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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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