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## Lectures D25-D26 : 3D Rigid Body Dynamics

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**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) or, 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 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 Stable Oscillatory 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 – 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**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 = 0Retrograde Precession**If and have opposite signs Dynamics 16.07 21