Welcome to elearning session on CONTROL ENGINEERING ME 55

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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 4 04.09.2006 CHAPTER
II MATHEMATICAL MODELING (Continued)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Recap of Session III

• 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)

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

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

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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• Motion in Mechanical Systems can be
• Translational
• Rotational or
• Combination of above
• Mechanical Systems can be of two types
• Translational Systems
• Rotational Systems

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Variables that describe motion Displacement Vel
ocity Acceleration
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Translational Systems Mass
Mass Property or means Kinetic energy is
stored
F Mass Acceleration m x
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Linear Spring
Elasticity (Spring) Property or means
Potential energy is stored Spring force is
proportional to displacement
Spring Force Stiffness displacement Fs k x

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

• Exists in all system and opposes motion
• Static Friction
• Coulomb Friction
• Viscous Friction

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Viscous Friction (Damping)
Viscous Damping Means by which energy is
absorbed Damping Force is proportional to
velocity
.
Fd C x
Damping Force Damping Coefficient Velocity
Fd C x
.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Rotational Systems Inertia Element
Torque Inertia Angular Acceleration T J ?
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Torsional Spring
Torque Torsional Stiffness Angular
displacement T kt ?
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Friction for Rotational System
Damping Torque Damping Coefficient Angular
velocity Td B ?
.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Degrees of Freedom (DOF)
Mathematical Models First approximation
Simple Model How simple it should be? What is
the basis?
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Degrees of Freedom (DOF)

• Minimum number of independent coordinates
  required to determine completely the position of
  all parts of a system at any instant of time
• Single DOF Systems
• Two DOF Systems
• Multi DOF Systems
• Discrete or Lumped Parameter Systems- Finite
  number of degrees of freedom
• Continuous or Distributed Systems- Infinite
  number of degrees of freedom

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Discrete or Lumped Parameter Systems Most
practical systems are studied by treating them as
lumped masses, springs and dampers Continuous
or Distributed Systems Accurate results are
obtained by increasing the number of masses,
springs and dampers
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Single (DOF) System
Simple Pendulum
Spring mass System
y
l
h
x
DOF 1, ? x2 y2 l2
DOF 1, x
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Tup
DOF 1, x1
Anvil and foundation block (m)
x1
Soil damping (c)
Soil stiffness (k)
Forging Hammer Modeled as Single DOF System
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Tup
DOF 2, x1, x2
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)
Forging Hammer Modeled as Two DOF System
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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mr
x1
kr
Legend t Tire v Vehicle
w Wheel r Rider s Strut
eq Equivalent
Cr
mv
x2
ks
Cs
mw
mw
x4
x3
kt
kt
DOF 4, x1, x2, x3, x4
Motor Bike Modeled as Four DOF System
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Legend t Tire v Vehicle
w Wheel r Rider s Strut
eq Equivalent
meq
keq
Ceq
x1
Motor Bike Modeled as Single DOF System
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Cantilever Beam
Continuous System with Infinite DOF
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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Translational systems Illustration 2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Points to Note
The displacement of the mass is to be considered
zero with the system at equilibrium The spring
is neither stretched nor compressed Sign
Convention Adopted sign convention must not be
changed during the course of the problem All the
forces / torque in the direction of the
displacement are considered positive Otherwise,
negative Behaviour of the system is independent
of the sign convention
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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The weight of the mass has no effect on the
differential equation when the reference is at
the equilibrium position Hence, the differential
equation is same as that of the previous system
in which weight was not a factor
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Spring Mass Damper Systems
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Spring Mass Damper Systems with an input (
Illustration 1)
.
.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Spring Mass Damper Systems with an input (
Illustration 2)
F (Input)
ky-F
F
ky
m
m
K
.
.
c y
c y
y (response)
m
From NSL ? F ma ?my - (ky-F) - c y
C
.
..
.
..
? my c y ky F
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Torsional or Rotational Systems Illustration 1
kt
b
J
From NSL
?
.
.
..
.
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Torsional or Rotational Systems with an input
Illustration 2
kt
Let ?i Input And ?0 Response Let (?i gt ?0)
J
b
?0
?i
kt?0
.
J
Kt?i
b?
From NSL J?0 kt (?i - ?0) - b ? J ?0 b?0
kt ?0 kt ?i
.
b?
J
Kt (?i - ?0)
.
..
..
.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Torsional or Rotational Systems with an input
Illustration 3
From NSL
b
J
T
?
.
..
.
J ? T - b ? J ? b ? T
J
b ?
T
.
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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(Modeling Contd.)
Example 1 For the system shown in figure obtain
the mathematical model if x1 and x2 are initial
displacements
Let an initial displacement X1 be given to mass
m1 and X2 to mass m2.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Free Body Diagrams
Let X2 gt X1
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Based on Newtons second law of motion ??
ma For mass m1 m1x1 - K1x1 K2
(x2-x1)   m1x1 K1x1 K2 x2 K2x1 0   m1x1
x1 (K1 K2) K2x2 ----- (1)
..
..
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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For mass m2 m2x2 - K3x2 K2 (x2
x1)   m2x2 K3 x2 K2 x2 K2 x1   m2 x2 x2
(K2 K3) K2x1 ----- (2)
..
..
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical models are
..
m1x1 x1 (K1 K2) K2x2 ----- (1) m2 x2 x2
(K2 K3) K2x1 ----- (2)
..
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Example 2 For a two DOF spring mass damper
system obtain the mathematical model where F is
the input x1 and x2 are responses
b2 (Damper)
k2
.
.
k2 x2
b2 x2
k2 x2
b2 x2
m2
m2
m2
x2
x2
.
.
k1 x1
b1 x1
.
b1 x2
k1 x2
.
k1
b1
.
.
k1 (x1-x2)
b1 (x1-x2)
b1 x1
b1 x2
k1 x1
k1 x2
m1
m1
x1
x1
m1
F
F
F
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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From NSL ?? ma For mass m1 m1x1 F - b1
(x1-x2) - k1 (x1-x2) --- (a)
..
.
.
For mass m2 m2 x2 b1 (x2-x1) k1 (x2-x1) -
b2 x2 - k2 x2 --- (b)
..
.
.
.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
42
Example 3 For the system shown in figure the
responses of mass m1 and m2 are y and z
respectively for an input of x.
k2
z
k
y
k1
x (input)
m2
m1
Frictionless rollers
c
Frictionless rollers
kz
kz
z
y
ky
k1 x
ky
k2z
.
m2
.
m1
cy
cy
k1y
.
.
cz
cz
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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K(z-y)
K(z-y)
K2z
m2
m1
K1(y-x)
.
.
.
.
c(z-y)
c(z-y)
For Mass m1
..
.
.
m1y - k1 (y-x) k (z-y) c (z-y) - k1y
k1x kz ky cz cy
.
.
..
.
.
?m1y cy y (k k1) cz kz k1x
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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K(z-y)
K(z-y)
K2z
m2
m1
K1(y-x)
.
.
.
.
c(z-y)
c(z-y)
For mass m2

..
.
.
m2z - k (z-y) c (z-y) k2z kz ky
cz cy k2z
.
.
.
..
.
?m2z cz (k k2)z cy ky
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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..
.
.
?m1y cy y (k k1) cz kz k1x
----- (1)
.
.
..
?m2z cz (k k2)z cy ky
----- (2)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Summary

• Types of Mechanical Systems
• Behaviour of Elementary parts of Mechanical
  Systems
• Degrees of Freedom
• Models of Translational and Rotational Systems

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