research article optimization and analysis of...

Research ArticleOptimization and Analysis of Centrifugal Pump consideringFluid-Structure Interaction

Yu Zhang1 Sanbao Hu2 Yunqing Zhang3 and Liping Chen3

1 Wuhan Second Ship Design and Research Institute Wuhan Hubei 430064 China2Hubei Key Laboratory of Advanced Technology of Automobile Parts Wuhan University of Technology Wuhan Hubei 430074 China3 Center for Computer-Aided Design School of Mechanical Science amp Engineering Huazhong University of Science amp TechnologyWuhan Hubei 430074 China

Correspondence should be addressed to Yunqing Zhang zhangyqhusteducn

Received 24 February 2014 Accepted 23 July 2014 Published 13 August 2014

Academic Editor Mohammad A Abido

Copyright copy 2014 Yu Zhang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents the optimization of vibrations of centrifugal pump considering fluid-structure interaction (FSI) A set ofcentrifugal pumps with various blade shapes were studied using FSI method in order to investigate the transient vibrationperformance The Kriging model based on the results of the FSI simulations was established to approximate the relationshipbetween the geometrical parameters of pump impeller and the root mean square (RMS) values of the displacement response atthe pump bearing block Hence multi-island genetic algorithm (MIGA) has been implemented to minimize the RMS value of theimpeller displacement A prototype of centrifugal pump has beenmanufactured and an experimental validation of the optimizationresults has been carried out The comparison among results of Kriging surrogate model FSI simulation and experimental testshowed a good consistency of the three approaches Finally the transient mechanical behavior of pump impeller has beeninvestigated using FSI method based on the optimized geometry parameters of pump impeller

1 Introduction

Centrifugal pumps provide the energy tomove fluids throughpiping systems including equipment piping and fittingsand through elevation changes in open systems Centrifugalpumps have been widely used in various industrial applica-tions such as oil and gas agriculture chemistry and marineindustry as well as metallurgy Because of the customersrsquoincreasing demands of high-quality pump optimizationdesign of centrifugal pump plays an important role in pumpindustry and there have been many efforts to optimize theperformance of centrifugal pump in recent years Anagnos-topoulos [1] proposed an optimization algorithm based onunconstrained gradient method to find the impeller geom-etry that could maximize the pump efficiency among a set ofblade angles Zhou et al [2] optimized the geometric shapeof the centrifugal impeller using orthogonal experimentmethod to improve the performance of the centrifugal pumpDerakhshan et al [3] presented the incomplete sensitivities

approach and genetic algorithms to obtain a higher efficiencyby redesigning the shape of impeller blades Papierski andBlaszczyk [4] decomposed the optimization design of cen-trifugal pump into two levels to maximize the efficiencyand simultaneously minimize required net positive suctionhead (NPSHr) These researches mainly focus on optimizingthe performance data of centrifugal pump such as headefficiency or NPSHr However the vibration performance isimportant especially for high-pressure centrifugal pump

The vibration that occurs while centrifugal pump workscan cause fatigue and damage of pump components andweaken the operation stability Vibrations of centrifugalpump have attracted interest of researchers For exampleHodkiewicz and Norton [5] investigated the influence ofdifferent flow rates on the vibration performance of double-suction centrifugal pump Guo and Maruta [6] presentedan experimental study of the pressure fluctuation and theimpeller vibration in a centrifugal pump with some vaneddiffusers Rodriguez et al [7] developed a theoretical analysis

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 131802 9 pageshttpdxdoiorg1011552014131802

2 The Scientific World Journal

approach to investigate the vibrating frequencies in thevibration of centrifugal pump induced by the rotor statorinteraction (RSI) Wang et al [8] studied the structuraldynamics characteristics and vibrations of pump volutecasing for a double-suction centrifugal pump using a fluid-structure coupling interface model

Studies agree in considering the fluid-structure interac-tion (FSI) as the source of the highest vibration levels in largecentrifugal pumps Moreover hydraulic excitation forces aredue to the FSI and cause pressure fluctuations mechanicalvibrations and alternating stresses in different componentsof centrifugal pump In recent years the application of FSItheory to centrifugal pumps became more popular and it iswell documented in literature [9ndash13]

Vibration performance is one of the most importantparameters in designing a centrifugal pump Actually exper-imental tests and CFD simulation are the two methodsperformed in order to obtain the centrifugal pump vibrationresponse However both of the two methods cannot beconsidered in optimizing the vibration performance of thepump using an iterative method Appropriate metamodelsmust be established between the decision variables andthe concerned objective functions Therefore metamodeltechnique demonstrates its superiority in the optimizationproblem of engineering

Kriging metamodels [14] were originally proposed bythe South African mining engineer named Danie GerhardusKrige With the rapid development of computer technologyKriging metamodels have been widely used in various fields[15ndash19] Kriging metamodel differs from other metamodelsbecause of the optimal unbiased prediction for the unknownresponse points [20] Compared to traditionally response sur-face methods Kriging shows its superiority in high dimen-sional nonlinear problems and prediction accuracy due tothe stochastic assumption [21] especially for multiobjectiveoptimization problems [22]

This paper presents an effective optimization methodbased on Kriging metamodel The presented method opti-mizes the vibration performance of the centrifugal pumpundergoing FSI phenomena which reasonably take advan-tages of the FSI simulation Kriging metamodel and experi-mental tests Although considerable researches were devotedto investigating the vibration performance of centrifugalpump it should be noted that there exists little literatureevidence on the vibration optimization of centrifugal pumpin particular that combines with FSI phenomenon Thesecond part of the paper deals with the study of the transientmechanical characteristics based on the optimized centrifu-gal pump using FSI method

2 Centrifugal Pump FSI Simulation Model

21 FSI Governing Equations In this study the fluid-struc-ture interaction (FSI) problemrsquos domain Ω consists of twosubdomains Ω119891 and Ω119904 with the boundaries as Γ119891 and Γ119904respectively The subscripts 119891 and 119904 denote the fluid partand solid part respectively The following section defines

the equations that govern the flow and the structural defor-mations of the pump

211 Fluid Flow Equations The fluid flowing through thecentrifugal pump is treated as incompressible and isothermalFor x isin Ω119891 the conservation of mass and Navier-Stokesequations governing the unsteady flow are respectivelywritten as

120597120588119891

120597119905+ nabla sdot (120588119891k119891) = 0

120588119891

120597k119891120597119905

+ 120588119891 (k119891 sdot nabla) k119891 minus nabla sdot 120590119891 = f119891

(1)

where 120588119891 is the fluid density k119891 denotes the fluid particlevelocity at time 119905 f119891 denotes the body forces per unit ofvolume on the fluid and 120590119891 is the stress tensor defined as

120590119891 = minus119901I + 120583 [nablak119891 + (nablak119891)119879] (2)

where 119901 is the pressure I denotes the unit tensor and 120583

represents the absolute viscosity

212 Structural Equations While working the centrifugalpump undergoes large deformation and rotation For x isin Ω119904the conservation of momentum for the solid deformation u119904is described through Lagrangian formulation

120588119904

1205972u1199041205971199052

= nabla sdot 120590119904 + f119904 (3)

where 120588119904 is the solid density 120590119904 represents the Cauchy stresstensor and f119904 denotes the body forces per unit volume on thesolid

Closure for (3) is found by evaluating the stress using therelevant constitutive relationsMoreover since the centrifugalpump is related to large deformation and rotation theconstitutive equations are described using a stress-strainrelationship

213 Interaction between Fluid and Solid As mentionedabove the FSI occurs during the running process of centrifu-gal pump Fluid pressure information transfers to the solidwhile displacements information of the solid transfers to theflow Furthermore on the no-slip fluid-structural interfacethe information exchange between the fluid and solid shouldfollow the equilibrium conditions

k =120597u120597119905

120590119891 sdot n = 120590119904 sdot nforallx isin Γ (4)

where Γ denotes the fluid-structural interface and Γ = Γ119891capΓ119904n represents the unit normal at the interface Γ

22 Decision Variables The working process of centrifugalpump involves vibrations FSI and energy conversion andloss As the ldquoheartrdquo component for a centrifugal pump

The Scientific World Journal 3

333

59

155

60

Figure 1 Main dimensions of centrifugal pumprsquos impellers (unitmm)

impeller plays an important role in all these phenomena andtransforms the mechanical energy into the kinetic energy ofthe fluid Moreover the geometry shape of impeller bladehas strong effect on pump performance including vibrationsThis paper focuses on optimizing the impeller blade tominimize the vibration response of the centrifugal pump

The recommended number of impeller blades for highhead centrifugal pumps is usually between five and seven Infact too many blades lead to higher friction losses and maycause low blade loading fewer blades may result in higherblade loading Turbulent dissipation losseswill rise because ofthe increased secondary flow and stronger deviation betweenblade and flow direction Therefore six blades are chosen

Figure 1 shows the main dimensional parameters of theimpellers of the centrifugal pump studied in this papernotably it is double suction type In addition among all kindsof centrifugal pumps the double suction centrifugal pumpsare widely used in industry for various applications due tolarge flux and high lift

Figure 2 shows the meridional section of the impellerDue to symmetry of the model the optimization of doublesuction impeller can be converted into the optimization of asingle suction type Hence as Figure 2 shows the meridionalsection of the single suction impeller is actually determinedby the solid line and dashed line 119865119866The solid line is parame-terized by quartic Bezier curve with five control points thefive decision variables for the pump impeller are 1205931 1205821 12059321205822 and 119897 Table 1 summarizes the boundaries of the decisionvariables Here 120582 is the relative position in the line segmentFor example taking line segment 119860119862 1205821 is the ratio of theline length 119860119861 to 119860119862 119897 is the length of line segment 119865119866

23 FSI Model Sample and Simulation Latin hypercubesampling (LHS) is a design of experiment (DOE) methodoriginally developed byMckay in 1979 LHS approach has thespace-filling character and can guarantee the sample pointscovering the entire design domains homogeneously Hence119 simple points and 30 test points have been obtain by LHS

E

D

AB

C

F

G

1205822

1205931

1205821

1205932

l

Figure 2 Meridional section of the pump impeller

Table 1 Decision variables and their boundaries

Decision variable Lower boundary Upper boundary1205931 (deg) 0 301205821 002 0981205932 (deg) 70 901205822 002 098119897 (mm) 145 195

method The simple points are the input data of Krigingsurrogate model while the test points are used to validate theaccuracy of the Kriging predictor

Table 2 summarizes the combinations of decision vari-ables in the sample points The FSI simulation models arebuilt based on these sample points Figure 3 shows one case ofFSI simulation models Figure 3(a) corresponds to a full FSImodel with solid and fluid parts Figure 3(b) is the cutawayview of the full FSI model and Figure 3(c) gives the detailedview of the tongue region The structural part consists ofpump volute casing impeller and impeller shaft while thefluid part is the liquid flowing through the structural partMoreover the fluid part is also called the hydraulic model ofcentrifugal pump

The calculation of structural part of the pump hasbeen carried out through computational structure dynamics(CSD) analysis performed using Abaqus FEA software Thepump volute casing and impeller are both made of alumi-num-bronze alloy the elastic modulus is 125000MPa thedensity is 7630Kgm3 and the Poissonrsquos ratio is 0327 Theimpeller shaft is made of alloy steel with elastic modulus of206000MPa density of 7800Kgm3 and Poissonrsquos ratio of03 The increment size of time step is set as 1 times 10

minus4 s andthe total simulation time is 6119879 where 119879 is the cycle of thepump corresponding to a changed angle of 60∘ Furthermorethe differential equation of the centrifugal pump at excitationstate by FSI can be expressed as

Mx (119905) + Cx (119905) + Kx (119905) = F (119905) (5)


Table 2 The sample points and corresponding results of FSI simulations

Serial number Decision variable Objective1205931 (deg) 1205821 1205932 (deg) 1205822 119897 (mm) RMS (mm)

1 000 0028 8576 0061 15771 054112 025 0183 7000 0223 18314 040683 051 0191 8729 0744 18441 042284 076 0728 7390 0752 18525 038675 102 0777 8322 0484 16110 041086 127 0199 8966 0109 15644 052717 153 0557 8780 0459 18907 052518 178 0085 7119 0427 16873 058129 203 0467 8424 0305 16195 0567210 229 0264 7508 0321 15517 0412811 254 0232 8051 0378 14797 0360312 280 0288 8153 0834 17763 04529

117 2949 0646 8864 0443 16364 03627118 2975 0817 8475 0516 16831 03667119 3000 0891 7610 0785 18949 05792

Structural part

Impeller shaft

Z

YX

(a) Full model

Fluid partImpellerZ

YX

(b) Cutaway view (c) Detail of the tongue region

Figure 3 One case of FSI simulation models

where 119905 is the time M C and K are the structural massmatrix structural damping matrix and stiffness matrixrespectively x(119905) x(119905) and x(119905) represent the accelerationvector velocity vector and displacement vector respectivelyF(119905) denotes the load vector of the node

The computational fluid dynamics (CFD) has been simu-lated using Fluent codeThe fluid is water with a temperatureof 20∘C density of 9982Kgm3 and viscosity of 1003 times

10minus3 Pasdots Table 3 lists the parameters for CFD simulations

The hydraulic models are established by the standard 119896 minus 120576

turbulence models and wall functions based on logarith-mic law which are consistent with the no-slip conditionStatic boundary condition and rotary boundary conditionare imposed on the boundary of volute flow domain andimpeller flow domain respectively Moreover the interactionbetween these two boundaries is taken into account throughthe moving mesh model The unsteady Reynolds-averagedNavier-Stokes (URANS) equations are calculated by finitevolume method (FVM) and the pressure-velocity couplingis solved by means of the SIMPLEC algorithm Second order

Table 3 Basic parameters for numerical simulations

Parameter ValueFlow rate 119876 2000 (m3h)Rotational speed 1400 rminNumber of blades 6Inlet operating pressure 1 (atm)

upwind discretizations are used for determine and diffusiveterms of the turbulencemodel equationsThe residual error isset as 1times 10

minus5 to judge whether the calculation is convergentIn addition the time step and total simulation time are set as1 times 10

minus4 s and 6119879 respectively in order to correspond withthe structural simulation part

The information exchange between solid (Abaqus) andfluid flow (Fluent) at the coupling interface is performedin the platform of MpCCI Figure 4 outlines the processof FSI simulation first the models of CSD and CFD areprepared independently such as the setting ofmaterial loads


CSDmodel

CFDmodel

Coupling interface

Scan Scan

CSDsolver

CFDsolver

FSIserver 1

FSIserver 2

FSIcontroller

Data Data

CSDresults

CFDresults

CFDpostprocessing

CSDpostprocessing

Preparation ofCSD and CFD

models

Preparation of FSImodel

Data transmission and

coupling calculation

Postprocessing ofresults

Read Read

Figure 4 The process of FSI simulation

and boundary conditions Then the pressure informationof the fluid is transferred to Abaqus for structure analysismeanwhile the displacements information of the structureare transferred to Fluent for fluid analysis and the infor-mation exchange at the coupling interface repeatedly untilthe calculation has converged Finally the results of FSIsimulation are post processed using both Abaqus and Fluent

As aforementioned this paper mainly focuses on opti-mizing the vibration performance of the centrifugal pumpusing FSI Hence the root mean square (RMS) value of thedisplacement response at the pump bearing block is chosenas the objective function which can be defined as follows

119880RMS = radic1

119873

119873

sum

119894=1

1198802119894

(6)

where 119873 is the total number of the time steps and 119880119894

denotes the displacement response of the 119894th time step andthe direction is the vertical direction of the bearing supportThe last column of Table 2 summarizes the results of 119880RMScalculated through FSI simulations at the sample pointswhich are the output data used to build the Kriging model

3 Kriging-Based Optimization

31 Kriging Approach Kriging predicts unknown values of arandom function based on all of the observed points [22]Moreover Kriging metamodels show global performancerather than local characteristics A combination of a global

model and localized departures of the form describes aKriging model thus

119910 (119909) =

119899

sum

119896=1

120573119896119891119896 (119909) + 119885 (119909) (7)

where 119910(119909) is the response function f(119909) = [1198911(119909) 119891119899(119909)]

119879 is the regression basis function 119899 is the number ofthe basis function and 120573 = [1205731 120573119899]

119879 is the regressioncoefficient 119885(119909) is assumed as a realization of an indepen-dent Gaussian random process with zero mean and spatialcorrelation function given by [23]

Co V [119885 (120591) 119885 (119909)] = 1205902119877 (120579 120591 119909) (8)

where 1205902 denotes the process variance 119877(120579 120591 119909) is the

correlation function between the points 120591 and 119909 and 120579 is theunknown correlation parameter Several types of correlationmodels such as linear correlation model and exponentialcorrelation model can be considered However the Gausscorrelation model adopted in this paper is more popular inKriging metamodels with the form

119877 (120579 120591 119909) = exp(minus

119898

sum

119895=1

120579119895(120591119895 minus 119909119895)2) (9)

where the quantities 120591119895 and 119909119895 respectively denote the 119895thcomponents of sample points 120591 and 119909 119898 is the dimension ofthe decision variables


The predicted value and estimation error at point 119909 arerespectively given by

119910 (119909) = f119879 (119909) + r119879 (119909)Rminus1 (Y minus F)

119904 (119909) = 1205902(1 + u119879(F119879Rminus1F)

minus1u minus r(119909)119879Rminus1r (119909))

(10)

where Y represents the response of the sample points u =

F119879Rminus1r(119909) minus f(119909) F is a vector which is composed bythe value of f(119909) at each sample point r119879(119909) denotes avector which represents the correlation between an unknownpoint and all known sample points In addition r119879(119909) =

[119877(120579 119909 1199091) sdot sdot sdot 119877(120579 119909 119909119873)] 119873 is the total number of thesample points R is an 119873 times 119873 symmetric correlation matrixwritten in the following form

R =[[

[

119877 (1199091 1199091) sdot sdot sdot 119877 (1199091 119909119873)

d

119877 (119909119873 1199091) sdot sdot sdot 119877 (119909119873 119909119873)

]]

]

(11)

Under the unbiased condition the unknown parameters120573 and 120590

2 can be estimated through

= (F119879Rminus1F)minus1F119879Rminus1Y

2=

1

119898(Y minus F)

119879Rminus1 (Y minus F)

(12)

As a matter of fact once the types of regression modeland correlation model have been chosen the correlationmatrix R and unknown parameters 120573 and 120590

2 all depend onthe correlation parameter 120579 Thus a Kriging metamodel iscompletely established only if the value of 120579 is determinedFurthermore themost commonly used approach to calculatethe value of correlation parameter 120579 is maximum likelihoodestimation (MLE) and the problem can be converted into anunconstrained global optimization problem as follows [20]

Minimize 120595 (120579) = 120590(120579)21003816100381610038161003816R120579

1003816100381610038161003816

1119898

Subject to 120579 gt 0

(13)

32 Modeling and Verification of the Kriging Model Krigingmetamodel is established according to Table 2The input datais the set of 119 sample points obtained through LHSmethodand the input variables are the impeller geometric parameters1205931 1205821 1205932 1205822 and 119897 shown in Figure 2The output data are theresults of FSI simulations corresponding to the sample pointsand the output variable is the RMS value of the displacementresponse 119880RMS Table 4 shows the parameters of the Krigingmetamodel 120573 1205902 and 120579

The Kriging metamodel can be applied to the vibrationoptimization only if the Kriging predictorrsquos estimated accu-racy is higher enough Otherwise the metamodel should berebuilt by adjusting the parameters An additional set of 30points obtained through LHS method is used as test pointsto verify the performance of Krigingrsquos predictor The FSIanalysis gives the RMS values of the displacement response

Table 4 The parameters of the Kriging model

Parameter Value120573 [2688119890 minus 5 minus0142 minus0062 minus0036 minus0093 0107]119879

1205902 000453120579 [8406 14718 18851 21512 21401]

0 5 10 15 20 25 30

035

04

045

05

055

06

065

Serial number

RMS

of d

ispla

cem

ent r

espo

nse (

mm

)

KrigingFSI

Figure 5 The results of RMS values at test points

corresponding with the test points In addition for the FSIsimulations based on the test points the basic parameters andboundary conditions are the same with the sample points

Figure 5 shows the results of the displacement responsersquosRMS values obtained by Kriging predictor and FSI simula-tions at the test points Results show that the predicted valuesof the Kriging metamodel correspond to the FSI simulationvaluesHence the vibration optimization of centrifugal pumpcan be performed based on the Kriging surrogate model

33 Optimization Based on the Surrogate Model The opti-mization problem of centrifugal pump in this paper can begiven as follows

Find X = [1205931 1205821 1205932 1205822 119897]119879

Minimize 119880RMS = 119891 (1205931 1205821 1205932 1205822 119897)

Subject to 0∘le 1205931 le 30

∘

002 le 1205821 le 098

70∘le 1205932 le 90

∘

002 le 1205822 le 098

145mm le 119897 le 195mm

(14)

where is 119891 the Kriging approximation of the displacementresponsersquos RMS values

The above-defined problem can be resolved throughmulti-island genetic algorithm (MIGA) amodified version ofgenetic algorithm (GA) MIGA decomposes the populationin one generation into several subpopulations The subpopu-lations are also called ldquoIslandsrdquo and the genetic operations are


Table 5 The parameter settings of MIGA

Parameters ValueSize of subpopulation 100Number of islands 10Number of generations 10Gene size 32Rate of crossover 10Rate of mutation 001Rate of migration 05Interval of migration 5Number of runs for the problem 30

executed on each ldquoIslandrdquo independently Furthermore thisindependency can prevent the optimization solution fromlocal optima Table 5 lists the detailed parameter settings usedfor MIGAThe experiments are carried out on a Desktop PCwith Intel Core 2 quadCPU and 325GBRAM all of the coreshave the speed of 266GHz Due to stochastic behavior ofMIGA algorithm at least 30 independent runs are requiredto provide the results with statistical confidence

4 Result and Discussion

41 Optimization Result and Validation Table 6 shows theoptimization result and average CPU timeThe RMS value ofthe displacement response improves to 03341mmMoreoverthe accuracies of Kriging metamodel and FSI simulationhave been further validated through experimental testsThusa prototype of centrifugal pump based on the geometricparameters in Table 6 has been produced as shown inFigure 6The pump is fixed on the test bench a 1MWelectricmachinery drives the pump impeller A displacement sensorinstalled at the pump bearing measures the displacementresponse of the bearing block The water circulates in a closeloop and the flow rate is constant The basic test parameterscorrespond to parameters of FSI simulations summarized inTable 3

Table 7 compares the results of Kriging predictor FSIsimulation and experimental test The results given by thethree methods well agree to each other The error of Krigingmetamodel is 31 the error of FSI simulation is 44and they are both less than 5 It is well known that theexperimental test plays an indispensable role in validating theoptimization design for centrifugal pump However manu-facturing a prototype pump or the experimental equipment isexpensive Furthermore due to the complexity of the modelthe FSI simulation is time-consuming Hence the optimizeddesign of the pump shouldminimize both elements costs andcalculation time

This research shows that the predictive ability of the Krig-ing model has been well justified both by FSI simulations andby experimental test Therefore the well validated surrogatemodel can completely replace time-consuming FSI simula-tions and substitute a great majority of expensive experimenttestsThat is the Kriging surrogatemodel provides great con-venience in studying the vibration performance of centrifugal

Figure 6 The prototype of centrifugal pump corresponding to theoptimization result

pump especially for accumulating the practical experience ofpump design Moreover the well validated surrogate modelcan benefit both the further development of centrifugal pumpmanufacturer and the improvement of the pump designerrsquosability Therefore the surrogate model method makes theinvestigation of pump performance easy which is of courseon the promise that the model accuracy is high enough

42 The Analysis of Impeller Mechanical Behavior throughFSI Themechanical characteristics of the pump impeller aresignificant for the working behaviors of centrifugal pumpsDuring the working process of a centrifugal pump theperiodic hydraulic loads imposed on the pump will lead tothe dynamic deformation of the impeller and impeller shaftMoreover the dynamic deformation will further influencethe flow field distribution The analysis of the mechanicalbehavior of the impeller is a typical FSI problem In generalthere aremainly three types of loads acting on pump impellercoupling pressure load from the fluid gravity and inertiaforce due to the circular motion However all these loadsare finally balanced by the support reaction of the bearingsand the input moment of the pump FSI method allowsinvestigating the dynamic force of the impeller and theinput moment and the calculation results are important tohighlight the mechanical properties of the centrifugal pumpFor example the analysis results can help in choosing theappropriate sizes and types of impeller shaft and bearings

Actually either the radial force of the pump impeller orthe input moment of the pump cannot be easily measuredbecause of the expensive measuring equipment and complexmultipoints installationWhen the simulation model is accu-rate FSI simulationmethod shows advantage in obtaining theradial force and input moment Furthermore as previouslymentioned in Section 41 the results obtained by Krigingpredictorrsquos FSI simulation and experiment test well agree toeach other The comparison indicates that the FSI simulationmodel is well validated and FSI simulation model leads toreliable results This research investigates the radial forceof the pump impeller and the input moment of the pumpthrough FSI method In addition the radial force and inputmoment are calculated based on the FSI simulation modelin Section 41 The basic settings of the FSI simulation are


Table 6 The result of optimization

1205931 (deg) 1205821 1205932 (deg) 1205822 119897 (mm) RMS (mm) Average time (s)2637 0938 8331 0934 15689 03341 1836

Table 7 Results of Kriging FSI simulation and experiment

Kriging FSI ExperimentRMS (mm) 03341 03296 03447

0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500

Figure 7 The radial force of the pump impeller

0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)

Figure 8 The input moment of the pump

unchanged such as the definitions of material propertiessimulation time step boundary condition and couplinginterface

Figures 7 and 8 show the results of dynamic radial forceof pump impeller and dynamic input moment of the pumprespectivelyThe time-dependent transient force andmomentin each time step are calculated by direct integrationmethodBoth the curve of radial force and moment indicate cyclicalfluctuation in general and six cycles corresponding to afull pump impeller revolution However as a result of thetongue region shown in Figure 3(c) there exists more or lessdisturbance on the wave crest or wave trough

5 Conclusions

This paper proposes a Kriging-based optimization methodfor the vibrations optimization of centrifugal pumps which

well integrates Kriging surrogate model FSI simulations andexperimental tests Moreover the proposed method over-comes the faults of expensive computation and cost and ithas been proved to be effective on improving pump vibrationperformance in terms of minimum cost and reduction ofdevelopment period

The Kriging surrogate model of pump vibration perfor-mance has been established based on the sample pointsand the results at the test points showed that the Krigingpredictor well agreed with the FSI simulationsThe final opti-mized decision variables have been obtained using MIGAa prototype has been manufactured according to optimizedvalues of geometrical parameters of the pump Experimentaltests carried out on prototype well agreed with the results ofKriging metamodel and FSI simulation

Furthermore based on the final optimized decisionvariables the dynamic mechanical performance of pumpimpeller was further investigated using FSI method Theresults showed that the radial force curve and moment curveexhibited cyclical fluctuation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by the National Natural ScienceFoundation of China (no 11172108) This financial support isgratefully acknowledged

References

[1] J S Anagnostopoulos ldquoA fast numerical method for flow analy-sis and blade design in centrifugal pump impellersrdquo Computersand Fluids vol 38 no 2 pp 284ndash289 2009

[2] L Zhou W Shi and S Wu ldquoPerformance optimization ina centrifugal pump impeller by orthogonal experiment andnumerical simulationrdquoAdvances inMechanical Engineering vol2013 Article ID 385809 7 pages 2013

[3] S Derakhshan B Mohammadi and A Nourbakhsh ldquoThecomparison of incomplete sensitivities and Genetic algorithmsapplications in 3D radial turbomachinery blade optimizationrdquoComputers and Fluids vol 39 no 10 pp 2022ndash2029 2010

[4] A Papierski and A Blaszczyk ldquoMultiobjective optimization ofthe semi-open impeller in a centrifugal pump by a multilevelmethodrdquo Journal of Theoretical and Applied Mechanics vol 49no 2 pp 327ndash341 2011

[5] M R Hodkiewicz and M P Norton ldquoThe effect of change inflow rate on the vibration of double-suction centrifugal pumpsrdquoProceedings of the Institution of Mechanical Engineers E Journalof Process Mechanical Engineering vol 216 no 1 pp 47ndash582002


[6] S Guo and YMaruta ldquoExperimental investigations on pressurefluctuations and vibration of the impeller in a centrifugal pumpwith vaned diffusersrdquo International Journal B vol 48 no 1 pp136ndash143 2005

[7] C G Rodriguez E Egusquiza and I F Santos ldquoFrequenciesin the vibration induced by the rotor stator interaction in acentrifugal pump turbinerdquo Journal of Fluids Engineering vol129 no 11 pp 1428ndash1435 2007

[8] F J Wang L X Qu L Y He and J Y Gao ldquoEvaluation offlow-induced dynamic stress and vibration of volute casing fora large-scale double-suction centrifugal pumprdquo MathematicalProblems in Engineering vol 2013 Article ID 764812 9 pages2013

[9] F K Benra ldquoApplication of fluidstructure interaction methodsto determine the impeller orbit curves of a centrifugal pumprdquo inProceedings of the 5th IASMEWSEAS International Conferenceon Fluid Mechanics and Aerodynamics pp 169ndash174 AthensGreece 2007

[10] A Fontanals Garcıa A D J Guardo Zabaleta M G CoussiratNunez and E Egusquiza Estevez ldquoNumerical study of thefluidmdashstructure interaction in the diffuser passage of a centrifu-gal pumprdquo in Proceedings of the 4th International Conferenceon Computational Methods for Coupled Problems in Science andEngineering pp 1ndash10 2011

[11] Q Jiang L Zhai L Wang and D Wu ldquoFluid-structureinteraction analysis on turbulent annular seals of centrifugalpumps during transient processrdquo Mathematical Problems inEngineering Article ID 929574 Art ID 929574 22 pages 2011

[12] S Yuan J Pei and J Yuan ldquoNumerical investigation onfluid structure interaction considering rotor deformation for acentrifugal pumprdquo Chinese Journal of Mechanical Engineeringvol 24 no 4 pp 539ndash545 2011

[13] J Pei S Yuan and J Yuan ldquoFluid-structure coupling effects onperiodically transient flow of a single-blade sewage centrifugalpumprdquo Journal ofMechanical Science andTechnology vol 27 no7 pp 2015ndash2023 2013

[14] J P C Kleijnen ldquoKriging metamodeling in simulation areviewrdquo European Journal of Operational Research vol 192 no3 pp 707ndash716 2009

[15] T W Simpson T M Mauery J J Korte and F MistreeldquoKriging models for global approximation in simulation-basedmultidisciplinary design optimizationrdquo AIAA Journal vol 39no 12 pp 2233ndash2241 2001

[16] S Sakata F Ashida and M Zako ldquoStructural optimizatiionusing Kriging approximationrdquo Computer Methods in AppliedMechanics and Engineering vol 192 no 7-8 pp 923ndash939 2003

[17] D Huang T T Allen W I Notz and N Zeng ldquoGlobaloptimization of stochastic black-box systems via sequentialkriging meta-modelsrdquo Journal of Global Optimization vol 34no 3 pp 441ndash466 2006

[18] E Davis and M Ierapetritou ldquoA kriging based method forthe solution of mixed-integer nonlinear programs containingblack-box functionsrdquo Journal of GlobalOptimization vol 43 no2-3 pp 191ndash205 2009

[19] H Li T Qiu B Zhu J Wu and XWang ldquoDesign optimizationof coronary stent based on finite element modelsrdquoThe ScientificWorld Journal vol 2013 Article ID 630243 10 pages 2013

[20] J D Martin ldquoComputational improvements to estimatingKriging metamodel parametersrdquo Journal of Mechanical Designvol 131 no 8 Article ID 084501 7 pages 2009

[21] A Matta M Pezzoni and Q Semeraro ldquoA Kriging-basedalgorithm to optimize production systems approximated by

analytical modelsrdquo Journal of Intelligent Manufacturing vol 23no 3 pp 587ndash597 2012

[22] M ZakerifarW E Biles andGW Evans ldquoKrigingmetamodel-ing in multiple-objective simulation optimizationrdquo Simulationvol 87 no 10 pp 843ndash856 2011

[23] R Jin W Chen and T W Simpson ldquoComparative studies ofmetamodelling techniques under multiple modelling criteriardquoStructural and Multidisciplinary Optimization vol 23 no 1 pp1ndash13 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



approach to investigate the vibrating frequencies in thevibration of centrifugal pump induced by the rotor statorinteraction (RSI) Wang et al [8] studied the structuraldynamics characteristics and vibrations of pump volutecasing for a double-suction centrifugal pump using a fluid-structure coupling interface model

Studies agree in considering the fluid-structure interac-tion (FSI) as the source of the highest vibration levels in largecentrifugal pumps Moreover hydraulic excitation forces aredue to the FSI and cause pressure fluctuations mechanicalvibrations and alternating stresses in different componentsof centrifugal pump In recent years the application of FSItheory to centrifugal pumps became more popular and it iswell documented in literature [9ndash13]

Vibration performance is one of the most importantparameters in designing a centrifugal pump Actually exper-imental tests and CFD simulation are the two methodsperformed in order to obtain the centrifugal pump vibrationresponse However both of the two methods cannot beconsidered in optimizing the vibration performance of thepump using an iterative method Appropriate metamodelsmust be established between the decision variables andthe concerned objective functions Therefore metamodeltechnique demonstrates its superiority in the optimizationproblem of engineering

Kriging metamodels [14] were originally proposed bythe South African mining engineer named Danie GerhardusKrige With the rapid development of computer technologyKriging metamodels have been widely used in various fields[15ndash19] Kriging metamodel differs from other metamodelsbecause of the optimal unbiased prediction for the unknownresponse points [20] Compared to traditionally response sur-face methods Kriging shows its superiority in high dimen-sional nonlinear problems and prediction accuracy due tothe stochastic assumption [21] especially for multiobjectiveoptimization problems [22]

This paper presents an effective optimization methodbased on Kriging metamodel The presented method opti-mizes the vibration performance of the centrifugal pumpundergoing FSI phenomena which reasonably take advan-tages of the FSI simulation Kriging metamodel and experi-mental tests Although considerable researches were devotedto investigating the vibration performance of centrifugalpump it should be noted that there exists little literatureevidence on the vibration optimization of centrifugal pumpin particular that combines with FSI phenomenon Thesecond part of the paper deals with the study of the transientmechanical characteristics based on the optimized centrifu-gal pump using FSI method

2 Centrifugal Pump FSI Simulation Model

21 FSI Governing Equations In this study the fluid-struc-ture interaction (FSI) problemrsquos domain Ω consists of twosubdomains Ω119891 and Ω119904 with the boundaries as Γ119891 and Γ119904respectively The subscripts 119891 and 119904 denote the fluid partand solid part respectively The following section defines

the equations that govern the flow and the structural defor-mations of the pump

211 Fluid Flow Equations The fluid flowing through thecentrifugal pump is treated as incompressible and isothermalFor x isin Ω119891 the conservation of mass and Navier-Stokesequations governing the unsteady flow are respectivelywritten as

120597120588119891

120597119905+ nabla sdot (120588119891k119891) = 0

120588119891

120597k119891120597119905

+ 120588119891 (k119891 sdot nabla) k119891 minus nabla sdot 120590119891 = f119891

(1)

where 120588119891 is the fluid density k119891 denotes the fluid particlevelocity at time 119905 f119891 denotes the body forces per unit ofvolume on the fluid and 120590119891 is the stress tensor defined as

120590119891 = minus119901I + 120583 [nablak119891 + (nablak119891)119879] (2)

where 119901 is the pressure I denotes the unit tensor and 120583

represents the absolute viscosity

212 Structural Equations While working the centrifugalpump undergoes large deformation and rotation For x isin Ω119904the conservation of momentum for the solid deformation u119904is described through Lagrangian formulation

120588119904

1205972u1199041205971199052

= nabla sdot 120590119904 + f119904 (3)

where 120588119904 is the solid density 120590119904 represents the Cauchy stresstensor and f119904 denotes the body forces per unit volume on thesolid

Closure for (3) is found by evaluating the stress using therelevant constitutive relationsMoreover since the centrifugalpump is related to large deformation and rotation theconstitutive equations are described using a stress-strainrelationship

213 Interaction between Fluid and Solid As mentionedabove the FSI occurs during the running process of centrifu-gal pump Fluid pressure information transfers to the solidwhile displacements information of the solid transfers to theflow Furthermore on the no-slip fluid-structural interfacethe information exchange between the fluid and solid shouldfollow the equilibrium conditions

k =120597u120597119905

120590119891 sdot n = 120590119904 sdot nforallx isin Γ (4)

where Γ denotes the fluid-structural interface and Γ = Γ119891capΓ119904n represents the unit normal at the interface Γ

22 Decision Variables The working process of centrifugalpump involves vibrations FSI and energy conversion andloss As the ldquoheartrdquo component for a centrifugal pump


333

59

155

60







E

D

AB

C

F

G

1205822

1205931

1205821

1205932

l








Mx (119905) + Cx (119905) + Kx (119905) = F (119905) (5)




1 000 0028 8576 0061 15771 054112 025 0183 7000 0223 18314 040683 051 0191 8729 0744 18441 042284 076 0728 7390 0752 18525 038675 102 0777 8322 0484 16110 041086 127 0199 8966 0109 15644 052717 153 0557 8780 0459 18907 052518 178 0085 7119 0427 16873 058129 203 0467 8424 0305 16195 0567210 229 0264 7508 0321 15517 0412811 254 0232 8051 0378 14797 0360312 280 0288 8153 0834 17763 04529

117 2949 0646 8864 0443 16364 03627118 2975 0817 8475 0516 16831 03667119 3000 0891 7610 0785 18949 05792

Structural part

Impeller shaft

Z

YX

(a) Full model

Fluid partImpellerZ

YX















CSDmodel

CFDmodel

Coupling interface

Scan Scan

CSDsolver

CFDsolver

FSIserver 1

FSIserver 2

FSIcontroller

Data Data

CSDresults

CFDresults

CFDpostprocessing

CSDpostprocessing


models





Read Read




119880RMS = radic1

119873

119873

sum

119894=1

1198802119894

(6)






119910 (119909) =

119899

sum

119896=1

120573119896119891119896 (119909) + 119885 (119909) (7)




Co V [119885 (120591) 119885 (119909)] = 1205902119877 (120579 120591 119909) (8)



119877 (120579 120591 119909) = exp(minus

119898

sum

119895=1

120579119895(120591119895 minus 119909119895)2) (9)





119904 (119909) = 1205902(1 + u119879(F119879Rminus1F)


(10)




R =[[

[

119877 (1199091 1199091) sdot sdot sdot 119877 (1199091 119909119873)

d

119877 (119909119873 1199091) sdot sdot sdot 119877 (119909119873 119909119873)

]]

]

(11)




2=

1

119898(Y minus F)


(12)



Minimize 120595 (120579) = 120590(120579)21003816100381610038161003816R120579

1003816100381610038161003816

1119898


(13)





1205902 000453120579 [8406 14718 18851 21512 21401]

0 5 10 15 20 25 30

035

04

045

05

055

06

065

Serial number

RMS

of d

ispla

cem

ent r

espo

nse (

mm

)

KrigingFSI





Find X = [1205931 1205821 1205932 1205822 119897]119879

Minimize 119880RMS = 119891 (1205931 1205821 1205932 1205822 119897)


∘

002 le 1205821 le 098

70∘le 1205932 le 90

∘

002 le 1205822 le 098

145mm le 119897 le 195mm

(14)




















0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500


0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)




5 Conclusions







Acknowledgment


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










333

59

155

60







E

D

AB

C

F

G

1205822

1205931

1205821

1205932

l








Mx (119905) + Cx (119905) + Kx (119905) = F (119905) (5)




1 000 0028 8576 0061 15771 054112 025 0183 7000 0223 18314 040683 051 0191 8729 0744 18441 042284 076 0728 7390 0752 18525 038675 102 0777 8322 0484 16110 041086 127 0199 8966 0109 15644 052717 153 0557 8780 0459 18907 052518 178 0085 7119 0427 16873 058129 203 0467 8424 0305 16195 0567210 229 0264 7508 0321 15517 0412811 254 0232 8051 0378 14797 0360312 280 0288 8153 0834 17763 04529

117 2949 0646 8864 0443 16364 03627118 2975 0817 8475 0516 16831 03667119 3000 0891 7610 0785 18949 05792

Structural part

Impeller shaft

Z

YX

(a) Full model

Fluid partImpellerZ

YX















CSDmodel

CFDmodel

Coupling interface

Scan Scan

CSDsolver

CFDsolver

FSIserver 1

FSIserver 2

FSIcontroller

Data Data

CSDresults

CFDresults

CFDpostprocessing

CSDpostprocessing


models





Read Read




119880RMS = radic1

119873

119873

sum

119894=1

1198802119894

(6)






119910 (119909) =

119899

sum

119896=1

120573119896119891119896 (119909) + 119885 (119909) (7)




Co V [119885 (120591) 119885 (119909)] = 1205902119877 (120579 120591 119909) (8)



119877 (120579 120591 119909) = exp(minus

119898

sum

119895=1

120579119895(120591119895 minus 119909119895)2) (9)





119904 (119909) = 1205902(1 + u119879(F119879Rminus1F)


(10)




R =[[

[

119877 (1199091 1199091) sdot sdot sdot 119877 (1199091 119909119873)

d

119877 (119909119873 1199091) sdot sdot sdot 119877 (119909119873 119909119873)

]]

]

(11)




2=

1

119898(Y minus F)


(12)



Minimize 120595 (120579) = 120590(120579)21003816100381610038161003816R120579

1003816100381610038161003816

1119898


(13)





1205902 000453120579 [8406 14718 18851 21512 21401]

0 5 10 15 20 25 30

035

04

045

05

055

06

065

Serial number

RMS

of d

ispla

cem

ent r

espo

nse (

mm

)

KrigingFSI





Find X = [1205931 1205821 1205932 1205822 119897]119879

Minimize 119880RMS = 119891 (1205931 1205821 1205932 1205822 119897)


∘

002 le 1205821 le 098

70∘le 1205932 le 90

∘

002 le 1205822 le 098

145mm le 119897 le 195mm

(14)




















0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500


0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)




5 Conclusions







Acknowledgment


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1 000 0028 8576 0061 15771 054112 025 0183 7000 0223 18314 040683 051 0191 8729 0744 18441 042284 076 0728 7390 0752 18525 038675 102 0777 8322 0484 16110 041086 127 0199 8966 0109 15644 052717 153 0557 8780 0459 18907 052518 178 0085 7119 0427 16873 058129 203 0467 8424 0305 16195 0567210 229 0264 7508 0321 15517 0412811 254 0232 8051 0378 14797 0360312 280 0288 8153 0834 17763 04529

117 2949 0646 8864 0443 16364 03627118 2975 0817 8475 0516 16831 03667119 3000 0891 7610 0785 18949 05792

Structural part

Impeller shaft

Z

YX

(a) Full model

Fluid partImpellerZ

YX















CSDmodel

CFDmodel

Coupling interface

Scan Scan

CSDsolver

CFDsolver

FSIserver 1

FSIserver 2

FSIcontroller

Data Data

CSDresults

CFDresults

CFDpostprocessing

CSDpostprocessing


models





Read Read




119880RMS = radic1

119873

119873

sum

119894=1

1198802119894

(6)






119910 (119909) =

119899

sum

119896=1

120573119896119891119896 (119909) + 119885 (119909) (7)




Co V [119885 (120591) 119885 (119909)] = 1205902119877 (120579 120591 119909) (8)



119877 (120579 120591 119909) = exp(minus

119898

sum

119895=1

120579119895(120591119895 minus 119909119895)2) (9)





119904 (119909) = 1205902(1 + u119879(F119879Rminus1F)


(10)




R =[[

[

119877 (1199091 1199091) sdot sdot sdot 119877 (1199091 119909119873)

d

119877 (119909119873 1199091) sdot sdot sdot 119877 (119909119873 119909119873)

]]

]

(11)




2=

1

119898(Y minus F)


(12)



Minimize 120595 (120579) = 120590(120579)21003816100381610038161003816R120579

1003816100381610038161003816

1119898


(13)





1205902 000453120579 [8406 14718 18851 21512 21401]

0 5 10 15 20 25 30

035

04

045

05

055

06

065

Serial number

RMS

of d

ispla

cem

ent r

espo

nse (

mm

)

KrigingFSI





Find X = [1205931 1205821 1205932 1205822 119897]119879

Minimize 119880RMS = 119891 (1205931 1205821 1205932 1205822 119897)


∘

002 le 1205821 le 098

70∘le 1205932 le 90

∘

002 le 1205822 le 098

145mm le 119897 le 195mm

(14)




















0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500


0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)




5 Conclusions







Acknowledgment


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










CSDmodel

CFDmodel

Coupling interface

Scan Scan

CSDsolver

CFDsolver

FSIserver 1

FSIserver 2

FSIcontroller

Data Data

CSDresults

CFDresults

CFDpostprocessing

CSDpostprocessing


models





Read Read




119880RMS = radic1

119873

119873

sum

119894=1

1198802119894

(6)






119910 (119909) =

119899

sum

119896=1

120573119896119891119896 (119909) + 119885 (119909) (7)




Co V [119885 (120591) 119885 (119909)] = 1205902119877 (120579 120591 119909) (8)



119877 (120579 120591 119909) = exp(minus

119898

sum

119895=1

120579119895(120591119895 minus 119909119895)2) (9)





119904 (119909) = 1205902(1 + u119879(F119879Rminus1F)


(10)




R =[[

[

119877 (1199091 1199091) sdot sdot sdot 119877 (1199091 119909119873)

d

119877 (119909119873 1199091) sdot sdot sdot 119877 (119909119873 119909119873)

]]

]

(11)




2=

1

119898(Y minus F)


(12)



Minimize 120595 (120579) = 120590(120579)21003816100381610038161003816R120579

1003816100381610038161003816

1119898


(13)





1205902 000453120579 [8406 14718 18851 21512 21401]

0 5 10 15 20 25 30

035

04

045

05

055

06

065

Serial number

RMS

of d

ispla

cem

ent r

espo

nse (

mm

)

KrigingFSI





Find X = [1205931 1205821 1205932 1205822 119897]119879

Minimize 119880RMS = 119891 (1205931 1205821 1205932 1205822 119897)


∘

002 le 1205821 le 098

70∘le 1205932 le 90

∘

002 le 1205822 le 098

145mm le 119897 le 195mm

(14)




















0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500


0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)




5 Conclusions







Acknowledgment


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119904 (119909) = 1205902(1 + u119879(F119879Rminus1F)


(10)




R =[[

[

119877 (1199091 1199091) sdot sdot sdot 119877 (1199091 119909119873)

d

119877 (119909119873 1199091) sdot sdot sdot 119877 (119909119873 119909119873)

]]

]

(11)




2=

1

119898(Y minus F)


(12)



Minimize 120595 (120579) = 120590(120579)21003816100381610038161003816R120579

1003816100381610038161003816

1119898


(13)





1205902 000453120579 [8406 14718 18851 21512 21401]

0 5 10 15 20 25 30

035

04

045

05

055

06

065

Serial number

RMS

of d

ispla

cem

ent r

espo

nse (

mm

)

KrigingFSI





Find X = [1205931 1205821 1205932 1205822 119897]119879

Minimize 119880RMS = 119891 (1205931 1205821 1205932 1205822 119897)


∘

002 le 1205821 le 098

70∘le 1205932 le 90

∘

002 le 1205822 le 098

145mm le 119897 le 195mm

(14)




















0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500


0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)




5 Conclusions







Acknowledgment


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























0 1 2 3 4 5 6

0500

100015002000250030003500

Time-periodic

Radi

al fo

rce (

N)

minus1000

minus500


0 1 2 3 4 5 6

0

Time-periodic

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

Mom

ent (

Nmiddotm

)




5 Conclusions







Acknowledgment


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article optimization and analysis of...

Documents