Jean-Marie Berthelot Mechanics of Rigid Bodies ISMANS Institute for Advanced Materials and Mechanics Le Mans, France
Jean-Marie Berthelot Mechanics of Rigid Bodies Jean-Marie Berthelot is an Emeritus Professor at the Institute for Advanced Materials and Mechanics (ISMANS), Le Mans, France. His current research is on the mechanical behaviour of composite materials and structures. He has published extensively in the area of composite materials and is the author of numerous international papers and textbooks, in particular a textbook entitled Composite Materials, Mechanical Behavior and Structural Analysis published by Springer, New York, in 1999.
Jean-Marie Berthelot Mechanichs of Rigid Bodies ISMANS Institute for Advanced Materials and Mechanics Le Mans, France
Preface The objective of this book is to develop the fundamental statements of the Mechanics of Rigid Bodies. The text is designed for undergraduate courses of Mechanical Engineering. The basic mathematical concepts are covered in the first part, thereby making the book self-contained. The different parts of the book are carefully developed to provide continuity of the concepts and theories. Finally the text has been established so as to construct chapter after chapter a unified procedure for analysing any mechanical system constituted of rigid bodies. The first part, Mathematical Basics, introduces the usual concepts needed in the study of mechanical systems: vector space R 3, geometric space, vector derivatives, curves. A chapter is devoted to torsors whose concept is the key of the book. The general notion of measure centre is introduced in this chapter. The second part, Kinematics, begins with the analysis of the motion of a point (kinematics of point). Particular motions are next considered, with a chapter related to motions with central acceleration. Next, the kinematics of a rigid body is studied: parameter of situation, kinematic torsor, analysis of particular motions. The change of reference system, which introduces the notion of entrainment has been excluded deliberately from this part. The notion of entrainment is not really assimilated by the studients at this level of the text. In fact this notion is implicitly introduced by using the concept of kinematic torsor. The change of reference system will be considered as a whole within the frame of Kinetics (Part 4). The last chapter analyses the kinematics of rigid bodies in contact. The third part, Mechanical Actions, introduces first the general concepts of the mechanical actions exerted on a rigid body or on a system of rigid bodies. Represented by torsors, the mechanical actions have general properties which are derived from the concepts considered previously for torsors. Thus, mechanical actions are classified as forces, couples and arbitrary actions. Gravitation and gravity are analysed. A chapter is devoted to the mechanical actions involved by the connections between rigid bodies, whose concept is the basis of the technological design of mechanical systems. The introduction of the power developed by a mechanical action simplifies greatly the restrictions imposed in the case of perfect connections (connections without friction). In the last chapter, the investigation of some problem of Statics will grow the reader familiar with the analysis of mechanical actions exerted on a body or a system of bodies. The fourth part, Kinetics of Rigid Bodies, introduces the tools needed to analyse the problems of Dynamics: operator of inertia, kinetic torsor, dynamic torsor and kinetic energy. Next, the problem of the change of reference system is considered. At this step, the reader has acquired the whole elements needed to analyse the problems of Dynamics of a rigid body or a system of rigid bodies. This analysis is developed in the fifth part Dynamics of Rigid Bodies. First, the general process for analysing a problem of Dynamics is established. Next, particular problems are considered. The process of analysis is always the same: kinematic analysis, kinetic analysis, investigation of the mechanical actions, deriving the equations of Dynamics as a consequence of the fundamental principle of dynamics, assumptions
vi Preface on the physical nature of connections between bodies, solving the equations of motion and the equations of connections. The designer will have to take an interest in the parameters of the motion as well as in the mechanical actions exerted at the level of connections to design the mechanical systems. The application of the fundamental principle of dynamics allows us to derive the whole equations of dynamics (equations of motion and equations of mechanical actions at the level of connections). However, designer which takes an interest only in the equations of motion needs a systematic tool for deriving these equations: the Lagrange s equations which are considered in the last chapter of part V. In general, the equations of motions of a body or of a system of rigid bodies are complex, and most of these equations can not be solved using an analytical process. Now, mechanical engineers dispose of numerical tools (numerical processes and microcomputers) needed to solve the motion equations, whatever the complexity of these equations may be. The sixth part, Numerical procedures for the Resolution of Motion Equations, is an introduction to the numerical processes used to solve equations of motion. Examples are considered. The correction of the exercises is reported at the end of the textbook. The writing has been developed extensively and structured in such a way to improve the capacity of the comprehension of the reader. At the end of the textbook, the designer will have all the elements which allow him to implement a complete and structured analysis of mechanical systems. June 2009, Le Mans, Jean-Marie BERTHELOT Note. The development of this textbook is based on a generalized use of the concept of torseur (in French). We think that this concept is not really used in the English textbooks. We will call this concept as torsor. In the textbook, the English formulation was thus transposed from the French formulation. The author would be highly grateful with whoever would bring any element likely to be able to make progress the development, and thus the comprehension, of the textbook.
Contents Preface v PART I Mathematical Basic Elements 1 Chapter 1 Vector Space 3 3 1.1 3 Definition of the Vector Space... 3 1.1.1 Vectors... 3 1.1.2 Vector Addition... 3 1.1.3 Multiplication by a Scalar... 4 1.2 3 Linear Dependence and Independence. Basis of... 5 1.2.1 Linear Combination... 5 1.2.2 Linear Dependence and Independence... 5 1.2.3 3 Basis of the Vector Space... 7 1.2.4 Components of a Vector... 7 1.3 Scalar Product... 8 1.3.1 Definition... 8 1.3.2 Magnitude or Norm of a Vector... 8 1.3.3 Analytical Expression of the Scalar Product in an Arbitrary Basis... 9 1.3.4 Orthogonal Vectors... 9 1.3.5 Orthonormal Basis.... 9 1.3.6 Expression of the Scalar Product in an Orthonormal Basis... 10 1.4 Vector Product... 10 1.4.1 Definition... 10 1.4.2 Analytical Expression of the Vector Product in an Arbitrary Basis... 11 1.4.3 Direct Basis... 11 1.4.4 Expression of the Vector Product in a Direct Basis.... 12 1.4.5 Mixed Product... 12 1.4.6 Property of the Double Vector Product... 12 1.5 3 Bases of the Vector Space... 13 1.5.1 Canonical Basis... 13 1.5.2 Basis Change... 13 Exercises... 16 Comments... 17 Chapter 2 The Geometric Space 18 2.1 The Geometric Space Considered as Affine to the Vector Space 3... 18
Contents ix 2.1.1 The Geometric Space... 18 2.1.2 Consequences... 19 2.1.3 Distance between Two Points... 20 2.1.4 Angle between Two Bipoints... 20 2.1.5 Reference Systems... 21 2.2 Subspaces of the Geometric Space: Line, Plane... 22 2.2.1 Line... 22 2.2.2 Plane... 23 2.2.3 Lines and Planes with Same Directions... 24 2.2.4 Orthogonal Lines and Planes... 25 2.3 Characterization of the Position a Point of the Geometric Space... 26 2.3.1 Coordinate Axes... 26 2.3.2 Direct Orthonormal Reference System.... 27 2.3.3 Cartesian Coordinates... 27 2.4 Plane and Line Equations... 29 2.4.1 Cartesian Equation of a Plane... 29 2.4.2 Cartesian Equation of a Line... 30 2.5 Change of Reference System... 31 2.5.1 General Case... 31 2.5.2 Refernce Systems with a Same Axis... 32 2.5.3 Arbitrary Reference Systems with the Same Origin... 34 Exercises... 37 Comments... 39 Chapter 3 Vector Function. Derivatives of a Vector Function 40 3.1 Vector Function of One Variable.... 40 3.1.1 Definition... 40 3.1.2 Derivative... 40 3.1.3 Properties of the Vector Derivative... 41 3.1.4 Examples... 42 3.2 Vector Function of Two Variables... 44 3.2.1 Definition... 44 3.2.2 Partial Derivatives... 44 3.2.3 Examples... 45 3.3 Vector Function of n Variables... 45 3.3.1 Definitions... 45 3.3.2 Examples... 46 Comments... 49 Chapter 4 Elementary Concepts on Curves 50 4.1 Introduction... 50 4.2 Curvilinear Abscissa. Arc Length of a Curve... 51 4.3 Tangent. Normal. Radius of Curvature.... 52 4.4 Frenet Trihedron... 52 Exercise... 54 Comments 54
x Contents Chapter 5 Torsors 55 5.1 Definition and Properties of the Torsors... 55 5.1.1 Definitions and notations... 55 5.1.2 Properties of the Moments... 56 5.1.3 Vector Space of Torsors... 56 5.1.4 Scalar Invariant of a Torsor... 57 5.1.5 Product of Two Torsors... 58 5.1.6 Moment of a Torsor about an Axis.... 58 5.1.7 Central Axis of a Torsor... 59 5.2 Particular Torsors. Resolution of an Arbitrary Torsor... 60 5.2.1 Slider... 60 5.2.2 Couple-Torsor... 62 5.2.3 Arbitrary Torsor... 63 5.2.4 Conclusions... 64 5.3 Torsors associated to a Field of Sliders Defined on a Domain of the Geometric Space... 64 5.3.1 Torsor Associated to a Finite Set of Points... 64 5.3.2 Torsor Associated to a Infinite Set of Points... 65 5.3.3 Important Particular Case. Measure Centre... 67 Exercises.... 70 Comments.... 72 PART II Kinematics 73 Chapter 6 Kinematics of Point 75 6.1 Introduction... 75 6.2 Trajectory and Kinematic Vectors of a Point... 75 6.2.1 Trajectory... 76 6.2.2 Kinematic Vectors... 77 6.2.3 Tangential and Normal Components of Kinematic Vectors... 78 6.2.4 Different Types of Motions... 79 6.3 Expressions of the Components of Kinematic Vectors as Functions of Cartesian and Cylindrical Coordinates... 81 6.3.1 Cartesian Coordinates... 81 6.3.2 Cylindrical Coordinates... 82 Exercises.... 83 Comments.... 83 Chapter 7 Study of Particular Motions 84 7.1 Motions with Rectilinear Trajectory... 84 7.1.1 General Considerations... 84 7.1.2 Uniform Rectilinear Motion... 85 7.1.3 Uniformly Varied Rectilinear Motion... 85 7.1.4 Simple Harmonic Rectilinear Motion.... 86 7.2 Motions with a Circular Trajectory... 87 7.2.1 General Equations... 87
Contents xi 7.2.2 Uniform Circular Motion... 88 7.2.3 Uniformly Varied Circular Motion... 89 7.3 Motions with a Contant Acceleration Vector... 90 7.3.1 General Equations... 90 7.3.2 Study of the case where the Trajectory is Rectilinear... 91 7.3.3 Study of the case where the Trajectory is Parabolic... 92 7.4 Helicoidal Motion... 94 7.5 Cycloidal Motion... 96 Exercises.... 98 Comments.... 99 Chapter 8 Motions with Central Acceleration 100 8.1 General Properties... 100 8.1.1 Definition... 100 8.1.2 A Motion with a Central Acceleration is a Plane Trajectory Motion... 100 8.1.3 Areal Velocity... 101 8.1.4 Area Law... 102 8.1.5 Expression of the Kinematic Vectors... 102 8.1.6 Polar Equation of the Trajectory... 102 ( ) T 2 8.1.7 Motions for which a ( M, t) = ω OM... 103 ( T ) OM 8.2 Motions with Central Acceleration for which a ( M, t) = K... 104 OM 3 8.2.1 Equations of the Trajectories... 104 8.2.2 Study of the Trajectories... 105 8.2.3 Velocity Magnitude at a Point of the Trajectory... 107 8.2.4 Elliptic Motion. Kepler s Laws... 108 Comments.... 110 Chapter 9 Kinematics of Rigid Body 111 9.1 General Considerations... 111 9.1.1 Notion of Rigid Body... 111 9.1.2 Locating a Rigid Body... 111 9.2 Relations between the Trajectories and the Kinematic Vectors of Two Points Attached to a Solid... 113 9.2.1 Relation between the Trajectories... 113 9.2.2 Relation between the Velocity Vectors... 114 9.2.3 Expression of the Instantaneous Vector of Rotation... 115 9.2.4 Kinematic Torsor... 116 9.2.5 Relation between the Acceleration Vectors... 117 9.3 Generalization of the Composition of Motions... 118 9.3.1 Composition of Kinematic Torsors... 118 9.3.2 Inverse Motions... 120 9.4 Examples of Solid Motions.... 121 9.4.1 Motion of Rotation about a Fixed Axis... 121 9.4.2 Translation Motion of a Rigid Body... 124
xii Contents 9.4.3 Motion of a Body Subjected to a Cylindrical Joint... 125 9.4.4 Motion of Rotation about a Fixed Point... 127 9.4.5 Plane Motion... 129 Exercises.... 134 Comments.... 136 Chapter 10 Kinematics of Rigid Bodies in Contact 137 10.1 Kinematics of Two Solids in Contact.... 137 10.1.1 Solids in Contact at a Point. Sliding... 137 10.1.2 Spinning and Rolling... 138 10.1.3 Conclusions... 139 10.1.4 Solids in Contact in Several points... 140 10.2 Transmission of a Motion of Rotation... 140 10.2.1 Général Elements... 140 10.2.2 Transmission by Friction... 141 10.2.3 Gear Transmission... 145 10.2.4 Belt Transmission... 148 Exercises.... 150 Comments.... 151 PART III The Mechanical Actions 153 Chapter 11 General Elements on the Mechanical Actions 155 11.1 Concepts Relative to the Mechanical Actions... 155 11.1.1 Notion of Mechanical Action... 155 11.1.2 Representation of a Mechanical Action... 155 11.1.3 Classification of the Mechanical Actions... 156 11.1.4 Mechanical Actions Exerting between Material Sets... 158 11.1.5 External Mechanical Actions Exerting on a Material Set... 158 11.2 Different Types of Mechanical Actions... 159 11.2.1 Physical Natures of the Mechanical Actions... 159 11.2.2 Environnement and Effective Actions... 159 11.3 Power and Work... 160 11.3.1 Definition of the Power... 160 11.3.2 Change of Reference System... 161 11.3.3 Potential Energy... 161 11.3.4 Work... 162 11.3.5 Power and Work of a Force... 163 11.3.6 Set of Rigid Bodies... 164 Exercises... 165 Comments.... 167 Chapter 12 Gravitation. Gravity. Mass Centre 169 12.1 Phenomenon of Gravitation... 169 12.1.1 Law of Gravitation 169
Contents xiii 12.1.2 Gravitational Field... 170 12.1.3 Action of gravitation induced by a Solid Sphere... 170 12.1.4 Action of gravitation induced by the Earth... 172 12.2 Action of Gravity... 173 12.2.1 Gravity Field Induced by the Earth... 173 12.2.2 Action of Gravity Exerted on a Material System... 174 12.2.3 Power Developed by the Action of Gravity... 175 12.3 Determination of Mass Centres... 177 12.3.1 Mass Centre of a Material System... 177 12.3.2 Mass Centre of the Union of Two Sets... 178 12.3.3 Mass Centre of a Homogeneous Set... 179 12.3.4 Homogeneous Bodies with Geometrical Symmetries... 180 12.4 Examples of Determination of Mass Centres... 181 12.4.1 Homogeneous Solid Hemisphere... 181 12.4.2 Homogeneous Solid with Complex Geometry... 182 12.4.3 Non-Homogeneous Solid... 183 Exercises... 184 Comments... 185 Chapter 13 Actions of Contact between Solids. Connections 186 13.1 Laws of Contact between Solids.... 186 13.1.1 Introduction.... 186 13.1.2 Contact in a Point... 186 13.1.3 Couples of Rolling and Spinning... 191 13.2 Connections... 192 13.2.1 Introduction.... 192 13.2.2 Classification of Connections... 193 13.2.3 Actions of Connection... 197 13.2.4 Connection without Friction... 198 13.2.5 Connection with Friction... 202 Comments.... 203 Chapter 14 Statics of Rigid Bodies 204 14.1 Introduction... 204 14.2 Law of Statics... 204 14.2.1 Case of a Rigid Body... 204 14.2.2 Case of a Set of Rigid Bodies... 205 14.2.3 Mutual Actions... 206 14.3 Statics of Wires or Flexible Cables... 207 14.3.1 Mechanical Action Exerted by a Wire or a Flexible Cable... 207 14.3.2 Equation of Statics of a Wire... 208 14.3.3 Wire or Flexible Cable Submitted to the Gravity... 209 14.3.4 Contact of a Wire with a Rigid Body... 210 14.4 Examples of Equilibrium... 212 14.4.1 Case of a Rigid Body... 212 14.4.2 Case of a System of Two Rigid Bodies... 217 Exercises... 222 Comments 223
xiv Contents PART IV Kinetics of Rigid Bodies 225 Chapter 15 The Operator of Inertia 227 15.1 Introduction to the Operator of Inertia... 227 15.1.1 Operator Associated to a Vector Product.... 227 15.1.2 Extending the Preceding Concept... 228 15.1.3 The Operator of Inertia... 229 15.2 Change of Coordinate System.... 230 15.2.1 Change of Origin... 230 15.2.2 Relations of Huyghens... 232 15.2.3 Diagonalisation of the Matrix of Inertia... 232 15.2.4 Change of Basis... 233 15.3 Moments of Inertia with respect to a point, an axis, a plane... 234 15.3.1 Definitions... 234 15.3.2 Relations between the Moments of Inertia... 235 15.3.3 Case of a Plane Solid... 235 15.3.4 Moment of Inertia with respect to an Arbitrary Axis... 236 15.4 Determination of Matrices of Inertia... 237 15.4.1 Solids with Material Symmetries... 237 15.4.2 Solids having a Symmetry of Revolution... 239 15.4.3 Solids with Spherical Symmetry... 241 15.4.4 Associativity... 242 15.5 Matrices of Inertia of Homogeneous Bodies.... 244 15.5.1 One-Dimensional Solids... 244 15.5.2 Two-Dimensional Solids... 245 15.5.3 Three-Dimensional Solids... 249 Exercises... 253 Comments... 254 Chapter 16 Kinetic and Dynamic Torsors. Kinetic Energy 255 16.1 Kinetic Torsor... 255 16.1.1 Definition... 255 16.1.2 Kinetic Torsor Associated to the Motion of a Body... 256 16.1.3 Kinetic Torsor for a Set of Bodies... 257 16.2 Dynamic Torsor... 258 16.2.1 Definition... 258 16.2.2 Dynamic Torsor Associated to the Motion of a Body... 258 16.2.3 Dynamic Torsor for a Set of Bodies... 259 16.2.4 Relation with the Kinetic Energy... 260 16.3 Kinetic Energy.... 260 16.3.1 Definition... 260 16.3.2 Kinetic Energy of a Body... 261 16.3.3 Kinetic Energy of a Set of Solids... 262 16.3.4 Derivative of the Kinetic Energy of a Solid with respect to Time... 262 Exercises... 263 Comments... 264
Contents xv Chapter 17 Change of Reference System 265 17.1 Kinematics of Change of Reference... 265 17.1.1 Relation between the Kinematic Torsors... 265 17.1.2 Relation between the Velocity Vectors. Velocity of Entrainment... 266 17.1.3 Composition of Acceleration Vectors... 268 17.2 Dynamic Torsors... 269 17.2.1 Inertia Torsor of Entrainment... 270 17.2.2 Inertia Torsor of Coriolis... 271 17.2.3 Relation between the Dynamic Torsors Defined relatively to Two Different References... 272 Comments... 273 PART V Dynamics of Rigid Bodies 275 Chapter 18 The Fundamental Principle of Dynamics and its Consequences 277 18.1 Fundamental Principle.... 277 18.1.1 Statement of the Fundamental Principle of Dynamics... 277 18.1.2 Class of Galilean Reference Systems... 277 18.1.3 Vector Equations Deduced from the Fundamental Principle... 278 18.1.4 Scalar Equations Deduced from the Fundamental Principle.... 279 18.2 Mutual Actions... 280 18.2.1 Theorem of Mutual Actions... 280 18.2.2 Transmission of Mechanical Actions... 281 18.3 Theorem of Power-Energy.... 281 18.3.1 Case of One Solid... 281 18.3.2 Case of a Set of Bodies... 282 18.3.3 Mechanical Actions with Potential Energy... 283 18.4 Application of the Fundamental Principle to the Study of the Motion of a Free Body in a Galilean Reference... 284 18.4.1 General Problem... 284 18.4.2 Particular Cases... 286 18.5 Application to the Solar System.... 288 18.5.1 Galilean Reference... 288 18.5.2 Motion of Planets... 290 18.5.3 The Earth in the Solar System... 290 Comments... 291 Chapter 19 The Fundamental Equation of Dynamics in Different References 293 19.1 General Elements... 293 19.1.1 Fundamental Equation of Dynamics in a Non Galilean Reference... 293 19.1.2 The Reference Systems used in Mechanics... 294 19.2 Fundamental Relation of Dynamics in the Geocentric Reference... 295 19.2.1 General Equations... 295
xvi Contents 19.2.2 Case of a Solid Located at the Vicinity of the Earth... 297 19.3 Fundamental Relation in a Reference Attached to the Earth... 298 19.3.1 Equations of Motion... 298 19.3.2 Action of Earthly Gravity... 299 19.3.3 Conclusions on the Equations of Dynamics in a Reference Attached to the Earth... 300 19.4 Equations of Dynamics of a Body with respect to a Reference whose the Motion is Known Relatively to the Earth... 301 Comments... 303 Chapter 20 General Process for Analysing a Problem of Dynamics of Rigid Bodies 304 20.1 Dynamics of Rigid Body... 304 20.1.1 General Equations... 304 20.1.2 General Process of Analysis... 305 20.2 Dynamics of a Set of Bodies... 306 20.3 Conclusion... 307 Comments... 308 Chapter 21 Dynamics of Systems with One Degree of Freedom Analysis of Vibrations 309 21.1 General Equations.... 309 21.1.1 Introduction... 309 21.1.2 Parameters of Situation... 310 21.1.3 Kinematics... 310 21.1.4 Kinetics... 310 21.1.5 Mechanical Actions Exerted on the Solid... 311 21.1.6 Application of the Fundamental Principle... 311 21.2 Vibrations without Friction... 313 21.2.1 Equation of Motion... 313 21.2.2 Free Vibrations... 313 21.2.3 Forced Vibrations. Steady State... 314 21.3 Vibrations with Viscous Damping... 318 21.3.1 Equation of Motion with Viscous Damping... 318 21.3.2 Free Vibrations... 318 21.3.3 Vibrations in the case of a Harmonic Disturbing Force... 324 21.3.4 Forced Vibrations in the case of a Periodic Disturbing Force... 331 21.3.5 Vibrations in the case of an Arbitrary Disturbing Force... 332 21.3.6 Forced Vibrations in the case of a Motion Imposed to the Support... 333 21.4 Vibrations with Dry Friction... 336 21.4.1 Equations of Motion... 336 21.4.2 Free Vibrations... 337 21.5 Equivalent Viscous Damping... 339 21.5.1 Introduction... 339 21.5.2 Energy Dissipated in the case of Viscous Damping... 340 21.5.3 Stuctural Damping... 340
Contents xvii 21.5.4 Dry Friction... 342 21.5.5 Fluid Friction... 343 21.5.6 Conclusion... 345 Exercises... 346 Comments... 346 Chapter 22 Motion of Rotation of a Solid about a Fixed Axis 347 22.1 General Equations.... 347 22.1.1 Introduction... 347 22.1.2 Parameters of Situation... 348 22.1.3 Kinematics... 349 22.1.4 Kinetics... 350 22.1.5 Mechanical Actions Exerted on the Sold... 351 22.1.6 Application of the Fundamental Principle of Dynamics.... 352 22.2 Examples of Motions of Rotation about an Axis... 354 22.2.1 Solid in Rotation Submitted only to the Gravity... 354 22.2.2 Pendulum of Torsion... 356 22.3 Problem of the Balancing of Rotors... 357 22.3.1 General Equations of an Unbalanced Solid in Rotation... 357 22.3.2 Mechanical Actions Exerted on the Shaft of Rotor... 360 22.3.3 Principle of the Balancing... 360 Exercises... 362 Comments... 364 Chapter 23 Plane Motion of a Rigid Body 365 23.1 Introduction.... 365 23.2 Parallelepiped Moving on an Inclined Plane... 365 23.2.1 Parameters of Situation and Kinematics... 365 23.2.2 Kinetics of the Motion... 366 23.2.3 Mechanical Actions Exerted on the Parallelepiped... 367 23.2.4 Equations Deduced from the Fundamental Principle... 368 23.2.5 Motion without Friction... 369 23.2.6 Motion with Dry Friction... 370 23.2.7 Motion with Viscous Friction... 371 23.3 Analysis of Sliding and Rocking of a Parallelepiped on an Inclined Plane... 372 23.3.1 Introduction... 372 23.3.2 Parameters of Situation and Kinematics... 373 23.3.3 General Equations... 374 23.3.4 Analysis of the Different Motions... 375 23.3.5 Conclusions... 379 23.4 Motion of a Cylinder on an Inclined Plane.... 380 23.4.1 Introduction... 380 23.4.2 Parameters of Situation and Kinematics... 381
xviii Contents 23.4.3 Mechanical Actions Exerted on the Cylinder... 382 23.4.4 General Equations... 383 23.4.5 Analysis of the Different Motions... 385 23.5 Conclusions... 387 Comments... 388 Chapter 24 Other Examples of Motions of Rigid Bodies 389 24.1 Solid in Translation.... 389 24.1.1 General Expressions of a Solid in Translation... 389 24.1.2 Free Solid in Translation... 391 24.2 Motion of a Solid Placed on a Wagon.... 392 24.2.1 Introduction... 392 24.2.2 Parameters of Situation... 393 24.2.3 Kinetics... 394 24.2.4 Analysis of the Mechanical Actions... 394 24.2.5 Equations of Dynamics... 395 24.2.6 Analysis of the Different Motions... 397 24.3 Coupled Motions of Two Solids... 402 24.3.1 Introduction... 402 24.3.2 Parameters of Situation and Kinematics... 403 24.3.3 Kinetics... 404 24.3.4 Analysis of the Mechanical Actions... 406 24.3.5 Equations Deduced from the Fundamental Principle of Dynamics.... 408 24.3.6 Analysis of the Equations Deduced from the Fundamental Principle... 409 Exercises... 411 Comments... 412 Chapter 25 The Lagrange Equations 413 25.1 General Elements... 413 25.1.1 Free Body and Connected Body... 413 25.1.2 Partial Kinematics Torsors... 413 25.1.3 Power Coefficients... 415 25.1.4 Perfect Connections... 415 25.2 Lagrange Equations Relative to a Rigid Body... 416 25.2.1 Introduction to the Lagrange Equations.... 416 25.2.2 Lagrange Equations... 417 25.2.3 Case where the Mechanical Actions Admit a Potential Energy... 418 25.3 Lagrange Equations for a Set of Rigid Bodies... 419 25.3.1 Lagrange Equations for Each Solid... 419 25.3.2 Lagrange Equations for the Set (D)... 420 25.3.3 Case where the Parameters of Situation are Linked... 421 25.4 Applications.... 422 25.4.1 Motion of a parallelepiped Moving on an Inclined Plane... 422 25.4.2 Coupled Motions of Two Solids... 423 25.4.3 Double Pendulum... 425 A.25 Appendix... 431
Contents xix Exercises... 434 Comments... 434 PART VI Numerical Methods for Solving Differential Equations. Application to Equations of Motion 435 Chapter 26 Numerical Methods for Solving First Order Differential Equations 437 26.1 General Elements... 437 26.1.1 Problem with Given Initial Conditions... 437 26.1.2 General Method of Resolution... 438 26.1.3 Euler Method... 438 26.2 Single-Step Methods... 440 26.2.1 General Elements... 440 26.2.2 Methods of Runge-Kutta Type... 442 26.2.3 Romberg Method... 446 26.3 Multiple-Step Methods... 449 26.3.1 Introduction to the Multiple-Step Methods... 449 26.3.2 Methods based on the Newton interpolation... 450 26.3.3 Generalization of the Multiple-Step Methods... 452 26.3.4 Examples of Multiple-Step Methods.... 453 26.3.5 Results... 454 Exercises... 456 Comments... 456 Chapter 27 Numerical Procedures for Solving the Equations of Motions 457 27.1 Equation of Motion with One Degree of Freedom... 457 27.1.1 Form of the Equation of Motion with One Degree of Freedom... 457 27.1.2 Principle of the Numerical Resolution... 457 27.1.3 Application to the case of the Motion of a Simple Pendulum... 458 27.2 Equations of Motions with Several Degrees of Freedom... 461 27.2.1 Form of the Equations of Motions with Several Degrees of Freedom... 461 27.2.2 Principle of the Numerical Resolution... 462 27.2.3 Trajectories and Kinematic Vectors... 462 27.3 Motions of Planets and Satellites... 463 27.3.1 Motion of a Planet about the Sun... 463 27.3.2 Motion of a Satellite around the Earth... 467 27.3.3 Launching and Motion of a Moon Probe... 468 27.4 Motion of a Solid on an Inclined Plane.... 469 27.5 Coupled Motion of Two Solids... 471 27.5.1 Equations of Motion... 471 27.5.2 Analytical Solving in the case of Low Amplitudes and in the Absence of Friction... 474 27.5.3 Numerical Computation of the Equations of Motion... 476
Exercises... 479 Comments... 480 PART VII Solutions of the Exercises 481 Chapter 1 3 Vector Space... 483 Chapter 2 The Geometric Space.... 486 Chapter 4 Elementary Concepts on Curves... 492 Chapter 5 Torsors... 494 Chapter 6 Kinematics of Point... 500 Chapter 7 Study of Particular Motions... 505 Chapter 9 Kinematics of Rigid Body... 509 Chapter 10 Kinematics of Rigid Bodies in Contact... 516 Chapter 11 General Elements on the Mechanical Actions.... 523 Chapter 12 Gravitation. Gravity. Mass Centre... 531 Chapter 14 Statics of Rigid Bodies... 538 Chapter 15 The Operator of Inertia... 548 Chapter 16 Kinetic and Dynamic Torsors. Kinetic Energy... 559 Chapter 21 Dynamics of Systems with One Degree of Freedom Analysis of Vibrations... 567 Chapter 22 Motion of Rotation of a Solid about a Fixed Axis... 571 Chapter 24 Other Examples of Motions of Rigid Bodies... 577 Chapter 25 The Lagrange Equations... 596