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Mechanical Engineer's Handbook



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Academic Press Series in Engineering
Series Editor
J. David Irwin

This a series that will include handbooks, textbooks, and professional reference
books on cutting-edge areas of engineering. Also included in this series will be singleauthored professional books on state-of-the-art techniques and methods in engineering. Its objective is to meet the needs of academic, industrial, and governmental
engineers, as well as provide instructional material for teaching at both the undergraduate and graduate level.
The series editor, J. David Irwin, is one of the best-known engineering educators in
the world. Irwin has been chairman of the electrical engineering department at
Auburn University for 27 years.

Published books in this series:
Control of Induction Motors
2001, A. M. Trzynadlowski

Embedded Microcontroller Interfacing for McoR Systems
2000, G. J. Lipovski

Soft Computing & Intelligent Systems
2000, N. K. Sinha, M. M. Gupta

Introduction to Microcontrollers
1999, G. J. Lipovski

Industrial Controls and Manufacturing
1999, E. Kamen

DSP Integrated Circuits
1999, L. Wanhammar

Time Domain Electromagnetics
1999, S. M. Rao

Single- and Multi-Chip Microcontroller Interfacing
1999, G. J. Lipovski

Control in Robotics and Automation

1999, B. K. Ghosh, N. Xi, and T. J. Tarn

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Auburn University


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Mechanical
Engineer's

Handbook
Edited by

Dan B. Marghitu
Department of Mechanical Engineering, Auburn University,
Auburn, Alabama

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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTER 1

Statics
Dan B. Marghitu, Cristian I. Diaconescu, and Bogdan O. Ciocirlan
1. Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . .
1.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Product of a Vector and a Scalar . . . . . . . . . . . . . . . . . .
1.4 Zero Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Resolution of Vectors and Components . . . . . . . . . . . . . .
1.8 Angle between Two Vectors . . . . . . . . . . . . . . . . . . . . .
1.9 Scalar (Dot) Product of Vectors . . . . . . . . . . . . . . . . . . .
1.10 Vector (Cross) Product of Vectors . . . . . . . . . . . . . . . . . .
1.11 Scalar Triple Product of Three Vectors . . . . . . . . . . . . . .
1.12 Vector Triple Product of Three Vectors . . . . . . . . . . . . . .
1.13 Derivative of a Vector . . . . . . . . . . . . . . . . . . . . . . . . .
2. Centroids and Surface Properties . . . . . . . . . . . . . . . . . . . . . .

2.1 Position Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Centroid of a Set of Points . . . . . . . . . . . . . . . . . . . . . .
2.4 Centroid of a Curve, Surface, or Solid . . . . . . . . . . . . . . .
2.5 Mass Center of a Set of Particles . . . . . . . . . . . . . . . . . .
2.6 Mass Center of a Curve, Surface, or Solid . . . . . . . . . . . .
2.7 First Moment of an Area . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Theorems of Guldinus±Pappus . . . . . . . . . . . . . . . . . . .
2.9 Second Moments and the Product of Area . . . . . . . . . . . .
2.10 Transfer Theorem or Parallel-Axis Theorems . . . . . . . . . .
2.11 Polar Moment of Area . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Principal Axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Moments and Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Moment of a Bound Vector about a Point . . . . . . . . . . . .
3.2 Moment of a Bound Vector about a Line . . . . . . . . . . . . .
3.3 Moments of a System of Bound Vectors . . . . . . . . . . . . .
3.4 Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.5 Equivalence . . . . . . . . . . . . . . . . . . . . . .
3.6 Representing Systems by Equivalent Systems
4. Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Equilibrium Equations. . . . . . . . . . . . . . . .
4.2 Supports. . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Free-Body Diagrams . . . . . . . . . . . . . . . . .
5. Dry Friction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Static Coef®cient of Friction . . . . . . . . . . . .
5.2 Kinetic Coef®cient of Friction . . . . . . . . . . .
5.3 Angles of Friction. . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dan B. Marghitu, Bogdan O. Ciocirlan, and Cristian I. Diaconescu
1. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Space and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Angular Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Kinematics of a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Position, Velocity, and Acceleration of a Point. . . . . . . . .
2.2 Angular Motion of a Line. . . . . . . . . . . . . . . . . . . . . . .
2.3 Rotating Unit Vector . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Straight Line Motion . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Curvilinear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Normal and Tangential Components . . . . . . . . . . . . . . .
2.7 Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Dynamics of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Newton's Second Law. . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Newtonian Gravitation . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Inertial Reference Frames . . . . . . . . . . . . . . . . . . . . . .
3.4 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Normal and Tangential Components . . . . . . . . . . . . . . .
3.6 Polar and Cylindrical Coordinates . . . . . . . . . . . . . . . . .
3.7 Principle of Work and Energy . . . . . . . . . . . . . . . . . . .
3.8 Work and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Conservative Forces . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Principle of Impulse and Momentum. . . . . . . . . . . . . . .
3.12 Conservation of Linear Momentum . . . . . . . . . . . . . . . .
3.13 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 Principle of Angular Impulse and Momentum . . . . . . . . .
4. Planar Kinematics of a Rigid Body . . . . . . . . . . . . . . . . . . . . .
4.1 Types of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Rotation about a Fixed Axis . . . . . . . . . . . . . . . . . . . . .
4.3 Relative Velocity of Two Points of the Rigid Body . . . . . .
4.4 Angular Velocity Vector of a Rigid Body. . . . . . . . . . . . .
4.5 Instantaneous Center . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Relative Acceleration of Two Points of the Rigid Body . . .

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Dynamics

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Table of Contents

4.7 Motion of a Point That Moves Relative to a Rigid Body
5. Dynamics of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Equation of Motion for the Center of Mass. . . . . . . . .
5.2 Angular Momentum Principle for a System of Particles.
5.3 Equation of Motion for General Planar Motion . . . . . .
5.4 D'Alembert's Principle . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dan B. Marghitu, Cristian I. Diaconescu, and Bogdan O. Ciocirlan
1. Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Uniformly Distributed Stresses . . . . . . . . . . . . . . . . . . . .
1.2 Stress Components . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Mohr's Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Triaxial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Elastic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Shear and Moment . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Normal Stress in Flexure. . . . . . . . . . . . . . . . . . . . . . . .
1.10 Beams with Asymmetrical Sections . . . . . . . . . . . . . . . . .
1.11 Shear Stresses in Beams . . . . . . . . . . . . . . . . . . . . . . . .
1.12 Shear Stresses in Rectangular Section Beams . . . . . . . . . .
1.13 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.14 Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. De¯ection and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Spring Rates for Tension, Compression, and Torsion . . . . .
2.3 De¯ection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 De¯ections Analysis Using Singularity Functions . . . . . . . .

2.5 Impact Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Castigliano's Theorem . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Long Columns with Central Loading . . . . . . . . . . . . . . . .
2.10 Intermediate-Length Columns with Central Loading . . . . . .
2.11 Columns with Eccentric Loading . . . . . . . . . . . . . . . . . .
2.12 Short Compression Members . . . . . . . . . . . . . . . . . . . . .
3. Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Endurance Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fluctuating Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Constant Life Fatigue Diagram . . . . . . . . . . . . . . . . . . . .
3.4 Fatigue Life for Randomly Varying Loads . . . . . . . . . . . . .
3.5 Criteria of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Mechanics of Materials

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CHAPTER 4

Theory of Mechanisms
1. Fundamentals . . . . . . . . . . . . . . . . .
1.1 Motions . . . . . . . . . . . . . . . . .
1.2 Mobility . . . . . . . . . . . . . . . . .
1.3 Kinematic Pairs . . . . . . . . . . . .
1.4 Number of Degrees of Freedom .
1.5 Planar Mechanisms. . . . . . . . . .
2. Position Analysis . . . . . . . . . . . . . . . .
2.1 Cartesian Method . . . . . . . . . . .
2.2 Vector Loop Method . . . . . . . . .
3. Velocity and Acceleration Analysis . . . .
3.1 Driver Link . . . . . . . . . . . . . . .
3.2 RRR Dyad . . . . . . . . . . . . . . . .
3.3 RRT Dyad. . . . . . . . . . . . . . . .
3.4 RTR Dyad. . . . . . . . . . . . . . . .
3.5 TRT Dyad. . . . . . . . . . . . . . . .
4. Kinetostatics . . . . . . . . . . . . . . . . . .
4.1 Moment of a Force about a Point
4.2 Inertia Force and Inertia Moment
4.3 Free-Body Diagrams . . . . . . . . .
4.4 Reaction Forces . . . . . . . . . . . .
4.5 Contour Method . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . .

CHAPTER 5

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190
190
190
191
199
200
202
202
208
211
212
212
214
215
216
223
223
224
227
228
229
241

Craciunoiu
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244
244
247
253
253
253
258
261
262
267
270
275
283
283
283

284
284
290
292
293
296

Machine Components
Dan B. Marghitu, Cristian I. Diaconescu,
1. Screws . . . . . . . . . . . . . . . . . . . . .
1.1 Screw Thread . . . . . . . . . . . .
1.2 Power Screws . . . . . . . . . . . .
2. Gears . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . .
2.2 Geometry and Nomenclature . .
2.3 Interference and Contact Ratio .
2.4 Ordinary Gear Trains . . . . . . .
2.5 Epicyclic Gear Trains . . . . . . .
2.6 Differential . . . . . . . . . . . . . .
2.7 Gear Force Analysis . . . . . . . .
2.8 Strength of Gear Teeth . . . . . .
3. Springs . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . .
3.2 Material for Springs . . . . . . . .
3.3 Helical Extension Springs . . . .
3.4 Helical Compression Springs . .
3.5 Torsion Springs . . . . . . . . . . .
3.6 Torsion Bar Springs . . . . . . . .
3.7 Multileaf Springs . . . . . . . . . .
3.8 Belleville Springs . . . . . . . . . .


and Nicolae
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Dan B. Marghitu


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4. Rolling Bearings . . . . . . . . . . . . . . . . . .
4.1 Generalities . . . . . . . . . . . . . . . .
4.2 Classi®cation . . . . . . . . . . . . . . .
4.3 Geometry. . . . . . . . . . . . . . . . . .
4.4 Static Loading . . . . . . . . . . . . . . .
4.5 Standard Dimensions . . . . . . . . . .
4.6 Bearing Selection . . . . . . . . . . . .
5. Lubrication and Sliding Bearings . . . . . . .
5.1 Viscosity . . . . . . . . . . . . . . . . . .
5.2 Petroff's Equation . . . . . . . . . . . .
5.3 Hydrodynamic Lubrication Theory .
5.4 Design Charts . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . .
CHAPTER 6

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297
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298
298
303
304
308
318
318
323
326
328
336


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3. Linear Systems with Finite Numbers of Degrees of Freedom
3.1 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . .
3.2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . .
3.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Analysis of System Model . . . . . . . . . . . . . . . . . . .
3.5 Approximative Methods for Natural Frequencies . . . .
4. Machine-Tool Vibrations . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The Machine Tool as a System . . . . . . . . . . . . . . .
4.2 Actuator Subsystems . . . . . . . . . . . . . . . . . . . . . .
4.3 The Elastic Subsystem of a Machine Tool . . . . . . . .
4.4 Elastic System of Machine-Tool Structure . . . . . . . . .
4.5 Subsystem of the Friction Process. . . . . . . . . . . . . .
4.6 Subsystem of Cutting Process . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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340
341
342
343
345
352
359
369
370
374
380
385
386
392
404
405
407
416
416
418

419
435
437
440
444

Theory of Vibration
Dan B. Marghitu, P. K. Raju, and Dumitru Mazilu
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Linear Systems with One Degree of Freedom . . . . .
2.1 Equation of Motion . . . . . . . . . . . . . . . . . .
2.2 Free Undamped Vibrations . . . . . . . . . . . . .
2.3 Free Damped Vibrations . . . . . . . . . . . . . . .
2.4 Forced Undamped Vibrations . . . . . . . . . . .
2.5 Forced Damped Vibrations . . . . . . . . . . . . .
2.6 Mechanical Impedance. . . . . . . . . . . . . . . .
2.7 Vibration Isolation: Transmissibility. . . . . . . .
2.8 Energetic Aspect of Vibration with One DOF .
2.9 Critical Speed of Rotating Shafts. . . . . . . . . .

CHAPTER 7

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Principles of Heat Transfer
Alexandru Morega
1. Heat Transfer Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 446
1.1 Physical Mechanisms of Heat Transfer: Conduction, Convection,
and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

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1.2 Technical Problems of Heat Transfer . . . . . .
2. Conduction Heat Transfer . . . . . . . . . . . . . . . . .
2.1 The Heat Diffusion Equation . . . . . . . . . . .
2.2 Thermal Conductivity . . . . . . . . . . . . . . . .
2.3 Initial, Boundary, and Interface Conditions . .
2.4 Thermal Resistance . . . . . . . . . . . . . . . . .
2.5 Steady Conduction Heat Transfer . . . . . . . .
2.6 Heat Transfer from Extended Surfaces (Fins)

2.7 Unsteady Conduction Heat Transfer . . . . . .
3. Convection Heat Transfer . . . . . . . . . . . . . . . . . .
3.1 External Forced Convection . . . . . . . . . . . .
3.2 Internal Forced Convection . . . . . . . . . . . .
3.3 External Natural Convection. . . . . . . . . . . .
3.4 Internal Natural Convection . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 8

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455
456
457
459
461
463
464
468
472
488
488
520
535
549
555

Nicolae Craciunoiu and Bogdan O. Ciocirlan
1. Fluids Fundamentals . . . . . . . . . . . . . . . . . . . . . . .
1.1 De®nitions . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Systems of Units . . . . . . . . . . . . . . . . . . . . .
1.3 Speci®c Weight . . . . . . . . . . . . . . . . . . . . . .
1.4 Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Vapor Pressure . . . . . . . . . . . . . . . . . . . . . .

1.6 Surface Tension . . . . . . . . . . . . . . . . . . . . . .
1.7 Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Bulk Modulus of Elasticity . . . . . . . . . . . . . . .
1.9 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Hydrostatic Forces on Surfaces . . . . . . . . . . . .
1.11 Buoyancy and Flotation . . . . . . . . . . . . . . . .
1.12 Dimensional Analysis and Hydraulic Similitude .
1.13 Fundamentals of Fluid Flow. . . . . . . . . . . . . .
2. Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Absolute and Gage Pressure . . . . . . . . . . . . .
2.2 Bernoulli's Theorem . . . . . . . . . . . . . . . . . . .
2.3 Hydraulic Cylinders . . . . . . . . . . . . . . . . . . .
2.4 Pressure Intensi®ers . . . . . . . . . . . . . . . . . . .
2.5 Pressure Gages . . . . . . . . . . . . . . . . . . . . . .
2.6 Pressure Controls . . . . . . . . . . . . . . . . . . . . .
2.7 Flow-Limiting Controls . . . . . . . . . . . . . . . . .
2.8 Hydraulic Pumps . . . . . . . . . . . . . . . . . . . . .
2.9 Hydraulic Motors . . . . . . . . . . . . . . . . . . . . .
2.10 Accumulators . . . . . . . . . . . . . . . . . . . . . . .
2.11 Accumulator Sizing. . . . . . . . . . . . . . . . . . . .
2.12 Fluid Power Transmitted . . . . . . . . . . . . . . . .
2.13 Piston Acceleration and Deceleration. . . . . . . .
2.14 Standard Hydraulic Symbols . . . . . . . . . . . . .
2.15 Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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560
560
560
560
561
562
562
562
562
563
564
565
565
568
572
572
573
575
578
579
580
592
595
598

601
603
604
604
605
606

Fluid Dynamics

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Table of Contents

2.16 Representative Hydraulic System . . . . . . . . . . . . . . . . . . . . . 607
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

Control
Mircea Ivanescu
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 A Classic Example . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Transfer Functions for Standard Elements . . . . . . . . . . .
3.2 Transfer Functions for Classic Systems . . . . . . . . . . . . .

4. Connection of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Steady-State Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Input Variation Steady-State Error . . . . . . . . . . . . . . . . .
6.2 Disturbance Signal Steady-State Error . . . . . . . . . . . . . .
7. Time-Domain Performance . . . . . . . . . . . . . . . . . . . . . . . . .
8. Frequency-Domain Performances . . . . . . . . . . . . . . . . . . . . .
8.1 The Polar Plot Representation . . . . . . . . . . . . . . . . . . .
8.2 The Logarithmic Plot Representation. . . . . . . . . . . . . . .
8.3 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. Stability of Linear Feedback Systems . . . . . . . . . . . . . . . . . . .
9.1 The Routh±Hurwitz Criterion . . . . . . . . . . . . . . . . . . . .
9.2 The Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Stability by Bode Diagrams . . . . . . . . . . . . . . . . . . . . .
10. Design of Closed-Loop Control Systems by Pole-Zero Methods .
10.1 Standard Controllers. . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 P-Controller Performance . . . . . . . . . . . . . . . . . . . . . .
10.3 Effects of the Supplementary Zero . . . . . . . . . . . . . . . .
10.4 Effects of the Supplementary Pole . . . . . . . . . . . . . . . .
10.5 Effects of Supplementary Poles and Zeros . . . . . . . . . . .
10.6 Design Example: Closed-Loop Control of a Robotic Arm .
11. Design of Closed-Loop Control Systems by Frequential Methods
12. State Variable Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13. Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1 Nonlinear Models: Examples . . . . . . . . . . . . . . . . . . . .
13.2 Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Stability of Nonlinear Systems . . . . . . . . . . . . . . . . . . .
13.4 Liapunov's First Method . . . . . . . . . . . . . . . . . . . . . . .
13.5 Liapunov's Second Method . . . . . . . . . . . . . . . . . . . . .
14. Nonlinear Controllers by Feedback Linearization . . . . . . . . . . .

15. Sliding Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Fundamentals of Sliding Control . . . . . . . . . . . . . . . . .
15.2 Variable Structure Systems . . . . . . . . . . . . . . . . . . . . .
A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Differential Equations of Mechanical Systems . . . . . . . . .

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612
613
614
616
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618
620
623
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624
628
631
632
633
637
639
640
641
648
649
650
651
656
660
661
664
669
672
678
678
681
685
688
689
691

695
695
700
703
703

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CHAPTER 9


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xii

Table of Contents

A.2 The Laplace Transform . . . . . . .
A.3 Mapping Contours in the s-Plane
A.4 The Signal Flow Diagram . . . . .
References . . . . . . . . . . . . . . . . . . .

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707
707
712
714

Differential Equations and Systems of Differential
Equations
Horatiu Barbulescu
1. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Ordinary Differential Equations: Introduction . . . . . . . . . . . .
1.2 Integrable Types of Equations . . . . . . . . . . . . . . . . . . . . . .
1.3 On the Existence, Uniqueness, Continuous Dependence on a
Parameter, and Differentiability of Solutions of Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . .
2. Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Integrating a System of Differential Equations by the
Method of Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Finding Integrable Combinations . . . . . . . . . . . . . . . . . . . .
2.4 Systems of Linear Differential Equations. . . . . . . . . . . . . . . .
2.5 Systems of Linear Differential Equations with Constant
Coef®cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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766
774
816
816

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

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APPENDIX

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The purpose of this handbook is to present the reader with a teachable text

that includes theory and examples. Useful analytical techniques provide the
student and the practitioner with powerful tools for mechanical design. This
book may also serve as a reference for the designer and as a source book for
the researcher.
This handbook is comprehensive, convenient, detailed, and is a guide
for the mechanical engineer. It covers a broad spectrum of critical engineering topics and helps the reader understand the fundamentals.
This handbook contains the fundamental laws and theories of science
basic to mechanical engineering including controls and mathematics. It
provides readers with a basic understanding of the subject, together with
suggestions for more speci®c literature. The general approach of this book
involves the presentation of a systematic explanation of the basic concepts of
mechanical systems.
This handbook's special features include authoritative contributions,
chapters on mechanical design, useful formulas, charts, tables, and illustrations. With this handbook the reader can study and compare the available
methods of analysis. The reader can also become familiar with the methods
of solution and with their implementation.
Dan B. Marghitu

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Preface


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Contributors

Horatiu Barbulescu, (715) Department of Mechanical Engineering,
Auburn University, Auburn, Alabama 36849
Bogdan O. Ciocirlan, (1, 51, 119, 559) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849
Nicolae Craciunoiu, (243, 559) Department of Mechanical Engineering,
Auburn University, Auburn, Alabama 36849
Cristian I. Diaconescu, (1, 51, 119, 243) Department of Mechanical
Engineering, Auburn University, Auburn, Alabama 36849
Mircea Ivanescu, (611) Department of Electrical Engineering, University
of Craiova, Craiova 1100, Romania
Dan B. Marghitu, (1, 51, 119, 189, 243, 339) Department of Mechanical
Engineering, Auburn University, Auburn, Alabama 36849
Dumitru Mazilu, (339) Department of Mechanical Engineering, Auburn
University, Auburn, Alabama 36849
Alexandru Morega, (445) Department of Electrical Engineering, ``Politehnica'' University of Bucharest, Bucharest 6-77206, Romania
P. K. Raju, (339) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849

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Numbers in parentheses indicate the pages on which the authors' contributions begin.


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1


Statics

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DAN B. MARGHITU, CRISTIAN I. DIACONESCU, AND
BOGDAN O. CIOCIRLAN

Department of Mechanical Engineering,
Auburn University, Auburn, Alabama 36849

Inside
1. Vector Algebra
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13

2

Terminology and Notation 2

Equality 4
Product of a Vector and a Scalar 4
Zero Vectors 4
Unit Vectors 4
Vector Addition 5
Resolution of Vectors and Components 6
Angle between Two Vectors 7
Scalar (Dot) Product of Vectors 9
Vector (Cross) Product of Vectors 9
Scalar Triple Product of Three Vectors 11
Vector Triple Product of Three Vectors 11
Derivative of a Vector 12

2. Centroids and Surface Properties
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12

3. Moments and Couples
3.1
3.2

3.3
3.4
3.5
3.6

12

Position Vector 12
First Moment 13
Centroid of a Set of Points 13
Centroid of a Curve, Surface, or Solid 15
Mass Center of a Set of Particles 16
Mass Center of a Curve, Surface, or Solid 16
First Moment of an Area 17
Theorems of Guldinus±Pappus 21
Second Moments and the Product of Area 24
Transfer Theorems or Parallel-Axis Theorems 25
Polar Moment of Area 27
Principal Axes 28

30

Moment of a Bound Vector about a Point 30
Moment of a Bound Vector about a Line 31
Moments of a System of Bound Vectors 32
Couples 34
Equivalence 35
Representing Systems by Equivalent Systems 36

1



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2

Statics

40

5. Dry Friction

46

4.1
4.2
4.3
5.1
5.2
5.3

Equilibrium Equations 40
Supports 42
Free-Body Diagrams 44
Static Coef®cient of Friction 47
Kinetic Coef®cient of Friction 47
Angles of Friction 48

References 49

1. Vector Algebra

1.1 Terminology and Notation

T

he characteristics of a vector are the magnitude, the orientation, and
the sense. The magnitude of a vector is speci®ed by a positive
number and a unit having appropriate dimensions. No unit is stated if
the dimensions are those of a pure number. The orientation of a vector is
speci®ed by the relationship between the vector and given reference lines
andaor planes. The sense of a vector is speci®ed by the order of two points
on a line parallel to the vector. Orientation and sense together determine the
direction of a vector. The line of action of a vector is a hypothetical in®nite
straight line collinear with the vector. Vectors are denoted by boldface letters,
for example, a, b, A, B, CD. The symbol jvj represents the magnitude (or
module, or absolute value) of the vector v. The vectors are depicted by either
straight or curved arrows. A vector represented by a straight arrow has the
direction indicated by the arrow. The direction of a vector represented by a
curved arrow is the same as the direction in which a right-handed screw
moves when the screw's axis is normal to the plane in which the arrow is
drawn and the screw is rotated as indicated by the arrow.
Figure 1.1 shows representations of vectors. Sometimes vectors are
represented by means of a straight or curved arrow together with a measure
number. In this case the vector is regarded as having the direction indicated
by the arrow if the measure number is positive, and the opposite direction if
it is negative.

Figure 1.1

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Statics

4. Equilibrium


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A bound vector is a vector associated with a particular point P in space
(Fig. 1.2). The point P is the point of application of the vector, and the line
passing through P and parallel to the vector is the line of action of the vector.
The point of application may be represented as the tail, Fig. 1.2a, or the head
of the vector arrow, Fig. 1.2b. A free vector is not associated with a particular
point P in space. A transmissible vector is a vector that can be moved along
its line of action without change of meaning.

Figure 1.2
To move the body in Fig. 1.3 the force vector F can be applied anywhere
along the line D or may be applied at speci®c points AY BY C . The force vector
F is a transmissible vector because the resulting motion is the same in all
cases.

Figure 1.3
The force F applied at B will cause a different deformation of the body
than the same force F applied at a different point C . The points B and C are
on the body. If we are interested in the deformation of the body, the force F
positioned at C is a bound vector.

Statics

3


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1. Vector Algebra


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4

Statics

Statics

The operations of vector analysis deal only with the characteristics of
vectors and apply, therefore, to both bound and free vectors.

1.2 Equality
Two vectors a and b are said to be equal to each other when they have the
same characteristics. One then writes
a ˆ bX

1.3 Product of a Vector and a Scalar
DEFINITION

The product of a vector v and a scalar s, s v or vs, is a vector having the
following characteristics:
1. Magnitude.

js vj  jvsj ˆ jsjjvjY
where jsj denotes the absolute value (or magnitude, or module) of the
scalar s.

2. Orientation. s v is parallel to v. If s ˆ 0, no de®nite orientation is
attributed to s v.
3. Sense. If s b 0, the sense of s v is the same as that of v. If s ` 0, the
sense of s v is opposite to that of v. If s ˆ 0, no de®nite sense is
attributed to s v. m

1.4 Zero Vectors
DEFINITION

A zero vector is a vector that does not have a de®nite direction and whose
magnitude is equal to zero. The symbol used to denote a zero vector is 0. m

1.5 Unit Vectors
DEFINITION

A unit vector (versor) is a vector with the magnitude equal to 1.

m

Given a vector v, a unit vector u having the same direction as v is obtained
by forming the quotient of v and jvj:
v
X
uˆ
jvj

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Equality does not imply physical equivalence. For instance, two forces
represented by equal vectors do not necessarily cause identical motions of

a body on which they act.


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1.6 Vector Addition

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The sum of a vector v 1 and a vector v 2 X v 1 ‡ v 2 or v 2 ‡ v 1 is a vector whose
characteristics are found by either graphical or analytical processes. The
vectors v 1 and v 2 add according to the parallelogram law: v 1 ‡ v 2 is equal to
the diagonal of a parallelogram formed by the graphical representation of the
vectors (Fig. 1.4a). The vectors v 1 ‡ v 2 is called the resultant of v 1 and v 2 .
The vectors can be added by moving them successively to parallel positions
so that the head of one vector connects to the tail of the next vector. The
resultant is the vector whose tail connects to the tail of the ®rst vector, and
whose head connects to the head of the last vector (Fig. 1.4b).
The sum v 1 ‡ …Àv 2 † is called the difference of v 1 and v 2 and is denoted
by v 1 À v 2 (Figs. 1.4c and 1.4d).

Statics

5

1. Vector Algebra

Figure 1.4
The sum of n vectors v i , i ˆ 1Y F F F Y n,
n

€
iˆ1

v i or v 1 ‡ v 2 ‡ Á Á Á ‡ v n Y

is called the resultant of the vectors v i , i ˆ 1Y F F F Y n.


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6

Statics

Statics

The vector addition is:
1. Commutative, that is, the characteristics of the resultant are indepen-

dent of the order in which the vectors are added (commutativity):
v1 ‡ v2 ˆ v2 ‡ v1X
2. Associative, that is, the characteristics of the resultant are not affected

by the manner in which the vectors are grouped (associativity):
v 1 ‡ …v 2 ‡ v 3 † ˆ …v 1 ‡ v 2 † ‡ v 3 X

3. Distributive, that is, the vector addition obeys the following laws of

distributivity:

s


n
€
iˆ1
n
€

iˆ1

si ˆ

vi ˆ

n
€
iˆ1
n
€
iˆ1

…vsi †Y for si Tˆ 0Y si P ‚
…s v i †Y for s Tˆ 0Y s P ‚X

Here ‚ is the set of real numbers.
Every vector can be regarded as the sum of n vectors …n ˆ 2Y 3Y F F F† of
which all but one can be selected arbitrarily.

1.7 Resolution of Vectors and Components
Let i1 , i2 , i3 be any three unit vectors not parallel to the same plane
ji1 j ˆ ji2 j ˆ ji3 j ˆ 1X

For a given vector v (Fig. 1.5), there exists three unique scalars v1 , v1 , v3 , such
that v can be expressed as
v ˆ v 1 i1 ‡ v2 i2 ‡ v3 i3 X

Figure 1.5
The opposite action of addition of vectors is the resolution of vectors. Thus,
for the given vector v the vectors v1 i1 , v2 i2 , and v3 i3 sum to the original
vector. The vector vk ik is called the ik component of v, and vk is called the ik
scalar component of v, where k ˆ 1Y 2Y 3. A vector is often replaced by its
components since the components are equivalent to the original vector.

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v


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Every vector equation v ˆ 0, where v ˆ v1 i1 ‡ v2 i2 ‡ v3 i3 , is equivalent
to three scalar equations v1 ˆ 0, v2 ˆ 0, v3 ˆ 0.
If the unit vectors i1 , i2 , i3 are mutually perpendicular they form a
cartesian reference frame. For a cartesian reference frame the following
notation is used (Fig. 1.6):

Statics

7

1. Vector Algebra

Figure 1.6

and
i c jY i c kY j c kX
The symbol c denotes perpendicular.
When a vector v is expressed in the form v ˆ vx i ‡ vy j ‡ vz k where i, j,
k are mutually perpendicular unit vectors (cartesian reference frame or
orthogonal reference frame), the magnitude of v is given by
q
jvj ˆ v2x ‡ v2y ‡ v2z X
The vectors v x ˆ vx i, v y ˆ vy j, and v z ˆ vy k are the orthogonal or rectangular component vectors of the vector v. The measures vx , vy , vz are the
orthogonal or rectangular scalar components of the vector v.
If v 1 ˆ v1x i ‡ v1y j ‡ v1z k and v 2 ˆ v2x i ‡ v2y j ‡ v2z k, then the sum of
the vectors is
v 1 ‡ v 2 ˆ …v1x ‡ v2x †i ‡ …v1y ‡ v2y †j ‡ …v1z ‡ v2z †v1z kX

1.8 Angle between Two Vectors
Let us consider any two vectors a and b. One can move either vector parallel
to itself (leaving its sense unaltered) until their initial points (tails) coincide.
The angle between a and b is the angle y in Figs. 1.7a and 1.7b. The angle
between a and b is denoted by the symbols (aY b) or (bY a). Figure 1.7c
represents the case (a, b† ˆ 0, and Fig. 1.7d represents the case (a, b† ˆ 180 .
The direction of a vector v ˆ vx i ‡ vy j ‡ vz k and relative to a cartesian
reference, i, j, k, is given by the cosines of the angles formed by the vector

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i1  iY i2  jY i3  k


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8


Statics

and the representative unit vectors. These are called direction cosines and
are denoted as (Fig. 1.8)
cos…vY i† ˆ cos a ˆ l Y cos…vY j† ˆ cos b ˆ mY cos…vY k† ˆ cos g ˆ nX
The following relations exist:
vx ˆ jvj cos aY vy ˆ jvj cos bY vz ˆ jvj cos gX

Figure 1.8

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Statics

Figure 1.7


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9

1. Vector Algebra

DEFINITION

Statics

1.9 Scalar (Dot) Product of Vectors
The scalar (dot) product of a vector a and a vector b is
a Á b ˆ b Á a ˆ jajjbj cos…aY b†X

For any two vectors a and b and any scalar s
…sa† Á b ˆ s…a Á b† ˆ a Á …sb† ˆ sa Á b

m

If
a ˆ ax i ‡ a y j ‡ a z k
b ˆ bx i ‡ by j ‡ bz kY
where i, j, k are mutually perpendicular unit vectors, then
a Á b ˆ ax b x ‡ ay b y ‡ az b z X
The following relationships exist:
i Á i ˆ j Á j ˆ k Á k ˆ 1Y
i Á j ˆ j Á k ˆ k Á i ˆ 0X
Every vector v can be expressed in the form
v ˆ i Á vi ‡ j Á vj ‡ k Á vkX
The vector v can always be expressed as
v ˆ vx i ‡ vy j ‡ vz kX
Dot multiply both sides by i:
i Á v ˆ vx i Á i ‡ vy i Á j ‡ vz i Á kX
But,
i Á i ˆ 1Y and

i Á j ˆ i Á k ˆ 0X

Hence,
i Á v ˆ vx X
Similarly,
j Á v ˆ vy

and


k Á v ˆ vz X

1.10 Vector (Cross) Product of Vectors
DEFINITION

The vector (cross) product of a vector a and a vector b is the vector (Fig. 1.9)
a  b ˆ jajjbj sin…aY b†n

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and