# Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties
## 2 The Model and Method
### Deterministic Metamaterial Beam Structure and Its Band Gap Calculation  
Before analyzing the spatial uncertainties in the metamaterial beam, this subsection briefly introduces the deterministic metamaterial beam structure and its band gap calculation method. As shown in Figure 2, a finite conventional MB structure with \(N_{u}\) unit cells is formed by periodically attaching a series of resonators to a uniform aluminum beam. The lattice constant is \(a\), and the width and thickness of the aluminum beam are denoted as \(b\) and \(h\), respectively. Each resonator contains a spring \(k\) and a mass \(m\). Each unit cell contains \(N_{e}\) Euler-Bernoulli beam elements. Thus, the uniform aluminum beam can be divided into \(N_{u}N_{e}\) beam elements. It is worth emphasizing that \(N_{e}\) must be even so that the spring mass resonator can be located just on the node. The stiffness matrix and mass matrix of the Euler-Bernoulli beam element are shown in Ref. [8]. After assembling these \(N_{u}N_{e}\) elements, the global stiffness matrix \(\mathbf{K}_{\mathrm{beam}}\) and global mass matrix \(\mathbf{M}_{\mathrm{beam}}\) of the aluminum beam can be obtained. It should be noted that node numbering is carried out from left to right along MB during assembly. Furthermore, the global dynamic stiffness matrix \(\mathbf{D}_{\text{beam}}\) of the aluminum beam can be obtained as follows:  
\[\mathbf{D}_{\text{beam}}=\mathbf{K}_{\text{beam}}-\omega^{2}\mathbf{M}_{\text {beam}} \tag{1}\]  
Here, \(\omega\) is the circular frequency. On the other hand, the influence of the spring mass resonator on the dynamic behavior of the MB can be obtained by the methodology described in Ref. [4]. The dynamic effect matrix of the spring mass resonator \(\mathbf{D}_{\text{R}}\) is defined as follows:  
\[\mathbf{D}_{\text{R}}(i,j)=\left\{\begin{array}{cc}\frac{1}{(1/k)-(1/\omega ^{2}m)}&i=j=N_{e}(2s-1)+1(s=1,2\cdots N_{u})\\ 0&\text{else}\end{array}\right. \tag{2}\]  
\(\mathbf{D}_{\text{R}}\) Has the same dimensions as \(\mathbf{D}_{\text{beam}}\). Obviously, the global dynamic stiffness matrix of the metamaterial beam \(\mathbf{D}\) is equal to \(\mathbf{D}_{\text{beam}}\) + \(\mathbf{D}_{\text{R}}\). The dynamic equation of the finite MB can be written in the following form:  
\[\mathbf{D}\mathbf{U}=\mathbf{F} \tag{3}\]  
where \(\mathbf{U}\) is the dynamic response vector and \(\mathbf{F}\) is the structural load vector  
The band gap frequency range can usually be determined by performing a vibration transmission analysis. The transmission coefficient of vibration \(T\) is defined as:  
\[T=20\times\log_{10}\left|\frac{u_{\text{out}}}{u_{\text{in}}}\right| \tag{4}\]  
In the above, \(u_{in}\) and \(u_{out}\) represent the displacements of the excitation point and the response point, respectively. The left and right ends of the metamaterial beam are selected as the excitation and response points, respectively. Therefore, \(u_{in}\) and \(u_{out}\) are located in the first and penultimate rows of the vector \(\mathbf{U}\), respectively. Assuming that the displacement of the excitation point is \(1\) (\(\mathbf{U}(1,1)=u_{in}=1\)), the implementation method is: (1) modify  
Figure 1: Flowchart of the method for solving the dynamic response of the interval MB structure.  
Figure 2: Conventional MB structure.  
\(\mathbf{D}(1,1)\) to a value far greater than the other elements in \(\mathbf{D}\), such as \(\mathbf{D}(1,1)=10^{50}\); (2) define \(\mathbf{F}\) as follows:  
\[\left\{\begin{array}{cc}\mathbf{F}(n,1)=10^{50},&n=1\\ \mathbf{F}(n,1)=0,&n\
eq 1\end{array}\right. \tag{5}\]  
Finally, solving Equation (3) yields \(u_{out}\). If \(\left|u_{\text{in}}\right|\) is greater than \(\left|u_{\text{out}}\right|\), namely, \(T\) is less than 0, it indicates that the flexural wave is attenuated when propagating through the metamaterial beam. Therefore, the frequency range wherein \(T\) is less than zero is the band gap.