Abstract
The hyper-chaotic least square method for finding all of the real solutions of nonlinear equations was proposed and the following displacement analysis on the 33rd non-plane 2-coupled–degree nine-link Barranov truss was completed. Four constrained equations were established by a vector method with complex numbers according to four loops of the mechanism, and four supplement equations were also established by increasing four variables and the relation of the sine and cosine functions. The established eight equations are those of the forward displacement analysis of the mechanism. In combining the least square method with hyper-chaotic sequences, a hyper-chaotic least square method based on utilizing a hyper-chaotic discrete system to obtain and locate initial points so as to find all the real solutions of the nonlinear questions was proposed. A numerical example was given. A comparison was also done with another means of finding a solution method. The results show that all of real solutions were quickly obtained, and it proves the correctness and validity of the proposed method.
Highlights
In the kinematic analysis and the innovative design of the plane mechanism, the planar basic kinematic chain is analyzed as an independent structure unit; in particular, the displacement analysis forms the most basic work
It is difficult to research the displacement analysis of a 9‐ link Barranov truss; the more difficult work is involved in the displacement analysis of the 9‐link Barranov truss, for which the coupling coefficient is 2
Combining the Newton downhill method with hyper‐chaotic sequences, a hyper‐ chaotic Newton‐downhill method based on utilizing a hyper‐chaotic discrete system to obtain and locate initial points in order to find all of the real solutions of the nonlinear questions was proposed [15]
Summary
In the kinematic analysis and the innovative design of the plane mechanism, the planar basic kinematic chain is analyzed as an independent structure unit; in particular, the displacement analysis forms the most basic work. The researched 9‐link Barranov truss is the 33rd non‐ planar basic kinematic chain in [1], which has a symmetrical structure, and the variables of position‐ closed equations are 3 or 4, so the elimination is very difficult. The chaotic sequence is a new method for obtaining all of the solutions for the real numbers of the mechanism by using the Newton iterative starting points of chaotic and hyper‐chaotic systems [11,12,13]. Combining the Newton downhill method with hyper‐chaotic sequences, a hyper‐ chaotic Newton‐downhill method based on utilizing a hyper‐chaotic discrete system to obtain and locate initial points in order to find all of the real solutions of the nonlinear questions was proposed [15]. The initial value is gained by adopting the hyper‐chaotic Hénon map, which is applied in the least square method for solving the 33rd Barranov truss. The calculation example shows that the proposed method is correct and effective
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