Although there are two linkages for every set of identical dimensions, as (16) indicates, only one of them will be the sought function generator。 In order to utilize the results of the design procedure of function generation for desirable slider positions and velocities, a suitable function like y = x0 ≤ x ≤ 1 is selected to linearize the whole process, essential for control。 When equation pair (3), after substitution of the linear function, is differentiated with respect to time (t ), the following relationships are obtained:

 where w and V are the angular and linear velocities of the crank and slider, respectively。 From (17) it is deduced that the required slider velocity is a linear function of the crank angular velocity, as written below:

 The actual slider velocity is determined by taking the time derivative of the displacement function G(s,ψ) in (1), according to chain rule:

 In order to perform a velocity analysis, the error in the slider-velocity is computed by taking the difference between (18), constant required value, and then actual value calculated by means of (19) and (3) at selected stations。 After realizing the synthesis procedure coupled with the evaluation of displacement and velocity error, it is a trivial issue to relate the resulting design to the desired position and velocity of the slider。 From (18), the necessary operational speed of the crank (w) is easily computed corresponding to the slider of the velocity V 。 For the desired position of the slider, it is a simple matter to place crankshaft center O at a point shifted backward by an amount equal to the difference between the desired and determined (s0) positions of the slider, with respect to Fig。 2。 If the synthesized mechanism is utilized for cutting, then an alternative way to obtain 

Fig。 3 Multi-loop slider-crank mechanisms

desirable output position is to fix the cutter to the slider at a point away from the coupler–slider joint by a distance equal to the difference between desirable distance and s0。

3 Extensions and implications

Formulation of the synthesis problem is based on only one slider-crank mechanism。 However, the methods involved in the synthesis process, namely Subdomain, Galerkin and Precision-point are supposed to yield more than one solution。 Thus, each resulting solution may be regarded as a module composing a multi-link mechanism。 Shown in Fig。 3 are two and four slider-crank modules connected in parallel with each other, where there are two and four solutions resulting from this synthesis process, respectively。 In this way, by varying the input data such as sub domains, weighting functions and precision-points, many slider-crank module designs can be obtained to form as many loops as there are available solutions resulting from the process, ending in the synthesized multi-loop mechanism。 The fact that many possible solutions can be obtained by means of the mentioned methods can be utilized to select proper designs satisfying a number of criteria。 For example, it may be desired that the crank make a full revolution。 In case the resulting module dimensions (x1, x2, x3) do not satisfy the rotateability conditions (e。g。 x2 − |x1| ≥ x3, x2 ≥ |x1| with reference to Fig。 2) input parameters may be varied to directly choose the appropriate module fulfilling the rotateability conditions。 Other criteria such as maximum displacement, time ratio between forward and return strokes, maximum acceleration, pressure angle etc, may be used to pick up the correct module out of many resulting solutions。

It would be interesting to investigate simultaneously the velocities of the slider in the designed module in forward and backward strokes of the mechanisms。 Figure 4 shows the positions of the crank and the slider, when the slider goes into (_s) region, where uniform motion is expected。 The formulae indicating to the start and end of the uniform motion in the direction of backward stroke are given below。 Fig。 4 Positions of the slider and crank in the forward and backward strokes