Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method

This paper addresses the elastic buckling and vibration characteristics of isotropic and orthotropic nanoplates using finite strip method. In order to consider small scale effect, Eringen’s nonlocal continuum elasticity is employed. The governing nanoplate equations are derived using the principle of virtual work while B3-spline finite strip method is applied to the buckling and vibration analyses. The buckling load and vibration frequency of graphene sheets, which are subjected to biaxial compression and pure shear loading, are determined whilst the effects of different parameters such as sheet size, nonlocal parameter, aspect ratio and boundary conditions are investigated. The interaction curves of the critical biaxial compression loading as well as the interaction curves of the critical uniaxial compression and shear loading are also obtained. It is shown that small scale effect plays considerable role in the analysis of small sizes plates.     

非均匀变厚度梁的动力响应的一般解

在本文中提出一个新方法——阶梯折算法来研究在任意载荷下任意非均匀和任意变厚度伯努利-欧拉梁的动力响应问题.研究了自由振动和强迫振动.新方法需要将区间离散为一定数目的元素,每个元素可看作是均匀和等厚度的.因此均匀、等厚度梁的一般解可在每个元素上应用.然后用初参数表示的整个梁的一般解使之满足相邻二元素间的物理和几何连续条件,这样就可以得到解析形式的自由振动的频率方程和解析形式的强迫振动的最终解,它化为求解二元线性代数方程,与离散元素的数目无关.现在的方法可推广应用至任意非均匀及任意变厚度有粘滞性和其他种类的梁以及其他结构元件问题上去.     

Forced vibration of plastic structures taking into account the elastic and damping properties of the material (systems with one degree of freedom)

A method, which takes into account the elastic properties and hysteresis losses of plastic materials, is proposed for calculating the forced vibrations of plastic structures with one degree of freedom. A solution of the equation of motion of the system is found in trigonometric form, and an algebraic equation for determining the vibration amplitude is obtained. In its general form the latter equation can be solved for special cases only; when it is required to determine the vibration amplitude in the general case, the successive approximations method is recommended, or rigorous solution of the algebraic equation for each specific case.Mekhanika Polimerov, Vol. 1, No. 3, pp. 101–106, 1965     

Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring

The free and forced vibrations of a Kelvin-Voigt viscoelastic beam, supported by a nonlinear spring are analytically investigated in this paper. The governing equations of motion along with the compatibility conditions are obtained employing Newton’s second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model. The method of multiple timescales, a perturbation technique, is employed which ultimately leads to approximate analytical expressions for vibration response, and provides better insight into how the system parameters influence the vibration response. Finally, the effect of system parameters on the linear and nonlinear natural frequencies, vibration responses and frequency-response curves of the system is characterized.     

阻尼振动系统的强迫响应

本文讨论阻尼振动系统(离散或连续系统)在周期外激励作用下的强迫响应;系统的阻尼阵不能对角化.导出了强迫响应一般解的显式解析表达式.利用这些解式,本文对某些振动现象作了较普遍的解析讨论.如,从一般角度讨论了单阻尼振动系统特有的“固定振幅点”现象;讨论了同相位外激励作用下产生同相位响应的条件等等.本文导出的解式对一大类系统仅含很低阶的矩阵求逆运算,因而在计算机数值计算上.本方法比之于现有方法,具有程序简单、耗时少和精度高等优点.本方法可平行应用于转子动力学的不平衡响应分析.     

汽车电磁悬架系统的Noether对称性及其应用

研究了含有电磁悬架汽车振动系统的Noether对称性,给出了系统的守恒量,并通过守恒量求得系统的对称性解.以能量形式,建立汽车不同振动形式下的Lagrange(拉格朗日)方程.选取位移坐标为广义坐标,研究了各种振动形式下系统的Noether对称性,并给出相应的Noether恒等式、Killing方程和广义Noether定理.研究系统守恒量,运用存在的守恒量,给出一种新的求解汽车振动系统响应的方法;并应用到具体的车体振动系统计算中,给出了系统在转弯、制动或加速等情况下的位移响应和速度响应曲线.     

Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates

We consider a dam process with a general (state dependent) release rule and a pure jump input process, where the jump sizes are state dependent. We give sufficient conditions under which the process has a stationary version in the case where the jump times and sizes are governed by a marked point process which is point (Palm) stationary and ergodic. We give special attention to the Markov and Markov regenerative cases for which the main stability condition is weakened. We then study an intermittent production process with state dependent rates. We provide sufficient conditions for stability for this process and show that if these conditions are satisfied, then an interesting new relationship exists between the stationary distribution of this process and a dam process of the type we explore here.Supported in part by The Israel Science Foundation, grant no. 372/93-1.     

在任意不定常温度场和任意法向动载荷联合作用下中心开孔圆底扁球壳的动力问题

本文得出了在任意不定常温度场和任意法向动载荷联合作用下中心开孔圆底扁球壳的动力问题的解析解.我们假设温度沿壳体厚度直线分布.在第一部分.我们研究了常用边界条件下的中心开孔圆底扁球壳的自由振动.作为例子,我们计算了一边缘夹紧的扁球壳的自然基频(m=0),所得结果与E.Reissner[1]的结果作了比较.频率方程的解法是钱伟长[2]提出来的.这将在附录3中介绍.在第二部分,我们研究了在任意谐温度场和任意谐法向动载荷联合作用下的中心开孔圆底扁球壳的强迫振动.在第三部分,我们研究了在任意不定常温度场和任意法向动载荷联合作用下的具有初始条件的上述壳体的强迫振动.在附录1和2中,我们讨论了如何用应力函数来表示位移边界条件和m=1情形的边界条件.     

On positivity and stability of a class of time-delay systems

In this paper, the properties of positivity and stability dependent on and independent of the delays as well as the closed-loop stabilization under linear feedback of continuous-time linear time-invariant multi-input multi-output dynamic systems subject to point constant delays are discussed. A main attention is paid to provide joint stability and positivity results which are shown to be conflictive objectives in the general case. Links with positive realness are given for the single-input single-output case. Illustrative examples are also given.     

The product representation of a locally dependent random graph

Directed graphs with random black and white colourings of edges such that the colours of edges from different vertices are mutually independent are called locally dependent random graphs. Two random graphs are equivalent if they cannot be distinguished from percolation processes on them if only the vertices are seen. A necessary and sufficient condition is given for when a locally dependent random graph is equivalent to a product random graph; that is one in which the edges can be grouped in such a way that within each group the colours of the edges are equivalent and between groups they are independent. As an application the random graph corresponding to a spatial general epidemic model is considered.     

Characterization of Trimmed NURBS Surface Boundaries Using Topological Criteria

This paper presents an approach based on topology for the determination of characteristics and properties of curves used in the trimming of NURBS surfaces. Through discrete subdivision and topological criteria, a method is presented to determine characteristics of the boundary; such as whether the set of trimming curves forms a set of closed loops, whether trimming curves contain singularities or self intersections, and whether the boundary is simply connected. A surface mesh partitionning the parameter space is used, formed of isoparametric lines in both parametric directions. Topological properties of the cells of this mesh and their intersections with the trimming curves allow to localize the boundary. Topological treatment of this localization allows to define the interior and exterior of the face, and to refine the boundary localization. Singularities and self intersections of the boundary as well as voids in the face are investigated through the study of topological properties of neighbors. As an application, an algorithm for point localization is presented that very rapidly allows to determine whether a given point in parameter space lies inside, on the boundary or outside of the trimmed surface.     

Three-Dimensional Maneuvering and Control of Flexible Robots

This paper develops a general approach to the three-dimensional maneuver and vibration control of a robot in the form of a chain of flexible links. The equations for the rigid-body maneuvering motions are derived by means of Lagrange equations in terms of quasi-coordinates and the equations for the elastic deformations by means of ordinary Lagrange equations. The equations of motion are derived for the full system simultaneously, using recursive equations to relate the motions of a given link to the motions of the preceding links in the chain. The maneuver is carried out by means of joint torques and the vibration is suppressed by means of point actuators dispersed throughout the links. The controls are designed by the Liapunov direct method. A numerical example demonstrates the theoretical developments.     

The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer

The effect of vertical vibration on the onset of Marangoni convection in a horizontal layer of a viscous incompressible uniform liquid with a free surface and a hard (solid) or soft (impermeable and stress-free) wall is investigated. In the case of harmonic vibration, a dispersion relation is constructed in explicit form using continued fractions. From this, equations are obtained for determining the critical values of the parameters for all three main types of loss of stability. Neutral curves of the monotonic and oscillatory instability are constructed, for fixed frequency and amplitude of the vibration, in the form of a graph of the Marangoni number against the wave number. The regions of parametric resonances, corresponding to synchronous and subharmonic modes are determined. The frequency values for which a high-frequency asymptotic form is reached are obtained. The long-wave Marangoni oscillatory instability is investigated, and it is shown that in this case the Marangoni numbers are negative and depend only on the Prandtl and Biot numbers.     

One-way and two-way couplings of CFD and structural models and application to the wake-body interaction

The current study focuses on the wake-body interaction of a circular cylinder, whose transverse free vibration is modeled by a mass-spring-damper system coupled to a computational fluid dynamics (CFD) model for the flow and wake. We first simulate the free vibration of the elastically-mounted cylinder and the wake, and analyze the transverse load it exerts on the cylinder and its phase with the vibration. We vary the damping by three orders of magnitude and examine the difference in the wake-body interaction for slightly-damped and highly-damped systems. We then use the spectral properties of the free vibration and use them to construct two different types of forced vibrations: one consists only of the fundamental component of the free vibration, and the other accounts for all spectral properties of it. We compare the wake load for each type to that corresponding to the free vibration. The forced vibrations correspond to a one-way coupling and the information is communicated from the CFD model to the structural model, whereas the free vibration corresponds to a two-way coupling of the models. By comparing the spectral properties of the wake load, including the phase relation of its components with the vibration, which we obtained for the free vibration and for the equivalent forced vibration, we identify the effects of the wake feedback. The findings show that a forced vibration does not reproduce exactly the wake load at small and intermediate levels of structural damping. As the damping increases, the vibration changes from being in-phase with the wake load to being 90° out-of-phase with it, corresponding to two different wake states, and the forced vibration gives wake load that is very close to the one occurring in the case of full wake-body interaction.     

Transfer matrix method for the free and forced vibration analyses of multi-step Timoshenko beams coupled with rigid bodies on springs

Multi-step Timoshenko beams coupled with rigid bodies on springs can be regarded as a generalized model to investigate the dynamic characteristics of many structures and mechanical systems in engineering. This paper presents a novel transfer matrix method for the free and forced vibration analyses of the hybrid system. It is modeled as a chain system, where each beam and each rigid body with its supporting spring are dealt with one element, respectively. The transfer equation of each element is deduced based on separation of variables method. The system overall transfer equation is obtained by substituting an element transfer equation into another. Then, the free vibration characteristics are acquired by solving exact homogeneous linear equations. To compute the forced vibration response with modal superposition method, the body dynamic equations and augmented eigenvectors are established, and the orthogonality of augmented eigenvectors is mathematically proved. Without high-order global dynamic equation or approximate spatial discretization, the free and forced vibration analyses of the hybrid system are achieved efficiently and accurately in this study. As an analytical approach, the present method is easy, highly stylized, robust, powerful and general for the complex hybrid systems containing any number of Timoshenko beams and rigid bodies. Four numerical examples are implemented, and the results show that this method is computationally efficient with high precision.     

均布载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程,研究了波纹扁壳在均布载荷作用下的非线性受迫振动问题．采用格林函数方法,将扁壳的非线性偏微分方程组化为非线性积分微分方程组．再使用展开法求出格林函数,即将格林函数展开为特征函数的级数形式,积分微分方程就成为具有退化核的形式,从而容易得到关于时间的非线性常微分方程组．针对单模态振形,得到了谐和激励作用下的幅频响应．作为算例,研究了正弦波纹扁球壳的非线性受迫振动现象．该文的解答可供波纹壳的设计参考．     

Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes

This study deals with the vibration analysis of zigzag and chiral rotating functionally graded carbon nanotubes (FG-CNT) invoking Love's shell theory using wave propagation approach. The frequency equation is formed in the eigenvalue form. It has been shown that with the increase of angular speed, frequencies of forwarding curve decrease and backward curve increase. The phenomena of frequency versus length-and height-to-radius ratios are noted as decreasing and increasing, respectively, for rotating CNTs. The backward and forward frequency curves of clamped-free are lower throughout the computation than the clamped-clamped zigzag and chiral carbon nanotube depending upon the rotating speed. MATLAB software is used to calculate the rotating (backward and forward) frequencies of SWCNTs and the frequency peaks in the present results show excellent stability across a wide range of parameters. Using geometrical and material parameters, the vibration results are given in tabular and graphical form. It is thus desirable to produce more precise estimations of the vibrational frequencies of CNTs. The present results are compared with earlier literature using simply supported boundary conditions and show a good coincidence.     

基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究

以纳米机器人等智能器件中的功能梯度纳米板结构为研究对象,基于非局部应变梯度理论,研究了其弯曲和屈曲问题.推导了一般情况下的功能梯度纳米板运动方程,弯曲和屈曲作为其特例可简化而成.分析了非局部尺度参数、材料特征尺度参数、梯度指数、纳米板尺寸等对弯曲挠度和临界屈曲载荷的影响.结果表明:不同高阶连续介质力学理论下的最大挠度都...     

Bifurcation and chaos of thin circular functionally graded plate in thermal environment

A ceramic/metal functionally graded circular plate under one-term and two-term transversal excitations in the thermal environment is investigated, respectively. The effects of geometric nonlinearity and temperature-dependent material properties are both taken into account. The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power law distribution of the volume fraction of the constituents. Using the principle of virtual work, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic forcing excitation and thermal load are derived. For the circular plate with clamped immovable edge, the Duffing nonlinear forced vibration equation is deduced using Galerkin method. The criteria for existence of chaos under one-term and two-term periodic perturbations are given with Melnikov method. Numerical simulations are carried out to plot the bifurcation curves for the homolinic orbits. Effects of the material volume fraction index and temperature on the criterions are discussed and the existences of chaos are validated by plotting phase portraits, Poincare maps. Also, the bifurcation diagrams and corresponding maximum Lyapunov exponents are plotted. It was found that periodic, multiple periodic solutions and chaotic motions exist for the FGM plate under certain conditions.     

Solutions fondamentales pour un milieu élastique à isotropie transverse

The general solution of an axisymmetric problem for a homogeneous medium is given for a surface concentrated loading, and from it a closed form solution for a point force is deduced. The infinite and semi-infinite spaces are considered for various boundary conditions.