TAN Lu,ZHOU Qi-dou,JI Gang,ZHANG Wei-kang

(Dept.of Naval Architecture and Ocean Engineering,Naval Univ.of Engineering,Wuhan 430033,China)

Study on the Influence of Bulkhead Arrangement Form on the Structural Acoustic Characteristics of Cylindrical Shell

TAN Lu,ZHOU Qi-dou,JI Gang,ZHANG Wei-kang

(Dept.of Naval Architecture and Ocean Engineering,Naval Univ.of Engineering,Wuhan 430033,China)

Abstract:The vibration response of cylindrical shells with periodic or aperiodic arrangement of bulkhead in typical force excitation was calculated by finite element method(FEM).Then,the mean square velocity level transfer functions were obtained by using the vibration response of different shells as the input.At the typical frequency,the wave number spectrum analysis method was applied to transform the vibration response of different shells from the spatial domain into the wavenumber domain.Therefore,the vibration of the shells can be separated into various simple harmonic waves,whose amplitude determines their respective contribution to the vibration.Finally,by comparing the mean square velocity level transfer functions and wavenumber spectrums of the different shells,the influence of different bulkhead arrangement forms on the structural acoustic characteristics of cylindrical shell was obtained.The results show that:for the vertical force excitation,the cylindrical shell with aperiodic arrangement of bulkhead has better structural acoustic characteristics,which however is not obvious for axial force excitation.In the case of the combination of the aperiodic arrangement of bulkhead and the measures of strengthening local structure,the cylindrical shell can obtain better structural acoustic characteristics in both vertical and axial force excitation.

Key words:cylindrical shell;periodic;aperiodic;wavenumber spectrum;structural acoustic characteristics

0 Introduction

Cylindrical shell with bulkhead is an important structural form of ocean engineering[1].The design of bulkhead arrangement form of cylindrical shell usually includes the structural strength,difficulties in design and construction,the space of internal equipments,but the improvement of structural acoustic characteristics of cylindrical shell has not been paid enough attention.The study of the characteristics of cylindrical shell with periodic or aperiodic arrangement of bulkhead can provide references for understanding and optimizing the structural acoustic characteristics of cylindrical shell.

So far,there have been many researches about the structural acoustic characteristics ofcylindrical shell,but few focus on the influence of the bulkhead arrangement.Bulkhead arrangement form can be divided into two categories:the periodic and the aperiodic.The studies on structural acoustic characteristics of periodic and aperiodic structure have focused on the seventy’s of the last century to the beginning of this century.They mainly use analysis method to study the simple structure,such as:periodic or aperiodic structure beam[2-5],periodic or aperiodic one-way stiffened plate[6-9].

The main conclusions of the vibration characteristics of periodic structure are:the vibration of simple periodic structure have alternately pass band and stop band in the frequency domain[10-11]due to the structure wave needs to satisfy the Bloch theory in periodic structure[12-13],and in the pass band structure wave can propagate without attenuation,while in the stop band structure wave is exponentially attenuated with the increase of propagation distance.The main conclusions of the vibration characteristics of the aperiodic structure are:the vibration of the aperiodic structure has the Anderson localization phenomenon[14-15],that the vibration energy of the structure gathers in the vicinity of the excitation source,and the amplitude of the vibration exponentially decays with the distance away from the excitation source.Although some important conclusions have been given by above researches,the study of the influence of bulkhead arrangement form on the structural acoustic characteristics of cylindrical shell has not been done.

In this paper,the structural acoustic characteristics of typical cylindrical shells which have periodic or aperiodic bulkheads are analyzed in vertical or axial force excitation.The FEM is used to calculate the vibration response of the typical shells,and the mean square velocity level transfer function(response of mean square velocity level in unit excitation)of the shell is obtained.To identify each harmonic wave component of the structure vibration,and to explain the mechanism of vibration response caused by the changes of structure due to wave,the wavenumber spectrum analysis method[16]is used to obtain the vibration power wavenumber spectrum of the shell.Based on the analysis of mean square velocity level transfer functions and wavenumber spectrums of the typical shells,the influence and physical mechanism of the bulkhead arrangement on the structural acoustic characteristics of cylindrical shell are summarized.And the measures combining the aperiodic arrangement of bulkhead with the strengthening of local structure is creatively raised to control the vibration and noise.The paper no longer introduces FEM and mean square velocity level transfer functions theory,but focuses on the wavenumber spectrum analysis theory of the cylindrical shell.

1 Wavenumber spectrum analysis theory of cylindrical shell

To transform the vibration response of cylindrical shell from the spatial domain into the wavenumber domain,the vibration response field with finite length in axial direction is extended to infinite.The assumption that there are infinite cylindrical baffles at the two ends of the finite shell[17]leads to the expression of the vibration response as:

where x and θ are axial and circumferential coordinates in cylindrical coordinate system respectively,and wRand wIare the real and imaginary parts of the normal displacement w of the shell,and L is the length of cylindrical shell,andThe Fourier series expansion of wRand wIis carried out in the circumferential direction,and the Fourier transform is carried out in the axial direction[16]:

where kxand n are axial and circumferential wavenumber,and the wavenumber of a single structure wave component is expressed as kx,（）n.Eq.(2)shows that the vibration response field of cylindrical shell is decomposed into a superposition of simple traveling waves with different axial and circumferential wavenumber.are the amplitude of the wave whose wavenumber marked as

The square of the normal velocity response of the shell in Eq.(1)is integrated on the surface of the cylindrical shell,and the vibration powerexpressed by normal velocity on the surface of the whole cylindrical shell is obtained,which is called normal velocity vibration power for short.

where a is radius of cylindrical shell,ρ0and c0are the density and sound velocity of the coupled fluid.The multiplier(the acoustic impedance ρ0c0)can transform equation dimension into power unit,which do not change the frequency characteristic of the vibration power.The fluid in this paper refers to the air.Combining the Eq.(2)and Eq.(3),and by using the Parseval equation,the function can be simplified into[16]:

The total normal velocity vibration power of the shell can be expressed as the superposition of the normal velocity vibration power of the wave components of the wavenumber

Therefore,according to the Eq.(4)and the Eq.(5),the normal velocity vibration powerof a single simple traveling wave of the wavenumbercan be acquired:

2 Structural acoustic characteristics of cylindrical shell with periodic or aperiodic arrangement of bulkhead in vertical force excitation

The principal dimension of typical cylindrical shell with different bulkhead arrangement form is shown in Fig.1.The thicknesses of the shell and bulkhead are 14 mm and 10 mm,respectively.The ribs are arranged periodically,and the distance is 600 mm apart from each other.Rib thickness is 7 mm and the height is 300 mm.There are two typical cylindrical shell models which are divided into 6 cabins.The bulkhead spacing of model 1(periodic arrangement of bulkheads)is 6 000 mm,and the bulkhead arrangement form of model 2(aperiodic arrangement of bulkheads)is shown in Fig.1.Two models are all made of steel,whose Young’s modulus is 2.05×105N/mm2,and the density is 7.80×10-6kg/mm3,and the Poisson’s ratio is 0.3.The two typical excitations are vertical force excitation and axial force excitation respectively which are also shown in Fig.1.

Fig.1 The cylindrical shell with aperiodic arrangement of bulkhead

Because the cabin 1(the first cabin in the left side of the cylindrical shell)is close to the excitation source,the vibration response of cabin 1 was not included in the mean square velocity level of the shell,otherwise the vibration characteristics of the shell away from the excitation source is not obvious.

Fig.2 Mean square velocity level transfer function of model 1 and model 2 in vertical force excitation

Fig.2 denotes the mean square velocity level transfer functionsof model 1 and model 2 in vertical force excitation.Three conclusions can be summed up from this figure.Firstly,the influence of bulkhead arrangement form on the shell vibration is relatedto the relative size of cabin length and structure wavelength.At low frequency,the structure wavelength is much longer than the length of the cabin,so the shell vibration is less influenced by bulkhead arrangement form.At high frequency,the structure wavelength is much smaller than the length of the cabin,so the vibration of the shell is mainly affected by the rib arrangement form but not bulkhead arrangement form.At middle frequency,the structure wavelength is similar to the length of the cabin,so bulkhead arrangement form has a great influence on the shell vibration.Secondly,the vibration of model 1 has alternately pass band and stop band in the frequency domain.According to the vibration shapes,the three spectrum peaks marked in Fig.2 represent three pass bands,whose corresponding vibration shape of the cabin is mainly based on 0.5,1,1.5 times the structure wave in axial direction,respectively.Thirdly,in the pass band,the value of mean square velocity level transfer function of model 2 is smaller than that of model 1.It indicates that,comparing with the periodic arrangement of bulkheads,the vibration of the shell with aperiodic arrangement of bulkhead shows the phenomenon of Anderson localization in which the amplitude of structure wave has significant attenuation.

Taking the typical peak frequency of 292 Hz as an example,the wavenumber spectrum analysis of the vibration of cylindrical shell with periodic or aperiodic arrangement of bulkhead is carried out.Fig.3 denotes the normal velocity vibration power wavenumber spectrum of model 1 and model 2 corresponding to circumferential wavenumber n=1 in 292 Hz.The horizontal coordinate of the figure is axial dimensionless wavenumber where

Fig.3 The normal velocity vibration power wavenumber spectrum of model 1 and model 2 corresponding to n=1 at 292 Hz

In Fig.3,there are a series of peaks in the normal velocity vibration power wavenumber spectrum of model 1.According to the dimensionless wavenumber,it can be found that the structure wavelength corresponding to each peak is equal to 1/m(m is positive integer,and the value of m corresponding to the peaks have been given in Fig.3)times the length of cabin,which indicates the ‘selection’ effect of the axial structure bending wave of cylindrical shell with periodic arrangement of bulkhead.

The largest spectrum peaks(m=1)of the transfer function of model 1 and model 2 correspond to the same dimensionless wavenumber.It suggests that the vibration shapes of the two models in axial direction are the same,of which the wavelength is equal to the length of the cabin,which is the main factor of the vibration attenuation of model 2.Fig.4 shows the vibration shapes of model 1 and model 2 in vertical force excitation in 292 Hz.It shows that,comparing with the periodic arrangement form of bulkheads,the matching relation of the phase ofstructure wave between cabins is broken by the aperiodic arrangement of bulkheads,therefore the shell vibration attenuated.

3 Structural acoustic characteristics of cylindrical shell with periodic or aperiodic arrangement of bulkhead in axial force excitation

Fig.5 denotes the mean square velocity level transfer functions of model 1 and model 2 in axial force excitation.It shows that the arrangement form of the bulkheads has little effect on the mean square velocity of the cylindrical shell.

Fig.5 Mean square velocity level transfer functions of model 1 and model 2 in axial force excitation

Fig.6 Vibration shapes of model 1 and model 2 in axial force excitation at 271 Hz

In Fig.7,the vibration of the two models is mainly composed of the waves whose dimensionless wavenumber are(0,43)and(4,239).Combined with Fig.6,the long structure wave of wavenumber(0,43)is mainly manifested in the right end of the shell.Its wavelength is approximately 3 times the length of cabin in axial.The bending wave of this kind is mainly generated by the longitudinal wave encountering the bulkheads.Since the arrangement form of bulkheads has little effect on the longitudinal waves,it has little influence on that bending wave.

On the other hand,the short structure wave of wavenumber(4,239)is mainly manifestedin the left end of the shell.Its wavelength is approximately 0.5 times the length of cabin in axial.Because the axisymmetric structure bending wave is repressed by bulkhead with strong radial stiffness,the amplitude of the wave of wavenumber n=4 has a great attenuation at the bulkhead 2(the second bulkhead from the left side of shell as shown in Fig.1).As a result,the difference of the two arrangement forms of bulkhead on the vibration of the shell is not obvious.In order to weaken the vibration of the whole shell,the length of cabin 1 should be as short as possible in the case of other design conditions permitted.

Fig.7 The normal velocity vibration power wavenumber spectrum of model 1 and model 2 corresponding to n=0,4 at 271 Hz

4 Structural acoustic characteristics of cylindrical shell with aperiodic arrangement of bulkhead and local structure strengthening

According to the analysis on structural acoustic characteristics of model 1 and model 2 in vertical and axial force excitation,the aperiodic arrangement of bulkheads is beneficial to the vibration control of the shell in vertical force excitation,but is not in axial force excitation.Considering the strong vibration of cabin 1 which closes to the excitation source,the structural acoustic characteristics of the shell with strengthened local structure and aperiodic arrangement of bulkhead are analyzed.

Fig.8 Mean square velocity level transfer function of model 1 and model 3 in vertical and axial force excitation

The cylindrical shell(called model 3)is strengthened by 8 longitudinal frames arranged uniformly along the circumferential direction in the cabin 1.The longitudinal frames are as long as the cabin 1,and their thicknesses and height are 14 mm and 300 mm,respectively.

Fig.8 denotes the mean square velocity level transfer functions of models 1-3 in vertical and axial force excitation.The results show that,comparing with the vibration of model 1,the vibration of the model 3 is greatly attenuated in both vertical force excitation and axial force excitation.For vertical force excitation,according to Figs.2-8,it is shown that local structure strengthening has no significant influence on the vibration of cylindrical shell with aperiodic arrangement of bulkhead.At that time,their structural acoustic characteristics are always better than the shell with periodic arrangement of bulkhead.For axial force excitation,because the axial stiffness of cabin 1 is stronger after strengthened by longitudinal frames,the amplitude of axisymmetric bending wave decreased significantly,which is the main reason of vibration attenuation of shell.

5 Conclusions

In this paper,the influence of the bulkhead arrangement form on the structural acoustic characteristics of cylindrical shells in two typical excitations is discussed.The main results are as follows:

(1)Only in the case that the structure wavelength is similar to the length of cabin,the vibration of shell can be influenced by bulkhead arrangement form in vertical force excitation.

(2)For vertical force excitation,the vibration of the shell with periodic arrangement of bulkhead has alternately pass band and stop band in the frequency domain.And the vibration of the shell with aperiodic arrangement of bulkhead has the phenomenon of Anderson localization resulting in the decrease of wave amplitude in pass band.

(3)For axial force excitation,bulkhead arrangement form has less influence on the vibration of the shell.There are two main reasons.Firstly,the axisymmetric bending waves are repressed by bulkheads which have strong radial stiffness.So,the length of the first cabin should be as short as possible in the case of other design conditions permitted.Secondly,the bulkhead arrangement form is not effective to control the bending waves generated by longitudinal waves when encountering bulkheads.

在小尺度空間中進(jìn)行水景設(shè)計(jì)應(yīng)堅(jiān)持人性化、實(shí)用性、生態(tài)性、參與性等設(shè)計(jì)原則與可持續(xù)建設(shè)構(gòu)想，不能一味地簡(jiǎn)單模仿.一般需要綜合考慮以下幾個(gè)方面：

(4)In the case of the combination of the aperiodic arrangement of bulkhead and the measures of strengthening local structure,the cylindrical shell can obtain better structural acoustic characteristics in vertical or axial force excitation.

[1]Huang Jiaqiang.Optimum design of long cabin stiffened cylindrical shell[J].Journal of Ship Mechanics,2016,20(1-2):137-141.(in Chinese)

[2]Mead D J.Wave propagation and natural modes in periodic systems:I.mono-coupled systems[J].Journal of Sound and Vibration,1975,40(1):1-18.

[3]Mead D J.Wave propagation and natural modes in periodic systems:II.multi-coupled systems,with and without damping[J].Journal of Sound and Vibration,1975,40(1):19-39.

[4]Bouzit D,Pierre C.An experimental investigation of vibration localization in disordered multi-span beams[J].Journal of Sound and Vibration,1995,187(4):649-669.

[5]Bansal A S.Collective and localized modes of mono-coupled multi-span beams with large deterministic disorders[J].The Journal of Acoustical Society of America,1997,102(6):3806-3809.

[6]Mead D J,Zhu D C,Bardell N S.Free vibration of an orthogonally stiffened flat plate[J].Journal of Sound and Vibration,1988,127(1):19-48.

[7]Mead D J,Parthan S.Free wave propagation in two-dimensional periodic plates[J].Journal of Sound and Vibration,1979,64(3):325-348.

[8]Eatwell G P.Free-wave propagation in an irregularly stiffened,fluid-loaded plate[J].Journal of Sound and Vibration,1983,88(4):507-522.

[9]Photiadis D M.Anderson localization of one-dimensional wave propagation on a fluid-loaded plate[J].The Journal of A-coustical Society of America,1992,91(2):771-780.

[10]Mead D J.A new method of analyzing wave propagation in periodic structures;applications to periodic Timoshenko beams and stiffened plates[J].Journal of Sound and Vibration,1986,104(1):9-27.

[11]Mead D J,Yaman Y.The response of infinite periodic beams to point harmonic forces:a flexural wave analysis[J].Journal of Sound and Vibration,1991,144(3):507-530.

[12]Ohlrich M.Forced vibration and wave propagation in mono-coupled periodic structures[J].Journal of Sound and Vibration,1986,107(3):411-434.

[13]Mead D J.Wave propagation in continuous periodic structures:research contributions from Southampton,1964-1995[J].Journal of Sound and Vibration,1996,190(3):495-524.

[14]Anderson P W.Absence of diffusion in certain random lattices[J].Physical Review,1958,109(5):1492-1505.

[15]Kazushige Ishii.Localization of eigenstates and transport Phenomena in the one-dimensional disordered system[J].Progress of Theoretical Physics Supplement,1973,53(53):77-138.

[16]Tan Lu,Ji Gang,Zhang Weikang,et al．Slender cylindrical vibration and radiation by use of wave-number domain approach[J].Journal of Naval University of Engineering,2013,25(3):66-71.(in Chinese)

[17]Wang C,Lai J C S.The sound radiation efficiency of finite length acoustically thick circular cylindrical shells under mechanical excitation I:Theoretical analysis[J].Journal of Sound and Vibration,2000,232(2):431-447.

分艙形式對(duì)圓柱殼結(jié)構(gòu)聲學(xué)性能影響研究

譚 路，周其斗，紀(jì) 剛，張緯康
（海軍工程大學(xué) 艦船工程系，武漢430033）

為研究艙壁布置形式對(duì)圓柱殼結(jié)構(gòu)聲學(xué)性能的影規(guī)律，文章應(yīng)用有限元方法對(duì)等間距與不等間距分艙圓柱殼模型的振動(dòng)響應(yīng)進(jìn)行了計(jì)算，以此為輸入可計(jì)算獲得殼體的均方法向速度級(jí)聲學(xué)傳遞函數(shù)，并在典型頻率處應(yīng)用波數(shù)譜分析方法將殼體空間域振動(dòng)場(chǎng)轉(zhuǎn)換到波數(shù)域上，對(duì)殼體的振動(dòng)進(jìn)行波形分離與量化，分析獲得了艙壁等間距布置與不等間距布置圓柱殼在典型激勵(lì)下的結(jié)構(gòu)聲學(xué)性能，并解釋其機(jī)理。研究表明：對(duì)垂向激勵(lì)，圓柱殼不等間距分艙相比等間距分艙具有較好的結(jié)構(gòu)聲學(xué)性能，但對(duì)軸向激勵(lì)，兩者區(qū)別不明顯。若將不等間距分艙與局部結(jié)構(gòu)補(bǔ)強(qiáng)措施相結(jié)合，則可使得圓柱殼在兩典型工況下均能獲得更優(yōu)的結(jié)構(gòu)聲學(xué)性能。

圓柱殼；等間距；不等間距；波數(shù)譜；結(jié)構(gòu)聲學(xué)

U661.42

A

譚 路（1989-），男，海軍工程大學(xué)博士研究生；周其斗（1962-），男，博士，海軍工程大學(xué)教授，博士生導(dǎo)師；紀(jì) 剛（1975-），男，博士，海軍工程大學(xué)副教授，碩士生導(dǎo)師；張緯康（1939-），男，博士，海軍工程大學(xué)教授，博士生導(dǎo)師。

10.3969/j.issn.1007-7294.2017.09.011

Article ID： 1007-7294（2017）09-1170-10

Received date：2017-04-15

Foundation item:Supported by National Defense Pre-Research Foundation of China(9140A14080512JB11165)

Biography：TAN Lu(1989-),male,Ph.D.,student of Naval University of Engineering,E-mail:512425568@qq.com;ZHOU Qi-dou(1962-),male,Ph.D.,professor/tutor.