## 1 Introduction  
Over the past two decades, there has been a growing concern regarding the noise generated by mechanical systems [1, 2, 3]. These reliable and commonly used systems are found extensively in sectors such as automotive and aerospace. For instance, delivery carts in the automotive industry, and cabin pressurization and air conditioning systems in aviation, contribute to substantial levels of noise, which can have adverse effects on human health and the environment [4, 5]. For instance, exposure to high levels of noise has been associated with significant human health consequences, including hearing impairment, chronic stress, and cardiovascular diseases [3]. The presence of vibrating components within these mechanical systems is responsible for generating continuous high levels of noise. Conventionally, this issue was addressed by proposing and developing noise--reducing devices known as acoustic filters. These filters are designed to decrease the level of noise to a permissible threshold for human hearing. Acoustic filters function by absorbing sound energy, resulting in a decreased level of noise. In industries such as automotive and aviation, traditional acoustic filters like glass fibers and mineral woods have been widely implemented over the last 50 years [4]. However, the production of these conventional filters has negative impacts on both human health and the environment. From a health perspective, manufacturing processes for glass fibers and mineral wood release harmful particles into the air, which can be inhaled leading to respiratory problems [6, 7]. On an environmental level, producingthese traditional acoustic filters requires extensive use of non--renewable resources while contributing to carbon emissions [7; 8; 9]. In recent years, a field known as periodic metamaterials has shown great promise in the area of vibration and acoustic attenuation. Metamaterials are specially engineered materials that possess unique properties not commonly found in natural substances. They offer possibilities for reducing vibrations and sound levels in mechanical systems. By harnessing their distinct mechanical and acoustical characteristics, metamaterials have the ability to manipulate and control the transmission of mechanical and sound waves. The key factor behind their effectiveness is their ability to create frequency ranges called bandgaps, within which they either block or attenuate waves passing through them. It is worth noting that these bandgaps differ between vibrational wave attenuation and acoustic wave attenuation within periodic metamaterials. Initial studies have focused on examining the attenuation of mechanical waves through the presence of bandgaps in periodic metamaterials. These periodic structures include hexagonal honeycomb, square, triangular, kagome lattice, and chiral--based lattices with varying degrees of porosity [10; 11; 12]. This investigation aims to analyze how changes in the topological parameters affecting porosity can impact both the width and location of these bandgaps. For instance, a triangular periodic metamaterial with different rates of porosity exhibits a single complete bandgap but covers different frequency ranges depending on its level of porosity [10]. Similarly, commercially used hexagonal honeycombs show that, as their level of porosity decreases, fewer bandgaps are formed [11]. These examples demonstrate the influence of topological parameters on bandgap formation and highlight the possibility of tuning bandgaps through adjustments to topology. Recent research has focused on advancing periodic metamaterials to enhance their ability to attenuate specific frequency ranges through programmable deformations or novel designs. The development of programmable periodic metamaterials has shown promising results as they enable dynamic control over bandgaps and allow for real--time adjustments to different wave frequencies.