Optimization of Factors Affecting Vibration Characteristics of …downloads.hindawi.com/journals/mpe/2019/1845056.pdf · 2019-07-30 · of Unbalance Response for Machine Motorized

Research ArticleOptimization of Factors Affecting Vibration Characteristicsof Unbalance Response for Machine Motorized Spindle UsingResponse Surface Method

Elhaj A I Ahmed12 and Li Shusen 1

1College of Mechanical and Electrical Engineering Northeast Forestry University Harbin 150040 China2Department of Mechanical Engineering Faculty of Engineering Science University of Nyala Nyala PO Box 155 Sudan

Correspondence should be addressed to Li Shusen lishusenzp126com

Received 4 October 2018 Revised 20 December 2018 Accepted 22 January 2019 Published 10 February 2019

Academic Editor Mohammed Nouari

Copyright copy 2019 Elhaj A I Ahmed and Li Shusen This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this study the response surface (RS) method and forced rotordynamic analyses together with Finite-Element-Analysis (FEA)have been established to optimize the factors affecting the vibration characteristics The spindle specification bearings locationscutting force and motor-rotor unbalance mass are proposed to represent the design factors and then they are utilized to developMachineMotorized Spindle (MMS)The FEA-basedDesign of Experiment (DOE) is adopted to simulate the output responses withthe input factors wherein these DOE design points are used to carry out the RS models to visualize more obvious factors affectingthe dynamic characteristicsofMMSThe sensitivities of these factors and their contributions to the vibration of imbalance responsehave been evaluated by using the RS models The simulation results show that the motor-rotor shaft inner diameter the distanceof the back bearing location and the rotating unbalance-mass are highly sensitive to the vibration characteristics compared to theother factors It is found that more than two-fifths of total vibration response amplitude has been conducted by induced rotatingimbalance mass The results also showed that the proposed factors optimization method is practicable and effective in improvingthe vibration response characteristics

1 Introduction

In the machined component the machining accuracy ismost critical and is related to several factors wherein thesefactors include thermal error positioning error tool-holderand rotating unbalance force induced error andmotion error[1] To improve the accuracy of machining these afore-mentioned errors have been significantly recognized andrewarded by the certain situation of art method for instancethe machine-tool movement errors [2] thermal induceddeformation [3] and deflection due to the cutting tool insmall milling [4] Optimization of parameters influencing thevibration characteristics of spindle under rotating unbalanceforce induced is the research foundation for lively balancingcontrol that is more important in vibration monitoring andcontrol Because of complex structures of MMS the opti-mization of parameters affecting its dynamic characteristics

due to motor-rotor unbalance induced is much difficult [5]Recently considerable researches have been carried out toimprove the vibration response characteristics of MMS Theeffects of design variables or operation parameters on systemdynamics response have been covered with the inclusivevariety of researches [6ndash12] These previous researches werefocused on improving the dynamic performances of high-speed (HS)motorized spindle through optimizing the designfactors The effect of operating parameters on the systemdynamics response is also enclosed by several kinds ofresearches [13ndash19] they focused on improving the machiningparameters that affect the surface roughness of differentmaterials machined on various CNCmachines A significanteffort has been exerted on the design optimization of amotorized-spindle-bearing system Lee et al [20] proposeda design optimization method to reduce the mass of a flexiblerotor supported in ball bearings via rotational velocities and

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1845056 12 pageshttpsdoiorg10115520191845056

2 Mathematical Problems in Engineering

ProtectionFlangeProtection

Flange

Drainage HoleDrainage HoleDrainage HoleDrainage Hole

Front BearingHousing

Stator Housing Coolant Inlet Coolant Outlet

StatorRear BearingHousing Hydraulic

CylinderFlange

EncoderRing

Oslash24

5m

m

Oslash24

6m

m

Oslash29

6m

m

Figure 1 The Section view of the MMS

load reliant on stiffness characteristics in constraints of thebearing fatigue life and system eigenvalues Lin Y-H and LinS-C [21] proposed the finite-element-method to develop thereduced-weight design of a spindle supported by oil-lubricantbearing under a frequency constraint using a successivequadratic programming method Lin C-W [22] developed amathematical model for an optimization problem to reducethe total costs of the tolerance and a predictable value of thedissipated quality under definite constraints They proposedan optimization technique that integrated the Monte-CarloSimulation and genetic-algorithm to simultaneously explorethe optimal design vectors and tolerance of bearings LinC-W [23] also applied Taguchi technique to recognize theoptimal values of design vectors for a robust HS spindlesystem with respect to the signalnoise ratio of systemFirst-Mode-Natural-Frequency (FMNF) under gyroscopicmoment effect Most of these studies were focused onimproving the dynamic characteristics through simultane-ouslyminimizing the material used in the structure andmax-imizing the FMNF of different spindle-systems Howeverthe studies on optimization of factors affecting the vibrationresponse characteristics of MMS under rotating unbalanceforce effects due to its complex dynamic model has beenrelatively lacking Usually the designer mostly focused onunforced vibration design optimization However the influ-ence of the gyroscopic moments and motor-rotor imbalanceforces on the dynamic response are slightly disregardedThis system design manner is unsatisfactory for balancingand control of the machine vibration In HS machining theextra loads created by rotating elements may well cause agreatest influence on the quality of the machined componentFor instance due to HS machining the gyroscopic momenteffects and rotating unbalance forces will generate morevibration for the MMS which directly affects the dynamic

performances of MMS Therefore to precisely predict theactual performance of MMS it is necessary to conduct thevibration design under rotating elements forces inducedIn the present work optimization of the factors affectingvibration characteristics under rotating unbalance force effectis presented The RS method and forced rotordynamic anal-yses along with FEA are used to determine the optimalcombinations of the factors that improve the vibration behav-iors of MMS The spindle specifications bearings locationscutting force and motor-rotor unbalance mass are proposedto represent the design factors and then they are utilizedto develop MMS The Central-Composed-Design (CCD)and Box-Behnken Design (BBD) approaches due to theirefficiency in providing considerable data in a least numberof required statistical experiments are adopted to generatethe DOE wherein these DOE design points data then areused to build theRSmodels To visualizemore obvious designfactors affecting the dynamic characteristics of MMS the RSevaluation is carried out in detail The sensitivities of designfactor on the vibration response and the optimization of theirlevels are covered by using these RS models Finally directreal solution of the finite element model (FEM) and literatureresults are used to examine the quality of the RS modelconsidered

2 Motorized Spindle System Specifications

Figure 1 shows the section view of lathe motorized spindlewhich was established to hold motorized spindle rather thanconventional-pulley drive L-series lathe spindle standard JISA2-6 [24] It is designed to withstand maximum 3700N ofcutting force and 7000 rpm of machining velocity and pulledby a 168kw motor coupled to the shaft with synchronous-motor Type 1FE1093-6WV The SKF-roller-bearings under

Mathematical Problems in Engineering 3

000 20000 (mm)10000

Figure 2 The developed finite element model of MMS withmeshing

the designation of 7220BECBY and NNCF5013CV werepresented to support spindle at bearings locations TheMMSshown in Figure 1 is developed as an example case to conductthe FEA-based DOE for RS evaluation In order to predict thevibration response under all sensational loads theworkpiecesholder (chuck standard B6151sc) is also considered in theFEM

21 Finite Element Model-Based Experimental Design TheMMSwithstanding various forces under operation conditionis utilized in this work wherein these forces include therotating unbalance force transmitted to the spindle via theelectric motor and the cutting-load applied at the spindle-nose Based on ANSYS SpaceClaim the 3D FEM for MMS isestablished In order to simplify the simulation process all theinduced masses for the chuck and motor-rotor are modeledas a point mass (MASS21 element) with remote point typeconnection under inertia load effects The bearings are mod-eled by using 2D elastic spring-damper-element (COMBI214)by ground-to-body type connection The bearings stiffnessis estimated according to the bearing specifications whichis 4x10E5Nmm for front bearing and is 35x10E6Nmm forthe back bearing The structural damping ratio is 005 Thebearing stiffness is adjusted as an invariable value since theeffect of a load of bearing and rotational speed on the stiffnessof bearing is disregarded To constrain degree of freedomon axial direction the displacement constraint is added toboth front bearings positions and back bearing positionAutomatic mesh division method is used to mesh the entirestructural model of MMSThe entire spindle model has beenmeshed by considering the magnitude and accuracy of thedesign the selected size of the mesh unit was 10 mm hencethe FEM meshing is shown in Figure 2 The material used inthis study is structural steel (E = 210GPa 120588 = 7850kgm3 and120592 = 03)

22 Unbalance Response Analysis To analyze the vibrationof unbalance response the motor-rotor unbalance forceinduced and cutting force are used for exciting the FEMThe unbalance force varied occasionally according to thesinusoidal rule as follows [5]

F (119905) = 1198721199031205962 sin120596119905 = 997888rarrΧ1205962 sin120596119905 (1)

where M r 120596 and X are represented rotor mass radiusof the rotor excitation frequency and amplitude response

Chip

Workpiece

t

-

ＭＮ

Ｈ

N

FR

CuttingTool

Figure 3 Merchant circle with cutting force diagram

respectively In this study the excitation force is defined byrotor unbalance mass and its radius in ANSYS mechanicalenvironment To calculate the cutting force we examined thesample case of 200HB AISI 4340 part of steel with a diameterof 110 mm length of 50 mm under machining conditionof 035 mmrev feed rate 6 mm depth of cut and 600 rpmspindle speed According to this case the components of theresultant cutting force can be found from the Merchant circleshown in Figure 3 as follows [24]

F119905 = ΤΒ119888N119904cos (120573 minus 120572)

sin (120601) cos (120601 + 120573 minus 120572)

F119891 = ΤΒ119888N119904sin (120573 minus 120572)

sin (120601) cos (120601 + 120573 minus 120572)

(2)

where T Bc and Ns represent chip thickness the widthof the cut in a radial direction and shear strength of thematerial respectivelyThe tangential force factors and specificcutting can be determined according to [24] In this studythe vibration responses for the stress strain and deformationamplitude at span between bearings were taken as outputresponses in the evaluation of MMS The definitions offactors used to control the output responses are definedin detail where their lower and upper levels are shown inTable 1The forced rotordynamic analyses for theMMS underexcitation of cutting force and motor-rotor unbalance massare established within the excitation frequency range of 60-600Hz by using the direct integration method

23 Design of Experiment The DOE is a method originallyadvanced for prototypical fitting with experimental data inwhich the affiliation between the factors their interactionsand the output response is defined [25] In RS methodthe DOE can be used to fit the simulated response data tomathematical models The RS method contains the numberof DOE types such as CCDs and BBD in which each typeis used to achieve specific functionality The main differenceof BBD from CCDs is three level quadratic designs in whichthe explored space is represented by [minus1 0 +1] For more


Table 1 The definition of design factors their lower and upper limits and the FEM nominal level

Definition Design factors Lower Level Nominal Level Upper LevelThe inner diameter of spindle shaft at the back bearings location Y1 [mm] 40 45 55The inner diameter of spindle shaft at the motor-rotor location Y2 [mm] 40 45 60The inner diameter of spindle shaft at the front bearings location Y3 [mm] 50 54 60The locations of back bearings set to the back end of the motor rotor Y4 [mm] 10 50 70The locations of first front bearings set to the front end of the motor rotor Y5 [mm] 20 50 60The locations of second front bearings set to the front end of the motor rotor Y6 [mm] 10 50 60Cutting Force Y7 [N] 3330 3700 4000Rotating Unbalance mass Y8 [kg] 0 6 10

quality experiment designs the CCDs and BBD due to theirefficiency in providing much data in a nominal number ofrequired statistical experiments have been adopted to run theexperiments These experiments were adjusted according toupper and lower limits of the factors explained in Table 1The numbers of experimental runs are conducted to find outthe responses for stress strain and total deformation (TD)at the bearings span as well as Structural Weight (SW) TheAuto-Define (AD) and Face-Cantered (FC) options are usedto perform the experiments of CCDs The experiments arecarried out as follows first 82 experimental runs for AD-CCD second 82 experimental runs for FC-CCD and third65 experimental runs for BBD Then the best results havebeen obtained after repeating the experiments many times foreach approach In the later sections the comparison betweenthese experiments designs is carried out to define the mostrefined RS model

3 Results and Discussions

After the design space is sampled in DOE the responsesdataset can be obtained through the FEM simulation Toevaluate the sensitivities of factors on the vibration responsesthe variation of input with output responses is graphicallypresented based on genetic-aggregation method For morerefined RS results the best-fit curve and its verifications arecarried out to evaluate the quality of the RS model Theinterpolation models that provided continuous variation ofthe responses with respect to the design factors are used to fitthe response surface modelThe goodness fit curves for stress(P39) strain (P40) and TD (P41) and their verification arerepresented in Figure 4 From Figures 4(a) 4(b) and 4(c) itcan be shown that the scatter charts presented are normalizedfor the values of each output response Among these modelsFigure 4(c) ismore precise from the view of verification pointand it can be clearly seen in Figure 4(d) after refinement

As results discussed for the quality of RS the BBDapproach is more significant than CCDs approaches from theview of the best-fit plots and verification points Thus theDOE models generated by CCD methods could be omittedfrom later RS evaluation The raw BBD design pointrsquos dataare provided in Table 2 in which the most design pointsobtained by BBD are described These BBD design pointsare used to build RS models for the sensitivities analysis inaddition to the RS optimizer wherein the RS optimizer drawsinformation from the RS model

31 Sensitivities Analysis The contributions of the variableson the dynamic response are graphically plotted in Figure 5It revealed that the higher input percentages show that thecorresponding design variable has a stronger effect on theoutput amplitude of stress P39 strain P40 and TD P41The factors Y2 Y4 and Y8 have the greatest effect on theoutput response whereas the other variables have the smallestinfluences on the output response From Figures 5(a) 5(b)and 5(c) it can be seen that the effect of factor Y8 on thevibration amplitude is the first factor Y4 is the second andfactor Y2 is the third Among these factors the vibrationresponse conducted by Y8 is the largest when it comparedto Y2 and Y4 For example in Figure 5(a) Y4 and Y8 aremore effective parameters for stress response In Figure 5(b)Y2 and Y8 are more significant parameter for strain responsewhile in Figure 5(c) their effect on TD response is as followsY8 Y4 and Y2 respectively Thus the rotating unbalance-mass (Y8) and the motor-rotor shaft inner diameter (Y2) arethe vital factors that entirely control the vibration responsesfor MMS

In order to obtain the specific contribution of factorsof Y4 Y2 and Y8 on the vibration responses the interpo-lation models that provided continuous variation of inputwith output responses are graphically plotted in Figures 6and 7 respectively As seen the effect of each factor onoutput response at different step points can be observedand the least output response always yields the optimumdynamic characteristics Thus the smallest amount of theoutput response will define the most favorable level ofeach factor Figure 6 shows that the relationship betweenthe output response with Y4 in Figures 6(a) and 6(c) islinear whereas in Figure 6(b) it is nonlinear Similarlyit can be shown from Figure 7 that the variation of theoutput response with Y2 and Y8 is approximately nonlin-ear where the vibration response is decreased or elevatedwith increasing the variable space along their differencelimits

As a result to enhance an optimum performance ofMMS the factors Y2 and Y4 should be kept at maximumpotential while the factor Y8 should be kept as minimumas possible From the view of RS evaluation the optimalcombinations of the response points observed are as followsY1 [475mm] Y2 [50mm] Y3 [55mm] Y4 [40mm] Y5[35mm] Y6 [40mm] Y7 [36652N] and Y8 [5kg] stressresponse [0108MPa] strain response [21E-7mmmm] andTD response [031mm]


Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01

Observed from Design Points0 01 02 03 04 05 06 07 08 09 1

Predicted vs Observed - Normalized Values

P39P40P41

Verification Points for P39Verification Points for P40Verification Points for P41

(a)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0



P39P40P41


(b)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41

Verification Points for P39Verification Points for P40Verification Points for P41(c)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0




(d)

Figure 4The output response simulated by the RS vs that observed from the DOE (a) Auto-Define-CCD (b) Face-Centred-CCD (c) BBDand (d) refined BBD

32 Optimization To find the optimum combination of vari-ables that improve the dynamic response the optimizationis carried out based on RS evaluations in Section 31 Theobjective function and constraint are defined according to(3) including SW and TD The two most popular RS opti-mizations are Multiobjective Genetic Algorithm (MOGA)and screening methods (SM) To obtain more refined resultsMOGA and SM have been utilized to increase the numbersof the RS optimization runs The Direct Optimization (DO)is also conducted to check the quality of RS optimizationsince it utilizes a real calculation rather than RS evaluationThe optimization configuration is adjusted to generate 2000samples initially and 100 samples per iteration and found theoptimum candidates point in a maximum of 20 iterations

Each optimization method has suggested three optimumcandidate points as summarized in Table 3 From the resultsshown in Table 3 it is noted that the candidatersquos points forRS are comparable to real solutions of DO except minorvariation among the results Based on the criteria of themaximum allowed cutting force rotating unbalance massand minimum SW the candidatesrsquo points highlighted by lowast inTable 3 are more significant Thus these marked candidatesrsquopoints are suggested to represent the optimal combinationof factors that improve the vibration behaviors of MMSAlso it can be showed that the SW is improved to 1945saving hence not only does the proposed method enhancethe design factors but also the material resource consumedcan be reduced

Find Y = (Y1Y2Y3 Y8) (3)


Table 2 The design points of BBD experiments data

Run Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Stress [MPa] Strain [mmmm] TD [mm] SW [kg]1 475 50 55 40 35 40 3665 5 010805 204E-07 0315214 17426142 40 50 50 40 35 20 3665 5 008361 408E-07 0308941 18186933 55 50 50 40 35 20 3665 5 005578 519E-07 0350467 17444914 40 50 60 40 35 20 3665 5 01347 135E-07 0307662 17308875 55 50 60 40 35 20 3665 5 011597 224E-07 034802 16566856 40 50 50 40 35 60 3665 5 008623 508E-07 0290122 18186937 55 50 50 40 35 60 3665 5 003871 435E-07 0321984 17444918 40 50 60 40 35 60 3665 5 004787 417E-07 0293676 17308879 55 50 60 40 35 60 3665 5 001637 426E-07 0324317 165668510 475 40 50 40 10 40 3665 5 014505 434E-07 0314232 192787211 475 40 60 40 10 40 3665 5 00999 247E-07 0315685 184006612 475 60 50 40 10 40 3665 5 006471 689E-07 0337772 160931413 475 60 60 40 10 40 3665 5 010354 461E-07 0337358 152150914 475 40 50 40 60 40 3665 5 014425 437E-07 0290244 192787215 475 40 60 40 60 40 3665 5 010226 196E-07 029403 184006616 475 60 50 40 60 40 3665 5 00559 661E-07 030426 160931417 475 60 60 40 60 40 3665 5 010669 362E-07 0304491 152150918 475 40 55 10 35 40 3665 0 012801 831E-07 0433314 188596519 475 60 55 10 35 40 3665 0 022891 112E-06 0418388 156740720 475 40 55 70 35 40 3665 0 005891 435E-07 0227901 188596521 475 60 55 70 35 40 3665 0 013504 692E-07 0275409 156740722 475 40 55 10 35 40 3665 10 02121 141E-06 0730907 188596523 475 60 55 10 35 40 3665 10 03791 186E-06 0708668 156740724 475 40 55 70 35 40 3665 10 009917 754E-07 0381429 188596525 475 60 55 70 35 40 3665 10 022209 114E-06 0461991 156740726 475 50 55 10 35 20 3330 5 018199 137E-07 0394835 174261427 475 50 55 70 35 20 3330 5 003595 571E-07 0205719 174261428 475 50 55 10 35 60 3330 5 016095 259E-07 0345363 174261429 475 50 55 70 35 60 3330 5 001503 355E-07 0195859 174261430 475 50 55 10 35 20 4000 5 018034 139E-07 0392291 174261431 475 50 55 70 35 20 4000 5 003717 567E-07 0204992 174261432 475 50 55 10 35 60 4000 5 015953 262E-07 0342976 174261433 475 50 55 70 35 60 4000 5 001501 356E-07 0195504 174261434 40 50 55 40 10 20 3665 5 010713 435E-07 0317481 177678635 55 50 55 40 10 20 3665 5 009067 506E-07 0363253 170258336 40 50 55 40 10 60 3665 5 007181 256E-07 0299475 177678637 55 50 55 40 10 60 3665 5 004018 155E-07 0332554 170258338 40 50 55 40 60 20 3665 5 011663 499E-07 0295696 177678639 55 50 55 40 60 20 3665 5 01021 473E-07 0332294 170258340 40 50 55 40 60 60 3665 5 007022 324E-07 0275881 177678641 55 50 55 40 60 60 3665 5 004082 174E-07 030406 170258342 475 50 50 40 10 40 3665 0 017247 546E-07 039225 178452143 475 50 60 40 10 40 3665 0 009173 749E-07 0393006 169671544 475 50 50 40 60 40 3665 0 01656 475E-07 0362988 178452145 475 50 60 40 60 40 3665 0 009054 899E-07 0366577 1696715- - - - - - - - - - - - -- - - - - - - - - - - - -61 55 50 55 70 35 40 3330 5 006956 276E-07 0211405 170258362 40 50 55 10 35 40 4000 5 006667 257E-07 0356704 1776786


Table 2 Continued

Run Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Stress [MPa] Strain [mmmm] TD [mm] SW [kg]63 55 50 55 10 35 40 4000 5 006236 462E-07 0371103 170258364 40 50 55 70 35 40 4000 5 004448 182E-07 020032 177678665 55 50 55 70 35 40 4000 5 006965 274E-07 0210732 1702583

Table 3 Initial design optimal candidate points and real solution or DO

DesignFactors

InitialDesign

Response Surface Optimization DO(SM)(MOGA) (SM)

1lowast 2 3 1 2 3lowastY1 [mm] 45 544 54 543 507 48 523 507Y2 [mm] 45 599 599 598 592 595 564 584Y3 [mm] 54 599 597 598 595 584 511 594Y4 [mm] 50 684 679 684 667 637 687 534Y5 [mm] 50 422 469 423 325 122 591 109Y6 [mm] 50 503 50 425 519 205 529 424Y7 [N] 3700 39379 39884 38598 35462 38359 36784 33024Y8 [kg] 6 52 53 55 33 47 45 067SW [kg] 18 145 146 145 151 152 154 154TD [mm] 0225 022 022 023 023 025 019 0086

Minimize SW =Nsumk=1

120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to

119879119863 le 026119898119898 119886119899119889 119873 119890119897119890119898119890119899119905119904 11989911990611989811988711989011990340 le 1198841 le 55 40 le 1198842 le 6050 le 1198843 le 60 10 le 1198844 le 7020 le 1198845 le 60 10 le 1198846 le 603330 le 1198847 le 4000 0 le 1198848 le 10

(4)

TheTD found in the proposed approach is relatively highbecause the simulated results are conducted in maximumexcitation frequency of 600Hz According to the designcriteria of this MMS (167Hz machining speed) the FMNFshould be 30 higher than operation frequency [26] Toexamine the TD at FMNF (200Hz) the real solution forRS analysis results has been carried out The solutions areconducted by inserting the optimal design points obtainedby MOGA and SM as well as DO into the current designpoints and then are updated in the FEM The FEA realsolving results for these optimizationmethods are graphicallyrepresented in Figure 8 From Figures 8(a) 8(b) and 8(c)it can be seen that the maximum TD is found in the rangefrom 0037mm to 0049mm hence the optimized FEMs fullymet the design requirements [28] Among these models theFEM in Figure 8(a) is more significant model that meetsthe cutting force requirements Furthermore the comparisonbetween the FEA and the RS analysis results is carried out inFigure 9 wherein the FEA denotes the initial design sincethe RS analysis refers to the design points 1lowast in Table 3 From

Figure 9 it is noted that these FRF results obtained are similarconfirming that the design points of BBD are accurately fittedto the RS models

To check the quality of FEM the experimentally validatedresults in [1] and the FEA results in [26 27] are used to verifythis FEM by calculating the dimensionless amplitude Thedimensionless amplitude is known as the ratio of dynamicTD to Static Deformation (SD) under external force In factthe SD of the spindle-shaft has been constrained by thetheory of the maximum SD hence it should be less than orequal to 00002 of the span between bearings for adequaterigidity [28] The dynamic TD amplitude is considered at theFMNF With assuming that SD is constrained by literaturethe dimensionless amplitude for optimal FEM and [1 2627] are reported in Table 4 From Table 4 it is noted thatthere are minor differences among these values due tothe difference in material properties boundary conditionetc As results discussed the proposed factors optimizationmethod is feasible and successful in improving the vibrationcharacteristics Thus it can be considered as a practical


35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)

Figure 5The contributions of overall variables on the dynamic response characteristics (a) stress response (b) strain response and (c) totaldeformation response


Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Deformation Response

Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)

Figure 6 The effect of the distance of the back bearing location on the vibration response characteristics (a) stress response (b) strainresponse (c) deformation response

Table 4 The validation experiment for dimensionless amplitude response of MMS

Item RS- Evaluation DO-Real Solution Ref [26] Ref [27] Ref [1]Dimensionless Amplitude 062 038 043 063 076

guide for improving the vibration of unbalance response forMMS

4 Conclusion

In this study the RS method and forced rotordynamicanalyses along with FEA are used to determine the optimalcombinations of factors that improve the vibration responsecharacteristics The spindle specification bearings locationscutting force and motor-rotor unbalance mass are proposedto represent the design factors and then they are utilized todevelop MMS The most popular RS designs of CCDs and

BBD methods are used to build the RS models To visualizemore visible variables affecting the dynamic behaviors thesensitivities analyses and optimization of these variables havebeen carried out based on RS models Finally the qualityof RS optimization has been verified by utilizing a directreal solution of the FEM as well as literature results Thesimulation results concluded the following

(1) A comparison between CCDs and BBD has demon-strated that the BBD is more efficient than the CCDsfrom the view of RS quality models

(2) Themotor-rotor shaft inner diameter Y2 the distanceof back bearing location Y4 and rotating unbalance


Stress Response031029027025023021019017015013011009

Stress Response [MPa]

03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Strain Response (times10

minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)

Figure 7 The effect of motor-rotor shaft inner diameter and rotating unbalance mass induced on the vibration response characteristics (a)stress response (b) strain response (c) deformation response

0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)

Figure 8 The dynamic TD response after optimization (a) MOGA (b) SM for RS and (c) SM for DO


FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600

FEA Steady-State FRFRS Optimization Steady-State FRF

Figure 9 The comparison between the FEA and RS analysis results

mass Y8 have the greatest effects on the dynamic ofunbalance response compared with the other factors

(3) It is found that more than two-fifths of total vibrationresponse is conducted due to motor-rotor unbalanceforce induced

(4) From the view of the verification results the factorsoptimization method is practicable and effective inimproving the vibration responses characteristics

Data Availability

The numerical simulation data files used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

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

Acknowledgments

This work was supported by the College of Mechanicaland Electrical Engineering Northeast Forestry UniversityHarbin 150040 China and the grant fund will be under theresponsibility of the corresponding author ldquoLi Shusenrdquo Theauthors also gratefully acknowledge the technical supportprovided by it during conducting this work

References

[1] P Huang W B Lee and C Y Chan ldquoInvestigation of the effectsof spindle unbalance induced error motion on machiningaccuracy in ultra-precision diamond turningrdquoThe InternationalJournal of Machine Tools and Manufacture vol 94 pp 48ndash562015

[2] L B Kong C F Cheung S To et al ldquoA kinematics andexperimental analysis of form error compensation in ultra-precision machiningrdquo The International Journal of MachineTools and Manufacture vol 48 no 12-13 pp 1408ndash1419 2008

[3] S J Zhang S To S JWang et al ldquoA review of surface roughnessgeneration in ultra-precision machiningrdquo The International

Journal of Machine Tools and Manufacture vol 91 pp 76ndash952015

[4] M Habibi B Arezoo andM Vahebi Nojedeh ldquoTool deflectionand geometrical error compensation by tool pathmodificationrdquoThe International Journal of Machine Tools and Manufacturevol 51 no 6 pp 439ndash449 2011

[5] J Xul X Zheng J Zhang et al ldquoVibration characteristics ofunbalance response for motorized spindle systemrdquo ProcediaEngineering vol 174 pp 331ndash340 2017

[6] S Jiang and S Zheng ldquoDynamic design of a high-speedmotor-ized spindle-bearing systemrdquo Journal of Mechanical Design vol132 no 3 article no 034501 5 pages 2010

[7] H Q Li and Y C Shin ldquoAnalysis of bearing configurationeffects on high speed spindles using an integrated dynamicthermo-mechanical spindle modelrdquo The International Journalof Machine Tools and Manufacture vol 44 no 4 pp 347ndash3642004

[8] C-W Lin and J F Tu ldquoModel-based design of motorizedspindle systems to improve dynamic performance at highspeedsrdquo Journal of Manufacturing Processes vol 9 no 2 pp 94ndash108 2007

[9] J Liu and X Chen ldquoDynamic design for motorized spindlesbased on an integrated modelrdquo The International Journal ofAdvancedManufacturing Technology vol 71 no 9-12 pp 1961ndash1974 2014

[10] S Park C Lee and Y Hwang ldquoLightweight design of 45000 rmin spindle using full factorial design and extremevertices design methodsrdquo Journal of Central South University ofTechnology vol 18 no 1 pp 153ndash158 2011

[11] C Lin ldquoOptimization of bearing locations for maximizing firstmode natural frequency of motorized spindle-bearing systemsusing a genetic algorithmrdquo Applied Mathematics vol 05 no 14pp 2137ndash2152 2014

[12] YDong Z Zhou andMLiu ldquoBearing preloadoptimization formachine tool spindle by the influencingmultiple parameters onthe bearing performancerdquo Advances in Mechanical Engineeringvol 9 no 2 pp 1ndash9 2017

[13] I Asilturk S Neseli and M A Ince ldquoOptimisation ofparameters affecting surface roughness of Co28Cr6Momedicalmaterial during CNC lathemachining by using the Taguchi andRSM methodsrdquoMeasurement vol 78 pp 120ndash128 2016

[14] D B Rao K V Rao and A G Krishna ldquoA hybrid approach tomulti response optimization of micro milling process param-eters using Taguchi method based graph theory and matrixapproach (GTMA) and utility conceptrdquoMeasurement vol 120pp 43ndash51 2018


[15] P-C Chen R-H Zhang Y Aue-u-lan et al ldquoMicromachiningmicrochannels on cyclic olefin copolymer (COC) substrateswith the taguchi methodrdquo Micromachines vol 8 no 9 p 2642017

[16] K Gok A Gok and Y Kisioglu ldquoOptimization of processingparameters of a developed new driller system for orthopedicsurgery applications using Taguchi methodrdquo The InternationalJournal of AdvancedManufacturing Technology vol 76 no 5-8pp 1437ndash1448 2014

[17] E Kuram and B Ozcelik ldquoMicro-milling performance of AISI304 stainless steel using Taguchi method and fuzzy logicmodellingrdquo Journal of Intelligent Manufacturing vol 27 no 4pp 817ndash830 2016

[18] H Mao L Jiao S Gao et al ldquoSurface quality evaluation inmeso-scale end-milling operation based on fractal theory andthe Taguchi methodrdquo The International Journal of AdvancedManufacturing Technology vol 91 no 1-4 pp 657ndash665 2017

[19] J Yang J Guan X Ye et al ldquoEffects of geometric and spindleerrors on the quality of end turning surfacerdquo Journal of ZhejiangUniversity-SCIENCE A vol 16 no 5 pp 371ndash386 2015

[20] D-S Lee and D-H Choi ldquoReduced weight design of a flexiblerotor with ball bearing stiffness characteristics varying withrotational speed and loadrdquo Journal of Vibration and Acousticsvol 122 no 3 pp 203ndash208 2000

[21] Y-H Lin and S-C Lin ldquoOptimal weight design of rotor systemswith oil-filmbearings subjected to frequency constraintsrdquo FiniteElements in Analysis and Design vol 37 no 10 pp 777ndash7982001

[22] C-W Lin ldquoSimultaneous optimal design of parameters andtolerance of bearing locations for high-speed machine toolsusing a genetic algorithm andMonteCarlo simulationmethodrdquoInternational Journal of Precision Engineering and Manufactur-ing vol 13 no 11 pp 1983ndash1988 2012

[23] C-W Lin ldquoAn application of Taguchimethod on the high-speedmotorized spindle system designrdquo Proceedings of the InstitutionofMechanical Engineers Part C Journal ofMechanical Engineer-ing Science vol 225 no 9 pp 2198ndash2205 2011

[24] A KutluDesign andDevelopment of a Lathe Spindle [MScThe-sis] KTH Industrial Engineering and Management MachineDesign 2016

[25] A Labdi andM Bouchetara ldquoOptimization of the thermal andmechanical behavior of an aeronautical brake disc using thedesign of experiments approachrdquoMechanics vol 24 no 2 2018

[26] D Liu H Zhang Z Tao et al ldquoFinite element analysis ofhigh-speed motorized spindle based on ANSYSrdquo The OpenMechanical Engineering Journal vol 5 no 1 pp 1ndash10 2011

[27] E A I Ahmed and L Shusen ldquoThe vibration of machinemotorized spindle based on forced rotordynamic analysis andresponse-surface methodrdquo Vibroengineering PROCEDIA vol21 pp 1ndash7 2018

[28] P H Joshi Machine Tools Handbook Tata McGraw-Hill Pub-lishing Company Limited New Delhi India 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


ProtectionFlangeProtection

Flange

Drainage HoleDrainage HoleDrainage HoleDrainage Hole

Front BearingHousing

Stator Housing Coolant Inlet Coolant Outlet

StatorRear BearingHousing Hydraulic

CylinderFlange

EncoderRing

Oslash24

5m

m

Oslash24

6m

m

Oslash29

6m

m

Figure 1 The Section view of the MMS

load reliant on stiffness characteristics in constraints of thebearing fatigue life and system eigenvalues Lin Y-H and LinS-C [21] proposed the finite-element-method to develop thereduced-weight design of a spindle supported by oil-lubricantbearing under a frequency constraint using a successivequadratic programming method Lin C-W [22] developed amathematical model for an optimization problem to reducethe total costs of the tolerance and a predictable value of thedissipated quality under definite constraints They proposedan optimization technique that integrated the Monte-CarloSimulation and genetic-algorithm to simultaneously explorethe optimal design vectors and tolerance of bearings LinC-W [23] also applied Taguchi technique to recognize theoptimal values of design vectors for a robust HS spindlesystem with respect to the signalnoise ratio of systemFirst-Mode-Natural-Frequency (FMNF) under gyroscopicmoment effect Most of these studies were focused onimproving the dynamic characteristics through simultane-ouslyminimizing the material used in the structure andmax-imizing the FMNF of different spindle-systems Howeverthe studies on optimization of factors affecting the vibrationresponse characteristics of MMS under rotating unbalanceforce effects due to its complex dynamic model has beenrelatively lacking Usually the designer mostly focused onunforced vibration design optimization However the influ-ence of the gyroscopic moments and motor-rotor imbalanceforces on the dynamic response are slightly disregardedThis system design manner is unsatisfactory for balancingand control of the machine vibration In HS machining theextra loads created by rotating elements may well cause agreatest influence on the quality of the machined componentFor instance due to HS machining the gyroscopic momenteffects and rotating unbalance forces will generate morevibration for the MMS which directly affects the dynamic

performances of MMS Therefore to precisely predict theactual performance of MMS it is necessary to conduct thevibration design under rotating elements forces inducedIn the present work optimization of the factors affectingvibration characteristics under rotating unbalance force effectis presented The RS method and forced rotordynamic anal-yses along with FEA are used to determine the optimalcombinations of the factors that improve the vibration behav-iors of MMS The spindle specifications bearings locationscutting force and motor-rotor unbalance mass are proposedto represent the design factors and then they are utilizedto develop MMS The Central-Composed-Design (CCD)and Box-Behnken Design (BBD) approaches due to theirefficiency in providing considerable data in a least numberof required statistical experiments are adopted to generatethe DOE wherein these DOE design points data then areused to build theRSmodels To visualizemore obvious designfactors affecting the dynamic characteristics of MMS the RSevaluation is carried out in detail The sensitivities of designfactor on the vibration response and the optimization of theirlevels are covered by using these RS models Finally directreal solution of the finite element model (FEM) and literatureresults are used to examine the quality of the RS modelconsidered

2 Motorized Spindle System Specifications

Figure 1 shows the section view of lathe motorized spindlewhich was established to hold motorized spindle rather thanconventional-pulley drive L-series lathe spindle standard JISA2-6 [24] It is designed to withstand maximum 3700N ofcutting force and 7000 rpm of machining velocity and pulledby a 168kw motor coupled to the shaft with synchronous-motor Type 1FE1093-6WV The SKF-roller-bearings under


000 20000 (mm)10000





F (119905) = 1198721199031205962 sin120596119905 = 997888rarrΧ1205962 sin120596119905 (1)


Chip

Workpiece

t

-

ＭＮ

Ｈ

N

FR

CuttingTool



F119905 = ΤΒ119888N119904cos (120573 minus 120572)

sin (120601) cos (120601 + 120573 minus 120572)

F119891 = ΤΒ119888N119904sin (120573 minus 120572)

sin (120601) cos (120601 + 120573 minus 120572)

(2)














Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


(a)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0



P39P40P41


(b)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0




(d)




Find Y = (Y1Y2Y3 Y8) (3)





Table 2 Continued



DesignFactors

InitialDesign




120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to


(4)





35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









000 20000 (mm)10000





F (119905) = 1198721199031205962 sin120596119905 = 997888rarrΧ1205962 sin120596119905 (1)


Chip

Workpiece

t

-

ＭＮ

Ｈ

N

FR

CuttingTool



F119905 = ΤΒ119888N119904cos (120573 minus 120572)

sin (120601) cos (120601 + 120573 minus 120572)

F119891 = ΤΒ119888N119904sin (120573 minus 120572)

sin (120601) cos (120601 + 120573 minus 120572)

(2)














Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


(a)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0



P39P40P41


(b)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0




(d)




Find Y = (Y1Y2Y3 Y8) (3)





Table 2 Continued



DesignFactors

InitialDesign




120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to


(4)





35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018



















Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


(a)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0



P39P40P41


(b)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0




(d)




Find Y = (Y1Y2Y3 Y8) (3)





Table 2 Continued



DesignFactors

InitialDesign




120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to


(4)





35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


(a)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0



P39P40P41


(b)

Pred

icte

d fro

m th

e Res

pons

e Sur

face 11

1

09

08

07

06

05

04

03

02

01

0

minus01



P39P40P41


Pred

icte

d fro

m th

e Res

pons

e Sur

face 1

09

08

07

06

05

04

03

02

01

0




(d)




Find Y = (Y1Y2Y3 Y8) (3)





Table 2 Continued



DesignFactors

InitialDesign




120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to


(4)





35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018












Table 2 Continued



DesignFactors

InitialDesign




120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to


(4)





35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









Table 2 Continued



DesignFactors

InitialDesign




120588119896Α 119896L 119908ℎ119890119903119890 120588 119894119904 119889119890119899119904119894119905119910 119860 119894119904 119886119903119890119886 119871 119894119904 119890119897119890119898119890119899119905 119897119890119899119892119905ℎ

Subjected to


(4)





35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









35 11

1

6

11

P39

Local Sensitivity

39

1

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(a)

11

22

892

40

7 5

P40

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

(b)

6 1

16

47

25

44

P41

Local Sensitivity

Y1Y3Y2Y4

Y6Y5Y8Y7

1

(c)



Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









Stre

ss R

espo

nse [

MPa

]01901801701601501401301201101

009008007006005004003002001

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stress Response

(a)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70

Stra

in R

espo

nse (

times10

minus7)

[mm

mm

^-1]

29

28

27

26

25

24

23

22

21

2

Strain Response

(b)

Y4 [mm]10 15 20 25 30 35 40 45 50 55 60 65 70


Def

orm

atio

n Re

spon

se [m

m]

03703603503403303203103

02902802702602502402302202102

(c)





4 Conclusion






Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









Stress Response031029027025023021019017015013011009


03

025

02

015

01

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

(a)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210


minus6)

[mm

mm

^-1]

15

1

05

Strain Response17

15

13

11

09

07

05

03

(times10minus6)

(b)

Y2 [mm]

40

45

50

5560

Y8[kg]

109

8 76543210

Deform

ation Response[m

m]


06

05

04

03

065

06

055

05

045

04

035

03

(c)


0049013 Max

00053703 Min

00441640039315003446500296160024767001991800150690010219

(a)

0037922 Max

00040452 Min

00341580030394002663002286600191020015338001157300078093

(b)

0029858 Max

000022961 Min

002656600232740019982001669001339800101060006813800035217

(c)



FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









FRF

(mm

)

03

025

02

015

01

005

0

Frequency (Hz)0 100 200 300 400 500 600






Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Optimization of Factors Affecting Vibration Characteristics of …downloads.hindawi.com/journals/mpe/2019/1845056.pdf · 2019-07-30 · of Unbalance Response for Machine Motorized