[edit]

\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\]

Elementary Rigid Body motion

Notation

• Basic notation
• Kronecker-delta symbol
• Einstein Summation Convention
• Einstein Summation Convention: examples

Vectors

• Vector product and vector spaces
• Vector cross-products, scalar triple products, and Levi-Cevita symbol
• Components of Vectors
• Linear Transformation and Adjoint

Tensors

• Motivation and Introduction
• Components/Matrix form
• Identity Tensor
• Null Tensor
• Dyadic products
• Representation of tensors as a sum of dyadics products
• Transpose, Symmetric and Skew-Symmetric Tensors, Symmetric and Skew-symmetric parts of a tensor
• Trace of a Tensor
• Product of Tensors

Differential geometry of curves

• Cartesian Coordinates
• Normal and Tangential Coordinates
• Cylindrical Coordinates

Calculus of variations

• Basic Notation

Kinematics

Rotation Tensors:

• Rotation tensors in general
  • Rotation axis
  • Components of a vectors thar are parallel and perpendicular to the rotation axis
  • Action of rotation on components of vectors thar are parallel and perpendicular to the rotation axis
• Rotation tensors in 3D
• Transpose, Symmetric, Skew Symmetric parts of R
• Finding Angle and Axis of Rotation given a Rotation Tensor
• Orthogonal Tensor
• Orthogonal and Proper Orthogonal transformation
• Rotations as Proper Orthogonal transformations
• Co-rotational basis
• Time derivative of the Co-rotational basis

Non-dimensional kinematics

• Non-dimensionalization
• Non-dimensional rigid body kinematics

Euler angles

• Euler angles
• Components of the rotation tensor in terms of Euler angles

Angular velocities

• Velocity field
• Axial vectors
• Angular velocity tensor and vectors
• Time derivative of angular velocity
• Composition of rotations
• Relative angular velocity vector

Kinetics

Geometric quantities related to a rigid body

• Center of Mass
• Euler and Inertia tensors
• Time derivative of the Inertia tensor

Momentum

• Linear momentum
• Angular momentum
• Time Derivative of Angular Momentum
• Solution to Euler Equations

Torque

• Torque

Balance Laws

• Balance of angular momentum
• Axially symmetric Top
  • Precession

Energy

• Kinetic Energy
• Potential Energy

Simple Applications

• Baseball, Frisbee flight, Magnus Force
• Whirling Rod, Instability (Markdown files)

Lagrangian Mechanics

Numerics

Mathematica Notes

• Mathematica Notes for ENGN 1370

Dynamic Euler Equation

• Lie-Group Integration Scheme
• Implicit Time integration scheme

Class projects 2018

• Paige