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ME 242 Lecture 33 04.27

# ME 242 Lecture 33 04.27 - ME 242 Dynamics Today's Key...

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1 ME 242 Dynamics April 27, 2009 Eric Wang Today’s Key Concepts Chapter 8: 3D motion of Rigid Bodies We will only do kinematics in 3D Spherical coordinates Angular velocity in 3D Angular acceleration in 3D Polar Coordinates a m = ˙ r r ˙ θ 2 ) e r + (2˙ r ˙ θ + r ˙ ˙ θ ) e θ r m / O = re r v m = ˙ r e r + r ˙ θ e θ Polar Coordinates in 3D X Y Z Video Clip The Force is Strong X Y Z

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2 Spherical Coordinates R m / O = Re R Introduce x -Z Plane R m / O = Re R = re r + R cos φ k 2 sets of polar coordinates r = R sin φ Coordinate Transformation Arrays cos θ 0 0 0 0 1 cos θ -sin θ sin θ e θ j i e r k k cos φ sin φ 0 -sin φ 0 cos φ 1 0 0 e θ e θ e r e φ e R k R m / O = Re R = R (sin φ e r + cos φ k ) = re r + R cos φ k Velocity & Acceleration a m = ( ˙ ˙ R R ˙ φ 2 R ˙ θ 2 sin 2 φ ) e R R m / O = Re R v m = ˙ Re R + R ˙ φ e φ + R ˙ θ sin φ e θ + 2 ˙ R ˙ φ + R ˙ ˙ φ R ˙ θ 2 sin(2 φ ) 2 e φ + 2 ˙ R ˙ θ sin φ + R ˙ ˙ θ sin φ + 2 R ˙ θ ˙ φ cos φ [ ] e θ Reduces to Polar φ
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