Lectures D25-D26 : 3D Rigid Body Dynamics 12 November 2004.
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Transcript of Lectures D25-D26 : 3D Rigid Body Dynamics 12 November 2004.
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Lectures D25-D26 :Lectures D25-D26 :3D Rigid Body Dynamics3D Rigid Body Dynamics
12 November 2004
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Dynamics 16.07 Dynamics D25-D26
OutlineOutline
• 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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Dynamics 16.07 Dynamics D25-D26
Equations of MotionEquations of Motion
Conservation of Linear Momentum
Conservation of Angular Momentum
or
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Dynamics 16.07 Dynamics D25-D26
Equations of Motion in Rotating CoordinatesEquations 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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Dynamics 16.07 Dynamics D25-D26
Equations of Motion in Rotating CoordinatesEquations 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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or,
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Dynamics 16.07 Dynamics D25-D26
Example: Parallel Plane MotionExample: Parallel Plane Motion
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Body fixed axis
Solve (3) for ωz, and then, (1) and (2) for Mx and My.
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Dynamics 16.07 Dynamics D25-D26
Euler’s EquationsEuler’s Equations
If xyz are principal axes of inertia
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Dynamics 16.07 Dynamics D25-D26
Euler’s EquationsEuler’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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Dynamics 16.07 Dynamics D25-D26
Example: Stability of Torque Free Motion Example: Stability of Torque Free Motion
Body spinning about principal axis of inertia,
Consider small perturbation
After initial perturbation M = 0
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Small
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Dynamics 16.07 Dynamics D25-D26
Example: Stability of Torque Free MotionExample: Stability of Torque Free Motion
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From (3) constantDifferentiate (1) and substitute value from (2),
or,
Solutions,
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Dynamics 16.07 Dynamics D25-D26
Example: Stability of Torque Free MotionExample: Stability of Torque Free Motion
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Growth Unstable
Oscillatory Stable
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Gyroscopic Motion
• Bodies symmetric w.r.t.(spin) axis
• Origin at fixed point O (or at G)
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic MotionGyroscopic Motion
• XY Z fixed axes• x’y’z body axes — angular velocity ω• xyz “working” axes — angular velocity Ω
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Euler AnglesGyroscopic Motion Euler Angles
– position of xyz requires and
– position of x’y’z requires , θand ψ
Relation between ( ) and ω,(and Ω )
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Precession
Nutation
Spin
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Euler AnglesGyroscopic Motion Euler Angles
Angular Momentum
Equation of Motion,
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Euler AnglesGyroscopic Motion Euler Angles
become
. . . not easy to solve!!
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Steady PrecessionGyroscopic Motion Steady Precession
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Steady PrecessionGyroscopic Motion Steady Precession
Also, note that
H does not change in xyz axes
External Moment
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Dynamics 16.07 Dynamics D25-D26
Gyroscopic Motion Steady PrecessionGyroscopic Motion Steady Precession
Then,
If
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precession velocity,spin velocity
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Dynamics 16.07 Dynamics D25-D26
Steady Precession with M = 0Steady Precession with M = 0
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constant
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Dynamics 16.07 Dynamics D25-D26
Steady Precession with M = 0 DirecSteady Precession with M = 0 Direct Precessiont PrecessionFrom x-component of angular momentum equation,
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If thensame sign as
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Dynamics 16.07 Dynamics D25-D26
Steady Precession with M = 0Steady Precession with M = 0Retrograde PrecessionRetrograde Precession
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and have opposite signs
If