Simulation of vibrations produced by localized faults in rolling elements of bearings in gearboxes

Transcription

1 5 th Autralaian ongre on Alie Mechanic, AAM ecemer 007, Briane, Autralia Simulation of viration rouce y localize fault in rolling element of earing in gearoxe N.Sawalhi 1 an R.B.Ranall 1 1 School of Mechanical an Manufacturing Engineering, The Univerity of New South Wale, Syney 05, Autralia Atract: A comine gear/earing moel ha een mae to tuy the interaction etween gear an earing in the reence of fault. The thirty-four egree-of-freeom moel of the Univerity of New South Wale (UNSW) gearox tet rig i a lume ma arameter moel, which ha the caacity to moel ifferent fault tye (localize inner/outer race fault an extene fault). Thi i in aition to it original caacity for moelling all an crack in the gear. The gear/earing moel take into conieration the liage in the earing, the Hertzian contact an the nonlinearity of the earing tiffne (time variant). Reult have earlier een ulihe for imulate local fault in the inner an outer race of the earing a well a for an extene inner race fault. For a comlete moelling of localize fault, the imulation moel ha now een uate to enale imulating fault in the rolling element. The moeling of the fault in the rolling element take into conieration the ifference in race curvature an rereent a new contriution to the eveloment of the moel. The agreement etween the rocee imulate an meaure ignal how the routne of the eveloe moel an it uitaility for teting new iagnotic an rognotic algorithm. Keywor: Enveloe analyi, gear/earing moel, Hertzian contact, localize fault, rolling element earing. 1 Introuction In orer to facilitate the eveloment of iagnotic an rognotic technique for rolling element earing in real ytem, a neceity exite to rouce imulation moel, where fault can e imlemente uner ifferent oerating conition, rather than waiting for thee to occur naturally, or alternatively having them eee for laoratory teting. Thi aer illutrate how to moel localize fault in rolling element in a gearox environment. The aer i arrange a follow: After reviewing the ynamic imulation moel of rolling element earing [1] in ection, the moelling of localize fault in rolling element i icue in ection 3. A rief review of the UNSW gear/earing moel [1,] i reente in ection 4. The exerimental an imulate reult for a localize all fault are comare an icue in ection 5. ynamic Simulation of rolling element earing Figure 1 illutrate the rolling element earing-eetal moel, which ha een erive an reente in [1]. The five egree-of-freeom moel (5 OF) i ae on Hertzian contact theory [3], which relate the raceway ilacement to the earing loa, an alo account for the liage etween the element. Force in the horizontal ( f ) an vertical ( f ) irection are given in (1). f x = k n n 1.5 γ δ coφ, f y = k = 1 = 1 x 1.5 y γ δ in φ (1) The loa eflection factor k een on the contact geometry an the elatic contact of the material, the exonent n = 1.5 for all earing an 1.1 for roller earing. The ucrit enote the all numer an varie etween 1 an n (total numer of rolling element). γ i a contact witch which account for the fact that comreion occur only for oitive value. Thi i efine in () a:

2 γ = 1, if δ > 0 0, otherwie () Figure 1 Five OF earing-eetal moel The overall contact eformation (contact comreion) for the th -rolling element δ i a function of the inner race ilacement relative to the outer race in the x an y irection ( x x ), y y ) ), the element oition φ (time varying) an the clearance (c ). Thi i given y: ( ( δ = ( x x )coφ + ( y y ) in φ c β (3) where φ (angular oition of the rolling element ) i efine in (4) a a function of the time increment t, the reviou cage oition φo an the cage ee ω c an account for the liage in the rolling element (uncertainty in all oition). reivou cage oition π ( 1) } 644 liage φ = + ω ct + φo + (0.5 ran ) φli n (4) β i a fault witch which imulate the contact lo at a efine angular oition ( φ ) over an angular itance of ( φ ). The eth of the efect wa efine a a function of φ o that a graual rather than intant lo/gain of contact i achieve. 3 Moelling localize fault in rolling element A all on one of the rolling element (rolling element k) rotate at the ame ee a the rolling element i.e. ω (all in frequency), which i efine in (5) a: in

3 ω in ω = 1 co( α ) (5) i the rolling element iameter an i the itch iameter. The oition of the all ( φ ) can e efine a in (6) φ = ω + φ (6) where in t φ o o : initial tarting location of the all. The oition of the all i aociate with the ee of the rolling element, which in turn i a function of the haft ee. Thi mean that the rolling element fault hare ome characteritic with fault on the inner race an exhiit a maor ifference from the fault aociate with the outer race (fixe location). The lo of contact i etecte for only the faulty rolling element k, an thi i one twice for each comlete rotation of that rolling element, i.e. when it i in touch with the inner race an with the outer race. The witch value an the erio of witching on ( φ ) will not e the ame for oth race, which reult from the ifference in curvature etween the two race an i alo a function of the rolling element iameter. The inner race will contact eeer an longer comare to the outer race. Intea of efining two all geometrie, the fault β witch can e moifie to how thi variation a ene y the ifferent race a in (7): 0, k 1, if 0 < φ < φo β = r r + i o if π < φ < π + φ ) = k (7) ( i 0, otherwie The erivation of the witch value i illutrate in Figure (not to cale), when the alle rolling element i in touch with the inner race. For thi cae, the total lo of contact - a a reult of creating a lot of with x - i the ummation of two value. The firt i the maximum eth the inner race will enter in the lot region ( i a calculate in (8), which rereent the movement of the inner race from it original contact oition - moving ownwar). The econ rereent the movement of the all from it original contact oition a a reult of creating the lot ( a reente in (9)- moving r uwar). Note that when the alle element i in touch with the outer race, the net change in the contact ( ) i the ifference etween the rolling element movement ( r ) an outer race movement ( o ) a hown in (9) an (10) reectively. Thi i ue to the fact that the outer race move ownwar an o oe the rolling element, which mean that the net change in the contact etween the original an the new contact oition can e otaine y utracting the two change. i = ri ri x (8) r = ( ) x (9)

4 Similarly o = ro ro x where r o + = (10) The angular with of the fault will alo een on the race contacting the efective roller an are given y (11) for the outer ( φ o ) an inner ( φi ) race contact. x φ o ( ra ) =, ro x φ i ( ra ) = (11) ri Figure Rolling element all eth a ene y the inner race (not to cale) 4 The UNSW ur gear tet rig moel A lume ma arameter moel (34 OF moel) wa contructe to rereent the ifferent arameter of the UNSW gearox tet rig moel (tiffne, aming an mae) in oth tranlational an rotational egree of freeom []. Both the 5-OF earing moel an the gear moel [4] were incororate in the overall moel. For a full an comlete ecrition of thi moel ee reference []. Matla an Simulink were emloye to olve the et of the equation of motion. The earing moel wa imlemente a an S-function (ecial function for Simulink ), which wa ynamically uate a the imulation roceee. The loa eenent non-linearity of the gearmeh tiffne wa imlemente a a look-u tale with cuic line interolation [4]. 5 Exerimental an imulate reult A notch fault wa introuce [5] into one of the rolling element (all) in a imilar way to thoe introuce earlier to oth inner an outer race uing electric ark eroion [5]. The electroe wa eigne in uch a way that the fault woul exten over an inclue angle of 60 o over the all urface. The generate ga in the all ha a rectangular cro-ection (0.5 mm in eth an 0.5 mm wie). In the earing imulation function, the with of the fault wa et to 0.5 mm, while the eth wa et to 10.7 m when the all touche the inner race an 7.4 m when it touche the outer race (ecaue of the reective convexity/concavity of the race). A reloa value of 3 m wa ue. 5.1 Time omain an ower ectrum comarion Figure 3 how the accelerometer time omain ignal for one comlete rotation of the gear for oth the imulate an meaure ignal. The attern of the imact in oth imulation an meaure ignal i the ame, excet that in the imulation ignal the imule are more oviou an le ame. Note that the ifference are much maller in ana filtere ignal given elow. To ee the effect of the rolling element fault on the ower ectra, oth faulty imulate an the meaure ignal are comare with their fault free counterart an the reult i lotte in Figure 4. The rolling element localize fault i no excetion to what ha een reviouly [1,] for the localize

5 fault in the inner an outer race. In oth the imulation an the exerimental cae, the localize all efect excite the frequency region aove 10 khz. In the imulation, thi i centre aroun the 15 khz, while in the meaure ignal, it excite a wie range of the tructure reonance a can e een y the circle art. Thee are reonile for the ifference in time ignal in Figure Enveloe analyi Figure 3 Time omain ignal (a) Simulate () Meaure The ignature of the rolling element fault i inecte through the filtere ignal (otaine via the SK filter) an their correoning quare enveloe ectra. Thee are lotte in Figure 5 for the imulate an exerimental cae, an how a much etter agreement etween the imulate an the actual ignal than the unfiltere ignal of Figure 3. The ominant frequency i (x all in frequency (BSF)) (5 Hz) a the fault imact twice uring one comlete rotation of the all (on the inner an the outer race), which i moulate y the cage ee (4 Hz). Note alo the exitence of the BSF comonent in the quare enveloe ectra for oth the imulate an the meaure ignal. Thi i ecaue of the variation of the imact everity etween the inner an outer race; ue to the ifference in race curvature. If the efect eth were moelle to e the ame for oth the race, only harmonic of (x BSF) woul aear in the quare enveloe ectra. Figure 4 Power ectra comarion (a) Simulate () Exerimental

6 Figure 5 Enveloe analyi for the localize rolling element fault. Left: imulate Right: exerimental. (a) Filtere ignal an () Square enveloe ectra 6 oncluion Thi aer reente a imulation moel for a localize fault in a rolling element in one of the earing of a ingle tage gearox. The imulate fault rouce ignal, which are reaonaly comarale to the meaure one. The imulate an meaure ignal reacte imilarly to ignal roceing technique normally alie to rolling element earing (ower ectrum comarion an enveloe analyi). The imilarity etween imulate an meaure ignal uecte to iagnotic technique ugget that the moel can e ue efficiently to imulate fault of ifferent ize an location an coul e quite ueful in the eveloment of rognotic metho. 7 Acknowlegment Thi work i uorte y the Autralian efence Science an Technology Organization (STO). Reference [1] Sawalhi, N., Ranall R.B an Eno H. (006): Simulating gear an earing interaction in the reence of fault: aer reente at the 7th IFToMM onference on Rotor ynamic, Vienna, Autria, 5-8 Setemer 006. [] Sawalhi, N. (007): iagnotic, rognotic an fault Simulation for rolling element earing. Ph iertation, UNSW, Syney. [3] Harri, F.J. (000): Rolling Bearing Analyi II, John Wiley, New York. [4] Eno, H. (005): A tuy of gear fault y imulation, an the eveloment of ifferential iagnotic technique. Ph iertation, UNSW, Syney. [5] Ho,. (1999): Bearing iagnotic an Self-Aative Noie ancellation. Ph iertation, UNSW, Syney.