Analysis on Chaotic Vibrations an Arch with Small Rize and a Cable

(1)讨论了小尺寸拱的非线性振动和混沌问题。基本运动方程通过伽辽金法求解，所得时间变量通过周期振动的谐波平衡法和混沌振动的龙格-库塔-吉尔法求解。采用单自由度方法。利用分岔图、庞加莱图和功率谱得到混沌振动。获得并讨论了非线性振动和混沌振动。在二分之一次谐波共振附近发现了混沌振动。得到了通过重复间歇性nT周期振动产生混沌的途径。(2)讨论了小尺寸拱的非线性振动和混沌。采用多自由度方法。针对不同的尺寸比和阻尼常数，讨论了第三模态的非线性自由振动和第一固有频率范围附近的非线性受迫振动。检验了高阶模态对非线性振动行为和第一模态混沌振动的影响。(3)研究了小尺寸矩形板的非线性振动特性。采用伽辽金法对用Marguerre方程描述初始偏转板大偏转的方程进行了分析。得到了不同尺寸比和高宽比下的一阶及以上节点的非线性自由振动和受迫振动。(4)研究了跨长、塔身形状、截面和斜拉索布置等结构特性对斜拉桥斜拉索局部振动的影响。对日本 28 座斜拉桥进行了检查。采用谐波平衡法分析了支撑激励下电缆的非线性振动。显示了各种参数的全局振动和局部振动、不稳定区域的宽度和参数共振幅度之间的关系。

(1) Nonlinear vibrations and chaos of an arch with a small rize are discussed. The basic equation of motion is solved by a Galerkin method and the resulting time variable is solved by the harmonic balance method for periodic vibrations and Runge-Kutta-Gill method for chaotic vibrations. The single-degree-of freedom approach is emplyed. Chaotic vibration is obtained by using bifurcation diagrams, Poincare map and power spectrums. Nonlinear vibrations and chaotic vibrations are obtained and discussed. Chaotic vibrations are found near the one-half subharmonic resonance. The route to chaos through repeating intermittently nT-periodic vibration is obtained.(2) Nonlinear vibrations and chaos of an arch with a small rize are discussed. The multiple-degrees-of freedom approach is emplyed. Nonlinear free vibration of the third mode and nonlinear forced vibrations near the first natural frequency range are discussed for various rize ratios and damping constants. The effect of the higher mode on nonlinear vibration behaviors and chaotic vibrations of the first mode is chcked.(3) The nonlinear vibration properties of a rectangular plate with a small rize are examined. The equations describing the large deflection of the initially deflected plate using the Marguerre equation are analyzed by a Galerkin method. Nonlinear free vibrations and forced vibrations of the first mode and higher nodes are obtained for various rize ratios and aspect ratios.(4) Effects of structural properties such as the span length, tower shape, the cross section and the stay cable arrangement on local vibrations of stay cables of cable stay bridges are studied. 28 cable stay bridges in Japan are examined. Nonlinear vibrations of the cable subjected to the support excitations are analyzed by the harmonic balance method. The relations between global vibrations and local vibrations, widths of unstable regions and amplitudes of parametric resonances are shown for various parameters.