非对称周期结构中耦合波的传播特性

陈荣,吴天行

(上海交通大学机械系统与振动国家重点实验室,上海200240)

摘要：为了揭示周期结构中纵向波和弯曲波的耦合作用，设计了对称和非对称周期结构。考虑子结构中的纵向和弯曲耦合运动，利用导纳法和传递矩阵法，得到了周期单元的传递方程。由于结构中存在多种波的耦合作用，在求解周期单元的传播系数时将出现变态矩阵，采用波型分组法，求得了周期结构中多种波型的传播系数。推导了半无限长和有限长周期结构在纵向力、横向力和弯矩作用下的动态响应。数值计算结果表明，对称周期结构中纵向波和弯曲波的带隙结构相互独立；非对称周期结构中纵向波和弯曲波的耦合明显改变了两种波的带隙结构，只有在两种波阻带重叠的频段内结构上的振动响应才存在衰减。

关键词：周期结构；传递矩阵法；Euler梁；耦合波；传播系数

中图分类号:TB115;TH113;O321文献标志码：A

基金项目：国家自然科学基金资助项目(51079118,51279148);武汉理工大学自主创新研究基金资助(135105006)

收稿日期：2014-05-12修改稿收到日期：2014-07-30

基金项目：国家自然科学基金资助(11072066);国家重点基础研究发展计划(973)：(2013CB733004)

收稿日期：2013-05-28修改稿收到日期：2014-02-11

Coupledwavepropagationinasymmetricperiodicstructures

CHEN Rong, WU Tian-xing(StateKeyLaboratoryofMechanicalSystemandVibration,ShanghaiJiaotongUniversity,Shanghai200240,China)

Abstract:Symmetric and asymmetric periodic structures were designed to investigate the coupling of longitudinal and flexural waves propagating in the structures. By use of mechanical mobility method and transfer matrix method, transfer matrices of the elements were derived in the consideration of the coupling of longitudinal and flexural wave motions. The multiple types of waves propagating in the periodic structures were divided into two categories to avoid the numerical difficulties in solving the case of ill-conditioned transfer matrix. The propagation constants of the longitudinal and flexural waves were calculated, and the harmonic responses of semi-infinite and finite periodic structure in symmetric and asymmetric arrangements were obtained. Numerical simulations reveal that longitudinal wave and flexural wave are uncoupled in symmetric periodic structure; the band structures of longitudinal and flexural wave are significantly influenced by the coupling of the two waves, and longitudinal and flexural vibration response are attenuated only in the zones where stop bands of the both waves overlap.

Keywords:periodicstructure;transfermatrixmethod;eulerbeam;couplingwave;propagationconstant

在实际工程中，许多结构如磁悬浮列车的承载导轨、多跨距桥梁、船舶结构、 钢筋混凝土都是由一定数量相同或类似的单元结构，以相同方式连接而成的，他们具有振动带隙特性。当一种弹性波在一维对称周期结构中传播时，某些频率范围内波不能通过，称之为带隙(bandgap)；……