Singularity and kinematic bifurcation analysis of pin-bar mechanisms using analogous stiffness method

Based on the analogy between bifurcation of equilibrium paths in structures and kinematic bifurcation of mechanisms, this paper proposes an analogous stiffness method to detect the singularity and kinematic bifurcation of mechanisms. The analogous stiffness in mechanisms is first defined as the derivative of the state variable with respect to the controlling variable. By investigating the value of analogous stiffness, the singularity can be classified into output singularity, input singularity and architectural singularity. And the kinematic characteristics of free joints at corresponding singularity configurations are expounded. The singular and kinematic bifurcation points of mechanisms can then be determined by solving analogous stiffness equations and compatibility equations simultaneously. Following that, the analytical criterion for finite motion of corresponding free joints at singularity configurations is derived from the second-order analysis of compatibility equations. The efficiency of the proposed method is finally illustrated by three typical examples.     

A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry

Stability conditions are the key to transform kinematically indeterminate structures into prestressed structures or deployable structures. From the viewpoint of symmetry, a necessary condition is presented for the stability of symmetric pin-jointed structures with kinematic indeterminacy. The condition is derived from the positive definiteness of the quadratic form of the tangent stiffness matrix. Numerical examples verify that the proposed necessary stability condition is in accord with the conventional theory of structural rigidity, and is considered to be more comprehensible. It is robust and easy to implement. Results show that a symmetric prestressed structure is guaranteed to possess integral prestress modes, if the necessary condition is satisfied. Further, a pin-jointed structure with fully symmetric mechanism modes is proved to be unstable, if it does not satisfy the condition.     

Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios

Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal “snaking” bifurcation curves, which are connected by an infinite number of “rung” segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above.     

一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.     

Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et?al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S ₄-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.     

Post-buckling behavior of a fixed arch for variable geometry structures

The behavior of a bistable strut for variable geometry structures was investigated in this paper. A fixed shallow arch subjected to a central concentrated load was used to study the equilibrium path of the bistable strut. Based on a nonlinear strain–displacement relationship, the critical loads for both the symmetric snap-through and asymmetric bifurcation buckling modes were obtained. Moreover, the principal of virtual work was also used to establish the post-buckling differential equilibrium equations of the arch in the horizontal and vertical directions. Therefore, the whole mechanical behavior before and after the buckling of fixed arches is investigated.     

Tensile stability of cylindrical membranes

The tensile stability of rotationally symmetric thin membranes composed of isotropic, incompressible and elastic materials is considered by investigating under what conditions the initial equilibrium configuration can bifurcate to another rotationally symmetric equilibrium mode.The general equilibrium equations of a rotationally symmetric membrane are first derived in cylindrical coordinates. The initially cylindrical membrane is studied. The classic solution, which assumes homogeneous deformations, is shown to be a special case of the general equations. Perturbation theory is employed to find the bifurcation points from the homogeneous mode.This study shows that, for the chosen boundary conditions, no rotationally symmetric equilibrium mode exists near the cylindrical mode except the cylindrical mode itself. This corresponds to all experimental data that the author is aware of. The initially cylindrical membrane either remains cylindrical or goes into a non-rotationally symmetric mode.     

含刚性体可展开结构的展开过程动力学分析方法

在可展结构中设置刚性体可以有效提高展开效率,保证形状精度,因此含刚性体可展结构具有较高的实用价值。刚性体按需要可为刚性曲板等,一般形状复杂且形式多样,含任意形状刚性体可展结构的展开过程动力学分析比较困难。本文以笛卡尔坐标系下节点坐标及位移为变量,利用广义逆矩阵建立了一种通用的刚性体动力学方程,给出了利用Newmark—...     

Non-linear analysis of unbraced frames of variable geometry

A non-linear analysis of unbraced, rigid-jointed, and elastically restrained (against rotation) portal frames of variable geometry is presented. The loading system consists of eccentric concentrated loads and a transverse uniformly distributed load. Among the important conclusions one may list the following: (a) symmetric portal frames, loaded symmetrically, buckle through a stable bifurcation (sway-buckling) from a symmetric bent equilibrium configuration, and (b) nonsymmetric portal frames do not buckle; their behavior is similar to that of imperfect cantilever columns.     

An eigenanalysis-based bifurcation indicator proposed in the framework of a reduced-order modeling technique for non-linear structural analysis

The Koiter–Newton method is a reduced order modeling technique which allows us to trace efficiently the entire equilibrium path of a non-linear structural analysis. In the framework of buckling the method is capable to handle snap-back and snap-through phenomena but may fail to predict reliably bifurcation branches along the equilibrium path. In this contribution we extend the original Koiter–Newton approach with a reliable and accurate bifurcation indicator which is based on an eigenanalysis of the reduced order tangent stiffness matrix. The proposed indicator has a negligible numerical effort since all computations refer to the reduced order model which is typically of very small dimension. The extension allows the identification of bifurcation points and a tracing of corresponding bifurcation branches in each sector of the equilibrium path. The performance of the method in terms of reliability, accuracy and computational effort is demonstrated with several examples.     

Homogeneous Equilibrium Configurations of a Hyperelastic Compressible Cube under Equitriaxial Dead-Load Tractions

Non-uniqueness, bifurcation and stability of homogeneous solutions to the equilibrium problem of a hyperelastic cube subject to equitriaxial dead-load tractions are investigated. Besides the basic and theoretical questions raised by the analysis, the study is motivated by the somewhat surprising feature of this nonlinear problem for which the symmetric load may give rise to asymmetric stable deformations. In reality, the equilibrium problem, formulated for general homogeneous compressible isotropic materials with polyconvex energy function, may exhibit primary and secondary bifurcations. A primary bifurcation occurs when there exist paths of equilibrium states that bifurcate from the primary path of three equal principal stretches. These bifurcation branches have two coinciding stretches and along them, through secondary bifurcations, other completely asymmetric bifurcation branches, which are characterized by all three stretches different, may risen. In this case, the cube transforms into an oblique parallelepiped. With increasing loads, they are also possible discontinuous paths of equilibria which evince prompt jumps in the deformation process. Of course, the set of asymmetric solutions admitted by the equilibrium problem depends on the specific form of the stored energy function adopted. In this paper, expressions governing the global development of asymmetric equilibrium branches are derived. In particular, conditions to have bifurcation points are individualized. For compressible neo-Hookean and Mooney-Rivlin materials a wide parametric analysis is carried out showing by means of graphs the most interesting branches. Finally, using the energy criterion, a detailed study is performed to assess the stability of the computed solutions.      

三浦折纸超材料结构数字化设计与模型验证

折纸结构在航空航天、柔性电子、汽车船舶和建筑结构等领域具有较好的应用前景. 三浦折纸单元沿三向拓展可构建出三浦折纸超材料结构, 具有高孔隙、可自锁、平面折展、负泊松比、形态可控等特性. 为了便于生成折纸超材料结构的复杂三维模型、推广应用于缓冲吸能结构及可展结构, 本文利用Matlab和Grasshopper软件, 发展了三浦折纸超材料结构的数字化设计方法, 利用数字化建模及3D打印技术, 实现了零厚度及非零厚度三维折纸模型的统一建模, 并开展了物理模型验证分析, 探讨了3D打印制作折纸超材料结构模型的优缺点; 推导了三浦折纸超材料的折痕长度、相对密度、折叠率等特性与几何参数的关系, 利用Abaqus/Explicit软件开展了结构准静态压缩过程分析与验证, 揭示相对密度对结构吸能指标的影响规律. 研究结果表明, 折纸超材料结构数字化设计方法高效、准确, 便于结构选型及优化分析, 所得三维模型结果与理论值吻合较好. 当胞元面板构型、面板厚度及结构折痕总长不变时, 相对密度较小的三浦折纸超材料结构具备更为优异的吸能效率.      

Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure

Based on the assumption of finite deformation, the Hamilton variational principle is extended to a nonlinear elastic Euler-type beam-column structure located on a nonlinear elastic foundation. The corresponding three-dimensional (3D) mathematical model for anaiyzing the nonlinear mechanical behaviors of structures is established, in which the effects of the rotation inertia and the nonlinearity of material and geometry are considered. As an application, the nonlinear stability and the post-buckling for a linear elastic beam with the equal cross-section located on an elastic foundation are analyzed.One end of the beam is fully fixed, and the other end is partially fixed and subjected to an axial force. A new numerical technique is proposed to calculate the trivial solution,bifurcation points, and bifurcation solutions by the shooting method and the Newton-Raphson iterative method. The first and second bifurcation points and the corresponding bifurcation solutions are calculated successfully. The effects of the foundation resistances and the inertia moments on the bifurcation points are considered.     

Stability of Geometrically Imperfect General-View Shells

An efficient numerical algorithm is created on the basis of the method of curvilinear meshes, the method of basis reduction, and the asymptotic Koiter theory. It enables us to study load curves to reveal singular (limit and bifurcation) points and to analyze their behavior at branch points. Such an approach allows analyzing the nonlinear deformation of shell structures of arbitrary form and their sensitivity to imperfections. The technique is illustrated by an example of a conic panel with imperfections in the shape     

Imperfection sensitivity analysis of hill-top branching with many symmetric bifurcation points

Imperfection sensitivity properties are derived for finite dimensional elastic conservative systems exhibiting hill-top branching at which arbitrary many bifurcation points coincide with a limit point. The critical load at a hill-top branching point is demonstrated to be insensitive to initial imperfections when all the bifurcation points are individually symmetric. Therefore, it is not dangerous to design a frame or truss so that many members buckle simultaneously at the limit point, although the notion of the danger of optimization by compound bifurcation is widespread.     

Imperfection sensitivity of optimal symmetric braced frames against buckling

Imperfection sensitivity characteristics of the non-linear buckling load factors of non-optimal and optimal symmetric frames are investigated. The frames are subjected to symmetric proportional vertical loads. The optimization problem is formulated under constraints on linear buckling load factors. Although the buckling load factors corresponding to sway and non-sway modes coincide at the optimum design, the non-sway-type asymmetric bifurcation point disappears as a result of geometrically non-linear analysis. Therefore, the maximum allowable load factors of perfect and imperfect systems should be determined by assigning displacement constraints. It is shown that quantitative evaluation is possible for imperfection sensitivity and mode interaction based on the higher order differential coefficients of the total potential energy even for frames of which the critical points should be numerically obtained. Numerical examples are presented to show that the properties of the non-sway bifurcation point are similar to those of a symmetric bifurcation point, and the interaction between sway and non-sway modes does not always lead to enhancement of imperfection sensitivity.     

面向压电智能结构精确变形的协同优化设计方法

智能结构集智能材料与传统材料于一体,能够实现结构的主动控制,在航空航天等领域具有巨大的应用潜力.由于其系统复杂且具有多场耦合效应,智能结构的整体式优化设计方法成为结构控制技术研究的关键之一.为了提高压电智能结构的整体性能和变形精度,提出了同时考虑压电驱动器布局(分布位置及角度)和基体结构拓扑构型的协同优化设计新方法.采用多点约束方法 (multi-point constraints,MPC)建立压电驱动器和基体结构的连接,定义一种与测量点目标位移相关的权重函数,以实现结构的精确变形控制.通过协同优化设计,压电驱动器可以获得最优的分布位置及角度,同时基体结构获得最优的拓扑构型,从而提升了压电智能结构系统的整体驱动性能和变形精度.通过进一步分析,研究了精确变形、体分比约束与结构优化构型和整体刚度的关系,以及优化结果中可能存在的传力路径畸变现象.数值算例的设计结果表明,采用协同优化设计方法,能够扩大结构的寻优空间,有效减小变形误差,实现压电智能结构的精确变形控制.     

Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies

We study the dynamics of a massless particle in an annular configuration of N bodies, N − 1 of which have equal masses m and are located in equal distances on a fictitious circle and one has mass βm and is located at the center of the circle. Our interest is focused on the bifurcation points from planar to three-dimensional families of symmetric periodic orbits in the above problem. We study numerically the evolution of these bifurcation points with respect to the variation of the mass parameter β. In particular we investigate the continuous evolution of bifurcation points for values of β from 2 up to 1000. The two distinct cases of the system’s behavior at β = 2 and 1000 are examined comparatively and various conclusions are drawn regarding the overall dynamical evolution of the three-dimensional system as the relative mass of the central body grows.     

Post-bifurcation of perfect and imperfect spherical elastic membranes

When a spherical elastic membrane is inflated it is well known that it may bifurcate into an aspherical mode after the pressure maximum is reached. Upon further inflation the spherical configuration is regained. Here we follow the developing aspherical solution path, for specific forms of strain-energy function, using a simple numerical method. For a realistic strain-energy function it is shown that the post-bifurcation solution curve connects the two bifurcation points. We also consider the inflation of imperfect spherical membranes and show that bifurcation may still occur. For the class of Ogden materials we investigate the asymptotic shape of arbitrary axisymmetric membranes.     

一类二维非自治离散系统中的复杂簇发振荡结构1）

簇发振荡是多时间尺度系统复杂动力学行为的典型代表,簇发振荡的动力学机制与分类问题是簇发研究的重要问题之一,但当前学者们所揭示的簇发振荡的结构大多较为简单.研究以非自治离散Duffing系统为例,探讨具有复杂分岔结构的新型簇发振荡模式,并将其分为两大类,一类经由Fold分岔所诱发的对称式簇发,另一类经由延迟倍周期分岔所诱发的非对称式簇发.快子系统的分岔表现为典型的含有两个Fold分岔点的S形不动点曲线,其上、下稳定支可经由倍周期（即Flip）分岔通向混沌.当非自治项（即慢变量）穿越Fold分岔点时,系统的轨线可以向上、下稳定支的各种吸引子（例如,周期轨道和混沌）进行转迁,因此得到了经由Fold分岔所诱发的各种对称式簇发;而当非自治项无法穿越Fold分岔点,但可以穿越Flip分岔点时,系统产生了延迟Flip分岔现象.基于此,得到了经由延迟Flip分岔所诱发的各种非对称簇发.特别地,文中所报道的簇发振荡模式展现出复杂的反向Flip分岔结构.研究结果表明,这与非自治项缓慢地反向穿越快子系统的Flip分岔点有关.研究结果丰富了离散系统簇发的动力学机理和分类.