p09fl23 - Lecture 23: Euler equations (26 Oct 09) A....

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Lecture 23: Euler equations (26 Oct 09) A. Review: Euler’s equations 1. Calculate the time derivative of angular momentum relative to the center of mass in terms of torques d ~ L dt | inertial = d ~ L dt | body + × ~ L = ~ Γ 2. Take the body-fixed axes to be the principal axes of the moment of inertia tensor (3 orthogonal vectors ˆ e ( s ) j , eigenvalues I j ), ~ L = I 1 ω 1 ˆ e 1 + ... , ˆ e 1 × ˆ e 2 = ˆ e 3 (and cyclic); t -derivative is in body frame. I 1 1 dt = ω 2 ω 3 ( I 2 - I 3 ) + Γ 1 3. Note that in this form the vector components are projected onto body- fixed time-dependent axes, especially when there is a torque. The solu- tion for motion with external torques requires a self-consistent analysis. Discussion at end of Sec. 26: “body-associated” axes rather than body- fixed axes arise in Problem 5.2. 4. then some examples from FW sec. 28. B. Compound pendulum 1. Rigid body constrained to rotate about a stationary fixed axis ˆ e 3 . 2. Locate the center-of-mass (cm) in the plane perpendicular to ˆ
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p09fl23 - Lecture 23: Euler equations (26 Oct 09) A....

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