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# SARVAJANIK COLLEGE OF ENGINEERING AND TECHNOLOGY

Applications of Laplace
Transforms
DEPARTMENT OF CIVIL ENGINEERING
YEAR II SEM III
Prepared By

NAME ENROLLMENT NO

## Parishi Dalal 130420106059

Basically, a Laplace transform
will convert a function in
some domain into a
function in another domain,
without changing the value
of the function.
For example
let’s take this function where
a Laplace transform is used
to convert a function of
time to a function of
frequency. Pierre Simon Laplace :
French mathematician and astronomer
L {cos(wt)} = w / (s² + w²)
 We use Laplace transform to convert equations having
complex differential equations to relatively simple
equations having polynomials. Since equations having
polynomials are easier to solve, we employ Laplace
transform to make calculations easier.
 With Laplace transform nth degree differential
equation can be transformed into an nth degree
polynomial. One can easily solve the polynomial to get
the result and then change it into a differential
equation using inverse Laplace transform.
The development of the logarithm was considered the
most important development in studying astronomy. In
much the same way, the Laplace transform makes it
much easier to solve differential equations.
A Laplace transform is a type of
integral transform.

 st
0
e f (t ) dt  F ( s )
F ( s ) is the Laplace transform of f (t ).
Write L  f (t )  F (s),
L  y (t )  Y ( s ),
L  x(t )  X ( s ), etc.
Control systems

Electrical
Nuclear Physics circuit
analysis Digital Signal Processing

Deflection
in beams
 Theamazing thing about using Laplace
transforms is that we can convert the whole
ODE initial value problem (I.V.P) into laplace
transformed functions of s, simplify the
algebra, find transformed solution F(s) the
undo the transform to get back to the
required solution f as a function of t.
 Laplace Transform systems have the extremely
important property that if the input to the system is
sinusoidal, then the output will also be sinusoidal at
the same frequency but in general with different
magnitude and phase. These magnitude and phase
differences as a function of frequency are known as
the frequency response of the system.
Using the Laplace transform, it is possible to convert a
system's time-domain representation into a
frequency-domain output/input representation, known
as the transfer function. In so doing, it also
transforms the governing differential equation into
an algebraic equation which is often easier to analyze.
 Frequency-domain methods are most often used for
analyzing LTI single-input/singleoutput (SISO) systems
A simple Laplace transform is conducted while
sending signals over two-way communication
medium (FM/AM stereos, 2 way radio sets, cellular
phones.)
 When information is sent over medium such as
cellular phones, they are first converted into
time-varying wave and then it is super imposed on
the medium. In this way, the information
propagates. Now, at the receiving end, to
decipher the information being sent, medium
wave’s time functions are converted to
frequency functions. This is a simple example of
Laplace transform in real life.
 Laplace transformation is crucial for the study of
control systems, hence they are used for the
analysis of HVAC(Heating, Ventilation and Air
Conditioning ) control systems, which are used in
all modern building and constructions.
 In order to get the true form of radioactive
decay, a Laplace transform is used.
 It makes studying analytic part of Nuclear
Physics possible.
 Traffic engineering is the application of Laplace Transform
to the quantification of speed control in the modelling of
road bumps with hollow rectangular shape. In many
countries the current practice used for lowering the
vehicle speed is to raise road bumps above the road
surface. If a hollow bump is used it may be and offers
other advantages over road bumps raised above the road
surfaces. The method models the vehicle as the classical
one-degree-of-freedom system whose base follows the
road profile, approximated by Laplace Transform.

##  Laplace transforms can be used in areas such as medical

field for blood-velocity/time wave form over cardiac cycle
from common femoral artery.
Laplace transforms are particularly effective
on differential equations with forcing functions
that are piecewise, like the Heaviside function,
and other functions that turn on and off.

A sawtooth function
Laplace Transforms in real life
An inverse Laplace transform is an improper
contour integral, a creature from the world
of complex variables.

## That’s why you don’t see them naked very often.

You usually just see what they yield, the output.

## In practice, Laplace transforms and inverse

Laplace transforms are obtained using tables
and computer algebra systems.
Other Applications
Semiconductor
mobility
Call completion in
wireless networks
Vehicle vibrations on
compressed rails
Behavior of magnetic
and electric fields
above the
atmosphere
 http://malinimath.blogspot.in/2009/07/laplace-
transform
 Control Tutorials for MATLAB and Simulink -
Introduction: System ...
 http://ctms.engin.umich.edu/CTMS/index.php?examp
le=Introduction..
 Differential Equations for Engineers Wei-Chau Xie
University of Waterloo
 http://jdebug.org/practical-applications-of-laplace-
transform
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