255

Vol.35,No.4(2013)255‐260

TribologyinIndustry

www.tribology.fink.rs

OptimalDesignofaCamMechanismwithTranslatingFlat‐FaceFollowerusing

GeneticAlgorithmI.Tsiafis

a,S.Mitsia,K.D.Bouzakisa,A.Papadimitriou

aa

AristotleUniversityofThessaloniki,DepartmentofMechanicalEngineering,Greece.

Keywords:

CamMechanismGeneticAlgorithmsContactStressOptimization

ABSTRACT

Theoptimumdesignofacammechanismisatimeconsumingtask,duetothenumerous alternatives considerations. In the presentwork, the problem ofdesignparameters optimization ofa cammechanismwith translating flat‐facefollower is investigated fromamulti‐objectivepointofview.Thedesignparameters, just likethecambasecircleradius,the follower facewidthandthefolloweroffsetcanbedeterminedconsideringasoptimizationcriteriatheminimizationof the cam size,of the input torqueandof the contact stress.During the optimization procedure, a number of constraints regarding thepressureangle,thecontactstress,etcaretakenintoaccount.Theoptimizationapproach,basedongeneticalgorithm,isappliedtofindtheoptimalsolutionswith respect to the afore‐mentioned objective function and to ensure thekinematicrequirements.Finally,thedynamicbehaviourofthedesignedcammechanismisinvestigatedconsideringthefrictionalforces.

©2013PublishedbyFacultyofEngineering

Correspondingauthor:

I.TsiafisAristotleUniversityofThessaloniki,DepartmentofMechanicalEngineering,GreeceE‐mail:tsiafis@eng.auth.gr

1. INTRODUCTIONThe optimal design of cam mechanism ishandled in many publications [1‐6], wherevariousconstraintsandmethodsareconsidered.Parts applied in cam systems are often coatedfor increasingtheirsuperficialhardnessandforreducing friction coefficient [7,8]. Α non‐linearprogramming technique with constraints,known as SUMT algorithm is used in [3] foroptimum synthesis of a disk cam mechanismwith swinging roller follower. In [4] the designparametersaredeterminedbytheminimizationof the maximum compressive stress at thecontact area of a cam‐disk mechanism with

translatingrollerfollower,wherethecamprofileis described with the aid of cubic splinefunctions. Tsiafis et al. present in [5] a multi‐objectiveprocedurebasedongeneticalgorithmstooptimizethedesignparametersofadisk‐cammechanismwitharollerfollower.In the present paper the problem of the designparametersoptimizationofacammechanismwitha reciprocating flat‐face follower is investigated,using multi‐objective optimization with geneticalgorithm. The design parameters for this type ofmechanismaretheradiusofthecambasecircle,thefollower face width and the follower offset. Theoptimization is achieved by the development of

RESEARCH

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programsusingthehighlevelcomputinglanguageMATLABwith the GA (genetic algorithm) toolboxapplication.Furthermore,thedynamicalanalysisofthe designed mechanism considering friction isinvestigated.2. MATHEMATICALFORMULATIONA cam mechanism with a translating flat‐facefollowerisshowninFig.1.Thecamisassumedtohave constant angular velocity. The profile of thecamcanbedeterminedconsideringthekinematicalanddynamicalrequirementsofthemechanism.The design parameters under optimization arethecambasecircleRb,thewidthfollowerfaceLandthefolloweroffseteasshowninFig.1.Theoptimizationofthedesignparametersofthecam mechanism can be achieved by theminimisation of the cam size, of the torquerequiredtodrivethecamandthecontactstressbetweenthecamandthefollower.

Fig.1.Cammechanismwithtranslatingflat‐facefollower.Therefore, it could be formulated as anoptimization problem, where the objectivefunction(F)takesintoaccountthecamsize(F1),the input torque(F2)and themaximumcontactstress(F3):

1 2 3F F F F (1)

with

b1F R L (2)

2

( P v )F = T

ω (3)

1 2

2 2 21 2

1 2

1 1

max

P'F σ

μ μρ

E E

(4)

whereTistheinputtorque,Pisthetotalnormalloadon the cam,v is the follower velocity,ω isthe camshaft angular velocity, σmax is themaximum contact stress between the followerandthecam,P’isthenormalloadperunitwidthof the contacting members, ρ is the radii ofcurvatureofthecam,μ1andμ2arePoisson’sratioforthecamandthefollowerrespectivelyandE1,E2arethemoduleofelasticityofthecamandthefollowerrespectively.The weighting factors α, β and γ are used inorder to scale the contribution of thecorresponding terms in the objective functionvalue.Theminimizationoftheobjectivefunctiondeterminestheoptimumvaluesoftheunknownparameters. During the optimization procedurethe following functional constraints areimposed:

a) The maximum value of the pressure anglemust be smaller than the maximumpermitted:δmax<δper.

Thepressureanglecanbecalculatedby[1]:

2 2

b

v eatan

s R e (5)

wheresisthefollowerdisplacement.

b) The maximum value of the contact stressmust be smaller than the materialpermissiblestrength:σmax<σper.

c) The offset e must satisfy the constraints:0<e<L/2ande<s.

d) Inorder toavoid the follower jamming theeccentricityamustfulfiltheconditions[1]:

2

21

20 )ξ(μb

μ

ba

(6)

and a<L/2, where the dimensions a, b and theparameterξareexplained inFig.1andμ is thecoefficientoffrictionbetweenthefollowerstemanditsguide.

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The distance a is calculated with the followingequation:

1/ 222ba r – R s (7)

with 1/ 22 2r x y ,wherexandyarethecam

profilecoordinates.3. PROPOSEDALGORITHM

In the present paper a multi‐objective geneticalgorithm (GA) method in MATLABprogramming environment is used to find theoptimalsolution.Theinputdataarethecammechanismtype,thekinematic and functional requirements, thevariablesboundsandthealgorithmparameters.In these parameters are included the initialparameters of the GA such as the populationsize, the crossover rate, the mutation rate, etc.and the number of the GA loops. Using theequation (1) the fitness function is defined,whichisusedinallstepsofthealgorithm.During the genetic algorithm, startingpopulations are randomly generated to setvariablesvalues,whichareusedtocalculatethefitnessfunctionvalue.Geneticalgorithm[9]usesselection, elitism, crossover and mutationprocedures tocreatenewgenerations.Thenewgenerationsconvergestowardsaminimumthatis not necessarily the global one. After somerepetitions when the maximum generations’number is achieved, the variables valuescorresponding to theminimum fitness functionvalue are selected as the optimum variablesvaluesofthegeneticalgorithm.An important issue in genetic algorithms is thetreatmentofconstraints.Foreachsolutionofthepopulation, the objective fitness values arecalculated. Furthermore, every solution ischeckedforconstraintsviolation.4. NUMERICALAPPLICATIONThe introduced methodology is applied to findthedesignparametersofacammechanismwithtranslatingflat‐facefollowerwherethefolloweroffsetissetequaltozero(e=0).

Figure2shows thekinematicrequirementspertransient region of the indicated in this figurefollowerdisplacementdiagram.

Fig.2.Kinematicrequirements.

Fig.3.Materialspropertiesandfunctionalrequirements.

The functional requirements and the materialpropertiesusedinthisinvestigationareinsertedinFig.3.Theparametersinvolvedinalltests,mainlyinGAprocedure,arethesameandselectedasoptimumsthrough many applied tests: population of

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individuals = 20, cross probability = 80 %, elitecount = 2 and the maximum number ofgenerationsis100.Considering kinematic requirements thedisplacement, velocity and acceleration of thefolloweraredetermined(Fig.4).

Fig.4.Thefollowermotiondiagrams.Ingeneral theweighting factorsα,βandγof thefitness function (1) are selected considering theimportanceoftheobjectivesthatmustbeachievedby themechanism.Ahigh valueof theweightingfactor α increases the importance of first part oftheobjectivefunction(F1)thatistoobtainasmallcam size. After several tests the followingweighting factors are chosen: α=0.1, β=0.1 andγ=0.8. Running the MATLAB codes with abovementioned parameters, the following designparametersareobtained:Rb=32.67mmandL=53.21mm. For constructedmechanism these parametersarefinallyset:Rb=35mmandL=50mm.

ThecamprofileisshowninFig.5.The3DmodelofthedesignedcammechanismisillustratedinFig.6.

Fig.5.Camprofile.

Fig.6.3Dmodelofthecammechanism.

5. FORCEANALYSISOFCAMMECHANISMCONSIDERINGFRICTIONFORCES

Inthissectionthedynamicforceanalysisofthedesigned mechanism considering the frictionforce between follower and its guide and thefrictionforcebetweencamandflatfacefollowerisinvestigated.The force transmission of a radial cam with areciprocatingflat‐facedfollowerisshowninFig.7,wherePistheexternalloadonthefollower,μisthecoefficientoffrictionbetweenthefollowerstem and its guide, μo the coefficient of friction

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betweenthecamandtheflatfacefolloweranddistheguidediameter.

Fig. 7. Force transmission of cam mechanism withtranslatingflat‐facefollower.From the equilibrium equations of horizontalandverticalforcesandmomentsaboutthepointA and assuming that difference the between

12

dμ N and 22

dμ N is negligible, the forces Fc,

N1andN2aredetermined[1]:

C

bPF

Γ (8)

01

a μ ξb PN

Γ (9)

02

1 a μ b ξ P

Ν Γ

(10)

with

02 1 2 Γ b aμ μμ b ξ

and

0 bP ms cs k s s F

wheremisthefollowermass,s, s and s arethedisplacement, velocity and acceleration of thefollower respectively, c is the dampingcoefficient, k is the spring constant, s0 is theinitial compression of the spring and Fb is thefollowerweight.Furthermore,thefrictionforcesarewrittenas:

0 0 Q μ F (11)

1 1 Q μN (12)

2 2Q μN (13)

and the cam shaft torque due to the friction isgivenby:

0 1 22 2

f b

d dT Q R s Q Q (14)

Fig.8.Constructedcammechanism.

Fig. 9. Friction forces of cam mechanism withtranslatingflat‐facefollower.

Fig.10.Inputtorquewithandwithoutfriction.

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In the designed and constructed mechanism(Fig. 8) the data used in dynamic analysis are:μ=0.78, μ0=0.15, m=1 kg, k=3004 N/m, s0=13mm,d=50mmandb=50mm.

The damping coefficient is1000

21000

k m

c ζ

with ζ=0.1. The spring constant k is chosenconsideringthespringforcegreaterthaninertiaforce corresponding to maximum deceleration,inordertoavoidthejumpphenomenon.Theparameterξisdeterminedwiththerelation:ξ=(15‐s)/b.Thediagramof friction forcesversuscamangleisillustratedinFig.9.In Fig. 10 is inserted the diagram of the inputtorquewithandwithoutfriction.6. CONCLUSION

In this paper the optimization of the designparameters of a cam mechanism with a flat‐faced follower is approached. For this task themulti‐objective optimization with geneticalgorithm is applied using the high levelprogramming language of MATLAB. Theoptimization satisfies constraints which aremade in order to operate a cam mechanismproperly. This procedure is automatic, givesresults fast and it appears to be reliable. Thefinal results provide useful information for acammechanismsynthesisandcanbeusedasabasis of final preference depending on theobjectivesthathavetobesucceeded.Subsequently, after the cam mechanismsynthesis, the applied friction forces arecalculated.Themostimportantconclusionisthefact that the friction forces are analogous withtheactionofthefollowermovement.Thismeansthat in theareasofdwell the friction forcesare

steady,whereasintheareasofriseorreturnthefrictionforcesalterinanalmostsimilarway.

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