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# Review of Mathematical Approach to Engineering Problems

Authors:
*Correspondence to Author:
Department of Mathematics, Sir M.
Visvesvaraya Institute of Technolo-
gy,Bangalore-562157, India
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
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(,,,) 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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Gangadharaiah, Y. H , AJCSA, 2017; 1:3
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
RC RC





   


   

   

(1.4)
Taking inverse Laplace transform, we get
(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. (3.13.2) 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
 Gere, J.M. and Timoshenko, S.P.,
Mechanics of Materials, Third SI Edition.
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 Popov, E., Engineering Mechanics of Solids.
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 J. E. Connor, J.E. and and Faraji, S.,
Fundamentals of Structural Engineering.
Berlin Heidelberg: Springer-Verlag, 2012.
 Hjelmstad, K.D., Fundamentals of Structural
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Springer-Verlag, 2005.
 White, R.E. and and Subramaniam, V.R.,
Computational Methods in Chemical
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 Keil, F., Mackens, W., Vo, H. And Werther,
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 Caldwell, J. and Ram, Y.M., Mathematical
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Lemmon, E.W., Thermodynamic Properties
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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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