and velocities simultaneously within the context of a single function generating module。 This is further supported by the fact that velocity and position of the slider are not independent of each other。 The basic reference is to the geometry of a single slider-crank mechanism, Fig。 2 as the fundamental module。

As the first step, it is necessary to represent the motion of the slider-crank mechanism with a single displacement function。 This is done by writing out the coordinates of the crank-pin and slider-pin joints within the closure of the loop

Fig。 2 Geometry of the slider-crank mechanism

in the x and y directions, Fig。 2, squaring and summing to eliminate the connecting rod angle, δ。 With the notation in Fig。 2, this yields the following equation:

To generate the function y = f (x) x0 ≤ x ≤ xn by means of the slider-crank, a one-to-one correspondence is established between linkage variables (s,ψ) and function variables (y, x) through the following linear analogy:

ψ0, s0 are initial crank angle and starting slider position measured rightward from the crank-center projection on the line of motion, respectively, Fig。 2。 (–) sign in (3) stands for slider displacement to the left, (+) to the right。 A review of (1)–(4) or the geometry of the slider-crank mechanism, Fig。 2, will reveal that, based on presumed values of the input and output scales (Rx,Rv) to secure physically meaningful results, there are at most five available parameters, namely (x1, x2, x3,ψ0, s0) or (z1, z2, z3,ψ0, s0), which can be used in the design process for the generation of the aforementioned function。

The condition on which the so-called Subdomain method [27] is based is that the average of the displacement function G(s,ψ) should be zero within selected subintervals of the function interval [x0, xn]。 The number of subintervals will be equal to the number of parameters taken into consideration i。e。 five here。 If the subinterval in question is [xi−1, xi] ∈ [x0, xn] for i = 1, 。 。 。 , 5 then mathematically Subdomain method is to mean the following:

In the case of Galerkin method [27] displacement function G(s,ψ) is required to be orthogonal to a set of weighting functions wi(x), the number of which is identical with that of parameters, defined on the same interval as the given function y(x) i。e。 [x0, xn]。 Galerkin method leads to the following set of equations:

For comparative purposes, if the problem is approached from the stand-point of Collocation [27], then the displacement function G(s,ψ) is set equal to zero at a number of points called collocation, precision or accuracy points, xi , which belong to the function interval [x0, xn] and are as many as there are parameters。 Precision-point or collocation method yields the following set:

(+) is associated with rightward, (–) with leftward slider movement。 However coefficients involved in (8) and (9) have quite different forms in each method as shown below。 In Subdomain method coefficients in question are defined as follows:

For Galerkin method, the following are valid;

In the case of Precision-point or collocation method, the coefficients are defined by the following expressions:

By elimination of z1, z2, z3 from (8), one arrives at a couple of equations which contain ψ0 and s0;

The definitions of coefficients in (14) are to be found in the

Appendix。

From (13) it is possible to solve s0 as a function of ψ0;

Now a scheme can be devised to secure all possible roots

(ψ0, s0) which render Ap1 and Ap2 zero values simultaneously。 This scheme starts with ψ0 = 0° and runs up to 360° by increments of DEL, an arbitrary length。 In the meantime s0(ψ0) is calculated by means of (15) for every ψ0 between 0 and 360°, and one of Ap1 and Ap2, whichever is used in the process, changes sign, intervals in which roots are to be found are located。 Then roots are estimated by the use of a linear interpolation in the interval with errors, according to experience, much lower than 0。1%。 Now any three equations of the set (8) will serve to determine z1, z2, z3 from which the remaining slider-crank dimensions x1, x2, x3 are to be obtained。 Structural error between the generated and prescribed functional values is calculated at desired stations through (3) and the following explicit expression for s from (1):