The bifurcation analysis on the circular functionally graded plate with combination resonances

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.     

粘弹性轴向运动梁的非线性动力学行为

本文研究了带有小脉动的轴向运动粘弹性梁的分岔及混沌现象。建立了系统的动力学模型。通过二阶Galerkin截断，把描述系统运动的偏微分方程离散化。利用数值方法分别分析了几种运动脉动频率时，梁随轴向运动脉动幅值，平均速度及粘弹性系数等几个参数变化时的运动分岔行为。利用Lyapunov指数识别系统的动力学行为，区分准周期振动和混沌运动。     

时变张力作用下轴向运动梁的分岔与混沌

轴向运动结构的横向参激振动一直是非线性动力学领域的研究热点之一. 目前研究较多的是轴向速度摄动的动力学模型,参数激励由速度的简谐波动产生. 但在工程应用中,存在轴向张力波动的运动结构较为广泛,而针对轴向张力摄动的模型研究较少. 本文研究了时变张力作用下轴向变速运动黏弹性梁的分岔与混沌. 考虑随着时间周期性变化的轴向张力,计入线性黏性阻尼,采用Kelvin模型的黏弹性本构关系,给出了梁横向非线性 振动的积分--偏微分控制方程. 首先应用四阶Galerkin截断方法将控制方程离散化,然后采用四阶Runge-Kutta方法计算系统的数值解,进而确定其动力学行为. 基于梁中点的横向位移和速度的数值结果,仿真了梁沿平均轴速、张力摄动幅值、张力摄动频率以及黏弹性系数变化的倍周期分岔与混 沌运动,并且通过计算系统的最大李雅普诺夫指数来识别其混沌行为. 结果表明:较小的平均轴速有助于梁的周期运动,梁在临界速度附近容易发生倍周期分岔与混沌行为. 随着张力摄动幅值的增大,梁的振动幅值的混沌区间不断增大. 较小的黏弹性系数和张力摄动频率更容易使梁发生混沌运动. 最后,给出时程图、频谱图、相图以及Poincaré 映射图来确定梁的混沌运动.      

Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam

This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.     

轴向运动导电板磁弹性非线性动力学及分岔特性

研究磁场环境中轴向运动导电薄板磁弹性动力学及分岔特性。考虑几何非线性因素,在给出薄板运动的动能、应变能及外力虚功的基础上,应用哈密顿变分原理,得到磁场中轴向运动薄板的非线性磁弹性振动方程,并给出洛伦兹电磁力的确定形式。针对横向磁场环境中条形板共振特性进行分析,应用多尺度法和奇异性理论,得到稳态运动下的分岔响应方程以及普适开折对应的转迁集。通过算例,分别得到以磁感应强度、轴向运动速度和激励力为分岔控制参数的分岔图、最大李雅普诺夫指数图和庞加莱映射图等计算结果,讨论不同分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,通过相应参数的改变可实现对系统复杂动力学行为的控制。     

热环境中功能梯度圆形薄板的混沌运动

针对陶瓷-金属功能梯度圆板,同时考虑几何非线性、材料物性参数随温度变化且材料组分沿厚度方向按幂律分布的情况,应用虚功原理给出了热载荷与横向简谐载荷共同作用下的非线性振动偏微分方程。在固支无滑动的边界条件下,通过引入位移函数,利用伽辽金方法得到了达芬型非线性动力学方程。利用Melnikov方法,给出了热环境中功能梯度圆板可能发生混沌运动的临界条件。通过数值算例,给出了不同体积分数指数和温度的同宿分岔曲线,平面相图和庞加莱映射图,讨论其对临界条件的影响,证实了系统混沌运动的存在。通过分岔图和与其相对应的最大李雅普诺夫指数图,分析了激励频率和激励幅值对倍周期分岔的影响及变化规律,发现系统可出现周期、倍周期和混沌等复杂动力学响应。     

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。     

磁场中轴向运动条形板强非线性超谐共振及混沌运动

针对磁场环境中周期外载作用下轴向运动导电条形板的非线性振动及混沌运动问题进行研究。应用改进多尺度法对横向磁场中条形板的强非线性振动问题进行求解,得到超谐波共振下系统的分岔响应方程。根据奇异性理论对非线性动力学系统的普适开折进行分析,求得含两个开折参数的转迁集及对应区域的拓扑结构分岔图。通过数值算例,分别得到以磁感应强度、轴向拉力、激励力幅值和激励频率为分岔控制参数的分岔图和最大李雅普诺夫指数图,以及反映不同运动行为区域的动力学响应图形,讨论分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,可通过相应参数的改变实现对系统复杂动力学行为的控制。     

Combination,principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.     

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton??s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincaré maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincaré sections.