1974) will be used here. The author
verified the equation development via Mathematica™ and devised a MATLAB™
implementation of this methodology.
Consider the transfer function estimate, )( ωjG , represented as a ratio of
two polynomials
LL
+⋅+⋅+⋅+⋅++⋅+⋅+⋅+⋅+= 4
43
32
21
44
33
2210
)()()(1)()()(
)(ωωωωωωωωω
jbjbjbjbjajajajaa
jG (C.1)
If we carry out the exponentiation of the ωj terms and group imaginary and real
parts, we have
)()(
)()1()()()( 4
52
314
42
2
45
231
44
220
ωω
ωτσωβα
ωωωωωωωωωωω
τσ
βα
jDjN
jj
bbbjbbaaajaaajG =
++=
+−++−+−++−=
444 3444 21 L444 3444 21 L
444 8444 76L
444 8444 76L
(C.2)
Note that β and τ are the frequency dependent parts of the numerator, N, and
denominator, D, respectively.
218
Next, consider the exact transfer function, )( ωjF , which represents a
perfect fit to the experimental data points. The polynomial form of the function is
not known, but we can express its value at each point as the sum of the real and
imaginary parts of the data points at each frequency
)()()( kkk jIRjF ωωω += (C.3)
The error, εk, between the values of the exact transfer function and the
estimate at a specific frequency, ωk, is
)()()()()(
k
kkkkk jD
jNjFjGjFωωωωωε −=−≡ (C.4)
If we select )( kjD ω to be a weighting function, and multiply both sides
of the above equation by this function, we have
)()()()( kkkkk jNjFjDjD ωωωεωε −=≡′ (C.5)
Recalling that
kkkk
kkkk
jjNjjNjjDjjD
βωαωωβαωτωσωωτσω
+=⇒+=+=⇒+=
)()()()(
(C.6)
substituting this into the equation for kε′, and separating real and imaginary parts,
we have
4444 34444 21444 3444 21)()(
)()(kk h
kkkkkkk
g
kkkkkkk RIjIRωω
βωτωσατωσε −++−−=′ (C.7)
The magnitude of the error at frequency ωk is
)()()()( 22kkkkk hgjhg ωωωωε +=+=′ (C.8)
219
In order to obtain an overall estimate of the error of the curve fit over m
data points, we can define a mean square error parameter, E, as shown below
[ ]∑∑==
−++−−=′=m
kkkkkkkkkkkkkk
m
kk RIIRE
1
22
1
2 )()( βωτωσατωσε (C.9)
We wish to minimize the overall error between the exact transfer function
and its estimate. Thus, we partially differentiate E with respect to each unknown
coefficient ai, bi and set each result equal to zero. Taking second partial
derivatives of this equation yields positive definite functions, thus we are assured
that the relative extrema are indeed minimum points.
At this point Sanathanan, et al. (1963) note the poor accuracy of this
method due to the use of the weighting function )( ωjD in the development of the
error parameter. Since D is dependent on frequency, their preliminary results
indicate that this approach does not provide a good fit of the data at lower
frequencies. This is due to the fact that the value of )( ωjD is larger at higher
frequencies, which tends to favor the accuracy of the curve fit at higher
frequencies, resulting in poor accuracy at lower frequencies. As a solution to this
problem, they suggest using the following iterative approach.
Recalling the expressions for the error between the values of the exact
transfer function and the estimate at a specific frequency, ωk, and the expression
in which the weighting function is applied
)()()()()(
k
kkkkk jD
jNjFjGjFωωωωωε −=−≡ (C.11)
)()()()( kkkkk jNjFjDjD ωωωεωε −=≡′ (C.12)
220
We can eliminate the effect of the weighting function by employing an
iterative procedure in which the weighted error at a particular iteration, kε′, is
divided by the value of the weighting function from the previous iteration, where
the subscript L represents the iteration number, as shown below
1)()()()(
)()(
−
−=′≡′′Lk
LkkLk
Lk
Lkk jD
jNjFjDjD ω
ωωωω
εε (C.13)
To ensure that this new error estimate is positive definite, we take the
square of its magnitude
2
1
2
)()()()(
−
−=′′Lk
LkkLkk jD
jNjFjDω
ωωωε (C.14)
If we define
21)(
1
−
=Lk
kLjD
Wω
(C.15)
Then
kLkkLLkkLkk WWjNjFjD 222 )()()( εωωωε ′=−=′′ (C.16)
We now define a new error parameter, E′, to replace the previous
parameter, E
∑∑==
′=′′=′n
kkLk
n
kkLkL WWE
1
2
1
2 εε (C.17)
Continuing with the minimization of this error parameter, we partially
differentiate E′ with respect to each unknown coefficient ai, bi and set each result
equal to zero.
221
Recalling the following
L
L
L
L
67
45
231
66
44
22
67
45
231
66
44
220
)(
1)(
)(
)(
ωωωττωωωσσ
ωωωββωωωαα
bbbbb
bbbb
aaaaa
aaaaa
n
n
n
n
−+−==−+−==
−+−==−+−==
(C.18)
we can carry out the partial differentiation of E′, as shown below
[ ]
[ ]
[ ][ ][ ][ ][ ][ ] 0)(2
0)(2
0)(2
0)(2
0)(2
0)(2
0)(2
0)(2
1
7
7
1
6
6
1
5
5
1
4
4
1
3
3
1
2
2
11
10
=−+=∂
′∂
=−−=∂
′∂
=−+−=∂
′∂
=−−−=∂
′∂
=−+=∂
′∂
=−−=∂
′∂
=−+−=∂
′∂
=−−−=∂
′∂
∑
∑
∑
∑
∑
∑
∑
∑
=
=
=
=
=
=
=
=
kL
m
kkkkkkkkk
m
kkLkkkkkkk
kL
m
kkkkkkkkk
m
kkLkkkkkkk
kL
m
kkkkkkkkk
m
kkLkkkkkkk
kL
m
kkkkkkkkk
kL
m
kkkkkkk
WIRaE
WIRaE
WIRaE
WIRaE
WIRaE
WIRaE
WIRaE
WIRaE
βωστωω
στωσω
βωστωω
στωσω
βωστωω
στωσω
βωστωω
ατωσ
(C.19)
222
[ ]
[ ][ ][ ][ ][ ][ ] 0)(2)(2
0)(2)(2
0)(2)(2
0)(2)(2
0)(2)(2
0)(2)(2
0)(2)(2
1
77
7
1
66
6
1
55
5
1
44
4
1
33
3
1
22
2
11
=−+−−−=∂
′∂
=−+−−−−=∂
′∂
=−++−−−=∂
′∂
=−++−−=∂
′∂
=−+−−−=∂
′∂
=−+−−−−=∂
′∂
=−++−−−=∂
′∂
∑
∑
∑
∑
∑
∑
∑
=
=
=
=
=
=
=
kL
m
kkkkkkkkkkkkkkkkkk
kL
m
kkkkkkkkkkkkkkkkkk
kL
m
kkkkkkkkkkkkkkkkkk
kL
m
kkkkkkkkkkkkkkkkkk
kL
m
kkkkkkkkkkkkkkkkkk
kL
m
kkkkkkkkkkkkkkkkkk
kL
m
kkkkkkkkkkkkkkkkkk
WRIRIRIbE
WRIIIRRbE
WRIRIRIbE
WRIIIRRbE
WRIRIRIbE
WRIIIRRbE
WRIRIRIbE
βωτωσωατωσω
βωτωσωατωσω
βωτωσωατωσω
βωτωσωατωσω
βωτωσωατωσω
βωτωσωατωσω
βωτωσωατωσω
(C.20)
Making the substitutions for α, β, σ, and τ, and moving terms that do not
include a polynomial coefficient (either ai or bi) to the right hand sides of the
equations, yields a system of algebraic equations in which the polynomial
coefficients, ai and bi, are the unknowns. This system of equations is shown in
matrix form below. These equations were taken out to the degree such that the
numerator and denominator polynomials of the transfer function curve fit could
be computed up to seventh order, if desired.
223
=
−−−−−−−−−−−
−−−−−−−−−−−
−−−−−−−−−−−
−−−−−−−−−−−
−−−−−−−−−−
−−−−−−−−−−
−−−−−−−−−−
−−−−−
0
0
0
0
0000000
0000000
0000000
0000000
00000000
00000000
00000000
00001
6
4
2
7
6
5
4
3
2
1
0
7
6
5
4
3
2
1
7
6
5
4
3
2
1
0
14121081413121110987
12108131211109876
12108612111098765
10861110987654
10864109876543
86498765432
864287654321
1413121110981412108
13121110987121086
1211109876121086
11109876510864
1098765410864
98765438642
87654328642
7654321642
U
U
U
TSTSTSTS
BBBBBBBAAAAAAAA
UUUUSTSTSTSTUUUTSTSTSTS
UUUUSTSTSTSTUUUTSTSTSTS
UUUUSTSTSTSTUUUTSTSTSTS
UUUUSTSTSTSTSTSTSTSTSTSTST
STSTSTSTSTSTST
STSTSTSTSTSTST
STSTSTSTSTSTST
λλλλλλλλ
λλλλλλλλ
λλλλλλλλ
λλλλλλλ
The above matrix equation can be cast in symbolic form as
[ ] CcoeffsM = (C.21)
Where
∑
∑
∑
∑
=
=
=
=
+=
=
=
=
m
kkLkk
iki
m
kkLk
iki
m
kkLk
iki
m
kkL
iki
WIRU
WIT
WRS
W
1
22
1
1
1
)()(
)(
)(
)(
ω
ω
ω
ωλ
(C.22)
C.2 CURVE FIT COMPUTATIONAL PROCEDURES
Before the computations begin, it is necessary to indicate the degrees of
the numerator and denominator polynomials individually. It may be necessary to
224
run different cases in which these degrees are varied, such that the best curve fit is
obtained. Sanathanan, et al note that choosing excessively high polynomial
degrees in some cases can yield transfer functions that correspond to unstable
systems. Thus, extreme care must be taken to verify the time-domain stability of
the estimated transfer function.
Given m experimental data points, we wish to determine the polynomial
ratio that best fits these data points. Each data point consists of the following
information:
frequency: The frequency of the input to the system. Frequencies should
be expressed in radians per second. Frequency response data are usually
presented using these units.
magnitude: The ratio of the magnitude of the input to the system
response. For this analysis, it is important for these values NOT to be
expressed on a logarithmic scale, i.e., in decibels (dB). If the frequency
response data are presented in this traditional form, the magnitudes should
be transformed from decibels into a “dimensionless ratio” scale before
proceeding. The formula for this transformation is
)20/(10 valuedBmagnitude = (C.23)
phase angle: The difference in phase between the input and the system
response. The phase angle should be expressed in radians. Frequency
response phase data are traditionally presented in units of degrees. The
phase angle data can be converted from degrees to radians using the
following formula
225
radians ⋅=3602π degrees (C.24)
Recall from the equation development in the previous section that the
frequency response data points represent a “perfect fit” at each frequency, i.e., the
exact transfer function, represented by )( ωjF . The real and imaginary parts of
the exact transfer function are calculated using the following formulas
)sin()()cos()(
anglephasemagnitudeIanglephasemagnitudeR
k
k
⋅=⋅=
ωω
(C.25)
The iterative procedure now begins with the calculation of the elements of
the matrix [M] and the column vector C using the formulas for λ, S, T, and U
given above. For the first iteration, the magnitude of the function )( kjW ω is
assumed to be equal to one. The matrix equation [ ] CcoeffsM = is solved for
the coefficients ai, bi. These coefficients are used to compute the numerator and
denominator values of the transfer function estimate, )(&)( kk jDjN ωω at each
frequency ωk. The difference between the values of exact transfer function and
the transfer function estimate at each frequency data point, kε ′′, are computed and
the error parameter E′is formed, using the equations defined in the previous
section.
If the error parameter exceeds the specified error tolerance, the iterative
procedure is repeated, beginning with the calculation of the elements of the matrix
[M] and column vector C. The values of these elements will be different than
those of the previous iteration, since the formulas used to calculate them depend
on the function )( kjW ω . For each iteration, W is dependent on the value of the
226
denominator of the transfer function estimate, )( kjD ω , from the previous
iteration.
If the error parameter falls below the specified error tolerance, no further
iterations are performed and the polynomial coefficients for that iteration are
deemed sufficient to represent the transfer function estimate.
227
MATLAB™ Script for Frequency Response Function Curve Fitting(filename: frf_fit.m)
clearclose all
% filename: FRF_fit.m
% This script performs a curve fit on frequency response% function (FRF, or transfer function) data points and% returns the coefficients of the numerator and denominator% polynomials.%% Tom Connolly% Last updated: 1.03.2000
% ---------------------------------------------------------
% Enter the orders of the numerator and denominator% polynomials. This script will compute up to and% including seventh order polynomials.
n=2; % order of the numerator & denominator polynomialsd=2;
% Enter error parameter threshold valuetol=0.0001;
% Specify the maximum number of iterations allowed.max_iter=200;
% ---------------------------------------------------------
% Load FRF data fileload frf4.out -asciiFRF=frf4;
% Extract frequency data (frf.m outputs in Hz)freq=FRF(:,1);
% Extract magnitude data (frf.m outputs in dimensionless% units)mag=FRF(:,2);
228
% Extract phase data (frf.m outputs in degrees)phase_deg=FRF(:,3);
% ---------------------------------------------------------
% convert phase angles into radiansphase_rad = 2*pi/360*phase_deg;
% Convert frequencies to radians/secw= 2*pi*freq;
% Convert mag to dB for FRF plotting purposes.dB=20*log10(mag);
% ---------------------------------------------------------
% Calculate real and imaginary parts of the FRF, R and I% respectivelyR= mag.*cos(phase_rad);I= mag.*sin(phase_rad);
% Calculate the complex value of the FRF at each frequency% (for use later)i=sqrt(-1);F= R + i*I;
% ---------------------------------------------------------
% initialize V, S, T, U arraysV=zeros(1,14);S=zeros(1,14);T=zeros(1,13);U=zeros(1,14);
% Initialize the denominator array, D.D=ones(length(mag),1);
% for the first iteration, the elements of the D array% (and thus the weighting functions for% each frequency) are taken to be equal to one
% Initialize the numerator array, N.N=zeros(length(mag),1);
229
% Initialize the error parameter E, and iteration counter.
check=tol;L=1;
% =========================================================% ========= Begin conditional loop ================% =========================================================
% Check to see if error tolerance has been met.
while (check >= tol & L <=max_iter)
% Save the value of the D array from the last iteration% to calculate the weighting function for current% iteration
D_prev=D;
% solve the matrix equation: [M]coeffs= C % [M]= matrix of V, S, T, U values % coeffs= column vector of polynomial coefficients,
% An and Bd % C= column vecotr of other S, T, U values % coeffs= [M]-1 * C
% Calculate the weighting function, W, based on D_prev
W = (abs(D_prev.*F)).^(-2);
% Calculate the needed V (lambda) values (even% coefficients only)
for h=2:2:(2*n) V(h)= sum(w.^(h) .* W); end
% Calculate the needed S values (even coefficients only) for h=2:2:(2*d) S(h)=sum(w.^(h) .* R .* W); end
230
% Calculate the needed T values (odd coefficients only) for h=1:2:(2*d+1) T(h)=sum(w.^(h) .* I .* W);
end
% Calculate the needed U values (even coefficients only) for h=2:2:(2*d) U(h)=sum(w.^(h) .* (R.^2 + I.^2) .* W); end
% Build the M matrix, the size of which depends on the% specified order of the polynomials, i.e., n and d. The% M matrix will be a square matrix with dimension n+1+d.
% It is easiest to generate the whole matrix in its four% separate quadrants and then take the needed partitions.
% Quadrant I (V values) dimension: 8x8
Q1=[1 0 -V(2) 0 V(4) 0 -V(6) 0 0 V(2) 0 -V(4) 0 V(6) 0 -V(8) V(2) 0 -V(4) 0 V(6) 0 -V(8) 0 0 V(4) 0 -V(6) 0 V(8) 0 -V(10) V(4) 0 -V(6) 0 V(8) 0 -V(10) 0 0 V(6) 0 -V(8) 0 V(10) 0 -V(12) V(6) 0 -V(8) 0 V(10) 0 -V(12) 0 0 V(8) 0 -V(10) 0 V(12) 0 -V(14)];
% Quadrant II (S and T values) dimension: 8x7
Q2=[ T(1) S(2) -T(3) -S(4) T(5) S(6) -T(7) -S(2) T(3) S(4) -T(5) -S(6) T(7) S(8) T(3) S(4) -T(5) -S(6) T(7) S(8) -T(9) -S(4) T(5) S(6) -T(7) -S(8) T(9) S(10)
231
T(5) S(6) -T(7) -S(8) T(9) S(10) -T(11) -S(6) T(7) S(8) -T(9) -S(10) T(11) S(12) T(7) S(8) -T(9) -S(10) T(11) S(12) -T(13) -S(8) T(9) S(10) -T(11) -S(12) T(13) S(14)]
% Quadrant III (S and T values) dimension: 7x8
Q3=[T(1) -S(2) -T(3) S(4) T(5) -S(6) -T(7) S(8) S(2) T(3) -S(4) -T(5) S(6) T(7) -S(8) -T(9) T(3) -S(4) -T(5) S(6) T(7) -S(8) -T(9) S(10) S(4) T(5) -S(6) -T(7) S(8) T(9) -S(10) T(11) T(5) -S(6) -T(7) S(8) T(9) -S(10) -T(11) S(12) S(6) T(7) -S(8) -T(9) S(10) T(11) -S(12) T(13) T(7) -S(8) -T(9) S(10) T(11) -S(12) -T(13) S(14)]
% Quadrant IV (U values) dimension: 7x7
Q4=[U(2) 0 -U(4) 0 U(6) 0 -U(8) 0 U(4) 0 -U(6) 0 U(8) 0 U(4) 0 -U(6) 0 U(8) 0 -U(10) 0 U(6) 0 -U(8) 0 U(10) 0 U(6) 0 -U(8) 0 U(10) 0 -U(12) 0 U(8) 0 -U(10) 0 U(12) 0 U(8) 0 -U(10) 0 U(12) 0 -U(14)];
% Take the needed partitions of each quadrant matrix
% need n+1 x n+1 partition of Q1 matrix M1=Q1(1:n+1,1:n+1);
% need n+1 x d partition of Q2 matrix M2=Q2(1:n+1,1:d);
% need d x n+1 partition of Q3 matrix M3=Q3(1:d,1:n+1);
% need d x d partition of Q4 matrix M4=Q4(1:d,1:d);
% Now put partitions together M=[M1 M2
232
M3 M4];
% Build the [C] column vector (upper and lower % partitions)
CU=[1 T(1) S(2) T(3) S(4) T(5) S(6) T(7)]'; CL=[0 U(2) 0 U(4) 0 U(6) 0]';
% put needed partitions together C=[CU(1:n+1); CL(1:d)];
% Solve the matrix equation for the coefficients coeffs=inv(M)*C;
% --------------------------- ERROR CALCULATION -----------
% Calculate Numerator for FRF at each frequency data point% for this iteration
for k=1:length(mag)
% calculate each polynomial term for r=1:n+1 N_terms(r)=coeffs(r)*(i*w(k))^(r-1); end
% sum the polynomial terms to get the numerator N(k)=sum(N_terms);end
% Calculate the Denominator for the FRF at each frequency% data point
for k=1:length(mag)
% calculate each polynomial term for r=1:d D_terms(r)=coeffs(n+1+r)*(i*w(k))^r; end
% sum the polynomial terms to get the denominator D(k)=sum(D_terms) + 1; % B(0)=1end
233
% Calculate e" at each frequency data point for this% iteration.for k=1:length(mag) eps(k)=(D(k)*F(k) - N(k))/D(k) ;end
% convert to a column vectorepsilon=eps';
% Store epsilon values for this each iteration in a history% matrixsave_epsilon(1:length(epsilon),L)=abs(epsilon);
% Calculate error parameter E for this iterationE(L)= sum((abs(epsilon)).^2 .* W);
% Set tolerance check value to most recent value of the% error parameter.check=E(L);
% Store coefficient values for each iteration in a history% matrixco(1:length(coeffs),L)=coeffs;
% ---------------------- Calculate relative error ---------
% Absolute error = (actual data point @ given frequency) –% (N/D @ given frequency)
for k=1:length(mag) absolute_err(k)= abs(F(k)) - abs(N(k)/D(k)); rel_err(k)= 100*absolute_err(k)/abs(F(k));end
% Store relative error values in a history matrixsav_rel_err(1:length(mag),L)=rel_err';
% Take the mean of the relative errors for each iteration mean_rel_err(L)= mean(rel_err);
% ---------------------------------------------------------
% Update iteration counterL=L+1;
234
end
% =========================================================% ====== End conditional loop =======================% =========================================================
% Print out the results to the screen
% Note that the value of the error parameter sometimes% doesn't decrease with each iteration. So, we need to% seek the lowest value of the error parameter, which% corresponds to the best fit.
% Find the minimum value of the error parameter E, and% its index% min_E= error paramter value, E% iteration_number= corresponding matrix index
[min_E,iteration_number]=min(abs(E)) disp('minimum normalized error') mean_rel_err(iteration_number)
iteration_number =100
% Display the coefficients that correspond to the% iteration with the minimum % error
disp('minimum E coefficients') results=co(:,iteration_number);
% Manipulate the coefficients such that they are% displayed in descending order for the MATLAB bode% function command below.
numr=results(1:n+1); denr=results(n+2:n+1+d);
for h=1:n+1 num(h)=numr(n+2-h); end
for h=1:d
235
den(h)=denr(d+1-h); end den(d+1)=1; % B(0)=1
% Print out the coefficients: Ai and Bi % % for h=1:n+1 % a=num2str(h-1); % sprintf('A%s = %15.13f',a,co(h,iteration_number)) % end % % for h=n+1:n+d % a=num2str(h-n); % sprintf('B%s =%15.13f',a,co(h+1,iteration_number)) % end
% Print out the number of iterations performed sprintf('%f iterations were performed',L-1)
% Print out the numerator and denominator polynomial % coefficients num den
% Check stability of TF by displaying the poles (roots% of the denominator)
roots_den=roots(den)
% ------------------ PLOTS ----------------
% Generate FRF data using the obtained FRF polynomial% coefficients. Then do a overlaid plot for comparison% purposes.
% form "sys" using "tf" (transfer function)
sys=tf(num,den);
% this syntax outputs the data for the bode plot[bmag,bphase] = bode(sys,w);
% compensate for (flatten) the 3-d array output from the% bode command above
for m=1:length(bmag)
236
magb(m)=bmag(1,1,m); phaseb(m)=bphase(1,1,m); end
bmag=magb; bphase=phaseb; bdB=20*log10(bmag);
% -------------------------------------------------------
subplot(3,1,1),semilogx(w,dB,'g*',w,bdB,'r') title('Comparison of magnitude ratios') ylabel('magnitude ratio (dB)') legend('original data', 'curve fit')
subplot(3,1,2),semilogx(w,phase_deg,'g*',w,bphase,'r') title('Comparison of phase angles') ylabel('phase angle (degrees)')
subplot(3,1,3),semilogx(w,sav_rel_err(:,iteration_number),'b-x') title('Values of relative error vs. frequency foriteration that corresponds to best fit') xlabel('frequency (rad/s)') ylabel('percent error')
% print numerator and denominator values directly to theplot! %txtn1=num2str(num); %txtd1=num2str(den); %txtn=sprintf('numerator coefficients: %s',txtn1); %txtd=sprintf('denominator coefficients: %s',txtd1); %text(10,-20,txtn) %text(10,-23,txtd)
figure(2) subplot(3,1,1),plot(E)
title('Error parameter at each iteration') xlabel('iteration number') ylabel ('error parameter - E')
237
subplot(2,1,2),plot(mean_rel_err) title('Normalized error averaged over all frequencies') xlabel('iteration number') ylabel('mean normalized err') figure(3) bode(num,den,1e0, 1e2)
238
Appendix D: Mathematica™ Output
The following is a typical Mathematica™ file that shows the derivation of
the theoretical transfer function from the transmission matrices and the
decomposition of the required filter impedance into basic impedances. The file
presented here is from the general case of the electromagnetic (EM) vehicle
suspension application.
239
240
241
242
243
244
245
246
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Vita
Thomas Joseph Connolly was born in New York City, New York on
September 24, 1965, the son of Richard and Leona Connolly. He attended
elementary and intermediate school in the New York City public school system
and attended Cardinal Spellman High School in the Bronx, where he graduated in
1983. Thomas did his undergraduate studies at the State University of New York
at Stony Brook, where he majored in Mechanical Engineering and minored in
Technology and Society. He received a Bachelor of Engineering Degree in May,
1988.
After completing his undergraduate studies, Thomas moved to Texas,
where he worked as a Reliability Engineer on the B-2 Stealth Bomber Program at
LTV Aircraft Products Group in Dallas. In July 1990, he moved to Houston to
work as a Systems Engineer on the Space Shuttle Program for Lockheed
Engineering and Sciences Company at the NASA Johnson Space Center. In
1992, he worked as a Systems Engineer on the Space Station Program for
Grumman Space Station Integration Division at program headquarters in the
Washington, D.C. area.
In June 1993, Thomas moved to Austin to begin his graduate studies at the
University of Texas. He received a Master of Science degree in Aerospace
Engineering in December 1995. Up until this time, he worked as a Research
Assistant and a Teaching Assistant for the Senior Design course in the
Department of Aerospace Engineering and Engineering Mechanics under the
guidance of Dr. Ronald Stearman. In 1996, Thomas began his studies toward a
Ph.D. degree in the Department of Mechanical Engineering, under the guidance of
254
Dr. Raul Longoria. During this time, he worked as a Research Assistant at the UT
Center for Electromechanics under the direction of Professor William Weldon.
For a portion of this time, he was also a teaching assistant for the System
Dynamics and Controls Laboratory course in the Department of Mechancial
Engineering. His academic interests include modeling of physical systems,
synthesis of engineering systems with active elements, nonlinear systems
analysis, and structural dynamics.
Since beginning graduate school, Thomas continued his study of the
Russian language at the undergraduate and graduate levels, under the instruction
of Dr. Thomas Garza in the Department of Slavic Languages. He traveled to St.
Petersburg and Moscow in the summer of 1997.
Permanent address: 2912 Gerber Place, New York, NY 10465
This dissertation was typed by the author.