Imperfection sensitivity in the nonlinear vibration of initially stresses functionally graded plates

In this paper, the nonlinear partial differential equations of nonlinear vibration for an imperfect functionally graded plate (FGP) in a general state of arbitrary initial stresses are presented. The derived equations include the effects of initial stresses and initial imperfections size. The material properties of a FGP are graded continuously in the direction of thickness. The variation of the properties follows a simple power-law distribution in terms of the volume fractions of the constituents. Using these derived governing equations, the nonlinear vibration of initially stressed FGPs with geometric imperfection was studied. The present approach employed a perturbation technique, the Galerkin method and the Runge–Kutta method. The perturbation technique was used to derive the nonlinear governing equations. The motion of imperfect FGPs was obtained by performing the Galerkin method and then solved by the Runge–Kutta method. Numerical solutions are presented for the performances of perfect and imperfect FGPs. The nonlinear vibration of a simply supported ceramic/metal FGP was solved. It is found that the initial stress, geometric imperfection and volume fraction index greatly change the behavior of nonlinear vibration.     

圆柱壳几何大变形非线性频率求解的渐近摄动法

为解决圆柱壳在工作状态中由几何大变形而引起的弱非线性振动问题，将渐近摄动法引入求解考虑几何非线性的薄壁圆柱壳振动频率。首先，应用Donnell＇s简化壳理论获得了考虑几何大变形情况下具有位移三次项的非线性频率方程，把位移及频率以非线性参数的幂级数形式展开，并令同次幂的非线性项系数相等，由此得到非线性频率一次近似值与初始振幅的一系列耦合代数方程，引入Galerkin＇s方法对非线性频率方程进行解耦正交并忽略其中的永年项，考虑了对应实数根，各阶频率对应的振幅间不存在相互耦合的内共振现象，最终在引入小参数后用摄动法求出了非线性频率的一次近似解。计算结果表明，几何非线性使薄壁圆柱壳产生硬化，其非线性频率升高，并同时讨论了线性、非线性频率与节径数及初始位移之间的关系。     

Nonlinear membrane model for large amplitude vibration of single layer graphene sheets

The nonlinear vibrational properties of single layer graphene sheets (SLGSs) are investigated using a membrane model. The nonlinear equation of motion is considered for the SLGSs by including the effects of stretching due to large amplitudes. The equation of motion is numerically solved utilizing the finite difference method for SLGSs with different initial and boundary conditions, sizes and pretensions. It is concluded that the nonlinear fundamental frequency of SLGSs increases by increasing the pretension and initial velocity. In addition, it is observed that an increase in the pretension weakens the effects of the initial velocity on the fundamental frequency, such that the fundamental frequency approximately becomes independent of the initial velocity. This is an important feature of the vibrating systems consisting of SLGSs which are used in the nano-electromechanical systems (NEMS), where resonators with a specific fundamental frequency and independent of the initial velocity are of interest.     

Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers

Nonlinear vibration analysis of thin circular pre-stressed functionally graded (FG) plate integrated with two uniformly distributed piezoelectric actuator layers with an initial nonlinear large deformation are presented in this paper. Nonlinear governing equations of motion are derived based on classical plate theory (CPT) with von-Karman type geometrical large nonlinear deformations. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectrically actuated circular FG plate are derived. An exact series expansion method is used to model the nonlinear electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages are studied and their nonlinear effects are evaluated. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FG plate on the dynamic characteristics of the structure.     

具有初始应力加劲板的非线性动力学特性

针对工程结构中常用的加劲板,研究了其在初始应力作用下的非线性振动特性。将母板与加劲肋分开考虑,其中母板按薄板理论考虑,加劲肋按Euler-Bernoulli梁理论考虑,根据母板与加劲肋的应力与应变关系建立系统的应变能表达式,同时结合系统的动能表达式,并利用Lagrange方程建立系统的非线性动力微分方程。运用椭圆函数求得加劲板单模态的非线性频率,采用同伦分析方法求解加劲板的3∶1内共振。通过参数分析,重点讨论初始应力变化对系统非线性动力特性的影响,并得出初始应力的存在对加劲板非线性动力特性的影响规律。     

斜拉桥拉索模态耦合非线性共振响应特性

考虑拉索垂度及几何非线性的影响,导出了斜拉索的面内外耦合非线性振动方程。通过Galerkin方法,将偏微分方程化为常微分方程,用多尺度法对耦合方程求解,得出了面内外一阶模态耦合振动特性。对典型斜拉桥拉索进行了计算,结果表明:拉索可能产生1:1内共振,共振频率区间及振动幅值与索的垂度大小有关;索内阻尼对共振特性也有较大的影响;由于非线性的影响,响应幅值与初始条件有关。     

架空电线在悬链状态下的非线性振动响应分析

本文推导了考虑初始弛度的输电线的非线性振动偏微分方程 ,并采用伽辽金过程而得到常微分方程。应用多尺度法和L -P(Lindstedt_poincare)法求得了受谐扰力和脉冲激励的输电线非线性振动响应的近似解。分析了初始弛度对响应的影响 ,得到了一些有益的结果。这些结果对于架设电线或维护输电线安全 ,都将是有益的参考。     

Stochastic modeling of ecosystem with two competing species

A nonlinear stochastic model describing two competing species of grass and woody vegetation is proposed. The nonlinear behaviors are caused by the interspecies and intraspecies competitions, and the stochastic variation describes the uncertainty in the cattle stocking rate. Two types of stochastic processes are proposed to model the variation part of the stocking rate. One is Gaussian white-noise process, and another is randomized sinusoidal process. The system nonlinear behaviors are investigated for the deterministic case and the cases with two different stochastic variations. Some key characteristics of the system dynamics are found to be different for cases with and without stochastic variations. With the stochastic variation, a single stable state in the deterministic system is diffused into a region of stable states, and a separatrix dividing the two attraction zones no longer exists. The system may follow different trajectories and lead to different outcomes, beginning from the same initial state. It is also found that the stationary probability of the system response depends on the initial conditions, a special phenomenon for the investigated nonlinear system. Although the white-noise process and the randomized sinusoidal process are quite different in nature, the qualitative behaviors of the system are similar. Furthermore, effects of the initial state and the intensity of the stochastic variations on the system behaviors are investigated.     

The nonlinear vibration of an initially stressed laminated plate

Nonlinear partial differential equations of motion for a laminated plate in a general state of non-uniform initial stress are presented in various plate theories. This study uses Lo’s displacement field to derive the governing equations. The higher-order terms in Lo’s theory can be disregarded, to obtain the equations of simpler forms and even other theories for laminated plate. These nonlinear partial equations are transformed to ordinary nonlinear differential equations using the Galerkin method. The Runge–Kutta method is used to obtain the ratio of nonlinear frequency to linear frequency. The numerical solutions of an initially stressed laminate plate based on various plate theories obtained by the Galerkin and Runge–Kutta method are presented herein. Using these equations with various theories, the nonlinear vibration behavior of laminated plate is studied. The results show that apparent discrepancies exist among the various displacement fields, which indicates the transverse shear strain, normal strain and initial stress state have great effect on the vibration behavior of laminate plate under nonlinear vibration.     

大跨度斜拉桥的施工非线性倒拆分析

以斜拉桥静载弯曲能量作为目标函数,综合考虑斜拉索垂度效应、梁柱效应和大位移效应,并采用混合法求解系统方程,确定斜拉索成桥时的初始张拉力。以由此确定的斜拉桥初始状态作为施工非线性倒拆分析的初始状态,倒拆分析的过程中需要考虑相应阶段的混凝土收缩、徐变影响,倒拆分析的结果作为斜拉桥施工阶段的斜拉索张拉力,即斜拉桥施工阶段的最终优化结果。该方法经过算例验证,取得了较好的结果,为斜拉桥的施工控制提供了依据。     

考虑刚度非线性软化的修正硬化土模型

基于硬化土模型(HS模型)只能描述土体应变硬化行为和刚度线性软化现象,提出了考虑刚度非线性软化的修正硬化土模型。修正模型采用与分段幂函数关系描述初始骨干曲线,并考虑了初始静偏应力和加载速率对应力-应变关系的影响。修正HS模型能较好地描述不同初始静偏应力和加载速率下土体的应变硬化、应变软化和刚度非线性软化等特征。最后,基于原状杭州软粘土的三轴试验,验证了修正模型的合理性。     

张力膜结构初始形态分析的曲面四边形单元

张力膜结构的初始形状不能随意选择,它必须符合平衡条件和建筑使用要求。根据几何非线性有限元理论,提出张力膜结构初始形态分析的8结点曲面四边形等参单元。通过建立曲线坐标,在应变的线性部分引入法向位移及单元曲率和扭率的影响,推导了张力膜结构的单元刚度矩阵和结点力列阵。采用完全的Newton-Raphson迭代法求解非线性方程组。数值算例表明该单元是一种高效、稳定和可靠的单元。     

气泡群振荡及噪声数值模拟研究

针对水中气泡群振荡及辐射噪声问题,基于均质混合流模型,考虑气泡动力学作用,建立数学模型;并采用拉格朗日有限体积法,对气泡群振荡过程及辐射噪声进行数值模拟。重点研究了初始气泡半径、含气率及气泡群尺度等对振荡辐射噪声的影响。结果表明,随着初始含气率、气泡群尺度增大,气泡辐射噪声频率逐渐减小、声压幅值逐渐增大即声强增大;初始气泡半径对辐射噪声频率影响较小,但声压幅值随其增大而减小。     

Comparison of linear and nonlinear analysis in estimating the Thomas model parameters for methylene blue adsorption onto natural zeolite in fixed-bed column

Comparison analysis of linear least square method and nonlinear least square method for estimating the kinetic parameters was made using the experimental column data of methylene blue (MB) adsorption onto zeolite at different flow rates and initial concentration. The data were fitted to Thomas model equations using linear and nonlinear regressive analysis, respectively. The error analysis was performed. Present investigation showed that the linear and nonlinear methods are both suitable to predict the breakthrough curves using Thomas model parameters and the nonlinear method is better.     

分缝结构的非线性响应分析

本文给出了模拟结构分缝张开和闭合的有限元模型，该模型可以模拟缝面初始压力、抗拉强度的影响，且具有形成方便，适用范围广等优点；文中导出了非线性有限元的计算公式，编制了相应的非线性有限元程序，并对两个块体接触问题和一个分缝拱肋结构的非线性动力响应进行了计算，研究了不同接缝参数、不同初始压力及基座是否设缝对结构非线性响应的影响，给出了有关结论。     

分数阶脉冲微分方程初值问题解的存在性

利用Green函数可以将分数阶微分方程初值问题转化为等价的积分方程.近来此方法被应用于讨论非线性分数阶微分方程初值问题解的存在性.本文讨论菲线性分数阶脉冲微分方程初值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Carathéodory条件,利用非紧性测度的性质和M(o)nch,8不动点定理证明解的存在性.     

径向载荷作用下复合材料圆柱壳的非线性动力屈曲

采用半解析法求解径向阶跃载荷作用下复合材料圆柱壳的非线性动力屈曲。基于一阶剪切变形理论，由Hamilton原理推导出包含横向剪切变形以及几何初缺陷的圆柱壳的非线性动力方程，位移及载荷沿周向采用级数展开，由Galerkin方法得到微分方程组，通过有限差分法求解；根据响应情况，由B—R准则判定屈曲，确定屈曲临界载荷。     

位势井及其对具异号源项波动方程的应用

本文研究具有两个异号非线性源项的波动方程的初边值问题。应用位势井方法,解决了不具正定能量情况下问题整体解的存在性问题。证明了对于非线性项的指数在一定条件下,问题存在整体弱解。从而拓展了已知结果。     

基于钢塔架敏感性的连续倒塌分析与数值模拟

通过引入Pandey提出的敏感性指标,给出钢塔架关键杆件的识别方法。基于备用荷载路径设计法,对钢塔架结构在风荷载作用下的连续倒塌进行线弹性静力、非线性静力、线弹性动力、非线性动力等4种分析。结果表明:结构动力响应的大小取决于风荷载作用方向及关键杆件出现的位置;线弹性静力分析是偏于保守,非线性动力分析是更加精确的;建议采用非线性动力方法对钢塔架进行连续倒塌评估。通过失效杆件的等效反力卸载过程,模拟钢塔架连续倒塌的动力过程,提出了等效反力的初始持续时间t₀(考虑初始状态)和卸载时间t_p的计算方法。     

Linearized model for error-compensated kinetic determinations without prior knowledge of reaction order or rate constant

This paper describes a new algorithm for calculation of reaction orders, rate constants, and initial and final values of detector signal from several signal vs time data points. The algorithm utilizes a linearized version of the rate equation and is intended primarily to provide initial estimates of these kinetic parameters for other curve-fitting methods. However, under some circumstances, the linearized model can provide sufficiently reliable results that subsequent processing by other methods is not needed. Simulated data with different levels of superimposed noise, data densities, reaction orders, rate constants, and signal change are used to evaluate the algorithm both for its primary purpose of providing initial estimates for other curve-fitting methods and as an independent method. Results are compared with those obtained with a nonlinear least-squares method and two initial-rate methods. The new algorithm provides less reliable results than those obtained by the nonlinear curve-fitting method for some situations (e.g. reaction orders greater than two, low data densities) but has the advantage that it is applicable to reaction orders at and near unity where the nonlinear method to which it is compared fails.