Vibration analysis on the rolling element bearing-rotor system of an air blower

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1 Journal of Mechancal Scence and echnology 6 (3) () 653~659 wwwsprngerlnkcom/content/738-9x DOI 7/s6---6 Vbraton analyss on the rollng element bearng-rotor system of an ar blower Wu Hao,, Zhou Qong, Zhang Zhmng and An Q,* School of Mechancal and Power Engneerng, East Chna Unversty of Scence and echnology, Shangha 37, Chna Shangha Insttute of Specal Equpment Inspecton and echnology esearch, Shangha, 333, Chna (Manuscrpt receved October 8, ; evsed August 3, ; Accepted October 5, ) Abstract Ar blowers are wdely used n ndustry he vbraton of the rollng bearng-rotor s a key factor n the blower s performance because t sgnfcantly nfluences the securty and workng lfe of the whole system In prevous research on the vbraton characterstcs of the ar blower, the supportng rollng element bearng was always smplfed as a partcle on a shaft wth radal stffness and dampng coeffcent Such smplfcaton neglects the effects of the bearng structure on the vbraton performance of the rotor system In ths paper, a numercal model of the bendng stffness of the tapered roller bearng was establshed through mechancs and deformaton analyss On the base of the model, a new MM (transfer matrx method) for bearng-rotor system was establshed; the new MM consders the nfluences of the bearng structure on the vbraton of the rotor system Furthermore, modal analyss on an ar blower rotor system was carred out by usng the new MM, and the mode shape, crtcal speed and unbalance response of the ar blower system were obtaned he same blower rotor was also analyzed by FEM to verfy the valdaton of the new MM, showng that the new method proposed n ths paper for vbraton characterstcs calculaton of an ar blower s credble Keywords: Ar blower; Bearng-rotor system; ransfer matrx method; Vbraton Introducton Ar blowers, wth rotors supported by rollng element bearng, are wdely used n ndustry In applcaton, ar blowers often face many dfferent problems whch wll nfluence the workng process of a factory One of the most common problems of ar blowers s vbraton he vbraton calculaton of ar blowers s a focus n the power engneerng feld he blower s vbraton manly orgnates from the oscllatons of the blower s shell and bearng-rotor system [-3], wth the bearng-rotor system as the prmary source he most mportant vbraton parameters of the ar blower rotor nclude crtcal speed, unbalanced response, etc o obtan these parameters, the stffness and dampng of the supportng rollng element bearng should frst be calculated, and then the governng equatons of the whole system can be establshed On these bases, the dynamc characterstcs of the ar blower rotor system can be revealed and the vbraton parameters can be obtaned Prevous researchers who focused on the vbraton of radal ball bearngs were ahnejat and Gohar [] hey bult a deep groove ball bearng model ncludng an extrapolated equaton hs paper was recommended for publcaton n revsed form by Assocate Edtor Ohseop Song * Correspondng author el: , Fax: E-mal address: anq@ecusteducn KSME & Sprnger wth effect of squeeze-flm hey suggested that the lmt cycle frequency and ampltude of the ball bearng s affected by the number of balls, appled load and radal nternal clearance Even n the presence of elastohydrodynamc lubrcatng flm between balls and the raceways, a peak of the ball passage frequency (BPF) appears n the spectrum An et al [5, 6] used fve-degree-of-freedom model of rollng element bearng and represented the bearngs by nonlnear sprngs consderng elastohydrodynamcs effects ahman et al [7] extended An s work by nvestgatng the free vbraton response of a horzontally mounted mult-bearng spndle system hey analyzed dfferent bearng settngs of the front bearng sets and varaton n axal preload hen they [8] showed BPF and ts harmoncs due to non-lnear Hertzan contact between the balls and the raceways for a defect free bearng Karacay et al [9] proposed a vbraton model of a ball bearng wth wavness consderng three degrees of freedom Wu et al [, ] obtaned the exact soluton of a snglespan non-unform moshenko beam carryng number of sprng-mass systems by usng lumped-mass and contnuousmass transfer matrx method, respectvely Yong et al [] presented transfer matrx solutons to study the axsymmetrc and non-axsymmetrc consoldaton of a multlayered sol system under arbtrary loadng ew et al [3] studed transent response on deep groove ball bearng-asymmetrc elastc rotor by transfer matrx me-

2 65 H Wu et al / Journal of Mechancal Scence and echnology 6 (3) () 653~659 Fg Confguraton of the ar blower thod He obtaned load dstrbuton on each roller element and solved knematc equatons for the bearng-rotor system by ange-kutta method u et al [] developed a dscrete tme transfer matrx method of a mult-body system to study general mult-body system dynamcs by combnng and expandng the transfer matrx method and the numercal ntegraton procedure Hseh et al [5, 6] developed a modfed transfer matrx method for analyzng the coupled lateral and torson vbratons of the symmetrc/asymmetrc rotor-bearng system wth an external perturbng torque From the above lteratures, t can be seen that n the past research about the vbraton characterstcs of a bearng-rotor system, the bearng was separated from the system and was treated as a pont wth stffness and dampng, whch neglects the nfluences of bearng structure on the bearng-rotor system In fact, the rollng element bearng structure nfluences the stffness of the shaft sgnfcantly In ths paper, the analytcal model for calculatng overall radal stffness and overall dampng of the tapered roller bearng, and the numercal model for calculatng bendng stffness of the tapered roller bearng are establshed On such bass, new MM and vbraton equatons of the bearng-rotor system takng nto consderaton rollng element bearng structures were establshed An actual ar blower was analyzed wth the new MM to show that the bearng structure sgnfcantly affects the vbraton characterstcs of a bearng-rotor system he rotor was also analyzed by FEM and results were verfed, whch shows that the new method proposed for vbraton characterstcs calculaton of ar blowers n ths paper s credble A new transfer matrx method Deformaton analyss of bearng-rotor system Fg shows an ar blower used n a power staton n Chna It s manly composed of a motor, pulley and bearng-rotor he most mportant part of the blower s ts rollng bearngrotor system Durng the vbraton analyss of bearng-rotor system, the foundaton s treated as rgd and ts vbraton s gnored Fg shows the geometry parameters of the rollng bearng-rotor system, whch s composed of an mpeller, pulley, shaft and a par of tapered roller bearngs (38) he mpeller, located n the mddle of the two bearngs, s composed of Fg Confguraton of bearng-rotor system of the ar blower Fg 3 Bendng deformaton of the tapered roller bearng-rotor system Fg Deformaton of the tapered roller bearng under bendng moment several bar peces of weght 65 kg and nerta 379 kg m ; the pulley located on the rght sde of the shaft has a weght of 5897 kg wth nerta 37 kgm and the rotor s workng speed s 6 rpm Accordng to lumped mass method, the beam s smplfed to be several dsks and beams, and the dsk s consdered to have no thckness and the beam s consdered to have no mass As Fg 3 shows, the ar blower rotor s supported by two tapered roller bearngs Under the acton of radal load, the shaft bends, and bendng moments by an external radal load wll act upon the two bearngs Under the radal load, the shaft wll produce radal elastc dsplacement and produce a tltng angle between the nner and outer raceways of bearngs As Fg shows, for the tapered rollng bearng, the contact

3 H Wu et al / Journal of Mechancal Scence and echnology 6 (3) () 653~ angle s α, concal angle s β and the angle between neghbor roller center lnes s ψ Under combned actons of radal load and bendng moment, the bearng wll deform Each rollng element s dvded nto n sectons unformly along the generatrx, and then they are analyzed separately accordng to the upper and lower half crcle δ () n () where x,, q when δ For the bottom roller, the deformaton on the th secton s (a) oad dstrbuton on sngle roller δ δr n () +, x [, ] () For the rollers n upper crcle, the deformaton on the th secton s ( ) ( ) δj δ cos jψ (3) For the rollers n the lower crcle, the deformaton on the th secton s ( ) ( ) δj δ cos jψ () Accordng to Hertz contact theory [7], stress dstrbuton along the roller generatrx under radal force and bendng moment can be expressed as: ( ) ( ) δ x Aq ( x) + Bq ( x)ln q ( x) j j j j δ x Aq ( x) + Bq ( x)ln q ( x) j j j j Mechancs analyss of bearng-rotor system he bendng moment actng on the tapered rollng bearng can be obtaned as M M+ M (6) where Z n M + D snα q j j n Z n + snα j j n () () M D q (5) where q j () and q j () can be obtaned by Eq (5) Because there s an mplct functon n Eq (5), bendng moment of bearng should be solved usng Newton-aphson teratve method for Eqs (5) and (6) When the tolerance between adjacent calculaton results of M s, the result can be consdered relable By the above equatons, the bendng moment and tltng angle can be obtaned Accordng to the defnton of stffness, the bendng stffness of the tapered roller bearng can be obtaned as: dm kr d (b) oad dstrbuton on bearng Fg 5 oad dstrbuton on roller and bearng Fg 5 shows the stress dstrbuton and the moment on sngle rollng element and the whole bearng 3 adal stffness and dampng of the tapered roller bearng Accordng to the authors past research [8, 9], radal stffness of tapered roller bearng s ΔFr Kc lm ΔFr Δδ Δδ lm ΔF 3 3 r ln( Fr +ΔFr) ln Fr Fr ( Fr +ΔFr) n+ mfr + mln( Fr +Δ Fr) + C ΔFr ΔFr 3 n + m + mln Fr + 3C Fr (8) and radal dampng of tapered roller bearng s (7)

4 656 H Wu et al / Journal of Mechancal Scence and echnology 6 (3) () 653~659 Kt and C as generates staton transfer ma- Settng trx Q Q Jω mω () Fg 6 New staton of bearng C + C C New transfer matrx method here s a tlt angle between nner and outer raceways besdes radal compresson; the tlt angle changes state vectors transfer relaton of roller bearng, as Fg 6 shows Bearng staton transfer relaton consderng bearng geometry structure s Y Y Y M M + kr, Q Q + ( k, + ωc mω ) Y t (9) () where Y,, M and V denote the transverse dsplacement, slope, bendng moment and shearng force at left sde of staton respectvely, whle Y,, M andv denote the same quanttes at the rght sde of staton respectvely k t and k r denote the bearng radal and bendng stffness respectvely, ω denotes the rotatng speed of the shaft and C denotes the radal dampng of the roller bearng Wrtng Eq () n a matrx form gves Y Y [ S ] M M Q Q where [ ] S s the transfer matrx of staton, k Jω [ S ] r, kt, + ωc mω () By Eqs ()-(3) the overall transfer matrx of the system can be obtaned: Y Y [ ] M M Q Q + where [ ] [ ] [ ] denotes the transfer matrx of secton Q s (3) From Eq (3) t can be seen that the state varables at the left sde of staton + may be obtaned from those at the left sde of staton Y eplacng by { δ } and repeatng applcaton of M Q transfer Eq (3) yelds the followng relatonshps: { δ} [ ] { δ} { δ} [ ] { δ} [ ] [ ] { δ} 3 { δ} [ ] { δ} [ ] [ ] [ ] { δ} { δ} [ ] { δ} [ ] [ ] [ ] [ ] { δ} n+ n n n 3 { δ} [ ] { δ} [ ] [ ] [ ] [ ] [ ] { δ} { δ} n+ s n+ n+ s n+ n 3 where [ s ] n + denotes transfer matrx of staton n + ; denotes the overall transfer matrx of the rotor system 3 3 [ s ] [ ] [ ] [ ] [ ] n+ n () By the overall transfer matrx, the relatonshp between the state varables at the left sde of the beam { δ } and those at the rght sde of the beam{ δ } n + can be obtaned Substtutng

5 H Wu et al / Journal of Mechancal Scence and echnology 6 (3) () 653~ Frst mode Second mode Fg 7 ransfer matrx model of bearng-rotor system boundary condtons M, Q M, Q nto Eq (3), leads to n+ n+ Modal dsplacement 5-5 Y Y + Non-trval soluton for Eq (5) requres that ( ω ) 3 3 Δ (5) (6) Shaft lenth(m) Fg 8 Mode shape accomplshed by MM x - Eq (6) s the frequency equaton of a beam and can be used for determnng the crtcal speed of the beam Correspondng to each natural frequency ω, there wll be a correspondng mode shape If the boundary condtons are gven by M Q and the ntal state varables are gven Y 3 3 by, then the rato of dsplacement and the M Q slope on any pont of shaft can be obtaned j VAU(m) FEQ(rpm) Fg 9 Unbalanced response calculated by the new MM 3 Modal analyss on the ar blower 3 Calculatng results of new MM Accordng to the lumped mass method, bearng-rotor system of the ar blower s smplfed as a combnaton of no thckness dsks and massless beams, and the bearng s smplfed as the combnaton of radal stffness, dampng and bendng stffness coeffcents Fg 7 shows the lumped mass model Accordng to numercal calculatng model, the bendng stffness of the tapered roller bearng s638 N m/rad Substtutng the bendng stffness value and lumped mass model of the ar blower nto the new MM, the vbraton characterstcs can be calculated Fg 8 shows the frst two modal shapes of the ar blower Accordng to MM, the frst crtcal speed s 67 rpm, and the second crtcal speed s 89 rpm Fg 9 shows the unbalance response ampltude of the center of ar blower s mpeller It can be seen that at the rotatng speed of 95 rpm, the frst unbalanced response ampltude at the center of the ar blower s 96 m, and at the rotatng speed of 79 rpm, the second unbalance response ampltude s 7 m Fg Fnte element model of the bearng-rotor system 3 Calculatng results by FEM Ansys s used for modal analyss to the same ar blower rotor Fg shows the FEM model of the ar blower rotor

6 658 H Wu et al / Journal of Mechancal Scence and echnology 6 (3) () 653~659 (a) Frst mode shape calculated by FEM Fg Unbalanced response calculated by FEM Fg shows unbalance response curves of the ar blower rotor It can be seen that the frst unbalance response ampltude s 9 m, the second response ampltude s almost the same Accordng to FEM, the frst crtcal speed s 5 rpm, and the second crtcal speed s 363 rpm Sold 5 s used, whch has 8 nodes and 6 freedoms Combne s used to smulate the tapered roller bearng and axal freedom of the system s lmted Snce the geometry s complex, the element shape n mesh process we used s Quad Standard method has been chosen to solve the problem In ths fnte element analyss, tapered roller bearngs are smulated by Combne, and axal freedom of the system s lmted he governng equatons of the whole FEM model s [ M ]{ x} + [ K]{ x} + [ C]{ x} { F} && & (7) K denotes stffness ma- where [ M ] denotes mass matrx, [ ] trx, [ C] denotes dampng matrx he governng equatons of each node s { u} { ϕ} cos( ω t) (b) Second mode shape calculated by FEM Fg Mode shape accomplshed by FEM (8) where { ϕ } denotes the th modal vectors, ω denotes the th natural frequency Submttng Eq (8) nto (7), the natural frequency can be obtaned hrough modal expanson, each modal shape curve can be obtaned Fg shows the frst two modal shapes of the ar blower rotor It can be seen that modal shapes obtaned by FEM are qute smlar to MM Both of them reflect the nfluences of the bearng structure on the rollng element bearng-rotor system Conclusons () he vbraton characterstcs of an ar blower rotor supported by tapered roller bearngs s studed hrough mechancs analyss, the numercal calculatng model for structure stffness of the tapered roller bearng s establshed and ntroduced nto the vbraton analyss for rollng element bearngrotor system Consderng structure stffness, radal stffness and dampng of the bearng, a new state vectors relatonshps between two ends of bearng s establshed On ths bass, a new MM s establshed () Modal analyss on an ar blower s carred out by smultaneous use of the new MM and FEM he frst crtcal speed accordng to MM(67) s 88% faster than that accordng to FEM(5), and the frst unbalanced response ampltude at the center of ar blower accordng to MM( 96 m ) s 67% greater than that accordng to FEM( 9 m ) Furthermore, Fg 8 (MM) and Fg (FEM) show smlar mode shapes consderng bearng structures he comparsons show that the new MM n ths paper s credble Acknowledgment hs work was supported by ECSU (East Chna Unversty of Scence and echnology) and SSEI (Shangha Insttute of Specal Equpment Inspecton and echnology esearch) Nomenclature E ν F r l n p : Young modulus, MPa : Posson s rato : adal force, N : oller length, m : otatng speed, rpm : ubrcate flm pressure, Pa

7 H Wu et al / Journal of Mechancal Scence and echnology 6 (3) () 653~ q : oad unt length along roller generatrx, N/ m A, A, A, m, nare mddle coeffcents, 3 where + tan β tan β A cos β + sn β tan β + tanα tan α snα A, cosα + snα tan α sn β + tan( α + β ) tan ( α + β ) sn( α + β ) A3 cos( α + β ) + sn( α + β ) tan ( α + β ) sn β 6cos( α + β) 86cos( α + β ) m ZlE cosα ZlE cos( α + β ) 6cos( α + β) DZlE ( A+ A) cosα n ln + 58 ZlE cosα 86cos( α + β) DAA ln ZlE cos( α + β) ZlE ( A3 A) cos( α + β) eferences [] Y Ohta and E Outa, Evaluaton and predcton of bladepassng frequency nose generated by a centrfugal blower, ASME Journal of urbo machnery, 8 (3) (996) [] H Sun and S ee, Numercal predcton of centrfugal compressor nose, Journal of Sound and Vbraton, 69 () () -3 [3] M Youns and F Bakr, Numercal and expermental study of unsteady flow n a centrfugal fan, Proc IMechE: Journal of Power and Energy, (7) (7) 5-36 [] H ahnejat and Gohar, he vbratons of radal ball bearngs, Pro of the Insttuton of Mechancal Engneers-Part C- Mechancal engneerng scence, 99 (C3) (985) 8-93 [5] An and H ahaejat, A fve degrees of freedom analyss of vbratons n precson spndles, Internatonal Journal of Machne ools and Manufacture, 3 () (99) -8 [6] An and H ahnejat, Vbraton modelng of rotatng spndles supported by lubrcated bearngs, Journal of rbology, () () [7] A K ahman and An, On the performance of multsupport spndle-bearng assembles, Proc of Insttuton of Mechancal Engneers-Part K-Journal of Mult-body Dynamcs, 6 () () 7-3 [8] N Akturk and M Uneeb, he effects of number of balls and preload on vbratons assocated wth ball bearngs, Journal of rbology, 9 (997) [9] Karacay and N Akturk, Vbratons of a grndng spndle supported by angular contact ball bearngs, Proc of Insttuton of Mechancal Engneers-Part K-Journal of Mult-body Dynamcs, (8) 6-7 [] J S Wu and C Chen, A lumped-mass MM for free vbraton analyss of a mult-step moshenko beam carryng eccentrc lumped masses wth rotary nertas, Journal of Sound and Vbraton, 3 (6) [] J S Wu and C Chen, A contnuous-mass MM for free vbraton analyss of non-unform beam wth varous boundary condtons and carryng multple concentrated elements, Journal of Sound and Vbraton, 3 (8) -3 [] Z Y A and Q S Wang, ransfer matrx solutons to axsymmetrc and non-axsymmetrc consoldaton of multlayered sols, Acta Mechanca, ()55-7 [3] A ew and N S Feng, On usng the transfer matrx formulaton for transent analyss of nonlnear rotor-bearng systems, Internatonal Journal of otatng Machnery, (6) () 5-3 [] X u and B He, Dscrete tme transfer matrx method for multbody system dynamcs, Multbody System Dynamcs, (5) 37-3 [5] S C Hseh and J H Chen, A modfed transfer matrx method for the coupled lateral and torsonal vbratons of symmetrc rotor-bearng systems, Journal of Sound and Vbraton, 89 (3) (6) [6] S C Hseh and J H Chen, A modfed transfer matrx method for the couplng lateral and torsonal vbratons of asymmetrc rotor-bearng systems, Journal of Sound and Vbraton, 3 (-5) (8) [7] H H Hertz, Hertz s mscellaneous papers, MacMllan, ondon, EN (896) [8] H Wu and J W Wang, Study on the calculatng method of radal stffness of tapered roller bearng, ubrcaton Engneerng, 33 (8) 39-3 [9] H Wu and J W Wang, Calculatng Method for dampng of apered oller Bearngs, Bearng () (9) 5-9 Hao Wu receved hs PhD degree n Mechancal Desgn and heory from East Chna Unversty of Scence and echnology, Shangha, Chna Hs current research nterests nclude nspecton of specal equpments and vbraton of elevators Q An s a professer n the Department of Mechancal and Power Engneerng, East Chna Unversty of Scence and echnology, Shangha, Chna Hs current research nterests nclude trbology and dynamcs of bearng-rotor system