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
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3. Equations of Motion Conservation of Linear Momentum
Conservation of Angular Momentum or
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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)
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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)
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6. Example: Parallel Plane Motion
Body fixed axis
Solve (3) for ωz, and then, (1) and (2) for Mx and My.
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7. Euler’s Equations If xyz are principal axes of inertia 6
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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).
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9. Example: Stability of Torque Free Motion
Body spinning about principal axis of inertia,
Consider small perturbation
After initial perturbation M = 0
Small
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10. Example: Stability of Torque Free Motion
From (3)
constant
Differentiate (1) and substitute value from (2),
or,
Solutions,
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11. Example: Stability of Torque Free Motion
Growth
Unstable
Oscillatory
Stable
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12. Gyroscopic Motion Bodies symmetric w.r.t.(spin) axis
Origin at fixed point O (or at G)
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13. Gyroscopic Motion XY Z fixed axes x’y’z body axes — angular velocity ω
xyz “working” axes — angular velocity Ω
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14. 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 )
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15. Gyroscopic Motion Euler Angles
Angular Momentum
Equation of Motion,
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16. Gyroscopic Motion Euler Angles
become
. . . not easy to solve!!
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17. Gyroscopic Motion Steady Precession
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18. Gyroscopic Motion Steady Precession
Also, note that
H does not change in xyz axes
External Moment
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19. Gyroscopic Motion Steady Precession
Then, If
precession velocity,
spin velocity
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20. Steady Precession with M = 0
constant
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21. Steady Precession with M = 0 Direct Precession
From x-component of angular momentum equation, If
then
same sign as
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22. Steady Precession with M = 0 Retrograde PrecessionIf
and have opposite signs
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