™ ! r e i s a E ng Making Ever ythi
s c i n a h c e M s s l a i r e t a M of Learn to: • Understand key mechanics concepts • Grasp principles of stress, strain, and deformation and their interactions • Solve indeterminate statics problems
James Ja mes H. Al Allen len III III,, PE, PE, PhD PhD Assistant Assista nt Profes Professor sor of Civil Engin Engineeri eering ng University of Evan Evansville sville
Chapter 1
Predicting Behavior with Mechanics of Materials In This Chapter Chapter ▶ Defining mechanics of materials ▶ Introducing stresses and strains ▶ Using mechanics of materials to aid in design
M
echanics of materials is one of the first application-based engineering classes you face in your educational career. It’s part of the branch of physics known as mechanics, which includes other fields of study such as rigid body statics and dynamics. Mechanics is an area of physics that allows you to study the behavior and motion of objects in the world around you.
Mechanics of materials uses basic statics and dynamics principles but allows you to look even more closely at an object to see how it deforms under load. It’s the area of mechanics and physics that can help you decide whether you really should reconsider knocking that wall down between your kitchen and living room as you remodel your house (unless, of course, you like your upstairs bedroom on the first floor in the kitchen). Although statics can tell you about the loads and forces that exist when an object is loaded, it doesn’t tell you how the object behaves in response to those loads. That’s where mechanics of materials comes in.
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Part I: Setting the Stage for Mechanics of Materials
Tying Statics and Mechanics Together Since the early days, humans have looked to improve their surroundings by using tools or shaping the materials around them. At first, these improvements were based on an empirical set of needs and developed mostly through a trial-and-error process. Structures such as the Great Pyramids in Egypt or the Great Wall of China were constructed without the help of fancy materials or formulas. Not until many centuries later were mathematicians such as Sir Isaac Newton able to formulate these ideas into actual numeric equations (and in many cases, to remedy misconceptions) that helped usher in the area of physics known as mechanics.
Mechanics Mechanics, and more specifically the core areas of statics and dynamics, are based on the studies and foundations established by Newton and his laws of motion. Both statics and dynamics establish simple concepts that prove to be quite powerful in the world of analysis. You can use statics to study the behavior of objects at rest (known as equilibrium ), such as the weight of snow on your deck or the behavior of this book as it lies on your desk. Dynamics, on the other hand, explains the behavior of objects in motion, from the velocity of a downhill skier to the trajectory of a basketball heading for a winning shot. What statics and dynamics both have in common is that at their fundamental level, they focus on the behavior of rigid bodies (or objects that don’t deform under load). In reality, all objects deform to some degree (hence why they’re called deformable bodies ), but the degree to which they deform d eform depends depe nds entirely on the mechanics of the materials themselves. Mechanics of materi- als (which is sometimes referred to as strength of materials or mechanics of mechanics that attempts to explain deformable bodies ) is another branch of mechanics the effect of loads on objects. The development of mechanics of materials over the centuries has been based on a combination of experiment and observation in conjunction with the development of equation-based theory. Famous individuals such as Leonardo da Vinci (1452–1519) and Galileo Galilei (1564–1642) conducted experiments on the behavior of a wide array of structural objects (such as beams and bars) under load. And mathematicians and scientists such as Leonhard Euler (1707–1783) developed the equations used to provide the basics for column theory. Mechanics of materials is often the follow-up course to statics and dynamics in the engineering curriculum because it builds directly on the tools and concepts you learn in a statics and dynamics course, and it opens the door to engineering design. And that’s where things get interesting.
Chapter 1: Predicting Behavior with Mechanics of Materials
Defining Behavior in Mechanics of Materials The fact that all objects deform under load is a given. Mechanics of materials helps you determine how much the object actually deforms. Like statics, mechanics of materials can be very methodical, allowing you to establish a few simple, guiding steps to define the behavior of objects in the world around you. You can initially divide your analysis of the behavior of objects under load into the study and application of two basic interactions: stress and strain. With the basic concepts of stress and strain, you have two mechanisms for determining the maximum values of stress and strain, which allow you to investigate whether a material (and the object it creates) is sufficiently strong while also considering how much it deforms. You can then turn your attention to specific sources of stress, which I introduce a little later in this chapter.
Stress Stress is the measure of the intensity of an internal load acting on a cross section of an object. Although you know a bigger object is capable of supporting a bigger load, stress is what actually tells you whether that object is big enough. This intensity calculation allows you to compare the intensity of the applied loads to the actual strength (or capacity ) of the material itself. I introduce the basic concept of stress in Chapter 6, where I explain the difference between the two types of stress, normal stresses and shear stresses. With this basic understanding of stress and how these normal and shear stresses can exist simultaneously within an object, you can use stress transfor- mation calculations (see Chapter 7) to determine maximum stresses (known as principal stresses ) and their orientations within the object.
Strain Strain is a measure of the deformation of an object with respect to its initial length, or a measure of the intensity of change in the shape of a body. Although stress is a function of the load acting inside an object, strain can occur even without load. Influences such as thermal effects can cause an object to elongate or contract due to changes in temperature even without a physical load being applied. For more on strain, turn to Chapter 12.
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Part I: Setting the Stage for Mechanics of Materials As with stresses, strains have maximum and minimum values (known as principal strains ), and they occur at a unique orientation within an object. I show you how to perform these strain transformations in Chapter 13.
Using Stresses to Study Behavior Stresses are what relate loads to the objects they act on and can come from a wide range of internal forces. The following list previews several of the different categories of stress that you encounter as an engineer:
✓ Axial: Axial stresses arise from internal axial loads (or loads that act along the longitudinal axis of a member). Some examples of axial stresses include tension in a rope or compression in a short column. For more on axial stress examples, turn to Chapter 8. ✓ Bending: Bending stresses develop in an object when internal bending moments are present. Examples of members subject to bending are the beams of your favorite highway overpass or the joists in the roof of your house. I explain more about bending stresses in Chapter 9. ✓ Shear: Shear stresses are actually a bit more complex because they can have several different sources. Direct shear is what appears when you try to cut a piece of paper with a pair of scissors by applying two forces in opposite direction across the cut line. Flexural shear is the result of bending moments. I discuss both of these shear types in Chapter 10. Torsion (or torque ) is another type of loading that creates shear stresses on objects through twisting and occurs in rotating machinery and shafts. For all things torsion, flip to Chapter 11.
Studying Behavior through Strains You can actually use strains to help with your analysis in a couple of circumstances:
✓ Experimental analysis: Strains become very important in experiments because, unlike stresses, they’re quantities that you can physically measure with instruments such as electromechanical strain gauges. You can then correlate these strains to the actual stresses in a material using the material’s properties. ✓ Deformation without load: Strain concepts can also help you analyze situations in which objects deform without being subjected to a load such as a force or a moment. For example, some objects experience changes in shape due to temperature changes. To measure the effects of temperature change, you must use the concepts of strain.
Chapter 1: Predicting Behavior with Mechanics of Materials
Incorporating the “Material” into Mechanics of Materials After you understand the calculations behind stress and strains, you’re ready to turn your attention to exploring the actual behavior of materials. All materials have a unique relationship between load (or stress) and deformation (or strain), and these unique material properties are critical in performing design. One of the most vital considerations for the stress-strain relationship is Hooke’s law (see Chapter 14). In fact, it’s probably the single most important concept in mechanics of materials because it’s the rule that actually relates stresses directly to strain, which is the first step in developing the theory that can tell you how much that tree limb deflects when you’re sitting on it. This relationship also serves as the basis for design and the some of the advanced calculations that I show you in Part IV.
Putting Mechanics to Work When you have the tools to analyze objects in the world around them, you can put them to work for you in specific applications. Here are some common mechanics of materials applications:
✓ Combined stresses: In some cases, you want to combine all those single and simple stress effects from Part II into one net action. You can analyze complex systems such as objects that bend in multiple directions simultaneously (known as biaxial bending ) and bars with combined shear and torsion effects. Flip to Chapter 15 for more. ✓ Displacements and deformations: Deformations are a measure of the response of a structure under stress. You can use basic principles based on Hooke’s law to calculate deflections and rotations for a wide array of scenarios. (See Chapter 16.) ✓ Indeterminate structures: For simple structures, the basic equilibrium equations you learn in statics can give you all the information you need for your analysis. However, the vast majority of objects are much more complex. When the equilibrium equations from statics become insufficient to analyze an object, the object is said to be statically indetermi- nate. In Chapter 17, I show you how to handle different types of these indeterminate systems by using mechanics of materials principles.
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Part I: Setting the Stage for Mechanics of Materials ✓ Columns: Unlike most objects that fail when applied stresses reach the limiting strength of the material, columns can experience a geometric instability known as buckling, where a column begins to bow or flex under compression loads. Chapter 18 gives you the lowdown on columns. ✓ Design: Design is the ability to determine the minimum member size that can safely support the stresses or deflection criteria. This step requires you to account for factors of safety to provide a safe and functional design against the real world. Head to Chapter 19 for more. ✓ Energy methods: Energy methods are another area of study that relates the principles of energy that you learned in physics to concepts involving stresses and strain. In Chapter 20, I introduce you to energy method concepts such as strain energy and impact.
Contents at a Glance Introduction ................................................................ 1 Part I: Setting the Stage for Mechanics of Materials ...... 7 Chapter 1: Predicting Behavior with Mechanics of Materials ...................................... 9 Chapter 2: Reviewing Mathematics and Units Used in Mechanics of Materials ...... 15 Chapter 3: Brushing Up on Statics Basics .................................................................... 25 Chapter 4: Calculating Properties of Geometric Areas ............................................... 41 Chapter 5: Computing Moments of Area and Other Inertia Calculations ................. 55
Part II: Analyzing Stress ............................................ 83 Chapter 6: Remain Calm, It’s Only Stress!.....................................................................85 Chapter 7: More than Meets the Eye: Transforming Stresses .................................... 99 Chapter 8: Lining Up Stress Along Axial Axes ............................................................ 131 Chapter 9: Bending Stress Is Only Normal: Analyzing Bending Members .............. 149 Chapter 10: Shear Madness: Surveying Shear Stress ................................................ 161 Chapter 11: Twisting the Night Away with Torsion ..................................................177
Part III: Investigating Strain .................................... 189 Chapter 12: Don’t Strain Yourself: Exploring Strain and Deformation .................... 191 Chapter 13: Applying Transformation Concepts to Strain .......................................201 Chapter 14: Correlating Stresses and Strains to Understand Deformation............ 215
Part IV: Applying Stress and Strain........................... 233 Chapter 15: Calculating Combined Stresses...............................................................235 Chapter 16: When Push Comes to Shove: Dealing with Deformations ................... 251 Chapter 17: Showing Determination When Dealing with Indeterminate Structures................................................................................... 273 Chapter 18: Buckling Up for Compression Members ................................................ 301 Chapter 19: Designing for Required Section Proper ties ........................................... 313 Chapter 20: Introducing Energy Methods ................................................................... 331
Part V: The Part of Tens ........................................... 343 Chapter 21: Ten Mechanics of Materials Pitfalls to Avoid........................................ 345 Chapter 22: Ten Tips to Solving Mechanics of Materials Problems ........................ 349
Index ...................................................................... 355
a s er ! E g in t y r e v E g Makin
™
s c i t p O Learn to: • se opt cs pr nc ples and dev ces properly • vo d common m stakes n work ng w th ypical optics problems • eterm ne mage locat ons and characteristics with simple calculations • Grasp the basic concepts behind lasers and laser applications
Galen Duree, Jr., PhD Pro essor o Physics and Optical Engineering Director, Center or Applied Optics Studies Rose-Hulman Institute of Technology
Chapter 1
Introducing Optics, the Science of Light In This Chapter ▶ Uncovering the basic properties of light ▶ Getting a glimpse of optics applications
L
ight is probably one of those things that you take for granted, kind of like gravity. You don’t know what it is or where it comes from, but it’s always there when you need it. Your sight depends on light, and the information you get about your environment comes from information carried by the light that enters your eye. Humans have spent centuries studying light, yet it remains something of a mystery. We do know many properties of light and how to use them to our benefit, but we don’t yet know everything. Therefore, optics is the continuing study of light, from how you make it to what it is and what you can do with it. In fact, optics consists of three fields: geometrical optics, physical optics, and quantum optics. As we learn more about light, we find new ways to use it to improve our lives. This chapter shines a little, well, light on light.
Illuminating the Properties of Light Because of an accidental mathematical discovery, light is called an elec- tromagnetic wave, a distinction indicating that light waves are made up of electric and magnetic fields. You’re probably used to thinking of light as the stuff your eyes can detect. For many people who work with light on a regular basis, however, the term light applies to all electromagnetic radiation, anything from ultra-low frequencies to radio frequencies to gamma rays.
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Part I: Getting Up to Speed on Optics Fundamentals Light has both wave and particle properties (as I discuss in the chapters in Part I), but you can’t see both at the same time. Regardless of the properties, light is produced by atoms and accelerating charges. You can choose from many different arrangements to produce light with the desired wavelength or frequency (basically, the color that you want). Optics covers every light source from light bulbs to radio transmissions. You have three ways to manipulate where light goes (that is, to make light do what you want): reflection, refraction, and diffraction, which I introduce in the following sections. You can use some basic equations to calculate the result of light undergoing all these processes. Optics then goes farther to investigate ways to find practical uses of these phenomena, including forming an image and sending digital data down a fiber.
Creating images with the particle property of light You most commonly see the particle property of light when you’re working with geometrical optics, or making images (see Part II). In this theory, the particles of light follow straight-line paths from the source to the next surface. This idea leads to the simplest type of imaging: shadows. Shadows don’t give you a lot of information, but you can still tell the shape of the object as well as where the light source is. Two important concepts in geometrical optics are reflection and refraction. Reflection describes light bouncing off a surface. Refraction deals with the bending of the path of the light as the light goes from one material to another. You can use these processes to create and modify images, and knowledge of these effects can also help you deal with factors called aber- rations, which cause an image to be blurry. You can also use the lenses and mirrors that work with refraction and reflection to eliminate the washed-out effect you sometimes get when creating an image; if you have too much light, all the images created wash each other out, so all you see is light.
Harnessing interference and diffraction with the wave property of light Physical optics, which I cover in Part III, looks at the wave properties of light. Interference (where two or more waves interact with each other) and diffrac- tion (the unusual behavior of waves to bend around an obstacle to fill the space behind it) are unique to waves.
Chapter 1: Introducing Optics, the Science of Light To explain optical interference (interference between light waves), you need to know about optical polarization. Optical polarization describes the orientation of the plane that the light wave’s electric field oscillates in. In optics, only the electric field matters in almost all interactions with matter, because the electric field can do work on charged particles and the magnetic field can’t. Several devices can change the polarization state so that light can be used for many different applications, including lasers and optical encoding. The wave property allows you to use interference to help measure many things, such as the index of refraction and surface feature height or irregularities. Specifically, several optical setups called interferometers use interference for measurement. Diffraction, the other unique wave phenomenon, determines resolution, which is how close two objects can be while still being distinguishable. Arrangements with many slits placed very close together create a diffraction grating, which you can use to help identify materials by separating the different colors of light the materials emit.
Using Optics to Your Advantage: Basic Applications Understanding the basic properties of light is one thing, but being able to do something practical with them is another. (Head to the earlier section “Illuminating the Properties of Light” for more on these basic characteristics.) Putting the fundamental knowledge to good use means developing optical instruments for a wide variety of uses, as I discuss in Part IV. Here’s just a taste of some of the practical applications of optical devices:
✓ Manipulating images: As I note earlier in the chapter, knowing how images are made and changed with different types of lenses or mirrors allows you to design simple optical devices to change what the images look like. Eyeglasses are designed and built t o correct nearsightedness or farsightedness, and a simple magnifying glass creates an enlarged image of rather small objects. Physical characteristics limit how large an image a simple magnifier can make, so you can build a simple microscope with two lenses placed in the right positions to provide greater enlargement of even smaller objects. To see things far away, you can build a telescope and to project an image onto a large screen, you can build a projector.
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Part I: Getting Up to Speed on Optics Fundamentals ✓ Developing lighting: You can use also use optics principles to design lighting sources for particular applications, such as specific task lighting, general area lighting, and decorative lighting. The development of incandescent light bulbs, compact fluorescent bulbs, and future devices such as light emitting diodes (LEDs) all start with knowledge of the optical properties of materials. ✓ Seeing where the eye can’t see: Optics, and particularly fiber optics, can send light into areas that aren’t directly in your line of sight, such as inside a collapsed building or a body. Fiber optics relies on knowledge of total internal reflection (see Chapter 4) to be able to trap light inside a small glass thread.
Expanding Your Understanding of Optics The fundamental principles of optics can tell you what will happen with light in different situations, but making something useful with these principles isn’t so easy. Applications of optics, including optical systems, combine two or more optical phenomena to create a desired output. Most applications of optics require knowledge about how optical principles work together in one system; making optical systems requires careful thought to make sure that the light behaves in the way you want it to when you look at the final result, whether that’s light from a particular source (such as a light bulb or laser) or an image from a telescope or camera. Why are such advanced applications important? Seeing how all the optics phenomena work together (often in subtle ways) is the point of optical engineering. Knowing how light interacts with different materials and being able to read this information has led t o advances in important fields such as medical imaging and fiber-optic communication networks.
Considering complicated applications Some optics applications, such as those in Part V, require combinations of many different optics principles to make useful devices. Cameras that record images require knowledge of image formation, focusing, and intensity control to make nice pictures. Holography and three-dimensional movies put depth perception and diffraction gratings to work. Many medical-imaging techniques exploit the effects of light and how light carries information.
Chapter 1: Introducing Optics, the Science of Light Lasers are a special light source with many uses. Because lasers are light, you have to understand how light works so that you can use them effectively and safely. Lasers today are involved in medical applications, various fabrication tasks, numerous quality control arrangements, optical storage discs such as CDs and DVDs, and a variety of military and law enforcement applications (but no laser guns yet). Complex imaging devices can also allow you to see in low- or no-light situations. Thermal cameras create images based on temperature differences rather than the amount of reflected light. The age-old arrangement of looking at the heavens requires modifications of the simple telescope to overcome some of the limitations of using refracting optics.
Adding advanced optics Advanced optics (see Part VI) covers phenomena that aren’t simply based on simple refraction. When the index of refraction — normally independent of the intensity of the light — changes with the intensity, weird things can happen, such as frequency conversion in crystals. The area of advanced optics that studies these effects is called nonlinear optics, and it has provided numerous new diagnostic capabilities and laser wavelengths. Another area of advanced optics is single photon applications. Single photon applications show some rather bizarre behavior associated with the fact that light is in an indeterminate state unless you make a measurement. This subject (also presented in Part VI) is the basis for new applications in secure communications and super-fast computing.
Paving the Way: Contributions to Optics The field of optics is full of contributions from students challenging the establishment and the established way of thinking. Part VII includes some experiments you can try to experience some of the optics principles presented in this book; building some simple optical devices lets you begin to discover the challenges of building optical systems. After all, experiments are the root of discovery, so Part VII also looks at some important optics breakthroughs and the people who performed them. All this information allows you to see how knowledge of optics advanced with contributions from newcomers to the field as well as established optical scientists. Using the basic principles outlined in this book, you’ll have enough
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Part I: Getting Up to Speed on Optics Fundamentals knowledge to be able to delve deeper into any optics subject you encounter in school, work, or just your curiosity. As optics technology progresses, you’ll have the basic background knowledge to tackle any of the technology paths that develop. After all, the f ield of optics benefited from contributions from many different levels; if that doesn’t motivate you to make the next significant contribution to the science of light, I don’t know what will.
Contents at a Glance Introduction ................................................................ 1 Part I: Getting Up to Speed on Optics Fundamentals ...... 7 Chapter 1: Introducing Optics, the Science of Light .....................................................9 Chapter 2: Brushing Up on Optics-Related Math and Physics................................... 15 Chapter 3: A Little Light Study: Reviewing Light Basics ............................................. 31 Chapter 4: Understanding How to Direct Where Light Goes .....................................45
Part II: Geometrical Optics: Working with More Than One Ray ............................................ 57 Chapter 5: Forming Images with Multiple Rays of Light ............................................. 59 Chapter 6: Imaging with Mirrors: Bouncing Many Rays Around ............................... 69 Chapter 7: Imaging with Refraction: Bending Many Rays at the Same Time ...........77
Part III: Physical Optics: Using the Light Wave ........... 95 Chapter 8: Optical Polarization: Describing the Wiggling Electric Field in Light .... 97 Chapter 9: Changing Optical Polarization ..................................................................113 Chapter 10: Calculating Re�ected and Transmitted Light with Fresnel Equations ............................................................................................... 131 Chapter 11: Running Optical Interference: Not Always a Bad Thing ......................143 Chapter 12: Diffraction: Light’s Bending around Obstacles ..................................... 161
Part IV: Optical Instrumentation: Putting Light to Practical Use .................................. 179 Chapter 13: Lens Systems: Looking at Things the Way You Want to See Them .....181 Chapter 14: Exploring Light Sources: Getting Light Where You Want It ................197 Chapter 15: Guiding Light From Here to Anywhere ................................................... 213
Part V: Hybrids: Exploring More Complicated Optical Systems.................................... 227 Chapter 16: Photography: Keeping an Image Forever ..............................................229 Chapter 17: Medical Imaging: Seeing What’s Inside You (No Knives Necessary!) ... 247 Chapter 18: Optics Everywhere: Exploring Other Medical, Industrial, and Military Uses ...................................................................... 259 Chapter 19: Astronomical Applications: Using Telescopes .....................................271
Part VI: More Than Just Images: Getting into Advanced Optics ................................... 285 Chapter 20: Index of Refraction, Part 2: You Can Change It! ....................................287 Chapter 21: Quantum Optics: Finding the Photon.....................................................301
Part VII: The Part of Tens ......................................... 311 Chapter 22: Ten Experiments You Can Do Without a $1-Million Optics Lab ......... 313 Chapter 23: Ten Major Optics Discoveries — and the People Who Made them Possible ....................................................................... 319
Index ...................................................................... 325
Table of Contents Introduction ................................................................. 1 About This Book .............................................................................................. 1 Conventions Used in This Book ..................................................................... 2 What You’re Not to Read ................................................................................ 3 Foolish Assumptions ....................................................................................... 3 How This Book Is Organized .......................................................................... 3 Part I: Getting Up to Speed on Optics Fundamentals ........................ 4 Part II: Geometrical Optics: Working with More Than One Ray ......4 Part III: Physical Optics: Using the Light Wave .................................. 4 Part IV: Optical Instrumentation: Putting Light to Practical Use ..... 4 Part V: Hybrids: Exploring More Complicated Optical Systems ...... 5 Part VI: More Than Just Images: Getting into Advanced Optics ...... 5 Part VII: The Part of Tens ...................................................................... 5 Icons Used in This Book .................................................................................5 Where to Go from Here ................................................................................... 6
Part I: Getting Up to Speed on Optics Fundamentals ....... 7 Chapter 1: Introducing Optics, the Science of Light . . . . . . . . . . . . . . . .9 Illuminating the Properties of Light ..............................................................9 Creating images with the particle property of light ........................10 Harnessing interference and diffraction with the wave property of light ...................................................... 10 Using Optics to Your Advantage: Basic Applications ............................... 11 Expanding Your Understanding of Optics .................................................. 12 Considering complicated applications ............................................. 12 Adding advanced optics .....................................................................13 Paving the Way: Contributions to Optics ................................................... 13
Chapter 2: Brushing Up on Optics-Related Math and Physics . . . . . .15 Working with Physical Measurements........................................................15 Refreshing Your Mathematics Memory ...................................................... 16 Juggling variables with algebra .......................................................... 16 Finding lengths and angles with trigonometry ................................18 Exploring the unknown with basic matrix algebra .......................... 21 Reviewing Wave Physics ..............................................................................26 The wave function: Understanding its features and variables ......26 Medium matters: Working with mechanical waves ......................... 28 Using wavefronts in optics .................................................................29
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Optics For Dummies Chapter 3: A Little Light Study: Reviewing Light Basics. . . . . . . . . . . .31 Developing Early Ideas about the Nature of Light ....................................31 Pondering the particle theory of light ............................................... 32 Walking through the wave theory of light ........................................ 32 Taking a Closer Look at Light Waves .......................................................... 33 If light is a wave, what’s waving? Understanding electromagnetic radiation ............................................................... 33 Dealing with wavelengths and frequency: The electromagnetic spectrum ......................................................36 Calculating the intensity and power of light ....................................36 Einstein’s Revolutionary Idea about Light: Quanta................................... 37 Uncovering the photoelectric effect and the problem with light waves..................................................38 Merging wave and particle properties: The photon ........................ 39 Let There Be Light: Understanding the Three Processes that Produce Light ..................................................................40 Atomic transitions ............................................................................... 40 Accelerated charged particles ........................................................... 41 Matter-antimatter annihilation........................................................... 42 Introducing the Three Fields of Study within Optics ................................ 42 Geometrical optics: Studying light as a collection of rays .............42 Physical optics: Exploring the wave property of light .................... 43 Quantum optics: Investigating small numbers of photons ............43
Chapter 4: Understanding How to Direct Where Light Goes . . . . . . . .45 Re�ection: Bouncing Light Off Surfaces .....................................................45 Determining light’s orientation .......................................................... 46 Understanding the role surface plays in specular and diffuse re�ection ................................................... 47 Appreciating the practical difference between re�ection and scattering .................................................................48 Refraction: Bending Light as It Goes Through a Surface .......................... 50 Making light slow down: Determining the index of refraction ....... 50 Calculating how much the refracted ray bends: Snell’s law ..........51 Bouncing light back with refraction: Total internal re�ection ....... 52 Varying the refractive index with dispersion ................................... 53 Birefringence: Working with two indices of refraction for the same wavelength ..........................................54 Diffraction: Bending Light around an Obstacle .........................................55
Table of Contents
Part II: Geometrical Optics: Working with More Than One Ray ............................................. 57 Chapter 5: Forming Images with Multiple Rays of Light. . . . . . . . . . . .59 The Simplest Method: Using Shadows to Create Images .........................60 Forming Images Without a Lens: The Pinhole Camera Principle ............ 62 Eyeing Basic Image Characteristics for Optical System Design ..............63 The type of image created: Real or virtual .......................................63 The orientation of the image relative to the object ........................63 The size of the image relative to the object .....................................64 Zeroing In on the Focal Point and Focal Length ........................................65 Determining the focal point and length ............................................65 Differentiating real and virtual focal points .....................................66
Chapter 6: Imaging with Mirrors: Bouncing Many Rays Around. . . . .69 Keeping it Simple with Flat Mirrors.............................................................69 Changing Shape with Concave and Convex Mirrors ................................. 70 Getting a handle on the mirror equation and sign conventions ...71 Working with concave mirrors ..........................................................72 Exploring convex mirrors ................................................................... 74
Chapter 7: Imaging with Refraction: Bending Many Rays at the Same Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77 Locating the Image Produced by a Refracting Surface ............................. 78 Calculating where an image will appear ...........................................78 Solving single-surface imaging problems..........................................80 Working with more than one refracting surface .............................. 83 Looking at Lenses: Two Refracting Surfaces Stuck Close Together .......85 Designing a lens: The lens maker’s formula .....................................85 Taking a closer look at convex and concave lenses ........................ 88 Finding the image location and characteristics for multiple lenses ...........................................................................89 D’oh, fuzzy again! Aberrations ...........................................................91
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Optics For Dummies
Part III: Physical Optics: Using the Light Wave ............ 95 Chapter 8: Optical Polarization: Describing the Wiggling Electric Field in Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97 Describing Optical Polarization ................................................................... 97 Focusing on the electric �eld’s alignment ........................................ 98 Polarization: Looking at the plane of the electric �eld ...................99 Examining the Different Types of Polarization ........................................100 Linear, circular, or elliptical: Following the vector path ..............100 Random or unpolarized: Looking at changing or mixed states ...107 Producing Polarized Light .......................................................................... 108 Selective absorption: No passing unless you get in line ............... 108 Scattering off small particles ............................................................ 109 Re�ection: Aligning parallel to the surface ..................................... 110 Birefringence: Splitting in two .......................................................... 111
Chapter 9: Changing Optical Polarization . . . . . . . . . . . . . . . . . . . . . . .113 Discovering Devices that Can Change Optical Polarization ..................113 Dichroic �lters: Changing the axis with linear polarizers ............114 Birefringent materials: Changing or rotating the polarization state ................................................. 117 Rotating light with optically active materials ................................121 Jones Vectors: Calculating the Change in Polarization ..........................121 Representing the polarization state with Jones vectors ..............121 Jones matrices: Showing how devices will change polarization .... 124 Matrix multiplication: Predicting how devices will affect incident light ................................................................. 126
Chapter 10: Calculating Reflected and Transmitted Light with Fresnel Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Determining the Amount of Re�ected and Transmitted Light ..............131 Transverse modes: Describing the orientation of the �elds ........ 132 De�ning the re�ection and transmission coef�cients ................... 133 Using more powerful values: Re�ectance and transmittance ...... 134 The Fresnel equations: Finding how much incident light is re�ected or transmitted .................................... 135 Surveying Special Situations Involving Re�ection and the Fresnel Equations ......................................................................136 Striking at Brewster’s angle .............................................................. 137 Re�ectance at normal incidence: Coming in at 0 degrees ............ 137 Re�ectance at glancing incidence: Striking at 90 degrees ............ 138 Exploring internal re�ection and total internal re�ection ............ 138 Frustrated total internal re�ection: Dealing with the evanescent wave ............................................... 139
Table of Contents Chapter 11: Running Optical Interference: Not Always a Bad Thing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 Describing Optical Interference................................................................. 143 On the fringe: Looking at constructive and destructive interference ........................................................144 Noting the conditions required to see optical interference ......... 145 Perusing Practical Interference Devices: Interferometers .....................146 Wavefront-splitting interferometers ................................................ 146 Amplitude-splitting interferometers................................................ 151 Accounting for Other Amplitude-Splitting Arrangements ...................... 154 Thin �lm interference ........................................................................ 154 Newton’s rings.................................................................................... 157 Fabry-Perot interferometer...............................................................158
Chapter 12: Diffraction: Light’s Bending around Obstacles . . . . . . . .161 From Near and Far: Understanding Two Types of Diffraction ............... 162 De�ning the types of diffraction ......................................................162 Determining which type of diffraction you see .............................. 163 Going the Distance: Special Cases of Fraunhofer Diffraction ................164 Fraunhofer diffraction from a circular aperture ............................ 165 Fraunhofer diffraction from slits......................................................167 Getting Close: Special Cases of Fresnel Diffraction ................................. 172 Fresnel diffraction from a rectangular aperture ............................173 Fresnel diffraction from a circular aperture ................................... 174 Fresnel diffraction from a solid disk ................................................ 175 Diffraction from Fresnel zone plates ...............................................175
Part IV: Optical Instrumentation: Putting Light to Practical Use ................................... 179 Chapter 13: Lens Systems: Looking at Things the Way You Want to See Them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181 Your Most Important Optical System: The Human Eye .......................... 181 Understanding the structure of the human eye ............................. 182 Accommodation: Flexing some muscles to change the focus .....183 Using Lens Systems to Correct Vision Problems ....................................185 Corrective lenses: Looking at lens shape and optical power....... 185 Correcting nearsightedness, farsightedness, and astigmatism ....... 186 Enhancing the Human Eye with Lens Systems ........................................190 Magnifying glasses: Enlarging images with the simple magni�er .............................................................. 191 Seeing small objects with the compound microscope .................192 Going the distance with the simple telescope ...............................194 Jumping to the big screen: The optical projector .........................195
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Optics For Dummies Chapter 14: Exploring Light Sources: Getting Light Where You Want It. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197 Shedding Light on Common Household Bulbs ........................................ 198 Popular bulb types and how they work .......................................... 198 Reading electrical bulb rates ............................................................ 201 Shining More-Ef�cient Light on the Subject: Light Emitting Diodes .....201 Looking inside an LED ....................................................................... 202 Adding color with organic light emitting diodes ........................... 203 LEDs on display: Improving your picture with semiconductor laser diodes.................................................204 Zeroing in on Lasers .................................................................................... 205 Building a basic laser system ........................................................... 206 Comparing lasers to light bulbs ....................................................... 211
Chapter 15: Guiding Light From Here to Anywhere. . . . . . . . . . . . . . . .213 Getting Light in the Guide and Keeping it There: Total Internal Re�ection .........................................................................213 Navigating numerical aperture: How much light can you put in? ...............................................................................214 Examining light guide modes ...........................................................215 Categorizing Light Guide Types ................................................................. 216 Fiber-optic cables ..............................................................................216 Slab waveguides ................................................................................. 220 Putting Light Guides to Work: Common Applications ............................ 221 Light pipes ..........................................................................................221 Telecommunication links .................................................................. 221 Imaging bundles ................................................................................. 224
Part V: Hybrids: Exploring More Complicated Optical Systems .................................... 227 Chapter 16: Photography: Keeping an Image Forever . . . . . . . . . . . . .229 Getting an Optical Snapshot of the Basic Camera ................................... 230 Lens: Determining what you see ...................................................... 231 Aperture: Working with f-number and lens speed .............................234 Shutter: Letting just enough light through ..................................... 236 Recording media: Saving images forever ........................................ 236 Holography: Seeing Depth in a Flat Surface .............................................237 Seeing in three dimensions ............................................................... 237 Exploring two types of holograms ................................................... 238 Relating the hologram and the diffraction grating ........................240 Graduating to 3-D Movies: Depth that Moves! .........................................243 Circular polarization..........................................................................243 Six-color anaglyph system ................................................................ 244 Shutter glasses ................................................................................... 244
Table of Contents Chapter 17: Medical Imaging: Seeing What’s Inside You (No Knives Necessary!) Necessary!) . . . . . . . . . . . . . . . . . . . . . . . . . . . .247 Shining Light into You and Seeing What Comes Out ..............................247 X-rays ................................................................................................... 248 Optical coherence tomography ....................................................... 250 Endoscopes ........................................................................................251 Reading the Light that Comes Out of You ................................................ 253 CAT scans ...........................................................................................254 PET scans ............................................................................................ 255 NMR scans ..........................................................................................256 MRI scans ............................................................................................ 257
Chapter 18: Optics Everywhere: Exploring Other Medical, Industrial, Industrial, and Military Uses . . . . . . . . . . . . . . . . . . . . . . . . .259 Considering Typical Medical Procedures Involving Lasers ................... 259 Removing stuff you don’t want: Tissue ablation............................260 Sealing up holes or incisions ...........................................................263 Purely cosmetic: Doing away with tattoos, varicose veins, and unwanted hair .............................................. 264 Getting Industrial: Making and Checking Products Out with Optics ....265 Monitoring quality control ...............................................................265 Drilling holes or etching materials ..................................................265 Making life easier: Commercial applications.................................. 266 Applying Optics in Military and Law Enforcement Endeavors .............. 267 Range �nders ...................................................................................... 267 Target designation ............................................................................. 268 Missile defense ................................................................................... 268 Night vision systems .........................................................................269 Thermal vision systems .................................................................... 270 Image processing ............................................................................... 270
Chapter 19: Astronomical Applications: Applications: Using Telescopes. Telescopes . . . . . . . .271 Understanding the Anatomy of a Telescope ............................................ 272 Gathering the light ............................................................................. 272 Viewing the image with an eyepiece ...............................................273 Revolutionizing Refracting Telescopes ....................................................274 Galilean telescope .............................................................................. 275 Kepler’s enhancement....................................................................... 276 Reimagining Telescope Design: Re�ecting Telescopes .......................... 277 Newtonian ........................................................................................... 277 Cassegrain........................................................................................... 278 Gregorian ............................................................................................279 Hybrid Telescopes: Lenses and Mirrors Working Together .................. 280 Schmidt ...............................................................................................280 Maksutov............................................................................................. 281 Invisible Astronomy: Looking Beyond the Visible .................................. .................................. 282 When One Telescope Just Won’t Do: The Interferometer...................... 283
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xviii Optics For Dummies Part VI: More Than Just Images: Getting into Advanced Optics .................... ............................... ................ ..... 285 Chapter 20: Index of Refraction, Part 2: You Can Change It! . . . . . . .287 Electro-Optics: Manipulating the Index of Refraction with Electric Fields ........................................................... 287 Dielectric polarization: Understanding the source of the electro-optic effect ..........................................288 Linear and quadratic: Looking at the types of electro-optic effects .................................................289 Examining electro-optic devices ...................................................... 293 Acousto-Optics: Changing a Crystal’s Density with Sound .................... 295 The acousto-optic acousto-optic effect: Making a variable diffraction diffraction grating ...295 Using acousto-optic devices............................................................. 296 Frequency Conversion: Affecting Light Frequency with Light ..............297 Second harmonic generation: Doubling the frequency ................297 Parametric ampli�cation: Converting a pump beam into a signal beam .................................................298 Sum and difference frequency mixing: Creating long or short wavelengths............................................. 299
Chapter 21: Quantum Optics: Optics: Finding the Photon . . . . . . . . . . . . . . . . . 301 Weaving Together Wave and Particle Properties ...................................301 Seeing wave and particle properties of light .................................. 302 Looking at wave and particle properties of matter ....................... 304 Experimental Evidence: Observing the Dual Nature of Light and Matter .....................................................................306 Young’s two-slit experiment, revisited ...........................................306 Diffraction of light and matter .......................................................... 307 The Mach-Zehnder interferometer .................................................. 308 Quantum Entanglement: Looking at Linked Photons .............................308 Spooky action: Observing interacting photons .............................308 Encryption and computers: Developing technology with linked photons ....................................................................... 309
Table of Contents
Part VII: The Part of Tens .................... .............................. .................... ............ .. 311 Chapter 22: Ten Experiments You Can Do Without a $1-Million $1-Million Optics Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313 Chromatic Dispersion with Water Spray .................................................. 313 The Simple Magni�er ..................................................................................314 Microscope with a Marble .......................................................................... 314 Focal Length of a Positive Lens with a Magnifying Glass .......................314 Telescope with Magnifying Glasses ..........................................................315 Thin Film Interference by Blowing Bubbles ............................................. 316 Polarized Sunglasses and the Sky..............................................................316 Sky..............................................................316 Mirages on a Clear Day ...............................................................................317 Spherical Aberration with a Magnifying Glass ......................................... 317 Chromatic Aberration with a Magnifying Glass ....................................... 318
Chapter 23: Ten Major Optics Discoveries — and the People Who Made them Possible . . . . . . . . . . . . . . . . . . . . . . 319 The Telescope (1610).................................................................................. 319 Optical Physics (Late 1600s) ...................................................................... 320 Diffraction and the Wave Theory of Light (Late 1600s)..........................320 1600s)..........................320 Two-Slit Experiment (Early 1800s) ............................................................ 321 Polarization (Early 1800s) ..........................................................................321 Rayleigh Scattering (Late 1800s) ...............................................................321 Electromagnetics (1861) ............................................................................. 322 Electro-Optics (1875 and 1893) .................................................................. 322 Photon Theory of Light (1905) ................................................................... 322 The Maser (1953) and The Laser (1960) ................................................... 323
Index ................... .............................. ..................... .................... ..................... ..................... .......... 325
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Chapter 1
Introducing Signals and Systems In This Chapter ▶ Figuring out the math you need for signals and systems work ▶ Determining the different types of signals and systems ▶ Understanding signal classifications and domains ▶ Checking out possible products with behavioral level modeling ▶ Looking at real products as signals and systems ▶ Using open-source computer tools to check your work
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hich came first: the signal or the system? Before you answer, you may want to know that by system, I mean a structure or design that operates on signals. You live and breathe in a sea of signals, and systems harness signals and put them to work. So which came first, you think? It may not really matter, but I’m guessing — as I smooth out a long imaginary philosopher-type beard — that signals came first and then began passing through systems. But I digress. The study of signals and systems as portrayed in this book centers on the mathematical modeling of both signals and systems. Mathematical modeling allows an engineer to explore a variety of product design approaches without committing to costly prototype hardware and software development. After you tune your model to produce satisfactory results, you can implement your design as a prototype. And at some point, real signals (and sometimes math-based simulations) test the system design before full implementation. When studying signals and systems, it’s easy to get mired in mathematical details and lose sight of the big picture — the functional systems of your end result. So try to remember that, at its best, signals and systems is all about designing and working with products through applied math. Math is the means, not the star of the show.
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Part I: Getting Started with Signals and Systems Two broad classes of signals are those that are continuous functions of time t and those that are discrete functions of time index n. Throughout this book, I separate information on continuous- and discrete-time signals and systems. In this chapter, I introduce simple continuous and discrete signals and the corresponding systems. I also point out some of the distinguishing characteristics of signal types. Before getting started, I want to mention that signals as functions of time are how most people experience the real world of computer and electronic engineering, yet transforming signals and systems to other domains — specifically, the frequency, s- , and z- domains — and back again is quite beneficial in some situations. I touch on the transformation of signals and systems in this chapter and dig into the details in Parts III and IV. In this chapter, I also cover the important role of computer tools in signals and systems problem solving and tell you how to use a few specific opensource programs. If you want to set up these freely available tools on your computer, you can follow along when I describe specific functions that enable you to check your work or work more efficiently — after you get a handle on core concepts and techniques.
Applying Mathematics Anyone aspiring to a working knowledge of signals and systems needs a solid background in math, including these specific concepts:
✓ Calculus of one variable ✓ Integration and differentiation ✓ Differential equations To actually implement designs that center on signals and systems, you also need a background in these subjects:
✓ Electrical/electronic circuits ✓ Computer programming fundamentals, such as C/C++ and Java ✓ Analysis, design, and development software tools ✓ Programmable devices Many signals and systems designers rely on modeling tools that use a matrix/ vector language or class library for numerics and a graphics visualization capability to allow for rapid prototyping. I use numerical Python for examples in this book; other languages with similar syntax include MATLAB and NI LabVIEW MathScript.
Chapter 1: Introducing Signals and Systems
Finding perspective on analog processing Once upon a time, the implementation path for signals and systems was purely analog circuit design. As technology has advanced, solutions based on digital signal processing (discrete-time signals and systems) through powerful low-cost and low-power digital hardware has become the mainstay. Digital hardware solutions are programmable and can be reconfigured through software updates after products ship.
The signals you’re likely to work with in the real world are analog in nature, but you’ll almost always process them digitally. Knowing programming languages is important in this environment. Yet analog signal processing is alive and well — it’s vital to your working knowledge of signals and systems — but the overall role of analog processing in current design is less formidable than it’s been in the past.
With so many electrical engineering solutions being software-based today — versus a matter of analog circuitry (see nearby sidebar “Finding perspective on analog processing”) — a system designer can also be the implementer. This leap requires only simulation code to be transformed into the implementation language, such as Verilog or C/C++. Working pencil-and-paper solutions for signals and systems coursework requires a good scientific calculator. I recommend a calculator that supports complex arithmetic operations, using the minimum number of keystrokes. At minimum, your calculator needs to have trig, log, and exponential functions for signals and systems work.
Getting Mixed Signals . . . and Systems Signals come in two flavors: continuous and discrete. It’s the same story with systems. In other words, some signals — and some systems — are active all the time; others aren’t. In this section, I describe continuous and discrete signals along with the corresponding systems. I also tell you how to classify certain signals and systems based on their most basic properties.
Going on and on and on Continuous-time signals and systems never take a break. When a circuit is wired up, a signal is there for the taking, and the system begins working — and doesn’t stop. Keep in mind that I use the term signal here loosely; any one specific signal may come and go, but a signal is always present at each and every time instant imaginable in a continuous-time system.
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Part I: Getting Started with Signals and Systems
Continuous-time signals Continuous signals function according to time t . A sinusoidal function of time is one of the most basic signals. The mathematical model for a sinusoid signal is , where A is the signal amplitude, is the signal frequency, and is the signal phase shift. The independent variable is time t . If you’re curious about the first peak of x ( t ) occuring at 3/16, notice that this occurs when the argument of the cosine is 0 — the is, or . I cover this signal in detail in Chapter 3, but to help you get acquainted, check out the plot of a sinusoid signal in Figure 1-1.
Figure 1-1: The plot of a sinusoidal signal.
The amplitude of this signal is 3, the frequency is 2 Hz, and the phase shift is rad.
Continuous-time systems Systems operate on signals. In mathematical terms, a system is a function or , that maps the input signal to output signal . operator, An example of a continuous-time system is the electronic circuits in an amplifier, which has gain 5 and level shift 2: . See a block diagram representation of this simple system in Figure 1-2.
Figure 1-2: A simple continuous time system model.
Building an amplifier that corresponds to this mathematical model is another matter entirely. You can create a simple electronic circuit, but it will have limitations that the math model doesn’t have. It’s up to you, as an electronic
Chapter 1: Introducing Signals and Systems engineer, to refine the model to accurately reflect the level of detail needed to assess overall performance of a design candidate.
Working in spurts: Discrete-time signals and systems Discrete-time signals and systems march along to the tick of a clock. Mathematical modeling of discrete-time signals and systems shows that activity occurs with whole number (integer) spacing, but signals in the real world operate according to periods of time, or the update rate also known as the sampling rate. Discrete-time signals, which can also be viewed as sequences, only exist at the ticks, and the systems that process these signals are, mathematically speaking, resting in the periods between signal activity. Systems take inputs and produce outputs with the same clock tick, generally speaking. Depending on the nature of the digital hardware and the complexity of the system, calculations performed by the system continue — between clock ticks — to ensure that the next system output is available at the next tick when a new signal sample arrives at the input.
Discrete-time signals Discrete-time signals are a function of time index n. Discrete-time signal , unlike continuous-time signal , takes on values only at integer number values of the independent variable n. This means that the signal is active only at specific periods of time. Discrete-time signals can be stored in computer memory because the number of signal values that need to be stored to represent a finite time interval is finite. The following simple signal, a pulse sequence, is shown in Figure 1-3 as a stem plot — a plot where you place vertical lines, starting at 0 to the sample value, along with a marker such as a filled circle. The stem plot is also known as a lollipop plot — seriously.
Figure 1-3: A simple discrete time signal.
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Part I: Getting Started with Signals and Systems The stem plot shows only the discrete values of the sequence. Find out more about discrete-time signals in Chapter 4.
Discrete-time systems A discrete-time system, like its continuous-time counterpart, is a function, , that maps the input to the output . An example of a discrete-time system is the two-tap filter:
The term tap denotes that output at time instant n is formed from two time instants of the input, n and n – 1. Check out a block diagram of a two-tap filter system in Figure 1-4.
Figure 1-4: A simple discrete time system model.
In words, this system scales the present input by 3/4 and adds it to the past value of the input scaled by 1/4. The notion of the past input comes about because is lagging one sample value behind . The term filter describes the output as an averaging of the present input and the previous input. Averaging is a form of filtering.
Classifying Signals Signals, both continuous and discrete, have attributes that allow them to be classified into different types. Three broad categories of signal classification are periodic, aperiodic, and random. In this section, I briefly describe these classifications (find details in Chapters 3 and 4).
Periodic Signals that repeat over and over are said to be periodic. In mathematical terms, a signal is periodic if
Chapter 1: Introducing Signals and Systems
The smallest T or N for which the equality holds is the signal period. The sinusoidal signal of Figure 1-1 is periodic because of the property of cosine. The signal of Figure 1-1 has period 0.5 seconds (s), which turns out to be the reciprocal of the frequency Hz. The square wave signal of Figure 1-5a is another example of a periodic signal.
Figure 1-5: Examples of signal classifications: periodic (square wave) (a), aperiodic (rectangular pulse) (b), and random (noise) (c).
Aperiodic Signals that are deterministic (completely determined functions of time) but not periodic are known as aperiodic. Point of view matters. If a signal occurs infrequently, you may view it as aperiodic. The rectangular pulse of duration shown in Figure 1-5b is an aperiodic signal.
Random A signal is random if one or more signal attributes takes on unpredictable values in a probability sense (you love statistics, right?).
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Part I: Getting Started with Signals and Systems The full mathematical description of random signals is outside the scope of this book, but here are two good examples of a random signal:
✓ The noise you hear when you’re between stations on an FM radio. See a waveform representation of this noise in Figure 1-5c. ✓ Speech: If you try to capture audio samples on a computer of someone speaking the word hello over and over, you’ll find that each capture looks a little different. Engineers working with communication receivers are concerned with random signals, especially noise.
Signals and Systems in Other Domains Most of the signals you encounter on a daily basis — in computers, in wireless devices, or through a face-to-face conversation — reside in the time domain. They’re functions of independent variable t or n. But sometimes when you’re working with continuous-time signals, you may need to transform away from the time domain ( t ) to either the frequency domain ( ) or the s- domain ( s ). Similarly, for discrete-time signals, you may need to transform from the discrete-time domain ( n ) to the frequency domain ( ) or the z- domain ( ). z Systems, continuous and discrete, can also be transformed to the frequency and s- and z -domains, respectively. Signals can, in fact, be passed through systems in these alternative domains. When a signal is passed through a system in the frequency domain, for example, the frequency domain output signal can later be returned to the time domain and appear just as if the timedomain version of the system operated on the signal in the time domain. This section briefly explores the world of signals and systems in the frequency, s- , and z- domains. Find more on these alternative domains in Chapters 13 and 14.
Viewing signals in the frequency domain The time domain is where signals naturally live and where human interaction with signals occurs, but the full information for a signal isn’t always visible in that space. Consider the sum of a two-sinusoids signal (as depicted in Figure 1-6):
Chapter 1: Introducing Signals and Systems
Figure 1-6: The frequency domain view for a sum of a twosinusoids signal.
The top waveform plot, denoted s1, is a single sinusoid at frequency f 1 and peak amplitude A1. The waveform repeats every period T 1 = 1/f 1. The second waveform plot, denoted s2, is a single sinusoid at frequency f 2 > f 1 and peak amplitude A2 < A1. The sum signal, s1 + s2, in the time domain is a squiggly line (third waveform plot), but the amplitudes and frequencies (periods) of the sinusoids aren’t clear here as they are in the first two plots. The frequency spectrum (bottom plot) reveals that is composed of just two sinusoids, with both the frequencies and amplitudes discernible. Think about tuning in a radio station. Stations are located at different center frequencies. The stations don’t interfere with one another because they’re separated from each other in the frequency domain. In the frequency spectrum plot at the bottom of Figure 1-6, imagine that f 1 and f 2 are the signals from two radio stations, viewed in the frequency domain. You can design a receiving system to filter s1 from s1 + s2. The filter is designed to pass s1 and block s2. (I cover filters in Chapter 9.) Use the Fourier transform to move away from the time domain and into the frequency domain. To get back to the time domain, use the inverse Fourier transform. (Find out more about these transforms in Chapter 9.)
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Part I: Getting Started with Signals and Systems
Traveling to the s- or z-domain and back From the time domain to the frequency domain, only one independent variable, , exists. When a signal is transformed to the s- domain, it becomes a function of a complex variable . The two variables (real and imaginary parts) describe a location in the s- plane. In addition to visualization properties, the s- domain reduces differential equation solving to algebraic manipulation. For discrete-time signals, the z- transform accomplishes the same thing, except differential equations are replaced by difference equations. Did you think going to the z- domain meant taking a nap? Details on difference equations begin in Chapter 7.
Testing Product Concepts with Behavioral Level Modeling Computer and electrical engineers provide society with a vast array of products — ranging from cellphones and high-definition televisions to powerful computers with high resolution displays that are small and lightweight. The mystery of how brilliant people come up with world-changing ideas may never be solved, but after an idea is out there, engineers work through a process that allows them to test, or model, potential solutions to find out whether the idea is likely to work in the real world. For products that rely on signal processing, engineers use signals and system modeling and analysis to reveal what’s possible. When you’re trying to quickly prove a solution approach, you’ll often turn to behavioral level modeling of certain elements of the overall system to avoid low-level implementation details. For example, a subsystem design may require knowledge of a signal parameter (such as amplitude or frequency) to function. At first, you may assume that the parameter is well known. Later, you add low-level details to estimate (not perfectly) the parameter. As your confidence and understanding grows, you represent the low-level details in the model and actual implementation becomes possible. Behavioral level modeling also applies when you need to model physical environments that lie outside a design but are needed to evaluate performance under realistic scenarios.
Chapter 1: Introducing Signals and Systems In this section, I describe the role of abstraction as a means to generate preliminary concepts and then work those concepts into a top-level design. The top-level design becomes a detailed plan as you work down to implementation specifics. Mathematical modeling is a thread running through the entire process, so you come to rely on it.
Staying abstract to generate ideas Behavioral level modeling isn’t void of hardware constraints and realities, but it requires a certain level of abstraction to allow preliminary concept solutions to materialize quickly. Behavioral level models depend on applied mathematics. In other words, computer and electronic engineers don’t frequently handle actual hardware and devices used for an implementation. The model of the hardware is what’s important at this point. The engineer’s job is to conceptualize systems and subsystems through a framework of mathematical concepts, and abstraction provides great creative freedom to explore the possibilities. Suppose you seek a new design for an existing system to improve performance. You hope to make such improvements with new device technology. You don’t want to get bogged down in all the details of how to interface this device into the current design, so you move up in abstraction with a model to quickly find out how much you can improve performance with a new design. If adequate improvement potential doesn’t exist, then you settle down and investigate other options. Rinse, lather, and repeat. Keep in mind that improved performance isn’t always the primary objective of signals and systems modeling. Sometimes, a design is driven by cost, availability of materials, manufacturing processes, and time to market, or some other consideration.
Working from the top down A design that relies on signals and systems starts from a top-level view and works down to the nitty-gritty details of final implementation. Analysis and simulation performed at the top level depends on behavioral level modeling. The model is ultimately broken into subsystems for testing and refinement, and then the system comes together again before implementation.
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Part I: Getting Started with Signals and Systems Typically, your task as an electronic engineer is to create some new or enhanced functionality for a computer- or electrical-based product. For example, you may need to support a new radio interface due to recent standard updates. At first, the changes may seem simple and straightforward, but as you dig into the work, you may begin to see that the changes require significant adjustments in signal processing algorithms. This means that the new radio interface will require a few totally new designs, so you need to model and simulate various implementation approaches to find out what’s likely to work best.
Relying on mathematics Many people write off signals and systems as a pile of confusing math, and they run for the hills. True, the math can be intimidating at first, but the rewards of seeing your finely crafted mathematical model lead the way to a shipping product is worth the extra effort — at least I think so. In the end, the math is on your side. It’s the only way to model concepts that function properly in the real world. My go-to approach when a problem seems unsolvable: Take it slow and steady. If a solution isn’t clear after you think about the problem for a while, walk away and come back to it later. Practice and experience with various problem-solving techniques and options help, so try to work as many types of problems as you can — especially in the areas you feel the most discomfort. Eventually, a solution reveals itself. When possible, verify your solutions by using computer analysis and simulation tools. In this book, I use Python with the numerical support and visualization capabilities of PyLab (NumPy, SciPy, matplotlib) and the IPython environment to perform number-crunching analysis and simulations. For problems involving more symbolic mathematics, I use the computer algebra system (CAS) provided by Maxima.
Exploring Familiar Signals and Systems I’m guessing you have some level of familiarity with consumer electronics, such as MP3 music players, smartphones, and tablet devices, and realize that these products rely on signals and systems. But you may take for granted the cruise control in your car. In this section, I point out the signals and systems framework in familiar devices at the block diagram level — a system diagram
Chapter 1: Introducing Signals and Systems that identifies the significant components inside rectangular boxes, interconnected with arrows that show the direction of signal flow. The block diagram expresses the overall concept of a system without intimate implementation details.
MP3 music player Signals and systems are operating in all the major peripherals of the music player — even in the processor. In reality, signals are in every part of the system, but I exclude pure digital signals in this example, so I don’t address memory. The processor runs an operating system (OS); under that OS, tasks perform digital signal processing (DSP) algorithms for streaming audio and image data. Note that this book is focused on one-dimensional signals only. Find a top-level block diagram of an MP3 device in Figure 1-7. All the peripheral blocks (the blocks that sit outside the processor block) contain a combination of continuous- and discrete-time systems. You stream digital music in real time from memory in a compressed format. The processor has to decompress the audio stream into signal sample values (a discrete-time signal) to send to the audio codec. The audio codec contains a digital-to-analog converter (DAC) that converts the discrete-time signal to a continuous-time signal. The Wi-Fi and Bluetooth radios (blocks with antennas) interface to the processor with digital data but interface to the antenna by using a continuoustime signal at a frequency of 2.4 GHz. The sensors’ block acquires analog signals from the environment, temperature, light level, and acceleration in three dimensions.
Figure 1-7: MP3 music player block diagram.
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Part I: Getting Started with Signals and Systems
Smartphone The structure of a smartphone is similar to an MP3 music player, but a smartphone also has a global positioning receiver (GPS) and multiband radio blocks that send and receive continuous-time signals from base stations (antenna sites) of a cellular network. The GPS receiver acquires signals from multiple satellites to get your latitude and longitude. The primary purpose of the GPS in most smartphones is to provide location information when placing an emergency call (E911). Check out a block diagram of a smartphone in Figure 1-8. Four antennas are shown, but only a single multiband antenna is employed in most models, so only a single antenna structure is really needed.
Figure 1-8: Smartphone block diagram.
The multiband cellular radio subsystem is thick with signals and systems. The multiband digital communications transmitter (tx) and receiver (rx) allows the smartphone to be backward compatible with older technologies as well as with the newest high-speed wireless data technologies. This transmitter and receiver enable the product to operate throughout the world. A smartphone is overflowing with signals and systems examples!
Automobile cruise control I think all new automobiles come equipped with a cruise control system now. This is good news because this feature may keep you from getting a speeding ticket when you’re driving long distances on the interstate. It’s also great for getting better gas mileage. But I’m no sales guy for cruise control. I just think this product is interesting from a signals and systems standpoint.
Chapter 1: Introducing Signals and Systems Figure 1-9 shows a block diagram of a cruise control system. Cruise control involves both electrical and mechanical signals and systems. The controller is electrical and the plant, the system being controlled, is the car. Wind and hills are disturbance signals, which thwart the normal operation of the control system. The controller puts out a compensating signal to the throttle to overcome wind resistance (an opposing force) and the force of gravity when going up and down hills. The error signal that follows the summing block is driven to a very small value by the action of the feedback loop. This means that the output velocity tracks the reference velocity. This is exactly what you want. For a more detailed look at cruise control, check out the case studies at www.dummies.com/extras/signalsandsystems .
Figure 1-9: Block diagram of an automobile cruise con trol system.
Using Computer Tools for Modeling and Simulation Today’s technology-based solutions are rarely built without the use of some form of computer tool. Signals and systems research and product development is no exception. Throughout this book, I show you how to solve problems by hand calculation and how to check your work with computer tools. Hand calculation is vital for building concepts. Computer tools help ensure that you don’t make mistakes. And why wouldn’t you use the best tools available for your work? A variety of commercial and open-source tools are available for signals and systems problem solving. Two broad categories are computer algebra system (CAS) programs, such as Mathematica, Maple, and Maxima, and those that excel at vector/matrix problem solving, such as MATLAB, NI LabVIEW MathScript, Octave, and Python. Both types of computer programs offer function libraries that are tailored to the needs of the signals and systems analysis and simulation.
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Part I: Getting Started with Signals and Systems The examples in this book feature two open-source tools:
✓ Scientific Python via PyLab and the shell IPython Python becomes scientific Python with the inclusion of NumPy and SciPy for vector/matrix number crunching and matplotlib for graphics.
✓ CAS Maxima via wxMaxima I’ve chosen open-source tools because I want to provide an easy on-ramp for users everywhere. Both Mac and Windows OS computers can run these software products via free downloads. Specifically for this book, I wrote the code module ssd.py , which provides additional signals and systems functions. After you import this module into your IPython session, you can run all the examples in this book. I prefer to use the QT console version of IPython (see www.ipython.org ). Similarly for wxMaxima, the notebook ssd.wxm contains all the example code from this book, organized by chapter.
Getting the software Python and IPython (including NumPy, SciPy, and matplotlib) from Enthought Python Distribution (EPD) is a free download for the 32-bit version ( www. enthought.com/products/epd_free.php ). Python(x,y) is also very good, especially under Windows ( http://code.google.com/p/pythonxy ). If you’re running Linux, in particular Ubuntu Linux, the Ubuntu Software Center is a good starting place. If you’re an experienced open-source user, you can do a custom install as opposed to the monolithic distributions. If you’re looking for a full integrated development environment (IDE) for debugging Python, I suggest the open-source IDE Eclipse ( www.eclipse.org ) with the plug-in PyDev ( http://pydev.org ). Eclipse is supported on Mac, Windows, and Linux. I developed the module ssd.py by using this setup. Find wxMaxima for Windows and Mac at http://andrejv.github.com/ wxmaxima. Under Ubuntu Linux, you can find wxMaxima in the Ubuntu Software Center. To get files specific to this book go to www.dummies.com/extras/ signalsandsystems for the Python code module ssd.py and the Maxima notebook ssd.wxm along with some tutorial screencasts and documents.
Chapter 1: Introducing Signals and Systems
Exploring the interfaces Take a quick tour of the interfaces of these computer programs when you get them installed. I provide a peek of how the program looks on the Mac in Figures 1-10, 1-11, and 1-12. The appearance and functionality for Windows is virtually the same.
Figure 1-10: The wxMaxima notebook interface to Maxima.
You can send Maxima plots to a file in a variety of formats or display them directly in the notebook, as shown in Figure 1-11.
Figure 1-11: The IPython QT console window.
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Part I: Getting Started with Signals and Systems You can write and debug functions right from the console window, as shown in Figure 1-12.
Figure 1-12: matplotlib plot window resulting from a call to plot (x,y) in IPython.
You can manipulate plots by using the controls you see at the bottom of the figure window. Plot cursors are also available. You can save plots from the command line or from the figure window. Many of the plots found in this book were created with matplotlib.
Seeing the Big Picture Figure 1-13 illustrates the content organization of this book as an unfolding of core topics, starting from the time domain and moving to the frequency domain before exploring the s- and z -domains. Continuous (left side) and discrete (right side) signals and systems topics parallel each other every step of the way — with some continuous- and discrete-time topics shared (center) within a few chapters. The last four chapters, which follow the z- domain chapter, emphasize applications, including signal processing, wireless communications, and control systems. I start with the time domain because this is where signals originate and where systems operate on signals (with the exception of transform domain processing, which is covered in Chapter 12). The frequency domain augments a base knowledge of both signals and systems and is important to grasping sampling theory, which leads to the processing of continuous-time signals in the discretetime domain. The s- and z -domain are the last of the core topics, but by no means are they any less important than the topics that come before them. The s- and z- domains are particularly powerful when working with linear time-invariant systems described by differential and difference equations.
Chapter 1: Introducing Signals and Systems After covering the core topics, you can appreciate the chapter that focuses on how to work across domains (Chapter 15). Get a taste of how signals and systems fit into the real world of electrical engineering by reading the case studies at www.dummies.com/extras/signalsandsystems . Take a look at the application examples to get inspired when you’re struggling to see the forest for the trees of the dense study of signals and systems.
Figure 1-13: Signals and systems topic flow.
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Contents at a Glance Introduction ................................................................ 1 Part I: Getting Started with Signals and Systems ........... 7 Chapter 1: Introducing Signals and Systems .................................................................. 9 Chapter 2: Brushing Up on Math ................................................................................... 29 Chapter 3: Continuous-Time Signals and Systems ...................................................... 51 Chapter 4: Discrete-Time Signals and Systems ............................................................ 77
Part II: Exploring the Time Domain.............................. 97 Chapter 5: Continuous-Time LTI Systems and the Convolution Integral ................. 99 Chapter 6: Discrete-Time LTI Systems and the Convolution Sum ........................... 119 Chapter 7: LTI System Differential and Difference Equations in the Time Domain ..................................................................................................... 149
Part III: Picking Up the Frequency Domain ............... 163 Chapter 8: Line Spectra and Fourier Series of Periodic Continuous-Time Signals ............................................................................................ 165 Chapter 9: The Fourier Transform for Continuous-Time Signals and Systems ..... 191 Chapter 10: Sampling Theory ....................................................................................... 219 Chapter 11: The Discrete-Time Fourier Transform for Discrete-Time Signals.......241 Chapter 12: The Discrete Fourier Transform and Fast Fourier Transform Algorithms ................................................................... 263
Part IV: Entering the s- and z-Domains ...................... 283 Chapter 13: The Laplace Transform for Continuous-Time....................................... 285 Chapter 14: The z -Transform for Discrete-Time Signals ........................................... 307 Chapter 15: Putting It All Together: Analysis and Modeling Across Domains ....... 327
Part V: The Part of Tens ........................................... 343 Chapter 16: More Than Ten Common Mistakes t o Avoid When Solving Problems.............................................................................................. 345 Chapter 17: Ten Properties You Never Want to Forget ............................................351
Index ...................................................................... 355
™ ! r e si a E g in Making Ever yth
s c i t a St
Learn to: • Grasp the study of statics for success in the classroom • Apply complex concepts such as vectors, internal and external forces, and free-body diagrams • Solve problems in every aspect of statics
James H. Allen III, PE, PhD, Assistant Professor of Civil Engineering University of Evansville
Chapter 1
Using Statics to Describe the World around You In This Chapter ▶ Defining statics and related studies ▶ Introducing vectors ▶ Exploring free-body diagrams ▶ Looking at specific applications of statics
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tatics is a branch of physics that is especially useful in the fields of engineering and science. Although general physics may describe all the actions around you, from the waving of leaves on a tree to the reflection of light on a pond, the field of statics is much more specific. In fact, statics is actually a part of most physics courses. So if you’ve ever taken a high school or college physics course, chances are that some of the information in this book may seem vaguely familiar. For example, one of the first areas you study in physics is often Newtonian mechanics, which is basically statics and dynamics. Physics classes typically cover a wide range of topics, basically because physics has a wide range of applications. Conversely, a statics course is much more focused (which doesn’t necessarily mean it’s simple). Whoever said that the devil is in the details may well have been talking about statics. Before you panic, close the book, and begin questioning why you ever thought you could understand statics, let me reassure you that just because statics isn’t always simple doesn’t mean it’s always difficult. If anything, statics does happen to be very methodical. If you follow some basic steps and apply some basic ideas and theory, statics actually can become a very straightforward application process. Now, about those details . . .
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Part I: Setting the Stage for Statics
What Mechanics Is All About The study of the world around you requires knowledge of many areas of physics, often referred to as mechanics. The mathematician Archimedes of Syracuse (287–212 BC) is often credited as being the first person to systematically study the behavior of objects by using mechanics and is attributed with saying “Give me a place to stand and I will move the Earth.” This statement, while rather grandiose for his time, proves itself to be at the very heart of the study of mechanics (and, more specifically, statics).
Mechanics refers to one of the core areas of physics, usually concentrated around the principles of Sir Isaac Newton and his basic laws of motion, and is an area of concentration that engineers and scientists often study in addition to basic physics classes. These courses develop the core curriculum for many basic engineering programs and are usually common classes across all disciplines. Specific engineering disciplines may require additional courses in each of these core areas to teach additional (and often more advanced) topics. One of these core areas is in the area of statics, which isn’t the study of how you should move across a shag carpet in order to apply a jolt of electricity to your younger siblings or how to implement the latest hygiene techniques to avoid those dreadful bad hair days. In this book, I define statics as the mechanical study of the behavior of physical objects that remain stationary under applied loads (which I discuss later in this chapter). The behavior of the floor beams in your house as you stand in the middle of your living room is an example of a static application. The area of dynamics , on the other hand, is the study of objects in motion. So as you walk down the hall, your behavior and its effect on your house becomes a dynamic application. The result of a car driving down a bumpy road, the flow of water through a creek, and the motion of those shiny little metallic balls that hang from strings and haunt/hypnotize you with their “clack, clack, clack” as they bounce off each other are all examples of dynamic behavior. Finally, you come to mechanics of materials (sometimes referred to as strength of materials ), which is yet another branch of mechanics that focuses on the behavior of objects in response to loads. This area of mechanics builds on concepts from both dynamics and statics.
Putting Vectors to Work One of the most basic tools to include in your basket of statics tricks is the knowledge of vectors, which I discuss in detail in Part II of this book. Think of vectors as being one of the major staples, such as rice or potatoes, of your
Chapter 1: Using Statics to Describe the World around You statics kitchen. Statics forms the foundation for a complete meal of engineering design. Vectors come in all shapes and forms, and you can use them for a wide variety of purposes, which I introduce you to in Chapters 4 and 5. But the vector discussion doesn’t end there. I also show you several different ways to mathematically work with vectors, including building the foundation for a vector’s equation (see Chapters 6, 7, and 8).
Peeking at a few vector types One of the first vectors you need to get familiar with is the position vector, which basically tells you how to get from one point to another. These vectors are very handy for giving directions, measuring distances, and creating other vectors; you can read about them in Chapter 5. The most common type of vector that you deal with has to do with loads, or forces (see the following section). Think of a force as being that number that pops up when you step on your bathroom scale that reminds you that you should have worked out last night instead of eating a second helping of cheesecake. The bigger that number gets, the bigger the force that is being applied to your scale. Forces are one of the major types of actions that can affect a body in statics.
Understanding some purposes of vectors One purpose of vectors is to help define direction. Many forces act along straight lines but aren’t necessarily acting at a distinct point. By creating a unit vector (a special type of vector with a specific length), you can define the direction of these lines without actually knowing the specific coordinates or location data; unit vectors also prove to be very useful for creating forces (another type of vector). Check out Chapter 5 for more on these vectors as well. You can also use vectors to define the rotational behaviors (or spinning effects) of an object, which I explain in Chapter 12. You can also combine multiple vectors to create a single combined vector, which can be useful for dealing with multiple forces. In addition, knowing how you can break down vectors into smaller vectors and calculate their size allows you to determine, say, how big a chair needs to be to support a given weight, including figuring out the size of its legs and even the number of legs necessary. In fact, for three-dimensional statics problems, vectors are practically mandatory. Chapters 7 and 8 deal with combining and breaking down vectors, respectively.
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Part I: Setting the Stage for Statics
Defining Actions in Statics In mechanics, you must become familiar with a large number of actions to be able to study how an object behaves, ranging from velocity and momentum in dynamics, to thermal effects, stress, and strain in mechanics. Fortunately, the types of effects in statics are contained in a fairly brief list:
✓ Forces: Forces are a type of load that causes an object to translate (move linearly) in the direction of the applied force. Forces can be spread out or acting at a single location, but they always cause an object to want to translate. You can use forces to measure the intensity of one object striking another, the weight of a car as it drives across a bridge deck, or the effect of water pressure on the side of a submarine. Flip to Chapters 9 and 10 for more on forces. ✓ Moments: Moments are a type of load that causes an object to rotate in space without translation. Moments are usually the result of some sort of twisting or spinning effect, such as a shaft attached to a motor, or a reaction from a second object that is acting on the other. For example, turning the handle of a wrench applies a moment to a bolt, which then causes it to rotate. Chapter 12 gives you the lowdown on moments. One of your biggest challenges in statics is how to accurately depict and determine the type of action or behavior being applied to a system. If an elephant sits on your favorite living room recliner, you can easily tell what the final outcome of that action will probably be: You now have a broken chair, and a trip to the furniture store is in your future. Although most people will wonder how you got an elephant in your living room in the first place, as a statics enthusiast you’re more interested in exploring the behavior of the elephant’s weight and determining how much force is transmitted through the seat, into the legs, and ultimately into the ground. This field is where your study of statics begins (don’t worry, no zoology or elephant anatomy knowledge is required). Because forces and moments are such an important part of statics, you need to be able to calculate them for different kinds of problems. In Part III, I show you how to calculate forces and moments in both two- and three-dimensional situations. Load effects in statics are typically classified into three basic categories:
✓ Concentrated forces: Concentrated forces (or forces that act at a single point) include the force from a ball as it’s thrown toward a wall, or even the force that your shoes exert on the floor from your self weight. I cover these forces more in Chapter 9. ✓ Distributed forces: Distributed forces are forces that are spread over an area and are used to represent a wide variety of forces on objects. The
Chapter 1: Using Statics to Describe the World around You weight of snow on the roof of your house or of soil pressure on your basement wall is a distributed load. Chapter 10 shows you how to determine their net effect (or the resultant ), and Chapter 11 illustrates how to determine the location where this resultant is acting.
✓ Concentrated moments: Concentrated moments are a type of load that causes a rotation effect on an object. The behavior of your hand on a door knob or a wrench on a nut is an example of rotational behaviors that are caused by moments. I describe the types of moments and how they are created in more detail in Chapter 12.
Sketching the World around You: Free-Body Diagrams The ability to draw a free-body diagram (or F.B.D., the picture representations of the problem you want to investigate) is vital when you start a static analysis because F.B.D.s depict the problem you’re trying to solve, and they help you write the equations you need for performing a static analysis. In fact, if you don’t get the F.B.D. completely correct, you may end up solving for a completely different problem altogether. The more you practice creating free-body diagrams, the more proficient you become. Free-body diagrams must feature various items, including dimensions, self weight, support reactions, and the various forces I discuss in Part III. (Head to Chapter 13 for a full checklist of required items.) You can also break a larger F.B.D. into additional diagrams; this tactic is useful because you can use these smaller diagrams to find information that helps you solve for a wide variety of effects, such as support reactions (physical restraints) and internal forces, that you may not notice on the larger drawing. You can find information on these topics in Chapter 14. When you’re working with F.B.D.s with multiple applied loads and supports, simplifying those diagrams can make your work a lot, well, simpler. Chapter 15 gives you several tricks for simplifying F.B.D.s; one of the most useful techniques is the principle of superposition, which allows you to quickly compute behaviors by simply adding the responses of the individual cases. You can also simplify your diagram by moving a force from one location on an object to another while preserving the original behavior; you can read more about this in Chapter 15 as well.
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Part I: Setting the Stage for Statics
Unveiling the Concept of Equilibrium Isaac Newton (1642–1727) helped establish the laws of motion and gravity (covered in Chapter 16) that are still used today. Equilibrium is a special case of Newton’s laws where acceleration of an object is equal to zero (that is, it isn’t experiencing an acceleration), which results in an object being in a stable or balanced condition. If you lean back in your chair such that it’s supported by two legs, you notice that you reach a special point where you remain somewhat balanced. (But don’t try this at home.) However, if you lean a little bit forward, the chair starts to rock forward and usually winds up safely back on the front two legs. This simple motion means that equilibrium hasn’t been maintained. If you lean too far back, the chair starts to lean backward and unless you catch yourself, you soon find yourself lying on the ground. But good news: While you’re lying on your back counting the little birds circling your head, you’ve actually arrived at a new equilibrium state. Although you can simplify statics down to three basic equilibrium relationships for two-dimensional problems (and six equations for three-dimensional problems, though they’re similar in concept), you can investigate a wide variety of problems with these relationships. Flip to Chapters 17 and 18 for more on equilibrium in two and three dimensions, respectively.
Applying Statics to the Real World So what’s an engineer to do after getting a handle on F.B.D.s, loads, equilibrium, and other statics trappings? Why, put them to use in actual applications, of course! Real-world statics is where all the conceptual info you read about becomes much more interesting and much more practical. You can employ statics concepts to a wide variety of applications; some of the most common ones include the following:
✓ Trusses: Trusses are systems of simple objects interconnected to create a single combined system. They’re commonly used in roof systems and as bridges that span large distances. In Chapter 19, I explain the basic assumptions of trusses and then illustrate the method of joints and the method of sections for analyzing forces within the truss. ✓ Beams and bending members: The majority of objects you work with in statics have up to three different types of internal forces (axial, shear, and moment, which I cover in Chapter 20). These internal forces are what engineers use to design structural members within a building. The
Chapter 1: Using Statics to Describe the World around You forces sometimes cause a member to deflect (move away from being parallel), creating a bending member . You analyze these bending members by using shear and moment diagrams, which you can also read about in Chapter 20.
✓ Frames and machines: Frames and machines, though similar to trusses, can experience similar behaviors to beams and bending members. In fact, a large number of structural objects and tools that you use on a daily basis are actually either a frame or a machine. For example, simple hand tools such as clamps, pliers, and pulleys are examples of simple machines. Frames are more general systems of members that you can use in framing for structures. Chapter 21 gives you the lowdown on working with frames and machines. ✓ Cable systems: Cable systems are a unique type of structure and can produce some amazing architectural bridges known as suspension bridges. In Chapter 22, I describe the assumptions behind cable systems and present the techniques you need to solve cable problems. ✓ Submerged surfaces: Submerged surfaces are objects that are subjected to fluid pressure, such as dams. Fluids can apply hydrostatic pressure and pressure from self weight to submerged surfaces, and I describe both of those in Chapter 23. A discussion of statics applications wouldn’t be complete without talking about friction, the resistance an object feels along a contact surface as it moves in a particular direction. The two main types of frictional behavior are sliding (where the object moves across the surface in response to a force) and tipping (where the object responds to a force by toppling over rather than moving across a surface). These friction forces are the source of a large number of strange behaviors and require you to make assumptions about a behavior and then use free-body diagrams and the equations of equilibrium to verify them. Chapter 24 is your headquarters for all things friction.
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Contents at a Glance Introduction ................................................................ 1 Part I: Setting the Stage for Statics .............................. 7 Chapter 1: Using Statics to Describe the World around You ....................................... 9 Chapter 2: A Quick Mathematics Refresher ................................................................. 17 Chapter 3: Working with Unit Systems and Constants ............................................... 31
Part II: Your Statics Foundation: Vector Basics ............ 39 Chapter 4: Viewing the World through Vectors...........................................................41 Chapter 5: Using Vectors to Better De�ne Direction .................................................. 51 Chapter 6: Vector Mathematics and Identities ............................................................ 69 Chapter 7: Turning Multiple Vectors into a Single Vector Resultant ........................ 79 Chapter 8: Breaking Down a Vector into Components ............................................... 95
Part III: Forces and Moments as Vectors .................... 107 Chapter 9: Applying Concentrated Forces and External Point Loads..................... 109 Chapter 10: Spreading It Out: Understanding Distributed Loads ............................ 123 Chapter 11: Finding the Centers of Objects and Regions .........................................135 Chapter 12: Special Occasions in the Life of a Force Vector: Moments and Couples ................................................................................................ 149
Part IV: A Picture Is Worth a Thousand Words (Or At Least a Few Equations): Free-Body Diagrams..... 167 Chapter 13: Anatomy of a Free-Body Diagram ........................................................... 169 Chapter 14: The F.B.D.: Knowing What to Draw and How to Draw It ...................... 185 Chapter 15: Simplifying a Free-Body Diagram ............................................................ 199
Part V: A Question of Balance: Equilibrium................ 207 Chapter 16: Mr. Newton Has Entered the Building: The Basics of Equilibrium ..... 209 Chapter 17: Taking a Closer Look at Two-Dimensional Equilibrium: Scalar Methods ............................................................................................................ 219 Chapter 18: Getting Better Acquainted with Three-Dimensional Equilibrium: Vector Methods .......................................................................................................... 229
Part VI: Statics in Action ......................................... 241 Chapter 19: Working with Trusses ..............................................................................243 Chapter 20: Analyzing Beams and Bending Members ............................................... 259 Chapter 21: Working with Frames and Machines ...................................................... 279 Chapter 22: A Different Kind of Axial System: Cable Systems ................................. 293 Chapter 23: Those Darn Dam Problems: Submerged Sur faces ................................ 309 Chapter 24: Incorporating Friction into Your Applications ..................................... 321
Part VII: The Part of Tens ......................................... 339 Chapter 25: Ten Steps to Solving Any Statics Problem ............................................. 341 Chapter 26: Ten Tips for Surviving a Statics Exam ................................................... 347
Index ....................................................................... 353
™ ! r ie s a E g in h Making Ever yt
s c i m a n y d o T herm
Learn to: • Master the concepts and principles of thermodynamics • Develop the problem-solving skills used by professional engineers • Ace your thermodynamics course
Michael Pauken, PhD Senior Mechanical Engineer, NASA’s Jet Propulsion Laboratory California Institute of Technology
Chapter 1
Thermodynamics in Everyday Life In This Chapter ▶ Seeing thermodynamics in the world around you ▶ Changing energy from one form to another ▶ Getting energy to do work and move heat for you ▶ Figuring out relationships, reactions, and mixtures (nothing personal) ▶ Inspiring you to save the world from an energy shortage
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hermodynamics is as old as the universe itself, and the universe is simply the largest known thermodynamic system. When the universe ends in a whimper and the total energy of the universe dissipates to nothingness, so will thermodynamics end. Broadly speaking, thermodynamics is all about energy: how it gets used and how it changes from one form to another. In many cases, thermodynamics involves using heat to provide work, as in the case of your automobile engine, or doing work to move heat, as in your refrigerator. With thermodynamics, you can find out how efficient things are at using energy for useful purposes, such as moving an airplane, generating electricity, or even riding a bicycle. The word thermodynamics has a Greek heritage. The first part, thermo, conveys the idea that heat is somehow involved, and the second part, dynamics, makes you think of things that move. Keep these two ideas in mind as you look at your world in terms of the basic laws of thermodynamics. This book is written to help you understand that thermodynamics is about turning heat into power, a concept that really isn’t so complicated after all.
Grasping Thermodynamics Many thermodynamic systems are at work in the natural world. That sun you see in the sky is the ultimate energy source for the earth, warming the air, the ground, and the oceans. Huge masses of air move over the earth’s surface. Giant currents of water swirl in the oceans. This movement and swirling happens because of the transformation of heat into work.
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Part I: Covering the Basics in Thermodynamics Energy takes many different forms — it can’t be created or destroyed, but it can change form. This statement is one of the fundamental laws of thermodynamics. Consider how energy changes form in storm clouds:
✓ Storm clouds have motion within them. ✓ Motion between moisture droplets in clouds rubbing against each other creates friction. ✓ Friction causes a buildup of static charge. ✓ When the charge becomes high enough, the clouds produce lightning. ✓ This electrical surge of energy can then start a fire on the ground, and before you know it, you have a combustion problem on your hands. Not only does energy change form, but matter (that is, a material or substance) also changes form in many thermodynamic systems. Storm clouds are formed by water evaporating into the air. As the water vapor reaches the colder parts of the atmosphere, it condenses to form clouds. Eventually, the amount of moisture the clouds contain becomes great enough to collect into droplets and form liquid water again, so it rains. One thing people have observed about energy is that it flows in a preferred direction. This observation is another fundamental law of thermodynamics. Heat flows from a hot object to a cold object. Wind blows from a region of high pressure to a region of low pressure. Some forms of energy are developed by forces of nature. Air bubbles move upwards in water against gravity because buoyancy forces them to rise. Water droplets fall in the atmosphere because the force of gravity pulls them toward the ground. Another brilliant observation about energy is that if you have absolutely no energy at all, you have no temperature. The concept of absolute zero temperature is a fundamental law of thermodynamics. I cover the changing forms of energy and matter and the fundamental laws that govern how these changes work in Part I.
Examining Energy’s Changing Forms Many clever people have observed the fundamental laws of thermodynamics in natural systems and applied them to create some wonderful ways of doing work by harnessing energy. Heat is used to generate steam or heat up air that moves a piston in a cylinder or spins a turbine. This movement is used to turn a shaft that can operate a lawn mower; move a car, a truck, a locomotive, or a ship; turn an electric generator; or propel an airplane. Other clever people have used thermodynamic principles to use work to move heat from one place to another. Refrigerators and heat pumps remove heat from one location to produce a desirable cooling or heating effect. The work required for this cooling shows up on your electric bill every month.
Chapter 1: Thermodynamics in Everyday Life In Part II, I show you how the fundamental laws of thermodynamics can tell you how much heat you need to provide to produce work that can be used to move a car, fly an airplane, or turn an electric generator. You can also use the laws of thermodynamics to find out how efficient something is at using energy. Energy is the basis of every thermodynamic process. When you use energy to do something, it changes form along the way. When you start your car, the battery causes the starter to turn. The battery is a big, heavy box of chemical energy. The battery’s job is to change chemical energy into electrical energy. An electric motor rotates (a form of kinetic energy) the engine, and the spark plugs fire. These sparks ignite fuel via a combustion process wherein the chemical energy from gasoline is turned into a form of thermal energy called internal energy. In the few seconds it takes to start your car, energy changes from chemical to electrical to kinetic to thermal or internal energy.
Kinetic energy A car battery provides electricity to operate your starter. As the motor turns, the electrical energy is converted into a form of mechanical energy called kinetic energy. Kinetic energy involves moving a mass so that it has velocity. The mass doesn’t have to be very large to have kinetic energy — even electrons have kinetic energy — but the mass has to be moving. Before you start the car, nothing in the engine is moving so it has no kinetic energy. After the engine is started, it has kinetic energy because of its moving pistons and rotating shafts. If the car is parked while the engine is running, the car as a “system” has no kinetic energy until the engine makes the car move.
Potential energy If you drive your car up a hill and park it there, you change the kinetic energy of the car into another form of energy called potential energy. Potential energy is only available with gravity. You must have a mass located at an elevation above some ground state. Potential energy gets its name from its potential to be converted into kinetic energy. You see this conversion process when you park on a hill and forget to apply the parking brake. Potential energy changes back into kinetic energy as your car rolls down the hill.
Internal energy When you apply the brakes to stop your car, you make energy change form again. You know the car has kinetic energy because it’s moving. Stopping the car changes all this kinetic energy into heat. Brake pads squeeze onto steel disks or steel drums, creating friction. Friction generates heat — sometimes a lot of heat. When materials heat up, another form of energy called internal energy increases. Have you ever smelled a burning odor while driving down
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Part I: Covering the Basics in Thermodynamics long hills? That odor indicates that someone used their brakes to slow down, and the brakes overheated. Do your brakes a favor: Shift into a lower gear and allow the engine to do the braking for you. When the engine is used as a brake, the kinetic energy of the moving car compresses the air in the cylinders, and the energy changes into internal energy because the air heats up from compression. All that internal energy just goes out the tailpipe.
Watching Energy and Work in Action Until the invention of the steam engine, man had to slug it out against nature with nature. Horses pulled coaches, mules pulled plows, sails moved ships, windmills ground grain, and water wheels pressed apples into cider that f ermented and made man feel happy for all his labors. The steam engine was able to replace these natural work sources and move coaches, plows, and ships, among many other things. For the first time, fire was harnessed to provide something more than just heat — it was used to do work. This use of heat to accomplish work is what Part III is all about. Over time, many different kinds of work machines were developed, theories were made, and experiments were done until a rational system of analyzing heat and work was developed into the field of thermodynamics.
Engines: Letting energy do work A heat engine is a machine that can take some source of heat — burning gasoline, coal, natural gas, or even the sun — and make it do work, usually in the form of turning a shaft. With a rotating shaft, you can make things move — think of elevators or race cars. Every heat engine uses four basic processes that interact with the surroundings to accomplish the engine’s job. These processes are heat input, heat rejection, work input, and work output. Take your automobile engine as an example of a heat engine. Here are the four basic processes that go on under the hood: 1. Work input
Air is compressed in the cylinders. This compression requires work from the engine itself. Initially, this work comes from the starter. As you can imagine, this process takes a lot of work, which is why they don’t have those crank handles on the front of cars any more. 2. Heat input
Fuel is burned in the cylinder, where the heat is added to the engine. The heated air in the cylinder naturally wants to increase in pressure and expand. The pressure and expansion move the piston down the cylinder.
Chapter 1: Thermodynamics in Everyday Life 3. Work output
As the expanding gas in the cylinder pushes the piston, work is output by the engine. Some of this work compresses the air in adjacent cylinders. 4. Heat rejection
The last process removes heat with the exhaust from the engine.
Refrigeration: Letting work move heat When Willis Carrier made air conditioners a popular home appliance, he did more than make people comfortable and give electric utilities a reason for growth and expansion. He brought thermodynamics into the home. Thermodynamics has been there all along, and you never realized it. Refrigerators, freezers, air conditioners, and heat pumps are all the same in thermodynamics. Only three basic processes involve energy interacting with the surroundings in what is known as the refrigeration cycle: 1. Heat input
Heat is absorbed from the cold space to keep it cold. 2. Work input
Work is added to the system to pump the heat absorbed from the cold space out to the hot space. 3. Heat rejection
Heat is rejected to the hot space. Actually, a fourth process takes place in most refrigeration cycles, but it doesn’t involve a change in energy. Instead of having a work-output process in the cycle like heat engines do, refrigerators simply utilize a pressure-reducing device in the system. Energy doesn’t change form in such a device.
Getting into Real Gases, Gas Mixtures, and Combustion Reactions Using energy to generate electric power, cool your house, fly a jet, or race cars around the Indianapolis Motor Speedway is the glamorous side of thermodynamics. But behind the movie stars are a supporting cast and crew of thermodynamic relationships (this is jargon for “mathematical equations”) for real gases, gas mixtures, and combustion reactions that make it all happen.
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Part I: Covering the Basics in Thermodynamics In Part IV, you discover the difference between a real gas and an ideal gas. There you see that real gases behave a bit differently than ideal gases. You also figure out the thermodynamic properties of a mixture of gases, such as water vapor and air for heating, air conditioning, and ventilating purposes. Lastly, you calculate how much energy you can get out of fuel in a combustion reaction to power your jet, your race car, or your lawn mower. If you want to sell jet engines to an aircraft manufacturer, you have to show that your engine burns fuel efficiently. To build a jet engine, you need to know how much energy a combustion reaction adds to an engine and how much the air in the engine heats up as a result of the combustion. To figure out the latter, you use thermodynamic relationships of real gases to calculate properties such as temperature, pressure, and energy.
Discovering Old Names and New Ways of Saving Energy As you learn about thermodynamics, you’ll run across a number of names. Some of the names may be familiar; others may be new to you. For example, when you get your electric bill, it tells you how many watt-hours of electricity you used last month. If you reheat yesterday’s leftover pizza, you set your oven to 350 degrees Fahrenheit. (Or, if you live outside the U.S., you set your oven to some temperature in degrees Celsius.) That big rig that’s riding your bumper on the highway burns diesel fuel. How did these terms — watt, Fahrenheit, Celsius, and Diesel — become part of our language? In Part V, you discover that these words (and six more) are actually the last names of characters bent on figuring out what energy is and how to harness it for the benefit of mankind (and maybe to line their pockets with some folding money). Pioneers in thermodynamics didn’t just work in the good old days; there are modern-day pioneers as well. The world’s demand for energy steadily increases while energy resources dwindle. Part V shows you ten ways innovative thinkers have improved energy consumption for automobiles, air conditioners, refrigerators, and electric power plants. Making a better future for all has motivated many people to think of better ways to use energy. Even Albert Einstein got a patent for making a better air-conditioning system (see Chapter 18). Maybe you’ll be inspired to create your own innovation and make a name for yourself in thermodynamics.