DYNAMICSYSTEMS.APPLICATIONSINTECHNICAL DumitruBĂLĂ1,a* 1CălugăreniStreet,DrobetaTurnuSeverin,România a*dumitru_bala@yahoo.com… [614015]

DYNAMICSYSTEMS.APPLICATIONSINTECHNICAL
DumitruBĂLĂ1,a*
1CălugăreniStreet,DrobetaTurnuSeverin,România
a*[anonimizat]
Keywords:stability,Leapunovfunction,primeintegralLagrangian,stabilitycharts
Abstract.Thepaperincludesthestudyofthestabilityofsomedynamicalsystemsgivenbysystems
ofdifferentialequations.Oftheannalysedsystems,somerepresentmechanicalvibratingsystems.
Introduction
Generally,themovementofamaterialpointorofasystemofmaterialpointscanbedescribedwith
thehelpofthedifferentialequations.Theproblemoftheintegrationofthedifferentialequationsorof
thesystemsofdifferentialequationsisnotalwayseasytosolve.That’swhy,inthiscaseitisvery
importantthequalitativeanalysisoftheseproblems.
Thequalitativeanalysisisachapterotthetheoryofthedifferentialequationsthatstudiesthe
behaviourofthesolutionsofaproblembythedatasoftheproblem,withoutknowingthosesolutions.
Notalwaysthesimpleintroductioninthecomputerofanequationcanleadtofavourableresults.
Sometimestheexistencetheoremsaremoreimportantthanthenon-existencetheorems.Ontheother
hand,theexistenceofmoresolutionscanleadattheblockofthenumericalcalculus,likeinthecase
oftheapparitionofabifurcation.
Thecompletestudyofsomeproblemsfordifferentialequationsconsistsin:thequantitative
analysis(methodsofinferenceofsomesolutions),thequalitativeanalysisandthenumericalsolving
onthecomputer.Wemustnoticethatthenumericalsolvingmustcomeafterthequalitative
analysis.Thequalitativeanalysisistheonethatfinallyshowswhichisthesetofallthesolutionsofa
problem,whatbehaviourofthesolutionswecanexpectconsiderringallthepossiblevaluesofallthe
datas.Onlyonesolutioncorrespondingtoafixsetofdataswillcorrespondtoonlyoneaspectofthis
evolution.
Theproductivityoftheworkatthemillingmachinesdependsdefintielybythethreadingcapacity
which,inmanycasesisearlylimitatedbytheinsufficientdynamicstability[1,2,3].Thelostofthe
dynamicalstabilityatathreadingbehaviour,sometimesmuchundertheonelimitatedbythepowerof
theengineofthedriveofthetool,itisbecausebothatthedesignofthetoolmachinesandatthe
designofthetechnologicalprocess,thedynamicbehaviourofthesystemtoolmachine–part–
device–tool,andespeciallyofthetoolmachineistakenintoconsiderrationinasimplifiedmanner.
Thebookofthemachine,deliveredtothebeneficiarytogetherwiththetoolmachine,mustinclude
[1,4]:
a)Thechartofunconditionedstabilityliftedbythreading.
b)Thechartsrepresentingthevariationofthedirectionalcoefficienciesandalsothechartsofthereal
partofthefrequentialamplitudecharacteristic–phaseofthemachine.
c)Recommandationsforavoidingthepossibledomainsofdynamicalinstability.
Therequestsonthegloballeveltoassureforthetool–machineastabledynamicbehaviour,has
becomeadailypreocupationandso,thecompetivityoftheindustryforthebuildingoftool–
machinescannotbeseenwithoutthestudyofthestability.
Thestudyofthestabilityofadynamicalsystemthatcandescribeamillingmachineforwheel
gear
Oneofthemeasuresthatmustbefulfilledinordertoincreasethethreadingcapacityforthemilling
machineforwheelgearFD-320isthediminuishingasmuchaspossibleofthevibrationsthatappear

inthethreadingprocess[4].Theincreaseofthethreadingcapacitybutalsoofthequalityofthe
workedsurfacesleadstotheincreaseoftheproductioncapacity.Thevibrationsofthesupplesystem
aredescribedbythesolutionsofthesystemofdifferentialequations(1)wherePS meCCJ,,are
constantrepresentingthemomentofinertiaandthelasttwoconstantofelasticity.Thechoseofthe
indexwasdonetakingintoconsiderationtheelectricalengine,thetoolandthepart[3,4].
Westartwiththecinematicsystem




.)()()()()()(
53 51
66535 31
44315 13
221
P S meS me
PPP S meP me
SSP S meP S
meme
CCCxxCxxC
JCxxxCCCxxCxxC
JCxxxCCCxxCxxC
JCxxx

(1)
WebuildtheLagrangeextensionfollowingthemethodofprof.dr.ConstantinUdriște[5].Forthiswe
intorducethevectorialfield“millingmachine”thathassixcomponents




,)()(),,,,,(),,,,()()(),,,,,(),,,,,()()(),,,,,(),,,,,(
53 51
65432166654321535 31
65432144654321315 13
654321226543211
P S meS me
PPP S meP me
SSP S meP S
meme
CCCxxCxxC
JCxxxxxxXxxxxxxxXCCCxxCxxC
JCxxxxxxXxxxxxxxXCCCxxCxxC
JCxxxxxxXxxxxxxxX
(2)
where
),,,,,(654321 XXXXXXX și ),,,,,(654321 xxxxxxx
Thesystem(1)isanautonomoussystemintheform
),,,,,(654321 xxxxxxXdtdx
ii 6,1i. (3)
Theequillibriumpointsofthesystem(1)areinformx=(a,0,a,0,a,0)whereaisaconstant.
Theffunctiongivenby

6
12
21
iiX f (4)
Representstheenergydensityassociatedtothevectorialfield“millingmachine”andtheeuclidian
structureij.Thegeommetricaldynamicassociatedtothesupplesystemisdescribedbythe
differentialsystemofseconddegree


jj
ij
ji
ii
dtdx
xX
xX
xf
dtxd) (22
,i,j=6,1 (5)
ThatprovestobeanEuler-Lagrangeextension.Inotherwords,thelagrangian

 6
12)(21
iiiXdtdxL (6)
or
fdtdxXdtdxLi
ii
ii   
 6
16
12)(21(6')
Determinsthisseconddegreesystemwithsixdegreesoffreedom,whosetrajectoriescontainalsothe
solutionsofthesystem(1).Dependingontheconstantvaluesthatappearinthesystemgivenby(1),
weapplythestabilitytheoremsLeapunovforautonomoussystems.AsaLeapunovfunctionwecan
taketheenergydensity,alangragianandaprimeintegrale.
Exemple2.uLsystem
Beinggiventhesystemofdifferentialequations,givenbytherelation(1).Thissystemdescribesa
technicalprocessandisknownastheuLsysteminthespecialityliterature[1,6].


213 3231 2121
xxcx xbxxxxaxaxx

(7)
WebuildtheLagrangeextensionusingthemethodofprof.Dr.ConstantinUdriste[5].Forthiswe
introducethevectorialfieldwhichhasthreecomponents.


213 3213231 321212 3211
),,(),,(),,(
xxcx xxxXbxxx xxxXaxaxxxxX
(8)
where
),,(321XXXX și ),,(321xxxx
The(1)systemisanautonomoussystemlikethis
),,(321xxxXdtdx
ii 3,1i. (9)
Theequilibriumpointsofthesystem(7)are
( bcbc,,b)and(- bc bc,,b).

Thefgivenfunction

3
12
21
iiX f (10)
representstheenergydensityassociatedwiththevectorialfieldandwiththeeuclidianstructureij.
Westudythestabilityofthe(7)systemon/3R.
WetakeasaLeapunovfunction
22),,(2
32
2
321xxxxxV  (11)
WeverifytheconditionswhichmustbefulfilledbytheVfunctionforthesystemtobestabile.We
verifythestabilityconditionsfromtheLeapunovtheoremforautonomoussystems.
TheVfunctionadmitspartialderivativesoffirstorderwhicharecontinuousandV(0,0,0)=0.Italso
0,0)(xxV .
2
32
2213
3231
212
1) () ()(),(
cx bxxxcxxVbxxxxVaxaxxVf gradV

Itisfulfilledthethirdcondition,too,ifwehavetheconditionsb≤0andc≥0.Withthisconditionsthe
systemisstabile.
Wecalculateforthesystem(7)f,LandH.
]) () () [(212
2132
2312
12 xxcx bxxx axax f  (12)
  dtdxxaxaxdtdx
dtdx
dtdxL1
21223 22 21)(])()()[(21) (231bxxxdtdx2
-(213xxcx )dtdx3+ ]) () () [(212
2132
2312
12 xxcx bxxx axax  (13)
])()()[(2123 22 21
dtdx
dtdx
dtdxH   ]) () () [(212
2132
2312
12 xxcx bxxx axax  (14)
Methodofstudy
Inordertostudythestabilityofthedynamicsystemsdescribedbythedifferentiateequations
systemsitisusedLeapunovfunction.ThereisnotageneralmethodofdetermingLeapunovfunction.
AsaLeapunovfunctionwecanconsideranenergydensity,alagrangianorahamiltonian.The
Leapunovfunctionmustverifythestabilityconditionsfromthestabilitytheoremsforautonomous
systems.Finallyitisobtainedtheconditionorconditionsnecessarytothesystemtobestableor
asimptomaticallystable.
Conclusions

ForthefirstcaseofthedynamicsystemstudiedwhencosideringasaLeapunovfunctionthe
lagrangianortheenergydensityitisstudiedthestabilityofthesystem.Theconditionofstabilitywill
dependonPS meCCJ,,constants.Theyhaveatechnicalsignificance,accordingtothepresentationof
thework.
Thelagrangianiscalculedwith(6)or(6)formulaandtheenergydensitywith(4)formula.
ConcerningtheseconddynamicsystemweconsiderasLeapunovfunctiontheVfunctiongiven
bytherelation(11)andhavingtheformenergy.ThestabilitycanbestudiedtakingasLeapunov
functionthefunctionsgivenbytherelations(12),(13)and(14).Fortheseconddynamicsystemthis
workdeterminedthestabilityconditions.
Forbothdynamicsystemsthestabilityconditionsdependontheconstantsappearinginthe
dynamicsystems.Theseconstantshaveatechnicalsignificance,havingasaresultthetechnical
characteristicswhichshouldbecarriedoutsothatthemechanicalsystemscouldbeperformant.
TheoriginalityoftheworkconsistsonthewayLeapunovfunctionisrealised.Thebetterthe
Leapunovfunctionchosenthemoreandbetterprecisedthestabilitydomainis.Atthemilling
machinesthestabilitydiagramsaregraphicalrepresentationshavingontheirabscisethevalueofthe
rotationofthemillandontheordonatethedepthofmilling.
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