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MOTIVATION FOR ENGINEERING MATHEMATICS
Whether we are standing outside in frigid temperatures the morning of the shopping nightmare known as Black Friday or arching over our keyboards ready to place a preorder, the goal is likely the same. A bold new technology has hit the market, and we have become ravenous for the chance to be one of the first to own it. The store opens its doors to the public, only for the display floor to be flooded by customers. After braving the crowd and waiting patiently, the clerk slides the device across the counter into our hands. Glancing over the package, the sexy tagline catches the eye, “engineered to satisfy.”
Clear across the world, a native of a country in dire poverty walks up to a water purification system, with jug in hand. Parched, she turns the handle to let the spigot release a gush of fresh water. After filling the jug to the brim, the woman turns to see writing on the belly of the tank, “engineered for a better world.”
These mottos appear to be common among a wide range of products, even razor blades and shampoos. What do these companies mean by “engineered?” Surely they are being dishonest, especially if we believe engineering only involves copious amounts of math. Although advanced products seemingly appear out of thin air, the underlying technical achievements of products consumed by the public exist only through the solving of engineering problems and basic research. Whether the product appears on the shelf in a local electronics store, in cyberspace, or in the heart of another country’s impoverished village, the concept does not change.
To clarify the ominous term, engineering, we will describe it as the process of using scientific and mathematical principles to solve problems.
Definition 1.1: Engineering is the process of using the principles of mathematics and science to solve problems.
As engineers, we are attempting to meet the needs of a client. The term client is a broad term for the end user(s)—the people who will be using the product. In the definition, the words “principles of mathematics” appear, hinting that math is one of our most powerful tools that we need to use in order to reach our final design. If we cannot understand the basic math needed to solve the classical engineering problems, there is little hope in making any further advancements. Employing mathematical methods may not be the primary activity during the development of a product; however, analyzing the resulting design’s performance and figuring out how to improve its functions will require applying theory. Therein lies the tug-of-war between what we deem to be engineering design and engineering analysis. The former, design, focuses on the creativity injected into the iterative steps taken to arrive at the final product, whereas the latter, analysis, relishes in the use of mathematics and science to perform the needed checks throughout the process. As this text focuses on the analysis portion of engineering, we need to understand what we mean by an engineering problem and how can we solve them.
1.1 ABSTRACTION IN ENGINEERING
The solving of physical problems in engineering is deeply rooted in the mathematical discoveries of the past few centuries. An obvious example is the development of calculus in order to solve and model dynamic problems (i.e., situations with moving objects). Other results are subtler, like the connection between so-called “imaginary numbers” and alternating current in a circuit. When considering complex situations, it is rarely helpful to dive in headfirst. Instead, we often attempt to derive meaning and assign relationships between things—this is the core of abstraction.
Definition 1.2: Abstraction is the process of extracting the essence of a mathematical concept or physical situation by removing any dependence on real-world constraints (when appropriate).
With respect to engineering, the idea of abstraction enables us to detach the problem from reality—in a sense—by keeping everything important and developing a viable solution. Abstraction occurs more frequently than we realize. Consider the analysis of a bridge; we can certainly build a bridge any way we please within the appropriate regulations, but we need to make sure it can bear the load of the traffic passing over it. Since we do not have a physical bridge to embed sensors throughout, collect measurements, and officially green-light the design, other methods are needed. We could implement our existing knowledge of physics and draw a free-body diagram (FBD), which we tend to draw frequently when solving a problem. Think of an FBD as an abstraction of a physical situation that acts as a rough working sketch. The lack of visual clutter allows us to analyze different quantities acting on an object—like the loads on the bridge—without overloading our senses.
FBDs enable us to ignore the finer details of an object or structure and focus only on the properties of interest to us. To create an FBD, we isolate the object we want to analyze by representing it as a dot (many things in engineering can be represented by a dot or a point—see Figure 1.1) or a basic outline of the figure if the dimensions are important, while keeping necessary forces, velocities, and so on.
Figure 1.1. Representing a weight by a dot or point
Example 1.1: Cantilever Beam and a Weight—In Figure 1.2, we have an abstraction of a cantilever beam (a beam that is rigidly attached to a support, like a wall, at one end) with a crate hanging from the unattached end. Our problem is determining the tension in the rope. The beam is not necessarily important because we are concerned with the rope and the crate, meaning we ignore the beam and represent the crate as a single point. Now, how can we think of the weight mathematically? Since the force due to the gravity is pulling the crate down, we can draw the weight of the crate as a downward arrow to preserve the direction and call it W. As long as this crate does not move, the tension in the rope is going to pull in the opposite direction in order to support the box; therefore, we can draw an arrow pointing upward and call it T. In this case, that is all we need!
Figure 1.2. Abstracting a cantilever beam with a weight
Drawing the picture is half the battle; we need to use mathematics in order to find the solution to our problem, what is the tension in the rope? Using the idea that nothing in the picture is moving, the rope should have a tension equal to weight of the crate—otherwise, the crate would move either down (if too little force was acting to hold it up) or up (if too much force was applied, causing “lift”). Therefore, T = W.
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The problem in Example 1.1 is a simple scenario from statics, and the answer may be intuitive to most; of course, not all engineering problems are this simplistic.
Definition 1.3: Statics is the branch of mechanics where the systems to be analyzed are in equilibrium; in other words, the systems and objects are not moving.
In this text, we will examine methods to solve problems like Example 1.1—along with more difficult scenarios—in a general section that we will call “solving engineering problems.”
Another abstraction we are concerned with is the mathematical modeling of various objects and interactions. In other words, how can we describe physical phenomena using the language of mathematics?
Definition 1.4: A mathematical model is a description of a process or system in the language of mathematics.
In addition to modeling objects, which can be done to an extent with FBDs and other techniques, actions like forces or disturbances are also important to describe mathematically. For instance, how do we model a strike of lightning—an instantaneous flash? How can we describe a hammer hitting a nail—a sudden impact that lasts a fraction of a second? What about a simple, quick, isolated knock on a table (Figure 1.3)?
Figure 1.3. Can all of these actions be described using the same mathematics?
Through abstraction, we can extrapolate that each of these actions share the concept of an instantaneous event of disruption to an otherwise stable situation; thus, we can deduce a single mathematical object can describe these ideas in a general sense (later on, we will define and explore this object, the Dirac delta function). For the purposes of this text, we will frame the content with respect to one of the most important abstractions in engineering, systems.
Definition 1.5: A system is a collection of parts that work together to form a whole.
1.2 ENGINEERING SYSTEMS AS A STARTING POINT
An engineering system can be simplistic or obscenely complex, but a straightforward method exists for reducing unwieldy systems down to easy-to-interpret diagrams. We often represent a system using a black-box diagram, as shown in Figure 1.4.
Definition 1.6: A black-box diagram is an abstraction of a process/system in terms of inputs and outputs without knowing the internal workings of the process/system.
The system could be anything: a circuit, a series of gears, or an entire automobile; regardless of the context, we can still represent any system as a black-box diagram.
Figure 1.4. A black-box diagram of a system
In the picture, we labeled only three objects: the input, the system itself, and the output. Copious amounts of engineering problems can be understood in this sense; however, the helpfulness of this viewpoint depends on the problem itself.
Example 1.2: A Phone as a System—We can imagine a system more practically by thinking of the process of making a phone call. First, consider the black box as the phone—our system. Now, we need to interact with this phone somehow in order to make a call, and this often involves typing in a phone number—this is our input. Once we enter the phone number and press the call button, the result is making a call—our output. This interaction is summarized in Figure 1.5.
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Figure 1.5. Idealizing a phone as a system
Example 1.3: The Spring and Mass—One of the simplest situations we can describe using the idea of a system in engineering is the “spring and mass.” In this scenario, a block is attached to a spring, which is then fixed to a wall—the entire apparatus is the system (Figure 1.6). Now, what is the input? Commonly, the input is an applied force on the block setting it into motion, like pulling the block away from the wall. The motion of the mass is considered the output of the system.
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The systems approach to solving engineering problems can be split into three guiding questions serving to split this text into digestible pieces. Our goal is to abstract problems in engineering in order to find the solutions, make predictions, and validate our design choices. With this in mind, we need to answer the following questions:
Figure 1.6. Simple spring and mass system
Question 1: How do we solve basic problems in engineering?
Question 2: How can we describe the inputs and outputs of a system mathematically?
Question 3: How can engineering systems be modeled mathematically in order to complement the inputs and outputs?
Naturally, attempting to take a streamlined path through the content is nearly impossible due to the interconnected nature of mathematics; therefore, we will attempt to split the content as cleanly as possible. We will begin with general methods used to solve problems in engineering, which will then spill over into describing inputs and outputs. Finally, we will bring the content to a bold finale where modeling engineering systems will be briefly discussed.