## 2 Materials and Methods
### Acoustic Radiation Formulation of Plate with Point Masses  
In this numerical simulation, we consider an annular circular plate in flexural vibration is set on flat rigid baffle having infinite extent as reported in Figure 1. We are neglecting the acoustic scattering of the edges of a vibrating structure in this investigation. Further, if P be considered as sound pressure amplitude, \(\mathrm{S_{s}}\) be considered as the surface of the sound source, \(q\) be considered as the Green methods function in free field. Furthermore, if \(l_{s}\) and \(l_{p}\) be considered as the position vectors of source and receiver and If the surface normal vector at \(l_{s}\) is taken as \(f\), then using by Rayleigh integral [10], structure sound radiation can be obtained by Equation (3)  
\[\mathrm{P}\left(l_{p}\right)=\int_{S_{s}}\left(\frac{\partial q}{\partial f} \left(l_{p},l_{s}\right)P\left(l_{p}\right)-\frac{\partial P}{\partial f} \left(l_{s}\right)q\left(l_{p},l_{s}\right)\right)ds\left(l_{s}\right) \tag{3}\]  
Consideration of the plane wave approximation to determine the sound pressure radiated from non-planar source in far and free field environment can be obtained by Equation (4)  
\[\mathrm{P}\left(l_{p}\right)=\frac{\rho_{0}c_{0}B}{4\pi}\int_{S_{s}}\frac{e^{ iB|l_{p}-l_{s}|U}\left(l_{s}\right)}{|l_{p}-l_{s}|}(1+\cos\eta)dS \tag{4}\]  
Further for our consideration, if \(\rho_{0}\) is considered as the mass density of air, \(c_{0}\) is considered as the speed of sound in air, \(B\) is considered as the corresponding acoustic wave number, and \(\dot{U}\) and \(\dot{u}\) is considered as both the corresponding vibratory velocity amplitude and spatial dependent vibratory velocity amplitude in the z direction at \(l_{s}\), then the modal sound pressure \(\mathrm{P_{mn}}\) from a normal plane [10], for an annular plate with (m, n)th mode is obtained from simplifying above Equation (4) with Hankel transform and is obtained Equations (5) and (6)  
\[\mathrm{P_{mn}}(R,\alpha,\beta)=\frac{\rho_{0}c_{0}Be^{iB_{mn}R_{d}}}{2R_{d}} \cos n\beta(-i)^{n+1}A_{n}\left[u(l)\right](1+\cos\eta) \tag{5}\]  
\[A_{f}\left[u(l)\right]=\int\limits_{0}^{\infty}u(l)J_{n}(B_{l}l)ld,Bl=B\sin \theta;R_{d}=\left|l_{p}-l_{s}\right| \tag{6}\]  
where \(J_{n}\) is considered as Bessel function of order \(n\), (\(\alpha\), \(\beta\)) are considered as the cone and azimuthal angles of the observation positions, \(\eta\) is considered as angle between the surface normal vector andthe vector from source position to receiver position, and \(A\) is considered as Hankel transform. Further from the far field condition, \(R_{d}\) in the denominator is approximated by \(R\) where \(R=\left|\left.I_{\text{p}}\right|\right.\) is considered to be radius of the sphere. The observation positions are represented by some points having equal angular increments (\(\Delta\varphi\), \(\Delta\alpha\)) on a sphere \(S_{\text{p}}\). Then, at all of the observation positions, the sound pressure is obtained by the above Equations (4)-(6). Where '\(\Delta\varphi^{\prime}\) represents the small increment in the circumferential direction of the plate. In far field for the (m, n)th mode, the modal sound power \(S_{\text{mn}}\)[10; 16] is obtained by Equation (7)  
\[S_{\text{mn}}=\left(D_{\text{mn}}S_{\text{v}}\right)_{s}=\frac{1}{2}\int\limits \limits_{0}^{2\pi}\int\limits_{0}^{\pi}\frac{\text{P}_{\text{mn}}^{2}}{\rho_{0} c_{0}}R^{2}\sin\alpha d\alpha d\beta \tag{7}\]  
where, we considered \(D_{\text{mn}}\), as the acoustic intensity and we considered \(S_{\text{v}}\) as area of the control surface.