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。     

高速冷轧机辊系非线性动力学模型及控制参数优化

综合运用轧制过程动态轧制力模型、轧机系统分段非线性弹性力模型和分段非线性摩擦力模型,建立了考虑非线性因素耦合的高速冷轧机辊系动力学模型,采用平均法求解幅频响应方程,研究了轧机系统主要参数对轧机垂直振动的影响。分析结果表明：动态轧制力一次项刚度越小,轧机系统固有频率越大;动态轧制力三次项刚度不为零时,轧机系统出现明显跳跃现象;非线性弹性力起主导作用时,易发生跳跃现象;非线性摩擦力越大,轧机系统振动幅值越小,分段特性减弱;外扰力幅值越大,振动幅值越大,振动越剧烈;线性阻尼越大,振动幅值越小且减小幅度越缓慢。     

各种板边条件下大挠度圆板自由振动的分岔解

计及几何非线性,在各种板边条件下,建立圆板自由振动的非线性动力学方程.采用Galerkin法,将圆板的非线性动力学偏微分方程简化成三种标准类型的Duffing方程.提出一类强非线性动力系统的两项谐波法,将描述动力系统的二阶常微分方程,化为以频率、振幅为变量的非线性代数方程组,考虑初始条件补充约束方程,构成频率、振幅为变量的封闭非线性代数方程组.利用Maple程序可以方便地求解.结果表明,两项谐波法不仅适合于对称振动问题,而且适合于非对称振动问题.     

Non-linear dynamic analysis of shallow spherical shells on elastic foundations

The present study investigates the effect of Winkler-Pasternak elastic foundation parameters on the nonlinear dynamic response of shallow spherical shells. The values of foundation parameters ( and ) have been determined for the minimaximum central response of the shallow shells for both the clamped as well as simply supported immovable edge conditions. Donnell type partial differential equations governing the moderately large amplitude behaviour of shallow spherical shells resting on Winkler-Pasternak elastic foundations under step pressure loading, have been analysed. The space and time-wise integrations of governing equations have been carried out using Chebyshev series and Houbolt techniques, respectively. It is also shown that the present analysis can be extended to study the dynamic buckling of shallow shells resting on elastic foundations.     

功能梯度材料板壳多场耦合主动控制仿真

利用力-电-热多场耦合的弱变分方程和加强假定应变法，建立了一用于温度梯度作用下功能梯度材料板壳力学性能分析的固体壳单元，该单元既可用作实体单元，又可模拟薄曲壳结构，在厚跨比非常小的情况下也能获得令人满意的精度。结合常位移-速度反馈控制算法，对基于压电传感器、驱动器的功能梯度材料板壳的热变形及振动的主动控制进行了数值仿真，探讨了组分材料体积分数幂指数分布、控制增益等对功能梯度材料力学性能和控制效果等的影响。算例表明了常位移-速度反馈控制算法的正确性和单元的高效性。     

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.     

Nonlinear resonance of the rotating circular plate under static loads in magnetic field

The rotating circular plate is widely used in mechanical engineering, meanwhile the plates are often in the electromagnetic field in modern industry with complex loads. In order to study the resonance of a rotating circular plate under static loads in magnetic field, the nonlinear vibration equation about the spinning circular plate is derived according to Hamilton principle. The algebraic expression of the initial deflection and the magneto elastic forced disturbance differential equation are obtained through the application of Galerkin integral method. By mean of modified Multiple scale method, the strongly nonlinear amplitude-frequency response equation in steady state is established. The amplitude frequency characteristic curve and the relationship curve of amplitude changing with the static loads and the excitation force of the plate are obtained according to the numerical calculation. The influence of magnetic induction intensity, the speed of rotation and the static loads on the amplitude and the nonlinear characteristics of the spinning plate are analyzed. The proposed research provides the theory reference for the research of nonlinear resonance of rotating plates in engineering.     

薄板结构动力学特性建模解析

运用边界元法研究薄板结构的动力学特性。从薄板自由振动的微分方程式出发,由相应的静态基本解建立薄板自由振动的边界控制方程,并依据边界上的约束条件,推导出薄板结构在边界和域内的动力学方程。为了避免求域内项的积分,将上述两个方程进行联立求解,快速、准确地得到薄板结构的频率方程表达式。数值计算和实验结果表明,该文方法具有良好的解析计算精度和较高的计算效率。     

一种Hilbert变换法在非线性系统分析中的应用

Hilbert变换是信号处理领域常用工具之一,将Hilbert变换法改进并应用到信号分解和非线性系统振动分析中。振动信号的多谐波性,使得Hilbert变换法提取信号的瞬时频率和瞬时相角可以通过滤波的方法分离成快变和慢变两部分,从而提取系统的振动分量;通过迭代计算,依次获得振动信号中所有谐波分量;将非线性振动方程用瞬态幅值相关的瞬态参数表示,从而求解系统的频响关系曲线方程。通过相应的数值模拟计算,验证了改进的Hilbert变换法在非线性振动分析的有 效性。     

Winkler地基上材料非线性矩形薄板1/3次亚谐共振

为研究Winkler地基上材料非线性矩形薄板受简谐激励的非线性振动,应用弹性力学理论建立其动力学方程,并由Galerkin方法将其转化为非线性振动方程。应用非线性振动的多尺度法求得系统1/3次亚谐共振的近似解,并进行数值计算。分析激励、几何参数、阻尼系数、非线性参数对共振响应曲线的影响。     

磁电弹薄板的非线性主共振

针对磁电弹性薄板,结合大挠度板理论,考虑几何非线性,建立了非线性无阻尼强迫振动的数学模型,应用改进的L—P法,对非线性无阻尼强迫振动问题进行求解,得到薄板稳定状态下的幅频响应方程,数值模拟了不同情况下系统的幅频响应曲线图。通过比较分析得出：板的厚度、外激励力、参数e的不同取值以及压电材料BaTi03、压磁材料CoFe204及磁电弹材料BaTi03／CoFe204与Terfenol—D／PZT均会对系统的幅频响应曲线产生影响,主要表现为对系统主共振区间,弹簧软硬特性,幅值跳跃现象的影响。这些结论在理论上可以更好地指导工程结构的设计。     

波纹管在温度载荷作用下的几何非线性分析

针对波纹管在一般弹性范围内可达到很大的位移，呈现较强非线性的特点，对温度载荷作用下的波纹管几何非线性问题进行有限元分析。基于Sanders小应变、中等转动薄壳理论，通过引入“伪载荷”而建立起一个形式简单、易于实施的迭代格式，并进行了数值求解。算例验证了理论推导和程序编制的正确性。文末对正温差、零温差和负温差作用下的波纹管进行应力分析，发现温度变化对波纹管波峰和波谷附近的应力有明显的影响，而对其环板中心附近的应力影响相对小些。     

Exact solutions for vibration of cylindrical shells with intermediate ring supports

This paper presents new exact solutions for vibration of thin circular cylindrical shells with intermediate ring supports, based on the Goldenveizer–Novozhilov shell theory (Theory of thin shells; The theory of thin elastic shells). An analytical method is proposed to study the vibration behaviour of the ring supported cylindrical shells. In the proposed method, the state-space technique is employed to derive the homogenous differential equation system for a shell segment and a domain decomposition approach is developed to cater for the continuity requirements between shell segments. Exact frequency parameters are presented in tables and design charts for circular cylindrical shells having multiple intermediate ring supports and various combinations of end support conditions. These exact vibration frequencies may serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shell problems.     

考虑载荷变化的准零刚度隔振器动态特性

为了研究载荷变化对准零刚度隔振器动态特性及隔振性能的影响,首先,分析具有分段非线性特性的准零刚度隔振器静力学特性;其次,引入光滑函数,应用增量谐波平衡法得到准零刚度隔振器的周期解,分析载荷与激励幅值对隔振器动态特性的影响;最后,定义准零刚度隔振器的力传递率,分析载荷与激励幅值对其隔振性能的影响。研究结果表明:当激励幅值较大时,载荷变化会导致响应的常数项在共振区附近出现交叉折叠的现象;载荷与激励幅值的变化会改变隔振系统的共振频率、传递率峰值以及有效隔振频带,但该准零刚度隔振器的隔振性能仍优于线性隔振器。     

波纹圆薄板的非线性振动

以圆柱正交异性板代替波纹板［１］，试图通过在圆柱正交异性Ｋａｒｍａｎ型静力方程中引入惯性力来反映波纹板的动力学问题。求解时以正交完备Ｌｅｇｅｎｄｒｅ多项式构成挠度试函数，由相容方程找出应力函数，然后用加权残数法使平衡方程的残差极小化，得出时间函数的Ｄｕｆｆｉｎｇ方程。一次近似求得了波纹板线性振动时的固有频率及非线性固有频率，并得到了强迫振动时的幅频关系。一些结果与有关文献进行了比较，表明本文的尝试是可行的，以Ｌｅｇｅｎｄｒｅ多项式构成挠度函数收敛快、精度高。     

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。     

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.     

薄板组合结构动力学特性解析研究

研究薄板组合结构的动力学特性。基于薄板自由振动问题的动态基本解,运用边界元法建立薄板在横向和面内方向的边界控制方程式,为了提高解析建模的精度和效率,根据耦合的思想推导出薄板组合结构的动力学方程,采用频率扫描的方法,准确、快速地求解出薄板组合结构的频率,并得到其相应的振型。数值算例表明本建模方法的可行性和良好的解析计算精度。