WORK-TO-HEAT TRANSDUCTION IN THERMO-FLUID SYSTEMS ENERGY-BASED MODELING IS BUILT ON THERMODYNAMICS - the fundamental science of physical processes. THERMODYNAMICS IS TO PHYSICAL SYSTEM DYNAMICS WHAT GEOMETRY IS TO MECHANICS. WHY SHOULD WE CARE ABOUT THERM
Convection bonds and "pseudo" bonds
Even in the simplest case of matter transport, power has two components, one due to the rate of work done, the other due to transported internal energy of the material. "Pseudo" bond graphs depict two distinct bonds. On
NODICITY
One of the important ways that physical system behavior differs between domains is the way elements may be connected. Electric circuit elements may be connected in series or in parallel - networks of arbitrary structure may be assembled This no
ENERGY-STORING COUPLING BETWEEN DOMAINS MULTI-PORT ENERGY STORAGE ELEMENTS Context: examine limitations of some basic model elements. EXAMPLE: open fluid container with deformable walls P=gh h=AV V = Cf P where Cf = A g
-fluid capacitor But when squeezed,
NONLINEAR MECHANICAL SYSTEMS (MECHANISMS) The analogy between dynamic behavior in different energy domains can be useful. Closer inspection reveals that the analogy is not complete. One key distinction of mechanical systems is the role of kinematics - the
EXAMPLE: THERMAL DAMPING
work in air sealed outlet
A BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. -mechanical work is done. The mechanical work done on the air is converted to heat. -the air tempera
INERTIAL MECHANICS Neville Hogan The inertial behavior of a mechanism is substantially more complicated than that of a translating rigid body. Strictly speaking, the dynamics are simple; the underlying mechanical physics is still described by Newton's law
LAGRANGE'S EQUATIONS (CONTINUED)
Mechanism in "uncoupled" inertial coordinates: (innermost box in the figure)
F = dp dt ; p = Mv
Mechanism in generalized coordinates: (middle box in the figure) = d/dt Ek*/; = I(); Ek*(,) = tI()
d L L * = with L(,) = Ek (,
Kinematic transformation of mechanical behavior
Neville Hogan
Generalized coordinates are fundamental
If we assume that a linkage may accurately be described as a collection of linked rigid bodies, their generalized coordinates are a fundamental requireme
MULTI-DOMAIN MODELING WHAT'S THE ISSUE? Why not just "write down the equations"? - standard formulations in different domains are often incompatible usually due to incompatible boundary conditions (choice of "inputs") EXAMPLE: SIMPLE FLUID SYSTEM Scenario
CAUSAL ANALYSIS
Things should be made as simple as possible - but no simpler. Albert Einstein How simple is "as simple as possible"? Causal assignment provides considerable insight.
EXAMPLE: AQUARIUM AIR PUMP
oscillatory motion in this direction coil leve
NETWORK MODELS OF TRANSMISSION LINES AND WAVE BEHAVIOR MOTIVATION: Ideal junction elements are power-continuous. Power out = power out instantaneously In reality, power transmission takes finite time. Power out power in Consider a lossless, continuous uni
NONLINEAR MECHANICAL SYSTEMS CANONICAL TRANSFORMATION S AND NUMERICAL INTEGRATION Jacobi Canonical Transformations A Jacobi canonical transformations yields a Hamiltonian that depends on only one of the conjugate variable sets. Assume dependence on new mo
EXAMPLE: THERMAL DAMPING
work in air sealed outlet
A BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. -mechanical work is done. The mechanical work done on the air is converted to heat. -the air tempera
Junction elements in network models. Classify by number of ports and examine the possible structures that result. Using only one-port elements, no more than two elements can be assembled.
Combining two two-ports yields another two-port.
At most two one-po
Massachusetts Institute of Technology Department of Mechanical Engineering
2.141 Modeling and Simulation of Dynamic Systems
INTRODUCTION
GOAL OF THE SUBJECT Methods for mathematical modeling of engineering systems Computational approaches are ubiquitous i
LINEARIZED ENERGY-STORING TRANSDUCER MODELS Energy transduction in an electro-mechanical solenoid may be modeled by an energy-storing multiport.
e= i
.
.
IC
F . x
Energy transduction in an electric motor may be modeled by a gyrator.
e= i
GY
F . x
But the
CO-ENERGY (AGAIN) In the linear case, energy and co-energy are numerically equal. -the value of distinguishing between them may not be obvious. Why bother with co-energy at all? EXAMPLE: SOLENOID WITH MAGNETIC SATURATION. Previous solenoid constitutive eq
EXAMPLE: ELECTROMAGNETIC SOLENOID A common electromechanical actuator for linear (translational) motion is a solenoid.
Current in the coil sets up a magnetic field that tends to center the movable armature.
Electromagnetic Solenoid
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Neville Hogan
O
Contact instability
Problem:
Contact and interaction with objects couples their dynamics into the manipulator control system This change may cause instability Example:
integral-action motion controller coupling to more mass evokes instability
Impedanc
Interaction Control
Manipulation requires interaction
object behavior affects control of force and motion
Independent control of force and motion is not possible
object behavior relates force and motion contact a rigid surface: kinematic constraint mov
EXAMPLE: IDEAL GAS MANY LOW-DENSITY GASES AT MODERATE PRESSURES MAY BE ADEQUATELY MODELED AS IDEAL GASES. Are the ideal gas model equations compatible with models of dynamics in other domains? AN IDEAL GAS IS OFTEN CHARACTERIZED BY THE RELATION PV = mRT P
REVIEW NETWORK MODELING OF PHYSICAL SYSTEMS EXAMPLE: VIBRATION IN A CABLE HOIST Bond graphs of the cable hoist models help to develop insight about how the electrical R-C filter affects the mechanical system dynamics. Equivalent mechanical system: velocit
REVIEW NETWORK MODELING OF PHYSICAL SYSTEMS
a.k.a. "lumped-parameter" modeling
EXAMPLE: VIBRATION IN A CABLE HOIST Problem The cage of an elevator is hoisted by a long cable wound over a drum driven through a gear-set by an electric motor. The motor is re
Stirling Engine
Marten Byl 12/12/02
1
x
Te R Th Tc
=0
Figure 1: Schematic of Stirling Engine with key variables noted.
Introduction
In the undergraduate class 2.670 at M.I.T., the students explore basic manufacturing tech niques by building a stirling eng
T h e Basic Bond Graph Primitives
Fundamental quantities and relations
P : p o w e r e: effort p: m o m e n t u m e
=
f: flow dpldt dq/dt Denotes
E : energy
q: displacement f Bond Graph Symbol
=
Electrical Network Icon
Typical M echanical Icon
power p
BLOCK DIAGRAMS, BOND GRAPHS AND CAUSALITY The main purpose of modeling is to develop insight. "Drawing a picture" of a model promotes insight. Why not stick with the familiar block diagrams? Block diagrams provide a picture of equations; -they portray ope
AMPLIFIERS A circuit containing only capacitors, amplifiers (transistors) and resistors may resonate. A circuit containing only capacitors and resistors may not. Why does amplification permit resonance in a circuit with only one kind of storage element?
A
NETWORK MODELS OF BERNOULLI'S EQUATION
The phenomenon described by Bernoulli's equation arises from momentum transport due to mass flow. EXAMPLE: A PIPE OF VARYING CROSS-SECTION.
section 1 Q1 section 2 A2 Q2 v2 P2
A 1
v1
P1
Assume: incompressible flow
Ideal asymmetric junction elements Relax the symmetry assumption and examine the resulting junction structure. For simplicity, consider two-port junction elements. As before, assume instantaneous power transmission between the ports without storage or dis