Comparison for the Effect of Different Attachment of Point Masses on Vibroacoustic Behavior of Parabolic Tapered Annular Circular Plate

Abstract

In this paper, a comparison for the effect of different arrangement of point masses on vibroacoustic behavior of parabolic tapered annular circular plate with different taper ratios are analyzed by keeping the total mass of the plate plus point masses constant. Three different arrangement of thickness variation are considered. The mathematical tool FEM using ANSYS is used to determine the vibration characteristic and both FEM and Rayleigh integral is used to determine the acoustic behavior of the plate. Further, Case II plate (parabolic decreasing increasing thickness variation) for all combination of point masses is found to have reduction in natural frequency parameter in comparison to other cases of parabolic tapered plate. In terms of acoustic behavior, sound power levels of different cases of plate with different point mass combination are observed. It is observed that the Case II plate with two point masses combination shows the highest sound power and the Case III plate for all cases of point mass combination is least prone to acoustic behavior. Furthermore, It is observed that at low forcing frequency average radiation efficiency of parabolic tapered plate for different arrangement of point masses is almost same, but at high forcing frequency average radiation increases for higher taper ratio. Finally, a brief discussion of peak sound power reduction and actuation for different arrangement of point masses with different taper ratios are provided.

1. Introduction

Plate with tapered annular circular plates with different combinations of point masses has many engineering applications. They are used in many structural components, i.e., building, design, diaphragms, and deck plates in launch vehicles, diaphragms of turbines, aircraft and missiles, naval structures, nuclear reactors, optical systems, construction of ships, automobiles and other vehicles, the space shuttle, etc. These tapered plates with different combinations of point masses are found to have greater resistance to bending, buckling, and vibration in comparison to plates of uniform thickness. It is interesting to know that tapered plates with different thickness variation have drawn the attention of most of the researchers in this field. However, tapered plates with different combination of point masses can alter the dynamic characteristic of structures with a change in stiffness. Hence, for practical design purposes, the vibration and acoustic characteristics of such tapered plates are equally important. In comparison to the present study, several existing works are presented where the researchers have investigated the vibration response [1,2,3,4,5,6,7,8,9] of circular or annular plates of tapered or uniform thickness. However, in terms of acoustic behavior, many researchers have contributed most. Lee and Singh [10] used the thin and thick plate theories to determine the sound radiation from out-of-plane modes of a uniform thickness annular circular plate. Thompson [11] used the Bouwkamp integral to determine the mutual and self-radiation impedances both for annular and elliptical pistons. Levine and Leppington [12] analyzed the sound power generation of a circular plate of uniform thickness using exact integral representation. Rdzanek and Engel [13] determined the acoustic power output of a clamped annular plate using an asymptotic formula. Wodtke and Lamancusa [14] minimized the acoustic power of circular plates of uniform thickness using the damping layer placement. Wanyama [15] studied the acoustic radiation from linearly-varying circular plates. Lee and Singh [16] used the flexural and radial modes of a thick annular plate to determine the self and mutual radiation. Cote et al. [17] studied the vibro acoustic behavior of an unbaffled rotating disk. Jeyraj [18] used an isotropic plate with arbitrarily varying thickness to determine its vibro-acoustic behavior using the finite element method (FEM). Ranjan and Ghosh [19] studied the forced response of a thin plate of uniform thickness with attached discrete dynamic absorbers. Bipin et al. [20] analyzed an isotropic plate with attached discrete patches and point masses with different thickness variation with different taper ratios to determine its vibro acoustic response. Lee and Singh [21] investigated the annular disk acoustic radiation using structural modes through analytical formulations. Rdzanek et al. [22] investigated the sound radiation and sound power of a planar annular membrane for axially-symmetric free vibrations. Doganli [23] determined the sound power radiation from clamped annular plates of uniform thickness. Nakayama et al. [24] investigated the acoustic radiation of a circular plate for a single sound pulse. Hasegawa and Yosioka [25] determined the acoustic radiation force used on the solid elastic sphere. Lee and Singh [26] used a simplified disk brake rotor to investigate the acoustic radiation through a semi-analytical method. Thompson et al. [27,28] analyzed the modal approach for different boundary conditions to calculate the average radiation efficiency of a rectangular plate. Rayleigh [29] determined the sound radiation from flat finite structures. Maidanik [30] analyzed the total radiation resistance for ribbed and simple plates using a simplified asymptotic formulation. Heckl [31] used the wave number domain and Fourier transform to analyze the acoustic power. Williams [32] determined the wave number as a series in ascending power to estimate the sound radiation from a planar source. Keltie and Peng [33] analyzed the sound radiation using the cross-modal coupling from a plane. Snyder and Tanaka [34] demonstrated the importance of cross-modal contributions for a pair of modes through total sound power output using modal radiation efficiency. Martini et al. [35] investigated the structural and elastodynamic analysis of rotary transfer machines by a finite element model. Croccolo et al. [36] determined the lightweight design of modern transfer machine tools using the finite element model. Martini and Troncossi [37] determined the upgrade of an automated line for plastic cap manufacture based on experimental vibration analysis. Pavlovic et al. [38] investigated the modal analysis and stiffness optimization: the case of a tool machine for ceramic tile surface finishing using FEM.

In this paper, a numerical simulation has been proposed for the vibroacoustic behavior of parabolic tapered plate with different attachment of point masses. The review of the literature suggested that the vibroacoustic behavior of plate with different point masses is not reported. Hence taking consideration of all these facts, this paper is focused on the vibroacoustic behavior of parabolic tapered annular circular plate keeping the mass of the plate plus point masses constant for all cases of parabolic thickness variation. Therefore, the numerical simulation is provided in this paper by taking consideration of different plates with different taper ratios under time varying harmonic excitations.

2. Materials and Methods

2.1. Free Vibration of Plate

In this simulation process, the vibration of the plate is performed and its modal characteristic is determined. The natural frequency along with the modes shape of the plate during modal analysis is obtained as

$$\left(\left[K\right]-{\omega }^2\left[M\right]\right){\psi }_{\mathrm{mn}}=0$$

(1)

In the above formulation, $\left[M\right]$ is said to be mass matrix and $\left[K\right]$ is said to be the stiffness matrix. The mode shape is represented by ${\psi }_{mn}$ and the corresponding natural frequency of the plate is represented by ω denoted as rad/sec. Further, ${\lambda }^2$ known as the non-dimensional frequency parameter which is obtained as

$${\lambda }^2=\omega a^2\sqrt{\frac{\rho h}{D}}$$

(2)

where, D is said to be the flexure rigidity $=\frac{\mathrm{E}{h}^3}{12\left(1-{\upsilon }^2\right)}$, a is said to be the outer radius, E is said to be the Young’s modulus of elasticity, $\upsilon$ is said to be Poisson’s ratio, h is said to be the thickness of the plate, and $\rho$ is said to be the density of plate.

2.2. 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, 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)

$$\mathrm{P}\left(l_p\right)={\displaystyle {\int }_{S_s}\left(\frac{\partial q}{\partial f}\left(l_p,l_s\right)P\left(l_p\right)-\frac{\partial P}{\partial f}\left(l_s\right)q\left(l_p,l_s\right)\right)}ds\left(l_s\right)$$

(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)

$$\mathrm{P}\left(l_{\mathrm{p}}\right)=\frac{{\rho }_0{c}_{0}B}{4\pi }{\displaystyle {\int }_{S_s}\frac{e^{iB\left|l_p-l_s\right|\dot{U}}(l_s)}{\left|l_p-l_s\right|}}\left(1+\cos \eta \right)dS$$

(4)

Further for our consideration, if ρ₀ is considered as the mass density of air, c₀ 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 ${\mathrm{P}}_{\mathrm{mn} }$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)

$${\mathrm{P}}_{\mathrm{mn}}(R,\alpha ,\beta )=\frac{{\rho }_0{c}_{0}B{e}^{i{B}_{\mathrm{mn}}{R}_d}}{2R_d}\cos n\beta {\left(-i\right)}^{n+1}{A}_n\left[\dot{u(l)}\right]\left(1+\cos \eta \right)$$

(5)

$$A_f\left[\stackrel{·}{u(l)}\right]={\displaystyle \underset{0}{\overset{\infty }{\int }}\stackrel{·}{u}}\left(l\right)J_n\left(B_{l}l\right)ldl,Bl=B\sin \theta ;R_d=\left|l_p-l_s\right|$$

(6)

where J_n is considered as Bessel function of order n, (α, β) are considered as the cone and azimuthal angles of the observation positions, $\mathsf{\eta }$ is considered as angle between the surface normal vector and the 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 = |$l_{\mathrm{p}}$| is considered to be radius of the sphere. The observation positions are represented by some points having equal angular increments ($\Delta \phi , \Delta \alpha$) on a sphere S_v. Then, at all of the observation positions, the sound pressure is obtained by the above Equations (4)–(6). Where ‘$\Delta \phi$′ 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_mn [10,16] is obtained by Equation (7)

$$S_{\mathrm{mn}}={\left(D_{\mathrm{mn}}{S}_v\right)}_s=\frac{1}{2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\pi }{\int }}\frac{{\mathrm{P}}_{\mathrm{mn}}^2}{{\rho }_0{c}_0}}}{R}^2\sin \alpha d\alpha d\beta$$

(7)

where, we considered D_mn, as the acoustic intensity and we considered S_v as area of the control surface. Furthermore, if σ_mn is considered as radiation efficiency of the plate, then radiation efficiency [10] is obtained by Equation (8)

$${\sigma }_{\mathrm{mn}}=\frac{S_{\mathrm{mn}}}{\left|{\dot{u}}_{\mathrm{mn}}^2\right|ts},\left|{\dot{u}}_{\mathrm{mn}}^2\right|ts=\frac{1}{2\pi (a^2-b^2)}{\displaystyle {\int }_b^a{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\dot{U}}^{2}d\phi dl}}$$

(8)

where, $\left|{\dot{u}}_{\mathrm{mn}}^2\right|$_ts is considered to be the spatially average r.m.s velocity for the two normal surfaces of the plate. If the plate thickness (h) effect is considered, then from the two normal surfaces of the plate at (Z = 0.5h and −0.5h), the modal sound power [16] due to the sum of two sound radiations is obtained by Equations (9)–(11)

$${\mathrm{P}}_{\mathrm{mn}}(R,\alpha ,\beta )=\left(1+\cos \alpha \right){\mathrm{P}}_{\mathrm{mn}}^s(R,\alpha ,\beta )+\left(1-\cos \alpha \right){\mathrm{P}}_{\mathrm{mn}}^o(R,\alpha ,\beta )$$

(9)

$${\mathrm{P}}_{\mathrm{mn}}^s(R,\alpha ,\beta )=\frac{{\rho }_0{c}_0{B}_{\mathrm{mn}}{e}^{i{B}_{\mathrm{mn}}R}}{2R}{e}^{-i{B}_{\mathrm{mn}}\left(\frac{h}{2}\right)\cos \alpha }\cos n\beta {(-i)}^{n+1}{A}_n\left[\dot{U(l)}\right]$$

(10)

$${\mathrm{P}}_{\mathrm{mn}}^0(R,\alpha ,\beta )=\frac{{\rho }_0{c}_0{B}_{\mathrm{mn}}{e}^{i{B}_{\mathrm{mn}}R}}{2R}{e}^{-i{B}_{\mathrm{mn}}\left(\frac{h}{2}\right)\cos \alpha }\cos n(\beta +\varphi ){(-i)}^{n+1}{A}_n\left[\dot{U(l)}\right]$$

(11)

where, for (m, n)th mode, $B_{\mathrm{mn}}$ is considered to be the corresponding acoustic wave number whereas s and o represent the source side and opposite to source side.

2.3. Thickness Variation of the Plate

In this numerical simulation processes, we considered three different parabolic thickness variation of plate for vibration analysis and are reported in Figure 2. The radial direction is taken for thickness variation and the total mass of the plate plus point masses are kept constant. The thickness is varied in radial direction and is given by h_x = h [1 −T_x {f(x)}ⁿ], where ‘h’ is the maximum thickness of the plate.

$$\mathrm{where}, f(x)={{\displaystyle \{}}_{1,x=a}^{0,x=b} \mathrm{and} f(x)=\frac{x-b}{a-b} \mathrm{where} b<x<a$$

(12)

In this study, the taper ratio (T_x) is obtained by Equation (13)

$${\mathrm{T}}_x=\left(1-\frac{h_{\min }}{h}\right)$$

(13)

The Case I plate (parabolically decreasing) thickness variation, Case II plate (parabolically decreasing–increasing) thickness variation, and Case III plate (parabolically increasing–decreasing) thickness variation of (Figure 2) are obtained by Equations (14)–(16)

$$h_x=h\left\{1-{\mathrm{T}}_x{\left(\frac{x-b}{a-b}\right)}^n\right\}$$

(14)

$$h_x=h\left\{1-{\mathrm{T}}_x{\left(1-abs\left(1-2\frac{(x-b)}{(a-b)}\right)\right)}^n\right\}$$

(15)

$$h_x=h\left\{1-{\mathrm{T}}_{x}abs{\left(1-2\frac{(x-b)}{(a-b)}\right)}^n\right\}$$

(16)

where, n = 2 for parabolic thickness variation. The total volume of the plate plus point masses as well as unloaded plate is kept constant and is given by the Equation (17)

$$\mathrm{Volume} = \pi (a^2-b^2)h={\displaystyle \underset{b}{\overset{a}{\int }}(a^2-b^2)h_{x}dx}$$

(17)

The process is based on the numerical simulation technique using FEM. The plate is modeled in ANSYS with Plane 185 with 8 brick nodes and having three degrees of freedom at each node. A Structural Mass 21 in ANSYS is added to locate the point mass at the nodes. The number of element and nodes for uniform unloaded plate comes out to be 5883 and 1664, respectively. The numerical results obtained using FEM are compared with the other existing literature. The structure is modeled as such that the volume of the uniform unloaded plate is equal to the volume of the plate with point mass and as a result the whole mass of the plate with point mass remains constant. For plates with different cases of thickness variation with different point masses, we tried to keep the mesh as close to the mesh of the uniform unloaded plate. For vibration analysis and for Case I plate with one point mass combination, the modal structure consists of 5726 elements with 1575 nodes. Similarly, for Case I plate with two point masses combinations, the modal structure will be consists of 5720 elements with 1537 nodes where as for Case I plate with four point masses combination, the modal structure will be consists of 5712 elements with 1517 nodes respectively. In this numerical simulation technique, the meshes of different cases of plate with different plate thickness and with different combination of point masses are not exactly equal to the uniform unloaded plate. However for plate with other combination of point masses with different parabolic thickness, variations of 1–2% of mesh as that of uniform unloaded plate are considered. Furthermore, around the plate for creating the acoustic medium environment FLUID 30 and FLUID 130 are used. FLUID 30 is used for fluid structure interaction. FLUID 130 elements are created by imposing a condition of infinite space around the source and to prevent the back reflection of sound waves to the source. For acoustic calculation, the number of elements and nodes for uniform unloaded plate comes out to be 14,680 and 3639, respectively. For Case I plate with one point mass the number of elements and nodes after proper convergence comes out to be 14,662 and 3627, respectively. Further, after proper convergence, for Case I plate with two point masses the number of element and nodes after proper convergence comes out to be 14,644 and 3615 respectively. For Case I plate with four point masses, the number of element and nodes after proper convergence found to be 14,620, and 3605 respectively. For other cases of plate with different point masses combination, again a variation of 1–2% of mesh as that of uniform unloaded plate is taken in this numerical simulation technique. Consider the plate is vibrating in air medium with air density ρ₀ = 1.21 kg/m³. At 20 °C, the speed of sound c₀ of air is assumed as 343 m/s. The plate with structural damping coefficient is taken as 0.01. Rayleigh integral is applied to determine acoustic power calculation and ANSYS is used as a tool for numerical simulation. In this paper, we considered a plate with outer radius ‘a’ and inner radius ‘b’ as shown in Figure 2.

In this numerical simulation processes, a comparison for the effect of natural frequency parameter, effect of sound power level, effect of average radiation efficiency, and effect of peak sound power level for different cases of parabolic tapered plate with different combinations of point masses are obtained. For all cases of tapered plate with point masses, the mass of the plate is kept constant. The out of plane (m, n)th modes are considered and the different taper ratios are varied in the range of 0.00–0.75 for this process of simulation. The plate is clamped at inner and free at the outer boundary. We are considered the three arrangements of plate with different combination of point masses as shown in Figure 3. The selection of different combinations of point masses are such that the mass of uniform unloaded plate is equal to mass of plate + point mass and in all cases total mass of the plate with point masses remains constant. The dimension and the material properties of an annular circular plate with point masses are reported in

Table 1. Different specific dimension and material properties of plate with point mass consider in this literature.

Dimension of the Plate with Point Mass	Isotropic Annular Circular Plate
Outer radius (a) m	0.1515
Inner radius (b) m	0.0825
Radii ratio, (b/a)	0.54
Thickness (h) m	0.0315
Thickness ratio, (h/a)	0.21
Density, ρ (kg/m3)	7905.9
Young’s modulus, E (GPa)	218

.

3. Validation of the Present Study

In this process of numerical simulation technique, the validation of modal frequency of thick annular isotropic plate is done with the published result of Lee et al. [10] as shown in

Table 2. Comparison and validation of natural frequency parameter λ² of clamped-free uniform annular circular plate with that of Lee et al. [10] at T_x = 0.00.

Plate	Mode	Non Dimensional Frequency Parameter, λ2
		H. Lee et al. [10]	Present Work
Uniform plate b/a = 0.54 h/a = 0.21	(0, 0)	13.61	13.49
	(0, 1)	13.43	13.50
	(0, 2)	15.28	14.12
	(0, 3)	16.81	16.67

. In [10], Lee et al. used the thick and thin plate theories to provide the solution for the natural frequency parameter of uniform annular circular plate. In our study, we have calculated our result using FEM by taking the same dimension of plate as that of Lee et al. Therefore, our result of simulation in this study has a good agreement with the published results [10] as reported in Table 2. For the acoustic power calculation, the computed analytical, numerical, and published experimental results [10] are considered as reported in Figure 4. From Figure 4, good agreements of computed acoustic results are seen to be obtained analytically and numerically in line with published experimentally results [10].

4. Result and Discussion

4.1. Effect of Natural Frequency Parameter (λ²) of Plate with Different Combinations of Point Masses with Different Taper Ratios

In this paper, the simulation results for the effect of natural frequency parameter (λ²) due to different combination of point mass are considered. The different parabolic thickness variations are taken where the analysis of the plate is done by keeping mass of the plate + point mass constant. We have considered the first four frequency parameter and hence the numerical comparison is done between the uniform unloaded plate and plate with different combination of point masses for uniform thickness at T_x = 0 as reported in

Table 3. Numerical comparison of different frequency parameter λ² with different modes of uniform unloaded plate for taper ratio T_x = 0.00 with that of different combinations of point masses.

Mode	Un-Loaded Plate	Plate with One Point Mass	% λ2	Plate with Two Point Masses	% λ2	Plate with Four Point Masses	% λ2
(0,0)	13.49	13.46	0.223	13.45	0.296	13.35	1.033
(0,1)	13.50	13.48	0.148	13.44	0.444	13.32	1.333
(0,2)	14.12	14.08	0.283	14.06	0.424	14.02	0.708
(0,3)	16.67	16.64	0.017	16.64	0.017	16.62	0.299

. We find from Table 3 that the effect of natural frequency parameter for the unloaded plate and the plate with different combination of point masses is almost same. For further simulation results, a comparison of percentage variation of frequency parameter with the modes are investigated for different cases of plate with point masses combination as reported in Figure 5. It is clear from Figure 5 that the (0,1) mode increases for two point mass and four point mass combinations but decreases for one point mass combinations. In our numerical simulation results, we see that there is abrupt decrease of (0, 3) mode for all point masses combinations due to more stiffness associated with these modes. In our numerical simulation we compare λ² with all modes both for unloaded plate and plate with four point masses combinations as reported in Figure 6. We find that the effect of natural frequency parameter due to four point masses shows the little decrease in the frequency parameter. This may happen due to more stiffness associated with this plate. Further, in this numerical simulation process, we compare

Table 4. Numerical comparison of different frequency parameter λ² with different modes of plate with one point mass combinations for different parabolic thickness variations and for different taper parameters (T_x).

Case	Mode	Natural Frequency Parameter, λ2
		Tx = 0.00	Tx = 0.25	Tx = 0.50	Tx = 0.75
I	(0,0)	13.4802	12.9904	12.4703	11.9305
	(0,1)	13.4942	12.9745	12.4442	11.8918
	(0,2)	14.1132	13.6135	13.0936	12.5502
	(0,3)	16.6624	16.0733	15.4611	14.8254
II	(0,0)	13.4811	12.8952	12.2718	11.6202
	(0,1)	13.4932	12.8774	12.2414	11.5752
	(0,2)	14.1134	13.5187	12.8943	12.2384
	(0,3)	16.6624	15.9610	15.2277	14.4605
III	(0,0)	13.4812	13.4891	13.4902	13.4904
	(0,1)	13.4943	13.4784	13.4808	13.4804
	(0,2)	14.1136	14.1109	14.1134	14.1125
	(0,3)	16.6625	16.6600	16.6624	16.6618

,

Table 5. Numerical comparison of different frequency parameter λ² with different modes of plate with two point masses combinations for different parabolic thickness variations and for different taper parameters (T_x).

Case	Mode	Natural Frequency Parameter, λ2
		Tx = 0.00	Tx = 0.25	Tx = 0.50	Tx = 0.75
I	(0,0)	13.4753	12.9768	12.4592	11.9285
	(0,1)	13.4825	12.9682	12.4392	11.8825
	(0,2)	14.0925	13.6092	13.0878	12.5325
	(0,3)	16.6532	16.0691	15.4592	14.8125
II	(0,0)	13.4768	12.8825	12.2685	11.6125
	(0,1)	13.4832	12.8785	12.2386	11.5624
	(0,2)	14.0965	13.5085	12.9186	12.2252
	(0,3)	16.6582	15.9528	15.2582	14.4518
III	(0,0)	13.4793	13.4758	13.5076	13.4721
	(0,1)	13.4825	13.4768	13.5195	13.4821
	(0,2)	14.0968	14.0952	14.1392	14.0926
	(0,3)	16.6582	16.6592	16.7002	16.6523

and

Table 6. Numerical comparison of different frequency parameter λ² with different modes of plate with four point masses combinations for different parabolic thickness variation and for different taper parameter (T_x).

Case	Mode	Natural Frequency Parameter, λ2
		Tx = 0.00	Tx = 0.25	Tx = 0.50	Tx = 0.75
I	(0,0)	13.4852	12.9877	12.4701	11.9301
	(0,1)	13.4902	12.9765	12.4443	11.8937
	(0,2)	14.1025	13.6040	13.0838	12.5412
	(0,3)	16.6635	16.0712	15.4619	14.8258
II	(0,0)	13.4842	12.8924	12.2715	11.6202
	(0,1)	13.4902	12.8805	12.2418	11.5762
	(0,2)	14.1035	13.5102	12.8858	12.2304
	(0,3)	16.6638	15.9618	15.2302	14.4612
III	(0,0)	13.4852	13.4820	13.4850	13.4852
	(0,1)	13.4906	13.4853	13.4902	13.4902
	(0,2)	14.1037	14.1008	14.1022	14.1032
	(0,3)	16.6638	16.6612	16.6639	16.6635

for natural frequency parameter (λ²) of plate with different combinations of point masses combinations for different cases of tapered plate. It is observed from the Table 4, Table 5 and Table 6 that Case II plate (parabolically decreasing—increasing thickness variation) reports the reduction in natural frequency parameter for all cases of thickness variations with different combinations of point masses in respect to Case I plate (parabolic decreasing thickness variation). This reduction of natural frequency parameter for Case II plate may be due to the less stiffness associated than that of Case I plate. It is found that due to more stiffness associated with Case III plate (parabolic increasing—decreasing thickness variation), it shows the almost equal effect of natural frequency parameter as that of uniform unloaded plate for all combination of point masses. However, for plate with different parabolically thickness variations with all cases of four point mass combinations, alteration of modes are observed at higher taper ratios.

4.2. Acoustic Radiation of Tapered Annular Circular Plate with Different Attachment of Point Masses with Different Taper Ratios

In this numerical simulation processes, the sound power level (dB, reference = ${10}^{-12}\mathrm{watts})$ of annular circular plate with different combinations of point masses is considered. The plate with different parabolic thickness variation is analyzed due to transverse vibration. The different taper ratios are taken as range from (0.00–0.25). A concentrated load of 1N is considered under time-varying harmonic excitations which are acted at different excitation location at different nodes. A harmonic frequency range of 0–8000 Hz is taken to determine the sound radiation characteristic. We considered the Case I plate with parabolic decreasing thickness variation as a convergence study. Figure 7 compares analytically and numerically the sound power level for Case I plate with four point masses combination for taper ratio, T_x = 0.75 and for different modes. On comparison of sound power, we observed a good agreement of computed results as depicted from Figure 7. In this numerical simulation, the numerical comparison of sound power level for Case I plate with different combinations of point masses for different taper ratios are reported in Figure 8, Figure 9 and Figure 10. From Figure 8, Figure 9 and Figure 10, it is investigated that for sound power level up to 20 dB, we do not get any design options for different taper ratios for plate with both one point mass and for two point masses combinations. However, for four point masses combination, we do not find any sound power level upto 30 dB. However, for sound power level up to 30 dB, we get all taper ratios, T_x = 0.00, 0.25, 0.50, and 0.75 as design options in frequency bands A and B for plate with one point mass combinationas reported in Figure 8. It is noteworthy that, for sound power level up to 50 dB, we get more design options for sound power levels in different frequency bands, i.e., C, D, and E as reported in Figure 8. From Figure 9, it is apparent that for sound power level up to 30 dB, then in frequency band A only taper ratio T_x = 0.00, 0.25, 0.50, and 0.75 are available design alternative for plate with two point mass combination. However, for sound power level up to 50 dB, we get wider frequency bands, B, C, and D for different taper ratios as reported in Figure 9. From Figure 10, it is investigated that for sound power level up to 40 dB is possible only in frequency bands A, B and C only with all taper ratios T_x = 0.00, 0.25, 0.50, and 0.75 and therefore is the available design alternative for plate with four point mass combination. However, for sound power level up to 50 dB, we get broader range of frequency denoted as D, E and F for all taper ratios as reported in Figure 10. It may be inferred from Figure 8, Figure 9 and Figure 10 that plate with different combinations of point masses plays a significant role in sound power reduction in different frequency bands. For plates with four point masses combinations, the lowest sound power is observed in comparison to one point mass and two point mass combinations. However stiffness contribution due to various taper ratios have very limited impact on sound power level reduction in comparison to that of modes and excitation locations of plate with different combination of point masses. From Figure 8, Figure 9 and Figure 10, it is observed that for excitation frequencies up to 2000 Hz, the effect of different combinations of point masses and stiffness variation due to different taper ratios do not have a significant effect on sound power radiation for clamped-free forcing boundary condition. However, when the excitation frequency increases beyond 2000 HZ and up to the first peak, Case I plate with one point mass combinationreports the higher sound power level only for a higher taper ratio. However, for Case I plate with two point masses and four point masses combinations, there are variations of the sound power level. This is due to variation of peaks due to different taper ratios at this forcing region. Beyond 2000 HZ, Case II plate with two point massescombinations is seen to have the highest sound power level. However, the sound power for Case III plate is found to be decreased for all combination of point masses. Different modes do influence the sound power peaks as evident from Figure 8, Figure 9 and Figure 10. Sound power level peak obtained for different modes (0, 0) and (0, 1) is investigated and it is observed that the dissimilar peak for (0, 0) and (0, 1) is observed for plate with different point masses. However, with increasing taper ratio, sound power levels do shift towards lower frequency for all combinations of point masses. It is observed that at higher forcing frequency beyond 4000 Hz, different taper ratios alter its stiffness for different cases of thickness variations. It is needless to mention that for higher frequency beyond 4000 Hz up to 8000 HZ, plate with different combination of point mass alter its stiffness at higher forcing frequency. The acoustic power curve is seen to intersect each other at this high forcing region.

Table 7. Comparison of peak sound power level and radiation efficiency of plate having different parabolically varying thickness with different combinations of point masses for T_x = 0.75.

Type	Plate Thickness Variation	Plate with One Point Mass		Plate with Two Point Masses		Plate with Four Point Masses
		SPL (dB)	RE (σmn)	SPL (dB)	RE (σmn)	SPL (dB)	RE (σmn)
Point masses	Case I	82	1.058	78	1.007	77	0.994
	Case II	81	1.045	83	1.079	77	0.994
	Case III	79	1.020	77	0.994	76	0.935

compares the peak sound power level of different parabolic tapered plate with different combinations of point masses for taper ratio T_x = 0.75. It is interesting to note that the lowest sound power of 76 dB is observed for plate with four point mass combinations among all different thicknesses and the highest power of 82 dB is observed for plate with one point mass combination. Figure 11, Figure 12 and Figure 13 compares Case I, Case II, and Case III for sound power level numerically for different combinations of point mass for taper ratio, T_x = 0.75. From Figure 11, Figure 12 and Figure 13 it is investigated that for excitation frequency up to 2000 Hz, plate with different parabolic thickness variations does not have contribute much on sound power radiation. However, beyond excitation frequency of 2000 HZ and up to the first peak, it is investigated that Case II plate with two point masses combination is very good sound radiator of sound power 83 dB in comparison to 82 dB of plate with one point mass combination. Case III plates with all combinations of point masses are seen to have poor sound radiation. Figure 14 compares radiation efficiency (σ_mn) analytically and numerically for Case I plate with four point masses combination having parabolically decreasing thickness variation and for taper ratio T_x = 0.75. It is observed that on comparison the results obtained for radiation efficiency matches well with each other as reported in Figure 14. Figure 15 compares the variation of radiation efficiency for Case I plate (parabolic decreasing thickness variation) with different arrangement of point masses for different taper parameter T_x. In this numerical simulation, it is found that for exciting frequencies up to 1000 HZ, the effect of radiation efficiency with different arrangement of point masses and for different taper ratios is independent of excitation frequency. However, at a given forcing frequency beyond 1000 HZ, higher taper ratios cause higher radiation efficiency as evident from Figure 15. It can also be seen that with increasing taper ratio sound power level peaks do shift towards lower frequency as reported in Figure 15. Moreover, beyond 2000 HZ, different taper ratios alter its stiffness at higher frequency and radiation efficiency curve tends to intersect each other at this high forcing region. It can also be seen that all radiation curves due to all combination of point masses tends converge in a frequency range of 6800–7200 HZ and clear peaks are seen at this frequency band. From Figure 15, it is noted that with increasing taper ratio the radiation efficiency increases for all combination of point masses. Among these radiation combinations, the highest radiation efficiency is shown by Case II plate with two point mass combinztions. The moderate radiation efficiency is seen to be observed for Case I plate with one point mass and two point masses combinations as reported in Table 7. However, at higher forcing frequencies, different parabolic tapered plate (Cases I, II, and III) with four point masses combination shows the least radiation efficiency as evident from Table 7. It is interesting to note that the lowest radiation efficiency (σ_mn) is shown by Case III plate. Thus, Case III plate may be considered a poor radiator among all the thickness variation with different combinations of and point masses. Figure 16 compares the radiation efficiency numerically for plates with different parabolic thickness variation two point masses combination for taper ratio T_x = 0.75. It is investigated that all cases of parabolic tapered plate contribute almost the same radiation efficiency as depicted from Figure 16. Figure 17 shows the numerical comparison of sound power level for plate with two point masses combination for different parabolic thickness variation and for taper ratio T_x = 0.75. It is observed that almost equal and increasing peak sound power level is seen for all cases of parabolic tapered plate. Hence, the stiffness variation due to different taper ratios has negligible effect on acoustic radiation as evident from Figure 16 and Figure 17.

4.3. Peak Sound Power Level Variation with Different Taper Ratios for All Combinations of Point Masses Attached to a Plate

In this numerical simulation, the peak sound power level is calculated. The different cases of plate with different parabolically varying thickness are considered. The different combinations of point mass are taken and different taper ratios are considered as shown in Figure 18. We are aiming at the highest peak sound power level for different combinations of point mass attached to a plate which is reported at first peak which corresponds to (0, 0) mode of the plate. In this numerical simulation, different cases of plate are investigated with point mass combination. For the Case I plate with one point mass combination, it is seen that peak sound power level increases for increasing value of taper ratio. For two point masses and four point masses combinations, there seems to be the variation of peak sound power levels for increasing value of taper ratio as evident from Figure 18. For Case I plate it is seen that peak is maximum for taper ratio, T_x = 0.75 for plate with one point mass combination and peak is minimum for taper ratio, T_x = 0.75 for four point masses combination. It is further noticed that for Case II plate the highest peak is seen for taper ratio, T_x = 0.75 for two point masses combination. Similarly, it is seen that for Case III plate lowest peak is observed for taper ratio, T_x = 0.75 for plate with four point masses combination. Thus from the simulation result, it is quite obvious that peak sound power level corresponds to (0, 0) mode is deeply affected by different combinations of point masses. It is observed that plate with different combinations of point masses with different taper ratios provide us design options for peak sound power level. As for example, for peak sound power reduction, taper ratios, T_x = 0.75 with four point mass combination and taper ratio, T_x = 0.75 with two point masses combination, for Case III plate may be the options. Similarly, for sound power actuation, taper ratio T_x = 0.75 with one point mass combination for Case I plate and two point masses combination for Case II plate may be the another alternative solution.

5. Conclusions

This paper represents a comparison of vibroacoustic behavior of different cases of parabolic tapered annular circular plate. The different combinations of point masses of tapered plate are considered. The clamped free boundary condition of the plate is taken where the mass of the unloaded plate and the mass of the plate plus point masses is kept constant. In this numerical simulation, it is investigated that Case II plate for all combination of point masses shows reduction in natural frequency parameter in comparison to Case I plate. This may happen perhaps due to less stiffness associated with the Case II plate. However, the natural frequency parameter for Case III plate is found to be same as that of uniform unloaded plate. For acoustic radiation behavior, it is noted that mode variation and all cases of parabolic tapered plate with different combinations of pointmasses have significant impact on sound power level. Whereas the sound power level is less contributed by the stiffness variation due to different taper ratios. Up to 50 dB, we get abroad range of frequencies as design options for all taper ratios, T_x = 0.00, 0.25, 0.50, and 0.75. This includes all cases of parabolic tapered plate with different combinations of point massesin different frequency bands. The numerical simulation results in minimum sound power level for all cases of thickness variation of plate with four point masses combination. On other hand, Case II plate reports the highest sound power level with two point masses combinations. It is interesting to note that Case III plate with all combination of point masses is seen to have the lowest sound power level among all variations and may be considered as the lowest sound radiator. Finally, design options for peak sound power level different combinations of point masses with different taper ratios are considered. For example, for peak sound power reduction, taper ratios T_x = 0.75 with four point masses combination and taper ratio T_x = 0.75 with two point masses combination for Case III plate may be the options. Similarly, for sound power actuation, taper ratio T_x = 0.75 with one point mass for Case I plate and two point masses combination for Case II plate may be the another solution.