Covers both holonomic and non-holonomic constraints in a study of the mechanics of the constrained rigid body.

* Covers all types of general constraints applicable to the solid rigid

* Performs calculations in matrix form

* Provides algorithms for the numerical calculations for each type of constraint

* Includes solved numerical examples

* Accompanied by a website hosting programs

Nicolae Pandrea, Mechanical engineer, PhD. eng., Professor at the University of Pite?ti. 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 St?nescu, Mechanical engineer, Mathematician, PhD. eng., PhD. math., Professor at the University of Pite?ti, 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).

The increasing complexity of mechanics problems means that engineers and scientists have to look beyond traditional approaches, and must now use numerical methods to obtain solutions. New algorithms will continue to be developed in the coming years, enabling even more complex problems to be solved.

Dynamics of the Rigid Solid with General Constraints by a Multibody Approach considers both the holonomic and non-holonomic constraints for the mechanics of the constrained rigid body. It takes a multibody system approach, performs calculations in matrix form and covers the matrix of constraints in detail. Algorithms for the numerical calculations are provided for the methods described and the theory is applied to solved numerical examples. It begins with presenting the elements of mathematical calculation that will be used throughout the book, before moving on to cover dynamics and kinematics. Matrix differential equations of motion are then presented, and equilibrium and generalized forces are discussed. Finally the motion of the solid rigid with constraints at given proper points, proper curves and a bounding surface is covered.

Key features:

• Covers all types of general constraints applicable to the solid rigid
• Performs calculations in matrix form
• Provides algorithms for the numerical calculations for each type of constraint
• Includes solved numerical examples
• Accompanied by a website hosting programs

Preface xi

1 Elements of Mathematical Calculation 1

1.1 Vectors: Vector Operations 1

1.2 Real Rectangular Matrix 4

1.3 Square Matrix 6

1.4 Skew Matrix of Third Order 10

Further Reading 12

2 Kinematics of the Rigid Solid 15

2.1 Finite Displacements of the Points of Rigid Solid 15

2.2 Matrix of Rotation: Properties 16

2.2.1 General Properties 16

2.2.2 Successive Displacements 17

2.2.3 Eigenvalues: Eigenvectors 18

2.2.4 The Expression of the Matrix of Rotation with the Aid of the Unitary Eigenvector and the Angle of Rotation 20

2.2.5 Symmetries: Decomposition of the Rotation into Two Symmetries 24

2.2.6 Rotations About the Axes of Coordinates 25

2.3 Minimum Displacements: The Chasles Theorem 27

2.4 Small Displacements 33

2.5 Velocities of the Points of Rigid Solid 34

2.6 The Angular Velocity Matrix: Properties 37

2.6.1 The Matrices of Rotation About the Axes of Coordinates 37

2.6.2 The Angular Velocity Matrix: The Angular Velocity Vector 38

2.6.3 The Matrix of the Partial Derivatives of the Angular Velocity 39

2.7 Composition of the Angular Velocities 41

2.8 Accelerations of the Points of Rigid Solid 42

Further Reading 43

3 General Theorems in the Dynamics of the Rigid Solid 45

3.1 Moments of Inertia 45

3.1.1 Definitions: Relations Between the Moments of Inertia 45

3.1.2 Moments of Inertia for Homogeneous Rigid Solid Bodies 47

3.1.3 Centers of Weight 47

3.1.4 Variation of the Moments of Inertia Relative to Parallel Axes 49

3.1.5 Variation of the Moments of Inertia Relative to Concurrent Axes 50

3.1.6 Principal Axes of Inertia: Principal Moments of Inertia 52

3.2 Momentum: The Theorem of Momentum 54

3.3 Moment of Momentum: The Theorem of Moment of Momentum 56

3.4 The Kinetic Energy of the Rigid Solid 57

Further Reading 58

4 Matrix Differential Equations of the Motion of Rigid Solid 61

4.1 The Differential Equations Obtained from the General Theorems 61

4.1.1 General Aspects 61

4.1.2 The Differential Equations 62

4.2 The Lagrange Equations in the Case of the Holonomic Constraints 63

4.3 The Equivalence between the Differential Equations Obtained from the General Theorems and the Lagrange Equations 65

4.3.1 The Equivalence for the First Component 65

4.3.2 The Equivalence for the Second Component 66

4.4 The Matrix Differential Equations for the Motion of the Constrained Rigid Solid 71

4.4.1 The Matrix of Constraints 71

4.4.2 The Lagrange Equations for Mechanical Systems with Constraints 73

4.4.3 The Mathematical Model of the Motion of Rigid Solid with Constraints 75

4.4.4 General Algorithm of Calculation 76

4.4.5 The Calculation of the Forces of Constraints 78

4.4.6 The Elimination of the Matrix of the Lagrange multipliers 80

Further Reading 85

5 Generalized Forces: The Equilibrium of the Rigid Solid 89

5.1 The Generalized Forces in the Case of a Mechanical System 89

5.2 The General Expressions of the Generalized Forces in the Case of Rigid Solid 90

5.2.1 The Case When at a Point Acts a Given Force 90

5.2.2 The Case When the Rigid Solid is Acted by a Torque of Given Moment 93

5.3 Conservative Forces 94

5.3.1 General Aspects 94

5.3.2 The Weight 96

5.3.3 The Elastic Force of a Spring 97

5.4 The Equilibrium of the Constrained Rigid Solid 98

5.4.1 The Equations of Equilibrium: Numerical Solution 98

5.4.2 The Case When the Functions of Constraints Introduce Auxiliary Coordinates (Pseudo-Coordinates) 100

5.5 The Equilibrium of the Heavy Rigid Solid Hanged by Springs 104

5.5.1 The Matrix Equation of Equilibrium 104

5.5.2 Numerical Solution 106

5.5.3 The Case When the Fixed Reference System Coincides to the Local Reference System a...