Research Article Optimization Design for Detonation ...downloads.hindawi.com/journals/mpe/2016/5036083.pdf · to carry out the test ight in accordance with airworthiness regulations

Research ArticleOptimization Design for Detonation Powerplant Based onLS-DYNA Simulation

Yongbao Ai and Ming Lu

College of Field Engineering PLA University of Science and Technology Nanjing Jiangsu 210007 China

Correspondence should be addressed to Ming Lu aybjackai163com

Received 28 February 2016 Revised 11 May 2016 Accepted 30 June 2016

Academic Editor Michael Vynnycky

Copyright copy 2016 Y Ai and M Lu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To ensure the accuracy of the work capacity of the detonation powerplant the explosive and shock process of detonationpowerplants was simulated with LS-DYNA Many maximum rising displacements of the cartridge indicating the work capacity ofthe device were obtained under different fit clearances of the device It was proved that fit clearances were the key factors affectingthe work capacity of the device and reasonable range for fit clearances was found Besides the objective function constraintcondition and optimization design variables of the Genetic Algorithm were determined according to the design indicators of thedetonation powerplantThe theoretical values of fit clearances of the optimization design of detonation powerplants were obtainedAt last the tests of thework capacity of the detonation powerplant and LS-DYNA simulation proved the rationality of the theoreticalvalues from the Genetic Algorithm providing an experimental proof for the accuracy design which could control the service doormovement accurately

1 Introduction

According to theCCAR-21-R3 ldquoProvisions for theApproval ofCivil Aviation Products and Partsrdquo of Civil AviationAdminis-tration ofChina enoughmeasuresmust be taken to safeguardthe life safety of the crew in flight tests The new regional jetindependently designed and manufactured by China needsto carry out the test flight in accordance with airworthinessregulations To make sure that the test flight crew are able toescape from the aircraft in an emergency situation the mul-tisection telescopic detonation powerplant which is the coredevice of the emergency escaping support system is made Itis the first case in China where accurate blasting technique isapplied to the civil aviation life-saving field [1] The workingprinciple sketch of the detonation powerplant is shown inFigure 1 Its function is to overcome about 6KN aerodynamicdrag applied on the surface of the service door and pushit toward the cabin in certain track and pose through thecooperation of the four sets of devices [2] providing anemergency barrier-free escaping tunnel to the crew Theresearch and manufacture of detonation powerplants open anew field for civil aviation life-saving During the movementthe service door needs to keep in certain track and pose

Namely the angle of the doorrsquos lateral overturning should notbe larger than 10∘ the movement distance is 2m the erroris smaller than 15 and no unacceptable harm to the poseof the aircraft or the crew is caused during the movementTherefore to ensure the accuracy of the work capacity of thedetonation powerplant and the maximal utilization rate ofthe gunpowder simulating the working process of detonationpowerplants with LS-DYNA and optimizing the design ofparameters through Genetic Algorithm have great meaning

At the early stage of the research and development of thedetonation powerplant in the work capacity test experimentit has found that there was great discrepancy in the workcapacity of the devices which does not comply with thetechnical indicator of the error of the work capacity beingless than 10 As a result the movement of the service doorcannot be controlled accurately To solve this problem andimprove the energy utilization rate of the gunpowder keyfactors that influence the work capacity of the detonationpowerplant are analyzed and studied [3] And it is foundthat the fit clearances inside the device are key factors Thesketch of the detonation powerplant is depicted in Figure 2including height of the cartridge119867

1 outer diameter119863

1 inner

diameter 1198891 and height of the cavity ℎ

1 height of the slide

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5036083 12 pageshttpdxdoiorg10115520165036083

2 Mathematical Problems in Engineering

Aerodynamicdrag

The moving directionof the service door

CabinCabinService door

Service door Cabin

Detonationpowerplant

1500

mm

Figure 1 Working principle of the detonation powerplant pushing door [2]

Cartridge

GunpowderSlide cylinder

Fixed cylinder

BaseElectric ignitor pedestal

Electric ignitor

1

D1

d1

d2

L4

D3

H1

H2

H3

h1

h2

d4

d998400998400

3

d998400

3

D998400998400

2

D998400

2

D998400998400

2

D998400

2

1205752

2

1205751

2

1205753

2

Figure 2 Physical model of the detonation powerplant

cylinder1198672 outer wall diameter1198631015840

2

outer lace diameter119863101584010158402

inner diameter 119889

2 and height of the outer lace ℎ

2 height of

the fixed cylinder1198673 outer diameter119863

3 inner wall diameter

1198891015840

3

and inner lace diameter119889101584010158403

length of the cavity inside base1198714and diameter 119889

4 the fit clearance between the outer wall of

the cartridge and the inner wall of the slide cylinder 1205751 the fit

clearance between the outer wall of the slide cylinder and thelace of the fixed cylinder 120575

2 the fit clearance between the lace

of the slide cylinder and the inner wall of the fixed cylinder1205753Through the simulations in Section 2 it is found that

the fit clearances between the outer wall of the cartridgeand the inner wall of the slide cylinder the outer wall ofthe slide cylinder and the lace of the fixed cylinder and thelace of the slide cylinder and the inner wall of the fixedcylinder are the key factors that affect the accuracy of thework capacity of the detonation powerplant Besides thereasonable range of these fit clearances is obtained which laysa foundation for the next research Section 3 optimizes thedesign of the three fit clearances of the detonation powerplantwith the Genetic Algorithm [4ndash6] figures out the law offit clearances affecting the work capacity of the detonation

powerplant and the energy utilization rate of the gunpowderand at last obtains the theoretical optimal values of the fitclearances In Section 4 to test the rationality of the valuesobtained in Section 3 a detonation powerplant work capacitytest experiment is carried out and the explosive and shockprocess of the detonation powerplant is simulated with LS-DYNA Results illustrate that within an acceptable errorrange the work capacity of the detonation powerplant withthe theoretical parameters can satisfy the design indicators

2 LS-DYNA Simulation

21 LS-DYNAModeling and Solution

211 Analysis of Modeling There are mainly two ways tosimulate the explosive process including

(1) Lagrange Algorithm in which the explosive elementis the eight-node entity element The explosive ele-ment and the element exploded could share the samenode or could be connected by the contact The firstis relatively faster than the second in computing

Mathematical Problems in Engineering 3

CartridgeSlide cylinderFixed cylinder

Base

Electric ignitor pedestal

CartridgeSlide cylinder

Fixed cylinder


Figure 3 3D model of the detonation powerplant

Air ring cylinder

Figure 4 The finite element mesh model of the detonation powerplant

(2) Arbitrary Lagrangian-Eulerian ALE in which theexplosive element is the Euler element and the ele-ment exploded is the Lagrange element The explo-sion between the two grids is simulated by the definedcoupling [7]

In Lagrange Algorithm there is a great chance that severedistortion happens to the explosive element thus stoppingthe computing processTherefore though ALE is slower thanLagrange Algorithm it can effectively avoid such problemscaused by severe distortion of the grids as computationaldivergence and unreliable computing results

212 Model Building A calculation physics model is builtwith Solidworks and then the model is imported in LS-DYNA which is presented in Figure 3 (On the left is theSolidworks 3D model on the right is the Ansys 3D model)

The explosive is calculated with Eulerian Algorithm anddepicted with MAT ELASTIC PLASTIC HYDEO materialmodel andPROPELLANT DETONATIONequation of statethe air is also calculatedwith EulerianAlgorithmbut depictedwith NULL material model and LINEAR POLYNOMIALequation of state the detonation powerplant is calculatedwith Lagrange Algorithm and depicted with RIGID materialmodel [8ndash10] A finite element model is got after meshingwhich is shown in Figure 4

The work done by detonation powerplants in pushing theservice doormainly has two parts (1)work done to overcomethe aerodynamic drag on the service door (2) work done toprovide kinetic energy to the service door which enables theservice door tomove specified distance in prescribed pose In

0 20 40 60 80 1000

100

200

300

400

500 LS-DYNA user input

T (ms)Min = 0

Max = 48481

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 5 1205751

= 0024 1205752

= 0067 and 1205753

= 0053 T-Z curve

the shock the cartridges of the detonation powerplant pushthe service door doing work to the outside According tothe design of the detonation powerplant work capacity testexperiment in Section 4 a load of 06 KN is applied in thecartridge to simulate the process

22 Simulation Results and Analyses Select a set of fitclearances of the cartridge slide cylinder and fixed cylinderfor designed detonation powerplants then output the time-displacement curve of the cartridge with LS-PrePost (seeFigure 5)Themaximumvalue of the displacement representsthe work capacity of the detonation powerplant By changingthe entity model constantly simulations of the explosive and


Table 1 Four sets of fit clearances unit mm

119860 119861 119862 119863

1205751

= 1205752

= 1205753

1205753

1205751

= 1205752

1205752

1205751

= 1205753

1205751

1205752

= 1205753

001 001

005

001

005

001

005

002 002 002 002003 003 003 003004 004 004 004005 005 005 005006 006 006 006007 007 007 007008 008 008 008009 009 009 009010 010 010 010

0 2 4 6 8 10

n

H0

650

600

550

500

450

400

H(m

m)

A

B

C

D

Figure 6 Variation curve of 40 sets maximum displacements of thecartridge in accordance with different clearance parameters

shock process of the detonation powerplant with different fitclearances (Table 1) can be realized As a result a series ofeffective data are obtained with which wave curves are drawnin Figure 6 Figure 6 shows that under different fit clearancesmaximum displacements of cartridge indicating the workcapacity of the detonation powerplant are different Thisillustrates that the fit clearances of the detonation powerplantare the key factors affecting the work capacity of the deviceand the energy utilization rate of the gunpowder After earlydesign it is clear that the work capacity needs to reach30275 J By taking the mass of the weight in Section 4 119872 =

60 kg into the formula 119882 = 119872119892119867 the most appropriatelifting height of the weight can be known 119867

0= 5149mm

In Figure 6 the height 1198670is represented by the straight line

parallel to the 119909-axis On the basis of fluctuations of curve 119860119861 119862 and 119863 near the line 119867

0 reasonable range of the three

fit clearances can be determined That is 002mm le 1205751

le

005mm 003mm le 1205752 1205753le 007mm

3 Optimization Design of Internal BallisticsParameters with Genetic Algorithm

31 Basic Hypothesis The internal ballistics zero dimensionmathematical model of the detonation powerplant is a spaceaveraging parameter model based on Lagrange hypothesis[11] On the basis of internal ballistics theory and practicalsituation of the detonation powerplant hypotheses as belowhave been made [12] (1) The burning of gunpowder followsthe geometry burning rule (2) The burning of gunpowderparticles follows the burning velocity rule (3) Gunpowdergas equation of state complies with the Nobel-Abel equation(4) Leave out the gas pressure gradient in the cavity ofthe detonation powerplant (5) Ingredients produced in theburning of gunpowder remain the same (6)The loss of heat iscorrected by decreasing gunpowder impetus 119891 or increasingratio of specific heat 119896 (7) Ignore the influence of assistantgunpowder charge on the performance of the detonationpowerplant internal ballistics (8) Ignore the influence of elec-tric igniters on the performance of the detonation powerplantat the moment of ignition

32 Determination of Objective Function In this section themathematical equations of the optimization design of thedetonation powerplant internal ballistics parameters are builtand the optimization design of internal ballistics is carriedout so that the design cycle is shortened and the designquality is improvedTheprocess of the detonation powerplantdoing work to outside can be divided into four stagesOn the basis of the characteristic of each stage and classicballistic theories such as internal ballistics gunpowder gasequation of state burning equation energy conservation lawand kinematic equation [13ndash17] the mathematical models ofinternal ballistics of the four stages are built

The first stage is the period from the ignition of gunpow-der to the time when the cartridge and slide cylinder start tomove In this period the gunpowder is burning in constantvolume and the gas pressure in the cavity produced by theburning of gunpowder gradually increases from zero to startpressure The constant volume equation of state gunpowdershape function Euler equation which represents the one-dimensional linear motion of the gas in device [18] andrelative gas leakage flow of this period are

Ψ =

1Δ minus 1120588119901

1198911199010+ 120572 minus 1120588

119901

Ψ = 120594119885 (1 + 120582119885)

120597120588

120597119905+

120597 (120588119876)

120597119911= 0

120597 (120588119876)

120597119905+

120597 (1205881198762

)

120597119911= 120588119891119911minus

120597119901

120597119911

120597

120597119905(120588119864) +

120597

120597119911(120588119864119876) = 120588119902 minus

120597 (119876119901)

120597119911+ 120588119876119891

119911


120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

=120587119905 [(119889

1+ 1205751) 1205751+ (1198892+ 1205752) 1205752+ (1198893minus 1205753) 1205753]

120596

(1)

The second stage is the period from the time when thecartridge and slide cylinder start tomove to the timewhen thegunpowder burns out In this period the cartridge and slidecylinder move along the axis of the fixed cylinder When thegunpowder burns out the gas pressure in the cavity reachesthemaximumThe power state function gunpowder burningequation equation of themovement of the cartridge and slidecylinder the kinematical equation that calculates the speedand distance of cartridgersquos and slide cylinderrsquosmovement andrelative gas leakage flow of this period are

Ψ = 120594119885 (1 + 120582119885)

119889119885

119889119905=

1205831119901119899

1198901

=119901119899

119868119896

1198782119901 = 120593119898

119889V119889119905

V =119889119897

119889119905

1198782119901 (119897Ψ+ 119897) = 119891120596 (Ψ minus 120578) minus (119896 minus 1) 120593

119898V2

2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(2)

The third stage is the period from the time when thegunpowder burns out to the time when the slide cylinderrsquosmovement stops In this period the gas of high temperatureand pressure continues to expand and do work to outsidepushing the cartridge and slide cylinder to move Meanwhilethe gas pressure inside the cavity starts to drop After the slidecylinder moves for a distance its lower lace strikes the upperlace of the fixed cylinder and it is stopped The kinematicalequation of the movement of the cartridge and slide cylinderenergy equation and relative gas leakage flow of this periodare

1198782119901 = 120593119898

119889V119889119905

1198782119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(3)

The fourth stage is the period from the time when themovement of the slide cartridge stops to the time when thecartridge separates from the slide cylinder In this periodthough the pressure of the gas keeps dropping it continues

to expand and push the cartridge to move along the innerwall of the slide cylinder Then the process of doing workfinishes until the cartridge separates from the slide cylinderThe kinematical equation of the movement of the cartridgethe energy equation and relative gas leakage flow of thisperiod are

1198781119901 = 120593119898

119889V119889119905

1198781119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

1198601119876119905

120596

(4)

Later the service door stops accelerating and gets aninitial velocity then it starts flat parabolic motion Whenthe service door touches the floor of the cabin it starts tospin around the horizontal centroidal axis and continues tolose speed until the speed reduces to zero Then the servicedoor falls on the floor The energy equation that transformsthe process above into the working process of devices in thedetonation powerplant work capacity test experiment is

119882 = 119872119892119867 = 119864max = 119898V2max (5)

There are many parameters involved in the design of thedetonation powerplant Among these parameters some aredynamic variables some are constant numbers some havea relatively big influence on the performance of the internalballistics of the detonation powerplant while some are thesecondary parameterswhich only have a little influence someare independent from each other and some have influence onone another with certain correlation among them

Optimizing the design variables must target the inde-pendent variables which have the most influence on theperformance of the device and can respond most sensitivelyFor three fit clearances and their influences on the workcapacity of the detonation powerplant being investigatedthe constraint conditions are the following (1) Taking thesetup space for the detonation powerplant and the structuralstrength of the parts into consideration set the maximumair pressure as 119901max = 150MPa (2) The volume of thegunpowder room is not only related to the setup spaceof the detonation powerplant but also closely related tothe fit sizes of the slide cylinder cartridge and the fixedcylinder On the basis of the results of parameter optimizationdesign in early stage set the volume of the gunpowder roomas 1610mm3 le 119881 le 1645mm3 (3) The fit clearancebeing too large or too small will affect the improving ofthe accuracy and consistency of the work capacity of thedetonation powerplant And it also has a direct influenceon the frictional resistance and gas leakage of the deviceAccording to Section 2 three fit clearances can be set as002mm le 120575

1le 005mm 003mm le 120575

2 1205753

le 007mm(4) The most appropriate rising height of the weight is 119867

0=

5149mm


Above all the objective function of the question about theoptimization design of the detonation powerplant internalballistics is

min 1003816100381610038161003816119867 (119881 1205751 1205752 1205753) minus 1198670

1003816100381610038161003816

119901 (119881 1205751 1205752 1205753) le 119901max

119881 isin [1610 1645]

1205751isin [002 005]

1205752 1205753isin [003 007]

(6)

The fitness function is

Fit (119891 (119909 119910)) =

119867max minus 119891 (119909 119910) 119891 (119909 119910) lt 119867max

0 119891 (119909 119910) ge 119867max(7)

119867max is the maximum approximated value of 119891(119909 119910) in theequation

33 Implementation Process of Genetic Algorithm The opti-mization design process of internal ballistics parameters withGenetic Algorithm of the detonation powerplant is shown inFigure 7

By employing MATLAB program optimal results ofthe detonation powerplant internal ballistics parameters areobtained according to the internal ballistics zero dimensionmathematical model of the device and Genetic AlgorithmThe size of the population has a direct effect on the con-vergence procedure and the efficiency of calculating If thepopulation is too large it will increase the calculating timegreatly if the population is too small the calculating processmight stop when a regional optimal result is obtained [19]The study focuses on the processing of the rising height ofthe weight and the maximal gas pressure and the populationchosen in this paper has 50 individuals The length ofchromosome depends on the precision of optimal resultsThemore precise optimal results are the longer the chromosomewill be The length of the three chromosomes that representthe three fit clearances in this paper is 18 and the searchingrange is 002mm le 120575

1le 005mm 003mm le 120575

2 1205753

le

007mm The length of the chromosome that represents thevolume of gunpowder room is 18 and the searching rangeis 1610mm3 le 119881 le 1645mm3 Maximum generation isthe condition that determines when the Genetic Algorithmshould be stopped Usually whether the algorithm shouldbe stopped or not depends on the running conditions ofalgorithm the convergence situation and the quality ofthe result The maximum generation in this paper is 800Parameters of optimization design of internal ballistics withGenetic Algorithm are shown in Table 2

According to the optimal results of the Genetic Algo-rithm when genetic revolution goes on to the 800th gener-ation the convergence is reached So choose the first gener-ation the 380th generation and the 800th generation of allthe 800 generations as representatives and the individuals ofthese generations are shown in Tables 3 4 and 5 respectively

Table 3 shows that individuals of the first generation areproduced by the program randomly Table 4 shows that when

The binary coding of the detonation powerplant internal ballistics parameters

First population

Model of the detonation powerplant internal ballistics

Feasible solution

Selection action

Crossover operation

Mutation operation

Fitness function

Yes or no evaluation on fitness function

Optimal solution of the detonation powerplant

internal ballistics objective function

Yes

No

Figure 7 Flow chart of Genetic Algorithm of the detonationpowerplant

Table 2 Table of parameters of genetic algorithm

Factors ValueThe size of population 50The length of chromosome 18Maximum generation 800Crossover probability 07Mutation rate 0001Generation gap 05

the revolution reaches the 380th generation the individualsshow the tendency of convergence which ismanifested in thetable as the parameters dividing into six areas with each areashowing the convergence intensively Table 5 shows that whenthe program evolves to the last generation the population inthe constraint situation reaches the best and the algorithmstops The optimal results are 119881 = 162955mm3 120575

1=

0034mm 1205752

= 0051mm and 1205753

= 0047mm After theoptimization the optimal rising height of the weight thatrepresents the work capacity of the detonation powerplant is


Table 3 Results of the first generation

119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)

Figure 8 The maximum height convergence curve of weight (a) and the volume convergence curve of gunpowder chamber (b)

Table 4 Results of the 380th generation

119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm01ndash10 1631388 0032 0047 0043 144398 53157811ndash18 1631286 0032 0048 0043 143896 52134519ndash26 1631184 0033 0048 0043 143995 52339127ndash34 1631081 0033 0049 0044 144169 52748535ndash42 1630979 0034 0049 0044 144276 52953243ndash50 1630774 0034 0050 0044 144102 525438


119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm01ndash50 1629552 0034 0051 0047 143116 515205

119867 = 51521mm Convergences of the all the generations areshown in Figures 8 and 9

It is shown in the convergence curve that the optimaldesign variable 119881 120575

1 1205752 1205753 and the objective function 119867

vibrate violently in the early stage of Genetic Algorithmcomputing With the increasing of iterations the range ofthe vibration becomes smaller When the iteration of thealgorithm reaches 500 the algorithmfinds the optimal resultsof the detonation powerplant internal ballistics parametersIn the solution procedure of the algorithm vibrations willhappen occasionally for the mutation operation which willnot influence the convergence of optimal results

4 Experiment on the Accuracy of WorkCapacity of the Detonation Powerplant

On the basis of the theoretical optimal parameters of fit clear-ances of the detonation powerplant obtained in Section 33detonation powerplants illustrated in Figure 10 aremade (On

the left is themechanical partwithout gunpowder and electricigniters On the right is the eight sets of assembled detonationpowerplants)

To test the accuracy of the work capability and consis-tency of the detonation powerplant the evaluation device ofwork capacity is made The device consisting of a baseboarda weight and two guide rods is depicted in Figure 11The baseboard is a 10mm thick steel plate on which thedetonation powerplant and guide rods are fixed the guiderods are two 1000mm long cylindrical rods which are 20mmin diameter The guide rods are fixed on the baseboardby thread connection whose axes are perpendicular to thesurface of the baseboard There are two graduated scalesattached to the rods indicating the length theweight is 60 kgIn themiddle of theweight there are two through-holeswhichare 25mm in diameter and the distance between the twoholes is 120mm The whole process of the work capacity testexperiment is recorded by a high-speed camera The resultsare shown in Figure 12


0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)

Figure 9 The convergence curve of 1205751

(a) 1205752

(b) and 1205753

(c)

Figure 10 Detonation powerplants


Guide rod

Weight

Baseboard Deflagration powerplant

Figure 11 The evaluation device of the work capacity of the detonation powerplant

(a) Early stage of detonation (b) Uplifted stage of weight (c) Falling stage of weight

Figure 12 Power capability assessment experiment process of the detonation powerplant

Table 6 Results of experiment on the accuracy of work capacity ofthe detonation powerplant

119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056

The requirements of materials the high-level machiningprecision and special eclectic igniters make the manufacturecost of the detonation powerplant very high In order toreduce the research cost this study tests the work capacityof the device under theoretical optimal parameters of fitclearances got from the Genetic Algorithm through the eightsets of experiment and LS-DYNA simulation of explosive andshock process of the deviceThe results of the experiment aredisplayed in Table 6 and the result of LS-DYNA simulationis shown in Figure 13

Remove the deviated data from the eighth experimentand analyze and compute the datum as follows The averagevalue of results is 119909 = sum

119899

119894

119909119894119899 = 5109mm the range is

119883 = 119909max minus 119909min = 25mm the relative error betweenthe experimental average and the theoretical value is 120578

1=

|119909 minus 119909119905|119909119905times 100 = 084 the relative error between

the simulation value and the theoretical value is 1205782

= |ℎ minus

119909119905|119909119905times 100 = 049 There are reasonable errors between

0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve

the experimental value and the theoretical value as wellas the simulation value and the theoretical value And theexperimental results have small fluctuation At last after theoptimization the work capacity of the detonation powerplantmeets the technical indicator that the error needs to be lessthan 10


5 Conclusions

Conclusions as below are drawn on the basis of LS-DYNAsimulation of the explosive and shock process of the detona-tion powerplant optimal design of Genetic Algorithm andrelated test experiments

(1) By simulating the explosive and shock process ofthe detonation powerplant with LS-DYNA the mainfactors that affect the accuracy of the work capacityof the device are found They are the fit clearancesbetween the outer wall of the cartridge and the innerwall of the slide cylinder the outer wall of the slidecylinder and the lace of the fixed cylinder and thelace of the slide cylinder and the inner wall of thefixed cylinder and the reasonable range of these threefit clearances is figured out which lays a foundationfor further optimization design of the detonationpowerplant

(2) The internal ballistics zero dimension mathematicalmodel of the detonation powerplant is used as theoptimal design model of the device internal ballisticsparameters in Genetic Algorithm and the objectivefunction optimization design variables and restraintcondition used in the study are appropriate As aresult optimal fit clearances as expected are found

(3) The device used in the detonation powerplant workcapacity test experiment is simple and appropriateand satisfies the accuracy required in theory And thework capacity of the detonation powerplant underthe optimal fit clearances obtained from GeneticAlgorithm is tested through experiment and LS-DYNA simulation And within an acceptable errorrange the results from the experiment are consistentwith the theoretical values of fit clearances whichsatisfies the design goal

In conclusion the analysis method used in this paper ismeaningful for further optimization design of the detonationpowerplant in both theory and reality Nowadays it has beenimplied to the test flight of airliner made in China

Nomenclature

Ψ The percentage of the gunpowder burnedΔ Density of gunpowder installed120588119901 Density of gunpowder

120572 Gunpowder gas covolume which is 05119885 Relative thickness of the gunpowder

burned120588 Density of gas inside the device119876 Flow velocity of gas in vertical direction119860 Area of clearance axial section119905 Independent variable time119911 Independent variable displacement1205831 Burning speed coefficient which is 02

119899 Burning speed index which is 082

119868119896 Total pressure impulse

119864 Total energy119872 Mass of the weight119867 Height of weightrsquos rising120582 120594 The shape feature and quantity of

gunpowder119897Ψ Reduction diameter of gunpowderrsquos free

volume120593 Calculated coefficient of secondary work

done by the device 120593 = 1205931+ 1205963119898

1205931= 12

119891 Gunpowder force which is 310 kJkg119891119911 Mass force per unit in vertical direction

119901 Gas pressure inside the device119882 Gravitational potential energy of the

weight119902 Heat passed to unit mass air in unit time120578 Relative air leakage flow119910 Total air leakage120596 Mass of gunpowder installedV Speed of cartridge119897 Displacement of cartridge119896 Adiabatic coefficient which is 12119898 Mass of equivalent mass entity1198901 Propellant web size which is 0068

1198781 Area of the cartridge axial section

1198782 Area of the slide cylinder axial section

119892 Gravity coefficientVmax Maximum value of the cartridgersquos speed119864max Maximum value of the cartridgersquos kinetic

energy1199010 Device start-up pressure 119901

0= (119898119892 + 119865)

Device 119865 initiates resistance force which isthe aerodynamic load applied on servicedoor in flying

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work described in this paper is financially supported bythe China National Aviation Holding Company under Grantno NJCX-RW-20100208The authors would like to gratefullyacknowledge this support

References

[1] H L Lu and Y Fei Escape Survival and Rescue in the AircraftAccidents National Defense Industry Press Beijing China2006

[2] C F Fan Y H Wang R Ma M J Duan and X Chang ldquoTheforce state analysis of the service door on an airlinerrdquo AdvancedMaterials Research vol 798 pp 325ndash327 2013

[3] J A Longridge Internal Ballistics Kessinger Pub CompanyWhitefish Mont USA 2008

[4] L DavisGenetic Algorithms and Simulated Annealing PittmanLondon UK 1987


[5] D E Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[6] S Forrest ldquoGenetic algorithms principles of natural selectionapplied to computationrdquo Science vol 261 no 5123 pp 872ndash8781993

[7] S Q Shi J K Kang M Wang et al The Engineer Implicationof ANSYSLS-DYNA in Explosion and Shock Architecture ampBuilding Press Beijing China 2011

[8] J O Hallquist LS-DYNA Theoretical Manual Livermore Soft-ware Technology Corporation Livermore Calif USA 1998

[9] J D Reid LS-DYNA Examples Manual Livermore SoftwareTechnology Corporation Livermore Calif USA 1998

[10] LSTCLS-DYNA Keyword Userrsquos Manual Livermore SoftwareTechnology Corporation Livermore Calif USA 2003

[11] R J Muzzy and C E Wooldridge ldquoInternal ballistic considera-tions in hybrid rocket designrdquo Journal of Spacecraft and Rocketsvol 4 no 2 pp 255ndash262 1967

[12] J Greig J Earnhart N Winsor et al ldquoInvestigation of plasma-augmented solid propellant interior ballistic processesrdquo IEEETransactions on Magnetics vol 29 no 1 pp 555ndash560

[13] P G Baer and J M Frankle ldquoThe simulation of interior ballisticperformance of guns by digital computer programrdquo Tech RepBRLMR-1183 BRL ARDC Ballistic Research LaboratoriesAberdeen Proving Ground Md USA 1962

[14] S Jaramaz D Mickovic and P Elek ldquoTwo-phase flows ingun barrel theoretical and experimental studiesrdquo InternationalJournal of Multiphase Flow vol 37 no 5 pp 475ndash487 2011

[15] M Sanford and T A DelGuidice Energy AbsorbingCountermass for Shoulder-Launched Rocket Weapon NTISADD019605 2000

[16] H Miura A Matsuo and Y Nakamura ldquoThree-dimensionalsimulation of pressure fluctuation in a granular solid propellantchamber within an ignition stagerdquo Propellants ExplosivesPyrotechnics vol 36 no 3 pp 259ndash267 2011

[17] J Nussbaum P Helluy J-M Herard and B Baschung ldquoMulti-dimensional two-phase flow modeling applied to interior bal-listicsrdquo Journal of Applied Mechanics vol 78 no 5 Article ID051016 9 pages 2011

[18] L G Vulkov ldquoOn the conservation laws of the compressibleeuler equationsrdquo Applicable Analysis vol 64 no 3-4 pp 255ndash271 1997

[19] L F Ding Z W Cheng and Z F Chen ldquoSimulation andresearch on the control parameters of genetic algorithmrdquoScience amp Technology Information vol 36 article 774 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Aerodynamicdrag

The moving directionof the service door

CabinCabinService door

Service door Cabin

Detonationpowerplant

1500

mm

Figure 1 Working principle of the detonation powerplant pushing door [2]

Cartridge

GunpowderSlide cylinder

Fixed cylinder


Electric ignitor

1

D1

d1

d2

L4

D3

H1

H2

H3

h1

h2

d4

d998400998400

3

d998400

3

D998400998400

2

D998400

2

D998400998400

2

D998400

2

1205752

2

1205751

2

1205753

2

Figure 2 Physical model of the detonation powerplant

cylinder1198672 outer wall diameter1198631015840

2

outer lace diameter119863101584010158402

inner diameter 119889

2 and height of the outer lace ℎ

2 height of

the fixed cylinder1198673 outer diameter119863

3 inner wall diameter

1198891015840

3

and inner lace diameter119889101584010158403

length of the cavity inside base1198714and diameter 119889

4 the fit clearance between the outer wall of

the cartridge and the inner wall of the slide cylinder 1205751 the fit

clearance between the outer wall of the slide cylinder and thelace of the fixed cylinder 120575

2 the fit clearance between the lace

of the slide cylinder and the inner wall of the fixed cylinder1205753Through the simulations in Section 2 it is found that

the fit clearances between the outer wall of the cartridgeand the inner wall of the slide cylinder the outer wall ofthe slide cylinder and the lace of the fixed cylinder and thelace of the slide cylinder and the inner wall of the fixedcylinder are the key factors that affect the accuracy of thework capacity of the detonation powerplant Besides thereasonable range of these fit clearances is obtained which laysa foundation for the next research Section 3 optimizes thedesign of the three fit clearances of the detonation powerplantwith the Genetic Algorithm [4ndash6] figures out the law offit clearances affecting the work capacity of the detonation

powerplant and the energy utilization rate of the gunpowderand at last obtains the theoretical optimal values of the fitclearances In Section 4 to test the rationality of the valuesobtained in Section 3 a detonation powerplant work capacitytest experiment is carried out and the explosive and shockprocess of the detonation powerplant is simulated with LS-DYNA Results illustrate that within an acceptable errorrange the work capacity of the detonation powerplant withthe theoretical parameters can satisfy the design indicators

2 LS-DYNA Simulation

21 LS-DYNAModeling and Solution

211 Analysis of Modeling There are mainly two ways tosimulate the explosive process including

(1) Lagrange Algorithm in which the explosive elementis the eight-node entity element The explosive ele-ment and the element exploded could share the samenode or could be connected by the contact The firstis relatively faster than the second in computing



Base



Fixed cylinder



Air ring cylinder







0 20 40 60 80 1000

100

200

300

400


T (ms)Min = 0

Max = 48481

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 5 1205751

= 0024 1205752

= 0067 and 1205753

= 0053 T-Z curve





119860 119861 119862 119863

1205751

= 1205752

= 1205753

1205753

1205751

= 1205752

1205752

1205751

= 1205753

1205751

1205752

= 1205753

001 001

005

001

005

001

005

002 002 002 002003 003 003 003004 004 004 004005 005 005 005006 006 006 006007 007 007 007008 008 008 008009 009 009 009010 010 010 010

0 2 4 6 8 10

n

H0

650

600

550

500

450

400

H(m

m)

A

B

C

D




0= 5149mm





le

005mm 003mm le 1205752 1205753le 007mm





Ψ =

1Δ minus 1120588119901

1198911199010+ 120572 minus 1120588

119901

Ψ = 120594119885 (1 + 120582119885)

120597120588

120597119905+

120597 (120588119876)

120597119911= 0

120597 (120588119876)

120597119905+

120597 (1205881198762

)

120597119911= 120588119891119911minus

120597119901

120597119911

120597

120597119905(120588119864) +

120597

120597119911(120588119864119876) = 120588119902 minus

120597 (119876119901)

120597119911+ 120588119876119891

119911


120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

=120587119905 [(119889

1+ 1205751) 1205751+ (1198892+ 1205752) 1205752+ (1198893minus 1205753) 1205753]

120596

(1)


Ψ = 120594119885 (1 + 120582119885)

119889119885

119889119905=

1205831119901119899

1198901

=119901119899

119868119896

1198782119901 = 120593119898

119889V119889119905

V =119889119897

119889119905


119898V2

2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(2)


1198782119901 = 120593119898

119889V119889119905

1198782119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(3)



1198781119901 = 120593119898

119889V119889119905

1198781119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

1198601119876119905

120596

(4)


119882 = 119872119892119867 = 119864max = 119898V2max (5)



1le 005mm 003mm le 120575

2 1205753


0=

5149mm



min 1003816100381610038161003816119867 (119881 1205751 1205752 1205753) minus 1198670

1003816100381610038161003816

119901 (119881 1205751 1205752 1205753) le 119901max

119881 isin [1610 1645]

1205751isin [002 005]

1205752 1205753isin [003 007]

(6)


Fit (119891 (119909 119910)) =

119867max minus 119891 (119909 119910) 119891 (119909 119910) lt 119867max

0 119891 (119909 119910) ge 119867max(7)




1le 005mm 003mm le 120575

2 1205753

le





First population


Feasible solution

Selection action

Crossover operation

Mutation operation

Fitness function




Yes

No





1=

0034mm 1205752

= 0051mm and 1205753




119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Base



Fixed cylinder



Air ring cylinder







0 20 40 60 80 1000

100

200

300

400


T (ms)Min = 0

Max = 48481

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 5 1205751

= 0024 1205752

= 0067 and 1205753

= 0053 T-Z curve





119860 119861 119862 119863

1205751

= 1205752

= 1205753

1205753

1205751

= 1205752

1205752

1205751

= 1205753

1205751

1205752

= 1205753

001 001

005

001

005

001

005

002 002 002 002003 003 003 003004 004 004 004005 005 005 005006 006 006 006007 007 007 007008 008 008 008009 009 009 009010 010 010 010

0 2 4 6 8 10

n

H0

650

600

550

500

450

400

H(m

m)

A

B

C

D




0= 5149mm





le

005mm 003mm le 1205752 1205753le 007mm





Ψ =

1Δ minus 1120588119901

1198911199010+ 120572 minus 1120588

119901

Ψ = 120594119885 (1 + 120582119885)

120597120588

120597119905+

120597 (120588119876)

120597119911= 0

120597 (120588119876)

120597119905+

120597 (1205881198762

)

120597119911= 120588119891119911minus

120597119901

120597119911

120597

120597119905(120588119864) +

120597

120597119911(120588119864119876) = 120588119902 minus

120597 (119876119901)

120597119911+ 120588119876119891

119911


120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

=120587119905 [(119889

1+ 1205751) 1205751+ (1198892+ 1205752) 1205752+ (1198893minus 1205753) 1205753]

120596

(1)


Ψ = 120594119885 (1 + 120582119885)

119889119885

119889119905=

1205831119901119899

1198901

=119901119899

119868119896

1198782119901 = 120593119898

119889V119889119905

V =119889119897

119889119905


119898V2

2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(2)


1198782119901 = 120593119898

119889V119889119905

1198782119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(3)



1198781119901 = 120593119898

119889V119889119905

1198781119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

1198601119876119905

120596

(4)


119882 = 119872119892119867 = 119864max = 119898V2max (5)



1le 005mm 003mm le 120575

2 1205753


0=

5149mm



min 1003816100381610038161003816119867 (119881 1205751 1205752 1205753) minus 1198670

1003816100381610038161003816

119901 (119881 1205751 1205752 1205753) le 119901max

119881 isin [1610 1645]

1205751isin [002 005]

1205752 1205753isin [003 007]

(6)


Fit (119891 (119909 119910)) =

119867max minus 119891 (119909 119910) 119891 (119909 119910) lt 119867max

0 119891 (119909 119910) ge 119867max(7)




1le 005mm 003mm le 120575

2 1205753

le





First population


Feasible solution

Selection action

Crossover operation

Mutation operation

Fitness function




Yes

No





1=

0034mm 1205752

= 0051mm and 1205753




119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860 119861 119862 119863

1205751

= 1205752

= 1205753

1205753

1205751

= 1205752

1205752

1205751

= 1205753

1205751

1205752

= 1205753

001 001

005

001

005

001

005

002 002 002 002003 003 003 003004 004 004 004005 005 005 005006 006 006 006007 007 007 007008 008 008 008009 009 009 009010 010 010 010

0 2 4 6 8 10

n

H0

650

600

550

500

450

400

H(m

m)

A

B

C

D




0= 5149mm





le

005mm 003mm le 1205752 1205753le 007mm





Ψ =

1Δ minus 1120588119901

1198911199010+ 120572 minus 1120588

119901

Ψ = 120594119885 (1 + 120582119885)

120597120588

120597119905+

120597 (120588119876)

120597119911= 0

120597 (120588119876)

120597119905+

120597 (1205881198762

)

120597119911= 120588119891119911minus

120597119901

120597119911

120597

120597119905(120588119864) +

120597

120597119911(120588119864119876) = 120588119902 minus

120597 (119876119901)

120597119911+ 120588119876119891

119911


120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

=120587119905 [(119889

1+ 1205751) 1205751+ (1198892+ 1205752) 1205752+ (1198893minus 1205753) 1205753]

120596

(1)


Ψ = 120594119885 (1 + 120582119885)

119889119885

119889119905=

1205831119901119899

1198901

=119901119899

119868119896

1198782119901 = 120593119898

119889V119889119905

V =119889119897

119889119905


119898V2

2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(2)


1198782119901 = 120593119898

119889V119889119905

1198782119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(3)



1198781119901 = 120593119898

119889V119889119905

1198781119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

1198601119876119905

120596

(4)


119882 = 119872119892119867 = 119864max = 119898V2max (5)



1le 005mm 003mm le 120575

2 1205753


0=

5149mm



min 1003816100381610038161003816119867 (119881 1205751 1205752 1205753) minus 1198670

1003816100381610038161003816

119901 (119881 1205751 1205752 1205753) le 119901max

119881 isin [1610 1645]

1205751isin [002 005]

1205752 1205753isin [003 007]

(6)


Fit (119891 (119909 119910)) =

119867max minus 119891 (119909 119910) 119891 (119909 119910) lt 119867max

0 119891 (119909 119910) ge 119867max(7)




1le 005mm 003mm le 120575

2 1205753

le





First population


Feasible solution

Selection action

Crossover operation

Mutation operation

Fitness function




Yes

No





1=

0034mm 1205752

= 0051mm and 1205753




119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

=120587119905 [(119889

1+ 1205751) 1205751+ (1198892+ 1205752) 1205752+ (1198893minus 1205753) 1205753]

120596

(1)


Ψ = 120594119885 (1 + 120582119885)

119889119885

119889119905=

1205831119901119899

1198901

=119901119899

119868119896

1198782119901 = 120593119898

119889V119889119905

V =119889119897

119889119905


119898V2

2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(2)


1198782119901 = 120593119898

119889V119889119905

1198782119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

(1198601+ 1198602+ 1198603) 119876119905

120596

(3)



1198781119901 = 120593119898

119889V119889119905

1198781119901 (1198971+ 119897) = 119891120596 (1 minus 120578) minus

(119896 minus 1)

2120593119898V2

120578 =119910

120596=

1198601119876119905

120596

(4)


119882 = 119872119892119867 = 119864max = 119898V2max (5)



1le 005mm 003mm le 120575

2 1205753


0=

5149mm



min 1003816100381610038161003816119867 (119881 1205751 1205752 1205753) minus 1198670

1003816100381610038161003816

119901 (119881 1205751 1205752 1205753) le 119901max

119881 isin [1610 1645]

1205751isin [002 005]

1205752 1205753isin [003 007]

(6)


Fit (119891 (119909 119910)) =

119867max minus 119891 (119909 119910) 119891 (119909 119910) lt 119867max

0 119891 (119909 119910) ge 119867max(7)




1le 005mm 003mm le 120575

2 1205753

le





First population


Feasible solution

Selection action

Crossover operation

Mutation operation

Fitness function




Yes

No





1=

0034mm 1205752

= 0051mm and 1205753




119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min 1003816100381610038161003816119867 (119881 1205751 1205752 1205753) minus 1198670

1003816100381610038161003816

119901 (119881 1205751 1205752 1205753) le 119901max

119881 isin [1610 1645]

1205751isin [002 005]

1205752 1205753isin [003 007]

(6)


Fit (119891 (119909 119910)) =

119867max minus 119891 (119909 119910) 119891 (119909 119910) lt 119867max

0 119891 (119909 119910) ge 119867max(7)




1le 005mm 003mm le 120575

2 1205753

le





First population


Feasible solution

Selection action

Crossover operation

Mutation operation

Fitness function




Yes

No





1=

0034mm 1205752

= 0051mm and 1205753




119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873 119881mm3 1205751

mm 1205752

mm 1205753

mm 119901maxMPa 119867mm1 1626477 0024 0067 0054 115312 4857052 1627091 0023 0066 0053 142642 6001933 1628012 0022 0065 0052 136398 5052384 1626988 0021 0067 0055 122432 4943455 1627909 0020 0066 0054 116675 4865496 1627205 0024 0065 0053 118125 4885967 1627192 0023 0067 0052 119987 4906438 1627546 0022 0066 0055 121015 4926909 1627821 0021 0065 0054 122689 49473610 1627735 0020 0067 0053 126507 49678311 1627332 0024 0066 0052 128674 49883012 1626875 0023 0065 0055 130435 50087713 1626798 0022 0067 0054 132895 50292314 1627809 0021 0066 0053 134453 50497015 1628576 0020 0065 0052 138012 50701816 1626308 0024 0067 0055 139897 50906417 1626957 0023 0066 0054 141476 51111018 1627948 0022 0065 0053 142889 51315719 1627827 0021 0067 0052 143114 51520420 1627813 0020 0066 0055 143423 51725121 1627144 0024 0065 0054 143464 51929822 1627002 0023 0067 0053 143896 52134523 1627982 0022 0066 0052 143995 52339124 1627627 0021 0065 0055 144102 52543825 1627635 0020 0067 0054 144169 52748526 1627240 0024 0066 0053 144276 52953227 1627458 0023 0065 0052 144398 53157828 1626862 0022 0067 0053 144501 53362529 1627738 0021 0066 0054 144599 53567230 1628329 0020 0065 0054 144701 53771931 1626609 0024 0067 0053 144734 53976632 1627312 0023 0066 0052 144868 54181233 1627459 0022 0065 0055 144953 54385934 1627142 0021 0067 0054 145013 54590635 1628203 0020 0066 0053 145457 55000136 1627612 0022 0065 0054 145896 55614037 1626861 0023 0066 0055 146201 56228038 1627275 0021 0067 0053 146834 56842139 1628411 0020 0066 0052 146987 57251440 1627002 0024 0065 0055 147099 57865441 1626836 0023 0067 0054 147609 58274842 1627866 0022 0066 0053 147875 58684243 1627624 0021 0066 0055 147963 59093544 1627871 0020 0067 0052 148023 59502945 1627472 0024 0065 0052 148341 59912246 1626672 0023 0067 0055 147961 58888747 1627740 0022 0066 0054 146001 56023348 1628009 0021 0066 0052 145634 55204649 1627479 0020 0067 0055 147001 57456150 1628232 0020 0065 0055 147053 576536


0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800200

300

400

500

600

700

800

900

n

H(m

m)

(a)

0 100 200 300 400 500 600 700 8001610

1615

1620

1625

1630

1635

1640

1645

n

V(m

m3)

(b)



119873 119881mm3 1205751

mm 1205752

mm 1205753



119873 119881mm3 1205751

mm 1205752

mm 1205753











0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800002

0025

003

0035

004

0045

005

n

1205751

(mm

)

(a)

0 100 200 300 400 500 600 700 800

004

0045

005

0055

006

0065

007

n

1205752

(mm

)(b)

0 200 400 600 800003

0035

004

0045

005

0055

006

0065

n

1205753

(mm

)

(c)


(a) 1205752

(b) and 1205753

(c)



Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Guide rod

Weight






119873 1 2 3 4 5 6 7 8119867mm 5103 5114 5096 5121 5112 5103 5114 5056



119899

119894

119909119894119899 = 5109mm the range is


1=



= |ℎ minus


0 20 40 60 80 1000

100

200

300

400

500


T (ms)Min = 0

Max = 51268

Z-r

igid

bod

y di

spla

cem

ent (

mm

)

Figure 13 1205751

= 0034 1205752

= 0051 and 1205753

= 0047 T-Z curve



5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusions






Nomenclature










1205931= 12








0= (119898119892 + 119865)


Competing Interests


Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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