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( 1) (w ) Ni ,w1--1 (f 2 ) Ni +1-11 -1 (f 2 ) k ,k = ( k1 - 1) xi + k1 - xi +1 xi + k1 -1 - xi
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(543) (544)
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(w ) (w ) k - 1 (f 2 - xi ) Ni , k1 -1 (f 2 ) ( xi + k1 - f 2 ) Ni +1, k1 -1 (f 2 ) + Ni(,w1) (f 2 ) = 1 k k1 - w - 1 xi + k1 -1 - xi xi + k1 - xi +1 ) M (j vk2 ( s2 ) ,
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( - ) ( ) M j vk21-1 ( s2 ) M j v -,1k2 -1 ( s2 ) , +1 = ( k2 - 1) y j + k2 - y j +1 y j + k2 -1 - yi
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(545)
(v ) (v ) k - 1 ( s2 - y j ) M j , k2 -1 ( s2 ) ( y j + k2 - s2 ) M j +1, k2 -1 ( s2 ) ) M (j vk2 ( s2 ) = 2 + , k2 - v - 1 y j + k2 -1 - yi y j + k2 - y j +1
(546)
The B-spline surface interpolation can also be expressed by matrix as
[ S1 (f2 , s2 )] = [ N (f2 )] [ p] [ M ( s2 )]
where S1 (f 21 , s21 ) S f , s ( ) [ S1 (f2 , s2 )] = 1 22 21 M S1 (f 2 n , s21 ) S1 (f 21 , s22 ) L S1 (f 21 , s2 m ) S1 (f 22 , s22 ) L S1 (f 22 , s2 m ) M O M S1 (f 2 n , s22 ) L S1 (f 2 n , s2 m ) n m
(547)
[ N (f 2 )] =
N1, k1 (f 21 ) L Nn , k1 (f 21 ) M O M N1, k (f 2 ) L Nn , k (f 2 ) 1 n n n n 1
p1, 1 L p1, m [ p] = M O M pn , 1 K pn , m n m
[ M ( s2 )] = M
M1, k2 ( s21 ) L M1, k2 ( s2 m ) O M Mm , k ( s2 ) L Mm , k ( s2 ) 2 1 2 m m m
Follower Motion Synthesis for the Three-Dimensional Cam Before examining the synthesis process, some basic requirements should be noted: 1 The number of motion constraints is not limited However, for the nonparametric forms of the B-spline surface interpolation used here, the pattern of the motion constraints must be a rectangular grid data form 2 To guarantee jerk continuity, the synthesized motion function in each parametric direction must be of at least degree = 4 This means that the order of the B-spline functions must be greater than or equal to 5 3 Since the pro le is a closed surface in the parametric direction of rotation, the motion function along that parametric direction, ie, f2, which is used to form the must
CAM MOTION SYNTHESIS USING SPLINE FUNCTIONS
be continuous everywhere, including the ends (as noted earlier) The general form of the B-spline surface interpolation can guarantee the continuity only of the internal region of the interpolated surface but not between the two ends Closed periodic B-spline functions produced from uniform knot sequences cited earlier solve this problem 4 No special consideration of continuity is required at the surface edges in the direction of translation so the multiple uniform knot sequence is used The systematic procedure for implementing the B-spline surface interpolation can be established as follows: 1 Set up the motion constraints in a n m rectangular grid If any data are absent in the rectangular grid, they can be lled in by using the B-spline interpolation applied along either the rotating or the translating directions 2 Select the appropriate order of the B-spline functions As shown in Eq (537), the proper value of the order is between two and the number of motion constraints Recall that the degrees of the functions are one less than the order 3 Construct the knot sequence according to the demand of each parametric direction The knot sequence, xI, for the closed periodic B-spline (along the rotation coordinate) can be obtained by f 2min i = 1 xi = f 2max - f 2min - 2 i n + 1 xi -1 + n The knot sequence in the translation coordinate can be found as s2min 1 j k2 s2 - s2min y j = y j -1 + max k2 + 1 j m m - k2 + 1 s2 m + 1 j m + k2 max
(548)
(549)
4 Determine the values of B-spline functions corresponding to motion constraints by using Eqs (538) to (541) 5 Collect the values of B-splines and motion constraints to form the matrices
[ N (f 2 )], [ S1 ], and [ M( s2 )]
6 Obtain the coef cient matrix [p] as follows:
[ p] = [ N (f 2 )]-1 [ S1 (f 2 , s2 )] [ M ( s2 )]-1
7 Evaluate the complete follower motion function by applying Eq (547) After the follower motion function is determined, kinematic properties for the velocity v1 (f2, s2), the acceleration A1 (f2, s2), and the jerk J1 (f2, s2) of the synthesized follower motion can be derived by differentiating Eq (537) When the angular and linear velocities of the cam are constant, the derivatives can be expressed as
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