kp 3224 chapt 01
TRANSCRIPT
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Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter 1
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INTRODUCTION TO SYSTEM DYNAMICS
In this section we establish some basic terminology and discuss
the meaning of the topic "system dynamics," its methodology,and its applications.
SYSTEM: The original meaning of the term is a combination of
elements intended to act together to accomplish an objective. For
example, a link in a bicycle chain is usually not considered to bea system. The system designer must focus on how all the
elements act together to achieve the system's intended purpose
INPUT/OUTPUT: In the system dynamics meaning of the
terms, an input is a cause; an output is an effect due to the input.
The behavior of a system element is specified by its input-
output relation, which is a description of how the output is
affected by the input. The input-output relation expresses the
cause-and-effect behavior of the element.
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Figure 1.1.1
Figure 1.1.1, can be in the form of a table of numbers, a graph, or a
mathematical relation. The input-output or causal relation is, fromNewton's second law, a = f / m. The input is f and the output is a.
input-output relations for the elements in the system provide a means
of specifying the connections between the elements.
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Static and Dynamics Elements
When the present value of an element's output depends
only on the present value of its e ay the element is astatic
element. For example, the current flowing through or
depends only on the present value of the applied voltage.
The resistor is thus element.
If an element's present output depends on past inputs,
we say it is a dynamic element. For example, the
present position of a bike depends on what its velocity
has been from the start.
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Input= is provided to a system that delivers an output.
System= component that act together developing
output from the input.
Component= individual units within the system.
System= static ifoutput is dependent only on
instantaneous input.
System= dynamic when the output is the function of
the history of the input.
System response= output from the system for a given
system input.
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MODELING OF SYSTEMS
Table 1.1.1 contains a summary of the methodology.
Simplifying the problem
sufficiently and applying the
appropriate fundamental principles
is called modeling, and theresulting mathematical description
is called a mathematical model, or
just a model. When the modeling
has been finished, we need to solve
the mathematical model to obtain
the required answer.
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Table 1.1.1
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DYNAMIC SYSTEM AND THEIR RESPONSE
1) Given the input and the system components,
determine the system response.
2) Given the input and desired output, determine a set
of system componentsthat can be used to achieve the
desired output.
Output of dynamic system= dependent on history of
input, may be time dependent. Output varies with time
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System response
Response of a system depends on many factor
including system components and how the system ismodeled.
Order of a system is a key factor in understanding the
dynamics such a system. Transfer function : relates order to a mathematical
property of a system
First order system : modeled by first order differentialequation
Second order system : is modeled by second order
differential equation
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Higher order system : modeled by a set of differential
equations; a system modeled by three second-order
differential equations is a six-order system.
Free response of system : response due to nonzero
initial conditions and occurs is the absence of any
other system input.
Force response : system is subject to a nonzero inputfor t> 0
General system (linear system) : sum of forced
response and the free response. Transient response : free response or to the system
response shortly after input is changed.
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Steady-state response : systems response after a long
period of time.
If the input is periodic, the steady-state response is
periodic.
Steadystate response of a linear system when it exist
is independent of initial conditions.
Equilibrium : the balance achieved between
competing forces.
The term is often used to describe the state of a
system when system variables do not change withtime. A mechanical system is in static equilibrium
when the resultant of external forces is zero.
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Mathematical modeling = process through which the
dynamic system is obtained.
Leads to development ofmathematical equations that
describe the behavior of the system.
Behavior of a dynamic systemusuallygoverned by a
different equations, an integral equation, an
integrodifferential equation or a set of differentialequations in which time is the independentvariable.
Dependent variables represent the system outputs.
Fi 1 1 2
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Figure 1.1.2
Figure 1.1.2 shows a robot arm,whose motion must be properly
controlled to move an object to a
desired position and orientation. To
do this, each of the several motorsand drive trains in the arm must be
adequately designed to handle the
load, and the motor speeds and
angular positions must be properly
controlled.
Fi 1 1 3
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Figure 1.1.3
Figure 1.1.3 shows a typical motor
and drive train for one arm joint.
Knowledge of system dynamics is
essential to design thesesubsystems and to control them
properly.
Fig re 1 1 4
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Figure 1.1.4
Figure 1.1.4 shows the mechanical
drive for a conveyor system. The
motor, the gears in the speed
reducer, the chain, the sprockets,and the drive wheels all must be
properly selected, and the motor
must be properly controlled for the
system to work well.
Figure 1 1 5
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Figure 1.1.5
(Figure 1.1.5).Active suspension
systems, whose characteristics can
be changed under computer
con-trol, and vehicle-dynamics
control systems are undergoing
rapid development, and their
design requires an understanding
of system dynamics.
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Dimension and Units Dimension = representation of how a physical
variable is expressed quantitatively 7 basic dimension.
= mass, length, time, temperature, electric current,
luminous intensity, amount of substance in moles.
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Table 1.2.1:UNITS
Table 1 2 2
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Table 1.2.2
D l i li d l
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Developing linear modelA linear model of a static system element has the form y = mx + b, where x
is the input and y is the outputof the element.
Example: The deflection of a cantilever beam is the distance its end moves in
response to a force applied at the end (Figure 1.3.1). The following table gives the
measured deflectionx that was produced in a particular beam by the given applied
force f. Plot the data to see whether a linear relation exists between f and x.
Figure 1 3 2
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Figure 1.3.2
Solution
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Solution
Common sense tells us that there must be zero beam
deflection if there is no applied force, so the curve
describing the data must pass through the origin. The
straight line shown was drawn by aligning a straightedge so
that it passes through the origin and near most of the data
points. The data lies close to a straight line, so we can use
the linear functionx = af to describe the relation. The value
of the constant a can be determined from the slope of theline, which is
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SYSTEM CLASSIFICATION
LINEAR SYSTEM = mathematical model involve
only linear differential equation
NONLINEAR SYSTEM = mathematical model
containnonlinear differential equation
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Linearization of Differential Equations
The modeling and response of linear systems, or
system governed by linear differential equations. Assumption are often made to render systems linear.
When appropriate, the equations may be linearized
using mathematical methods to approximate thenonlinear equation by a linear equation.
eg.1.4
CO O S S S
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CONTROL SYSTEMS
Dynamic system are design their response ispredictable when subject to defined input.
Eg. Output variables for a heating and air conditioning
system is the temperature of the room to service.INPUT = temperature of room to be heated / cooled
(thermostat setting)
> when the input change, system responddynamically to change the room temperature
Th i diti i t t b d i ith
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The air conditioning system must be design with a
sensor to determine the room temperature and
compare it to the thermostat temperature.
When the two temperatures are different, the system
must respond so that the output temperature
dynamically approaches the input temperature.
A system that senses its output and responds to adifference between input and output is called a
feedback control system. The feedback is design to
provide stable response to unpredictable changes insystem input.
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Mathematical Modeling of Dynamic System
Mathematical modeling can be used to achieve one of
the three objectives.. 1) system analysis is used to determine the outptut for
a specific system.
2) system design is used to determine the systemcomponents and their parameters such that a specific
system output is achieved.
3) system synthesis is the determination of system
components and their parameters to achieve a specific
performance for a variety of system inputs.
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Procedure for the Mathematical Modeling
STEP 1: define the system to be modeled. Identify the
input to the system and what will constitute output. STEP 2: the assumption under which the modeling
occurs must be identified and stated.
Implicit assumption & explicit assumption STEP 3: system components are identified and their
behavior quantified.
STEP 4: variables and parameters are verified. STEP 5: applicable physicallaws are applied.
STEP 6: initial condition
STEP 7 th ti l l i i f d t
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STEP 7: mathematical analysis is performed to
determine the time independent solution for the
dependent variables.
STEP 8: the systemoutput is determined from the
mathematical solution obtain in STEP 7
STEP 9: the model is validated
Figure 1.3.3
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Figure 1.3.4
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Figure 1.3.5
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Figure 1.3.6
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Figure 1.4.1
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Figure 1.4.2
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Figure 1.4.3
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Figure 1.4.4
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Figure 1.4.5
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Figure 1.4.6
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Figure 1.4.7
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Figure 1.4.8
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Figure 1.4.9
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Figure 1.4.10
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Figure 1.5.1
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Figure 1.5.2
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Figure 1.6.1
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Figure 1.6.2
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Figure 1.6.3
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