| Description |
1 online resource (xix, 540 pages) : illustrations |
| Series |
Mechanical engineering series |
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Mechanical engineering series (Berlin, Germany)
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| Contents |
880-01 Preface -- [pt]. 1. Linear algebra -- Prologue -- 1. Vector spaces -- 1.1. Vectors -- 1.1.1. The "algebra" of vector spaces -- 1.2. The basis of a vector space -- 1.2.1. Spanning sets -- 1.2.2. Linear independence -- A test for linear independence of n-tuples : reduction to Echelon form -- 1.2.3. Bases and the dimension of a vector space -- Theorems on dimension -- 1.3. The representation of vectors -- 1.3.1. N-tupe representations of vectors -- 1.3.2. Representations and units -- 1.3.3. isomorphisms among vector spaces of the same dimension -- 2. Linear transformations on vector spaces -- 2.1. Matrices -- 2.1.1. The "partitioning" and rank of matrices -- The rank of a matrix -- 2.1.2. Operations on matrices -- Inner product -- Transpose of a matrix product -- Block multiplication of partitioned matrices -- Elementary operations through matrix products -- 2.2. Linear transformations -- Domain and range of [linear] transformation and their dimension -- 2.2.1. Linear transformations : basis and representation -- Dyadics -- 2.2.2. Null space of a linear transformation -- Dimension of the null space -- Relation between dimensions of domain, range, and null space -- 2.3. Solution of linear systems -- "Skips" and the null space -- 2.3.1. Theory of linear equations -- Homogeneous linear equations -- Non-homogeneous linear equations -- 2.4. Linear operators -- differential equations |
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880-01/(Q 9. Integrals of motion -- 9.1. Integrals of the motion -- 9.2. Jacobi's integral -- an energy-like integral -- 9.3. "Ignorable coordinates" and integrals -- 10. Hamiltonian dynamics -- 10.1. The variables -- Solution for q⁺ј q, p; t) -- 10.2. The equations of motion -- 10.2.1. Legendre transformations -- 10.2.2. Q and p as Lagrangian variables -- 10.2.3. An important property of the Hamiltonian -- 10.3. Integrals of the motion -- 10.4. Canonical transformations -- 10.5. Generating functions -- 10.6. Transformation solution of Hamiltonians -- 10.7. Separability -- 10.7.1. The Hamilton-Jacobi equation -- 10.7.2. Separable variables -- Special case -- ignorable coordinates -- 10.8. Constraints in Hamiltonian systems -- 10.9. Time as a coordinate in Hamiltonians -- Epilogue -- Index |
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3. Special case -- square matrices -- The "algebra" of square matrices -- 3.1. The inverse of a square matrix -- Properties of the inverse -- 3.2. The determinant of a square matrix -- Properties of the determinant -- 3.3. Classification of square matrices -- 3.3.1. Orthogonal matrices -- rotations -- 3.3.2. The orientation of non-orthonormal bases -- 3.4. Linear systems : n equations in n unknowns -- 3.5. Eigenvalues and eigenvectors of a square matrix -- 3.5.1. Linear independence of eigenvectors -- 3.5.2. The Cayley-Hamilton theorem -- 3.5.3. Generalized eigenvectors -- 3.5.4. Application of eigenvalues/eigenvectors -- 3.6. Application -- basis transformations -- 3.6.1. General basis transformations-- Successive basis transformations -- 3.6.2. Basis rotations -- 3.7. Normal forms of square matrices -- 3.7.1. Linearly independent eigenvectors -- diagonalization -- Diagonalization of real symmetric matrices -- 3.7.2. Linearly dependent eigenvectors -- Jordan normal form |
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880-02 [pt]. 2. 3-D rigid body dynamics -- Prologue -- 4. Kinematics -- 4.1. Motion of a rigid body -- 4.1.1. General motion of a rigid body -- Differentials -- 4.1.2. Rotation of a rigid body -- Differential rigid body rotation -- Angular velocity and acceleration -- Time derivative of a unit vector with respect to rotation -- 4.2. Euler angles -- 4.2.1. Direction angles and cosines -- Vector description -- Coordinate system description -- 4.2.2. Euler angles -- Vector description -- Coordinate system description -- 4.3. Moving coordinate systems -- 4.3.1. Relative motion : points -- 4.3.2. Relative motion : coordinate systems -- Time derivatives in rotating coordinate systems -- Applications of theorem 4.3.1 -- Rotating coordinate system equations -- Distinction between the "A/B" and "rel" quantities -- The need for rotating coordinate systems -- 4.4. Machine kinematics -- 4.4.1. Motion of a single body -- A useful trick -- The non-slip condition -- The instantaneous center of zero velocity-- 4.5.2. Kinematic constraints imposed by linkages -- Clevis connections -- Ball-and-socket connections -- 4.4.3. Motion of multiple rigid bodies ("machines") -- Curved interconnections -- General analysis of universal joints |
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880-02/(Q 9. Integrals of motion -- 9.1. Integrals of the motion -- 9.2. Jacobi's integral -- an energy-like integral -- 9.3. "Ignorable coordinates" and integrals -- 10. Hamiltonian dynamics -- 10.1. The variables -- Solution for q⁺ј (q, p; t) -- 10.2. The equations of motion -- 10.2.1. Legendre transformations -- 10.2.2. Q and p as Lagrangian variables -- 10.2.3. An important property of the Hamiltonian -- 10.3. Integrals of the motion -- 10.4. Canonical transformations -- 10.5. Generating functions -- 10.6. Transformation solution of Hamiltonians -- 10.7. Separability -- 10.7.1. The Hamilton-Jacobi equation -- 10.7.2. Separable variables -- Special case -- ignorable coordinates -- 10.8. Constraints in Hamiltonian systems -- 10.9. Time as a coordinate in Hamiltonians -- Epilogue -- Index |
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5. Kinetics -- 5.1. Particles and systems of particles -- 5.1.1. Particle kinetics -- Linear momentum and its equation of motion -- Angular momentum and its equation of motion -- Energy -- A caveat regarding conservation -- 5.1.2 Particle system kinetics -- Kinetics relative to a fixed system -- Kinetics relative to the center of mass -- 5.2. Equations of motion for rigid bodies -- 5.2.1. angular momentum of a rigid body -- the inertia tensor -- Properties of the inertia tensor -- Principal axes -- 5.2.2. Equations of motion -- Forces/moments at interconnections -- Determination of the motion of a system -- 5.2.3. A special case -- the gyroscope -- Gyroscope coordinate axes and angular velocities -- Equations of motion -- Special case -- moment-free gyroscopic motion -- General case -- gyroscope with moment -- 5.3. Dynamic stability -- 5.4. Alternatives to direct integration -- 5.4.1. Energy -- Kinetic energy -- Work -- Energy principles -- 5.4.2. Momentum -- 5.4.3 Conservation application in general -- Epilogue |
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[pt]. 3. Analytical dynamics -- Prologue -- 6. Analytical dynamics : perspective -- 6.1. Vector formulations and constraints -- 6.2. Scalar formulations and constraints -- 6.3. Concepts from virtual work in statics -- 7. Lagrangian dynamics : kinematics -- 7.1. Background : position and constraints -- Categorization of differential constraints -- Constraints and linear independence -- 7.2. Virtual displacements -- 7.3. Kinematic vs. kinetic constraints -- 7.4. Generalized coordinates -- Derivatives a r and v with respect to generalized coordinates and velocities -- 8. Lagrangian dynamics : kinetics -- 8.1. Arbitrary forces : Euler-Lagrange equations -- Notes on the Euler-Lagrange equations-- 8.2. conservative forces : Lagrange equations -- Properties of the Lagrangian -- 8.3. Differential constraints -- 8.3.1. algebraic approach to differential constraints -- 8.3.2. Lagrange multipliers -- Interpretation of the Lagrange multipliers -- 8.4. Time as a coordinate |
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9. Integrals of motion -- 9.1. Integrals of the motion -- 9.2. Jacobi's integral -- an energy-like integral -- 9.3. "Ignorable coordinates" and integrals -- 10. Hamiltonian dynamics -- 10.1. The variables -- Solution for q̇ (q, p; t) -- 10.2. The equations of motion -- 10.2.1. Legendre transformations -- 10.2.2. Q and p as Lagrangian variables -- 10.2.3. An important property of the Hamiltonian -- 10.3. Integrals of the motion -- 10.4. Canonical transformations -- 10.5. Generating functions -- 10.6. Transformation solution of Hamiltonians -- 10.7. Separability -- 10.7.1. The Hamilton-Jacobi equation -- 10.7.2. Separable variables -- Special case -- ignorable coordinates -- 10.8. Constraints in Hamiltonian systems -- 10.9. Time as a coordinate in Hamiltonians -- Epilogue -- Index |
| Summary |
As the name implies, Intermediate Dynamics: A Linear Algebraic Approach views "intermediate dynamics"--Newtonian 3-D rigid body dynamics and analytical mechanics--from the perspective of the mathematical field. This is particularly useful in the former: the inertia matrix can be determined through simple translation (via the Parallel Axis Theorem) and rotation of axes using rotation matrices. The inertia matrix can then be determined for simple bodies from tabulated moments of inertia in the principal axes; even for bodies whose moments of inertia can be found only numerically, this procedure allows the inertia tensor to be expressed in arbitrary axes--something particularly important in the analysis of machines, where different bodies' principal axes are virtually never parallel. To understand these principal axes (in which the real, symmetric inertia tensor assumes a diagonalized "normal form"), virtually all of Linear Algebra comes into play. Thus the mathematical field is first reviewed in a rigorous, but easy-to-visualize manner. 3-D rigid body dynamics then become a mere application of the mathematics. Finally analytical mechanics--both Lagrangian and Hamiltonian formulations--is developed, where linear algebra becomes central in linear independence of the coordinate differentials, as well as in determination of the conjugate momenta. Features include: o A general, uniform approach applicable to "machines" as well as single rigid bodies. o Complete proofs of all mathematical material. Similarly, there are over 100 detailed examples giving not only the results, but all intermediate calculations. o An emphasis on integrals of the motion in the Newtonian dynamics. o Development of the Analytical Mechanics based on Virtual Work rather than Variational Calculus, both making the presentation more economical conceptually, and the resulting principles able to treat both conservative and non-conservative systems |
| Bibliography |
Includes bibliographical references and index |
| Notes |
English |
| In |
Springer e-books |
| Subject |
Dynamics.
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Algebras, Linear.
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kinetics (dynamics)
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Ingénierie.
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Algebras, Linear
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Dynamics
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| Form |
Electronic book
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| ISBN |
9780387280592 |
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0387280596 |
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0387283161 |
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9780387283166 |
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