1. GUJARAT TECHNOLOGICAL UNIVERSITY
SARVAJANIK COLLEGE OF ENGINEERING AND TECHNOLOGY
Applications of Laplace Transforms
DEPARTMENT OF CIVIL ENGINEERING
YEAR II SEM III

2. Prepared By NAME ENROLLMENT NO Margi Dave 130420106012
Dipika Chaudhari
Dhruv Jani
Jenifar Modi
Aashna Patel
Deep Patel
Parishi Dalal

3. What is Laplace transform ?
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. L {cos(wt)} = w / (s² + w²)
Pierre Simon Laplace :
French mathematician and astronomer

4. Why do we use Laplace Transform ?
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.

5. is the Laplace transform of
A Laplace transform is a type of
integral transform.
is the Laplace transform of
Write

6. Various applications of laplace transforms
Control systems
Various applications of laplace transforms
Electrical circuit analysis
Nuclear Physics
Digital Signal Processing
Deflection in beams

7. The amazing 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.
I.V.P
Algebraic Expression
Inverse
Laplace
transform
Laplace transform
Soln. to IVP
Algebraic Equation

8. Use of laplace transform
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

9. Application in digital signal processing
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.

11. 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.

12. Use of Laplace transforms in nuclear physics
In order to get the true form of radioactive decay, a Laplace transform is used.
It makes studying analytic part of Nuclear Physics possible.

14. Mathematical Modelling of the Road Bumps Using Laplace Transform
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.
Application in medical field
Laplace transforms can be used in areas such as medical field for blood-velocity/time wave form over cardiac cycle from common femoral artery.

15. A sawtooth function 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. 1 1 1 t
A sawtooth function

16. 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.

17. Other Applications Semiconductor mobility
Call completion in wireless networks
Vehicle vibrations on compressed rails
Behavior of magnetic and electric fields above the atmosphere

18. Thank you

19. References : http://malinimath.blogspot.in/2009/07/laplace- transform
Control Tutorials for MATLAB and Simulink - Introduction: System ...
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Differential Equations for Engineers Wei-Chau Xie University of Waterloo
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