Elastic Rod Intended Dynamics Model And Dynamic Model

Elastic rod, an important classical mechanical model widely used in many engineering systems such as undersea cables, fibers and biological tissues in 20th century, have been attracted great attention during the last 40 years as a useful model in describing DNA. And Lots of achievements on this topic have been obtained recently.In this paper, mathematical modeling and numerical simulation method on elastic rods are studied, including the surface model of an elastic rod, the quasi-Hamilton equations and symplectic method of Kirchhoff rod, and the existence of solution of the equations and its numerical stimulation. The main results includes:(1) Based on the assumption of a Kirchhoff rod and Using Kirchhoff analogue technique, a differential equation system is raised to describe the surface of a super slender elastic rod.(2) Quaternion is introduced to set up the quasi-dynamical model of a super-slender elastic rod. Preceding results, though raised long ago, were described by Euler angles (ψ,θ,φ). The singularity of Euler angles atθ= kπmake its numerical treatment very difficult. In this paper, by introducing quaternion to replace the Euler angles in the quasi-dynamical models, we derived the quasi-Hamilton equations of Kirchhoff rod. So symplectic method can be introduced in its numerical computation.(3) According to the existence of solution of Lax equations, we proved the existence of solution of dynamic rod equations. Under the condition that the elastic energy function holds quadratic form, the numerical stimulation of dynamic rod is given.