MIT16_07F09_Lec30

MIT16_07F09_Lec30 - J. Peraire, S. Widnall 16.07 Dynamics...

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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L30- 3D Rigid Body Dynamics: Tops and Gyroscopes 3D Rigid Body Dynamics: Euler Equations in Euler Angles In lecture 29, we introduced the Euler angles as a framework for formulating and solving the equations for conservation of angular momentum. We applied this framework to the free-body motion of a symmetrical body whose angular momentum vector was not aligned with a principal axis. The angular moment was however constant. We now apply Euler angles and Eulers equations to a slightly more general case, a top or gyroscope in the presence of gravity. We consider a top rotating about a fixed point O on a at plane in the presence of gravity. Unlike our previous example of free-body motion, the angular momentum vector is not aligned with the Z axis, but precesses about the Z axis due to the applied moment. Whether we take the origin at the center of mass G or the fixed point O , the applied moment about the x axis is M x = Mgz G sin , where z G is the distance to the center of mass.. Initially, we shall not assume steady motion, but will develop Eulers equations in the Euler angle variables (spin), (precession) and (nutation). 1 Referring to the figure showing the Euler angles, and referring to our study of free-body motion, we have the following relationships between the angular velocities along the x,y,z axes and the time rate of change of the Euler angles. The angular velocity vectors for , and are shown in the figure. Note that these three angular velocity vectors are not orthogonal, giving rise to some cross products when the angular velocities i are calculated about the three principal axes. x = sin sin + cos (1) y = sin cos sin (2) z = cos + (3) 3D Rigid Body Dynamics: Eulers Equations We consider a symmetric body, appropriate for a top, for which the moments of inertia I xx = I yy = I and I zz = I . The angular momentum is then H x = I x (4) H y = I y (5) H z = I z (6) For the general motion of a three-dimensional body, we have Eulers equations in body-fixed axes which rotate with the body so that the moment of inertia is constant in time. In this body-fixed coordinate system, the conservation of angular momentum is H = d ([ I ] { } ) = AppliedMoments (7) dt 2 d Since we have chosen to work in a rotating coordinate system so that dt I = 0, we must pay the price, applying Coriolis theorem to obtain the time derivative of the angular velocity vector in the rotating coordinate system H = d H + H , (8) dt resulting in the Euler equations expressed in the x,y,z coordinate system moving with the body. In general, we must rotate with the total angular velocity of the body, so that the governing equation for the conservation of angular momentum become, with = ....
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This note was uploaded on 11/06/2011 for the course AERO 112 taught by Professor Widnall during the Fall '09 term at MIT.

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MIT16_07F09_Lec30 - J. Peraire, S. Widnall 16.07 Dynamics...

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