[PDF] On Noteworthy Applications of Laplace Transform in Real Life PDF IJISRT20MAR091.pdf

The Laplace Transform has several applications in the field of science and technology In this paper we will discuss about applications of Laplace Transform in real

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Volume 5, Issue 3, March 2020 International Journal of Innovative Science and Research Technology

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On Noteworthy Applications of Laplace

Transform in Real Life

P. C. Jadhav, S. S. Sawant, O. S. Kunjir, T. A. Karanjkar (Sinhgad Academy of Engineering, Pune)

Abstract:- Mathematics is a methodical application of matter. It is so said because the subject makes a man methodical or

more systematic. To justify & validate research findings, various mathematical tools are used. Laplace transform plays a vital

role in wide field of science & technology which can be considered as a shortcut for complex calculations. This paper provides

solid foundation of what Laplace transform is and its properties and its application in various fields which can further be

useful in real life as well.

MSC: 34A08; 34C10; 26A33

Keywords:- Laplace Transform, Mass Spring Damper System, Chemical Pollution, Transfer Function.

I. INTRODUCTION INTEGRAL TRANSFORM

Let K(s, t) be a function of two variables

function f(s) defined by the integral (assumed to be convergent) is called the Integral transform of the function F(t) and is denoted by L{F(t)}

The function ܭ

A. Laplace Transform:

If the Kernel ܭ

The f(s) defined by the above equation is called the Laplace Transform of the function F(t) and is also denoted by L{F(t)} or

F(s).

B. Existance of Laplace Transforms:

՜λ, then Laplace Transform of F(t)

that is F(s) exist ׊ discuss about applications of Laplace Transform in real life.

C. Properties of Laplace Transform

¾ Linearity Property: -

Where a and b are constants

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¾ Change of scale property: -

¾ First shifting property: - if L {F(x)} = f(s) then

¾ Laplace transform of derivatives: -

II. APPLICATIONS

A. Mass Spring Damper System

control of the car & comfort of occupants. The spring allows the wheels to

move up to absorb bumps in the road & reduce jolting, while the dampers prevent bouncing up & down Consider the mechanical

system as shown in figure.

Fig (1.1)

The generalized equation for the system can be formulated as ܨ = ݉ݔࡇ + ܾ b= damping coefficient k= spring coefficient x= displacement

F= Resultant force

Taking Laplace transform throughout

The generalized equation for the system can be formulated as = ࡇ + ࡆ + where m= mass of system

b= damping coefficient k= spring coefficient x= displacement

F= Resultant force

The generalized equation for the system can be formulated as = ࡇ + ࡆ + where m= mass of system

b= damping coefficient k= spring coefficient x= displacement

F= Resultant force

From fig (1.1), 2 + 4 + 3 = 10 sin

Taking Laplace transform throughout

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L [2

2 ] + 4 [ ] + 3 [] = 10 L[ sin ]

By (II) & assuming initial conditions ,

(2 + 4 + 3)() = 10 2 + 2

Taking = 1

10 () = (2 + 1)(2 + 4 + 3)

Solving it by partial fraction,

1 1 1 1 +

() = 10 [ 4 + + 1 20 + + 3 5 10] 2 + 1

Taking inverse Laplace transform,

1 3 1 1

= 10 [ 4 20 cos + 5 ] 10

Hence,

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Graph of solution of the system

Fig. (1.2)

Depending upon the mass, spring coefficient and damper coefficient, different responses to the system can be recorded. It is

necessary to analyze the mass-spring-damper system mathematically to be able to size your spring, damper and the mass of the object

you want to stabilize and to be able to describe the reaction for a given system.

B. Chemical Pollution in a Reservoir

Water Pollution due to contaminants has become serious threat to environment as well as to human health. Normally pollution I

large reservoirs commonly occurs on a time dependent scale in which system is not in steady state condition for pollutants. The basic

idea is, The formulation that governs time dependent concentration of an aqueous species in a reservoir is,

H1: Volume of reservoir is constant

H2: Flow rate remains constant

H3 : Reaction rate remains constant

H4 : Pollutant is uniformly distributed in reservoir

H5 : Input & output of water is same By H4 & H5 ,

As, M(t) = Vܥ

dM(t) dt = ܥܳ

QM(t) V

ܥܳ = ܸ0 ܥܳ

Assuming only fresh water is coming in,

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Applying Laplace transform & solving,

(ݐ) = (0)ܸ݁ Ex.

How much time would it take for pollutant to reach acceptable level if volume of lake is 25 × 106݉3,

Flow of fresh water is 1.5 × 106݉3, initial concentration of contaminant is 106݌ܽ

֜ (ݐ) = (0)ܸ݁

Solving it with given data,required time can be calculated as ݐ = 11.55 units approximately

Hence, such a model can be prepared to overcome water pollution which has serious ill effects over human health.

C. Transfer Function of Control System

Fig (3.1)

The tank shown in figure is initially empty. A constant flow rate Qin is added for t>0. The rate at which flow leaves the tank

(Qout) = CH.

A = cross sectional area M = Mass of fluid

Hence mass flow rate

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To construct a differential equation for head H we know, mass flow rate into tank is equal to mass in flow rate mass out flow

rate. i.e. A *݀ܪ (since Qout = CH)

Hence Qin = A *݀ܪ

Taking Laplace Transform on both sides

L[ܳ] = ܣ *ܮ [݀ܪ

From property (d),

= 1 But we know,ݑݐ = CH Applying Laplace transform,

From Equation (1) and (2)

1+(ܵܥ

which represents transfer function of control system. Hence using this transfer function we can control the water level in tank.

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III. CONCLUSION

In this paper we have tried focusing on such unusual applications of Laplace Transform which may resolve many practical

problems in day to day life in easier way.

chemicals in water which may in turn be beneficial to human life. Also the Transfer function derived by using Laplace Transform may

help us to regulate water which is very important natural resource.

REFERENCES

[1]. J. K. Goyal [2]. national Journal of Trend in Research and Development, Volume 3(1), ISSN: 2394-9333,2016. [3]. rnal of the Egyptian Mathematical Society (2015) 23, 102107. [4]. Sci. Revs. Chem. Commun.: 2(3), 2012,

264-271 ISSN 2277-2669.

[5]. ENGINEERING [6]. -ISSN: 2395-0056 Volume: 05 Issue: 05 |

May-2018 ISSN: 2395-0072.

[7]. s Class Spring 99.
quotesdbs_dbs17.pdfusesText_23

[PDF] [PDF] On Noteworthy Applications of Laplace Transform in Real Life

ISSN No:-2456-2165

On Noteworthy Applications of Laplace

Transform in Real Life

MSC: 34A08; 34C10; 26A33

I. INTRODUCTION INTEGRAL TRANSFORM

Let K(s, t) be a function of two variables

The function ܭ

A. Laplace Transform:

If the Kernel ܭ

B. Existance of Laplace Transforms:

՜λ, then Laplace Transform of F(t)

C. Properties of Laplace Transform

¾ Linearity Property: -

Where a and b are constants

ISSN No:-2456-2165

¾ Change of scale property: -

¾ Laplace transform of derivatives: -

II. APPLICATIONS

A. Mass Spring Damper System

Fig (1.1)

F= Resultant force

Taking Laplace transform throughout

F= Resultant force

F= Resultant force

From fig (1.1), 2 + 4 + 3 = 10 sin

Taking Laplace transform throughout

ISSN No:-2456-2165

2 ] + 4 [ ] + 3 [] = 10 L[ sin ]

By (II) & assuming initial conditions ,

Taking = 1

10 () = (2 + 1)(2 + 4 + 3)

Solving it by partial fraction,

1 1 1 1 +

Taking inverse Laplace transform,

1 3 1 1

Hence,

ISSN No:-2456-2165

Graph of solution of the system

Fig. (1.2)

B. Chemical Pollution in a Reservoir

H1: Volume of reservoir is constant

H2: Flow rate remains constant

H3 : Reaction rate remains constant

H5 : Input & output of water is same By H4 & H5 ,

As, M(t) = Vܥ

QM(t) V

ܥܳ = ܸ0 ܥܳ

Assuming only fresh water is coming in,

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Applying Laplace transform & solving,

֜ (ݐ) = (0)ܸ݁

C. Transfer Function of Control System

Fig (3.1)

A = cross sectional area M = Mass of fluid

Hence mass flow rate

ISSN No:-2456-2165

Hence Qin = A *݀ܪ

Taking Laplace Transform on both sides

L[ܳ] = ܣ *ܮ [݀ܪ

From property (d),

From Equation (1) and (2)

1+(ܵܥ

ISSN No:-2456-2165

III. CONCLUSION

REFERENCES

264-271 ISSN 2277-2669.

May-2018 ISSN: 2395-0072.