International Journal of Acoustics and Vibration

102 Pages·2016·19.29 MB·English

International Journal of
EDITORIALOFFICE
Acoustics and Vibration
EDITOR-IN-CHIEF
MalcolmJ.Crocker
AquarterlypublicationoftheInternationalInstituteofAcousticsandVibration
MANAGINGEDITOR
MarekPawelczyk
Volume 21, Number 2, June 2016
ASSOCIATEEDITORS
DariuszBismor
NickolayIvanov
EDITORIAL
ZhuangLi
TheICSV23—fromAncienttoModernAcoustics
ASSISTANTEDITORS
KonstantinosVogiatzis . . . . . . . . . . . . . . . . . . . . . . . 136
TeresaGlowka
SebastianKurczyk
ARTICLES
KarolJablonski
Dynamic Stability Analysis of a Circularly Tapered Rotating Beam
EDITORIALASSISTANT
SubjectedtoAxialPulsatingLoadandThermalGradientunderVarious
JillGloverRoberts
BoundaryConditions
MacieFranklin RashmitaParidaandPusparajDash . . . . . . . . . . . . . . . . . . . 137
Method for Evaluating the Statistical Relationship between Sound
Pressure Level and Noise Annoyance Based on a Nonlinear Time Series
RegressionModelandanExperiment
HisakoOrimoto,AkiraIkutaandYeguiXiao . . . . . . . . . . . . . . . . 145
EDITORIALBOARD
Multi-objective Optimization of a Multi-chamber Perforated Muffler
JorgeP.Arenas UsinganApproximateModelandGeneticAlgorithm
Valdivia,Chile ShuguangZuo,KaijunWeiandXudongWu . . . . . . . . . . . . . . . . 152
JonathanD.Blotter Developing Vibration Equations of an Orthotropic Wrapped Shell,
Provo,USA ConsideringResidualStressEffects;AMathematicalApproach
AmiraliSadeqi,MojtabaMahzoonandMohammadHassanKadivar . . . . . . 164
LeonidGelman
Vibration Analysis of Cracked Beams Using Adomian Decomposition
Cranfield,UK
MethodandNon-BaselineDamageDetectionviaHigh-PassFilters
QiboMao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
SamirGerges
Floriano´polis,Brazil Vibration Analysis of Non-homogenous Orthotropic Visco-elastic
RectangularPlateofParabolicallyVaryingThicknesswithThermalEffect
ColinH.Hansen PoojaSinghalandArunKumarGupta . . . . . . . . . . . . . . . . . . . 178
Adelaide,Australia
CharacterisationofMajorFaultDetectionFeaturesandTechniquesforthe
HughHunt Condition-BasedMonitoringofHigh-speedCentrifugalBlowers
SamerGowid,RogerDixonandSaudGhani . . . . . . . . . . . . . . . . 184
Cambridge,England
Experimental and Theoretical Approach to Generalized Empirical Data-
DanMarghitu
basedModelofNoiseinCeilingFan
Auburn,USA RupeshV.BhortakeandBimeshKumar . . . . . . . . . . . . . . . . . . 192
ManoharLalMunjal AnInvestigationofInitialShockCellFormationinTurbulentCoandaWall
Bangalore,India Jets
CarolineP.Lubert,ChristianR.SchwantesandRichardJ.Shafer . . . . . . . 199
KazuhideOhta
Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of
Fukuoka,Japan
Thin Circular Functionally Graded Plates Using Iterative and Explicit
GoranPavic AnalyticalSolution
RachidElKaak,KhalidElBikriandRhaliBenamar . . . . . . . . . . . . 209
Villeurbanne,France
Two-Temperature Generalized Thermoelastic Infinite Medium with
SubhashSinha
CylindricalCavitySubjectedToTimeExponentiallyDecayingLaserPulse
Auburn,USA EmanA.N.Al-Lehaibi . . . . . . . . . . . . . . . . . . . . . . . . . 222
AbouttheAuthors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
INFORMATION
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KonstantinosVogiatzis
vibration, as well as environmental noise, vibration, sound-
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136 http://dx.doi.org/10.20855/ijav.2016.21.1E80 InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016
Dynamic Stability Analysis of a Circularly Tapered
Rotating Beam Subjected to Axial Pulsating Load
and Thermal Gradient under Various Boundary
Conditions
RashmitaParida
DepartmentofMechanicalEngineering,CollegeofEngineeringandTechnology,BhubaneswarTechnoCampus,
Ghatikia,Bhubaneswar,Orissa,India-751003
PusparajDash
DepartmentofMechanicalEngineering,VSSUTBurla,Sambalpur,Orissa,India-768018
(Received13August2013;accepted8October2015)
Thedynamicstabilityofacircularlytaperedrotatingbeamsubjectedtoapulsatingaxialexternalexcitationwith
thermalgradientwasstudiedforallpossiblecombinationsofclamped, guided, pinned, fixed, andfreeboundary
conditions. TheequationsofmotionandassociatedboundaryconditionswereobtainedusingtheextendedHamil-
ton’sprinciple.Thentheseequationsofmotionandtheassociatedboundaryconditionswerenon-dimensionalised.
A set of Hill’s equations were obtained from the non-dimensional equations of motion by the application of the
extendedGalerkinmethod. ThezonesofparametricinstabilitywereobtainedusingSaito-Otomiconditions. The
effectsofvariousboundaryconditions,thermalgradient,taper,androtationalspeedontheregionsofparametric
instability were investigated and presented through a series of graphs. The results reveal that increasing rota-
tional speed and taper have stabilizing effects, whereas increasing thermal gradient has a destabilizing effect for
allboundaryconditionsofthebeam.
NOMENCLATURE Ω UniformangularvelocityΩofthe
beamaboutz’-axis
A(x),A(ξ) Areaofagenericsectionofthebeam
Ω Rotationalspeedparameter
0
A Crosssectionalareaattheendx=l
1 ω Excitationfrequency
C Hubradius
0 ω Dimensionlessfundamentalnaturalfrequency
0
c Dimensionlesshubradius,=C /l
0 0 Θ Non-Dimensionalexcitationfrequency,=ω/c
d(x),d(ξ) Diameterofagenericsectionofthebeam
Ψ Referencetemperature
0
d Diameterattheend,x=l
1 Ψ Temperatureattheend,ξ =1
1
E(x),E(ξ) Young’smodulusatagenericsection
E Young’smodulusattheend,x=l
1
I(x),I(ξ) Momentofinertiaatagenericsection 1. INTRODUCTION
I Momentofinertiaattheend,x=l
1
The stability analysis and dynamic behaviour of a rotating
l Lengthofthebeam
cantileverbeamwithaxialorientationperpendiculartotheaxis
m(ξ) Massdistributionfunction
of spin is very essential for its practical applications such as
P Staticaxialload
0
turbomachineryblades,rotorbladesofhelicopter,aircraftpro-
P Dynamicaxialload
1
pellers, flexible appendages of spacecraft, satellite antennas,
p(τ) Dimensionlessload
and robotic manipulators, to name a few. In some cases the
p Dimensionlessstaticaxialload
0
structureshavetooperateunderelevatedtemperatures. Alin-
p Dimensionlessdynamicaxialload
1
earrelationisobservedbetweentheYoung’smodulusandthe
S(ξ) Momentofinertiadistributionfunction
temperatureofmostengineeringmaterials.
T(ξ) Elasticitymodulusdistributionfunction
t Time Anamplenumberofpublicationsareavailableregardingthe
w(x,t) Transversedeflectionofthebeam designandanalysisofrotatingstructures. Theflexuralvibra-
γ Coefficientofthermalexpansionofthe tions of a rotating cantilever beam with a tip mass at the free
beammaterial endhasbeenstudiedbyBhat.1 Heproposedthebeamcharac-
δ Thermalgradientparameter teristic orthogonal polynomials in the Rayleigh-Ritz method.
a∗ Diametertaperparameter LiuandYeh2 haveinvestigatedtheinfluenceofrestrainedpa-
η Dimensionlesstransversedeflection,=w/l rameters on the Eigen frequencies of rotating uniform and
ξ Dimensionlesslength,=x/l non-uniform beams with a restrained base using Galerkin’s
τ Dimensionlesstime,=ct method. The stability analysis of a rotating shaft due to pul-
ρ Densityofthebeammaterial satingtorqueappliedatitsendhasbeenstudiedbyUngerand
InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016 (pp. 137(cid:21)144) http://dx.doi.org/10.20855/ijav.2016.21.2402 137
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
Brull.3 They found that instability arising from the combi-
nation resonance has the most adverse effect. Kammer and
Schlack4adoptedtheKrylov-Bogoliubov-Mitropolskii(KBM)
perturbation techniqueto solve theproblem of instabilitydue
to the time dependent angular velocity in a rotating Euler
beam. Namachchivaya5 investigated the dynamic stability of
rotating shaft under the excitation ofcombined harmonic and
stochastic load and derived the stability conditions explicitly
for the first and second order moments considering the shaft
as a two degrees of freedom system. Bauer and Eidel6 stud-
ied the effects on vibration and buckling of a rotating Euler
Figure1.SystemConfiguration.
beam of uniform cross section because of its spin speed, hub
radiusandaspectratioconsideringanorientationperpendicu- dimensionaltemperaturegradientatthefreeendhasbeenre-
lar to the axis of rotation. The dynamic stability analysis of ported. Theeffectoftherotationparameter,geometricparam-
rotating Timoshenko beam with a root flexibility using finite eters, taperparameter, andthethermalgradientonthenondi-
elementmethodwasinvestigatedforthefirsttimebyAbbas.7 mensionalstaticbucklingloadszonesandalsoontheparamet-
Ishida et al.8 studied the vibration and stability of a rotating ricinstabilityzonesareinvestigated.
shaftunderasinusoidalaxialforceassumingafourdegreesof
freedomsystem. Thestabilityofataperedcantileverbeamon 2. FORMULATION OF THE PROBLEM
Winkler foundation subjected to a follower force was studied
byLee.9 Hefoundthatthecriticalflutterloadsofbothtapered 2.1. System Configuration
beamsandbeamsofauniformcrosssectionareunaffectedby
Arotatingtaperedcantileverbeamoflengthl setoffadis-
thepresenceofviscousdampingintheelasticfoundation. Lin
tance C from the axis of rotation which rotates at a uniform
andChen10studiedthedynamicstabilityofrotatingcomposite 0
angularvelocityΩaboutaverticalz’-axisiscapableofoscil-
beamsusingfiniteelementmethod. Tanetal.12 discussedthe
latinginthex–z plane. Thebeamisorientedalongthex-axis
instability of a spinning pre-twisted beam under compressive
perpendiculartotheaxisofrotationasshowninFig.1. Apul-
axialloadsassumingEuler-Bernoullibeamtheory(EBT)and
satingaxialforceP(t) = P +P cosωtisappliedattheend
assumedmodemethod. Banerjeeetal.12 appliedthedynamic 0 1
x = C +l ofthebeamalongthepointofC.G.ofthecross-
0
stiffnessmethodtoanalysethefreevibrationofarotatingtaper
sectionintheaxialdirection,withωbeingthefrequencyofthe
beamthatfollowsEBT.Theyderivedsomeexplicitanalytical
appliedload, tbeingthetime, andP andP beingthestatic
0 1
expressions and used the Wittrick-Williams algorithm for the
anddynamicloadamplitudes,respectively.
solution. ShahbaandRajasekaran13usedthedifferentialtrans-
Thefollowingassumptionsaremadeforderivingtheequa-
formelementmethod(DTEM)anddifferentialquadratureel-
tionsofmotion:
ement method (DQEL) of lower order to solve the equations
offreevibrationandstabilityoftaperedEuler-Bernoullibeam a) Thematerialofthebeamishomogeneous&isotropicin
madeofaxiallyfunctionallygradedmaterial. Yangetal.14 de- nature.
veloped a finite element model to study the free vibration of
b) Thedeflectionsofthebeamaresmallandthetransverse
arotatinguniformEuler-Bernoullibeam. Theyconsideredthe
deflection w(x,t) is the same for all points of a cross-
couplingofaxialandtransversevibrationandofelasticdefor-
mationsandrigidmotion. Nayaketal.15 investigatedthesta- section.
bility of a sandwich beam on viscoelastic supports subjected
c) ThebeamobeysEuler-Bernoullibeamtheory.
to a pulsating axial load with temperature gradient. Soltani
et al.16 proposed a numerical solution based on power series d) Extensionaldeflectionofthebeamisneglected.
method to derive the critical buckling loads and frequency of
freevibrationsfortaperedthinbeams. Bulut17 studiedthedy- e) A steady one-dimensional temperature gradient is as-
sumedtoexistalongthecentrallengthofthebeam.
namicstabilityofparametricallyexcitedrotatingtaperedbeam
and found that the number of instability zones increases with
f) Extensionandrotaryinertiaeffectsarenegligible.
thetaperratio.
Asurveyofliteraturerevealsthatsomeworkhasbeendone Theexpressionsforpotentialenergy,kineticenergyandwork
onparametricinstabilityanddynamicstabilityofasymmetric doneareasfollows:
rotating beam, parametric instability of a non-uniform beam
l
with thermal gradient resting on a Pasternak foundation, and 1(cid:90) (cid:18)∂2w(cid:19)2
V = E(x)I(x) dx
thatofasymmetricsandwichbeamfordifferentboundarycon- 2 ∂x2
ditions. However, no work has been done to study the static 0
anddynamicstabilityofarotatingtaperedbeamwiththermal l x(cid:48)
1(cid:90) (cid:90) (cid:18)∂w(cid:19)2
gradientundervariousboundaryconditions. Thusthepresent + ρA(x)Ω2(C +x) dx; (1)
2 0 ∂x
work mainly deals with a theoretical study of a rotating ta-
0 0
pered beam with a pulsating load and thermal gradient under
various boundary conditions. The static and dynamic stabil- 1(cid:90)l (cid:18)∂w(cid:19)2 1(cid:90)l
ityofarotatingtaperedcantileverbeamwhichisfixedatone T = ρA(x) dx+ ρΩ2A(x)w2dx; (2)
2 ∂t 2
endandsubjectedtoanaxialpulsatingloadandasteady,one-
0 0
138 InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
1 l (cid:18)∂w(cid:19)2 where f (τ) is an unknown function of time and η (ξ) is a
W = ∫P(t) dx; (3) r r
P 2 ∂x coordinate function to be chosen so as to satisfy as many of
0
the boundary conditions in Eq. (8) as possible. It is further
wherew(x,t)istransversedeflectionofthebeam.
assumedthatη (ξ)canberepresentedbyasetoffunctions(9)
r
The application of the extended Hamilton’s principle gives
whichsatisfytheconditionsobtainedfromEq.(7)bydeleting
thefollowingequationofmotionandboundaryconditions:
the terms containing ω and p(τ). It is further assumed that
0
coordinate functions for the various boundary conditions can
t2
δ∫(T −V −W )=0; (4) beapproximatedbytheonesgiveninTable1.
P
t1 Substitutionoftheseriesofsolutionsinthenondimensional
equationsofmotionandsubsequentapplicationofthegeneral
[E(x)I(x)w ] +ρA(x)w +ρΩ2I(x)w Galerkin method leads to the following matrix equations of
,xx ,xx .tt ,xx
motion:
−[N(x )w ] +P (t)w =0; (5)
1 ,x ,x ,xx (cid:110) (cid:111)
[M] f¨ +[K]{f}−{p [H]−p cosθτ[H]}{f}
0 1
(cid:104) (cid:105)
whereN(x ) = 1ρA(x)Ω2 (C +l)2−(C +x(cid:48))2 . The
1 2 0 0 ={0}; (10)
boundaryconditionsatx=C andx=C +lare
0 0
wheref¨= ∂2f and{f}={f ,............,f }T.Thevar-
∂τ2 1 N
[E(x)I(x)w,xx],x+P (t)w,x =0; (cid:82)1
ious matrix elements are given by m(ξ)η (ξ)η (ξ)dξ =
i j
[E(x)I(x)w ] =0;
,xx x=l 0
M ;
w =0. (6) ij
,t (cid:82)1(cid:26) S(ξ)T (ξ)ηi(cid:48)(cid:48)(ξ)ηj(cid:48)(cid:48)(ξ)+ (cid:27)dξ =K ;
In the above expression w,x = ∂∂wx, w,xx = ∂∂2xw2, w,t = ∂∂wt, 0 Ω20[q(ξ)−rg]ηi(cid:48)(ξ)ηj(cid:48)(ξ) ij
w,tt = ∂∂2tw2 . (cid:82)1ηi(cid:48)(ξ)ηj(cid:48)(ξ)dξ =Hij;and
Introducingthedimensionlessparameters, 0
ξ = xl,η = wl,c0 = Cl0,τ =ct; ∵i,j =1,2,.........,N.
(cid:16) (cid:17)
∵c2 = Eρ(Ax()xI)(lx4) ; 2.3. Regions of Dynamic Instability
∂w = ∂η and(cid:0)∂w(cid:1)2 =(cid:16)∂η(cid:17)2; Let [L] be the modal matrix of [M]−1[K]. Then by the
∂x ∂ξ ∂x ∂ξ
(cid:16) (cid:17)2 (cid:16) (cid:17)2 introduction of the linear coordinate transformation, {f} =
∂∂2xw2 = 1l ∂∂2ξη2 and ∂∂2xw2 = l12 ∂∂2ξη2 ; [L]{v},{v}beinganewsetofgeneralizedcoordinatesyields,
∂w =cl(cid:16)∂η(cid:17)and(cid:0)∂w(cid:1)2 =c2l2(cid:16)∂η(cid:17)2; {v¨}+(cid:2)ω2(cid:3){v}+p cos(θτ)[B]{v}={0}; (11)
∂t ∂τ ∂t ∂τ n 1
p(τ)= PE(=1tI)l12,p(τ)=p0+p1cosθτ; where[ωn2]isaspectralmatrixcorrespondingto[m]−1[k]and
()(cid:48) = ∂(),()∂(),etc. [B]=−[L]−1[M]−1[H][L].
∂ξ ∂τ
Thenondimensionalequationofmotionandboundarycondi- Equation(11)canbewrittenas,
tionscanbewrittenas
m=N
(cid:88)
[S(ξ)T (ξ)η(cid:48)(cid:48)](cid:48)(cid:48)+m(ξ)η¨+(cid:2)r Ω2+p(τ)(cid:3)η(cid:48)(cid:48) v¨n+ωn2vn+p1cos(θτ) bnmun =0,
g 0 m=1
−Ω2[q(ξ)η(cid:48)](cid:48) =0; (7) n=1,2,......,N (12)
0
and Equation(12)representsasystemofN coupledHill’sequa-
tionswithcomplexcoefficients.
{[S(ξ)T (ξ)η(cid:48)(cid:48)](cid:48)+p(τ)η(cid:48)}ξ=1 =0; Here, ωn and bnm are complex quantities, given by ωn =
[S(ξ)T (ξ)η(cid:48)(cid:48)] =0; ωn,R+jωn,I;bn,m =bnm,R+jbnm,I.
ξ=1
The boundaries of the region of instability of simple and
η(0,τ)=0;
combinationresonancesareobtainedusingthefollowingcon-
η(cid:48)(0,τ)=0. (8) ditionsbySaito&Otomi.18
Intheaboveexpressionsr = I(ξ);Ω2 = ρA(ξ)Ω2l4; Case(A):Simpleresonance
g A1l2 0 E1I1
Ω2q(ξ) = N(x1)l2; A(ξ) = A m(ξ),E(ξ) = E T (ξ), Inthiscase,theregionsofinstabilityaregiven:Whendamp-
0 E1I1 1 1 ingispresent.
I(ξ)=I S(ξ).
1
(cid:118) (cid:16) (cid:17)
2.2. Approximate Solution (cid:12)(cid:12)(cid:12)(cid:12)2θ −ωµ,R(cid:12)(cid:12)(cid:12)(cid:12)< 41(cid:117)(cid:117)(cid:116)P21 b2µµω,R2 +b2µµ,I −16ωµ2,I. (13)
Theapproximatesolutiontothenondimensionalequations µ,R
ofmotionisassumedas
And,fortheundampedcase,
η(ξ,τ)=(cid:88)N ηr(ξ)fr(τ); (9) (cid:12)(cid:12)(cid:12)(cid:12)2θ −ωµ,R(cid:12)(cid:12)(cid:12)(cid:12)< 41(cid:12)(cid:12)Pω1bµµ,R(cid:12)(cid:12). (14)
r=1 µ,R
InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016 139
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
Table1.Coordinatefunctions.
Endarrangement Coordinatefunctioni=1,2,......,r
P-P η(ξ)=sin(πiξ)
G-P η(ξ)=cos{(2i−1)πξ/2}
C-P η(ξ)=2(i+2)ξ(i+1)−(4i+6)ξ(i+2)+2(i+1)ξ(i+3)
C-C η(ξ)=ξ(i+1)+2ξ(i+2)+ξ(i+3)
C-CUR η(ξ)=(i+3)(i+2)2(i+1){ξ(i+1)−2ξ(i+2)+ξ(i+3)}
C-F η(ξ)=(i+2)(i+3)ξ(i+1)−2i(i+3)ξ(i+2)+i(i+1)ξ(i+2)
forµ=1,2,......,N.
Case(B):Combinationresonanceofthesumtype
This type of resonance occurs when µ (cid:54)= v; µ,v =
1,2,......,N andtheregionsofinstabilityaregiven:
Forthedampedcase,
(cid:12) (cid:12)
(cid:12)ω 1 (cid:12)
(cid:12)(cid:12)2 − 2(ωµ,R+ωv,R)(cid:12)(cid:12)<
(cid:118)
ω√µ,I +ωv,I(cid:117)(cid:117)(cid:116) ωµ,RPω21v,R (bµv,Rbvµ,R+bµv,Ibvµ,I) . (15)
8 ωµ,Iωv,I −16ωµ,Iωv,I
Fortheundampedcase,
(cid:115)
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)ω2 − 12(ωµ,R+ωv,R)(cid:12)(cid:12)(cid:12)< P41 bωµv,Rbωvµ,R. (16) αFi∗gu=re2)2.. Stabilitydiagramforpinned-pinnedbeamwithΩ0 =5(δ =0.1,
µ,R v,R
Consequently, the mass distribution m(ξ) and the moment
ofinertiadistributionS(ξ)aregivenbytherelations
Case(C):Combinationresonanceofthedifferencetype
m(ξ)=[1+α∗(1−ξ)]2;
This type of resonance occurs when µ < v, (µ,v =
S(ξ)=[1+α∗(1−ξ)]4.
1,2,......,N)andtheregionsofinstabilityaregiven:
Forthedampedcase, The temperature above the reference temperature at any
point ξ from the end of the beam is assumed to be Ψ =
(cid:12) (cid:12)
(cid:12)ω 1 (cid:12) Ψ (1−ξ). Choosing Ψ = Ψ , the temperature at the end
(cid:12)(cid:12)2 − 2(ωv,R−ωµ,R)(cid:12)(cid:12)< ξ 0= 1, as the reference0tempera1ture, means the variation of
(cid:118) modulusofelasticityofthebeamisdenotedby
ω√µ,I +ωv,I(cid:117)(cid:117)(cid:116) ωµ,RP¨1ω2v,R (−bµv,Rbvµ,R+bµv,Ibvµ,I) . E(ξ)=E1[1−γΨ1(1−ξ)],0(cid:54)γΨ1 <1;
8 ωµ,Iωv,I −16ωµ,Iωv,I E(ξ)=E1T (ξ),
where γ is the coefficient of thermal expansion of the beam
(17)
material, δ = γΨ is the thermal gradient parameter, and
1
Fortheundampedcase, T (ξ)=[1−δ(1−ξ)],whereδisthethermalgradientalong
thelengthofthebeam.
(cid:115)
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)ω2 − 12(ωv,R−ωµ,R)(cid:12)(cid:12)(cid:12)< P41 −ωbµµv,R,Rωbvv,µR,R (18) bouTnhdeardyyncoamndiictiostnasbhilaitsybeaennalaynsiaslyosfedthaesfsoylslotewms:for various
Dynamic stability analysis of the circularly tapered rotating If, for the change in value of a parameter, the width of the
beam with axial pulsating load and thermal gradient under instabilityzoneincreasesorthezoneispulleddownorshifted
various boundary conditions has been carried out by using towardsalowerexcitationfrequencyregion,thestabilityofthe
Eqs. (14), (16), and (18). From them, regions of instability systemworsens. Otherwise,ifwiththechangeinthevalueof
areobtainedforvariouscases. the parameter, the width of the instability zone decreases or
ispulleduporshiftedtowardsthehigherexcitationfrequency
3. NUMERICAL RESULTS AND region, orifthenumberofinstabilityzonesreduces, thenthe
DISSCUSSIONS stabilityofthesystemimproves. Withtheseaboveconditions,
theeffectofvariousparameteronthedynamicstabilityofthe
Numerical results were obtained for various values of the systemhavebeenanalysed.
parameters like rotation parameter, geometric parameter, ta-
The regions of parametric instability of a beam with vari-
per parameter, and thermal gradient. The linearly tapered
ousboundaryconditionsfortwodifferentvaluesofrotational
cantilever beam with a circular cross-section is assumed to
speedparametersareshowninFigs.2,3,5,6,8,9,11,12,14,
have a diameter varying according to the relation d(ξ) =
15,17,and18.
d [1+α∗(1−ξ)],
1
whered isthediameterofthebeamattheendξ =1,andα∗
1
isthediametertaperparameter.
140 InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
Figure3.Stabilitydiagramforpinned-pinnedbeamwithΩ0 =15(δ=0.1, Figure6. Stabilitydiagramforguided-pinnedbeamwithΩ0 =5(δ =0.1,
α∗=2). α∗=2).
Figure4. Stabilitydiagramforpinned-pinnedbeamwithΩ0 =5(δ =0.4, Figure7. Stabilitydiagramforguided-pinnedbeamwithΩ0 =5(δ =0.4,
δ=0.8,α∗=1). δ=0.8α∗=1).
Figure5. Stabilitydiagramforguided-pinnedbeamwithΩ0 =2(δ =0.1, Figure8.Stabilitydiagramforclamped-pinnedbeamwithΩ0=5(δ=0.1,
α∗=2). α∗=2).
InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016 141
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
Figure9.Stabilitydiagramforclamped-pinnedbeamwithΩ0=15(δ=0.1, Figure 12. Stability diagram for clamped-clamped beam with Ω0 = 15
α∗=2). (δ=0.1,α∗=2).
Figure10.Stabilitydiagramforclamped-pinnedbeamwithΩ0=5(δ=0.4, Figure 13. Stability diagram for clamped-clamped beam with Ω0 = 5
δ=0.8,α∗=1). (δ=0.4,δ=0.8,α∗=1).
Figure 11. Stability diagram for clamped-clamped beam with Ω0 = 5 Figure14.Stabilitydiagramforclamped-CURbeamwithΩ0 =5(δ=0.1,
(δ=0.1,α∗=2). α∗=2).
142 InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
Figure15.Stabilitydiagramforclamped-CURbeamwithΩ0=15(δ=0.1, Figure17. Stabilitydiagramforclamped-freebeamwithΩ0 =5(δ =0.1,
α∗=2). α∗=2).
δFi=gu0re.81,6α.∗S=tab1i)li.tydiagramforclamped-CURbeamwithΩ0 =5(δ=0.4, αFi∗gu=re21).8.Stabilitydiagramforclamped-freebeamwithΩ0 =15(δ=0.1,
clamped-free conditions, whereas it has a destabilizing effect
Combination resonances of the difference-type do not oc-
onguided-pinnedandclamped-pinnedbeams.
curinanyofthecasesunderconsideration. Whileanincrease
The influence of taper parameter and thermal gradient pa-
in the value of Ω reduces the width of the first-order simple
0 rameter on the principal region of instability is shown in
resonance zones of clamped-clamped beam, it widens some
Figs. 4, 7, 10, 13, 16, and 19. The figures show the effect
of the combination resonance zones and shifts all the regions
oftwovaluesofthethermalgradientparameterδontheinsta-
to higher excitation frequencies. The combination resonance bilityregionsfortaperparameterα∗ =1foralltheconsidered
regions at Θ = (ω + ω ) of a clamped-clamped beam re-
3 1 boundary conditions. It has been observed that, for all cases,
duce in width due to the increase in angular velocity. For a
theinstabilityregionsexperienceaslightincreaseinwidthand
beam with clamped-free end conditions, a higher rotational
shift towards lower excitation frequencies with an increase in
speed reduces the span of most of the instability regions and
thevalueofδ.
makesthebeamlesssusceptibletoperiodicforcesbyrelocat-
ingthemathigherfrequencies,whereasariseintherotational
4. CONCLUSION
speedincreasesthespanofmostoftheinstabilityregionsofa
clamped-pinnedbeamandrelocatesthemathigherexcitation A computational analysis of the dynamic stability of a ta-
frequencies.Ontheotherhand,itrepositionsthosearound2ω1 pered cantilever beam with pulsating axial load and thermal
and (ω1 +ω2) at lower frequencies and reduces the width of gradient under various boundary conditions was considered.
the ones at Θ = 2ω2, 2ω3 and (ω3 +ω1). With increase in The programming was developed using MATLAB. The fol-
the angular velocity of pinned-pinned beam, most of the res- lowingaretheconclusionsdrawnfromthestudy.
onance zones are widened, but those near Θ = 2ω and 2ω
2 3 The dynamic stability of a rotating tapered beam under a
are reduced in span. Further, while the instability regions in
pulsating axial load was investigated for all possible combi-
thevicinitiesof2ω shiftstolowerfrequencies,allothersare
1 nations of clamped, guided, pinned, and free boundary con-
repositionedathigherones.
ditions. Results reveal that, a higher rotational speed make a
From the figures, it is observed that increase in rota- clamped-freebeamlesssensitivetoperiodicforces.Whilerise
tional speed parameter stabilizes the beams with pinned- intheangularvelocityreducestheintensityofthesimpleres-
pinned,clamped-clamped,clamped-clampedunrestrained,and onancesofclamped-clampedbeam,itincreasestheseverityof
InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016 143
R.Parida,etal.: DYNAMICSTABILITYANALYSISOFACIRCULARLYTAPEREDROTATINGBEAMSUBJECTEDTOAXIALPULSATING...
7 Abbas,B.A.H.DynamicstabilityofarotatingTimoshenko
beamwithaflexibleroot,J.SoundVib.,108,25–32(1986).
http://dx.doi.org/10.1016/s0022-460x(86)80308-x
8 Ishida, Y., Ikeda, T., Yamamoto, T., and Esaka, T. Para-
metricallyExcitedOscillationsofaRotatingShaftUndera
PeriodicAxialForce,JSMEInt.J.,31(4),698–704(1988).
http://dx.doi.org/10.1299/jsmec1988.31.698
9 Lee, H. P. Dynamic stability of a tapered cantilever beam
on an elastic foundation subjected to a follower force,
Int. J. Solids Structures, 33 (10), 1409–1424, (1996).
http://dx.doi.org/10.1016/0020-7683(95)00108-5
10 Lin, C. Y. and Chen, L. W. Dynamic stability of rotating
compositebeamswithaviscoelasticcore,Compos.Struct.,
Figure19. Stabilitydiagramforclamped-freebeamwithΩ0 =5(δ =0.4, 58 (2), 185–194, (2002). http://dx.doi.org/10.1016/s0263-
δ=0.8,α∗=1). 8223(02)00127-7
some of the combination resonances. It is also observed that 11 Tan,T.H.,Lee,H.P.andLeng,G.S.B.Parametricinsta-
clamped-pinnedandpinned-pinnedbeamsmayeitherstabilize
bilityofspinningpretwistedbeamssubjectedtosinusoidal
ordestabilizewithanincreaseinrotationalspeed.
compressiveaxialloads,Compos.Struct.,66(6),745–764,
An increase in taper parameter reduces the widths of the (1998).http://dx.doi.org/10.1016/s0045-7949(98)00002-9
principalregionsofinstabilityandshiftsthemtowardshigher
excitation frequencies, thus making the beam less sensitive 12 Banerjee, J. R., Su, H., and Jackson, D. R. Free vibra-
to periodic forces. However, an increase in thermal gradient tion of rotating tapered beams using the dynamic stiff-
widens the principal regions of instability, shifting them to- nessmethod, J.SoundVib., 298(4–5)1034–1054, (2006).
wardslowerexcitationfrequencies, therebymakingthebeam http://dx.doi.org/10.1016/j.jsv.2006.06.040
moresensitivetoperiodicforces. Thus,itmaybeinferredthat
13 Shahba,A.andRajasekaran,S.Freevibrationandstability
increasingtaperhasstabilizingeffectsonthebeams, whereas
oftaperedEuler–Bernoullibeamsmadeofaxiallyfunction-
increasing temperature gradient has a destabilizing effect on
ally graded materials, Appl. Math. Model., 36 (7), 3094–
thebeamsofallcases.
3111,(2012).http://dx.doi.org/10.1016/j.apm.2011.09.073
REFERENCES 14 Yang, J. B., Jiang, L .J., and Chen, D. C. Dy-
namicmodellingandcontrolofarotatingEuler–Bernoulli
1 Bhat, R. B. Transverse vibrations of a rotating uni- beam, J. Sound Vib., 274 (3–5), 863–875, (2004).
form cantilever beam with tip mass as predicted by us-
http://dx.doi.org/10.1016/s0022-460x(03)00611-4
ing beam characteristic orthogonal polynomials in the
Rayleigh-Ritzmethod,J.SoundVib.,105,199–210,(1986). 15 Nayak,S.,Bisoi,A.,Dash,P.R.,andPradhan,P.K.Static
http://dx.doi.org/10.1016/0022-460x(86)90149-5 stability of a viscoelastically supported asymmetric sand-
wichbeamwiththermalgradient, Int.J.Adv.Struct.Eng.,
2 Liu, W. H. and Yeh, F.-H.Vibrations of non-uniform 6(3),Article:65,(2014).http://dx.doi.org/10.1007/s40091-
rotating beams, J. Sound Vib.,119, 379–384, (1987). 014-0065-2
http://dx.doi.org/10.1016/0022-460x(87)90463-9
16 Soltani, M., Asgarian, B., and Mohri, F. Elastic instability
3 Unger, A. and Brull, M. A. Parametric instabil- andfreevibrationanalysesoftaperedthin-walledbeamsby
ity of a rotating shaft due to pulsating torque, J. thepowerseriesmethod,J.Constr.SteelRes.,96,106–126,
Appl. Mech., Trans. ASME, 48, 948–958, (1981). (2014).http://dx.doi.org/10.1016/j.jcsr.2013.11.001
http://dx.doi.org/10.1115/1.3157761
17 Bulut, G. Effect of taper ratio on paramet-
4 Kammer, D.C.andSchlack, Jr., A.L.DynamicResponse ric stability of a rotating tapered beam, Eur.
of a Radial Beam With Nonconstant Angular Velocity, J. J. Mech. A Solids, 37, 344–350, (2013).
Vib.Acoust.,Stress,Reliab.Des.,Trans.ASME,109,138– http://dx.doi.org/10.1016/j.euromechsol.2012.08.007
143,(1987).http://dx.doi.org/10.1115/1.3269405
18 Saito, H. and Otomi, K. Vibration and stability of elasti-
5 Namachchivaya, N. S. Mean square stability of a ro- callysupportedbeamscarryinganattachedmassunderax-
tating shaft under combined harmonic and stochas- ialandtangentialloads,J.SoundVib.,62,257–266,(1976).
tic excitations, J. Sound Vib., 133, 323–336 (1989). http://dx.doi.org/10.1016/0022-460x(79)90025-7
http://dx.doi.org/10.1016/0022-460x(89)90929-2
6 Bauer, H. F. and Eidel, W. Vibration of a rotating uni-
form beam, Part II: Orientation perpendicular to the
axis of rotation, J. Sound Vib., 122, 357–375 (1988).
http://dx.doi.org/10.1016/s0022-460x(88)80360-2
144 InternationalJournalofAcousticsandVibration,Vol. 21,No. 2,2016