Engineering Mechanics STATICS & DYNAMICS SECOND EDITION Francesco Costanzo Department of Engineering Science and Mechanics Penn State University Michael E. Plesha Department of Engineering Physics University of Wisconsin-Madison Gary L. Gray Department of Engineering Penn State University Science and Mechanics Connect i Learn I Succeed
force, STATICS TABLE OF CONTENTS Preface xiv 2.3 Cartesian Representation of Vectors in Three Dimensions 66 1 Introduction 1 1.1 Engineering and Statics 1 1.2 A Brief History of Statics. 3 Galileo Galilei 4 Isaac Newton 4 1.3 Fundamental Principles. 5 Newton's laws of motion 7 1.4 Force 7 1.5 Units and Unit Conversions 9 Dimensional homogeneity and unit conversions 9 Prefixes 10 Angular measure 10 Small angle approximations 12 Accuracy of calculations 13 1.6 Newton's Law of Gravitatioa 16 Relationship between specific weight and density 17 1.7 Failure. 21 Chapter Review. 23 2 Vectors: Force and Position 29 2.1 Basic Concepts 29 Introduction - and tides 29 position, vectors, Denoting vectors in figures 31 Basic vector operations 32 Performing vector operations 34 Resolution of a vector into vector components 34 2.2 Cartesian Representation of Vectors in Two Dimensions. 48 Introduction - Cartesian representation and a walk to work 48 Unit vectors 48 Cartesian coordinate system 49 Cartesian vector representation 49 Addition of vectors using Cartesian components 51 Position vectors 52 Right-hand Cartesian coordinate system 66 Cartesian vector representation 66 Direction angles and direction cosines 66 Position vectors 67 Use of position vectors to write expressions for force vectors 68 Some simple structural members 68 2.4 Vector Dot Product 84 Dot product using Cartesian components 85 Determination of the angle between two vectors 85 Determination of the component of a vector in a particular direction 86 Determination of the component of a vector perpendicular to a direction 87 2.5 Vector Cross Product 101 Cross product using Cartesian components 102 Evaluation of cross product using determinants 103 Determination of the normal direction to a plane 105 Determination of the area of a parallelogram 105 Scalar triple product 105 Chapter Review. 116 Equilibrium of Particles 125 3.1 Equilibrium of Particles in Two Dimensions 125 Free body diagram (FBD) 126 Modeling and problem solving 130 Cables and bars 131 Pulleys 133 Reactions 134 3.2 Behavior of Cables, Bars, and Springs 151 Equilibrium geometry of a structure 151 Cables 151 Bars 152 Modeling idealizations and solution of F = 0 152 Springs 152 Vll
3.3 Equilibrium of Particles in Three Dimensions 166 Reactions 166 Solution of algebraic equations 166 Summing forces in directions other than x, y, or z 167 3.4 Engineering Design. 181 > i Objectives of design 181 Particle equilibrium design problems 182 Chapter Review. 195 4 Moment of a Force and Equivalent Force Systems 203 4.1 Moment of a Force. Scalar approach 204 Vector approach 205 Varignon's theorem 207 203 Which approach should I use: scalar or vector? 208 4.2 Moment of a Force About a Line 220 Vector approach 220 Scalar approach 221 4.3 Moment of a Couple 232 Vector approach 233 Scalar approach 233 Comments on the moment of a couple 233 Equivalent couples 234 Reactions 272 Free body diagram (FBD) 274 Alternative equilibrium equations 276 Gears 277 Examples of correct FBDs 277 Examples of incorrect and/or incomplete FBDs 279 5.3 Equilibrium of Bodies in Two Dimensions Additional Topics. 298 Why are bodies assumed to be rigid? 298 Treatment of cables and pulleys 298 Springs 299 Superposition 300 Supports and fixity 300 Static determinacy and indeterminacy 301 Two-force and three-force members 303 5.4 Equilibrium of Bodies in Three Dimensions. 322 Reactions 322 More on bearings 322 Scalar approach or vector approach? 324 Solution of algebraic equations 324 Examples of correct FBDs 325 Examples of incorrect and/or incomplete FBDs 327 5.5 Engineering Design 344 Codes and standards 346 Design problems 347 Chapter Review. 353 Equivalent force systems 234 Resultant couple moment 235 6 Moments as free vectors 235 Structural Analysis and Machines. 363 Structures and machines 363 4.4 Equivalent Force Systems. Transmissibility of a force 246 Equivalent force systems 247 Some special force systems 248 Wrench equivalent force systems 250 Why are equivalent force systems called equivalent? 250 246 6.1 Truss Structures and the Method of Joints.. 364 When may a structure be idealized as a truss? 365 Method ofjoints 365 Zero-force members 367 Typical truss members 369 6.2 Truss Structures and the Method of Sections 380 Chapter Review. 263 Treatment of forces that are not at joints 382 Static determinacy and indeterminacy 383 5 Equilibrium of Bodies 271 5.1 Equations of Equilibrium 271 5.2 Equilibrium of Rigid Bodies in Two Dimensions 272 Design considerations 384 6.3 Trusses in Three Dimensions 399 Stability of space trusses and design considerations 400 viii
center 6.4 Frames and Machines 408 Analysis procedure and free body diagrams (FBDs) 408 Examples of correct FBDs 409 Examples of incorrect and/or incomplete FBDs 411 Chapter Review. 426 7 Centroids and Distributed Force Systems.. 431 7.1 Centroid 431 Introduction - of gravity 431 Centroid of an area 433 Centroid of a line 434 Centroid of a volume 435 Unification of concepts 435 Which approach should I use: composite shapes or integration? 435 8.2 Internal Forces in Straight Beams 503 Determination of V and M using equilibrium 503 Shear and moment diagrams 503 8.3 Relations Among Shear, Moment, and Distributed Force. 514 Relations among V, M, and w 514 Determination of V and M using integration 515 Which approach should I use? 516 Tips and shortcuts for drawing shear and moment diagrams 517 Design considerations 518 Chapter Review. 527 Friction 533 9.1 Basic Concepts 533 Finer points: surfaces and lines in three A brief history of tribology 533 dimensions 436 7.2 Center of Mass and Center of Gravity 449 Center of mass 449 Center of gravity 450 7.3 Theorems of Pappus and Guldinus 461 Area of a surface of revolution 461 Volume of a solid of revolution 462 Proof of the Pappus-Guldinus theorems 462 7.4 Distributed Forces, Fluid and Gas Pressure Loading 468 Distributed forces 468 Distributed forces applied to beams 468 Fluid and gas pressure 469 Forces produced by fluids 471 Forces produced by gases 473 A simple experiment 534 Coulomb's law of friction 535 Coefficients of friction 535 Dry contact versus liquid lubrication 537 Angle of friction 537 Problems with multiple contact surfaces 537 Wedges 538 Coulomb's law of friction in three dimensions 538 Design considerations 538 9.2 Problems with Multiple Contact Surfaces.. 551 Determination of sliding directions 551 9.3 Belts and Cables Contacting Cylindrical Surfaces. 560 Equilibrium analysis 560 Chapter Review. 567 Chapter Review. 486 10 Moments of Inertia 573 8 Internal Forces 493 8.1 Internal Forces in Structural Members... 493 Why are internal forces important? 493 Internal forces for slender members in two dimensions 494 Internal forces for slender members in three dimensions 495 Determination of internal forces 495 10.1 Area Moments of Inertia 573 An example test scores 573 An example beam loading 574 Definition of area moments of inertia 574 What are area moments of inertia used for? 575 Radius of gyration 576 Evaluation of moments of inertia using integration 577 ix
10.2 Parallel Axis Theorem 585 Evaluation of moments of inertia using Use of parallel axis theorem in integration 586 composite shapes 597 Use of parallel axis theorem for composite Chapter Review. 612 shapes 586 10.3 Mass Moments of Inertia. 593 A Technical Writing A-1 An example figure skating 593 Definition of mass moments of inertia 593 B Answers to Selected Problems A-5 What are mass moments of inertia used for? 594 Radius of gyration 595 Credits. C-1 Parallel axis theorem 595 Evaluation of moments of inertia using integration 596 Index 1-1 x
ma DYNAMICS TABLE OF CONTENTS Preface xiv 2.8 Motion in Three Dimensions 142 Cylindrical coordinates 142 1 Introduction to Dynamics 1 1.1 The Newtonian Equations 1 1.2 Fundamental Concepts in Dynamics 5 Space and time 5 Force, mass, and inertia 5 Particle and rigid body 6 Vectors and their Cartesian representation 6 Useful vector "tips and tricks" 9 Spherical coordinates 143 Cartesian coordinates 144 Chapter Review. 155 3 Force and Acceleration Methods for Particles 169 3.1 Rectilinear Motion. 169 Applying Newton's second law 169 Units 10 1.3 Dynamics and Engineering Design. 26 System modeling 27 / Force laws 171 Equation(s) of motion 173 Inertial reference frames 173 Degrees of freedom 174 2 Particle Kinematics 29 2.1 Position, Velocity, Acceleration, and Cartesian Coordinates. 29 Position vector 30 Trajectory 30 Velocity vector and speed 30 Acceleration vector 32 3.2 Curvilinear Motion 192 Newton's second law in 2D and 3D component systems 192 3.3 Systems of Particles 217 Engineering materials one atom at a time 217 Newton's second law for systems of particles 217 Chapter Review. 234 Cartesian coordinates 32 2.2 Elementary Motions 47 Rectilinear motion relations 47 Circular motion and angular velocity 49 2.3 Projectile Motion. 68 2.4 The Time Derivative of a Vector 80 Time derivative of a unit vector 80 Time derivative of an arbitrary vector 81 2.5 Planar Motion: Normal-Tangential Components 92 Normal-tangential components 92 2.6 Planar Motion: Polar Coordinates. 105 Polar coordinates and position, velocity, and acceleration 105 2.7 Relative Motion Analysis and Differentiation of Geometrical Constraints 121 Relative motion 121 Differentiation of geometrical constraints 122 4 Energy Methods for Particles 241 4.1 Work-Energy Principle for a Particle 241 Work-energy principle and its relation with F 241 Work of a force 243 4.2 Conservative Forces and Potential Energy.. 257 Work done by the constant force of gravity 257 Work of a central force 257 Conservative forces and potential energy 258 Work-energy principle for any type of force 260 When is a force conservative? 260 4.3 Work-Energy Principle for Systems of Particles. 281 Internal work and work-energy principle for a system 281 Kinetic energy for a system of particles 282 4.4 Power and Efficiency. 298 Power developed by a force 298 xi
Efficiency 298 Chapter Review. 305 Vector approach 452 Differentiation of constraints 453 Instantaneous center of rotation 453 5 Momentum Methods for Particles 313 5.1 Momentum and Impulse 313 Impulse-momentum principle 313 Conservation of linear momentum 316 5.2 Impact 335 Impacts are short, dramatic events 335 Definition of impact and notation 335 Line of impact impacting objects 335 and contact force between 6.3 Planar Motion: Acceleration Analysis 474 Vector approach 474 Differentiation of constraints 474 Rolling without slip: acceleration analysis 475 6.4 Rotating Reference Frames 494 The general kinematic equations for the motion of a point relative to a rotating reference frame 494 Coriolis component of acceleration 498 Chapter Review. 512 Impulsive forces and impact-relevant FBDs 336 Coefficient of restitution 336 7 Unconstrained direct central impact 338 Unconstrained oblique central impact 338 Impact and energy 339 5.3 Angular Momentum 361 Moment-angular momentum relation for a particle 361 Angular impulse-momentum for a system of particles 362 Euler's first and second laws of motion 365 Newton-Euler Equations for Planar Rigid Body Motion 521 7.1 Newton-Euler Equations for Bodies Symmetric with Respect to the Plane of Motioa 521 Linear momentum: translational equations 521 Angular momentum: rotational equations 522 Graphical interpretation of the equations of motion 526 7.2 Newton-Euler Equations: Translation 529 5.4 Orbital Mechanics 385 Determination of the orbit 385 Energy considerations 391 5.5 Mass Flows 401 Steady flows 401 Variable mass flows and propulsion 404 Chapter Review. 422 7.3 Newton-Euler Equations: Rotation About a Fixed Axis 539 7.4 Newton-Euler Equations: General Plane Motioa 553 Newton-Euler equations for general plane motion 553 Chapter Review. 575 6 Planar Rigid Body Kinematics. 433 6.1 Fundamental Equations, Translation, and Rotation About a Fixed Axis 433 Crank, connecting rod, and piston motion 433 Qualitative description of rigid body motion 434 General motion of a rigid body 435 Elementary rigid body motions: translations 437 Elementary rigid body motions: rotation about a fixed axis 438 Planar motion in practice 439 6.2 Planar Motion: Velocity Analysis. 452 8 Energy and Momentum Methods for Rigid Bodies. 583 8.1 Work-Energy Principle for Rigid Bodies... 583 Kinetic energy of rigid bodies in planar motion 583 Work-energy principle for a rigid body 585 Work done on rigid bodies 585 Potential energy and conservation of energy 586 Work-energy principle for systems 588 Power 588 8.2 Momentum Methods for Rigid Bodies 618 Impulse-momentum principle for a rigid body 618 Xll
Angular impulse-momentum principle for a rigid body 619 8.3 Impact of Rigid Bodies 639 Rigid body impact: basic nomenclature and assumptions 640 Classification of impacts 640 Central impact 640 Eccentric impact 642 Constrained eccentric impact 643 Chapter Review. 656 9 Mechanical Vibrations 663 Summing angular velocities 722 10.2 Three-Dimensional Kinetics of Rigid Bodies 738 Newton-Euler equations for three-dimensional motion 738 Kinetic energy of a rigid body in three-dimensional motion 743 Chapter Review 759 A Mass Moments of Inertia A-1 Definition of mass moments and products of inertia A-1 How are mass moments of inertia used? A-3 9.1 Undamped Free Vibration 663 Oscillation of a railcar after coupling 663 Standard form of the harmonic oscillator 665 Linearizing nonlinear systems 666 Energy method 667 9.2 Undamped Forced Vibration 681 ' Radius of gyration A-4 Parallel axis theorem A-4 Principal moments of inertia A-6 Moment of inertia about an arbitrary axis A-9 Evaluation of moments of inertia using composite shapes A-10 Standard form of the forced harmonic oscillator 681 9.3 Viscously Damped Vibration 695 Viscously damped free vibration 695 Viscously damped forced vibration 698 Chapter Review.. 714 B Angular Momentum of a Rigid Body A-11 Angular momentum of a rigid body undergoing three-dimensional motion A-ll Angular momentum of a rigid body in planar motion A-14 C Answers to Even-Numbered Problems A-15 10 Three-Dimensional Dynamics of Rigid Bodies 721 10.1 Three-Dimensional Kinematics of Rigid Bodies 721 Credits. C-1 Index 1-1 Computation of angular accelerations 722»«XIII