StudyonFluid-StructureCouplingVibrationof...

Research ArticleStudy on Fluid-Structure Coupling Vibration ofCompressor Pipeline

Jia Wu 1 Chunjie Li 2 Shuiying Zheng 2 and Jingheng Gao 3

1School of Mechanical Engineering Zhejiang University Hangzhou 310027 China2Institute of Chemical Machinery Zhejiang University Hangzhou 310027 China3Hangzhou Jizhi Mechatronic Co Ltd Hangzhou 310030 China

Correspondence should be addressed to Shuiying Zheng zhengshuiyingzjueducn

Received 26 March 2019 Revised 17 June 2019 Accepted 17 July 2019 Published 7 August 2019

Academic Editor Huu-Tai +ai

Copyright copy 2019 Jia Wu et al +is 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

In practical engineering pipeline vibration is often not caused by a single factor but by a combination of many factors A fluid-structure coupling method is proposed in this paper and used to study the vibration of the compressor pipeline under theinteraction of pipeline structure and airflow in it +e method is based on structured grids so that the displacements of grid nodescan be calculated accurately at each time step +e results of transient calculation show that when the given inlet mass flow rate isconstant and there is no other disturbance the pressure fluctuation and the vibration of pipeline structure will occur by usingfluid-structure coupling and the vibration frequencies are consistent with the third- and fifth-order structural natural frequenciesMoreover the higher the pressure in the pipe the greater the fluid-structure coupling vibration In addition the fluid-structurecoupling vibration not only occurs in the studied pipeline but also propagates to distant downstream pipeline Comparing theabove results with experimental results it is found that the results of fluid-structure coupling calculation are in agreement with theactual situation which shows that the method is reasonable and reliable and can be applied to engineering

1 Introduction

Compressor pipeline vibration is a hidden danger thatcannot be ignored for safe production of factories Strongpipeline vibration affects the normal operation of devicesand causes great harm Many studies have been done on thevibration of compressor pipeline +ey are mainly dividedinto three aspects

Firstly it is the study of airflow pulsation in the pipeline+e vibration of the compressor pipeline encountered inproduction is mostly caused by air pulsation especially inreciprocating compressor Reciprocating compressor ischaracterized by intermittent and periodic suction andexhaust and thus it will stimulate the inlet and outlet of thepipeline to produce intense air pulsation which has a seriousimpact on the performance and work Only by thoroughlystudying the mechanism of airflow pulsation establishing areasonable analysis model of airflow pulsation accuratelypredicting the airflow pulsation in the compressor pipeline

system and reasonably designing the structural parametersof the pipeline can the airflow pulsation and its influence becontrolled in the minimum range Skudrzyk [1] in order tostudy the fluid characteristics in the pipeline used the planewave theory to study the dynamic characteristics of the gascolumn Hayama et al [2] revised the assumption of planewave theory of pipeline in resonance zone which greatlyreduced the relative error of the revised calculation resultsand basically met the design requirements of practical en-gineering Maclaren et al [3] considering environmentalfactors proposed conservative and characteristic equationswith more perfect mathematical models and more accuratecalculation methods which were suitable for one-di-mensional unsteady flow andmore close to the actual flow ofgas in the pipeline Durant et al [4] carried out vibrationmeasurement induced by airflow in a straight pipe andobtained the spectral density curve of air pulsation at dif-ferent Mach numbers the velocity spectral density of vi-bration and the sound pressure value radiated outward In

HindawiShock and VibrationVolume 2019 Article ID 8624324 12 pageshttpsdoiorg10115520198624324

addition the API Standard 618 of American PetroleumInstitute specifies the design criteria of gas flow pulsationand pipeline vibration for reciprocating compressors used inthe petrochemical industry

Secondly the dynamic characteristics and dynamic re-sponses of pipeline structures are analyzed and studiedPractice shows that even though the airflow fluctuation inthe pipeline has been controlled within a very small rangemechanical resonance can still be caused by the un-reasonable design of the pipeline structure resulting instrong vibration of the pipeline +erefore the structuraldynamic characteristics of the pipeline must be carefullystudied Paıdoussis and Issid [5] regarded the pipeline as abeam model and derived the motion equation Afterconsidering the pipeline as a beam model Rayleignmethod Dunkerley method Ritz method and Galerkinmethod can be used to solve the low-order structuralnatural frequencies of simple pipeline For pipelines withcomplex spatial structures the low-order structural naturalfrequencies can be solved by the finite element methodtransfer matrix method and impedance analysis method[6] Irie et al [7] used the Timshenko beam model to derivethe transfer matrix method for vibration and stability ofconveying fluid pipeline Lesmez et al [8] had done pio-neering work on the transfer matrix method for vibrationof the space pipeline system In the process of derivationthe method of separating variables is used and the ei-genvalue is included in the final characteristic matrix of thepipeline system

Finally the fluid-structure coupling vibration of thepipeline is studied +e fluid flow in the pipeline will lead tothe vibration of the pipeline structure and the vibration ofthe pipeline structure will in turn affect the motion state ofthe fluid in the pipeline +erefore coupling vibrationbetween fluid and structure is produced in the pipeline+efluid-structure coupling vibration has a great impact on thedynamic characteristics of the pipeline Wiggert et al[9 10] established 4-equation model and 14-equationmodel for the study of fluid-structure interaction vibrationof straight pipes Erath et al [11] studied the effect of waterhammer with fluid-structure interaction on the vibrationresponse and characteristics of pipe systems Tijsseling andLavooij [12] used the characteristic line method to analyzethe dynamic response of the pipeline system under theaction of the water hammer +e characteristic line methodis suitable for the simple pipeline system However for thecomplex pipeline system the finite element method hashigh accuracy and efficiency Sreejith et al [13] used thefinite element method to analyze the fluid-structure cou-pling dynamic response of pipeline under variable velocityPittard et al [14] used FLUENTsoftware to establish a largeeddy model and studied the influence of fluid pressurefluctuation on pipeline vibration characteristics Menteret al [15] used ANSYS MFX fluid-structure multiphysicalfield coupling scheme based on elastic dynamic meshdeformation technology to calculate the finite length elastictube +e calculated vibration pattern and the discharge ofthe lagging vortices of the elastic tube agreed well with the

theoretical structure In addition many experiments havebeen carried out to revise and validate the fluid-structurecoupling theory Vardy et al [16ndash18] studied the coupledvibration of branched pipes valves 90 degree curved pipesand single-elbow pipe systems by means of experimentsZiada et al [19] carried out an experimental comparativestudy on the fluid-acoustic coupling theory of T-shapedpipes Duan et al [20] used experiments to study Y-shapedpipes similar to T-shaped pipes found the gas-solid two-phase flow law in this type of pipes and obtained valuableexperimental data

+e purpose of this paper is to study the vibration of thecompressor pipeline under the interaction of pipelinestructure and airflow by using a proposed fluid-structurecoupling method and then to verify the effectiveness of thismethod by comparing with experimental results

2 Fluid-Structure Coupling Method

To simulate the transient calculation of pipeline vibration afluid-structure coupling method combining three-di-mensional transient flow field simulation with structuraldynamic calculation is proposed Transient flow field sim-ulation is solved with the mesh movement method +emotion equation used in structural dynamic calculationadopts the modal order reduction method +e data ex-change between fluid dynamics and structural dynamics isrealized by data files which makes quasicoupling calculationpossible +e fluid-structure coupling is described by theweak coupling method +is means that the fluid andstructural equations are computed in every time step andthe data transferred at the fluid-structural interface are usedas the boundary condition of the two domains One ad-vantage of the weak coupling method is to make full use ofthe mature computational fluid dynamics (CFD) softwarewithout rewriting the code +e flow chart of fluid-structurecoupling transient calculation for pipeline vibration isshown in Figure 1

+e transient flow field of pipeline is simulated byFLUENT software +e fluid forces are obtained by in-tegrating the fluid pressures through the application of user-defined function (UDF) With the fluid boundary conditiontransferred to a data file the displacement is obtained bysolving the motion equation After the structural boundarycondition is transferred to the data file the CFD code is thenapplied to update the grid position based on the proposedmesh movement method +is iterative procedure is re-peated until the vibration curve is stable

21 Modal Order Reduction Modal order reduction is toreplace the original complex high-order systemmodel with alow-order model which can maintain the main character-istics of the original system It is required that the results ofthe low-order model can be close to those of the originalsystem

+e motion equation of the pipeline system is

[M] euroq1113864 1113865 +[C] _q1113864 1113865 +[K] q1113864 1113865 f1113864 1113865 (1)

2 Shock and Vibration

where [M] is the mass matrix [C] is the damping matrix [K]is the stiffness matrix q is the vector of degrees of freedomand f is the external excitation vector

Because the low-frequency vibration of pipeline isstudied in this paper and the influence of higher than the

fifth natural frequency on the vibration of pipeline is smallthe first five modes of pipeline system are mainly consideredAccording to the modal order reduction the structuralmotion equation of the pipeline can be simplified as follows

M1 0 0 0 0

0 M2 0 0 0

0 0 M3 0 0

0 0 0 M4 0

0 0 0 0 M5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ηmiddotmiddot1ηmiddotmiddot2ηmiddotmiddot3ηmiddotmiddot4ηmiddotmiddot5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

C1 0 0 0 0

0 C2 0 0 0

0 0 C3 0 0

0 0 0 C4 0

0 0 0 0 C5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

_η1_η2_η3_η4_η5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

K1 0 0 0 0

0 K2 0 0 0

0 0 K3 0 0

0 0 0 K4 0

0 0 0 0 K5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η1η2η3η4η5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ϕT1 xf yf zf( 1113857 middot Ff





⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(2)

where M1 M2 M3 M4 and M5 are the first five modalmasses C1 C2 C3 C4 and C5 are the first five modaldamping K1 K2 K3 K4 and K5 are the first five modalstiffness η is the modal generalized coordinate and Ff is theforce acting on the pipeline and the coordinates of actionpoint are xf yf and zf φ1 φ2 φ3 φ4 and φ5 are the first fivemodes

+e displacement of each point in the pipeline can beobtained by the inverse transformation of the modal gen-eralized coordinate and its expression is as follows

q(x y z t) ϕ1 ϕ2 ϕ3 ϕ4 ϕ51113858 1113859

η1(t)

η2(t)

η3(t)

η4(t)

η5(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(3)

22 Dynamic Mesh Updating During the transient analysisof pipeline vibration if the fluid domain involves irregular

Establish fluid-structure coupling transientcomputation model for pipeline

Simulate transientflow field

Calculate fluid forces actingon pipeline

Calculate the displacementsof structure

Call structural dynamiccomputation model

Complete dynamic mesh updating

Calculate nexttime step

Yes

No

Stop

Data files

Fluid forces(Fx Fy Fz)

Transient structuraldynamic calculation

Solve motion equation bymodal order reduction

method

Displacements(∆x ∆y ∆z)

Figure 1 Flow chart of fluid-structure coupling transient calculation for pipeline vibration

Shock and Vibration 3

movement of the pipe dynamic mesh updating will be adifficult problem Without proper movement of grid nodesthe calculation of transient flow field cannot be carried out+erefore a reliable dynamic mesh method is needed tomodel the flow in which the shape of the region varies withtime

221 StudiedModel +e experimental apparatus of a pistonair compressor piping system is shown in Figure 2 +edischarge volume of the air compressor is 10m3min andthe rated pressure is 08MPa +ere is an air storage tank atthe outlet of the compressor to make the air supply morestable +e connection between the pipe and the supportingstructure constrains the translations of the pipe but therotations are not fully constrained +e pipeline systemconsists of a main pipe and a branch pipe and the end of thebranch pipe is closed +e geometry of the computationalmodel and the distribution of measuring points (MP) areshown in Figure 3 All the cross sections of the main andbranch pipes are circular with nominal diametersDm andDbof 65mm +e standard k-epsilon model is used to describethe fluid flow in pipeline Compressed air supplied to themeasuring pipeline by the piston compressor is used as afluid +e compressibility factor is calculated to be about0988 Since it is close to 1 the medium can be regarded asideal gas +e inlet boundary condition is set to mass-flow-inlet and the outlet boundary condition is set to pressure-outlet +e pressure-velocity coupling is treated with theSIMPLE scheme and the terms in the solution equations arediscretized by the second-order upwind scheme +e hybridinitialization method is used when t 0 second +e timestep size is 0001 second and the time stepping method isfixed+e computational time is 8 seconds and the pressure-based solver is chosen for the present analysis

222 Fluid Region Division +e pipeline system includes astraight pipe tee junction and elbow +eir mesh sizes aredifferent and the rules of mesh node movement are alsodifferent Based on the above considerations the fluid regionis divided into six parts which are shown in Figure 4 +eseparts are numbered 0sim6

223 Grid Motion Figure 5 shows the XY plane of region1 Node K represents the initial position of any node inregion 1 and node Kprime represents the current position ofnode K after the grid moves Region 1 is regarded as aslender beam and its cross section remains flat after de-formation In the initial state the cross section number ofeach node in region 1 is given and the calculation ofcumulative displacement of each node is set up when thegrid moves In the current state the coordinates of node Kprimeare read and the coordinates of the initial position K arecalculated according to the accumulated displacement+en the number of the cross sections is queried by thecoordinates and the displacement corresponding to the nexttime step is allocated to node Kprime to complete the update of

the motion +e grid motion algorithm developed for region1 applies to region 3 region 4 and region 6

Region 2 can be considered as a rigid body and all itsnodes move with the center node as a whole

Region 5 is divided into several subdomains Figure 6shows the XY plane of region 5 Node P represents theinitial position of any node in region 5 and node Pprimerepresents the current position of node P after the gridmoves In the initial state the coordinates of the center of thecircle O are recorded the cross section of each node in theregion 5 is numbered and the calculation of the cumulativedisplacement of each node is set up when the grid moves Inthe current state the coordinates of the center of the circleOprimeare calculated according to the cumulative displacementRead the coordinates of node Pprime calculate its azimuth angleαprime and judge its cross section number according to theazimuth angle +en the cumulative displacement is in-quired according to the number and the coordinates of theinitial position P are obtained +en the initial azimuthangle α of the node is calculated by using the coordinates ofnode P and center O and the initial cross section numberand corresponding displacement of the next time step areobtained +e displacement is allocated to node Pprime to updatethe motion

224 Mesh Updating Process +e process of mesh updatingat the current time requires the following steps

(1) +e coordinates of all nodes and their cross sectionnumbers in the initial state are stored and the cal-culation of cumulative displacement of each node isset up

(2) All subregions mentioned in Section 222 areaccessed in turn ensuring that all grid nodes in thefluid region are looped over +e coordinates of thenodes will be compared with the results in step (1) todetermine the position of the nodes the next time

(3) +e displacement is calculated and assigned to thenode according to its position thus completing themotion updating

CompressorAir storage tank

Support Support Support

ElbowBranch pipe

Main pipe

Figure 2 Experimental apparatus of pipeline vibration


During the mesh updating process the total number andtopological relationship of the grid nodes remain un-changed After all nodes are looped over at the current timetheir coordinates are all known and can be calculated ac-curately at the next time+erefore mesh updating moves inan orderly and controllable direction Even after tens ofthousands of times of movement the mesh after updatingcan still maintain high quality thus ensuring the smoothoperation of transient flow field calculation

3 Numerical Results

31 Determination of Plane Wave Acoustic Frequency Apulse excitation test was carried out to determine the planewave acoustic frequencies of the pipeline +e magnitudeand duration of the pulse excitation are as follows from001 s to 002 s the mass flow rate is 04 kgs at other timesthe mass flow rate is 02 kgs+e test was completed withoutfluid-structure coupling and the fluid domain remainedstationary +e code of pulse excitation was imported intothe inlet boundary condition through UDF +e time-do-main waveform under excitation is shown in Figure 7(a)and the corresponding spectrum is shown in Figure 7(b)Figure 7(a) is a decaying pulse response curve Damping iscaused by flow resistance and friction between gas mole-cules It can be seen from the spectrum that the first threeacoustic frequencies of plane wave are 825Hz 275Hz and330Hz respectively

32 Interaction between Structural Natural Frequency andPlane Wave Acoustic Frequency To study the interactionbetween structure natural frequencies and plane waveacoustic frequencies of the pipeline seven cases are selectedIn these seven cases only the values of modulus of elasticityhave been changed to ensure that their mode shapes are thesame which is convenient for the following study +estructural natural frequencies of these seven cases are listedin Table 1 +e reasons for choosing these seven cases are asfollows Firstly the acoustic frequencies of the pipeline areobtained by the pulse excitation method +en Case 5 is

Inlet Outlet

Support Branch pipeMain pipe

Closed

MP3MP4

MP5

MP1 MP2

Dm

Db

528Dm

231Dm 154Dm 92Dm

854Dm

302Db

Elbow 31Db

Figure 3 Geometry of the computational model and distribution of measuring points

1 2 3

4

X

Y

Z5

6

Figure 4 Region division

Current state

Initial state

Kprimey

xK

Figure 5 XY plane of region 1

Current state

Pprime

Oprime

Oα

αprime

P

Initial state



selected to make its first-order structural natural frequencyclose to the first-order acoustic frequency Case 1 is selectedto make its second-order structural natural frequency closeto the first-order acoustic frequency Case 3 is selected tomake its fourth-order structural natural frequency close tothe frequency doubling of the first-order acoustic frequencyCase 7 is selected to make its fourth-order structural naturalfrequency close to the second-order acoustic frequencieswhile ensuring that all frequencies are in the low-frequencyrange (below 40Hz) Finally considering the continuity ofthe research data Cases 2 4 and 6 are supplemented+rough these seven cases the potential relationship be-tween structural natural frequencies and acoustic frequen-cies is explored

+e corresponding first five order mode shapes areshown in Figure 8 +e first- and fourth-order mode shapesare on the YZ plane and the second third and fifth-ordermode shapes are on the XY plane

Based on the fluid-structure coupling method of pipelinevibration mentioned in this paper the transient calculationunder these seven cases is carried out respectively +ecalculation results of MP4 in Case 5 are shown in Figure 9From Figure 9(a) it can be seen that the pressure oscillationis maintained +is is due to fluid-structure coupling Wheninitializing the given initial condition is not the value of thesteady state so there will be a transient response process Inthis process the pressure fluctuation of the fluid in the pipewill lead to the vibration of the pipeline structure and thevibration of the pipeline structure will in turn affect the

motion of the fluid in the pipe +is results in a continuousexcitation in which the vibration of some frequency com-ponents (Figure 9(c)) is stimulated +erefore the pressureoscillation can be maintained in the final steady state As canbe seen from Figure 9(b) there are three frequency com-ponents 825Hz 170Hz and 285Hz However fromFigure 9(c) it can be seen that as time goes on the frequencyof 825Hz is attenuated and finally only two frequencycomponents are consistent with the third- and fifth-orderstructural natural frequencies of Case 5 +is is because thevibration displacement direction of the two structural nat-ural frequencies coincides with the direction of the forceformed by the pressure wave on the pipe wall thus stim-ulating the vibration of the pipe

It is noteworthy that the vibration displacement di-rection of the second-order structural natural frequency isalso the same as the direction of the force formed by thepressure wave but it is not stimulated in Figure 9(b) Inorder to study this phenomenon the pressure difference ofthe branch pipe in X and Y directions is calculated re-spectively As can be seen from Figure 10 the pressuredifference in the y direction is much larger than that in the xdirection Moreover as can be seen from Figure 8 the vi-bration of the second-order structural natural frequency is inthe x direction of the branch pipe the vibration of the third-order structural natural frequency is mainly in the y di-rection of the main pipe and a little in the y-direction of thebranch pipe and the vibration of the fifth-order structuralnatural frequency is mainly in the y-direction of the branch

Pres

sure

(Pa)

25000

20000

15000

10000

5000

0

ndash5000

ndash10000

ndash15000

ndash20000

Time (s)00 05 10 15 20 25 30 35 40

(a)

Pres

sure

(Pa)

825Hz

275Hz330Hz

Frequency (Hz)

1400

1200

1000

800

600

400

200

00 10 20 30 40 50 60 70 80 90 100

(b)

Figure 7 Time-domain waveform and spectrum of plane wave (a) time-domain waveform under excitation (b) spectrum

Table 1 Structural natural frequencies of seven cases

First-order (Hz) Second-order (Hz) +ird-order (Hz) Fourth-order (Hz) Fifth-order (Hz)Case 1 61 83 131 148 209Case 2 65 88 138 157 221Case 3 69 93 144 165 233Case 4 74 100 153 176 250Case 5 85 115 170 200 285Case 6 90 121 178 210 301Case 7 120 162 221 269 395


pipe +erefore the vibration of the fifth-order structuralnatural frequency is larger

+e analysis process of other cases is the same as aboveIn order to avoid redundancy the figures of their calculationresults are no longer presented in the paper Table 2 showsthe peak-peak value and main frequency components ofpipeline vibration in these seven cases under fluid-structureinteraction It can be seen from the table that in any case thefrequency components of pipeline vibration are consistentwith the third- and fifth-order structural natural frequenciesAlthough the first-order structural natural frequency isgenerally considered to be the easiest to be stimulated it isnot excited after steady state because the resultant force ofpressure wave acting on the z-direction of the pipe wall is 0In addition when the vibration frequency is close to theplane wave acoustic frequency the amplitude of the

vibration increases and when the vibration frequency is faraway from the plane wave acoustic frequency the amplitudeof the vibration decreases significantly

33 Effect of Different Pressures in Pipe In order to study theeffect of pressure on pipeline vibration under fluid-structureinteraction the vibration of pipeline was calculated underfour different pressures +e results are shown in Figure 11It can be seen from the figure that the greater the pressure inthe pipe the more vibration the pipe will be arousedMoreover in the case of nonresonance stable inlet flow andno external disturbance this phenomenon also occurs +ereason is that when the pressure wave encounters the closedbranch it will produce a great impact force which will leadto strong vibration of the pipeline structure

DisplacementStep = 1Sub = 1

(a)


(b)


(c)


(d)


(e)

Figure 8 First five order mode shapes of the pipeline structure (a) first-order mode shape (b) second-order mode shape (c) third-ordermode shape (d) fourth-order mode shape and (e) fifth-order mode shape


34 Influences onDistantDownstreamPipeline +e pressurefluctuations downstream of the pipeline with and withoutfluid-structure coupling are given in Figure 12 It is not

difficult to see that pipeline vibration has no effect on thedownstreamwithout fluid-structure coupling but in the caseof fluid-structure coupling the vibration will propagate

600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)

Figure 9 Calculation results of MP4 in Case 5 (a) time-domain waveform (b) spectrum (c) wavelet transform

60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)

Figure 10 Pressure difference of the branch pipe in two directions (a) x direction (b) y-direction


downstream When there are tee junctions elbows andclosed valves it will form an exciting force in a certaindirection to stimulate the vibration there

4 Experimental Results

Vibration velocity sensors and pressure sensors werearranged in the measuring pipeline and the measuringpoints are shown in Figure 3 Vibration signals from thevibration velocity sensors and pressure signals from thepressure sensors were recorded on PC data acquisitionsoftware by using a data acquisition board +e schematic ofmeasuring system is shown in Figure 13 +e samplingfrequency was 1000Hz and the sampling time was 8 s +etime-domain data were transformed into the frequency-domain data by fast-Fourier transform (FFT)

41 Determination of Structural Natural Frequencies Anexcitation test was carried out to determine structuralnatural frequencies of the pipeline A force hammer is usedto strike the pipeline quickly which is equivalent to a pulseexcitation to the pipeline structure Although the magnitudeand duration of the excitation are unknown it does notaffect the acquisition of structural natural frequencies +eycan be obtained by spectrum analysis of the measured vi-bration signals under excitation +e time-domain wave-form of MP1 under excitation is shown in Figure 14(a) andthe corresponding spectrum is shown in Figure 14(b) It canbe seen from the spectrum that there is a strongly dominantfrequency of 151Hz which is a certain order structuralnatural frequency Using the same method excitation testswere also carried out at other measuring points In order toavoid redundancy the figures of excitation test results atother measuring points are no longer given in the paperTable 3 shows the structural natural frequencies measured atall measuring points As can be seen from Table 3 the firstfive structural natural frequencies of the pipeline system are46Hz 84Hz 151Hz 214Hz and 259Hz respectively

42 Vibration Comparison under Different PressuresVibration measurements under six different pipeline pres-sures were carried out +e six kinds of pipeline pressure are01MPa 02MPa 03MPa 04MPa 05MPa and 056MParespectively Vibration data are obtained and analyzed ateach measuring point under each kind of pipeline pressure

Figure 15 shows the time-domain waveform and spec-trum of MP4 at a pressure of 02MPa As can be seen fromFigure 15(b) the main frequency components are 165Hzand 249Hz By comparing them with the structural naturalfrequencies described in Section 41 it can be seen that theyare consistent with the third- and fifth-order structuralnatural frequencies In addition by comparing the timehistories of experimental measurement and numerical cal-culation it can be seen that Figure 9(a) has a transientresponse process while Figure 15(a) does not +e reasonsare as follows In numerical calculation the initial conditiongiven for initialization is not the value of the steady state sothere will be a transient response process In this process thepressure fluctuation of the fluid in the pipe will lead to thevibration of the pipeline structure and the vibration of thepipeline structure will in turn affect themotion of the fluid inthe pipe Vibration that is inconsistent with the direction of

Table 2 Peak-peak value andmain frequency components of sevencases

Peak-peak value (Pa) Main frequencycomponents (Hz）

Case 1 300 130 210Case 2 340 1375 225Case 3 410 145 235Case 4 480 155 2525Case 5 250 170 285Case 6 240 1775 3025Case 7 10 220 395

1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20

Figure 11 Vibration waveforms of MP3 in Case 7 under fourdifferent pressures

300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)

Without fluid-structure couplingWith fluid-structure coupling Case 7With fluid-structure coupling Case 5

Pres

sure

(Pa)

Figure 12 Pressure fluctuations of MP2 with and without fluid-structure coupling


Vibration velocitysensors

Pressure sensors

MP1

MP2

MP3

MP4

MP5

Data acquisitionboard PC

data acquisition soware

Figure 13 Schematic of the measuring system

0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)

Figure 14 Time-domain waveform and spectrum of MP1 (a) time-domain waveform under excitation (b) spectrum

Table 3 Structural natural frequencies measured at all measuring points

First-order (Hz) Second-order (Hz) +ird-order (Hz) Fourth-order (Hz) Fifth-order (Hz)MP1 mdash mdash 151 mdash mdashMP2 mdash mdash mdash 214 mdashMP3 mdash 84 mdash mdash mdashMP4 46 mdash mdash mdash 259MP5 46 mdash mdash mdash 259

1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)

Figure 15 Time-domain waveform and spectrum of MP4 02MPa (a) time-domain waveform (b) spectrum


the force generated by the pressure wave will attenuate to 0and vibration that is consistent with the direction of theforce generated by the pressure wave will maintain and tendto oscillate stably However in the experiment the vibrationmeasurement is carried out after the compressor is openedand operated for a period of time so there is no transientprocess +e steady-state experiment is similar to the steady-state oscillation process after 3 seconds in Figure 9(a) Itshould be noted that the experiment results also containnoise signals +is steady-state oscillation process also needsto be solved by the transient solution method +e purposeof this paper is to study the fluid-structure coupling vi-bration of the pipeline under steady oscillatory state+erefore by comparing the experimental results with thenumerical results under steady oscillatory state it can beseen that the vibration frequencies are consistent with thethird- and fifth-order structural natural frequencies

Table 4 shows the peak-peak value and main frequencycomponents of MP4 under different pressures It can be seenfrom the table that no matter how much pipe pressure is thefrequencies of pipeline vibration are the structural naturalfrequencies which is consistent with the results of numericalcalculation in Section 32 Furthermore the higher the pipepressure the greater the vibration which is in agreementwith the results of numerical calculation in Section 33 +esame conclusions are reached at other measuring points

5 Conclusions

(1) +e fluid-structure coupling method proposed isreasonable and reliable by comparing numericalresults with experimental results and can be appliedto engineering

(2) +rough the fluid-structure coupling calculation it isfound that the pressure fluctuation and the vibrationof pipeline structure will occur when the inlet flow isstable and there is no external disturbance +e vi-bration frequencies are consistent with the third- andfifth-order structural natural frequencies and thefirst-order structural natural frequency is not exciteddue to the direction of force

(3) +e higher the pressure in the pipe the greater thefluid-structure coupling vibration It is useless to addaccumulators in the high-pressure pipeline Al-though the inlet flow is steady there will still bevibration in the pipeline

(4) +e fluid-structure coupling vibration not only oc-curs in the studied pipeline but also propagates todistant downstream pipeline When there are teejunctions elbows and closed valves it will form anexciting force in a certain direction to stimulate thevibration there

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Key Research andDevelopment Program of China (no 2016YFC0801200)

References

[1] E Skudrzyk e Foundations of Acoustics Springer-VerlagBerlin Germany 1971

[2] S Hayama Y Mohri and T Watanabe ldquoResonant ampli-tudes of pressure pulsation in pipelines 1st report resonantamplitudes in case of a single sinusoidal flow inputrdquo Bulletinof JSME vol 20 no 146 pp 955ndash962 1977

[3] J F T Maclaren A B Tramschek A Sanjines andO F Pastrana ldquoA comparison of numerical solutions of theunsteady flow equations applied to reciprocating compressorsystemsrdquo Journal of Mechanical Engineering Science vol 17no 5 pp 271ndash279 1975

[4] C Durant G Robert P J T Filippi and P-O MatteildquoVibroacoustic response of a thin cylindrical shell excited by aturbulent internal flow comparison between numericalprediction and experimentationrdquo Journal of Sound and Vi-bration vol 229 no 5 pp 1115ndash1155 2000

[5] M P Paıdoussis and N T Issid ldquoDynamic stability of pipesconveying fluidrdquo Journal of Sound and Vibration vol 33no 3 pp 267ndash294 1974

[6] S S Chen ldquoVibrations of continuous pipes conveying fluidrdquoin Flow-Induced Structural Vibrations pp 663ndash675 SpringerBerlin Germany 1974

[7] T Irie G Yamada and I Takahashi ldquoVibration and stabilityof a non-uniform Timoshenko beam subjected to a followerforcerdquo Journal of Sound and Vibration vol 70 no 4pp 503ndash512 1980

[8] M W Lesmez D C Wiggert and F J Hatfield ldquoModalanalysis of vibrations in liquid-filled piping systemsrdquo Journalof Fluids Engineering vol 112 no 3 pp 311ndash319 1990

[9] D C Wiggert R S Otwell and F J Hatfield ldquo+e effect ofelbow restraint on pressure transientsrdquo Journal of FluidsEngineering vol 107 no 3 pp 402ndash406 1985

[10] D CWiggert F J Hatfield and S Stuckenbruck ldquoAnalysis ofliquid and structural transients in piping by the method ofcharacteristicsrdquo Journal of Fluids Engineering vol 109 no 2pp 161ndash165 1987

[11] W Erath B Nowotny and J Maetz ldquoModelling the fluidstructure interaction produced by a waterhammer duringshutdown of high-pressure pumpsrdquo Nuclear Engineering andDesign vol 193 no 3 pp 283ndash296 1999

Table 4 Peak-peak value and main frequency components of MP4under different pressures

Peak-peak value (cms)Main frequencycomponents

(Hz)01MPa 008 166 24902MPa 012 165 24903MPa 015 165 24804MPa 018 164 24605MPa 020 164 246056MPa 021 164 246


[12] A S Tijsseling and C S W Lavooij ldquoWaterhammer withfluid-structure interactionrdquo Applied Scientific Researchvol 47 no 3 pp 273ndash285 1990

[13] B Sreejith K Jayaraj N Ganesan C PadmanabhanP Chellapandi and P Selvaraj ldquoFinite element analysis offluid-structure interaction in pipeline systemsrdquo Nuclear En-gineering and Design vol 227 no 3 pp 313ndash322 2004

[14] M T Pittard R P Evans R D Maynes and J D BlotterldquoExperimental and numerical investigation of turbulent flowinduced pipe vibration in fully developed flowrdquo Review ofScientific Instruments vol 75 no 7 pp 2393ndash2401 2004

[15] F Menter P Sharkey S Yakubov andM Kuntz ldquoOverview offluid-structure coupling in ANSYS-CFXrdquo in 25th InternationalConference on Offshore Mechanics and Arctic EngineeringHamburg Germany June 2006

[16] A E Vardy D Fan and A S Tijsseling ldquoFluid-structureinteraction in a T-piece piperdquo Journal of Fluids and Structuresvol 10 no 7 pp 763ndash786 1996

[17] A S Tijsseling A E Vardy and D Fan ldquoFluid-structureinteraction and cavitation in a single-elbow pipe systemrdquoJournal of Fluids and Structures vol 10 no 4 pp 395ndash4201996

[18] A S Tijsseling ldquoAn overview of fluid-structure interactionexperiments in single-elbow pipe systemsrdquo Journal of Zhe-jiang University-SCIENCE A vol 20 no 4 pp 233ndash242 2019

[19] S Ziada KW Mclaren and Y Li ldquoFlow-acoustic coupling inT-junctions effect of T-junction geometryrdquo Journal of Pres-sure Vessel Technology vol 131 no 4 article 041302 2009

[20] G B Duan Z M Liu G L Chen S G Hu and J ZhaoldquoExperimental investigation of gas-solid two-phase flow inY-shaped pipelinerdquo Advanced Powder Technology vol 21no 4 pp 468ndash476 2010


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addition the API Standard 618 of American PetroleumInstitute specifies the design criteria of gas flow pulsationand pipeline vibration for reciprocating compressors used inthe petrochemical industry

Secondly the dynamic characteristics and dynamic re-sponses of pipeline structures are analyzed and studiedPractice shows that even though the airflow fluctuation inthe pipeline has been controlled within a very small rangemechanical resonance can still be caused by the un-reasonable design of the pipeline structure resulting instrong vibration of the pipeline +erefore the structuraldynamic characteristics of the pipeline must be carefullystudied Paıdoussis and Issid [5] regarded the pipeline as abeam model and derived the motion equation Afterconsidering the pipeline as a beam model Rayleignmethod Dunkerley method Ritz method and Galerkinmethod can be used to solve the low-order structuralnatural frequencies of simple pipeline For pipelines withcomplex spatial structures the low-order structural naturalfrequencies can be solved by the finite element methodtransfer matrix method and impedance analysis method[6] Irie et al [7] used the Timshenko beam model to derivethe transfer matrix method for vibration and stability ofconveying fluid pipeline Lesmez et al [8] had done pio-neering work on the transfer matrix method for vibrationof the space pipeline system In the process of derivationthe method of separating variables is used and the ei-genvalue is included in the final characteristic matrix of thepipeline system

Finally the fluid-structure coupling vibration of thepipeline is studied +e fluid flow in the pipeline will lead tothe vibration of the pipeline structure and the vibration ofthe pipeline structure will in turn affect the motion state ofthe fluid in the pipeline +erefore coupling vibrationbetween fluid and structure is produced in the pipeline+efluid-structure coupling vibration has a great impact on thedynamic characteristics of the pipeline Wiggert et al[9 10] established 4-equation model and 14-equationmodel for the study of fluid-structure interaction vibrationof straight pipes Erath et al [11] studied the effect of waterhammer with fluid-structure interaction on the vibrationresponse and characteristics of pipe systems Tijsseling andLavooij [12] used the characteristic line method to analyzethe dynamic response of the pipeline system under theaction of the water hammer +e characteristic line methodis suitable for the simple pipeline system However for thecomplex pipeline system the finite element method hashigh accuracy and efficiency Sreejith et al [13] used thefinite element method to analyze the fluid-structure cou-pling dynamic response of pipeline under variable velocityPittard et al [14] used FLUENTsoftware to establish a largeeddy model and studied the influence of fluid pressurefluctuation on pipeline vibration characteristics Menteret al [15] used ANSYS MFX fluid-structure multiphysicalfield coupling scheme based on elastic dynamic meshdeformation technology to calculate the finite length elastictube +e calculated vibration pattern and the discharge ofthe lagging vortices of the elastic tube agreed well with the

theoretical structure In addition many experiments havebeen carried out to revise and validate the fluid-structurecoupling theory Vardy et al [16ndash18] studied the coupledvibration of branched pipes valves 90 degree curved pipesand single-elbow pipe systems by means of experimentsZiada et al [19] carried out an experimental comparativestudy on the fluid-acoustic coupling theory of T-shapedpipes Duan et al [20] used experiments to study Y-shapedpipes similar to T-shaped pipes found the gas-solid two-phase flow law in this type of pipes and obtained valuableexperimental data

+e purpose of this paper is to study the vibration of thecompressor pipeline under the interaction of pipelinestructure and airflow by using a proposed fluid-structurecoupling method and then to verify the effectiveness of thismethod by comparing with experimental results

2 Fluid-Structure Coupling Method

To simulate the transient calculation of pipeline vibration afluid-structure coupling method combining three-di-mensional transient flow field simulation with structuraldynamic calculation is proposed Transient flow field sim-ulation is solved with the mesh movement method +emotion equation used in structural dynamic calculationadopts the modal order reduction method +e data ex-change between fluid dynamics and structural dynamics isrealized by data files which makes quasicoupling calculationpossible +e fluid-structure coupling is described by theweak coupling method +is means that the fluid andstructural equations are computed in every time step andthe data transferred at the fluid-structural interface are usedas the boundary condition of the two domains One ad-vantage of the weak coupling method is to make full use ofthe mature computational fluid dynamics (CFD) softwarewithout rewriting the code +e flow chart of fluid-structurecoupling transient calculation for pipeline vibration isshown in Figure 1

+e transient flow field of pipeline is simulated byFLUENT software +e fluid forces are obtained by in-tegrating the fluid pressures through the application of user-defined function (UDF) With the fluid boundary conditiontransferred to a data file the displacement is obtained bysolving the motion equation After the structural boundarycondition is transferred to the data file the CFD code is thenapplied to update the grid position based on the proposedmesh movement method +is iterative procedure is re-peated until the vibration curve is stable

21 Modal Order Reduction Modal order reduction is toreplace the original complex high-order systemmodel with alow-order model which can maintain the main character-istics of the original system It is required that the results ofthe low-order model can be close to those of the originalsystem

+e motion equation of the pipeline system is

[M] euroq1113864 1113865 +[C] _q1113864 1113865 +[K] q1113864 1113865 f1113864 1113865 (1)





M1 0 0 0 0

0 M2 0 0 0

0 0 M3 0 0

0 0 0 M4 0

0 0 0 0 M5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

C1 0 0 0 0

0 C2 0 0 0

0 0 C3 0 0

0 0 0 C4 0

0 0 0 0 C5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

_η1_η2_η3_η4_η5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

K1 0 0 0 0

0 K2 0 0 0

0 0 K3 0 0

0 0 0 K4 0

0 0 0 0 K5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η1η2η3η4η5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(2)



q(x y z t) ϕ1 ϕ2 ϕ3 ϕ4 ϕ51113858 1113859

η1(t)

η2(t)

η3(t)

η4(t)

η5(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(3)









Yes

No

Stop

Data files




method

















ElbowBranch pipe

Main pipe




3 Numerical Results



Inlet Outlet


Closed

MP3MP4

MP5

MP1 MP2

Dm

Db

528Dm

231Dm 154Dm 92Dm

854Dm

302Db

Elbow 31Db


1 2 3

4

X

Y

Z5

6


Current state

Initial state

Kprimey

xK


Current state

Pprime

Oprime

Oα

αprime

P

Initial state








Pres

sure

(Pa)

25000

20000

15000

10000

5000

0

ndash5000

ndash10000

ndash15000

ndash20000

Time (s)00 05 10 15 20 25 30 35 40

(a)

Pres

sure

(Pa)

825Hz

275Hz330Hz

Frequency (Hz)

1400

1200

1000

800

600

400

200

00 10 20 30 40 50 60 70 80 90 100

(b)










(a)


(b)


(c)


(d)


(e)





600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





M1 0 0 0 0

0 M2 0 0 0

0 0 M3 0 0

0 0 0 M4 0

0 0 0 0 M5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

C1 0 0 0 0

0 C2 0 0 0

0 0 C3 0 0

0 0 0 C4 0

0 0 0 0 C5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

_η1_η2_η3_η4_η5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

K1 0 0 0 0

0 K2 0 0 0

0 0 K3 0 0

0 0 0 K4 0

0 0 0 0 K5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

η1η2η3η4η5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦






⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(2)



q(x y z t) ϕ1 ϕ2 ϕ3 ϕ4 ϕ51113858 1113859

η1(t)

η2(t)

η3(t)

η4(t)

η5(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(3)









Yes

No

Stop

Data files




method

















ElbowBranch pipe

Main pipe




3 Numerical Results



Inlet Outlet


Closed

MP3MP4

MP5

MP1 MP2

Dm

Db

528Dm

231Dm 154Dm 92Dm

854Dm

302Db

Elbow 31Db


1 2 3

4

X

Y

Z5

6


Current state

Initial state

Kprimey

xK


Current state

Pprime

Oprime

Oα

αprime

P

Initial state








Pres

sure

(Pa)

25000

20000

15000

10000

5000

0

ndash5000

ndash10000

ndash15000

ndash20000

Time (s)00 05 10 15 20 25 30 35 40

(a)

Pres

sure

(Pa)

825Hz

275Hz330Hz

Frequency (Hz)

1400

1200

1000

800

600

400

200

00 10 20 30 40 50 60 70 80 90 100

(b)










(a)


(b)


(c)


(d)


(e)





600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia















ElbowBranch pipe

Main pipe




3 Numerical Results



Inlet Outlet


Closed

MP3MP4

MP5

MP1 MP2

Dm

Db

528Dm

231Dm 154Dm 92Dm

854Dm

302Db

Elbow 31Db


1 2 3

4

X

Y

Z5

6


Current state

Initial state

Kprimey

xK


Current state

Pprime

Oprime

Oα

αprime

P

Initial state








Pres

sure

(Pa)

25000

20000

15000

10000

5000

0

ndash5000

ndash10000

ndash15000

ndash20000

Time (s)00 05 10 15 20 25 30 35 40

(a)

Pres

sure

(Pa)

825Hz

275Hz330Hz

Frequency (Hz)

1400

1200

1000

800

600

400

200

00 10 20 30 40 50 60 70 80 90 100

(b)










(a)


(b)


(c)


(d)


(e)





600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



3 Numerical Results



Inlet Outlet


Closed

MP3MP4

MP5

MP1 MP2

Dm

Db

528Dm

231Dm 154Dm 92Dm

854Dm

302Db

Elbow 31Db


1 2 3

4

X

Y

Z5

6


Current state

Initial state

Kprimey

xK


Current state

Pprime

Oprime

Oα

αprime

P

Initial state








Pres

sure

(Pa)

25000

20000

15000

10000

5000

0

ndash5000

ndash10000

ndash15000

ndash20000

Time (s)00 05 10 15 20 25 30 35 40

(a)

Pres

sure

(Pa)

825Hz

275Hz330Hz

Frequency (Hz)

1400

1200

1000

800

600

400

200

00 10 20 30 40 50 60 70 80 90 100

(b)










(a)


(b)


(c)


(d)


(e)





600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







Pres

sure

(Pa)

25000

20000

15000

10000

5000

0

ndash5000

ndash10000

ndash15000

ndash20000

Time (s)00 05 10 15 20 25 30 35 40

(a)

Pres

sure

(Pa)

825Hz

275Hz330Hz

Frequency (Hz)

1400

1200

1000

800

600

400

200

00 10 20 30 40 50 60 70 80 90 100

(b)










(a)


(b)


(c)


(d)


(e)





600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







(a)


(b)


(c)


(d)


(e)





600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




600

450

300

150

0

ndash150

ndash300

ndash450

Time (s)

Pres

sure

(Pa)

00 05 10 15 20 25 30 35 40

(a)

Frequency (Hz)

825Hz

170Hz

285Hz

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

(b)

1 2 3 4 5 6 7 8Time (s)

0

5

10

15

20

25

30

35

40

Freq

uenc

y (H

z)

200

400

600

800

1000

1200

1400

(c)


60

40

20

0

ndash20

ndash40

ndash60

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

(a)

Time (s)

Pres

sure

diff

eren

ce (P

a)

0500 10 15 20 25 30 35 40

600

400

200

0

ndash200

ndash400

ndash600

(b)












1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











1000

750

500

250

0

ndash250

ndash500

ndash750

ndash1000

Time (s)

0MPa1MPa

2MPa4MPa

Pres

sure

(Pa)

00 02 04 06 08 10 12 14 16 18 20


300

200

100

0

ndash100

ndash20000 05 10 15 20 25 30 35

Time (s)


Pres

sure

(Pa)




Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Pressure sensors

MP1

MP2

MP3

MP4

MP5




0 1 2 3 4 5 6 7 8ndash8

ndash6

ndash4

ndash2

0

2

4

6

8

10

Vibr

atio

n ve

loci

ty (c

ms

)

Time (s)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

151Hz

000

005

010

015

020

025

Vibr

atio

n ve

loci

ty (c

ms

)

(b)




1 2 3 4 5 6 7 80Time (s)

ndash015

ndash010

ndash005

000

005

010

015

Vibr

atio

n ve

loci

ty (c

ms

)

(a)

0 5 10 15 20 25 30 35 40Frequency (Hz)

165Hz

249Hz

0000

0003

0006

0009

0012

0015

0018

Vibr

atio

n ve

loci

ty (c

ms

)

(b)





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




5 Conclusions





Data Availability




Acknowledgments


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


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