基于共形几何代数的一种9杆巴氏桁架位移分析

Displacement Analysis of a Nine-Link Baranov Truss Based on Conformal Geometric Algebra

基于共形几何代数,提出了一种9杆巴氏桁架位移分析的新几何建模和计算方法。在共形几何代数框架下,根据点、球、面、点对等基本几何元素的表示方法以及三角形面积的有向性,通过几何体的相交、分离和对偶运算,建立2个位移约束方程式;然后,通过一步结式消元得到该问题的一元54次方程,无增根无漏根。所提方法的优势在于2个位移约束方程的推导脱离了坐标系,且约束方程数的减少简化了方程组的消元过程。数值实例表明了所提方法的正确性,为其他9杆巴氏桁架的位移分析求解理论提供了一种新思路。

Based on conformal geometry algebra,a novel geometric modeling and computing method for the displacement analysis of a nine-link Baranov truss is proposed.Under the frame of conformal geometric algebra,in terms of the representations of the basic geometric elements such as point,sphere,surface,and point pair and the area sign of the triangle,two constraint equations are formulated by the intersection,dissection and dual operation.Then,a 54th-degree univariate equation without extraneous roots and roots loss is derived by one-step resultant elimination.The advantage of the proposed method lies in that the derivation of two constraint equations is free of coordinate and the elimination procedure is simplified due to the reduction of the number of constraint equations.At last,a numerical example is given to validate the correctness of the method.The proposed method provides a new sight for the theoretical solution of the displacement analysis for other nine-link Baranov trusses.