Research on Power Flow Transmission through Elastic Structure into … · 2019. 12. 12. · AdvancesinAcousticsandVibration Y X Z L Y L x L Z p E D F D S 0 =1000kg/G 3 c 0 =1500m/s

Research ArticleResearch on Power Flow Transmission through Elastic Structureinto a Fluid-Filled Enclosure

Rui Huo 12 Chuangye Li 12 Laizhao Jing 12 andWeikeWang 12

1School of Mechanical Engineering Shandong University Jinan 250061 China2Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Shandong University) Ministry of EducationJinan 250061 China

Correspondence should be addressed to Rui Huo huoruisdueducn

Received 30 October 2017 Revised 8 February 2018 Accepted 15 March 2018 Published 2 May 2018

Academic Editor Kim M Liew

Copyright copy 2018 Rui Huo et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The work of this paper is backgrounded by prediction or evaluation and control of mechanical self-noise in sonar array cavity Thevibratory power flow transmission analysis is applied to reveal the overall vibration level of the fluid-structural coupled systemThrough modal coupling analysis on the fluid-structural vibration of the fluid-filled enclosure with elastic boundaries an efficientcomputational method is deduced to determine the vibratory power flow generated by exterior excitations on the outside surfaceof the elastic structure including the total power flow entering into the fluid-structural coupled system and the net power flowtransmitted into the hydroacoustic field Characteristics of the coupled natural frequencies and modals are investigated by anumerical example of a rectangular water-filled cavity with five acoustic rigid walls and one elastic panel Influential factors ofpower flow transmission characteristics are further discussed with the purpose of overall evaluation and reduction of the cavitywater sound energy

1 Introduction

11 Background The work of this paper is backgrounded byprediction or evaluation and control of mechanical self-noisein sonar array cavity The mechanical self-noise which iscaused by structural vibration of sonar cavityrsquos wall mightsignificantly weaken the detection performance of sonar atlower frequencies [1 2] The sources of mechanical self-noise might be multiple such as vibrating machines on theship which diffuse vibration energy or second excitation ofstructure-borne sound However it is essential to compre-hend the characteristics of interaction between the enclosedwater sound field and its elastic boundary structures for thepurpose of prediction evaluation and control of interiorhydroacoustic noise [3]

The subjects of cabin noise in various flight vehiclesand automobiles are more familiar in the investigation offluid-structural coupled vibration of acoustoelastic enclosurewhich mainly focus on characteristics of sound transmissionthrough the elastic wall into interior sound field resulting

from exterior air-borne sound [4] In these cases weakcoupling has been commonly assumed because of the lowdensity of air and high stiffness of cabin wall which meansthat the cavityrsquos interior sound pressure would have littleinfluence on the vibration of cavity wall and modals ofinterior sound field would also be affected very lightly [5] Incontrast a much stronger coupling might be present when awater sound field takes place of the air [6]

The sound pressure is most commonly used to representthe property of sound field in the study of acoustic-structuralcoupling of acoustoelastic enclosureThe ratio of sound pres-sure at the outside surface of the elastic cavity wallboard tothat at internal surface which is defined as ldquonoise reductionrdquois applied to evaluation of sound transmission characteristics[4 7] Since the sound pressure would change greatly atdifferent points of the soundfield the value of noise reductionwould also be very different and a comprehensive measurefor example power flow would be expected for an overallevaluation of vibration level of the enclosed sound fieldThe power flow has been validated and widely utilized as

HindawiAdvances in Acoustics and VibrationVolume 2018 Article ID 5273280 16 pageshttpsdoiorg10115520185273280

2 Advances in Acoustics and Vibration

a comprehensive measure for evaluation of overall level ofvibration energy of vibration isolation systems mounted onflexible foundations [8] which could also be explained asaverage sound power when applied to sound field analy-sis

In this paper through modal coupling analysis on thefluid-structural vibration of the water-filled enclosure withelastic boundaries an efficient computational method isdeduced to determine the vibratory power flow generatedby exterior excitations on the outside surface of the elasticstructure including the total power flow entering into thefluid-structural coupled system and the net power flowtransmitted into the hydroacoustic field Characteristics ofthe coupled natural frequencies and modals are investigatedby a numerical example of a rectangular water-filled cavitywith five acoustic rigid walls and one elastic panel Influentialfactors of power flow transmission characteristics are furtherdiscussed with the purpose of overall evaluation and reduc-tion of the cavity water sound energy

12 Theoretical Development There has been a continuouseffort for decades on investigation of fluid-structural mech-anism of closed sound field with flexible boundaries It hasbeen recommended that the locally reactive acoustic normalimpedance was the earlier theory to understand the soundabsorption caused by the interaction between a reverberationroom and its surrounding walls [9] Later attention was paidto the modal coupling between the enclosed sound fields andthe flexible walls to reveal the more complicated mechanismdemonstrated by experimental results which could not beinterpreted by the locally reactive theory [10 11]

The modal responses of acoustoelastic enclosures werefirst developed by Dowell et al [12 13] by applying Greenrsquosfunction to the inhomogeneous wave differential equationof the enclosed sound field and applying the classical modaland eigenvalue theorem to the simultaneous fluid-structuraldifferential equations to result in a resolution of couplingmodals There are still other resolution methods for the sameacoustoelasticity equations which could be referred to suchas Laplace transformation [14] and Ritz series [15] In generalDowellrsquos method is based on the familiar uncoupled acousticenclosure modes and structural modes could be moreeasily implemented and has been successfully applied to theinvestigation of variety of fluid-structural interaction systems[16 17] Beginning with the ldquomodal coupling methodrdquo Panand Bies gave an insight analysis of the weak-coupled andwell-coupled modals and their decay characteristics of arectangular panel-cavity coupled system [18 19] Davis putforward a method for approximate estimation of the couplednatural frequencies of acoustoelastic enclosures by ldquocouplingcoefficientrdquo [20]

Other important developments might lie in the field ofdiscrete numerical techniques such as FEMBEM for fluid-structural vibration analysis However these methods areusually preferred in the investigation of irregularly shapedcavities and targeting specific engineering problemsAnd thatwould be beyond the discussion of this paper which wouldmainly focus on a general theoretical evaluation method forthe overall vibration level of a fluid-filled enclosure through

vibratory power flow calculation especially based onDowellrsquosmodal coupling theory

2 Theory

21 Equations of Fluid-Structural Coupled Vibration Con-sider that a fluid-filled enclosure occupies a volume 119881 Itsboundary 119863 = 119863119878 + 119863119865 where 119863119865 = 0 represents the flexi-ble area of the surrounding wall and 119863119878 (might be zero)represents the acoustic rigid area

The fluid inside the enclosure satisfied the wave equationand associated boundary condition

1198700nabla2119901 (120590 119905) minus 1205880 1205972119901 (120590 119905)1205971199052 = 0 (120590 isin 119881) (1)

120597119901 (120590 119905)120597119899 = minus1205880119886119899 (120590 119905) (120590 isin 119863119865) (2)

120597119901 (120590 119905)120597119899 = 0 (120590 isin 119863119878) (3)

where 119901(120590 119905) is the sound pressure at point 120590(119909 119910 119911) isin 119881119886119899(120590 119905) is the acceleration of the flexible wall in the normaldirection 119899 (positive outward) 1205880 and1198700 are the equilibriumfluid density and fluid volume stiffness respectively

If119863119865 = 0 (1) hasmodal solutions119865119860119903(120590)sdotexp(119895120596119860119903119905) 119903 =0 1 2 where120596119860119903 is the 119903th acoustical natural frequency inthe condition of rigid boundary and119865119860119903(120590) is the correspond-ing natural mode with orthogonality as follows

∭119881

119865119860119903 (120590) 119865119860119904 (120590)120588011988820 dV = 0 119903 = 119904119872119860119903 119903 = 119904

∭119881

[nabla119865119860119903 (120590)]119879 sdot [nabla119865119860119904 (120590)]1205880 dV = 0 119903 = 1199041205962119860119903119872119860119903 119903 = 119904

(4)

where 1198880 = radic11987001205880 is the acoustic velocity of the fluid 119872119860119903is the 119903th acoustical modal mass in the condition of rigidboundary and nabla119865119860119903(120590) = [120597119865119860119903120597119909 120597119865119860119903120597119910 120597119865119860119903120597119911]119879 isthe column gradient vector of modal function 119865119860119903(120590)

Consider the solution of (1) with119863119865 = 0 being in the formof modal superposition that is

119901 (120590 119905) = infinsum119903=0

119865119860119903 (120590) 119875119903 (119905) = [F (120590)]119879 P (119905) (120590 isin 119881) (5)

where F(120590) and P(119905) are column vectors of modal function119865119903(120590) and its corresponding modal coordinate 119875119903(119905) respec-tively that is F(120590) = [1198651198600(120590) 1198651198601(120590) 1198651198602(120590) ]119879 and P(119905) =[1198750(119905) 1198751(119905) 1198752(119905) ]119879

After substituting (5) into (1) multiply both sides of theresultant equation with a left-multiplication matrix (vector)F(120590) and finally integrating the equation over volume 119881 oneobtains

∭119881F (120590) [nabla2119901 (120590 119905)] dVminus 111988820∭119881

F (120590) [F (120590)]119879 P (119905) dV = 0 (6)

Advances in Acoustics and Vibration 3

By applying Greenrsquos theorem to the first term of aboveequation one has

111988820∭119881F (120590) [F (120590)]119879 P (119905) dV

+ ∭119881[nablaF (120590)]119879 nablaF (120590)P (119905) dV

= ∯119863F (120590) 120597119901 (120590 119905)120597119899 d119860

(7)

where nablaF(120590) is the gradient matrix of modal function 119865119903(120590)nablaF(120590) = [nabla1198651198600(120590) nabla1198651198601(120590) nabla1198651198602(120590) ]Now substitute the boundary condition equations (2)sim(3)

and orthogonality equation (5) into (7)

M119860 [P (119905) +Ω2119860P (119905)] = minus∯119863F (120590) 119886119899 (120590 119905) d119860 (8)

whereM119860 andΩ119860 are diagonal matrices of acoustical modalmasses and natural frequencies respectively that is M119860 =diag[119872119860011987211986011198721198602 ] andΩ119860 = diag[1205961198600 1205961198601 1205961198602 ]

The flexible boundary of the cavity is assumed to be thin-wall structures where linear partial differential equationswould be adopted to fit the thin-wall structuresrsquo vibrationsuch that

119878119908 (120590 119905) + 119898119861 1205972119908 (120590 119905)1205971199052 = 119901119860 (120590 119905) minus 119901119861 (120590 119905)(120590 isin 119863119865)

(9)

where 119878 is a linear differential operator representing struc-tural stiffness 119898119861 is structural mass per unit area 119901119860(120590 119905)and 119901119861(120590 119905) are excitations on the surface of the thin-wallstructures due to the cavity acoustics and external dynamicalforces (intensity of pressure) respectively 119908(120590 119905) is thedisplacement response of the thin-wall structures which isdefined in the normal direction of 119863119865

The solution of (9) could be expressed as

119908 (120590 119905) = infinsum119895=1

119882119861119895 (120590) 119861119895 (119905) = [W (120590)]119879 B (119905)(120590 isin 119863119865)

(10)

where119882119861119895(120590) is the 119895th modal function that is defined on119863119865and concerned with the property of the thin-wall structuresin vacuo and 119861119895(119905) is the modal coordinate correspondingto 119882119861119895(120590) W(120590) and B(119905) are column vectors of 119882119861119895(120590)and 119861119895(119905) respectively that is W(120590) = [1198821198611(120590)1198821198612(120590)1198821198613(120590) ]119879 and B(119905) = [1198611(119905) 1198612(119905) 1198613(119905) ]119879

By substituting (10) into (9) and using the orthogonalityof 119882119861119895(120590) there would be a modal differential function asfollows

M119861 [B (119905) +Ω2119861B (119905)] = Q119860 (119905) + Q119861 (119905) (11)

where M119861 and Ω119861 expressed as M119861 = diag[119872119861111987211986121198721198613 ] and Ω119861 = diag[1205961198611 1205961198612 1205961198613 ] are diagonal matrices

of the modal masses and natural frequencies respectivelyand 120596119861119895 and 119872119861119895 = ∬

119863119865119898119861[119882119861119895(120590)]2d119860 (119895 = 1 2 3 )

represent the 119895th natural frequency and modal mass of thethin-wall structures in vacuo respectively Q119860(119905) and Q119861(119905)are column vectors of the general forces due to 119901119860(120590 119905)and 119901119861(120590 119905) loaded on the thin-wall structures in vacuorespectively and

Q119860 (119905) = ∬119863119865

W (120590) 119901119860 (120590 119905) d119860Q119861 (119905) = minus∬

119863119865

W (120590) 119901119861 (120590 119905) d119860(12)

The right-hand term of (8) and the first term Q119860(119905) onthe right hand of (11) are of the fluid-structural interactionbetween the sound field inside the cavity and its flexible wallsSubstituting (10) into the right-hand term of (8) and takingnotice of 119886119899(120590 119905) = 1205972119908(120590 119905)1205971199052 at 120590 isin 119863119865 and 119886119899(120590 119905) = 0at 120590 isin 119863119878 one could define a coupling matrix L as follows

L = ∬119863119865

F (120590) [W (120590)]119879 d119860 = [119871119903119895](119903 = 0 1 2 119895 = 1 2 3 )

(13)

where 119871119903119895 denotes the element of the coupling matrix L at the119903th row and the 119895th columnAnd (8) turns into

M119860 [P (119905) +Ω2119860P (119905)] = minusL sdot B (119905) (14)

Dealing with Q119860(119905) one could express 119901119860(120590 119905) in (12)with (5) and (11) would become

M119861 [B (119905) +Ω2119861B (119905)] = L119879 sdot P (119905) + Q119861 (119905) (15)

22 Modal Analysis In order to carry out a modal analysisabout the fluid-structural vibration system governed by (14)and (15) letQ119861(119905) = 0 and suppose that there exist vibrationsolutions as follows

P (119905) = (radicM119860)minus1 sdot 120594119860 sdot exp (119895120596119905) B (119905) = (Ω119861radicM119861)minus1 sdot 120594119861 sdot exp (119895120596119905)

(16)

where radicM119860 = diaglfloorradic1198721198600 radic1198721198601 radic1198721198602 rfloor and radicM119861 =diaglfloorradic1198721198611 radic1198721198612 radic1198721198613 rfloor are square roots of the diago-nal acoustical modal matrix M119860 and the diagonal structuralmodal matrixM119861 respectively 120594119860 = [1205941198600 1205941198601 1205941198602 ]119879 and120594119861 = [1205941198611 1205941198612 ]119879 are column vectors of fluid-structuralcoupled modal shape coefficients related to the cavity soundfield and the flexible boundary structures respectively

Substituting (16) into (14) and (15) an eigenvalue problemcould be obtained as

A120594 = [A11 A12A21 A22

][120594119860120594119861

] = 1205962 [120594119860120594119861

] = 1205962120594 (17)


where 120594 could be named as vector of fluid-structural coupledmodal shape coefficients and A is a symmetric characteristicmatrix and Arsquos partitioned matrices could be calculated byA11 = Ω2119860 + (radicM119860)minus1LMminus1119861 L119879(radicM119860)minus1 A12 = A11987921 =minus(radicM119860)minus1LΩ119861(radicM119861)minus1 and A22 = Ω2119861

Equation (17) would give eigenvalues of matrix A that is1205961198621198962 = Ω1198621198962(1 + 119895120578119862119896) and the accompanying eigenvectors[120594(119896)]119879 = [120594119860(119896)119879 120594119861(119896)119879] = [1205941198600(119896) 1205941198601(119896) 1205941198602(119896) 1205941198611(119896) 1205941198612(119896) 1205941198613(119896) ] 119896 = 0 1 2 where Ω119862119896 corre-sponds to the 119896th fluid-structural natural frequency 120578119862119896is the loss factor associated with the 119896th damped normalmode which might be resulting from the introduction ofa complex stiffness of the flexible boundary or a complexvolume stiffness 1198700 of the fluid in consideration of thedamping properties of the fluid-structural system It shouldalso be noticed that [120594(119896)] might be complex vectors when1205961198621198962 are complex numbers

The fluid-structural coupled modal functions of thecavityrsquos sound field and the flexible boundaries would beexpressed as

119865119862119896 (120590) = [F (120590)]119879 (radicM119860)minus1 120594(119896)119860 (120590 isin 119881) 119882119862119896 (120590) = [W (120590)]119879 (Ω119861radicM119861)minus1 120594(119896)119861 (120590 isin 119863119865)

(18)

23 Vibratory Power FlowTransmission In the condition thatthe flexible boundary structures of the cavity are subjectedto a harmonic exterior excitation that is 119901119861(120590 119905) = 119875119861(120590) sdotexp(119895120596119905) letP119861 = minus∬

119863119865W(120590)sdot119875119861(120590)d119860P119861 indicates that the

amplitudes of general forces belong to the uncoupled flexiblestructures The steady responses of the fluid-structural cou-pling cavity would be

119901 (120590 119905) = [F (120590)]119879 (radicM119860)minus1 Χ119860HTP119861 exp (119895120596119905)= 119875 (120590 120596) sdot exp (119895120596119905) (120590 isin 119881)

(19)

119908 (120590 119905) = [W (120590)]119879 (Ω119861radicM119861)minus1 Χ119861HTP119861 exp (119895120596119905)= 119882 (120590 120596) sdot exp (119895120596119905) (120590 isin 119863119865)

(20)

where 119875(120590 120596) and 119882(120590 120596) denote the amplitudes of theharmonic sound pressure in the cavity and harmonic dis-placement of the thin-wall structuresΧ119860 andΧ119861 arematricescomposed of arrays of eigenvectors of the characteristicmatrix A that is Χ119860 = [120594119860(0)120594119860(1)120594119860(2) ] and Χ119861 =[120594119861(0)120594119861(1)120594119861(2) ] And

H (120596) = (minus1205962M119862 +Ω1198622 sdot M119862)minus1 (21)

T = minusX119867119860 (radicM119860)minus1 LMminus1119861 + X119867119861Ω119861 (radicM119861)minus1 (22)

whereH could be named as the complex frequency responsematrix of the fluid-structural coupled cavity and T is a

transformation matrix to transform the general force P119861into its fluid-structural expression (the derivation of thematrices H and T has been explained via (A5)sim(A8) inAppendix A2) M119862 = X119867119860 sdot Χ119860 + X119867119861 sdot Χ119861 is a diagonalmatrix of the fluid-structural coupled modal masses andΩ119862 = diag[1205961198620 1205961198621 1205961198622 ] is a diagonalmatrix of the fluid-structural coupled natural frequencies The superscript ldquo119867rdquodenotes Hermitian transposition of matrices

The power flow (density) inputted by exterior excitation119901119861(120590 119905) into the fluid-structural system is

119901in (120590 120596) = 1205962120587 int12059621205870

Re 119901119861 (120590 119905)sdot Reminus120597119908 (120590 119905)120597119905 d119905

= minus1205962 Re 119895119882 (120590 120596) sdot 119875lowast119861 (120590) (120590 isin 119863119865) (23)

The total power flow input is

119875in (120596) = ∬119863119865

119901in (120590 119905) d119860 = minus1205962 Re119895P119879119861T119879H (120596)sdot X119879119861 (Ω119861radicM119861)minus1∬

119863119865

W (120590) sdot 119875lowast119861 (120590) d119860 = 1205962sdot Re119895P119879119861T119879H (120596) sdot X119879119861 (Ω119861radicM119861)minus1 Plowast119861

(24)

where the superscript ldquolowastrdquo denotes conjugation of complexnumbers

The power flow (density) transmitted through the fluid-structural interaction boundary of the cavity into theenclosed sound field is

119901tr (120590 120596) = 1205962120587 int12059621205870

Re 119901 (120590 119905)1003816100381610038161003816120590isin119863119865sdot Reminus120597119908 (120590 119905)120597119905

10038161003816100381610038161003816100381610038161003816120590isin119863119865 d119905= 1205962 Re 119895119882lowast (120590 120596) sdot 119875 (120590 120596)|120590isin119863119865

(25)

The total transmission power flow is

119875tr (120596) = ∬119863119865

119901tr (120590 120596) d119860= 1205962 Re∬

119863119865

[F (120590)]119879 sdot Λ (120596) sdot W (120590) d119860

= 1205962 Reinfinsum119903=0

infinsum119895=1

119871119903119895 sdot 120582119903119895 (120596)

(26)

where Λ(120596) = 119895(radicM119860)minus1X119860H(120596) sdotTP119861P119867119861 T

119867[H(120596)]lowastX119867119861 (Ωlowast119861radicM119861)minus1 = [120582119903119895(120596)] might benamed as a power transmission matrix and 120582119903119895(120596) denotesthe element of matrix Λ(120596) at the 119903th row and the 119895thcolumn


Y

X

Z

LY

Lx

LZ

pE

DFDS

0 = 1000 kgＧ3

c0 = 1500 ms

O

Figure 1 A panel-cavity coupled system

3 Numerical Simulation and Analysis

31 Simulation Model A panel-cavity coupled system shownin Figure 1 consists of a rectangularwater-filled roomwith fiverigid walls and one simply supported plate subject to exteriorharmonic distributed force (pressure) 119901119864(120590 119905)

In discussion of the distribution shape of exterior exci-tation the plane harmonic wave incident is a commonassumption Suppose that119901119864(120590 119905) = 119875119864sdotexp[119895(120596119905minus1198961015840119909 sin 120579)]where 119875119864 is the amplitude 1198961015840 is the wave number and 120579is the incident angle (119901119864 is uniform along the 119910 direction)119901119864 has a perpendicular component 119901119861(120590 119905) = 119875119864 cos 120579 sdotexp(minus1198951198961015840119909 sin 120579) sdot exp(119895120596119905) = 119875119861(120590) sdot exp(119895120596119905) Generally119875119861(120590) would be a complex function and have infinite varietyof distribution shapes when the wave frequency velocityand incident angle changed Figure 2 shows one example ofdistribution shape of 119875119861(120590) with wave frequency 119891 = 500Hzvelocity 119888 = 344ms and incident angle 120579 = 1205873 It istrue that the modal coupling method is valid in dealing withthose variant distribution shapes of 119875119861(120590) However somespecific analysis on the special case with 120579 = 0 that is auniform119875119861 over the plate surface would also give indicationsof general significance The uniform excitation had beenadopted by other authors previously [7 15] And moreovertaking sonar array cavities as examples they are regularlymounted on related ship structures through rubber blanketsthe uniform structural excitation assumption would be abasic consideration

For the water-filled rectangular room the natural fre-quencies and modal functions of the sound field with rigidboundaries are determined by

1205962119860119903 = 11987001205880 [(119898119860119903120587119871119909 )2 + (119899119860119903120587119871119910 )2 + (119897119860119903120587119871119911 )2] (27)

119865119860119903 (119909 119910 119911)= cos(119898119860119903120587119909119871119909 ) cos(119899119860119903120587119910119871119910 ) cos(119897119860119903120587119911119871119911 ) (28)

where forall(119898119860119903 119899119860119903 119897119860119903) isin 1198733 119903 isin 119873 and let 120596119860119903 arrange in asequence |12059611986002| lt |12059611986012| lt |12059611986022| lt sdot sdot sdot

For a simply supported plate its natural frequencies andmodal functions are determined by

1205962119861119895 = 1205874119864ℎ212120588 (1 minus 1205832) (11989821198611198951198712119909 + 11989921198611198951198712119910 )2

(29)

119882119861119895 (119909 119910 0) = sin(119898119861119895120587119909119871119909 ) sin(119899119861119895120587119910119871119910 ) (30)

where 119864 120588 ℎ and 120583 are Youngrsquos modulus mass densitythickness and Poissonrsquos ratio of the plate respectivelyforall(119898119861119895 119899119861119895) isin 1198732 119895 isin 119873+ let 120596119861119895 arrange in a sequence|12059611986112| lt |12059611986122| lt |12059611986132| lt sdot sdot sdot

The geometrical and material properties adopted fornumerical computation are as follows 119871119909 = 04m 119871119910 =06m 119871119911 = 07m and ℎ = 0005m 1205880 = 1000 kgm3 and1198700 = 225times109 Patimes(1+10minus4 j)119864 = 20times1011 Patimes(1+10minus3 j)120588 = 78 times 103 kgm3 and 120583 = 028 (steel)32 Modal Analysis There is a convergence investigationabout the fluid-structural coupled natural frequencies result-ing from (17) at first In Table 1 with a fixed number of platemodals involved in calculation the convergence of couplednatural frequencies could be observed by increasing thenumber of water sound modals involved The convergencecould also be observed by increasing the number of platemodals involved in Table 2 It could be suggested that themodal coupling method could achieve good convergence insolving the fluid-structural coupled problem described hereAnd it could also be observed that the convergence at lowerfrequencies is more rapid than that at higher frequenciesand the solution precision would be more dependent onaccounting for more water sound modals However there isno need to carry out a high accuracy calculation here for atheoretical qualitative analysis and in the later part of thispaper 50 plate modals and 500 water sound field modalsare taken into consideration by which totally 550 coupledmodals could be revealed And also because there is no needto list all 550 modes here only partial data (the first severalmodes) are listed in Tables 1 and 2

Figure 3 gives a comparison of sound pressure solutionsbetween the modal coupling approach and FEM (by thesoftware of LMS VirtualLab Acoustics) as a theoretical val-idation verification The differences between the two resultsin Figure 3(a) are due to modal truncation more modes areinvolved in the FEMBEM software

The fluid-structural coupled natural frequencies resultingfrom (17) are compared with those of sound field with rigidboundaries obtained by (27) and simply supported steel plateobtained by (29) which are listed in Table 3 As mentionedabove 550 coupled modals have been obtained by involving50 plate modals and 500 water sound field modals in modalcoupling calculation but only the first 11 coupled modalfrequencies are presented

As a whole it could be concluded that the fluid-structuralfrequencies are very different from those of the water soundfield (with rigid boundaries) and the flexible boundary plate(in vacuo) which means that there would be a strong


Table 1 Coupled natural frequencies with different number of water sound modals involved

Fluid-structural coupled natural frequency 119891119862119896 (Hz)Number of water sound modals involved 50 200 500 1000 2000Number of plate modals involved 50 50 50 50 50119896

0 0 0 0 0 01 1068 1031 1015 1009 10032 2185 2084 2060 2039 20253 2344 2227 2176 2155 21344 2911 2709 2647 2622 25975 4476 4151 4047 4002 39596 4554 4207 4091 4045 39987 4761 4573 4518 4471 44398 5619 5431 5372 5334 53059 6267 5815 5662 5590 552310 6825 6177 5994 5921 5847

Table 2 Coupled natural frequencies with different number of plate modals involved

Fluid-structural coupled natural frequencies 119891119862119896 (Hz)Number of water sound modals involved 500 500 500 500 500Number of plate modals involved 25 50 100 200 500119896

0 0 0 0 0 01 1015 1031 1015 1015 10152 2061 2084 2060 2060 20603 2176 2227 2175 2175 21754 2647 2709 2647 2647 26475 4047 4151 4046 4046 40466 4092 4207 4091 4091 40917 4518 4573 4516 4516 45168 5372 5431 5371 5371 53719 5662 5815 5660 5660 566010 5995 6177 5993 5993 5993

Table 3 Natural frequencies of the water sound field plate and fluid-structural coupled cavity

Natural frequencies of water sound field Natural frequencies of plate Fluid-structural couplednatural frequencies

119903 (119898119860119903 119899119860119903 119897119860119903) 119891119860119903 (Hz) 119895 (119898119861119895 119899119861119895) 119891119861119895 (Hz) 119896 119891119862119896 (Hz)0 (0 0 0) 0 0 01 (0 0 1) 10714 1 (1 1) 1080 1 10152 (0 1 0) 12500 2 (1 2) 2076 2 20603 (0 1 1) 16463 3 (2 1) 3322 3 21764 (1 0 0) 18750 4 (1 3) 3737 4 26475 (0 0 2) 21429 5 (2 2) 4319 5 40476 (1 0 1) 21595 6 (2 3) 5980 6 40917 (1 1 0) 22535 7 (1 4) 6063 7 45188 (0 1 2) 24808 8 (3 1) 7059 8 53729 (1 1 1) 24952 9 (3 2) 8056 9 566210 (0 2 0) 25000 10 (2 4) 8305 10 5994


010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part

Figure 2 One example of nonuniform distribution shape of 119875119861(120590) (wave frequency 119891 = 500Hz velocity 119888 = 344ms and incident angle120579 = 1205873)

matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

(a) Comparison of sound pressure solutions

Pressure (nodal values)1Occurrence 540

649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)

(b) Cloud picture of boundary sound pressure by VL(540Hz)

Figure 3 Comparison of sound pressure solutions between the modal coupling approach and FEM

coupling between the enclosed water sound field and itselastic surrounding structures If the fluid in the cavity wasair (1205880 asymp 129 kgm3 1198880 asymp 344ms) it could be found thatthe coupled natural frequencies 119891119862119896 would approximately beequal to either someuncoupled structural natural frequencies119891119861119895 or some uncoupled cavity acoustical natural frequencies119891119860119903 and that is a situation of weak coupling The motionof each subsystem in a weakly coupled system will not beessentially different from that of the uncoupled systemsHowever if the density of the medium in the cavity is muchdenser than air such as water the coupling may turn out tobe strong and big deformation of the resulting modes fromthe uncoupled panel and the cavity modes may be expected[18] In this sense the strong coupling could be judged by any

unneglectable departure of coupled natural frequencies fromevery natural frequency of the uncoupled flexible structuresand cavity

Except 1198911198600 = 0 the natural frequencies of the watersound field are much higher than those of the simplysupported plate and the fluid-structural coupled naturalfrequencies are inclined to come to be rather lower Itseemed that one might carry out a comparison betweenthe coupled natural frequencies and those of plate and thechange regulation of differences of adjacent fluid-structuralnatural frequencies119891119862(119896+1)minus119891119862119896 is similar to those119891119861119895minus119891119861(119895minus1)of the plate However the frequency distribution of couplednatural frequencieswould turn to be lower andmore crowdedas the order 119896 increased

8 Advances in Acoustics and VibrationW

B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz

Figure 4 Some uncoupled plate modals

In order to reveal modal coupling characteristics Figures4 and 5 show several uncoupled plate modals 119882119861119895(120590) whichare expressed by (30) and fluid-structural coupled platemodals119882119862119896(120590)which are determined by (29) throughmodalcoupling calculationThe figure shows that1198821198620 is almost thesame as 1198821198611 while it is true according to 1198821198620rsquos expressionand that is just an example of weak-coupledmodal It seemedthat the coupledmodal shape1198821198623 is similar to the uncoupledmodal shape 1198821198614 however they are quite different in factaccording to the expression of 1198821198623 and that is a strongcoupled modal Phenomena of strong modal coupling areobvious when inspecting 1198821198627 and 1198821198628 shown in Figure 5

The similar couplings have also happened to the acous-tical cavity modals And moreover except that the coupledacoustical modal 1198651198620 is practically equal to 1198651198600 that is therigid body modal of the uncoupled cavity sound field allother fluid-structural coupled acoustical cavity modals arecomposed in a strong couplingmanner that is they are linearcombinations of several119865119860119903 the uncoupled acousticalmodalsof the rigid wall cavity As 119865119862119903(120590) is defined in three-dimen-sional space and it is inconvenient to plot it by a planar figureFigure 6(a) illustrates one coupled acoustical modal shape in

the plane 119911 = 1198711199112 and Figure 6(b) illustrates the appur-tenant participant coefficients of the uncoupled acousticalcavity modals in the constitution of coupled acoustical cavitymodal

33 Power Flow Transmission In the simulation model ofFigure 1 the vibratory power flow inputted by exteriorexcitation into the whole fluid-structural coupled system andthe enclosed water sound field that is 119875in calculated by (24)and 119875tr calculated by (26) is dissipated by system dampingAnd thus the higher or lower power flow level would be acomprehensive indicator to measure the vibration level orenergy level of the panel-cavity coupled system and watersound field

Figure 7 shows the spectrum of input power flow 119875in andtransmitted power flow119875tr inwhich the drop between119875in and119875tr is the dissipation power of the platersquos damping Because theexterior excitation is symmetric (uniform 119875119861 as mentionedin Section 31) only symmetric modals are present and thespectrumpeaks at 0Hz 2176Hz 4518Hz and 5372Hz couldbe associated with the modals shown in Figures 5 and 6There is some similarity between 119875in or 119875tr and the water


010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz

(c) 1198821198627 asymp 00074(minus0281198821198611 minus 0261198821198614 +1198821198618 + 01411988211986111 + 00711988211986112)

WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0

(d) 1198821198628 asymp 00062(minus0111198821198611 minus 0161198821198614 minus1198821198618 + 01811988211986111 + 00811988211986112)

Figure 5 Some coupled plate modals

86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104

10 20 30 40 50 60 70 80 90 1000Modal serial number

minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents

(b) Participant coefficients of the uncoupled cavity modals

Figure 6 Simple illustration of one coupled cavity modal shape


minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)

Figure 7 Spectrum of 119875in and 119875tr

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)

Figure 8 Spectrum of water sound pressure

sound pressure at the center of the platersquos interior surfacethat is 119875((1198711199092 1198711199102 0) 120596) (refer to (21)) which is shown inFigure 8 However the power flow is of evaluation of soundpower

As a theoretical investigation hypothesize that the mate-rial property parameters of the plate that is Youngrsquos modulus119864 mass density 120588 Poissonrsquos ratio 120583 and damping loss factorcould be altered independently In Figure 9(a) transmittedpower flows are compared under different Youngrsquos modulusof the elastic plate where the value of 724 times 1010 Pa is byreference to aluminium When the platersquos elasticity modulusdecreases the platersquos natural frequencies would decreasesimultaneously and thiswouldmove the fluid-structural cou-pled natural frequencies into lower frequency ranges which

has been predicted by Table 3Thus the alteration of elasticitymodulus of plate would lead to a phenomenon of ldquofrequencyshiftingrdquoThe119875tr spectrumwith smaller platersquos elasticitymod-ulus could be regarded as a contraction of that with greaterelasticity modulus toward lower frequencies and the peaksof 119875tr would occur at relatively lower frequencies and becomemore crowded However since the variation range of Youngrsquosmodulus would be limited in practice its influence on powerflow transmissionmight not be very serious and it could alsobe observed that reduction of Youngrsquos modulus might bringabout a benefit of slight reduction of 119875trrsquos peak valleys

In order to make an inspection of the alteration of differ-ences between 119875in and 119875tr with different plate elasticity mod-ulus Figure 9(b) shows the spectra of power flow ratio PR =10 log(119875in119875tr) The peaks of PR would always appear aroundthe resonance frequencies except at1198911198620 which could be easilyexplained by the fact that the platersquos damping consumes moreenergy when system resonances take place In the lowestfrequency range around 1198911198620 = 0 the plate consumes littleenergy and therefore119875tr asymp 119875in and PR asymp 0 It should be notedthat greater PRs might not imply lower levels of transmittedpower flow 119875tr the fact is probably just the opposite becausethe power flow input 119875in might be in much higher levels atthe same time In this sense minority of PR peak numberswould be a good design for noise isolation which requires agreater plate elasticity modulus And instead high PR valuesbetween adjacent resonance peaks of the PR spectra would beof real benefit for the purpose of 119875tr attenuation which couldbe discovered through a synthesized analysis of the figuresshown

In Figure 10(a) different mass density values were givento the elastic plate in which the value of 2770 kgm3 is byreference to aluminium A significant feature of the spectra inthe figure is that changes in platersquos mass density would appar-ently change the average level of transmitted power flow 119875trAccording to (29) the increase of platersquos mass density wouldcause decreasing of platersquos natural frequencies and ldquofrequencyshiftingrdquo of fluid-structural coupled modals as shown in Fig-ure 10(a) similar to the situation of decreasing Youngrsquos mod-ulus in Figure 9 However the increase of platersquos mass densitywould increase its modal masses at the same time Andthat is the reason why 119875trrsquos average level is cut down eventhough more resonance modals would come into being inthe relative lower frequency band It could also be explainedby the fact that a heavier vibrating mass would generatea greater reduction in dynamic force (or pressure) trans-mission Figure 10(b) is about the power flow ratio PR PRcould not effectively reveal the apparent 119875tr reduction atnonresonance frequencies such as 800Hzsim1200Hz becausethe transmitted power flow 119875tr is very close to the power flowinput 119875in at those frequencies

Another important influential factor that should be paidattention is the damping loss factor of the plate Figure 11(a)demonstrates a conflictive situation where smaller dampingloss factor would increase the peaks of transmitted powerflow at resonance frequencies while at the broadband non-resonance frequencies smaller damping would be beneficialto reduction of transmitted power flow To explain this resultone might make an analogy with the vibration isolation


200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

E = 724 times 1010 ０ times (1 + 10minus3 Ｄ)



(a) Transmitted power flow

minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

E = 724 times 1010 ０ times (1 + 10minusE = 20 times 1011 ０ times (1 + 10minus3 Ｄ)

3 Ｄ)

(b) Power flow ratio

Figure 9 Effect of platersquos elasticity modulus on power flow transmission

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ

= 1 times 104 ＥＡＧ3

3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3


Figure 10 Effect of platersquos mass density on power flow transmission

theory If the elastic plate was considered as some kind ofelastic isolator which was specially designed to attenuate thetransmission of exterior excitation energy into the water-filled cavity the damping would increase the power trans-mission and would not be expected except for attenuationof resonance peak Through illustration of power flow ratioPR as that in Figure 11(b) it could be confirmed that greaterdamping could be used to obstruct energy transmissionat resonance frequencies whereas smaller damping would

be favorable in the nonresonance frequency ranges whichwould not be counted in power flow ratio

Poissonrsquos ratio 120583would affect the power flow transmissionthe same way Youngrsquos modulus does as shown in Figure 9Referring to (29) increasingYoungrsquosmodulus could be equiv-alent to increasing Poissonrsquos ratio However the variationscope of Poissonrsquos ratio is much smaller than that of Youngrsquosmodulus and the power flow transmission would be affectedmore byYoungrsquosmodulus than byPoissonrsquos ratio And for this


200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)




Figure 11 Effect of damping loss factor of plate on power flow transmission

reason no additional repetitive figures would be put forwardhere

4 Conclusions

Backgrounded by evaluation or control of mechanical self-noise in sonar array cavity transmitted power flow or soundpower input calculation is carried out by modal couplinganalysis on the fluid-structural vibration of the fluid-filledenclosure with elastic boundaries Power flow transmissionanalysis is presented through a numerical simulation exam-ple of water-filled rectangular panel-cavity coupled systemDetailed discussion is carried out about power flow transmis-sion characteristics affected by variation of material propertyparameters of cavityrsquos elastic boundary structure aiming atreduction of water sound level inside the cavity From theresults one could draw the following conclusions

(1) Power flow or sound power transmission analysiscould be a valuable method for evaluation or prediction ofwater sound level in dealing with strong coupled vibrationproblems of water-filled acoustoelastic enclosure systems

(2) The fluid-structural coupled natural frequencies of awater-filled acoustoelastic enclosurewould be greatly affectedby interaction between the hydroacoustic field and its sur-rounding elastic boundaries There would be a tendency thatthe fluid-structural coupled natural frequencies turn to besmaller in value and more crowded in frequency distributionthan those of elastic boundary structures and inner watersound field in rigid boundary condition

(3) Decreasing elasticity modulus or Poisonrsquos ratio ofwater sound cavityrsquos thin-wall structures would cause thefrequency distribution of system modals to contract towardlower frequency ranges and result in more power flowtransmission peaks in the lower frequency band while slight

reduction of peak valleys of transmitted power flowcould alsobe expected on the other hand

(4) Denser material could be beneficial to an apparentattenuation of average level of power flow transmissioninto the water-filled enclosure even though it would beaccompanied with a decrease of system natural frequencies

(5) Smaller inner damping of enclosurersquos thin-wall struc-tures which were distinguished from that specially designedfor sound absorption destination and only for the purposeof suppression of resonance peaks might be proposed toattenuate the average level of power flow transmission

Appendix

A Derivation Procedures of RelatedMatrices Equations

A1 Eigenvalue Problem Equations (14) and (15) could beeasily verified to be equivalent to the following simultaneousmodal differential equations which are fundamental inDow-ellrsquos modal coupling method

119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865

119882119861119895119901119861d119860 and allother symbols have been mentioned previously


Through some simple algebraic algorithm (14) and (15)could be grouped together and written as

[PB] + [Ω2119860 + Mminus1119860 LM

minus1119861 L119879 minusMminus1119860 LΩ2119861

minusMminus1119861 L119879 Ω2119861

][PB]

= [minusMminus1119860 LMminus1119861 Q119861Mminus1119861 Q119861

] (A2)

While carrying out a modal analysis one could simply letP = 1205941015840119860 sdot exp(119895120596119905) B = 1205941015840119861 sdot exp(119895120596119905) andQ119861 = 0 thus

1205962 [12059410158401198601205941015840119861

] = [Ω2119860 + Mminus1119860 LMminus1119861 L119879 minusMminus1119860 LΩ2119861

minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)

The above equation raises an eigenvalue problem but itis nonstandard because the matrix A1015840 is asymmetric Thetransformation expressed by (16) would result in a standardeigenvalue problem which has been expressed by (17) wherethe matrix A is symmetric And moreover one could provethat

A = [(radicM119860)minus1 L (radicM119861)minus1 Ω119860minusΩ119861 O][(radicM119861)minus1 L119879 (radicM119860)minus1 minusΩ119861

Ω119860 O]

= [(radicM119860)minus1 L (radicM119861)minus1 Ω119860minusΩ119861 O][(radicM119860)minus1 L (radicM119861)minus1 Ω119860minusΩ119861 O

]119879 (A4)

whereO denotes a zero matrix with proper dimension rankBefore any damping factors are introduced (A4) proves

that the matrix A is positive semidefinite that is all itseigenvalues would be nonnegative That would provide amathematical guarantee that the eigenvalue calculation ofmatrixAwould not fail at any occurrences of negative squarenatural frequencies

By substituting any 119896th eigenvalue of matrix A and itsaccompanying eigenvector which have been symbolled with1205961198621198962 and [120594(119896)]119879 = [120594119860(119896)119879 120594119861(119896)119879] respectively into (16)(ie let 120596 = 120596119862119896 and 120594119860 = 120594119860(119896) and 120594119861 = 120594119861(119896)) the 119896thfluid-structural coupled modal of the cavity sound field andits flexible boundary structures which have been symbolledwith 119865119862119896 and119882119862119896 respectively are obtained by taking out thetime-independent part of P(119905) and B(119905) that is (18)A2 Harmonic Solution In the situation of forced vibrationthat is Q119861 = 0 there is a standard decoupling procedurefor (A2) by applying an eigenmatrix X119867 = [X119860119867X119861119867]which is composed of the series of Arsquos eigenvectors wherethe detailed definitions of X119860 and X119861 have been mentionedpreviously in Section 22 after (19) and (20) and note thatHermitian transposition of matrices has been employed herein consideration of the introduction of damping loss factorsReplacing the real elasticity modulus of the flexible bound-ary structures or real volume stiffness of the cavity soundfield with a complex elasticity modulus or complex volumestiffness is a regular method when damping factors are to beinvolved in analysis which has been declared in Sections 2231 and 33 the method has also been adopted by many otherauthors

Now let

[P (119905)B (119905)] = [

[(radicM119860)minus1 O

O (Ω119861radicM119861)minus1]]

[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)

where 120572(119905) is a column vector variable to take place of P(119905)and B(119905)

After substituting the above equation into (A2) multiplyboth sides of the resultant equation with a left-multiplicationmatrix X119867Tminus10 one has

X119867Tminus10 T0X sdot + X119867Tminus10 A1015840T0X sdot 120572

= X119867Tminus10 [minusMminus1119860 LMminus1119861 Q119861Mminus1119861 Q119861

] (A6)

Take notice that Tminus10 A1015840T0 = A and X119867X and X119867AX are

all diagonalmatrices according to the theory of orthogonalityof eigenvectors In fact the diagonal elements of matrixM119862 = X119867X are just the modal masses of the fluid-structuralcoupled modals and the diagonal elements of matrix Ω2119862 =Mminus1119862 (X119867AX) are the complex fluid-structural coupled naturalfrequencies these two matrices have already been defined inSection 22 after (20) With this knowledge (A2) or (A6) isturned into its decoupled expression

+Ω2119862120572 = Mminus1119862 X119867Tminus10 [minusMminus1119860 LMminus1119861 Q119861

Mminus1119861 Q119861] (A7)

Obviously the transient or steady solution of the aboveequation under arbitrary deterministic Q119861(119905) could beobtained by the method of convolution integral Only har-monic solution is concerned here Referring to (12) when119901119861(120590 119905) is harmonic Q119861(119905) is harmonic too And supposethat Q119861(119905) = P119861 sdot exp(119895120596119905) and 120572(119905) = 120572 exp(119895120596119905) by sub-stituting them into (A7) the time-dependent part exp(119895120596119905)


would be eliminated and the time-independent part of 120572(119905)is obtained that is

120572 = (minus1205962I +Ω2119862)minus1Mminus1119862 X119867Tminus10 [minusMminus1119860 LMminus1119861 P119861Mminus1119861 P119861

]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O

O Ω119861radicM119861][minusMminus1119860 LMminus1119861 P119861

Mminus1119861 P119861]

= H (120596)sdot [minusX119867119860 (radicM119860)minus1 LMminus1119861 + X119867119861Ω119861 (radicM119861)minus1]P119861= H (120596) sdot T sdot P119861

(A8)

where I is a unit matrix of proper dimension rank

A3 Power Flow Formulation The basic formulation forpower flow computation is

119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)

sdot Re 119881 exp (119895120596119905 + 119895120593) d119905 = 12119865119881 cos120593= 12Re [119865 exp (119895120596119905)]lowast [119881 exp (119895120596119905 + 119895120593)]= 12Re [119865 exp (119895120596119905)] [119881 exp (119895120596119905 + 119895120593)]lowast

(A9)

where 119865 cos120596119905 or 119865 exp(119895120596119905) is a harmonic excitation force119881 cos120596119905 or119881 exp(119895120596119905+119895120593) is the harmonic velocity responseat the action point of the excitation force and 120593 is the phasedifference between the excitation force and velocity response

By applying the above formulation (23) would be appar-ent And because 119875119861(120590) is of 120590-dependent distribution over119863119865 119901in(120590 120596) in (23) is of distribution of power flow densityEquation (25) is an integral of 119901in over 119863119865 to count up thetotal power flowThe integral is initiated by quotation of (22)and (23) as follows

119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865

1205962 Re 119895 [119882 (120590 120596)]119879 sdot 119875lowast119861 (120590) d119860= minus1205962 Re∬

119863119865

119895119879119861T119879 [H (120596)]119879

sdot X119879119861 [(Ω119861radicM119861)minus1]119879W (120590) sdot 119875lowast119861 (120590) d119860 (A10)

Note that in the last expression of the right hand ofthe above equation the matrices P119861 T H(120596) X119861 Ω119861 andM119861 are all 120590-independent only W(120590) sdot 119875lowast119861 (120590) would join inthe integral in (A10) The integral would result in (P119861)lowast inwhich P119861 has been defined in the beginning paragraph ofSection 23 and the superscript ldquolowastrdquo denotes conjugation ofcomplex numbers Other matters need attention H(120596) Ω119861andM119861 are all diagonalmatrices that is [H(120596)]119879 = H(120596) and[(Ω119861radicM119861)minus1]119879 = (Ω119861radicM119861)minus1 After taking into account allabove factors (24) is realized

The obtainment of (25) is similar to that of (23) in whichthe cavity sound pressure at the interior surface of the thin-wall structures takes place of the exterior excitation And alsothe integral of119901tr(120590 120596) is processed similar to that of119901in(120590 120596)at the beginning by quotation of (22) and (25)

119875tr (120596) = ∬119863119865

1205962 Re 119895119882lowast (120590 120596) sdot 119875 (120590 120596)|120590isin119863119865 d119860= ∬119863119865

1205962 Re 119895119875 (120590 120596)sdot [119882 (120590 120596)]11986710038161003816100381610038161003816120590isin119863119865 d119860 = 1205962sdot Re∬

119863119865

119895 [F (120590)]119879 (radicM119860)minus1

sdot Χ119860HTP119861P119867119861 T119867H119867X119867119861 [(Ω119861radicM119861)minus1]119867

sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879

sdot [119895 (radicM119860)minus1

sdot Χ119860HTP119861P119867119861 T119867HlowastX119867119861 (Ωlowast119861radicM119861)minus1]

sdot W (120590) d119860

(A11)

The basis of the above transformation includes the fol-lowing W(120590) is real H and Ω119861 are diagonal and M119861 is realand diagonal It could be examined that the product of thematrices between [F(120590)]119879 andW(120590) in the last expression ofthe right hand of the above equation which has been definedas Λ(120596) in (26) is 120590-independent And the expansion of theproduct [F(120590)]119879Λ(120596)W(120590) is

F119879ΛW = [1198651198600 (120590) 1198651198601 (120590) 1198651198602 (120590) sdot sdot sdot] [[[12058211 (120596) 12058212 (120596) sdot sdot sdot12058221 (120596) 12058222 (120596) sdot sdot sdot d

]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)


Table 4 Dimension ranks of related matrices with R cavity acoustical modals and N thin-wall structural modals

Matrices Dimension rank Matrices Dimension rank Matrices Dimension ranknabla119865119860119903 3 times 1 F 119877 times 1 P 119877 times 1nablaF 3 times 119877 M119860 119877 times 119877 Ω119860 119877 times 119877W 119873 times 1 B 119873 times 1 M119861 119873 times 119873Ω119861 119873 times 119873 Q119860 119873 times 1 Q119861 119873 times 1L 119877 times 119873 120594119860120594119860(119896)1205941015840119860 119877 times 1 120594119861120594119861(119896)1205941015840119861 119873 times 1120594120594(119896) (119877 + 119873) times 1 AA1015840 (119877 + 119873) times (119877 + 119873) A11 119877 times 119877A12 119877 times 119873 A21 119873 times 119877 A22 119873 times 119873X119860 119877 times (119877 + 119873) X119861 119873 times (119877 + 119873) X (119877 + 119873) times (119877 + 119873)S (119877 + 119873) times 1 H (119877 + 119873) times (119877 + 119873) T (119877 + 119873) times 119873S119876 119873 times 1 M119862 (119877 + 119873) times (119877 + 119873) Ω119862 (119877 + 119873) times (119877 + 119873)P119861 119873 times 1 Λ 119877 times 119873 T0 (119877 + 119873) times (119877 + 119873)120572120572 (119877 + 119873) times 1

Since 120582119903119895(120596) is 120590-independent when the integral of[F(120590)]119879Λ(120596)W(120590) over119863119865 is proceeding only 119865119860119903(120590)119882119861119895(120590)is involved and that would result in 119871119903119895 which has beendefined in (13) or (A1) And at last 119875tr(120596) is formulated by(26)

B Dimension Ranks of Related Matrices underModal Truncation

Modal truncation has to be implemented in the applicationof modal coupling method because there are no numericaltechniques that could provide a computation in infinitymanner by nowAnd because of themodal convergence prop-erty modal truncation errors could be controlled properlySuppose that there are totally 119877 cavity acoustical modals and119873 thin-wall structural modals to be accounted for the fluid-structural modal coupling dimension ranks of associatedmatrices mentioned in this paper are listed in Table 4

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] H Guo D Luo M Chen and F Zhou ldquoPrediction methodfor low frequency of self-noise in submarinersquos fore body sonarplatform areardquo Ship Science and Technology vol 27 no 4 pp74ndash77 2005

[2] M Yu J Ye Y Wu and S Lu ldquoPrediction and control methodof self-noise in shiprsquos sonar domesrdquo Journal of Ship Mechanicsvol 6 no 5 pp 80ndash94 2002

[3] T Musha and T Kikuchi ldquoNumerical calculation for determin-ing sonar self noise sources due to structural vibrationrdquoAppliedAcoustics vol 58 no 1 pp 19ndash32 1999

[4] S Narayanan andR L Shanbhag ldquoSound transmission throughelastically supported sandwich panels into a rectangular enclo-surerdquo Journal of Sound and Vibration vol 77 no 2 pp 251ndash2701981

[5] G B Warburton ldquoVibration of a cylindrical shell in an acousticmediumrdquo Journal of Mechanical Engineering Science vol 3 no1 pp 69ndash79 1961

[6] J-M David andMMenelle ldquoValidation of a medium-frequen-cy computational method for the coupling between a plate anda water-filled cavityrdquo Journal of Sound and Vibration vol 265no 4 pp 841ndash861 2003

[7] J Yang X Li and J Tian ldquoTheoretical analysis of sound trans-mission through viscoelastic sandwich panel backed by acavityrdquo ACTA Acustica vol 24 no 6 pp 617ndash626 1999

[8] J Yang Y P Xiong and J T Xing ldquoVibration power flow andforce transmission behaviour of a nonlinear isolator mountedon a nonlinear baserdquo International Journal of Mechanical Sci-ences vol 115-116 pp 238ndash252 2016

[9] P M Morse ldquoSome aspects of the theory of room acousticsrdquoThe Journal of the Acoustical Society of America vol 11 no 1 pp56ndash66 1939

[10] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquoThe Journal of the Acoustical Society of America vol 91no 2 pp 949ndash956 1992

[11] S R Bistafa and J W Morrissey ldquoNumerical solutions of theacoustic eigenvalue equation in the rectangular roomwith arbi-trary (uniform) wall impedancesrdquo Journal of Sound and Vibra-tion vol 263 no 1 pp 205ndash218 2003

[12] E H Dowell G F Gorman III and D A Smith ldquoAcoustoe-lasticity General theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[13] E H Dowell ldquoComment on lsquoOn Dowellrsquos simplification foracoustic cavity-structure interaction and consistent alterna-tivesrsquordquoThe Journal of the Acoustical Society of America vol 133no 3 pp 1222ndash1224 2013

[14] R W Guy ldquoThe response of a cavity backed panel to externalairborne excitation A general analysisrdquo The Journal of theAcoustical Society of America vol 65 no 3 pp 719ndash731 1979

[15] J H Ginsberg ldquoDerivation of a Ritz series modeling techniquefor acoustic cavity-structural systems based on a constrainedHamiltons principlerdquo The Journal of the Acoustical Society ofAmerica vol 127 no 5 pp 2749ndash2758 2010


[16] V B Bokil and U S Shirahatti ldquoA technique for the modalanalysis of sound-structure interaction problemsrdquo Journal ofSound and Vibration vol 173 no 1 pp 23ndash41 1994

[17] N Kang and A Raman ldquoAeroelastic flutter mechanisms of aflexible disk rotating in an enclosed compressible fluidrdquo Journalof Applied Mechanics vol 71 no 1 pp 120ndash130 2004

[18] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo The Journal ofthe Acoustical Society of America vol 87 no 2 pp 691ndash707 1990

[19] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashexperimental partrdquo The Journalof the Acoustical Society of America vol 87 no 2 pp 708ndash7171990

[20] R B Davis ldquoA simplified approach for predicting interactionbetween flexible structures and acoustic enclosuresrdquo Journal ofFluids and Structures vol 70 pp 276ndash294 2017

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a comprehensive measure for evaluation of overall level ofvibration energy of vibration isolation systems mounted onflexible foundations [8] which could also be explained asaverage sound power when applied to sound field analy-sis

In this paper through modal coupling analysis on thefluid-structural vibration of the water-filled enclosure withelastic boundaries an efficient computational method isdeduced to determine the vibratory power flow generatedby exterior excitations on the outside surface of the elasticstructure including the total power flow entering into thefluid-structural coupled system and the net power flowtransmitted into the hydroacoustic field Characteristics ofthe coupled natural frequencies and modals are investigatedby a numerical example of a rectangular water-filled cavitywith five acoustic rigid walls and one elastic panel Influentialfactors of power flow transmission characteristics are furtherdiscussed with the purpose of overall evaluation and reduc-tion of the cavity water sound energy

12 Theoretical Development There has been a continuouseffort for decades on investigation of fluid-structural mech-anism of closed sound field with flexible boundaries It hasbeen recommended that the locally reactive acoustic normalimpedance was the earlier theory to understand the soundabsorption caused by the interaction between a reverberationroom and its surrounding walls [9] Later attention was paidto the modal coupling between the enclosed sound fields andthe flexible walls to reveal the more complicated mechanismdemonstrated by experimental results which could not beinterpreted by the locally reactive theory [10 11]

The modal responses of acoustoelastic enclosures werefirst developed by Dowell et al [12 13] by applying Greenrsquosfunction to the inhomogeneous wave differential equationof the enclosed sound field and applying the classical modaland eigenvalue theorem to the simultaneous fluid-structuraldifferential equations to result in a resolution of couplingmodals There are still other resolution methods for the sameacoustoelasticity equations which could be referred to suchas Laplace transformation [14] and Ritz series [15] In generalDowellrsquos method is based on the familiar uncoupled acousticenclosure modes and structural modes could be moreeasily implemented and has been successfully applied to theinvestigation of variety of fluid-structural interaction systems[16 17] Beginning with the ldquomodal coupling methodrdquo Panand Bies gave an insight analysis of the weak-coupled andwell-coupled modals and their decay characteristics of arectangular panel-cavity coupled system [18 19] Davis putforward a method for approximate estimation of the couplednatural frequencies of acoustoelastic enclosures by ldquocouplingcoefficientrdquo [20]

Other important developments might lie in the field ofdiscrete numerical techniques such as FEMBEM for fluid-structural vibration analysis However these methods areusually preferred in the investigation of irregularly shapedcavities and targeting specific engineering problemsAnd thatwould be beyond the discussion of this paper which wouldmainly focus on a general theoretical evaluation method forthe overall vibration level of a fluid-filled enclosure through

vibratory power flow calculation especially based onDowellrsquosmodal coupling theory

2 Theory

21 Equations of Fluid-Structural Coupled Vibration Con-sider that a fluid-filled enclosure occupies a volume 119881 Itsboundary 119863 = 119863119878 + 119863119865 where 119863119865 = 0 represents the flexi-ble area of the surrounding wall and 119863119878 (might be zero)represents the acoustic rigid area

The fluid inside the enclosure satisfied the wave equationand associated boundary condition

1198700nabla2119901 (120590 119905) minus 1205880 1205972119901 (120590 119905)1205971199052 = 0 (120590 isin 119881) (1)

120597119901 (120590 119905)120597119899 = minus1205880119886119899 (120590 119905) (120590 isin 119863119865) (2)

120597119901 (120590 119905)120597119899 = 0 (120590 isin 119863119878) (3)

where 119901(120590 119905) is the sound pressure at point 120590(119909 119910 119911) isin 119881119886119899(120590 119905) is the acceleration of the flexible wall in the normaldirection 119899 (positive outward) 1205880 and1198700 are the equilibriumfluid density and fluid volume stiffness respectively

If119863119865 = 0 (1) hasmodal solutions119865119860119903(120590)sdotexp(119895120596119860119903119905) 119903 =0 1 2 where120596119860119903 is the 119903th acoustical natural frequency inthe condition of rigid boundary and119865119860119903(120590) is the correspond-ing natural mode with orthogonality as follows

∭119881

119865119860119903 (120590) 119865119860119904 (120590)120588011988820 dV = 0 119903 = 119904119872119860119903 119903 = 119904

∭119881

[nabla119865119860119903 (120590)]119879 sdot [nabla119865119860119904 (120590)]1205880 dV = 0 119903 = 1199041205962119860119903119872119860119903 119903 = 119904

(4)

where 1198880 = radic11987001205880 is the acoustic velocity of the fluid 119872119860119903is the 119903th acoustical modal mass in the condition of rigidboundary and nabla119865119860119903(120590) = [120597119865119860119903120597119909 120597119865119860119903120597119910 120597119865119860119903120597119911]119879 isthe column gradient vector of modal function 119865119860119903(120590)

Consider the solution of (1) with119863119865 = 0 being in the formof modal superposition that is

119901 (120590 119905) = infinsum119903=0

119865119860119903 (120590) 119875119903 (119905) = [F (120590)]119879 P (119905) (120590 isin 119881) (5)

where F(120590) and P(119905) are column vectors of modal function119865119903(120590) and its corresponding modal coordinate 119875119903(119905) respec-tively that is F(120590) = [1198651198600(120590) 1198651198601(120590) 1198651198602(120590) ]119879 and P(119905) =[1198750(119905) 1198751(119905) 1198752(119905) ]119879

After substituting (5) into (1) multiply both sides of theresultant equation with a left-multiplication matrix (vector)F(120590) and finally integrating the equation over volume 119881 oneobtains

∭119881F (120590) [nabla2119901 (120590 119905)] dVminus 111988820∭119881

F (120590) [F (120590)]119879 P (119905) dV = 0 (6)



111988820∭119881F (120590) [F (120590)]119879 P (119905) dV


= ∯119863F (120590) 120597119901 (120590 119905)120597119899 d119860

(7)



M119860 [P (119905) +Ω2119860P (119905)] = minus∯119863F (120590) 119886119899 (120590 119905) d119860 (8)



119878119908 (120590 119905) + 119898119861 1205972119908 (120590 119905)1205971199052 = 119901119860 (120590 119905) minus 119901119861 (120590 119905)(120590 isin 119863119865)

(9)



119908 (120590 119905) = infinsum119895=1

119882119861119895 (120590) 119861119895 (119905) = [W (120590)]119879 B (119905)(120590 isin 119863119865)

(10)



M119861 [B (119905) +Ω2119861B (119905)] = Q119860 (119905) + Q119861 (119905) (11)



119863119865119898119861[119882119861119895(120590)]2d119860 (119895 = 1 2 3 )


Q119860 (119905) = ∬119863119865

W (120590) 119901119860 (120590 119905) d119860Q119861 (119905) = minus∬

119863119865

W (120590) 119901119861 (120590 119905) d119860(12)


L = ∬119863119865

F (120590) [W (120590)]119879 d119860 = [119871119903119895](119903 = 0 1 2 119895 = 1 2 3 )

(13)




M119861 [B (119905) +Ω2119861B (119905)] = L119879 sdot P (119905) + Q119861 (119905) (15)



(16)



A120594 = [A11 A12A21 A22

][120594119860120594119861

] = 1205962 [120594119860120594119861

] = 1205962120594 (17)






(18)





(19)


(20)







119901in (120590 120596) = 1205962120587 int12059621205870

Re 119901119861 (120590 119905)sdot Reminus120597119908 (120590 119905)120597119905 d119905



119875in (120596) = ∬119863119865


119863119865


(24)



119901tr (120590 120596) = 1205962120587 int12059621205870



(25)


119875tr (120596) = ∬119863119865

119901tr (120590 120596) d119860= 1205962 Re∬

119863119865

[F (120590)]119879 sdot Λ (120596) sdot W (120590) d119860

= 1205962 Reinfinsum119903=0

infinsum119895=1

119871119903119895 sdot 120582119903119895 (120596)

(26)




Y

X

Z

LY

Lx

LZ

pE

DFDS

0 = 1000 kgＧ3

c0 = 1500 ms

O






1205962119860119903 = 11987001205880 [(119898119860119903120587119871119909 )2 + (119899119860119903120587119871119910 )2 + (119897119860119903120587119871119911 )2] (27)

119865119860119903 (119909 119910 119911)= cos(119898119860119903120587119909119871119909 ) cos(119899119860119903120587119910119871119910 ) cos(119897119860119903120587119911119871119911 ) (28)



1205962119861119895 = 1205874119864ℎ212120588 (1 minus 1205832) (11989821198611198951198712119909 + 11989921198611198951198712119910 )2

(29)

119882119861119895 (119909 119910 0) = sin(119898119861119895120587119909119871119909 ) sin(119899119861119895120587119910119871119910 ) (30)









0 0 0 0 0 01 1068 1031 1015 1009 10032 2185 2084 2060 2039 20253 2344 2227 2176 2155 21344 2911 2709 2647 2622 25975 4476 4151 4047 4002 39596 4554 4207 4091 4045 39987 4761 4573 4518 4471 44398 5619 5431 5372 5334 53059 6267 5815 5662 5590 552310 6825 6177 5994 5921 5847



0 0 0 0 0 01 1015 1031 1015 1015 10152 2061 2084 2060 2060 20603 2176 2227 2175 2175 21754 2647 2709 2647 2647 26475 4047 4151 4046 4046 40466 4092 4207 4091 4091 40917 4518 4573 4516 4516 45168 5372 5431 5371 5371 53719 5662 5815 5660 5660 566010 5995 6177 5993 5993 5993



119903 (119898119860119903 119899119860119903 119897119860119903) 119891119860119903 (Hz) 119895 (119898119861119895 119899119861119895) 119891119861119895 (Hz) 119896 119891119862119896 (Hz)0 (0 0 0) 0 0 01 (0 0 1) 10714 1 (1 1) 1080 1 10152 (0 1 0) 12500 2 (1 2) 2076 2 20603 (0 1 1) 16463 3 (2 1) 3322 3 21764 (1 0 0) 18750 4 (1 3) 3737 4 26475 (0 0 2) 21429 5 (2 2) 4319 5 40476 (1 0 1) 21595 6 (2 3) 5980 6 40917 (1 1 0) 22535 7 (1 4) 6063 7 45188 (0 1 2) 24808 8 (3 1) 7059 8 53729 (1 1 1) 24952 9 (3 2) 8056 9 566210 (0 2 0) 25000 10 (2 4) 8305 10 5994


010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part


matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)



649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)







B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




111988820∭119881F (120590) [F (120590)]119879 P (119905) dV


= ∯119863F (120590) 120597119901 (120590 119905)120597119899 d119860

(7)



M119860 [P (119905) +Ω2119860P (119905)] = minus∯119863F (120590) 119886119899 (120590 119905) d119860 (8)



119878119908 (120590 119905) + 119898119861 1205972119908 (120590 119905)1205971199052 = 119901119860 (120590 119905) minus 119901119861 (120590 119905)(120590 isin 119863119865)

(9)



119908 (120590 119905) = infinsum119895=1

119882119861119895 (120590) 119861119895 (119905) = [W (120590)]119879 B (119905)(120590 isin 119863119865)

(10)



M119861 [B (119905) +Ω2119861B (119905)] = Q119860 (119905) + Q119861 (119905) (11)



119863119865119898119861[119882119861119895(120590)]2d119860 (119895 = 1 2 3 )


Q119860 (119905) = ∬119863119865

W (120590) 119901119860 (120590 119905) d119860Q119861 (119905) = minus∬

119863119865

W (120590) 119901119861 (120590 119905) d119860(12)


L = ∬119863119865

F (120590) [W (120590)]119879 d119860 = [119871119903119895](119903 = 0 1 2 119895 = 1 2 3 )

(13)




M119861 [B (119905) +Ω2119861B (119905)] = L119879 sdot P (119905) + Q119861 (119905) (15)



(16)



A120594 = [A11 A12A21 A22

][120594119860120594119861

] = 1205962 [120594119860120594119861

] = 1205962120594 (17)






(18)





(19)


(20)







119901in (120590 120596) = 1205962120587 int12059621205870

Re 119901119861 (120590 119905)sdot Reminus120597119908 (120590 119905)120597119905 d119905



119875in (120596) = ∬119863119865


119863119865


(24)



119901tr (120590 120596) = 1205962120587 int12059621205870



(25)


119875tr (120596) = ∬119863119865

119901tr (120590 120596) d119860= 1205962 Re∬

119863119865

[F (120590)]119879 sdot Λ (120596) sdot W (120590) d119860

= 1205962 Reinfinsum119903=0

infinsum119895=1

119871119903119895 sdot 120582119903119895 (120596)

(26)




Y

X

Z

LY

Lx

LZ

pE

DFDS

0 = 1000 kgＧ3

c0 = 1500 ms

O






1205962119860119903 = 11987001205880 [(119898119860119903120587119871119909 )2 + (119899119860119903120587119871119910 )2 + (119897119860119903120587119871119911 )2] (27)

119865119860119903 (119909 119910 119911)= cos(119898119860119903120587119909119871119909 ) cos(119899119860119903120587119910119871119910 ) cos(119897119860119903120587119911119871119911 ) (28)



1205962119861119895 = 1205874119864ℎ212120588 (1 minus 1205832) (11989821198611198951198712119909 + 11989921198611198951198712119910 )2

(29)

119882119861119895 (119909 119910 0) = sin(119898119861119895120587119909119871119909 ) sin(119899119861119895120587119910119871119910 ) (30)









0 0 0 0 0 01 1068 1031 1015 1009 10032 2185 2084 2060 2039 20253 2344 2227 2176 2155 21344 2911 2709 2647 2622 25975 4476 4151 4047 4002 39596 4554 4207 4091 4045 39987 4761 4573 4518 4471 44398 5619 5431 5372 5334 53059 6267 5815 5662 5590 552310 6825 6177 5994 5921 5847



0 0 0 0 0 01 1015 1031 1015 1015 10152 2061 2084 2060 2060 20603 2176 2227 2175 2175 21754 2647 2709 2647 2647 26475 4047 4151 4046 4046 40466 4092 4207 4091 4091 40917 4518 4573 4516 4516 45168 5372 5431 5371 5371 53719 5662 5815 5660 5660 566010 5995 6177 5993 5993 5993



119903 (119898119860119903 119899119860119903 119897119860119903) 119891119860119903 (Hz) 119895 (119898119861119895 119899119861119895) 119891119861119895 (Hz) 119896 119891119862119896 (Hz)0 (0 0 0) 0 0 01 (0 0 1) 10714 1 (1 1) 1080 1 10152 (0 1 0) 12500 2 (1 2) 2076 2 20603 (0 1 1) 16463 3 (2 1) 3322 3 21764 (1 0 0) 18750 4 (1 3) 3737 4 26475 (0 0 2) 21429 5 (2 2) 4319 5 40476 (1 0 1) 21595 6 (2 3) 5980 6 40917 (1 1 0) 22535 7 (1 4) 6063 7 45188 (0 1 2) 24808 8 (3 1) 7059 8 53729 (1 1 1) 24952 9 (3 2) 8056 9 566210 (0 2 0) 25000 10 (2 4) 8305 10 5994


010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part


matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)



649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)







B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







(18)





(19)


(20)







119901in (120590 120596) = 1205962120587 int12059621205870

Re 119901119861 (120590 119905)sdot Reminus120597119908 (120590 119905)120597119905 d119905



119875in (120596) = ∬119863119865


119863119865


(24)



119901tr (120590 120596) = 1205962120587 int12059621205870



(25)


119875tr (120596) = ∬119863119865

119901tr (120590 120596) d119860= 1205962 Re∬

119863119865

[F (120590)]119879 sdot Λ (120596) sdot W (120590) d119860

= 1205962 Reinfinsum119903=0

infinsum119895=1

119871119903119895 sdot 120582119903119895 (120596)

(26)




Y

X

Z

LY

Lx

LZ

pE

DFDS

0 = 1000 kgＧ3

c0 = 1500 ms

O






1205962119860119903 = 11987001205880 [(119898119860119903120587119871119909 )2 + (119899119860119903120587119871119910 )2 + (119897119860119903120587119871119911 )2] (27)

119865119860119903 (119909 119910 119911)= cos(119898119860119903120587119909119871119909 ) cos(119899119860119903120587119910119871119910 ) cos(119897119860119903120587119911119871119911 ) (28)



1205962119861119895 = 1205874119864ℎ212120588 (1 minus 1205832) (11989821198611198951198712119909 + 11989921198611198951198712119910 )2

(29)

119882119861119895 (119909 119910 0) = sin(119898119861119895120587119909119871119909 ) sin(119899119861119895120587119910119871119910 ) (30)









0 0 0 0 0 01 1068 1031 1015 1009 10032 2185 2084 2060 2039 20253 2344 2227 2176 2155 21344 2911 2709 2647 2622 25975 4476 4151 4047 4002 39596 4554 4207 4091 4045 39987 4761 4573 4518 4471 44398 5619 5431 5372 5334 53059 6267 5815 5662 5590 552310 6825 6177 5994 5921 5847



0 0 0 0 0 01 1015 1031 1015 1015 10152 2061 2084 2060 2060 20603 2176 2227 2175 2175 21754 2647 2709 2647 2647 26475 4047 4151 4046 4046 40466 4092 4207 4091 4091 40917 4518 4573 4516 4516 45168 5372 5431 5371 5371 53719 5662 5815 5660 5660 566010 5995 6177 5993 5993 5993



119903 (119898119860119903 119899119860119903 119897119860119903) 119891119860119903 (Hz) 119895 (119898119861119895 119899119861119895) 119891119861119895 (Hz) 119896 119891119862119896 (Hz)0 (0 0 0) 0 0 01 (0 0 1) 10714 1 (1 1) 1080 1 10152 (0 1 0) 12500 2 (1 2) 2076 2 20603 (0 1 1) 16463 3 (2 1) 3322 3 21764 (1 0 0) 18750 4 (1 3) 3737 4 26475 (0 0 2) 21429 5 (2 2) 4319 5 40476 (1 0 1) 21595 6 (2 3) 5980 6 40917 (1 1 0) 22535 7 (1 4) 6063 7 45188 (0 1 2) 24808 8 (3 1) 7059 8 53729 (1 1 1) 24952 9 (3 2) 8056 9 566210 (0 2 0) 25000 10 (2 4) 8305 10 5994


010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part


matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)



649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)







B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Y

X

Z

LY

Lx

LZ

pE

DFDS

0 = 1000 kgＧ3

c0 = 1500 ms

O






1205962119860119903 = 11987001205880 [(119898119860119903120587119871119909 )2 + (119899119860119903120587119871119910 )2 + (119897119860119903120587119871119911 )2] (27)

119865119860119903 (119909 119910 119911)= cos(119898119860119903120587119909119871119909 ) cos(119899119860119903120587119910119871119910 ) cos(119897119860119903120587119911119871119911 ) (28)



1205962119861119895 = 1205874119864ℎ212120588 (1 minus 1205832) (11989821198611198951198712119909 + 11989921198611198951198712119910 )2

(29)

119882119861119895 (119909 119910 0) = sin(119898119861119895120587119909119871119909 ) sin(119899119861119895120587119910119871119910 ) (30)









0 0 0 0 0 01 1068 1031 1015 1009 10032 2185 2084 2060 2039 20253 2344 2227 2176 2155 21344 2911 2709 2647 2622 25975 4476 4151 4047 4002 39596 4554 4207 4091 4045 39987 4761 4573 4518 4471 44398 5619 5431 5372 5334 53059 6267 5815 5662 5590 552310 6825 6177 5994 5921 5847



0 0 0 0 0 01 1015 1031 1015 1015 10152 2061 2084 2060 2060 20603 2176 2227 2175 2175 21754 2647 2709 2647 2647 26475 4047 4151 4046 4046 40466 4092 4207 4091 4091 40917 4518 4573 4516 4516 45168 5372 5431 5371 5371 53719 5662 5815 5660 5660 566010 5995 6177 5993 5993 5993



119903 (119898119860119903 119899119860119903 119897119860119903) 119891119860119903 (Hz) 119895 (119898119861119895 119899119861119895) 119891119861119895 (Hz) 119896 119891119862119896 (Hz)0 (0 0 0) 0 0 01 (0 0 1) 10714 1 (1 1) 1080 1 10152 (0 1 0) 12500 2 (1 2) 2076 2 20603 (0 1 1) 16463 3 (2 1) 3322 3 21764 (1 0 0) 18750 4 (1 3) 3737 4 26475 (0 0 2) 21429 5 (2 2) 4319 5 40476 (1 0 1) 21595 6 (2 3) 5980 6 40917 (1 1 0) 22535 7 (1 4) 6063 7 45188 (0 1 2) 24808 8 (3 1) 7059 8 53729 (1 1 1) 24952 9 (3 2) 8056 9 566210 (0 2 0) 25000 10 (2 4) 8305 10 5994


010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part


matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)



649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)







B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0 0 0 0 0 01 1068 1031 1015 1009 10032 2185 2084 2060 2039 20253 2344 2227 2176 2155 21344 2911 2709 2647 2622 25975 4476 4151 4047 4002 39596 4554 4207 4091 4045 39987 4761 4573 4518 4471 44398 5619 5431 5372 5334 53059 6267 5815 5662 5590 552310 6825 6177 5994 5921 5847



0 0 0 0 0 01 1015 1031 1015 1015 10152 2061 2084 2060 2060 20603 2176 2227 2175 2175 21754 2647 2709 2647 2647 26475 4047 4151 4046 4046 40466 4092 4207 4091 4091 40917 4518 4573 4516 4516 45168 5372 5431 5371 5371 53719 5662 5815 5660 5660 566010 5995 6177 5993 5993 5993



119903 (119898119860119903 119899119860119903 119897119860119903) 119891119860119903 (Hz) 119895 (119898119861119895 119899119861119895) 119891119861119895 (Hz) 119896 119891119862119896 (Hz)0 (0 0 0) 0 0 01 (0 0 1) 10714 1 (1 1) 1080 1 10152 (0 1 0) 12500 2 (1 2) 2076 2 20603 (0 1 1) 16463 3 (2 1) 3322 3 21764 (1 0 0) 18750 4 (1 3) 3737 4 26475 (0 0 2) 21429 5 (2 2) 4319 5 40476 (1 0 1) 21595 6 (2 3) 5980 6 40917 (1 1 0) 22535 7 (1 4) 6063 7 45188 (0 1 2) 24808 8 (3 1) 7059 8 53729 (1 1 1) 24952 9 (3 2) 8056 9 566210 (0 2 0) 25000 10 (2 4) 8305 10 5994


010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part


matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)



649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)







B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



010

2030

4050

020

4060

80minus1

minus05

0

05

1

xy

PB

(a) Real part0

1020

3040

50

020

4060

80minus05

0

05

1

xy

PB

(b) Imaginary part


matlabVL

100 200 300 400 500 600 700 800 900 10000Frequency (Hz)

minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)



649

365

805

minus204

minus488

minus772

minus106

minus134

minus162

minus191

minus219

On Boundary

(Ｇ2)







B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



B1

010

2030

4050

020

4060

80

02

04

0

08

06

1

yx

(a) 1198821198611 = sin(120587119909119871119909) sin(120587119910119871119910) 1198911198611 asymp 1080Hz

WB4

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(b) 1198821198614 = sin(120587119909119871119909) sin(3120587119910119871119910) 1198911198614 asymp 3734Hz

010

2030

4050

020

4060

80minus1

minus05

0

05

1

WB8

yx

(c) 1198821198618 = sin(3120587119909119871119909) sin(120587119910119871119910) 1198911198614 asymp 7059Hz

WC11

010

2030

4050

020

4060

80minus1

minus05

0

05

1

yx

(d) 11988211986111 = sin(120587119909119871119909) sin(5120587119910119871119910) 11989111986111 asymp 9052Hz








010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



010

2030

4050

020

4060

800

0005

001

0015

002

0025

WC0

yx

fC0 = 0 Hz

(a) 1198821198620 asymp 00248(1198821198611 + 0051198821198614 + 0021198821198618 + 00111988211986111)

minus002

minus001

0

001

002

WC3

010

2030

4050

020

4060

80

yx

fC3 asymp 2176 Hz

(b) 1198821198623 asymp 00132(minus0211198821198611 +1198821198614 + 0041198821198618 + 00511988211986111 + 00311988211986112)

minus001

minus0005

0

0005

001

WC7

010

2030

4050

020

4060

80

yx

fC7 asymp 4518 Hz


WC8

010

2030

4050

020

4060

80

yx

fC8 asymp 5372 Hz

minus001

minus0005

0005

001

0



86

865

87

875

88

010

2030

4050

020

4060

80

yx

times104

FC8(x

yL

z2

)

(a) 1198651198628(119909 119910 1198711199112) 1198911198628 asymp 5372Hz

times104


minus6

minus4

minus2

0

2

4

6

8

Mod

al p

artic

ipan

t coe

ffici

ents




minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

minus150

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

PＣＨ

PＮＬ

PＣＨ

andPＮＬ

(dB)


minus150

minus100

minus50

0

50

100

150

Soun

d pr

essu

re (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

P(Lx2 Ly 0)

P(Lx2 Ly Lz2)









200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)





minus10

0

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)


3 Ｄ)



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)

= 2770 ＥＡＧ

= 7800 ＥＡＧ


3

3


0

90

minus10

10

20

30

40

50

60

70

80

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

= 2770 ＥＡＧ

= 7800 ＥＡＧ3

3







200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)

minus600

minus550

minus500

minus450

minus400

minus350

minus300

minus250

minus200

PＮＬ

(dB)




minus20

0

20

40

60

80

100

120

140

160

PR (d

B)

200 400 600 800 1000 1200 1400 1600 1800 20000Frequency (Hz)






4 Conclusions








Appendix



119872119860119903119903 + 1198721198601199031205962119860119903119875119903 = minusinfinsum119895=1

119871119903119895119895 (119903 = 0 1 2 )

119872119861119895119895 + 1198721198611198951205962119861119895119861119895 = infinsum119903=0

119871119903119895119875119903 + 119876119861119895(119895 = 1 2 )

(A1)

where 119871119903119895 = ∬119863119865

119865119860119903119882119861119895d119860 119876119861119895 = minus∬119863119865




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




[PB] + [Ω2119860 + Mminus1119860 LM


minusMminus1119861 L119879 Ω2119861

][PB]


] (A2)


1205962 [12059410158401198601205941015840119861


minusMminus1119861 L119879 Ω2119861

][12059410158401198601205941015840119861

]

= A1015840 [12059410158401198601205941015840119861

] (A3)



Ω119860 O]


]119879 (A4)




Now let

[P (119905)B (119905)] = [



[X119860X119861

]120572 (119905)

= T0 sdot X sdot 120572 (119905) (A5)





] (A6)




Mminus1119861 Q119861] (A7)





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





]= (minus1205962M119862 +Ω2119862M119862)minus1

sdot [X119867119860 X119867119861 ] [radicM119860 O


Mminus1119861 P119861]


(A8)



119875119865119881 = 1205962120587 int21205871205960

119865 cos120596119905 sdot 119881 cos (120596119905 + 120593) d119905= 1205962120587 int2120587120596

0Re 119865 exp (119895120596119905)


(A9)



119875in (120596) = minus∬119863119865

1205962 Re 119895119882 (120590 120596) sdot 119875119861 (120590) d119860= minus∬

119863119865


119863119865

119895119879119861T119879 [H (120596)]119879




119875tr (120596) = ∬119863119865



119863119865

119895 [F (120590)]119879 (radicM119860)minus1


sdot W (120590) d119860 = 1205962 Re∬119863119865

[F (120590)]119879



sdot W (120590) d119860

(A11)



]]]

[[[1198821198611 (120590)1198821198612 (120590)

]]]

= infinsum119903=0

infinsum119895=1

120582119903119895 (120596) sdot 119865119860119903 (120590) sdot 119882119861119895 (120590) (A12)









References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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