sometimes referred to as point-position reduction method [7]。 Since the degree of accuracy attained by construction techniques depends highly upon the skills of the draftsman, graphical synthesis often leads to poor results。 Accuracy requirements are better met by analytical methods。 Especially upon the wide-spread use of computers after 1960, these methods have become popular and indispensible。 One of the well-established methods [5], which handles four accuracy points is sometimes referred to as

Precision-point approach founded on the idea of precision point, coined firstly by Freudenstein [21], where position error is zero。 An analytical approach taking into account the fifth-order approximation of an ideal function has been suggested in [22]。 A numerical iteration technique has been proposed

with a supposed convergence criteria by Wu [23]。 An overlay technique without reference to complicated mathematics is demonstrated as an educational tool to illustrate the synthesis of slider-crank mechanisms [24]。 In another work [25], complex number method has been used to synthesize an off-set five- bar slider-crank mechanism to perform five position function generation in two different phases。 Threeprecision position synthesis of the mechanism has been applied to the kinematic design of toggle clamps in [26]。 Although function generation establishes a framework for designing an economical cutting system by making use of the intrinsically available geometric parameters, it is not sufficient by itself, unless supported by additional thoughts。 Knowing that the slider-crank mechanism is essentially nonlinear, linearization is necessary in order to control slider

velocity during cutting。 Thus, if a linear function is imposed on the displacement relationships between the input crank and output slider members of the mechanism, then any desirable uniform velocity is to be obtained on the slider corresponding to a uniform rotational speed of the crank。 In this way, following the function synthesis of the slider-crank mechanism, necessary crank angular velocity can be picked up for a given cutting linear velocity of the slider。 Another point to be supplemented to the process of function generation is the efficiency consideration。 Since cutting occurs in a

limited but well-defined portion of the forward stroke, functional efficiency of cutting over one cycle of motion is to increase when for a determined interval of the backward stroke uniform velocity can also generated。 This brings the question of methodology in mind, with many inherent methoddependent

parameters making it possible to choose appropriate designs from a multitude of solutions satisfying several criteria at the same time。

The basic contribution of this work is the unique geometric design process by which control over the position and velocity of slider is obtained with reasonable errors through the application of Subdomain and Galerkin methods [27] to the slider-crank module in a unified way。 The classical precision-point approach has also been included in a unique style into the methodology to broaden the spectrum of designs and also for comparative purposes。 Although the formulation of the design process is realized on a single slidercrank mechanism, regarded as one module, it is possible to assemble multi-loop mechanisms with identical desirable position and velocity characteristics simply because several solutions result from the process。 It is also shown that considerable uniformity in velocities can be achieved in the resulting designs during both forward and return strokes simultaneously。 A number of examples is presented to demonstrate the effectiveness and efficiency of the methodology。

2 Theory

There are seemingly different quantities in engineering, which turn out to be purely geometric in essence。 For instance, velocity, instantaneous centers for velocity, normal acceleration are basically kinematic quantities depending on time and space。 It is well known that they can be determined on the basis of geometry [4, 6]。 Likewise, although mechanical advantage and kinetostatic analyses in machinery are force-related concepts, they can also be estimated on geometric grounds [4, 6, 26]。 It is more convenient to find the solution of an engineering problem when it is broken down to simpler forms called modules。 Departing from such a perspective, the fundamental proposition is to solve the problem of designing slider-crank mechanisms and their combinations for desirable slider positions