# An Alternative Series Solution for Free Vibration Analysis of Asymmetric L-Shaped Membranes
## 1 Introduction  
Membranous structures are widely used in civil, mechanical, and biomedical applications owing to the advantages of easy folding, as well as being inflatable and lightweight [1]. In principle, one may view the natural vibration characteristics of membranes as exceptional cases of thin-plate vibrations. Nowadays, vibrating membranes are highly relevant to the design and fabrication of microelectromechanical systems (MEMSs) [2], sensors/ transducers [3, 4], nanoindenters [5], and atomic force microscopes [6]. In addition, membranes play a vital role in the development of space satellite technology, such as ultra-large membrane mirrors [7], solar sails [8], and gossamer spacecraft [9]. For simple geometric shapes such as rectangles, circles, ellipses, and triangles, the exact analytical solutions to the free vibration problems of elastic homogeneous membranes can be easily found in some engineering mathematics [10], vibration [11], or acoustics [12] reference books. However, when the geometric shape is slightly irregular or more complex, resorting to semi-analytical, mesh-free, or mesh-based numerical solutions (e.g., [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]) generally becomes necessary due to the relative difficulty of obtaining exact analytical solutions. In past research related to the free vibration of membranes, the geometric configurations of membranes are diverse. The homogeneous \"symmetric\" L-shaped membrane is notably popular among these various shapes (e.g., [24, 25, 26, 27]). The earliest related study dates back to the 1960s, with a semi-analytical solution provided by [PERSON] et al. [28], whose solution is often used as a benchmark for comparing other numerical methods (e.g., [29, 30, 31, 32, 33]). Apart from the works of [28, 34], most solutions are numerical (e.g., [35, 36, 37]). As for the \"asymmetric\" L-shaped membranes, there are even fewer studies, with the only recent semi-analytical series solution given by [PERSON] et al. [38]. Unfortunately, to the best ofthe authors' knowledge, no exact analytical solutions are yet available for symmetric or asymmetric L-shaped membranes. Pursuing enhanced performance and adaptability in MEMSs has led to significant advancements in device design. A critical focus in this arena is the development of piezoresistive pressure sensors with tunable operating frequencies, a feature vital for adapting to diverse applications and environmental conditions. The sensitivity of these sensors, primarily determined by the structure of the pressure-sensitive membrane, has been the subject of extensive research and optimization efforts. Traditional approaches have utilized composite membranes with center boss or island structures to increase stiffness [39]. At the same time, more recent innovations have explored the benefits of peninsula-structured membranes [40] and hollow stiffening structures [41]. Building upon these foundations, the L-shaped composite membrane [38] has emerged as a promising solution, offering a unique combination of geometric optimization and tunability. This study revisits the issue of free vibration in asymmetric L-shaped membranes, previously explored by [PERSON] et al. [38]. The region-matching technique (RMT) remains the primary solution method used to derive a semi-analytical series solution. The RMT hinges on combining the domain decomposition method and the method of separation of variables (MSV), each leveraging its strengths. The former overcomes the inherent limitation of the MSV, which can only be applied when the geometric contours are comfortable with a separable coordinate system, while the latter uses eigenfunctions that satisfy most boundary conditions, ensuring convergent and stable computed results. The present study introduces an advancement in the analysis of asymmetric L-shaped membranes through the application of a three-segmented domain decomposition (3-SDD) strategy, marking a notable departure from the two-segmented domain decomposition (2-SDD) strategy employed by [PERSON] et al. [38].