MEASURES AND SYMPTOMS OF THE WEAR MARGIN IN …

QUARTERLYISSN 1232-9312 3/2017 (106)

PROBLEMY EKSPLOATACJI

Journal of MachineC o n s t r u c t i o n and Maintenance

Piotr BIELAWSKIMaritime University of Szczecin, Faculty of Marine Engineering [email protected]

MEASURES AND SYMPTOMS OF THE WEAR MARGIN IN FUNCTIONAL UNIT NODES OF PRODUCTION SYSTEM ITEMS

Key words: wear margin, functional unit, thermal node, hydrodynamic node, tribological node, coefficient k, resistance coefficient, Sommerfeld number.

Abstract: Remedial actions aimed a production system items should be taken after unacceptable differences occur between the desired and actual values of item functional unit characteristic measures. The author justifies the need to use the term ‘wear margin of a functional unit node’. Three basic models of nodes are presented, and selected models for each node are described taking into account their kinematics, the shape of solids, and the physicochemical properties of fluids. Substitute measures of the wear margin and measures of diagnostic symptoms of hydrodynamic, thermal, and tribological nodes are described and justified.

Miary i symptomy potencjału eksploatacyjnego węzłów zespołów funkcjonalnych obiektów systemu produkcyjnego

Słowa kluczowe: potencjał eksploatacyjny, zespół funkcjonalny, węzeł cieplny, węzeł hydrodynamiczny, węzeł tribologiczny, współczynnik k, współczynnik oporu, liczba Sommerfelda.

Streszczenie: Działania sanacyjne na obiektach systemu produkcyjnego powinny być następstwem wystąpienia nieakcep-towalnych różnic pożądanych i rzeczywistych wartości miar cech zespołów funkcjonalnych obiektów. W pracy uzasadniono potrzebę posługiwania się pojęciem „potencjał eksploatacyjny węzła zespołu funkcjonalnego”. Przedstawiono trzy podstawowe modele węzłów i dla każdego opisano wybrane modele szczegółowe uwzględniające kinematykę, kształt ciał stałych i wła-ściwości fizykochemiczne płynów. Opisano i uzasadniono miary zastępcze potencjału eksploatacyjnego i miary symptomów diagnostycznych węzłów: hydrodynamicznego, cieplnego i tribologicznego.

Introduction

If we assume that an enterprise by itself definesdesired values of that enterprise’s characteristics, thenquality characteristics of a production system item and thevaluesofthesecharacteristicsmeasuresshouldresultfrom the decomposition of the characteristics and desired values of production system characteristic measures.Actualvaluesofcharacteristicsmaydifferfromdesiredvalues. The differencesmay be due tomanufacturingerrorsandwearcausedby theuseof items.When thedifferences become unacceptable, remedial actionshave to be taken in relation to such items. In [1, 2],

algorithmsofremedialactionsarediscussed,alongwiththe introduction of such concepts as functional unit wear margin and functional unit node.

Production system items are products of other production systems. The life cycle of a productunderstood as an item of a production system in the social market economy consists of the design,manufacture, operation, and scrapping stages. Actualcharacteristics and characteristicmeasurevaluesof anitemaredefinedatthestageofdevelopment(design)ofan itemasaproduct, anddesirablevaluesofmachinecharacteristic measures should stem from the desired qualitycharacteristicsoftheenterprisecompany.

p. 117–126

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Designingofanitemconsistsinpreparingtheitem’sspecification in the form of design and constructiondocumentation.Thespecificationsdefinethefollowing:– The ranges of characteristic measure values

describingtheitem’sload(torque,rotaryspeed);– Thedesiredrangesofcharacteristicmeasurevalues

ofthestructuralmaterialoftheitem’selements;– Thedesiredrangesofcharacteristicmeasurevalues

of the item’s elements (macrostructure, form anddimensionsofelements);

– Thedesiredrangesofcharacteristicmeasurevaluesforbuilt-inunitsandelementsofanitem(backlash,preloadofbolts,pressfit,tightness,unbalance,etc.);and,

– The desired value ranges of the characteristicmeasuresoftheworkingfluid(e.g.,viscosityofoil).

It is assumed that if all those measures have a valuein thedesired range, then the itemasawholewillreceive desired characteristics with values fallingwithinthedesiredrange.Characteristics values rendered to an item in

manufacturingchangedue towear, the resultofusingan item. The effects of wear are incorporated in thecharacteristics describing the production system. Suchcharacteristics are not listed in item specification.Notallthecharacteristicsfeaturesofanitemaresubjecttochange.Therefore,itisconvenienttousetheconceptof‘wearmarginofanitem’.Thewearmarginofanitemcanbedefinedasasetofitem’scharacteristicmeasures,whosevalues• aretheresultofitemmanufacturing,• decreaseduetowearduringtheiruse,and• canberestoredduringanoverhaulormaintenance.

The problems with using the term ‘wear margin of amachine’arethefollowing:– Machine specifications do not include on-line

measurable characteristics directly describing the wearmargin.

– Many characteristics are necessary to describe the item’swearmargin.Duetoalargenumberofcharacteristicsnecessary

to describe wear margin, it is purposeful to establishsubstitutemeasures,i.e.measuresthatcanbedeterminedon the basis of characteristic measures of built-in elements,workingfluids, load,orrelativemotion.Theproblemmaybesolvedbyusingtheterm‘wearmarginofafunctionalunit’ofanitem,forthefollowingreasons:– The wear margin of an item is the resultant of wear

marginsofitsfunctionalunits.– Presence in the item of functional units implementing

partial functions means that the rate of degradation of wear margins of each functional unit may be differentandmayrequireaseparateremedialaction.

– Combining items into aggregates is done by joiningelementsofselectedfunctionalunits.‘New’functional units are created that are not listed in the specifications.

A prerequisite to use the concept of wear margin of functionalunitsisthedecompositionofdesiredvaluesofcharacteristicmeasuresofanitemintodesiredvaluesoffunctionalunitwearmarginmeasures.

Directmeasurement of the values ofwearmarginmeasures of operating functional units generally is not possible. It isnecessary tomake indirectmeasurementsusing signals emitted by functional unit elements.Elements of a functional unit form nodes.A functionalunitcanberepresentedasthesumofnodes,whereoneelementofafunctionalunitmaybeapartofseveralnodes.Inthesimplestcase,afunctionalunitcanbeasinglenode.There are three basic models of functional unit nodes:1. Hydrodynamic node: two solids that constitute

achannel separatedbyafluid.Thefluid is affectedbyhydrodynamicload,whichforcesanddeterminesfluidmotionrelativetothesolids.

2. Tribologicalnode: twosolidsareseparatedbyfluid.Atleastoneofthebodiesissubjectedtomechanicalload,whichforcesanddeterminestherelativemotionofthesolids.

3. Thermalnode:twofluidsareseparatedbyasolid.Thefluidsareatleastunderthermalload.Heatistransferredfromonefluidtoanotherfluidthroughasolid.Theliteratureonthesubjectprovidesmanydetailed

modelsoftheabovebasicmodels,incorporatingtheshapeof solids andphysicochemicalpropertiesoffluids. It ispurposeful to analyse existing physical and mathematical descriptions and to indicate those quantities that can be adopted as substitute measures of the wear margin of agivennodeandidentifyassociateddiagnosticsymptommeasures.

1. Measures and symptoms of hydrodynamic node wear margin

Aspecificmodelofthenode‘loadedfluidactingontwosolids’isdevelopedinfluidmechanicsas‘flowoffluidsinpressurepipes/flowoffluidsinaroundcross-sectionpipe’.Fluidflowinpipesismodelledasaone-dimensional flow: values of quantities describing theflowdependonone coordinateofposition.Accordingto[3],theconceptofone-dimensionalflowisquitewellverifiableforafullydevelopedturbulentflow,becausethe velocity profile is then relatively flat and globalquantities, suchas theflux,volume,orkineticenergy,virtuallydonotdependonvelocitydistribution.

A pipeline section can be composed of straight pipe sectionsanddesiredbuilt-inelements:elbows,branchedpipes, cross-section changes, and instruments.Built-incomponentsintermsofhydrodynamicsarecontractions,flown-roundbodies,oracombinationofbothand,dueto energy losses, are regarded as obstacles. Obstaclesmay also be undesired obstructions created due to manufacturing and installation errors (e.g., weldingof pipelines) and as a result ofwear.Characteristics ofa straight section stem from characteristics of elements

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making up that section and characteristics obtained after building them in: inside diameter d [m], length l [m],absolute roughness of internal surface k (peak height of asperities)[µm],andrelativeroughnessk/d. In thecaseofembeddedelements,theircharacteristicsarequantitiesdescribingtheshapeanddimensionsoftheelement.Fluidcanbea liquidoragas. Incompressiblefluid isafluidwhose density in a specific range of temperature andpressure is negligibly small [3].Real fluidswithMachnumberupto0.3(fortheairupto100 m/s=360 km/h)areconsideredasincompressiblefluids[4].Inconsiderationsofincompressiblefluids,Bernoulli’sequationisimportant.Fluidcharacteristicsincludekinematicviscosityν [m2/s]and density ρ [kg/m3].Characteristics offluid load andmotionarefluidflowratew,meanflowratewm[m/s],andgeodetic height g[m].

Duringtheflowinapipeline,energylossesoccur.Energylossesare theresultoffrictionin thefluidandfriction between the pipeline wall and the fluid. Themechanismoflossesbetweenafluidandawall[4]arethe thermal movement of molecules at velocitywb in thedirectionperpendiculartothedirectionoftheflux,whichcausesfluidmoleculestocollidewitheachother.Molecules in the valleys of material asperities losevelocityw.Molecules diffusing back to the fluxmustbeacceleratedagaintothespeedoffluidfluxw. Energy losses in a pipeline are presented as pressure losses betweentwocross-sections1and2[4],Fig.1.

Pressure losses in a straight section of a pipeline are describedbyDarcy-Weisbachformula:

∆p ld

wm= λ ρ2

2 (1)

The number λ is called the linear resistance coefficient or coefficient of friction. It depends on theReynoldsnumberRe=wl/v,Table1.

Table 1. The value of the linear coefficient of resistance depending on the type of fluid flow

Re value Formula

≤2320 λ =64Re ;λ = f(Re)

2320<Re<105 λ =

0 3164

,Re ;λ = f(Re)

65d/k <Re<1300d/k

λ

λ

=

+

1

2 2 51 0 272

log ,Re

,/d k ;λ = f(Re,k/d)

Re>1300d/k

1 1 74 2 2λ= −, log k

d ;λ = f (k/d)

Source:Authorsonthebasisof[3,4].

Losses on obstacles are taken into account by means ofcoefficientζ. Thecoefficient ζ comprises additional losses compared to the losses of a straight section of the pipeline[4].

∆p wp

m= ζρ2

2 (2)

The values of coefficient ζ are calculated experimentally.

Disturbances of the flux occur generally on theobstacle as well as within sections before and after it (after the obstacle – in a section 10 to 30d in case of turbulentmotion)[4],Fig.1.

Fig. 1. Pressure loss between cross sections 1 and 2: P – obstacle, lP – the length of the obstacle, ΔpP – loss at the obstacle (loss focused locally), (p’1–p’2) – loss at the straight section (loss evenly distributed), (p’1–p”2) – the total loss at the section lo

Source:[5].

When the flow is laminar, in areas after any ofthe two types of obstacle, (contraction and a solid)turbulencesareformed,characterisedbysmallandlargeeddies[4],Fig.2.

Fig. 2. Areas of turbulence after an obstacle in laminar flow: 1 – pipeline wall, 2 – orifice, 3 – turbulence (eddies) behind the orifice, 4 – flown-round solid body 5 – turbulence (eddies) after a flown-round body

Source:[5].


According to [6], everynon-stationaryprocess ina gas or liquid is a source of turbulences that spread in awavemotion.Eachobstacle,therefore,isasourceofnoise–continuousacousticemission.

Whenaliquidflowsthroughanorifice,thepressurein the orifice can theoretically take any low value.Practically, however, thepressurevalue is limited andmaynotfallbelowthepressureofevaporation(boiling)correspondingtothetemperatureoftheflowingliquid.The drop of pressure to values close to evaporationpressure is accompanied by the emission of gases and vapours from the liquid, the phenomenon known as cavitation [3]. Cavitation causes wear and acousticemission.

In the case of flown-round bodies, at adequatelylarge Reynolds numbers, vortex shedding can occur.We can determine the frequency of vortex sheddingexperimentally. The results are presented in the formof a Str – Re diagram [4]. Strouhal number (St, Str)is a dimensionless number (dimensionless time) that describes unstable flows in which accelerations havea dominant role:

,

Str = ⋅d fwm

(3)

where f–thefrequencyofvortexshedding.

Thephenomenonofvortexsheddingcausesburstacousticemission.

The publication [6] contains a semi-empiricalrelation determining total acoustic power No [W]generated by an obstacle which is passed by a medium surrounding a pipeline:

,

N k p dco =

∆ 3 2

2 3ρ (4)

where Δp[Pa]–totalpressurelosscausedbyabodyinthepipeline;k – constant (for air k = 2.5·10–4);c – speed ofsoundinthesurroundingmedium[m/s].

2. Measures and symptoms of thermal node wear margin

If the principal load of a node consisting of a solid separatedbytwofluidsisthermalload,suchanodecanbe called a thermal node. Heat flux is a measure ofthermal load, Q . Heat flux is defined as the ratio of

elementary amount of heat dQ to time length dt of the heat transfer Q = dQ/dt[W].

A solid body/barrier can have a multilayerstructure,e.g.,twocylindersconnectedbypressfit.Onthemacro scale, a solid body in the form ofmachineelement material is non-homogeneous; it has a core,surfacelayer,andfilms.Thebarriersurfacemaybeofdifferentshape.Intheknownmodelsofthermalnodesthefollowingaredistinguished[7,8]:– Aplane/flat barrier/wall (e.g., of a heat exchanger,

slidebearingguide,thermalinsulator,etc.;and,– Cylinder (cylinder surface)/cylindrical barrier: heat

exchanger‘tube-in-tube’,shellandhousingofaslidebearingetc.

Heat transfer from a hot source through a barrier toacoldsourcetakesplacebyconduction,convection,and radiation. The participation of each phenomenonis specific to a particular node. Each of the abovephenomena is characterised by a different drop in temperature in thedirectionof theheatflow.Figure3presents the temperature drop for the node, inwhich,in Zones 1 and 3, heat transfer is mainly throughconvection.

Fig. 3. The temperature drop for the node liquid–barrier–liquid: d – heat transfer path, dp – barrier thickness, TG – hot source temperature (temperature in the fluid flux), Tz – cold source temperature, T1, T2 – temperature at liquid-solid interface, 1, 3 – convection, 2 – heat conduction

Source[9].

Heatfluxpassing throughaflatbarrier,Fig.3, ismodelled [7, 8] asone-dimensional,fixed temperatureheatconduction,sincetemperaturechangesinonlyonedirection,localtemperatureisconstantintime.

QdA T T

p= −( )λ

1 2 (5)

where λ – proportionality factor called heat conduction coefficientorthermalconductivity[W/mK](inclinationof the line T = f(d) is proportional to the value of λ);A – surface area of the barrier measured perpendicular toheatflowdirection[m2];(T1 – T2) = ΔT temperature differenceonbothsidesofthebarrier[K].


Foramultilayerwall,atasteadyflowthroughaflatbarrier,thefollowingapplies:

dp = d1 + d2 + dn(6)

Q A T

dii

i

n=

=∑∆

λ1(7)

The literatureon the subjectprovidesappropriateformulasforothershapesofthebarrier.

The coefficient λ [W/mK] is a quantitycharacterizingagivenmedium/materialintermsoftheabilitytoconduct/insulateheat.Itisamaterialproperty,whichisaconstantdeterminedexperimentally.

The process of heat transfer from the flux to thebarrier(andviceversa)iscalledconvectiveheattransfer,describedbyNewton’slaw[7,8]:

, Q A T= α∆ (8)

where ΔTtemperaturedifferencebetweenthefluidandthewallforexample,fromFigure3,respectively:Q A T TG G1 1= −( )α ; Q A T TZ Z3 2= −( ))α ;

α–convectiveheattransfercoefficient[W/m2K].

HeattransfercoefficientresultsfromthedefinitionofNusseltnumber[7,8]:

αλ

=NLu p

(9)

where λp – thermal conductivity of stationary liquid; L – characteristic length of a thermal node (in the case of channels–hydraulicdiameter,intheothercases–lengthof thefluidflux);Nu – Nusselt number (dependent on othernumbersofsimilarity:Reynolds,Prandtl,Grashof,Froud).

It follows from definitions describing Nusseltnumber thatheat transfercoefficientα depends on the following:– Thephysicalpropertiesofbarrierandfluidmaterials;– Thegeometriesofthebarrierandthefluidflux;– Thepropertiesofthebarriersurface(profile);and,– Thefluidvelocityrelativetothebarrier.

Theprocessofheattransferfromthefluidfluxofhighertemperaturethroughabarriertothefluxofalowertemperature is called heat penetration,ortransmission.Heatfluxpenetrating throughflatbarriers isdescribedasfollows[7,8]:

, Q Ak T= ∆ (10)

where k–overallheattransfercoefficient[W/m2K].

Forexample,fromFigure3:

ΔT = TG – TZ (11)

k d

G

i

ii

n

Z

=+ +

=∑1

1 11α λ α

(12)

The temperature on the barrier surface at a steady heat exchange is not equal, and the temperaturedistribution is dependent on the direction of the fluxtransferringheat.Intube-in-tubenodes,thetemperaturechangesalongthecylindergeneratrices.Itisadvisableto know the temperature along the relevant generatrixonbothsurfacesofthebarrier.Theaveragetemperaturedifference ΔT is determined. The logarithmic meantemperaturedifferenceΔTmisauniversalmeasureoftheaveragetemperatureΔT.Themethodfordeterminingtheaveragelogarithmictemperaturedifferenceispresentedafter [7], inFigure4,asanexampleofacounter-flowheatexchanger.

Fig. 4. The temperature as a function of the fluid flux path in a node with fluid counter flows under adiabatic conditions: lw – heat exchanger length; T’G, T’Z – inlet temperature of the hot and cold flux; T”G, T”Z – hot and cold outlet fluid temperatures

Source:[9].

∆

∆ ∆∆∆

T T TTT

m =−

max min

max

minln

(13)

TheaveragelogarithmictemperaturedifferenceΔTm is,accordingto[8],independentofflowdirection.ΔTmax andΔTminaredifferencesbetweenfluidtemperaturesonagivensideoftheexchanger.

The study [10] indicates a possibility of anexperimental determination of k–overallheattransfercoefficient. It follows from theheat/energybalanceofa thermal node that

, Q Q Q Ak Tm1 2 3= = = ∆ (14)

( )


becauseheatfluxinafluidfloware

Q m c TG vG G1 = ∆ (15)

.

Q m c TZ vZ Z3 = ∆(16)

Therefore,

,k m c T

A Tm c TA T

G vG G

m

Z vZ Z

m= = ∆

∆∆

∆ (17)

wheremG,mZ – fluxofhotandcoldfluid,cvG,cvZ – specificheatatconstantvolumeofhot

andcoldfluid.

3. Measures and symptoms of wear margin of tribological nodes

Deliberately constructed tribological nodes servetoefficientlyandsafelytransferforcesand/ormoments,and enable the motion of one mechanically loaded elementrelativetoanotherelement,wheretheelementsare separated by liquid. Load excites the motion ofone or both elements and has a component normal to thedirectionofmotion.Thereisrelativemotionalongthegapand/orperpendiculartothegap.Separationbya liquid occurs as the result of the following:– Thelubricatingwedgebetweentwoelementsand/or– The effect of liquid extrusion from the gap between

elements.Intermsofpurpose/function:

– A stationary element of the trobological node may supportajournalofrotorshaftorrotordisc(e.g.,ofapiston).

– Bothmovingelementsmaycarrythemomentfromone rotor to the other (they are elements of rotor discs).

– A functional node may contain many nodes connected in parallel and/or series (roller bearing,apairofgears).

– The effect of oil wedge occurs in nodes where the followingexists[4]:

– A narrow clearance, or gap, exists between solidbodies.

– Alongthegap,thereisrelativevelocityofwallsu.– Thewallsaregeometricallyconvergent.

Inaconvergentgap,Fig.5,thepressurebuildsup[4].Accordingto[11],oil,thankstothephenomenonofwetting solids by liquids, is takenby an elementwithgreaterspeedandtransportedtothegap,whichnarrowsin the direction of movement. As a result of super-pressureformed, theseparatingliquidflowsoutof thegapinthedirectionofmovementandsideways(atthe

beginning and end of the sliding surface atmospheric pressuremust prevail). Such a node is a combinationof a friction pump and pressure accumulator with an overallefficiencynotexceeding1/3 (much lower thanpositive-displacementpumps).

Intheliteratureonthesubject,e.g.[12]:– Thecoefficientoffrictionμdecisiveforfrictionwork

WRisgivenasdependentontheminimumthicknessof the gap ho and substitute roughness Raz.

– Theminimumthicknessof thegap(min. thicknessofoilfilm)isconsideredtobeproportionaltoHerseynumber He.

µη

= ( ) =f h R h H H uqo a o e ez

, , ~ , (17)

where η – dynamic viscosity of separating liquids;R R Ra a az

= +1 2

2 2,Raz – substitute measure of surface

roughness of interacting walls; q – unit load of tribological node; Ra1, Ra2 – arithmetic means of the profiledeviationfromthemean lineofWalls1and2,makingupthetribologicalnode.

Fig. 5. A convergent gap of the node and distribution of fluid velocity in the gap: 1, 2 – node forming elements moving at speeds w1 and w2, 3 – separating liquid, l – gap length, ho – minimum thickness of the gap, P – normal force loading the node

Source:Authoronthebasisof[4].

Aconvergentgapmaybeformedbythefollowing:– The elastic strain of the mating surfaces under the

operatingload;– Theself-aligningfixingofoneelement(slidebase);

or,– The eccentric positioning of thejournalandashell.

As a result of local deformations caused by largeHertzpressure,elementsnominally incongruentbecomecongruentinthecontactarea,andduetohighpressure, the viscosity increases significantly. Theresult is that the product of u·η becomes large enough to create a very thin lubricating film separating thematingsurfaces.

Thegapneednotbedesignedascongruent,becausein longitudinal bearings, the sliding surface settles


(withinthegap)automatically.Maximumpressurepmax in this bearing is proportional to the quotient ηlu/ho

2[4].Theconvergentgaparisesinradialslidebearings

withanoff-centredshellandjournal,Fig.6.

Fig. 6. Radial slide bearing: a – stationary journal, b – slow turnover of the journal at start-up, c – journal position at fluid friction, O1 – journal centre, O2 – shell centre, P – load, ω – angular velocity, Ω = R – r radius of the backlash circle, ho – minimum thickness of the oil film


Withconstantvaluesofload,bearingbacklashandlubricant viscosity, as the rotary speed increases thefollowingoccur[13]:– Thejournalmovesinthedirectionoppositetothe

turning element, then breaks off from the shell,Fig.6b.

– After break-off, the journal centre initiallymovesin the direction of rotation, then having achievedacertainrotaryspeed,startstolifttowardstheshellcentre,Fig.6c.

– As the speed increases, the journal moves onaroughlycircularpath.Whentheangularvelocityapproaches infinity, the journal centre coincideswiththeshellcentre,Fig.7.Thepositionofjournalcentrerelativetotheshell

depends on the equilibrium between the load and reactionforceoftheoilwedge.TheequilibriumpositionisinfluencedmainlybythedimensionlessSommerfeldnumber So,Table2:

SoPbd

=ψηω

2

(18)

TheSommerfeldnumber isasimilaritycriterionof hydrodynamic slide bearings working on the oil wedgeprinciple.Whenbearingsaresimilar, i.e.havethesamerelativelengthb/dandthesamejournal/shellangle, and the same Sommerfeld number, then theirworkingconditionsaresimilar,i.e.theyshowthesamejournalpositionintheshellandthesamecoefficientoffriction[13].

Table 2. Characteristics and measures of the wear margin of a radial slide bearing

Characteristics of

Absolute and relative measures

Dimensionless quantity

shape of node elements

internal diameter D of the shelljournaldiameterdbacklash s = D – drelativebacklashψ = s/Dbearing length b

So Pbd

=ψηω

2

movement angularvelocityω

fluid kinematic viscosityη

load force Punit load q = P/bd

Source:Author.

Variousauthorsuserelativequantitiesformodellingradialbearings.Thedependenceofrelativeeccentricityεandrelativeoilfilmthickness(1 –ε)onSommerfeldnumber Soforvariousb/d are presented in the form of diagrams[14].

Forconstantvalueanddirectionoftheloadandforagivenrotaryspeed,thetrajectoryofthejournalcentreisapoint,ifthefollowingistrue:– The resultant load is a harmonic force, and the

trajectorybecomesacircle,Fig.7.


– The harmonic load, when the shaft or bearing areanisotropic,makesthetrajectorybecomeanellipse.

– A periodic non-harmonic load produces a more complexshapeofthetrajectory.

– Thetrajectoryradiusdependsnotonlyonload,butalso on the mode of the whole rotor and a position of thebearingrelativetotherotorvibrationnodes.

Fig. 7. A trajectory of journal centre: Ω – radius of backlash circle, A – backlash circle, B – path of average values of journal centre, C – path of variable component of trajectory, 1 – detachment of the journal from the shell 2 – average trajectory for a specific speed, 3 – theoretical position of journal centre for speeds towards infinity

Source:Author.

Forbearingcalculations,itisassumedthatthefluxinanarrowgapofthebearingislaminar,particlesgetrelativelylowaccelerations,andfluxlinesaredeterminedby the gap shape.At high speeds or low viscosities,additional effects occur, causing the formation ofvorticesand,consequently,turbulentflux.Thisleadstoanapparentincreaseinviscosity,anincreaseinoilfilmthickness,andahighercoefficientoffriction[15].

The efficiency of machine bearings capable ofworking in a wide speed range is a function of rotary speed.Thedependenceof thecoefficientof friction inabearingon rotaryspeed isdescribedby theStribeckcurve.

AmodifiedStribeckcurveispresentedinFigure8.Itindicatesadecreaseinthecoefficientoffrictionduringuse at constant nominal speed (section 3–4) and anincreaseinthecoefficientoffrictionatzerorotaryspeed(6–1).Thefallinsection3–4isduetooiltemperaturerising to nominal bearing temperature and the resulting decreaseinviscosity(changeofSommerfeldnumber).

Anincreaseinthefrictioncoefficientatsection6–1isexplained by the fact that total extrusion of oil from the gaptakesplaceafteraspecifictimeaftermovementisstopped[16].

Fig. 8. The coefficient of friction as a function of rotary speed during runup and rundown: 1–2–3 runup of a bearing-supported machine; 3–4 work till constant temperature; 4–5–6 machine rundown; 6–7 extrusion of oil from non-working bearing; A – mixed friction; B – fluid friction; ωmin – minimum speed; ωn – rated speed; μa – coefficient of friction during the run-up for ωx; μd – coefficient of friction during the rundown for ωx


Slidebearingwearresultsinthefollowing:– Increasedbacklashand/or– Adecreaseinviscosity.

Increased backlash leads to an increase in Sommerfeldnumber.Adecrease inviscosity increasesSommerfeld number. The inverse of the Sommerfeldnumber1/So is a substitute measure of the radial slide bearingwearmargin[17].

The impact of wear margin on the minimum thicknessofoilfilmcanberepresented in theformofadiagramcontainingacollectionofcurvesforconstantvaluesofbacklash,load,andviscosity,Fig.9.Limitsoftheminimumoilfilmthicknessho may be established as follows:– ho minduetoariskofbearingseizure,and– ho maxduetofrictionlosses.

Thetrajectoryofthejournalcentreismeasurable.Systems for measuring trajectories, described instandards [18], enabledirectoscilloscopeobservationsof the trajectory [19] and measurements of varyingtrajectories. It is possible to use such systems formeasuring the distance (radius) from the shell centre to the average trajectory for a given speed.However,this requires a special calibration of the system and a special algorithm combining an increase in distance withachangeofthenodewearmargin.

Journal of Machine Construction and Maintenance | PROBLEMY EKSPLOATACJI | 3/2017 12�12�

Vibrationsofnodeelementscanbeasymptomofbearingwearmargin.Thepressurebuildup ina liquidbetween elements of a tribological node means that both elementsareaffectedbyoppositeforces,i.e.theproductof the average pressure buildup and the surface areaoccupiedbythe‘buildup’.Ifthenodeloadisvariable,the pressure and forces resulting from the existence of pressure are variable.Variable forces acting on theelement generate its vibrations. Values of elementvibrationmeasuresdependonthefollowing:– Thevaluesofpressuremeasures,and– Transmittance between the point of vibration

measurement and the point at which a force from pressurebuildupisapplied.Atavariableload,besidestheoilwedgeeffect,the

effectofextrusiontakesplace.Thepureeffectofextrusionoccurswhenaloadingforcechangesitssign[20].

Thetransitionfromfluidfrictiontomixedfrictionbringsaboutanadditionalsymptom–acousticemission.

Summary and conclusions

An analysis of the existing physical and mathematical descriptions and of available researchresults permits one to indicate those quantities that can be adopted as substitute measures of the wear margin of a given node and identify related diagnostic symptommeasures.

A pipeline section is an example of a hydrodynamic node. The wear margin of a pipeline reduces due toprocessessuchaserosion,cavitationerosion,corrosion,particulatematterdepositsontheinsidewalls.

The pipeline wear leads to changes in pipeline diameter, increased roughness of the pipes, andchanged profiles of the built-in elements. This causeschangesofvaluesofwearmarginsubstitutemeasures:Reynoldsnumber,relativeroughnessand,consequently,coefficientsofresistance.

Coefficients of resistance are diagnosable. Thesymptoms are pressure drops between two pipeline cross-sectionsandmeanflowrates.Pressuredropsareassociated with the dissipation of pressure energy to heatenergyandelasticenergyofwaves.Thedissipationof energy takes place in areas of turbulence: areas of small vortices are formed and large vortices are shedaway. The density of dissipated energy distribution isthegreatestonfluidfluxobstacles.Theobstaclescanbearesultoflocalwear.Vortexformationandsheddingisaccompaniedbyacousticemission.Thelevelofacousticemissionismeasurable.Theincreasedlevelofemissionsis due to a decrease in the wear margin of a pipeline or as aresultofnewobstaclesformedinthepipeline.

Aheatexchanger isanexampleof thermalnode.Asubstitutemeasureofaheatexchanger,overallheattransfercoefficientk,dependsonthecharacteristicsofmaterialsandfluids,theshapeofbuilt-inelements,thequantitiesdescribingmotionandload.Theoverallheattransfercoefficientcanrangeinvaluefromk1 to k2:

k1≤k≤k2

where k1–100%ofnodewearmargin, k2–0%ofnodewearmargin.

Thermal and erosivewear results in a change ofcharacteristicsdecisiveforthewearmarginofathermalnode. The actual value of the overall heat transfercoefficientkchanges.

Thecoefficientk as a substitute measure of the wear marginofathermalnodeisdiagnosable.Themeasureofsymptom of thermal node wear margin is the product ofhotorcoldfluidfluxmG/ZandtheratioΔTG/Z /ΔTm,changes of the temperature ΔT of hot or cold fluid,respectively, to the average logarithmic temperaturedifferenceΔTm.

A radial slide bearing is an example of a tribological node.Inradialslidebearings,thefollowingaretrue:– The location of the journal centre relative to the

shellandminimumthicknessofoilfilminabearingdepend on the dimensionless quantity known as SommerfeldnumberSo.

– Quantities determining Sommerfeld number aremeasures of the wear margin characteristics of the slidebearing.

– WearincreasesSommerfeldnumber,andtheinverseof Sommerfeld number 1/So decreases along with increasedbacklashandloadandreducedviscosity.

– TheinverseofSommerfeldnumber1/So is a substitute measureoftheradialslidebearingwearmargin.

– AsdependentonSommerfeldnumber, thedistancebetweenthe journalcentreandtheshellcentrecanbe a measure of the symptom of slide bearing wear margin(Inthiscase, it isnecessarytocalibratethemeasuringcircuit).

Fig. 9. The relation between thickness of the oil film and bearing wear margin: Son – nominal value of Sommerfeld number, Soe – limit value of Sommerfeld number, index n – nominal values, index e – limit (extreme) values

Source:Author.

.


– If a node is under periodically variable load, thesymptomsmay be vibrations of a rotating or non-rotatingelementofthenode.Forspecificsolutionstostructuralfunctionalunits,

wecancreatemodelsthatareacombinationofspecificmodels. The wear margin of a functional unit can bedescribedbydifferentcriteriaofsimilarity/coefficientsofproportionalityrelatedtosymptomsofvaryingformsof energy and various shapes of themeasured signal.Oneandthesamesignalcanbeacarrierofmanywearmarginsymptoms.

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