# An Alternative Series Solution for Free Vibration Analysis of Asymmetric L-Shaped Membranes
## 2 Formulations  
Figure 1a illustrates an asymmetric L-shaped membrane initially stretched and clamped along its perimeter. The thin, elastic membrane is assumed to be homogeneous in material composition. The geometric configuration is defined by four parameters: \(a\) and \(b\) represent the lengths of the horizontal and vertical arms of the 'L' shape, respectively; \(d\) is the width of the vertical arm; and \(h\) is the height of the horizontal arm. The tension force per unit length is denoted as \(T\). The mass per unit area is constant and represented by \(\rho\). The concave corner of the membrane serves as the reference point for the Cartesian coordinate system \((x,y)\). Following the 3-SDD strategy, two auxiliary line boundaries, \(S_{h}\) and \(S_{\wp}\), were introduced to segment the geometry of the problem into three distinct regions: three bounded areas designated as regions 1, 2, and 3 (see Figure 1a). Within these delineated regions, the membrane transverse deflections \(u_{j}\) that remain small must conform to the governing Helmholtz equation, which is expressed as follows:  
\[\
abla^{2}u_{j}+k^{2}u_{j}=0,\ j=1,\ 2,\ 3, \tag{1}\]  
where \(\
abla^{2}\) is the Laplacian operator, \(k=\omega\sqrt{\rho/T}\) is the wavenumber, and \(\omega\) is the angular natural vibration frequency. The subscript \(j\) indicates the specific number. The time-harmonic factor is understood throughout this section. The zero displacement boundary conditions on the edge of region 1 are as follows:  
\[u_{1}(x,y)=0,\ x=-d,\ 0\leq y\leq b, \tag{2}\]  
\[u_{1}(x,y)=0,\ y=b,\ -d\leq x\leq 0, \tag{3}\]  
\[u_{1}(x,y)=0,\ x=0,\ 0\leq y\leq b, \tag{4}\]  
The clamped boundary conditions on the edge of region 2 are as follows:  
\[u_{2}(x,y)=0,\ y=-h,\ -d\leq x\leq 0, \tag{5}\]  
\[u_{2}(x,y)=0,\ x=-d,\ -h\leq y\leq 0, \tag{6}\]  
The fixed boundary conditions on the edge of region 3 are as follows:  
\[u_{3}(x,y)=0,\ y=-h,\ 0\leq x\leq a, \tag{7}\]  
\[u_{3}(x,y)=0,\ x=a,\ -h\leq y\leq 0, \tag{8}\]  
\[u_{3}(x,y)=0,\ y=0,\ 0\leq x\leq a. \tag{9}\]  
Using the MSV, the displacement fields in regions 1, 2, and 3 can be derived, respectively, as follows [43; 44]:  
\[u_{1}(x,\ y)=\sum\limits_{n=1}^{\infty}A_{n}\sin\Bigl{(}\frac{n\pi}{d}x \Bigr{)}\frac{\sinh[(y-b)\xi_{n}]}{\sinh(b\ \xi_{n})}, \tag{10}\]  
\[u_{1}(x,y)=0,\ x=-d,\ 0\leq y\leq b, \tag{2}\]0  
\[ \begin{array}{c}+\sum\limits_{n=1}^{\infty}C_{n}\sin\Bigl{(}\frac{n\pi}{d}x \Bigr{)}\frac{\sinh[(y+h)\xi_{n}]}{\sinh(h\ \xi_{n})}\end{array} , \tag{11}\]  
\[u_{3}(x,\ y)=\sum\limits_{n=1}^{\infty}D_{n}\frac{\sinh[(\ x-a)\ \varphi_{n}]}{ \sinh(a\varphi_{n})}\sin\Bigl{(}\frac{n\pi}{h}y\Bigr{)}, \tag{12}\]  
with  
\[\xi_{n}=\sqrt{\Bigl{(}\frac{n\pi}{d}\Bigr{)}^{2}-k^{2}}, \tag{13}\]\[\varphi_{n}=\sqrt{\left(\frac{n\pi}{h}\right)^{2}-k^{2}}, \tag{14}\]  
where the expansion coefficients \(A_{n}\), \(B_{n}\), \(C_{n}\), and \(D_{n}\) are undetermined. It should be noted that Equations (10)-(12) satisfy the governing Equation (1) and the majority of boundary conditions surrounding the membrane edge, except those on the \(S_{h}\) and \(S_{v}\).