Lagrangian Mechanics

Lagrangian Mechanics explains the subtleties of analytical mechanics and its applications in rigid body mechanics. The authors demonstrate the primordial role of parameterization, which conditions the equations and thus the information obtained; the essential notions of virtual kinematics, such as the virtual derivative and the dependence of the virtual quantities with respect to a reference frame; and the key concept of perfect joints and their intrinsic character, namely the invariance of the fields of compatible virtual velocities with respect to the parameterization. Throughout the book, any demonstrated results are stated with the respective hypotheses, clearly indicating the applicability conditions for the results to be ready for use. Numerous examples accompany the text, facilitating the understanding of the calculation mechanisms. The book is mainly intended for Bachelor's, Master's or engineering students who are interested in an in-depth study of analytical mechanics and its applications.

Auteur

Anh Le van is Professor at the University of Nantes, France, and teaches structural mechanics at the Faculty of Science. His research at the GeM laboratory (Institute for Research in Civil and Mechanical Engineering) focuses on membrane structures in large deformations.

Rabah Bouzidi is Associate Professor at the University of Nantes, France. He also teaches structural mechanics, and researches membrane structures in large deformations at the GeM laboratory.

Contenu

Preface xi

1 Kinematics 1

1.1 Observer Reference frame 1

1.2 Time 2

1.2.1 Date postulate 2

1.2.2 Date change postulate 2

1.3 Space 3

1.3.1 Physical space 3

1.3.2 Mathematical space 4

1.3.3 Position postulate 4

1.3.4 Typical operations on the mathematical space E 6

1.3.5 Position change postulate 7

1.3.6 The common reference frame R0 9

1.3.7 Coordinate system of a reference frame 12

1.3.8 Fixed point and fixed vector in a reference frame 14

1.4 Derivative of a vector with respect to a reference frame 15

1.5 Velocity of a particle 17

1.6 Angular velocity 17

1.7 Reference frame defined by a rigid body: Rigid body defined by a reference frame 19

1.8 Point attached to a rigid body: Vector attached to a rigid body 19

1.9 Velocities in a rigid body 20

1.10 Velocities in a mechanical system 22

1.11 Acceleration 24

1.11.1 Acceleration of a particle 24

1.11.2 Accelerations in a mechanical system 24

1.12 Composition of velocities and accelerations 24

1.12.1 Composition of velocities 24

1.12.2 Composition of accelerations 25

1.13 Angular momentum: Dynamic moment 25

2 Parameterization and Parameterized Kinematics 27

2.1 Position parameters 27

2.1.1 Position parameters of a particle 27

2.1.2 Position parameters for a rigid body 28

2.1.3 Position parameters for a system of rigid bodies 32

2.2 Mechanical joints 33

2.3 Constraint equations 33

2.4 Parameterization 37

2.5 Dependence of the rotation tensor of the reference frame on the retained parameters 39

2.6 Velocity of a particle 41

2.7 Angular velocity 44

2.8 Velocities in a rigid body 45

2.9 Velocities in a mechanical system 47

2.10 Parameterized velocity of a particle 48

2.10.1 Definition 48

2.10.2 Practical calculation of the parameterized velocity 49

2.11 Parameterized velocities in a rigid body 50

2.12 Parameterized velocities in a mechanical system 51

2.13 Lagrange's kinematic formula 52

2.14 Parameterized kinetic energy 54

3 Efforts 57

3.1 Forces 57

3.2 Torque 59

3.3 Efforts 60

3.4 External and internal efforts 61

3.4.1 External effort 61

3.4.2 Internal effort 62

3.5 Given efforts and constraint efforts 62

3.6 Moment field 64

4 Virtual Kinematics 67

4.1 Virtual derivative of a vector with respect to a reference frame 67

4.2 Virtual velocity of a particle 70

4.3 Virtual angular velocity 75

4.4 Virtual velocities in a rigid body 81

4.4.1 The virtual velocity field (VVF) associated with a parameterization 81

4.4.2 Virtual velocity field (VVF) in a rigid body 82

4.5 Virtual velocities in a system 83

4.5.1 VVF associated with a parameterization 83

4.5.2 VVF on each rigid body of a system 84

4.5.3 Virtual velocity of the center of mass 84

4.6 Composition of virtual velocities 85

4.6.1 Composition of virtual velocities of a particle 85

4.6.2 Composition of virtual angular velocities 86

4.6.3 Composition of VVFs in rigid bodies 87

4.7 Method of calculating the virtual velocity at a point 88

5 Virtual Powers 91

5.1 Principle of virtual powers 91

5.2 VP of efforts internal to each rigid body 92

5.3 VP of efforts 92

5.4 VP of efforts exerted on a rigid body 94

5.4.1 General expression 94

5.4.2 VP of zero moment field efforts exerted upon a rigid body 94

5.4.3 Dependence of the VP of efforts on the reference frame 94

5.5 VP of efforts exerted on a system of rigid bodies 95

5.5.1 General expression 95 5.5...