Method for vibration and acoustic radiation analysis of underwater stiffened plates with acoustic coatings

Based on the effective medium theory as well as the three-dimensional viscoelastic theory, a semi-analytical model of the three-dimensional underwater vibration and acoustic radiation with cavity-type stiffened plate structures covered with acoustic coatings is established to solve the issues related to underwater vibration and acoustic radiation calculation of the stiffened plate and shell structure with complex acoustic coatings. By comparing the far-field acoustic radiation characteristics of the proposed semi-analytical model and the finite element numerical model, this research verifies the validity and accuracy of the proposed model. On such a basis, this research preliminarily investigates the coupling effect exerted by stiffeners and parent plates in stiffened plates as well as the vibroacoustic characteristics of acoustic coatings, thereby providing a valuable idea for the prediction of vibroacoustic characteristics concerning underwater structures.


Introduction
To reduce the radiated noise of the submarine, acoustic coatings with damping, acoustical insulation and decoupling functions are typically laid on the surface of the submarine hull.It is worth noting that the evaluation of vibration reduction and noise reduction performance of an acoustic coating is generally based on the change in vibroacoustic characteristics of plate and shell structures before and after being covered with acoustic coatings.For this reason, the research on vibration [10] and acoustic radiation of underwater plate-shell composite structures with acoustic coatings serves as an essential aspect in the research field of vibroacoustic characteristics of underwater structures in recent years, which has received extensive attention from scholars at home and abroad [1][2][3][4][5][6] .
As a whole, the relevant published literature reveals three prevalent research methods.Specifically, the first one is the analytical method.Based on equivalently processing some parameters (such as density, wave number, impedance, etc.) of the acoustic coating with complex structure into uniform layers by the layered medium theory, this method further implements the corresponding analytical modeling and solution according to laminated plate and shell theory, coupled fluid-solid theory, elasticity theory, continuous boundary conditions of displacement and stress at the interface and other factors.The second one can be summarized as a numerical method represented by the finite element/boundary element method and statistical energy analysis method.Through the grid-based division of acoustic coatings, parent plate and shell structures and exterior flow fields, this method utilizes the normal continuity condition of vibration and acoustic quantity on the surface of the structure to perform coupled fluid-solid modeling and calculation.Lastly, the third one is the experimental testing method.By virtue of collecting the vibration displacement of the sample surface as well as the underwater acoustic radiation pressure by a vibration sensor and hydrophone respectively, this method can determine the underwater vibroacoustic characteristics of the structure through further data processing.Overall, it is mainly divided into two categories, encompassing the small-sample acoustic tube test and the large-sample anechoic tank test [7][8][9][10] .
Admittedly, the existing research failed to deeply explore the underwater vibroacoustic characteristics of three-dimensional stiffened plates covered with acoustic coatings involving complex acoustic structures.In this regard, this paper intends to model and analyze the underwater vibroacoustic performance related to this.To this end, this research initially employs the effective medium theory to equate the covering layer to a uniform material.Secondly, based on the elasticity theory, this research establishes a semi-analytical model of underwater vibration and acoustic radiation for stiffened plate structures covered with acoustic coatings.By comparing the far-field acoustic radiation values of the proposed semi-analytical model and finite element numerical model, this research ultimately verifies the validity and accuracy of the proposed model.On such a basis, this research preliminarily investigates the coupling effect exerted by stiffeners and parent plates in stiffened plates as well as the vibroacoustic characteristics of acoustic coatings.Figure 1 illustrates the underwater acoustic transmission model under the vertical incidence of plane waves with the cylindrical cavities within the acoustic coating in a periodic horizontal arrangement pattern.The wave propagation theory in the layered medium is leveraged to calculate the acoustic pressures p1 and p2 as well as the velocities v1 and v2 at the interface between the structure and the incident acoustic field/transmitted acoustic field.Furthermore, the effective medium theory (EMM) [3] is employed to determine the equivalent parameters concerning the acoustic coating.The parameter of the Reynolds number are described in Table 1.

Vibration and acoustic radiation model of acoustic coatings and stiffened plates
Figure 2 depicts the vibration and acoustic radiation model of a finite stiffened plate simply supported on four sides with an acoustic coating containing a transverse cylindrical cavity under the excitation, in which the ambient field on one side of the parent plate is air, whereas the acoustic radiation environment on the other side of the acoustic coating is water.Notably, the simple harmonic excitation is placed in the center of the parent plate with the whole structure being embedded in the infinite rigid baffle.According to the classical plate-shell theory [7][8][9] , the motion equation of the substrate plate is:  ( ) ( )  is the bending stiffness of the substrate, where 1 E and 1 v are Young's modulus and Poisson's ratio of the substrate respectively.Assuming that the fluid in this model is compressible and inviscid, the time-harmonic factor e iωt is considered.Then the sound pressure p in the fluid satisfies the wave equation.(2) 0 c is the speed of sound.

Verification of the calculation model
The parameter settings for the parent plate and the material containing a transverse cylindrical cavity acoustic coating are described in Table 2.The thickness h1 and the side length L of the parent plate are equal to 9 mm and 0.6 m respectively.The thickness h2 of the decoupling layer is set to 50 mm.The cavity radius r is set to 5 mm.The transverse spacing d is set to 0.03 m and the magnitude of the excitation is set to 1 N.
The equivalent density and equivalent acoustic velocity of the acoustic coating based on the inversion of the effective medium theory are presented in Figures 3 and 4    Given that the parent plate is an orthotropic stiffened plate and the material parameters of the stiffeners of the stiffened plate are the same as those of the parent plate, the spacing of the stiffeners satisfies lx = ly = 0.15 m.The width a of its cross-section is set to 0.006 m, the height b is set to 0.012 m and the acoustical power contrast value is set to 1 × 10-12 W. acoustical power of the homogeneous coating and the coating with a transverse cavity, with the acoustical power contrast value taken as 1 × 10-12 W.

Validity verification of the proposed calculation method
As can be seen from Figure 5(a) Homogeneous acoustic coating and Figure 5(b) Acoustic coating containing a transverse cylindrical cavity, the theoretical calculation results align well with the finite element simulation results within the [0, 2000] Hz frequency band.Concurrently, the proposed theoretical method can reflect the acoustic radiation of stiffened plates after being covered with acoustic coatings.

Analysis of coupling effect exerted by parent plates and stiffeners
By changing the structural parameters and arrangement mode of the stiffeners on the surface of the stiffened plate, this research explores the law of the coupling effect exerted by the parent plate and the stiffeners, provided that the parameter setting of the acoustic coating is consistent with that in Table 1 and the acoustical power contrast value is taken as 1 × 10-12 W.

Distribution modes of stiffeners of the stiffened plate.
It should be noted that the distribution modes of stiffeners on the stiffened plate mainly encompass the orthogonal distribution mode and the unidirectional distribution mode.Based on this, this research compares the radiated acoustical power of the structure under the foregoing two distribution modes.As can be seen from Figure 6(a) Homogeneous acoustic coating and Figure 6(b) Acoustic coating containing a transverse cavity, the unidirectional distribution mode achieves a relatively good noise reduction effect near the frequency of 650 Hz within the [50, 2000] Hz frequency band.In contrast, the orthogonal distribution mode is helpful to eliminate some acoustical power models in the case of the unidirectional distribution mode, to expand the frequency bandwidth of the acoustical power valley, and therefore to showcase more excellent vibration attenuation and noise reduction effects.

Distribution spacing of stiffeners on the stiffened plate.
Given the influence of the distribution spacing of stiffeners on the radiated acoustical power of the structure, taking the unidirectional/orthogonal stiffened plates with homogeneous decoupling acoustic coating as examples, their stiffener spacing is taken as 0.12 m, 0.15 m, 0.2 m, and 0.3 m, respectively.Among them, the distribution of the stiffeners of the orthogonally stiffened plates follows the orthogonal distribution of 4 × 4, 3 × 3, 2 × 2 and 1 × 1 respectively.In contrast, the distribution of the stiffeners of the unidirectional stiffened plates follows the uniform unidirectional distribution of 4, 3, 2 and 1 respectively.As can be seen from Figure 7(a) Orthogonal stiffened plates and Figure7(b) Unidirectional stiffened plates, the stiffener spacing exerts a certain degree of influence on the acoustic radiation, which is embodied in the following two aspects.On the one hand, in the calculated frequency band, the acoustical power model does not change significantly with the increasing spacing of stiffeners, but it shows a trend of shifting to low frequency.Moreover, a larger stiffener spacing leads to a larger deviation value.On the other hand, in the case where the stiffener spacing is taken as 0.2 m, the acoustical power model in the [1000, 2000] Hz frequency band exhibits the most significant change with the curve of the acoustical power being inconsistent with the overall law.
Specifically, with the increase of the stiffener spacing, the value of acoustic radiation power valley near 600 Hz frequency of the unidirectional stiffened plate increases continuously.In contrast, in the case where the stiffener spacing is taken as 0.2 m, the peak value of acoustical power of the orthogonally stiffened plates in the [1000, 2000] Hz frequency band is higher than that in other stiffener spacing cases.In particular, its acoustical power model at 1100 Hz shows the most significant degree of deviation to low frequency.

Structural parameters of stiffeners of the stiffened plate.
Given the influence of structural parameters of stiffeners on the radiated acoustical power of the structure, taking the unidirectional/orthogonal stiffened plates with a spacing of 0.15 m, which are covered with homogeneous decoupling acoustic coatings, as examples, this research changes the size of their rectangular cross-sections only by changing the width and height of their cross-sections to achieve the comparison of their radiated acoustical power.Regarding the specific parameter setting, the fixed width a is set to 0.006 m, whereas the height b is taken as 0.006 m, 0.009 m, 0.012 m and 0.015 m respectively.By contrast, the fixed height b is set to 0.012 m, whereas the width a is taken as 0.003 m, 0.006 m, 0.009 m and 0.012 m respectively.As can be seen from Figure 9(a) Height of the cross-section and Figure 9(b) Width of the crosssection, the cross-sectional size of the stiffeners of the unidirectional stiffened plates exerts a certain influence on the acoustic radiation within the calculated frequency band.Specifically, first of all, the peak and valley frequencies of its acoustical power tend to shift to high frequency with the increasing cross-sectional size of the stiffeners in the whole frequency band with a larger size indicating a larger deviation value.Secondly, within the [500, 900] Hz frequency band, the peak value of acoustical power increases slightly with the increasing width and height of the cross-section of the stiffeners, while the valley value of acoustical power decreases correspondingly.Thirdly, within the [1000, 1350] Hz frequency band, the peak value of acoustical power decreases slightly with the increasing width and height of the cross-section of the stiffeners, while the valley value of acoustical power increases correspondingly.Lastly, within the [1600, 2000] Hz frequency band, the valley value of acoustical power decreases with the continuous increase of the width and height of the cross-section of the stiffeners.

Analysis of vibro-acoustic characteristics of the acoustic coating with a transverse cylindrical cavity
With the homogeneous coating as a benchmark, this research further discusses the underwater vibroacoustic characteristics of the acoustic coating with a transverse cylindrical cavity from three perspectives, encompassing noise reduction, vibration suppression and vibration isolation.Notably, the structural and material parameters of the homogeneous coating, as well as the acoustic coating with a transverse cylindrical cavity, are depicted in Table 2, while other parameters of the stiffened plate and external fluid are consistent with those described in "Section III.A."

Noise reduction characteristic.
The noise reduction characteristic of the acoustic coating is reflected by the attenuation degree of its acoustical power.The radiated acoustical power of the underwater stiffened plate covered with decoupling acoustic coating is illustrated in Figure 10. Figure 10 depicts the acoustic radiation curves of ordinary sheets and stiffened plates before and after being covered with homogeneous coatings and acoustic coatings with a transverse cylindrical cavity.
As can be seen from Figure 10, in the case of the stiffened plate with a homogeneous coating, the acoustical power model of the stiffened plate is considerably shifted to high frequency under the influence of the decoupling effect exerted by the coating with the acoustical power almost effectively attenuated in the whole frequency band.On the other hand, in the case of the stiffened plate covered with the acoustic coating containing a transverse cylindrical cavity, the radiated acoustical power of the stiffened plate is further attenuated within the range of [200, 1500] Hz under the influence of the decoupling effect exerted by the cavity, but it is not effectively attenuated at the peak of 750 Hz.Additionally, a new model of the acoustical power curve appears near 200 Hz in the case of the stiffened plate with the acoustic coating containing a transverse cylindrical cavity.This implies that the transverse cylindrical cavity structure is beneficial to further improve the noise reduction effect of the acoustic coating in the middle and low-frequency bands.

Vibration suppression characteristic.
In addition, the mechanism related to the vibration reduction and noise reduction exerted by the acoustic coating is partly manifested in its suppression of the vibration of the parent plate.In this regard, the measurement index of the vibration amplitude of the parent plate is the dry-surface mean square vibration velocity of the acoustic coating.
Figure 11(a) which represents the comparison of the vibration levels of the dry-surface mean square vibration velocity, depicts the dry-surface mean square vibration velocity curves of ordinary sheets and stiffened plates before and after being covered with homogeneous coatings and acoustic coatings with a transverse cylindrical cavity.In contrast, Figure 11(b) which represents the comparison of the drysurface mean square vibration velocity insertion losses, illustrates the dry-surface mean square vibration velocity insertion loss curves of stiffened plates with homogeneous coatings and acoustic coatings with a transverse cylindrical cavity.As can be seen from Figure 11 (a), compared with the acoustical power curve shown in Figure 10, in the case of the stiffened plate with a homogeneous coating, the coupling effect exerted by the stiffened plate and the coating leads to a new first-order vibration model around 200 Hz.Within the whole frequency band, the resonant frequency of the vibration model of the stiffened plate is significantly shifted to high frequency with the frequency of the fifth-order resonant peak value even exceeding the calculated frequency range.Moreover, except for the third-order resonant peak value, the resonant peak value at each order is attenuated to some extent.Nevertheless, in the case of the stiffened plate with an acoustic coating containing a transverse cylindrical cavity, the resonant frequency of the parent plate is almost unchanged compared with that of the stiffened plate with a homogeneous coating, and only the peak values at the second-order and fourth-order resonant frequencies increase.Among them, the increase in the fourth-order resonant frequency is particularly significant.
As can be seen from Figure 11 (b), within the [1180, 1300] Hz frequency band, the dry-surface mean square vibration velocity insertion loss of the stiffened plate with homogeneous coatings is higher than that of the stiffened plate with acoustic coatings containing a transverse cylindrical cavity.However, the results of the two are roughly the same in other frequency bands, indicating that the contribution of the transverse cylindrical cavity structure to the vibration suppression of the acoustic coating is extremely limited.

Vibration isolation characteristic.
The attenuation of vibration during the process of transmission within the coating acts as one of the reasons why the coating plays a role in reducing vibration and noise.In this regard, the vibration index of the contact surface between the coating and the fluid is the wetted-surface mean square vibration velocity, while the measurement index of the vibration attenuation during the transmission process is the vibration transmission loss.
Figure 12 (a) indicates the dry-surface and wetted-surface mean square vibration velocity curves of stiffened plates with homogeneous coatings and acoustic coatings containing a transverse cylindrical cavity, respectively.In addition, Figure 12  As can be seen from Figure 12 (a), no significant difference is observed regarding the dry-surface mean square vibration velocity between the stiffened plates with homogeneous coatings and acoustic coatings containing a transverse cylindrical cavity.Notably, in the frequency bands around 200 Hz and 1250 Hz, the dry-surface mean square vibration velocity of stiffened plates covered with acoustic coatings containing a transverse cylindrical cavity is even higher than that of those covered with homogeneous coatings.Nevertheless, the wetted surface mean square vibration velocity of stiffened plates covered with acoustic coatings containing a transverse cylindrical cavity is lower than that of those covered with homogeneous coatings within the whole frequency band.Moreover, the frequencyband width of the resonant peak value around 1250 Hz is shorter than that of those covered with homogeneous coating, demonstrating that the vibration attenuation of stiffened plates covered with acoustic coatings containing a transverse cylindrical cavity is greater than that of those covered with homogeneous coatings.Admittedly, the foregoing conclusion can be confirmed more intuitively by comparing the vibration transmission loss curves presented in Figure 12 (b).As can be seen from Figure 12 (b), the transmission loss of stiffened plates covered with acoustic coatings containing a transverse cylindrical cavity is almost higher than that of those covered with homogeneous coatings within the whole frequency band, and its amplitude variation within the [400, 2000] Hz frequency band is not only small but also uniform.In contrast, stiffened plates covered with homogeneous coatings present prominent valley intervals in the frequency band around 1250 Hz.This further indicates that the transverse cylindrical cavity structure is beneficial to further improve the vibration isolation effect of the coating in the middle and low-frequency bands, thereby exerting a certain degree of influence on the models at the resonant peak value.

Influence of structural parameters on vibro-acoustic characteristics of acoustic coatings containing a transverse cylindrical cavity
Figure 13 presents the cross-sectional geometry of a single primitive cell of the coating containing a transverse cylindrical cavity, where a represents the lattice constant and d represents the cavity diameter, with h and hg representing the thickness of the coating as well as the cavity cover plate respectively.This section will discuss the influence of various structural parameters on the noise reduction and vibration isolation performance of the coating containing a transverse cylindrical cavity while ensuring that the thickness of the coating is unchanged.Figure 14 illustrates the changes in radiated acoustical power levels and vibration transmission losses of the structure after the lattice constant is changed.
As can be seen from Figure 14, in terms of noise reduction performance, the change in the lattice constant generates a fairly limited effect on the peak values of the first-order, second-order and thirdorder acoustic radiation models, while the peak values of the fourth-order and fifth-order acoustical powers increase correspondingly with the increase of lattice constant, showing a worse noise reduction effect.Meanwhile, within the whole frequency band, a larger lattice constant implies a larger acoustical power valley, demonstrating a worse noise reduction effect.
On the other hand, in terms of vibration isolation performance, a larger lattice constant implies a smaller vibration transmission loss within the whole frequency band, demonstrating a worse vibration isolation effect.

Cavity radius.
On the premise of keeping the lattice constant and the thickness of the cavity cover plate unchanged, this research further analyzes the influence of the changes in cavity radius on the vibroacoustic performance of the coating.Among them, the increase in cavity radius leads to a decrease in cavity wall thickness in the case where the lattice constant remains unchanged.
Figure 15 denotes the radiated acoustical power levels and vibration transmission losses of the cavity at different radii.As can be seen from Figure 15, a larger cavity radius within the whole frequency indicates a smaller equivalent stiffness and density of the coating, a larger punching rate, and a larger impedance mismatch between the plate and water, resulting in a smaller radiated acoustical power, a larger vibration transmission loss, as well as a better vibration attenuation and noise reduction effect.

Thickness of the cavity cover plate.
Under the condition of keeping the cavity size and lattice constant unchanged, this research changes the thickness of the cavity cover plate by changing the relative position of the cavity in the vertical direction within the coating, thus analyzing the influence of the thickness change in the cavity cover plate on the vibroacoustic performance of the coating.
Figure 16 presents the radiated acoustical power levels and vibration transmission losses of the cavity cover plate under different thickness conditions.As can be seen from Figure 16, under the condition of solely changing the thickness of the cavity cover plate, the radiated acoustical power curves of the three sizes coincide with the vibration transmission loss curves.Notably, the law that a larger thickness of the cavity cover plate leads to a larger acoustical power and a smaller vibration transmission loss is only satisfied around 1180 Hz.This implies that solely changing the thickness of the cavity cover plate generates limited influence on

Conclusions
To sum up, based on the effective medium theory, the underwater classical plate and shell theory, the three-dimensional elasticity theory, etc., this paper proposes a semi-analytical and semi-numerical calculation and analysis model for underwater vibroacoustic characteristics of a finite stiffened plate structure with an acoustic coating containing a transverse cylindrical cavity by using finite element analysis software Comsol.Relevant findings indicate that the proposed model can accurately predict the underwater vibration and acoustic radiation characteristics of a finite stiffened plate structure with an acoustic coating containing a transverse cylindrical cavity.Meanwhile, this research analyzes the law of the coupling effect exerted by the parent plate and the stiffeners of the stiffened plate, as well as the influence of the structural parameters of the coating on the underwater vibroacoustic characteristics.Simply put, this research provides a concise and effective method for the analysis of underwater vibroacoustic characteristics as well as vibration attenuation and noise reduction mechanisms of coatings with complex acoustic structures.

Figure 1 .
Figure 1.Plane wave projection model of acoustic coatings.

Figure 2 .
Figure 2. Vibro-acoustic model of the stiffened plate covered with the acoustic coating.

1 
and 1 h are the density and thickness of the substrate, w is the shear displacement, ( ) , f x y is the external point exciting force exerted on the plate per unit area and ( ) , , 0 z xy  is the normal force exerted by the covering layer on the substrate. respectively.

Figure 3 .
Figure 3. Equivalent density of the acoustic coating containing a transverse cylindrical cavity.

Figure 4 .
Figure 4. Equivalent acoustic velocity of the acoustic coating containing a transverse cylindrical cavity.

Figure 5 .
Figure 5. Radiated acoustical power of stiffened plate structures covered with acoustic coatings.To verify the validity of the calculation model, the classical analytical method (AM), the model algorithm established in this paper and the finite element method (FEM) are leveraged to calculate the

Figure 6 .
Figure 6.Effect of the stiffener distribution modes on the radiated acoustical power.

Figure 7 .
Figure 7. Effect of the stiffener distribution spacing on the radiated acoustical power.

Figure 8 .Figure 9 .
Figure 8.Effect of the cross-section size of the stiffeners of the orthogonal stiffened plates on the radiated acoustical power.

Figure 10 .
Figure 10.Comparison of Radiated Acoustical Power Levels of Stiffened Plates before and after Being Covered with Acoustic Coatings.

Figure 11 .
Figure 11.Comparison of the dry-surface mean square vibration velocity and its insertion loss of stiffened plates before and after being covered with decoupling acoustic coating.
(b) illustrates the vibration transmission loss curves of stiffened plates with homogeneous coatings and decoupling acoustic coatings containing a transverse cylindrical cavity, respectively.(a) (b) Figure 12.Comparison of the dry-surface and wetted-surface mean square vibration velocity as well as the vibration transmission losses of stiffened plates covered with homogeneous coatings and acoustic coatings containing a transverse cylindrical cavity.

Figure 13 .
Figure 13.Cross-sectional view of a single primitive cell of the coating containing a transverse cylindrical cavity.

3. 4 . 1 Figure 14 .
Figure 14.Effect of changes in the lattice constant on vibro-acoustic characteristics of the acoustic coating.

Figure 15 .
Figure 15.Effect of the Changes in the Cavity Radius on Vibro-acoustic Characteristics of the Acoustic Coating.

Figure 16 .
Figure 16.Effect of the Changes in the Thickness of the Cavity Cover Plate on Vibro-acoustic Characteristics of the Acoustic Coating.
.1088/1742-6596/2756/1/012008 13 the vibroacoustic performance of the coating when the punching rate of the coating containing a transverse cylindrical cavity remains unchanged.

Table 1 .
Reynolds number re and inlet velocity u∞.

Table 2 .
Material parameters for the acoustic coating and parent plate.