二元复合材料板的自由振动: 半解析法

二元复合材料板是超材料板结构中常见的单元之一. 针对由材料参数相差两个量级的基体和嵌入体组成的二元复合材料板, 提出结构自由振动的半解析模型, 并对其振动特性进行了研究. 基于区域分解法和二元材料的分布, 将二维平板分解成两个子区域. 通过在振型函数中附加区域试函数, 来描述复合材料板面内刚度突变引起局部位移和转角的非光滑性. 基于二元复合材料板的基本边界条件和两子区连接处的变形协调条件, 构造了新的振型函数. 基于经典薄板理论, 利用带特殊试函数的里兹法, 求得不同几何构型下二元复合材料板的固有频率和振型, 并研究了嵌入体的尺寸和位置对结构振动特性的影响规律. 通过收敛分析并与有限元仿真结果对比, 验证了本文方法的准确性. 研究结果表明: 传统的全局试函数在分析具有振动局部化的模态时会得到不准确的结果, 而附加区域试函数可以显著提高里兹法的收敛速度以及结果的准确性; 嵌入体位置对低阶固有频率的作用不明显, 却能显著改变低阶振型节线的分布和振动局部化发生的区域.      

可重构力学超材料的设计与波动特性研究

力学超材料中的弯曲梁双稳态结构由于其主动调控性强且调控精度高等优点近年来受到广泛关注.文章利用中心受压弯曲梁的不稳定性设计了六角型双稳态结构,首先建立了等效弯曲梁模型,基于梁变形微分方程及能量最低原理探明了结构双稳态特性的产生基理,之后利用有限元数值计算研究了结构几何参数对其整体力学性能的影响,分别得到了具备自恢复及双稳态性能的结构几何参数范围,绘制了几何参数与力学性能之间的相图.同时,可重构结构的可控变形能力有助于调整整体的色散特性,利用数值仿真研究了具备双稳态特性的结构在拉伸和压缩两种构型下的色散关系,对比分析了不同结构几何参数及构型转变对结构产生的带隙位置及范围的影响,之后对由不同构型单胞组成的周期性结构进行了频响分析来验证带隙计算的准确性.通过六角型可重构结构的力学特性、色散特性研究及频响分析表明可以通过结构几何参数的设计实现对结构整体性能的主动调控,为可逆向设计的弹性波超材料结构研究分析提供了一条可靠路径.     

热环境下功能梯度夹层双曲壳自由振动分析

功能梯度夹层双曲壳结构广泛应用在航空航天、海洋工程等领域中,对于该类结构的动力学特性研究非常重要.论文以热环境下功能梯度夹层双曲壳为研究对象,在三阶剪切变形理论的基础上,考虑横向拉伸作用的影响提出了一种新的位移场,假设材料的物性参数与温度有关,且沿厚度方向表示为幂律函数.利用Hamilton原理得到简支边界条件下功能梯度夹层双曲壳三维振动系统动力学方程,利用Navier法求得两种不同夹层类型的系统固有频率.研究了几何物理参数和温度场对功能梯度夹层双曲壳自由振动固有频率的影响.     

载流线圈非均匀感应磁场中圆板的磁弹性固有振动

针对处于载流圆线圈平面内非均匀感应磁场中的导电圆板,基于Kirchhoff薄板理论,给出了磁场中圆板的磁弹性横向振动基本方程。根据电磁理论,推得载流线圈感应非均匀磁场强度的椭圆积分表达式,并导出了圆板所受电磁力计算式。通过位移函数的设定并应用伽辽金法,得到了圆板的磁弹性轴对称振动微分方程及固有频率的表达式。通过算例,绘制了周边夹支和简支两种边界条件下圆板的磁弹性固有振动特性曲线图,分析了两种边界条件下固有频率、阻尼比、电磁力矩随线圈载流强度、圆板半径、板厚等参数的变化规律。结果表明,线圈的载流强度对圆板的振动和电磁特性有显著影响,即电流增大,磁感应强度及阻尼比随之显著增大,圆板固有频率随之显著减小。     

热环境下功能梯度夹层双曲壳三维自由振动分析

功能梯度夹层双曲壳结构广泛应用在航空航天、海洋工程等领域中,对于该类结构的动力学特性研究非常重要。本文以热环境下功能梯度夹层双曲壳为研究对象,在三阶剪切变形理论的基础上,考虑横向拉伸作用的影响提出了一种新的位移场,假设材料的物性参数与温度有关,且沿厚度方向表示为幂律函数。利用Hamilton原理得到简支边界条件下功能梯度夹层双曲壳三维振动系统动力学方程,利用Navier法求得两种不同夹层类型的系统固有频率。研究了几何物理参数和温度场对功能梯度夹层双曲壳自由振动固有频率的影响。     

矩形夹层板的非线性磁弹性随机振动

以表层较薄、夹心较软的四边简支矩形夹层板为研究对象,分析其在磁场环境中的非线性磁弹性随机振动问题。根据板壳磁弹性基本理论、夹层板的弯曲振动理论、连续体的随机振动理论,利用伽辽金积分法得到了在电磁场中受横向随机载荷作用时四边简支矩形夹层板的非线性磁弹性随机振动方程;并利用FPK方程法解出了四边简支矩形夹层板非线性随机振动位移响应和速度响应的方差、位移响应和速度响应的概率密度等多个数字特征。最后针对具体算例,通过数值模拟讨论了电磁参数、功率谱密度参数、板的几何尺寸的变化对各数字特征的影响。由数值模拟结果可知,调节随机激励、磁场强度、板的几何尺寸的大小能有效地控制结构随机振动产生振动位移的概率。     

智能材料MFC驱动双稳态板的跳变特性研究

智能可变形结构能感知周围环境的变化并做出适当的反应,其主要原理是利用智能材料层产生应变以达到驱动主体结构发生变形。含有智能材料压电纤维MFC的双稳态复合材料层合板可以通过智能驱动实时调整主体结构的形状,实现稳态构型之间的跳变。本文对中心点固支的MFC压电智能驱动双稳态复合材料层合板的稳态构型、跳变规律及其影响因素进行了系统地研究。研究内容主要包括：含MFC的双稳态层合板的稳态构型、分叉行为和驱动跳变分析。利用理论模型分析了层合板的长宽比、厚度尺寸、铺层方式和MFC粘贴位置对双稳态板分叉行为的影响,并预测了双稳态层合板的临界尺寸以及驱动跳变的临界电压。双稳态板的跳变是一个大变形的非线性行为,实现和控制跳变行为的研究是其应用的基础。本文的研究结果为新型压电材料驱动双稳态板的结构设计提供了一定的理论参考。     

用连续质量有限元素法求解动力系统的固有特性

1.问题的提出在求解动力系统的固有特性时,现在常用的有限元素法的核心在于对所研究的元素假定位移(插值)函数.它们通常是与振动频率无关的纯几何函数.应用这种函数导得的“质量矩阵”和“刚度矩阵”去求解系统的固有特性,阶数是有限的,误差随阶次增高而加大.就频率而言通常只有前半数能满足工程使用要求,振型及其导数     

亚音速气流中复合材料悬臂板的非线性振动响应研究

随着材料科学的发展,越来越多的新型材料应用到了工程实践中.在气流激励的作用下,对于以航空航天工程为背景、采用复合材料的板壳结构的非线性动力学问题仍是动力学领域的研究热点.本文研究了复合材料悬臂板在亚音速气流条件下的非线性振动和响应.根据理想不可压缩流体的流动条件和Kutta–Joukowski升力定理,基于升力面理论,利用涡格法计算了三维有限长平板机翼上的亚音速气动升力.将亚音速气动力施加到复合材料悬臂板上,利用Hamilton原理,考虑Reddy三阶剪切变形理论并引入冯·卡门非线性应变位移关系,建立了有限长平板的非线性动力学微分方程.利用有限元方法考察了不同几何参数下层合板悬臂板的固有特性,通过比较不同材料和几何参数的线性系统的固有频率,得到不同比例的内共振关系.利用Galerkin方法将偏微分方程截断为两自由度非线性常微分方程,在这里考虑了1:2的内部共振关系并利用多尺度法进行了摄动分析.对应多个选取参数,得到了频率响应曲线.结果展示了硬化弹簧型行为和跳跃现象.     

各向异性矩形板和环扇形板横向自由振动的一种通用解法

提出各向异性矩形板和环扇形板在弹性边界约束下横向自由振动的通用解法。对于各向异性环扇形板，引入径向对数坐标简化其基本理论。两种不同形状板的几何参数和势能可建立统一的表达式，基于改进Fourier级数和Hamilton原理，从而实现板自由振动问题的统一求解。两种形状板自由振动问题的通用解法具有广泛适用性、高精度和高效性。其收敛性和精度得益于位移的改进Fourier级数的表达，可消除初始横向位移函数及其导数在整个区域内的潜在不连续。所提方法的这些特征通过若干数值算例得到验证。     

非对称铺层复合材料层合板的双稳态特性半解析研究

非对称铺层的复合材料层合板在存在热残余应力的情况下，具有双稳态性质．层合板的两个稳态之间仅需要一个适当的激励就可以互相转化，因此该结构在变体飞机上应用广泛．基于经典层合板理论，本文引入几何大变形建立了具有双稳态性质的复合材料层合板的能量泛函，提出了一个高阶的位移场函数，用瑞利里兹法推导出一组关于位移场函数系数的非线性方程组．结合牛顿迭代法和消元法求解非线性方程组，得到了层合板面外位移场．同时利用有限元软件ABAQUS，对复合材料层合板双稳态机理进行了数值模拟．选取了几组具有代表性的铺层进行计算，以有限元结果为基准，比较了本文的位移场结果与前人的结果，验证了高阶位移场函数的准确性．     

Pseudo-nonlinear dynamic analysis of buckled pipes

In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations.     

State-vector equation with damping and vibration analysis of laminates

Based on the modified mixed Hellinger-Reissner(H-R)variational principle for elastic bodies with damping,the state-vector equation with parameters is directionally derived from the principle.A new solution for the harmonic vibration of simply supported rectangular laminates with damping is proposed by using the precise integration method and Muller method.The general solutions for the free vibration of underdamping,critical damp and overdamping of composite laminates are given simply in terms of the linear damping vibration theory.The effect of viscous damping force on the vibration of com- posite laminates is investigated through numerical examples.The state-vector equation theory and its application areas are extended.     

Snap-through of unsymmetric fiber-reinforced composite laminates

An approximate theory based on assumed strain and displacement fields, the Rayleigh–Ritz technique, and virtual work is used to predict the snap-through forces and moments for three families of unsymmetric fiber-reinforced composite laminates. Unsymmetric laminates generally have two stable equilibrium configurations when cooled from their elevated cure temperature, and it is the moment required to snap the laminate from one stable configuration to the other that is the subject of this paper. A simple force-controlled experiment is described which is used to measure the snap-through moment and the characteristics of the configuration change, by way of strains, in four laminates. The correlation between predicted results and experimental measurements is quite good, both in terms of moment levels and in terms of strain response.     

纤维曲线铺放的复合材料层合板的自由振动分析

假设曲线纤维的方向角沿板的长度方向按照线性规律变化,导出了纤维曲线铺设时的参考路径,将参考路径沿板的宽度方向平移可得一种曲线纤维增强复合材料单层板,当这种纤维曲线铺设的单层板对称铺放时即可得相应的曲线纤维增强复合材料层合板．基于弹性薄板的小挠度理论,建立了曲线纤维增强复合材料层合板自由振动问题的基本方程,采用微分求积法进行数值求解,得到了层合板的自振频率及相应的振型．与已有文献计算结果的比较,验证了本文计算结果的正确性．通过数值算例分析了微分求积法求解本问题时的收敛性,研究了纤维铺放路径和边界条件的不同对曲线纤维增强复合材料层合板频率及振型的影响,研究结果可为该种结构的设计提供一定的参考．     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

动能梯度环形截面梁的横向自由振动

对功能梯度材料制成的环形截面梁,假设材料的物性参数沿壁厚方向按幂率变化,基于Lagrange函数和Hamilton 原理,建立了该梁横向自由振动的Hamilton 对偶方程组. 采用辛方法求解了Hamilton 矩阵的辛本征问题,得到了简支、两端固定、悬臂和左端固定右端铰支4 种约束的FGM(functionally gradedmaterials)环形截面梁的固有频率和振型函数. 算例给出了这4 种约束的FGM 环形截面梁前8 阶无量纲固有频率随材料体积分数的变化规律,分析了材料体积分数对FGM 环形截面梁固有频率的影响.     

有限长压电层合简支板自由振动的三维精确解

基于三维弹性理论和压电理论，导出了有限长矩形压电层合简支板的动力学方程及相应的边界条件，给出了一种求解压电层合板自由振动三维精确解的方法；分析了正、逆向压电效应对层合板振动频率的影响．本文所述的方法和结果对于求解其他三维动态问题，验证、比较其他简化模型、有限元计算结果以及工程应用都有指导意义．     

Modeling and analysis of dynamic characteristics of multi-stable waterbomb origami base

Origami has recently received wide attention, and the study on its dynamic characteristics remains a nascent field. The waterbomb origami is a common subtype of origami, and its base structure is treated as a bi-stable configuration in the literature. The systematical framework for modeling, simulation and dynamic analysis of the vibration for the waterbomb origami base is established in this paper. In the presented model, the motion of the waterbomb origami base is divided into two working patterns according to its geometric characteristic. The nonlinear governing equation of motion of the waterbomb origami base is formulated based on the Lagrange’s equation. The base’s free and forced responses can be calculated by using the fourth-order Runge–Kutta method. The developed model is validated by the results predicted by the simulation in ADAMS. With the developed theoretical framework, the base’s vertical effective stiffness and natural frequency of its linearized system are discussed to reveal their programmability with respect to the base’s structure and design parameters. Remarkably, the bifurcations of its equilibria, including the pitchfork, transcritical and (special) saddle-node bifurcations, are analyzed. Unlike the bi-stable configuration reported in the literature, the mono- and tri-stable configurations can also be realized by the base due to gravity. Furthermore, the complex nonlinear dynamic behaviors, including chaos, are revealed.