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*Correspondence to Author:

Gangadharaiah, Y. H

Department of Mathematics, Sir M.

Visvesvaraya Institute of Technolo-

gy,Bangalore-562157, India

How to cite this article:

Gangadharaiah, Y. H .Review of

Mathematical Approach to Engi-

neering Problems. American Jour-

nal of Computer Sciences and Ap-

plications, 2017; 1:3.

eSciPub LLC, Houston, TX USA.

Website: http://escipub.com/

Gangadharaiah, Y. H , AJCSA, 2017; 1:3

American Journal of Computer Sciences and Applications

(ISSN:2575-775X)

Research Article AJCSA, 2017, 1:3

Review of Mathematical Approach to Engineering Problems

Mathematics is widely used in every engineering fields. In this

paper, several examples of applications of mathematics in me-

chanical, chemical, optimization and electrical engineering are

discussed. Laplace transform mathematical tool is applied to

solve problems. Applications here are the real ones found in the

engineering elds, which may not be the same as discussed in

many mathematics text books. The purpose of this paper is to

relate mathematics to engineering eld.

Keywords: Laplace transform ; Mechanical; Mechanical; Elec-

trical; Optimization

Gangadharaiah, Y. H

Department of Mathematics, Sir M. Visvesvaraya Institute of Technology,Bangalore-562157, India

ABSTRACT

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Gangadharaiah, Y. H , AJCSA, 2017; 1:3

1.Introduction

Mathematical modeling has always been an

important activity in science and engineering.

The formulation of qualitative questions about

an observed phenomenon as mathematical

problems was the motivation for and an integral

part of the development of mathematics from

the very beginning. Although problem solving

has been practiced for a very long time, the use

of mathematics as a very effective tool in

problem solving has gained prominence in the

last 50 years, mainly due to rapid

developments in computing. Computational

power is particularly important in modeling

engineering systems, as the physical laws

governing these processes are complex.

Besides heat, mass, and momentum transfer,

these processes may also include chemical

reactions, reaction heat, adsorption,

desorption, phase transition, multiphase flow,

etc. This makes modeling challenging but also

necessary to understand complex interactions.

All models are abstractions of real systems and

processes. Nevertheless, they serve as tools

for engineers and scientists to develop an

understanding of important systems and

processes using mathematical equations. In all

engineering context, mathematical modeling is

a prerequisite for: design and scale-up; process

control; optimization; mechanistic

understanding; evaluation/planning of

experiments; trouble shooting and diagnostics;

determining quantities that cannot be

measured directly; simulation instead of costly

experiments in the development lab; feasibility

studies to determine potential before building

prototype equipment or devices.

Mathematics is the background of every

engineering fields. Together with physics,

mathematics has helped engineering develop.

Without it, engineering cannot evolved so fast

we can see today. Without mathematics,

engineering cannot become so fascinating as it

is now. Linear algebra, calculus, statistics,

differential equations and numerical analysis

are taught as they are important to understand

many engineering subjects such as fluid

mechanics, heat transfer, electric circuits and

mechanics of materials to name a few.

However, there are many complaints from the

students who find it difficult to relate

mathematics to engineering. After studying

differential equations, they are expected to be

able to apply them to solve problems in heat

transfer, for example. However, the truth is

different. For many students, applying

mathematics to engineering problems seems to

be very difficult. Many examples of engineering

applications provided in mathematics textbooks

are often too simple and have assumptions that

are not realistic. See([8],[9],[10],[11]) for a good

textbook which discusses mathematical

modelling with real life applications. A lot of

problems solved using Maple and MATLAB are

given in [12,13,14]. The purpose of this paper

is to show some applications of mathematics to

various engineering fields. The applications

discussed do not need advanced mathematics

so they can be understood easily.

2. Mathematical Approach to Engineering

Problems

In this section we discussed four engineering

problems, first problem is about electrical

circuit problem, the second on the mixing

solutions in two tanks , Optimization Problem

and the last on the stability problem .

2. 1. Electrical circuit

To find the current in the RC-circuit in Figure1.

If a single rectangular wave with voltage is

applied as a input . The circuit is assumed to

be quiescent before the wave is applied. The

input in terms of unit step function is given by

0

v t V u t a u t b

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Figure.1 RC-circuit with input

vt

Applying KVL to above circuit, we get

0

0

1t

Ri t i d V u t a u t b

C

(1.1)

Taking Laplace transform, we get

0

1as bs

Is ee

RI s V

C s s s

(1.2)

0

1as bs

ee

R I s V

sC s s

(1.3)

011

as bs

Vee

Is Rss

RC RC

(1.4)

Taking inverse Laplace transform, we get

11

0t a t b u

RC RC

V

i t e u t a e u t b

R

(1.5)

is the required current in the RC-circuit.

2.2. Mixing Problem Involving Two

Tanks

Tank in Figure.2 initially contains 100 gal

of pure water. Tank initially contains 100 gal

of water in which 150 lb of salt are dissolved.

The inflow into is from and containing 6 lb of

salt from the outside. The inflow into is 8

gal/min from. The outflow from is, as shown

in the figure. The mixtures are kept uniform

by stirring. Our aim is find the salt contents

1

xt

and

2

xt

in tanks T1 and T2.

2.3. Optimization Problem

(Minimization of drag-to-lift ratio)

Airplane pilots share a challenge with

flying birds: How far can they go. What is

their range for a fixed amount of fuel? Still

better, can they maximize their range? It

turns out that for a given amount of fuel, the

speed that maximizes the range is the one

that maximizes the aerodynamic quantity,

called the lift-to-drag ratio, or, conversely,

minimizes its inverse, the drag-to-lift ratio.

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Figure.2 Mixing Problem Involving Two Tanks

Setting up the model.

/ min - / minTime rate of change Inflow Outflow

For tank

1

T

:

'

1 2 1

28

6

100 100

x t x t x t

(2.1)

For tank

2

T

:

'

2 2 1

82

100 100

x t x t x t

(2.2)

with initial conditions are

12

0 0, 0 150xx

.

By taking the Laplace transform we get

12

6

0.08 0.02s L x t L x t s

(2.3)

12

0.08 0.08 150L x t L x t

(2.4)

We solve this algebraically for

12

L x t and L x t

and we write the solutions in terms of partial

fractions,

19 0.48 100 62.5 37.5

0.12 0.04 0.12 0.04

s

L x t s s s s s s

(2.5)

2

2150 12 0.48 100 125 75

0.12 0.04 0.12 0.04

ss

L x t s s s s s s

(2.6)

By taking the inverse transform we arrive at the solution

0.12 0.04

1100 62.5 37.5

tt

x t e e

(2.7)

0.12 0.04

2100 125 75

tt

x t e e

(2.8)

are the required salt contents in T1 and T2

Figure. 3 a typical jet with a free-body

diagram superposed. The plane is climbing at

an angle, α, at a speed, V, relative to the

ground. The climb or flight direction angle, α,

is zero for level flight, and positive for

ascending flight and negative for descending

flight. The free-body diagram shows the

forces that act to support the plane and move

it forward, as described in the aerodynamic

literature. The plane’s weight, W, is

supported by a lift (force), L, that is

perpendicular to the flight path. The engines

provide a thrust, T, that moves the plane

along the flight path by overcoming the drag

(force), D, that also acts along the flight path,

albeit it in a direction that retards flight. The

plane’s wing has a surface area, S, and

span, b.

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Figure. 3. A typical jet with a superposed free-body diagram showing the aerodynamic

forces acting.

Lift force and drag force is given by

2

1

2L

L SV C

(3.1)

And

2

1

2D

D SV C

(3.2)

where CL and CD are the corresponding lift and

drag coefficients. (We should note that the

drag-velocity relation is more complicated when

planes fly closer to the speed of sound, due to

drag produced by compressibility effects either

on rapidly rotating propellers or on the wings of

jet aircraft). The makeup of the CL and CD

coefficients and their relationship provide, the

complexity we will see in our search for an

optimum flight speed. But first we need to do a

little equilibrium analysis because taken

superficially, equations. (3.1–3.2) suggest that

the drag-to-lift ratio L/D is independent of the

speed V, so how could it be minimized with

respect to V?

We sum the forces superposed on the plane in

Figure . 3 in the x and y directions:

cos sin cos 0

x

F T L D

(3.3)

And

sin cos sin 0

t

F T L D W

(3.4)

If the climb angle, α, is assumed to be small,

Using the approximations

32

sin ........ cos 1 ........

62

and

equations (3.3-3.4) can be simplified and

solved to show that the lift L is,

2

1W

LW

(3.5)

which means that the drag-to-lift ratio is simply

DD

LW

(3.6)

Equation(3.5) clearly shows that the lift force

supports the plane’s weight, while equation

(3.6) provides a speed-dependent ratio of the

drag force to the weight. Now we return to the

drag coefficients because that is the logical

step for casting the D/L ratio in terms of the

plane’s speed, V.

It turns out that the drag coefficient is

expressed as a sum of two terms,

0

2

2.

L

DD

kSC

CC b

(3.7)

The first term represents the parasite or friction

drag caused by shear stresses resulting from

the air speeding over and separating from the

wing. The second term is the induced drag : it

is independent of the air viscosity and is

created by wings of finite span (i.e., real wings!)

because of momentum changes needed to

produce lift, according to Newton’s second law.

Note that the induced drag is proportional to

the square of the lift coefficient,

2

L

C

.

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Now we can combine equations. (3.1) and (3.6)

to write the drag-to-lift ratio as

2

2D

SV C

D

LW

(3.8)

after which we can further combine. equations.

(3.1), (3.5) and (3.7) to rewrite (3.5) . (3.8) as

22

01 02

DC V C V

L

(3.9)

With the constants

01 02

C and C

defined as

0

01 02 2

2

,

2

D

SC kW

CC

Wb

(3.10)

Thus, the objective function or cost for this

optimization problem is defined in equations.

(3.9), and its coefficients as presented in

equation (3.10) are simply constants reflecting

the values of the problem’s physical parameters:

ρ, S, W, the wing span, b, the parasite drag

coefficient,

0D

C

and a dimensionless shape

constant, k .

The extreme value of this unconstrained

optimization problem is then found by the

standard calculus approach, that is,

3

01 02

2 2 0

dD C V C V

dV L

(3.11)

which has the following extreme value:

14

02

01 02 min

min 01

2 2 .

C

DC C at V

LC

(3.12)

With the aid of equation (3.10), the minimum

drag-to-lift ratio can then be written in its final

form

0

2

min

2D

kSC

D

Lb

(3.13)

This is a classical result in aerodynamics.

Further, it is also easily demonstrated at this

minimum D/L ratio occurs only when the

parasite drag and the induced drag are equal

and, consequently, independent of the plane

weight W.

2.4. Stability Problem

Many smaller portable tape recorders have a

capacitor microphone built in, since such a

system is simple and robust. It works on the

principle that if the distance between the plates

of a capacitor changes then the capacitance

changes in a known manner, and these

changes induce a current in an electric circuit.

This current can then be amplified or stored.

The basic system is illustrated in Figure.4

There is a small air gap (about 0.02 mm)

between the moving diaphragm and the fixed

plate. Sound waves falling on the diaphragm

cause vibrations and small variations in the

capacitance C ; these are certainly sufficiently

small that the equations can be linearized.

.

Figure.4 Capacitor microphone

We assume that the diaphragm has mass m

and moves as a single unit so that its motion is

one-dimensional. The housing of the

diaphragm is modelled as a spring and-dashpot

system. The plates are connected through a

simple circuit containing a resistance and an

imposed steady voltage from a battery. Figure

illustrates the model. The distance x (t) is

measured from the position of zero spring

tension, F is the imposed force and f is the

force required to hold the moving plate in

position against the electrical attraction.

The mechanical motion is governed by

Newton’s equation

2

20

d x dx

m x f F

dt dt

(4.1)

and the electrical circuit equation gives

q

E RI C

(4.2)

The variation of capacitance C with x is given

by the standard formula

0

Ca

Cxa

(4.3)

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Figure.5 Capacitor microphone model.

where a is the equilibrium distance between the

plates. The force f is not so obvious,

2

0

1

2q

fCa

(4.4)

It is convenient to write the equations in the

first-order form

dx

vdt

(4.5)

0

dv

m v x f F

dt

(4.6)

0

q a x

dq

RE

dt C a

(4.7)

Furthermore, it is convenient to non-

dimensionalize the equations. With distance

and velocity, for the time and the charge using

standard non-dimensionalization procedure by

neglecting prime,

2

10

, , , 2

ka

t x v q

X V Q

aC ka

And equations are

'0

RC k

XV

(4.8)

'2

0

RF

V X V Q

C m ka

(4.9)

'0

2

0

12

EC

Q Q X C ka

(4.10)

There are four non-dimensional parameters:

the external force divided by the spring force

gives the first, G = F/ka ; the electrical force

divided by the spring force gives the second,

22

202

0

2

EC

DC ka

; and the remaining two are

0

RC

A

and

0

R

BCm

The final equations are

' ' 2 '

,1X A V B V X V Q G and Q Q X D

(4.11)

In equilibrium, with no driving force, G = 0 and

' ' ' 0V X V Q

, so that

20

1

QX

Q X D

(4.12)

on eliminating Q, weget

22

1X X D

(4.13)

There are two physically satisfactory

equilibrium solutions

10

3X

and

1

13

X

,

and the only question left is whether they are

stable or unstable. Using standard stability

analysis

Get the only solution that can possibly be

stable is the one for which

1

3

X

and other

solution is unstable.

Having established the stability of one of the

positions of the capacitor diaphragm, the next

step is to look at the response of the

microphone to various inputs. The

characteristics can most easily be checked by

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Gangadharaiah, Y. H , AJCSA, 2017; 1:3

looking at the frequency response, which is the

system response to an individual input G = b

jwt

e

as the frequency ω varies. This will give

information of how the electrical output

behaves and for which range of frequencies the

response is reasonably flat. The essential point

of this example is to show that a practical

vibrational problem gives a stability problem.

3. Conclusions

In this paper, four of applications of

mathematics for different engineering fields

have been presented. The problems are from

real life and solved different techniques. It is

expected that the problems presented in this

paper can motivate reader to understand

mathematics better. Mathematics should be

enjoyable as it has helped engineering evolved.

References

[1] Gere, J.M. and Timoshenko, S.P.,

Mechanics of Materials, Third SI Edition.

Dordercht: Springer Science Business

Media, 1991.

[2] Popov, E., Engineering Mechanics of Solids.

New Jersey: Prentice-Hall, 1990.

[3] J. E. Connor, J.E. and and Faraji, S.,

Fundamentals of Structural Engineering.

Berlin Heidelberg: Springer-Verlag, 2012.

[4] Hjelmstad, K.D., Fundamentals of Structural

Mechanics, Second Edition. New York:

Springer-Verlag, 2005.

[5] White, R.E. and and Subramaniam, V.R.,

Computational Methods in Chemical

Engineering with Maple. Springer-Verlag.

Berlin Heidelberg, 2010.

[6] Keil, F., Mackens, W., Vo, H. And Werther,

J., Scientific Computing in Chemical

Engineering. Springer-Verlag, Berlin

Heidelberg, 1996.

[7] Caldwell, J. and Ram, Y.M., Mathematical

Modelling, Springer Science Business

Media. Dordercht, 1999.

[8] Braun, M., Differential Equations and Their

Applications, Springer Science Business

Media. New York, 1993.

[9] Erwin Kreyszing., Advanced Engineering

Mathematics : John Wiley & Sons, 2014.

[10] Glyn James., Advanced Modern

Engineering Mathematics: Pearson

Education Limited, 2011.

[11] Popov, E., Engineering Mechanics of

Solids. New Jersey: Prentice-Hall, 1990.

[12] Jacobsen, R.T., Penoncelo, S.G. and

Lemmon, E.W., Thermodynamic Properties

of Cryogenic Fluids. Springer Science

Business Media, New York, 1997.

[13] Reid, R.C., Prausnitz, J.M. and Poling,

B.E., The Properties of Gases and Fluids.

McGraw-Hill Inc., New York, 1987.

[14] Gander, W. And Hrebicek, J., Solving

Problems in Scientific Computing Using

Maple and MATLAB. Springer-Verlag,

Berlin Heidelberg, 2014

[15] K. M. Heal, M. L. Hansen, and K. M.

Rickard, Maple V Learning Guide. Springer-

Verlag, New York, 1998.

[16] R. M. Corless, Essential Maple, , Springer-

Verlag, New York, 1995.

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Gangadharaiah, Y. H , AJCSA, 2017; 1:3

Appendix

Properties of Laplace Transform:

some of the important properties of Laplace transform which will be used in its applications are

discussed below.

1. Definition of a Laplace Transform

F(s)L f t

f t

0

estdt

and

tfsFL

1

2 Linearity: The Laplace transform of the sum, or difference, of two signals in time domain is

equal to the sum, or difference, of the transforms of each signals, that is,

1 2 1 2

L C f t C g t C L f t C L g t

3. Differentiation: If the function

ft

is piecewise continuous so that it has continuous

derivative

1n

ft

of order

1n

and a sectionally continuous derivative

n

ft

in every finite

interval

0,

, then

1 2 1

0 0 0

n n n n

L f t s L f t s f s f f

3. Integration:

0

tFs

L f t dt s

4. Laplace transform of Unit step signal

as

e

L u t a s

5. Second shifting theorem:

as

L f t a u t a e L f t

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