调和力作用下刚体运动动力学研究

S. Hogan
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引用次数: 186

摘要

本文详细分析了最简单、应用最广泛的刚性块体受调和力模型。该块具有极其复杂的动力学,揭示了许多不同类型的响应。找到了所有阶的对称单碰撞次谐波轨道,并给出了它们所处的参数空间区域。特别是发现了非对称轨道的周期倍级联,它最终产生明显的非周期或混沌响应。说明了对初始条件的敏感性,这导致渐近动力学预测的不确定性。然而,在实际地震中,瞬态反应可能是最重要的。为此,引入了最大暂态域的概念。从这个角度来看,尽管吸引力的渐近域具有混沌的性质,但响应显示出相当有序和可预测的。结果表明,在建立一套商定的初始条件之前,安全问题不能令人满意地解决。在初始条件正确的情况下,木块在非常高的加速度下可以存活,在非常低的加速度下可以倒塌。在试图重现给定结构的反应时,还考虑使用实际地震记录。如果目前的结论适用于一般激励,那么记录中的小错误可能会产生很大的反应差异。目前的方法包括轨道稳定技术和详细的数值计算。这些结果得到了来自电子模拟电路的令人鼓舞的定性一致的支持。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the dynamics of rigid-block motion under harmonic forcing
In this paper the simplest and most widely used model of a rigid block undergoing harmonic forcing is analysed in detail. The block is shown to possess extremely complicated dynamics, with many different types of response being revealed. Symmetric single-impact subharmonic orbits of all orders are found and regions of parameter space in which they occur are given. In particular, period-doubling cascades of asymmetric orbits are found, which ultimately produce an apparently non-periodic or chaotic response. Sensitivity to initial conditions is illustrated, which leads to uncertainty in the prediction of the asymptotic dynamics. Nevertheless, the transient response may be the most important in connection with real earthquakes. To this end, the concept of the domain of maximum transients is introduced. In this light the response is shown to be quite ordered and predictable, despite the chaotic nature of the asymptotic domain of attraction. It is shown that safety issues cannot be satisfactorily resolved until an agreed set of initial conditions is established. It appears that blocks may survive under very high accelerations and topple at very low accelerations provided the initial conditions are correct. Consideration is also given to the use of actual earthquake recordings in attempting to reproduce responses in given structures. If the present conclusions carry over to general excitations, then small errors in recordings may produce large differences in response. The present methods include orbital stability techniques together with detailed numerical computations. These results are backed up by encouraging qualitative agreement from an electronic analogue circuit.
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