NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
A Brief Review of Laplace Transforms
Dr Bishakh Bhattacharya
Dr.
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc .
Module 2- Lecture 8
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
This Lecture Contains
This Lecture Contains
 Forward Laplace Transform
 Laplace Transform of Differential Equation
 Inverse Laplace Transform Inverse Laplace Transform
Poles and Zeros of a Transfer Function
An Assignment Problem to Solve
Joint Initiative of IITs and IISc .
Module 2- Lecture 8
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Introduction
•
Laplace Transformation is a very useful tool for analysis of a dynamic system
in Frequency-Domain. This transformation helps to transform differential
equations into the form of algebraic equations which is easier to manipulate.
manipulate
•
A time-domain signal f(t) which may represent a forcing function or the
response of a system may be transformed into frequency domain by using the
following transformation:


L f (t )  F ( s ) 
Where s = σ + jω, is a complex variable.
Example:
f(t)
( ) = A e-pt u(t)
()

F ( s)   A e
0

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 pt
f (t ) e  st dt
0

e u (t ) dt  A  e ( s  p )t dt
 st
A ( s  p ) t
e
s p
0

t 0

A
s p
3
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Some useful Laplace
p
Transforms
No.
f(t), t≥ 0
F(s)
1
(t)
1
2
u(t)
1/s
3
t
1/s2
4
t2
2!/s3
5
e-at
1/(s+a)
6
Sin at
a/(s2 + a2)
7
Cos at
s/(s2 + a2)
8
(1-at) e-at
s/(s+a)2
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4
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Laplace
p
Transform of a Differential Equation
q
Applying the same principle on a differential equation one can obtained an
algebraic equation. Consider a second order mechanical system represented by
the following differential equation:
m
d 2x
dt 2
c
dx
 kx  F sin(t )
dt
Applying Laplace transform and assuming zero initial condition the above
equation could be transformed as
m s 2 X ( s )  cs X ( s )  k X ( s ) 
F
s2  2
Denoting the right hand side of the above equation as F(s) , one can express the
ratio of F(s) and frequency-domain response X(s) as
T (s) 
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X (s)
1

F ( s ) m s 2  cs  k
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Transfer Function of a SDOF system
y
Here, we are considering a SDOF system with system parameters m (mass), c (damping)
and k (stiffness).
T(s)
( ) is also known as transfer function of the system.
y
In a block diagram
g
form this
can be represented as
F(s)
T(s)
X(s)
T ( s) 
1
ms 2  cs  k
The response of a system in time domain could be obtained by carrying out
inverse Laplace Transformation of the transfer function.
function The inverse Laplace
Transform is written as
  j
1
1
st
L [T ( s )] 
T
(
s
)
e
ds

2j   j
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Inverse Laplace
p
Transform
However, the relationship stated is seldom used. If T(s) is rational, one commonly
uses the method of partial fraction expansion. Consider a rational function T(s)
expressed as:
b1s m  b2 s m 11  ...  bm 1
T ( s) 
s n  a1s n 1  ...  an
Factoring the numerator and denominator polynomials one can also write
m
T (s)  K
 (s  z )
i
i 1
n
 (s  p )
i
i 1
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Poles and Zeros of the Transfer Function
•
•
Corresponding to the numerator polynomial, zi’s are referred as the zeros of
the transfer function while the roots of the numerator polynomial pi’s are
k
known
as the
h poles
l off the
h transfer
f function.
f
i
Now, the transfer function T(s) may be expressed as
T (s) 
•
Cn
C1
C
 2  ... 
s  p1 s  p 2
s  pn
where,
Ci  ( s  pi ) T ( s )
•
s  pi
Finally, the response of the system may be expressed as
n
x(t )   C i e pi t
i 1
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Examples:
A system is being excited by a step function at time t = 3 s and by a delta function
at time t = 5 s. Find out the Laplace Transform of these two functions by using the
Transformation table. Now, find out the response of a SDOF system (mass 1kg,
stiffness 10N/m and damping 1 N/m/s while subjected to the above excitations.
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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 2- Lecture 8
Special
p
References for this lecture
 System Dynamics and Response: Graham Kelly, Cengage Learning
 System Dynamics for Engineering Students: Nicolae Lobontiu, Academic Publisher
 Control Systems Engineering: Norman S Nise, John Wiley & Sons
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