INTELLIGENT POWERTRAIN DESIGN

Jimmy C. Mathews

Advisors: Dr. Joseph Picone Dr. David Gao

The BOND GRAPH Methodology for Modeling of Continuous Dynamic Systems and its Application in Powertrain Design

INTELLIGENT POWERTRAIN DESIGN

Page 2 of 42Intelligent Powertrain Design

Outline

• Dynamic Systems and Modeling

• Bond Graph Modeling Concepts

Introduction and basic elements of bond graphs

Causality and state space equations

• System Models and Applications using the Bond Graph Approach

Electrical Systems

Mechanical Systems

• The Generic Modeling Environment (GME) and Bond Graph Modeling

• Some Future Concepts


• Dynamic Systems Related sets of processes and reservoirs (forms in which matter or energy exists) through which material or energy flows, characterized by continual change.

• Common Dynamic Systems electrical, mechanical, hydraulic, thermal among numerous others.

• Real-time Examples moving automobiles, miniature electric circuits, satellite positioning systems

• Physical systems Interact, store energy, transport or dissipate energy among subsystems

• Ideal Physical Model (IPM) The starting point of modeling a physical system is mostly the IPM.

• To perform simulations, the IPM must first be transformed into mathematical descriptions, either using Block diagrams or Equation descriptions

• Downsides – laborious procedure, complete derivation of the mathematical description has to be repeated in case of any modification to the IPM [3].

Dynamic Systems and Modeling


Computer Aided Modeling and Design of Dynamic Systems

• Basic Concepts

Physical System

Schematic Model

Classical Methods, Block Diagrams OR Bond

Graphs

Simulation and Analysis Software

Output

Data Tables & Graphs

STEP 1: Develop an ‘engineering model’

STEP 2: Write differential equations

STEP 3: Determine a solution

STEP 4: Write a program

The Big Question??

Differential Equations

GME + Matlab/Simulink

Fig 1. Modeling Dynamic Systems [1]


Bond Graph Methodology

• Invented by Henry Paynter in 1961, later elaborated by his students Dean C. Karnopp and Ronald C. Rosenberg

• An abstract representation of a system where a collection of components interact with each other through energy ports and are placed in a system where energy is exchanged [2]

• A domain-independent graphical description of dynamic behavior of physical systems

• Consists of subsystems which can either describe idealized elementary processes or non-idealized processes [3]

• System models will be constructed using a uniform notations for all types of physical system based on energy flow

• Powerful tool for modeling engineering systems, especially when different physical domains are involved

• A form of object-oriented physical system modeling

Fig 2. Subsystems of a bond graph [3]


• Bond Graphs

• Conserves the physical structural information as well as the nature of sub-systems which are often lost in a block diagram.

• When the IPM is changed, only the corresponding parts of a bond graphs have to be changed. Amenable to modification for ‘model development’ and ‘what if?’ situations.

• Use analogous power and energy variables in all domains, but allow the special features of the separate fields to be represented.

• The only physical variables required to represent all energetic systems are power variables [effort (e) & flow (f)] and energy variables [momentum p (t) and displacement q (t)].

• Dynamics of physical systems are derived by the application of instant-by-instant energy conservation. Actual inputs are exposed.

• Linear and non-linear elements are represented with the same symbols; non-linear kinematics equations can also be shown.

• Provision for active bonds. Physical information involving information transfer, accompanied by negligible amounts of energy transfer are modeled as active bonds.

The Bond Graph Modeling Formalism


The Bond Graph Modeling Formalism (contd..)

• A Bond Graph’s Reach

Electrical

Mechanical Translation

Mechanical Rotation

Hydraulic/Pneumatic

Chemical/Process Engineering

Thermal

MagneticFigure 3. Multi-Energy Systems Modeling using Bond Graphs



• Introductory Examples

• Electrical Domain

Power Variables:

Electrical Voltage (u) & Electrical Current (i)

Power in the system: P = u * i

Constitutive Laws: uR = i * R

uC = 1/C * (∫i dt)

uL = L * (di/dt); or i = 1/L * (∫uL dt)

Fig 4. A series RLC circuit [4]

Fig. 5 Electric elements with power ports [4]

Represent different elements with visible ports (figure 5)

To these ports, connect power bonds denoting energy exchange

The voltage over the elements are different

The current through the elements is the same


The R – L - C circuit

The common current becomes a “1-junction” in the bond graphs.

Note: the current through all connected bonds is the same, the voltages sum to zero


Fig 6. The RLC Circuit and its equivalent Bond Graph [4]

1



Fig 7. The Spring Mass Damper System and its equivalent Bond Graph [4]

• Mechanical Domain

Mechanical elements like Force, Spring, Mass, Damper are similarly dealt with.

Power variables: Force (F) & Linear Velocity (v)

Power in the system: P = F * v

Constitutive laws: Fd = α * v Fs = KS * (∫v dt) = 1/CS * (∫ v dt)Fm = m * (dv/dt); or v = 1/m * (∫Fm dt); Also, Fa = force

The common velocity becomes a “1-junction” in the bond graphs. Note: the velocity of all connected bonds is the same, the forces sum to zero)


Analogies!

Lets compare! We see the following analogies between the mechanical and electrical elements:

• The Damper is analogous to the Resistor.• The Spring is analogous to the Capacitor, the mechanical compliance corresponds with the

electrical capacity.• The Mass is analogous to the Inductor.• The Force source is analogous to the Voltage source.• The common Velocity is analogous to the loop Current.

Notice that the bond graphs of both the RLC circuit and the Spring-mass-damper system are identical. Still wondering how??

• The bond graph modeling language is domain-independent.• Each of the various physical domains is characterized by a particular conserved quantity.

Table 1 illustrates these domains with corresponding flow (f), effort (e), generalized displacement (q), and generalized momentum (p).

• Note that power = effort x flow in each case.



f

flow

e

effort

q = ∫f dt

generalized displacement

p = ∫e dt

generalized momentum

Electromagnetic i

current

u

voltage

q = ∫i dt

charge

λ = ∫u dt

magnetic flux linkage

Mechanical Translation

v

velocity

f

force

x = ∫v dt

displacement

p = ∫f dt

momentum

Mechanical Rotation ω

angular velocity

T

torque

θ = ∫ω dt

angular displacement

b = ∫T dt

angular momentum

Hydraulic / Pneumatic

φ

volume flow

P

pressure

V = ∫φ dt

volume

τ = ∫P dt

momentum of a flow tube

Thermal T

temperature

FS

entropy flow

S = ∫fS dt

entropy

Chemical μ

chemical potential

FN

molar flow

N = ∫fN dt

number of moles

Table 1. Domains with corresponding flow, effort, generalized displacement and generalized momentum




• Foundations of Bond Graphs

Based on the assumptions that satisfy basic principles of physics; a. Law of Energy Conservation is applicable b. Positive Entropy production c. Power Continuity

• Closer look at Bonds and Ports

Power port or port: The contact point of a sub model where an ideal connection will be connected; location in a system where energy transfer occurs

Power bond or bond: The connection between two sub models; drawn by a single line (Fig. 8)

Bond denotes ideal energy flow between two sub models; the energy entering the bond on one side immediately leaves the bond at the other side (power continuity).

A Bef

(directed bond from A to B)

Fig. 8 Energy flow between two sub models represented by ports and bonds [4]

Energy flow along the bond has the physical dimension of power, being the product of two variables

effort and flow called power-conjugated variables

Power bond viewed as interaction of energy and bilateral signal flow



• Bond Graph Elements – 9 elements

Drawn as letter combinations (mnemonic codes) indicating the type of element.

C storage element for a q-type variable, e.g. capacitor (stores charge), spring (stores displacement)

L storage element for a p-type variable, e.g. inductor (stores flux linkage), mass (stores momentum)

R resistor dissipating free energy, e.g. electric resistor, mechanical friction

Se, Sf sources, e.g. electric mains (voltage source), gravity (force source), pump (flow source)

TF transformer, e.g. an electric transformer, toothed wheels, lever

GY gyrator, e.g. electromotor, centrifugal pump

0, 1 0 and 1 junctions, for ideal connection of two or more sub-models



• Storage Elements

Two types; C – elements & I – elements; q–type and p–type variables are conserved quantities and are the result of an accumulation (or integration) process; they are the state variables of the system.

C – element (capacitor, spring, etc.)

q is the conserved quantity, stored by accumulating the net flow, f to the storage element

Resulting balance equation dq/dt = f

Fig. 9 Examples of C - elements [4]

An element relates effort to the generalized displacement

1-port element that stores and gives up energy without loss



I – element (inductor, mass, etc.)

p is the conserved quantity, stored by accumulating the net effort, e to the storage element.

Resulting balance equation dp/dt = e

Fig. 10 Examples of I - elements [4]

For an inductor, L [H] is the inductance and for a mass, m [kg] is the mass. For all other domains, an I – element can be defined.



R – element (electric resistors, dampers, frictions, etc.)

R – elements dissipate free energy and energy flow towards the resistor is always positive.

Algebraic relation between effort and flow, lies principally in 1st or 3rd quadrant.e = r * (f)

Fig. 11 Examples of Resistors [4]

If the resistance value can be controlled by an external signal, the resistor is a modulated resistor, with mnemonic MR. E.g. hydraulic tap



Fig. 12 Examples of Sources [4]

Sources (voltage sources, current sources, external forces, ideal motors, etc.)

Sources represent the system-interaction with its environment. Depending on the type of the imposed variable, these elements are drawn as Se or Sf.

Fig. 13 Example of Modulated Voltage Source [4]

When a system part needs to be excited by a known signal form, the source can be modeled by a modulated source driven by some signal form (figure 13).



Transformers (toothed wheel, electric transformer, etc.)

An ideal transformer is represented by TF and is power continuous (i.e. no power is stored or dissipated). The transformations can be within the same domain (toothed wheel, lever) or between different domains (electromotor, winch).

e1 = n * e2 & f2 = n * f1

Efforts are transduced to efforts and flows to flows; n is the transformer ratio.

Fig. 14 Examples of Transformers [4]



Gyrators (electromotor, pump, turbine)

An ideal gyrator is represented by GY and is power continuous (i.e. no power is stored or dissipated). Real-life realizations of gyrators are mostly transducers representing a domain-transformation.

e1 = r * f2& e2 = r * f1

r is the gyrator ratio and is the only parameter required to describe both equations.

Fig. 15 Examples of Gyrators [4]



Junctions

Junctions couple two or more elements in a power continuous way; there is no storage or dissipation at a junction.

0 – junction

Represents a node at which all efforts of the connecting bonds are equal. E.g. a parallel connection in an electrical circuit.

The sum of flows of the connecting bonds is zero, considering the sign.

0 – junction can be interpreted as the generalized Kirchoff’s Current Law.

Equality of efforts (like electrical voltage) at a parallel connection.

Fig. 16 Example of a 0-Junction [4]



1 – junction

Is the dual form of the 0-junction (roles of effort and flow are exchanged).

Represents a node at which all flows of the connecting bonds are equal. E.g. a series connection in an electrical circuit.

The efforts of the connecting bonds sum to zero.

1- junction can be interpreted as the generalized Kirchoff’s Voltage Law.

In the mechanical domain, 1-junction represents a force-balance, and is a generalization of Newton’ third law.

Additionally, equality of flows (like electrical current) through a series connection.

Fig. 17 Example of a 1-Junction [4]



Some Miscellaneous Stuff!

Power Direction: The power is positive in the direction of the power bond. If power is negative, it flows in the opposite direction of the half-arrow.

Typical Power flow directions

R, C, and I elements have an incoming bond (half arrow towards the element)

Se, Sf: outgoing bond

TF– and GY–elements (transformers and gyrators): one bond incoming and one bond outgoing, to show the ‘natural’ flow of energy.

These are constraints on the model!



• Causal Analysis

Causal analysis is the determination of the signal direction of the bonds

Establishes the cause and effect relationships between the bonds

Indicated in the bond graph by a causal stroke; the causal stroke indicates the direction of the effort signal.

The result is a causal bond graph, which can be seen as a compact block diagram.

Causal analysis covered by modeling and simulation software packages that support bond graphs; Enport, MS1, CAMP-G, 20 SIM

Fig. 18 Causality Assignment [4]


Causality Type

Elements Representation Interpretation

Fixed

Se

Sf

Constrained TF

OR

Se ef

e

Sef

Sf ef

e

Sff

TF

e1 e2

f2f1n

e1 e2

f2f1TFn

e1 e2

f2f1TFn

TF

e1 e2

f2f1n

Causal Constraints: Four different types of constraints need to be discussed prior to following a systematic procedure for bond graph formation and causal analysis



Causality Type

Elements Representation

Constrained

GY OR

0 Junction

1 Junction

Preferred

C Integral Causality (Preferred) Derivative Causality

L Integral Causality (Preferred) Derivative Causality

e1 e2

f2f1GYr

e1 e2

f2f1GYr


0

1

OR any other combination where exactly one bond brings in the effort variable

OR any other combination where exactly one bond has the causal stroke away from the junction

C C

L L


Causality Type

Elements Representation

IndifferentR OR

R R


Some notes on Preferred Causality (C, I)

Causality determines whether an integration or differentiation w.r.t time is adopted in storage elements. Integration has a preference over differentiation because:

1. At integrating form, initial condition must be specified.

2. Integration w.r.t. time can be realized physically; Numerical differentiation is not physically realizable, since information at future time points is needed.

3. Another drawback of differentiation: When the input contains a step function, the output will then become infinite.

Therefore, integrating causality is the preferred causality. C-element will have effort-out causality and I-element will have flow-out causality


• Electrical Circuit # 1 (R-L-C) and its Bond Graph model

0 0 0

0 1 0 1 0

+-

U0

U1U2

U3

Examples

U1 U2 U3

0: U12 0: U23


0: U12

0 1 0 1 0Se : UC : C

0: U23

R : R I : L

U1 U3U2

Examples (contd..)

1Se : U

C : C

R : R

I : L


Examples (contd..)

The Causality Assignment Algorithm:

1Se : U

C : C

R : R

I : L

1. 2.

1Se : U

C : C

R : R

I : L

1Se : U

C : C

R : R

I : L

3.


Examples (contd..)

• Electrical Circuit # 2 and its Bond Graph modelR1

R2R3

C1

C2

L1

R1 R2 R3

C2C1L1

• A DC Motor and its Bond Graph model


Examples (contd..)

• A Drive Train Schematic and its Bond Graph model

Bond Graph without Drive Shaft Compliance [9]

Bond Graph with Drive Shaft Compliance [9]

SE

TF

1 TF 0

τL ωL

τR ωR

ωi

Differential Ratio

Transmission Ratio

A Drive Train Schematic [9]

SF

TF

1 TF 0

τL ωL

τR ωR

ωi

Drive Shaft Compliance

0

C


Examples (contd..)

• Schematic for Tire and Suspension and their Bond Graph model

Suspension model for one wheel and anti-roll bar

Bond Graph of a wheel-tire system – Longitudinal Dynamics [9]

Bond Graph of a wheel-tire system – Transverse Dynamics [9]

Schematic of a tire and suspension [9]

Bond Graph of a wheel-tire system – Vertical Dynamics [9]


Generation of Equations from Bond Graphs

1Se : U

C : C

R : R

I : L1

2

4

3

Fig. 19 Bond Graph of a series RLC circuit

• A causal bond graph contains all information to derive the set of state equations.

• Either a set of Ordinary first-order Differential Equations (ODE) or a set of Differential and Algebraic Equations (DAE).

• Write the set of mixed differential and algebraic equations.• For a bond graph with n bonds, 2n equations can be

formed, n equations each to compute effort and flow or their derivatives.

• Then, the algebraic equations are eliminated, to get final equations in state-variable form.

For the given RLC circuit, Se = e1= U;

e2 = R * f2;

(de3/dt) = (1/C) * f3;

(df4/dt) = (1/L) * e4;

f1 = f4; f2 = f4; f3 = f4;

e4 = e1 - e2 - e3

Hence, e1 - e2 - e3 = U – (R * f2) – e3 = U – (R * f4) – e3

(df4/dt) = (1/L) * (U – (R * f4) – e3) - - - - - - - (i)


Also, (de3/dt) = (1/C) * f3 = (1/C) * f4 - - - - - - - - (ii)

In matrix form, (dx/dt) = Ax + Bu

(de3/dt) 0 1/C e3 0

= + U

(df4/dt) -1/L -R/L f4 1/L

Generation of Equations from Bond Graphs (contd..)


The Bond Graph Metamodeling Environment in GME


Applications in GME Metamodeling Environment

• RLC Circuit


• DC Motor

Applications in GME Metamodeling Environment (contd..)


Applications in GME Metamodeling Environment (contd..)

DC Motor model


Future Concepts

• Defining the GME Approach for analysis of Bond Graphs [1]

Conventional Approach Probable GME / Matlab Approach

1. Determination of Physical System and specifications from the requirements.

2. Draw a functional Block Diagram.

3. Transform the physical system into a schematic.

4. Use Schematic and obtain a mathematical model, a block diagram or a state representation.

5. Reduce the block diagram to a close loop system.

6. Analyze, design and test.

1. Identify the physical system elements and represent a word Bond Graph.

2. Represent a bond graph model in GME.

3. GME interpreters generate equations in a suitable form (e.g. state-space variable matrix form) suitable for analysis in Matlab.

4. Use Matlab, to analyze, design and test.


Future Concepts (contd..)

Fig 20. The Simulation Generation Process [7]

Bond Graph Interpreters in GME ??

• Creating Bond Graph Interpreters


• Advanced Bond Graph Techniques

Expansion of Bond Graphs to Block Diagrams

Bond Graph Modeling of Switching Devices

Hierarchical modeling using Bond Graphs

Use of port-based approach for Co-simulation

Future Concepts (contd..)


References

1. Granda J. J, “Computer Aided Design of Dynamic Systems” http://gaia.csus.edu/~grandajj/

2. Wong Y. K., Rad A. B., “Bond Graph Simulations of Electrical Systems,” The Hong Kong

Polytechnic University, 1998

3. http://www.ce.utwente.nl/bnk/bondgraphs/bond.htm

4. Broenink J. F., "Introduction to Physical Systems Modeling with Bond Graphs,"

University of Twente, Dept. EE, Netherlands.

5. Granda J. J., Reus J., "New developments in Bond Graph Modeling Software Tools: The

Computer Aided Modeling Program CAMP-G and MATLAB," California State

University, Sacramento

6. http://www.bondgraphs.com/about2.html

7. Vashishtha D., “Modeling And Simulation of Large Scale Real Time Embedded Systems,” M.S.

Thesis, Vanderbilt University, May 2004

8. Hogan N. "Bond Graph notation for Physical System models," Integrated

Modeling of Physical System Dynamics

9. Karnopp D., “System Dynamics: Modeling and simulation of mechatronic systems”

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INTELLIGENT POWERTRAIN DESIGN