In order to overcome the disadvantage of conventional precise point methods,optimization approaches and numerical atlas,and further improve the accuracy and efficiency of rigid-body guidance synthesis of planar four-bar linkages with prescribes timing,a novel analytical approach with Fourier series was presented to solve the rigid-body guidance synthesis problem of planar four-bar linkages.Firstly,according to the periodicity of the linkage rotation angle function,the series mathematical formula of the linkage rotation angle function was established by using Fourier series theory.The harmonic parameters of the linkage rotation angle function were obtained by DFT.As a result,the linkage rotation angle function was expressed as the summation of a Fourier series with the input angle as the variable.The harmonic parameters relationship between the linkage rotation angle function and the rigid-body guidance rotation angle function was obtained by analysing the internal connection between them in the planar motion.Then,according to the theory of complex vector,the vector loop equation of planar four-bar linkages was established.The linkage rotation angle function was formulated according to the Fourier series,and was substituted into a vector loop equation.By using eliminated element method,the vector loop equation was changed into a complex number equation that contained only the fundamental dimensions of mechanism，the harmonic parameters of the linkage rotation angle function and the input angle.With regard to the properties of the complex exponent，the mathematic expression containing the fundamental dimensions of mechanism and the harmonic parameters of the linkage rotation angle function was obtained.Accordingly,the relationship between the fundamental dimensions of mechanism and the harmonic parameters of the rigid-body guidance rotation angle function was obtained by variable substitution in terms of the previous harmonic parameters relationship.Based on this relationship,the new design equations for rigid-body guidance synthesis of planar four-bar linkages were established.A cubic equation that contained only the fundamental dimensions of mechanism and the harmonic parameters of the rigid-body guidance rotation angle function was obtained by dialytic elimination.Through solving the cubic equation,a general formula was derived for the rigid-body guidance synthesis problem of planar four-bar linkages using the harmonic parameters of the rigid -body guidance rotation angle function.After the fundamental dimensions of mechanism are determined,the harmonic parameters of position of rigid-body guidance were obtained by DFT.According to the harmonic parameters relationship between the linkage rotation angle function and the position of rigid-body guidance,the formula,which can compute the real size and installing dimensions of the linkage mechanism,was obtained.Based on the aforementioned theory,the procedure of solving rigid-body guidance synthesis problem by the proposed method can be obtained.As a result,a new analytical method for rigid-body guidance synthesis of planar four-bar linkages with prescribes timing was established.Based on the synthesis procedure,the computer programs have been developed for the proposed method by MATLAB.An example was provided to verify the validity and feasibility of the proposed method.Verification results showed that the proposed approach can overcome the shortage of precise point method and directly to solve rigid-body guidance synthesis problems for planar four-bar linkages with no limitations on the number of precision points.Compared with numerical atlas and optimization approaches,the proposed method avoided the use of extensive numerical atlas databases and optimal initial solutions,and obtained the results by solving the equation.Therefore,this approach has the characteristics of high accuracy,fast solution velocity and high repeatability,and is suitable for computer programming.The research of this approach provided the theory basis for development of convenient synthesis software.
