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

12 November 2004

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**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

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**Equations of Motion Conservation of Linear Momentum**

Conservation of Angular Momentum or Dynamics 16.07 2

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**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

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**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

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**Example: Parallel Plane Motion**

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

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

Dynamics 16.07 6

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**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

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**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

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**Example: Stability of Torque Free Motion**

From (3) constant Differentiate (1) and substitute value from (2), or, Solutions, Dynamics 16.07 9

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**Example: Stability of Torque Free Motion**

Growth Unstable Oscillatory Stable Dynamics 16.07 10

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**Gyroscopic Motion Bodies symmetric w.r.t.(spin) axis**

Origin at fixed point O (or at G) Dynamics 16.07 11

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**Gyroscopic Motion XY Z fixed axes x’y’z body axes — angular velocity ω**

xyz “working” axes — angular velocity Ω Dynamics 16.07 12

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**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

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**Gyroscopic Motion Euler Angles**

Angular Momentum Equation of Motion, Dynamics 16.07 14

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**Gyroscopic Motion Euler Angles**

become . . . not easy to solve!! Dynamics 16.07 15

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**Gyroscopic Motion Steady Precession**

Dynamics 16.07 16

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**Gyroscopic Motion Steady Precession**

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

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**Gyroscopic Motion Steady Precession**

Then, If precession velocity, spin velocity Dynamics 16.07 18

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**Steady Precession with M = 0**

constant Dynamics 16.07 19

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**Steady Precession with M = 0 Direct Precession**

From x-component of angular momentum equation, If then same sign as Dynamics 16.07 20

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**Steady Precession with M = 0 Retrograde Precession**

If and have opposite signs Dynamics 16.07 21

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