Dynamic Modelling of Mechanical Systems Notes | EduRev

: Dynamic Modelling of Mechanical Systems Notes | EduRev

 Page 1


NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dynamic Modelling of Mechanical Systems
Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering gg
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Page 2


NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dynamic Modelling of Mechanical Systems
Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering gg
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Hints of the Last Assignment g
The Governing EOM may be written as:
0 ) (
) ( ) (
. ..
.
1 1 2 1 1
..
1 1
? ? ? ?
B K K M
t f x B x x K x M
a
Now, you may consider the following states for the system:
0 ) (
2 2 2 2 1 2 1 2 2
? ? ? ? ? x B x K x x K x M
?
?
?
?
?
?
?
?
?
.
1
1
x
x
X
C h d d O i f fi d O d bi h
?
?
?
?
?
?
?
?
.
2
2
x
x
X
2
Covert the two second order ODEs into four first order ODEs  and obtain the state space 
representation.
Joint Initiative of IITs and IISc -Funded by MHRD
Page 3


NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dynamic Modelling of Mechanical Systems
Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering gg
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Hints of the Last Assignment g
The Governing EOM may be written as:
0 ) (
) ( ) (
. ..
.
1 1 2 1 1
..
1 1
? ? ? ?
B K K M
t f x B x x K x M
a
Now, you may consider the following states for the system:
0 ) (
2 2 2 2 1 2 1 2 2
? ? ? ? ? x B x K x x K x M
?
?
?
?
?
?
?
?
?
.
1
1
x
x
X
C h d d O i f fi d O d bi h
?
?
?
?
?
?
?
?
.
2
2
x
x
X
2
Covert the two second order ODEs into four first order ODEs  and obtain the state space 
representation.
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
This Lecture Contains
?ModellingofaMechanical System ?Modelling of a Mechanical System
?Basic Elements of a Mechanical System y
?Examples to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
Page 4


NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dynamic Modelling of Mechanical Systems
Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering gg
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Hints of the Last Assignment g
The Governing EOM may be written as:
0 ) (
) ( ) (
. ..
.
1 1 2 1 1
..
1 1
? ? ? ?
B K K M
t f x B x x K x M
a
Now, you may consider the following states for the system:
0 ) (
2 2 2 2 1 2 1 2 2
? ? ? ? ? x B x K x x K x M
?
?
?
?
?
?
?
?
?
.
1
1
x
x
X
C h d d O i f fi d O d bi h
?
?
?
?
?
?
?
?
.
2
2
x
x
X
2
Covert the two second order ODEs into four first order ODEs  and obtain the state space 
representation.
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
This Lecture Contains
?ModellingofaMechanical System ?Modelling of a Mechanical System
?Basic Elements of a Mechanical System y
?Examples to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Mechanical Systems
Mechanical systems are generally modeled as a lumped 
parametersystem suchthatadistributed systemlikeabeam parameter system, such that a distributed system like a beam 
could be considered to be a system consisting of an array of 
rigid inertiaelements linked by a combination of  mass-less 
springand dashpot elements. 
The inertia elements represent the kinetic energy stored in  p gy
the system; springs the potential energy and dashpots the 
energy that gets dissipated from a mechanical system in the 
formofheat/soundetc form of heat/sound etc . 
Joint Initiative of IITs and IISc -Funded by MHRD
Page 5


NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Dynamic Modelling of Mechanical Systems
Dr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering gg
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Hints of the Last Assignment g
The Governing EOM may be written as:
0 ) (
) ( ) (
. ..
.
1 1 2 1 1
..
1 1
? ? ? ?
B K K M
t f x B x x K x M
a
Now, you may consider the following states for the system:
0 ) (
2 2 2 2 1 2 1 2 2
? ? ? ? ? x B x K x x K x M
?
?
?
?
?
?
?
?
?
.
1
1
x
x
X
C h d d O i f fi d O d bi h
?
?
?
?
?
?
?
?
.
2
2
x
x
X
2
Covert the two second order ODEs into four first order ODEs  and obtain the state space 
representation.
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
This Lecture Contains
?ModellingofaMechanical System ?Modelling of a Mechanical System
?Basic Elements of a Mechanical System y
?Examples to Solve
Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
Mechanical Systems
Mechanical systems are generally modeled as a lumped 
parametersystem suchthatadistributed systemlikeabeam parameter system, such that a distributed system like a beam 
could be considered to be a system consisting of an array of 
rigid inertiaelements linked by a combination of  mass-less 
springand dashpot elements. 
The inertia elements represent the kinetic energy stored in  p gy
the system; springs the potential energy and dashpots the 
energy that gets dissipated from a mechanical system in the 
formofheat/soundetc form of heat/sound etc . 
Joint Initiative of IITs and IISc -Funded by MHRD
NPTEL >> Mechanical Engineering >>  Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5
For translatory mechanical systems, inertia is represented by 
mass ‘ m’, while for rotational systems this is represented by 
moment of inertia ‘J’.
Consider a rotor of mass ‘m’, rotating about it’s centroidal 
axis. The moment of inertia will be defined as:
dm r J
m
?
?
2
Where ‘r’ denotes the distance of an elemental mass dm 
from the centroidal axis. For a rotation about an axis which is 
at a distance ‘d’ from the centroidal axis, following parallel-
axis theorem the moment of inertia could be expressed as:
2
d J J
2
d m J J
new
? ?
Joint Initiative of IITs and IISc -Funded by MHRD
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