受圆柱面约束弹性杆的平衡与稳定性

讨论受圆柱面约束的圆截面弹性杆的平衡与稳定性。以描述截面姿态的欧拉角为变量,建立受约束弹性杆的平衡方程。利用方程的初积分导出约束力、截面内力及挠性线的解析表达式。作为特殊的平衡状态,讨论杆的螺旋线平衡的存在条件。用相平面法分析螺旋线平衡的稳定性,导出解析形式的稳定性条件。     

Stability of equilibrium states in a simple system with unilateral contact and Coulomb friction

The aim of this paper is to study the stability of equilibrium states in a mechanical system involving unilateral contact with Coulomb friction. Since the assumptions made in classical stability theorems are not satisfied with this class of systems, we return to the basic definitions of stability by studying the time evolution of the distance between a given equilibrium and the solution of a Cauchy problem where the initial conditions are in a neighborhood of the equilibrium. It was recently established that the dynamics is well posed in the case of analytical data. In the present study, we focus in particular on the stability of the equilibrium states under a constant force and deal only with a simple mass-spring system in .     

Characteristic stability parameter of the axisymmetric equilibrium surface of a capillary liquid

In [1] the question of stability of the equilibrium state of a capillary liquid in weak force fields was reduced to determination of conditions such that the smallest eigenvalue λ* of a certain boundary problem would be positive. In [2] it was shown that λ* is a monotonic function of the parameter χ, dependent on the form of the vessel. The basic properties of the function λ*(χ) were also described. In the present study, these properties are used to study the general problem of stability of an axisymmetric liquid surface. A method for calculation of the critical values of the parameter χ and construction of the maximum stability region is given. Special attention is given to the cases of complete weightlessness, and action of gravitational and centrifugal forces. Critical values of the parameter χ are presented for these cases either graphically or analytically, which, given the shape of the vessel, permits evaluation of the stability of any of the family of axisymmetric equilibrium surfaces. We note that in the case of action by gravitational forces χ values for certain equilibrium surfaces were obtained by Barnyak.     

边坡三维极限平衡法的通用形式

基于以下假定条件:(1) 稳定系数定义为材料的强度折减系数;(2) 土体为刚体,底滑面服从Mohr-Columb强度破坏准则;(3) 微条柱底部法向力dNz的作用点处于条柱底部中点;(4)滑面剪力与底滑面和xoz平面交线的夹角为θ。本文建立了边坡三维极限平衡法的通用形式,通过给定不同的限制条件,可分别得到三维普通条分法 、三维简化毕肖普法 、三维简化简布法 、三维Spencer法 等三维极限平衡的具体算法。     

States of Equilibrium and Secondary Loss of Stability of a Straight Rod Loaded by an Axial Force

A general analytical solution of the problem of postcritical deformation of a straight incompressible rod loaded by an axial force is given. The bending of the rod is studied under various boundary conditions, and new states of equilibrium the occurrence of which is due to the secondary loss of stability are found. It is shown that, for simply supported and clamped rods, the solution bifurcates when the ends meet.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

Hydrstatics in weak force fields

Equilibrium forms of a liquid surface in weak gravitation fields have been studied in [1], As noted in [1], not all the equilibrium forms may be realized in practice, since they are not always stable. Below we consider the problem of the stability of the equilibrium state of an ideal, incompressible liquid under the influence of surface-tension forces and a potential mass force field. In solving this problem we use the principle of minimal system potential energy. The stability condition is formulated in terms of the eigenvalues of the linear boundary-value problem which arises in considering the question of the potential energy minimum. This general condition is applied to the axisymmetric problem, and, in particular, to the problem of the stability of a liquid suspended in a cylindrical vessel.In conclusion, the author wishes to thank M. A. Belyaeva for carrying out the calculations and N. D. Kopachevskii and A. D. Myshkis for their interest in the study and helpful suggestions.     

Dynamic stability of shallow arch with elastic supports—application in the dynamic stability analysis of inner winding of transformer during short circuit

In this paper, the dynamic stability of a shallow arch with elastic supports subjected to impulsive load is used as a theoretical model to investigate the dynamic stability problem of inner windings of power transformer under short-circuit condition. Firstly, the series solution representing the equilibrium configurations of a shallow arch is obtained by solving the corresponding non-linear integration-differential equation. The local stability of each equilibrium configuration is discussed, and the sufficient condition for stability of the shallow arch system as well as the critical load against snap-through is obtained. Secondly, the equivalent relation between short-circuit load and impulsive one, and the electrical forces transferred pattern between the coils of inner windings are assumed. Then the results of the shallow arch model are applied to the case of the inner winding of transformer and the formulas for computing critical electromagnetic force and the dynamic stability criterion of the inner windings are established. Finally, examples are offered and the theoretical results are shown to agree well with the experimental ones.     

Secondary loss of stability of an euler rod

The solution of the classical problem of the postcritical behavior of a compressed, simply supported rod is considered. The stability of the well-known elastic solution that emanates from the first critical point is analyzed and new branches of the states of equilibrium are found. The value of the force that corresponds to the secondary loss of stability is determined. Chaplygin Siberian Aviation Institute, Novosibirsk 630054. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 184–185, November–December, 1999.     

Effect of interface stresses on the image force and stability of an edge dislocation inside a nanoscale cylindrical inclusion

Dislocation mobility and stability in inclusions can affect the mechanical behaviors of the composites. In this paper, the problem of an edge dislocation located within a nanoscale cylindrical inclusion incorporating interface stress is first considered. The explicit expression for the image force acting on the edge dislocation is obtained by means of a complex variable method. The influence of the interface effects and the size of the inclusion on the image force is evaluated. The results indicate that the impact of interface stress on the image force and the equilibrium positions of the edge dislocation inside the inclusion becomes remarkable when the radius of the inclusion is reduced to nanometer scale. The force acting on the edge dislocation produced by the interface stress will increase with the decrease of the radius of the inclusion and depends on the inclusion size which differs from the classical solution. The stability of the dislocation inside a nanoscale inclusion is also analyzed. The condition of the dislocation stability and the critical radius of the inclusion are revised for considering interface stresses.     

受均布载荷压杆后屈曲形态的数值计算

对受均布载荷压杆的屈曲及后屈曲行为进行了分析.基于杆的大变形理论,考虑杆的轴向伸长,建立了受均布载荷作用下细长压杆的几何非线性平衡方程.采用打靶法和解析延拓法数值求解非线性两点边值问题,得到了杆的后屈曲平衡路径和平衡构形.     

弹性支持扁拱动力稳定性分析和变压器内线圈短路动稳定分析

本文采取脉冲载荷激励下弹支扁拱弹性动力稳定性的理论模型分析电力变压器内线圈短路动稳定问题.首先导出了脉冲载荷激励下弹支扁拱非线性运动积分微分方程级数解,给出了各平衡位置(奇点)的局部稳定(不稳定)性证明,导出弹支扁拱抗跳跃失稳的稳定性充分条件、判别公式和临界载荷;其次导出了短路载荷与脉冲载荷的相当关系,考虑内线圈各线匝(铜条)短路电动力到内线匝的传递,给出了内线圈短路动稳定临界电动力计算公式和动稳定判别方法;最后作出与实验结果的比较和算例.     

Sufficient conditions for stability of Euler elasticas

Based on the conjugate point theory in calculus of variations, sufficient conditions on stability of all Euler elasticas for a column clamped at one end and guided at the other are proposed in this paper. For the Euler elasticas, an initial value problem is solved numerically by the Euler iteration. Sufficient conditions are satisfied that Euler elasticas with one half wave are stable and the others with more than one half wave are unstable in post-buckling. As load is smaller than the Euler critical force, it is sufficient theoretically that straight shape is stable. As load exceeds the Euler critical force, it is sufficient theoretically that straight shape is unstable.     

Post-divergence and post-flutter analysis of sub-tangentially loaded and damped Beck column using a nonlinear FE procedure

Both post-divergence and post-flutter behaviors of damped Beck columns subjected to a sub-tangentially follower force are rigorously explored using an exact co-rotational frame element. First a linear stability theory of the damped Beck column is summarized using a stability map. A geometrically nonlinear frame element based on the co-rotational formulation is then formulated including mass matrix, Rayleigh damping matrix, and load-correction stiffness matrix due to circulatory forces. The dynamic FE analysis is performed using Newmark integration method. Finally a Beck column model is parametrically analyzed in order to investigate non-linear stability characteristics of the internally and externally damped non-conservative system. In particular, interesting nonlinear behaviors of Beck column quite different from those predicted by the linear theory are reported through static and dynamic nonlinear analysis with variation of sub-tangentiality of the follower force.     

Optimal shape of a twisted and compressed rod

We formulate and solve the problem of determining the shape of an elastic rod stable against buckling and having minimal volume. The rod is loaded by a concentrated force and a couple at its ends. The equilibrium equations are reduced to a single nonlinear second-order equation. The eigenvalues of the linearized version of this equation determine the stability boundary. By using Pontryagin's maximum principle we determine the optimal shape of the rod.     

On the stability of an elastic column positioned on an elastic half space

Summary Stability of a heavy elastic column loaded by a concentrated force at the top is analysed. It is assumed that the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full non-linear system of equilibrium equations. Also, the type of bifurcation at the lowest eigenvalue is examined and shown that it could be both super-and sub-critical. The post-critical shape of the column is determined by numerical integration of the equilibrium equations. Received 13 June 1998; accepted for publication 12 November 1998     

Equilibrium and stability of a z pinch in a multipole magnetic field

In the theoretical investigation of the dynamic stabilization of a current-carrying plasma filament by a high-frequency multipole magnetic field it is usually assumed that the cross section of the filament has a circular form in equilibrium [1, 2]. This considerably simplifies the calculations but it does not correspond to reality, since the surface of the plasma must be fluted in the multipole field. An attempt to estimate the influence of the deformation of the filament cross section on its stability against bending in the special case of quadrupole field was made in [3], in which the parameters were determined of the elliptical cross section corresponding to a plasma filament with current in a quadrupole field and an expression was found for the electrodynamic force acting on the filament in the case of long-wavelength kink perturbations. However, this force was calculated incorrectly in [3]. In the present paper a study is made of the equilibrium and stability of a current-carrying plasma filament against kink perturbations in the general case of a multipole stabilizing field. Under the assumption that the flute depth is small, the equilibrium form of the cross section of the current-carrying plasma filament in the multipole magnetic field is found and the components of the force exerted by the field on the perturbed filament are calculated. It is shown that the external field interacts with the current in the perturbed filament only in the case of a quadrupole field. The results are discussed in connection with the problem of multipole dynamical stabilization of a z pinch against kink perturbations.     

Buckling instabilities in coupled nano-layers

We study the dynamic buckling of a pair of dissimilar Euler-Bernoulli beams subject to compressive edge loading whose transverse displacements are coupled through non-linear interactions, a problem motivated by the mechanics of graphene layers. The transverse coupling models van der Waals interaction and is derived from a Lennard-Jones 12-6 potential. The beams are assumed to be a fixed distance apart at their ends, although this distance is not necessarily equal to the equilibrium distance as identified from the Lennard-Jones potential. Therefore, the equilibrium configuration is not necessarily straight. Via a Galerkin method, the governing equations are reduced to a system that can be used to calculate equilibrium configurations as well as the stability of these configurations. We show that the buckling instability in this model is significantly affected by the presence of the interaction force as well as the separation of the graphene layers at the boundaries.     

Stability for manifolds of equilibrium state of generalized Hamiltonian system with additional terms

For a generalized Hamiltonian system with additional terms, stability for the manifolds of the equilibrium state is presented. Equilibrium equations, disturbance equations and the first approximate equations of the system are given. A theorem for the stability of the manifolds of the equilibrium state of a general autonomous system is used for the generalized Hamiltonian systems with additional terms, and three propositions on the stability of the manifolds of the equilibrium state of the system are obtained. An example is given to illustrate the application of the method and results. At last, we study the stability for manifolds of the equilibrium state of the Euler equations of a rigid body subjected to external moments of force, by using of the method in this paper.     

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.