MAE 3401 – Modeling and Simulation

SECTION 2: BOND GRAPH FUNDAMENTALS

K. Webb MAE 3401

Bond Graphs ‐ Introduction

As engineers, we’re interested in different types of systems: Mechanical translational Mechanical rotational Electrical Hydraulic

Many systems consist of subsystems in different domains, e.g. an electrical motor

Common aspect to all systems is the flow of energy and powerbetween components

Bond graph system models exploit this commonality Based on the flow of energy and power Universal – domain‐independent Technique used for deriving differential equations from a bond graph

model is the same for any type of system

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Power and Energy Variables3

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Bonds and Power Variables

Systems are made up of components Power can flow between components We represent this pathway for power to flow with bonds

A and B represent components, the line connecting them is a bond Quantity on the bond is power

Power flow is positive in the direction indicated – arbitrary Each bond has two power variables associated with it

Effort and flow

The product of the power variables is power

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Power Variables

Power variables, e and f, determine the power flowing on a bond The rate at which energy flows between components

DomainEffort Flow

PowerQuantity Variable Units Quantity Variable Units

General Effort – Flow – ∙

Mechanical Translational Force N Velocity m/s ∙

Mechanical Rotational Torque N‐m Angular

velocity rad/s ∙

Electrical Voltage V Current A ∙

Hydraulic Pressure Pa (N/m2) Flow rate m3/s ∙

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Energy Variables

Bond graph models are energy‐based models Energy in a system can be:

Supplied by external sources Stored by system components Dissipated by system components Transformed or converted by system components

In addition to power variables, we need two more variables to describe energy storage: energy variables

Momentum Displacement

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Momentum

Momentum – the integral of effort

≡so

For mechanical systems:

which, for constant mass, becomes

Newton’s second law

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Displacement

Displacement – the integral of flow

≡so

For mechanical systems:

The definition of velocity

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Energy Variables

Displacement and momentum are familiar concepts for mechanical systems All types of systems have analogous energy variables We’ll see that these quantities are useful for describing energy storage

DomainMomentum Displacement

Quantity Variable Units Quantity Variable Units

General Momentum – Displacement –

Mechanical Translational Momentum N‐s Displacement m

Mechanical Rotational

Angular momentum N‐m‐s Angle rad

Electrical Magnetic flux V‐s Charge C

Hydraulic Hydraulic momentum Γ N‐s/m2 Volume m3

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Energy – Kinetic Energy

Energy is the integral of power

We can relate effort to momentum

So, if it is possible to express flow as a function of momentum, , we can express energy as a function of momentum,

This is kinetic energy Energy expressed as a function of momentum

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Energy – Potential Energy

Energy is the integral of power

We can relate flow to displacement

So, if it is possible to express effort as a function of displacement, , we can express energy as a function of displacement,

This is potential energy Energy expressed as a function of displacement

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Energy – Mechanical Translational

For a mechanical translational system

and

, 1

so 1

1

12

12

12 . .

We can also express force and energy as a function of displacement

,

so

12 . .

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Energy – Electrical

For an electrical system

and

, 1

so 1 1

12

12

12 .

We can also express voltage and energy as a function of charge

, 1

so1 1

12

12

12 .

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Energy – Summary

For some system components, flow can be expressed as a function of momentum These components store energy as a function of momentum This is kinetic energy or magnetic energy

For other components, effort can be expressed as a function of displacement These components store energy as a function of displacement This is potential energy or electrical energy

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One‐Port Bond Graph Elements15

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System Components

System components are defined by how they affect energy flow within the system – they can:

1. Supply energy2. Store energy

a) As a function of – kinetic or magnetic energyb) As a function of – potential or electrical energy

3. Dissipate energy4. Transform or convert energy

Different bond graph elements for components in each of these categories Categorized by the number of ports ‐ bond attachment points

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Active One‐Port Elements

External sources that supply energy to the system Effort Source

Supplies a specific effort to the system E.g., force source, voltage source, pressure source

Flow Source Supplies a specific flow to the system E.g., velocity source, current source, flow source

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Passive One‐Port Elements

One‐port elements categorized by whether they dissipate energy or store kinetic or potential energy

Three different one‐port elements: Inertia Capacitor Resistor

Same three elements used to model system components in all different energy domains

Each defined by a constitutive law A defining relation between two physical quantities – two of the four energy and power variables

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Inertia

Inertia – a component whose constitutive law relates flow to momentum

1

where is the relevant inertia of the component Inertias store energy as a function of momentum A kinetic energy storage element

Domain Inertia Parameter Symbol Units

General Inertia ‐

Translational Mass kg

Rotational Moment of inertia Kg‐m2

Electrical Inductance H

Hydraulic Hydraulic inertia Kg/m4

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Inertia

Bond graph symbol for an inertia:

Physical components modeled as inertias:

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Inertia – Energy Storage

Constitutive law:1

Stored energy:. .

. .1

2

Mechanical:

. . 212

Electrical:

. . 212

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Inertia – Constitutive Law

Constitutive law for an inertia can be expressed in linear, integral, or derivative form:

or

Domain Linear Integral Derivative

General1 1

Translational1 1

Rotational1 1

Electrical1 1

Hydraulic1Γ

1

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Capacitors

Capacitor – a component whose constitutive law relates effort to displacement

1

where is the relevant capacitance of the component Capacitors store energy as a function of displacement A potential‐energy‐storage element

Domain Capacitance Parameter Symbol Units

General Capacitance ‐

Translational Compliance 1/ m/N

Rotational Rotational compliance 1/ rad/N‐m

Electrical Capacitance F

Hydraulic Hydraulic capacitance m5/N

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Capacitor

Bond graph symbol for a capacitor:

Physical components modeled as capacitors:

Note that spring constants are the inverse of capacitance or compliance

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Capacitor – Constitutive Law

Constitutive law:1

Stored energy:. .

. .1

2

Mechanical:

. . 2/12

Electrical:

. .∙2

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Capacitor – Constitutive Law

Constitutive law for a capacitor can be expressed in linear, integral, or derivative form:

or

Domain Linear Integral Derivative

General1 1

Translational 1

Rotational 1

Electrical1 1

Hydraulic1V

1

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Resistors

Resistor – a component whose constitutive law relates flow to effort

or ∙

where is the relevant resistance of the component Resistors dissipate energy A loss mechanism

Domain Resistance Parameter Symbol Units

General Resistance ‐

Translational Damping coefficient N‐s/m

Rotational Rotational damping coeff. N‐m‐s/rad

Electrical Resistance Ω

Hydraulic Hydraulic resistance N‐s/m5

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Resistor

Bond graph symbol for a resistor:

Physical components modeled as resistors:

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Resistor – Power Dissipation

Constitutive law:1

or ∙

Power dissipation:

∙

Mechanical:

Electrical:

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Resistor – Constitutive Law

Constitutive law for a resistor can express flow in terms of effort, or vice‐versa:

or ∙

Domain Flow Effort

General1 ∙

Translational1 ∙

Rotational1

∙

Electrical1 ∙

Hydraulic1 ∙

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Viscous vs. Coulomb Friction

We’ve assumed a specific type of mechanical resistance – viscous friction A linear resistance Realistic? – Sometimes

Can we model coulomb friction as a resistor?

Yes, if the constitutive law relates effort ( ) and flow ( )

It does – velocity determines direction of the friction force

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N‐Port Bond Graph Elements32

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Multi‐Port Elements ‐ Junctions

So far, we have sources and other one‐port elements These allow us to model things like this:

Want to be able to model multiple interconnected components in a system Need components with more than one port Junctions: 0‐junction and 1‐junction

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0‐Junctions

0‐junction – a constant effort junction All bonds connected to a 0‐junction have equal effort Power is conserved at a 0‐junction

Constant effort:

Power is conserved:

so

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0‐Junctions

Constant‐effort 0‐junction translates to different physical configurations in different domains

Mechanical translational Constant force – components connected in series

Electrical Constant voltage – components connected in parallel

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1‐Junctions

1‐junction – a constant flow junction All bonds connected to a 1‐junction have equal flow Power is conserved at a 1‐junction

Constant flow:

Power is conserved:

so

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1‐Junctions

Constant‐flow 1‐junction translates to different physical configurations in different domains

Mechanical translational Constant velocity – components connected in parallel

Electrical Constant current – components connected in series

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Cascaded 0‐Junctions

Equal efforts, flows sum to zero

(1)

0 (2)

(3)

Substitute (2) into (1)

(4)

Substitute (3) into (4)

Can collapse the cascade to a single 0‐junction

Internal bond directions are irrelevant

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Cascaded 1‐Junctions

Equal flow, efforts sum to zero

(1)

(2)

0 (3)

Substitute (2) into (1)

(4)

Substitute (3) into (4)

Can collapse the cascade to a single 1‐junction

Internal bond directions are irrelevant

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Two‐Port Bond Graph Elements40

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Two‐Port Bond Graph Elements

Two‐port elements: Transformer Gyrator

Transmit power Two ports – two bond connection points Power is conserved:

Ideal, i.e. lossless, elements

May provide an interface between energy domains E.g. transmission of power between mechanical and electrical subsystems

Bonds always follow a through convention One in, one out

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Transformer

Transformers – relate effort at one port to effort at the other and flow at one port to flow at the other Efforts and flows related through the transformer modulus, m

Constitutive law:

and

Power is conserved, so1

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Relation of efforts, 1 and 2 Balance the moments:

Relation of flows, and Equal angular velocity all along lever arm:

Transformer – Mechanical

Bond graph model:

Include effort‐to‐effort or flow‐to‐flow relationship

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Relation of efforts, 1 and 2 Voltage scales with turns ratio:

Relation of flows, and Current scales with the inverseof the turns ratio:

Power is conserved

Transformer – Electrical

Bond graph model:

Include effort‐to‐effort or flow‐to‐flow relationship

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Gyrator

Gyrators – effort at one port related to flow at the other Efforts and flows related through the gyrator modulus, r Constitutive law:

and

Power is conserved so1

Gyrator modulus relates effort and flow – a resistance Really, a transresistance

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Gyrator ‐ Example

Ideal electric motor Electrical current, a flow, converted to torque, an effort

Current and torque related through the motor constant,

Power is conserved relationship between voltage and angular velocity is the inverse