Section 2 Bond Graph Fundamentalsweb.engr.oregonstate.edu/~webbky/MAE3401_files/Section 2 Bond...
Transcript of Section 2 Bond Graph Fundamentalsweb.engr.oregonstate.edu/~webbky/MAE3401_files/Section 2 Bond...
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