Shock and Vbraton 18 (2011) 53 62 53 DOI 10.3233/SAV-2010-0573 IOS Press Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng Xuejun L, Ypng Shen and Songla Wang Hunan Unversty of Scence and Technology, Hunan Provncal Key Laboratory of Health Mantenance for Mechancal Equpment, Hunan, Chna Receved 8 February 2010 Revsed 5 May 2010 Abstract. The large-scale rotary machne wth mult-supportng, such as rotary kln and rope layng machne, s the key equpment n the archtectural, chemstry, and agrculture ndustres. The body, rollers, wheels, and bearngs consttute a chan multbody system. Axs lne deflecton s a vtal parameter to determne mechancs state of rotary machne, thus body axal vbraton needs to be studed for dynamc montorng and adjustng of rotary machne. By usng the Rccat transfer matrx method, the body system of rotary machne s dvded nto many subsystems composed of three elements, namely, rgd dsk, elastc shaft, and lnear sprng. Multple wheel-bearng structures are smplfed as sprngs. The transfer matrces of the body system and overall transfer equaton are developed, as well as the response overall moton equaton. Taken a rotary kln as an nstance, natural frequences, modal shape, and response vbraton wth certan exctng axs lne deflecton are obtaned by numercal computng. The body vbraton modal curves llustrate the cause of dynamcal errors n the common axs lne measurement methods. The dsplacement response can be used for further measurement dynamcal error analyss and compensaton. The response overall moton equaton could be appled to predct the body moton under abnormal mechancs condton, and provde theory gudance for machne falure dagnoss. Keywords: Mult-supportng, axs lne, body vbraton, ansotropy stffness, Rccat transfer matrx method 1. Introducton The larger-scale rotary machnes, such as rotary kln, rope layng machne, cylnder drer, and coolng machne, are wdely used for materal calcnaton, strandng, dryng, and coolng n the archtectural, chemstry, and agrculture ndustres. Ther man bodes of these machnes are supported by many groups of wheel- bearng supportng structures, as shown n Fg. 1. These machnes are generally large scale, over-weght, and statcally ndetermnate. Axs lne deflecton s a vtal parameter to determne mechancs state of rotary machne, that s well nvestgated n Refs. [1 4]. Lttle axs lne deflecton can cause large dfference load dstrbuted between each wheel. It s reported that when the axs lne deflecton s ± 10 mm, the body stress ncreases three tmes, and the load appled on the supportng wheel ncreases one tme [2,5]. Body axal vbraton are excted when asymmetrc geometry occurs caused by abrason and thermal expanson, machne runs abnormally, and drvelne vbrates. Ths brngs dynamc stress to all parts and causes severe hazard to the machne. Therefore, body axal vbraton of the large-scale rotary machne wth mult-supportng should be pad suffcent attenton. In practce, as for a rotary kln, axal lne measurement s every- Correspondng author: Ypng Shen, E-mal: ypng1011@163.com. ISSN 1070-9622/11/$27.50 2011 IOS Press and the authors. All rghts reserved
54 X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng Fg. 1. The sketch of the large-scale rotary machne wth mult-supportng. Fg. 2. Dscrete model wth N subsystems. day radcal task to mantan the equpment [6 10]. However, dynamcal measurement error s found to be mportant for effcent and accurate adjustment. Thus, t s necessary to perform dynamcal modelng and analyss of ths knd of rotary machne. The body, rollers, wheels, and bearngs of rotary machne consttute a chan multbody system. A lot of methods for chan multbody system dynamcs have been studed by many researchers [11,12]. The transfer matrx method (TMM) has been developed for a long tme and has been appled wdely n structure mechancs and multbody system dynamcs [13]. The advantage of TMM s that the global dynamcs equatons of the system are not needed and the matrx order does not ncrease wth the number of elements. Holzer frstly used TMM to solve the problems of torson vbraton of rods [14]. Prohl appled TMM for rotor system dynamcs [15]. Horner proposed Rccat transfer matrx method (Rccat TMM) to mprove numercal stablty [16]. Dokansh proposed the fnte element transfer matrx method to solve plate structure vbraton problem [17]. Kumar and Sankar developed dscrete tme transfer matrx method(dttmm) for tme varant system dynamcs [18]. In recently years, many researchers contnuously developed TMM to a wde varety of engneerng problems [19 27]. In ths nvestgaton, Rccat transfer matrx method s used to analyze the dynamcs of the body system wth multple wheel-bearng supportng structures. Ths paper s organzed as follows. In Secton 2, the body s dvded nto many subsystems wth the deduced stffness coeffcents of multple wheel-bearng supportng structures, and transfer matrces and overall equaton are establshed. In Secton 3, the Rccat transform equaton s ntroduced to mprove numercal stablty for computng natural frequences and modal shape of the body system. The response overall transfer equaton s gven n Secton 4. In Secton 5, a rotary kln as an example s presented wth the numercal results, and then some conclusons are gven. 2. Dscretzaton of the large-scale rotary machne wth mult-supportng The body, rollers, wheels, and bearngs of the large-scale rotary machne can be consdered as a chan multbody system. By usng TMM, the body system may be dvded nto a certan number of subsystems, whch can be represented by varous elements ncludng rgd dsks, elastc shafts, and lnear sprngs, as shown n Fg. 2. The head of rotary machne s the coordnate system orgn, the straght axs lne s x axes, the vertcal drecton of the supportng secton s y axes, and the horzontal drecton s z axes. For any subsystem, all elements are lmted to a plane moton, y and z plane.
X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng 55 Fg. 3. The geometry and mechanc model of wheel-bearng supportng structure. 2.1. Stffness of multple wheel-bearng supportng structures Multple wheel-bearng supportng structures are represented by lnear sprngs n system dscrete model, as shown n Fg. 2. For rotary machne, each supportng structure s composed of a roller set around the body, two wheels located on the left and the rght of roller at 30 0 angle wth vertcal lne, and four shafts and bearngs, as shown n Fg. 3(a). The supportng structures are smplfed as sprngs, as shown n Fg. 3(b). For any subsystem composed of a rgd body wth lnear sprngs(shown n Fg. 3(c)), m represents the weght of the rgd dsk, k ay and k az denote the supportng stffness of the rght wheels n y and zaxs, k by and k bz are the supportng stffness of the left wheels n y and z axs, k eq s the equvalent stffness of the journal bearng, m l s the wheel weght. The equvalent stffness of supportng structures can be deduced by usng ther dfference moton equaton. For the mechanc model of the supportng structure n Fg. 3(c), the equvalent stffness of the supportng structure s ansotropy. By gnorng the couplng terms of y and z drecton, the mass and stffness can be wrtten n matrx form [ ] [ ] [ ] [ ] ml 0 kaz 0 kbz 0 keq 0 M l = ; K 0 m a = ; K l 0 k b = ; K = (1) ay 0 k by 0 k eq In the supportng structure, wheel, shaft and journal bearngs can be consdered as a rgd dsk wth a elastc sprng, because the journal bearng wth overload and low speed can not generate ol flm [18,19]. Therefore, the stffness k eq can be easly wrtten as k eq = k l k p /(k l + k p ) where k l s the shaft stffness, k p s the bearng stffness. Supposed the dsplacement vector of body s D = [y, z] T, the dsplacement vectors of the left and rght wheel are D l = [y l, z l ] T and D r = [y r, z r ] T, respectvely. The moton of the supportng structure can be expressed as M l D + K a D l + K b D r K eq (D D l D r ) = 0 (3) For natural vbraton of the body system, the soluton of the above equaton can be assumed as D = De st D l = D l e st D r = D r e st (4) When neglectng damper effects, S = β, β s the natural frequency of the body system, β k (k = 1, 2,...) denotes k th natural frequency. Then, the equvalent stffness matrces of supportng structure can be expressed as K sa = K eq [K eq + K a + S 2 M l ] 1 [K a + S 2 M l ] K sb = K eq [K eq + K b + S 2 M l ] 1 [K b + S 2 M l ] (5) where K sa, K sb denote the complex equvalent stffness matrces of the th left and rght wheel-bearng structure, respectvely. Let K denotes the total stffness of the two sdes supportng structures, as yelded as K = K sa + K sb (2) (6)
56 X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng Fg. 4. Schematc dagram of the subsystem and elements. 2.2. State vectors, transfer equatons and transfer matrces of elements The state vectors of any element end are defned as Z = [z, θ y, M y, Q z, y, θ z, M z, Q y ] T (7) where y, z are the pont coordnates of the body rotary center, θ y, θ z are correspondng angular dsplacements, M y,m Z are the correspondng nternal torques, Q y,q Z are the correspondng nternal forces, subscrpt s the subsystem ndces, and superscrpt T denotes the matrx transpose. The transfer equaton and the transfer matrx T between the dvded body subsystems can be wrtten as Z = T Z 1 The transfer equaton descrbes the mutual relatonshp between the state vectors at two ends of the th subsystem. For rotary machne, any subsystem s composed of three elements, namely, a rgd dsk wth a lnear sprng, and an elastc shaft. Fgure 4 shows a subsystem combned three elements, a rgd dsk wth a elastc sprng, and an elastc shaft respectvely. Accordng to D Alembert s prncple, the dynamc equaton of a dsk wth a sprng (Seen n Fg. 4(b)) can be wrtten as Q y,+1 = Q y, m S 2 y k yy, y Q z,+1 = Q z, m S 2 z k zz, z M y,+1 = M y, I S 2 θ y, M z,+1 = M z, I S 2 θ z, where k yy, k zz are the total stffness of the supportng structure n y and z axs, namely dagonal terms of K, I s the moment of nerta. Therefore, transfer matrx for ths element can be wrtten as Z +1 = D Z (10) D = 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 I S 2 1 0 0 0 0 0 m S 2 k zz 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 I S 2 1 0 0 0 0 0 m S 2 k yy 0 0 1 (8) (9) (11)
X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng 57 As rotary machne generally operates at low speed, the gyroscopc effects are not ncluded n the above equaton. For the elastc shaft wthout mass (Seen n Fg. 4(c)), transfer matrx B s wrtten as 1 l l / l 3/ 6 0 0 0 0 0 1 l/ / 0 0 0 0 0 0 1 l 0 0 0 0 B = 0 0 0 1 0 0 0 0 0 0 0 0 1 l / l 3/ 6 (12) 0 0 0 0 0 1 l/ / 0 0 0 0 0 0 1 l 0 0 0 0 0 0 0 1 where l s the correspondng length of elastc shaft, E s the elastc modulus, J s second moment of area. Therefore, for any subsystem composed of a dsk wth a sprng, and a elastc shaft (Seen n Fg. 4(a)), transfer matrx can be obtaned accordng to the mechancs relatonshp of materal ( ) t11 t T = B D = 12 (13) t 21 t 22 t 12 = t 21 = 0 ( ) 1 l3 6 ms 2 + K zz l + IS2 l2 l 3 ( ) t 11 = l2 ms 2 + K zz 1 + l IS2 l l ( ) ms 2 + K zz IS 2 1 l ( ) ms 2 + K zz 0 0 1 6 ( ) 1 l3 6 ms 2 + K yy l + IS2 l2 l 3 ( ) t 22 = l2 ms 2 + K yy 1 + l IS2 l l ( ) ms 2 + K yy IS 2 1 l ( ) ms 2 + K yy 0 0 1 6 3. Rccat transfer equaton of the large-scale rotary machne wth mult-supportng The overall transfer equaton of the body system can be obtaned by multplyng the transfer matrces of the subsystems Z N = T all Z 0 T all = T N T N 1...T...T 2 T 1 where T all s the overall transfer matrx of the body system, Z 0,Z N are the state vectors of boundares of the body, subscrpt denotes the subsystem ndces. Substtutng boundares state vectors nto Eq. (14), the natural frequency equaton can be obtaned. Then solvng the natural frequency equaton, the natural frequences can be ganed. Usng the transfer equatons of subsystem Eq. (8) for each natural frequences, the state vector of any end of subsystem can be obtaned easly. However, wth the ncrease of the number N of the subsystems, roundng errors are magnfed by multplyng transfer matrces, even make the computaton results nvald. The Rccat transfer matrx method s an mportant way to mprove the numercal stablty [16,19,22]. When usng Rccat transfer matrx method, the state vectors are dvded nto two parts Z = [f,e] T = [M y, Q z, M z, Q y, y, θ z, z, θ y ] T (15) where f ncludes the state varables that are zero on the boundary of the body systems, e ncludes the other unknown varables. (14)
58 X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng Therefore, Eq. (8) can be yelded as [ ] [ ] [ ] f u11 u = 12 f e u 21 u 22 e u 11 = +1 u 21 = 1 l 0 0 0 1 0 0 0 0 1 l 0 0 0 1 l l 3 6 0 0 0 0 0 0 0 0 l l(ms 2 + K zz IS 2 0 0 u 12 = ( ) ms 2 + K zz 0 0 0 0 0 l(ms 2 + K yy IS 2 0 0 ( ) ms 2 + K yy 0 l 3 6 ( ) 1 l3 6 ms 2 + K yy l + IS2 0 0 ( ) u 22 = l2 ms 2 + K yy 1 + l IS2 0 0 ( ) 0 0 2 1 l3 6 ms 2 + K zz l + ( ) IS2 0 0 l2 ms 2 + K zz 1 + l IS2 (16) Introducng the Rccat transform f = R e (17) Substtutng t nto Eq. (16), we obtan e = [u 21 R + u 22 ] 1 e +1 (18) f +1 = [u 11 R + u 12 ] [u 21 R + u 22 ] 1 e +1 (19) By comparng Eqs (19) wth (17), the transfer formula can be found as R +1 = [u 11 R + u 12 ] [u 21 R + u 22 ] 1 (20) Accordng to the boundary condtons, matrces f,e, and Ron boundares can be determned as f 0 = 0, e 0 0, R 0 = 0, f N = 0, e N 0 (21) By usng Eq. (20) repeatedly, the matrx R at any subsystem can be obtaned. At the last subsystem Nof the boundary can be substtuted nto Eq. (17), and we can get f N = R N e N Accordng to the boundary condtons, we get the natural frequency equaton R N = R 11 R 12 R 21 R 22 = 0 N (22) (23) By solvng Eq. (23), the natural frequences can be obtaned. The dchotomy method s adopted to search the natural frequences β k (k = 1, 2...n)whch s approxmately satsfed Eq. (23). For each natural frequency β k, usng the overall transfer equaton of the body system and the transfer equatons of elements, the state vector of any pont as well as modal shape of the body system can be obtaned easly.
X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng 59 4. Response vbraton analyss of the large-scale rotary machne wth mult-supportng In practce, the body vbraton of rotary machne s excted by nteror factors causng axs lne deflecton and exteror forces. The nteror factors, common asymmetrc geometry of major parts caused by manufacture, abrason, and expanson dfference, wll result n dynamcal change of rotary centre, thus the correspondng element state vector s changed [8,10,28]. The exteror forces are ntroduced from drvelne vbraton, materal varety, abnormal operaton, and machne start/stop, etc. [29,30]. The above exctaton wll ntroduce ether dsplacement vectors ( y, z) or force vectors ( Q y, Q z ) to correspondng element state. The body response can also be analyzed by usng Rccat-TMM. The state vector of the other end of the excted element s rewrtten as [ f e ] +1 = [ ] u11 u 12 u 21 u 22 [ f e ] + [ f ē where f and ē s the exctng correspondng vectors. The Rccat transform s yelded as f = R e + p Substtutng t nto Eq. (24) ] R +1 = [u 11 R + u 12 ] [u 21 R + u 22 ] 1 (26) p +1 = [u 11 p + f] R +1 [u 21 p + ē] (27) e = [u 21 R + u 22 ] 1 e +1 [u 21 R + u 22 ] 1 [u 21 p + f] (28) By usng Eqs (26) and (27) repeatedly, the matrx p at any subsystem can be obtaned. Consderng the last element end Nof the boundary,r N and p N are substtuted nto Eq. (25), then we can get the overall moton equaton f N = R N e N + p N = 0 Equaton (29) ndcates the relatonshp between the state varables at the boundares of the body system. In general, the unknown e N can be obtaned by solvng the above equaton by replacng β wth the rotary speed of the body. Accordng to the transfer matrx Eq. (16), the response values of state vectors can be obtaned. (24) (25) (29) 5. Practcal applcaton 5.1. Dynamcs model of rotary kln We computed the natural frequences of the body system of rotary kln, as well as the response wth axs lne deflecton of the supportng structure. The rotary kln s shown n Fg. 5(a). The body length and radus are 100m and 4m respectvely, the total weght s 950 10 4 N. The kln has 5 supportng structures, located 6.3 m, 24.3 m, 44.1 m, 67.5 m, and 89.1 m, from kln head, respectvely. The kln s dvded nto varous elements accordng to body structure, as shown n Fg. 5(b). Any subsystem s composed of rgd dsks, elastc shafts, and lnear sprngs. The dynamcs model of the body system s a chan multbody system composed of 12 subsystems, namely, 12 rgd body elements, 12 elastc shaft elements, and 5 lnear sprng elements. Supposed 5 mm axs lne deflecton on the thrd supportng structure Y drecton s appled for response analyss of the body system, as shown n Fg. 5(b). Accordng to the materal and geometry propertes of the rotary kln, the values for the computatonal parameters of transfer matrx equatons are as follows. l 1 = 5.3 m, l 3 = 16 m, l 5 = 17.8 m, l 7 = 3.5 m, l 8 = 17.9 m, l 10 = 19.6 m, l 12 = 9.9 m, = l 4 = l 6 = l 9 = l 11 = 2 m; m 1 = 580.5 KN, m 2 = 343 KN, m 3 = 1902.5 KN, m 4 = 440 KN, m 5 = 1661.5 KN, m 6 = m 9 = 370 KN, m 7 = 600 KN, m 8 = 1432 KN, m 10 = 1581.5 KN, m 11 = 260 KN, m 12 = 247.5 KN; I = 4*m ; 1 = 3 = 5 = 8 = 10 = 12 = 125 GN. m.m, 2 = 4 = 6 = 7 = 9 = 11 = 250 GN. m.m; Ansotropy stffness of the supportng structures n y and z drecton are gven as follows. y drecton: k yy1 = 6.5 10 4 N/mm, k yy3 = 36 10 4 N/mm, k yy5 = 39.08 10 4 N/mm, k yy8 = 22.46 10 4 N/mm, k yy10 = 3.78 10 4 N/mm z drecton: k zz1 = 3.76 10 4 N/mm, k zz3 = 20.78 10 4 N/mm, k zz5 = 20.56 10 4 N/mm, k zz8 = 12.96 10 4 N/mm, k zz10 = 2.18 10 4 N/mm
60 X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng Table 1 Frst fve order natural frequences β k of kln (r/mn) Modal β 1 β 2 β 3 β 4 β 5 TMM 28.5 51.5 60.7 145.4 148.5 Rccat TMM 28.5 51.4 59.8 145.2 147.5 Fg. 5. The sketch of rotary kln and dynamcs model of the body system. 5.2. Natural frequences of rotary kln Fg. 6. Frst three modal curve of the body system. The results of the natural frequences β k (k = 1, 2, 3,...) of the body system are shown n Table 1. The frst three modal curves are shown n Fg. 6. It can be seen clearly from Table 1 that the results of natural frequences get by Rccat-TMM and by TMM have good agreements, whch mutually valdatng the dynamcs model, numercal accuracy n ths paper. As rotary speed of rotary kln s about 3 r/mn, t s known form Table 1 that the kln do not need to pass resonance frequency, so there s no resonance when start/stop the machne. The frst three orders modal curves are shown n Fg. 6. The practcal track s ellpse wth long axs y and short axs z. 5.3. Dynamc response of kln body system The dynamcs response of the body system wth 5 mm axs lne deflecton on the thrd supportng structure Y drecton s smulated wth Rccat TMM. In order to know the axs lne vbraton condton, t s mportant to know
X. L et al. / Dynamc modelng and analyss of the large-scale rotary machne wth mult-supportng 61 Fg. 7. Dsplacement response of the body system. the dsplacement response, as shown n Fg. 7. Some conclusons are drawn as follows: (1) The frst three orders modal curves are polygonal lnes. Wth the ncrease of the natural frequences, each polygonal lne become steeper, and body vbraton become stronger. (2) Vbraton values n supportng structure poston are opposte to the others postons, and vbraton nodes are not n supportng postons. Therefore, dynamcal errors of the common axs lne measurement system are produced as ther measurements generally conduct n the supportng postons. Modal analyss results provde mportant theory to develop accuracy axs lne measurement system wth dynamcal error compensaton. (3) Vbraton ampltudes n the mddle of elastc bodes lmted between two supportng structures are too large to gnore. Fatgue damage should nclude these dynamc stresses n structure strength analyss and desgn. Furthermore, body vbraton wll brng exctng forces to other parts, whch may accelerate uneven fatgue damage. 6. Conclusons The dynamcs of the large-scale rotary machne wth multple wheel-bearng structures s nvestgated n ths paper. Based on the Rccat transfer matrx method, the transfer matrces and overall transfer equaton are developed to calculate natural frequences, and response overall moton equaton s establshed for response analyss. Multple wheel-bearng structures are smplfed as lnear sprngs, and ther ansotropy equvalent stffness are deduced. Taken a rotary kln as an nstance, natural frequences, modal curves, and response vbraton are obtaned by solvng Eqs (23) and (29). The body vbraton modal curves llustrate the cause of dynamcal errors n common axs lne measurement methods. The dsplacement response can be used for further measurement dynamcal error analyss and compensaton. The response overall moton equaton could be appled to predct the body moton under abnormal mechancs condton, and provde theory gudance for machne falure dagnoss. Acknowledgements The authors would lke to acknowledge the support of the Natonal Hgh-tech R&D Program 863, Grant NO.2007AA04Z415, the Natonal Natural Scence Foundaton of Chna, Grant NO.50675066, and Scentfc Research Fund of Hunan Provncal Educaton Department, Grant NO.09C407.
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