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Tham khảo tài liệu 'advanced engineering dynamics 2010 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 74 Rigid body motion in three dimensions Fig. 4.14 Binet diagram ellipsoid principal axes 1 2 3 sphere radius 1.95 Fig. 4.15 Binet diagram ellipsoid principal axes 1 2 3 sphere radius 2.05 Fig. 4.16 Binet diagram ellipsoid principal axes 3 2 1 sphere radius 2.6 remains constant. A similar situation occurs when tidal effects are present. In both cases it is assumed that variations from the nominal shape are small. The general effect is that Binet s ellipsoid will slowly shrink. For the case of rotation about the 3 axis the intersection curve reduces and the motion remains stable but in the case Euler s angles 75 of rotation close to the 1 axis the intersection curve will slowly increase in size leading to an unstable condition. Rotation close to the 2 axis is unstable under all conditions. 4.10 Euler s angles The previous sections have been concerned for the most part with setting up the equations of motion and looking at the properties of a rigid body. Some insight to the solution of these equations was gained by means of Poinsot s construction for the case of torque-free motion. The equations obtained involved the components of angular velocity and acceleration but they cannot be integrated to yield angles because the co-ordinate axes are changing in direction so that finite rotation about any of the body axes has no meaning. We are now going to express the angular velocity in terms of angles which can uniquely define the orientation of the body. Such a set are Euler s angles which we now define. Figure 4.17 shows a body rotating about a fixed point o or its centre of mass . The XYZ axes are an inertial set with origin o. The xyz axes are in the general case attached to the body. If the body has an axis of symmetry then this is chosen to be the z axis. Starting with XYZ and the xyz coincident we impose a rotation of 0 about the z axis. There then follows a rotation of 0 about the new X axis the x axis and finally we give a rotation of 0 about the final z axis. The