非周期系统的局域化现象研究  

Localization phenomena in disordered periodic systems

作　　者：李宗威 纪刚[1] 周其斗[1] LI Zongwei;JI Gang;ZHOU Qidou(College of Naval Architecture and Ocean Engineering,Naval University of Engineering,Wuhan 430033,China)

机构地区：[1]海军工程大学舰船与海洋学院,湖北武汉430033

出　　处：《中国舰船研究》2020年第3期88-94,共7页Chinese Journal of Ship Research

摘　　要：[目的]结构非周期性产生的局域化效应可通过局域化因子定量描述,为研究其产生机理及具体表达式,需查明局域化因子的变化规律。[方法]从波动分析角度出发,分析周期结构通频带和止频带的产生原理,对比研究周期结构与非周期结构不同的传递特性。基于Furstenberg定理,给出非周期局域化因子的数学表达式。以弹簧质量链模型为实例,导出相应周期系统的通频带和止频带以及非周期系统的局域化因子数学表示式,以此研究局域化因子随频率的变化规律。基于蒙特卡罗方法的数值仿真,验证相关结论的正确性。[结果]结果表明,局域化程度与导致周期结构成为非周期结构的扰动方差相关;局域化因子具有随机分布特性,但随着单元数量的增加而向极限逼近,且极限值可预报。[结论]研究结果有助于理解一般非周期结构的振动局域化现象,并对指导工程振动控制具有一定的参考作用。[Objectives]The localization effect produced by a disordered structure can be quantitatively described by localizing factors.In order to study the mechanism of the effect and the expression of localizing factors,the rules of the localizing factors must be ascertained.[Methods]From the perspective of waves,this paper analyzes the generation principle of the pass band and stop band of periodic structures,and explains the reason for the localization of disordered periodic structures.It compares the transfer characteristics of periodic and disordered periodic structures,and gives the calculation method for localizing factors based on Furstenberg's Theorem.Taking the mass-spring chain model as an example,this paper derives the pass-band and stop band of periodic systems,as well as the mathematical expressions of localizing factors for disordered periodic systems,studies the frequency variation of localizing factors and verifies them via numerical simulation based on the Monte Carlo method.[Results]The results show that the degree of localization is related to the perturbation variance.The localizing factors have random distribution characteristics,but they approach the limit with the increase in the number of units,which can be predicted.[Conclusions]This paper can provide references for understanding the vibration localization of generally disordered periodic structures,and guiding vibration control in engineering.

关 键 词：非周期结构 Furstenberg定理 局域化因子 蒙特卡罗方法 

分 类 号：U661.44[交通运输工程—船舶及航道工程]