# Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties
## 4 Conclusions  
In this paper, a metamaterial beam model with spatially varying interval density was first established. Then, the interval dynamic equation could be obtained by integrating the decomposition result of the interval field based on KL expansion into the standard finite element formulation. Next, an IPFEM was applied to obtain the bounds of the dynamic response interval vector. Finally, an IVTA was carried out to intuitively reflect the uncertainties in the transmission coefficient of vibration. Meanwhile, the MCM and VM were also presented as reference methods.  
For convenience, the frequency range wherein the upper bound of the interval transmission coeffcient \(\overline{T}\) is less than 0 is called the safe band gap. No matter how the interval density field fluctuates, the elastic wave within the safe band gap cannot propagate through the metamaterial beam. In other words, in engineering, if researchers only know the specific ranges of uncertainties in the MB structures but cannot determine the exact values, they can use the method in this paper to deal with the uncertainties and calculate the safe band gap. Then, using the safe band gap range instead of the deterministic band gap range can ensure that the metamaterial beam always provides vibration attenuation characteristics and is safe enough during application processes.  
The numerical results showed that the interval density field of the basic beam could only affect the ending frequency of the safe band gap. On the other hand, the uncertainties of the resonators affected both the beginning frequency and ending frequency of the safe band gap. When both the uncertainties in the density of the basic beam and the uncertainties in the resonators were considered, the safe band gap range could be reduced by 8.87% compared to the deterministic band gap without considering uncertainties. In the numerical examples presented in this paper, the beginning frequency calculated by the IPFEM was basically the same as the reference solutions, but the ending frequency was always higher than the reference solutions. In other words, the IPFEM could not obtain a conservative safe band gap frequency range. However, the relative errors of the IPFEM were basically less than 6%, and its calculation precision could basically meet the engineering requirements. This paper only considered a one-dimensional metamaterial beam structure. Next, the authors of this paper will try to study the uncertainties in two-dimensional or three-dimensional metamaterial structures based on the secondary development platform of finite element software.  
Conceptualization, F.H. and Z.S.; methodology, F.H. and X.F.; software, F.H., Z.Z. and X.F.; validation, F.H.; formalism analysis, F.H.; investigation, F.H.; resources, Z.S. and D.Q.; data curation, F.H.; writing--original draft preparation, F.H.; writing--review and editing, Z.S.; visualization, F.H.; supervision, Z.S.; project administration, Z.S. and D.Q. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was funded by the National Natural Science Foundation of China (grant No. 12272172), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (No. 22 KJB580005), the Youth Talent Promotion Project from China Association for Science and Technology (2022 QNRC001), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.