pos_vel_acc

pos_vel_acc - Position A B A B r r r = Operator rule for...

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Unformatted text preview: Position: A B A B / r r r + = Operator rule for differentiation: p ω p p × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ xyz XYZ dt d dt d Explanation: To find the derivative of a vector p with respect to time in the XYZ coordinate system we may ask the observer in the xyz to differentiate p with respect to t and then add the correction p ω × . The vector ω is the angular velocity of the coordinate system xyz. x y ω X Y A O B A r B r A B / r x y ω X Y A O B x y ω X Y A O B A r B r A B / r Velocity: XYZ A B XYZ A XYZ B dt d dt d dt d ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ / r r r But B XYZ B dt d v r = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ A XYZ A dt d v r = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ A B A B A B xyz A B XYZ A B dt d dt d / / / / / r ω v r ω r r × + = × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ We therefore have A B A B A B / / r ω v v v × + + = Acceleration: ( ) XYZ A B XYZ A B XYZ A XYZ B dt d dt d dt d dt d ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ / / r ω v v v But B XYZ B dt d a v = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ A XYZ A dt d a v = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ A B A B A B xyz A B XYZ A B dt d dt d / / / / / v ω a v ω v v × + = × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ( ) ( ) ( ) ( ) A B A B A B A B xyz A B XYZ A B r dt d dt d / / / / / / r ω ω v ω ω r ω ω r ω r ω × × + × + × = × × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × & We therefore have ( ) A B A B A B A B A B r / / / / 2 × + × + × × + + = ω v ω r ω ω...
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This note was uploaded on 12/02/2011 for the course ME 4133 taught by Professor Ram during the Fall '06 term at LSU.

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pos_vel_acc - Position A B A B r r r = Operator rule for...

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