Lectures D25-D26 :
3D Rigid Body Dynamics
12 November 2004
Outline
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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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Equations of Motion
Conservation of Linear Momentum
Conservation of Angular Momentum
or
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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)
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Equations of Motion in Rotating Coordinates
or,
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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Example: Parallel Plane Motion
Body fixed axis
Solve (3) for ωz, and then, (1) and (2) for Mx and My.
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Euler’s Equations
If xyz are principal axes of inertia
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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).
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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
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Example: Stability of Torque Free Motion
From (3)
constant
Differentiate (1) and substitute value
from (2),
or,
Solutions,
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Example: Stability of Torque Free Motion
Growth
Unstable
Oscillatory
Stable
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Gyroscopic Motion
• Bodies symmetric w.r.t.(spin) axis
• Origin at fixed point O (or at G)
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Gyroscopic Motion
• XY Z fixed axes
• x’y’z body axes — angular velocity ω
• xyz “working” axes — angular velocity Ω
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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 )
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Gyroscopic Motion
Euler Angles
Angular Momentum
Equation of Motion,
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Gyroscopic Motion
Euler Angles
become
. . . not easy to solve!!
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Gyroscopic Motion
Steady Precession
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Gyroscopic Motion
Steady Precession
Also, note that
H does not change in xyz axes
External Moment
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Gyroscopic Motion
Steady Precession
Then,
If
precession velocity,
spin velocity
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Steady Precession with M = 0
constant
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Steady Precession with M = 0
Direct Precession
From x-component of angular momentum equation,
If
then
same sign as
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Steady Precession with M = 0
Retrograde Precession
If
and
have opposite signs
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