# Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties
## 3 Results and Discussion
### Metamaterial Beam with Interval Density Field  
In this subsection, the density of a basic beam is considered to contain spatial uncertainties. The midpoint of the interval density field is \(\rho_{0}=2730\) kg/m\({}^{3}\). The number of terms in KL expansion is \(N=10\). The interval field parameters \(C_{B}\) and \(l_{B}\) are equal to 0.05 and 0.8\(L_{\text{beam}}\), respectively. The other parameters are the same as in Section 3.1. Thus, an IVTA can be carried out, and its results are displayed in Figure 5. For comparison, Figure 5 also shows the deterministic IVTA results (midpoint line), i.e., the calculation results in Figure 4. The yellow line, blue line, and red line represent the coefficient \(\overline{T}\) calculated by IPFEM, VM, and MCM, respectively. The four curves in Figure 5 basically coincide at the low frequency range. However, when the frequency gradually approaches the ending frequency of the band gap, the coefficient \(\overline{T}\) deviates obviously from the deterministic analysis results. For convenience, the frequency range where the upper bound of the interval transmission coeffcient \(\overline{T}\) is less than 0 is referred to as the safe band gap. No matter how the interval density field fluctuates, the metamaterial beam structure can always provide vibration attenuation in the frequency range of the safe band gap. For example, the safe band gap obtained by the two reference methods (MCM and VM) ranged from 29.4452 Hz to 67.7894 Hz (see the purple rectangular frame in Figure 5), which reduced the frequency range of the safe band gap by 5.37% compared to the deterministic band gap range. However, in fact,  
Figure 4: Vibration transmission of a finite MB.  
near the ending frequency, the curves representing the calculation results of the VM and MCM did not coincide exactly. For the IPFEM, the safe band gap ranged from 29.4452 Hz to 69.3760 Hz (see the light blue frame in Figure 5). In order to validate the calculation precision of the IPFEM, the relative error of the IPFEM with respect to the MCM was also calculated as follows:  
\[\delta=\frac{\left|f_{\text{P}}-f_{\text{M}}\right|}{f_{\text{M}}}\times 100\% \tag{27}\]  
where \(f_{\text{P}}\) represents the beginning frequency or ending frequency of the safe band gap calculated by the IPFEM, and \(f_{\text{M}}\) denotes the beginning frequency or ending frequency of the safe band gap calculated by the MCM. That is to say, in the numerical example shown in Figure 5, the relative error of the beginning frequency of the safe band gap is 0%, and the relative error of the ending frequency is 2.34%. At the same time, the calculated result curve of the IPFEM in Figure 5 also had a certain deviation from the two reference methods near the ending frequency. The reason for this was that there was a natural frequency near the ending frequency, and the nonlinearity was strengthened near the natural frequency, which led to the gradual deviation of the calculation results of various interval methods [27; 28]. In other words, due to the spatial uncertainties of the density of the basic beam, the safe band gap frequency range was narrower than the deterministic band gap frequency range (29.4452-69.9634 Hz). Specifically, only the ending frequency of the safe band gap is affected by the spatial uncertainties of the density of a basic beam, but the beginning frequency is not. This is mainly because, in the deterministic metamaterial beam structure, the beginning frequency is \(\frac{1}{2\pi}\sqrt{\frac{k}{m}}\) and the ending frequency is \(\frac{1}{2\pi}\sqrt{\frac{k}{m}\left(1+\frac{m}{\rho 5a}\right)}\)[6]. Obviously, the uncertainties of the density of a basic beam only affect the ending frequency.