Mathematics or particularly applied mathematicsis widely used in every engineering fields. In this paper, several examples ofapplications of mathematics in mechanical, chemical, and electrical engineeringare discussed.
Applications here are the real ones found in the engineeringfields, which may not be the same as discussed in many mathematics textbooks.The purpose of this paper is to relate mathematics to engineering subject. Manyengineering students find it difficult to solve engineering problems which needmathematics a lot. The students have studied mathematics before (calculus,linear algebra, numerical analysis) but when they study engineering subjectswhich involve mathematics they often cannot relate mathematics to thosesubjects. It is hoped that through examples given, engineering students can bemotivated to understand their engineering problems better. Also, it is expectedthat mathematics lecturers can be encouraged to provide mathematics problemswhich are more related to engineering fields.
KEY WORDS :- Differentialequations, 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. Togetherwith 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 taughtas they are important to understand many engineering subjects such as fluidmechanics, heat transfer, electric circuits and mechanics of materials to namea few. However, there are many complaints from the students who find itdifficult to relate mathematics to engineering.
After studying differentialequations, they are expected to be able to apply them to solve problems in heattransfer, for example. However, the truth is different. For many students,applying mathematics to engineering problems seems to be very difficult. Manyexamples of engineering applications provided in mathematics textbooks areoften too simple and have assumptions that are not realistic.
See 1 for agood textbook which discusses mathematical modelling with real lifeapplications. A lot of problems solved using Maple and MATLAB are given in 2. Content Analysis of electric ACcircuitsWe live inan electrified world: electro-technology is used everywhere and living todaywithout electricity is virtually impossible even for the shortest durationof time. Electricpower is generated both at central power plants andby dispersed units such as wind turbine farms or large solar cell arrays, andby means of large and complex electric networks the power istransmitted/distributed from production units to consumers. Electro-technologyis also a fundamental prerequisite for the waywe 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 ofelectro-technology is of vital importance, and a very important tool is circuit analysis. A good understanding of this topic enables us to analyze and todesign electric circuits and systems, irrespective of whether we considertransmission of power in the hundreds of megawatts scale — or we considerlow-level signals within integrated electronics.Circuit analysisuses mathematics to analyze the network, and as an example, considerthe network shown in Figure 1 below. This networkconsists of three elements: a voltage sourceu(t), a resistor R and an inductance L. This simple circuit may modele.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 theequation (1) which is a first-order differential equation. In electrical engineeringthe forcing function u(t) is often a sinusoidal function: (2)where Ua is the amplitude and ? is the angularfrequency for the voltage. For example, the electric outlet our homeshas and The question is now: how do we find the current i(t)? Solutionusing trigonometric identitiesSince 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 Iaand so that (1) is fulfilled.
Using , we get (4) where . Substitutionof (4) into (1) yields (5)Whichimplies (6) Equating coefficients gives (7)and thislinear 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 aftersome 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 currentin the circuit) requires a lot of tedious calculations, even for a very simple electriccircuit. More complexcircuits will requirea lot more work using this procedure, but fortunately — as illustrated below — an alternative method using complex numbers and complexfunctions is available. Solutionusing complex forcing functionThestarting point is Euler’s identity where jdenotes the complex unit.
Suppose for a moment that the voltage source inFigure 1 is the (unrealizable) complex-valued voltage defined by