An R(x)-orthonormal theory for the vibration performance of a non-smooth symmetric composite beam with complex interface

作     者：Chein-Shan Liu Bo-Tong Li 

作者机构：College of Mechanics and MaterialsHohai UniversityNanjing 210098China Center of Excellence for Ocean EngineeringDepartment of Mechanical and Mechatronic EngineeringTaiwan Ocean UniversityKeelung 202-24China School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijing 100083China 

出 版 物：《Acta Mechanica Sinica》 (力学学报（英文版）)

年 卷 期：2019年第35卷第1期

页      面：228-241,I0007页

核心收录：

学科分类：08[工学] 0802[工学-机械工程] 0701[理学-数学] 0801[工学-力学（可授工学、理学学位）] 0702[理学-物理学] 

基　　金：The Thousand Talents Plan of China Fundamental Research Funds for the Central Universities for the financial support to the first author are highly appreciated supported by the Fundamental Research Funds for the Central Universities 

主　　题：Symmetric composite beams R(x)-orthogonality of second-order derivatives of boundary functions R(x)-orthonormal theory Non-smooth/discontinuous interface Sequentially closed-from natural frequencies 

摘      要：A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A :=[aij ],aij= 0 if i + j is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of A. We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.