Mathematics or particularly applied mathematics
is widely used in every engineering fields. In this paper, several examples of
applications of mathematics in mechanical, chemical, and electrical engineering
are discussed. Applications here are the real ones found in the engineering
fields, which may not be the same as discussed in many mathematics textbooks.
The purpose of this paper is to relate mathematics to engineering subject. Many
engineering students find it difficult to solve engineering problems which need
mathematics a lot. The students have studied mathematics before (calculus,
linear algebra, numerical analysis) but when they study engineering subjects
which involve mathematics they often cannot relate mathematics to those
subjects. It is hoped that through examples given, engineering students can be
motivated to understand their engineering problems better. Also, it is expected
that mathematics lecturers can be encouraged to provide mathematics problems
which are more related to engineering fields.

KEY WORDS :- Differential
equations, linear algebra, trigonometry, complex numbers, Euler’s identity, Matrices,
mass balance, first order reaction, system of differential equations, entropy,
chemical equation.

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INTRODUCTION

 

Mathematics is the background of every engineering fields. Together
with physics, mathematics has helped engineering develop. Without it,
engineering cannot have 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 1 for a
good textbook which discusses mathematical modelling with real life
applications. A lot of problems solved using Maple and MATLAB are given in 2. 

Content

Analysis of electric AC
circuits

We live in
an electrified world: electro-technology is used everywhere and living today
without electricity is virtually impossible even for the shortest duration
of time. Electric
power is generated both at central power plants and
by dispersed units such as wind turbine farms or large solar cell arrays, and
by means of large and complex electric networks the power is
transmitted/distributed from production units to consumers. Electro-technology
is also a fundamental prerequisite for the way
we today distribute and store information: electronic circuits are used everywhere in our society to convey information. Living without electro-technology is not an option.

From an electrical engineering point of view, detailed knowledge of
electro-technology is of vital importance, and a very important tool is circuit analysis. A good understanding of this topic enables us to analyze and to
design electric circuits and systems, irrespective of whether we consider
transmission of power in the hundreds of megawatts scale — or we consider
low-level signals within integrated electronics.

Circuit analysis
uses mathematics to analyze the network, and as an example, consider
the network shown in Figure 1 below. This network
consists of three elements: a voltage source
u(t), a resistor R and an inductance L. This simple circuit may model
e.g. an electric furnace, an electric motor,
a loud-speaker, etc.

R

   i(t)

 

 

                                               u(t)                                                          L

 

 

 

Figure 1: RL series circuit.

 

Using Kirchhoff’s laws, the resistive-inductive circuit fulfills the
equation

                           

                                                  
                  (1)

 

which is a first-order differential equation. In electrical engineering
the forcing function u(t) is often a sinusoidal function:

                           

                                                                                   
(2)

where Ua is the amplitude and ? is the angular
frequency for the voltage. For example,

the electric outlet our homes
has

 and

The question is now: how do we find the current i(t)?                                                                             

 

 

 

  Solution
using trigonometric identities

Since the input (the forcing function) is on the form (2), we may guess that the current
(the forced response,
stationary solution) is given by

                                                                                 (3)

Hence,
we need to find the constants Ia
and

 so that (1) is fulfilled.

Using

, we get

 (4)

where

.

 

Substitution
of (4) into (1) yields

       (5)

Which
implies

(6)

   Equating coefficients gives

   

                                                                                  (7)

and this
linear set of equations may be used to find

and

. Completing the algebra leads to

             

                                                  (8)

To get (8) on the format (3), we use the identity

 and after
some calculations we finally get

))                                                                       (9)        

Note how the current amplitude Ia and the phase angle

        may be read seen from (9).

This example shows that finding the solution (in this case the current
in the circuit) requires a lot of tedious calculations, even for a very simple electric
circuit. More complex
circuits will require
a lot more work using this procedure, but fortunately — as illustrated below — an alternative method using complex numbers and complex
functions is available.

 

Solution
using complex forcing function

The
starting point is Euler’s identity

 where j
denotes the complex unit. Suppose for a moment that the voltage source in
Figure 1 is the (unrealizable) complex-valued voltage defined by

 

x

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