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.

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