功能梯度矩形板的非线性自由振动

研究了功能梯度矩形薄板的非线性自由振动问题.采用幂律分布规律描述功能梯度材料沿厚度的梯度性质,基于von Kámán理论,建立了功能梯度薄板的非线性振动控制方程.应用Bubnov-Galerkin法得到了功能梯度矩形薄板的单模态非线性振动的时域常微分方程,借助其势能函数分析了系统的周期振动状态.采用Lindstedt-Poincaré法和Runge-Kutta法分别获得了功能梯度矩形薄板单模态非线性周期振动的摄动解和数值解.研究表明:功能梯度薄板非线性振动控制方程中包含表征拉弯耦合效应的控制项,这导致其常微分方程中出现二次项;系统振幅在板横向的正负两个方向上是不相等的,其振动存在关于板中面的不对称性.     

某纳米级机械振荡器的非线性自由振动特性分析

主要对某纳米级振荡器的非线性自由振动特性进行了初步的研究。基于Lennard-Jones势能推导了单位面积的平板与无限长平板间的势能。为计算方便，引入无量纲化变量，给出了纳米级振荡器非线性自由振动的状态方程和Hamilton函数。根据负的Lennard-Jones力曲线和弹簧力曲线的相交特性，对非线性自由振动的平衡位置特性进行了分析，指出了影响平衡位置特性的若干重要参量。对给定参量的非线性自由振动的相位图特性、振动周期等进行了定量及定性研究。     

高维含参系统的非线性振动

提出了一种适于高维一般含参系统的改进平均法，分别给出了该系统在自治及非自治情况下的非线性振动近似解的计算公式，并用该法研究了ＰＤ控制器及ＰＩＤ控制器作用下的磁力轴承转子系统的非线性振动问题．     

颇具特色的教材《振动力学》

由高等教育出版社出版的普通高等教育“九五”教育部重点教材《振动力学》,是经教育部工程力学专业教学指导委员会审定出版的．该书的首位作者是上海交通大学博士生导师刘延柱教授．《振动力学》第一个显著特色是在内容体系编排上将线性振动和非线性融为一体．为教学上便于接受,在基本部分内仅用直观的定性分析方法（如相平面法）和简单的近似解析方法（如谐波平衡法）讨论非线性振动．单自由度系统自由振动和受迫振动两章统一处理线性振动和非线性振动,将前者作为后者的特例,自激振动一章着重从物理概念上说明非线性现象．多自由度系统…     

求单自由度非线性振动数值解的Taylor展开法

本文针对具体单自由度非线性振动系统建立了便于计算机实现的形式，求数值解Ｔａｙｌｏｒ展开法，给出实例与Ｒｕｎｇｅ－Ｋｕｔｔａ法进行比较，并应用于混沌解的计算．     

横向磁场中轴向运动铁磁梁的磁弹性主共振

研究在外激励力与磁场作用下轴向运动铁磁梁的磁弹性非线性主共振问题．基于弹性理论和电磁理论,给出梁的动能和弹性势能表达式,根据哈密顿原理,推导出磁场中轴向运动铁磁梁的磁弹性双向耦合非线性振动方程．通过伽辽金积分法进行离散,得出两端简支边界条件下铁磁梁磁弹性非线性强迫振动方程．应用多尺度法对方程进行求解,得出幅频响应方程．最后通过算例,给出铁磁梁的幅频特性曲线、振幅-磁感应强度和振幅-外激励力曲线并进行分析．结果显示,在幅频响应曲线中铁磁梁的轴向运动速度、外激励力、轴向拉力越大,共振振幅越大;而磁感应强度越大,振幅越小．     

斜拉桥拉索-阻尼器系统非线性响应分析

考虑索的抗弯刚度、垂度及几何非线性的影响，得出了索一阻尼器系统的空间非线性振动偏微分方程，用中心差分法将微分方程在空间内离散，导出了系统的非线性振动常微分方程组。结合Newmark法及虚拟力法提出了一种用于求解非线性振动瞬态响应的杂交分析算法。并以典型的斜拉桥拉索为研究对象，给出了数值算例，并与Runge—Kutta直接积分法进行了比较，说明了杂交算法的准确性及有效性。     

强非线性动力系统周期解分析

给出一类强非线性动力系统周期解存在性，唯一性和稳定性的简易差别法以及周期解的摄动法。本差别法把问题归结为干扰力在相应的未扰系统振动周期上的功函数及其导数的讨论，其限制条件比现有结果弱。本摄动法可以认为是经典Ｌｉｎｄｓｔｅｄｔ－Ｐｏｉｎｃａｒｅ（Ｌ－Ｐ）法在强非线性振动系统的推广。它与Ｌ－Ｐ法的主要区别在于假设系统的振动频率为相角的非线性函数。     

应用于具有二次,三次非线性系统的增量谐波平衡法

本文导出了适用于具有二次、三次非线性的微分方程组的增量谐波平衡法，研究了扁拱的相加型和相减型的联合共振问题以及二自由度系统的强非线性振动问题，算例表明，增量谐波平衡法是一个求解多自由度系统强非线性振动的有效的半解析的数值方法。     

附磁阶梯变厚度悬臂梁压电俘能器的理论建模及分析

论文建立了一种附磁阶梯变厚度压电悬臂梁的动力学模型并分析了系统的俘能特性。基于Euler-Bernoulli梁理论分段建立系统能量函数并引入非线性磁势能,利用Lagrange方程建立了系统机电耦合动力学方程;利用数值方法分析了磁间距对系统振动特性的影响,此外还研究了系统单稳态和双稳态响应,探讨了厚度比、长度比、磁间距和外激励幅值对系统动力学响应和俘能特性的影响。结果表明,磁间距是影响系统势能的主要因素,调节磁间距可使系统产生单稳态和双稳态响应,从而有效提高俘能器俘能特性;与传统等截面悬臂梁压电俘能器相比,通过优化结构参数,附磁阶梯变厚度悬臂梁压电俘能器能够发生明显的非线性振动现象,实现宽频带振动能量采集。     

Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.     

Modeling rock fragmentation by coupling Voronoi diagram and discretized virtual internal bond

The rock fragmentation involves the inter-block and the intra-block fracture. A simulation method for rock fragmentation is developed by coupling Voronoi diagram (VD) and discretized virtual internal bond (DVIB). The DVIB is a lattice model that consists of bonds. The VD is used to generate the potential block structure in the DVIB mesh. Each potential block may contain any number of bond cells. To characterize the inter-block fracture, a hyperelastic bond potential is employed for the bond cells that are cut by the VD edges. While to characterize the intra-block fracture, an elastobrittle bond potential is adopted for the bonds in a block. By this method, both the inter-block and intra-block fracture can be well simulated. The simulation results suggest that this method is a simple and efficient approach to rock fragmentation simulation with block smash.     

Nonlinear Schrodinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrodinger equations with combined power-type nonlinearities and harmonic potential.By constructing a variational problem the potential well method is applied.The structure of the potential well and the properties of the depth function are given.The invariance of some sets for the problem is shown.It is proven that,if the initial data are in the potential well or out of it,the solutions will lie in the potential well or lie out of it,respectively.By the convexity method,the sharp condition of the global well-posedness is given.     

A variational approach to coarse graining of equilibrium and non-equilibrium atomistic description at finite temperature

The aim of this paper is the development of equilibrium and non-equilibrium extensions of the quasicontinuum (QC) method. We first use variational mean-field theory and the maximum-entropy (max-ent) formalism for deriving approximate probability distribution and partition functions for the system. The resulting probability distribution depends locally on atomic temperatures defined for every atom and the corresponding thermodynamic potentials are explicit and local in nature. The method requires an interatomic potential as the sole empirical input. Numerical validation is performed by simulating thermal equilibrium properties of selected materials using the Lennard-Jones (LJ) pair potential and the embedded-atom method (EAM) potential and comparing with molecular dynamics results as well as experimental data. The max-ent variational approach is then taken as a basis for developing a three-dimensional non-equilibrium finite-temperature extension of the QC method. This extension is accomplished by coupling the local temperature-dependent free energy furnished by the max-ent approximation scheme to the heat equation in a joint thermo-mechanical variational setting. Results for finite-temperature nanoindentation tests demonstrate the ability of the method to capture non-equilibrium transport properties and differentiate between slow and fast indentation.     

Analysis of stress-strain relations by use of an anisotropic hardening plastic potential

A method of analyzing plastic behavior by use of an anisotropic hardening plastic potential is proposed. The plastic potential surface in deviatoric stress space is assumed to be the same as the equi-plastic-strain surface. Stress-strain relations in combined loading and in multi-axial cyclic loading are calculated by use of the anisotropic hardening plastic potential and the normality rule of the plastic strain increment vector to the plastic potential surface, which are experimentally determined or confirmed by subjecting thinwalled tubular test specimens of $\text{60}\text{40}$ brass to combined axial load, internal pressure and torsion. The calculated results agree fairly well with the experimental observations.     

Finite difference method of transinet nonlinear free surface wave problems

A finite difference method is developed for computing the two-dimensional transient potential flow generated by an impulse on the free surface. Both the dynamic and kinematic free surface conditions are considered in nonlinear version. the primary features of the present paper include the use of special coordinates transformations so that the geometry of the flow field is transformed into a time-invariant region, presents an iteration process, by which the velocity potential is computed as the solution of a Poisson equation, the application of fast Fourier transform (FFT) technique results in a tri-diagonal system of equations which can be readily solved by the Thomas algorithm, the computing time is significantly reduced. Thus an efficient technique for handling the transient potential problems is well justified. The feasibility of the present method has been verified by two examples including different initial disturbances respectively.     

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method.The invariance of some sets under the flow of these problems and the vac- uum isolation of solutions are obtained by introducing a family of potential wells.Then the threshold result of global existence and nonexistence of solutions are given.Finally, the problem with critical initial conditions are discussed.     

Buckling instabilities in coupled nano-layers

We study the dynamic buckling of a pair of dissimilar Euler-Bernoulli beams subject to compressive edge loading whose transverse displacements are coupled through non-linear interactions, a problem motivated by the mechanics of graphene layers. The transverse coupling models van der Waals interaction and is derived from a Lennard-Jones 12-6 potential. The beams are assumed to be a fixed distance apart at their ends, although this distance is not necessarily equal to the equilibrium distance as identified from the Lennard-Jones potential. Therefore, the equilibrium configuration is not necessarily straight. Via a Galerkin method, the governing equations are reduced to a system that can be used to calculate equilibrium configurations as well as the stability of these configurations. We show that the buckling instability in this model is significantly affected by the presence of the interaction force as well as the separation of the graphene layers at the boundaries.     

Reducing numerical diffusion in interfacial gravity wave simulations

We demonstrate how the background potential energy is an excellent measure of the effective numerical diffusion or antidiffusion of an advection scheme by applying several advection schemes to a standing interfacial gravity wave. All existing advection schemes do not maintain the background potential energy because they are either diffusive, antidiffusive, or oscillatory. By taking advantage of the compressive nature of some schemes, which causes a decrease in the background potential energy, and the diffusive nature of others, which causes an increase in the background potential energy, we develop two background potential energy preserving advection schemes that are well‐suited to study interfacial gravity waves at a density interface between two miscible fluids in closed domains such as lakes. The schemes employ total variation diminishing limiters and universal limiters in which the limiter is a function of both the upwind and local gradients as well as the background potential energy. The effectiveness of the schemes is validated by computing a sloshing interfacial gravity wave with a nonstaggered‐grid Boussinesq solver, in which QUICK is employed for momentum and the pressure correction method is used, which is second‐order accurate in time. For scalar advection, the present background potential energy preserving schemes are employed and compared to other TVD and non‐TVD schemes, and we demonstrate that the schemes can control the change in the background potential energy due to numerical effects. Copyright © 2005 John Wiley & Sons, Ltd.     

不同理论下广义压电热弹性问题的有限元求解

基于G-L和L-S广义压电热弹性理论研究了无限大厚压电板在上下表面受到条带状热冲击时的广义压电热弹性问题。在时间非常短的情况下，为避免积分变换求解带来的精度丢失，采用有限元方法对问题在时间域进行直接求解，获得压电板在热冲击作用下的温度、位移、应力及电势等，并将结果与经典压电热弹性理论进行比较。结果表明，直接求解方法可以准确描述热在介质中以有限的速度传播。