摘要
作为晶格模型的一维非线性单摆链,在振动激励下,会出现各种模式的包络波,但是理论上不能预言这些起始波动的模式和所需要的激励条件.在Floquet线性不稳定分析的理论框架里,对振动激励下的单摆链进行了起始不稳定性的解析分析.通过数值计算,发现下谐波模式的激励幅度远低于同频模式,解释了起始响应模式为1/2分频的下谐波模式的实验结果.得到在各种激励频率下,起始波形、时间响应模式和起始激励加速度幅值之间的关系,实验结果支持了理论计算.
Instabilites of the one dimensional nonlinearly coupled pendulum chain, a model of lattice, will arise under vertical vibration. Nonlinear waves including soliton and chaos have been observed in such a chain. However, up to now, the modes of the onset instabilities together with their driving conditions are still kept unknown. In this article, the onset instability, a wave driven by the lowest amplitude of the excitation, has been analyzed for the chain in the theoretical framework of Floquet instability analysis. The Floquet method has been extended to the case of the discrete system. The numerical computation shows that the driving amplitude for a sub-harmonic (half driving frequency) response mode is much lower than that for a harmonic response mode, which explains the fact that the observed solitons, always oscillates at the half driving frequency. The relations between the onset pattern together with its response mode with time, and the driving acceleration amplitude have also been obtained at different driving frequency. Therefore, for a given driving frequency, we can theoretically predict both the spatial distribution and temporal response of the onset instability. In the experiment, we constructed an experimental chain made of 47 pendulums. The waveforms and responses of the onset instabilities were measured at different driving frequency. Comparing the results with the theoretical ones, we find that the experimental data during the medium wave-number are out of the theoretical expectation. In other words, the theory developed here can well agree with the experimental observations in the small and large wave-number regimes. The long-wave approximation used in the Floquet instability analysis makes the prediction fail in the medium wave-number regime. Fortunately, the theory is still suitable to describe the onset waves with large wave-numbers because they are patterns with small wave-numbers in phase mismatch mode.
出处
《南京大学学报（自然科学版）》
CAS
CSCD
北大核心
2005年第2期198-203,共6页
Journal of Nanjing University（Natural Science）
基金
国家自然科学基金(10374050)
国家教教育部跨世纪优秀人才计划基金
国家重点基础研究发展规划(973)"非线性科学中的若干前沿问题"
