受轴向激励弹性支承梁的稳定性分析

对于广泛存在的弹性支撑梁,首次呈现支承弹簧刚度对轴向激励下梁横向振动稳定性的影响.应用Hamilton原理,建立了两端由线性弹簧支撑的受轴向激励梁的动力学控制方程.通过解析方法计算了受轴向压力梁的固有频率,得到了支撑弹簧刚度与系统固有频率和临界轴力的关系.Galerkin截断后,通过多尺度法和Runge-Kutta法,计算得到了梁参激振动稳态响应的半解析与数值解.讨论了激励幅值、支撑弹簧刚度、平均轴力对系统非线性响应幅值及软硬特性的影响.利用Routh-Hurwitz稳定性判据,求得系统的参激稳定边界,着重讨论了支撑弹簧刚度、阻尼系数的影响.研究发现,边界支撑弹簧的刚度可以显著改变受轴向激励梁的参激稳定边界.因此,研究结果将为广泛存在受到轴向激励结构的设计提供指导.     

四边简支矩形薄板的双Hopf分岔分析

基于奇异性理论,研究了主参数共振-1∶3内共振情形下参数激励与外激励联合作用下四边简支矩形薄板的双Hopf分岔问题.考虑弱阻尼和弱激励的情形,得到了四边简支矩形薄板的分岔方程,给出了四边简支矩形薄板在参数平面μ-σ1上的分岔图.对参数激励与外激励联合作用下四边简支矩形薄板的阻尼系数、外激励、参数激励以及调谐参数进行不同的取值,通过数值模拟得到了四边简支矩形薄板平衡解将发生Hopf分岔,并分岔出周期解,薄板系统的非线性振动形式为周期运动.当四边简支矩形薄板的参数满足给定条件时,我们得到薄板的1∶3共振双Hopf分岔.随后,四边简支矩形薄板将会呈现概周期振动.     

时滞反馈作用下压电梁的参数共振分析

基于Hamilton原理建立了受控压电梁的参数振动方程,研究了轴向激励压电梁时滞速度反馈控制的主参数共振.采用非线性振动的多尺度法研究了压电梁的亚谐波主参数共振,并对其稳定域进行分析.通过算例分析得到了不同时滞,控制增益,轴力影响下压电梁参数共振的稳定域和时程曲线.结果表明:时滞值增大,系统承受轴向力减小,相对于控制增益则反之.同时,随着轴向力增大,在一定范围内主动阻尼增加,时滞反馈能有效降低响应幅值.随着时滞值增大,减振效果变差.     

Variational formulation of optimization against loss of dynamic stability

The variational formulation of optimization problems for a simply supported beam, subject to a longitudinal force, periodic in time is given in this paper. Objective functions characterizing the parametric resonance are introduced. The paper deals with the problem of finding the best possible tapering of the beam, i.e. the shape of the beam, which maximizes or minimizes one of the objective functions, under the constraint of constant volume. In some cases, the optimization problems under circumstances of parametric resonance resolve into optimization problems with respect to natural frequency.     

Multicriteria optimization of parametrically excited systems against loss of dynamic stability

The Mathieu-Hill differential equation is often encountered in engineering and physics problems. The stability of periodic response of vibrating nonlinear systems can be also examined by means of the variational Mathieu-Hill equation. Therefore the problems of the optimization of parametrically excited systems described with Mathieu-Hill equations against a loss of stability are important.In this paper, a formulation of monocriterion and multicriteria optimization problems for mechanical elements like bars or plates, subjected to a parametrically exciting force, periodic in time is given.In multicriteria optimization the scalar objective functions characterizing the parametric resonance are introduced. The paper deals with the problem of finding the control function—function of the shape (the area of the cross-section of the beam or the thickness of the plate) which maximizes or minimizes the objective function under the constraint of constant volume. In some cases, the optimization problems under conditions of parametric resonance resolve into optimization problems with respect to natural frequency.The examples of optimization against loss of stability are solved and analysed in the state of parametric periodic resonance.     

Architectures for vibration-driven micropower generators

Several forms of vibration-driven MEMS microgenerator are possible and are reported in the literature, with potential application areas including distributed sensing and ubiquitous computing. This paper sets out an analytical basis for their design and comparison, verified against full time-domain simulations. Most reported microgenerators are classified as either velocity-damped resonant generators (VDRGs) or Coulomb-damped resonant generators (CDRGs) and a unified analytical structure is provided for these generator types. Reported generators are shown to have operated at well below achievable power densities and design guides are given for optimising future devices. The paper also describes a new class-the Coulomb-force parametric generator (CFPG)-which does not operate in a resonant manner. For all three generators, expressions and graphs are provided showing the dependence of output power on key operating parameters. The optimization also considers physical generator constraints such as voltage limitation or maximum or minimum damping ratios. The sensitivity of each generator architecture to the source vibration frequency is analyzed and this shows that the CFPG can be better suited than the resonant generators to applications where the source frequency is likely to vary. It is demonstrated that mechanical resonance is particularly useful when the vibration source amplitude is small compared to the allowable mass-to-frame displacement. The CDRG and the VDRG generate the same power at resonance but give better performance below and above resonance respectively. Both resonant generator types are unable to operate when the allowable mass frame displacement is small compared to the vibration source amplitude, as is likely to be the case in some MEMS applications. The CFPG is, therefore, required for such applications.     

桥式吊车系统的伪谱最优控制设计

对于桥式吊车系统的最优控制问题,根据实际的工况要求,性能指标有时不一定是标准的二次形式.同时,在实际的控制问题中,状态和控制输入往往会受到一些边界条件和路径过程中的约束.针对这一问题,本文应用Chebyshev伪谱优化算法来处理,它可以处理状态和控制约束的非线性最优化问题以及一个非标准的目标函数.首先对桥式吊车系统模型进行一系列的坐标变换,将其转变为上三角系统形式的误差模型.然后将桥式吊车最优控制问题转化成具有一系列代数约束的参数优化问题,即非线性规划问题.通过求解离散化后的参数优化问题,得到桥式吊车的最优控制律.本文还给出了Chebyshev伪谱最优解的可行性和一致性分析.最后,在仿真研究中验证该控制器的有效性.     

Resonance of a nonlinear forced system with two-frequency parametric and self-excitations

The resonance of a single-degree-of-freedom forced nonlinear system with two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms and the excitation force with close frequencies is examined. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The behaviour of steady-state responses and their stability are studied. The effects of damping, and magnitudes of self-excitation, parametric excitation and the excitation force are analyzed. The theoretical results are verified by numerical integration results of the governing equation and the modulation equations.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method

The eigenvalue problems of clamped-free and hinged-hinged Bernoulli-Euler beams on elastic foundation with a single edge crack, axial loading and excitation force were numerically formulated using the differential quadrature method (DQM). Appropriate boundary conditions accompanied the DQM to transform the partial differential equation of a Bernoulli-Euler beam with a single edge crack into a discrete eigenvalue problem. The DQM results for the natural frequencies of cracked beams agree well with other literature values. The sampling point number effect, the location of the crack effect and the depth of the crack effect on the accuracy variation of calculated natural frequencies are presented by using two elements in this work. The effects of axial loading, foundation stiffness, opening crack and closing crack are also studied.     

On the problem of spacing of oil wells and control of their production rates

A statement and a numerical solution of the optimization problem for sites of spacing of wells (well clusters) and modes of their operation in an oil field are presented. The mathematical problem represents a parametric problem of optimal control of the distributed system with concentrated sources, which is described by partial differential equations. Similar problems arise in the development and control of technological processes of various assignments and the design of control systems of technical objects. For its numerical solution, the initial problem is reduced to a finite-dimensional optimization problem with constraints of special features. Formulas of a gradient of the functional of the reduced problem are derived. Results of the solution of control problems are given.     

基于MEMS的参量放大的设计与实现

参量放大效应是一种非线性振动放大效应,利用该效应可以极大地增加MEMS陀螺和加速度计等谐振式传感器的振幅,从而提高谐振式传感器检测灵敏度和信噪比。根据理论分析和有限元仿真结果,设计了一种MEMS参量放大结构。该结构采用SOG工艺加工,在950Pa左右的气压环境下对该结构进行了实验研究,并得到了该结构参激电压、相位与放大倍数之间的关系。实验结果表明：参量放大效应的实测结果与理论分析结果吻合很好,当参数激励电压为5.7V时,参量放大倍数超过50倍,大幅提高了结构的振幅,为以后应用参量放大效应提高MEMS惯性仪表性能提供了一种新方法。     

Optimal actuation of nonlinear resonant systems

In this study, we examine a new approach for actuation of dynamical systems with minimum work and maximum amplitude while maintaining constraints on the actuation force. Two methodology issues are addressed in the paper: sensitivity analysis about the nonlinear transient response and exploration of the strongly nonlinear relationship between the two objectives and the actuation design variables. The optimization analysis is carried out on lightly damped Duffing systems. The formulation of the optimization problem is found ideally suited to resolve the difficulty of dependence of response on initial conditions. The tradeoff curve of work and amplitude is computed. The optimal actuation strove to compensate for the limited force amplitude by an abrupt change in the force in time. Finally the optimization procedure is demonstrated on the kinematic design of hovering insect flight. The optimal design gives 50% reduction in power consumption, with a larger cutback for an actuation with high acceleration.     

Shape preserving design of vibrating structures using topology optimization

In several engineering components, the shape of some functional surfaces needs to be preserved in order to avoid losing performance or even its functionality when subjected to loads. This is particularly important when tight tolerances are required for operational conditions in some regions. If the deformation significantly affects product functionality, it is interesting to use a shape preserving design technique. This will often reduce deformation in a local region. To achieve that, we deal with topology optimization of elastic, continuum structures with Rayleigh damping, subjected to time-harmonic, design-independent external dynamic loading with prescribed excitation frequency, amplitude and spatial distribution. In topology optimization for vibrating structures, the obtained design should often have its resonance frequencies driven far away from the given excitation frequency in order to avoid resonance and to reduce vibration levels. In this work, we explore harmonic vibration problems with the excitation frequency lower than the first resonance frequency of the initial structure. Dynamic compliance minimization is used to improve dynamic response of the structure. An additional local dynamic compliance constraint is used to define the shape preserving problem, thus, reducing deformation in specific regions of a part named shape preserving region (SPR). A commercial FE code (ANSYS^?) is used to solve the finite element problem. The optimization Method of Moving Asymptotes (MMA) is used with the modified Solid Isotropic Material with Penalization (SIMP) material interpolation scheme. The effectiveness of this technique is presented using 2D plane structures. Coherent results were achieved using the proposed optimization formulation. It is possible to observe significant decrease on local deformation, at expense of little increase on global dynamic compliance.     

MEMS氮化钛谐振式压力传感器研究

提出了一种基于TiN双谐振梁结构的MEMS谐振式压力传感器,利用TiN谐振梁的谐振频率与被测压力的关系进行压力测量。针对传感器的敏感结构建立了数学模型,通过理论分析和ANSYS有限元软件的模拟,得出了谐振梁长度的变化范围和谐振梁的结构参数,并确定了TiN谐振梁在矩形压力膜上方的最佳位置。这对传感器的结构设计具有重要的指导意义。     

Algorithms for point-wise state variable constraints in structural optimization

This paper presents algorithms to treat the point-wise state variable constraints in finite dimensional and distributed parameter structural optimization problems. The idea is to impose such constraints at all local maximum points, or over a small region around the maximum points. Therefore, methods for design sensitivity analysis to handle the constraints at some particular point for the distributed parameter problem are presented. The direct differentiation and adjoint variable methods are employed to derive the design sensitivity expressions. For the finite dimensional problem the new idea is easily carried out in optimization process. Simply supported and clamped beams are optimized using the new approach. These are modeled with nonuniform beam elements. Comparisons of finite dimensional and distributed parameter problems are also made.     

参数激励和外激励联合作用下薄板的非线性动力学

研究了在参数激励和外激励联合作用下四边简支矩形薄板的非线性动力学.基于von Karman理论,推导出了在参数激励和外激励联合作用下四边简支矩形薄板的动力学方程.利用Galerkin法对偏微分方程进行三阶离散,得到一个三自由度的常微分方程.考虑1:2:4内共振-主参数共振-1/2亚谐共振的情况,利用多尺度法得到了薄板系统的六维的平均方程.最后,采用数值方法研究了薄板的周期和混沌运动.结果发现外激励对薄板的混沌运动是敏感的.     

Determination of optimal actuator forces and positions in smart structures using adjoint method

The problem of optimal design of structures with active support is analyzed in the paper. The sensitivity expressions with respect to the generalized force and the position of actuator are derived by the adjoint structure approach. Next, the optimality conditions are formulated by means of an introduced Lagrangian function. The problem of introduction of a new actuator is also considered and the condition of modification is expressed by means of the topological derivative. The obtained sensitivity formula, optimality conditions and modification conditions are applied in the optimization algorithm with respect to the number, positions and generalized forces of the actuators. Numerical examples of optimal control of beams illustrate the procedure proposed in the paper.     

Finite element displacement method for large amplitude free flexural vibrations of beams and plates

A finite element method to determine the nonlinear frequency of beams and plates for large amplitude free vibrations is presented. The equation of motion is characterized by the basic stiffness, mass, geometrical stiffness and the associated inplane force matrices. The procedures for solving the system equations of motion are discussed, and the explicit formulations of the geometrical stiffness and inplane force matrices of a rectangular plate element are given. Examples of large amplitude free oscillations for rectangular plates and beams with various boundary conditions are given. Characteristics of convergence are investigated. In all the cases where comparisons with previous investigations are made, good agreement has been obtained. It indicates that the present method will give results entirely adequate for engineering purposes.     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.