ace con Body Cone Instantaneous a=ws XWp Axis of Rotation ngular acceleration can be determined using Coriolis theorem, and is given by a=ws xwp(see lecture D12 for details) General motion In the general case, the displacement of a rigid body is determined by a translation plus a rotation about some axis. This result is a generalization of Eulers theorem, which is sometimes known as Chasles' theorem In practice, this means that six parameters are needed to define the position of a 3D rigid body. For instance, ve could choose three coordinates to specify the position of the center of mass, two angles to define the axis of rotation and an additional angle to determine the magnitude of the rotation. Unlike the motion about a fixed point, it is not always possible to define an instantaneous axis of rotation. Consider, for instance, a body which is rotating with angular velocity w and, at the same time, has a translational velocity parallel to w. It is clear that, in this case, all the points in the body have a non-ze locity and therefore an instantaneous center of rotation cannot be defined It turns out that, in some situations, the motion of the center of mass of a 3D rigid body can be determined independent of the orientation. Consider, for instance, the motion of an orbiting satellite in free fight. I this situation, the sum of all external forces on the satellite does not depend on the satellite's attitude, and therefore, it is possible to determine the position without knowing the attitude. In more complex situations, however, it may be necessary to solve simultaneously for both the position of the center of mass and the ttitude The velocity, Up, and acceleration, ap, of a point, P, in the rigid body can be determined if we know the velocity, vo, and acceleration, ao, of a point in the rigid body, o, as well as the body's angular velocity, w ding expressions, given below, are particular cases of the relative motion xpressions derived in lecture D12, UO′+×T P O +×(u×rp) (4)Instantaneous Axis of Rotation Body Cone Space Cone The angular acceleration can be determined using Coriolis’ theorem, and is given by α = ωs×ωp (see lecture D12 for details). General Motion In the general case, the displacement of a rigid body is determined by a translation plus a rotation about some axis. This result is a generalization of Euler’s theorem, which is sometimes known as Chasles’ theorem. In practice, this means that six parameters are needed to define the position of a 3D rigid body. For instance, we could choose three coordinates to specify the position of the center of mass, two angles to define the axis of rotation and an additional angle to determine the magnitude of the rotation. Unlike the motion about a fixed point, it is not always possible to define an instantaneous axis of rotation. Consider, for instance, a body which is rotating with angular velocity ω and, at the same time, has a translational velocity parallel to ω. It is clear that, in this case, all the points in the body have a non-zero velocity, and therefore an instantaneous center of rotation cannot be defined. It turns out that, in some situations, the motion of the center of mass of a 3D rigid body can be determined independent of the orientation. Consider, for instance, the motion of an orbiting satellite in free flight. In this situation, the sum of all external forces on the satellite does not depend on the satellite’s attitude, and, therefore, it is possible to determine the position without knowing the attitude. In more complex situations, however, it may be necessary to solve simultaneously for both the position of the center of mass and the attitude. The velocity, vP , and acceleration, aP , of a point, P, in the rigid body can be determined if we know the velocity, vO′ , and acceleration, aO′ , of a point in the rigid body, O′ , as well as the body’s angular velocity, ω, and acceleration, α. The corresponding expressions, given below, are particular cases of the relative motion expressions derived in lecture D12, vP = vO′ + ω × r ′ P (3) aP = aO′ + ω˙ × r ′ P + ω × (ω × r ′ P ) . (4) 5