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Rigid Body Dynamics: Equations, Euler Angles, Precession

Learn the fundamental equations of motion for rigid bodies, explore Euler angles, and understand steady precession. This lecture covers gyroscopic motion, stability of torque-free motion, and Euler equations. Practice examples included.

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Rigid Body Dynamics: Equations, Euler Angles, Precession

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

  2. 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

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

  4. 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

  5. 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) or, Dynamics 16.07 4

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

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

  8. 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

  9. 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

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

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

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

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

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

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

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

  17. Gyroscopic Motion Steady Precession Dynamics 16.07 16

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

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

  20. Steady Precession with M = 0 constant Dynamics 16.07 19

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

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

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