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Lectures D25-D26 : 3D Rigid Body Dynamics 12 November 2004

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Dynamics 16.07Dynamics D25-D26 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 1

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Dynamics 16.07Dynamics D25-D26 Equations of Motion Conservation of Linear Momentum Conservation of Angular Momentum or 2

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Dynamics 16.07Dynamics D25-D26 Equations of Motion in Rotating Coordinates Angular Momentum Time variation -Non-rotating axes XY Z ( I changes) big problem! - Rotating axes xyz ( I constant) 3

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Dynamics 16.07Dynamics D25-D26 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) 4 or,

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Dynamics 16.07Dynamics D25-D26 Example: Parallel Plane Motion 5 Body fixed axis Solve (3) for ω z, and then, (1) and (2) for Mx and My.

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Dynamics 16.07Dynamics D25-D26 Eulers Equations If xyz are principal axes of inertia 6

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Dynamics 16.07Dynamics D25-D26 Eulers 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). 7

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Dynamics 16.07Dynamics D25-D26 Example: Stability of Torque Free Motion Body spinning about principal axis of inertia, Consider small perturbation After initial perturbation M = 0 8 Small

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Dynamics 16.07Dynamics D25-D26 Example: Stability of Torque Free Motion 9 From (3)constant Differentiate (1) and substitute value from (2), or, Solutions,

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Dynamics 16.07Dynamics D25-D26 Example: Stability of Torque Free Motion 10 GrowthUnstable Oscillatory Stable

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Bodies symmetric w.r.t.(spin) axis Origin at fixed point O (or at G ) 11

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion XY Z fixed axes xyz body axes angular velocity ω xyz working axes angular velocity Ω 12

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Euler Angles – position of xyz requires and – position of xyz requires, θ and ψ Relation between ( ) and ω,(and Ω ) 13 Precession Nutation Spin

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Euler Angles Angular Momentum Equation of Motion, 14

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Euler Angles become... not easy to solve!! 15

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Steady Precession 16

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Steady Precession Also, note that H does not change in xyz axes External Moment 17

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Dynamics 16.07Dynamics D25-D26 Gyroscopic Motion Steady Precession Then, If 18 precession velocity, spin velocity

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Dynamics 16.07Dynamics D25-D26 Steady Precession with M = 0 19 constant

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Dynamics 16.07Dynamics D25-D26 Steady Precession with M = 0 Direct Precession From x-component of angular momentum equation, 20 If then same sign as

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Dynamics 16.07Dynamics D25-D26 Steady Precession with M = 0 Retrograde Precession 21 and have opposite signs If

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