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

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## Abstract

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 non-smooth/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 \({{\varvec{A}}}:=[a_{ij}]\), \(a_{ij}=0\) if \(i+j\) is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of \({{\varvec{A}}}\) 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 \({{\varvec{A}}}\). We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.

## Keywords

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

## Notes

### Acknowledgements

The Thousand Talents Plan of China (Grant A1211010) and Fundamental Research Funds for the Central Universities (Grant 2017B05714) for the financial support to the first author are highly appreciated. Dr. Liu is grateful to the twelfth Guanghua Engineering Science and Technology Prize. The work of Botong Li is supported by the Fundamental Research Funds for the Central Universities (Grant FRF-TP-17-020A1).

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