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L25 3D Rigid Body Kinematics

# L25 3D Rigid Body Kinematics - J Peraire S Widnall 16.07...

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J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25 - 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 di ff erent 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, θ 1 and θ 2 , not only would the combined rotation θ be di ff erent from θ 1 + θ 2 , but in general θ 1 + θ 2 = θ 2 + θ 1 . This situation is illustrated in the figure below, in which we consider a 3D rigid body undergoing two 90 o rotations about the x and y axis. 1

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It is clear that the result of applying the rotation in x first and then in y is di ff erent 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. This result can also be understood by considering the rotation of axes by a coordinate transformation. Consider a transformation ( x ) = [ T 1 ]( x ), and a subsequent coordinate transformation ( x ) = [ T 2 ]( x ). The ( x ) coordinate obtained by the transformation ( x ) = [ T 1 ][ T 2 ]( x ) will not be the same as the coordinates obtained by the transformation ( x ) = [ T 2 ][ T 1 ]( x ), in other words, order matters. Angular Velocity About a Fixed Point On the other hand, if we consider infinitesimal rotations only, it is not di cult to verify that they do indeed behave as vectors. This is illustrated in the figure below, which considers the e ff ect of two combined infinitesimal rotations, d θ 1 and d θ 2 , on point A . ( figure reproduced from J.L. Meriam and K.L. Kraige, Dynamics, 5th edition, Wiley) As a result of d θ 1 , point A has a displacement d θ 1 × r , and, as a result of d θ 2 , point A has a displacement d θ 2 × r . The total displacement of point A can then be obtained as d θ × r , where d θ = d θ 1 + d θ 2 . Therefore, it follows that angular velocities ω 1 = θ ˙ 1 and ω 2 = θ ˙ 2 can be added vectorially to give ω = ω 1 + ω 2 . This
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L25 3D Rigid Body Kinematics - J Peraire S Widnall 16.07...

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