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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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MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

MULT. INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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