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

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.