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chap9_notes_RCRRRR

# chap9_notes_RCRRRR - RCRRRR Spatial Mechanism Summary of...

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1 RCRRRR Spatial Mechanism – Summary of How to Obtain θ 1 Group 2 Spatial Mechanism given: α 12 , α 23 , α 34 , α 45 , α 56 , α 61 a 12 , a 23 , a 34 , a 45 , a 56 , a 61 S 1 , S 2 , S 3 , S 4 , S 6 θ 6 (input angle) find: θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , S 5 . Want to get four equations of the form: (a i x m 2 + b i x m + d i ) x n + (e i x m 2 + f i x m + g i ) = 0 i = 1..4 (9.2) where the coefficients a 1 through g 4 are quadratic in x 1 . The first pair of equations is derived from the following subsidiary tan-half-angle laws for a spherical hexagon: X ¯ 4 - X 612 = (Y ¯ 4 - Y 612 )x 3 , (9.19) (X ¯ 4 + X 612 )x 3 = -(Y ¯ 4 + Y 612 ) . (9.20) which are converted to s 34 (X ¯ 4 - X 612 ) = (c 34 Z 612 - s 34 Y 612 - c 45 )x 3 , (9.23) s 34 (X ¯ 4 + X 612 )x 3 = -(c 34 Z 612 + s 34 Y 612 - c 45 ) . (9.24) Secondary tan-half angle laws are now created realizing that . 2 x + 1 = 2 tan + 1 2 1 = d x d 2 3 3 2 3 3  θ θ 6 R R R R R C 5 4 3 2 1

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2 After much algebraic manipulation the result is {[a 34 (Z 612 -c 34 c 45 ) - s 34 c 34 Z 0612 + s 34 2 Y 0612 + S 3 s 34 2 X 612 ] s 45 - a 45 s 34 + a 45 s 34 c 45 (c 34 Z 612 - s 34 Y 612 )} x 3 + (9.44) a 45 c 45 s 34 2 X 612 + S
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chap9_notes_RCRRRR - RCRRRR Spatial Mechanism Summary of...

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