On the nonlinear vibration of heated corrugated circular plates with shallow sinusoidal corrugations

The large amplitude free vibration of corrugated circular plates with shallow sinusoidal corrugations under uniformly static temperature changes is investigated. Based on the nonlinear bending theory of thin shallow shells, the governing equations for corrugated plates are established from Hamilton's principle. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich averaging method following an assumed harmonic time mode. Then by introducing the Green's function, the resulting dynamic compatible equation and corresponding boundary conditions are converted into equivalent integral equations. Taking the central maximum amplitude of the plate as the perturbation parameter, the perturbation-variation method is used to dynamic equilibrium equation with the aid of Computer Algebra Systems, Maple, from which, the third-order approximate characteristic relation of frequency vs. amplitude for nonlinear vibration of heated corrugated plates is obtained, and the frequency–amplitude characteristic curve is plotted for some specific values of temperature and geometrical parameters. It is found that the rise in temperature will decrease the frequency and vice versa. The nonlinear effect weakens when corrugations become deeper and dense. The present method can easily be expanded for the analysis of nonlinear vibration problem for other heated thin plates and shells.     

横向磁场中轴向运动条形导电板的组合共振

给出了磁场中轴向运动条形导电薄板的动能、应变能以及电磁力表达形式,应用哈密顿变分原理,推导出了轴向运动导电板的非线性磁弹性振动微分方程。通过位移函数的设定并应用伽辽金积分法,得到横向磁场中对边简支边界约束轴向运动条形板的达芬型磁弹性振动方程。利用多尺度法进行求解,得到组合共振发生时确定共振幅值的幅频响应方程,并给出定常稳定解的判定条件。通过数值算例,得到轴向运动速度、磁感应强度、激励力和轴向拉力等参量不同时的振幅变化规律曲线图以及系统振动的时程响应图和相图,分析了不同参量对共振幅值和非线性特征的影响,并对系统呈现的概周期和混沌运动行为变化规律进行了分析。     

Moving-load dynamic problems: A tutorial (with a brief overview)

This tutorial is dedicated to the study of structural dynamics problems caused by moving loads. Through a simple example of a simply supported beam traversed by a moving mass, several fundamental concepts peculiar to moving-load problems are introduced. The necessary mathematics involved is presented. The analytical procedure is also presented for a circular plate excited by a rotating oscillator. Then numerical results of a circular beam spinning about its longitudinal axis excited by an axially moving surface load are provided. A variety of moving-load problems are briefly reviewed with some published papers and books to help readers quickly get into problems of their interests. Readers are expected to get a flavour of what moving-load problems are about, what general methods are available and what research has been done from studying this tutorial. Knowledge of partial differential equations and vibration theory of beams and plates is required in order to understand this tutorial.     

Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators

This paper deals with the nonlinear vibration and dynamic response of simply supported shear deformable cross-ply laminated plates with piezoelectric actuators subjected to mechanical, electrical and thermal loads. The material properties are assumed to be independent of the temperature and electric field. Theoretical formulations are based on the higher order shear deformation plate theory and general von Kármán-type equation, which includes thermo-piezoelectric effects. Due to the bending and stretching coupling effects, a nonlinear static problem is first solved to determine the pre-vibration deformation caused by temperature field and control voltage. By adding an incremental dynamic state to the pre-vibration state, the equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of hybrid laminated plates. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply laminated plates. The results presented show the effects of temperature rise, applied voltage and stacking sequence on the nonlinear vibration and dynamic response of the plates.     

旋转机械耦合传动系统参激振动分析与控制

以一类旋转机械传动系统的扭振为研究对象,考虑轴系时变刚度变化和非线性摩擦阻尼力的影响,根据Lagrange原理建立了一类旋转机械传动系统的耦合非线性动力学方程,利用多尺度方法推导出了系统的平均方程,并根据Hopf分岔理论给出了系统发生Hopf分岔的充要条件及周期运动稳定性的判别方法,分析了参数共振和内共振条件下系统的分岔特性。为抑制传动系统因亚临界Hopf分岔引起的失稳振荡,引入非线性反馈控制,对系统的Hopf分岔点进行转移并控制周期运动的稳定性和幅值。研究结果为非线性旋转机械参激振动的分析及控制提供了有效方法。     

Closed form solutions of axisymmetric bending of circular plates having non-linear variable thickness

A new analytical method for evaluation of elastic stresses and deformations in axisymmetric plates having variable thickness according to a power of a linear function, either solid or annular, subjected to symmetrical bending due to lateral loads either distributed on upper surface or distributed along the inner or the outer edges. The proposed procedure is based on two independent integrals of the hypergeometric differential equation describing the rotation field and constitutes the generalization of the one found in the literature. This method allows to study a wide range of plates, be they solid or annular, converging or diverging with linear or non-linear thickness function, convex, concave or linear tapered, without the restrictions of the known procedures. Analytical results obtained by using this method utterly match both theoretical results which may be obtained in the specific case known (constant-thickness circular plate, linear variable thickness annular circular plate) and numerical results obtained by using FEA.     

集中载荷作用下双模量圆板弯曲计算

双模量圆板在集中载荷作用下发生弯曲变形时,会形成各向同性的拉伸和压缩区。因此,可把双模量圆板看成两种各向同性材料组成的层合板,采用弹性理论建立了双模量圆板在集中载荷作用下的静力平衡方程,利用此静力平衡方程确定了双模量圆板的中性面位置。在此基础上,建立了集中载荷作用下双模量圆板的弯曲微分方程,求得了圆板的挠度表达式。通过算例讨论分析了双模量对圆板弯曲变形的影响,得到了圆板材料拉压弹性模量相差较大时,其挠度计算不宜采用相同弹性模量经典薄板理论,而应该采用双模量薄板理论的结论。     

Large amplitude vibrations of thin elastic plates by the method of conformal transformation

A unified method for investigating large amplitude vibrations of thin elastic plates of any shape under clamped edge boundary conditions is presented, based on Von Karman governing equations generalised to the dynamical case. The conformal mapping technique is introduced and the domain is conformally transformed on to the unit circle. The deflection function is chosen beforehand in conformity with the prescribed boundary conditions and the stress function is solved taking only the first term of the mapping function. The transformed differential equations are solved by the Galerkin procedure to obtain the second order nonlinear differential equation for the unknown time function. The time equation is readily solved in terms of Jacobian elliptic functions. Frequency of linear and nonlinear oscillations as well as static nonlinear case are analysed for plates of circular, and regular polygonal shape. Results obtained are compared with other known results. From the comparative study of different results it is observed that the first term approximation of the mapping function yields fairly accurate results with less computational effort.     

A study on free vibration of a spinning disk

Without a logical jump, we have derived the governing equation for free vibration of a spinning circular disk by using the variational formulation based upon the Kirchhoff plate theory and von Karman strain one. It has been found during the derivation that the governing equation is theoretically valld under the assumption that in plane deflections are steady and axisymmetric, and that internal forces are linearized while the strains remain nonlinear. The natural frequencies and the critical speeds of a freely spinning disk are obtained approximately and their dependencies on the spinning speed, mode number, and natural freqeuncy of the stationary disk are analyzed.     

磁致伸缩效应在圆管中激励纵向导波的理论和试验研究

分析了导波在圆管中传播的模态,探讨了在圆管中激励纵向导波的磁致伸缩作用力模型。纵向导波由平行于圆管轴向的磁致伸缩力的作用而产生,磁致伸缩力除与被检测材料的特性有关外,主要取决于设置的静态偏置磁场和交变磁场的耦合作用。根据该模型设计了在圆管中激励纵向导波的磁致伸缩传感器,试验中获得了纵向导波,并通过磁致伸缩传感器对圆管分别作用不同的静态偏置磁场和交变磁场,试验研究两者对激励的纵向导波的影响。结果表明：作用于圆管上的静态偏置磁场与激励的纵向导波幅值呈抛物线关系,在低偏置磁场作用下,纵向导波幅值随着偏置磁场的增加而增大,在高偏置磁场作用下,纵向导波幅值随着偏置磁场的增加反而减小。导波在圆管中传播时,存在多模式和频散特性,交变磁场大小与激励的纵向导波幅值之间呈非线性关系。     

磁场中轴向运动导电板磁弹性主共振分析

给出轴向运动薄板动能、应变能以及电磁力虚功的表达形式。应用哈密顿变分原理,推得横向磁场中轴向运动条形导电薄板的非线性磁弹性振动方程。针对对边简支边界约束条件,通过位移函数的设定并应用伽辽金积分法,得到三阶位移展开形式下轴向运动板的非线性振动微分方程组。利用多尺度法对系统的主共振问题进行求解,分别得到三种频率关系条件下关于稳态解的幅频响应方程。依据李雅普诺夫稳定性理论对解的稳定性进行分析,得到相应的稳定性判别式。通过数值算例,得到轴向速度、磁感应强度、激励力幅值及板厚不同时的振幅变化规律曲线图,分析不同参量对共振幅值和非线性特征的影响,并对不同频率关系进行比较。     

线性分布载荷作用下双模量圆板的计算分析

双模量圆板在线性载荷作用下,会形成各向同性的拉伸区和压缩区.可将双模量圆板看成两种各向同性材料组成的层合板,采用弹性理论建立双模量圆板在外载荷作用下的静力平衡方程,利用此静力平衡方程即可确定双模量圆板的中性面位置.建立双模量圆板在外载荷作用下的弯曲微分方程,求得双模量圆板在线性分布载荷作用下的挠度方程.通过算例讨论分析双模量对圆板弯曲变形的影响,得到圆板材料拉压弹性模量相差较大时,其挠度计算不宜采用相同弹性模量经典薄板理论,而应该采用双模量薄板理论的结论.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

考虑动态刚度、传递误差及齿侧间隙的齿轮系统谐振分析

在所建立的考虑啮合动态刚度、传递误差、齿侧间隙等多种非线性因素的直齿圆柱齿轮系统非线性动力学分析模型中,将时变啮合刚度按5次谐波展开,齿侧间隙按3次多项式拟合,运用多尺度法对该模型进行了分析,推导了系统主共振和谐波共振响应下的频率响应方程,绘制了相应的频率响应曲线,分析了系统中的静态栽荷和动态栽荷以及阻尼对谐振响应幅值和频率的不同影响作用。     

带硬涂层圆盘构件的固有振动特性的精确解法

研究了在广义弹性简支边界条件下的具有硬涂层的圆盘构件的自由振动的量纲一固有频率的精确解.首先利用多铁性多层圆盘的解析分析的多层板弹性理论,导出带硬涂层的圆盘结构的状态方程,其中以位移、电势、磁势、应力、电位移和磁感应强度为状态变量.利用有限Hankel变换和传播矩阵法,得到考虑压电和压磁效应的带硬涂层的圆盘的量纲一固有频率的精确解.根据算例结果,比较了压电、压磁两类硬涂层材料在单面涂层、双面涂层和不同涂层厚度的结构配置下的固有频率变化规律.     

载流圆板的磁弹性随机振动

分析载流圆板在磁场中的磁弹性随机振动问题。根据薄板、薄壳体的磁弹性运动方程和连续体的随机振动理论,导出在磁场环境中圆板的磁弹性随机振动方程。对磁场中的载流圆板进行随机响应分析,得到圆板位移响应的自相关函数、功率谱密度函数及均值函数等数字特征。通过具体算例,对处于外加磁场中通入平稳随机电流的导电圆板,计算其位移响应的功率谱密度函数。     

耦合场中弹性圆形薄板在混沌状态下的应力分析

在给出圆薄板的非线性电磁弹性耦合运动基本方程及电磁力表达式的基础上,得到在横向稳恒磁场、环向电流和机械载荷共同作用下简支圆薄板的振动方程,并采用四阶Runge-Kutta法求解方程,得出圆板的弹性变形及应力状态.通过变化磁感应强度和电流密度,使系统由周期状态进入混沌状态;由分岔图与最大Lyaponov指数图判定系统是否进入混沌状态;并讨论磁场与电流对系统应力状态的影响.     

The load capacity of a squeeze film between curved porous rotating circular plates

The squeeze film between rotating circular plates is analysed. The curved upper plate approaches the flat non-porous plate normally. The Reynolds equations are uncoupled by using the Morgan—Cameron approximation and closed-form solutions are obtained. Expressions for the pressure and load capacity of the bearing are given. The effect of rotating fluid inertia is to reduce the load capacity of the bearing.     

考虑静强度要求的旋转直板叶片试验模型相似设计方法

针对旋转直板叶片动力学相似试验模型的设计问题,提出一种考虑满足静强度要求的相似模型设计方法。以叶盘和单个直板叶片固联组成的旋转结构为研究对象,应用板壳振动理论建立振动方程,并将旋转直板叶片所受惯性力和气动力等效为作用在旋转叶片上的静态外载荷。在对叶片进行应力分析得到强度设计准则的基础上,采用方程分析法建立动力学相似关系。理论证明了当模型和原型的材料不同时,须采用不完全几何相似的试验模型才能满足强度要求的结论。并通过设计实例,采用数值仿真的方法验证了所提出考虑满足静强度要求的动力学相似模型设计方法的有效性和正确性。该研究结果为旋转直板类构件动力学相似模型的设计进行理论补充。     

圆盘式磁流变离合器特性分析

磁流变离合器是通过磁流变液的剪切应力传递转矩的器件.介绍了一种园盘式磁流变离合器的工作原理,用Bingham 模型描述了磁流变液随外加磁场变化的流变特性,基于Navior-Stokes方程,分析了磁流变液在两圆盘间的流动,得到了离合器传递的转矩和输出角速度的方程,为离合器的设计奠定了理论基础.研究结果表明:离合器传递的转矩和输出角速度可由外加磁场连续控制.