Mechanics of Rigid Bodies

Transcription

1 Jean-Marie Berthelot Mechanics of Rigid Bodies ISMANS Institute for Advanced Materials and Mechanics Le Mans, France

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3 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.

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5 Jean-Marie Berthelot Mechanichs of Rigid Bodies ISMANS Institute for Advanced Materials and Mechanics Le Mans, France

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7 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

8 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.

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10 Contents Preface v PART I Mathematical Basic Elements 1 Chapter 1 Vector Space Definition of the Vector Space Vectors Vector Addition Multiplication by a Scalar Linear Dependence and Independence. Basis of Linear Combination Linear Dependence and Independence Basis of the Vector Space Components of a Vector Scalar Product Definition Magnitude or Norm of a Vector Analytical Expression of the Scalar Product in an Arbitrary Basis Orthogonal Vectors Orthonormal Basis Expression of the Scalar Product in an Orthonormal Basis Vector Product Definition Analytical Expression of the Vector Product in an Arbitrary Basis Direct Basis Expression of the Vector Product in a Direct Basis Mixed Product Property of the Double Vector Product Bases of the Vector Space Canonical Basis Basis Change Exercises Comments Chapter 2 The Geometric Space The Geometric Space Considered as Affine to the Vector Space

11 Contents ix The Geometric Space Consequences Distance between Two Points Angle between Two Bipoints Reference Systems Subspaces of the Geometric Space: Line, Plane Line Plane Lines and Planes with Same Directions Orthogonal Lines and Planes Characterization of the Position a Point of the Geometric Space Coordinate Axes Direct Orthonormal Reference System Cartesian Coordinates Plane and Line Equations Cartesian Equation of a Plane Cartesian Equation of a Line Change of Reference System General Case Refernce Systems with a Same Axis Arbitrary Reference Systems with the Same Origin Exercises Comments Chapter 3 Vector Function. Derivatives of a Vector Function Vector Function of One Variable Definition Derivative Properties of the Vector Derivative Examples Vector Function of Two Variables Definition Partial Derivatives Examples Vector Function of n Variables Definitions Examples Comments Chapter 4 Elementary Concepts on Curves Introduction Curvilinear Abscissa. Arc Length of a Curve Tangent. Normal. Radius of Curvature Frenet Trihedron Exercise Comments 54

12 x Contents Chapter 5 Torsors Definition and Properties of the Torsors Definitions and notations Properties of the Moments Vector Space of Torsors Scalar Invariant of a Torsor Product of Two Torsors Moment of a Torsor about an Axis Central Axis of a Torsor Particular Torsors. Resolution of an Arbitrary Torsor Slider Couple-Torsor Arbitrary Torsor Conclusions Torsors associated to a Field of Sliders Defined on a Domain of the Geometric Space Torsor Associated to a Finite Set of Points Torsor Associated to a Infinite Set of Points Important Particular Case. Measure Centre Exercises Comments PART II Kinematics 73 Chapter 6 Kinematics of Point Introduction Trajectory and Kinematic Vectors of a Point Trajectory Kinematic Vectors Tangential and Normal Components of Kinematic Vectors Different Types of Motions Expressions of the Components of Kinematic Vectors as Functions of Cartesian and Cylindrical Coordinates Cartesian Coordinates Cylindrical Coordinates Exercises Comments Chapter 7 Study of Particular Motions Motions with Rectilinear Trajectory General Considerations Uniform Rectilinear Motion Uniformly Varied Rectilinear Motion Simple Harmonic Rectilinear Motion Motions with a Circular Trajectory General Equations... 87

13 Contents xi Uniform Circular Motion Uniformly Varied Circular Motion Motions with a Contant Acceleration Vector General Equations Study of the case where the Trajectory is Rectilinear Study of the case where the Trajectory is Parabolic Helicoidal Motion Cycloidal Motion Exercises Comments Chapter 8 Motions with Central Acceleration General Properties Definition A Motion with a Central Acceleration is a Plane Trajectory Motion Areal Velocity Area Law Expression of the Kinematic Vectors Polar Equation of the Trajectory ( ) T Motions for which a ( M, t) = ω OM ( T ) OM 8.2 Motions with Central Acceleration for which a ( M, t) = K OM Equations of the Trajectories Study of the Trajectories Velocity Magnitude at a Point of the Trajectory Elliptic Motion. Kepler s Laws Comments Chapter 9 Kinematics of Rigid Body General Considerations Notion of Rigid Body Locating a Rigid Body Relations between the Trajectories and the Kinematic Vectors of Two Points Attached to a Solid Relation between the Trajectories Relation between the Velocity Vectors Expression of the Instantaneous Vector of Rotation Kinematic Torsor Relation between the Acceleration Vectors Generalization of the Composition of Motions Composition of Kinematic Torsors Inverse Motions Examples of Solid Motions Motion of Rotation about a Fixed Axis Translation Motion of a Rigid Body

14 xii Contents Motion of a Body Subjected to a Cylindrical Joint Motion of Rotation about a Fixed Point Plane Motion Exercises Comments Chapter 10 Kinematics of Rigid Bodies in Contact Kinematics of Two Solids in Contact Solids in Contact at a Point. Sliding Spinning and Rolling Conclusions Solids in Contact in Several points Transmission of a Motion of Rotation Général Elements Transmission by Friction Gear Transmission Belt Transmission Exercises Comments PART III The Mechanical Actions 153 Chapter 11 General Elements on the Mechanical Actions Concepts Relative to the Mechanical Actions Notion of Mechanical Action Representation of a Mechanical Action Classification of the Mechanical Actions Mechanical Actions Exerting between Material Sets External Mechanical Actions Exerting on a Material Set Different Types of Mechanical Actions Physical Natures of the Mechanical Actions Environnement and Effective Actions Power and Work Definition of the Power Change of Reference System Potential Energy Work Power and Work of a Force Set of Rigid Bodies Exercises Comments Chapter 12 Gravitation. Gravity. Mass Centre Phenomenon of Gravitation Law of Gravitation 169

15 Contents xiii Gravitational Field Action of gravitation induced by a Solid Sphere Action of gravitation induced by the Earth Action of Gravity Gravity Field Induced by the Earth Action of Gravity Exerted on a Material System Power Developed by the Action of Gravity Determination of Mass Centres Mass Centre of a Material System Mass Centre of the Union of Two Sets Mass Centre of a Homogeneous Set Homogeneous Bodies with Geometrical Symmetries Examples of Determination of Mass Centres Homogeneous Solid Hemisphere Homogeneous Solid with Complex Geometry Non-Homogeneous Solid Exercises Comments Chapter 13 Actions of Contact between Solids. Connections Laws of Contact between Solids Introduction Contact in a Point Couples of Rolling and Spinning Connections Introduction Classification of Connections Actions of Connection Connection without Friction Connection with Friction Comments Chapter 14 Statics of Rigid Bodies Introduction Law of Statics Case of a Rigid Body Case of a Set of Rigid Bodies Mutual Actions Statics of Wires or Flexible Cables Mechanical Action Exerted by a Wire or a Flexible Cable Equation of Statics of a Wire Wire or Flexible Cable Submitted to the Gravity Contact of a Wire with a Rigid Body Examples of Equilibrium Case of a Rigid Body Case of a System of Two Rigid Bodies Exercises Comments 223

16 xiv Contents PART IV Kinetics of Rigid Bodies 225 Chapter 15 The Operator of Inertia Introduction to the Operator of Inertia Operator Associated to a Vector Product Extending the Preceding Concept The Operator of Inertia Change of Coordinate System Change of Origin Relations of Huyghens Diagonalisation of the Matrix of Inertia Change of Basis Moments of Inertia with respect to a point, an axis, a plane Definitions Relations between the Moments of Inertia Case of a Plane Solid Moment of Inertia with respect to an Arbitrary Axis Determination of Matrices of Inertia Solids with Material Symmetries Solids having a Symmetry of Revolution Solids with Spherical Symmetry Associativity Matrices of Inertia of Homogeneous Bodies One-Dimensional Solids Two-Dimensional Solids Three-Dimensional Solids Exercises Comments Chapter 16 Kinetic and Dynamic Torsors. Kinetic Energy Kinetic Torsor Definition Kinetic Torsor Associated to the Motion of a Body Kinetic Torsor for a Set of Bodies Dynamic Torsor Definition Dynamic Torsor Associated to the Motion of a Body Dynamic Torsor for a Set of Bodies Relation with the Kinetic Energy Kinetic Energy Definition Kinetic Energy of a Body Kinetic Energy of a Set of Solids Derivative of the Kinetic Energy of a Solid with respect to Time Exercises Comments

17 Contents xv Chapter 17 Change of Reference System Kinematics of Change of Reference Relation between the Kinematic Torsors Relation between the Velocity Vectors. Velocity of Entrainment Composition of Acceleration Vectors Dynamic Torsors Inertia Torsor of Entrainment Inertia Torsor of Coriolis Relation between the Dynamic Torsors Defined relatively to Two Different References Comments PART V Dynamics of Rigid Bodies 275 Chapter 18 The Fundamental Principle of Dynamics and its Consequences Fundamental Principle Statement of the Fundamental Principle of Dynamics Class of Galilean Reference Systems Vector Equations Deduced from the Fundamental Principle Scalar Equations Deduced from the Fundamental Principle Mutual Actions Theorem of Mutual Actions Transmission of Mechanical Actions Theorem of Power-Energy Case of One Solid Case of a Set of Bodies Mechanical Actions with Potential Energy Application of the Fundamental Principle to the Study of the Motion of a Free Body in a Galilean Reference General Problem Particular Cases Application to the Solar System Galilean Reference Motion of Planets The Earth in the Solar System Comments Chapter 19 The Fundamental Equation of Dynamics in Different References General Elements Fundamental Equation of Dynamics in a Non Galilean Reference The Reference Systems used in Mechanics Fundamental Relation of Dynamics in the Geocentric Reference General Equations

18 xvi Contents Case of a Solid Located at the Vicinity of the Earth Fundamental Relation in a Reference Attached to the Earth Equations of Motion Action of Earthly Gravity Conclusions on the Equations of Dynamics in a Reference Attached to the Earth Equations of Dynamics of a Body with respect to a Reference whose the Motion is Known Relatively to the Earth Comments Chapter 20 General Process for Analysing a Problem of Dynamics of Rigid Bodies Dynamics of Rigid Body General Equations General Process of Analysis Dynamics of a Set of Bodies Conclusion Comments Chapter 21 Dynamics of Systems with One Degree of Freedom Analysis of Vibrations General Equations Introduction Parameters of Situation Kinematics Kinetics Mechanical Actions Exerted on the Solid Application of the Fundamental Principle Vibrations without Friction Equation of Motion Free Vibrations Forced Vibrations. Steady State Vibrations with Viscous Damping Equation of Motion with Viscous Damping Free Vibrations Vibrations in the case of a Harmonic Disturbing Force Forced Vibrations in the case of a Periodic Disturbing Force Vibrations in the case of an Arbitrary Disturbing Force Forced Vibrations in the case of a Motion Imposed to the Support Vibrations with Dry Friction Equations of Motion Free Vibrations Equivalent Viscous Damping Introduction Energy Dissipated in the case of Viscous Damping Stuctural Damping

19 Contents xvii Dry Friction Fluid Friction Conclusion Exercises Comments Chapter 22 Motion of Rotation of a Solid about a Fixed Axis General Equations Introduction Parameters of Situation Kinematics Kinetics Mechanical Actions Exerted on the Sold Application of the Fundamental Principle of Dynamics Examples of Motions of Rotation about an Axis Solid in Rotation Submitted only to the Gravity Pendulum of Torsion Problem of the Balancing of Rotors General Equations of an Unbalanced Solid in Rotation Mechanical Actions Exerted on the Shaft of Rotor Principle of the Balancing Exercises Comments Chapter 23 Plane Motion of a Rigid Body Introduction Parallelepiped Moving on an Inclined Plane Parameters of Situation and Kinematics Kinetics of the Motion Mechanical Actions Exerted on the Parallelepiped Equations Deduced from the Fundamental Principle Motion without Friction Motion with Dry Friction Motion with Viscous Friction Analysis of Sliding and Rocking of a Parallelepiped on an Inclined Plane Introduction Parameters of Situation and Kinematics General Equations Analysis of the Different Motions Conclusions Motion of a Cylinder on an Inclined Plane Introduction Parameters of Situation and Kinematics

20 xviii Contents Mechanical Actions Exerted on the Cylinder General Equations Analysis of the Different Motions Conclusions Comments Chapter 24 Other Examples of Motions of Rigid Bodies Solid in Translation General Expressions of a Solid in Translation Free Solid in Translation Motion of a Solid Placed on a Wagon Introduction Parameters of Situation Kinetics Analysis of the Mechanical Actions Equations of Dynamics Analysis of the Different Motions Coupled Motions of Two Solids Introduction Parameters of Situation and Kinematics Kinetics Analysis of the Mechanical Actions Equations Deduced from the Fundamental Principle of Dynamics Analysis of the Equations Deduced from the Fundamental Principle Exercises Comments Chapter 25 The Lagrange Equations General Elements Free Body and Connected Body Partial Kinematics Torsors Power Coefficients Perfect Connections Lagrange Equations Relative to a Rigid Body Introduction to the Lagrange Equations Lagrange Equations Case where the Mechanical Actions Admit a Potential Energy Lagrange Equations for a Set of Rigid Bodies Lagrange Equations for Each Solid Lagrange Equations for the Set (D) Case where the Parameters of Situation are Linked Applications Motion of a parallelepiped Moving on an Inclined Plane Coupled Motions of Two Solids Double Pendulum A.25 Appendix

21 Contents xix Exercises Comments PART VI Numerical Methods for Solving Differential Equations. Application to Equations of Motion 435 Chapter 26 Numerical Methods for Solving First Order Differential Equations General Elements Problem with Given Initial Conditions General Method of Resolution Euler Method Single-Step Methods General Elements Methods of Runge-Kutta Type Romberg Method Multiple-Step Methods Introduction to the Multiple-Step Methods Methods based on the Newton interpolation Generalization of the Multiple-Step Methods Examples of Multiple-Step Methods Results Exercises Comments Chapter 27 Numerical Procedures for Solving the Equations of Motions Equation of Motion with One Degree of Freedom Form of the Equation of Motion with One Degree of Freedom Principle of the Numerical Resolution Application to the case of the Motion of a Simple Pendulum Equations of Motions with Several Degrees of Freedom Form of the Equations of Motions with Several Degrees of Freedom Principle of the Numerical Resolution Trajectories and Kinematic Vectors Motions of Planets and Satellites Motion of a Planet about the Sun Motion of a Satellite around the Earth Launching and Motion of a Moon Probe Motion of a Solid on an Inclined Plane Coupled Motion of Two Solids Equations of Motion Analytical Solving in the case of Low Amplitudes and in the Absence of Friction Numerical Computation of the Equations of Motion

22 Exercises Comments PART VII Solutions of the Exercises 481 Chapter 1 3 Vector Space Chapter 2 The Geometric Space Chapter 4 Elementary Concepts on Curves Chapter 5 Torsors Chapter 6 Kinematics of Point Chapter 7 Study of Particular Motions Chapter 9 Kinematics of Rigid Body Chapter 10 Kinematics of Rigid Bodies in Contact Chapter 11 General Elements on the Mechanical Actions Chapter 12 Gravitation. Gravity. Mass Centre Chapter 14 Statics of Rigid Bodies Chapter 15 The Operator of Inertia Chapter 16 Kinetic and Dynamic Torsors. Kinetic Energy Chapter 21 Dynamics of Systems with One Degree of Freedom Analysis of Vibrations Chapter 22 Motion of Rotation of a Solid about a Fixed Axis Chapter 24 Other Examples of Motions of Rigid Bodies Chapter 25 The Lagrange Equations