# Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties
## 1 Introduction  
Acoustic metamaterials are artificial periodic materials with unique band gap properties [1; 2]. The band gap represents the range of frequencies in which elastic waves are always attenuated when propagating through acoustic metamaterials. Therefore, acoustic metamaterials can provide a new research direction for vibration control. Metamaterials are formed by the periodic arrangement of several specially designed microstructures, and it is the locally resonant mechanism of such microstructures that leads to the generation of band gaps [3]. Hence, the band gap is usually near the natural resonance frequency of the microstructure. So far, researchers have constructed diverse metamaterial beam (MB) structures by periodically adding various forms of microstructures to basic beams. [PERSON] et al. [4] formed a basic model of a metamaterial beam by periodically attaching a series of resonators to a homogeneous beam. It was found that both the resonant and the Bragg scattering band gaps existed in the model. [PERSON] et al. [5] attached multioscillators periodically to a basic beam to obtain multiple band gaps. Some researchers have proposed reducing the band gap frequency range of metamaterial beams by introducing a negative stiffness mechanism into the resonators [6; 7; 8]. [PERSON] et al. [9] studied the wave characteristics of a double-layer metamaterial beam structure using the transfer matrix method and found that its vibration reduction effect was better than that of the traditional metamaterial beam. In addition, there are other resonators of different forms, such as beam-like resonators [10], continuum-beam resonators [11], and 2-DOF type resonators [12], which can all contribute to the formation of unique band gaps. The above-mentioned methods for calculating band gaps were all based on the spatial periodicity of metamaterial structures. However, due to some uncontrollable factors, thedesign parameters are often affected by spatial uncertainties. Such spatial uncertainties are often difficult to measure and break the assumption of spatial periodicity to some extent. When statistical data are seriously lacking, one can simply consider that the value of the uncertain parameter only fluctuates within a certain known interval range, and the corresponding interval model becomes the most appropriate method for analyzing uncertainties. In the interval model, the interval field methods that can be applied to deal with spatial uncertainties fall into three main categories: (1) the explicit interval field formulation, (2) interval fields based on Karhunen-Loeve (KL) expansion, and (3) interval fields based on convex models. Following explicit formulation, an interval field can be interpreted as the superposition of base functions scaled by interval coefficients [13]. The interval coefficients are uncertain, but the base functions are deterministic, which ensures the spatial continuity of the uncertain parameters. The specific form of the base functions often depends on the engineering experience and intuitive judgment of the researchers. [PERSON] et al. [14; 15; 16; 17] proposed using KL decomposition to obtain the base functions. KL expansion quantifies the spatial dependency of the interval field by defining the spatial dependency function. Recently, some scholars have used convex models to describe interval field problems. Similarly, the convex model uses the auto-dependence function to describe the spatial dependence of the interval field [18; 19; 20; 21]. The commonly used convex models include the ellipsoidal model [22], multidimensional parallelepiped model [23], and improved multidimensional parallelepiped model [24]. [PERSON] et al. [25] and [PERSON] et al. [26] used the interval analysis technique and stochastic analysis technique, respectively, to study the uncertainties in acoustic metamaterials. However, there have been no studies on spatial uncertainties in acoustic metamaterials.