Structure-borne Sound Attenuation in A Multi-corner Structure with Attached Blocking Masses

(School of Naval Architecture,Ocean&Civil Eng.,Shanghai Jiao Tong University,Shanghai 200030,China)

1 Introduction

The attenuation of structure-borne sound in built-up structures such as those of a ship has been an issue of great interest for years to researchers concerned with mitigation of vibration and noise.Vibration at high level not only affects operation of machinery and even causes structural fatigue destruction but also becomes the source of structure-borne sound,which is the main offender of ship cabin noise.With the increasing demand for better habitability on board,it is of significance to investigate the mechanism of structure-borne sound attenuation in complex structures.

Elastic waves will be reflected at any discontinuity on traveling in structure and only part of its energy would be transmitted to the downstream section.This process is referred to as‘attenuation of structure-borne sound’by Cremer and Heckl[1],who provided the idea for preventing vibration energy from traveling without decay.Ship structure consists basically of steel plates and beams with very low loss factor and such structure must be not only statically stable but also watertight.It is not feasible to reduce vibration transmission by means of elastic connection for ship structure.Consequently,some rigid methods,e.g.setting discontinuity and attaching blocking masses may be considered as the effective way for structure-borne sound attenuation.

The concept of blocking mass,which is usually a beam of relatively large mass and stiffness with rectangular cross section,rigidly attached at the corner interface of plates or mounted straight on a flat plate to block transmission of elastic wave,is also introduced early in Cremer and Heckl’s book[1]on structure-borne sound.In this regard,there are several papers published in literature.Liu Honglin and Wang Deyu analyzed the vibration of plate structure with blocking mass by FEM(Finite Element Method)and made computations of corresponding sound radiation from the structure.Liu Jianhua and Jin Xianding studied the attenuation of flexural(bending)wave transmission with single blocking mass or parallel multi-blocking masses mounted on an infinite flat plate by wave approach[2-3]and the experimental results match well with theoretical analysis in their work.But all above works are restricted to situations of blocking mass attached on single plate and no corner is involved in the structure.However,the problem is complicated for two plates joined at angles in that there are both bending and longitudinal waves in two plates coupled at the corner[4].For this reason,the authors of this paper carried out analyses on attenuation of structure-borne sound at corner interface of two semi-infinite plates with attached blocking mass,taking both bending and longitudinal waves into consideration[5-6].Some new non-dimensional numbers are introduced to simplify equations which govern transmission and reflection.Numerical results revealed that blocking mass attached at the corner is effective for attenuation at high frequencies and transmission loss depends mainly on the weight of blocking mass,which was validated by some simplified experiments.

Considering a common ship structure,structure-borne waves generated by machineries in engine room will encounter several corners along the way of propagating to the superstructure.So wave analysis in multi-corner structure with attached blocking masses is carried out in this paper.

2 Theoretical analysis

2.1 Expressions of wave motion in multi-corner structure

The multi-corner structure in this paper is modeled as a built-up structure with N corner interfaces consisting of N+1 thin plates of different material and thickness(the left part of Fig.1).The plates are numbered from plate 0 to plate N with finite length li(i=0,1,2,…N),which are supposed to be much longer than the wavelength of the waves propagating in these plates to get rid of the effect of near field.Set N+1 local coordinate systems on these plates in turn and make positive xi-direction along the length of plate i,as shown in the right part of Fig.1.Thus the plates are joined end to end at two parallel boundaries xi=0 and xi=li(i=0,1,2,…N-1).The two free boundaries x0=0 and xN=lNare modeled as ‘reflection-free’ boundaries which absorb most of the incident waves.For more attenuation of structure-borne sound,blocking masses(not shown in Fig.1)may be attached to the corner interface.

Plane incident waves generated at x0=0 by line excitation propagate along the plates of the structure one by one via corner interface.For a given frequency,the incident waves in plate 0 may be expressed in the form of transverse and longitudinal velocity as

For the plate of finite length joined at its two boundaries with other plates,primary waves transmitted from the ‘neighbor’ plates will be reflected by the two boundaries back and forth repeatedly,and secondary waves propagating in both positive and negative xi-direction are generated time and again.Consequently,the wave field in the plate is composed of an infinite number of components,which are also partly transmitted back to the ‘neighbor’ plates when impinging on the joints.Meanwhile,to ensure dynamic equilibrium at the corner interface of two plates,part of incident bending wave will be converted to transmitted and reflected longitudinal ones and vice versa[7],along with evanescent near-field generated at the interface.For the reason that near-field waves can only affect transverse motion in the vicinity of corner interface and have almost no influence on vibration energy transmission,they are always neglected in the analysis provided that the length of the plate is much longer than the wavelength.Supposing the total(sum of the resultant components in both positive and negative directions)transverse and longitudinal velocities in plate i（ i=0,1,2…N）is expressed as where v+y,iand v-y,iare the transverse velocities at xi=0 and xi=lifor resultant(sum of the primary and all secondary components)bending wave propagating in positive and negative xi-direction respectively,v+x,iand v-x,ithe corresponding longitudinal velocities for resultant longitudinal waves,respectively.

Thus there are altogether 4(N+1)unknown velocities to be solved for the whole structure.The dynamic equilibrium condition at the ith(i=0,1,2…N)corner interface gives:

The 4N independent equations about the resultant wave components are deduced by applying Eqs.(5)～(8)to all corners from i=1 to i=N.It is noted that the ‘reflection-free’ boundary assumption at x0=0 and xN=lNassociated with Eqs.(1)and(2)gives:

Then the rest 4N velocities can be solved if incident wave in plate 0 is known.And the total transverse and longitudinal velocities at any position in plate i（ i=0,1,2…N）are ready to be obtained through Eqs.(3)and(4).

For a certain frequency,the spatial averaged vibration energy density is deduced as:

where ρiis the density of plate i,[]*denotes the complex conjugate.

Then the energy density in a certain frequency band is calculated as:

where ω1and ω2are the lower and upper cutoff frequency of the frequency band respectively.

2.2 Transmission efficiency and transmission loss in double-corner structure

The approach discussed above is then applied to investigate vibration energy transmission through the simplest multi-corner structure i.e.double-corner structure consisting of three plates with blocking masses attached at the two corner interfaces,as shown in Fig.2.

Incident waves,either bending or longitudinal,generated at x0=0 propagate throughout the whole structure via each plate and generate secondary waves when encountering corner interfaces.Under conditions of incident waves expressed in Eqs.(1)and(2)and the ‘reflection-free’boundaries at x0=0 and x2=l2,the transverse and longitudinal velocities in plate 2 are written as:

where TBB,TBLare bending and longitudinal transmission coefficient from plate 0 to plate 2 for bending wave incidence,numerically equal to corresponding bending and longitudinal velocities at x2=0 caused by unit velocity incidence of bending wave in plate 0,and TLB,TLLthe corresponding coefficient for longitudinal wave incidence respectively.By applying Eqs.(5)～(8)to both corners,the above transmission coefficients are solved as:

representing the resultant bending or longitudinal waves propagating in the positive x1-direction at x1=0 caused by unit primary waves transmitted from plate 0 to plate 1,in which:

representing secondary bending or longitudinal waves at x1=0 generated by unit primary waves after traveling one ‘round trip’in plate 1,which means the process that wave starting from the first corner is reflected at the second corner and travels back to the first one.

It is noted that velocity fields in the structure are always complex.It is more meaningful to consider vibration transmission in terms of energy(or power)other than velocity.So transmission efficiencies τBB,τBL,τLBand τLLrepresenting the ratios of vibration energy outputted from x2=l2to those inputted at x0=0 are introduced.They are expressed by using corresponding transmission coefficients from Eqs.(17)～(20)as follows:

where P represents the power flow(subscript I for input and O for output),density per unit area mi″=ρi·hi,in which hiis the thickness,CB,iand CL,ithe corresponding group velocities of bending or longitudinal wave respectively.

As mentioned earlier,attenuation of structure-bore sound is the main aim.And transmission loss(TL)is more commonly used to evaluate structure-borne sound attenuation in engineering for the sake of intuition and convenience of measurement.TL is defined by using transmission efficiency as in which τ may be chosen as any one of the transmission efficiencies expressed in Eqs.(29)～(32)to gain the corresponding transmission loss.

2.3 Notes

(1)For the reason that the transmission and reflection coefficients for corners consisting of two semi-infinite places used in Eqs.(5)～(8)are deduced under Kirchhoff’s plate theory[8],plates discussed in this paper should be restricted to thin plates.This requires λB>6h which sets an upper frequency limit as[1]

As for frequencies higher than fmax,shear deformations in the cross section should be taken into consideration so that equations will be deduced with Mindlin’s plate theory for thick plates[9].But for ship structures,fmaxdoes not add much inconvenience for the reason that thin plate assumption can be satisfied in most cases.For example,the common thickness of steel plate used in ship structure is about 20mm for which fmaxwill be 13.5kHz and that is high enough for general purpose of structure-borne sound investigation.

(2)The wave number kBand kLin discussion are those without consideration of damping.If the internal distributed damping in the structure is included,kBand kLshould only be replaced by[1]:

where η is the structural loss factor,cBand cLare phase velocities for bending and longitudinal waves without damping respectively.

3 Numerical investigation

Fig.3 to Fig.6 show the computational results of transmission losses in double-corner structure.All three plates of the structure are steel plates of 15mm in thickness and 1 000mm in length.Both corners are at the angle of π/4 with two attached steel beams of rectangular cross section 100mm×100mm as the blocking masses.The results are also compared with those of the same structure without blocking masses.

It is seen from the comparison of TL curves that:

(1)All TL curves fluctuate with frequency,which indicates transmission loss in doublecorner structure depends greatly on whether the intermediate plate(plate 1)responds at resonance.As has been discussed before,secondary wave will be generated at x1=0 after any wave finishing a ‘round trip’ in plate 1.If the primary and secondary waves are in phase with each other,then the resultant velocity in the positive x1-direction will be the largest and more vibration energy will be transmitted to plate 2,corresponding to the vales in figures.And if the pri-mary and secondary waves at the first corner are out of phase or,in other words,counteract with each other,less vibration energy will be transmitted to plate 2,corresponding to the peaks in the figures.

(2)Though fluctuating acutely,values of TL in double-corner structure with attached blocking masses are larger than those without blocking mass at high frequencies,especially for TLBB,which indicates blocking masses attached to double-corner structure are effective for structure-borne sound attenuation at high frequencies.

(3)TL curves with and without attached blocking masses have little difference at low fre-quencies except in the vicinity of certain special frequency.

(4)Each TL curve without attached blocking masses has a vale with negative value of TL in the vicinity of 50Hz,which is more evident for TLBBand TLBL.This vale corresponds to the first resonant frequency for bending of plate 1,at which all primary and secondary bending waves at x1=0 are in phase with each other.In this case,the first corner acts as a kind of amplifier,which is bad for structure-borne sound attenuation.But value of TL at the vale is significantly increased in case of attached blocking masses.This indicates that blocking masses improve attenuation of vibration energy at resonant frequency of plate 1.

(5)Difference between values of TL with and without attached blocking masses for TLBBand TLLBis larger than that for TLBLand TLLLat high frequencies,which indicates blocking masses are more effective for bending wave attenuation comparing with longitudinal one.

4 Experiment

An experiment on a simplified test sample analogous to the hull of a ship was carried out to validate the results from numerical computation.The test sample is a symmetrical structure consisting of six steel plates(2mm in thickness)numbered from①to⑥as shown in Fig.7.A rigid beam is attached to the joint of plate①and②at the centre line to ensure plane wave produced in plates of the sample by point excitation[6].The beam also divides the test sample into two double-corner subsystems on both sides i.e.plates①-③-⑤ without blocking masses on the left and plates②-④-⑥ with blocking masses on the right.The blocking masses attached on the right side subsystem are two steel beams of rectangular cross section 20mm×20mm.The sample was placed upside down with the free ends of plates⑤and⑥embedded in two sand boxes by 300mm to guarantee ‘reflection-free’ boundaries as shown in Fig.8.

For the reason that machinery excitation on its foundation produces predominantly bending waves into the ship structure[10],only bending wave incidence is considered here.On testing,a broad band point excitation force exerted on the rigid beam generates incident bending waves of the same magnitude into both sides.And both transverse and longitudinal vibration re-sponse at four measurement points on plates①,②,⑤ and⑥ respectively were recorded and processed.This procedure was repeated for 9 times with locations of the measurement points changed for each time.The measurement points on the right side subsystem are shown in Fig.7(right)and those on the left side are allocated symmetrically.

The measurement results from 9 measurement points on each plate are averaged and converted into transmission loss in 1/1 octave band with central frequency from 31.5Hz to 4kHz.Comparisons are shown in Fig.9 and Fig.10 and it is seen that:

(1)TLBBand TLBLmeasured in double-corner subsystem with blocking masses show almost the same trends as predicted.

(2)The values of measured TLBBand TLBLwith blocking masses are higher than those without blocking masses above 1kHz,which validate that blocking masses are effective for attenuation of structure-borne sound especially at high frequencies.

(3)Certain discrepancy is observed in Fig.9,which may be caused by the following factors:

(a)Plane wave assumption is no longer valid by point excitation on the rigid beam at frequencies much higher than the first mode(or resonant)frequency of the beam[5-6].

(b)The blocking masses attached to the test sample are not ideally rigid.And the elastic deformations of them can not be neglected at high frequencies where the wave length is very short.

(c)Under conditions of the free ends of the sample embedded in sand boxes,there are still some reflections remained,especially at low frequencies,which lower the transmission loss measured.

(d)Near field effect at the joint of two plates is neglected in numerical computation,which causes discrepancy especially at low frequencies.

(4)The discrepancy for TLBLis smaller than that for TLBBat high frequencies,which indicates violation of normal incidence of plane wave assumption and deformations of steel beam have less effect on longitudinal wave than bending one.This phenomenon is also observed in the simplified experiment on single corner structure[5].

(5)The computed first resonant frequency for bending of plate③(intermediate plate)is about 20Hz and resonance of the plate makes negative effect on transmission losses.So TLBB(even be negative from measurement)and TLBLmeasured at 31.5Hz 1/1 octave in the left side double-corner subsystem without blocking masses are much lower than those at 63Hz.And about 6dB increasing of TLBBand TLBLare observed after attached blocking masses,which validate the results from numerical investigation.

4 Conclusions

The work in this paper provides a theoretical and experimental basis for applying blocking masses on structure-borne sound attenuation on board of a ship.The expressions of wave motions in multi-corner structure are deduced by wave approach.And the transmission losses in double-corner structure are investigated as well.It is concluded that

(1)The transmission losses from bending wave incidence measured in the experiment matches well with those from prediction,which validate the effectiveness of wave approach used in prediction of structure-borne sound attenuation.

(2)Large increasing of TLBBand TLBLin double-corner structure with blocking masses is measured comparing with those without blocking masses especially at high frequencies.This indicates blocking masses attached to double-corner structure act as a kind of low pass filter,which provides more transmission loss at high frequencies.

(3)Energy transmission through double-corner structure depends mainly on whether the intermediate plate responds at resonance.And it is found that attenuation of vibration energy can be increased at the first resonant frequency of the intermediate plate in double-corner structure with attached blocking masses,which is also validated in the experiment.

(4)The deformation of the beam used as blocking mass and the violation of plane wave assumption should be taken into consideration if more accurate prediction is demanded at high frequencies.So more refined modeling is needed for further research.

[1]Cremer L,Heckl M,Ungar E E.Structure-borne sound[M].Second edition.Berlin:Springer-Verlag,1988.

[2]Liu Jianhua,Jin Xianding,Li Xiaobin.Attenuation of the plate flexural wave transmission through a vibration isolation mass[J].Journal of Ship Mechanics,2006,10(6):131-137.

[3]Liu Jianhua,Jin Xianding,Li Xiaobin.Impediment to structure-borne sound propagation from several paralleling arranged vibration isolation mass[J].Journal of Shanghai Jiaotong University,2003,37(8):1205-1208.

[4]Bercin A N.An assessment of the effects of in-plane vibrations on the energy flow between coupled plates[J].Journal of Sound and Vibration,1996,191(5):661-680.

[5]Che Chidong,Chen Duanshi.Attenuation effect of blocking mass attached at corner interface on transmission from plane longitudinal wave to bending wave[J].Journal of Ship Mechanics.(accepted and will be published)

[6]Chen Duanshi,Che Chidong.Analysis of vibration transmission at the corner interface of two plates for reduction of structure-borne sound[C].The Thirteenth International Congress on Sound and Vibration(ICSV 13),2006.

[7]Langley R S,Heron K H.Elastic wave transmission through plate/beam junctions[J].Journal of Sound and Vibration,1990,143(2):241-253.

[8]Lowe P G.Basic principles of plate theory[M].Glasgow:Surrey University Press,1982.

[9]Mindlin R D.Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates[J].Journal of Applied Mechanics,1951,18:31-38.

[10]Grice R M,Pinnington R J.A method for the vibration analysis of built-up structures,Part I:Introduction and analytical analysis of the plate-stiffened beam[J].Journal of Sound and Vibration,2000,230:825-849.