非线性弹性梁的动态次谐分岔与混沌运动

本文讨论非线性弹性梁在周期微扰作用下的动力学行为，文中将说明梁根据各种不同的外作用和结构本射的特性而出现的各种可能的运动过程和力学行为。利用ｅｌｎｉｋｏｖ方法，给出了不同特征的梁受周期载荷作用后，系统发生次谐分岔的条件，及同窠轨道或异轨道破裂后混沌运动发生的条件，并给出了具体算例。     

参激非线性振子混沌运动的实验研究

本文对一端固定一端滑动受轴向简谐载荷的屈曲梁的混沌运动进行了实验。研究了其基本参数共振了主参数共振两种情况。实验发现混沌运动的形式随着激励频率、激励幅值和阻尼的变化而变化。给出了油沌响应的谱和Ｐｏｉｎｃａｒｅ映射。     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

二次非线性粘弹性圆板的2/1+3/1超谐解

计及材料的非线性弹性和粘性性质，研究了圆板在简谐载荷作用下的2／1＋3／1超谐解，导出了相应的非线性动力方程。提出一类强非线性动力系统的叠加－叠代谐波平衡法。将描述动力系统的二阶常微分方程，化为基本解为未知函数的基本微分方程；及分岔解为未知函数的增量微分方程。通过叠加－迭代谐波平衡法得出了圆板的2／1＋3／1超谐解。对叠加迭代谐波平衡法和数值积分法进行了比较，两者结果吻合很高。并且讨论了2／1＋3／1超谐解的渐近稳定性。     

参激屈曲梁的倍周期分岔和混沌运动的实验研究

本文对一端固定一端滑动承受轴向简谐载荷的屈曲梁的非线性响应进行了实验。研究了其基本参数共振和主参数共振两种情况，揭示了系统的倍周期分岔和混沌运动等复杂动力学行为。在某些情况下，混沌吸引子和周期吸引子共存，另一些则存在间歇混沌。给出了响应的时间历程、相图、频率谱和Ｐｏｉｎｃａｒｅ映射     

静载荷对夹层圆板超谐波共振的影响

研究静载荷作用下夹层圆板的超谐波共振问题.基于Hoff型夹层板理论,给出了静载荷作用下夹层圆板的非线性动力学方程.应用Galerkin法推导了静载荷作用下夹层圆板的轴对称非线性振动方程.运用多尺度法分别对系统的三次超谐波问题和二次超谐波问题进行了求解,并依据Lyapunov稳定性理论得到了系统稳态运动的稳定性判据.通过算例,得到了周边简支约束下夹层圆板三次超谐波共振和二次超谐波共振的幅频响应曲线图、振幅-静载荷响应曲线图、振幅-激励力幅值响应曲线图;研究了不同参数对系统振幅的影响规律,并对解的稳定性进行了分析.     

载流圆板在磁场中的随机稳定性分析

本文根据大挠度板壳力学基础理论和电磁弹性力学理论,建立了载流圆板的非线性磁弹性随机振动力学模型,采用伽辽金变分法将其变换成非线性常微分动力学方程．通过拟不可积哈密顿系统的平均理论将该方程等价为一个一维伊藤随机微分方程．通过计算该方程的最大Lyapunov 指数判断该系统的局部随机稳定性,并进一步采用基于随机扩散过程的奇异边界理论判断该系统的全局稳定性．最后通过讨论该系统的稳态概率密度函数图的形状变化讨论了该动力系统的随机Hopf分岔的变化规律,并采用数值模拟对理论分析进行了验证．     

Bifurcation and chaos of the circular plates on the nonlinear elastic foundation

According to the large amplitude equation of the circular plate on nonlinear elastic foundation , elastic resisting force has linear item , cubic nonlinear item and resisting bend elastic item. A nonlinear vibration equation is obtained with the method of Galerkin under the condition of fixed boundary. Floquet exponent at equilibrium point is obtained without external excitation. Its stability and condition of possible bifurcation is analysed. Possible chaotic vibration is analysed and studied with the method of Melnikov with external excitation . The critical curves of the chaotic region and phase figure under some foundation parameters are obtained with the method of digital artificial.     

The Subharmonic Bifurcation of a Viscoelastic Circular Cylindrical Shell

In this paper the nonlinear dynamic behavior of a viscoelastic circular cylindrical shell under a harmonic excitation applied at both ends is studied. The modified Flugge partial differential equations of motion are reduced to a system of finite degrees of freedom using the Galerkin method. The equations are solved by the Liapunov–Schmidt reduction procedure. In order to study 1/2 and 1/4 subharmonic parametric resonance of the shell, the transition sets in parameter plane and bifurcation diagrams are plotted for a number of situations. Results indicate that, for certain static loads, the shell may display jumps due to the presence of dynamic periodic load with small amplitude. Additionally, different physical situations are identified in which periodic oscillating phenomena can be observed, and where 1/4 subharmonic parametric resonance is simpler than the 1/2-one.     

伪弹性TiNi合金圆薄板的准静态力学行为实验研究

采用阴影云纹和应变片方法对伪弹性TiNi合金圆板在固支条件下的准静态力学行为进行了实验研究,得到了载荷位移曲线、全场离面位移和局部应变等数据.载荷位移曲线呈现非线性、滞回耗能和无残余变形的特性,表明试样已经发生马氏体相变.应变测量显示,相变局限于加载中心较小区域,相变区内,环向应变大于径向应变,且拉伸侧应变大于压缩侧的应变.有限元模拟揭示出相变区内两侧表层的相变范围、相变铰区和马氏体相含量的不对称分布规律.     

Effect of Nonlinear Stiffness on the Motion of a Flexible Pendulum

In this paper, we study the effect of a harmonicforcing function and the strength of a nonlinearityon a two-degrees-of-freedom system namely, an elasticpendulum, with internal resonance (for examplenonlinearly elastic springs). The equations can alsobe used to model the coupling between a ship's pitchand roll. The system considered here is modeled by amass hanging from a spring that is pinned at one endto the ground. The mass is free to move in the radialdirection, is also free to rotate about the pin joint, and subject to a periodic forcing function. Theforcing function used in this paper is in thetangential direction. The amplitude of the forcingfunction is used here as the control parameter and thesystem's dynamics are studied through the variation ofthis parameter.The first part of the paper is dedicatedto establishing the route by which the motion of thesystem goes from a periodic attractor to a chaoticattractor. It was found that the route to chaos alwaysbegins with a secondary Hopf bifurcation followed byconsecutive torus-doubling bifurcations, ending withtorus breaking.A comparison was also made between the use of a linear spring, a weakly nonlinear spring, and astrongly nonlinear spring.This comparison showed that althoughthe route to chaos was not altered, the bifurcationsleading to chaos and the chaotic motion itselfoccurred at different frequency regimes. We observedthat the nonlinearity could aid the stabilizationof the periodicattractor beyond the previously seenthreshold of instability. Yet, if the strength of thenonlinearity is sufficiently large, it can lead tochaos in frequency regimes where chaos was notobserved previously. The strongly nonlinear systemshowed chaotic behavior for frequency regimes thatdisplayed only periodic motion for both the linearsystem and the weakly nonlinear system. The route tochaos for these frequency ranges was also found to bedifferent from that previously studied. This leads usto the hypothesis that chaos in this range was due tothe nonlinearity of the spring and not the coupling effect.     

气动载荷下旋转运动圆板磁弹性主共振分析

基于Kirchhoff薄板理论与哈密顿原理,建立旋转运动导电圆板的磁-气动弹性非线性动力学方程．根据电磁场基本原理得到旋转运动圆板所受电磁力表达式,同时采用一种简化的气动模型以描述作用于板上的气动载荷．基于贝塞尔函数形式振型函数的选取,应用伽辽金法得到旋转圆板的磁气动弹性轴对称非线性振动微分方程．应用多尺度法推导出主共振下系统的幅频响应方程,并依据Lyapunov方法得到系统稳态运动稳定性判据．通过算例,得到周边夹之约束下圆板主共振的幅频特性曲线图,以及振幅随磁感应强度和激励力幅值的变化曲线图;阐述了不同参数对系统共振幅值的影响规律,并对解的稳定性进行了分析．     

功能梯度压电圆板轴对称自由振动问题精确解

将功能梯度压电圆板的位移变量和电势变量写为分离变量的形式，由压电动力学平衡方程导出以位移、电势及其一阶导数为状态变量的状态方程，考虑周边固支接地的边界条件，导出了求解功能梯度压电圆板自振频率精确解的方程。将方程退化至一般的非梯度纯弹性圆板的形式，求解其自振频率，得到的结果与相应的理论解完全吻合，从而验证了本文方法的正确性。更进一步地对梯度函数沿板厚以指数形式变化的功能梯度压电圆板的自振频率进行了计算，并得到了梯度化对板自振频率的影响规律。     

Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow

Turbo-machineries, as key components, have wide applications in civil, aerospace, and mechanical engineering. By calculating natural frequencies and dynamical deformations, we have explained the rationality of the series form for the aerodynamic force of the blade under the subsonic flow in our earlier studies. In this paper, the subsonic aerodynamic force obtained numerically is applied to the low pressure compressor blade with a low constant rotating speed. The blade is established as a pre-twist and presetting cantilever plate with a rectangular section under combined excitations, including the centrifugal force and the aerodynamic force. In view of the first-order shear deformation theory and von-Kármán nonlinear geometric relationship, the nonlinear partial differential dynamical equations for the warping cantilever blade are derived by Hamilton's principle. The second-order ordinary differential equations are acquired by the Galerkin approach. With consideration of 1:3 internal resonance and 1/2 sub-harmonic resonance, the averaged equation is derived by the asymptotic perturbation methodology. Bifurcation diagrams, phase portraits, waveforms, and power spectrums are numerically obtained to analyze the effects of the first harmonic of the aerodynamic force on nonlinear dynamical responses of the structure.