Acoustic Radiation Damping of the Submerged Rectangular Plate

REN Hui-juan,SHENG Mei-ping

(1.Dept.of Physics and Electronic Engineering,Xianyang Normal University,Xianyang 712000,China; 2.Dept.of Marine,Northwestern Polytechnical University,Xi’an 710072,China)

Acoustic Radiation Damping of the Submerged Rectangular Plate

REN Hui-juan1,SHENG Mei-ping2

(1.Dept.of Physics and Electronic Engineering,Xianyang Normal University,Xianyang 712000,China; 2.Dept.of Marine,Northwestern Polytechnical University,Xi’an 710072,China)

For a rectangular plate submerged in water,which is drived by an point harmonic force and embed into the infinite rigidity baffle with simply supported boundary condition,the vibration velocity expression is obtained by revising the mass and damping for a rectangular plate in air.According to the definition of the damping loss factor,the expression of the radiation damping is obtained from the the angle of the energy.The relation of the radiation damping with the modal radiation effency is found.The results show that the radiation damping changes with the increasing of the wave number ratio,the radiation damping of the plate with one side submerged in water is well above the one submerged in air with two sides in the middle and high frequency band,and the more the thickness,the smaller the radiation damping is.

the rectangular plate;radiation damping;attached mass

0 Introduction

The structure will compress the fluid medium and forms acoustic wave into the far field when it vibrates in the medium.Then part of the mechanical energy of the vibrating structure will propagate into the medium in the form of the acoustic energy.The radiation damping is defined as the ratio of the acoustic energy and the total mechanical energy.

The acoustic radiation damping is an important parameter,and will influence the radiation efficency of the vibrating structure.The early acoustic radiation research is developed from the supersonic fluid environment of the plate and frame structure.Muhlstein[1]offered experiment measurement method of the acoustic radiation damping of a plate in the supersonic fluid. Later,Wallace[2]presented the acoustic radiation damping of a rectangular plate in the air. Kriegsman and Scandrett[3]estimated the sound radiation damping from the view of coupling of structure and sound.Fu[4]compared the acoustic radiation damping of the plate in the light fluid and the heavy fluid with the method of the finite element and boundary element.All of these researchs offer much help for the acoustic radiation damping from different angles.Thispaper adopts a new method to obtain the expression of the sound radiation damping of a rectangular plate submerged in water.The attached mass and the radiation damping are used to revised the velocity of the plate in air to obtain the velocity of the plate in water,then the expression of the radiation damping is deduced from the angle of the energy.This paper will lay the foundation for the research of its averaged acoustic radiation efficiency.

1 The sound power of the rectangular plate submerged in water

When a structure vibrates in the medium,acoustic field will form in the medium and influences the structure itself.Thus the attached mass and radiation damping come into being[5].

A rectangular plate submerged to water with simply supported boundary condition is embed into the infinite rigidity baffle(shown in Fig.1).The plate is partitioned into infinite small patches.The coordinates（x,y）and（x′,y′）describe the location of the random two small patches ds and ds′.The relative location of the patch ds′to ds is described with（h,θ）,in which h is the distance of ds and ds′,and θ is the included angle of BA and h.When the plate is drived by the harmonic point force F ejωtat the location（x0,y0）,considering the influence of the attached mass and radiation damping, the velocity amplitude of the point（x,y）can be expressed as

Fig.1 Integral sketch map of the surface in a rectangular plate

where ρwis the density of water,kwis the wave number.When the two sides of the plate submerged in water,the attached mass of the plate will become two times than the plate with one side submerged in water.

The acoustic pressure of the point（x,y）is described as

thus the sound intensity of the point（x,y）can be expressed as

Substituting Eqs.(1)and(3)into Eq.(4),and integrating for the Eq.(4),then the modal sound power radiated by the whole plate is given by

Eq.(5)includes the double integral.That is becauseitself in Eq.(3)include the integral.In order to obtain the sound power,one must integrate again for Eq.(4),then the double integral comes into being.

2 The radiation damping of the rectangular plate submerged in water

When the plate submerged in water vibrates,the maximum kinetic energy of the patch ds can be expressed as

When the rectangular plate submerged in water vibrates,the mechanical loss will generate,and the mechanical energy will convert into sound energy.According to the definition of the loss factor,the radiation damping of the plate vibrating in the（m,n）mode can be expressed as[4]

Substituting Eqs.(5)and(7)into Eq.(8),the radiation damping of the plate submerged in water is given by

3 The modal radiation efficiency and the relation with the radiation damping

When the plate driving by the harmonic point force F ejωtat the locationin order to acquire the average value of the square of speed’s modulus,an average is exerted over all possible response positions[6],thus the modal mean square velocity can be expressed as

where Wmnwis the modal sound power.From Eq.(5)we can see Wmnwis only related to（x0,y0）. As the same reason,the spatial average is exerted over all possible excitation positions[6],thus the spatially averaged modal sound power can be obtained as

Thus the modal radiation efficiency of the plate submerged in water is given by

Substituting Eq.(13)into Eq.(9),the radiation damping of the plate submerged in water can also be expressed as

From Eq.(14),we can see that the radiation damping is relative to not only the modal radiation efficiency,but also the attached density.Because the attached mass of a plate with one side submerged in water and another side submerged in air is different to the plate with two sides all submerged in water,the radiation damping of them is different.

Eqs.(9)and(14)provide the expression of the radiation damping of the rectangular plate submerged in water.If the medium is air,the radiation damping of the plate can be obtained by replacing all the parameters in water with that in air in Eqs.(9)and(10).Because the attached mass of the plate with two sides all in air is almost zero,the radiation damping of theplate with two sides all in air can be expressed as

where ρ0is the density of the air,σmnis the modal radiation efficiency of the plate in air,k0is the wave number in air.Eq.(15)is the exact radiation damping of a rectangular plate in air in Ref.[3].That means Eqs.(9)and(14)are valid and they are suit for the rectangular plate emerged in any fluid by revising the parameter about the fluid.

4 Numerical simulation

Fig.2 The radiation damping of the first 12 order modes of a rectangular plate with one side submerged in water -*-（1,1）-o-(2,1)-v-(3,1)---*---（2,2）---o---(3,2)---v---(4,2) -.-.*-.-.(3,3)-.-.o-.-.(4,3)-.-.v-.-.(5,3)…*…(4,4)…o…(5,4)…v…(6,4)

Fig.3 compares the radiation damping of the rectangular plates with one side submerged in water with that two sides all submerged in air for several modes.The size of the two plates is as above.It can be seen that in the middle and higher frequency band the radiation damp-ing of the rectangular plate with one side submerged in water is well above that two sides all submerged in air,while in the low frequency band,it has no obvious law.

Fig.3 Comparison of radiation damping of a plate in water with that in air—with one side submerged in water and another side submerged in air……with two sides all submerged in air

Fig.4 compares the radiation damping of three rectangular plates with one side submerged in water.Their thicknesses are 35 mm,25 mm and 15 mm,respectively.It can be found that the more the thickness,the smaller the radiation damping is.That is because the radiation damping is inversely proportional to its thickness.

Fig.4 Comparison of radiation damping of a plate with different thickness—15 mm-.-.-.25 mm……35 mm

5 Conclusions

It has great significance to study the acoustic radiation damping of a structure.According to the definition of the acoustic radiation damping,radiation damping expression is derived from the angle of the energy for a rectangular plate submerged in water.The deduction is well agreed with related references.The numerical simulation shows that with increasing of the wave number ratio,the radiation damping linearly increases for kw<>kmn,it approachs to 1.The radiation damping of the rectangular plate with one side submerged in water and another side submerged in air is well above the plate submerged in air with two sides in the middle and high frequency band,while in the low frequency band,it has no obvious law.The more the thickness,the smaller the radiation damping is.

[1]Muhlstein L Jr．Experimental evaluations of the aerodynamics damping of skin panels at low supersonic Mach numbers[C]. In proceedings of the 13th structures．Structural dynamics and materials conference．AIAA,1972:72-402.

[2]Wallace C E．The acoustic radiation damping of the modes of a rectangular panel[J]．J Acoust．Soc．Am.,1987,81(6): 1787-1794．

[3]Kriegsmann G A,Scandrett M A．Assessment of a new radiation damping model for structural acoustic interactions[J]．Journal of the Acoustical Society of America,1989,86:788-794.

[4]Fu Xihua.The research on acoustic radiation damping of the stiffened structure[D].Dalian:The master’s degree thesis of the Dalian University of Techenology,2007.

[5]Ren Huijuan.Study of the vibration and sound radiation of the underwater plate structure[D].Xi’an:The doctor’s degree thesis of the Northwestern Polytechnical University,2013.

[6]Xie G,Thompson D J,Jones C J C.The radiation efficiency of baffled plates and strips[J].Journal of Sound and Vibration,2005,280:181-209.

浸水矩形板的声辐射阻尼研究

任惠娟1，盛美萍2

（1.咸阳师范学院物理与电子工程学院，陕西咸阳712000；2.西北工业大学航海学院，西安710072）

对空气中简支边界矩形板在点简谐力激励下的振速响应公式从附加质量和声辐射阻尼的角度进行修正，得到了浸水矩形板的振速响应公式。从浸水矩形板振速公式出发，结合声辐射阻尼的定义，从能量的角度推导了矩形板声辐射阻尼的表达式，给出了其与模态辐射效率之间的关系，文中推论与已发表有关文献相比较一致性良好。研究表明：辐射阻尼随着波数比的变化而变化；在中高频段，单面临水矩形板的辐射阻尼远高于其在空气中时的辐射阻尼；板越厚，其声辐射阻尼越小。

矩形板；辐射阻尼；附加质量

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任惠娟（1972-），女，博士，咸阳师范学院物理与电子工程学院副教授；盛美萍（1970-），女，西北工业大学航海学院教授，博士生导师。

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10.3969/j.issn.1007-7294.2016.12.011

1007-7294（2016）12-1619-07

Received date:2016-07-19

Foundation item:Supported by the Natural Science Foundation of Shaanxi Province(2011JM1017)；the Science Research Project of the Education Committee of the Shaanxi Province(2013JK0618)

Biography：REN Hui-juan(1972-),female,associate professor,E-mail:304429666@qq.com; SHENG Mei-ping(1970-),female,professor/tutor,E-mail:smp@nwpu.edu.cn.