## 2 Fundamental Theory
### Analysis Process of the Wave Method  
The ABH beam is a special type of non-uniform structure, and its analysis process is extremely complex. Therefore, we extend the wave method [27] to the analysis of non-uniform beam structures, particularly in fractional-order systems. The wave method proves particularly suitable for analyzing vibration problems in this model, as its recursive formulation inherently incorporates continuity conditions for displacement, rotation, shear force, and bending moment during derivation. This characteristic renders it well suited to address the vibration continuity requirements between the two equations governing distinct regions of the structure. The transmission relation for vibrating waves in non uniform structures is derived in the following part. Suppose that \(w(x,t)=W(x)e^{i\omega t}\) and \(\varrho(x,t)=F(x)e^{i\omega t}\), the kinetic equation in the form of time-invariant separation is derived:  
\[EI(x)\frac{\partial^{2}W(x)}{\partial x^{4}}-\rho\omega^{2}A(x)W(x)=F(x)\ (0 \leq x<x_{0}) \tag{12}\] \[\big{[}E_{2}pH_{hh}(x)+E_{1}pH_{h}(x)+\mathbf{\eta}_{1}bH_{h}(x)(i \omega)^{\mathbf{\imath}}\big{]}\frac{\partial^{4}W(x)}{\partial x^{4}}-\rho\omega^ {2}A(x)W(x)=F(x)\ (x_{0}\leq x\leq L)\]  
As shown in Figure 4, the structure needs to be divided into a sufficient number of uniform beam micro-segments, where the interface of the micro-segments can be considered as a discontinuity feature. Therefore, the wave method is a semi-analytical method and the numerical calculation accuracy can be guaranteed when the number of segments is adequate. As shown in Figure 5, the \(W(x)\) in Equation (12) can be converted into the following form:  
\[W_{i+1}(x)= \begin{bmatrix}1&1\end{bmatrix} \mathbf{\eta}_{i}^{-}+ \begin{bmatrix}1 &1\end{bmatrix} \mathbf{\eta}_{i}^{-}\ \ (i=0,1,2,\cdots,n), \tag{13}\]  
where \(n\) is the number of segments; \(\mathbf{\eta}_{i}^{+}\) and \(\mathbf{\eta}_{i}^{-}\) are the vibration waves in the \(i\)th interface. \[\mathbf{\eta}_{i}^{+}= \begin{bmatrix}a_{1}e^{-ik_{i}x}\\ a_{2}e^{-ik_{i}x}\end{bmatrix} ,\ \mathbf{\eta}_{i}^{-}= \begin{bmatrix}a_{3}e^{ik_{i}x}\\ a_{4}e^{ik_{i}x}\end{bmatrix} ,\]  
where \(a_{ji}\) (\(j=1,2,3,4\)) are the vibrational wave coefficients; \(k_{i}\) is the wave number of the \(i\)th segment. Figure 4: Analysis technological process of ABH structure. Figure 5: Distribution of vibration waves of the discontinuous beam. The general equivalent iterative computational equations for vibration waves on non-uniform beams can be derived from the vibration wave transfer relationship on beams. \[ \begin{array}{l}\eta_{i}^{+}=T_{i1}^{+}\xi_{0}^{+}+R_{i,1}^{-}\eta_{i}^{-}\\ \xi_{0}^{-}=R_{i,1}^{+}\xi_{0}^{+}+T_{i,1}^{-}\eta_{i}^{-}\end{array} , \tag{14}\]  
where the iterative formulas for the equivalent transmission and reflection matrices \(\mathbf{T}_{i,1}^{+}\), \(\mathbf{T}_{i,1}^{-}\), \(\mathbf{R}_{i,1}^{+}\) and \(\mathbf{R}_{i,1}^{-}\) in the first \(i\) segment are as follows:  
\[ \begin{array}{l}\mathbf{T}_{i,1}^{+}=\mathbf{T}_{i}^{+}f_{i}\Big{(}\mathbf{I}-\mathbf{R}_{i-1,1}^{-}\mathbf{f}_{i}\mathbf{R}_{i}^{+}f_{i}\Big{)}^{-1}\mathbf{T}_{i-1,1}^{+}\\ \mathbf{R}_{i,1}^{-}=\mathbf{R}_{i}^{-}+\mathbf{T}_{i}^{+}f_{i}\Big{(}\mathbf{I}-\mathbf{R}_{i-1,1}^ {-}\mathbf{f}_{i}\mathbf{R}_{i}^{+}f_{i}\Big{)}^{-1}\mathbf{R}_{i-1,1}^{-}\mathbf{f}_{i}\mathbf{T}_ {i}^{-}\\ \mathbf{R}_{i,1}^{+}=\mathbf{R}_{i-1,1}^{+}+\mathbf{T}_{i-1,1}^{-}\mathbf{f}_{i}\mathbf{R}_{i}^{+} \mathbf{f}_{i}\Big{(}\mathbf{I}-\mathbf{R}_{i-1,1}^{-}\mathbf{f}_{i}\mathbf{R}_{i}^{+}\mathbf{f}_{i} \Big{)}^{-1}\mathbf{T}_{i-1,1}^{+}\\ \mathbf{T}_{i,1}^{-}=\mathbf{T}_{i-1,1}^{-}\mathbf{f}_{i}\mathbf{T}_{i}^{-}+\mathbf{T}_{i-1,1}^{-} \mathbf{f}_{i}\mathbf{R}_{i}^{+}\mathbf{f}_{i}\Big{(}\mathbf{I}-\mathbf{R}_{i-1,1}^{-}\mathbf{f}_{i}\bm {R}_{i}^{+}\mathbf{f}_{i}\Big{)}^{-1}\mathbf{R}_{i-1,1}^{-}\mathbf{f}_{i}\mathbf{T}_{i}^{-}\end{array} ,\]  
where \(f_{i}\) is the transfer matrix between the (\(i\)-1)th and \(i\)th interfaces, \(\mathbf{f}_{i}=