Modeling and analysis of nonlinear mechanics of a super-thin elastic rod

Nonlinear mechanics for a super-thin elastic rod with the biological background of DNA super-coiling macromolecules is an interdisciplinary research area of classical mechanics and molecular biology. It is also a subject of dynamics and elasticity because elastic bodies are analyzed via the theory of dynamics. It is in frontiers of general mechanics （dynamics and control）. This dissertation is devoted to model a constrained super-thin elastic rod and analyze its stability in equilibrium. The existing research results are summarized. Analytical mechanics is systematically applied to model the elastic rod. The Schroedinger equation for complex curvatures or complex bending moments is, respectively, extended from the case of circular crosssections to that of non-circular ones. The equilibrium of a rod constrained on a surface is investigated.

On dynamics of elastic rod based on exact Cosserat model

The analysis of kinematics and dynamics of an elastic rod with circular cross section is studied on the basis of exact Cosserat model under consideration of the tension and shear deformation of the rod. The dynamical equations of a rod with arbitrary initial shape are established in general form. The dynamics of a straight rod under axial tension and torsion is discussed as an example. In discussion of static stability in the space domain the Greenhill criteria of stability and the Euler load are corrected by the influence of tension and shear strain. In analysis of dynamical stability in the time domain it is shown that the Lyapunov and Euler stability conditions of the rod in space domain are the necessary conditions of Lyapunov＇s stability in the time domain. The longitudinal, torsional and lateral vibrations of a straight rod based on exact model are discussed, and an exact formula of free frequency of lateral vibration is obtained. The free frequency formulas of various simplified models, such as the Rayleigh beam, the Kirchhoff rod, and the Timoshenko beam, can be seen as special cases of the exact formula under different conditions of simplification.

Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage

A non-local solution for a functionally graded piezoelectric nano-rod is pre- sented by accounting the surface effect. This solution is used to evaluate the charac- teristics of the wave propagation in the rod structure. The model is loaded under a two-dimensional （2D） electric potential and an initially applied voltage at the top of the rod. The mechanical and electrical properties are assumed to be variable along the thick- ness direction of the rod according to the power law. The Hamilton principle is used to derive the governing differential equations of the electromechanical system. The effects of some important parameters such as the applied voltage and gradation of the material properties on the wave characteristics of the rod are studied.

Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables： time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.

ANALYSIS OF THERMAL POST-BUCKLING OF HEATED ELASTIC RODSTHEFRACTIO

Based on the nonlinear geometric theory of extensible rods, an exact mathematical model of thermal post_buckling behavior of uniformly heated elastic rods with axially immovable ends is developed, in which the arc length s(x) of axial line and the longitudinal displacement u(x) are taken as the basic unknown functions. This is a two point boundary value problem of first order ordinary differential equations with strong non_linearity. By using shooting method and analytical continuation, the nonlinear boundary value problems are numerically solved. The thermal post_buckled states of the rods with transversely simply supported and clamped ends are obtained respectively and the corresponding numerical data tables and characteristic curves are also given.

Mei symmetry and conserved quantities in Kirchhoff thin elastic rod statics

We investigate the application of the Mei symmetry analysis in finding conserved quantities for the thin elastic rod statics. By using the Mei symmetry analysis, we have obtained the Jacobi integral and the cyclic integrals for a thin elastic rod with intrinsic twisting for both the cases of circular and non-circular cross sections. Our results can be easily reduced to the results without the intrinsic twisting that have been reported. Through calculation, we find that the Noether symmetry can be more directly and easily used than the Mei symmetry in finding the first integrals for the thin elastic rod. These first integrals will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the elastic rod model for DNA.

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear SchrSdinger （NLS） equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.

Solution of nonlinear wave equation of elastic rod

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia are studied. A nonlinear wave equation is derived. The equation is solved by the method of full approximation.

Perturbation of symmetries for super-long elastic slender rods

This paper analyses perturbations of Noether symmetry, Lie symmetry, and form invariance for super-long elastic slender rod systems. Criterion and structure equations of the symmetries after disturbance are proposed. Considering perturbation of all infinitesimal generators, three types of adiabatic invariants induced by perturbation of symmetries for the system are obtained.

Influence of Wires Properties on Warp-Knitted Loop Geometry in Plane Based on Theory of Elastic Rod

Based on classic theory of elastic rod,the warp-knitted loop geometry in plane is independent of yarn properties,while there is a certain gap between the geometrical model and the actual fabrics.According to this problem,further analysis of loop geometry is done based on the theory of elastic rod with theoretical calculation and experiments.The theoretical analysis found that the distance between the contacted points at the loop root affected the loop geometry,and the distance was affected by the ratio of bending rigidity and the friction between yarns.The experiments,forming simple loop by taking the yarn as an elastic rod,found that the bending rigidity affected the loop geometry.Then the relationships between warp-knitted loop geometry in plane of metallic fabrics and wires properties were studied.The results show that metallic fabrics are more suitable for the theory of elastic rod;the friction and bending rigidity of wire yarns affect the loop geometry in plane.Also,the elongation of yarn affects the loop geometry in the actual warp-knitted fabric.

Elastic-plastic deformation of sandwich rod on elastic basis

Sandwich composite material possesses advantages of both light weight and high strength. Although the mechanical behaviors of sandwich composite material with the influence of single external environment have been intensively studied, little work has been done in the study of mechanical property, in view of the nonlinear behavior of sandwich composites in the complicated external environments. In this paper, the problem about the bending of the three-layer elastic-plastic rod located on the elastic base, with a compressibly physical nonlinear core, has been studied. The mechanical response of the designed three-layer elements consisting of two bearing layers and a core has been examined. The complicated problem about curving of the three-layer rod located on the elastic base has been solved. The convergence of the proposed method of elastic solutions is examined to convince that the solution is acceptable. The calculated results indicate that the plasticity and physical nonlinearity of materials have a great influence on the deformation of the sandwich rod on the elastic basis.

Structure properties and Noether symmetries for super-long elastic slender rod

DNA is a nucleic acid molecule with double-helical structures that are special symmetrical structures attracting great attention of numerous researchers. The super-long elastic slender rod, an important structural model of DNA and other long-train molecules, is a useful tool in analysing the symmetrical properties and the stabilities of DNA. This paper studies the structural properties of a super-long elastic slender rod as a structural model of DNA by using Kirchhoff＇s analogue technique and presents the Noether symmetries of the model by using the method of infinitesimal transformation. Baaed on Kirchhoff＇s analogue it analyses the generalized Hamilton canonical equations. The infinitesimal transfornaationa with rcspect to the radial coordinnte, the gonarnlizod coordinates, and the Cluasi-momenta of 5he model are introduced. The Noether gymmetries and conserved qugntities of the model are obtained.

General investigation for longitudinal wave propagation under magnetic field effect via nonlocal elasticity

In this paper, the propagation of longitudinal stress waves under a longitu- dinal magnetic field is addressed using a unified nonlocal elasticity model with two scale coefficients. The analysis of wave motion is mainly based on the Love rod model. The effect of shear is also taken into account in the framework of Bishop＇s correction. This analysis shows that the classical theory is not sufficient for this subject. However, this unified nonlocal elasticity model solely used in the present study reflects in a manner fairly realistic for the effect of the longitudinal magnetic field on the longitudinal wave propagation.

An elastic-plastic analysis of short-leg shear wall structures during earthquakes

Short-leg shear wall structures are a new form of building structure that combine the merits of both frame and shear wall structures. Its architectural features, structure bearing and engineering cost are reasonable. To analyze the elastic-plastic response of a short-leg shear wall structure during an earthquake, this study modified the multiple-vertical-rod element model of the shear wall, considered the shear lag effect and proposed a multiple-vertical-rod element coupling beam model with a new local stiffness domain. Based on the principle of minimum potential energy and the variational principle, the stiffness matrixes of a short-leg shear wall and a coupling beam are derived in this study. Furthermore, the bending shear correlation for the analysis of different parameters to describe the structure, such as the beam height to span ratio, short-leg shear wall height to thickness ratio, and steel ratio are introduced. The results show that the height to span ratio directly affects the structural integrity; and the short-leg shear wall height to thickness ratio should be limited to a range of approximately 6.0 to 7.0. The design of short-leg shear walls should be in accordance with the ＂strong wall and weak beam＂ principle.

考虑轴向大拉伸下弹性模量对聚酯系泊动态响应的影响

Owing to the particularity of a polyester fiber material,the polyester mooring undergoes large axial tensile deformation over long-term use.Large axial tensile deformation significantly impacts the dynamic response of the mooring system.In addition,the degrees of large axial tension caused by different elastic moduli are also different,and the force on the mooring line is also different.Therefore,it is of great significance to study the influence of elastic modulus on the dynamic results of the mooring systems under large axial tension.Conventional numerical software fails to consider the axial tension deformation of the mooring.Based on the theory of slender rods,this paper derives the formula for large axial tension using the method of overall coordinates and overall slope coordinates and provides the calculation programs.Considering a polyester mooring system as an example,the calculation program and numerical software are used to calculate and compare the static and dynamic analyses to verify the reliability of the calculation program.To make the force change of the mooring obvious,the elastic moduli of three different orders of magnitude are compared and analyzed,and the dynamic response results after large axial tension are compared.This study concludes that the change in the elastic modulus of the polyester mooring changes the result of the vertex tension by generating an axial tension.The smaller the elastic modulus,the larger the forced oscillation motion amplitude of the top point of the mooring line,the more obvious the axial tension phenomenon,and the smaller the force on the top of the polyester mooring.

大齿轮齿距偏差测量仪弹性测杆机构仿真及优化

为了实现大齿轮齿距偏差测量难的问题,研制出一种大齿轮齿距偏差测量仪.针对测量仪中的弹性测杆机构的弹性变形影响测量精度的问题,对其组成及工作原理进行介绍并使用ANSYS Workbench进行仿真计算.根据计算结果对其误差进行分析并优化,以提高该仪器测量精度.经过仿真分析,改变铰链厚度可以减小其测量误差,为测量仪今后的误差分析提供了一定的理论基础.

弹簧柔性运动仿真分析方法研究

汽车悬架弹簧是悬挂系统的重要组成部分,其主要作用为支撑车体和缓冲冲击。由于弹簧的实际运动包络获取较为困难,汽车弹簧运动包络仿真行业普遍运用三维软件绘制弹簧运动过程中的若干个形态,但是此方法未能考虑弹簧的材料特性和受力,故制作的包络准确性不高。针对以上问题,文章提出了一种弹簧的柔性运动仿真分析方法,考虑了弹簧的形态、受力及材料特性,可有效提升弹簧包络制作精度和效率。对下摆臂及相关部件减重降本、性能提升、紧凑布置等方面有重要的意义。

Stability and vibration of a helical rod with circular cross section in a viscous medium

The stability and vibration of a thin elastic helical rod with circular cross section in a viscous medium are discussed. The dynamical equations of the rod in the viscous medium are established in the Frenet coordinates of the centreline with the Euler angles describing the attitudes of the cross section as variables. We have proved that the Lyapunov and Euler conditions of stability of a helical rod in the space domain are the necessary conditions for the asymptotic stability of the rod in the time domain. The free frequencies and damping coefficients of torsional and flexural vibrations of the helical rod in the viscous medium are calculated.

Stability and vibration of a helical rod constrained by a cylinder

The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler＇s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov＇s and Euler＇s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form.