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lecturenotes3.20100204.4b6af9c3336a75.13877321

lecturenotes3.20100204.4b6af9c3336a75.13877321 - s =0 = F...

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3 Generalized forces De fi nition 16 Let C ( s ) be a curve that passes through C 0 at s = 0 . The virtual work along C ( s ) and evaluated at s = 0 equals the work performed by all physical forces acting on the mechanism under a virtual displacement corresponding to a change δ s in s . The virtual power along C ( s ) and evaluated at s = 0 equals the rate of change of virtual work with respect to s evaluated at s = 0 . De fi nition 17 For every chart ( C , ξ ) containing C 0 , there exists a row matrix F ξ (independent of C ( s ) ), such that the virtual work is given by F ξ · d ds ξ ( C ( s )) ¯ ¯ ¯ ¯ s =0 · δ s and the virtual power is given by F ξ · d ds ξ ( C ( s )) ¯ ¯ ¯ ¯ s =0 . De fi nition 18 The components of F ξ are the generalized forces with respect to the coordinate functions of the chart ( C , ξ ) . Proposition 19 Consider two coordinate charts ( C 1 , ξ 1 ) and ( C 2 , ξ 2 ) on the collection of allowable con fi gu- rations, such that C 1 C 2 6 = and ξ 1 and ξ 2 are related by the coordinate transformation ξ 2 = f ( ξ 1 ) . Let C ( s ) be a curve that passes through C 0 C 1 C 2 at s = 0 . It follows that F ξ 2 · d ds ξ 2 ( C ( s )) ¯ ¯ ¯
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Unformatted text preview: s =0 = F ξ 2 · ∂ ξ 1 f ( ξ 1 ( C )) · d ds ξ 1 ( C ( s )) ¯ ¯ ¯ ¯ s =0 and thus F ξ 1 = F ξ 2 · ∂ ξ 1 f ( ξ 1 ( C )) . 6 Proposition 20 Consider a coordinate chart ( C , ξ ) , where ξ = ( ξ 1 , ξ 2 ) T and F ξ = ( F 1 , F 2 ) . Suppose that ξ 1 = g ( ξ 2 ) on a restricted collection ˜ C ⊂ C of allowable con f gurations, such that ³ ˜ C , ξ 2 ´ is a chart. Let ˜ C ( s ) be a curve in ˜ C that runs through C at s = 0 . It follows that F ξ · d ds ξ ³ ˜ C ( s ) ´ ¯ ¯ ¯ ¯ s =0 = ( F 1 , F 2 ) · μ ∂ ξ 2 g ( ξ 2 ( C )) · d ds ξ 2 ( C ( s )) ¯ ¯ s =0 d ds ξ 2 ( C ( s )) ¯ ¯ s =0 ¶ = ¡ F 1 · ∂ ξ 2 g ( ξ 2 ( C )) + F 2 ¢ · d ds ξ 2 ( C ( s )) ¯ ¯ ¯ ¯ s =0 and thus F ξ 2 = F 1 · ∂ ξ 2 g ( ξ 2 ( C )) + F 2 . 7...
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