A new closed-form solution to inverse static force analysis of a spatial three-spring system

Abstract

In this paper, a new closed-form solution to the inverse static force analysis of a spatial three-spring system is presented. The system is formed by three springs each of which connects the ground at one end and joins a common point at the other. When a known force is applied to the common point of the system, the goal of inverse static analysis is to determine all the equilibrium configurations. A system of three polynomial equations in three variables is derived based on the geometric constraint and static force balancing. A 20 by 20 Dixon resultant matrices firstly derived from these three polynomials and then reduced to an 18 by 18 matrix. A 46th-degree univariate polynomial equation is yielded from the above 18 by 18 matrix. By further analysis, we found that 24 roots were degenerated and only the remaining 22 roots are the ones for the three-spring system. The result agrees with previous results. At last, two numerical examples are given to verify the elimination procedure. The presented algebraic elimination solution reveals some intrinsic geometry nature of this challenging problem.