超细长弹性杆动力学仿真的曲面处理算法

超细长弹性杆动力学研究在DNA的平衡、稳定性等问题的研究中有重要的应用。为了便于超细长弹性杆动力学研究中数值结果图形后处理以及研究表面接触等问题的需要,需要建立弹性杆的表面模型和相应算法。本文利用Kirchhoff弹性杆模型的动力学比拟技巧,建立了描述超细长弹性杆曲面的常微分/积分方程组,利用Adames方法和递推方法设计了方程的数值解法,并给出了超长弹性杆的数值仿真结果的图形处理的计算实例。     

A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations

The paper is concerned with a hybrid finite element formulation for the geometrically exact dynamics of rods with applications to chaotic motion. The rod theory is developed for in-plane motions using the direct approach where the rod is treated as a one-dimensional Cosserat line. Shear deformation is included in the formulation. Within the elements, a linear distribution of the kinematical fields is combined with a constant distribution of the normal and shear forces. For time integration, the mid-point rule is employed. Various numerical examples of chaotic motion of straight and initially curved rods are presented proving the powerfulness and applicability of the finite element formulation.     

Looping mechanics of rods and DNA with non-homogeneous and discontinuous stiffness

DNA molecules may form loops in response to binding with regulatory proteins that control the expression of genes. While DNA looping is a widely accepted gene regulatory mechanism, basic questions regarding the mechanics of the looping process remain open. The present paper contributes a computational rod model that accounts for non-homogeneous and discontinuous changes in stiffness to support the analysis of DNA looping. We pursue this objective in two steps. First, we illustrate the effects of non-homogeneous stiffness on the looping of generic rods under pure torsion. Results computed for this idealized case support our intuition that elastic deformation and strain energy localize in ‘soft’ regions and that equilibrium bifurcations are sensitive to non-homogenous stiffness. Next, we extend the formulation to describe the combined bending, torsion and compression induced on DNA by the LacR protein. We demonstrate that while moderate stiffness variations have only modest influence on LacR-mediated DNA looping, highly localized softening (e.g., ‘kinkable’ or ‘melted’ subdomains) may substantially reduce the energetic cost of looping and profoundly affect loop geometry.     

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.     

On the geometric conditions for multiple stable equilibria in clamped arches

Curved structures, such as beams, arches, and panels are capable of exhibiting snap-through buckling behavior when loaded laterally, that is they can exhibit multiple stable equilibria, sometimes after any external loading is removed. This is a consequence of highly nonlinear force-deflection relations with perhaps multiple crossings of the zero-force axis for typical equilibrium paths. However, the propensity to maintain a stable snapped-through equilibrium position (in addition to the nominally unloaded equilibrium configuration) after the load is removed depends on certain geometric properties. A number of clamped arches are used to illustrate the relation between geometry (essentially the shape) and corresponding equilibrium configuration(s), and especially those conditions for which the initial equilibrium configuration is the only stable shape possible. Furthermore, related results are obtained when a change in the thermal environment may cause a system to exhibit a stable snapped-through equilibrium even when the system at ambient thermal conditions does not. Some representative examples are produced using a 3D printer for verification purposes.     

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.     

Multivalued solutions of the spatial problems of nonlinear deformation of thin curvilinear rods

We have developed a finite-element model to study the spatial deformation of elastic rods at large displacements. A numerical algorithm for constructing multivalued nonlinear solutions in the presence of many bifurcation and limit points is formulated. Results of a study of the stability supercritical equilibrium forms of elastic rods, which were supported experimentally, are reported. Chaplygin Siberian Aviation Institute, Novosibirsk 630051. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 141–149, March–April, 1998.     

On the multi-mode behavior of vibrating rods attached to nonlinear springs

Nonlinear Dynamics - This paper investigates the harmonic response of vibrating rods with an array of nonlinear springs. The proposed analysis is multi-mode in the sense that the response functions...     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

Analysis of nonlinear solutions with many singular points in problems of spatial deformation of rods

We consider an algorithm for obtaining numerical solutions of the geometrically nonlinear problems of spatial deformation of elastic rods in the presence of many singular points. The questions of the construction of bifurcation solutions and the stability of the found states of equilibrium are discussed. The results of a study of the nonlinear deformation and stability of a ring in the spatial formulation, which are supported by experimental data, are given. Chaplygin Siberian Aviation Institute, Novosibirsk 630051. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 148–153, November–December, 1998.     

Compensation of torsion in rods by piezoelectric actuation

This paper considers the compensation of torsional deformations in rods with the help of thin integrated piezoelectric actuator layers. A laminated orthotropic rod is considered, for which the material properties of each layer are assumed to be homogenous. For the sake of a generalization, the piezoelectric actuation is expressed in terms of eigenstrains. The main scope is the derivation of a distribution of eigenstrains that is able to completely compensate the angle of twist caused by external torsional moments. Saint Venant’s theory of torsion for laminated orthotropic rods is extended for the presence of eigenstrains, which is performed by introducing an additional warping function. It is shown that the actuating torsional moment is a function of the eigenstrains and the additional warping function. For the example of a rectangular cross section, an analytic solution for the actuating moment and the additional warping function is presented. The results are verified by three-dimensional finite-element computations showing a very good accordance with the theoretical results over a large parameter range.     

On the nonlinear stability of relative equilibria of the full spacecraft dynamics around an asteroid

The full dynamics of a spacecraft around an asteroid, in which the gravitational orbit–attitude coupling is considered, has been shown to be of great value and interest. Nonlinear stability of the relative equilibria of the full dynamics of a rigid spacecraft around a uniformly rotating asteroid is studied with the method of geometric mechanics. The non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are given in the differential geometric method. A classical kind of relative equilibria of the spacecraft is determined from a global point of view, at which the mass center of the spacecraft is on a stationary orbit, and the attitude is constant with respect to the asteroid. The conditions of nonlinear stability of the relative equilibria are obtained with the energy-Casimir method through the semi-positive definiteness of the projected Hessian matrix of the variational Lagrangian. Finally, example asteroids with a wide range of parameters are considered, and the nonlinear stability criterion is calculated. However, it is found that the nonlinear stability condition cannot be satisfied by spacecraft with any mass distribution parameters. The nonlinear stability condition by us is only the sufficient condition, but not the necessary condition, for the nonlinear stability. It means that the energy-Casimir method cannot provide any information about nonlinear stability of the relative equilibria, and more powerful tools, which are the analogues of the Arnold’s theorem in the canonical Hamiltonian system with two degrees of freedom, are needed for a further investigation.     

On the delicate state of instability of a vertical riser transporting fluid

The variation in the dynamic characteristics of a flexible riser as the riser transitions from a vertical riser to a catenary-type riser is investigated. It is well known that the straight configuration of a flexible vertical riser conveying fluid destablizes in a divergence-type instability once the velocity of the transporting fluid exceeds a critical speed. As expected, the instability persists if a slight horizontal offset is introduced at the hang-off point. However, as demonstrated in this paper, if a finite horizontal offset is introduced then the instability vanishes and the resulting static configuration of the catenary-type riser is stable regardless of the transport speed of the fluid.     

Vibration of rods with a concentrated mass in the presence of non-conservative forces

Summary Separable and non-selfadjoint boundary-value problems representing the vibration of linearly elastic unidimensional systems are considered. The elastic system is modelled as a continuous distributed-parameter system where singularities in the mass distribution function can be neatly taken into account. Specifically, extending Green's function approach, the free vibration, stability and forced vibration of fixed-free rods with a tip mass and under the action of uniformly distributed non-conservative loads have been investigated analytically.Support of this research through the NSERC Grant No. OGPIN013 is greatly appreciated     

On Substitution Principles in Ideal Magneto-Gasdynamics by Means of Lie Group Analysis

The equations governing the flow of an inviscid thermally non-conducting fluid of infinite electrical conductivity in the presence of a magnetic field and subject to no extraneous forces are considered within the framework of Lie group analysis. It is shown how to recover and extend some results, known in literature as substitution principles, by conveniently requiring the invariance of the basic governing equations under a one-parameter Lie group of point transformations. Moreover, a new substitution principle for the equations ruling the plane motion of a fluid with adiabatic index Γ = 2 subjected to a transverse magnetic field is given. Some applications of the results are also given.     

On the approximation of acceleration waves in rods

This paper employs an approximate form of analysis based on the assumption of plane stress to find the transport equation and corresponding evolution law governing the intensity of acceleration wave propagation in an elastic rod of slowly varying area of cross-section. The result is then extended to include the case of slightly bent rods. In each of these cases it is shown that for a medium in which the strain energy function Σ(p) is such that d³Σ/dp³ ≠ 0, with p the displacement gradient, the acceleration wave intensity is governed by a Bernoulli equation. The work is concluded by showing that the analysis may also be applied to the case of a composite rod comprising an arbitrary number of homogeneous isotropic plane layers normal to the direction of acceleration wave propagation.     

细长结构几何非线性分析的子结构方法

将细长结构沿长度方向划分为多个子结构,并在每个子结构上建立一个随结构一起运动的连体基,则结构内任意点的位移可分解为连体基的转动和相对于连体基的小位移。利用细长结构这样的变形特征,本文详细讨论了连体基的转动,给出了与连体基选择方式相协调的节点位移及其虚变分表达式,并将子结构内部位移凝聚到了边界节点上。在此基础上,提出了一种细长结构几何非线性分析的子结构方法,可在不损失计算精度的前提下大幅度降低求解规模,从而提高了计算效率。数值算例验证了所提方法的有效性。     

On the analysis of 2D nonlinear gravity waves with a fully nonlinear numerical model

A fully nonlinear numerical method, developed on the basis of Euler equations, is used to study the dynamics of nonlinear gravity waves, mainly in the aspects of the propagation of Stokes wave with disturbed sidebands, the evolution of one wave packet and the interaction of two wave groups. These cases have previously been studied with the higher order spectral method, which will be an approximately fully nonlinear scheme if the order of nonlinearity is not large enough, while the present method in the case of the 2D model has an integration scheme that is exact to the computer precision. As expected, in most cases the results are consistent between these two numerical models and it is confirmed again that this fully nonlinear numerical model is also capable of maintaining a high accuracy and good convergence, particularly in the long-term evolutionary process.