A computer-applicable linear mathematical model has been developed to determine Burmester’s curves for infinitely close relative positions (cubic of stationary curvature), which indirectly uses Carter-Hall’s circle. By varying a free parameter and using elements of kinematic and analytical geometry, an incomparably simpler solution is achieved than that obtained by the third-degree equations of the Burmester's curves for stationary curvature. The mathematical model for the synthesis of four-bar linkages includes and a condition for the pressure angle, whereupon is uniquely defined the kinematic diagram of the mechanism. Of the pressure angle, the reactions of the forces in the kinematic pairs and the force sizing of the mechanism depend. The model would facilitate the engineers in the synthesis of four-bar linkages by generating a function approximating a given function in the vicinity of a given position, where the two functions have four infinitely close common points (3rd-order approximation). An example of the synthesis of a four-bar linkage illustrates the application of the model, which is linear - it includes only equations of straight lines written in Cartesian coordinates, which is why it is convenient for computer calculations.

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Introduction.The generation of polynomial power laws of motion for the synthesis of cam mechanisms is complicated by the need to determine the coefficients of power polynomials. The study objective is to discover a rational capability of generating рower law swith arbitrary terms number under s with an rbitrary number of terms under the synthesis of cam mechanisms.Materials and Methods.A unified formula for determining the values of coefficients of power polynomials with any number of integers and/or non-integer exponents is derived through the so-called transfinite mathematical induction.  Results.A unified formula for determining the values of coefficients, which gives correct results for any number of even and/or odd exponents, is presented. The correctness of the derived formula is validated by the results on the multiple checks for different numbers, even and odd values of the exponents of quinquinomial and hexanomial power functions.  Discussion and Conclusions. A unified formula for determining the values of coefficients of power polynomials makes it possible to rationally define the laws of motion without finite and infinite spikes in the synthesis of elastic cam-lever systems. This provides a rational determination of the laws of motion without finite and infinite spikes in the synthesis of elastic cam-lever systems, and simple verification of the accuracy of the results obtained. The functions are particularly suitable for the synthesis of polydyne cams, as well as cams, since one polynomial can be used throughout the entire geometric mechanism cycle.

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