Dynamics of the Rigid Solid with General Constraints by a Multibody Approach

Nicolae Pandrea, Mechanical engineer, PhD. eng., Professor at the University of Piteºti. Member of the Academy of Technical Sciences in Romania, has published 250 papers in Romania, U.S.A. and Europe and 7 books. Member of the Romanian Society of Acoustics. Winner of the "Traian Vuia" prize of the Romanian Academy. Co-author of the book Numerical Analysis with Applications in Mechanics and Engineering (Wiley, 2013).Nicolae-Doru Stãnescu, Mechanical engineer, Mathematician, PhD. eng., PhD. math., Professor at the University of Piteºti, has published 200 papers in Romania, Europe and U.S.A. and 11 books. Member of the International Institute of Acoustics and Vibrations, U.S.A, Member of the Société des Ingénieurs des l'Automobile, France. Winner of the "Traian Vuia" prize of the Romanian Academy. Co-author of the book Numerical Analysis with Applications in Mechanics and Engineering (Wiley, 2013).

• Preface xi1 Elements of Mathematical Calculation 11.1 Vectors: Vector Operations 11.2 Real Rectangular Matrix 41.3 Square Matrix 61.4 Skew Matrix of Third Order 10Further Reading 122 Kinematics of the Rigid Solid 152.1 Finite Displacements of the Points of Rigid Solid 152.2 Matrix of Rotation: Properties 162.2.1 General Properties 162.2.2 Successive Displacements 172.2.3 Eigenvalues: Eigenvectors 182.2.4 The Expression of the Matrix of Rotation with the Aid of the Unitary Eigenvector and the Angle of Rotation 202.2.5 Symmetries: Decomposition of the Rotation into Two Symmetries 242.2.6 Rotations About the Axes of Coordinates 252.3 Minimum Displacements: The Chasles Theorem 272.4 Small Displacements 332.5 Velocities of the Points of Rigid Solid 342.6 The Angular Velocity Matrix: Properties 372.6.1 The Matrices of Rotation About the Axes of Coordinates 372.6.2 The Angular Velocity Matrix: The Angular Velocity Vector 382.6.3 The Matrix of the Partial Derivatives of the Angular Velocity 392.7 Composition of the Angular Velocities 412.8 Accelerations of the Points of Rigid Solid 42Further Reading 433 General Theorems in the Dynamics of the Rigid Solid 453.1 Moments of Inertia 453.1.1 Definitions: Relations Between the Moments of Inertia 453.1.2 Moments of Inertia for Homogeneous Rigid Solid Bodies 473.1.3 Centers of Weight 473.1.4 Variation of the Moments of Inertia Relative to Parallel Axes 493.1.5 Variation of the Moments of Inertia Relative to Concurrent Axes 503.1.6 Principal Axes of Inertia: Principal Moments of Inertia 523.2 Momentum: The Theorem of Momentum 543.3 Moment of Momentum: The Theorem of Moment of Momentum 563.4 The Kinetic Energy of the Rigid Solid 57Further Reading 584 Matrix Differential Equations of the Motion of Rigid Solid 614.1 The Differential Equations Obtained from the General Theorems 614.1.1 General Aspects 614.1.2 The Differential Equations 624.2 The Lagrange Equations in the Case of the Holonomic Constraints 634.3 The Equivalence between the Differential Equations Obtained from the General Theorems and the Lagrange Equations 654.3.1 The Equivalence for the First Component 654.3.2 The Equivalence for the Second Component 664.4 The Matrix Differential Equations for the Motion of the Constrained Rigid Solid 714.4.1 The Matrix of Constraints 714.4.2 The Lagrange Equations for Mechanical Systems with Constraints 734.4.3 The Mathematical Model of the Motion of Rigid Solid with Constraints 754.4.4 General Algorithm of Calculation 764.4.5 The Calculation of the Forces of Constraints 784.4.6 The Elimination of the Matrix of the Lagrange multipliers 80Further Reading 855 Generalized Forces: The Equilibrium of the Rigid Solid 895.1 The Generalized Forces in the Case of a Mechanical System 895.2 The General Expressions of the Generalized Forces in the Case of Rigid Solid 905.2.1 The Case When at a Point Acts a Given Force 905.2.2 The Case When the Rigid Solid is Acted by a Torque of Given Moment 935.3 Conservative Forces 945.3.1 General Aspects 945.3.2 The Weight 965.3.3 The Elastic Force of a Spring 975.4 The Equilibrium of the Constrained Rigid Solid 985.4.1 The Equations of Equilibrium: Numerical Solution 985.4.2 The Case When the Functions of Constraints Introduce Auxiliary Coordinates (Pseudo-Coordinates) 1005.5 The Equilibrium of the Heavy Rigid Solid Hanged by Springs 1045.5.1 The Matrix Equation of Equilibrium 1045.5.2 Numerical Solution 1065.5.3 The Case When the Fixed Reference System Coincides to the Local Reference System at the Equilibrium Position 108Further Reading 1096 The Motion of the Rigid Solid with Constraints at Given Proper Points 1136.1 General Aspects: Classification 1136.2 Mathematical Aspects: Notations 1146.2.1 The Case of the Motion Depending on Only the Generalized Coordinates XO, YO, ZO, ψ, θ, φ 1146.2.2 The Case of the Constraints Depending on the Pseudo-Coordinates Too 1156.2.3 Relations of Calculation Necessary for the Numerical Algorithm 1156.3 The Study of the Rigid Solid with a Fixed Point 1166.4 The Rigid Solid with Two Fixed Points (the Rotational Motion of the Rigid Solid) 1186.5 The Rigid Solid with a Given Point Situated on a Fixed Surface 1216.5.1 The Case When the Surface is Defined by an Implicit Equation F X,Y,Z = 0 1216.5.2 The Case When the Surface is Defined by Parametric Equations 1236.6 The Rigid Solid with Several Points Situated on Fixed Surfaces (Curves) 1256.6.1 The Case When the Surfaces are Defined by Implicit Equations 1256.6.2 The Case When the Surfaces are Defined by Parametric Equations 1266.7 The Rigid Solid with a Fixed Point and with Another Point Situated on a Fixed Surface 1276.7.1 The Case When the Fixed Surface is Defined by an Implicit Equation 1276.7.2 The Case When the Fixed Surface is Defined by Parametric Equations 1296.8 The Rigid Solid with Two Given Points Situated on a Fixed Curve 1306.8.1 The Case When the Curve is Defined by Two Implicit Equations 1306.8.2 The Case When the Curve is Defined by Parametric Equations 1316.8.3 The Helical Motion of the Rigid Solid 132Further Reading 1337 The Motion of the Rigid Solid with Constraints on Given Proper Curves 1357.1 General Aspects: Classification 1357.2 The Rigid Solid Supported at Fixed Points on Given Proper Curves 1367.2.1 Notations 1367.2.2 The Matrix of Constraints 1377.3 The Rigid Solid at Which Given Proper Curves Support with Sliding on Fixed Curves 1387.3.1 Notations 1387.3.2 The Simple Contact between the Curves 1397.3.3 The Tangency Contact between Spatial Curves 1437.3.4 Contact with Sliding between Planar Curves (Rolling with Sliding on the Plan) 1447.4 Rolling without Sliding of a Curve on a Fixed Curve 1477.4.1 The General Case for Spatial Curves 1477.4.2 The Rolling Without Sliding of a Curve on a Fixed Curve in the Plan 1487.5 The Motion of the Rigid Solid at Which the Curves Jointed to It Support with Sliding on Fixed Surfaces 1517.5.1 The Case of a Single Curve 1517.5.2 The Case of the Supporting with Sliding by Curves on Surface 1547.6 The Rolling without Sliding of a Disk Bounded by a Spatial Curve on a Fixed Surface 1577.6.1 The Matrix Differential Equation of Motion 1577.6.2 The Forces at the Contact Point 1597.7 The Rolling without Sliding of a Planar Circle Disk on a Horizontal Plan 1607.8 The Rolling without Sliding of a Planar Elliptic Disk on a Horizontal Plan 1687.9 The Rolling without Sliding of a Hyperboidic Curve on a Horizontal Plan 1757.9.1 Hyperboidic Curves 1757.9.2 The Matrix Differential Equation of Motion 1767.10 The Rolling without Sliding of a Planar Circle Disk on a Cylindrical Surface with Horizontal Generatrices 1847.11 The Rolling without Sliding of Two Curves of a Rigid Solid on a Fixed Surface 1927.11.1 General Aspects 1927.11.2 The Differential Equations of Motion 1957.11.3 The Algorithm of Numerical Calculation 1967.12 The Rolling without Sliding of an Axle with Wheels (Disks) with Angular Deviations on a Horizontal Plan 1977.13 The Rolling without Sliding of an Axle with Disks on a Hyperbolic Paraboloid 2047.13.1 General Aspects 2047.13.2 The Initial Position 2067.13.3 The Differential Equations 207Further Reading 2148 The Motion of the Rigid Solid with Constraints on the Bounding Surface 2178.1 General Aspects: Classification 2178.2 The Rigid Solid Supported at Fixed Points 2188.2.1 The Matrix of Constraints 2188.2.2 The Matrix Differential Equation of Motion 2208.2.3 The Algorithm of Calculation 2218.3 The Rigid Solid Supported with Sliding on Fixed Curves 2368.3.1 The Matrix of Constraints 2368.3.2 The Matrix Differential Equation of Motion 2398.3.3 The Reactions 2398.3.4 The Algorithm of Calculation 2408.4 The Rolling without Sliding of the Rigid Solid on Two Fixed Curves 2448.4.1 General Considerations 2448.4.2 The Differential Equations of Motion 2468.4.3 The Algorithm for the Numerical Calculation 2488.5 The Rolling without Sliding of a Rigid Solid on a Fixed Surface 2548.5.1 The Matrix of Constraints 2548.5.2 The Matrix Differential Equation of Motion 2568.6 The Rolling without Sliding of a Toroidal Wheel on a Horizontal Plan 2578.6.1 The Equations of Torus 2578.6.2 The Tangency Conditions 2588.6.3 The Initial Conditions 2588.6.4 The Differential Equations of Motion 2608.7 The Rolling without Sliding of a Rigid Solid Supported on Two Fixed Surfaces 2658.7.1 General Aspects 2658.7.2 The Differential Equations of Motion 2678.7.3 The Determination of the Forces of Constraints 2698.7.4 The Rolling without Sliding of an Ellipsoid Acted only by its Own Weight on Two Plans 2708.8 The Rolling without Sliding of a Rigid Solid Supported at Two Points on a Fixed Surface 2918.8.1 General Aspects 2918.8.2 The Differential Equations of Motion: The Calculation of the Forces of Constraints 293Further Reading 294Appendix 297Index 315