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Rotating frame analysis of rigid body dynamics in space phasor variables
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View: Figures


Image of Fig. 1.
Fig. 1.

Orthographic views of orthogonal transformations. In each view the axis of rotation comes out of the plane of the paper and is shown as a circle in the relevant style. The -system is fixed in the lab frame with gravity along the − direction. The origin is at the top's pivot. The first rotation (left panel) is through angle about the -axis to form the -axes (these axes are not used in the analysis that follows). The next step (middle panel) consists of a rotation through angle θ about the -axis and the set of axes thus formed is denoted by . The axis of symmetry of the top lies along the -axis and the top can be seen in profile view. The final step (right panel) is a rotation through angle ψ about the -axis to get the -axes. The gray background highlights the fact that the rotation takes place in the plane of the top's spin. The light gray color has been consistently used to indicate the top here and in Fig. 2 ; moreover the , , and axes are shown in a darker shade of gray to highlight the fact that they are fixed in the top's frame. We note that and are both principal axes for the top; however, the latter frame carries the entire angular velocity of the top while the former has only a portion of it.

Image of Fig. 2.
Fig. 2.

Three-dimensional view of the top. The axes and angles are as shown in Fig. 1 . We should remember that the rotations through and are not coplanar. As described in the text, the rotation is according to the -convention of Goldstein (Ref. ).

Image of Fig. 3.
Fig. 3.

This figure shows the trajectory of the tip of a top satisfying Eqs. (15) and (16) with the specific set of initial conditions described in the text. The trajectory lies on the surface of a sphere, shown here in light gray, with the trajectory itself in dark gray. The lab frame axes 1, 2, and 3 are the suitable axes in which  = 0 and the motion has a simple description. The origin of the axes is at the top's pivot. Axes 1 and 2 lie in the equatorial plane and are shown as dashed lines, while 3 comes directly out of the page through the line of viewing. The angles and are defined with respect to 1 2 3. A second choice of lab-frame axes has been denoted by primes. Of these, is shown as an axis that makes a certain inclination with 3. The trajectory is very difficult to describe in terms of Euler angles defined with respect to the primed axes.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Rotating frame analysis of rigid body dynamics in space phasor variables