A theoretical model for fault diagnosis of localized bearing defects under non-weight-dominant conditions

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1 Journal of Phyi: Conferene See PAPER OPEN ACCESS A theoretial model for fault agnoi of loalized beang defet under non-weight-minant ontion To ite thi artile: Q K Han and F L Chu 2015 J. Phy.: Conf. Ser Related ontent - Shannon wavelet petrum analyi on trunated vibration ignal for mahine inipient fault detetion Jie Liu - A leakage-free reonane pare deompoition tehnique for beang fault detetion in gearboxe Shazali Oman and Wilon Wang View the artile online for update and enhanement. Thi ontent wa wnloaded from IP addre on 25/07/2018 at 05:01

2 A theoretial model for fault agnoi of loalized beang defet under non-weight-minant ontion Q K Han, F L Chu 1 The State Key Laboratory of Tbology, Tinghua Univety, Beiing , China hufl@tinghua.edu.n Abtrat. Fault agnoi of loalized beang defet under non-weight-minant ontion i tued in thi paper. A theoretial model with eight degree of freem i etablihed, onideng two tranvere vibration of the rotor and beang raeway and one high-frequeny reonant degree of freem. Both the Hertzian ontat between rolling element and raeway, beang learane, unbalane fore and elf-weight of rotor are taken into aount in the model. The loalized defet in both inner and outer raeway are modeled a half inuoidal wave. Then, the theoretial model i olved numeally and the vibrational repone are obtained. Through envelope analyi, the fault harateti frequenie of inner/outer raeway defet for vaou ontion, inlung the weight-minant ontion and non-weight-minant ontion, are preented and ompared with eah other. 1. Introdution Rolling element beang are important omponent of rotating mahinery. Loalized defet are frequently found in inner/outer raeway. The growth of loalized defet an aue the fault of rolling beang and even evere aident to the rotating mahinery if undeteted early. For year, extenive effort have been devoted to the modelling of loalized defet and then pret their effet upon the vibrational repone of the beang ytem [1-3]. Mot urrent tue foued on the beang ytem under the weight-minant ontion [4-6]. The load zone due to the elf-weight of the rotor i time-invaant. The rolling element peoally pa through the load zone and impat with the inner/outer raeway defet. The fault harateti frequenie for the defet ould be preted aorngly. However, the elf-weight might not be the minant load in many rotating mahinery, uh a the vertial rotor, lightly weight rotor, and et.al. In thee rotor, the unbalane fore i the minant load, and aue the load zone of the rolling beang vae with time. In thi ae, the impat between the rolling element and raeway defet beome more ompliate. Obviouly, the fault harateti frequenie would ffer tintly with that of the weight-minant ae. Therefore, fault agnoi of loalized beang defet under non-weight-minant ontion i tued in thi paper. A theoretial model with eight degree of freem i etablihed, onideng two tranvere vibration of the rotor and beang raeway and one high-frequeny reonant degree of freem. Both the Hertzian ontat between rolling element and raeway, beang learane, unbalane fore and elf-weight of rotor are taken into aount in the model. The loalized defet in both inner and outer raeway are modelled a half inuoidal wave. Then, the theoretial model i 1 To whom any orrepondene hould be addreed. Content from thi work may be ued under the term of the Creative Common Attbution 3.0 liene. Any further tbution of thi work mut maintain attbution to the author() and the title of the work, ournal itation and DOI. Publihed under liene by Ltd 1

3 olved numeally and the vibrational repone are obtained. Through envelope analyi, the fault harateti frequenie of inner/outer raeway defet for vaou ontion, inlung the weightminant ontion and non-weight-minant ontion, are preented and ompared with eah other. Finally, ome onluion are ummazed. 2. Theoretial model for rotor-beang ytem with loalized beang defet 2.1. Nonlinear retong fore of rolling beang with loalized defet The hemati agram for a deep grove ball beang i hown in Fig. 1. Generally, the outer rae i fixed on the pedetal. The inner rae, onneted with rotor, rotate under ontant angular peed. The oornate for dynami analyi i O X Y. There are two degree of freem for the inner rae and rotor, i.e. x, y. The ma and damping are m,, and the tiffne i provided by the nonlinear Hertz deformation between the ball and raeway. The outer rae and pedetal alo have two degree of freem, i.e. x, y. The orreponng ma, tiffne and damping are m, k,. p p p p p Beide, in order to imulate the high frequeny reonant repone of the rolling beang, a pngma-damping ytem ( mr, kr, r ) with high reonant frequeny i added to the model [3], a hown in Fig. 1. Figure 1. Defetive beang model The loation angle of the th ball i t bt 2 1 Nb, where Nb the number of ball, b the orbital peed of the ball. Under pure rolling ontion, the orbital peed of deep grove rolling beang i expreed by b 1 db Db 2, in whih d b the ball ameter and Db the pith ameter of the beang. Aorng to the geomet relation hown in Fig. 1, the ontat deformation for the the th ball at time t i o x x y y in (1) p p l Where l denote the raal learane. A loalized defet in the outer rae i onidered own in Fig. 1. The defet i modelled by a half inuoidal wave, and it loation, ize and depth are, repetively, expreed by o. Similarly, the loation, ize and depth for the inner rae defet are denoted by i. When 2

4 the ball i ontat with the defet, the ontat deformation hould be mofied to take into aount the adtional learane indued by the defet. For the rolling beang with outer rae defet, the ontat deformation ould be rewtten a x xp o y y p in l ho in (2) x xp o y y p in l ele A the outer rae i fixed, o the loation angle of the defet i alo ontant. However, a the inner rae i rotating, o the inner rae defet alo rotate with time. It loation angle hould be t i, in whih i the initial angle with repet to the X axi at time t 0. Thu, when the inner rae defet i onidered, the ontat deformation ould be expreed imilar with Eq. (2), while the parameter o hould be ubtituted by i. Aorng to the Hertz ontat theory, the ontat fore between the inner rae and ball i given by 3/2 Q K, in whih K the ontat tiffne the fore oeffiient. For 0 1 ; otherwie 0. The ontat tiffne K i related to the geometry and mateal property of the ontat pair, and ould be olved from Har monograph. Conideng the number of ball in ontat, the total retong fore of the rolling beang in the two axe are expreed a follow F F Nb 3/2 bx K o 1 Nb 3/2 by K in Equation of motion for the rotor-beang ytem A gid rotor ytem upported by two deep grove rolling beang i hown in Fig. 2. Thu, the equation of motion for the ytem ould be deved a follow 2 m x x F F m e o t bx1 bx2 u u m y y F F m e in t 2 by1 by2 u u m x x k x F 0 p1 p1 p1 p1 p1 p1 bx1 0 m y y y k k y k y F 0 p1 p1 p1 r1 p1 r1 r1 p1 r1 p1 r1 r1 by1 m y y y k y y r1 r1 r1 r1 p1 r1 r1 p1 m x x k x F 0 p2 p2 p2 p2 p2 p2 bx2 0 m y y y k k y k y F 0 p2 p2 p2 r 2 p2 r 2 r 2 p2 r 2 p2 r 2 r 2 by2 m y y y k y y r 2 r 2 r 2 r 2 p2 r 2 r 2 p2 In whih xpi, ypi, x ( i 1, 2 ) repreent the degree of freem for the beang 1 and 2, mpi, m,, and k, k ( i 1,2 ), repetively, denote the ma, damping and tiffne of beang 1 and 2. pi pi Obviouly, onideng the degree of freem of the rotor, pedetal igh-frequeny png-ma ytem, there are eight degree of freem for the rotor-beang ytem. Due to the nonlinear retong fore indued by the rolling beang, Eq. (4) ould only be olved by the numeal integration method. In the following etion, we will how the fault feature of the dynami repone for the ytem under vaou load ontion, inlung the non-weight-minant ontion. (3) (4) 3

5 Figure 2. Rotor-beang model 3. Simulation and uion In the imulation, the parameter for the rotor-beang ytem are hown in Table. 1. Two type of load ontion are onidered: (1) only weight ontion (ut the weight-minant ontion); (2) only unbalaned fore ontion (ut the non-weight-minant ontion). In the following analyi, the envelope petra for the rolling beang without defet, with inner rae defet and with outer rae defet are obtained to how the value and tbution of fault frequenie. Table 1. Sytem parameter Parameter Value Parameter Value Rotor ma (Kg) 3 Pedetal ma (Kg) Rotor damping (N/) 1200 Pedetal tiffne (N/m) 15e6 Rotor tiffne (N/m) 1e6 Pedetal damping (N/) 2210 Unbalaned ma (Kg) 0.01 Spng ma (Kg) 1 Ma eentity (m) 5e-4 Spng damping (N/) 9500 Beang pith ameter (m) Spng tiffne (N/m) 8e9 Ball ameter (m) Contat tiffne (N/m 3/2 ) Number of ball 8 8e9 Raal learane (m) 1e Only rotor elf-weight ontion (weight-minant ontion) The unbalaned fore exitation i ignored in thi etion in order for ompaon with the following analyi. Rotor peed i 20 rad/, and the loalized defet parameter are aumed to be 0.1rad, 1e 3rad o 1e 3m for the outer rae defet; 0.1rad 1e 3rad and hi 1e 3m for the inner rae defet. The envelope petra for the ytem with outer rae defet and inner rae defet are, repetively, hown in Fig. 3. It i hown that the fault harateti frequenie are onitent with urrent reult. For the outer rae defet, the harateti frequenie ould be expreed by nf o ( n 1,2,3,... ), where fo Nbb denote the ball paing frequeny. When the inner rae ha a loalized defet, one an ee from Fig. 3(b) that the harateti frequenie are the 4

6 ombination of inner rae fault frequeny with rotating peed, i.e. nfi m ( n 0,1,2,..., m...2 1,0,1,2,... ). Thee frequenie are alo identified and marked in Fig. 3. Figure 3. Envelop petra for the ytem under only weight ontion: (a) outer rae defet; (b) inner rae defet Only unbalaned fore exitation ontion (non-weight-minant ontion) For the lightly rotor, epeially the vertial rotor, it weight might be mall enough and even ould be ignored in the analyi. In thi etion, only the unbalaned fore i onidered in the imulation. The envelop petra for the ytem with outer rae defet and inner rae defet are preent in Fig. 4. Compared with the weight-minant ontion (Fig. 3), one an ee that the fault frequenie ffer tintly. For the outer rae defet, it i hown from Fig. 4(a) that the fault frequenie are relatively omplex. They ould be ummazed a: nfo m ( n 0,1,2,..., m...2 1,0,1,2,... ). When the inner rae defet i onider, a hown in Fig. 4(b), it i hown that the fault frequenie are related to the inner rae frequeny and odd time of rotating peed, i.e. nfi m ( n 0,1,2,..., m...4 2,0,2,4,... ). Although the fault frequenie are till the ombination of inner rae frequeny and rotating peed, the fferene i that only the odd time of rotating peed are found. 5

7 Figure 4. Envelop petra for the ytem under only unbalaned fore ontion: (a) outer rae defet; (b) inner rae defet. 4. Conluion The fault harateti of a rolling beang with loalized defet under non-weight-minant ontion are tued. Through envelope analyi, the fault harateti frequenie of inner/outer raeway defet for vaou ontion, inlung the weight-minant ontion and non-weightminant ontion, are preented and ompared with eah other. It i hown that the fault frequenie for the non-weight-minant ontion ffer tintly with that of the weight-minant ontion. Expement on a vertial rotor with defetive rolling beang are being onduted to vefy the theoretial reult. Referene [1] Sopanen J and Mikkola A 2003 Dynami model of a deep-groove ball beang inlung loalized and tbuted defet. Part 1: theory Pro. Intn Meh. Engr Part K: J. Multi-body Dynami [2] Zeki K and Hira K 2003 Simulation and analyi of vibration ignal generated by rolling element beang with defet Tbology International [3] Sawalhi N and Randall R B 2008 Simulating gear and beang interation in the preene of fault Part I. The ombined gear beang dynami model and the imulation of loalied beang fault Mehanial Sytem and Signal Proeing [4] Rafanania A, Abbaiona S, Farhianfarb A 2009 Nonlinear dynami modeling of urfae defet in rolling element beang ytem Journal of Sound and Vibration [5] Ahtekar A, Sadeghi F and Stake L-E 2010 Surfae defet effet on beang dynami Pro. IMehE Vol. 224 Part J: J. Engineeng Tbology [6] Patel V. N., Tann N and Pandey R K 2010 A Dynami Model for Vibration Stue of Deep Groove Ball Beang Conideng Single and Multiple Defet in Rae Journal of Tbology