16.07 Dynamics Fall 2004 Lecture D22-3d Rigid Body Kinematics In this lecture, we consider the motion of a 3D rigid body. We shall see that in the general three dimensional case, the angular velocity of the body can change in magnitude as well as in direction, and, as a consequence, the motion is considerably more complicated than that in two dimensions Rotation about a fixed point We consider first the simplified situation in which the 3d body moves in such a way that there is always a point, O, which is fixed. It is clear that, in this case, the path of any point in the rigid body which is at a distance r from O will be on a sphere of radius r that is centered at O. We point out that the fixed point O is not necessarily a point in rigid body(the second example in this notes illustrates this point) Euler's theorem states that the general displacement of a rigid body, with one fixed point is a rotation about some axis. This means that any two rotations of arbitrary magnitude about different axes can always be combined into a single rotation about some axis. At first sight, it seems that we should be able to express a rotation as a vector which has a direction along the axis of rotation and a magnitude that is equal to the angle of rotation. Unfortunately, if we consider two such rotation vectors, 01 and 62, not only would the combined rotation 8 be different from 81+02 but in general 81+82#02+01. This situation is illustrated in the figure below, in which we consider a 3D rigid body undergoing two 90 rotations about the z and y axis. It is clear that the result of applying the rotation in a first and then in y is different from the result obtained by rotating first in y and then in Therefore, it is clear that finite rotations cannot be treated as vectors, since they do not satisfy simple vector operations such as the parallelogram vector addition latJ. Peraire 16.07 Dynamics Fall 2004 Version 1.1 Lecture D22 - 3D Rigid Body Kinematics In this lecture, we consider the motion of a 3D rigid body. We shall see that in the general three dimensional case, the angular velocity of the body can change in magnitude as well as in direction, and, as a consequence, the motion is considerably more complicated than that in two dimensions. Rotation About a Fixed Point We consider first the simplified situation in which the 3D body moves in such a way that there is always a point, O, which is fixed. It is clear that, in this case, the path of any point in the rigid body which is at a distance r from O will be on a sphere of radius r that is centered at O. We point out that the fixed point O is not necessarily a point in rigid body (the second example in this notes illustrates this point). Euler’s theorem states that the general displacement of a rigid body, with one fixed point is a rotation about some axis. This means that any two rotations of arbitrary magnitude about different axes can always be combined into a single rotation about some axis. At first sight, it seems that we should be able to express a rotation as a vector which has a direction along the axis of rotation and a magnitude that is equal to the angle of rotation. Unfortunately, if we consider two such rotation vectors, θ1 and θ2, not only would the combined rotation θ be different from θ1 + θ2, but in general θ1 + θ2 6= θ2 + θ1. This situation is illustrated in the figure below, in which we consider a 3D rigid body undergoing two 90o rotations about the x and y axis. It is clear that the result of applying the rotation in x first and then in y is different from the result obtained by rotating first in y and then in x. Therefore, it is clear that finite rotations cannot be treated as vectors, since they do not satisfy simple vector operations such as the parallelogram vector addition law. 1