## 2 A Mathematical Framework and Its Corresponding Solution for a Symmetric ABH Beam
### The Solution of the Model Using the GEM  
Given that the ABH beam is subjected to a harmonic external excitation with the circle frequency \(\omega\), it follows that the structure's steady-state response will also be harmonic. These can be mathematically represented as  
\[\mathbf{f(t)}=\mathbf{F}e^{i\omega t},\quad\mathbf{F}=F_{t}\cdot\{\varphi_{1}(x_{f}), \cdots,\varphi_{n}(x_{f})\}, \tag{15}\]  
\[\mathbf{a(t)}=\mathbf{A}e^{i\omega t},\quad\mathbf{A}=[a_{1},a_{2},\cdots,a_{n}], \tag{16}\]  
where \(\mathbf{F}\) represents the amplitude vector of the external force and \(\mathbf{A}\) the amplitude vector of the response. Substituting Equations (15) and (16) into Equation (12) yields  
\[[\mathbf{K}-\omega^{2}\mathbf{M}]\mathbf{A}=\mathbf{F}. \tag{17}\]  
When there is no external force applied, it translates into a generalized eigenvalue problem characterized by the presence of damped angular eigenfrequencies  
\[\omega_{d}^{2}=\omega_{n}^{2}(1+i\eta) \tag{18}\]  
where \(\omega_{n}\) is the natural frequency of the system and \(\eta\) denotes the modal loss factor, derived from the modal strain energy method (MSEM) [39], which employs a complex Young's modulus, as detailed in Equation (2). The modal loss factor is the ratio of energy dissipated per cycle to the maximum energy stored within a cycle. It represents the viscoelastic properties of a material; a higher loss factor indicates greater viscosity, while a lower loss factor indicates greater elasticity. The modal loss factor plays a crucial role in evaluating how a structure dissipates energy when subjected to dynamic forces. It is especially valuable for determining the energy absorption capacity of damping layers due to the ABH effect.  
The key to solving Equation (17) is to determine the shape functions in Equations (4) and (5). Research presented in [17,21,29] indicates that employing power series and polynomial expansions for shape functions may result in matrix singularities and poorly conditioned problems, thereby potentially reducing the solution's accuracy. This is particularly relevant in the context of the ABH beam, where there are significant variations in wave numbers and wave velocities at the beam's ends. The Gaussian function is proposed as the shape function due to its effectiveness in reducing the likelihood of singularities in the mass matrix calculations. The Gaussian function has the standard form below:  
\[g(x)=\frac{1}{\sigma\sqrt{2\pi}}exp\bigg{(}-\frac{(x-\mu)^{2}}{2\sigma^{2}} \bigg{)} \tag{19}\]The function is symmetric with respect to the line \(x=\mu\), and it reaches its peak at \(x=\mu\). The value of \(\sigma\) controls the width of the function. The larger the \(\sigma\), the flatter the function and the more spread out the distribution; the smaller the \(\sigma\), the sharper the function and the more concentrated the distribution. In this paper, we choose the following series of Gaussian functions as the shape functions:  
\[\varphi_{i,k}(x)=\frac{1}{\sqrt{2\pi}}2^{\frac{a}{2}}exp\left(-\frac{(2^{a}x- k)^{2}}{2}\right), \tag{20}\]  
where the integer \(a\) serves as a scaling factor, while the integer \(k\) acts as a translation factor. By selecting appropriate values for \(a\) and \(k\) in Equation (20), the vibration of the entire ABH beam can be recovered [30]. \(a\) acts as a regulatory parameter that dictates the precision of the model. With an increasing number of recoverable modes as the value of \(a\) gets larger, the computational cost correspondingly rises. According to the suggestion in [30], we limit the range of \(k\) to \([-8,8]\) for the ABH beam considered in this paper. \(a\) is determined by the following relation:  
\[a\geq ceil\left(\log_{2}\left(\frac{8}{x_{b4}}\right)\right), \tag{21}\]  
where \(ceil(x)\) denotes the smallest integer that is greater than or equal to \(x\). For every given \(a\), the range of \(k\) is limited to  
\[k\in[-4,ceil(2^{a}x_{b4})+4] \tag{22}\]