The Application of Mathematics in the Engineering Disciplines

David Reeping, Kenneth J. Reid

• Publisher: Momentum Press
• Year: 2018
• Print ISBN: 9781606509074
• eBook ISBN: 9781606509081
• Topic: Technology & Engineering
• Subtopic: Engineering General
• Index: Technology & Engineering

This text serves as the companion text to Introductory Engineering Mathematics, which introduces common mathematical concepts we see in engineering, including trigonometry, calculus, and functions. This text assumes a level of mathematics of a high school senior, plus some elements from the introductory text. Additional concepts we see in engineering are also introduced: specifically, matrices, differential equations, and some introduction to series. The concepts are introduced by examples rather than strict mathematical derivation. As a result, this text likely will not be an effective substitute for a differential equations course, but by illustrating the implementation of differential equations, it can be a companion to such a course. We primarily use historical events as examples (including failures) to illustrate the use of mathematics in engineering and the intersection of the disciplines. We hope you develop an appreciation for how to apply these concepts, and find a new lens through which to view engineering successes (and failures).

CHAPTER 1

MODELING SYSTEMS IN ENGINEERING

If art can capture the human experience with a few brushstrokes, mathematical models capture the raw details of the same experience using equations and formulas. Anything that we would quantify (make measurable) or explain using numbers is a candidate for modeling. Something that is purely qualitative, meaning something that is not measured using numbers, can also be mathematically modeled, strangely enough.

What is a model? A model is a representation of an item, a process, an event, a person, etc. It may be a three-dimensional representation of a physical structure, or it may be a simplified process meant to represent a more complex one. Models are extraordinary helpful tools for cutting costs and increasing the efficiency and effectiveness of our products and processes. NASA certainly does not operate on a trial and error basis by launching multimillion dollar space shuttles and hoping the journey is successful. When the government is prepared to introduce a new program, we may hear that the program is expected to save (or cost) so many billions of dollars. To find this number, we can: either implement the program and let it run for a few years and see what happens, or develop a model and predict what we expect will happen. This is an example of a mathematical model.

Why else would we want to make a model? Many fields of research and businesses require a mathematical model to make decisions. These can take the form of statistical models, an offshoot of mathematical models that involve statistics, where the main goal is to produce something that can make accurate predictions. It is essentially the company genie that can only grant one big wish and possibly do a few tricks.

More complex models are based off experimental data. After a certain number of trials, we may be pretty confident that you could come up with a model to predict what will happen. That way, we can make an accurate prediction of the results without having to collect more data. This is particularly attractive once you start talking about outrageously expensive, time-intensive massive experiments that may or may not be feasible to run in real life. In the cases of real-world data, no perfect model exists, we can only aim to create and improve the models to a degree they are useful.

1.1 A NOTE ON CREATING MATHEMATICAL MODELING

We’ll start nice and easy by creating and interpreting a common mathematical model to make approximations of functions; however, the process of generating a mathematical model generalizes to just about any modeling challenge. Much like learning a second language, there is a learning curve in translating English to Math. Luckily, there are some tricks that we can use to ease into these translations.

1. Read and understand the problem. This step should be obvious, but we should always read a problem until we completely understand what is asked. Not interpreting the problem correctly is where most mistakes begin.

2. Once you understand the problem, pick out important information. While this may be a more acquired skill, identifying given information and necessary assumptions is crucial. Sometimes, extraneous information—data or descriptions that are not relevant or not helpful—may be included in the problem description. In the real world, extraneous information is abundant and can take many for...

Table of contents

• Cover
• Half Title Page
• Title Page
• Copyright Page
• Contents
• List of Figures
• List of Tables
• Acknowledgments
• Chapter 1 Modeling Systems in Engineering
• Chapter 2 Differential Equations in Engineering
• Chapter 3 Describing Systems Using Mathematics
• Chapter 4 Analyzing Failure in Systems
• About the Authors
• Index