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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L29 3D Rigid Body Dynamics 3D Rigid Body Dynamics: Euler Angles The diﬃculty of describing the positions of the bodyfixed axis of a rotating body is approached through the use of Euler angles: spin ψ ˙ , nutation θ and precession φ shown below in Figure 1. In this case we surmount the diﬃculty of keeping track of the principal axes fixed to the body by making their orientation the unknowns in our equations of motion; then the angular velocities and angular accelerations which appear in Euler’s equations are expressed in terms of these fundamental unknowns, the positions of the principal axes expressed as angular deviations from some initial positions. Euler angles are particularly useful to describe the motion of a body that rotates about a fixed point, such as a gyroscope or a top or a body that rotates about its center of mass, such as an aircraft or spacecraft. Unfortunately, there is no standard formulation nor standard notation for Euler angles. We choose to follow one typically used in physics textbooks. However, for aircraft and spacecraft motion a slightly different one is used; the primary difference is in the definition of the ”pitch” angle. For aircraft motion, we usually refer the motion to a horizontal rather than to a vertical axis. In a description of aircraft motion, ψ would be the ”roll” angle; φ the ”yaw” angle; and θ the ”pitch” angle. The pitch angle θ would be measured from the horizontal rather than from the vertical, as is customary and useful to describe a spinning top. 1 Figure 1: Euler Angles In order to describe the angular orientation and angular velocity of a rotating body, we need three angles. As shown on the figure, we need to specify the rotation of the body about its ”spin” or z bodyfixed axis, the angle ψ as shown. This axis can also ”precess” through an angle φ and ”nutate” through an angle θ . To develop the description of this motion, we use a series of transformations of coordinates, as we did in Lecture 3. The final result is shown below. This is the coordinate system used for the description of motion of a general threedimensional rigid body described in bodyfixed axis. To identify the new positions of the principal axes as a result of angular displacement through the three Euler angles, we go through a series of coordinate rotations, as introduced in Lecture 3. 2 x We first rotate from an initial X, Y, Z system into an x , y , z system through a rotation φ about the Z, z axis. The angle φ is called the angle of precession. ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎜ ⎜ ⎜ ⎝ x y ⎟ ⎟ ⎟ ⎠ = ⎜ ⎜ ⎜ ⎝ cosφ sinφ X X ⎜ ⎜ ⎜ ⎝ ⎟ ⎟ ⎟ ⎠ ⎟ ⎟ ⎟ ⎠ = [ T 1 ] ⎜ ⎜ ⎜ ⎝ ⎟ ⎟ ⎟ ⎠ ....
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 Fall '09
 widnall
 Dynamics, Angular Momentum, Rigid Body, Rotation, Euler angles, Euler, Ixx ωx

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