Good Afternoon Welcome to elearning session on CONTROL ENGINEERING ME 55

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Good Afternoon Welcome to e-learning session
onCONTROL ENGINEERING (ME 55)
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ByDr. B.K. Sridhara HeadDepartment of
Mechanical EngineeringThe National Institute of
EngineeringMysore 570 008
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Session 3 30.08.2006 CHAPTER
II MATHEMATICAL MODELING
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Recap (Session I II)

• Control Systems
• Types of Control Systems
• Open Loop and Closed Loop Control System
• Terminologies
• Block Diagram Representation of Control System

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Examples of Control Systems

• WATTS Speed Governor
• PUMA 560 Series robot arm
• Missile Launching and Guidance System
• Automatic Aircraft Landing System

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Figure 2.1 WATTS SPEED GOVERNOR
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Figure 2.2 PUMA 560 SERIES ROBOT ARM
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Figure 2.3 MISSILE LAUNCHING AND GUIDANCE SYSTEM
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Figure 2.4 AUTOMATIC AIRCRAFT LANDING SYSTEM
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Requirements for a Control System

• Examples discussed are over-simplified
• Illustrate universality of control principles
• Requirements are many and depend on the system
  under consideration

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Major Requirements

• 1. Stability
• Stable Systems
• Response to input must reach and maintain some
  useful value with in a reasonable period of time
• Unstable Systems
• Unstable control systems produce persistent or
  even violent oscillations of the output
• Output will be driven to some extreme limiting
  value

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• 2. Accuracy
• Accurate within the specified limits
• Capable of reducing any error to some tolerable
  limits
• Zero error is not really accomplishable due to
  inherent imperfections in the system components
• Many control systems do not require extreme
  accuracy
• Accuracy is relative matter with limits based
  upon a the particular application

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Cost of the control system increases rapidly if
  increased accuracy is demanded

3. Speed of Response

• A CS must complete its response to some input
  within an acceptable period of time
• A CS as no value if the time required to respond
  fully to some input is far greater than the time
  interval between inputs

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Steps involved in Design of Control Systems

• Selection of proper controllers or actuators
• Selection of proper feedback elements (sensors)
• Developing Physical Mathematical Models for the
  system/plant, actuators and sensors
• Design based on the models developed in step 3
  and other specifications
• Simulating system performance and fine tuning
• Prototype Building or testing

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Modeling of Control Systems

• A CS is a physical system which is Dynamic in
  Nature
• Analysis of dynamic characteristics is an
  essential part in its design
• Requires Modeling of the system
• Modeling involves two steps
• Physical Model Mathematical Model

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Physical Model

• First step to develop a mathematical model is to
  represent the CS as a physical model
• A CS can be made up of Mechanical, Electrical,
  Hydraulic, Thermal Pneumatic Systems
• No Physical system can be represented in its full
  intricacies. Idealizing assumptions are always
  made for the purpose of analysis and synthesis
• An idealized physical system is called a physical
  model

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Model

• Mathematical representation of the physical model
• MM represents the dynamics of the system very
  accurately or at least fairly well
• MM are obtained by using the physical laws
  governing a particular system
• Eg Newtons Laws Mechanical systems
• Kirchoffs Laws Electrical systems

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• A given physical model may lead to different
  mathematical models depending upon the choice of
  the variable and coordinate systems
• A MM is not unique to a given system
• A particular MM which gives greater insight in to
  the dynamic behaviour of the system is selected
• Mathematical models of most CS are differential
  equations
• When MM are solved for various input conditions
  dynamic response (behaviour) of the system is
  obtained

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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We can answer these questions

• Whether the system is stable?
• Whether the system is accurate?
• What is the speed of response?

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Models Caution

• More accurate MM, more complex they are
• Compromise between simplicity of the model and
  the accuracy
• As a first approximation build a simplified model
  to get a general feeling for the solution
• Improved model may then be built for a complete
  analysis

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Modeling of Mechanical Systems

• Practical systems Complex Difficult to consider
  all details for mathematical analysis
• Only most important features are considered to
  predict the behaviour of the system under
  specified conditions
• Often, overall behaviour of a complex system can
  be determined by considering a very simple model

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• The model should include enough details of the
  system
• What are these?
• Physical properties? Mechanical properties?
  Chemical composition? Geometry of parts?
• They are Elasticity, Mass / Inertia and Damping
• Why only these three?

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Dynamic behaviour involves alternate
  transformation of potential energy into kinetic
  energy and vice versa
• If damping is present some energy is dissipated
  in each cycle as seen in the following examples

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Simple Pendulum
Spring mass System
l
Umax,T 0, V0
Umax,T 0, V0
h
x
Datum
Tmax, Vmax, U0
mg
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Physical models are built by means of the
  following idealized elements
• Mass or Inertia elements
• Elastic or spring elements
• Damping elements

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Mass and elastic elements are energy storage
  elements
• Damping elements are energy absorbing
  (dissipative) elements
• Physical Model Spring-Mass-Damper System

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Elementary parts of a Mechanical system

• A means for storing potential energy (spring or
  elasticity)
• A means for storing kinetic energy (mass or
  inertia)
• A means by which energy is gradually dissipated
  (Damper)

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Examples
Tup
Anvil
Elastic pad
Foundation block
Soil
Figure 2.5 FORGING HAMMER
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Tup
Anvil and foundation block (m)
x1
Soil damping (c)
Soil stiffness (k)
Figure 2.6 PHYSICAL MODEL 1
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Tup
Anvil (m1)
x1
Damping of elastic pad (c1)
Stiffness of elastic pad (k1)
Foundation block (m2)
x2
Stiffness of soil (k2)
Damping of soil (c2)
Figure 2.7 PHYSICAL MODEL 2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Examples (Contd.)
Figure 2.8 MOTOR CYCLE WITH RIDER
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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m
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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mr
mv mr
Legend t Tire v Vehicle
w Wheel r Rider s Strut
eq Equivalent
kr
Cr
2ks
2Cs
mv
ks
Cs
Cs
2mw
mw
mw
2kt
kt
kt
MODEL 3
MODEL 4
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Models Governing Differential
Equations

• Equations - that describe the dynamic behaviour
  of the system
• How to obtain them?
• Use principles of dynamics such as Newtons
  second law of motion, DAlemberts principle,
  Principle of conservation of energy, etc
• Linear or non linear depending upon the behaviour
  of the components of the system

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Procedure

• Consider the physical model
• Draw the Free Body Diagrams (FBD)
• FBD-isolate the mass indicating all the external,
  reactive and inertia forces
• Apply Newtons Second Law of Motion to get the
  mathematical model
• ? F ma for translatary motion
• ? T I a for rotary motion

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Spring Mass Systems (Damping Assumed Zero)
Translational systems Illustration 1
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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..

• mx kx 0
• Differential Equation for the system which, when
  solved gives an expression for the displacement
  of mass as a function of time

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Translational systems Illustration 2
x
k
kx
m
m
Friction less surface
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Summary

• Examples of Control Systems
• Requirements for a Control Systems
• Steps involved in Design of Control Systems
• Modeling of Control Systems
• Modeling of Mechanical Systems
• Spring Mass System (GDE)

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THANK YOU
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore