Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics / R.M. Brannon.

Περιλαμβάνει βιβλιογραφικές παραπομπές.

1. Introduction -- 2. Notation and tensor analysis essentials -- 2.1. Linear fractional transform -- 2.2. Visualizing rotations

3. Orthogonal basis and coordinate transformations -- 3.1. Superimposed rotations -- 3.2. Basis rotations

4. Rotation operations -- 4.1. Why apparent inconsistency in placement of the negative sign?

5. Axis and angle of rotation -- 5.1. Euler-Rodrigues formula -- 5.2. Computing the rotation tensor given axis and angle -- 5.3. Corollary to the Euler-Rodrigues formula : existence of a preferred basis -- 5.4. Computing axis and angle given the rotation tensor

6. Rotations contrasted with reflections -- 7. Quaternion representation of a rotation -- 7.1. Shoemake's form -- 7.2. Relationship between quaternion and axis/angle forms

8. Dyadic form of an invertible linear operator -- 8.1. Special case : lab basis -- 8.2. Special case : dyadic form of a rotation operation -- 8.3. Constructing a rotation that will transform one specified vector to another specified vector -- 8.4. Constructing a rotation from knowledge of initial and final 'marker' locations in a body

9. Sequential rotations -- 9.1. The distinction between fixed and follower axes -- 9.2. Roll, pitch, yaw : sequential rotations about fixed (laboratory) axes -- 9.3. Euler angles : sequential rotations about 'follower' axes

10. Series expression for a rotation -- 10.1. Cayley transformations

11. Spectrum of a rotation -- 12. Polar decomposition -- 12.1. Difficult definition of the deformation gradient -- 12.2. Intuitive definition of the deformation gradient -- 12.3. The Jacobian of the deformation -- 12.4. Invertibility of a deformation -- 12.5. Sequential deformations -- 12.6. Matrix analysis version of the polar-decomposition theorem -- 12.7. Polar decomposition--a hindsight intuitive interpretation -- 12.8. Variational interpretation of the polar decomposition -- 12.9. A more rigorous (classical) presentation of the polar-decomposition theorem -- 12.10. The 'fast' way to do a polar decomposition in two dimensions -- 12.11. Scaling properties of a polar decomposition -- 12.12. Classic method for obtaining a polar decomposition in 3D -- 12.13. Another iterative polar decomposition in 3D

13. Strain measures -- 13.1. One-dimensional strain measures -- 13.2. Three-dimensional strain definitions

14. Remapping, advecting, or interpolating rotations -- 14.1. Proposal 1 : Map and re-compute the polar decomposition -- 14.2. Proposal 2 : Discard the 'stretch' part of a mixed rotation -- 14.3. Proposal 3 : Advect the pseudo-rotation vectors -- 14.4. Proposal 4 : Mix the quaternions -- 14.5. Advection enhancement strategy #1 : solve the compatibility equations -- 14.6. Mixing enhancement strategy #2 : Lagrangian tracers

15. Rates and other derivatives of rotation -- 15.1. The 'spin' tensor -- 15.2. The angular velocity vector -- 15.3. Angular velocity in terms of axis and angle of rotation -- 15.4. Derivatives of rotation with respect to angle and axis -- 15.5. Difference between vorticity and polar spin -- 15.6. The (commonly mis-stated) Gosiewski's theorem -- 15.7. Rates of sequential rotations -- 15.8. Rates of simultaneous rotations -- 15.9. Integration of rotation rates

16. Variations of tensor-valued functions of scalars and vectors -- 16.1. A motivational example -- 16.2. A comment about rates of proper functions -- 16.3. The time rate of a principal function of a symmetric tensor -- 16.4. Time rate of the logarithmic strain

17. Statistics of random orientation -- 17.1. Elementary probability and statistics refresher -- 17.2. Uniformly random unit vectors--the theory -- 17.3. Uniformly random unit vectors--alternative implementation -- 17.4. 'Centroidally random' unit vectors -- 17.5. 'Nautical' visualization of a rotation -- 17.6. Uniformly random rotations -- 17.7. A basic algorithm for generating a uniformly random rotation -- 17.8. Generalization to generate transversely isotropic orientation distributions -- 17.9. Alternative algorithm for generating a uniformly random rotation -- 17.10. Shoemake's algorithm for uniformly random rotations

18. Introduction to material and tensor symmetries -- 18.1. Anisotropy classification via group theory -- 18.2. Quantifying and visualizing orientations

19. Frame indifference -- 19.1. A 3D spring--who expected it would be this hard!? -- 19.2. Introduction to frame indifference -- 19.3. Kinematics changes under superimposed rigid motion -- 19.4. Mechanics principles frame change

20. Tensor symmetry (not material symmetry) -- 20.1. What is isotropy of a tensor? -- 20.2. Isotropic second-order tensors in 3D space -- 20.3. Isotropic second-order tensors in 2D space -- 20.4. Isotropic fourth-order tensors in 3D -- 20.5. The isotropic part of a fourth-order tensor -- 20.6. Tensor transverse isotropy -- 20.7. Material transverse isotropy

21. Scalars and invariants -- 22. PMFI for incremental constitutive models -- 22.1. A frame-indifferent spring rate equation -- 22.2. The PMFI in general -- 22.3. PMFI in rate forms of the constitutive equations -- 22.4. Co-rotational rates (convected, Jaumann, polar, etc) -- 22.5. Lie derivatives and reference configurations -- 22.6. Frame indifference is only an essential (not final) step

23. Rigid-body mechanics -- 23.1. Rate of rotation -- 23.2. The slope-intercept of rigid motion -- 23.3. The point-slope description of rigid motion -- 23.4. Velocity and angular velocity for rigid motion -- 23.5. Time rate of a vector embedded in a rigid body -- 23.6. Acceleration for rigid motion -- 23.7. Important properties of a rigid body -- 23.8. Linear momentum of a rigid body -- 23.9. Angular momentum of a rigid body -- 23.10. Kinetic energy of a rigid body -- 23.11. Newton's equation (balance of linear momentum) -- 23.12. Euler's equation (balance of angular momentum)

24. Pseudo-body force for spinning problems -- 24.1. Kinematics of superimposed rotation (general analysis) -- 24.2. Fiducial body force for superimposed rigid motion

25. Computer graphics visualization -- 25.1. Orientation of the body -- 25.2. Mapping from the body to the screen -- 25.3. Mapping from the screen to the virtual visible surface -- 25.4. Changing the screen image of a body

26. Voigt and Mandel components -- 26.1. An introductory 3D example -- 26.2. Voigt components (inefficient and error prone!) -- 26.3. Mandel components (nice!) -- 26.4. Voigt components of fourth-order minor-symmetric tensors -- 26.5. Mandel components of fourth-order minor-symmetric tensors -- 26.6. Mandel components of fourth-order general tensors -- 26.7. Fourth-order linear transformations -- 26.8. Spectral analysis of fourth-order tensors

27. Higher-order rotations -- 27.1. Rotators : fourth-order rotations in Mandel form -- 27.2. Fourth-order 'focused identity' (projection) tensors -- 27.3. Focused rotations -- 27.4. Components of focused identities and elided projectors -- 27.5. Single-plane fourth-order rotations -- 27.6. Preferred basis for single-plane rotation -- 27.7. Double-plane fourth-order rotations -- 27.8. Multi-plane fourth-order rotations -- 28. Closing remarks.

Whilst vast literature is available for the most common rotation-related tasks such as coordinate changes, most reference books tend to cover one or two methods, and resources for less-common tasks are scarce. Specialized research applications can be found in disparate journal articles, but a self-contained comprehensive review that covers both elementary and advanced concepts in a manner comprehensible to engineers is rare. Rotation, Reflection, and Frame Changes surveys a refreshingly broad range of rotation-related research that is routinely needed in engineering practice. By illustrating key concepts in computer source code, this book stands out as an unusually accessible guide for engineers and scientists in engineering mechanics.