an effect in lowering the peak structural error。 Corresponding errors involved in the displacement and the velocities

Fig。 8 Velocity error distributions in generating y = x 0 ≤ x ≤ 1 by Galerkin method (Table 4)

Table 5 Slider-crank as y = x 0 ≤ x ≤ 1 generator by Precision-point method

Design No。 _ψ DEL (°) Precision-points Peak Error

1 80 1。5 0, 0。45, 0。75, 0。90, 1。0 0。234E−1

2 70 1。5 0, 0。45, 0。75, 0。90, 1。0 0。315E−1

3 60 1。5 0, 0。45, 0。75, 0。90, 1。0 0。139E−1

4 50 0。5 0, 0。45, 0。75, 0。90, 1。0 −0。112E−1

Table 6 Slider-crank dimensions for designs in Table 5

Design No。 x1 x2 x3 ψ0 s0

1 −0。5072 2。8091 −2。3029 2。55 1。0690

2 −0。5683 3。2417 −2。6747 9。24 1。1293

3 0。5189 4。1719 −3。6530 335。20 −1。0855

4 0。6753 4。9967 −4。3715 335。54 −1。3086

generated for the slider in the designs are shown in Figs。 9 and 10。 As can be seen from Fig。 9, some of the error curves are not representable in the scale used for the other methods。 Although a few of designs by Galerkin and Precision-point methods seem close to each other, in general three methods produce different solutions。 This is a very advantageous situation, since it provides sufficient number of alternative designs to pick from in constructing either multi-loop mechanism or in considering additional criteria for their choice。 So far, the solutions resulting from the same design problem have been analyzed with respect to the indicated motion direction, i。e。 +_ψ counter clockwise −_ψ clockwise。 However, as mentioned previously, it is noteworthy to look at the position (S) and velocity (V ) error distributions in the backward stroke of the slider-crank mecha-论文网

Fig。 9 Structural error distributions in generating y = x 0 ≤ x ≤ 1 by Precision-point method (Table 6)

Fig。 10 Velocity error distributions in generating y = x 0 ≤ x ≤ 1 by Precision-point method (Table 6)

nism。 S-error and V -error curves pertaining to Subdomain, Galerkin, Precision-point methods are depicted in Figs。 11, 12, 13, 14, 15, 16。 It is observed that overall performances in achieving uniform velocities in the backward strokes for the resulting designs, except a few cases, are acceptable in the sense that the V -errors are less than 15%。 On the other hand, there are interesting quality solutions like the ones observed in the design-3, 4 of Galerkin method, Fig。 14 and design-3 of Precision-point method, Fig。 16。

Fig。 11 Structural error distributions for designs by Subdomain method (Table 2) in backward stroke

Fig。 12 Velocity error distributions for designs by Subdomain method (Table 2) in backward stroke

Fig。 13 Structural error distributions for designs by Galerkin method (Table 4) in backward stroke

Fig。 14 Velocity error distributions for designs by Galerkin method (Table 4) in backward stroke

Fig。 15 Structural error distributions for designs by Precision-point method (Table 6) in backward stroke

Fig。 16 Velocity error distributions for designs by Precision-point method (Table 6) in backward stroke

5 Conclusions This paper shows that the problem of designing slider-crank mechanisms for desirable slider positions and velocities can be solved based on the fact that position and the kinematic quantity velocity can be determined purely on geometric grounds。 It is further demonstrated that upon posing it as a function-generation problem with new methodology supported by the linearization, quite a number of quality designs resulting from the process makes it possible to realize the satisfaction of design criteria including effectiveness and efficiency in both forward and backward strokes, the construction of multi-loop mechanisms, rotateability conditions etc。 The new methodology involves unification of the so-called Subdomain, Galerkin methods together with the classical collocation method where average error in selected subintervals, orthogonality of error to selected weighting functions and errors themselves at selected stations be set to zero, respectively in a unique style。 By varying several inherent parameters of the methodology it is possible to optimize the solutions。 Although the crank has been considered as the driving link while the slider is the driven link in this work, due to the linearization process it is possible to utilize the results presented here in the applications where the roles of the input and output links are reversed i。e。 the slider and the crank being driving and driven links respectively。