mathematicalmodelingofmultiplequalitycharacteristicsofa...

Research ArticleMathematical Modeling of Multiple Quality Characteristics of aLaser Microdrilling Process Used in Al7075SiCp Metal MatrixComposite Using Genetic Programming

Mohammed Yunus and Mohammad S Alsoufi

Department of Mechanical Engineering College of Engineering and Islamic Architecture Umm Al-Qura University Al-AbdiahMakkah 24231 Saudi Arabia

Correspondence should be addressed to Mohammed Yunus yunusmohammedrediffmailcom

Received 16 August 2018 Accepted 12 November 2018 Published 2 January 2019

Academic Editor Aiguo Song

Copyright copy 2019 Mohammed Yunus and Mohammad S Alsoufi )is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

)e conventional method for machining metal matrix composites (MMCs) is difficult on account of their excellent characteristicscompared with those of their source materials Modern laser machining technology is a suitable noncontact method formachiningoperations of advanced engineering materials due to its novel advantages such as higher productivity ease of adaptation toautomation minimum heat affected zone (HAZ) green manufacturing decreased processing costs improved quality reducedwastage removal of finishing operations and so on)eir application includes hole drilling in an aircraft engine components suchas combustion chambers nozzle guide vanes and turbine blades made up of MMCs which meet quality standards that determinetheir suitability for service use )is paper presents a derived mathematical model based on evolutionary computation methodsusing multivariate regression fitting for the prediction of multiple characteristics (circularity taper spatter and HAZ) ofneodymium yttrium aluminum garnet laser drilling of aluminum matrixsilicon carbide particulate (AlSiCp) MMCs usinggenetic programming Laser drilling input factors such as laser power pulse frequency gas pressure and pulse width are utilizedFrom a training dataset different genetic models for multiple quality characteristics were obtained with great accuracy duringsimulated evolution to provide a more accurate prediction compared to empirical correlations

1 Introduction

Metal matrix composites (MMCs) are substances which blenda tough metallic matrix with a hard ceramic reinforcementpossessing excellent features such as high strength to wearratio high modulus and wear and corrosion resistance [1]MMCs are broadly used in the fields of the aerospace au-tomotive electronics and metallic industries MMCs com-prise a metal as a base material (matrix) and hard ceramicparticles such as B4C SiC and Al2O3 as reinforcement (longfibers short whiskers or particulates in irregular or sphericalshapes) )e properties of MMCs are judged by matrix re-inforcement and interface between them [2] )ey are amaterial which is difficult to machine due to the presence ofhard ceramic particles [3] Most of the research in the ma-chining of AlSiCp MMCs has focused on turning andmilling whereas drilling has been given less attention

11 Laser Drilling )e laser drilling system is growing ex-ponentially to suit the alternative program for achieving themajor demands of aerospace automobile metallic andelectric industrial potential especially microhole drilling indifferent components such as watches and turbine bladesfuselages printed circuit boards and so on [3] Pulsed NdYAG laser microhole drilling has gained popularity in recentyears to be used as an indispensable tool for microholedrilling of components for technologically advanced in-dustries Laser microhole drilling processes are successfullyemployed to both conductive and nonconductive materialsto remove by evaporation and the amount of molten ma-terial removed depends on the penetration of laser energygenerated from a sequence of laser pulses at the same place[4] Laser drilling in the production industry has been as-sociated with the various advantages of a high rate of ma-chining noncontact method hence there is no damage to

HindawiModelling and Simulation in EngineeringVolume 2019 Article ID 1024365 15 pageshttpsdoiorg10115520191024365

tool or tool wear increased product quality and less wastageHighly reflective materials require a low amount of laserpower with a short wavelength of NdYAG (compared withCO2 lasers) [5] While it possesses several advantages and iswidely approved by advanced industries it also has somedefects such as taperness noncircular holes HAZ recastlayer and so on [6]

12 Genetic Programming Genetic programming (GP) isthe most effective tool used to predict the behaviour ofvarious processes and for the formation of empiricalmodeling Normally GP supports the process of evolution innaturemdashDarwinrsquos theory of ldquosurvival of the fittestrdquomdashto findthe best solution to an assigned problem GP is known as thegeneralized form of the genetic algorithm (GA) and has beenextensively studied [7ndash9] In GP the model is represented byterminals and functions A well-known implementation ofGP is in symbolic regression and is used to determine themathematical expression for a given set of variables andfunctions )e function is generated using a Boolean op-erator (AND OR NOT or ANDNOT) nonlinear operators(sin cos tan exp tanh and log) and from basic mathe-matical operators (+ minus and times) )e fitness function iscalculated as the error between the actual value and thepredicted value of the symbolic expressions In GP indi-vidual terms are randomly initialized and the population isprogressed to find out the optimal solutions through variousoperations such as reproduction crossover and mutation)e reproduction process produces children as an input tothe next generation by replicating a fraction of the parentselected by the current generation Individuals havinghighest fitness values in the population are selected as theparent and used for reproduction Normally the crossoveroperation produces children by exchanging some parts oftheir selected parents )e crossover operation is dividedinto two types (subtree and node crossover) However thesubtree crossover has shown more significant effect than thenode crossover

A variety of methods are available to develop relation-ships between inputs and outputs for evaluating outputsunder varying input conditions without experimental work)ese forecasting output data are derived in the form ofequations using artificial neural network statistical re-gression methods like linear regression response surfaceANOVA etc with limited accuracy To obtain higher ac-curacy at about 993 to 998 for the mathematical modelderived the effective method available is genetic pro-gramming which is an application of machine learning orartificial intelligence using inbuilt algorithms )e use of theGP approach with experimental data to develop a mathe-matical model involving laser microhole drilling input andoutput parameters is presented in this work )ese math-ematical models can be used to study the production of high-quality microdrills of the pulsed NdYAG laser used inAl7075SiCp MMCs by minimizing the microhole drillingdefects such as the degree of taperness of a hole spatterand heat affected zone width and to maximize a hole cir-cularity [10] )e various drilling input parameters involved

are pulse power (v0) pulse frequency (v1) assistance of gaspressure (v2) and pulse width (v3) [11]

2 Materials and Methods

In the present work using the stir casting technique anMMC consisting of aluminum alloy 7075 as a base metalreinforced with particulates of silicon carbide having a sizeof 40ndash50 μm for 10 volume fractions is produced )emicroholes were drilled onto MMC plates using the pulsedNdYAG laser beam system (Model JK300D) and the testswere carried out at its maximum power capacity of 16 kWVarious input parameters such as laser input power (v0)pulse frequency (v1) assistance of gas pressure (v2) andpulse width (v3) at different levels have been selected )eexperimental results for various levels of input factors oflaser microhole drilling of MMC (Al707510SiCp) plate of2mm thick are shown in Tables 2ndash5 For a smaller diameterhole laser microdrilling is preferred especially when thematerials are very hard extra thin or made of glass andcomposites )e quality of these holes mainly depends onheat affected zone (HAZ) of hole walls taperness formedwhen the hole is enlarged circularity to maintain the uni-form dimension of the circular hole and spatter occurring atthe ends of a hole during resolidification of material [12 13]

)e input parameters of different levels can improve thequality of drilled holes )e quality of holes was determinedusing optical measuring microscope OLYMPUS STM6 on acut sectioned hole sample by measuring circularity spatterHAZ and taper characteristics and in each experimentalrun the laser was drilled at a spot size of 180 μm [10]Various studies have been conducted to investigate the ef-fects of input control parameters on the defects developed bythe laser microdrilling process [14 15] Due to an internalfocusing of the issue of a laser drilling process hole tapernessand noncircularity affect the quality of holes [16] Normallythere is spatter accumulation due to an incomplete sus-pension of removed MMC at the drilling zone whichresolidifies and adheres around the whole circumferenceHence it is advisable to produce high-quality circularmicrodrilled holes with the minimum amount of taperspatter and HAZ width )e above-said qualities are to bedetermined by the level of input parameters which requires amathematical model to study the process condition capableof producing the desired quality of product However thismodel is to be derived in such a way that all the charac-teristics measuring qualities are quickly measurable simul-taneously [11]

3 Genetic Programming Methodology

Various stages of GP involved in the flowchart are as followsFor the analysis of multiple characteristics of neo-

dymium yttrium aluminum garnet laser drilling of alumi-num matrixsilicon carbide particulate MMCs such ascircularity taper spatter and HAZ data accumulations ofexperiments were carried out as per stages involved inFigure 1 )e accumulated data were randomized usingnotitia DiscipulusTM software and were provided to the

2 Modelling and Simulation in Engineering

software in three groups viz training validation and ap-plied [17] )e three sets of data are generated by experi-mental results provided Higher numbers of experimentrunning around more than 100 are required for best solu-tion Test runs were conducted to determine desirable pa-rameters generating an optimal solution in the minimumpossible time Initially the tests were carried out at the defaultparameter settings such as population size crossover rateDSS subset size and so on and later varied to find optimumvalues [18])e different parameters involved in achieving thefinal mathematical model satisfying the results are tabulatedin Table 1 It shows the flow of set required to achieve the finalmodel which could provide us a mathematical model satis-fying the above conditions of the quantity involved [19]

31RegressionandFitnessMeasurement Regression analysisis a stochastic method in which symbolic regression findsboth the working model of output (or target) function andits inputs (or fixed coefficients) or at least an approximation(error measurement fit by linear or square) Fitness mea-surement indicates how far the output value predicted by theGP concurs with the experimental value

32 Correlation Coefficient rR )e linear correlation co-efficient refers to measuring the strength and direction of alinear relationship held between two variables (output Mand input N) where r value lies such that minus1 lt r lt +1 [8])e signs indicate a strong positive linear correlation be-tween M and N or perfect fit if r is close to +1 with anincrease of M values N values also increase whereas r isclose to minus1 if M and N have a strong negative linear cor-relation such that with an increase of M values N valuesdecrease For r 0 there is no linearweak correlation and avalue approaching zero represents a random nonlinearrelationship between the M and N )e square of the cor-relation coefficient provides you with the coefficient ofdetermination r2 to find the proportion of the variance ofoutput that is predictable from the inputs )is helps us to

determine how certain one can be inmaking predictions froma defined model r2 defined from the ratio of the illustratedvariation to the total variation in the range of 0 lt r2 lt 1signifies the strength of the linear correlation between P andQor represents the percentage of the data which is closest to theline of best fit [20] If r 0977 then r2 0994 which meansthat 994 of the total variation in Q can be explained by thelinear relationship between P and Q and the remaining 06of the variation in Q continues unexplained [21]

33Factors Involved inGPModeling )e various parametersinvolved in modeling GP are tabulated in Table 1 It showsthe flow of set required to achieve the final model whichcould provide you with a mathematical model satisfying theabove conditions of the quantity involved

Architecture-altering operations choose an architecture-altering operation from the availablerepertoire of such operations and create one new offspring program for the new population by applying

the chosen architecture-altering operation to one selected program

Create new individual program(s) for the population by applying the following genetic operations withspecified probabilities by reproduction cross over and mutation

Execution of each program in the population or selects one or two individual program(s) from thepopulation to ascertain its fitness (explicitly or implicitly) using the problemrsquos fitness measure

Iterations following substeps (called a generation) on the population carried out until the terminationcriterion is satisfied

Random generation of an initial population (called generation zero) of individual computer programsconsisting of the available functions and terminals

Figure 1 Stages involved in genetic programming

Table 1 Parameter setting for genetic programming

Parameters Value assignedPopulation size (P) 500Number of generations 1000Maximum depth of tree 6Maximum generation 50Functional set Multiply plus minus divideTerminal set (v1 v2 v3 v4 v5 v6 minus10 10)Number of runs 110Mutation rate 010

Crossover rate 75 nonhomologous25 homologous

Reproduction rate 005

Fitness r2)e square root of the sum of the squareof absolute value of the differences

(errors) between the programrsquos outputand the observed data

Termination

An individual emerges whose sum ofabsolute errors is less than specified

(a) required number of runs becompleted or (b) required correlation

coefficient is obtainedTerminal set T P random-constants

Modelling and Simulation in Engineering 3

4 Results and Discussion

)e selection of precise instructions from set F andavailable terminal genes from set f(0) play a vital role in GPmodeling and evolutionary process will build a mathe-matical model (ie organism) such that it is as fit as for theprediction of results )e model consists of both in-structions and function genes behaving similarly to thecharacter of computer programs differing in appearanceand dimensions [20 22] Previously extensive work ondeveloping mathematical models was also carried outusing linear regression second order and higher orderequations using response surface methods ANOVA boxartificial neural network and fuzzy logic method where themodels of these methods develop the amount of error isvery high compared to experimental values upto 3 digitnumbers [24] )ese analyses require high value of digitnumbers not lower and decimal numbers for output valuescompared to values of inputs to accomplish accuracy)ese methods are suitable for small number of inputsotherwise erroneous empirical relation would be gener-ated )e above nonGP methods may use limited dataaccording design matrix and orthogonal array of tablesand the equation will be developed by using optimal inputvalues [24]

Using GreyndashTaguchi method the overall performancecharacteristic in NdYAG laser microdrilling of aluminawas calculated using grey relational grade (GRG) to finda optimal parameter set for getting highest GRG )eoptimal value of GRG was 09172 which indicates theeffectiveness of the proposed approach and the highestvalue of GRG of 08989 for confirmation experiment

)e overall quality feature is improved by 203 at optimalcondition with HAZ width by 878 and the holetaper worsened by 214 )is shows that in multileveloptimization of performance characteristics cannotreach simultaneously optimum value instead they have tocompromise between various performance characteristicsto accomplish the optimum value pertaining to overallperformance Both the performance characteristics im-prove at optimum level with GRG of 01521 (1988) )etaperness (from 00491 to 00476 rad) and HAZ width(from 02180 to 01683mm) are decreased at the same timeFrom the results we can see that optimum results canprovide correlations at the accuracy level of 85 if outputsare limited to two numbers By increasing output qualitycharacteristics as well as input parameters and number ofexperiments it is very difficult to arrive at optimal set andcorresponding correlations obtained by regression anal-ysis )erefore GP-based solutions maintain an accuracylevel beyond statistical analysis irrespective of number ofoutputs and inputs [25]

)e model used experimental measurements collectedby converting into three independent datasets such astraining validation and applied Independent input vari-ables used are pulse power (v0) pulse frequency (v1) as-sistance of gas pressure (v2) and pulse width (v3) andcircularity spatter heat affected zone and taper as thedependent output variable Various models for outputs aredeveloped by GP using the training dataset [23] Bestmathematical models obtained from GP simulation aregiven by equations (1)ndash(4)

circularity 2GF2 minus v0

v0minus1745

v0+ 0904515 (1)

spatter 1 +N

2[ (K + L)minus(4Lminus 19K + L) + 2 [(4Lminus 19K) + L]05 ]minus 0551113896 1113897

2

minus 1152⎡⎣ ⎤⎦

4

(2)

taper V PminusN(NminusM + LM) + V(MminusN)3

+(OV)(MminusN)3

1113872 11138731113872 1113873

(MminusN)3 minusV(F + G + 1113874 Q F + H2Q2 minus 11113872 1113873

025Q2

+ 2H2Q2+ 41113874 1113875

3Qminus 3minus 2Fminus 2Gminus 2 F + H2Q2 minus 11113872 1113873025

Q2+ 2H2Q2

+ HQ + 51113874 1113875minus 2v3 + 0731113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minusH2Q2 minus 11113874 1113875

(MminusN)3 PminusN(NminusM + LM) + V(MminusN)

3+(OV)(MminusN)

31113872 11138731113872 1113873 + 105

(3)

heat affected zone (122LI)minus 0682

v1v31113890 1113891minus 0079minus

122F

Gminus05356G

v01113890 1113891 +

(122LI)minus 0682v1

(4)

Appendixes A B C and D show further details relatedto the derivation of the circularity spatter taper and heataffected zone respectively

Comparison of the experimental and predicted outputsusing the mathematical model obtained fromGP is shown inTables 2ndash5 for circularity taper spatter and HAZ of pulsed


NdYAG laser microhole drilling of MMC componentsErrors in determining predicted outputs are very few andthe percentage of error is less than plusmn1 which shows thatresults obtained from a GP mathematical model are highlyacceptable

Furthermore Figure 2 shows the regression fit for thepercentage of microdrilling process parameters of the MMCs

Figure 3 shows the experimental-predicted relationship ofcircularity HAZ taper and spatter with the normal distri-bution behaviour )e models generated by the geneticprogramming perform better based on statistical terms andthe historical dataset (training validation and applied data) toexhibit a better predictive capacity on the experimentaldataset)e analysis of themathematical expression of the GP

Table 2 Comparison between experimental and predicted values of circularity at R2 9968

No PP (W) (v0) PF (Hz) (v1) AGP (kgcm2) (v2) PW (ms) (v3)Circularity (mm)

Error Experimental GP

1 250 210 12 04 0908 0908066 minus66E minus 052 210 210 8 02 09032 0903074 00001263 240 220 12 04 0911 0911254 minus0000254 210 210 8 06 0953 0953124 minus0000125 250 220 8 06 09425 0941842 00006586 210 250 12 06 0978 0978011 minus11E minus 057 210 230 10 04 0948 0945836 00021648 230 210 10 06 0957 0958292 minus0001299 210 230 10 03 0923 0922506 000049410 240 210 11 03 0892 089308 minus00010811 220 210 9 03 0926 0926227 minus00002312 230 230 12 03 092 0920843 minus00008413 240 220 12 03 09205 0920919 minus00004214 220 250 8 03 0896 0895994 611E minus 0615 230 210 10 05 0947 094698 196E minus 0516 240 230 8 04 0899 0898971 294E minus 0517 240 210 11 02 0892 0893078 minus00010818 220 240 12 06 0974 0973441 000055919 240 250 10 06 0954 0955555 minus00015520 250 250 11 04 092 0919789 000021121 250 240 10 02 0894 0893536 000046422 230 250 9 05 0953 0949875 000312523 210 250 12 05 0958 0963079 minus00050824 250 240 10 03 0892 0893537 minus00015425 250 220 8 05 0938 0933247 000475326 250 250 11 03 0895 0903726 minus00087327 230 230 12 02 0893 0892581 000041928 230 220 11 02 089 0892581 minus00025829 240 240 9 06 095 0949358 000064230 220 240 12 02 0893 0892039 000096131 250 230 9 02 0896 0896958 minus00009632 220 220 10 05 0951 0950144 000085633 240 240 9 05 095 0943263 000673734 210 220 9 02 09 0896232 000376835 220 250 8 02 0902 0896907 000509336 250 210 12 05 0948 094813 minus00001337 220 220 10 04 0936 0923214 001278638 230 240 8 03 0901 090405 minus00030539 250 230 9 06 09475 0947726 minus00002340 230 250 9 04 0911 0928516 minus00175241 240 250 10 02 0895 0895507 minus00005142 220 210 9 04 091 090617 00038343 240 230 8 05 094 0936347 000365344 210 240 11 05 0951 0959169 minus00081745 220 230 11 05 095 0954643 minus00046446 220 230 11 06 097 0967844 000215647 230 240 8 04 0901 0906892 minus00058948 230 220 11 06 0966 0964493 000150749 210 240 11 04 093 0943397 minus0013450 210 220 9 03 0922 0918336 0003664PP pulse power (W) PF pulse frequency (Hz) PW pulse width (ms) AGP assist gas pressure (Kgcm3)


Model suggests specific laser output quality characteristics forthe experimental system under various control factors thatcan be associated with the performance of a laser microdrilledhole in MMCs )e models generated by genetic pro-gramming allow representation of the experimental datawithout a detailed knowledge of the phenomenon In addi-tion their study allows us to obtain a deeper insight into therelevant factors in describing the quality phenomenon for

instance changes in any of the factors observed in quality ofthe microhole produced that might be associated withchanges in the performance of hole production in MMC

5 Conclusions

In this present work new models of the circularity spatterheat affected zone and taper of MMCs drilling properties at

Table 3 Comparison between experimental and predicted values of HAZ at R2 9988

No PP (W) (v0) PF (Hz) (v1) AGP (kgcm2) (v2) PW (ms) (v3)HAZ (mm)


1 240 230 8 04 004 419E minus 02 minus0001882 230 210 10 05 0112 0107232 00047683 230 230 12 03 0068 713E minus 02 minus0003284 210 230 10 03 0068 717E minus 02 minus0003685 220 240 12 02 0081 802E minus 02 00007896 240 220 12 03 0078 782E minus 02 minus0000237 220 210 9 04 0088 875E minus 02 00004538 250 220 8 06 0104 0103178 00008229 210 210 8 02 0065 680E minus 02 minus00030510 250 210 12 04 01 997E minus 02 000031111 210 230 10 04 0069 704E minus 02 minus00013912 240 220 12 04 0073 720E minus 02 00010313 230 220 11 06 0085 846E minus 02 000041714 220 250 8 03 0102 0101978 221E minus 0515 220 230 11 06 0089 867E minus 02 000227216 220 220 10 05 0094 950E minus 02 minus00009817 240 240 9 06 0105 0102995 000200518 240 210 11 02 0085 812E minus 02 000384719 220 250 8 02 00747 728E minus 02 000189320 230 250 9 05 01 0103908 minus00039121 210 220 9 03 0082 805E minus 02 000151222 210 250 12 05 00909 914E minus 02 minus00005123 210 250 12 06 00921 928E minus 02 minus00007124 220 210 9 03 0101 0101721 minus00007225 240 240 9 05 01 010225 minus00022526 230 250 9 04 0094 911E minus 02 000288527 240 250 10 06 0096 956E minus 02 000040628 250 210 12 05 0097 983E minus 02 minus00013129 230 230 12 02 0076 774E minus 02 minus0001430 220 230 11 05 0092 906E minus 02 000137931 210 240 11 04 0072 725E minus 02 minus00004932 230 210 10 06 0095 925E minus 02 000254833 250 250 11 03 0078 772E minus 02 000082634 250 220 8 05 0105 0104177 000082335 210 220 9 02 0084 768E minus 02 000716836 220 240 12 06 0095 920E minus 02 000302537 230 240 8 04 0055 467E minus 02 000828138 210 210 8 06 01 979E minus 02 000205439 250 230 9 02 007 745E minus 02 minus00044840 240 250 10 02 0095 829E minus 02 001209841 240 210 11 03 008 790E minus 02 000102542 230 240 8 03 009 887E minus 02 000130943 210 240 11 05 0084 904E minus 02 minus00063844 250 240 10 03 0065 774E minus 02 minus00124145 250 230 9 06 0095 990E minus 02 minus00040246 250 250 11 04 008 0081854 minus00018547 230 220 11 02 0077 00783 minus0001348 220 220 10 04 0075 007292 00020849 250 240 10 02 0091 882E minus 02 000277650 240 230 8 05 009 0090104 minus00001


different laser operating parameters are generated usingDiscipulus GP software and C programming Using GP-based mathematical models quality of laser drilled holeproperties for MMCs involving various laser input pa-rameters is determined quickly by substitution of laser inputconditions and without conducting any experiments theactual results can be predicted )e comparison between thenew GP-based model and the experimental results indicated

that the new model is more accurately close to plusmn 0006 to0009 )erefore the new model can be considered analternative method to estimate the mechanical propertieswhen the experimental measurements or correlations arenot available )e correctness of solutions achieved by GPdepends on correlated evolutionary parameters thenumber of experimental results and their level of accuracyTo improve the structure of the model during evolution

Table 4 Comparison between experimental and predicted values of taper at R2 9943

No PP (W) (v0) PF (Hz) (v1) AGP (kgcm2) (v2) PW (ms) (v3)Taper (deg)


1 230 250 9 04 35 348E + 00 00205392 230 240 8 04 3702 368349 0018513 210 220 9 03 3815 3816499 minus000154 230 210 10 05 375 372E + 00 00347385 240 250 10 02 4135 412E + 00 00170456 210 240 11 04 345 343E + 00 00232497 230 230 12 03 3925 386E + 00 00643398 220 240 12 06 35 3487621 00123799 220 240 12 02 412 413E + 00 minus00097910 240 220 12 03 38 381E + 00 minus00070911 230 240 8 03 38 379E + 00 001077112 220 250 8 03 3962 396E + 00 000665913 230 220 11 02 416 416E + 00 minus00003714 240 220 12 04 3775 3757363 001763715 250 230 9 06 38 3770205 002979516 210 210 8 06 42 4200192 minus00001917 230 220 11 06 375 3718309 003169118 220 250 8 02 405 410E + 00 minus00483119 250 210 12 04 41 396E + 00 013864920 250 220 8 05 35 3522696 minus0022721 240 240 9 05 34 341E + 00 minus00102122 210 210 8 02 41 410E + 00 00016223 240 230 8 05 3505 3523799 minus0018824 210 230 10 04 358 3633331 minus00533325 220 210 9 03 36 3774172 minus01741726 250 250 11 03 38 3846722 minus00467227 210 240 11 05 3307 3441274 minus01342728 250 240 10 02 41 4105488 minus00054929 220 230 11 05 33 3441037 minus01410430 240 250 10 06 38 381E + 00 minus00065431 230 210 10 06 3305 3638603 minus0333632 230 250 9 05 338 341E + 00 minus00309333 230 230 12 02 405 4110147 minus00601534 210 220 9 02 4 4125376 minus01253835 250 230 9 02 415 4123817 002618336 240 240 9 06 375 3748872 000112837 240 210 11 03 39 386E + 00 004329438 250 210 12 05 365 3501807 014819339 210 250 12 06 325 347E + 00 minus02163440 210 250 12 05 345 349592 minus00459241 220 220 10 05 36 3572536 002746442 240 210 11 02 395 411E + 00 minus01627643 250 220 8 06 36 4114297 minus0514344 220 220 10 04 38 3694675 010532545 250 250 11 04 39 3655343 024465746 240 230 8 04 36 3615975 minus00159747 220 230 11 06 34 3677536 minus02775448 220 210 9 04 35 348E + 00 002319349 250 240 10 03 38 380E + 00 000456650 230 250 9 04 35 348E + 00 0020539


Table 5 Comparison between experimental and predicted values of spatter at R2 993

No PP (W) (v0) PF (Hz) (v1) AGP (kgcm2) (v2) PW (ms) (v3)Spatter (mm)


1 220 220 10 04 0055 549E minus 02 0000132 230 220 11 06 0044 442E minus 02 minus0000183 210 250 12 06 0044 434E minus 02 00006484 240 220 12 03 0043 432E minus 02 minus0000225 230 240 8 03 004 436E minus 02 minus0003656 210 230 10 03 0042 421E minus 02 minus73E minus 057 250 240 10 02 0065 650E minus 02 318E minus 058 230 230 12 02 0043 431E minus 02 minus78E minus 059 240 240 9 06 0072 716E minus 02 000040610 230 220 11 02 0045 450E minus 02 minus16E minus 0511 250 230 9 02 0068 682E minus 02 minus00002412 220 240 12 06 0052 523E minus 02 minus00002813 210 250 12 05 0043 429E minus 02 636E minus 0514 220 210 9 04 0059 594E minus 02 minus00004515 210 220 9 02 00445 0044101 000039916 220 240 12 02 005 502E minus 02 minus00001917 230 210 10 05 0072 703E minus 02 000166118 230 240 8 04 0071 697E minus 02 000126519 240 210 11 02 0056 554E minus 02 000056620 230 250 9 05 0074 725E minus 02 000152421 240 230 8 04 0075 0064522 001047822 220 250 8 03 0074 0071546 000245423 250 210 12 05 0046 518E minus 02 minus00058124 210 240 11 04 0043 420E minus 02 000099325 230 210 10 06 0049 521E minus 02 minus00030726 250 250 11 03 0065 550E minus 02 001000327 210 230 10 04 004 428E minus 02 minus00027728 210 210 8 06 0043 0042277 000072329 210 240 11 05 0045 0044648 000035230 220 230 11 06 0057 540E minus 02 000304531 240 220 12 04 0042 429E minus 02 minus00009332 240 250 10 02 0066 664E minus 02 minus00004233 230 230 12 03 0041 427E minus 02 minus00017134 230 250 9 04 0069 684E minus 02 0000556

R2 = 9968

0 5 10 15 20 25 30 35 40 45 50 55088089090091092093094095096097098099

Circ

ular

ity (m

m)

Counts

Experimental Predicted

(a)

R2 = 9988


0 5 10 15 20 25 30 35 40 45 50 55004

005

006

007

008

009

010

011

012

HA

Z (m

m)

Counts

(b)

Figure 2 Continued


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)

Figure 2 Regression fit for the percentage of microdrilling process parameters of the MMCs (a) Experimental-predicted value of cir-cularity (b) Experimental-predicted value of HAZ (c) Experimental-predicted value of taper (d) Experimental-predicted value of spatter

088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)

Circularity mm (experimental)

(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)

HAZ mm (experimental)

(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)

Taper deg (experimental)

(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)

Spatter mm (experimental)

(d)

Figure 3 Experimental-predicted relationship of (a) circularity (b) HAZ (c) taper and (d) spatter


more information was supplied through experimentalmeasurements )erefore the proposed mathematicalmodel has verified its results are adaptable up to reliabilityof 993 to 998 to forecast experimental results In thetesting stage the GP model gives the same result as wasfound out during the experiment with the reliability of centpercent )e GP approach has thus proved to be a highlyskilled and advantageous tool for recognizing correlations

in data when no proper theoretical or other methods arepossible or available

Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A

A3 (v0v1v3)2 minus(7114340000)1113872 11138732A2 minus 211138731113872 11138730741113960 1113961v21113872 11138731113872 1113961 + 1921113872 11138732

+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)

D minus (4B)3 minus 1351113960 11139614v321113872 1113873minus

2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C

DC2 +382Dv3v2C2 minus 2lowast v1 + v0( 1113857DC

E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2

(1C) + 382Dv3v2C2 minus 2v1 + v0+ 2v0minus 15F + 11v21113888 1113889108 ndash v1 (A7)

B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961minus A2 minus 056( )1113966 11139674 minus 0442

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961minus A2 minus 056( )1113966 11139674

1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)

E D 4[( Bminus(1 + v0B) D)]2 minus 0256minusD1113966 11139671113872 11138731113960 1113961

4[( Bminus(1 + v0B) D)]2 minus 0256minusD1113966 11139672minus 11113876 1113877

(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus

096Bminus((1 + v0)B)

1113890 1113891 +(28minus v2)2E1113960 1113961

[ Bminus(1 + v0B) + D[Cminus(096 Bminus(1 + v0B) )]]2 + v31113966 1113967 (B6)

G 2(28minus v2)2E1113960 1113961

[ Bminus(1 + v0B) + D[Cminus(096 Bminus(1 + v0B) )]]2+ 2v3 + 2E + 2 (B7)

H 2(28minus v2)2E

[ Bminus(1 + v0B) + D[Cminus(096 Bminus(1 + v0B) )]]2 + 2v3 + 3E + 3(B8)

x H2G 05

F + G (B9)

I H + H( v2[(F + G) + x] + 2[2x] 2[2x]minus[(F + G) + x] )

[(F + G) + x]minus[x]1113888 1113889 (B10)

J ( [(F + G) + x]minus[2x] + 2 (HIH) +( IHH [[(F + G) + x]minus[2x]]) minus 0446) minus 0316)2

[(F + G) + x]minus[2x] + 2H

IH+( IHH [[(F + G + x)]minus[2x]])1113882 1113883minus 0446

(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH

[[(F + G) + x]minus[2x]]1113896 1113897minus 0446 + J (B12)

L 12698minusK

Kminus

H

IH+

( (IHH[[(F + G) + x]minus[2x]]) minus 0446)

[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705

1113896 1113897 + v0minus 2 (K + L)minus4lowast Lminus 19

K + L1113882 1113883 + 2lowast

4lowastLminus 19K + L

1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+

( (IHH)[[(F + G) + x]minus[2x]] minus 0446))

[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873

2V2 (K + L)minus(4Lminus 19K + L) + 2 [(4Lminus 19)(K + L)]051113960 11139611113960 11139611113966 1113967

(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)

((10000lowast v0 + 542v3minus 460707433240896051609600)(9007199254740992v3minus 9382184668271360000))

+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)minus 0001v3+ ((2263)(10000v0 + 542v3minus 460707433240896051609600(9007199254740992v3minus 9382184668271360000))

+ (0001v3minus 0001)(1001v3minus 0001)0519528v3minus 17555 + 99891113872 1113873minus 00911113857

(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015

451lowast 1015v2minus 9lowast 1012v3 + 451lowast 1015minusA

+4526

10000v0 + 542v3minus 461lowast 1023 +(0001v3ndash1000)(1001v3ndash1000)05(19526v3minus 1755) + 1000minus 0001

(C4)

D 1113874(4526)111387410000v0 + 542v3minus 4607lowast 10221113872 11138731113882 9lowast 1012v3minus 94lowast 10181113872 1113873 +(0001v3minus 1000)(1001v3minus 1000)05

middot (19526v3minus 1755) + 10000minus 901lowast 1015 45lowast 1015v2minus 9lowast 1012v3 + 45lowast 10151113872 11138731113872 1113873

minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)

E 6Cminus Aminus 0001v3minus 1018lowast 10151113872 111387311138742252lowast 1016v0 + 122lowast 1016v3minus 1038lowast 1036 1043lowast 1018v3minus 94lowast 10181113872 1113873

+ 2252v3 ndash 225lowast 10181113872 1113873(1001v3minus 1000)05

(19526v3minus 1755) + 2252lowast 1019 ndash 4062lowast 1031

1041v2minus 203lowast 1027v3 + 102lowast 10301113872 1113873 ndash 45lowast 1015 45lowast 1014v2minus 9lowast 1012v3 + 45lowast 10151113872 1113873minus 226310000v0

+ 19526v3 ndash 461lowast 1021 9lowast 1012v3minus 94lowast 10181113872 1113873 +(0001v3minus 0001)(1001v3minus 0001)05

middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)

F 451lowast 10151113872 1113873 45lowast 1014v2minus 1043lowast 1018v3 + 451lowast 10151113872 1113873minus 0001v31113872 1113873 +(2263) 10000v0 + 542v3minus 46lowast 10231113872 1113873

1043lowast 1018v3ndash94lowast 10181113872 1113873 +(0001v3minus 0001)(1001v3minus 0001)05

(19527v3minus 17555 + 9989)

minus v3 + E2 451lowast 10151113872 1113873 45lowast 1015v2ndash1043lowast 1018v3 + 451lowast 10151113872 1113873minus 0001v31113872 1113873 +(2263)

111387410000v0 + 542v3minus 461lowast 1027 1043lowast 1018v3minus 94lowast 10181113872 1113873 + 1113874(0001v3minus 0001)(1001v3minus 0001)05

middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)

G minus ( v3 + E2 451lowast 10151113872 1113873 45lowast 1015v2ndash901lowast 1012v3 + 451lowast 10141113872 11138731113872 1113873minus 0001v3 +(2263)10000v0 + 542v3

minus 461lowast 10271113872 1113873 901lowast 1012v3minus 94lowast 10181113872 11138731113872 1113873 +(0001v3minus 0001)(1001v3minus 0001)05

middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q

8 minus 0731113872 1113873QminusQ2A ndash 902lowast 1015

45lowast 1015v2 ndash 901lowast 1012v3 + 451lowast 10151113872 1113873minus (4526) 10000v0 + 542v3minus 461lowast 10271018 lowast (105v3 ndash 94)1113872 11138731113872 1113873

+ (0001v3minus 1000)(1001lowast v3minus 1000)05

(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)


H v3 + E2 451lowast 10151113872 1113873 45lowast 1015v2minus 1126lowast 1014v31113872 1113873125 + 451lowast 10151113872 11138731113872 11138731113872 1113873minus(v31000)1113872 1113873 +(2263)

10000lowast v0 + 542v3minus 461lowast 10221018 lowast ((105v3ndash94))1113872 11138731113872 1113873 + ((0001v3minus 0001)(1001v3minus 0001)05

middot 109v3ndash899lowast 1081113872 11138731113857 (5121169 + 10000)2 Q8 minus 0731113872 1113873Q + F1113872 11138731113872 1113873

minus1

(C9)

I 11138746Aminus 36Cminus 8Fminus 8G + 111387411138748v3minus1113874 8Qminus 16Fminus 8G + 8minus 8 (HQ)2 minus 21113872 1113873

025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q

2+ HQ + 51113875minus 8v3 + 292111387511138743Qminus 2Fminus 2G + 3minus 2 (HQ)

2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)

2 minus 111387411138744Q + 4minus 4Fminus 4G F +(HQ)2 ndash 11113872 1113873

05

middot Q2

+ 4(HQ)2

+ 81113875 3Q + 3minus 2Fminus 2Gminus 2 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)

J 2Aminus 12Cminus 2Fminus 2G + 2I + 0004v3minus 2 F + H2Q

2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ

2Aminus 12Cminus 2Fminus 2G + 2I +(v3250)minus 2 F + H2Q2 minus 1( )025

Q2 + 2H2Q2 + 41113872 1113873

(C11)

K (2Qminus 2Fminus 2G + 2Iminus 4) F + H2Q

2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)

(2Qminus 2Fminus 2G + 2Iminus 4) F + H2Q2 minus 1( )025

Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J

(QminusFminusG + I) F + H2Q2 minus 1( )025

Q2 + 2H2 Q2 + 4( )

(C13)

M Kminus L +(2Qminus 2Fminus 2Gminus 4) F + H2Q025Q2 + 2H2Q2 + 4( 1113857

K

minus(2Qminus 2Fminus 2Gminus 4) F + H2Q2 minus 1( 1113857

025Q2 + 2H2Q2 + 41113873

K Kminus L +(2Qminus 2Fminus 2Gminus 4) F + H2Q2 minus 1( )025

Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)

N (2Qminus 2Fminus 2Gminus 4) F + H2Q2 minus 1( 1113857

025Q2 + 2H2Q2 + 41113872 1113873

KKminus L + 2Qminus 2Fminus 2G ndashQ2 minus 4( )025

Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)

O (minusN(NminusM +(LM))) F + G +(2Qminus 2Fminus 2Gminus 4) F + H2Q

2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875

middot 1113874 (3Qminus 2Fminus 2Gminus 5) F + H2Q

2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875

(3Qminus 2Fminus 2Gminus 5) F + H2Q

2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)

P O (MminusN)3V1113872 11138731113872 1113873 middot 111387411138741113874F + G + 1113874 (2Qminus 2Fminus 2Gminus 4) F + H

2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875

(3Qminus 2Fminus 2Gminus 5) F +(HQ)2 minus 11113872 1113873

025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)


D Heat Affected Zone (HAZ)

A 138v3(1630651v2v3 + 147828v1v2)

(v0v1)2minus1394v3v0v12minus414v32

v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus

101v0v12minus3v3v1v0minus v2 + 029 (D2)

C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2

minus3366v0v12minus092v3v1v0

+ 1091113890 11138911113896 1113897

2

middot 092 minus54355v2v3 + 49276v1v2

(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875

54355v2v3 + 49276v1v2(v0v1)21113872 1113873minus 3366v0v12( )minus(v3v1v0) + 1091113960 1113961 (D5)

F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)

middot54355v2v3 + 49276v1v2(v0v1)21113872 1113873minus 3366v0v12( 1113857minus(v3v1v0) + 1091113960 1113961

v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2

minus 1⎡⎣ ⎤⎦

4

minus 1⎛⎝ ⎞⎠

4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)

H D + 0741D + v3minus [0741D + v3]2 minus 11113966 1113967

2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877

+(3384(Gv0)minus(6FG)minus 0066)

D + 0741D + v3minus [0741D + v3]2 minus 11113966 11139672

+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2

(v0v1)2minus3366v0v12minus

v3v1v0

+ 1091113890 1113891

minusF(3384(Gv0)minus(6FG)minus 0066)


+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)

J (8514Gv0minus 151FGminus 01662)


+ 11113882 1113883 + Eminus 1692lowast (Gv0)1113876 1113877lowast (FG)

(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865

[(FG)minus 044(Gv0)]I2 +

v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2

v0minus 2JIH3 + 2IHv2( ) [ ]I( )

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability

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

Conflicts of Interest

)e authors have no conflicts of interest

Authorsrsquo Contributions

MY and MSA participated in the conceptualization of theresearch prepared the data for experimentation and theanalysis and interpretation of the data MY used Geneticprogram Discipulus software for the analysis includingwriting equations MY drafted the manuscript and figureswere sketched by MSA MY and MSA revised the manu-script for intellectual content and approved the finalmanuscript

References

[1] L H Manjunatha et al ldquoDevelopment and comparativestudies of aluminum-based carbon nano tube metal matrixcomposites using powder metallurgy and stir casting tech-nologyrdquo International Journal of Scientific and EngineeringResearch vol 8 no 2 pp 521ndash526 2017

[2] M Yunus J F Rahman and S Ferozkhan ldquoGenetic pro-gramming approach for the prediction of thermal charac-teristics of ceramic coatingsrdquo International Journal ofIndustrial Engineering Research and Development vol 2 no 1pp 69ndash79 2011

[3] A S Kuar B Doloi and B Bhattacharyya ldquoModelling andanalysis of pulsed NdYAG laser machining characteristicsduring micro-drilling of zirconia (ZrO2)rdquo InternationalJournal of Machine Tools and Manufacture vol 46 no 12pp 1301ndash1310 2006

[4] A Manna and B Bhattacharayya ldquoA study on machinabilityof AlSiC-MMCrdquo Journal of Materials Processing Technologyvol 140 no 1ndash3 pp 711ndash716 2003

[5] K H Leong ldquoDrilling with lasersrdquo in Industrial Laser So-lutions for Manufacturing pp 36ndash42 Pennwell Tulsa OKUSA 2000

[6] J Meijer ldquoLaser beam machining (LBM) state of the art andnew opportunitiesrdquo Journal of Materials Processing Tech-nology vol 149 no 1ndash3 pp 2ndash17 2004

[7] M Yunus J F Rahman and S Ferozkhan ldquoEvaluation ofmachinability characteristics of industrial ceramic coatingsusing genetic programming based approachrdquo InternationalJournal of Mechanical Engineering and Technology vol 2no 2 pp 126ndash137 2011

[8] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge MAUSA 1992

[9] M Yunus and M S Alsoufi ldquoPrediction of mechanicalproperties of plasma sprayed thermal barrier coatings (TBCs)with genetic programming (GP)rdquo International Journal ofEngineering Trends and Technology vol 47 no 3 pp 139ndash1452017

[10] M Ghoreishi D K Y Low and L Li ldquoComparative statisticalanalysis of hole taper and circularity in laser percussiondrillingrdquo International Journal of Machine Tools and Man-ufacture vol 42 no 9 pp 985ndash995 2002

[11] M V Lakshmi and M L Chaitanya ldquoApplication of taguchibased grey relational analysis for evaluating optimal param-eters of laser micro-drilling Al7075SiCp metal matrixcompositerdquo International Journal of Mechanical Engineeringand Technology vol 5 no 1 pp 16ndash22 2015

[12] C Bagger and F O Olsen ldquoPulsed mode laser cutting ofsheets for tailored blanksrdquo Journal of Materials ProcessingTechnology vol 115 no 1 pp 131ndash135 2001

[13] S Bandyopadhyay J K Sarin Sundar G Sundararajan andS V Joshi ldquoGeometrical features and metallurgical charac-teristics of NdYAG laser drilled holes in thick IN718 andTindash6Alndash4V sheetsrdquo Journal of Materials Processing Technol-ogy vol 127 no 1 pp 83ndash95 2002

[14] B S Yilbas and Z Yilbas Parameters Affecting Hole Geometryin Laser Drilling of Nimonic 75 SPIE Los Angeles CA USA1987

[15] D K Y Low L Li A G Corfe and P J Byrd ldquoSpatter-freelaser percussion drilling of closely spaced array holesrdquo In-ternational Journal of Machine Tools andManufacture vol 41no 3 pp 361ndash377 2001

[16] C M Sharp R E Mueller J Murthy M H McCay andJ Cutcher ldquoA novel anti-spatter technique for laser drillingapplications to surface texturingrdquo in Laser MaterialsProcessing pp 41ndash50 Laser Institute of America San DiegoCA USA 1997

[17] M Brezocnik M Kovacic and M Ficko ldquoPrediction ofsurface roughness with genetic programmingrdquo Journal ofMaterials Processing Technology vol 157-158 pp 28ndash362004

[18] J R Koza Genetic Programming II (Automatic Discovery ofReusable Programs) )e MIT Press Cambridge MA USA1994

[19] M Brezocnik and M Kovacic ldquoIntegrated genetic pro-gramming and genetic algorithm approach to predict surfaceroughnessrdquo Materials and Manufacturing Processes vol 18no 3 pp 475ndash491 2003

[20] L Gusel and M Brezocnik ldquoModeling of impact toughness ofcold formed material by genetic programmingrdquo Computa-tional Materials Science vol 37 no 4 pp 476ndash482 2006

[21] S R Choi D-M Zhu and R A Miller ldquoEffect of sintering onmechanical and physical properties of plasma-sprayed ther-mal barrier coatingsrdquo NASATM-2004-212625 NASAWashington DC USA 2004

[22] Y S Chang K S Park and B Y Kim ldquoNonlinear model forECG R-R interval variation using genetic programming ap-proachrdquo Future Generation Computer Systems vol 21 no 7pp 1117ndash1123 2005

[23] M Brezocnik and L Gusel ldquoPredicting stress distribution incold-formed material with genetic programmingrdquo In-ternational Journal of Advanced Manufacturing Technologyvol 23 no 7-8 pp 467ndash474 2004

[24] M Yunus and M S Alsoufi ldquoMathematical modelling of afriction stir welding process to predict the joint strength oftwo dissimilar aluminium alloys using experimental data andgenetic programmingrdquo Modelling and Simulation in Engi-neering vol 18 pp 1ndash18 2018

[25] A S Kuar B Acherjee D Ganguly and S Mitra ldquoOpti-mization of NdYAG laser parameters for microdrilling ofalumina with MultiqualityCharacteristics via greyndashtaguchimethodrdquo Materials and Manufacturing Processes vol 27no 3 pp 329ndash336 2012


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tool or tool wear increased product quality and less wastageHighly reflective materials require a low amount of laserpower with a short wavelength of NdYAG (compared withCO2 lasers) [5] While it possesses several advantages and iswidely approved by advanced industries it also has somedefects such as taperness noncircular holes HAZ recastlayer and so on [6]

12 Genetic Programming Genetic programming (GP) isthe most effective tool used to predict the behaviour ofvarious processes and for the formation of empiricalmodeling Normally GP supports the process of evolution innaturemdashDarwinrsquos theory of ldquosurvival of the fittestrdquomdashto findthe best solution to an assigned problem GP is known as thegeneralized form of the genetic algorithm (GA) and has beenextensively studied [7ndash9] In GP the model is represented byterminals and functions A well-known implementation ofGP is in symbolic regression and is used to determine themathematical expression for a given set of variables andfunctions )e function is generated using a Boolean op-erator (AND OR NOT or ANDNOT) nonlinear operators(sin cos tan exp tanh and log) and from basic mathe-matical operators (+ minus and times) )e fitness function iscalculated as the error between the actual value and thepredicted value of the symbolic expressions In GP indi-vidual terms are randomly initialized and the population isprogressed to find out the optimal solutions through variousoperations such as reproduction crossover and mutation)e reproduction process produces children as an input tothe next generation by replicating a fraction of the parentselected by the current generation Individuals havinghighest fitness values in the population are selected as theparent and used for reproduction Normally the crossoveroperation produces children by exchanging some parts oftheir selected parents )e crossover operation is dividedinto two types (subtree and node crossover) However thesubtree crossover has shown more significant effect than thenode crossover

A variety of methods are available to develop relation-ships between inputs and outputs for evaluating outputsunder varying input conditions without experimental work)ese forecasting output data are derived in the form ofequations using artificial neural network statistical re-gression methods like linear regression response surfaceANOVA etc with limited accuracy To obtain higher ac-curacy at about 993 to 998 for the mathematical modelderived the effective method available is genetic pro-gramming which is an application of machine learning orartificial intelligence using inbuilt algorithms )e use of theGP approach with experimental data to develop a mathe-matical model involving laser microhole drilling input andoutput parameters is presented in this work )ese math-ematical models can be used to study the production of high-quality microdrills of the pulsed NdYAG laser used inAl7075SiCp MMCs by minimizing the microhole drillingdefects such as the degree of taperness of a hole spatterand heat affected zone width and to maximize a hole cir-cularity [10] )e various drilling input parameters involved

are pulse power (v0) pulse frequency (v1) assistance of gaspressure (v2) and pulse width (v3) [11]

2 Materials and Methods

In the present work using the stir casting technique anMMC consisting of aluminum alloy 7075 as a base metalreinforced with particulates of silicon carbide having a sizeof 40ndash50 μm for 10 volume fractions is produced )emicroholes were drilled onto MMC plates using the pulsedNdYAG laser beam system (Model JK300D) and the testswere carried out at its maximum power capacity of 16 kWVarious input parameters such as laser input power (v0)pulse frequency (v1) assistance of gas pressure (v2) andpulse width (v3) at different levels have been selected )eexperimental results for various levels of input factors oflaser microhole drilling of MMC (Al707510SiCp) plate of2mm thick are shown in Tables 2ndash5 For a smaller diameterhole laser microdrilling is preferred especially when thematerials are very hard extra thin or made of glass andcomposites )e quality of these holes mainly depends onheat affected zone (HAZ) of hole walls taperness formedwhen the hole is enlarged circularity to maintain the uni-form dimension of the circular hole and spatter occurring atthe ends of a hole during resolidification of material [12 13]

)e input parameters of different levels can improve thequality of drilled holes )e quality of holes was determinedusing optical measuring microscope OLYMPUS STM6 on acut sectioned hole sample by measuring circularity spatterHAZ and taper characteristics and in each experimentalrun the laser was drilled at a spot size of 180 μm [10]Various studies have been conducted to investigate the ef-fects of input control parameters on the defects developed bythe laser microdrilling process [14 15] Due to an internalfocusing of the issue of a laser drilling process hole tapernessand noncircularity affect the quality of holes [16] Normallythere is spatter accumulation due to an incomplete sus-pension of removed MMC at the drilling zone whichresolidifies and adheres around the whole circumferenceHence it is advisable to produce high-quality circularmicrodrilled holes with the minimum amount of taperspatter and HAZ width )e above-said qualities are to bedetermined by the level of input parameters which requires amathematical model to study the process condition capableof producing the desired quality of product However thismodel is to be derived in such a way that all the charac-teristics measuring qualities are quickly measurable simul-taneously [11]

3 Genetic Programming Methodology

Various stages of GP involved in the flowchart are as followsFor the analysis of multiple characteristics of neo-

dymium yttrium aluminum garnet laser drilling of alumi-num matrixsilicon carbide particulate MMCs such ascircularity taper spatter and HAZ data accumulations ofexperiments were carried out as per stages involved inFigure 1 )e accumulated data were randomized usingnotitia DiscipulusTM software and were provided to the




















Termination











v0minus1745

v0+ 0904515 (1)

spatter 1 +N


2

minus 1152⎡⎣ ⎤⎦

4

(2)


+(OV)(MminusN)3

1113872 11138731113872 1113873


025Q2

+ 2H2Q2+ 41113874 1113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minus 2v3 + 0731113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minusH2Q2 minus 11113874 1113875


3+(OV)(MminusN)

31113872 11138731113872 1113873 + 105

(3)


v1v31113890 1113891minus 0079minus

122F

Gminus05356G

v01113890 1113891 +

(122LI)minus 0682v1

(4)














5 Conclusions


















R2 = 9968

0 5 10 15 20 25 30 35 40 45 50 55088089090091092093094095096097098099

Circ

ular

ity (m

m)

Counts


(a)

R2 = 9988


0 5 10 15 20 25 30 35 40 45 50 55004

005

006

007

008

009

010

011

012

HA

Z (m

m)

Counts

(b)

Figure 2 Continued


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)


088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)


(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)


(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)


(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)


(d)





Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




















Termination











v0minus1745

v0+ 0904515 (1)

spatter 1 +N


2

minus 1152⎡⎣ ⎤⎦

4

(2)


+(OV)(MminusN)3

1113872 11138731113872 1113873


025Q2

+ 2H2Q2+ 41113874 1113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minus 2v3 + 0731113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minusH2Q2 minus 11113874 1113875


3+(OV)(MminusN)

31113872 11138731113872 1113873 + 105

(3)


v1v31113890 1113891minus 0079minus

122F

Gminus05356G

v01113890 1113891 +

(122LI)minus 0682v1

(4)














5 Conclusions


















R2 = 9968

0 5 10 15 20 25 30 35 40 45 50 55088089090091092093094095096097098099

Circ

ular

ity (m

m)

Counts


(a)

R2 = 9988


0 5 10 15 20 25 30 35 40 45 50 55004

005

006

007

008

009

010

011

012

HA

Z (m

m)

Counts

(b)

Figure 2 Continued


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)


088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)


(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)


(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)


(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)


(d)





Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








v0minus1745

v0+ 0904515 (1)

spatter 1 +N


2

minus 1152⎡⎣ ⎤⎦

4

(2)


+(OV)(MminusN)3

1113872 11138731113872 1113873


025Q2

+ 2H2Q2+ 41113874 1113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minus 2v3 + 0731113875


Q2+ 2H2Q2

+ HQ + 51113874 1113875minusH2Q2 minus 11113874 1113875


3+(OV)(MminusN)

31113872 11138731113872 1113873 + 105

(3)


v1v31113890 1113891minus 0079minus

122F

Gminus05356G

v01113890 1113891 +

(122LI)minus 0682v1

(4)














5 Conclusions


















R2 = 9968

0 5 10 15 20 25 30 35 40 45 50 55088089090091092093094095096097098099

Circ

ular

ity (m

m)

Counts


(a)

R2 = 9988


0 5 10 15 20 25 30 35 40 45 50 55004

005

006

007

008

009

010

011

012

HA

Z (m

m)

Counts

(b)

Figure 2 Continued


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)


088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)


(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)


(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)


(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)


(d)





Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












5 Conclusions


















R2 = 9968

0 5 10 15 20 25 30 35 40 45 50 55088089090091092093094095096097098099

Circ

ular

ity (m

m)

Counts


(a)

R2 = 9988


0 5 10 15 20 25 30 35 40 45 50 55004

005

006

007

008

009

010

011

012

HA

Z (m

m)

Counts

(b)

Figure 2 Continued


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)


088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)


(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)


(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)


(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)


(d)





Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













R2 = 9968

0 5 10 15 20 25 30 35 40 45 50 55088089090091092093094095096097098099

Circ

ular

ity (m

m)

Counts


(a)

R2 = 9988


0 5 10 15 20 25 30 35 40 45 50 55004

005

006

007

008

009

010

011

012

HA

Z (m

m)

Counts

(b)

Figure 2 Continued


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)


088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)


(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)


(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)


(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)


(d)





Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


R2 = 9943


0 5 10 15 20 25 30 35 40 45 50 5532

34

36

38

40

42

44

Tape

r (de

g)

Counts

(c)

R2 = 9930


0 5 10 15 20 25 30 350035

0040

0045

0050

0055

0060

0065

0070

0075

0080

Spat

ter (

mm

)

Counts

(d)


088 090 092 094 096 098088

090

092

094

096

098 012

6 3 6 3 6 101 3

136

36

29

632

Circ

ular

ity m

m (p

redi

cted

)


(a)

004 006 008 010 012004

006

008

010

012 0 1 1 5 9 10 14 91

201

1510

148

0

HA

Z m

m (p

redi

cted

)


(b)

32 33 34 35 36 37 38 39 40 41 42 433334353637383940414243 1

4 47 5 5

85 3

71

011

4777

20

111

Tape

r deg

(pre

dict

ed)


(c)

004 005 006 007 0080040

0045

0050

0055

0060

0065

0070

0075 0

134 2 4 0

5 5 1

14

1

6

3

1

5

4

0

Spat

ter

mm

(pre

dict

ed)


(d)





Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Appendix

A Circularity

A ( ( 0954v0 +(2v2 + 01644)2(v2 + 01644)

2 minus ( 4503599627370496( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575minus(2v2 + 01644)(v2 + 01644)11138571113857

middot ( 2(2v2 + 01644)2(v2 + 01644)

2 minus ( 9007199254740992( 2v3minus v0 +(4v2 + 06576)(v2 + 01644)

+(2v2 + 01644)(v2 + 01644)1113857211138574720808237398575 + 2v0v31113857

minus1

(A1)

B A


+ 2A1113882 1113883 + v11113874 1113877v3v11113874 1113875

+ 09045(A2)

C 1

Bminus 2B7( )1113890 1113891lowast 2B

71113966 1113967minus 1351113960 11139612v31113960 1113961 (A3)


2Bminus 4B

61113874 1113875 + 2C1113882 1113883minus 111113874 11138750241113876 1113877 + 1081113882 1113883v3 (A4)

E 382Dv3v2C2 minus(2v1 + v0 + 2v2)

2DC +

382Dv3v2C2 minus 2v1 + v0( 1113857

E2 + v2 (A5)

F C


E2+ 382Dv3v2C

2 minus 2v1 + v0 (A6)

G E2


B Spatter

A 02454(8272 02454 + v2 minus 07335 + v3)8 minus 0181113960 1113961

02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961 (B1)

B 02454(8272 02454 + v2 minus 07335 + v3)8 minus 017321113960 1113961

+1


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(B2)

C 1 + v0

B1113882 1113883

202454

1


1113874 1113875minus 04421113882 11138832

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(B3)


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


D 02454


1113874 1113875minus 04421113882 11138832 minusC + 096

(B4)



(B5)

F Bminus(1 + v0)

B1113896 1113897 + D Cminus


1113890 1113891 +(28minus v2)2E1113960 1113961


G 2(28minus v2)2E1113960 1113961


H 2(28minus v2)2E


x H2G 05

F + G (B9)


[(F + G) + x]minus[x]1113888 1113889 (B10)


[(F + G) + x]minus[2x] + 2H


(B11)

K [(F + G) + x]minus[2x] + 2H

IH+

IHH


L 12698minusK

Kminus

H

IH+


[[(F + G) + x]minus[2x]] (B13)

M 24Lminus 19K + L

1113876 111387705


K + L1113882 1113883 + 2lowast


1113876 111387705

1113896 11138971113890 1113891 (B14)

N H

IH+


[[(F + G) + x]minus[2x]]

+M(M + 2v2minus 0153)21113872 1113873


(B15)

C Taper

MminusNminusL

M+

N

NminusM +(LM)1113888 1113889

4

V (C1)


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Q (Aminus 6C + 0002v3 + 1)A

((4508098723398239)(4503599627370496v2minus 9007199254740992v3 + 4512597819425982))minus 0001v3 +(2263)


+ (0001v3minus 0001)(1001v3minus 0001)05

(19528v3minus 17555 + 9989)minus 00911113872 1113873(4508098723398239)



(C2)

B 0001v3minusA +9016197446796478

(4503599627370496v2minus 9007199254740992v3 + 4512597819425982)

+4526

(10000v0 + 542v3minus(460707433240896051609600(9007199254740992v3minus 938)))

(C3)

C B2901lowast 1015


+4526


(C4)



minusA +(Q + 1)05

(Q + 4C)minus 0001111388311138751113875minus1

(C5)






middot (19526v3minus 17555 + 9989) + D05

Q1113872 11138731113875minus1

(C6)



(19527v3minus 17555 + 9989)



middot (19527v3minus 17555 + 9989)111387521113875 Q

8 minus 0731113872 1113873Q

(C7)



middot (1000000000v3minus 899000000)(5121169) + 998911138572 Q




(19526v3minus 17555) + 9989 + 00011113872 1113873E2

(C8)





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





minus1

(C9)


025Q

2+ 4H

2Q

2+ 81113874 11138753Q + 3minus 2Fminus 2G

minus 2 F +(HQ)2 minus 11113872 1113873

025Q

2+ 2H

2Q


2 minus 11113872 1113873025

middot Q2

+ 2(HQ)2

+ HQ + 51113875minus1

1113875minus 4 F +(HQ)2 ndash 11113872 1113873

025Q

2+ 8(HQ)


05

middot Q2

+ 4(HQ)2


025Q

2+ 2(HQ)

2+ HQ + 51113874 11138751113875

minus1+ 2Q + 12541113875

05

(C10)


2 minus 11113872 111387314

Q2

+ 2H2Q

2+ 41113874 1113875

12

minusJ


Q2 + 2H2Q2 + 41113872 1113873

(C11)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 405

minus(2J)


Q2 + 2H2Q2 + 41113872 1113873

(C12)

L J


Q2 + 2H2Q2 + 41113872 111387315minus

J


Q2 + 2H2 Q2 + 4( )

(C13)


K


025Q2 + 2H2Q2 + 41113873


Q2 + 2H2Q2 + 4K( )1113872 1113873

(C14)


025Q2 + 2H2Q2 + 41113872 1113873


Q2 + 2H2Q2 + 4( )K( )1113872 1113873+

(2LM)

Mminus(LM) (C15)


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ 41113874 11138751113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQ + 51113872 1113873minus(2v3 + 073)1113874 1113875


2 minus 11113872 1113873025

Q2

+ 2H2Q

2+ HQminusH

2Q

2 minus 11113872 11138733

1113874 11138751113875minus1

(C16)


2Q

225+ 0004H

2Qv3 + 11113872 1113873

2+ 41113874 1113875


025Q

21113874 11138751113875 + 2(HQ)

2 minus 11113875025

Q2

+ 2(HQ)2

+ HQ + 51113875 ndash (HQ)2 minus 11113875

minus1

(C17)



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



A 138v3(1630651v2v3 + 147828v1v2)


v1v0minus 138v3v2 + 04v3 + 1725 (D1)

B (23264v2v3 + 2109v1v2)

v0v1minus3366

v1+

(1630651v2v3 + 147828v1v2)

(v0v1)2minus


C 2731844A

v32minus 2v01113890 1113891

54355v2v3 + 49276v1v2(v0v1)2


+ 1091113890 11138911113896 1113897

2


(v0v1)2+

31v0v12

+092v3v1v0

+ 092A1113890 1113891minus 0121

(D3)

D B + 2Cminus(54355v2v3 + 49276v1v2)

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138971113888 1113889minus 1 (D4)

E [0741D + v3]2 minus 11113966 1113967

2minus 321113874 1113875


F minus54355v2v3 + 49276v1v2

(v0v1)2+3366v0v12

+v3

v1v0+ A1113896 11138978C(Eminus v3)


v0⎧⎨

⎩

⎫⎬

⎭

2

(D6)

G 11 (1minusF)4 minus 11113966 1113967

2minus 11113876 1113877 + F1113882 1113883

2minus 1 + F1113888 1113889

2


4


4

minus 11⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

2

(D7)


2+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877(FG)

(D8)

I 54355v2v3 + 49276v1v2


v3v1v0

+ 1091113890 1113891



+ 11113882 1113883 + Eminus 1692(Gv0)1113876 1113877G

(D9)




(D10)

K 251minus 2JIH3 + 2I(Hv2)1113858 11138591113864 1113865


v2v0

+ 2JIH3

+2I

(Hv2)1113896 1113897 (D11)

L v2 2v1 K

Iv01113960 11139612 minus 11113882 1113883

2


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠minus v0minus 126

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 2JIH3

+2I

(Hv2)1113896 1113897 (D12)


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Data Availability






References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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