THE ENGINEERING THE HANDBOOK - SECOND EDITION I pptx

1 Force-System Resultants and Equilibrium1.1 Force-System Resultants Concurrent Force Systems • Moment of a Force • Couple • Resultants of a Force and Couple System • Distributed Load

E NGINEERING

T H E

H A N D B O O K

SECOND EDITION

The Electrical Engineering Handbook Series

Series Editor

Richard C Dorf

University of California, Davis

Titles Included in the Series

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Forthcoming Titles

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1 Engineering—Handbooks, manuals, etc I Dorf, Richard C.

TA151.E424 2004

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in the practice of the profession whether in industry, education, or government The goal of this prehensive handbook is to replace a myriad of books with one highly informative, well-organized,definitive source of fundamental knowledge.

com-Organization

The fundamentals of engineering have evolved to include a wide range of knowledge, substantial empiricaldata, and a broad range of practice The focus of the handbook is on the key concepts, models, andequations that enable the engineer to analyze, design, and predict the behavior of complex devices,circuits, instruments, systems, structures, plants, computers, fuels, and the environment While data andformulae are summarized, the main focus is the provision of the underlying theories and concepts andthe appropriate application of these theories to the field of engineering Thus, the reader will find thekey concepts defined, described, and illustrated in order to serve the needs of the engineer over many years With equal emphasis placed on materials, structures, mechanics, dynamics, fluids, thermodynamics,fuels and energy, transportation, environmental systems, circuits and systems, computers and instru-ments, manufacturing, aeronautical and aerospace, and economics and management as well as mathe-matics, the engineer should encounter a wide range of concepts and considerable depth of exploration

of these concepts as they lead to application and design

The level of conceptual development of each topic is challenging, but tutorial and relatively mental Each of the more than 200 chapters is written to enlighten the expert, refresh the knowledge ofthe mature engineer, and educate the novice

funda-The information is organized into 30 major sections funda-The 30 sections encompass 232 chapters, andthe Appendix summarizes the applicable mathematics, symbols, and physical constants

Each chapter includes three important and useful categories: defining terms, references, and furtherinformation Defining terms are key definitions, and the first occurrence of each term defined is indicated1586_C000.fm Page v Thursday, May 20, 2004 3:04 PM

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in boldface in the text The definitions of these terms are summarized as a list at the end of each chapter.The references provide a list of useful books and articles for follow-up reading Finally, further information

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Locating Your Topic

Numerous avenues of access to information contained in the handbook are provided A complete table

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The index can also be used to locate key definitions The page on which the definition appears foreach key (defining) term is clearly identified in the index

The Engineering Handbook, Second Edition is designed to provide answers to most inquiries and directthe inquirer to further sources and references We hope that this handbook will be referred to often andthat informational requirements will be satisfied effectively

Acknowledgments

This handbook is testimony to the dedication of the associate editors, the publishers, and my editorialassociates I particularly wish to acknowledge at CRC Press Nora Konopka, Publisher; Helena Redshaw,Project Development Manager; Liz Spangenberger, Administrative Assistant; and Susan Fox, ProjectEditor

Richard C Dorf

Editor-in-Chief1586_C000.fm Page vi Thursday, May 20, 2004 3:04 PM

© 2005 by CRC Press LLC

Richard C Dorf, professor of electrical and computerengineering at the University of California, Davis, teachesgraduate and undergraduate courses in electrical engineer-ing in the fields of circuits and control systems He earned

a Ph.D in electrical engineering from the U.S Naval graduate School, an M.S from the University of Colorado,and a B.S from Clarkson University Highly concernedwith the discipline of engineering and its wide value tosocial and economic needs, he has written and lecturedinternationally on the contributions and advances in engi-neering and their value to society

Post-Professor Dorf has extensive experience with educationand industry and is professionally active in the fields ofrobotics, automation, electric circuits, and communica-tions He has served as a visiting professor at the University

of Edinburgh, Scotland; the Massachusetts Institute ofTechnology; Stanford University; and the University ofCalifornia, Berkeley

A Fellow of The Institute of Electrical and Electronics Engineers, Dr Dorf is widely known to theprofession for his Modern Control Systems, Tenth Edition (Prentice Hall, 2004) and Introduction to Electric Circuits, Sixth Edition (Wiley, 2004) He is the Editor-in-Chief of the Technology Management Handbook

(CRC Press, 1999), the Engineering Handbook, Second Edition (CRC Press, 2004), and CRC Handbook

of Engineering Tables (CRC Press, 2004)

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4 Reactions

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9 Pressure Vessels

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10 Axial Loads and Torsion

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11 Fracture Mechanics

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12 Dynamics of Particles: Kinematics and Kinetics

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13 Dynamics of Rigid Bodies: Kinematics and Kinetics

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14 Free Vibration, Natural Frequencies, and Mode Shapes

20 Linkages and Cams

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21 Tribology: Friction, Wear, and Lubrication

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22 Machine Elements

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23 Crankshaft Journal Bearings

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27 Earthquakes and Their Effects

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41 Pumps and Fans

45 The First Law of Thermodynamics

47 The Thermodynamics of Solutions

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55 Refrigeration and Cryogenics

65 Other Separation Processes

William C Corder and Simon P Hanson

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68 Internal Combustion Engines

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69 Gas Turbines

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70 Nuclear Power Systems

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74 Steam Turbines and Generators

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84 Operations and Environmental Impacts

87 Shallow Water and Deep Water Engineering

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XV Water Resources Engineering

XVI Linear Systems and Models

98 Transfer Functions and Laplace Transforms

101 Linear State–Space Models

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111 Filters (Passive)

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112 Power Distribution

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113 Grounding, Shielding, and Filtering

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121 A/D and D/A Converters

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122 Superconductivity

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123 Embedded Systems-on-Chips

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124 Electronic Data Analysis Using PSPICE and MATLAB

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128 Counters and State Machines (Sequencers)

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133 Transforms and Fast Algorithms

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134 Digital Filters

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135 Analog and Digital Communications

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136 Coding

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137 Computer Communication Networks

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138 Satellites and Aerospace

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139 Mobile and Portable Radio Communications

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140 Optical Communications

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141 Digital Image Processing

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142 Complex Envelope Representations for Modulated Signals

145 Programming Languages

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146 Input/Output Devices

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147 Memory and Storage Systems

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148 Nanocomputers, Nanoarchitectronics, and NanoICs

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149 Software Engineering

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150 Human–Computer Interface Design

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151 Sensors and Transducers

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152 Measurement Errors and Uncertainty

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165 Photogrammetry and Topographic Mapping

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166 Surveying Computations

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168 Surveying Applications for Geographic Information Systems

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169 Remote Sensing

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172 Nyquist Criterion and Stability

176 Robots and Controls

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177 State Variable Feedback

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183 Managing for Value

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184 Design, Modeling, and Prototyping

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185 Materials Processing and Manufacturing Methods

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186 Machine Tools and Processes

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187 Ergonomics/Human Factors

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188 Pressure and Vacuum

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192 Computer Integrated Manufacturing: A Data Mining Approach

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197 Propulsion Systems

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198 Aircraft Performance and Design

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199 Spacecraft and Mission Design

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202 Present Worth Analysis

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203 Project Analysis Using Rate-of-Return Criteria

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204 Project Selection from Alternatives

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205 Depreciation and Corporate Taxes

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206 Financing and Leasing

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210 Project Evaluation and Selection

XXIX Materials Engineering

I Statics

Force-System Resultants • Equilibrium

Centroid of a Plane Area • Centroid of a Volume • Surface Forces • Line Forces • Calculation of Surface Area and Volume of a Body with Rotational Symmetry • Determination of Centroids

Area Moments of Inertia • Mass Moments of Inertia1586_book.fm Page 1 Friday, May 7, 2004 3:56 PM

1

Force-System Resultants and Equilibrium1.1 Force-System Resultants

Concurrent Force Systems • Moment of a Force • Couple •

Resultants of a Force and Couple System • Distributed Loadings

Statics is a branch of mechanics that deals with the equilibrium of bodies, that is, those that are either

at rest or move with constant velocity In order to apply the laws of statics, it is first necessary to understandhow to simplify force systems and compute the moment of a force In this chapter these topics will bediscussed, and some examples will be presented to show how the laws of statics are applied

1.1 Force-System Resultants

Concurrent Force Systems

Force is a vector quantity that is characterized by its magnitude, direction, and point of application.When two forces F1 and F2 are concurrent they can be added together to form a resultant FR= F1+ F2using the parallelogram law,Figure 1.1 Here F1 and F2 are referred to as components of FR Successiveapplications of the parallelogram law can also be applied when several concurrent forces are to be added;however, it is generally simpler to first determine the two components of each force along the axes of acoordinate system and then add the respective components For example, the x, y, z (or Cartesian)components of F are shown in Figure 1.2 Here, i, j, k are unit vectors used to define the direction of thepositive x, y, z axes, and F x, F y, F z are the magnitudes of each component By vector addition, F=F x i+

Fy j + F zk When each force in a concurrent system of forces is expressed by its Cartesian components,the resultant force is therefore

(1.1)where SF x, SF y, SF z represent the scalar additions of the x, y, z components, respectively

FR=ÂF xi+ÂF yj+ÂF zk

Russell C Hibbeler

University of Louisiana at Lafayette

1586_book.fm Page 1 Friday, May 7, 2004 3:56 PM

Moment of a Force

When a force F acts on a body, it will cause both external and internal effects on the body These effectsdepend upon where the force is located For example, if F acts at point A on the body in Figure 1.3, itwill cause a specific translation and rotation of the body However, if F is applied to some other point,

B, which lies along the line of action of F, then the external effects regarding the motion of the bodyremain unchanged, although the body’s internal effects will be different This effect of sliding a forcealong its line of action is called the principle of transmissibility If the force acts at point C, which isnot along the line of action AB, then both the external and internal effects on the body will change Thedifference in external effects — notably the difference in the rotation of the body — occurs because ofthe distance d that separates the lines of action of the two positions of the force

This tendency for the body to rotate about a specified point O or axis as caused by a force is a vectorquantity called a moment. By definition, the magnitude of the moment is

where d is the moment arm or perpendicular distance from the point to the line of action of the force,

as in Figure 1.4 The direction of the moment is defined by the right-hand rule, whereby the curl of theright-hand fingers follows the tendency for rotation caused by the force, and the thumb specifies thedirectional sense of the moment In this case, MO is directed out of the page, since F produces counter-clockwise rotation about O It should be noted that the force can act at any point along its line of actionand still produce the same moment about O

FIGURE 1.1 Addition of forces by parallelogram law. FIGURE 1.2 Resolution of a vector into its x, y, z

com-ponents.

force line of action

F F

F

O d

F

1586_book.fm Page 2 Friday, May 7, 2004 3:56 PM

Sometimes the moment arm d is geometrically hard to determine To make the calculation easier, the

force is first resolved into its Cartesian components and then the moment about point O is determined

using the principle of moments, which states that the moment of the force about O is equal to the sum of

the moments of the force’s components about O. Thus, as shown in Figure 1.5, we have M O=Fd=F x y+F y x

The moment about point O can also be expressed as a vector cross product of the position vector r,

directed from O to any point on the line of action of the force and the force F, as shown in Figure 1.6 Here,

If r and F are expressed in terms of their Cartesian components, then as in Figure 1.7 the Cartesian

components for the moment about O are

(1.4)

Couple

A couple is defined as two parallel forces that have the same magnitude and opposite directions and are

separated by a perpendicular distance d, as in Figure 1.8 The moment of a couple about the arbitrary

(1.5)

Here the couple moment MC is independent of the location of the moment point O Instead, it depends

only on the distance between the forces; that is, r in the above equation is directed from any point on

the line of action of one of the forces (-F) to any point on the line of action of the other force F The

external effect of a couple causes rotation of the body with no translation, since the resultant force of a

couple is zero

Resultants of a Force and Couple System

A general force and couple-moment system can always be replaced by a single resultant force and couple

moment acting at any point O As shown in Figure 1.9(a) and Figure 1.9(b), these resultants are

(1.6)

(1.7)

where SF = F1 + F2 + F3 is the vector addition of all the forces in the system, and SMO = (r1 ¥ F1) + (r2 ¥

F2) + (r3 ¥ F3) + M1 + M2 is the vector sum of the moments of all the forces about point O plus the sum

of all the couple moments This system may be further simplified by first resolving the couple moment

into two components — one parallel and the other perpendicular to the force FR, as in Figure

1.9(b). By moving the line of action of FR in the plane perpendicular to M^ a distance d = M^/FR, so that

FR creates the moment M^ about O, the system can then be represented by a wrench, that is, a single

force FR and collinear moment M||, Figure 1.9(c)

Note that in the special case of q = 90º, Figure 1.9(b), M|| = 0 and the system then reduces to a single resultant force FR having a specified line of action This will always be the case if the force system is eitherconcurrent, parallel, or coplanar

Distributed Loadings

When a body contacts another body, the loads produced are always distributed over the surface area ofeach body If the area on one of the bodies is small compared to the entire surface area of the body, theloading can be represented by a single concentrated force acting at a point on the body However, if theloading occurs over a large surface area — such as that caused by wind or a fluid — the distribution ofload must be taken into account The intensity of this surface loading at each point is represented as apressure and its variation is defined by a load-intensity diagram On a flat surface, the load intensity

diagram is described by the loading function p = p(x, y), which consists of an infinite number of parallel

forces, as in Figure 1.10 Applying Equation (1.6) and Equation (1.7), the resultant of this loading andits point of application ( ) can be determined from

(1.8)

(1.9)

Geometrically, F R is equivalent to the volume under the loading diagram, and its location passesthrough the centroid or geometric center of this volume Often in engineering practice, the surface loading

is symmetric about an axis, in which case the loading is a function of only one coordinate, w = w(x).

Here the resultant is geometrically equivalent to the area under the loading curve, and the line of action

of the resultant passes through the centroid of this area

Besides surface forces as discussed above, loadings can also be transmitted to another body withoutdirect physical contact These body forces are distributed throughout the volume of the body A common

( , )( , )

( , )( , )

x–

– y

example is the force of gravity The resultant of this force is termed the weight; it acts through the body’s

center of gravity and is directed towards the center of the earth

1.2 Equilibrium

Equations of Equilibrium

A body is said to be in equilibrium when it either is at rest or moves with constant velocity For purposes

of analysis, it is assumed that the body is perfectly rigid, meaning that the particles composing the bodyremain at fixed distances from one another both before and after applying the load Most engineeringmaterials deform only slightly under load, so that moment arms and the orientation of the loadingremain essentially constant For these cases, therefore, the rigid-body model is appropriate for analysis.The necessary and sufficient conditions to maintain equilibrium of a rigid body require the resultantexternal force and moment acting on the body to be equal to zero From Equation (1.6) and Equation(1.7), this can be expressed mathematically as

(1.10)

(1.11)

If the forces acting on the body are resolved into their x, y, z components, these equations can be

written in the form of six scalar equations, namely,

(1.12)

Actually, any set of three nonorthogonal, nonparallel axes will be suitable references for either of theseforce or moment summations

If the forces on the body can be represented by a system of coplanar forces, then only three equations

of equilibrium must be satisfied, namely,

(1.13)

Here the x and y axes lie in the plane of the forces and point O can be located either on or off the body.

Free-Body Diagram

Application of the equations of equilibrium requires accountability for all the forces that act on the body.

The best way to do this is to draw the body’s free-body diagram This diagram is a sketch showing an

outlined shape of the body and so represents it as being isolated or “free” from its surroundings On thissketch it is necessary to show all the forces and couples that act on the body Those generally encounteredare due to applied loadings, reactions that occur at the supports and at points of contact with otherbodies, and the weight of the body Also one should indicate the dimensions of the body necessary for

F F M

x y O

  Â

=

=

=

000

computing the moments of forces Once the free-body diagram has been drawn and the coordinate axesestablished, application of the equations of equilibrium becomes a straightforward procedure.

When a body is in contact with a rough surface, a force of resistance called friction is exerted on the

body by the surface in order to prevent or retard slipping of the body This force always acts tangent tothe surface at points of contact with the surface and is directed so as to oppose the possible or existingmotion of the body If the surface is dry, the frictional force acting on the body must satisfy the equation

fixed support fixed support

θ

F θ

The equality F = m s N applies only when motion between the contacting surfaces is impending Here N

is the resultant normal force on the body at the surface of contact, and ms is the coefficient of staticfriction, a dimensionless number that depends on the characteristics of the contacting surfaces Typicalvalues of ms are shown in Table 1.2 If the body is sliding, then F = m k N, where m k is the coefficient of kineticfriction, a number that is approximately 25% smaller than those listed in Table 1.2

Constraints

Equilibrium of a body is ensured not only by satisfying the equations of equilibrium, but also by its beingproperly held or constrained at its supports If a body has more supports than are needed for equilibrium,

it is referred to as statically indeterminate, since there will be more unknowns than equations of

equilib-rium For example, the free-body diagram of the beam in Figure 1.11 shows four unknown support

reactions, A x , A y , M A , and B y, but only three equations of equilibrium are available for solution [Equation(1.13)] The additional equation needed requires knowledge of the physical properties of the body anddeals with the mechanics of deformation, which is discussed in subjects such as mechanics of materials

A body may be improperly constrained by its supports When this occurs, the body becomes unstableand equilibrium cannot be maintained Either of two conditions may cause this to occur — when thereactive forces are all parallel (Figure 1.12) or when they are concurrent (Figure 1.13)

In summary, then, if the number of reactive forces that restrain the body is a minimum — and theseforces are not parallel or concurrent — the problem is statically determinate, and the equations ofequilibrium are sufficient to determine all the reactive forces

Internal Loadings

The equations of equilibrium can also be used to determine the internal resultant loadings in a member,

provided the external loads are known The calculation is performed using the method of sections, which

TABLE 1.2 Typical Values for

Coefficients of Static Friction