基于动力松弛法的松弛索杆体系找形分析

索杆张力结构施工成形分析需要解决一个松弛态索杆系统的平衡形态求解问题,该问题可归结为一个给定构件原长的受荷索杆机构系统的找形问题。文中利用动力松弛法进行该类松弛索杆体系的找形分析。由于不建立刚度矩阵,避免了体系几何不稳定性所引起的刚度矩阵奇异问题。该方法采用悬链线索单元模型,可以考虑索大垂度效应。文中还推导了反映索原长和内力之间关系的悬链线索单元协调方程。通过一个正交松弛索网算例分析,考察了该找形方法的计算精度和收敛性。最后还模拟了一个索穹顶的施工成形过程,表明了该找形方法用于索杆张力结构施工成形分析的有效性。     

Numerical form-finding of tensegrity structures

A novel and versatile numerical form-finding procedure that requires only a minimal knowledge of the structure is presented. The procedure only needs the type of each member, i.e. either compression or tension, and the connectivity of the nodes to be known. Both equilibrium geometry and force densities are iteratively calculated. A condition of a maximal rank of the force density matrix and minimal member length, were included in the form-finding procedure to guide the search of a state of self-stress with minimal elastic potential energy. It is indeed able to calculate novel configurations, with no assumptions on cable lengths or cable-to-strut ratios. Moreover, the proposed approach compares favourably with all the leading techniques in the field. This is clearly exemplified through a series of examples.     

Analysis of clustered tensegrity structures using a modified dynamic relaxation algorithm

Tensegrities are spatial, reticulated and lightweight structures that are increasingly investigated as structural solutions for active and deployable structures. Tensegrity systems are composed only of axially loaded elements and this provides opportunities for actuation and deployment through changing element lengths. In cable-based actuation strategies, the deficiency of having to control too many cable elements can be overcome by connecting several cables. However, clustering active cables significantly changes the mechanics of classical tensegrity structures. Challenges emerge for structural analysis, control and actuation. In this paper, a modified dynamic relaxation (DR) algorithm is presented for static analysis and form-finding. The method is extended to accommodate clustered tensegrity structures. The applicability of the modified DR to this type of structure is demonstrated. Furthermore, the performance of the proposed method is compared with that of a transient stiffness method. Results obtained from two numerical examples show that the values predicted by the DR method are in a good agreement with those generated by the transient stiffness method. Finally it is shown that the DR method scales up to larger structures more efficiently.     

力密度找形分析方法及计算机实现

本文通过对力密度法的分析研究，概括了力密度法的基本原理和找形分析的过程，用矩阵形式给出了力密度的基本列式。通过对各个矩阵基本元素分析，找到了力密度矩阵的组装规律和一维变带宽存储的规则及实现算法和过程，并给出了详细的力密度矩阵的装配过程。通过基于C 语言的系统框架的程序，实现了该基于一维变带宽存储的力密度找形分析方法。最后，通过几个数值算例表明了，力密度法找形理论的先进性，及本文提出的算法过程实现的可行性和正确性。由于本文的算法是基于效率比较高的一维变带宽压缩存储格式，因此，可以实现大型索膜结构的找形分析。     

基于动态松弛法的柔索驱动并联机器人位置正解分析

根据悬链线解析表达式推导出悬索两端固定时索端拉力与索长之间的关系,用于求解特定长度的驱动柔索对处于某一位姿的动平台的作用力.动态松弛法可以从任意的初始状态开始,通过简单迭代而使系统达到静平衡状态.将该方法应用到柔索驱动并联机器人的位置正解分析中,可以得到与特定索长对应的动平台位姿.以舱索系统为例验证了方法的正确性.     

悬链线索单元算法的改进

H．B．Jayaraman在20世纪80年代推导的悬链线索元有限元法计算精度高，特别适用于精度要求比较高的大型索结构。但是，当索原长Lu的取值与悬索两节点之间的直线长度相近时，迭代不易收敛，甚至发散。提出了当该迭代不收敛时，应采用的迭代策略。计算结果表明，该方法准确，计算精度高，可供悬索结构设计、施工时参考。     

Energy coupled–dissipation control for 3-dimensional overhead cranes

Handling loads with small swings is difficult for a 3-dimensional overhead crane due to its hard nonlinearity. Moreover, the nonlinear dynamics increases the complexity of the required feedback, thus making the closed-loop system sensitive to a variation in the cable length that negatively influences the damping feature. To address these problems, a significant storage function characterized by the desired damping is constructed based on passivity. Consequently, a nonlinear controller is delivered by enforcing the coupled–dissipation inequality, thus drastically increasing the damping of the closed-loop system. In particular, new coupled–dissipation signals are fabricated to augment the coupling between the trolley movement and the payload sway. Due to its very simple structure that excludes the cable length, the proposed controller is robust to unknown cable lengths and easy to implement. In the frame of the Lyapunov theory, LaSalle’s invariance principle is applied to illustrate the corresponding stability. The effectiveness of the proposed control on improving the system performance is verified through simulation results.     

索牵引并联机器人静态平衡位置的正解研究

研究了一种大跨度但运动缓慢的索牵引并联机器人的位置正解问题,此时可认为系统处于静平衡状态,但必须考虑柔索垂度的影响。索牵引并联机器人的静态平衡位置的正解就是确定机器人末端执行器的位置和姿态,使得系统能在给定的柔索长度下处于静平衡状态。这涉及到系统的非线性力学分析而不仅仅是一个几何问题。本文采用悬链线方程描述柔索的力学特性,结合索端位移与机器人末端执行器位移之间的关系推导出机器人的末端执行器位移与索长变化之间的增量关系,据此提出一种迭代策略完成正解计算。数值算例验证了本文方法的有效性。     

A procedure for the static analysis of cable structures following elastic catenary theory

The paper proposes an unitary strategy for the static analysis of general cable nets under conservative loads. A form-finding is first performed in order to initialize the successive non linear analysis. The numerical procedures carried on in both steps, form finding and structural analysis of the net, employ a three dimensional elastic catenary element. Equilibrium conditions at internal nodes and kinematic compatibility at the end nodes of each cable are used to derive the global equations of the net. When the pre-stresses are high and the topology of the net is involved, an accurate initializing solution is essential for the convergence of the successive numeric non linear structural analysis (performed by Newton method). The numerical applications highlight the capability of the proposed procedure to solve three dimensional problems with taut and slack cables, out of plane distributed forces (modeling wind loads), point loads along the cables. The contemporary presence of cables and compression truss elements is also considered testing the effectiveness of the method in the analysis of tensegrity structures.     

Form-finding of a new kind of tensegrity tori using overlapping modules

In the past decades, the form-finding of tensegrity structures of regular geometric shapes, such as cylindrical tensegrities, polyhedral tensegrities, spherical tensegrities and so on, has been systematically studied. However, seldom studies on the form-finding of tensegrity tori have been reported. Considering the potential applications of the tensegrity tori in a number of fields, including architecture, sculpture, and other relevant fields, this paper carries out an exploration on a new kind of tensegrity tori. The topology of the new kind of tensegrity tori is based on the well-known cylindrical tensegrities and overlapping between every two adjacent tensegrity modules is allowed. Incorporating the singular value decomposition of equilibrium matrix with a force-finding algorithm, a general procedure for determining the feasible configurations for the new kind of tensegrity tori is proposed. Parametric analyses on several typical forms of the tensegrity tori are conducted and the feasible ranges of the design parameters and applicability of the feasible configurations are discussed.     

A computational framework for the form-finding and design of tensegrity structures

A new computational framework is proposed for the form-finding and design of tensegrity structures with or without super-stability. The form-finding of tensegrities is formulated as two unconstrained minimisation problems where their objective functions are defined based on eigenvalues of a modified force density matrix. The Nelder–Mead simplex method is then used to solve the minimisation problems. Furthermore, another efficient method is suggested for the interactive form-finding and design of tensegrities with geometrical and force constraints. Examples of the form-finding of tensegrities are presented and the results obtained are compared and contrasted with those analytical results documented in the literature, to verify the accuracy and efficiency of the developed methods.     

Three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot with non-negligible cable mass and elasticity

This study investigates the three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot by considering cable mass, elasticity, and mass of end-effector. According to these models, optimization of cable tensions and cable lengths using fminimax solver is performed to determine the static stiffness. Static and dynamic stiffness of the cables are obtained with simulations. Results show that analyses in three-dimension are very important to measure the actual performance of the cable driven parallel robot. This demonstrates potential for general applicability and motion of the cable driven parallel robot.     

Real solutions of the direct geometrico-static problem of under-constrained cable-driven parallel robots with 3 cables: a numerical investigation

This paper addresses the direct geometrico-static problem of under-constrained cable-driven parallel robots with 3 cables. The task at hand consists in finding all equilibrium configurations of the end-effector when the cable lengths are assigned. This problem is known to admit 156 solutions in the complex field, but the upper bound on the number of real solutions is as yet an open issue. Finding this bound is the objective of the paper. For this purpose, three numerical approaches are developed, namely a continuation procedure adapted from an algorithm originally proposed by Dietmaier and two evolutionary techniques based on a genetic algorithm and particle swarm optimization. In all cases, a number of sets of robot parameters for which the direct geometrico-static problem provides at the most 54 real configurations is found. The coherence of the obtained results leads to conjecture that the achieved bound is tight. However, formal proof is yet to be discovered.     

Unwinding characteristics of thin cables for inner and outer dispensers

When thin cables of materials such as yarn and tether are unwound from a spool dispenser, various balloon shapes are generated depending on the initial tensile force at the guide-eyelet point. The shapes are dependent on the properties of the cable because the point of unwinding depends on the thickness of the cable. In this paper, the unwinding characteristics are analyzed for inner and outer dispensers by using three types of thin cable. The dimensionless steady-state equation of motion is first derived from Hamilton’s principle for an open system and the perturbation scheme. The second-order differential equation is then solved by means of shooting method because the boundary conditions at the lift-off point are not fully sufficient. Several parameters such as cable diameter and initial tensile force are checked to study their effects on the balloon shapes. For a given material, the larger is the diameter of the cable, the higher is the air-drag coefficient on the cable, and the effect of the centrifugal force is weaker than that of the fluid resistance. Moreover, the maximum radius of the balloon and the total length of the control volume are small when the cable is thick.     

On the aerostatic divergence of suspension bridges: A cable-length-based criterion for the stiffness degradation

Motions of a suspension bridge immersed in turbulent wind flows, including both mean and stochastic, can under certain conditions lead to the degradation of the structure׳s torsional stiffness and ensuing aerodynamic torsional divergence (ATD). It is the incompressibility of the main cables that determines their effectiveness as supporting members be limited in a tension state, and dropping out of either one cable from a tension state could result in the vanish of the torsional stiffness contributed by the whole cable system. Although the physical conception is quite explicit, evaluation of the critical threshold of ATD of a suspension bridge, when experiencing stochastic aerodynamic responses, can be complicated by the deformations of the towers, the lateral oscillations of the deck and cables, and other high-order modes participating in the vibration. In view of this, a criterion based on the tracking of the length of the main cables is presented in this study. It is shown that, with the numerical example, the advent of ATD, the degradation of the system׳s stiffness, as well as some striking post-ATD behaviors, can be well explained with the proposed model by tracking the cable lengths.     

Experimental and Analytical Investigation of Carbon Fiber Cable Vibration Damping

For structures deployed in space using cables where vibration damping is critical for structural stability, cable damping is significant to structural performance. To provide a better understanding of damping mechanism of carbon fiber cables, this paper describes the tests of cable damping under different experimental configurations (different cable length, tension and type), and presents an analytical method for modeling and therefore predicting cable damping. The method is developed using simplified but physically realistic assumptions on material constitutive properties and geometric compatibility conditions, and considered the contact forces and friction between helical wires. The results of the proposed method and several related issues are discussed and compared with those from experiments. These results show that the proposed method is useful and applicable for predicting cable damping value and its variation with cable tension, length and type of the cables.     

斜拉索振动控制中MR阻尼器选型的研究

以全索全时段振动响应的均方根(RMS)评价MR阻尼器对斜拉索的减振效果。计算结果表明MR阻尼器型号是影响斜拉索减振效果的最主要因素。斜拉索的减振效果在选用合适的MR阻尼器时达到最佳。进而研究了MR阻尼器型号与阻尼器安装位置、施加的电压、斜拉索基频(张力、索长、质量)、激励荷载(类型、频率、幅值)等各种因素的关系,为MR阻尼器合理选型提供了优化设计的方法。型号选用主要是与斜拉索基频和MR阻尼器安装位置有关。在引起索基频变化的因素中,索质量对型号的选取影响最大;而索长对型号影响不大。对于索质量较大、张力较大、MR阻尼器安装位置较低、外界激励较大、频谱特征多变、低频为主时需要较强的MR阻尼器。进一步研究表明,半主动控制与开环控制的最优MR阻尼器型号有较好的一致性,因此半主动控制所选用的MR阻尼器型号可参照被动控制时最优MR阻尼器型号。     

Algebraic tensegrity form-finding

This paper concerns the form-finding problem for general and symmetric tensegrity structures with shape constraints. A number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form-finding problem. The concept of a tensegrity invariance (similarity) transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations. This result provides the basis for a new tensegrity form-finding tool. The generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives. State-of-the-art numerical algorithms are applied to solve several example problems. Examples are given for tensegrity plates, shell-class symmetric tensegrity structures and structures generated by applying similarity transformation.     

A novel method of determining the sole configuration of tensegrity structures

A novel analysis method is presented for form-finding of tensegrity structures. The spectral decomposition of the force density matrix and the singular value decomposition of the equilibrium matrix are performed iteratively to find the feasible sets of nodal coordinates and force densities. An algorithm of determining the sole configuration of free-form tensegrities is provided by specifying an independent set of nodal coordinates, which indicates the geometrical and mechanical properties of the structures can be at least partly controlled by the proposed method. Several illustrative examples are presented to demonstrate the efficiency and robustness in finding self-equilibrium configurations of tensegrity structures.     

Effect of hoisting cable elasticity on anti-sway controllers of quay-side container cranes

Oscillation frequency of crane payloads is the main and most important factor in crane anti-sway control systems design. In the summer of 2005, a Smart Sway Control system (SSC) was installed on a 65-ton quay-side container crane at Jeddah Port. During the calibration phase of the installation, it was observed that heavy payloads combined with the dynamic stretch of the hoist cables had a significant impact on the configuration of the hoisting mechanism and the pattern of oscillation. This introduced considerable change in the oscillation frequency of the payload, which resulted in a significant impact on the performance of the anti-sway control system. Empirical formulas had to be used to compensate for the change in the frequency approximation used in the controller algorithm. In this work, an analytic approximation of the oscillation frequency of the hoisting mechanism of a quay-side container crane is developed, which takes into consideration the elasticity of the hoisting cables. A parametric study is performed to investigate the extent of the effect of the hoisting cables stretch on the system behavior for a typical range of payload masses and cable lengths. The performance of the delayed feedback control system used in the SSC controller is simulated on an elastic cables model using both the elastic and rigid cable frequency approximations.