Lagrangian Mechanics

Name: Lagrangian Mechanics | An Advanced Analytical Approach
Brand: Wiley
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An Advanced Analytical Approach

Anh Le Van Rabah Bouzidi(Autor*in)

Wiley (Verlag)

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Erschienen am 3. Juni 2019

326 Seiten

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978-1-119-62971-9 (ISBN)

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Inhalt

Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Contents
Preface
The format adopted in this book
Synopsis
Acknowledgments
1. Kinematics
1.1. Observer - Reference frame
1.2. Time
1.2.1. Date postulate
1.2.2. Date change postulate
1.3. Space
1.3.1. Physical space
1.3.2. Mathematical space
1.3.3. Position postulate
1.3.4. Typical operations on the mathematical space E
1.3.5. Position change postulate
1.3.6. The common reference frame R0
1.3.7. Coordinate system of a reference frame
1.3.8. Fixed point and fixed vector in a reference frame
1.4. Derivative of a vector with respect to a reference frame
1.5. Velocity of a particle
1.6. Angular velocity
1.7. Reference frame defined by a rigid body: Rigid body defined by a reference frame
1.8. Point attached to a rigid body: Vector attached to a rigid body
1.9. Velocities in a rigid body
1.10. Velocities in a mechanical system
1.11. Acceleration
1.11.1. Acceleration of a particle
1.11.2. Accelerations in a mechanical system
1.12. Composition of velocities and accelerations
1.12.1. Composition of velocities
1.12.2. Composition of accelerations
1.13. Angular momentum: Dynamic moment
2. Parameterization and Parameterized Kinematics
2.1. Position parameters
2.1.1. Position parameters of a particle
2.1.2. Position parameters for a rigid body
2.1.3. Position parameters for a system of rigid bodies
2.2. Mechanical joints
2.3. Constraint equations
2.4. Parameterization
2.5. Dependence of the rotation tensor of the reference frame on the retained parameters
2.6. Velocity of a particle
2.7. Angular velocity
2.8. Velocities in a rigid body
2.9. Velocities in a mechanical system
2.10. Parameterized velocity of a particle
2.10.1. Definition
2.10.2. Practical calculation of the parameterized velocity
2.11. Parameterized velocities in a rigid body
2.12. Parameterized velocities in a mechanical system
2.13. Lagrange's kinematic formula
2.14. Parameterized kinetic energy
3. Efforts
3.1. Forces
3.2. Torque
3.3. Efforts
3.4. External and internal efforts
3.4.1. External effort
3.4.2. Internal effort
3.5. Given efforts and constraint efforts
3.6. Moment field
4. Virtual Kinematics
4.1. Virtual derivative of a vector with respect to a reference frame
4.2. Virtual velocity of a particle
4.3. Virtual angular velocity
4.4. Virtual velocities in a rigid body
4.4.1. The virtual velocity field (VVF) associated with a parameterization
4.4.2. Virtual velocity field (VVF) in a rigid body
4.5. Virtual velocities in a system
4.5.1. VVF associated with a parameterization
4.5.2. VVF on each rigid body of a system
4.5.3. Virtual velocity of the center of mass
4.6. Composition of virtual velocities
4.6.1. Composition of virtual velocities of a particle
4.6.2. Composition of virtual angular velocities
4.6.3. Composition of VVFs in rigid bodies
4.7. Method of calculating the virtual velocity at a point
5. Virtual Powers
5.1. Principle of virtual powers
5.2. VP of efforts internal to each rigid body
5.3. VP of efforts
5.4. VP of efforts exerted on a rigid body
5.4.1. General expression
5.4.2. VP of zero moment field efforts exerted upon a rigid body
5.4.3. Dependence of the VP of efforts on the reference frame
5.5. VP of efforts exerted on a system of rigid bodies
5.5.1. General expression
5.5.2. Dependence of the VP of the efforts on the reference frame
5.5.3. VP of zero moment field efforts exerted on a system of rigid bodies
5.5.4. VP of inter-efforts between the rigid bodies of a system
5.5.5. The specific case of the inter-efforts between two rigid bodies
5.6. Summary of the cases where the VV and VP are independent of the reference frame
5.7. VP of efforts expressed as a linear form of the q?*i
5.8. Potential
5.8.1. Definition
5.8.2. Examples of potential
5.9. VP of the quantities of acceleration
6. Lagrange's Equations
6.1. Choice of the common reference frame R0
6.2. Lagrange's equations
6.3. Review and the need to model joints
6.4. Existence and uniqueness of the solution
6.5. Equations of motion
6.6. Example 1
6.7. Example 2
6.7.1. Reduced parameterization
6.7.2. Total parameterization
6.7.3. Comparing the two parameterizations
6.8. Example 3
6.8.1. Independent parameterization
6.8.2. Total parameterization
6.8.3. Comparing the two parameterizations
6.9. Working in a non-Galilean reference frame
7. Perfect Joints
7.1. VFs compatible with a mechanical joint
7.1.1. Definition
7.1.2. Generalizing the definition of a compatible VVF
7.1.3. Example 1 for VVFs compatible with a mechanical joint
7.1.4. Example 2
7.1.5. Example 3: particle moving along a hoop rotating around a fixed axis
7.2. Invariance of the compatible VVFs with respect to the choice of the primitive parameters
7.2.1. Context
7.2.2. Relationships between the real quantities resulting from the two parameterizations
7.2.3. Relationships between the virtual quantities resulting from the two parameterizations
7.2.4. Identity between the VVFs associated with the two parameterizations and compatible with a mechanical joint
7.2.5. Example
7.3. Invariance of the compatible VVFs with respect to the choice of the retained parameters
7.3.1. Context
7.3.2. Relationships between the real quantities resulting from the two parameterizations
7.3.3. Relationships between the virtual quantities resulting from the two parameterizations
7.3.4. Identity between the VVFs associated with the two paramaterizations and compatible with a mechanical joint
7.3.5. Example 1
7.3.6. Example 2
7.3.7. Example 3: particle moving along a hoop rotating around a fixed axis
7.4. Invariance of the compatible VVFs with respect to the choice of the parameterization
7.5. Perfect joints
7.5.1. Definition of a perfect joint
7.5.2. Example 1
7.5.3. Example 2
7.5.4. Example 3
7.5.5. Example 4
7.6. Example: a perfect compound joint
7.6.1. Perfect combined joint
7.6.2. Superimposition of two perfect elementary joints
8. Lagrange's Equations in the Case of Perfect Joints
8.1. Lagrange's equations in the case of perfect joints and an independent parameterization
8.1.1. Lagrange's equations
8.1.2. Review
8.1.3. Particular case
8.2. Lagrange's equations in the case of perfect joints and in the presence of complementary constraint equations
8.2.1. Lagrange's equations with multipliers
8.2.2. Practical calculation using Lagrange's multipliers
8.2.3. Review
8.2.4. Remarks
8.3. Example: particle on a rotating hoop
8.3.1. Independent parameterization
8.3.2. Reduced parameterization no. 1
8.3.3. Reduced parameterization no. 2
8.3.4. Calculation of the engine torque
8.4. Example: rigid body connected to a rotating rod by a spherical joint (no. 1)
8.5. Example: rigid body connected to a rotating rod by a spherical (joint no. 2)
8.5.1. Total parameterization
8.5.2. Independent parameterization
8.6. Example: rigid body subjected to a double contact
8.6.1. Preliminary analysis
8.6.2. Independent parameterization
8.6.3. Reduced parameterization
9. First Integrals
9.1. Painlevé's first integral
9.1.1. Painlevé's lemma
9.1.2. Painlevé's first integral
9.2. The energy integral: conservative systems
9.2.1. Energy considerations in addition to the energy integral
9.3. Example: disk rolling on a suspended rod
9.4. Example: particle on a rotating hoop
9.5. Example: a rigid body connected to a rotating rod by a spherical joint (no. 1)
9.5.1. First integrals via Newtonian mechanics
9.6. Example: rigid body connected to a rotating rod by a spherical joint (no. 2)
9.7. Example: rigid body subjected to a double contact
9.7.1. Using Newtonian mechanics to find a first integral
10. Equilibrium
10.1. Definitions
10.1.1. Absolute equilibrium
10.1.2. Parametric equilibrium
10.2. Equilibrium equations
10.2.1. List of equations and unknowns
10.2.2. The explicit presence of time in equilibrium equations
10.3. Equilibrium equations in the case of perfect joints and independent parameterization
10.3.1. List of equations and unknowns
10.4. Equilibrium equations in the case of perfect joints and in the presence of complementary constraint equations
10.4.1. List of equations and unknowns
10.5. Stability of an equilibrium
10.6. Example: equilibrium of a jack
10.7. Example: equilibrium of a lifting platform
10.8. Example: equilibrium of a rod in a gutter
10.9. Example: existence of ranges of equilibrium positions
10.10. Example: relative equilibrium with respect to a rotating reference frame
10.11. Example: equilibrium in the presence of contact inequalities
10.12. Calculating internal efforts
10.13. Example: internal efforts in a truss
10.13.1. Tension force in bar A'A
10.13.2. Tension force in bar B'B
10.14. Example: internal efforts in a tripod
11. Revision Problems
11.1. Equilibrium of two rods
11.1.1. Analysis using Newton's law
11.2. Equilibrium of an elastic chair
11.3. Equilibrium of a dump truck
11.4. Equilibrium of a set square
11.5. Motion of a metronome
11.5.1. Equation of motion
11.5.2. First integral
11.5.3. The case of small oscillations
11.5.4. Case where the base S may slide without friction on the table T
11.5.5. First integral
11.6. Analysis of a hemispherical envelope
11.6.1. Studying the static equilibrium
11.6.2. Studying the oscillatory motion
11.7. A block rolling on a cylinder
11.7.1. Studying the equilibrium
11.7.2. Dynamic analysis
11.7.3. Case of small oscillations
11.8. Disk welded to a rod
11.8.1. First parameterization
11.8.2. First integral
11.8.3. Second parameterization
11.8.4. Third parameterization
11.9. Motion of two rods
11.9.1. Equations of motion
11.9.2. Relative equilibrium
11.10. System with a perfect wire joint
11.10.1. Lagrange's equations
11.10.2. First integral
11.11. Rotating disk-rod system
11.11.1. Independent parameterization
11.11.2. Total parameterization
11.11.3. Engine torque
11.12. Dumbbell
11.12.1. Equations set
11.12.2. First integrals
11.12.3. Analysis with specific initial conditions
11.12.4. Relative equilibrium
11.13. Dumbbell under engine torque
11.13.1. Independent parameterization
11.13.2. Painlevé's first integral
11.13.3. Total parameterization
11.14. Rigid body with a non-perfect joint
11.14.1. Equations set
11.14.2. Solving the equations set
11.14.3. Power of the engine and work dissipated through friction
Appendix 1: Tensors
Representative matrix of a tensor
Transpose of a tensor
Symmetric tensor: skew-symmetric tensor
Identity tensor
Product of two tensors
Inverse tensor
Orthogonal tensor
Tensor product
Appendix 2: Typical Perfect Joints
A2.1. Point contact between two rigid bodies
A2.2. Ball-and-socket joint (or spherical joint)
A2.3. Cylindrical joint
A2.4. Pivot (or hinged joint)
A2.5. Prismatic or sliding joint
A2.6. Helical joint (or screw joint)
Bibliography
Index
Other titles from iSTE in Mechanical Engineering and Solid Mechanics
EULA

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