The Influence on Modal Parameters of Thin Cylindrical Shell under Bolt Looseness Boundary

16 Pages·2016·7.1 MB·English

Hindawi Publishing Corporation
Shock and Vibration
Volume 2016, Article ID 4709257, 15 pages
http://dx.doi.org/10.1155/2016/4709257
Research Article
The Influence on Modal Parameters of Thin Cylindrical Shell
under Bolt Looseness Boundary
HuiLi,MingweiZhu,ZhuoXu,ZhuoWang,andBangchunWen
SchoolofMechanicalEngineeringandAutomation,NortheasternUniversity,No.3-11WenhuaRoad,HepingDistrict,
Shenyang110819,China
CorrespondenceshouldbeaddressedtoHuiLi;[email protected]
Received20July2015;Revised3October2015;Accepted5October2015
AcademicEditor:DanielMorinigo-Sotelo
Copyright©2016HuiLietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which
permitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
The influence on modal parameters of thin cylindrical shell (TCS) under bolt looseness boundary is investigated. Firstly, bolt
loosenessboundaryoftheshellisdividedintotwotypes,thatis,differentboltloosenessnumbersanddifferentboltlooseness
levels,andnaturalfrequenciesandmodeshapesarecalculatedbyfiniteelementmethodtoroughlymastervibrationcharacteristics
ofTCSundertheseconditions.Then,thefollowingmeasurementsandidentificationtechniquesareusedtogetprecisefrequency,
damping,andshaperesults;forexample,noncontactlaserDopplervibrometerandvibrationshakerwithexcitationlevelbeing
preciselycontrolledareusedinthetestsystem;“preexperiment”isadoptedtodeterminetherequiredtighteningtorqueandverify
fixedconstraintboundary;thesmall-segmentFFTprocessingtechniqueisemployedtoaccuratelymeasurenaturefrequencyand
laserrotatingscanningtechniqueisusedtogetshaperesultswithhighefficiency.Finally,basedonthemeasuredresultsobtained
bytheabovetechniques,theinfluenceonmodalparametersofTCSundertwotypesofboltloosenessboundariesisanalyzedand
discussed.Itcanbefoundthatboltloosenessboundarycansignificantlyaffectfrequencyanddampingresultswhichmightbe
causedbychangesofnonlinearstiffnessanddampingandinboltloosenesspositions.
1.Introduction orconstrainedbymanybolts,itsconstrainteffectivenesswill
inevitablybeweakenedundercomplexexternalload[11,12],
Thin cylindrical shell (TCS) has long been an important especiallywhenaboltorsomeboltscomeloose;thelooseness
structural component due to its high stiffness to weight conditionwillnotonlydecreaseconstraintstrengthbutalso
and strength to weight ratios, which is widely used in the increase interface abrasion and the probability of fatigue
engineering fields, such as aircraft casings, pipes and ducts, failure, and, once the bolt fracture happens, it will severely
rotarydrumsingranulator,andaircraftengine[1–5].Modal affect the function of TCS and could even lead to serious
parametersofTCSaremainlycomposedofnaturalfrequen- accident[13].
cies,modeshapes,anddampingratio;theseparametersare Atpresentgreateffortshavebeenmadetostudyvibration
thebasisoffurtherstudyonvibrationcharacteristicsofTCS character of TCS under complex and diverse boundary
[6–8],whichareofgreatimportancetotheoreticalmodeling, condition by scholars and researchers, for example, bolt
responseprediction,vibrationreductionoptimization,vibra- looseness,elasticity,andothercomplexboundaryconditions,
tion mechanism research, structural damage identification, andmanyencouragingresearchresultshavebeenobtained.
and so forth. Generally, TCS is working at a very harsh Forexample,Forsberg[14]studiedtheinfluenceofboundary
environment, such as thermal expansion and contraction, conditions on the modal characteristics of thin cylindrical
strong vibration, and uneven exciting force generated by shells. Totally 16 possible sets of homogeneous boundary
turbulentairflow,whichmayeasilymakeTCSoperateunder conditions were specified independently at each end of the
elasticityorloosenessboundary[9].Besides,duetodifferent shell,andthesesetsofconditionswerediscussedindetail.It
installation forms, the connected or constraint end of TCS hasbeenfoundthatevenforlongcylinders(lengthtoradius
maynotbewellfixed[10].Forexample,ifTCSisconnected ratio of 40 or more) the minimum natural frequency may
2 ShockandVibration
differ by more than 50% depending upon whether 𝑢 = 0 analysis results of the shell under more complex boundary
or the longitudinal stress resultant 𝑁𝑥 = 0 at both ends. condition can not be effectively verified, let alone validate
Koga [15] studied the effects of boundary conditions on someadvancedshelltheories.
the free vibrations of TCS and a simple formula for the Thisresearchcombinedtheorywithexperimenttoinves-
naturalfrequencywasderivedasanasymptoticsolutionfor tigate the influence on modal parameters of TCS under
the eigenvalue problems of the breathing vibrations, whose BLB.Firstly,boltloosenessboundaryoftheshellisdivided
accuracy was sequentially examined by a comparison with intotwotypes,thatis,differentboltloosenessnumbersand
numerical solutions and experimental results. The results different bolt looseness levels, and natural frequencies and
showedthattheformulawasaccurateenoughforengineering mode shapes of TCS under these conditions are calculated
anditwasapplicableunderanypossiblecombinationsofthe byfiniteelementmethod(FEM)inSection2;thusvibration
boundaryconditionsforthesimplysupported,theclamped, characteristics of TCS under the above two types of bolt
andthefreeendsoftheshell.Sofiyevetal.[16]proposedan loosenessboundariescanberoughlymastered.Thenwego
analyticalproceduretostudythefreevibrationcharacteristics ontosetupexperimentsystemtoaccuratelymeasuremodal
oflaminatedthincircularcylindricalshellsrestingonelastic parameters of TCS, and the corresponding test procedures
foundation. They found that natural frequency was as a and identification techniques which are suitable for the
functionoftheshelldisplacementamplitude,anditwasalso thin walled shell are also proposed in Section 3. Finally in
closerelatedtotheeffectofelasticfoundation,nonlinearity, Section4,basedontheaccuratemeasureddata,theinfluence
and number and ordering of layers of the shell. Liang and on natural frequencies, mode shapes, and damping ratios
Zhang [17] studied stiffness optimization of TCS under of TCS under two types of bolt looseness boundaries is
elasticityboundarycondition.Theexplicitformulaofinitial analyzedanddiscussedindetails.Thisresearchcanprovide
parametersolutionofvariablethicknessshellwasderivedby dynamicmodelingserviceforTCSundercomplexboundary
transfer matrix method, and the optimization process was condition, provide experimental data for effective selection
transformedintoaconstraintnonlinearsolvingprocess;thus of boundary parameters in the theoretical model, and also
the objective function can be successfully obtained by the provideanimportantreferenceforanalysisanddiagnosisof
steppedreductionmethod.Dongetal.[18]investigatedthe boltloosenessfaultofTCS.
influences of bolt looseness in missile clamping support on
vibrationcharacteristicsofthecylindricalshellstructureby
2.VibrationCharacteristicAnalysisofTCS
base excitation technique. Different steady response signals
underBoltLoosenessBoundary
were obtained by varying bolt pretightening force of the
attachment bolts in the structure, and power spectrum
In this section, in order to deeply understand vibration
densitydiagramsofsignalscorrespondingtodifferentstatus
characteristic of TCS, FEM is used to calculate vibration
ofthestructurewereanalyzedandanextractionmethodof
characteristicofTCS,suchasnaturalfrequenciesandmode
thespectralmomentloosenesswasproposedtodistinguish
shapes, under two types of bolt looseness boundaries with
fault characteristics of bolt looseness. Dong et al. [19] also
different looseness numbers and different looseness levels,
experimentallystudiedhowtomonitortheattachmentbolt
respectively. Although the resulting frequencies and shapes
loosenessinaclampedcylindricalshellstructure,andwavelet
may inevitably contain some calculation errors, they are
transformwasusedtoanalyzedamagecharacteristicofbolt
helpfulforustodeterminemeasuredfrequencyrange,build
loosenessbasedonthestructuralaccelerationresponsedata.
experimentalmodel,understandgeographicdistributionsof
Zhouetal.[20]usedthewavepropagationmethodtosolve
somenodesornodallines,andsoforth.
theequationsofmotionofTCSunderelastic-supportbound-
ary condition, and the elastic-support boundary condition
2.1.ResearchObject. TheTCSstudiedinthispaperisshown
was specified in terms of 8 independent sets of distributed
in Figure 1 and dimension parameters are listed in Table 1.
springs which have arbitrary stiffness values. Besides, the
ThematerialofTCSisstructuralsteelwithelasticmodulusof
effectsonnaturefrequenciesoftherestrainingspringswere
3
212Gpa,Poisson’sratioof0.3,andthedensityof7850kg/m .
also studied for a range of stiffness values and different
Its length is 95mm with 144mm external radius and an
geometrical characteristics of the shells, and it was found
averagethicknessof2mm.Thereistheextensionedgewith
thattherestrainingstiffnesscandrasticallyaffectfrequency
150mm external radius and 3mm thickness on this shell
parameters of TCS. Sun [21] used transfer matrix method
which is machined to be clamped by a clamping-ring with
tostudythefreevibrationofTCSunderelasticityboundary
eight M8 bolts, so that it can be certain that the shell is in
condition, taking into account elastic connection stiffness,
cantileverboundarycondition.Then,wecanloosensomeor
and the transfer matrix of global variables of TCS under
allofboltswithcertaintighteningtorquebytorquewrench,
elasticity boundary condition was obtained by multiplying
sothatwecananalyzevibrationcharacteristicofTCSunder
transfermatrixofstatevariablesintheconnectionboundary
boltloosenessboundary.
withtheoneoftheshellitself.
However,mostofresearchesdonebytheabovescholars
and researchers are mainly based on theory or simulation; 2.2.VibrationCharacteristicAnalysisofTCSunderBLBwith
experimentalstudiesontheinfluenceonmodalparameters Different Looseness Numbers. Finite element model of the
ofTCSunderboltloosenessboundary(BLB)arestillscarce. TCS under BLB with different looseness numbers is estab-
And as a lack of the related test conclusion, theoretical lishedwithANSYSParametricDesignLanguage(APDL)in
ShockandVibration 3
Thin
cylindrical
shell
Clamping-ring
M8 bolt
Extension edge
Figure1:TCSanditsclampedcircularringusedinboltloosenessboundary.
Table1:Dimensionparametersofthincylindricalshell.
Thicknessof
Length Thickness Internalradius Externalradius Extensionedgeradius
extensionedge
(mm) (mm) (mm) (mm) (mm)
(mm)
95 2 142 144 150 3
and the resulting natural frequencies and mode shapes are
obtained by Block Lanczos method, as listed in Tables 2
and 3, respectively. Then, use different numbers of spring
elements to simulate different bolt looseness numbers and
calculate the corresponding frequencies and shape results,
according to the sequence of loosening 1 bolt, 2 bolts, 3
bolts,and4bolts,fromsequenceItosequenceIVasshown
in Figure 3, and the calculated results are given in Table 2
Y
X andTable3.Additionally,thefrequenciesdifferencesunder
Z the constraint boundary (or no loose boundary) and bolt
loosenessboundarywithdifferentloosenessnumbersarealso
comparedinTable2,andFigure4givestherelationsbetween
naturalfrequencyandmodeshapeunderBLBwithdifferent
loosenessnumbers.Itshouldbenotedthat,duetothespeci-
ficityofthefirstmodeshape,itisdifficulttousetraditional
numberofaxialhalf-waves,𝑚,andcircumferentialwaves,𝑛,
todescribeitsvibrationshape;thusthefirstspecialfrequency
Figure2:FiniteelementmodeloftheTCSunderBLBwithdifferent and shape result are not compared under BLB in Table 2
loosenessnumbers. (the calculated results are only for roughly understanding
vibrationcharacteristicofTCSwhentheconstraintboltsare
loose;theyinevitablycontainsomecalculationerrorsbecause
ANSYS software, as seen in Figure 2. SOLID186 element constraint stiffness and damp parameters in bolt looseness
is used to create the model of the shell which consists of positionsarehardtosimulatewithoutexperimentaltest,and
6480 nodes and 960 elements, and MATRIX27 element is theyarenotthefocusofthispaper).
usedasspringelementtosimulateboltloosenessboundary ItcanbefoundfromtheaboveanalysisresultsinTables
with different looseness numbers, whose stiffness can be 2 and 3 and Figure 4 that (I) bolt looseness will result in
adjusted in the 𝑥, 𝑦, and 𝑧 direction. Firstly, use 8 spring the decrease of natural frequencies of TCS, and with the
elements with stiffness value of 1 × 108 in the above three increasing of bolt looseness numbers the frequency results
directionstosimulatethefree-clampedboundarycondition, will further decrease; (II) high order natural frequencies of
4 ShockandVibration
Table2:NaturalfrequenciesofTCSunderBLBwithdifferentloosenessnumbers.
Loosen Loosen Loosen Difference Loosen Difference
Nolooseness DifferenceI DifferenceII
Modal 𝐴 1bolt (𝐵−𝐴)/𝐴 2bolts (𝐶−𝐴)/𝐴 3bolts III 4bolts IV
order 𝐵 𝐶 𝐷 (𝐷−𝐴)/𝐴 𝐸 (𝐸−𝐴)/𝐴
(Hz) (%) (%)
(Hz) (Hz) (Hz) (%) (Hz) (%)
1 — 786.2 — 678.0 — 591.9 — 554.4 —
2 976.5 974.2 −0.2 935.0 −4.3 831.6 −14.8 758.9 −22.3
3 1068.7 1065.0 −0.3 1043.2 −2.4 1035.4 −3.1 1013.0 −5.2
4 1313.8 1298.5 −1.2 1248.4 −5.0 1193.3 −9.2 1146.8 −12.7
5 1535.9 1461.0 −4.9 1310.0 −14.7 1275.1 −17.0 1269.0 −17.4
6 1603.2 1597.6 −0.3 1629.0 1.6 1591.1 −0.8 1403.4 −12.5
7 1841.9 1743.4 −5.3 1681.1 −8.7 1655.8 −10.1 1588.3 −13.8
8 2000.8 1995.6 −0.3 1991.7 −0.5 1988.0 −0.6 1984.9 −0.8
Table3:ModeshapesofTCSunderBLBwithdifferentloosenessnumbers.
Loosen Loosen Loosen Loosen
Modal Nolooseness
order 𝐴(𝑚,𝑛) 𝐵1(b𝑚o,l𝑛t) 𝐶2(b𝑚o,lt𝑛s) 𝐷3(b𝑚ol,t𝑛s) 𝐸4(b𝑚o,lt𝑛s)
1 — (Special) (Special) (Special) (Special)
2 (1,5) (1,5) (1,5) (1,5) (1,5)
3 (1,6) (1,6) (1,6) (1,6) (1,6)
4 (1,7) (1,7) (1,4) (1,4) (1,4)
5 (1,4) (1,4) (1,7) (1,7) (1,7)
6 (1,8) (1,8) (1,8) (1,8) (1,3)
7 (1,3) (1,3) (1,3) (1,3) (1,8)
8 (1,9) (1,9) (1,9) (1,9) (1,9)
er10
b
m 9
u
n 8
e
av 7
w
I Loosen 1 bolt al 6
nti 5
II Loosen 2 bolts ere 4
III Loosen 3 bolts umf 3
I Circ 2700 900 1100 1300 1500 1700 1900 2100
IV II IV Loosen 4 bolts
III
Natural frequency (Hz)
Figure 3: Schematic of bolt loosening sequence under BLB with
No looseness Loosen 3 bolts
differentloosenessnumbers. Loosen 1 bolt Loosen 4 bolts
Loosen 2 bolts
Figure4:Therelationbetweennaturalfrequencyandmodeshape
the shell, for example, the 8th natural frequency, basically
underBLBwithdifferentloosenessnumbers.
willnotbeaffectedbyBLBwithdifferentloosenessnumbers,
andthefrequencydifferencerelatedtotheBLBandnoloose
condition is less than 0.8%; (III) with the increase of bolt
looseness number, low order mode shapes of TCS will be 2.3.VibrationCharacteristicAnalysisofTCSunderBLBwith
changed,buthighordermodeshapeswillstillbeunchanged; Different Looseness Levels. Similarly, SOLID186 element is
(IV)althoughfrequenciesandshapesofTCSwillbechanged usedtocreatethemodeloftheshellwhichconsistsof6480
under BLB, the changing trend of natural frequencies with nodes and 960 elements, and MATRIX27 element is used
modeshapesisconstantwhenthenumberofaxialhalf-waves asspringelementtosimulateboltloosenessboundarywith
𝑚 = 1, which shows that frequency values are up after the differentloosenesslevels;forexample,setthestiffnessvalue
decline with the increase of the number of circumferential of Matrix27 element in 𝑥, 𝑦, and 𝑧 direction to 0.75 × 108,
waves 𝑛, and usually frequency values related to 𝑛 > 8 are 0.25×108,and0.25×107,respectively(namely,25%looseness,
higherthan𝑛=2∼7. 75% looseness, and 97.5% looseness), as seen in Figure 5.
ShockandVibration 5
Table4:NaturalfrequenciesofTCSunderBLBwithdifferentloosenesslevels.
25% Difference 97.5% Difference
Nolooseness DifferenceI 75%
Mordoedral 𝐴 loos𝐵eness (𝐵−𝐴)/𝐴 looseness𝐶 (𝐶−I𝐴I)/𝐴 loos𝐷eness (𝐷−II𝐴I)/𝐴
(Hz) (%) (Hz)
(Hz) (%) (Hz) (%)
1 — 965.2 — 940.8 — 858.1 —
2 976.5 973.2 −0.3 956.0 −2.1 898.6 −8.0
3 1068.7 1067.4 −0.1 1056.0 −1.2 1035.4 −3.1
4 1313.8 1312.9 −0.1 1308.9 −0.4 1293.1 −1.6
5 1535.9 1530.7 −0.3 1504.3 −2.1 1382.8 −10.0
6 1603.2 1602.8 0 1601.5 −0.1 1594.9 −0.5
7 1841.9 1835.4 −0.4 1802.5 −2.1 1650.2 −10.4
8 2000.8 2000.5 0 1999.3 −0.1 1993.5 −0.4
I Fixed boundary No looseness
II Looseness boundary Looseness 97.5%
k
k×8 III Looseness 75%
IV Looseness 25%
Figure5:SchematicofdifferentboltloosenessleveladoptedbyFEM.
Table5:ModeshapesofTCSunderBLBwithdifferentlooseness er 9
b
levels. m
u 8
n
Nolooseness 25% 75% 97.5% e 7
Modal 𝐴 looseness looseness looseness wav 6
Order (𝑚,𝑛) 𝐵(𝑚,𝑛) 𝐶(𝑚,𝑛) 𝐷(𝑚,𝑛) ntial 5
1 — (Special) (Special) (Special) ere 4
2 (1,5) (1,5) (1,5) (1,5) umf 3
c
3 (1,6) (1,6) (1,6) (1,6) Cir 2
800 1000 1200 1400 1600 1800 2000 2200
4 (1,7) (1,7) (1,7) (1,4)
Natural frequency (Hz)
5 (1,4) (1,4) (1,4) (1,3)
6 (1,8) (1,8) (1,8) (1,7) No looseness 50% looseness
25% looseness 97.5% looseness
7 (1,3) (1,3) (1,3) (1,8)
8 (1,9) (1,9) (1,9) (1,9) Figure6:Therelationbetweennaturalfrequencyandmodeshape
underBLBwithdifferentloosenesslevels.
Firstly, use 8 spring elements with stiffness value of 1 ×
108 in above three directions to simulate the free-clamped undertheconstraintboundary(ornolooseboundary)and
boundary condition, and the resulting natural frequencies bolt looseness boundary with different looseness levels are
andmodeshapesareobtainedbyBlockLanczosmethod,as also compared in Table 4, and Figure 6 gives the relations
listed in Tables 4 and 5. Then, use 0.75 × 108, 0.25 × 108, betweennaturalfrequencyandmodeshapeunderBLBwith
and 0.25 × 107 of 8 spring elements to simulate different differentloosenesslevels.
bolt looseness levels, according to the sequence of 25% ItcanbefoundfromtheaboveanalysisresultsinTables
looseness,75%looseness,and97.5%loosenesstocalculatethe 4 and 5 and Figure 6 that (I) bolt looseness will result in
correspondingfrequenciesandshaperesults,fromsequence the decrease of natural frequencies of TCS, and with the
ItosequenceIIIasshowninFigure5,whicharealsogiven increasingofboltloosenesslevelsthefrequencyresultswill
in Tables 4 and 5. Additionally, the frequencies differences further decrease; (II) high order natural frequencies of the
6 ShockandVibration
Table6:ThedisadvantagesofdifferentvibrationexcitationdevicesformodaltestofTCSunderunderboltloosenessboundary.
ModalparametersofTCS
Excitation
device Natural Mode Dampingratio Disadvantage
frequency shape
Pulseexcitationlevelcannotbe
preciselycontrolledandthe
Hammer √ √ × excitationforcevariesforeach
measurement,anddoublehitcan
oftenleadtotesterrors.
Therelatedforcesensorwill
bringaddedmassandstiffnessto
Electromagnetic × × × TCS,whichwillseverelyaffect
exciter
testresultsofdampingand
naturalfrequency.
Theexcitationenergyof
piezoelectricceramicexciteris
Piezoelectric √ √ × ofteninsufficientforTCS,which
ceramicexciter willresultinpoorresponse
signalwithlowlevelofsignal
noiseratio.
Excitationfrequenciesarenot
thathigh,whichareoftenlimited
Vibrationshaker √ √ √ to1Hz∼3000Hz,andthetest
proceduresareoften
complicated.
shell, for example, the 8th natural frequency, basically will 3.1. Test System of Modal Parameters of TCS under Bolt
not be affected by BLB with different looseness levels, and Looseness Boundary. On the one hand, due to light weight,
the frequency difference related to the BLB and no loose closed modes, low level, and complicated local vibration of
condition is less than 0.4%; (III) with the increase of bolt TCS, traditional accelerometer will bring added mass and
looseness levels, low order mode shapes of TCS will be stiffness to the tested shell [22], which will severely affect
changed,buthighordermodeshapeswillstillbeunchanged; the tested frequency and damping results, so laser Doppler
(IV)althoughfrequenciesandshapesofTCSwillbechanged vibrometerisusedasnoncontactresponsesensortomeasure
under BLB, the changing trend of natural frequencies with thevibrationandfrequencyinformationoftheshell.Onthe
modeshapesisconstantwhenthenumberofaxialhalf-waves otherhand,differentexcitationtechniquesalsowillresultin
𝑚 = 1, which shows that frequency values are up after the test error, so the disadvantages of four common vibration
decline with the increase of the number of circumferential excitation devices are compared in Table 6; combining the
waves 𝑛, and usually frequency values related to 𝑛 > 8 are proposedtestmethodin[23],vibrationshakerisfinallycho-
higherthan𝑛=2∼7. senasexcitationsourcewithexcitationlevelbeingprecisely
controlled,andtestsystemofmodalparametersofTCSunder
boltloosenessboundaryisgiveninFigure7.Theinstruments
used in the test are as follow: (I) Polytec PDV-100 laser
3.TestSystemandMethod of Doppler vibrometer; (II) king-design EM-1000F vibration
ModalParametersofTCSunder shakersystems;(III)LongWeiPS-305DMDCpowersupply;
∘
BoltLoosenessBoundary (IV)AslongJGA25DCgearedmotor;(V)45 rotationmirror
∘
and45 fixedmirror;(VI)LMSSCADASMobileFront-End
InSection2,vibrationcharacteristicofTCSunderBLBand andDellnotebookcomputer.
its influence is analyzed. But due to the complexity of bolt In these devices, LMS SCADAS Mobile Front-End and
looseness boundary, the real influence of such boundary Dell notebook computer are responsible for recording and
onmodalparameters,especiallythedampingcharacteristics saving response signal from laser Doppler vibrometer. Dell
of the shell can not be accurately analyzed by simulation notebook computer with Intel Core i7 2.93GHz processor
method. Therefore, it is necessary to employ experimental and4GRAMisusedtooperateLMSTest.Lab10Bsoftware
test to investigate on the influence on modal parameters of and store measured data. For the frequency and damping
boltconstrainedshellunderdifferentloosenessboundaries. test, sine sweep excitation is conducted with a closed loop
In this section, experiment system is firstly established to controlviaaccelerometeronthecountertopofthevibration
∘
accurately measure modal parameters of the shell, and the shaker, and point 1, point 2, and point 3 (being 180 with
correspondingtestproceduresandidentificationtechniques eachother)areusedtogetresponsesignalbyadjustinglaser
whicharesuitableforthethinwalledshellarealsoproposed. point and average is used as the final results. In this test,
ShockandVibration 7
Laser Doppler
Response(m/s)−5050Time (s)100 1 2Poin3ts exScwiteaetipo n Point 1 45v∘ifibrxoemd meteirrror
ResponseM(m/s)Fr02ee0qauesnucyr (Hezd1)0 0d0a1t2aPo i3nts Vibration shaker PoThinitn 2 cylindricPalo sinhte l3l DC power supply
Response test 45∘rotation mirror
Data Response
DC geared motor
processing signal
LMS SCADAS Laser Doppler Vibration shaker
Computer mobile front-end vibrometery
Natural frequency and damping test Mode shape test
Figure7:SchematicoftestsystemofmodalparametersofTCSunderboltloosenessboundary.
naturalfrequencycanbepreciselydeterminedthrougheach arebig,morethan5∼20Hz,weneedtoincreasetorquevalue
resonantpeakinfrequencydomain,anddampingratiocan andtorepeatpreexperimentsseveraltimes.
alsobeidentifiedbythehalf-powerbandwidthmethodwhich
is calculated by measuring the bandwidth of the frequency 3.2.2. Measure Modal Parameters of TCS under Fixed Con-
curve(orapproximately3dB)downfromtheresonantpeak. straintBoundary. Thisstepinvolvesthreedifferentmeasure-
For mode shape test, laser rotating scanning technique is mentsandidentificationtechniques.Firstly,usingsinesweep
used to get shape results of TCS. Firstly, employ one of excitation by vibration shaker to test natural frequencies of
natural frequencies of TCS to drive the tested shell under TCSandinordertogetprecisefrequencyresults,thesmall-
theresonancestatebyvibrationshaker,andthenDCpower segmentFFTprocessingtechniqueisemployedtodealwith
supplyisusedtoprovidestablevoltageandcurrentforDC themeasuredsweepsignal.Thetimedomainsignalinvolving
∘
geared motor, and the motor is to drive the 45 rotation the 3rd natural frequency of the tested shell is shown in
∘
mirror to complete a set of cross-sectional scan with 360 Figure 8(a). If FFT processing technique is directly applied
circumferentialcoverageforthetestedshell,andinthisway onthissweepsignal,wecanobtainitsfrequencyspectrum,
modeshapesdataatcertainmodecanbeobtainedinashorter as seen in Figure 8(b), and the frequency of the response
amountoftimethantraditionaltestmethods. peakis1024.8Hz.However,ifthewholetimeofsweepsignal
canbedividedintomallsegments,andweconductFFTon
eachsegmentofthem(inthisexample,itis1swithrespect
3.2. Test Method of Modal Parameters of TCS under Bolt
tothewholetimeof68s),theresultingfrequencyspectrum,
Looseness Boundary. In this section, the test procedures of
as seen in Figure 8(c), is plotted through the combination
modal parameters and the related identification techniques
of the response peak of each segment (also treated with
whicharesuitableforTCSunderboltloosenessboundaryare
interpolationandsmoothing);thefrequencyvaluerelatedto
proposed,asseeninthefollowingfourkeysteps.
the peak is 1025.7Hz, which is truly accurate result of the
3rdnaturalfrequency.Therefore,fortime-dependentsweep
3.2.1. Accurately Determine Tightening Torque under Fixed signalof TCS, it is necessary to use the small-segment FFT
ConstraintBoundary. BecausemodalparametersofTCSare processingtechniquetoaccuratelygetfrequencyresults.
closelyrelatedtoconstraintboundary,inactualtest,wemust Then, use the half-power bandwidth technique to iden-
ensurethatoneendofthetestedshelliseffectivelyclamped; tify each damping ratio of TCS from the frequency spec-
tothisend,atorquewrenchisusedtodeterminethelevelof trumobtainedbysmall-segmentFFTprocessingtechnique.
tighteningtorqueontheM8boltsofclamping-ring,asseen Becausetheresonantpeakinthespectrumisalreadyknown,
inFigure1,andthe“preexperiment”isadoptedtodetermine wecanidentifytwohalf-powerbandwidthpointsbymeasur-
the required tightening torque as well as verify whether or ingthebandwidthofthefrequencycurve(orapproximately
not the tested shell is under fixed constraint boundary. For 3dB)downfromtheresonantpeak,consequentlyaccording
instance,itshouldbedoneatleastthreetimestotestnatural to the damping formula to calculate the corresponding
frequencies, and every time the same level of torque value damping results based on the MATLAB program. Figure 9
should be applied on M8 bolts. If test results of the first 3 gives time waveform and frequency spectrum for the third
naturalfrequenciesunderthreepreexperimentsarecloseto naturalfrequencyanddampingratioofTCSat3measuring
each other (e.g., 1∼3Hz), we will regard this torque value points,and, in orderto improveaccuracy offrequencyand
as the determined tightening torque under fixed constraint damping results, the final results are obtained by averaging
boundary.Ifthedifferencesbetweeneachnaturalfrequency thetestresultsatthesepoints.
8 ShockandVibration
4 1
1024.8
0.08
3
m/s) m/s) 0.06
de ( 2 de (
u u
plit plit 0.04
m m
A A
1
0.02
0 0
0 20 40 60 1010 1015 1020 1025 1030 1035
Time (s) Frequency (Hz)
(a) Therawsweepsignal (b) ThespectrumbydirectFFT
0.08
1025.7
0.06
m/s)
de ( 0.04
u
plit
m
A
0.02
0
990 1000 1010 1020 1030 1040 1050
Frequency (Hz)
(c) Thespectrumbysmall-segmentFFT
Figure8:The3rdnaturalfrequencyofCTCSobtainedbydifferentFFTprocessingtechniques.
5
m/s) m/s) 2 3
Velocity (−500 Time (s5)0 100 1 Me2asuring p oint3 Velocity (100100 Fre1q0u3e0ncy (Hz1)060 1090 1 M2easuring point
(a) Timewaveform (b) Frequencyspectrum
Figure9:TimewaveformandfrequencyspectrumforthethirdnaturalfrequencyanddampingratioofTCSat3measuringpoints.
Finally, use each of natural frequency to excite TCS at loosenboltsontheclamping-ring,accordingtothesequence
resonancestate,andgraduallyobtaineachmodeshapewith of loosening 1 bolt, 2 bolts, 3 bolts, and 4 bolts to conduct
obviousreductionintimecostsbylaserrotatingscanmethod. modal parameter test, as seen in Figure 10. It should be
noted that the excitation level and the position of the three
3.2.3.MeasureModalParametersofTCSunderBLBwithDif- measuringpointsmustbethesameastheonesunderfixed
ferent Looseness Numbers. After finishing the measurement constraint boundary, and the same test methods, such as
workunderfixedconstraintboundary,usetorquewrenchto thesmall-segmentFFTprocessingtechnique,thehalf-power
ShockandVibration 9
8
7
er 6
ord 5
al 4
d
o 3
M
2
Figure10:Schematicofboltloosenesssequence. 1
800 1000 1200 1400 1600 1800 2000
Natural frequency (Hz)
bandwidthtechnique,andthelaserrotatingscantechnique,
are employed to get natural frequencies, mode shapes, and No looseness Loosen 3 bolts
damping ratios of TCS under BLB with different looseness Loosen 1 bolt Loosen 4 bolts
numbers. Loosen 2 bolts
Figure 11: Scattergram of natural frequencies of TCS under BLB
3.2.4. Measure Modal Parameters of TCS under BLB with withdifferentloosenessnumbers.
Different Looseness Levels. In this step, firstly it is needed
to restore boundary condition to the fixed state, so the
8
same level of tightening torque which is determined by the 7
preexperimentinthefirststepisagainusedtotighteneight 6
er
M8 bolts on the clamping-ring. Then, set the torque value d 5
or
to 75%, 50%, and 25% of this tightening torque, according al 4
to the sequence of 25% looseness, 50% looseness, and 75% od 3
M
loosenesstocarryoutexperimentaltest.Applyingthesame 2
1
excitationlevelandusingthesametestmethod,wecanget
thecorrespondingfrequency,damping,andshaperesultsof 0.1 0.3 0.5 0.7 0.9 1.1 1.3
TCSunderBLBwithdifferentloosenesslevels. Damping ratio (%)
No looseness Loosen 3 bolts
4.TheInfluenceAnalysisofModalParameters
Loosen 1 bolt Loosen 4 bolts
ofTCSunderBoltLoosenessBoundary Loosen 2 bolts
Figure12:ScattergramofdampingratiosofTCSunderBLBwith
In this section, on the basis of both theoretical and exper-
differentloosenessnumbers.
imental results, that is, the simulation results calculated by
FEMinSection2andtheaccuratemeasureddataobtained
bythetestsystemandtestmethodinSection3,theinfluence
onnaturalfrequencies,modeshapes,anddampingratiosof theclampedendoftheshell(restrictedbytheheightofDC
TCSundertwotypesofboltloosenessboundariesisanalyzed gearedmotoritselfbutdonotaffectthetestresultswhenthe
anddiscussedindetail. numberofaxialhalf-waves𝑚=1).
The measured frequency, damping, and shape results
under BLB with different looseness numbers are listed in
4.1.TestResultsofModalParametersofTCSunderBLBwith
Tables 7, 8, and 9, respectively. Besides, in order to clearly
DifferentLoosenessNumbers. Accordingtotheproposedtest
describe the effect degree and trend of the shell under
method and procedures in Section 3, point 1, point 2, and
∘ different bolt looseness boundaries, the scattergrams of
point 3 are used as the response points, which are 180
natural frequencies and damping ratios of TCS related to
with each other and in the same cross section of the shell
different bolt looseness number are also given, as shown in
the axial distance from this section to free end of TCS is
Figures 11 and 12, and Figure 13 gives the relations between
about 5mm, as seen in Figure 7. For the natural frequency
naturalfrequencyandmodeshapeunderBLBwithdifferent
and damping test, the following setups and parameters are
loosenessnumbers.
chosen: (I) excitation level of 1g; (II) sweep rate of 1Hz/s;
(III)frequencyresolutionof0.125Hz;(IV)Hanningwindow
forsweepresponsesignalwithupwardsweepdirection;(V) 4.2.TheInfluenceAnalysisofModalParametersunder
frequencyrangeof0–2048Hz.Forthemodeshapetest,the BLBwithDifferentLoosenessNumbers
following setups and parameters are chosen: (I) excitation
level of 1g˜3g; (II) frequency resolution of 0.125Hz; (III) 4.2.1. The Influence on Natural Frequencies of TCS. From
rectangularwindowforstableresponsesignal;(IV)sampling Table 7 and Figure 11, the following can be found. (I) For
frequency of 12800Hz; (V) rotated scan speed of 2r/min. the most modes of TCS, the increase of bolt looseness
Besides,thefirst 8modeshapesofTCSareobtainedinthe number would reduce the natural frequencies of TCS, but
test, and each mode shape is assembled from two sets of if only 1 bolt is loosened, the decreased degree of natural
cross-sectional scans; one is in the section which includes frequenciesofTCSisnotobvious.Withtheincreaseofbolt
point1,point2,andpoint3andtheotherisabout25mmto loosenessnumber,thedecreaseddegreeoffrequencyvalues
10 ShockandVibration
Table7:NaturalfrequenciesofTCSunderBLBwithdifferentloosenessnumbers.
Loosen Loosen Loosen Loosen Effect
Nolooseness Effectdegree Effectdegree Effectdegree
Mordoedral 𝐴 1b𝐵olt (𝐵−𝐴)/𝐴 2b𝐶olts (𝐶−𝐴)/𝐴 3b𝐷olts (𝐷−𝐴)/𝐴 4b𝐸olts (𝐸d−eg𝐴re)/e𝐴
(Hz) (%) (%) (%)
(Hz) (Hz) (Hz) (Hz) (%)
1 906.3 888.7 −1.9 857.9 −5.3 857.5 −5.4 827.9 −8.6
2 980.8 980.0 0 947.1 −3.4 920.6 −6.1 886.5 −9.6
3 1025.7 1051.2 2.5 1021.0 −0.5 951.8 −7.2 927.9 −9.5
4 1072.3 1175.3 9.6 1147.0 7.0 1042.2 −2.8 1045.6 −2.5
5 1274.0 1218.6 −4.3 1235.0 −3.1 1145.1 −10.1 1143.8 −10.2
6 1312.8 1309.1 −0.3 1292.6 −1.5 1236.2 −5.8 1233.3 −6.1
7 1613.8 1612.2 −0.1 1613.0 0 1613.2 0 1612.3 −0.1
8 1996.3 1990.2 −0.3 1989.1 −0.4 1988.0 −0.4 1989.6 −0.3
Table8:DampingratiosofTCSunderBLBwithdifferentloosenessnumbers.
Loosen Loosen Loosen Loosen
Nolooseness Effectdegree Effectdegree Effectdegree Effectdegree
Mordoedral 𝐴 1b𝐵olt (𝐵−𝐴)/𝐴 2b𝐶olts (𝐶−𝐴)/𝐴 3b𝐷olts (𝐷−𝐴)/𝐴 4b𝐸olts (𝐸−𝐴)/𝐴
(%) (%) (%) (%) (%)
(%) (%) (%) (%)
1 0.20 0.13 −35.0 0.12 −40.0 0.18 −10.0 0.19 −5.0
2 0.76 0.37 −51.3 0.35 −53.9 0.31 −59.2 0.34 −55.3
3 1.14 0.53 −53.5 0.69 −39.5 0.76 −33.3 0.41 −64.0
4 0.27 0.89 229.6 0.81 200.0 0.82 203.7 0.85 214.8
5 0.74 1.23 66.2 0.99 33.8 1.40 89.2 1.36 83.8
6 0.37 0.22 −40.5 0.24 −35.1 0.85 129.7 0.91 145.9
7 0.39 0.33 −15.4 0.11 −71.8 0.15 −61.5 0.17 −56.4
8 0.30 0.28 −6.7 0.21 −30.0 0.21 −30.0 0.15 −50.0
9 frequencies are basically not affected by BLB with different
ave 8 looseness numbers. (III) For some modes, their natural
ntial wber 67 farreeqrueeanchcieeds(wloiollsegnoinugp1∼w2hbenolttsh)e;folorwexeramlopolese,ntheessmnauxmimbuemrs
ereum 5 increased degree of the 3rd and the 4th naturalfrequencies
mfn 4
u of TCS can increase to 9.6%. Because the corresponding
c 3
Cir 2 shaperesultsarenotchanged,theincreasedfrequencyvalues
800 1000 1200 1400 1600 1800 2000 mightbecausedbythechangesofnonlinearstiffnessinbolt
Natural frequency (Hz) loosenesspositions.
No looseness Loosen 3 bolts
Loosen 1 bolt Loosen 4 bolts
Loosen 2 bolts 4.2.2.TheInfluenceonDampingRatiosofTCS. FromTable8
and Figure 12, the following can be found. (I) For most
Figure13:Therelationbetweennaturalfrequencyandmodeshape
modesofTCS,boltloosenesswouldreducethedampingof
underBLBwithdifferentloosenessnumbers.
TCS. For example, for different bolt looseness number, the
decreasedrangeofdampingresultsisabout5%∼70%,butthe
decreaseddegreeofdampingisnotproportionaltothebolt
is becoming more obvious than the ones under the fixed loosenessnumber.(II)ForasmallpartofmodesofTCS,the
state. For example, the 1st and 2nd natural frequencies are dampingresultswillrisewiththeincreaseofboltlooseness
decreased nearly to 10%. (II) The increase of bolt looseness number, especially when the looseness number goes up to
number would also result in the decrease of high order of someextent,forexample,loosening3∼4bolts;theresulting
natural frequencies, but the decreased degree is very small. dampingvaluesrelatedtothe4thandthe6thmodearenearly
Takingthe7thand8thnaturalfrequenciesforexample,they 1∼2 times larger than the ones under no loose condition,
onlydecreaseto0.4%comparedwiththeonesunderthefixed whichmaybelargelycausedbythechangesofmodeshape
state.Therefore,itcanbeconcludedthathighordernatural orincreasedinterfacefrictioninboltloosenesspositions.