Theoretical and Numerical Estimation of Vibroacoustic Behavior of Clamped Free Parabolic Tapered Annular Circular Plate with Different Arrangement of Stiffener Patches

This paper compares the vibroacoustic behavior of a tapered annular circular plate having different parabolic varying thickness with different combinations of rectangular and concentric stiffener patches keeping the mass of the plate and the patch constant for a clamped-free boundary condition. Both numerical and analytical methods are used to solve the plate. The finite element method (FEM) is used to determine the vibration characteristic and both Rayleigh integral and FEM is used to determine the acoustic behavior of the plate. It is observed that a Case II plate with parabolic decreasing–increasing thickness variation for a plate with different stiffener patches shows reduction in frequency parameter in comparison to other cases. For acoustic response, the variation of peak sound power level for different combinations of stiffener patches is investigated with different taper ratios. It is investigated that Case II plate with parabolic decreasing–increasing thickness variation for an unloaded tapered plate as well as case II plate with 2 rectangular and 4 concentric stiffeners patches shows the maximum sound power level among all variations. However, it is shown that the Case III plate with parabolically increasing–decreasing thickness variation with different combinations of rectangular and concentric stiffeners patches is least prone to acoustic radiation. Furthermore, it is shown that at low forcing frequency, average radiation efficiency with different combinations of stiffeners patches remains the same, but at higher forcing frequency a higher taper ratio causes higher radiation efficiency, and the radiation peak shifts towards the lower frequency and alters its stiffness as the taper ratio increases. Finally, the design options for peak sound power actuation and reduction for different combinations of stiffener patches with different taper ratios are suggested.


Introduction
Tapered annular circular plates with different combinations of rectangular and concentric patches 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 tapering plates with different combinations of rectangular

Analytical and Numerical Formulation for Acoustic Radiation from Tapered Annular Circular Plate
It is considered that an annular circular plate of inner radius 'b' and outer radius 'a' in flexural vibration is set on flat rigid baffle having infinite extent as reported in Figure1. Acoustic scattering of the edges of a vibrating structure is neglected in this study. Let P be the sound pressure amplitude, Ss be the surface of the sound source, q be the Green methods function in free field, ls and lp be the position vectors of source and receiver and the surface normal vector at ls be f; then structure sound radiation can be obtained by the Rayleigh integral [10] as given by Equation

Analytical and Numerical Formulation for Acoustic Radiation from Tapered Annular Circular Plate
It is considered that an annular circular plate of inner radius 'b' and outer radius 'a' in flexural vibration is set on flat rigid baffle having infinite extent as reported in Figure 1. Acoustic scattering of the edges of a vibrating structure is neglected in this study. Let P be the sound pressure amplitude, S s be the surface of the sound source, q be the Green methods function in free field, l s and l p be the position vectors of source and receiver and the surface normal vector at l s be f ; then structure sound radiation can be obtained by the Rayleigh integral [10] as given by Equation (3): 4 of 31 The sound pressure, radiated from non-planar source in far and free field environment based on plane wave approximation can be expressed by Equation (4): P l p = ρ 0 c 0 B 4π S s e iB|l p −l s |U (l s ) l p − l s (1 + cos η)dS (4) Let ρ 0 be the mass density of air, c 0 be the speed of sound in air, B be the corresponding acoustic wave number, and . U and . u be the corresponding vibratory velocity amplitude and spatial dependent vibratory velocity amplitude in the z direction at l s , then from a normal plane [10], the modal sound pressure P mn for an annular plate with (m, n) th mode is obtained from simplifying Equation (4) with Hankel transform and is expressed by Equations (5) and (6): where X n is Bessel function of order n, (α, β) are the cone and azimuthal angles of the observation positions, respectively, η is the angle between the surface normal vector and the vector from source position to receiver position, and A is the Hankel transform. According to the far field condition, R d in the denominator is approximated by R where R = |l p | is considered to be radius of the sphere. Consider that on a sphere S v the observation positions are represented by some points having equal angular increments (∆ϕ, ∆α). If '∆ϕ' represents the small increment in the circumferential direction of the plate, at all of the observation positions, the sound pressures is given by Equations (4)- (6). The modal sound power S mn for the (m, n) th mode [10,16] from the far-field is given by Equation (7): where the acoustic intensity is represented by D mn and area of the control surface is represented by S v . The radiation efficiency σ mn of the plate [10] is given by Equation (8): where, for the two normal surfaces of the plate the spatially average r.m.s velocity is represented as . u 2 mn ts . Considering the plate thickness (h) effect, the sum of sound radiations [16] from two normal surfaces of the plate at (Z = 0.5 h and −0.5 h) will represent the modal sound power, which can be given by Equations (9)-(11): P mn (R, α, β) = (1 + cos α)P s mn (R, α, β) + (1 − cos α)P o mn (R, α, β) (9) P s mn (R, α, β) = ρ 0 c 0 B mn e iB mn R 2R e −iB mn ( h 2 ) cos α cos nβ(−i) n+1 A n · U(l) P 0 mn (R, α, β) = ρ 0 c 0 B mn e iB mn R 2R e −iB mn ( h 2 ) cos α cos n(β + φ)(−i) n+1 A n · U(l) (11) where, the corresponding acoustic wave number of the (m, n) th mode is represented by B m,n , s and o in Equations (10) and (11) represent the source side and the opposite to the source side. For numerical analysis, we have used ANSYS (ANSYS, Inc., Canonsburg, PA, USA) as a tool. The plates with rectangular and concentric patches are modeled in ANSYS with Plane 185 with 8 brick nodes and having three degrees of freedom at each node. The mesh is not exactly equal to all the cases of different thickness variation and stiffener. The number of element and nodes for uniform unloaded plates ends up being 5883 and 1664, respectively. For plates with different cases of thickness variation with different stiffener we tried to keep the mesh as close to the mesh of the unloaded plate. For vibration analysis and for a Case I plates with 1 rectangular stiffener, the modal structure consists of 5685 elements with 1638 nodes whereas for Case I plates with 1 concentric stiffener, it has 5524 elements and 1618 nodes. With other combinations of rectangular and concentric stiffener with different parabolic thickness, a variation of 5% of the mesh from that of the unloaded plate is considered. The numerical results obtained using FEM are compared with the existing literature. The structure is modeled as such that the total volume of the plate plus stiffeners is equal to the total volume of the uniform unloaded plate. As a result the whole mass of the plate plus stiffener is equal to the whole mass of the uniform unloaded plate. So for all the cases of plate, the mass will be constant. FLUID 30 and FLUID130 elements are used to create the acoustic medium environment around the plate. For fluid-structure interaction FLUID 30 is used. For the surface on outer sphere, 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 analysis, the number of element and nodes for a uniform unloaded plate ends up being 14,680 and 3639, respectively. For a Case I plate with 1 rectangular stiffener, and after proper convergence of modeling, the numbers of elements and nodes ends up being 14,124 and 3465 respectively while for Case I plate with 1 concentric stiffener, the numbers of elements and nodes found to be 13,934 and 3345 respectively. Again, for other combination of rectangular and concentric stiffeners with different parabolic thickness, a variation of 5% of mesh from that of the unloaded plate is taken. Consider the air medium where the plate is vibrating with air density ρ 0 = 1.21 kg/m 3 . At 20 • C, the speed of sound c 0 of air is taken as 343 m/s. The structural damping coefficient of the plate is assumed as 0.01.

Thickness Variation of the Plate
In this study, three different parabolic thickness variations of plates is considered for analysis and is reported in Figure 2. The radial direction is considered for thickness variation by keeping the total mass of the plate plus patch constant. In the radial direction the plate thickness is given by h where 'h' is the maximum thickness of the plate where, The taper parameter or taper ratio (T x ) is given by the equation: The Case I plate of ( Figure 2 with parabolically decreasing thickness variation is given by the equation: The Case II plate (parabolically decreasing-increasing) and Case III plate (parabolically increasing-decreasing) thickness variation of ( Figure 2) are given by the equations: where, n = 2 for parabolic thickness variation. The total volume of the plate plus patches as well as the unloaded plate is kept constant and is given by the equation: In this paper, a comparison for the effect of frequency parameters, effect of sound power levels, average radiation efficiency and peak sound power level is obtained for parabolic tapered plates. The out of plane (m, n) th modes in Z direction for the plate with different attachment of rectangular and concentric stiffener patches at different positions with different parabolically tapered varying thickness is considered. The plate is made tapered with different taper ratios of 0.25, 0.50 and 0.75. The mass of the plate plus rectangular or concentric patches are kept constant for this analysis. The inner clamped and outer free boundary condition is taken. Three arrangements of plates with different combinations of rectangular or concentric stiffener patches are considered as shown in Figure 3. The selection of different combinations of rectangular or concentric stiffener patches are such that the mass of the unloaded plate is equal to mass of the rectangular or concentric stiffener patches plus plate and in all three cases mass of the plate with stiffener patches remains constant. The specifications and the material properties of an annular circular plate with attached rectangular and concentric stiffener patches are reported in Table 1. Rayleigh integral has been used for sound power calculation and ANSYS has been used as a tool for computation.
In this paper, a comparison for the effect of frequency parameters, effect of sound power levels, average radiation efficiency and peak sound power level is obtained for parabolic tapered plates. The out of plane (m, n) th modes in Z direction for the plate with different attachment of rectangular and concentric stiffener patches at different positions with different parabolically tapered varying thickness is considered. The plate is made tapered with different taper ratios of 0.25, 0.50 and 0.75. The mass of the plate plus rectangular or concentric patches are kept constant for this analysis. The inner clamped and outer free boundary condition is taken. Three arrangements of plates with different combinations of rectangular or concentric stiffener patches are considered as shown in Figure 3. The selection of different combinations of rectangular or concentric stiffener patches are such that the mass of the unloaded plate is equal to mass of the rectangular or concentric stiffener patches plus plate and in all three cases mass of the plate with stiffener patches remains constant. The specifications and the material properties of an annular circular plate with attached rectangular and concentric stiffener patches are reported in Table 1. Rayleigh integral has been used for sound power calculation and ANSYS has been used as a tool for computation.

Validation of Natural Frequency Parameter and Acoustic Power Calculation
In this paper, the natural frequency parameter of a uniform unloaded annular circular plate is validated with the published result of Lee et al. [10] and is reported in Table 2. In reference [10], Lee et al. provide the solution for the natural frequency parameter of a uniform annular circular plate by Thick and thin plate theories. In our study we have calculated our result using FEM by taking the same dimension of plate as that of Lee et al. From Table 2, it is clearly understand that in this paper the results obtained are almost equal to the published results [10]. For the acoustic power calculation, the computed analytical, numerical and published experimental results [10] are considered as reported in Figure 4. From Figure 4, a good agreement of computed acoustic results is seen to be obtained analytically and numerically in line with published experimentally results [10].

Validation of Natural Frequency Parameter and Acoustic Power Calculation
In this paper, the natural frequency parameter of a uniform unloaded annular circular plate is validated with the published result of Lee et al. [10] and is reported in Table 2. In reference [10], Lee et al. provide the solution for the natural frequency parameter of a uniform annular circular plate by Thick and thin plate theories. In our study we have calculated our result using FEM by taking the same dimension of plate as that of Lee et al. From Table 2, it is clearly understand that in this paper the results obtained are almost equal to the published results [10]. For the acoustic power calculation, the computed analytical, numerical and published experimental results [10] are considered as reported in Figure 4. From Figure 4, a good agreement of computed acoustic results is seen to be obtained analytically and numerically in line with published experimentally results [10].

Effect of Natural Frequency Parameter (λ 2 ) of Plate with Different Combinations of Rectangular and Concentric Stiffener Patches with Different Taper Ratios
In this paper, the effect of the natural frequency parameter (λ 2 ) is investigated for annular plates with different attachment of rectangular and concentric stiffener patches for different positions. The analysis is made for annular plates with different cases of parabolic thickness variations keeping the mass of the plate plus patches constant. Table 3 compares the first four natural frequency parameters numerically of a uniform unloaded plate for taper ratio, Tx = 0.00, with different attachments of rectangular and concentric stiffener patches along with the percentage variation of λ 2 . It is clear from Table 3 that the plate with different arrangements of rectangular stiffener patches has the same effect of natural frequency parameters as that of the unloaded plate for taper ratio, Tx = 0.00. However, for concentric stiffener patches, the natural frequency parameter decreases with addition of patches and minimum for 4 concentric stiffener patches. In Table 3, the negative variation of frequency parameter is calculated by . Figures 5 and 6 show the comparison of negative % variation of natural frequency parameter with different modes for a plate with taper ratio, Tx = 0.00 and with different attachment of rectangular and concentric stiffener patches. From Figure 5 it is shown that due to less stiffness associated with these modes, the (0, 2) mode of plate with 4 rectangular stiffener patches and (0, 0) mode of plate with 1 rectangular stiffener patch show the lowest percentage variation of λ 2 . However, due to greater stiffness associated with the (0, 1) mode of plate with 1 rectangular stiffener patch, it showed the highest percentage variation of λ 2 . Furthermore, from Figure 6 it is observed that for concentric stiffener patches the (0, 1) mode of plate with all combinations of patches shows the highest value of percentage variation of λ 2 due to greater

Effect of Natural Frequency Parameter (λ 2 ) of Plate with Different Combinations of Rectangular and Concentric Stiffener Patches with Different Taper Ratios
In this paper, the effect of the natural frequency parameter (λ 2 ) is investigated for annular plates with different attachment of rectangular and concentric stiffener patches for different positions. The analysis is made for annular plates with different cases of parabolic thickness variations keeping the mass of the plate plus patches constant. Table 3 compares the first four natural frequency parameters numerically of a uniform unloaded plate for taper ratio, T x = 0.00, with different attachments of rectangular and concentric stiffener patches along with the percentage variation of λ 2 . It is clear from Table 3 that the plate with different arrangements of rectangular stiffener patches has the same effect of natural frequency parameters as that of the unloaded plate for taper ratio, T x = 0.00. However, for concentric stiffener patches, the natural frequency parameter decreases with addition of patches and minimum for 4 concentric stiffener patches. In Table 3, the negative variation of frequency parameter is calculated by Figures 5 and 6 show the comparison of negative % variation of natural frequency parameter with different modes for a plate with taper ratio, T x = 0.00 and with different attachment of rectangular and concentric stiffener patches. From Figure 5 it is shown that due to less stiffness associated with these modes, the (0, 2) mode of plate with 4 rectangular stiffener patches and (0, 0) mode of plate with 1 rectangular stiffener patch show the lowest percentage variation of λ 2 . However, due to greater stiffness associated with the (0, 1) mode of plate with 1 rectangular stiffener patch, it showed the highest percentage variation of λ 2 . Furthermore, from Figure 6 it is observed that for concentric stiffener patches the (0, 1) mode of plate with all combinations of patches shows the highest value of percentage variation of λ 2 due to greater stiffness associated with this mode. However, for all the remaining modes (0, 0), (0, 2) and (0, 3) of plates with concentric stiffener patches the stiffness decreases and as a result the percentage variation of λ 2 decreases associated with these modes. Figure 7 shows the numerical comparison of natural frequency parameters λ 2 with modes for an unloaded plate and for a plate with 4 rectangular and 4 concentric stiffener patches for taper ratio, T x = 0.00. It is clear from Figure 7 that the effect of natural frequency parameter due to 4 rectangular stiffener patches is almost same as that of the unloaded plate. However, a plate with 4 concentric stiffener patches shows little decrease in the frequency parameter due to greater stiffness associated with this plate with concentric patch. Tables 4-6 numerically compare the first four natural frequency parameter λ 2 of a plate with different combinations of rectangular and concentric stiffener patches for different cases of thickness variation with different taper ratios. It is observed from Tables 4-6 that the natural frequency parameter for a plate with concentric patches for all thickness variations reduces more in comparison to rectangular patches with increasing taper ratios. This may be due to the lower stiffness of the plate associated with concentric patches. Furthermore, it is observed that the frequency parameter for a Case II plate (parabolically decreasing-increasing thickness variation) for all cases of thickness variation with different combinations of rectangular and concentric stiffener patches reduces more in comparison to a Case I plate (parabolic decreasing thickness variation). This is due to the lower stiffness associated with the Case II plate than that of the Case I plate. It is further investigated that the effect of the frequency parameter for a Case III plate with different attachment of rectangular and concentric stiffener patches (parabolic increasing-decreasing thickness variation) is almost same as that of uniform unloaded plate due to more stiffness associated with the Case III plate. However, for all cases of different rectangular and concentric stiffener patches, plate with different parabolically thickness variations alters its modes at higher taper ratios. Table 3. Numerical comparison of first four natural frequency parameter λ 2 of uniform unloaded plate with different attachment of rectangular and concentric stiffener patches for taper ratio, T x = 0.00.

Acoustic Response Solution of Tapered Annular Circular Plate with Different Combination of Rectangular and Concentric Stiffener Patches with Different Taper Ratios
In this paper, the sound power level (dB, reference = 10 −12 watts) of an annular circular plate with a different attachment of rectangular and concentric stiffener patches is estimated. The plate for the sound power level is analyzed for all cases of different parabolic thickness variation due to transverse vibration. The taper ratio is maintained from a range (0.00-0.75). The sound power level is investigated by applying 1 N concentrated load under time-varying harmonic excitations at different excitation locations at different nodes, and a harmonic frequency range of (0-8000) HZ is taken to determine the sound radiation characteristic. The Case I plate with parabolic decreasing thickness variation is taken as a convergence study. Figures 8 and 9 compare the sound power level for a Case I plate obtained analytically and numerically for the taper ratio T x = 0.75 for 4 rectangular stiffener patches and 4 concentric stiffener patches, respectively, for different modes. A good agreement of computed results is seen in the comparison of sound power as depicted from Figures 8 and 9. Figures 10-12 shows the numerical comparison of sound power level for Case I plate with different combinations of rectangular stiffener patches for different taper ratios and for different modes under forced excitation. From Figure 10, a sound power level up to 30 dB is seen, and we do not get any broad range of frequencies for different taper ratios for plate with 1 rectangular stiffener patch. However, for a sound power level up to 40 dB, we get all taper ratios, T x = 0.00, 0.25, 0.50 and 0.75, with a broad range of frequencies in frequency band A only with 1 rectangular stiffener.

Acoustic Response Solution of Tapered Annular Circular Plate with Different Combination of Rectangular and Concentric Stiffener Patches with Different Taper Ratios
In this paper, the sound power level (dB, reference = 10 watts of an annular circular plate with a different attachment of rectangular and concentric stiffener patches is estimated. The plate for the sound power level is analyzed for all cases of different parabolic thickness variation due to transverse vibration. The taper ratio is maintained from a range (0.00-0.75). The sound power level is investigated by applying 1 N concentrated load under time-varying harmonic excitations at different excitation locations at different nodes, and a harmonic frequency range of (0-8000) HZ is taken to determine the sound radiation characteristic. The Case I plate with parabolic decreasing thickness variation is taken as a convergence study. Figures 8 and 9 compare the sound power level for a Case I plate obtained analytically and numerically for the taper ratio Tx = 0.75 for 4 rectangular stiffener patches and 4 concentric stiffener patches, respectively, for different modes. A good agreement of computed results is seen in the comparison of sound power as depicted from Figures 8  and 9. Figures 10-12 shows the numerical comparison of sound power level for Case I plate with different combinations of rectangular stiffener patches for different taper ratios and for different modes under forced excitation. From Figure 10, a sound power level up to 30 dB is seen, and we do not get any broad range of frequencies for different taper ratios for plate with 1 rectangular stiffener patch. However, for a sound power level up to 40 dB, we get all taper ratios, Tx = 0.00, 0.25, 0.50 and 0.75, with a broad range of frequencies in frequency band A only with 1 rectangular stiffener.       Patches are as reported in Figure 10. It is noteworthy that for sound power level up to 50 dB, we get a broader range of frequencies for sound power level in different frequency bands, i.e., B, C and D, as reported in Figure 10. From Figure 11, it is apparent that for a sound power level up to 20 dB, we do not get any broad range of frequencies for plate with 2 rectangular stiffener patches. However for a sound power level up to 30 dB, we get the broad range of frequencies in frequency band A only with taper ratio, Tx = 0.00, 0.25, 0.50 and 0.75 as available design alternative. For a sound power level up to 50 dB, we get more broad range of frequencies for the sound power level in different frequency bands, i.e., B and C, as reported in Figure 11. From Figure 12, it is shown that for a sound power level up to 10 dB, we do not get any broad range of frequencies for plate with 4 rectangular stiffener patches. But for a sound power level up to 20 dB, we get the broad range of frequencies in frequency  Patches are as reported in Figure 10. It is noteworthy that for sound power level up to 50 dB, we get a broader range of frequencies for sound power level in different frequency bands, i.e., B, C and D, as reported in Figure 10. From Figure 11, it is apparent that for a sound power level up to 20 dB, we do not get any broad range of frequencies for plate with 2 rectangular stiffener patches. However for a sound power level up to 30 dB, we get the broad range of frequencies in frequency band A only with taper ratio, Tx = 0.00, 0.25, 0.50 and 0.75 as available design alternative. For a sound power level up to 50 dB, we get more broad range of frequencies for the sound power level in different frequency bands, i.e., B and C, as reported in Figure 11. From Figure 12, it is shown that for a sound power level up to 10 dB, we do not get any broad range of frequencies for plate with 4 rectangular stiffener patches. But for a sound power level up to 20 dB, we get the broad range of frequencies in frequency Patches are as reported in Figure 10. It is noteworthy that for sound power level up to 50 dB, we get a broader range of frequencies for sound power level in different frequency bands, i.e., B, C and D, as reported in Figure 10. From Figure 11, it is apparent that for a sound power level up to 20 dB, we do not get any broad range of frequencies for plate with 2 rectangular stiffener patches. However for a sound power level up to 30 dB, we get the broad range of frequencies in frequency band A only with taper ratio, T x = 0.00, 0.25, 0.50 and 0.75 as available design alternative. For a sound power level up to 50 dB, we get more broad range of frequencies for the sound power level in different frequency bands, i.e., B and C, as reported in Figure 11. From Figure 12, it is shown that for a sound power level up to 10 dB, we do not get any broad range of frequencies for plate with 4 rectangular stiffener patches. But for a sound power level up to 20 dB, we get the broad range of frequencies in frequency bands A only with all taper ratios, T x = 0.00, 0.25, 0.50 and 0.75 and, therefore, this is the available design alternative. However, for a sound power level up to 40 dB, we get more design options for the sound power level in different frequency bands, i.e., B, C and D as reported in Figure 12. Furthermore, Figures 13-15 show the numerical comparison of the sound power level for a Case I plate with different combinations of concentric stiffener patches for different taper ratios. From Figure 13, it is seen that for a sound power level up to 30 dB, we do not get any design options for different taper ratios for the plate with both 1 concentric stiffener patch and 4 concentric stiffener patches; and for 2 concentric stiffener patches, we do not find any sound power level upto 10 dB. However, for a sound power level up to 40 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 1 concentric stiffener patch combinationas reported in Figure 13. It is noteworthy that for a sound power level up to 50 dB, we get more design options for the sound power level in different frequency bands, i.e., C, D and E as reported in Figure 13. From Figure 14, it is apparent that for a sound power level up to 20 dB, then in frequency band A only taper ratio T x = 0.00, 0.25, 0.50 and 0.75 are available design alternatives for a plate with the 2 concentric stiffener patches combination. But for sound power level up to 30 dB, we get wider frequency bands, B, C and D for different taper ratios as reported in Figure 14. From Figure 15, it is seen that for a sound power level up to 40 dB is possible only in frequency bands A only with all taper ratios, T x = 0.00, 0.25, 0.50 and 0.75 and, therefore, this is the available design alternative for plate with 4 concentric stiffener patches combination. However, for a sound power level up to 60 dB, we get a broader range of frequency denoted as B and C for all taper ratios as reported in Figure 15. bands A only with all taper ratios, Tx = 0.00, 0.25, 0.50 and 0.75 and, therefore, this is the available design alternative. However, for a sound power level up to 40 dB, we get more design options for the sound power level in different frequency bands, i.e., B, C and D as reported in Figure 12. Furthermore, Figures 13-15 show the numerical comparison of the sound power level for a Case I plate with different combinations of concentric stiffener patches for different taper ratios. From Figure 13, it is seen that for a sound power level up to 30 dB, we do not get any design options for different taper ratios for the plate with both 1 concentric stiffener patch and 4 concentric stiffener patches; and for 2 concentric stiffener patches, we do not find any sound power level upto 10 dB. However, for a sound power level up to 40 dB, we get all taper ratios, Tx = 0.00, 0.25, 0.50 and 0.75 as design options in frequency bands A and B for plate with 1 concentric stiffener patch combinationas reported in Figure  13. It is noteworthy that for a sound power level up to 50 dB, we get more design options for the sound power level in different frequency bands, i.e., C, D and E as reported in Figure 13. From Figure  14, it is apparent that for a sound power level up to 20 dB, then in frequency band A only taper ratio Tx = 0.00, 0.25, 0.50 and 0.75 are available design alternatives for a plate with the 2 concentric stiffener patches combination. But for sound power level up to 30 dB, we get wider frequency bands, B, C and D for different taper ratios as reported in Figure 14. From Figure 15, it is seen that for a sound power level up to 40 dB is possible only in frequency bands A only with all taper ratios, Tx = 0.00, 0.25, 0.50 and 0.75 and, therefore, this is the available design alternative for plate with 4 concentric stiffener patches combination. However, for a sound power level up to 60 dB, we get a broader range of frequency denoted as B and C for all taper ratios as reported in Figure 15.    It may be inferred from Figures 10-15 that a plate with different combinations of rectangular and concentric stiffener patches plays a significant role in sound power reduction in different frequency bands. A plate with 4 rectangular stiffener patches combination causes maximum sound power level reduction in comparison to 1 rectangular stiffener patch and 2 rectangular stiffener patch combinations for a Case I plate; whereas, for a plate with 4 concentric stiffener patches the lowest sound power is observed in comparison to other combinations. However, the stiffness contribution due to various taper ratios has a very limited impact on sound power level reduction in comparison to that of modes and excitation locations of plate with different combinations of rectangular and concentric stiffener patches. Furthermore, from Figures 10-15, it is observed that for an excitation frequency up to 2000 HZ, the effect of different combinations of rectangular and concentric stiffener  It may be inferred from Figures 10-15 that a plate with different combinations of rectangular and concentric stiffener patches plays a significant role in sound power reduction in different frequency bands. A plate with 4 rectangular stiffener patches combination causes maximum sound power level reduction in comparison to 1 rectangular stiffener patch and 2 rectangular stiffener patch combinations for a Case I plate; whereas, for a plate with 4 concentric stiffener patches the lowest sound power is observed in comparison to other combinations. However, the stiffness contribution due to various taper ratios has a very limited impact on sound power level reduction in comparison to that of modes and excitation locations of plate with different combinations of rectangular and concentric stiffener patches. Furthermore, from Figures 10-15, it is observed that for an excitation frequency up to 2000 HZ, the effect of different combinations of rectangular and concentric stiffener It may be inferred from Figures 10-15 that a plate with different combinations of rectangular and concentric stiffener patches plays a significant role in sound power reduction in different frequency bands. A plate with 4 rectangular stiffener patches combination causes maximum sound power level reduction in comparison to 1 rectangular stiffener patch and 2 rectangular stiffener patch combinations for a Case I plate; whereas, for a plate with 4 concentric stiffener patches the lowest sound power is observed in comparison to other combinations. However, the stiffness contribution due to various taper ratios has a very limited impact on sound power level reduction in comparison to that of modes and excitation locations of plate with different combinations of rectangular and concentric stiffener patches. Furthermore, from Figures 10-15, it is observed that for an excitation frequency up to 2000 HZ, the effect of different combinations of rectangular and concentric stiffener patches and stiffness variation due to different taper ratios do not have a significant effect on sound power radiation for clamped-free boundary condition. However, when the excitation frequency increases beyond 2000 HZ and up to the first peak, the sound power level is higher for only higher taper ratios for a Case I plate with both 1 rectangular stiffener and 1 concentric stiffener patch, and variation of sound power level due to variation of peaks for different taper ratios is observed for a plate with both 2 rectangular stiffener and 2 concentric stiffener patches and for 4 rectangular stiffener and 4 concentric stiffener patch combinations. For a Case II plate, beyond a forcing frequency 2000 HZ, the highest sound power level is associated with plate for 2 rectangular stiffener patches combinations; while for a Case III plate, the sound power level is seen to decrease for all combinations of rectangular stiffener patches with increasing taper ratio. Similar effect is observed for plate with concentric stiffener patches where plate with 2 concentric stiffener patches is seen to have highest radiation power. Again for case III plate the sound power is seen to be decreased for all combination of concentric stiffener patches. Furthermore, different modes do influence the sound power peaks as evident from Figures 10-15. Sound power level peak obtained for different modes (0, 0) and (0, 1) almost remain same for different taper ratios. However, no such sound power similarity of modes (0, 0) and (0, 1) is observed for plates with concentric stiffener patches. The sound power level does shift towards a lower frequency with increasing taper ratio for all combinations of rectangular and concentric stiffener patches. For a higher frequency beyond 4000 HZ, it is observed that different taper ratios alter its stiffness at higher forcing frequency for different cases of thickness variation. It is noteworthy that for a higher frequency beyond 4000 HZ and up to 8000 HZ, a plate with different combination of rectangular and concentric stiffener patches alters its stiffness at higher forcing frequency and the acoustic power curve tends to intersect each other at this high forcing region. Table 7 compares the peak sound power level of a plate having different parabolically varying thickness with different combinations of rectangular and concentric stiffener patches for a taper ratio T x = 0.75. It is interesting to note that a plate with 4 rectangular stiffener patches combination shows the lowest peak sound power level among all cases of thickness variations, and the lowest peak sound of 77 dB is obtained for Case III plate whereas the highest peak sound power level of 84 dB is obtained for a Case II plate with 2 rectangular stiffener patches combination. Similar effect is again observed for plate with concentric stiffener patch combination. The lowest sound power of 76 dB is observed for the plate with the 4 concentric stiffener patches combination, and the highest power of 83 dB is observed for plate with 1 concentric stiffener patch combination. Figures 16-21 shows the numerical comparison of sound power levels for Case I, Case II and Case III plates for different combinations of rectangular and concentric stiffener patches for taper ratio T x = 0.75. From Figures 16-21, it is observed that for all cases of thickness variation and for excitation frequency up to 2000 HZ, different parabolic thickness variation does not have any significant effect on sound power radiation. Furthermore, from Figures 16-18 it is seen that beyond excitation frequency of 2000 HZ and up to the first peak, a Case II plate with 2 rectangular stiffener patches shows the highest radiation power of 84 dB in comparison to a radiation power of 82 dB for a Case I plate with 1 rectangular stiffener patch combination. However, at this forcing frequency of 2000 HZ case III plate remains unaffected and shows the lowest peak sound level for all cases of thickness variations and so it is suggested that Case III plate is the lowest sound power radiator among all cases of thickness variation with different combinations of rectangular stiffener patches. Again, a similar effect is observed for plate with the concentric stiffener patches combination. Beyond 200 HZ, Case II plate with 2 concentric stiffener patches is a very good sound radiator of sound power 83 dB in comparison to 82 dB of plate with 1 stiffener patch combination. A Case III plate with all combination of concentric stiffener patches is found to be a poor sound radiator.            Figures 24 and  25 show the variation of radiation efficiency with different taper ratios Tx for different combination of rectangular and concentric stiffener patches for a Case I plate with parabolically decreasing thickness variation. It is seen that for all combinations of rectangular and concentric stiffener patches, the effect of radiation efficiency due to different taper ratios is independent of exciting frequency up to 1000 HZ, but at a given forcing frequency a higher taper ratio causes higher radiation efficiency beyond 1000 HZ. However, sound power level peaks do shift towards a lower frequency as taper ratio increases. For higher frequency beyond 2000 HZ, different taper ratios alter its stiffness at higher forcing frequency and the radiation efficiency curves tend to intersect each other at this high forcing region. It is interesting to note that the radiation curve tends to unity in the frequency band 6800-7200 HZ and a clear peak is seen at this frequency band for all combination of rectangular and concentric stiffener patches. Furthermore, from Figures 24 and 25 it is seen that the radiation efficiency increases with the taper ratio for all combinations of rectangular and concentric stiffener patches. Out of these combinations, the Case II plate with 2 rectangular stiffener patches and 2 concentric stiffener patches delivers the highest radiation efficiency whereas Case I plate with both 1 rectangular and 1 concentric stiffener patch and 4 rectangular and 4 concentric stiffener patches is seen to be a moderate radiator as depicted in Table 7. However, at higher forcing frequency, it is seen that both a plate with 4 rectangular and 4 concentric stiffener patches with all cases of thickness variation (Cases I, II and III) shows the least radiation efficiency for all combinations of rectangular stiffener patches, as evident from Table 7.Therefore, it is interesting to mention that a Case III plate shows the lowest radiation efficiency (σmn) for all cases of thickness variations and is a poor radiation emitter among all the thickness variations with different combinations of rectangular and concentric stiffener patches. Figure 26 shows the numerical comparison of radiation efficiency for a plate with both 4 rectangular and 4 concentric stiffener patches for taper ratio Tx = 0.75. It is found that the plate shows almost the same radiation efficiency as depicted from Figure 26. Figure 27 shows the numerical comparison of the sound power level for a plate with 4 rectangular and 4 concentric stiffener patches for taper ratio Tx = 0.75. It is observed that both the plates show almost the same peak for taper ratio Tx = 0.75. Hence, the effect of stiffness variation along with the modes has negligible effect for both the combinations.  Figures 24 and 25 show the variation of radiation efficiency with different taper ratios T x for different combination of rectangular and concentric stiffener patches for a Case I plate with parabolically decreasing thickness variation. It is seen that for all combinations of rectangular and concentric stiffener patches, the effect of radiation efficiency due to different taper ratios is independent of exciting frequency up to 1000 HZ, but at a given forcing frequency a higher taper ratio causes higher radiation efficiency beyond 1000 HZ. However, sound power level peaks do shift towards a lower frequency as taper ratio increases. For higher frequency beyond 2000 HZ, different taper ratios alter its stiffness at higher forcing frequency and the radiation efficiency curves tend to intersect each other at this high forcing region. It is interesting to note that the radiation curve tends to unity in the frequency band 6800-7200 HZ and a clear peak is seen at this frequency band for all combination of rectangular and concentric stiffener patches. Furthermore, from Figures 24 and 25 it is seen that the radiation efficiency increases with the taper ratio for all combinations of rectangular and concentric stiffener patches. Out of these combinations, the Case II plate with 2 rectangular stiffener patches and 2 concentric stiffener patches delivers the highest radiation efficiency whereas Case I plate with both 1 rectangular and 1 concentric stiffener patch and 4 rectangular and 4 concentric stiffener patches is seen to be a moderate radiator as depicted in Table 7. However, at higher forcing frequency, it is seen that both a plate with 4 rectangular and 4 concentric stiffener patches with all cases of thickness variation (Cases I, II and III) shows the least radiation efficiency for all combinations of rectangular stiffener patches, as evident from Table 7.Therefore, it is interesting to mention that a Case III plate shows the lowest radiation efficiency (σ mn ) for all cases of thickness variations and is a poor radiation emitter among all the thickness variations with different combinations of rectangular and concentric stiffener patches. Figure 26 shows the numerical comparison of radiation efficiency for a plate with both 4 rectangular and 4 concentric stiffener patches for taper ratio T x = 0.75. It is found that the plate shows almost the same radiation efficiency as depicted from Figure 26. Figure 27 shows the numerical comparison of the sound power level for a plate with 4 rectangular and 4 concentric stiffener patches for taper ratio T x = 0.75. It is observed that both the plates show almost the same peak for taper ratio T x = 0.75. Hence, the effect of stiffness variation along with the modes has negligible effect for both the combinations.

Peak Sound Power Level Variation with Different Taper Ratios for All Combinations of Rectangular and Concentric Stiffener Patches Attached to a Plate
Peak sound power level for a plate was estimated for annular plates with different attachment of rectangular and concentric stiffener patches. The peak sound was considered for plates with different parabolically varying thickness. The different taper ratios were taken as reported in Figures  28 and 29, respectively. Furthermore, peak sound power level for different attachments of rectangular and concentric stiffener patches attached to a plate is reported at the first peak which corresponds to

Peak Sound Power Level Variation with Different Taper Ratios for All Combinations of Rectangular and Concentric Stiffener Patches Attached to a Plate
Peak sound power level for a plate was estimated for annular plates with different attachment of rectangular and concentric stiffener patches. The peak sound was considered for plates with different parabolically varying thickness. The different taper ratios were taken as reported in Figures  28 and 29, respectively. Furthermore, peak sound power level for different attachments of rectangular and concentric stiffener patches attached to a plate is reported at the first peak which corresponds to

Peak Sound Power Level Variation with Different Taper Ratios for All Combinations of Rectangular and Concentric Stiffener Patches Attached to a Plate
Peak sound power level for a plate was estimated for annular plates with different attachment of rectangular and concentric stiffener patches. The peak sound was considered for plates with different parabolically varying thickness. The different taper ratios were taken as reported in Figures 28 and 29, respectively. Furthermore, peak sound power level for different attachments of rectangular and concentric stiffener patches attached to a plate is reported at the first peak which corresponds to (0, 0) mode of the plate. From Figure 28 it is seen that for a Case I plate with 1 rectangular stiffener patch combination, peak sound power level increases for increasing value of taper ratio whereas for 2 rectangular stiffener patches and 4 rectangular stiffener patches combinations, variations of peak sound power levels are observed for an increasing value of taper ratio. For a Case I plate, the maximum peak sound power level is obtained for taper ratio T x = 0.75 for plate with 1 rectangular stiffener patch combination. Furthermore, it is seen that peak is minimum for taper ratio, T x = 0.25 and maximum for taper ratio, T x = 0.50 for plate with 4 rectangular stiffener patches combination, whereas for a plate with 2 rectangular stiffener patches combination, the peak is at a minimum for taper ratio T x = 0.25 and maximum for taper ratio T x = 0.75. Similarly, for a Case II plate, the maximum peak sound power level is obtained for taper ratio T x = 0.75 and minimum peak is seen for taper ratio T x = 0.50 for a plate with 2 rectangular stiffener patches combination. Also, for a Case II plate, it is interesting to note that peak sound power level increases for increasing value of taper ratio for plate with 1 rectangular stiffener patch combination and the peak is seen to be maximum for a taper ratio T x = 0.75. However, for a plate with 4 rectangular stiffeners patches combination, the peak is seen to be at a minimum for taper ratio T x = 0.75 and maximum for taper ratio T x = 0.50. Furthermore, it is investigated that for case III plate, peak sound power level decreases for increasing value of taper ratio for all combinations of rectangular stiffener patches attached to a plate. For a Case III plate, the maximum peak of the sound power level is obtained for taper ratio T x = 0.25 for a plate with 2 rectangular stiffener patches combination and minimum peak is observed for taper ratio T x = 0.75 for a plate with 4 rectangular stiffener patches combination. From Figure 29, it is observed that for a Case I plate peak is maximum for taper ratio, T x = 0.75 for plate with 1 concentric stiffener patches and minimum for taper ratio T x = 0.50 for 2 concentric stiffener patches. For a Case II plate the highest peak is seen for taper ratio T x = 0.75 for 2 concentric stiffener patches. Similarly, for case III plate lowest peak is observed for taper ratio, T x = 0.75 for plate with 4 concentric stiffener patches. However, from Figures 28 and 29, it is necessary to mention that for a Case II unloaded tapered plate the highest peak sound power level is seen for taper ratio T x = 0.75. Furthermore, it is also observed that the peak sound power level increases for case I plate for taper ratio T x = 0.75 and peak sound power level decreases for a Case III plate for taper ratio, T x = 0.75.
It is thus quite obvious that different combinations of rectangular and concentric stiffener patches have a significant impact on peak sound power level corresponding to the (0, 0) mode. Furthermore, different combinations of rectangular and concentric stiffener patches with different taper ratios provide us design options for peak sound power level. For example, for peak sound power reduction, taper ratio T x = 0.75 with 4 rectangular stiffener patches and 4 concentric stiffener patches combination, as well as taper ratio T x = 0.50 with 2 rectangular stiffener patches combination, for a Case III plate may be the options. Similarly, for sound power actuation, taper ratio T x = 0.75 with 1 rectangular stiffener patch and 1 concentric stiffener patch combination for a Case I plate and 2 rectangular stiffener patches and 4 concentric stiffener patches combination for a Case II plate may be the alternative solution. However, for an unloaded tapered plate, it can be added that for taper ratio T x = 0.75 for a Case III plate may be considered as a poor sound emitter and a taper ratio T x = 0.75 for a Case I and Case II plate may be considered as the highest sound emitter.

Conclusions
A comparison is made of vibroacoustic behavior of tapered annular circular plates having different parabolically varying thickness with different combinations of rectangular and concentric stiffeners patches keeping the mass of the plate plus patch constant for a clamped-free boundary condition. It is observed that due to lower stiffness associated with the Case II plate, the nondimensional frequency parameter of a Case II plate reduces more in comparison to a Case I plate. However, for a Case III plate, the non-dimensional frequency parameter is same as that of unloaded plate. In response to acoustic behavior, it is observed that different combinations of rectangular and concentric stiffener patches and modes variation have significant impacts on the sound power level in comparison to the stiffness variation due to the taper ratio. It is observed that for a sound power level up to 50 dB, and for a plate with different parabolically varying thickness, we get all taper ratios, Tx = 0.00, 0.25, 0.50 and 0.75, with a broad range of frequencies as design options in different frequency bands for different combinations of rectangular and concentric stiffener patches. It is further shown that a plate with 4 rectangular and 4 concentric stiffener patches combination shows the minimum sound power level for all cases of thickness variation, whereas the highest power is obtained for a

Conclusions
A comparison is made of vibroacoustic behavior of tapered annular circular plates having different parabolically varying thickness with different combinations of rectangular and concentric stiffeners patches keeping the mass of the plate plus patch constant for a clamped-free boundary condition. It is observed that due to lower stiffness associated with the Case II plate, the non-dimensional frequency parameter of a Case II plate reduces more in comparison to a Case I plate. However, for a Case III plate, the non-dimensional frequency parameter is same as that of unloaded plate. In response to acoustic behavior, it is observed that different combinations of rectangular and concentric stiffener patches and modes variation have significant impacts on the sound power level in comparison to the stiffness variation due to the taper ratio. It is observed that for a sound power level up to 50 dB, and for a plate with different parabolically varying thickness, we get all taper ratios, T x = 0.00, 0.25, 0.50 and 0.75, with a broad range of frequencies as design options in different frequency bands for different combinations of rectangular and concentric stiffener patches. It is further shown that a plate with 4 rectangular and 4 concentric stiffener patches combination shows the minimum sound power level for all cases of thickness variation, whereas the highest power is obtained for a Case II plate with 2 rectangular and concentric stiffener patches combination. It is interesting to note that a Case III plate has the lowest sound power level among all variations and is seen to be the lowest sound radiator. Further different combinations of rectangular and concentric stiffener patches with different taper ratios provide us design options for peak sound power level. For example, for peak sound power reduction, taper ratios, T x = 0.75 with a 4 rectangular stiffener patches 4 concentric stiffener patches combination, and taper ratio T x = 0.50 with a 2 rectangular stiffener patches combination for a Case III plate may be the options. Similarly, for sound power actuation, a taper ratio T x = 0.75 with 1 rectangular stiffener patch and 1 concentric stiffener patch combination for a Case I plate and 2 rectangular stiffener patches and 4 concentric stiffener patches combination for a Case II plate may be an alternative solution. Furthermore, for unloaded tapered plates, it can be added that taper ratio T x = 0.75 for a Case III plate may be considered as a poor sound emitter and a taper ratio T x = 0.75 for Case I and Case II plates may be considered as the highest sound emitter.

Funding:
The work was carried out in the Indian Institute of Technology (ISM) Dhanbad, India. The APC will be funded by the corresponding author only.

Conflicts of Interest:
The authors declare no conflicts of interest.