Lectures D25-D26 : 3D Rigid Body Dynamics 12 November 2004

Outline Review of Equations of Motion Rotational Motion Equations of Motion in Rotating Coordinates Euler Equations Example: Stability of Torque Free Motion Gyroscopic Motion Euler Angles Steady Precession Steady Precession with M = 0 Dynamics 16.07 1

Equations of Motion Conservation of Linear Momentum Conservation of Angular Momentum or Dynamics 16.07 2

Equations of Motion in Rotating Coordinates Angular Momentum Time variation Non-rotating axes XY Z (I changes) big problem! - Rotating axes xyz (I constant) Dynamics 16.07 3

Equations of Motion in Rotating Coordinates xyz axis can be any right-handed set of axis, but . . . will choose xyz (Ω) to simplify analysis (e.g. I constant) Dynamics 16.07 4

Example: Parallel Plane Motion Body fixed axis Solve (3) for ωz, and then, (1) and (2) for Mx and My. Dynamics 16.07 5

Euler’s Equations If xyz are principal axes of inertia 6 Dynamics 16.07 6

Euler’s Equations Body fixed principal axes Right-handed coordinate frame Origin at: Center of mass G (possibly accelerated) Fixed point O Non-linear equations . . . hard to solve Solution gives angular velocity components . . . in unknown directions (need to integrate ω to determine orientation). Dynamics 16.07 7

Example: Stability of Torque Free Motion Body spinning about principal axis of inertia, Consider small perturbation After initial perturbation M = 0 Small Dynamics 16.07 8

Example: Stability of Torque Free Motion From (3) constant Differentiate (1) and substitute value from (2), or, Solutions, Dynamics 16.07 9

Example: Stability of Torque Free Motion Growth Unstable Oscillatory Stable Dynamics 16.07 10

Gyroscopic Motion Bodies symmetric w.r.t.(spin) axis Origin at fixed point O (or at G) Dynamics 16.07 11

Gyroscopic Motion XY Z fixed axes x’y’z body axes — angular velocity ω xyz “working” axes — angular velocity Ω Dynamics 16.07 12

Gyroscopic Motion Euler Angles Precession Nutation Spin – position of xyz requires and – position of x’y’z requires , θand ψ Relation between ( ) and ω,(and Ω ) Dynamics 16.07 13

Gyroscopic Motion Euler Angles Angular Momentum Equation of Motion, Dynamics 16.07 14

Gyroscopic Motion Euler Angles become . . . not easy to solve!! Dynamics 16.07 15

Gyroscopic Motion Steady Precession Dynamics 16.07 16

Gyroscopic Motion Steady Precession Also, note that H does not change in xyz axes External Moment Dynamics 16.07 17

Gyroscopic Motion Steady Precession Then, If precession velocity, spin velocity Dynamics 16.07 18

Steady Precession with M = 0 constant Dynamics 16.07 19

Steady Precession with M = 0 Direct Precession From x-component of angular momentum equation, If then same sign as Dynamics 16.07 20

Steady Precession with M = 0 Retrograde Precession If and have opposite signs Dynamics 16.07 21