Analytical model for low-frequency transmission loss calculation of membranes loaded with arbitrarily shaped masses

An analytical model for the transmission loss calculation of thin rectangular and circular membranes loaded with rigid masses of arbitrary shape, the so-called membrane-type acoustic metamaterials, is presented. The coupling between the membrane and the added masses is introduced by approximating the continuous interaction force with a set of discrete point forces. This results in a generalized linear eigenvalue problem that is solved for the eigenfrequencies and eigenvectors of the coupled system. The concept of the effective surface mass density is employed to calculate the low-frequency transmission loss using the obtained eigenpairs. The proposed model is verified using numerical data from a finite element model and the convergence behavior of the point matching approach is investigated using Richardson extrapolation. Finally, a method based upon the grid convergence index for estimating the error that is introduced due to the point matching approach is presented.