非惯性参考系中弹性薄板的动力学分析：参数激励与强迫激励联 …

本文研究了非惯性参考系中弹性薄板的大范围运动与大变形运动相互耦合时的非共振分岔，在建立了该动力系统运动控制方程的基础上，利用多尺度法得到了参数激励与强迫激励联合作用下辈在惯性参考系中弹性薄板非共振时的分岔响庆方程及其在反动和几何尺寸两个分岔参数影响下的空间分岔集，讨论了该动力系统的稳定性，并给出了它的非共振分岔响应曲线。     

非惯性参考系中弹性薄板动力系统在纵横振动相互耦合时的全局分 …

讨论非惯性参考系中弹性薄板动力系统１：１内共振时的全局分岔及其混沌性质。首先对系统的奇点进行了分析，进而得到了奇点附近同宿轨的参数方程，再用Ｍｅｌｉｎｉｋｏｖ方法研究了系统的同宿轨分岔及其混沌运动。研究表明，对各种不同共振情形，系统将由同宿轨分岔过渡到混沌运动。最后用数值仿真证实了理论分析的结果。     

非惯性参考系中弹性薄板动力系统在纵横振动相互耦合时的全局分岔及混沌性质

讨论非惯性参考系中弹性薄板动力系统１∶１内共振时的全局分岔及其混沌性质．首先对系统的奇点进行了分析，进而得到了奇点附近同宿轨的参数方程，再用Ｍｅｌｎｉｋｏｖ方法研究了系统的同宿轨分岔及其混沌运动．研究表明，对各种不同共振情形，系统将由同宿轨分岔过渡到混沌运动．最后用数值仿真证实了理论分析的结果．     

具有非轴对称刚度转轴的分岔

研究具有非轴对称刚度转轴的１／２亚谐共振和分岔,首先用Ｈａｍｉｌｔｏｎ原理导出运动微分方程,这是刚度系数周期性变化的参数激励方程,然后用多尺度法求得平均方程分岔响应方程和定常解,最后用奇异性理论分析分岔响应方程和定常解的稳定性,得到了局部分岔集和不同区域的不同分岔响应曲线。     

简谐荷载作用下弹性地基上不可伸长梁的2次超谐共振响应研究

基于已建立的弹性地基上不可伸长梁的非线性动力学模型,利用梁的量纲归一化运动方程和多尺度方法求得梁2次超谐共振的幅频响应方程和位移的二次近似解。进而,运用梁的幅频响应曲线对其超谐共振响应特性进行研究,同时分析了弹性地基模型、Winkler参数、外激励幅值、边界条件等对该共振响应的影响效应。结果表明:弹性地基模型中剪切参数的引入增大了梁2次超谐共振响应的幅值和多值区域;弹性地基Winkler参数的增加会抑制系统的共振响应,但同时会增加系统动力响应的软弹簧特性;在外激励幅值较小的情况下,系统共振响应未展现出明显的非线性特征;边界约束对弹性地基剪切参数作用于梁2次超谐共振响应的效应有显著影响,可在一定程度上改变系统响应幅值及多值区域。     

强非线性系统的亚谐共振解和其转迁集

本文通过非线笥时间变换，引入共振关系式，求出了强非线性振动系统主共振解和亚谐解。进而求得Ｄｕｆｆｉｇｎ方程主共振解以及从主共振取１／３亚谐分岔的转迁集，与ＩＨＢ（Ｉｎｃｃｒｍｅｎｔａｌ Ｈａｒｍｏｎｉｌ Ｂａｌａｎｃｃ）方法的结果比较表明，两者吻合良好。     

拱型结构在参、强激励下的非线性振动分析

利用数值解析法研究了拱型结构在参数激励与强迫激励联合作用下的非线性振动特性。得到了轴向力与固有频率的关系及轴向力对发生主共振,１／２亚谱共振的影响,由于１／２亚谐共振是高频激振低频响应,是最危险区域,应得到足够的重视,为工程设计提供了可靠的理论依据。     

用IHB法分析双谐波激励下铰接塔-油轮系统的非线性动力学特性

研究了考虑平方阻尼情况下,铰接塔-油轮系统在双谐波激励下的非线性动力学特性.将该系统简化为单自由度分段线性恢复力,含平方阻尼的运动学分析模型,建立了铰接装载塔系统的分段非线性动力学方程.采用增量谐波平衡法获得系统周期解,使用Floquet理论判断系统的运动稳定性,结合路径跟踪法跟踪系统响应曲线,获得了系统所有可能的亚谐、谐波、组合谐波共振运动.分析了不对称恢复刚度比值对系统亚谐、组合谐波共振和对系统运动倍周期分岔点的影响,比较了考虑平方阻尼和不考虑平方阻尼情况下系统非线性动力学特性,得到了系统的一些重要的非线性动力学特点.     

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。     

Duffing系统的主--亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.     

大挠度微曲矩形薄板的动力性质

本文导出了由位移表示的带初始挠度的粘弹性大挠度矩形薄板的运动方程。在简支边界条件下,应用分歧理论和Melnikov方法,研究在周期外力作用下系统的动力行为。给出了出现次谐分枝和马蹄态的条件。     

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.     

面内剪切非线性对复合材料层合板参数共振的影响

利用ＶｏｎＫａｒｍａｎ薄板大挠度理论和Ｈａｈｎ－Ｔｓａｉ本构方程研究了四边简支、一对边受压缩动载荷的（０／９０）。对称铺设正交各向异性矩形层合板的参数振动问题。     

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。     

Fundamental and subharmonic resonances in a system with a ‘1-1’ internal resonance

The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a 1–1 internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.     

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。     

Subharmonic Melnikov method of six-dimensional nonlinear systems and application to a laminated composite piezoelectric rectangular plate

In this paper, the bifurcations of subharmonic orbits are investigated for six-dimensional non-autonomous nonlinear systems using the improved subharmonic Melnikov method. The unperturbed system is composed of three independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The key problem at hand is the determination of the sufficient conditions on some of the periodic orbits for the unperturbed system to generate the subharmonic orbits after the periodic perturbations. Using the periodic transformations and the Poincaré map, an improved subharmonic Melnikov method is presented. Two theorems are obtained and can be used to analyze the subharmonic dynamic responses of six-dimensional non-autonomous nonlinear systems. The subharmonic Melnikov method is directly utilized to investigate the subharmonic orbits of the six-dimensional non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Using the subharmonic Melnikov method, the bifurcation function of the subharmonic orbit is obtained. Numerical simulations are used to verify the analytical predictions. The results of the numerical simulation also indicate the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.