On the elastica solution of a T3 tensegrity structure

Stability studies of a T3 tensegrity structure are performed. This structure is composed of three slender struts interconnected by six nonlinear elastic tendons and is prestressed. The struts are governed by linear constitutive laws and are allowed to buckle. Since tensegrity is used for modeling structures with quite large deformations, for example the cytoskeleton, and bifurcation theory—valid for small solutions of the nonlinear equations—does not directly apply, a general procedure for studying the stability behavior of the particular tensegrity model based upon the elastica theory is presented. The reference placement is defined by the prestress, and the equilibrium placements are defined by the applied force and moment.     

Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment

We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency–energy domain by computing forced frequency–energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.     

Stability of a tensegrity structure: application to cell mechanics

Stability studies of a tensegrity structure, used as a model for cell deformability, are performed. This structure is composed of 6 slender struts interconnected by 24 linearly elastic cables. The cables and the struts are governed by linear constitutive laws. The struts are allowed to buckle. Adapting experimental evidence, the struts have already buckled at the reference placement due to the prestress of the tendons. A general procedure for studying the stability behavior of the particular tensegrity model is presented. The reference placement is defined by the prestress, and the equilibrium placements are defined for any three-dimensional applied forces.     

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.      

Nonlinear resonances in an axially excited beam carrying a top mass: simulations and experiments

In this paper, nonlinear resonances in a coupled shaker-beam-top mass system are investigated both numerically and experimentally. The imperfect, vertical beam carries the top mass and is axially excited by the shaker at its base. The weight of the top mass is below the beam’s static buckling load. A semi-analytical model is derived for the coupled system. In this model, Taylor-series approximations are used for the inextensibility constraint and the curvature of the beam. The steady-state behavior of the model is studied using numerical tools. In the model with a single beam mode, parametric and direct resonances are found, which affect the dynamic stability of the structure. The model with two beam modes not only shows an additional second harmonic resonance, but also reveals some extra small resonances in the low-frequency range, some of which can be identified as combination resonances. The experimental steady-state response is obtained by performing a (stepped) frequency sweep-up and sweep-down with respect to the harmonic input voltage of the amplifier-shaker combination. A good correspondence between the numerical and experimental steady-state responses is obtained.     

Geometric and material nonlinear analysis of tensegrity structures

A numerical method is presented for the large deflection in elastic analysis of tensegrity structures including both geometric and material nonlinearities.The geometric nonlinearity is considered based on both total Lagrangian and updated Lagrangian formulations,while the material nonlinearity is treated through elastoplastic stress-strain relationship.The nonlinear equilibrium equations are solved using an incremental-iterative scheme in conjunction with the modified Newton-Raphson method.A computer program is developed to predict the mechanical responses of tensegrity systems under tensile,compressive and flexural loadings.Numerical results obtained are compared with those reported in the literature to demonstrate the accuracy and efficiency of the proposed program.The flexural behavior of the double layer quadruplex tensegrity grid is sufficiently good for lightweight large-span structural applications.On the other hand,its bending strength capacity is not sensitive to the self-stress level.     

数字状张拉整体结构的构型设计与力学性能模拟

针对大型张拉整体结构的设计问题,选取四棱柱状张拉整体结构和截角正八面体状张拉整体结构作为基本胞元,采用节点连接节点的方式建立球柱组合式数字状张拉整体结构,并使用基于结构刚度矩阵的大变形非线性数值求解方法对其进行力学性能分析.在两类胞元满足各自的自平衡条件和稳定性条件的前提下,组合得到的数字状张拉整体结构亦处于自平衡稳定状态,搭建了实物模型进行验证.以数字8状张拉整体结构为例,模拟研究了结构承受自重等分布载荷和单轴拉压等端部载荷时的静力学响应,以及结构无阻尼振动时的固有频率和模态等动力学性能.结果表明,结构在自重作用下的变形行为受初始预应力、压杆密度和拉索刚度的影响较大,对其进行合理配置方可确保结构具有足够刚度抵抗自重;结构在单轴拉压作用下呈现非线性的载荷-位移曲线,拉伸刚度随变形量的增大而增大,压缩刚度随变形量的增大而减小;结构的固有频率随初始预应力的增大而增大,而模态振型基本不变.研究结果丰富了大型张拉整体结构的外形种类,有望推动此类结构在土木建筑、结构材料等领域的应用.      

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.     

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.     

Nonlinear network modeling of multi-module floating structures with arbitrary flexible connections

Multiple floating modules connected by flexible connectors can be viewed as a network structure. A standard modeling process for multi-module floating structures in arbitrary topology is presented by using network theory. A three-dimensional model is developed using the linear wave theory, dynamic model of single floating module, constitutive model of flexible connectors and model of a mooring system. As a typical application, a floating airport model is established and further its nonlinear dynamic responses and connector loads are analyzed. Numerical results show that the traditional linear analysis may underestimate the actual results. The methodology applied in this paper is extensible to many engineering problems with network structures alike.     

基于数值子结构方法的结构弹塑性分析

结合震害调研及数值分析可知,结构最终失效可能仅由部分关键构件破坏引起,大部分构件仍处于弹性或小变形状态。因此为提高计算效率,在结构全过程分析中一致采用非线性单元建模并非必要,同时为准确考虑关键构件的非线性响应,本文提出一种新的数值子结构建模策略。进入弹塑性状态后,针对一般钢构件或钢筋混凝土构件采用动态替换子结构方法在单元或截面层次将其替换成非线性单元或非线性截面,并基于OpenSees平台开发了两类新单元予以实现;针对可能发生严重损伤的关键构件,采用隔离子结构方法将其隔离并建立精细化分析模型,考虑主、子结构间不同尺度边界耦合,并推导了切线刚度的传递关系,采用Client/Server技术在OpenSees平台开发了一类新的接口单元予以实现主、子结构之间的信息传递。为验证新开发单元的合理性,分别以钢及钢筋混凝土平面框架结构为例,采用纤维单元、动态替换子结构方法以及隔离子结构方法建模进行静、动力分析。计算结果表明,采用本文提出的动态替换子结构方法与常规建模方法的计算结果完全吻合并且可大幅缩短计算耗时,随着荷载水平的增大,结构中受到动态替换的构件比例急剧增大,计算效率提高程度略有降低,但仍远高于常规模型;采用本文提出的接口单元可准确传递主、子结构间的界面信息,为隔离数值子结构方法在结构弹塑性分析中的应用提供了基础。     

Self-stress design of tensegrity grid structures with exostresses

A numerical method is presented for initial self-stress design of tensegrity grid structures with exostresses, which is defined as a linear combination of the coefficients of independent self-stress modes. A discussion on proper division of the number of member groups for the purpose of existence of a single integral feasible self-stress mode has been explicitly given. Dummy elements to transform the tensegrity grid structure with statically indeterminate supports into self-stressed pin-jointed system without supports are employed. The unilateral properties of the stresses in cables and struts are taken into account. Evaluation of the stability for the structure is also considered. Several numerical examples are presented to demonstrate the efficiency and robustness in searching initial single integral feasible self-stress mode for tensegrity grid structures.     

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of three-to-one (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the Euler?CBernoulli beam model, the governing nonlinear PDE of the beam??s transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximate-analytical solution techniques, namely the method of Multiple Time Scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis. The obtained results at this stage using four different methods which are all in time?Cspace domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are obtained. Furthermore, the effect of axial compressive force in the dynamic analysis of such a beam is studied. Finally, under three-to-one-internal resonance condition the NNMs of the beam and the steady-state stability analysis are performed. Then the effect of changing the values of different parameters on the beam??s dynamic response is also considered. Moreover, 3-D plots of stability analysis in the steady-state condition and the beam??s amplitude frequency response curves are presented.     

薄膜结构流固耦合的CFD数值模拟研究

基于弱耦合分区求解策略,在CompaqVisualFortran6．5环境下搭建了薄膜结构三维流固耦合效应的CFD数值模拟平台。程序采用模块化编程思想,主要包含几何建模、流体分析、结构分析和数据交换四个模块。其中几何建模模块采用自行编制的膜结构找形分析程序,流体分析模块采用经过二次开发的计算流体力学软件FLUENT6．0,结构分析模块采用自行编制的膜结构动力分析程序MDLFX;在数据交换模块中,编制了基于薄板样条法的插值计算程序,以实现流固交界面上不同区域网格间的数据传递问题,编制了基于代数法和迭代法的动网格变形程序,以实现流固耦合运算中的动网格更新。基于该软件平台,对单向柔性屋盖和鞍形膜结构屋盖进行了流固耦合数值模拟,验证了方法的有效性。     

Experimental and numerical analysis of nonlinear dynamics of rotor–bearing–seal system

Nonlinear dynamics and stability of the rotor–bearing–seal system are investigated both theoretically and experimentally. An experimental rotor–bearing–seal device is designed and corresponding tests are carried out. The experimental rotor system is simplified as the Jeffcott rotor. The nonlinear oil–film forces are obtained under the short bearing theory and Muszynska nonlinear seal force model is used. Numerical method is utilized to solve the nonlinear governing equations. Bifurcation diagrams, waterfall plots, Poincaré maps, spectrum plots and rotor orbits are drawn to analyze various nonlinear phenomena and system unstable processes. Theoretical results from numerical analysis are in good agreement with results from experiments. Conclusions are drawn and prove that this study will contribute to the further understanding of nonlinear dynamics and stability of the rotor system with the fluid-induced forces from oil–film bearings and the seals.     

Selection of prestress for optimal dynamic/control performance of tensegrity structures

This paper concerns prestress optimization of a tensegrity structure for its optimal LQR performance. A linearized dynamic model of the structure is derived in which the force-density variables that parameterize the prestress of the structure appear linearly. A feasible region for these parameters is defined in terms of the extreme directions of the prestress cone. A numerical method for computing this basis for a structure prestress cone is proposed. The problem is solved using a gradient method that provides a monotonic decrease of the objective function inside the feasible region. A numerical example of a cantilevered planar tensegrity beam is shown.     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

基于ED-LSTM的智能汽车神经网络横向动力学建模与控制

车辆动力学建模过程中通常会进行简化和假设, 导致模型在某些工况下无法准确反映车辆的实际动态特性, 影响控制精度甚至安全性. 鉴于此, 该文提出了一种基于数据驱动的非线性建模与控制方法, 建立了新型神经网络车辆横向动力学多步预测模型, 实现了智能汽车对参考轨迹的跟踪控制. 首先, 在分析车辆单轨模型并考虑轮胎非线性和纵向负载转移的基础上, 基于编码器?解码器结构设计神经网络横向动力学模型. 其中, 使用串行排列来扩展微分方程描述不完全的动力学信息, 隐藏层神经元学习车辆的高度非线性和强耦合特性, 进而提高模型全局计算精度. 利用所构建的数据集进行模型训练和测试, 结果表明, 相比于物理模型, 所提出的模型在不同路面附着系数条件下均具有更高的建模精度, 具有隐式预测路面摩擦条件能力. 其次, 利用提出的模型设计轨迹跟踪控制算法, 根据车辆稳态转向假设, 计算所需的前轮转向角和稳态质心侧偏角, 将稳态质心侧偏角纳入基于路径误差的转向反馈中, 实现参考轨迹跟踪控制. 最后, 使用CarSim/Simulink联合仿真及HIL实验测试进行不同工况试验的对比分析, 对所提出的基于神经网络模型的控制算法进行评价, 结果表明, 该模型能够实现智能汽车在高速下精确的跟踪控制效果, 并具有良好的横向稳定性.      

Static and dynamic behaviors of belt-drive dynamic systems with a one-way clutch

This paper focuses on the nontrivial equilibrium and the steady-state periodic response of belt-drive system with a one-way clutch and belt flexural rigidity. A nonlinear piecewise discrete–continuous dynamic model is established by modeling the motions of the translating belt spans as transverse vibrations of axially moving viscoelastic beams. The rotations of the pulleys and the accessory are also considered. Furthermore, the transverse vibrations and the rotation motions are coupled by nonlinear dynamic tension. The nontrivial equilibriums of the belt-drive system are obtained by an iterative scheme via the differential and integral quadrature methods (DQM and IQM). Moreover, the periodic fluctuation of the driving pulley is modeled as the excitation of the belt-drive system. The steady-state periodic responses of the dynamic system are, respectively, studied via the high-order Galerkin truncation as well as the DQM and IQM. The time histories of the system are numerically calculated based on the 4th Runge–Kutta time discretization method. Furthermore, the frequency–response curves are presented from the numerical solutions. Based on the steady-state periodic response, the resonance areas of the dynamic system are obtained by using the frequency sweep. Moreover, the influences of the truncation terms of the Galerkin method, such as 6-term, 8-term, 10-term, 12-term, and 16-term, are investigated by comparing with the DQM and IQM. Numerical results demonstrate that the one-way clutch reduces the resonance responses of the belt-drive system via the torque-transmitting directional function. Furthermore, the comparisons in numerical examples show that the investigation on steady-state responses of the belt-drive system with a one-way clutch and belt flexural rigidity needs 16-term truncation     

基于非线性状态空间辨识的气动弹性模型降阶

高维、非线性气动弹性系统的模型降阶是当前气动弹性力学与控制领域的研究热点之一.然而国内外现有的非线性模型降阶方法仍存在辨识算法复杂、精度有待提高等问题.本研究提出了一种基于非线性状态空间辨识的跨音速气动弹性模型降阶方法. 首先,该方法基于非定常空气动力的单位脉冲响应数据,采用特征系统实现算法对非线性状态空间模型的线性动力学部分进行系统辨识. 其次,引入状态和控制输入的非线性函数, 采用优化算法对非线性函数的系数矩阵进行优化,进而得到考虑非线性效应的空气动力降阶模型.为了验证该降阶模型在预测跨音速气动弹性力学行为的精确性,本文以三维机翼为研究对象,分别从基于非线性降阶模型的气动力辨识、跨声速颤振边界计算和极限环振荡预测三方面进行了算例验证,并与现有的模型降阶方法进行了对比, 进一步说明本文所提出方法的有效性.研究结果表明, 该降阶模型对上述三类问题的计算精度与直接流-固耦合方法相吻合,可用于高效预测飞行器跨声速气动弹性力学行为.