## 2 A Mathematical Framework and Its Corresponding Solution for a Symmetric ABH Beam
### The Governing Equation  
In this context, the displacement vector of the beam represents the movement along the \(x\)-axis and \(z\)-axis at every point along the beam, including the damping layer. It should be noted that the assumption presupposes an ideal adhesion between the damping layer and the primary beam, which is crucial for maintaining displacement continuity. \(w(x,t)\) denotes the flexural displacement of any point on the beam in relation to time \(t\). According to Euler-Bernoulli beam theory, the displacement field of the beam can be expressed as follows:  
\[[-z\frac{\partial w}{\partial x},w]. \tag{3}\]  
Assume that \(w(x,t)\) can be expanded as  
\[w=\sum_{k}^{n}a_{k}(t)\varphi_{k}(x)=\mathbf{a}^{T}\boldsymbol{\varphi}, \tag{4}\]  
where \(\boldsymbol{a}\) is a series of time-dependent coefficients to be determined and \(\boldsymbol{\varphi}\) the admissible shape functions  
Figure 1: The diagrammatic representation of a symmetric ABH beam. \[\mathbf{a(t)} =[a_{1}(t),a_{2}(t),\cdots,a_{n}(t)]^{T}, \tag{5}\] \[\mathbf{\varphi(x)} =[\varphi_{1}(x),\varphi_{2}(x),\cdots,\varphi_{n}(x)]^{T}.\]  
From classical Lagrange theory, the Lagrange quantity \(L\) for the system depicted in Figure 1 is given by  
\[L=E_{k}-E_{s}+W, \tag{6}\]  
where \(E_{k}\) and \(E_{s}\) are the kinetic energy and potential energy of the structure, respectively, and \(W\) is the work done on the ABH structure by external forces \(F_{t}\),  
\[E_{k}=\frac{1}{2}\int\rho_{b}\Big{(}\frac{\partial\ w}{\partial\ t}\Big{)}^{2 }dV, \tag{7}\]  
\[E_{s}=\frac{1}{2}\int E_{b}I\Bigg{(}\frac{\partial^{2}\ w}{\partial\ x^{2}} \Bigg{)}^{2}dx+E_{e}, \tag{8}\]  
\[E_{e}=\frac{1}{2}K_{1}[\Big{(}\frac{\partial\ w(x_{b4},t)}{\partial\ x} \Big{)}^{2}+\Big{(}\frac{\partial\ w(0,t)}{\partial\ x}\Big{)}^{2}]+\frac{1}{ 2}K_{2}[w^{2}(x_{b4},t)+w^{2}(0,t)], \tag{9}\]  
\[W=F_{t}\,w(x_{f},t), \tag{10}\]  
where \(E_{e}\) is the potential energy generated by the four springs at the boundary of the ABH beam, and \(\rho_{b}\) and \(E_{b}\,I\) are the density and the flexural rigidity of the beam, respectively. The integration process in Equations (7) and (8) must encompass both the beam and its associated damping layers. From [29], the damping layers are assumed to be fully integrated with the beam and are conceptualized as integral components of the system, characterized by their inherent material attributes, namely, the modulus of elasticity \(E_{d}\) and the density \(\rho_{d}\). Substituting Equations (4)-(10) into the Lagrange equation  
\[w=\sum_{k}^{n}a_{k}(t)\varphi_{k}(x)=\mathbf{a}^{T}\boldsymbol{\varphi}, \tag{4}\]0  
one can obtain the following governing equations of the ABH beam in matrix form:  
\[w=\sum_{k}^{n}a_{k}(t)\varphi_{k}(x)=\mathbf{a}^{T}\boldsymbol{\varphi}, \tag{4}\]1  
where \(\mathbf{M}\) and \(\mathbf{K}\) stand for the mass and stiffness matrices, \(\mathbf{f(t)}=F_{t}\,\mathbf{\varphi}(x_{f})\) for the external force vector, and \(\mathbf{a(t)}\) for the response vector, as defined in Equation (5). The matrix \(\mathbf{M}\) can be broken down into three components that correspond to the uniformity \((\mathbf{M}_{uni})\), the ABH elements of the beam \((\mathbf{M}_{ABH})\), and the damping layers \((\mathbf{M}_{d})\), while the matrix \(\mathbf{K}\) can be dissected into four distinct segments, each associated with the aforementioned trio of components \((\mathbf{K}_{uni},\mathbf{K}_{ABH},\mathbf{K}_{d})\) as well as the springs \((\mathbf{K}_{e})\). The expressions of matrices \(\mathbf{M}\) and \(\mathbf{K}\) are given below:  
\[ \begin{split}\mathbf{M}&=\int\rho_{b}\,A(x)\mathbf{\varphi }\,\mathbf{\varphi}^{T}dx=\mathbf{M}_{uni}+\mathbf{M}_{ABH}+\mathbf{M}_{d}\\ &=2[\int_{0}^{x_{b1}}\rho_{b}\,h_{b}(x)\mathbf{\varphi}\,\mathbf{\varphi }^{T}dx+\int_{x_{b1}}^{x_{b2}}\rho_{b}\,h(x)\mathbf{\varphi}\,\mathbf{\varphi}^{T}dx+ \int_{x_{d1}}^{x_{b2}}\rho_{d}\,h_{d}(x)\mathbf{\varphi}\,\mathbf{\varphi}^{T}dx]\end{split}