Non-linear dynamic analysis of tensegrity structures using a co-rotational method

A new formulation is presented for the non-linear dynamic analysis of space truss structures. The formulation is based on the dynamics of 3D co-rotational rods. In the co-rotation method, the rigid body modes are assumed to be separated from the total deformations at the local element level. In this paper a new co-rotational formulation is proposed based on the direct derivation of the inertia force vector and the tangent dynamic matrix. A closed-form equation is derived for the calculation of the inertia force, the tangent dynamic matrix, the mass matrix and the gyroscopic matrix. The new formulation is used to perform dynamic analysis of example tensegrity structures. The developed formulation is applicable to tensegrity structures with non-linear effects due to internal mechanisms or geometric non-linearities, and is applied to two numerical examples. The efficiency of the proposed approach is compared to the conventional Lagrangian method, and savings in computation of about 55%, 54% and 37% were achieved.     

张拉整体结构的动力学等效建模与实验验证

为了实现张拉整体结构高效动力学计算, 并考虑其大范围运动中柔性杆局部动态屈曲, 提出了一种受压细长杆动力学降阶模型, 采用五节点弹/扭簧集中质量离散模型等效连续杆的静力学和动力学特性. 首先, 通过静力学等效分析推导了弹簧拉压刚度和扭簧弯曲刚度表达式, 可准确预测杆件受压屈曲和近似预测其后屈曲行为. 第二, 通过动能等效分析推导了集中质量表达式, 可准确预测杆在线速度场下的运动. 第三, 通过弯曲振动固有模态等效分析确定弯曲刚度和节点质量的分布参数, 合适的分布参数取值组合可将降阶模型前两阶固有频率相对误差均降低至1%以内. 第四, 在全局坐标系下建立张拉整体结构瞬态动力学方程, 并利用静力凝聚法实现方程高效迭代求解. 最后, 分别对球形张拉整体结构准静态压缩、模态分析和碰撞动力学进行仿真和实验对比分析, 证明了提出的动力学降阶模型可有效预测张拉整体结构的静力学行为、固有振动特性及瞬态动力学响应, 并分析了结构参数变化对其力学特性的影响规律. 本文提出的动力学等效建模与计算方法, 可望用于软着陆行星探测器、大型可展开空间结构及点阵材料等复杂张拉整体系统的动力学分析与控制.      

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.     

Damage assessment of tensegrity structures using piezo transducers

This paper presents the application of surface-bonded piezo-transducers for damage assessment of tensegrity structures through dynamic strain measurement and electro-mechanical impedance (EMI) technique. The two techniques are first applied on a single module tensegrity structure, 1 m×1 m in size and their damage diagnosis results compared. A single piezoelectric-ceramic (PZT) patch bonded on a strut measures the dynamic strain during an impact excitation of the structure. Damage is identified from the changes in global frequencies of the structure obtained from the PZT patch’s response. This is compared with the damage identified using the EMI technique, which is a signature based technique and operates at frequencies of the order of kHz. The dynamic strain approach, which requires commonly available hardware, is found to exhibit satisfactory performance vis-à-vis the EMI technique for damage assessment of tensegrity structures. The damage diagnosis exercise is then extended to a tensegrity grid structure, 2 m×2 m size, fabricated using galvanized iron (GI) pipes and mild steel wire ropes. The damage is localized using changes in natural frequencies observed experimentally using the dynamic strain approach and the corresponding mode shapes of the undamaged structure derived numerically. The dynamic strain approach is found to be very expedient, displays competitive performance and is at the same time cost effective for damage assessment of tensegrity structures.     

基于Gauss伪谱法的3D刚体摆姿态最优控制

研究Gauss伪谱法求解3D刚体摆姿态最优控制问题．针对其最优姿态控制问题,既要满足由任意位置运动到平衡位置姿态运动规划问题,又要满足系统含有动力学约束的力学模型问题,提出基于四元数来描述3D刚体摆的数学模型,建立3D刚体摆姿态的动力学和运动学方程,为了解决3D刚体摆在平衡位置处的姿态最优控制问题,设计基于Gauss伪谱算法的最优姿态开环控制器,得到了3D刚体摆的姿态最优控制轨迹,得到满足的可行解,通过仿真实验验证了其开环解在平衡位置的控制姿态最优性．     

Steady-state dynamics of a 3D tensegrity structure: Simulations and experiments

This paper considers a modeling and analysis approach for the investigation of the linear and nonlinear steady-state dynamics of a base excited 3D tensegrity module carrying a top mass. The tensegrity module contains three compressive members, which may buckle and six cables (tendons). First, a dynamic model of the system is derived using Lagrange’s equation with constraints. The buckling modeling of the compressive members is based on the assumed-mode method with a single mode discretization. The tendons are modeled as piecewise linear springs, which can only take tensile forces. This research focusses on the dynamic stability of the tensegrity structure by defining the geometrical and material properties in such a way that the system is just below the static stability boundary. Static and linear dynamic analysis is performed. In the nonlinear steady-state analysis, frequency-amplitude plots, power spectral density plots, bifurcation point continuation diagrams, and Poincaré maps are presented. A tensegrity structure is designed and manufactured and an experimental set-up is realized in order to validate the model by comparing experimentally and numerically obtained responses. In the validation stage, the numerical results are based on an amplifier-shaker-tensegrity structure model. It can be concluded that the numerical results match partly quantitatively and partly qualitatively with the experimentally obtained responses.     

Mechanical response of a tensegrity structure close to its integrity limit submitted to external constraints

The response to an external constraint of a symmetrical tensegrity structure made of elastic and rigid elements has been studied by numerical experiments. Two non-linear effects have been found when the structure is close to its integrity limit above which it collapses. The first one is that the mechanical power response of the tensegrity structure can be modulated according to the magnitude of the applied force. This effect indicates that the structure may act as a mechanical power amplifier. The second one is that a slightly prestressed tensegrity structure can offer a greater resilience to an applied force than more prestressed equivalent structures. This paradoxical stiffening effect indicates that increasing the prestress may not always be the most efficient way to keep the stability of the structure.     

Structural morphology of tensegrity systems

The coupling between form and forces, their structural morphology, is a key point for tensegrity systems. In the first part of this paper we describe the design process of the simplest tensegrity system which was achieved by Kenneth Snelson. Some other simple cells are presented and tensypolyhedra are defined as tensegrity systems which meet polyhedra geometry in a stable equilibrium state. A numerical model giving access to more complex systems, in terms of number of components and geometrical properties, is then evoked. The third part is devoted to linear assemblies of annular cells which can be folded. Some experimental models of the tensegrity ring which is the basic component of this “hollow rope” have been realized and are examined.     

Design methods of rhombic tensegrity structures

As a special type of novel flexible structures, tensegrity holds promise for many potential applications in such fields as materials science, biomechanics, civil and aerospace engineering. Rhombic systems are an important class of tensegrity structures, in which each bar constitutes the longest diagonal of a rhombus of four strings. In this paper, we address the design methods of rhombic structures based on the idea that many tensegrity structures can be constructed by assembling one-bar elementary cells. By analyzing the properties of rhombic cells, we first develop two novel schemes, namely, direct enumeration scheme and cell-substitution scheme. In addition, a facile and efficient method is presented to integrate several rhombic systems into a larger tensegrity structure. To illustrate the applications of these methods, some novel rhombic tensegrity structures are constructed.     

Topology design of tensegrity structures via mixed integer programming

This paper presents a numerical method for finding a tensegrity structure based on the ground structure method. We first solve a mixed integer programming (MIP) problem which maximizes the number of struts over the self-equilibrium condition of axial forces and the discontinuity condition of struts. Subsequently we solve the minimization problem of the number of cables in order to remove redundant self-equilibrium modes, which is also formulated as an MIP. It is regarded to be advantageous that our method does not require any connectivity information of cables and struts to be known in advance, while the obtained tensegrity structure is guaranteed to satisfy the discontinuity condition of struts rigorously.     

Form-finding of tensegrity structures via genetic algorithm

We propose an efficient method for the form-finding of tensegrity structures. The force densities of each tensegrity are obtained by the minimisation of a particular objective function, leading to a semi-positive definite force density matrix (a super-stable tensegrity) with a required rank deficiency. A genetic algorithm is used as a global search technique for the minimisation. The geometry of a tensegrity is subsequently formed based on those eigenvectors of the force density matrix corresponding to zero eigenvalues. Furthermore, two other methods are introduced to convert the asymmetrical geometry obtained from the main algorithm into its symmetrical counterparts. This transformation in geometry is performed by finding a suitable linear combination of the mentioned eigenvectors. Examples from well-known tensegrities including prismatic, truncated tetrahedron, expandable octahedron and truncated icosahedron tensegrities are studied using the present method, and the results obtained are compared with those documented in the literature to verify the efficiency of the present method.     

Equilibrium conditions of a tensegrity structure

This paper characterizes the necessary and sufficient conditions for tensegrity equilibria. Static models of tensegrity structures are reduced to linear algebra problems, after first characterizing the problem in a vector space where direction cosines are not needed. This is possible by describing the components of all member vectors. While our approach enlarges (by a factor of 3) the vector space required to describe the problem, the advantage of enlarging the vector space makes the mathematical structure of the problem amenable to linear algebra treatment. Using the linear algebraic techniques, many variables are eliminated from the final existence equations.     

On the solution of the Darboux problem

We consider the problem of determining the angular position of a rigid body in space from its known angular velocity and initial position (the Darboux problem) in quaternion setting. For an arbitrary angular velocity vector of the body, we present a solution based on Lappo-Danilevskii’s recursion relations [1]. New special cases of solvability of the Darboux problem in closed form are obtained.     

Deployment of tensegrity structures

In this paper we present a strategy for tensegrity structures deployment. The main idea is to use a certain set of equilibria to which the undeployed and deployed configurations belong. In the state space this set is represented by an equilibrium manifold. The deployment is conducted such that the deployment trajectory is close to this equilibrium manifold.     

Stability conditions of prestressed pin-jointed structures

Pin-jointed structures are first classified to trusses, tensile structures, and tensegrity structures in view of their respective stability properties. A sufficient condition for stability of an equilibrium state is derived for tensegrity structures. The condition is based on the bilinear forms of the linear and geometrical stiffness matrices considering the flexibility of members. The stability is defined by the positive definiteness of the tangent stiffness matrix, whereas the definition of prestress-stability is based on the geometrical stiffness matrix and the infinitesimal mechanisms. Numerical examples verify that the so-called super-stability condition might not be satisfied by a stable tensegrity structure, and that a prestress-stable structure can be unstable if the prestresses are moderately large.     

基于虚拟力概念的柔性臂空间机器人模糊退步自适应控制算法设计

讨论了漂浮基柔性臂空间机器人系统的动力学模拟、运动轨迹跟踪控制算法设计及柔性振动主动抑制。采用多体动力学建模方法并结合假设模态法,建立了漂浮基柔性臂空间机器人的系统动力学模型。基于该模型,针对系统惯性参数未知情况,提出了刚性运动基于模糊基函数网络自适应调节的退步控制算法,以完成柔性臂空间机器人载体姿态及机械臂各关节铰的协调运动。然后,为了主动抑制系统柔性振动,运用虚拟力的概念,构造了同时反映柔性模态和刚性运动轨迹的混合期望轨迹,通过改造原有的控制算法,提出了基于虚拟力概念的模糊退步自适应控制算法;这样不但保证了之前刚性运动控制方案对模型不确定的鲁棒性,而且能主动抑制柔性振动,从而提高了轨迹跟踪性能。理论分析及数值仿真算例均表明了控制方法的可行性。     

Effective mass density of rigid sphere-reinforced elastic composites

An analytical model is presented to study metaelastic dynamic behaviors of rigid sphere-reinforced elastic random composites. The model is based on the concept that the deviation of the displacement field of embedded rigid spheres from the displacement field of the composite is responsible for novel metaelastic dynamic behaviors of stiff sphere-reinforced composites. Compared to the existing models, the present model offers a simple general method to analyze various 3D metaelastic dynamic problems of rigid sphere-reinforced random composites, and its validity and efficiency are demonstrated by comparing the predicted results with known experimental or numerical data on several typical steel/glass/lead-polymer composites. Several basic dynamic problems of rigid sphere-reinforced composite beams and rods are investigated, and some metaelastic dynamic phenomena (such as vibration isolation, localized buckling, and natural frequency within the bandgap caused by an attached concentrated mass) of stiff sphere-reinforced composites are demonstrated.

     

Influence of elastic compliance of links on the dynamics and accuracy of a manipulating robot with rotational and translational joints

We pose problems of dynamic and kinematic control of spatial motions for multilink manipulating robot with elastic links and with rotational and translational joints. These problems are reduced to solving a system of ordinary and partial differential equations of hybrid type in the independent variables. We use a numerical integration algorithm for such systems which was developed earlier for manipulating robots with elastic links of anthropomorphic type. We discuss the difficulties arising in mathematical simulation of manipulating robots with simultaneously rotational and translational joints and with elastic links. To perform a comparative analysis and estimate the positional accuracy for the center of mass of the weight transported by the manipulator, we pose problems of dynamic and kinematic control of spatial motions of a manipulating robots with rigid links and with rotational and translational joints. The resolving equations obtained in this case are based on the Lagrange formalism of the second kind. By way of example, we present the solution of dynamic control problems for elastic and rigid two-link manipulators with one translational and two rotational joints.     

On the Dynamic Interaction and Cross-Interaction of 2D Rigid Structures with Orthotropic Elastic Media Possessing General Principal Axes Orientation

A direct boundary element method (BEM) implementation for the dynamic interaction analysis in the frequency domain of 2D rigid structures with elastic orthotropic media is presented. The BEM implementation is based on non-singular full-space influence functions. The rigid structure response is obtained by applying equilibrium and kinetic compatibility conditions. The method is applied to the analysis of the dynamic response of a rigid tunnel in a half-space with various elasticity principal axes inclinations and to the analysis of two rigid rectangular galleries in a half space with various distances between them.     

Dynamic analysis of a rigid-flexible inflatable space structure coupled with control moment gyroscopes

The vibration generated by the inflatable structure after deployment has a great impact on the performance of the payloads. In this paper, the influence of the control moment gyroscopes (CMGs) on the dynamic responses and characteristics of an inflatable space structure is studied, based on the flexible multibody dynamics in a combination of the absolute nodal coordinate formulation (ANCF) and the natural coordinate formulation (NCF). Firstly, the ANCF and NCF are used to accurately describe the large deformations and large overall motions of flexible inflatable tubes and rigid satellites, respectively. Then, instead of modeling gyroscopic flexible bodies, this paper pioneers a rigid body dynamic model of the CMG in detail by using the NCF modeling scheme, which can be attached to and coupled with any flexible bodies without any assumptions. Then, the orbital dynamic equations of the inflatable space structure coupled with distributed CMGs are obtained by considering the effects of Coriolis force, centrifugal force, and gravity gradient through coordinate transformation. The dynamic characteristics of the inflatable space structure are also analyzed by deriving the eigenvalue problem of a flexible multibody system. Finally, the accuracy of the CMG dynamic model is verified via a classic heavy top example. Several numerical examples are presented to study the influence of the magnitudes and directions of the rotor angular momentum of the CMGs on the dynamic responses and characteristics of the inflatable space structure.