Research Article Multiresponse Optimization of a Compliant...

Research ArticleMultiresponse Optimization of a Compliant Guiding MechanismUsing Hybrid Taguchi-Grey Based Fuzzy Logic Approach

Thanh-Phong Dao12

1Division of Computational Mechatronics Institute for Computational Science Ton Duc Thang UniversityHo Chi Minh City 700000 Vietnam2Faculty of Electrical amp Electronics Engineering Ton Duc Thang University Ho Chi Minh City 700000 Vietnam

Correspondence should be addressed toThanh-Phong Dao daothanhphongtdteduvn

Received 1 November 2015 Revised 4 April 2016 Accepted 12 April 2016

Academic Editor Chunlin Chen

Copyright copy 2016 Thanh-Phong Dao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Because of a lack of precisely miniaturized guiding mechanism micronanoindentation devices are now difficult to integrateinside scanning electron microscope A compliant guiding mechanism (CGM) with a compact size serving as an accuratepositioning platform is hence proposed in this paper The displacement and first natural frequency are the most importantquality responses of CGM The geometric parameters of CGM and applied force play a vital role in determining those responsesThe experiment plan is firstly designed by Taguchirsquos 119871

27orthogonal array A hybrid approach of Taguchi-grey based fuzzy logic

is then developed to optimize two responses simultaneously The grey relational analysis based on the fuzzy logic is used toachieve a grey-fuzzy reasoning grade (GFRG) that combines all the quality characteristics The GFRG serving as a performanceindex determines optimal parameter levels Analysis of variance is conducted to assess the significant parameters affecting theresponses The confirmation results revealed that the 195 120583m displacement of the CGM was many times greater than that of theprevious mechanisms It could be concluded that the quality responses of CGM can be significantly improved through the hybridoptimization approach The proposed methodology has great applications for related compliant mechanisms and engineeringsciences Taking benefits of a compact structure into account CGM has a great ability for micronanoindentation device insidescanning electron microscope

1 Introduction

Micronanoindentation technique now serves as an efficienttool to probe the mechanical properties of materials It cantest the hardness crack length fracture onset creep Youngrsquosmodulus crack propagation shear band formation plasticdeformation pile-up sink-in and so forth This techniquehas been commonly used in various fields ofmaterial sciencesemiconductor biomechanics biomedicines surface engi-neering civil and micronanoelectromechanical systems

For example the mechanical properties and bucklingof single-crystal nanolines were tested under indentation[1] The deduced elastic property of carbon nanotubes wasalso investigated via indentation [2] The indentation wasutilized to determine the fatigue strength of S38C steel [3]Via finite element simulations the pile-up on accuracy ofsharp indentation test was demonstrated [4] The Vickers

microhardness of boron suboxide ceramics was measuredvia indentation [5] The microindentation for human toothenamel and dentine was performed [6] The creep stressexponent from indentation data was studied [7]The fracturetoughness was investigated through indentation [8] Theindentation simulation was conducted for reverse analyses indepth-sensing [9] Few years later an in situ nanoindentationdevice was designed [10] The Vickers microindentationwas studied to assess the materials microhardness [11] Thebehavior of bone was simulated via indentation [12] Ciofuand Cretu [13] discovered the behaviors of ultramid plasticmaterials via microindentation The microindentation wasapplied for estimating the fracture toughness of Al6061-T6[14] The fracture and plastic deformation were monitoredusing indentation [15] The Berkovich Vickers and conicalindentation tests were compared via numerical simulations[16] The deformation behavior and mechanical properties of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5386893 17 pageshttpdxdoiorg10115520165386893

2 Mathematical Problems in Engineering

polycrystalline and single crystal alumina were studied viananoindentation [17]

It is known that the indentation technique offers manyadvantages over conventional tests such as reduced costeconomy in material and time simplified experimenta-tion capacity to discover an inhomogeneous surface formicrostrength variations and residual stress [1ndash17] How-ever most of available indentation instruments have com-plicated structures frameworks and large volumes Thoseinstruments are therefore extremely difficult to insert insidethe scanning electron microscope (SEM) the transmissionelectronmicroscope (TEM) the image analysis and so forthbecause the SEM and TEM have a small volume chambershort working distance electromagnetic sensing vacuumenvironment and vibration sensing [18ndash20]

Regarding the positioning accuracy a precise guidingmechanism is actually necessary for indentation device Thismechanism aims to drive a sample towards a fixed indenterIn the other words it should possess a positioning capacitywithin the range of a few hundreds of micronanometersTo fulfill that requirement designing a precise guidingmechanism with a compact size is now difficult chances foracademic scientists engineer and manufacturers

Up till now traditional rigid-link mechanisms cannotmeet the accurate positioning requirements due to friction inkinematic pairs need for lubricant maintenance assemblybacklash and so on On the contrary compliant mechanismgains themobility from the deflection of flexure hinges ratherthan from kinetic joints It offers several extra advantagessuch as smooth displacement no backlash zero friction noneed for lubricant no noise part-count reduction simplifiedmanufacturing processes increased precision and reliabilityreduced wear and weight decreased assembly time andmaintenance and monolithic structure manufactured byelectrical discharged machining or wire electrical dischargedmachining [21] As a result of the accurate requirementcompliantmechanism is a potential candidate for positioningsystem

From such analysis a compliant guiding mechanism(CGM) is designed in this study Taking advantage of acompact size into account this mechanism can serve asa precise positioning platform inside a microindentationdevice The microindentation device is composed of keycomponents such as CGM axis coarse positioner indenterholder displacement sensor and load cell The CGM is themost important element to construct a good indentationsystem

Theworking efficiency of CGM is assessed with consider-ation of two quality characteristics such as the displacementand first natural frequency Because the CGM has a largedisplacement the indentation depth is then broadenedMoreover the CGM has a high first natural frequency theresponding speed of the indentation device is also fasterIn compliant mechanisms a large displacement is alwaysconflicted with a high frequency [21] From another aspectthese quality responses may be affected by controllableparameters such as the force exerting the CGM and thegeometric parameters A suitable optimization approach isactually needed to achieve a satisfaction for both responses

simultaneouslyThe optimization process of CGM is based ondesign of experiments and thus experiment-based optimiza-tion methods are taken into consideration in this study

Recently the Taguchi method has been extensively usedfor optimizing in various engineering fields due to decreasednumber of experiments [22] Even though the Taguchimethod was successfully proven for optimizing a singleperformance characteristic [23 24] but it failed for opti-mizing multiple performance characteristics [25 26] As aresult a hybrid approach is needed to resolve multiresponseoptimization problems

In the past initialized by Deng [27] the grey relationalanalysis has been useful technique in dealing with poorincomplete and partial knowndata [28]Thehybrid Taguchi-based grey relational analysis was discovered for optimizingmultiple characteristics in different fields of manufacturing[29ndash32] An earlier method the fuzzy logic was originatedby Zadeh [33] which was used for dealing with uncertainand vague information This theory was developed for manydifferent fields For instance Chen et al [34] constructedthe probabilistic fuzzy logic for the partial feedback of therobust control of quantum systems Also Chen and Xiao[35] applied the probabilistic fuzzy logic for mobile robotsand for the mobile-robot reactive navigation [36] And thena neural-fuzzy formation controller was investigated for themultiagent systems [37] however the fuzzy logic rules maynot be easily amendable to dynamic changes of a process[38 39]

To compensate the defect of fuzzy logic the grey rela-tional analysis was integrated with the fuzzy logic in themultivariate system to obtain the best quality responses [40ndash44] The grey relational analysis has three types such as thelower-the better the higher-the better and the nominal-the better which contain certain degree of uncertainty andvagueness better than the fuzzy logic For example the greytheory was successfully applied for multiobjective optimiza-tion of milling [45] injection-molded part with a thin shellfeature [46] CNC turning process [47] and so forth Takingthe advantages of the Taguchi method into account greyrelational analysis and fuzzy logic a hybrid approach ofthe Taguchi-grey based fuzzy logic was applied usefully formultiresponse optimization problems [48] However thishybrid approach has not been developed for compliantmech-anisms yet especially for a compliant guiding mechanism ofindentation device

The objective of this study is to propose a multiresponseoptimization for a compliant guiding mechanism (CGM)The CGM with flexure hinges serves as a precise positioningmechanism It is a major part of indentation device insidethe SEM A hybrid approach of the Taguchi-grey basedfuzzy logic is developed to maximize the displacement andfirst natural frequency of CGM simultaneously The forceexerting CGM and geometric parameters of flexure hingeare considered as design variables The experiment layout isassigned by Taguchirsquos 119871

27orthogonal array As suggested by

the Taguchi method the response table and response graphare constructed to determine the optimal parameter levelsFurthermore the analysis of variance (ANOVA) is used to

Mathematical Problems in Engineering 3

F

L

T

W

y

x

R

Effector

Figure 1 Proposed model of compliant guiding mechanism (CGM)

recognize the significant parameter affecting the responsesFinally the experiments are conducted to validate the optimalresults

2 Design Analysis andOptimization Statement

21 Design Description of Guiding Compliant MechanismRegarding micronanoindentation devices inside SEM aprecise guiding mechanism is really needed to bring asample towards a fixed indenter As mentioned compliantmechanisms can fulfill the requirements of precise guidingmechanism [21]

In this study a compliant guidingmechanism (CGM)wasdesigned via the use of flexure hinges The CGM is aimed atserving as a precise positioning platformThe CGM includeda fixed base an end effector and three flexure hinges withrectangular cross sections as depicted in Figure 1 The CGMwas constructed to achieve the single-axis translation alongthe 119909-direction Comparedwith other hinges the rectangularcross-sectional flexure hinges were chosen for CGM becauseof making a larger displacement [21] Because the CGM hasa compact size the indentation device is therefore capable ofintegrating inside the SEM chamber

As seen in Figure 1 under the force 119865 exerted on theeffector from an appropriate actuator the CGMwould trans-late along the 119909-direction To obtain a single-axis translationfor the proposed CGM the following constraints should besatisfied

119882 ge 10119879 (1)

119871 ge 125119879 (2)

where 119882 is the width 119871 is the length and 119879 is the thicknessof the flexure hinges

The constraint in (1) was the locking of the correspondingtransverse flexion of the CGM while the constraint in (2)allowed flexibility of the CGM along the direction of appliedforce However dynamic loads frequently encountered dur-ing its stroke lead to decreased life On the other handa high stress concentration at the fixed end of the flexurehinges led to reduced fatigue life and increased mechanicalfailure [21] To reduce the stress concentration and improveperformances the CGMwas designed with a filleted radius 119877at the fixed ends of the flexure hinges In this paper all flexurehinges were designed with the same dimensions Moreoverto avoid and limit parasitic motion along the 119910-direction asymmetric structure for the CGM was suggested

The CGM monolithically fabricated via wire electricaldischarged machining process can reduce an extra assemblyfor the indentation device The performance efficiency ofindentation device is assessed by a broad displacement anda fast responding speed As explained when the CGM has alarge displacement the indentation depth will be improvedBesides the CGM has a high first natural frequency theresponding speed of the device is also faster The qualityresponses of CGM can be enhanced by an appropriateoptimization methodology On the other hand the optimaldesign of flexure hingersquos parameters is needed to achieve thebest quality responses of CGM

Material of aluminum alloy 6061-T651 was chosen for theGCM due to its excellent mechanical properties such as thehigh yield strength of 780MPa Youngrsquos modulus of 210GPalight density of 7850 kgm3 and Poissonrsquos ratio of 03

To sum up the CGM design was based on the principleof the compliant mechanism with flexure hinge The qualityresponses of CGM were measured via the displacement andfirst natural frequency To improve those characteristics thegeometric parameters of flexure hinge would be optimized bya hybrid optimization approach

22 Analysis of Parasitic Motion The parasitic motion is anundesired error for any single-axis positioning mechanismbecause it reduces the positioning accuracy of indentation


device Hence this error would be regarded during the designphase

A finite element analysis (FEA) is nowwidely used to real-ize performance characteristics of complex structures beforereal manufacturing Similarly this study would evaluate thecorrelation between the displacement along the 119909-axis andthe parasitic motion error along the 119910-axis via FEA throughANSYS software package [49]

To perform this analysis the eleven simulations wereconducted by ANSYS 2015 And then the displacement andparasitic motion error were collected The ratio () of theparasitic error (119901e) to the displacement (119889

119894) was calculated

The results showed that these ratios were less than 14 asseen in Table 1 To sum up the design of CGM was wellsatisfied with (1) and (2) It could be concluded that theCGM is capable of translating along the 119909-direction wellTheCGM could serve as an excellent positioning platform forindentation device

23 Effect of Design Variables on Quality Responses Consid-ering the effects of design parameters to the quality responsesof CGM many FEA simulations were carried out Usually toillustrate those effects a parameterrsquos value will be changed inthe specific range while the others are remained constantlyHowever this way actually spent much time because ofconstructing amodel in a CAD software and then thatmodelis imported into ANSYS to be analyzed

Unlike such a way the response surface method (RSM)in ANSYS [49] was adopted as a statistical regression modelto illustrate the effect of design variables on the displacementand the first natural frequency

In the first phase the 3D model of CGM was directlycreated in the design modeler module of ANSYS 2015Subsequently the five input parameters consisting of the force119865 length 119871 width 119882 thickness 119879 and filleted radius 119877 weremarked as parametric input variables At the same time thedisplacement and first natural frequency were also marked asparametric output variables To perform this computationalanalysis assuming that the force was in the range at 1800Nto 1804N the length was in the range from 400mm to404mm the width was in the range from 300mm to304mm the thickness was in the range from 22mm to26mm and the filleted radius was in the range from 04mmto 08mm

To realize a good regression model many criteria havebeen regarded Based on the computational results via RSMas in Table 2 the regression model for the computationalanalysis was evaluated as follows (1) 119877

2 is a coefficient ofdetermination (best value = 1) (2) MRR is a maximumrelative residual (best value = 0) (3) RMSE is a root meansquare error (best value = 0) (4) RRMSE is a relative rootmean square error (best value = 0) (5) RMAE is a relativemaximum absolute error (best value = 0) and (6) RAAEis a relative average absolute error (best value = 0) Theresults revealed that these parameters are almost the best forthe computational regression model in this study Hence therelationship between the input and the output parameterswasestablished properly

Table 1 Displacement parasitic error and ratio of 119901e119889119894

Number Displacement119889119894(mm)

Parasiticerror 119901e(mm)

Ratio of 119901e119889119894 ()

1 011232 000141 1262 012978 000155 1193 012896 000148 1154 013090 000162 1245 013091 00016 1226 012891 000154 1197 012823 000150 1178 013104 000171 1309 008983 000111 12410 019508 000241 12411 012970 000162 125

Table 2 Goodness of fit of the regression model

Parameters Value1198772 1

MRR 0RMSE 10minus12

RRMSE 0RMAE 0RAAE 0

Table 3 Coded levels of the design variables

Factors Unit Coded levelsminus2 minus1 0 1 2

119865 N 1800 18005 18010 18015 18040119871 mm 400 4010 4020 4030 4040119882 mm 300 3010 3020 3030 3040119879 mm 220 230 240 250 260119877 mm 040 050 060 070 080

The range unit of each input parameter is not the sameeven their values are unlike Thus to facilitate illustratingthe relationship between inputs and outputs on the sameplot each level of all input parameters was coded to besimilar The coded levels for the design variables are givenin Table 3 with consideration of the effects of the designvariables on the output parameters in only one plot Asshown in Figure 2 when each of the five design variableschanged its value the displacement would be varied Thethickness especially strongly affected the displacement Theresults revealed that all the five design variables had a directinfluence on displacement Using the same computationalmethod the input parameters also relatively affected thefrequency as seen in Figure 3

Most of the curves of Figures 2 and 3 were the nonlinearprofiles It could be concluded that the significant parametersthe force 119865 the length 119871 the width 119882 the thickness 119879and the filleted radius 119877 directly affected the two quality


Force

LengthWidthThicknessFilleted radius

005

01

015

02

025

Disp

lace

men

t (m

m)

minus1 0 1 2minus2Coded levels

Figure 2 Relationship between the input parameters and displace-ment

910912914916918920922

minus2 minus1 0 1 2

Freq

uenc

y (H

z)

Coded levels

Force


Figure 3 Relationship between the input parameters and frequency

responses Hence these parameters would be considered infurther optimization process

24 Optimization Statement The objective of the CGMwas to provide a large displacement and a high first nat-ural frequency These quality responses would be achievedthrough a suitable optimization process The multiresponseoptimization problem for CGM was briefly formulated asfollows

Find

119865 119871 119879119882 119877 (3)

Maximize displacement

1199101(119865 119871 119879119882 119877) (4)

Maximize first natural frequency

1199102(119865 119871 119879119882 119877) (5)

subject to constraints

120590max le

120590119910

SF(6)

within ranges

1800N le 119865 le 1804N

400mm le 119871 le 404mm

300mm le 119882 le 304mm

22mm le 119879 le 26mm

04mm le 119877 le 08mm

(7)

where the objective functions 1199101and 119910

2are measures of the

displacement and the first natural frequency respectivelyThey depend on applied force119865 length 119871 width119882 thickness119879 and filleted radius 119877 120590max is the maximum equivalentstress 120590

119910is the yield strength of the proposed material and

SF is the safety factorUsually any compliant mechanism also operates in the

elastic area of a specificmaterial [21] To avoid plastic failuresa safety factor as large as possible should be selected to be safeat the hinges In this study a safety factor of 3 was chosen forthe CGMThe design variables were assigned in the range of(7) due to the following reasons (i) if the length 119871 is lowerthan 400mm the displacement will be decreased and if 119871is larger than 404mm the size of CGM will be increased(ii) if the width 119882 is greater than 304mm the mass of theCGM can increase and if 119882 is smaller than 300mm thestiffness of CGM will be decreased (iii) if the thickness 119879

is thicknesses is lower than 04mm the displacement willbe increased but the frequency decreased and if 119879 is largerthan 08mm the proposed mechanism will become stifferthe displacement will be decreased (iv) the range of filletedradius 119877 is selected to avoid the stress concentration at theflexure hinge To sumup the lower bounds for the parameterswere selected to guarantee a good compliance and the upperbounds of the geometric dimensions were set to achieve acompact structure

3 Methodology

In general mathematical models should be formulated priorto an optimization process however if modeling errorsfrom mathematical models become large optimal resultswill be unacceptable To eliminate that limitation an opti-mization process for CGM was based on the experimentalprocedure With miniaturized number of experiments theTaguchi method was the most appropriate robust approachAs mentioned the Taguchi method only optimized a singleobjective it was hence coupled with others to optimizemultiple objectives

In this study a hybrid Taguchi-grey based fuzzy logicapproach was developed to serve as an effective multire-sponse optimization tool for the CGM It got a fuzzy rulesprocedure rather than making traditional grey relationalanalysis (GRG) estimation for grey relational analysis Usingthe GRG estimation the optimal quality responses wereobtained [29 30] On the contrary the fuzzy inference system(FIS) was used in this study to estimate the GRG FISswere also defined as fuzzy-rule-based systems fuzzy modelsand fuzzy associative memories Figure 4 shows a systematic


Define problem

Define control factors and quality characteristics

Design of experiments

Data collection

Analysis of signal to noise (SN) ratio for each quality characteristic

Data preprocessing or normalization

Determine deviation sequence

Determine grey relational coefficient

Defuzzifier

Interface engine

The response table and response graph of grey-fuzzy grade

Calculate ANOVA for grey-fuzzy grade

Predict the optimal factor level

Confirmation experiments

Grey relational analysis

Taguchi method

Fuzzy knowledge base

Fuzzifier

Membershipfunction

Fuzzy rules

Determine grey-fuzzy grade

Fuzzy logic analysis

Figure 4 The flowchart of multiresponse optimization approach

flowchart for the multiresponse optimal procedure usinga hybrid Taguchi-grey based fuzzy logic approach Theoptimization process was divided into key steps as follows

Firstly Steps 1ndash5 were based on Section 23 and theTaguchi method to identify optimization problem design ofexperiments and measurement of quality characteristics

Secondly Steps 6ndash8 were applied to the grey relationalanalysis The grey relational analysis was carried out throughsequence groups

Lastly Steps 9ndash13 were a development of the grey-fuzzy reasoning analysis The fuzzy logic system was hereinintegrated with the grey relational analysis The fuzzier usedthe membership functions to fuzzify the grey relationalcoefficient (GRC) of each quality performance The fuzzyrules (if-then control rules) were generated and defuzzifierconverted fuzzy predicted value into a grey-fuzzy reasoninggrade (GFRG) The response graph and table response wereestablished and then the optimal parameter levels were deter-mined Analysis of variance was employed to determine themost significant parameter affecting the quality responsesThen the optimal grey-fuzzy reasoning grade was identifiedThefinal phasewas a validation of the optimal results throughconfirmation experiments

Step 1 (define problem) The purpose of this study was theoptimal design for compliant guiding mechanism The qual-ity objectives ofCGMwere tomaximize the displacement andmaximize the first natural frequency simultaneously

Step 2 (define control factors and quality characteristics) Asprevious analysis the force 119865 length 119871 width 119882 thickness119879 and filleted radius 119877 were the important factors affectingthe quality objectives (the displacement and first naturalfrequency)

Step 3 (design of experiments using the orthogonal array ofTaguchi method) To investigate the entire design param-eters with a small number of experiments the Taguchimethod with special orthogonal array was employed forthe experimental layout as compared with a full factorialapproach [22]

Step 4 (experimental data collection) Data of the displace-ment and first natural frequency were measured by manyexperiments in real

Step 5 (analysis of signal to noise (119878119873) ratio for each qualitycharacteristic) The 119878119873 ratio (120578) for various responses wascalculated via the Taguchi method According to the Taguchimethod [22] three types of characteristic are consideredas follows the lower-the better the higher-the better andthe nominal-the better Because the displacement and firstnatural frequency were expected to obtain the maximumvalues the higher-the better was hence selected which wasdescribed as follows [28]

120578 = minus10 log(1

119899

119899

sum

119894=1

1

1199102

119894

) (8)


where 120578 is the signal to noise ratio in decibels 119910119894is the

measured response value of 119894th experiment 119899 denotes thenumber of experiments Equation (8) was applied for bothresponses

Step 6 (data preprocessing or normalization) To avoid theeffect of different units and to reduce the variability datapreprocessing or normalization was a necessary step inthe grey relational analysis [42ndash44] Data preprocessingwas transferring of the original sequence to a comparablesequence in the range from zero to one whichmeans that thegiven data sequence is transferred into a dimensionless datasequence

In this study the normalized 119878119873 ratio 119911119894(119896) value for 119899

experiments was calculated to adjust the measured values ondifferent scales to a common scale The type of the higher-the better was used for both quality responses hence thenormalization equation for the 119878119873 ratio for both qualityresponses was computed as follows

119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)

where 119911119894(119896) is the normalized 119878119873 value for the 119896th response

(119896 = 1 2 119898) in the 119894th experiment (known as thecomparability sequence of 119878119873 data) 120578

119894(119896) indicates the

estimated 119878119873 value (known as the original sequence of 119878119873ratio data) and max 120578

119894(119896) and min 120578

119894(119896) are the largest and

smallest values of 120578119894 respectively

Step 7 (determine deviation sequence) Before calculat-ing the grey relational coefficient the absolute deviationsequence 119911

119900(119896) of the reference sequence and comparability

sequence 119911119894(119896) must be determined The absolute deviation

sequence was calculated by the following equation [42ndash44]

Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)

where Δ119900119894(119896) is the absolute difference sequence between

119911119900(119896) and 119911

119894(119896) 119911119900(119896) is the reference sequence that presents

the ideal value (optimal value generally equal to 1 in anormalized sequence) for the 119896th response and 119911

119894(119896) is the

comparability sequence

Step 8 (determine grey relational coefficient) In the greyrelational analysis the grey relational coefficient (GRC) wascalculated to give the relationship between the optimal andactual normalized experimental results [42ndash44] In this studythe grey relational coefficient expressed the relationshipbetween the best and the actual normalized 119878119873 ratios Thegrey relational coefficient was computed as follows [42ndash44]

120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)

where 120574119894(119896) is the grey relational coefficient Δmin is the

smallest value of Δ119900119894 Δmax is the largest value of Δ 119900119894 and 120575

is the distinguishing coefficient (0 le 120575 le 1) for adjusting theinterval of 120574

119894(119896) In this study 120575 of 05 was set for the average

distribution due to the moderate distinguishing effects andgood stability of outcomes

Step 9 (grey-fuzzy reasoning analysis) In general the basicstructure of a fuzzy logic unit includes a fuzzifier member-ship functions a fuzzy rule base an inference engine and adefuzzifier This study used the GRC value of displacementand the GRC value of first natural frequency as the inputvariables for the fuzzy inference system (FIS)

Next the trapezoidal membership function was appliedfor fuzzification to achieve fuzzy sets The fuzzy rules werethen created in the inference engine to conduct fuzzy reason-ing and obtain fuzzy values Finally the trapezoidal mem-bership function was applied for defuzzification to obtaina multiresponse performance index (MPI) Fuzzy inputvariables and output variable used trapezoidal membershipfunctionsThe equation of trapezoidalmembership functionswas described as follows [50]

120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)

where 120583119860represents the membership functions of the fuzzy

set 119886 119887 119897 and 119903 denote parameters and 119909 is variableThe fuzzy rule consisting of a group of if-then control

rules was constructed for optimization process in this studyThe fuzzy rule was developed with two inputs 119909

1denotes

the grey relation coefficient value of displacement and 1199092

represents the grey relation coefficient value of first naturalfrequency and one output variable 119910 is denoted as MPIBased on the previous 27 experiments as well as to enhanceinterpretability of fuzzy algorithm and to refine classifiersthis study established the nine fuzzy subsets for GRC ofdisplacement tiny (T) very small (VS) small (S) small-medium (SM) medium (M) medium-large (ML) large (L)very large (VL) and huge (H) the three fuzzy subsets forGRC of frequency small (S) medium (M) and large (L) andthe nine fuzzy subsets for the MPI tiny (T) very small (VS)small (S) small-medium (SM) medium (M) medium-large(ML) large (L) very large (VL) and huge (H) The 27 fuzzyrules for two inputs and one output were formed according tothe concurrence that a large grey relational coefficient wouldbe a better process response These rules were developed toexpress the inference relationship between input and outputA set of rules were written for activating the FIS and theFIS was evaluated to predict the grey-fuzzy reasoning grade(GFRG) for all 27 experiments The linguistic fuzzy rule wasdescribed as follows

Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862

119894(119894 = 1 2 27) are fuzzy sets modeled

using membership functions 120583119860 120583119861 and 120583

119862 respectively

while 1199091and 119909

2are the input variables and 119910 is the output

variableIn this paper the max-min compositional operation of

Mamdani was adopted to perform the calculation of fuzzylogic reasoningTheMamdani implicationmethod was usedand the fuzzy logic reasoning of these rules then achieved afuzzy output The membership function 120583

1198620(119910) of the output

of the fuzzy logic reasoning could be expressed as follows[50]

1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611

(1199092)) or sdot sdot sdot

or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)

where and is the minimum operation and or is the maximumoperation

Finally the fuzzy output was converted into an absolutevalue using the defuzzification method In this study thecenter of gravity method for defuzzification was utilized totransform the fuzzy inference output 120583

1198620(119910) into a nonfuzzy

value 1199100(MPI) MPI was then used to search for the optimal

parameters In general the value of MPI lies within the rangeof 0 and 1This study carried out the proposed steps in toolboxMATLAB (2014a) with the Mamdani method The nonfuzzyvalue 119910

0 the MPI was calculated by following equation [51]

MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)

Step 10 (the response table and response graph of grey-fuzzyreasoning grade) Through the Taguchi method the optimalparameters were determined via constructing the responsetable and response graph of grey-fuzzy reasoning grade(GFRG)

Step 11 (calculate ANOVA for grey relational reasoninggrade) Analysis of variance (ANOVA) was used to estimatethe significant contribution of each factor on the grey rela-tional reasoning grade

Step 12 (predict the optimal grey-fuzzy reasoning grade underoptimum parameters) The optimal grey-fuzzy reasoninggrade was predicted considering the effect of all parametersor the most significant parameter The estimated mean of thegrey relational grade could be determined as

120583GFRG = GFRGm +

119902

sum

119904=1

(GFRGo minus GFRGm) (15)

where 120583GFRG is the optimal GFRG value of predicted meanGFRGm is the total mean of grey-fuzzy reasoning gradeGFRGo is the optimal mean grey-fuzzy reasoning gradefor each level of factor and 119902 is the number of significantparameters affecting the grey-fuzzy reasoning grade

In order to judge the closeness of the observed valueto the predicted value the confidence interval value of thepredicted value for the optimum factor level combinationwas determined The 95 confidence interval of confirma-tion experiments (CICE) was calculated using the followingexpression [43]

CICE = plusmnradic119865120572(1 119891e) 119881e (

1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at

95 (1 minus 120572) confidence interval against degrees of freedom 1and degrees of freedom of error 119891e119881e is the variance of error(get fromANOVA) 119899eff = 119899(1+119889) 119899 is the total trial numberin the orthogonal array 119889 is the total degrees of freedom offactors associated in estimates of the mean 120583GFRG and 119877e isthe number of repetitions for the confirmation experiments

Step 13 (confirmation experiments) Many prototypes werefabricated via wire electrical discharged machining processThe confirmation experimentations were then carried out tovalidate the optimal results

4 Results and Discussion

41 Experimental Measurement of Quality Responses Therelationship between the design variables and quality char-acteristics of CGM was discussed to determine the optimaldesign variables First the composition of various factorsand level values was designed through the Taguchi methodas given in Table 4 As mentioned CGM required a broaddisplacement and a high first natural frequency this studytherefore selected the type of the higher-the better for tworesponses according to theTaguchimethodWith five processparameters and three levels the experimental plan wasdesigned via using the orthogonal array 119871

27 as seen in

Table 5 The 27 experiments were conducted to measure thedisplacement and first natural frequency The experimentalinstruments were installed on a vibration isolated optical


Table 4 Controlling factors and their levels

Symbol Processparameters Level 1 Level 2 Level 3

119865 Force (N) 1800 1802 1804119871 Length (mm) 400 402 404119882 Width (mm) 300 302 304119879 Thickness (mm) 22 24 26

119877Filleted radius

(mm) 04 06 08

Table 5 Experimental layout using an 11987127orthogonal array

Trialnumber Force F Length L WidthW Thickness T Filleted

radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08

table (DAEIL systems model DVIO-I-1209M-100t Korea)to avoid any unexpected vibrations The prototypes werefabricated via using wire electrical discharged machiningprocess

The experimental set-up of displacement is demonstratedin Figure 5 The calibration instruments included a forcegage (Lutron tension and compression maximum 196N DC9V adapter model NF-9500 Taiwan) to adjust the forcevalue and an indicator (MITUTOYO code number 7010sJapan) The prototype was mounted on a base fixed on the

vibration isolation system table The measuring head of theindicator was adjusted to touch the end effector of the CGMThe indicator would display the value of displacement inmillimetersThe valuewould be recorded in each experimentFour experiments were conducted on the prototype and thefinal displacement result was the average value of the fourexperiments

The experimental photograph of first natural frequencyis shown in Figure 6 The measurements of the first naturalfrequency within the range of 500Hz to 5 kHz were per-formed to evaluate the dynamic characteristics of the mech-anism A modal hammer (model 9722A2000-SN 2116555)from KISTLER was used to apply the excitation to themechanism and the frequency response was measured byusing an accelerator (model 4744892) from KISTLER Theaccelerator was attached at location that was opposite theexcitation position of the hammer A modal analyzer (modelNI USB 9162) from National Instruments was utilized indata acquisition and analysis At the end of the hammer aforce sensor was attached to measure the applied force fromthe hammer CUTPRO software was installed in a computerto analyze the data The experiments were repeated fivetimes All experimental data and signal to noise ratios of tworesponses were collected and calculated as seen in Table 6

42 Multiresponse Optimization The principle of optimiza-tion process in this paper was conducted as follows the119878119873 ratios were calculated for each response by using (8)and the normalized 119878119873 values were firstly computed via(9) The deviation sequence was determined via (10) andthe grey relational coefficient (GRC) of each response wasthen determined by (11) Subsequently the grey relationalcoefficients (GRCs) would be imported into fuzzy logicsystem to achieve a grey-fuzzy reasoning grade (GFRG)through (12)ndash(14) Last via the Taguchi method the responsetable and response graph ofGFRGwere established to achievethe optimal parameters

Recalling (8)ndash(11) again 119896 in bracket (119896) represented thevarious responses The displacement response was corre-sponding to 119896 = 1 while the frequency response was corre-sponding to 119896 = 2 Specifically 119911

119894(1)was the normalized 119878119873

ratio of displacement and 119911119894(2)was the normalized 119878119873 ratio

of frequency Δ119900119894(1) and Δ

119900119894(2) were deviation sequences

of displacement and frequency respectively 120574119894(1) and 120574

119894(2)

were grey relational coefficients of deviation sequences ofdisplacement and frequency respectively In this paper Δminwas equal to zero (the smallest value of Δ

119900119894) Δmax was equal

to one (the largest value of Δ119900119894)

In this paper two inputs and one output were used forthe fuzzy logic systemThe inference engine (Mamdani fuzzyinference system) performed fuzzy reasoningwith fuzzy rulesto generate a fuzzy value These fuzzy rules were presentedin the form of ldquoif-thenrdquo control rule as in the proposedmethodology As mentioned in Section 3 the nine fuzzysubsets were used for GRC of displacement the three fuzzysubsets for GRC of frequency and the nine fuzzy subsets forthe output GFRG The 27 fuzzy rules were then establishedin a matrix form for this study as given in Table 7 Thecorrespondingmembership functions for the two inputs were


Jig

Screw

Slider

Compliant guiding mechanism

IndicatorForce gage

Figure 5 Experimental measurement of the displacement response

Computer

Modal analyzer

Compliant guiding mechanism (CGM)

Accelerator

Hammer

Figure 6 Experimental measurement of the first natural frequency response

1205831199091

and 1205831199092 respectively The membership function of the

output was 120583119910 The membership functions of the GRC of

displacement and GRC of frequency were plotted in Figures7(a) and 7(b) respectively The membership function ofoutput GFRG was given in Figure 8

Fuzzy rules were directly derived based on the fact ofldquothe higher-the betterrdquo characteristic The rule based fuzzylogic reasoning procedure was developed as seen in Figure 9As developed the twenty-seven rows represented twenty-seven fuzzy rules and the two first columns denoted twoinput variables that is GRC for displacement and frequencyrespectively The last column gave the defuzzified grey-fuzzy

reasoning grade (GFRG) By tracking maximum-minimumcompositional operation the fuzzy reasoning of these rulesyielded a fuzzy output Finally the defuzzifier convertedthe fuzzy predicted value into a GFRG by using MATLAB(R2014a) fuzzy logic toolbox

To sum up Table 8 provides the difference sequencesgrey relational coefficients and GFRG for two responsesThehigher value of GFRGmeans that the comparability sequencehad a stronger correlation to the reference sequence

Based on the Taguchi method the average grey-fuzzyreasoning grade for each input parameter level was calculatedinTable 9The response graph for averageGFRGat parameter


Table 6 Experimental results and SN ratios of two responses

Trial number 1199101(mm) SN of 119910

11199102(Hz) SN of 119910

2

1 012978 minus177358 91569820 5923502 012896 minus177909 91886598 5926503 013090 minus176612 91154112 5919554 013091 minus176605 91464157 5922505 012891 minus177943 91579683 5923606 012823 minus178402 92412678 5931467 013104 minus176519 91344788 5921378 008983 minus209316 109303575 6077279 019508 minus141957 76034755 57620210 012970 minus177412 91569820 59235011 012985 minus177312 91569820 59235012 011579 minus187266 96874378 59724213 011621 minus186951 96672673 59706114 011531 minus187627 96880407 59724715 011581 minus187251 96678696 59706616 011642 minus186794 96389005 59680517 011727 minus186163 96030546 59648218 011602 minus187093 96394997 59681119 011679 minus186519 96036520 59648720 014555 minus167398 86916617 58782121 014650 minus166832 86633494 58753722 014505 minus167696 86921885 58782623 014591 minus167183 86638743 58754224 014649 minus166838 86798192 58770225 014713 minus166460 86586430 58749026 014589 minus167195 86803470 58770727 014662 minus166761 86591688 587495

Table 7 Fuzzy rules in a matrix form

Grey-fuzzy reasoninggrade (GFRG)

GRC of frequencyS M L

GRC of displacementT T (rule 1) VS (rule 2) S (rule 3)VS VS (rule 4) S (rule 5) SM (rule 6)S S (rule 7) SM (rule 8) ML (rule 9)SM S (rule 10) SM (rule 11) M (rule 12)M SM (rule 13) M (rule 14) ML (rule 15)ML SM (rule 16) M (rule 17) ML (rule 18)L ML (rule 19) ML (rule 20) L (rule 21)VL ML (rule 22) L (rule 23) VL (rule 24)H ML (rule 25) L (rule 26) H (rule 27)

level was plotted as in Figure 10 The results from Table 9and Figure 10 indicated that the optimal input parameterlevel is F1L3W3T2R2 corresponding to the applied force atlevel 1 (1800N) the length at level 3 (404mm) the widthat level 3 (304mm) the thickness at level 2 (24mm) andthe filleted radius at level 2 (06mm) The results predictedthat the maximum displacement is equal to 0195mm andthe maximum first natural frequency is equal to 76034Hz

Table 8 Data of each sequence difference sequence grey relationalcoefficient and GFRG

Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287

Table 9 Response table for the mean GFRG for each parameterlevel

Factor Mean of GFRG for each level Max-min RankLevel 1 Level 2 Level 3

119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5

Total mean value of GFRG is 0288

Theoptimal resultswere corresponding to the 9th experimentwith a highest GFRG of 0625 (see Tables 6 and 8) It wasalso proven that the best quality characteristics are capableof corresponding to the highest GFRG

43 ANOVA for Grey-Fuzzy Reasoning Grade ANOVA wasperformed to understand the influence of design parameterson the quality characteristics Table 10 shows that factor 119865

(applied force) has the most significant influence on thetarget characteristics with percentage contribution of 5186followed by factor 119871 (length) with percentage 560 Factors


T VS S SM M ML L VL H

03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)

Figure 7 Membership function of two inputs (a) GRC of displacement 120574119894(1) and (b) GRC of frequency 120574

119894(2)


00

1

1

Mem

bers

hip

valu

e

Figure 8 Membership function of output (GFRG)

Table 10 ANOVA for GFRG

Factors DOF SS 119881 119865 119875 ()119865 2 00588 00294 108931 5186119871 2 00063 00032 11710 560119882 2 00020 10minus4 03657 175119879 2 00018 10minus4 03318 160119877 2 00013 10minus4 02452 120Error 16 00431 00027 3799Total 26 100

Significant at 95 confidence level

119882 (width) 119879 (thickness) and 119877 (filleted radius) have lowestinfluences with 175 160 and 120 respectively

44 Predict the Optimal Grey-Fuzzy Grade under OptimumParameters In this study the optimal grey-fuzzy reasoninggrade was predicted considering the effect of all controllingparameters at optimal level F1L3W3T2R2 The estimatedmean of the GFRG was determined by (15) and then 95confidence interval of confirmation experiments (CICE) wascalculated using (16) The expected mean value of confirma-tion experiments was obtained as 120583GFRG = 0423

The confirmation value 120583confirmation of the GFRG shouldfall within the range as follows

120583GFRG minus CICE le 120583confirmation le 120583GFRG + CICE

0189 le 120583confirmation le 0657

(17)

Table 11 Confirmation results

Responses Predicted value FEM value ErrorDisplacement 0195mm 0203mm 41Frequency 76034Hz 78086Hz 26GFRG 0625 0654 46

45 Confirmation Experiments

451 Optimal Confirmation By using optimal combina-tion F1L3W3T2R2 four confirmation experiments were per-formed to validate the optimal results The experimentalresults of Table 11 revealed that the confirmation GFRGresults fall within 95 of the CICE (see (17))

Four models at optimal combination F1L3W3T2R2 wereconstructed in Solidworks and conducted by finite elementmethod (FEM) in ANSYS to validate the optimal resultsThe results showed that the error between FEM result andpredicted value is less than 5 as seen in Table 11 Itmeans that the proposed methodology is reliable to predictthe optimal parameters of CGM Reasons may come fromthe following (1) manufacturing error leads to dimensionalerrors between the analysis model and the prototype and (2)

the used parameters in the simulation may be different fromthe actual materials

As mentioned a larger displacement would result ina broader indentation depth for any indentation deviceHence the displacement of various guiding mechanisms forindentation devices was searched for the purpose of compar-ison Compared with the existing guiding mechanisms forindentation devices in the literature review the displacementof CGM had the following highlights as seen in Table 12 thedisplacement of CGM was 875 times larger than that of [15]1525 times larger than that of [19] and 12 times higher thanthat of [20] It could be concluded that the displacement ofCGM is tremendously improved via the hybrid Taguchi-greybased fuzzy logic approach

452 Validation of Stress Constraint Figure 11 demonstratesthe experimental photo to measure the actual strain of theCGM The calibration instruments included a force gage(Lutron tension and compression maximum 196N DC9V adapter model NF-9500 Taiwan) to adjust the appliedforce a sensor gage (KFG 5-120-C1-11L1M2R KYOWA


GRC of displacement GRC of frequency GFRG

03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

Figure 9 Fuzzy logic reasoning procedure

Japan) a CPU (computer processing unit ASUS) DAQ (dataacquisition National International Instrument Japan) anda screen (ASUS) The prototype of CGM using optimalcombination F1L3W3T2R2wasmounted onto a base that wasfixed onto the vibration isolation system table There werethree sensor gages A sensor gage was glued at each flexurehinge of the CGM to measure the actual strain Each flexurehinge was measured separately

The experiment for each flexure hinge was repeated fourtimes The force gage was adjusted to gradually reach a valueof 180N Data from the sensor gage was transferred intothe DAQ instrument which generated data into the CPULABVIEW software was set up in the CPU so that the strainwave and value of the strain were displayed on the screenThestrain value after each experimental measure was recordedThe final actual strain of the CGM was the average value of


Table 12 Comparison the displacement of GCM with previousmechanisms

Mechanisms Displacement (120583m)The CGM 195Reference [15] 12Reference [19] 15Reference [20] 20

F1 F2 F3 L1 L2

Parameter levelL3 W1W2W3 T1 T2 T3 R1 R2 R3

ForceLengthThickness

WidthFilleted radius

02

022

024

026

028

03

032

034

Mea

n of

GFR

C

Figure 10 Response graph of grey-fuzzy reasoning grade

all the experiments The actual stress value of the CGM wascalculated using the following equation

120590 = 120576 times 119864 (18)

where 120590 is the actual stress 120576 is the mean actual strain and 119864

is Youngrsquos modulus of the proposed materialThe results indicated that the maximum real stress

(4063MPa) is still much lower than the yield strength ofaluminum alloy 6061-T651 (780MPa) This actual stress wasclose to the FEM equivalent stress (3854MPa) with anerror of 51 It could be concluded that the maximumstress satisfied the constraint in (6) This could deduced thatthe flexure hinge is safe during the working process Theoptimization process was ended

5 Preliminary Application forMicroindentation Testing

A preliminary model of microindentation device was devel-oped as depicted in Figure 12(a) It consists of the followingcomponents (1) compliant guiding mechanism (CGM)(2) load sensor (3) stage (4) specimen (5) indenter (6)

indenter holder (7) displacement sensor (8) reflector (9)coarse positioner and (10) base The specimen is driven bythe proposed CGM towards the fixed indenter The coarse

positioner is driven by a stepper motor to realize a coarseadjustment of the specimen During the indentation tests theload and displacement are measured by the load sensor anddisplacement sensor through the reflector respectively Thebase makes a tilt angle 120572 of 15 degrees from the platform ofSEM to provide a good observation angle during indentation

At the beginning the specimen is positioned at a suitablelocation by the coarse positioner and then the CGM isactivated by the applied force and the specimen will movetoward the indenter and then the indentation process isfinished As seen in Figure 12(b) the proposedCGM is drivenby an appropriate actuator in the 119909-axis to realize the preciseloading and unloading of the indenter During the testing theload sensor and the displacement sensor are used to measurethe applied load and the displacement response

Because of the 195 120583m broad displacement and the76034Hz high first-order frequency the proposed mech-anism has potential applications under large indentationdepth-and-high frequency conditions It could be concludedthat the CGM is an excellent candidate for indentation deviceinside the SEM A real indentation testing will be devoted forprospective work

6 Conclusions

This study has attempted to optimize the multiple qual-ity responses for a compliant guiding mechanism (CGM)The CGM was designed for future micronanoindentationdevices inside a scanning electron microscope (SEM) Itaimed to guide the specimen towards the fixed indenter

The displacement and first natural frequency were con-sidered as the most important quality responses of CGMThe force exerted on CGM and geometric parameters offlexure hinges were the design variables Through parasiticerror analysis in FEM the CGM had an excellent behaviorof single-axis translation along the 119909-direction

Firstly the effect of design variables on the two responseswas analyzed via the response surface methodology insideANSYS The results indicated that the force length widththickness and filleted radius are significantly affecting thetwo responses

Secondly the experimental layout was then conducted bythe 119871

27orthogonal array of Taguchi method Many proto-

types of CGM were fabricated via wire electric dischargedmachining process And then the experimentations werecarried out to collect data of two responses

Lastly a hybrid approach of Taguchi-grey based fuzzylogic was developed to optimize the displacement and firstnatural frequency simultaneously At the beginning phasethe 119878119873 ratios of two responses were calculated and thenormalized 119878119873 values were subsequently computed Thedeviation sequence was determined and the grey relationalcoefficient (GRC) of each response was then calculatedThe GRCs were then imported into fuzzy logic system toachieve a grey-fuzzy reasoning grade (GFRG) Based on theTaguchi method the response table and response graph ofGFRG were constructed and the optimal design parameterswere achieved ANOVA was performed to determine thesignificant parameters affecting the grey-fuzzy reasoning


Screen

CPU

Force gageStrain gage

Screw

Jig Slider


DAQ

Strainwave

Figure 11 Photo of the experimental strain

Stage of scanning electron microscope

(10) Base

(6) Indenter holder(7) Displacement sensor(8) Reflector(9) Coarse positioner

(2) Load sensor(3) Stage(4) Specimen(5) Indenter

(1) Compliant guiding mechanism (CGM)

(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)

Figure 12 (a) Preliminary model of indentation device and (b) 2D view


grade (GFRG) The results revealed that the applied force isthe most significant factor

The confirmation results indicated that the 195 120583m dis-placement of the CGMwasmany times greater than the othermechanisms in the literature review The quality responsesof the CGM could be extremely enhanced through theproposed approach The efficiency of methodology has beensuccessfully proven by experiments and simulations It isuseful for related compliant mechanisms and engineeringsciences Taking the advantages of a compact structure intoaccount the proposed mechanism can construct a compactmicronanoindentation device inside SEM

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] M K Kang B Li P S Ho and R Huang ldquoBuckling ofsingle-crystal silicon nanolines under indentationrdquo Journal ofNanomaterials vol 2008 Article ID 132728 11 pages 2008

[2] L Liu G Cao and X Chen ldquoMechanisms of nanoindentationonmultiwalled carbon nanotube and nanotube clusterrdquo Journalof Nanomaterials vol 2008 Article ID 271763 12 pages 2008

[3] J Gao G Dai J Zhao H Li L Xu and Z Zhu ldquoInfluence ofindentation on the fatigue strength of carbonitrided plain steelrdquoAdvances inMaterials Science and Engineering vol 2015 ArticleID 492693 9 pages 2015

[4] Z Tang Y Guo Z Jia Y Li and Q Wei ldquoExamining theeffect of pileup on the accuracy of sharp indentation testingrdquoAdvances inMaterials Science and Engineering vol 2015 ArticleID 528729 10 pages 2015

[5] R Machaka T E Derry I Sigalas andM Herrmann ldquoAnalysisof the indentation size effect in the microhardness measure-ments in B

6Ordquo Advances in Materials Science and Engineering

vol 2011 Article ID 539252 6 pages 2011[6] L C Hui J C Pei W H Bo and H K Jao ldquoYoungrsquos moduli

of human tooth measured using micro-indentation testsrdquo LifeScience Journal vol 9 pp 172ndash177 2012

[7] J Dean J Campbell G A Smith and T W Clyne ldquoCriticalassessment of the stable indenter velocity method for obtainingthe creep stress exponent from indentation datardquo Acta Materi-alia vol 80 pp 56ndash66 2014

[8] T H Zhang Y H Feng R Yang and P Jiang ldquoA method todetermine fracture toughness using cube-corner indentationrdquoScripta Materialia vol 62 no 4 pp 199ndash201 2010

[9] J M Antunes J V Fernandes L F Menezes and B M Cha-parro ldquoA new approach for reverse analyses in depth-sensingindentation using numerical simulationrdquo Acta Materialia vol55 no 1 pp 69ndash81 2007

[10] H Huang and H Zhao ldquoIn situ nanoindentation and scratchtesting inside scanning electronmicroscopes opportunities andchallengesrdquo Science of AdvancedMaterials vol 6 no 5 pp 875ndash889 2014

[11] S Balos L Sidjanin and B Pilic ldquoIndentation size effectin autopolymerized and microwave post treated poly(methylmethacrylate) denture reline resinsrdquo Acta Polytechnica Hungar-ica vol 11 no 7 pp 239ndash249 2014

[12] J Z Zhang M M Michalenko E Kuhl and T C OvaertldquoCharacterization of indentation response and stiffness reduc-tion of bone using a continuum damage modelrdquo Journal ofThermalMechanical Behavior of BiomedicalMaterials vol 3 pp189ndash202 2010

[13] C Ciofu and G Cretu ldquoSome aspects concerning the microindentation on testing of ultramid plastic materialsrdquo Interna-tional Journal of ModernManufacturing Technologies vol 6 pp2067ndash3604 2014

[14] S Amiri N Lecis AManes andMGiglio ldquoA study of amicro-indentation technique for estimating the fracture toughness ofAl6061-T6rdquo Mechanics Research Communications vol 58 pp10ndash16 2014

[15] R Rabe J-M Breguet P Schwaller et al ldquoObservation of frac-ture and plastic deformation during indentation and scratchinginside the scanning electron microscoperdquoThin Solid Films vol469-470 pp 206ndash213 2004

[16] N A Sakharova J V Fernandes J M Antunes and M COliveira ldquoComparison between Berkovich Vickers and conicalindentation tests a three-dimensional numerical simulationstudyrdquo International Journal of Solids and Structures vol 46 no5 pp 1095ndash1104 2009

[17] W G Mao Y G Shen and C Lu ldquoDeformation behaviorand mechanical properties of polycrystalline and single crystalalumina during nanoindentationrdquo Scripta Materialia vol 65no 2 pp 127ndash130 2011

[18] R Ghisleni K Rzepiejewska-Malyska L Philippe P Schwallerand J Michler ldquoIn situ SEM indentation experiments Instru-ments methodology and applicationsrdquo Microscopy Researchand Technique vol 72 no 3 pp 242ndash249 2009

[19] H Huang H Zhao Z Ma et al ldquoDesign and analysis of theprecision-driven unit for nano-indentation and scratch testrdquoJournal of Manufacturing Systems vol 31 no 1 pp 76ndash81 2012

[20] H Zhao H Huang Z Yang ZMa and Z FanDesign Analysisand Experiments of a Novel in situ SEM Indentation Device vol12 chapter 12 INTECH 2012

[21] L L Howell Compliant Mechanisms John Wiley amp Sons NewYork NY USA 2001

[22] R S Kumar K Sureshkumar and R Velraj ldquoOptimization ofbiodiesel production fromManilkara zapota (L) seed oil usingTaguchi methodrdquo Fuel vol 140 pp 90ndash96 2015

[23] H Tanyildizi andM Sahin ldquoApplication of Taguchi method foroptimization of concrete strengthened with polymer after hightemperaturerdquo Construction and Building Materials vol 79 pp97ndash103 2015

[24] T Du W Du K Che and L Cheng ldquoParametric optimiza-tion of overlapped helical baffled heat exchangers by Taguchimethodrdquo Applied Thermal Engineering vol 85 pp 334ndash3392015

[25] P Asokan R R Kumar R Jeyapaul and M Santhi ldquoDevel-opment of multi-objective optimization models for electro-chemicalmachining processrdquo International Journal of AdvancedManufacturing Technology vol 39 no 1-2 pp 55ndash63 2008

[26] Y S Tarng S C Juang andCH Chang ldquoTheuse of grey-basedTaguchi methods to determine submerged arc welding processparameters in hardfacingrdquo Journal of Materials Processing Tech-nology vol 128 no 1ndash3 pp 1ndash6 2002

[27] J L Deng ldquoGrey information spacerdquoThe Journal of Grey Systemvol 1 no 2 pp 103ndash117 1989

[28] S Ilo C Just and F Xhiku ldquoOptimisation of multiple qual-ity characteristics of hardfacing using grey-based TaguchimethodrdquoMaterials amp Design vol 33 no 1 pp 459ndash468 2012


[29] R K Pandey and S S Panda ldquoMulti-performance optimizationof bone drilling using Taguchi method based on membershipfunctionrdquoMeasurement vol 59 pp 9ndash13 2015

[30] S Roy A K Das and R Banerjee ldquoApplication of Grey-Taguchi basedmulti-objective optimization strategy to calibratethe PM-NHC-BSFC trade-off characteristics of a CRDI assistedCNG dual-fuel enginerdquo Journal of Natural Gas Science andEngineering vol 21 pp 524ndash531 2014

[31] U Caydas and A Hascalik ldquoUse of the grey relational analysisto determine optimum laser cutting parameters with multi-performance characteristicsrdquo Optics amp Laser Technology vol40 no 7 pp 987ndash994 2008

[32] M-J Tsai and C-H Li ldquoThe use of grey relational analysis todetermine laser cutting parameters forQFNpackageswithmul-tiple performance characteristicsrdquo Optics amp Laser Technologyvol 41 no 8 pp 914ndash921 2009

[33] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[34] C Chen G G Rigatos D Dong and J Lam ldquoPartial feedbackcontrol of quantum systems using probabilistic fuzzy estimatorrdquoin Proceedings of the 48th IEEE Conference on Decision andControl Held Jointly with the 28th Chinese Control Confer-ence (CDCCCC rsquo09) pp 3805ndash3810 IEEE Shanghai ChinaDecember 2009

[35] C Chen and T Xiao ldquoProbabilistic fuzzy control of mobilerobots for range sensor based reactive navigationrdquo IntelligentControl and Automation vol 2 pp 77ndash85 2011

[36] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[37] Y-H Chang C-L Chen W-S Chan H-W Lin and C-WChang ldquoFuzzy formation control and collision avoidance formultiagent systemsrdquoMathematical Problems in Engineering vol2013 Article ID 908180 18 pages 2013

[38] Y-F Tzeng and F-C Chen ldquoMulti-objective optimisation ofhigh-speed electrical discharge machining process using aTaguchi fuzzy-based approachrdquo Materials and Design vol 28no 4 pp 1159ndash1168 2007

[39] A K Pandey and A K Dubey ldquoTaguchi based fuzzy logicoptimization of multiple quality characteristics in laser cuttingof Duralumin sheetrdquo Optics and Lasers in Engineering vol 50no 3 pp 328ndash335 2012

[40] N-M Liu J-T Horng and K-T Chiang ldquoThemethod of grey-fuzzy logic for optimizing multi-response problems during themanufacturing process a case study of the light guide plateprinting processrdquo International Journal of Advanced Manufac-turing Technology vol 41 no 1-2 pp 200ndash210 2009

[41] R K Pandey and S S Panda ldquoOptimization of bone drillingparameters using grey-based fuzzy algorithmrdquo Measurementvol 47 no 1 pp 386ndash392 2014

[42] Y-S Yang and W Huang ldquoA grey-fuzzy Taguchi approach foroptimizing multi-objective properties of zirconium-containingdiamond-like carbon coatingsrdquo Expert Systems with Applica-tions vol 39 no 1 pp 743ndash750 2012

[43] T RajmohanK Palanikumar and S Prakash ldquoGrey-fuzzy algo-rithm to optimise machining parameters in drilling of hybridmetal matrix compositesrdquo Composites Part B Engineering vol50 pp 297ndash308 2013

[44] A K Pandey andAKDubey ldquoMultiple quality optimization inlaser cutting of difficult-to-laser-cut material using grey-fuzzy

methodologyrdquo International Journal of Advanced Manufactur-ing Technology vol 65 no 1 pp 421ndash431 2013

[45] H S Lu J Y Chen and C T Chung ldquoThe optimal cuttingparameter design of rough cutting process in side millingrdquoJournal of Achievements inMaterials andManufacturing vol 29pp 183ndash186 2008

[46] K-T Chiang and F-P Chang ldquoApplication of greyndashfuzzy logicon the optimal process design of an injection-molded part witha thin shell featurerdquo International Communications in Heat andMass Transfer vol 33 no 1 pp 94ndash101 2006

[47] C Ahilan S Kumanan and N Sivakumaran ldquoMulti-objectiveoptimisation of CNC turning process using grey based fuzzylogicrdquo International Journal of Machining and Machinability ofMaterials vol 5 no 4 pp 434ndash451 2009

[48] J-H Yeh and T-N Tsai ldquoOptimizing the fine-pitch copper wirebonding process with multiple quality characteristics using agrey-fuzzy Taguchi methodrdquo Microelectronics Reliability vol54 no 1 pp 287ndash296 2014

[49] ANSYS ANSYS Workbench Userrsquos Guide ANSYS CanonsburgPa USA 2015

[50] M J Wierman Fuzzy Sets Fuzzy Logic and Control Center ofMathematics of Uncertainty Inc Omaha Nebraska 2008

[51] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall New Delhi India 3rd edition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


polycrystalline and single crystal alumina were studied viananoindentation [17]

It is known that the indentation technique offers manyadvantages over conventional tests such as reduced costeconomy in material and time simplified experimenta-tion capacity to discover an inhomogeneous surface formicrostrength variations and residual stress [1ndash17] How-ever most of available indentation instruments have com-plicated structures frameworks and large volumes Thoseinstruments are therefore extremely difficult to insert insidethe scanning electron microscope (SEM) the transmissionelectronmicroscope (TEM) the image analysis and so forthbecause the SEM and TEM have a small volume chambershort working distance electromagnetic sensing vacuumenvironment and vibration sensing [18ndash20]

Regarding the positioning accuracy a precise guidingmechanism is actually necessary for indentation device Thismechanism aims to drive a sample towards a fixed indenterIn the other words it should possess a positioning capacitywithin the range of a few hundreds of micronanometersTo fulfill that requirement designing a precise guidingmechanism with a compact size is now difficult chances foracademic scientists engineer and manufacturers

Up till now traditional rigid-link mechanisms cannotmeet the accurate positioning requirements due to friction inkinematic pairs need for lubricant maintenance assemblybacklash and so on On the contrary compliant mechanismgains themobility from the deflection of flexure hinges ratherthan from kinetic joints It offers several extra advantagessuch as smooth displacement no backlash zero friction noneed for lubricant no noise part-count reduction simplifiedmanufacturing processes increased precision and reliabilityreduced wear and weight decreased assembly time andmaintenance and monolithic structure manufactured byelectrical discharged machining or wire electrical dischargedmachining [21] As a result of the accurate requirementcompliantmechanism is a potential candidate for positioningsystem

From such analysis a compliant guiding mechanism(CGM) is designed in this study Taking advantage of acompact size into account this mechanism can serve asa precise positioning platform inside a microindentationdevice The microindentation device is composed of keycomponents such as CGM axis coarse positioner indenterholder displacement sensor and load cell The CGM is themost important element to construct a good indentationsystem

Theworking efficiency of CGM is assessed with consider-ation of two quality characteristics such as the displacementand first natural frequency Because the CGM has a largedisplacement the indentation depth is then broadenedMoreover the CGM has a high first natural frequency theresponding speed of the indentation device is also fasterIn compliant mechanisms a large displacement is alwaysconflicted with a high frequency [21] From another aspectthese quality responses may be affected by controllableparameters such as the force exerting the CGM and thegeometric parameters A suitable optimization approach isactually needed to achieve a satisfaction for both responses

simultaneouslyThe optimization process of CGM is based ondesign of experiments and thus experiment-based optimiza-tion methods are taken into consideration in this study

Recently the Taguchi method has been extensively usedfor optimizing in various engineering fields due to decreasednumber of experiments [22] Even though the Taguchimethod was successfully proven for optimizing a singleperformance characteristic [23 24] but it failed for opti-mizing multiple performance characteristics [25 26] As aresult a hybrid approach is needed to resolve multiresponseoptimization problems

In the past initialized by Deng [27] the grey relationalanalysis has been useful technique in dealing with poorincomplete and partial knowndata [28]Thehybrid Taguchi-based grey relational analysis was discovered for optimizingmultiple characteristics in different fields of manufacturing[29ndash32] An earlier method the fuzzy logic was originatedby Zadeh [33] which was used for dealing with uncertainand vague information This theory was developed for manydifferent fields For instance Chen et al [34] constructedthe probabilistic fuzzy logic for the partial feedback of therobust control of quantum systems Also Chen and Xiao[35] applied the probabilistic fuzzy logic for mobile robotsand for the mobile-robot reactive navigation [36] And thena neural-fuzzy formation controller was investigated for themultiagent systems [37] however the fuzzy logic rules maynot be easily amendable to dynamic changes of a process[38 39]

To compensate the defect of fuzzy logic the grey rela-tional analysis was integrated with the fuzzy logic in themultivariate system to obtain the best quality responses [40ndash44] The grey relational analysis has three types such as thelower-the better the higher-the better and the nominal-the better which contain certain degree of uncertainty andvagueness better than the fuzzy logic For example the greytheory was successfully applied for multiobjective optimiza-tion of milling [45] injection-molded part with a thin shellfeature [46] CNC turning process [47] and so forth Takingthe advantages of the Taguchi method into account greyrelational analysis and fuzzy logic a hybrid approach ofthe Taguchi-grey based fuzzy logic was applied usefully formultiresponse optimization problems [48] However thishybrid approach has not been developed for compliantmech-anisms yet especially for a compliant guiding mechanism ofindentation device

The objective of this study is to propose a multiresponseoptimization for a compliant guiding mechanism (CGM)The CGM with flexure hinges serves as a precise positioningmechanism It is a major part of indentation device insidethe SEM A hybrid approach of the Taguchi-grey basedfuzzy logic is developed to maximize the displacement andfirst natural frequency of CGM simultaneously The forceexerting CGM and geometric parameters of flexure hingeare considered as design variables The experiment layout isassigned by Taguchirsquos 119871

27orthogonal array As suggested by

the Taguchi method the response table and response graphare constructed to determine the optimal parameter levelsFurthermore the analysis of variance (ANOVA) is used to


F

L

T

W

y

x

R

Effector







119882 ge 10119879 (1)

119871 ge 125119879 (2)





















Ratio of 119901e119889119894 ()

1 011232 000141 1262 012978 000155 1193 012896 000148 1154 013090 000162 1245 013091 00016 1226 012891 000154 1197 012823 000150 1178 013104 000171 1309 008983 000111 12410 019508 000241 12411 012970 000162 125



MRR 0RMSE 10minus12

RRMSE 0RMAE 0RAAE 0



119865 N 1800 18005 18010 18015 18040119871 mm 400 4010 4020 4030 4040119882 mm 300 3010 3020 3030 3040119879 mm 220 230 240 250 260119877 mm 040 050 060 070 080




Force


005

01

015

02

025

Disp

lace

men

t (m

m)



910912914916918920922

minus2 minus1 0 1 2

Freq

uenc

y (H

z)

Coded levels

Force





Find

119865 119871 119879119882 119877 (3)


1199101(119865 119871 119879119882 119877) (4)


1199102(119865 119871 119879119882 119877) (5)


120590max le

120590119910

SF(6)

within ranges

1800N le 119865 le 1804N

400mm le 119871 le 404mm

300mm le 119882 le 304mm

22mm le 119879 le 26mm

04mm le 119877 le 08mm

(7)








3 Methodology




Define problem



Data collection





Defuzzifier

Interface engine






Taguchi method


Fuzzifier

Membershipfunction

Fuzzy rules













120578 = minus10 log(1

119899

119899

sum

119894=1

1

1199102

119894

) (8)







119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)





119894(119896) and min 120578







Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)


119911119900(119896) and 119911



119894(119896) is the



120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)








120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)




1denotes



Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F

L

T

W

y

x

R

Effector







119882 ge 10119879 (1)

119871 ge 125119879 (2)





















Ratio of 119901e119889119894 ()

1 011232 000141 1262 012978 000155 1193 012896 000148 1154 013090 000162 1245 013091 00016 1226 012891 000154 1197 012823 000150 1178 013104 000171 1309 008983 000111 12410 019508 000241 12411 012970 000162 125



MRR 0RMSE 10minus12

RRMSE 0RMAE 0RAAE 0



119865 N 1800 18005 18010 18015 18040119871 mm 400 4010 4020 4030 4040119882 mm 300 3010 3020 3030 3040119879 mm 220 230 240 250 260119877 mm 040 050 060 070 080




Force


005

01

015

02

025

Disp

lace

men

t (m

m)



910912914916918920922

minus2 minus1 0 1 2

Freq

uenc

y (H

z)

Coded levels

Force





Find

119865 119871 119879119882 119877 (3)


1199101(119865 119871 119879119882 119877) (4)


1199102(119865 119871 119879119882 119877) (5)


120590max le

120590119910

SF(6)

within ranges

1800N le 119865 le 1804N

400mm le 119871 le 404mm

300mm le 119882 le 304mm

22mm le 119879 le 26mm

04mm le 119877 le 08mm

(7)








3 Methodology




Define problem



Data collection





Defuzzifier

Interface engine






Taguchi method


Fuzzifier

Membershipfunction

Fuzzy rules













120578 = minus10 log(1

119899

119899

sum

119894=1

1

1199102

119894

) (8)







119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)





119894(119896) and min 120578







Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)


119911119900(119896) and 119911



119894(119896) is the



120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)








120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)




1denotes



Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Ratio of 119901e119889119894 ()

1 011232 000141 1262 012978 000155 1193 012896 000148 1154 013090 000162 1245 013091 00016 1226 012891 000154 1197 012823 000150 1178 013104 000171 1309 008983 000111 12410 019508 000241 12411 012970 000162 125



MRR 0RMSE 10minus12

RRMSE 0RMAE 0RAAE 0



119865 N 1800 18005 18010 18015 18040119871 mm 400 4010 4020 4030 4040119882 mm 300 3010 3020 3030 3040119879 mm 220 230 240 250 260119877 mm 040 050 060 070 080




Force


005

01

015

02

025

Disp

lace

men

t (m

m)



910912914916918920922

minus2 minus1 0 1 2

Freq

uenc

y (H

z)

Coded levels

Force





Find

119865 119871 119879119882 119877 (3)


1199101(119865 119871 119879119882 119877) (4)


1199102(119865 119871 119879119882 119877) (5)


120590max le

120590119910

SF(6)

within ranges

1800N le 119865 le 1804N

400mm le 119871 le 404mm

300mm le 119882 le 304mm

22mm le 119879 le 26mm

04mm le 119877 le 08mm

(7)








3 Methodology




Define problem



Data collection





Defuzzifier

Interface engine






Taguchi method


Fuzzifier

Membershipfunction

Fuzzy rules













120578 = minus10 log(1

119899

119899

sum

119894=1

1

1199102

119894

) (8)







119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)





119894(119896) and min 120578







Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)


119911119900(119896) and 119911



119894(119896) is the



120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)








120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)




1denotes



Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Force


005

01

015

02

025

Disp

lace

men

t (m

m)



910912914916918920922

minus2 minus1 0 1 2

Freq

uenc

y (H

z)

Coded levels

Force





Find

119865 119871 119879119882 119877 (3)


1199101(119865 119871 119879119882 119877) (4)


1199102(119865 119871 119879119882 119877) (5)


120590max le

120590119910

SF(6)

within ranges

1800N le 119865 le 1804N

400mm le 119871 le 404mm

300mm le 119882 le 304mm

22mm le 119879 le 26mm

04mm le 119877 le 08mm

(7)








3 Methodology




Define problem



Data collection





Defuzzifier

Interface engine






Taguchi method


Fuzzifier

Membershipfunction

Fuzzy rules













120578 = minus10 log(1

119899

119899

sum

119894=1

1

1199102

119894

) (8)







119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)





119894(119896) and min 120578







Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)


119911119900(119896) and 119911



119894(119896) is the



120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)








120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)




1denotes



Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Define problem



Data collection





Defuzzifier

Interface engine






Taguchi method


Fuzzifier

Membershipfunction

Fuzzy rules













120578 = minus10 log(1

119899

119899

sum

119894=1

1

1199102

119894

) (8)







119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)





119894(119896) and min 120578







Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)


119911119900(119896) and 119911



119894(119896) is the



120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)








120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)




1denotes



Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119911119894(119896) =

120578119894(119896) minusmin 120578

119894(119896)

max 120578119894(119896) minusmin 120578

119894(119896)

(9)





119894(119896) and min 120578







Δ119900119894(119896) =

1003817100381710038171003817119911119900 (119896) minus 119911119894(119896)

1003817100381710038171003817 (10)


119911119900(119896) and 119911



119894(119896) is the



120574119894(119896) =

Δmin + 120575ΔmaxΔ119900119894(119896) + 120575Δmax

(11)








120583119860(119909 119886 119897 119903 119887) =

(119909 minus 119886)

(119897 minus 119886)119886 le 119909 le 119897

1 119897 lt 119909 lt 119903

(119909 minus 119887)

(119903 minus 119887)119903 le 119909 le 119887

(12)




1denotes



Rule 1 if 1199091is 1198601and 119909

2is 1198611 then 119910 is 119862

1else

Rule 2 if 1199091is 1198602and 119909

2is 1198612 then 119910 is 119862

2else

Rule 3 if 1199091is 1198603and 119909

2is 1198613 then 119910 is 119862

3else

Rule 4 if 1199091is 1198604and 119909

2is 1198614 then 119910 is 119862

4else

Rule 5 if 1199091is 1198605and 119909

2is 1198615 then 119910 is 119862

5else

Rule 6 if 1199091is 1198606and 119909

2is 1198616 then 119910 is 119862

6else

Rule 7 if 1199091is 1198607and 119909

2is 1198617 then 119910 is 119862

7else

Rule 8 if 1199091is 1198608and 119909

2is 1198618 then 119910 is 119862

8else


Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Rule 9 if 1199091is 1198609and 119909

2is 1198619 then 119910 is 119862

9else

Rule 10 if 1199091is 11986010and 119909

2is 11986110 then 119910 is 119862

10else

Rule 11 if 1199091is 11986011and 119909

2is 11986111 then 119910 is 119862

11else

Rule 12 if 1199091is 11986012and 119909

2is 11986112 then 119910 is 119862

12else

Rule 13 if 1199091is 11986013and 119909

2is 11986113 then 119910 is 119862

13else

Rule 14 if 1199091is 11986014and 119909

2is 11986114 then 119910 is 119862

14else

Rule 15 if 1199091is 11986015and 119909

2is 11986115 then 119910 is 119862

15else

Rule 16 if 1199091is 11986016and 119909

2is 11986116 then 119910 is 119862

16else

Rule 17 if 1199091is 11986017and 119909

2is 11986117 then 119910 is 119862

17else

Rule 18 if 1199091is 11986018and 119909

2is 11986118 then 119910 is 119862

18else

Rule 19 if 1199091is 11986019and 119909

2is 11986119 then 119910 is 119862

19else

Rule 20 if 1199091is 11986020and 119909

2is 11986120 then 119910 is 119862

20else

Rule 21 if 1199091is 11986021and 119909

2is 11986121 then 119910 is 119862

21else

Rule 22 If 1199091is 11986022and 119909

2is 11986122 then 119910 is 119862

22else

Rule 23 If 1199091is 11986023and 119909

2is 11986123 then 119910 is 119862

23else

Rule 24 If 1199091is 11986024and 119909

2is 11986124 then 119910 is 119862

24else

Rule 25 If 1199091is 11986025and 119909

2is 11986125 then 119910 is 119862

25else

Rule 26 If 1199091is 11986026and 119909

2is 11986126 then 119910 is 119862

26else

Rule 27 If 1199091is 11986027and 119909

2is 11986127 then 119910 is 119862

27

where 119860119894 119861119894 and 119862



119862 respectively

while 1199091and 119909




1198620(119910) of the output


1205831198620

(119910) = (1205831198601

(1199091) and 1205831198611


or (12058311986027

(1199091) and 12058311986127

(1199092))

(13)







MPI =sum1199101205831198620

(119910)

sum1205831198620

(119910) (14)





119902

sum

119904=1





1

119899eff+

1

119877e) (16)

where 120572 = risk = 005 119865120572(1 119891e) = 119865

005(1 119891e) is the 119865-ratio at





27 as seen in







(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(mm) 04 06 08



radius R1 1800 400 300 22 042 1800 400 302 24 063 1800 400 304 26 084 1800 402 300 24 065 1800 402 302 26 086 1800 402 304 22 047 1800 404 300 26 088 1800 404 302 22 049 1800 404 304 24 0610 1802 400 300 24 0811 1802 400 302 26 0412 1802 400 304 22 0613 1802 402 300 26 0414 1802 402 302 22 0615 1802 402 304 24 0816 1802 404 300 22 0617 1802 404 302 24 0818 1802 404 304 26 0419 1804 400 300 26 0620 1804 400 302 22 0821 1804 400 304 24 0422 1804 402 300 22 0823 1804 402 302 24 0424 1804 402 304 26 0625 1804 404 300 24 0426 1804 404 302 26 0627 1804 404 304 22 08












119894(2)






Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Jig

Screw

Slider


IndicatorForce gage


Computer

Modal analyzer


Accelerator

Hammer


1205831199091











11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11199102(Hz) SN of 119910

2








Number 119911119894(1) 119911

119894(2) Δ

119900119894(1) Δ

119900119894(2) 120574

119894(1) 120574

119894(2) GFRG

1 04744 05122 05256 04878 04875 05062 03032 04663 05218 05337 04782 04837 05111 02953 04855 04997 05145 05003 04929 04999 03124 04856 05091 05144 04909 04929 05046 03145 04658 05125 05342 04875 04834 05063 02936 04589 05375 05411 04625 04803 05195 02907 04869 05055 05131 04945 04935 05027 03148 00000 10000 10000 00000 03333 10000 02509 10000 00000 00000 10000 10000 03333 062510 04736 05122 05264 04878 04872 05062 030211 04751 05122 05249 04878 04879 05062 030412 03274 06674 06726 03326 04264 06005 023513 03320 06617 06680 03383 04281 05964 023714 03220 06676 06780 03324 04244 06007 023115 03276 06618 06724 03382 04265 05965 023316 03344 06536 06656 03464 04289 05907 023617 03437 06433 06563 03567 04324 05836 024018 03299 06537 06701 03463 04273 05908 023219 03384 06435 06616 03565 04305 05837 023620 06223 03685 03777 06315 05697 04419 028921 06307 03596 03693 06404 05752 04384 028822 06179 03687 03821 06313 05668 04420 028923 06255 03597 03745 06403 05718 04385 028824 06306 03648 03694 06352 05751 04404 028925 06362 03581 03638 06419 05789 04379 028726 06253 03650 03747 06350 05716 04405 028927 06318 03582 03682 06418 05759 04379 0287



119865 0333 0250 0282 0083 1119871 0285 0274 0307 0033 4119882 0280 0275 0310 0035 3119879 0268 0319 0278 0051 2119877 0275 0306 0284 0031 5







03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











03333 10

1M

embe

rshi

p va

lue

(a)

0

1

Mem

bers

hip

valu

e

03333 1

S M L

(b)


119894(2)


00

1

1

Mem

bers

hip

valu

e










(17)











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











03333 1 03333 1 0 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27







F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












F1 F2 F3 L1 L2




02

022

024

026

028

03

032

034

Mea

n of

GFR

C



120590 = 120576 times 119864 (18)










6 Conclusions









Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Screen

CPU


Screw

Jig Slider


DAQ

Strainwave



(10) Base




(10)

(6)

(7)(8)

(9)

(2) (3)(4)

(5)(1)

120572 = 15∘

(a)

F

Y

X

(b)





Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Competing Interests


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Multiresponse Optimization of a Compliant...

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