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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Free Vibration of Plate

#### 2.2. Acoustic Radiation Formulation of Plate with Point Masses

_{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)

_{0}is considered as the mass density of air, c

_{0}is considered as the speed of sound in air, B is considered as the corresponding acoustic wave number, and $\dot{U}$ and $\dot{u}$ is considered as both the corresponding vibratory velocity amplitude and spatial dependent vibratory velocity amplitude in the z direction at l

_{s}, then the modal sound pressure ${\mathrm{P}}_{\mathrm{mn}}$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)

_{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)

_{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)

_{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)

#### 2.3. Thickness Variation of the Plate

_{x}= h [1 −T

_{x}{f(x)}

^{n}], where ‘h’ is the maximum thickness of the plate.

_{x}) is obtained by Equation (13)

_{0}= 1.21 kg/m

^{3}. At 20 °C, the speed of sound c

_{0}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.

## 3. Validation of the Present Study

## 4. Result and Discussion

#### 4.1. Effect of Natural Frequency Parameter (λ^{2}) of Plate with Different Combinations of Point Masses with Different Taper Ratios

^{2}) 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. 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 λ

^{2}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, Table 5 and Table 6 for natural frequency parameter (λ

^{2}) 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

_{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 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

_{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

_{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.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, C.M.; Hong, G.M.; Tan, T.J. Elasting buckling of tapered circular plates. Comput. Struct.
**1995**, 55, 1055–1061. [Google Scholar] [CrossRef] - Gupta, A.P.; Goyal, N. Forced asymmetric response of linearly tapered circular plates. J. Sound Vib.
**1999**, 220, 641–657. [Google Scholar] [CrossRef] - Vivio, F.; Vullo, V. Closed form solutions of axisymmetric bending of circular plates having non-linear variable thickness. Int. J. Mech. Sci.
**2010**, 52, 1234–1252. [Google Scholar] [CrossRef] - Sharma, S.; Lal, R.; Neelam, N. Free transverse vibrations of non-homogeneous circular plates of linearly varying thickness. J. Int. Acad. Phys. Sci.
**2011**, 15, 187–200. [Google Scholar] - Wang, C.Y. The vibration modes of concentrically supported free circular plates. J. Sound Vib.
**2014**, 333, 835–847. [Google Scholar] [CrossRef] - Liu, T.; Kitipornchai, S.; Wang, C.M. Bending of linearly tapered annular Mindlin plates. Int. J. Mech. Sci.
**2001**, 43, 265–278. [Google Scholar] [CrossRef] - Duana, W.H.; Wang, C.M.; Wang, C.Y. Modification of fundamental vibration modes of circular plates with free edges. J. Sound Vib.
**2008**, 317, 709–715. [Google Scholar] [CrossRef] - Gupta, U.S.; Lal, R.; Sharma, S. Vibration of non-homogeneous circular Mindlin plates with variable thickness. J. Sound Vib.
**2007**, 302, 1–17. [Google Scholar] [CrossRef] - Kang, J.H. Three-dimensional vibration analysis of thick circular and annular plates with nonlinear thickness variation. Comput. Struct.
**2003**, 81, 1663–1675. [Google Scholar] [CrossRef] - Lee, H.; Singh, R. Acoustic radiation from out-of-plane modes of an annular disk using thin and thick plate theories. J. Sound Vib.
**2005**, 282, 313–339. [Google Scholar] [CrossRef] - Thompson, W., Jr. The computation of self- and mutual-radiation impedances for annular and elliptical pistons using Bouwkamp integral. J. Sound Vib.
**1971**, 17, 221–233. [Google Scholar] [CrossRef] - Levine, H.; leppington, F.G. A note on the acoustic power output of a circular plate. J. Sound Vib.
**1988**, 21, 269–275. [Google Scholar] - Rdzanek, W.P., Jr.; Engel, W. Asymptotic formula for the acoustic power output of a clamped annular plate. Appl. Acoust.
**2000**, 60, 29–43. [Google Scholar] [CrossRef] - Wodtke, H.W.; Lamancusa, J.S. Sound power minimization of circular plates through damping layer placement. J. Sound Vib.
**1998**, 215, 1145–1163. [Google Scholar] [CrossRef] - Wanyama, W. Analytical Investigation of the Acoustic Radiation from Linearly-Varying Circular Plates. Doctoral Dissertation, Texas Tech University, Lubbock, TX, USA, 2000. [Google Scholar]
- Lee, H.; Singh, R. Self and mutual radiation from flexural and radial modes of a thick annular disk. J. Sound Vib.
**2005**, 286, 1032–1040. [Google Scholar] [CrossRef] - Cote, A.F.; Attala, N.; Guyader, J.L. Vibro acoustic analysis of an unbaffled rotating disk. J. Acoust. Soc. Am.
**1998**, 103, 1483–1492. [Google Scholar] [CrossRef] - Jeyraj, P. Vibro-acoustic behavior of an isotropic plate with arbitrarily varying thickness. Eur. J. Mech. A/Solds
**2010**, 29, 1088–1094. [Google Scholar] [CrossRef] - Ranjan, V.; Ghosh, M.K. Forced vibration response of thin plate with attached discrete dynamic absorbers. Thin Walled Struct.
**2005**, 43, 1513–1533. [Google Scholar] [CrossRef] - Kumar, B.; Ranjan, V.; Azam, M.S.; Singh, P.P.; Mishra, P.; PriyaAjit, K.; Kumar, P. A comparison of vibro acoustic response of isotropic plate with attached discrete patches and point masses having different thickness variation with different taper ratios. Shock Vib.
**2016**, 2016, 8431431. [Google Scholar] - Lee, M.R.; Singh, R. Analytical formulations for annular disk sound radiation using structural modes. J. Acoust. Soc. Am.
**1994**, 95, 3311–3323. [Google Scholar] [CrossRef] - Rdzanek, W.J.; Rdzanek, W.P. The real acoustic power of a planar annular membrane radiation for axially-symmetric free vibrations. Arch. Acoust.
**1997**, 4, 455–462. [Google Scholar] - Doganli, M. Sound Power Radiation from Clamped-Clamped Annular Plates. Master’s Thesis, Texas Tech University, Lubbock, TX, USA, 2000. [Google Scholar]
- Nakayama, I.; Nakamura, A.; Takeuchi, R. Sound Radiation of a circular plate for a single sound pulse. Acta Acust. United Acust.
**1980**, 46, 330–340. [Google Scholar] - Hasegawa, T.; Yosioka, K. Acoustic radiation force on a solid elastic sphere. J. Acoust. Soc. Am.
**1969**, 46, 1139–1143. [Google Scholar] [CrossRef] - Lee, H.; Singh, R. Determination of sound radiation from a simplified disk brake rotor using a semi-analytical method. Noise Control Eng. J.
**2000**, 52. [Google Scholar] [CrossRef] - Squicciarini, G.; Thompson, D.J.; Corradi, R. The effect of different combinations of boundary conditions on the average radiation efficiency of rectangular plates. J. Sound Vib.
**2014**, 333, 3931–3948. [Google Scholar] [CrossRef] - Xie, G.; Thompson, D.J.; Jones, C.J.C. The radiation efficiency of baffled plates and strips. J. Sound Vib.
**2005**, 280, 181–209. [Google Scholar] [CrossRef] - Rayleigh, J.W. The Theory of Sound, 2nd ed.; Dover: New York, NY, USA, 1945. [Google Scholar]
- Maidanik, G. Response of ribbed panels to reverberant acoustic fields. J. Acoust. Soc. Am.
**1962**, 34, 809–826. [Google Scholar] [CrossRef] - Heckl, M. Radiation from plane sound sources. Acustica
**1977**, 37, 155–166. [Google Scholar] - Williams, E.G. A series expansion of the acoustic power radiated from planar sources. J. Acoust. Soc. Am.
**1983**, 73, 1520–1524. [Google Scholar] [CrossRef] - Keltie, R.F.; Peng, H. The effects of modal coupling on the acoustic power radiation from panels. J. Vib. Acoust. Stress Reliab. Des.
**1987**, 109, 48–55. [Google Scholar] [CrossRef] - Snyder, S.D.; Tanaka, N. Calculating total acoustic power output using modal radiation efficiencies. J. Acoust. Soc. Am.
**1995**, 97, 1702–1709. [Google Scholar] [CrossRef] - Martini, A.; Troncossi, M.; Vincenzi, N. Structural and elastodynamic analysis of rotary transfer machines by Finite Element model. J. Serb. Soc. Comput. Mech.
**2017**, 11, 1–16. [Google Scholar] [CrossRef] - Croccolo, D.; Cavalli, O.; De Agostinis, M.; Fini, S.; Olmi, G.; Robusto, F.; Vincenzi, N. A Methodology for the Lightweight Design of Modern Transfer Machine Tools. Machines
**2018**, 6, 2. [Google Scholar] [CrossRef] - Martini, A.; Troncossi, M. Upgrade of an automated line for plastic cap manufacture based on experimental vibration analysis. Case Stud. Mech. Syst. Signal Process.
**2016**, 3, 28–33. [Google Scholar] [CrossRef] - Pavlovic, A.; Fragassa, C.; Ubertini, F.; Martini, A. Modal analysis and stiffness optimization: The case of a tool machine for ceramic tile surface finishing. J. Serb. Soc. Comput. Mech.
**2016**, 10, 30–44. [Google Scholar] [CrossRef]

**Figure 1.**Acoustic radiation due to the vibration modes of unbaffled tapered plate in Z direction with different combination of attached point masses enclosed in a sphere.

**Figure 3.**Plate with different combinations of point masses with (0, 2) modes. (

**a**) Plate with one point mass. Mass of point mass = 0.1 times mass of the plate. (

**b**) Plate with two point masses. Mass of each point mass = 0.05 times mass of the plate. (

**c**) Plate with four point masses. Mass of each point mass = 0.025 times mass of the plate.

**Figure 4.**Analytical, experimental and numerical comparison of sound power level of unloaded plate having uniform thickness for taper ratio T

_{x}= 0.00.

**Figure 5.**Comparison of % variation of natural frequency parameter with modes for uniform plate with different combinations of point masses.

**Figure 6.**Comparison of variation of natural frequency parameter with modes for unloaded plate and for plate with four point masses.

**Figure 7.**Analytical and numerical comparison of sound power level for Case I plate with four point masses having parabolic decreasing thickness variation for taper ratio T

_{x}= 0.75.

**Figure 8.**Numerical comparison of sound power level for Case I plate with one point mass having parabolic decreasing thickness variation with different taper ratio T

_{x}.

**Figure 9.**Numerical comparison of sound power level for Case I plate with two point masses having parabolic decreasing thickness variation with different taper ratio T

_{x}.

**Figure 10.**Numerical comparison of sound power level for Case I plate with four point masses having parabolic decreasing thickness variation with different taper ratio T

_{x}.

**Figure 11.**Numerical comparison of sound power level for Case I plate having parabolic decreasing thickness variation for different combinations of point masses for taper ratio T

_{x}= 0.75.

**Figure 12.**Numerical comparison of sound power level for Case II plate having parabolic decreasing increasing thickness variation for different combinations of point masses for taper ratio T

_{x}= 0.75.

**Figure 13.**Numerical comparison of sound power level for Case III plate having parabolic increasing decreasing thickness variation for different combinations of point masses for taper ratio T

_{x}= 0.75.

**Figure 14.**Analytical and numerical comparison of radiation efficiency (σ

_{mn}) for Case I plate with four point masses having parabolic decreasing thickness variation for taper ratio T

_{x}= 0.75.

**Figure 15.**Numerical comparison of radiation efficiency (σ

_{mn}) of Case I plate for parabolically decreasing thickness variation with different combination of point masses combinations for taper ratio T

_{x}= 0.75.

**Figure 16.**Numerical comparison of radiation efficiency (σ

_{mn}) for Case I plate with two point masses combinations having different parabolic thickness variation for taper ratio T

_{x}= 0.75.

**Figure 17.**Numerical comparison of sound power level (dB) for Case I plate with two point masses combinations having different parabolic thickness variation for taper ratio T

_{x}= 0.75.

**Figure 18.**Peak sound power level (dB) for (

**a**) Case I, (

**b**) Case II, and (

**c**) Case III plate having different parabolic thickness variation with different combination of point masses.

**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/m^{3}) | 7905.9 |

Young’s modulus, E (GPa) | 218 |

**Table 2.**Comparison and validation of natural frequency parameter λ

^{2}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, λ^{2} | |
---|---|---|---|

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 |

**Table 3.**Numerical comparison of different frequency parameter λ

^{2}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 | % λ^{2} | Plate with Two Point Masses | % λ^{2} | Plate with Four Point Masses | % λ^{2} |
---|---|---|---|---|---|---|---|

(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 |

**Table 4.**Numerical comparison of different frequency parameter λ

^{2}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, λ^{2} | |||
---|---|---|---|---|---|

T_{x} = 0.00 | T_{x} = 0.25 | T_{x} = 0.50 | T_{x} = 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 λ

^{2}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, λ^{2} | |||
---|---|---|---|---|---|

T_{x} = 0.00 | T_{x} = 0.25 | T_{x} = 0.50 | T_{x} = 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 |

**Table 6.**Numerical comparison of different frequency parameter λ

^{2}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, λ^{2} | |||
---|---|---|---|---|---|

T_{x} = 0.00 | T_{x} = 0.25 | T_{x} = 0.50 | T_{x} = 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 |

**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 (σ_{mn}) | SPL (dB) | RE (σ_{mn}) | SPL (dB) | RE (σ_{mn}) | ||

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 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chatterjee, A.; Ranjan, V.; Azam, M.S.; Rao, M.
Comparison for the Effect of Different Attachment of Point Masses on Vibroacoustic Behavior of Parabolic Tapered Annular Circular Plate. *Appl. Sci.* **2019**, *9*, 745.
https://doi.org/10.3390/app9040745

**AMA Style**

Chatterjee A, Ranjan V, Azam MS, Rao M.
Comparison for the Effect of Different Attachment of Point Masses on Vibroacoustic Behavior of Parabolic Tapered Annular Circular Plate. *Applied Sciences*. 2019; 9(4):745.
https://doi.org/10.3390/app9040745

**Chicago/Turabian Style**

Chatterjee, Abhijeet, Vinayak Ranjan, Mohammad Sikandar Azam, and Mohan Rao.
2019. "Comparison for the Effect of Different Attachment of Point Masses on Vibroacoustic Behavior of Parabolic Tapered Annular Circular Plate" *Applied Sciences* 9, no. 4: 745.
https://doi.org/10.3390/app9040745