MEAM 535 Advanced Dynamics

Fall 1999

Summary of Week #3

1. Essentials of vector analysis
   • Basic vectors in a chosen coordinate syetm: the triad
   • Essential and non-essential (but convenient) features of basis vectors
   • Components (measure numbers) of a vector for a chosen triad
   • Representation of vectors as 3X1 arrays
   • Usefulness of array representation in computing dot and cross products
   • Skew-symmetric matrix form of a vector for carrying out cross-product as multiplication of a matrix and a vector
   • Free vectors vs. vectors that are fixed in space
2. Rotation matrices
   • Transformation of a vector components from one basis vectors to another, rotation matrix, and direction cosines
   • Properties of the rotation matrices: orthogonality, determinant = 1, composite rotations, etc.
   • Finite rotations do not commute.
   • A rotation matrix has three independent quantities: three angles of rotation
   • Body-fixed and space-fixed rotations
3. Derivative of a vector in a moving frame
   • Geometric derivation
   • Transport term = omega X r
4. Velocity and acceleration of a particle in a moving frame
   • Velocity has three components: velocity of the origin of the moving frame + velocity of the particel with respect to the moving frame + transport term
   • Acceleration has five terms: acceleration of the origin of the moving frame + acceleration of the particle with respect to the moving frame + angular acceleration X r + Coriolis acceleration + transport term of the transport term in velocity.
5. Angular acceleration
   Why is it convenient to express angular acceleration in the inertial or the moving body frame?
6. An example problem: a particle translating in a straight slot in a rotating disk
7. Two points on the same rigid body: velocity and acceleration
8. Compostion of angular velocity vectors with simpel proof
9. What is an angular velocity vector?
   1-2-3 body fixed rotations
   Infinetesimal rotations
   Skew-symmetric matrix form of angular velocity
   Definition of angular velocity vector: why don't we say (d/dt) of theta?
10. Derivation of the transport term using the definition of the angular velocity.
11. Angular velocity vector is ...
    • a nonholonomic quantity, i.e., you cannot integrate it, i.e., you cannot get as a derivative of something unless it is a simple rotation about one axis.
    • a free vector
    • is a defined quantity.
12. Kane's definition of the angular velocity vector Derivation using dyadic notation
13. Dyadic notation
    • Dyads and dyadics
    • Dot product of a dyad and a vector
    • Unit dyadic
    • Matrix representation of a dyad
    • Symmetric and skw-symmetric dyads
    • Vector representation of skew-symmetric dyad <
14. Derivation of Kane's definition of the angular velocity vector
    • Derivative of a vector in a moving frame
    • Angular velocity dyad
    • Proving that angular velocity dyad is skew-symmetric
    • Vector form of angular velocity dyad gives Kane's definition of angular velocity vector