旋转复合材料薄壁轴的不平衡非线性弯曲振动

研究几何非线性复合材料薄壁轴在偏心激励作用下的非线性振动特性。在轴的应变位移关系中引入Von Kármán几何非线性,基于Hamilton原理和变分渐进法(VAM)导出复合材料传动轴的拉-弯-扭耦合非线性振动偏微分方程组。为了着重研究轴的横向弯曲非线性振动特性,在上述模型中忽略轴向变形和扭转变形,得到轴的横向弯曲非线性振动偏微分方程,其中考虑了黏滞外阻和内阻的影响。采用Galerkin法,将偏微分方程转离散化为常微分方程,在此基础上利用四阶Runge-Kutta法对常微分方程组进行数值模拟,获得位移时间响应图、相平面图和功率谱图,研究了外阻、内组、偏心距和转速对非线性振动响应的影响,发现旋转复合材料薄壁轴存在混沌运动。     

内外共振联合作用下索－梁组合结构非线性振动分析

研究内共振和外共振联合作用下的索－梁组合结构非线性振动问题。利用Ｈａｍｉｌｔｏｎ原理推导索－梁组合结构非线性动力学方程,同时考虑索的垂度以及由梁和索之间模态耦合引起的非线性影响。利用Ｇａｌｅｒｋｉｎ方法将索－梁组合结构非线性运动偏微分方程离散为一组常微分方程。最后对数学模型进行数值计算,得到了不同内外共振联合作用下梁和索的模态时程曲线。研究表明,梁的稳态运动呈现周期性振荡,而索在不同的内外共振联合作用下,分别呈现出混沌或周期性振荡,并且索和梁之间持续的模态交替现象只能在特定的内外共振下出现。     

基于Magnus-Gaussian 截断的铣削系统稳定性的半离散分析法

机床铣削系统的动力学可以通过一微分差分方程组来描述。对该动力学模型进行稳定性分析,可以确定稳定的铣削参数域及稳定性图。半离散法作为一种有效的半解析稳定性分析法,在对铣削系统动力学模型进行零阶半离散化时的误差级数较大,往往需要进行多步长的运算才能获得比较精确的稳定性分析结果。本文通过引入Magnus-Gaussian 截断法,推导出了基于Magnus-Gaussian 截断的零阶半离散稳定性分析法。用该方法对铣床切削稳定性进行了分析,并获得了各种工况下的铣床稳定性图。结果表明：该方法比原来的零阶半离散稳定性分析法能够更快的收敛于稳定解,从而节省了稳定性分析的运算时间。     

平动矩形贮箱刚-液耦合非线性动力学研究

首先应用H-O原理建立了矩形贮箱刚-液耦合系统平动的耦合动力学模型,在贮箱水平运动情况下,给出满足边界条件的速度势函数和液面波高的级数表达式,采用伽辽金法离散,将动力学模型转化为常微分方程组,在给定贮箱运动规律和给定外力规律两种形式下,分析了刚-液耦合系统固有频率变化规律,并应用多尺度法对系统的一阶主共振进行解析分析,研究了液体稳态解的幅频曲线,均发现跳跃及软、硬特性随液深转换的现象,在给定水平外力下,得到液体稳态解的同时还可得到刚体稳态解,两者定性性态相同。最后用数值法验证了解析解的正确性。     

行程敏感式液阻减振器动力学特性的三维流-固耦合有限元仿真分析

建立了缸壁槽形行程敏感式减振器较精细的三维流-固耦合动力学仿真分析模型,利用基于ALE流-固耦合、滑移网格流-流耦合的同步化迭代流-固耦合计算方法及动网格控制方法求解减振器内部的稳态及瞬态流-固耦合动力学响应,进行了部分实验验证。量化分析了缸壁槽形行程敏感式减振器的高速阻尼特性、冲击阻尼特性和活塞偏置、缸壁槽通流面积及结构变化对阻尼特性的影响,结果表明:活塞沿压缩行程方向偏置后的阻尼特性是有利的,而沿伸张行程方向偏置的状态是不利的;缸壁槽的附加泄流作用可削减减振器在地面冲击激励下阻尼力的瞬间冲击峰值;多缸壁槽形行程敏感式减振器的阻尼特性表现出单条缸壁槽作用结果的叠加效应;缸壁槽对阻尼力的减弱比例随活塞速度幅值的减小而增大;缸壁槽在高速工况下被开启及关闭的瞬态过程中伴随有阻尼力波动现象,可通过适当延长缸壁槽端部收缩段加以改善。这些分析结论对于此类液阻减振器的变阻尼特性匹配设计具有重要的价值。     

电梯悬挂系统在建筑摇晃引起的位移激励下的横向振动分析

高层建筑在各种水平荷载作用下引起的摇晃对高速运行的电梯产生位移激励,这种激励是引起电梯悬挂系统横向振动的重要因素。以高速运行的电梯悬挂系统的横向振动为研究对象,应用Hamilton原理建立了电梯悬挂系统在建筑物摇晃引起的两点位移激励下的横向振动偏微分方程,并用Galerkin方法对运动方程进行离散化求解。数值算例计算了建筑物以不同频率摇晃时电梯悬挂系统的横向振动响应,并研究了轿厢弹簧刚度、集中阻尼和绳索分布阻尼对系统振动的影响。数值结果表明：当轿厢自振频率、绳索固有频率和建筑物摇晃引起的惯性力频率接近时,悬挂系统产生共振;通过改变弹簧刚度和阻尼系数,系统振动能得到不同程度的抑制。     

参数激励下碳纳米谐振器的非线性振动研究

该文研究了两端固支情况下的碳纳米管谐振器在直交流电激励作用下的非线性振动响应。考虑外激励频率接近其固有频率的主共振情况下以及几何非线性,忽略静电力的非线性、范德瓦耳斯力和初始静变形的影响,运用欧拉伯努利连续梁理论,通过哈密尔顿原理得到非线性控制方程。然后利用伽略金方法离散得到降阶方程,最后应用摄动分析方法获得其振动响应的幅频曲线。主要分析了阻尼、直交流电荷载、几何三次非线性项对响应的影响。结果表明几何三次非线性项直接影响碳纳米管谐振器在外激励作用的振动响应以及碳纳米管在交流电荷载作用下的振动响应相对于直流电荷载更加敏感,并在一定程度上发生软化行为等。     

The influence of the intermolecular surface forces on the static deflection and pull-in instability of the micro/nano cantilever gyroscopes

In this paper, the effects of van der Waals and Casimir forces on the static deflection and pull-in instability of a micro/nano cantilever gyroscope with proof mass at its end are investigated. The micro/nano gyroscope is subjected to coupled bending motions which are related by base rotation and nonlinearities due to the geometry and the inertial terms. It is actuated and detected by capacitance plates which are placed on the proof mass. The extended Hamilton principle is used to find the equations governing the static behavior of the clamp-free micro/nano gyroscopes under electrostatic, Casimir and van der Waals forces. The equations of static motion are discritized by Galerkin’s decomposition method. The nonlinear equilibrium equations are solved analytically using homotopy perturbation method (HPM). The static response of the micro/nano gyroscopes to variations in the DC voltage across the drive and sense electrodes is obtained and the effects of different parameters on pull-in instability are investigated. The presented results can be used for accurate estimations of the instability and performance of the micro/nano gyroscopes.     

Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials

This paper presents a new technique for studying the stability properties of dynamic systems modeled by delay‐differential equations (DDEs) with time‐periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the ‘infinite‐dimensional Floquet transition matrix U’. Two different formulas for the computation of the approximate U, whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second‐order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time‐periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance‐modulated turning. The results indicate that this method is an effective way to study the stability of time‐periodic DDEs. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton’s principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli–Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin’s method and the fourth-order Runge–Kutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.     

A study of early-time response in dynamically loaded visco-elastic composites

Spalling is an important failure mode which triggers delamination, thus affects through-thickness integrity of a laminate and hinders the later integral plate action.The aim of the present study is to model the propagation of one-dimensional waves caused by a short-duration dynamic load through a visco-elastic medium. Two types of viscous effects are considered and described by means of partial differential equations. Four pulse load shapes are considered and four cases analysed. A higher order Lagrangian finite element is used to model the wave propagation and the weak-form Galerkin method is adopted to solve the differential equations. Numerical solutions are compared to analytical ones (where they exist) and excellent temporal and spatial correlation is achieved.It is found that damping leads to a decrease in peak stresses and strains by up to 11% for 5% of critical damping, even during the direct loading phase. It is shown that the inclusion of strain-rate did not have an effect on strains but led to an increase in stresses by almost 100%. The inclusion of damping and strain-rate effects increased stress values by up to 70% compared to the non-viscous cases, rendering strain-rate effects more pronounced than damping effects.     

单闭室复合材料薄壁梁的结构阻尼

研究单闭室复合材料薄壁梁的结构阻尼特性。基于变分渐进法（VAM) 和Hamilton原理,分别建立薄壁梁的截面力位移关系和运动方程;采用Galerkin法对薄壁梁进行自由振动分析;在获得薄壁梁振动模态矢量的基础上,根据最大应变能理论,对薄壁梁的模态阻尼性能进行预测,并且将阻尼预测的结果与现有的有限元计算结果进行对比,验证了本文阻尼分析模型的有效性。进一步针对周向均匀刚度配置(CUS)和周向反对称刚度配置（CAS）两种构型复合材料薄壁箱形梁以及一个翼型截面梁,进行阻尼计算,揭示了纤维铺层角和截面宽高比等参数的影响     

Nonlinear resonant response of imperfect extensible Timoshenko microbeams

This paper investigates the nonlinear size-dependent dynamics of an imperfect Timoshenko microbeam, taking into account extensibility. Based on the modified couple stress theory, the nonlinear equations of motion for the longitudinal, transverse, and rotational motions are derived via Hamilton’s energy method. A high-dimensional finite degree-of-freedom system of ordinary differential equations is obtained by the application of the Galerkin scheme. This set of equations is solved through use of the pseudo-arclength continuation method. A stability analysis is conducted via use of the Floquet theory. The resonant motion characteristics of the microbeam are examined by plotting the frequency-response and force-response curves. The effect of system parameters on the resonant response of the system is highlighted.     

Element-free characteristic Galerkin method for Burgers’ equation

A new meshfree method named element-free characteristic Galerkin method (EFCGM) is proposed for solving Burgers’ equation with various values of viscosity. Based on the characteristic method, the convection terms of Burgers’ equation disappear and this process makes Burgers’ equation self-adjoint, which ensures that the spatial discretization by the Galerkin method can be optimal. After the temporal discretization, element-free Galerkin method is then applied to solve the semi-discrete equation in space. Moreover, the process is fully explicit at each time step. In order to show the efficiency of the presented method, one-dimensional and two-dimensional Burgers’ equations are considered. The numerical solutions obtained with different values of viscosity are compared with the analytical solutions as well as the results by other numerical schemes. It can be easily seen that the proposed method is efficient, robust and reliable for solving Burgers’ equation, even involving high Reynolds number for which the analytical solution fails.     

Performance evaluation of algorithms for mildly nonlinear elliptic problems

A number of numerical methods for mildly nonlinear elliptic boundary value problems on general domains is presented. The discretization procedures considered are: a fourth-order FFT-type method, collocation using Hermite bicubic splines and Galerkin with linear triangular as well as quadratic quadrilateral isoparametric elements. The linearized collocation and Galerkin equations are solved by various direct methods available in the ELLPACK system. A comparative study of the above equation solvers is presented for different domain geometries and compilers. The evaluation of software for the general mildly nonlinear elliptic equations is performed over 36 instances from a population of 16 parametrized problems with ‘real world’ and ‘mathematical’ behaviour. The performance data suggests that collocation is an effective method for such general problems, while Galerkin with quadratic quadrilateral isoparametric elements is uniformly superior to the one with linear elements.     

薄壁圆柱壳在流体脉动激励下的振动特性分析

摘要：为深入研究薄壁圆柱壳在流体脉动激励下的运动特性,应用Donnell简化壳理论,考虑阻尼、结构非线性和附加质量的影响,建立了薄壁圆柱壳在流体脉动激励下的非线性振动方程。基于Galerkin方法将偏微分方程转化为方便求解的常微分方程,利用多尺度法求解了系统主共振的一次近似解,得到了系统稳态响应的转迁集与分岔图,并通过奇异性分析,得到了系统工作稳定性和可靠性的结构参数区域。对薄壁圆柱壳在流体作用下的振动特性进行了数值模拟和实验研究,考察了阻尼系数、脉动频率、液体深度等对系统动力学特性的影响。研究表明,考虑阻尼、结构非线性和附加质量的非线性振动方程更能体现薄壁圆柱壳在流体脉动激励下完整的动力学特性,同时系统中存在多种分岔行为。     

复合材料悬臂板系统的多脉冲同宿轨道

针对复合材料悬臂板系统的非线性动力学行为进行了分析。模型考虑高阶横向剪切效应、几何大变形和横向阻尼的影响,基于Reddy的高阶剪切变形理von Karman的大变形理论,利用Hamilton原理,Galerkin离散和多尺度法得到系统横向位移的平均方程。应用广义Melnikov方法研究了复合材料悬臂板的非线性混沌动力学行为。分析了在共振带附近,复合材料悬臂板系统存在的Shilnikov类型多脉冲跳跃同宿轨道。最后结合数值模拟,进一步揭示系统存在多脉冲跳跃现象。     

Nonlinear dynamics of composite laminated cantilever rectangular plate subject to third-order piston aerodynamics

This paper presents the analysis of the nonlinear dynamics for a composite laminated cantilever rectangular plate subjected to the supersonic gas flows and the in-plane excitations. The aerodynamic pressure is modeled by using the third-order piston theory. Based on Reddy’s third-order plate theory and the von Kármán-type equation for the geometric nonlinearity, the nonlinear partial differential equations of motion for the composite laminated cantilever rectangular plate under combined aerodynamic pressure and in-plane excitation are derived by using Hamilton’s principle. The Galerkin’s approach is used to transform the nonlinear partial differential equations of motion for the composite laminated cantilever rectangular plate to a two-degree-of-freedom nonlinear system under combined external and parametric excitations. The method of multiple scales is employed to obtain the four-dimensional averaged equation of the non-automatic nonlinear system. The case of 1:2 internal resonance and primary parametric resonance is taken into account. A numerical method is utilized to study the bifurcations and chaotic dynamics of the composite laminated cantilever rectangular plate. The frequency–response curves, bifurcation diagram, phase portrait and frequency spectra are obtained to analyze the nonlinear dynamic behavior of the composite laminated cantilever rectangular plate, which includes the periodic and chaotic motions.     

输电线路防舞阻尼器系统参数分析及设计研究EI北大核心CSCD

针对目前覆冰导线舞动频发现状,提出了一种通过在导线靠近输电塔位置处设置阻尼器来减振耗能从而实现输电线路舞动抑制的方法。基于Hamilton原理运用多阶伽辽金函数推导得到了导线-阻尼器系统的广义运动方程,并以某750 kV单档八分裂输电线路为例进行运动方程特征值分析,得到了导线-阻尼器系统的动力特性,探索了导线的垂度参数、阻尼器安装位置、阻尼系数及刚度系数等对系统等效阻尼比的影响,阻尼器安装位置越靠近跨中,系统的最大阻尼比提升效果越明显,两侧对称安装阻尼器可以有效地减小最优阻尼系数。采用数值算例和有限元数值仿真技术比较了粘滞阻尼器与负刚度阻尼器(NSD)对导线系统的减振效果,并针对NSD提出了参数的优化设计方法。研究表明:相比传统阻尼器,NSD可以在较低阻尼系数下有效地提高系统各阶的最大阻尼比,且能够明显降低导线系统的自振频率;系统所能达到的一阶最大阻尼比对NSD安装位置的变化不敏感;通过有限元仿真证实了,基于NSD的输电导线阻尼器设置方案相较于传统阻尼器方案具有更好的防舞性能。     

Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads

This paper presents a simple analytical approach to investigate the stability of functionally graded plates under in-plane compressive, thermal and combined loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for functionally graded plates are derived by using the classical plate theory taking into account both geometrical nonlinearity in von Karman sense and initial geometrical imperfection. The resulting equations are solved by Galerkin procedure to obtain explicit expressions of postbuckling load–deflection curves. Stability analysis of a simply supported rectangular functionally graded plate shows the effects of the volume fraction index, plate geometry, in-plane boundary conditions, and imperfection on postbuckling behavior of the plate.