# 13: Rigid-body Rotation

- Page ID
- 9634

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

- 13.1: Introduction to Rigid-body Rotation
- Rotating reference frame.

- 13.2: Rigid-body Coordinates
- Body-fixed coordinate system.

- 13.3: Rigid-body Rotation about a Body-Fixed Point
- Spinning top.

- 13.4: Inertia Tensor
- Specifies the inertial properties of the rotating body.

- 13.5: Matrix and Tensor Formulations of Rigid-Body Rotation
- These formulations dramatically simplify solving problems involving rigid-body rotation.

- 13.6: Principal Axis System
- The inertia tensor is a real symmetric matrix. A property of real symmetric matrices is that there exists an orientation of the coordinate frame, with its origin at the chosen body-fixed point O , such that the inertia tensor is diagonal. The coordinate system for which the inertia tensor is diagonal is called the Principal axis system which has three perpendicular principal axes.

- 13.7: Diagonalize the Inertia Tensor
- Determine the eigenvalues and eigenvectors for solution.

- 13.8: Parallel-Axis Theorem
- The values of the components of the inertia tensor depend on both the location and the orientation about which the body rotates relative to the body-fixed coordinate system. The parallel-axis theorem is valuable for relating the inertia tensor for rotation about parallel axes passing through different points fixed with respect to the rigid body. For example, one may wish to relate the inertia tensor through the center of mass to another location that is constrained to remain stationary.

- 13.9: Perpendicular-axis Theorem for Plane Laminae
- Rigid-body rotation of thin plane laminae objects is encountered frequently. Examples of such laminae bodies are a plane sheet of metal, a thin door, a bicycle wheel, a thin envelope or book. Deriving the inertia tensor for a plane lamina is relatively simple because there are limits on the possible relative magnitude of the principal moments of inertia.

- 13.10: General Properties of the Inertia Tensor
- The inertial properties of a body for rotation about a specific body-fixed location is defined completely by only three principal moments of inertia irrespective of the detailed shape of the body. As a result, the inertial properties of any body about a body-fixed point are equivalent to that of an ellipsoid that has the same three principal moments of inertia. The symmetry properties of this equivalent ellipsoidal body define the symmetry of the inertial properties of the body.

- 13.13: Euler Angles
- Relate angles in the body-fixed frame to the space-fixed frame.

- 13.14: Angular Velocity
- In body-fixed and space-fixed frame.

- 13.15: Kinetic energy in terms of Euler angular velocities
- In the body-fixed frame.

- 13.16: Rotational Invariants
- Relate body-fixed observables to space-fixed properties.

- 13.17: Euler’s equations of motion for rigid-body rotation
- Expressed in body-fixed frame.

- 13.18: Lagrange equations of motion for rigid-body rotation
- Euler's equations of motion.

- 13.20: Torque-free rotation of an inertially-symmetric rigid rotor
- The spinning symmetric top.

- 13.21: Torque-free rotation of an asymmetric rigid rotor
- Spinning asymmetric top.

- 13.22: Stability of torque-free rotation of an asymmetric body
- Dynamical motion.

- 13.23: Symmetric rigid rotor subject to torque about a fixed point
- Gyroscope and spinning top.

- 13.24: The Rolling Wheel
- Non-slipping motion.

- 13.25: Dynamic balancing of wheels
- Bearing forces.

- 13.26: Rotation of Deformable Bodies
- High diver maneuvers.

*Thumbnail: Proper Euler angles geometrical definition. The xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes (N) is shown in green. (CC BY 3.0; Lionel Brits).*