# Derivation and Validation of Bandgap Equation Using Serepentine Resonator
## 1 Introduction  
A metamaterial is defined as an artificial material with wave manipulation that does not exist in nature. Numerous studies have been conducted in various fields, such as electromagnetic [1; 2; 3], optical [4; 5; 6], and acoustic [7; 8; 9; 10] to understand the characteristics of metamaterials. Among them, acoustic metamaterials have attracted significant attention in the vibro-acoustic field [11; 12; 13; 14; 15] due to their unique characteristics based on mechanism. The bandgap, one of the important characteristics of acoustic metamaterials, blocks the propagation of free bending waves of the host structure in a specific frequency region. Therefore, bandgap materials have a significant attenuation performance and are widely used in noise-, vibration-, and harshness-treatment procedures [16; 17; 18; 19; 20; 21].  
A bandgap can be generated by periodically arranging structures, such as resonators and scatterers, and inertial amplification. These structures make the wavenumber in the bandgap region imaginary, and consequently the bending wave becomes an evanescent wave and becomes dissipated, and the response of the host structure decreases in the bandgap region [22; 23; 24; 25]. Therefore, an equation for the bandgap can be derived even though the wavenumber of the bending wave in the host structure becomes imaginary. The equation is derived from the resonance frequency of the periodic structure and mass ratio between the host and periodic structures [26]. Since it is difficult to control the mass ratio, the resonance frequency of the resonator plays an important role in the formulationof the equation. Thus, the parameters of the periodic structures are important design parameters for determining the bandgap, and the bandgap can be controlled by adjusting the parameters.  
In spite of the wave manipulation properties of metamaterials, there is no adequate method to formulate the bandgap yet. Most metamaterials set the attenuation frequency region and determine the unit structure design in which the bandgap is implemented using high levels of simulation. Although this method is the most accurate, the simulations are time-consuming. To overcome this limitation, we have derived the equation of the bandgap mathematically to implement the bandgap in the desired frequency region by controlling the structural parameters. This equation helps to modularize the parameters of the bandgap region and reduce the time required for structural design.  
In this study, the experimental analysis is implemented using a serpentine spring composed of meandering beams as a resonator to generate the bandgap. Serpentine springs are primarily used in various research fields due to their high stretchability and increased utility in narrow spaces [27; 28; 29; 30]. In addition, the presence of various parameters in the serpentine spring provides adequate freedom to design the bandgap. Hence, we have derived the bandgap equation of the serpentine resonator and analytically verified it using a commercial software COMSOL. The relative errors between the numerical and analytical results are also analyzed. Additionally, the trend of the relative error when the design variables are modified is also identified to determine the most optimal design conditions. Finally, the results obtained are validated experimentally for similar design conditions and the theoretical equation is verified through the experiment. The experimental results show that the bandgap can be derived by an equation and the bandgap of the serpentine resonator can be successfully implemented.  
This paper is organized as follows. Section 2 presents the mathematical derivation of the equation of the bandgap for a serpentine resonator. Section 3 formulates the numerical derivation of the bandgap and analyzes the relative error. Section 4 describes the experimental setup for the validation and demonstrates the performance of the serpentine resonator. Finally, Section 5 presents the conclusions with a discussion and suggestions for future work.