载流悬臂柱壳的热磁弹性分析

在薄壳的几何方程、物理方程、运动方程和电动力学方程基础上,建立了载流柱壳热磁弹性耦合方程。采用线性化方法,得到了关于热磁弹性问题的线性迭代方程组,整理成含有8个基本未知函数的标准Cauchy型。考虑电磁场的焦耳热效应,引入热平衡方程及广义Ohm定律,得到了柱壳的温度场,并且推导得到了温度场积分特征值。讨论了载流悬臂柱壳应力、温度及变形随外加电磁参量的变化规律,并通过实例证实了可以通过改变电、磁、力场的参数来实现对板壳的应力、应变、温度的控制。     

三边简支载流薄板在交变磁场中的稳定性分析

针对三边简支、一边自由的载流矩形薄板,利用马丢方程解的稳定性,研究在交变磁场与机械载荷共同作用下的磁弹性稳定性问题.在导出载流薄板在电磁场与机械载荷共同作用下的磁弹性动力稳定方程的基础上,应用伽辽金原理将方程整理为马丢方程的标准形式.利用马丢方程系数的本征值关系,得出载流薄板磁弹性动力失稳临界状态的判别方程.通过具体算例,给出该矩形薄板在交变磁场中,动力失稳临界状态与相关参量之间的关系曲线及变化规律,并与均匀磁场中情形相比较.研究结果表明:变化电磁场的性质和大小,改变通电电流参数,均可改变电磁力的状态,从而达到控制载流薄板稳定性的目的.     

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。     

对边简支载流矩形薄板在电磁场中的稳定性分析

针对对边简支、另一对边固定载流矩形薄板,利用Mathieu方程解的稳定性,研究在电磁场与机械荷载共同作用下的磁弹性稳定性问题。在导出载流薄板在电磁场与机械荷载共同作用下的磁弹性动力稳定方程的基础上,应用Galerkin原理将稳定方程整理为Mathieu方程的标准形式,并将其归结为对Mathieu方程的求解问题。利用Mathieu方程系数、 的本征值关系,得出载流薄板磁弹性动力稳定临界状态的判别方程,并给出当 为小激励时的稳定区域图,以及Mathieu方程稳定解区域和非稳定解区域的分界。最后通过具体数值算例,给出该矩形薄板的磁弹性动力失稳临界状态与相关参量之间的关系曲线。研究结果表明,变化电磁场和通电电流的参数,可以改变电磁力的状态,从而达到控制载流薄板的变形,应力、应变状态及其稳定性的目的。     

四边简支载流矩形薄板的磁弹性动力屈曲分析

在载流薄板的磁弹性非线性运动方程、物理方程、几何方程、洛仑兹力表达式及电动力学方程的基础上,导出四边简支载流矩形薄板在电磁场与机械载荷共同作用下的磁弹性动力屈曲方程.应用Galerkin原理将该屈曲方程整理为Mathieu方程的标准形式,并利用Mathieu方程解的稳定区域与非稳定区域的分界,即方程系数的本征值关系,得出磁弹性问题屈曲临界状态的判别方程.通过具体算例,给出四边简支矩形板的磁弹性动力屈曲方程以及屈曲临界状态与相关参量之间的关系曲线,并对计算结果及其变化规律进行分析讨论.     

载流线圈中导电圆板的磁弹性固有振动

针对处于平行共轴三线圈和球形载流线圈产生磁场中的导电圆板,基于Kirchhoff薄板理论,给出磁场中圆板的磁弹性横向振动基本方程。根据电磁理论,导出线圈产生均匀磁场区域中圆板所受电磁力的表达式。通过位移函数的设定并应用伽辽金法,得到圆板的磁弹性轴对称振动微分方程及固有频率的表达式。通过算例,绘制了周边夹支和简支两种边界条件下圆板的磁弹性固有振动特性曲线图,分析了两种边界条件下固有频率、阻尼比、电磁力矩随线圈载流强度、线圈匝数和板厚等参数的变化规律。     

周边固定载流壳体的热磁弹性分析

针对磁场环境中周边固定载流壳体的热磁弹性问题进行研究。通过运动方程、几何方程、物理方程及电动力学方程,给出了载流壳体在机械场、电磁场、温度场作用下的基本方程。建立了包含10个基本未知量的非线性偏微分方程组,采用Newmark稳定有限等差式,获得可以利用离散正交法求解的标准型。对于周边固定壳体,导出了Lorentz力表达式、温度场和温度场积分特征值。分析了载流壳体应力、温度及变形随外加电磁参量的变化规律,通过实例计算,证实了改变电场、磁场、力场的参数能够实现对板壳应力、应变、温度的控制。为改变在电磁场环境中板壳结构的工作状态及强度研究提供理论分析和数值计算方法。     

横向磁场中载流薄板模态截断问题研究

根据板壳理论与磁弹性理论,在不考虑感应电场,仅考虑机械场与磁场的相互耦合作用情况下,在薄板非线性运动方程、物理方程、几何方程、洛仑兹力表达式及电动力学方程的基础上,建立在横向磁场和机械载荷共同作用下的大挠度载流矩形薄板的非线性运动方程.以三边简支一边自由矩形薄板为例,给出板的单模态运动方程和双模态运动方程.利用数学计算软件,采用四阶Ronger-Kuta数值方法,模拟非线性系统在单、双模态位移模式下的时程图、庞加莱截面图、相轨迹图.讨论电磁参数和外载荷参数对板在单、双模态位移模式下的非线性行为的影响及差异.结果表明,这些参数对薄板的运动有较大的影响,通过调整电磁参数可控制系统的混沌运动,以实现对系统运动特性的控制.     

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。     

横向磁场中矩形导电薄板的内共振特性分析

在给出薄板的电磁弹性运动基本方程及电磁力表达式的基础上,得到横向磁场作用下矩形薄板的磁弹性振动方程。针对一边固定三边简支的矩形薄板,通过位移模态展开并利用Galerkin法分离时间和空间变量,得到两自由度内共振非线性振动微分方程。采用多尺度法对模态方程组进行求解,得到了系统1∶3内共振情况下前两阶振动模态相互耦合的特征方程。通过算例,得到了关于系统内共振幅值的时程响应图和相图,分别讨论了无阻尼情况下系统初值、板厚以及有阻尼情况下磁场强度对内共振特性的影响,结果表明系统呈现明显的非线性内共振特征。     

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。     

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.     

On the nonlinear vibration of heated bimetallic shallow shells of revolution

The axisymmetrically nonlinear free vibration of a bimetallic shallow shell of revolution under uniformly distributed static temperature changes is investigated. Based on the nonlinear bending theory of thin shallow shells, the governing equations are established in forms similar to those of classical single-layered shells theory by redetermination of reference surface of coordinate. 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. The resulting equations, which form a nonlinear two-point boundary value problem, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude are obtained successfully. A detailed parametric study is conducted involving shell geometry and temperature parameters. The effects of these variables on the frequency-amplitude characteristics are plotted and discussed.     

轴向运动矩形板的谐波共振与稳定性分析

针对轴向运动矩形薄板的非线性振动问题,在给出薄板运动的动能和应变能的基础上,应用哈密顿变分原理,推得几何非线性下轴向运动薄板的非线性振动方程。通过位移函数和应力函数的设定,并应用伽辽金积分法,得到四边简支边界约束条件下受横向激励载荷作用轴向运动薄板的达芬型振动方程。利用多尺度法对系统的非线性谐波共振问题进行求解,得到稳态运动下关于共振幅值的幅频响应方程。依据李雅普诺夫运动稳定性理论对定常解的稳定性进行分析,得到解的稳定性判别式。通过数值算例,得到不同横向载荷和轴向速度下共振幅值的变化规律曲线图以及对应的相图,讨论分岔点变化以及倍周期运动规律,分析横向激励载荷和轴向运动速度对系统非线性动力学行为的影响。     

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.     

磁场中轴向变速运动载流梁的参强联合共振

研究磁场环境中轴向变速运动载流梁在简谐激励作用下的参强联合共振问题,应用弹性力学理论、电磁场基本理论以及哈密顿变分原理,得到轴向变速运动载流梁的非线性磁弹性耦合振动方程。利用伽辽金积分法对其进行时间变量和空间变量的离散化,进而运用多尺度法以及坐标变换的方法求得系统主共振-主参数共振的幅频响应方程。通过算例,得到了系统随不同参数变化的幅频响应曲线图、时间历程图、相轨迹图、庞加莱映射图和共振系统的动相平面轨迹图,分析了轴向速度、轴向拉力、磁感应强度、电流密度及强迫激励对系统主共振-主参数共振特性的影响,结果表明系统呈现典型的非线性振动特征和复杂的动力学行为。     

Non-linear analysis of skew thin plate by finite difference method

This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed.     

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.