Rigid Body Dynamics

Rigid Body Dynamics is a fundamental branch of mechanical engineering that analyzes the motion of objects under the action of forces, based on the key idealization that the object is a "rigid body"—meaning it does not deform or change shape. This field investigates the relationship between the forces and torques applied to an object and its resulting translational (linear) and rotational (angular) motion. By applying principles like Newton's laws of motion and Euler's equations, engineers can predict and control the dynamic behavior of systems ranging from vehicles and aircraft to robotic manipulators and machinery components.

1. Introduction to Rigid Body Dynamics
   1. Fundamental Concepts
      1. Definition of a Rigid Body
      2. Particle vs. Rigid Body
      3. The Rigid Body Assumption
      4. Kinematics vs. Kinetics
      5. Degrees of Freedom
         1. Translational Degrees of Freedom
         2. Rotational Degrees of Freedom
      6. Constraints in Rigid Body Motion
         1. Holonomic Constraints
         2. Non-holonomic Constraints
   2. Mathematical Preliminaries
      1. Vector Algebra Review
         1. Scalars and Vectors
         2. Vector Addition and Subtraction
         3. Scalar Multiplication
         4. Dot Product
         5. Cross Product
         6. Triple Scalar Product
         7. Triple Vector Product
      2. Coordinate Systems
         1. Cartesian Coordinates
            1. Position Vectors in Cartesian Coordinates
         2. Polar Coordinates
            1. Conversion between Cartesian and Polar
         3. Cylindrical Coordinates
            1. Conversion between Cartesian and Cylindrical
         4. Path Coordinates
            1. Tangential Unit Vectors
            2. Normal Unit Vectors
      3. Time Derivatives of Vectors
         1. Derivatives in a Fixed Frame
         2. Derivatives in a Rotating Frame
            1. Transport Theorem
            2. Angular Velocity Vector