用单杆附加方程法进行导杆机构精确位置综合

Precision position synthesis of guide-bar linkages using monad auxiliary equations

提出用单杆附加方程构建机构的运动综合方程式.通过增加单杆矢量分量和附加方程,将含角度变量的机构运动综合方程式转换为一般的多项式方程组,并且显著减少了综合方程组的总次数和混合体积(又称BKK界).如果从单杆矢量环路线性方程组中求出相应的单矢量分量公式,代入对应的附加方程,也可方便地推导出消除了中间变量的双杆组、三杆组公式.讨论了在四杆导杆机构中的刚体导引、函数生成和轨迹生成等综合问题上的应用,推导出相应的综合方程组.对综合方程的推导及对其总次数和混合体积的分析表明单杆附加方程法的方程组总次数接近于二、三杆组法,且其具有公式推导方便、模块性强等优点,理论上比双杆组法、三杆组法更容易用计算机来自动建立任意机构的综合方程.

The monad auxiliary equation method is presented to model the polynomial systems for kinematic synthesis of linkages. The kinematic synthesis equations of mechanisms originally derived from the closed vector loop, which contained the independent angular unknowns, were transformed skillfully to the polynomial systems by adding the component variables of monad vector and auxiliary equations, and then the total degrees and mixed volumes （or BKK bounds） of the systems were dramatically decreased. If the x and y components of the monads were yielded from the linear equations constructing by monad vectors in a closed vector loop and substituted into the corresponding auxiliary equations, the polynomial systems for dyad or triad were derived. The method was applied to the problems of motion, function and path synthesis for guide-bar linkages and their three forms of synthesis systems grouped were developed. The process to construct the synthesis systems and analyses of their total degrees and mixed volumes showed that the total degrees of the equations constructed by the method were really close to those of equations grouped by dyad and triad vectors, but the formulae of the former were derived much conveniently and the modularity about monad＇s formulae and structures was also better than the latter. Hence, the monad method may be theoretically adapted to automated modeling and solving the synthesis equations for any linkages in a computer.