Dynamic Similarity Design of Geared Rotor System in Five-Shaft Integrally Centrifugal Compressor

Transcription

1 Dynamc Smlarty Desgn of Geared Rotor System n Fve-Shaft Integrally Centrfugal Compressor Hao Zhang 1, Gaoshan We, Xu Chen, Qngka Han 1,* 1. Collaboratve Innovaton Center of Maor Machne Manufacturng n Laonng, Dalan Unversty of Technology, Dalan,1164, Chna. School of Mechancal Engneerng, Dalan Unversty of Technology, Dalan, 1164, Chna Abstract: Geared rotor system s the core component of mult-shaft ntegrally centrfugal compressor, as well as the part wth hgh falure rate. Consderng the problems of hgh cost of experment, and longtme cycle when drect usng prototype machne, t s necessary to test ts characterstc through a test rg wth the same structure and same dynamcs. Ths paper takes the rotor system of a fve-shaft ntegrally centrfugal compressor as the research obect. Through theoretcal analyss and smlar desgn theory, the desgn method of the fve rotor systems are studed, and a smlar l of the geared rotor system n ntegrally centrfugal compressor s obtaned. At last, the valdty of the scale l s confrmed by comparatve analyss, whch provdes expermental bass for the future expermental research. Keywords: Integrally centrfugal compressor; Geared rotor system; Dynamc smlarty desgn; test rg 1 Introducton Integrally centrfugal compressor s an advanced equpment wdely used n energy, petrochemcal and other felds. As the core component of ntegrally centrfugal compressor, ts rotor system conssts of a number of rotors meshng through gears, and long tme workng above crtcal speed wth hgh load. Its dynamc characterstcs are complex and cause hgh falure rate. The dynamc characterstcs of the rotor system have drect nfluence on the dynamc characterstcs of the whole system. At present, due to the lmtatons of the theoretcal l, expermental analyss s stll the best way to study the dynamc characterstcs of the rotor system. For a long tme, the theoretcal calculatons are often used on the dynamc desgn of the gear rotor system but wth few expermental data supported, whch makes the desgned system have a serous securty rsk and cause maor economc losses [1]. Wehrman[] ponted out that the dynamc problem of the rotor system s the key to * Correspondng author: Qngka Han (hanqngka@dlut.edu.cn)

2 the desgn of the whole compressor; Gruntfest[3] descrbed that under hgh temperature operaton the bearng of tltng tle solved the nstablty phenomenon of rotor system of mult-axes gear compressor and subsynchronous vbraton of a rotor system through eddy current brake. Moore[4] proposed an dea that the stablty of the rotor system would not be reduced wth the ncrease of load. In order to better study the rotor dynamc characterstcs of the fve-axes gear assembly centrfugal compressor and solve the problem of dffcult system testng, a correspondng expermental platform should be bult usng smlar desgn theory. At present, Dynamc smlarty theory has been wdely appled n large scale structural dynamc test ls. Harrs[5] ntroduced the applcaton of smlarty prncple and smlar test l n structural analyss of brdges and buldngs. Cho[6] proposed a dstorton l test method for the dstorton smlar l test of composte structure system. Chnese scholar Hu Pemn [7] analyzed the smlar l tests of rotor torsonal vbraton characterstcs, and the results showed that for a rotor system wth rgd ends at both ends, when ts geometrc sze was scaled down to ts orgnal 1/n, ts frequency characterstcs and vbraton s were bascally unchanged, and the natural frequency ncreased to ts orgnal n tmes. LUO Zhong[8,9] systematcally studed the theory of smlarty desgn, and proposed a dynamc smlarty desgn method for mult-axes rotor bearng system dstorton test l based on the transfer matrx method and senstvty analyss, and the method was Verfed. At present, due to the larger sze, compact structure, complex workng envronment and hgh rotatng speed of ntegrally centrfugal compressor, usng prototypes drectly for testng costs too much tme as well as money. Therefore, only theoretcal analyss and numercal smulaton are used n practcal engneerng desgn for theoretcal gudance. It s necessary to establsh a test rg wth smlar structural and smlar dynamcs. In ths paper, based on theoretcal analyss and smlar desgn theory, the desgn method of the scale test l for the ntegrally centrfugal compressor rotor system s studed, and the dynamc scale smlarty l of geared rotor system s obtaned, whch can provde experment bass for the dynamc characterstcs research of ntegrally centrfugal compressor. The lng and smlarty desgn method.1 The structural of the geared rotor system n ntegrally centrfugal compressor The dynamc l of the gear system n ntegrally centrfugal compressor wth fve shafts s shown n Fg.1. The system conssts of fve parallel rotors meshng wth helcal gears and supported wth bearngs at each end. The nput axs s connected to the steam turbne trough the couplng, and t can also be connected to the ntermedate shaft usng an addtonal gearbox. 1

3 O3 K5L, C 5L K 5R, C 5R O -F 4 F 4 K 4R, C 4R -F 3 F 3 M K R, C R K 4L, C 4L -F K L, C L F O1 I F 1 -F 1 K 1R, C 1R K 3R, C 3R K 1L, C 1L K 3L, C 3L The K L, C L Fg.1 The dynamc l of the geared rotor system and the K R, C R are used to stand for the stffness and dampng coeffcent matrx of the left and the rght supportng bearng of rotor ; the meshng force of the gear.. The lng method of the geared rotor system F s the For those rotor-bearng systems coupled by gear, the fnte element method can be used to splt the parts of the system nto dfferent fnte elements for lng and theoretcal analyss. There are several assumptons for the gear couplng rotor system lnear lng: a. Bearng blocks and foundatons are consdered as rgd; b. Assumng that the deformaton of the shaft n the process of operaton s very tny, and the non-lnear effects whch caused by the axal deformaton can be gnored, regardng the axs as a lnear elastc axs; c. Consderng the mpeller, wheel, drvng gear and axal sectons whch have an larger sze mutaton as rgd dsks; d. Approxmatng the bearng to the lnear sprng dampng element by neglectng the nonlnear characterstc of the bearng; e. Approxmate the gear meshng elements as lnear sprng dampng elements by gnorng the nonlnear factors such as tooth profle error, tooth sde gap of the gear. In ths l, each node has sx degrees of freedom. The dfferental moton equaton of the geared rotor system can be wrtten as MX ( C G) X KX F (1) where, X s the dsplacement vector of system node, M s the system mass matrx, C s the system dampng matrx whch ncludes the bearng dampng and the nternal

4 dampng of materals, G s the system gyroscopc moment matrx, s the rotor speed, K s the system stffness matrx, F s the force vector..3 The smlarty desgn method of the rotor system The smlarty desgn s a certan knd of specal method whch made the physcal quantty of the desgn system s proportonal to the physcal quantty correspondng to the same phenomenon of the orgnal system. Accordng to the smlarty prncple of complex system, the smlarty of the whole system wll be guaranteed when the ndependent subsystems and the subsystems connectng them are smlar at the same tme. Therefore, for the geared rotor system n Fg.1, the smlarty desgn of rotor system can be acheved by the sngle smlar desgn for each rotor. For the rotor system, the dfferental vbraton equaton can be descrbed as [11] : where, mx () y y y y t EI a a m p xe () x x x t x t t s the the mass of the unt length of the rotor, E s the elastc modulus of rotatng shaft, I s the secton moment of nerta of the shaft, ax () s the moment of nerta of unt length relatve to rotaton axs ( the moment of nerta of the axs can be gnored), y s the synthetc deflecton of the axs, px () s the strength of the rotor unbalance force, p x me x, ( )= ( ) ex () s the mbalance curve, s the rotatonal frequency of the rotor. As supplement to the Eq. (), the expresson of the nclnaton angle and normal stress of the axs elastc lne can be wrtten as d y/ dx (3) M / W where, M* s the bendng moment, W s the coeffcent of flexural secton. When desgnng the l, selectng those values whch can actually be changed ndependently of each other as ndependent values, then the scale of the remanng values can be found n the smlarty ndex correspondng to the crtera. On the premse of meetng the requrements of rotor dynamcs, t s also necessary to smplfy the structure of the complex rotor. The the crtcal speed ncreases wth the ncrease of the flexural stffness of the shaft secton, the senstvty of the th crtcal speed to the bendng stffness EI of the th shaft secton can be wrtten as N y y 1 3 l y y 1 1 l y y ( EI) π Nl In ths formula, 1 l 3 * ( 1,,3, ; 1,,, N 1) (4) s the length of the th axs, N s the number of nodes, the y, y and the, 1 respectvely means the regularzed lnear dsplacement and the angular dsplacement ampltude value of the left and rght nodes of the th shaft at the manly th vbraton.

5 The nfluence of shaft length on crtcal speed s exst n both postve and negatve sde. The senstvty of the th crtcal speed to the length of the th axs can be expressed as N 18 EI 4 9 y y 1 6l y y 1 1 l 1 1 l π N l ( 1,,3, ; 1,,, N 1) (5) The crtcal speed decreases wth the ncrease of the mass of nodes, the susceptblty of the the th crtcal speed to the mass of the th node and the moment of nerta can be expressed as J p J d N N y m J N J p d N ( 1,,3, ; 1,,, N 1) If the qth crtcal speed needs to be changed, the change amount s The exstng n parameters T (6) p 1,,,n N N1 N N q (7) ( )s allowed to have an approprate adustments, assume the change of the parameters s : So there s where, J s the Jacob matrx of the order T P1 P P n P (8) JP = N (9), concretely can be wrtten as formula qn (1), and the matrx elements are the senstvty of the crtcal speed to each parameter. N1 N1 N1 p1 p p n N N N N1, N,, Nq J= p1 p pn p1, p,, pn (1) Nq Nq Nq p1 p p n Accordng to the Eq.(9) and Eq.(1), f q n, the Eq.(9) could have nfntely many solutons. Meanwhle, the senstvty calculaton formulas of the Eqs. (), (5) and (6) are obtaned by gnore the trace above the second order, thus multple optmzaton to reduce error s necessary. 3 Desgn and verfcaton of the ntegrally centrfugal compressor test rg 3.1 Smlar desgn of the ntegrally centrfugal compressor test rg 4

6 The smlar desgn of the rotor system s acheved by ndvdually smlar desgns of each rotor. The desgn data of each rotor of the prototype are shown n Table 1, where I, M, O1, O and O3 stand for Input shaft, Intermedate shaft, Output shaft 1, Output shaft and Output shaft 3 respectvely. Table 1. Man parameters of fve-axs gear assembly compressor ptch dameter Transmsson rato Workng Revoluton 1st crtcal speed nd crtcal speed D(mm) n(r/mn) Hz (r/mn) Hz (r/mn) Hz I M O O O The smlar l of the rotor system should satsfy smlar prncples. a. Geometrcal smlarty. The spatal arrangement angle of each parallel shaft rotor s smlar to that of the prototype.the pressure angle and tlt angle of gear are consstent wth the prototype; b. Knematc smlarty. The transmton gear rato of each gear s smlar to that of the prototype. c. Dynamcal smlarty. The workng speeds of the nput shaft and the ntermedate shaft are below the 1 st crtcal speed, the operatng speed of the output shaft 1 s between ts 1 st crtcal and nd crtcal speed, and the operatng speeds of the output shaft and the output shaft 3 are both between the nd crtcal and the 3 rd crtcal. Accordng to the dynamc smlarty desgn method of dstorton l of multsegment rotor system, the materals whch are consstent wth that of the prototype rotor to do smlarty desgn are used.then do dynamc lng to obtan the dstorton l. At last, accordng to the Eq.(1), dynamc correcton s done to make the crtcal speed smlarty rato approxmately satsfy the equaton. The result s shown n Fgure. Fg.. Smlar desgn results for a fve-axs compressor rotor system 5

7 3. Numercal verfcaton of the dynamc characterstcs of the test rg By the method of unaxal smlarty verfcaton, the vbraton s and Campbell dagrams of the nput shaft, ntermedate shaft, and three output shafts of the prototype and scale l are calculated. The results are shown n Tables -6. Table. Comparson between ntrnsc characterstcs of the prototype nput shaft and the scale l nput shaft prototype Scale l The 1 st vbraton campbell dagram Workng frequency 工作转频 n(r/mn) Workng frequency 工作转频 n(r/mn) Table 3. Comparson of nherent characterstcs of the prototype ntermedate shaft and scale l ntermedate shaft prototype Scale l The 1 st vbraton 6

8 Campbell dagram Workng frequency 工作转频 n(r/mn) Workng frequency 工作转频 n(r/mn) Table 4. Comparson of the nherent characterstcs of the prototype output shaft 1 and scale l output shaft 1 The 1 st vbraton prototype Scale l The nd vbraton Campbell dagram 4 Workng frequency 工作转频 n(r/mn) Workng frequency 工作转频 n(r/mn) Table 5. Comparson of nherent characterstcs of prototype output shaft and scale l output shaft The 1 st vbraton prototype Scale l 7

9 The nd vbraton Campbell dagram 4 Workng frequency 工作转频 Workng frequency 工作转频 n(r/mn) n(r/mn) Table 6. Comparson of Intrnsc Characterstcs of Prototype Output Shaft 3 and Scale Model Output Shaft 3 prototype Scale l The 1 st vbraton The nd vbraton Campbell dagram 4 Workng 工作转频 frequency n(r/mn) n(r/mn) Accordng to Table ~6, the vbraton and the Campbell dagram of the prototype rotor s bascally consstent wth that of ts smlar rotor below the operatng speed, and there s a good smlar relatonshp betweeen the two rotors. Accordng to the smlarty prncple of the complex system, the test-rg rotor system obtaned Workng frequency 工作转频 8

10 through smlar desgn can reflect the dynamc characterstcs of the prototype rotor system. The man parameters of each rotor n the obtaned rotor system are shown n Table 7. Table 7. Man parameters of the test rg Number of teeth Gear Dameter Transmsson rato Work Revoluton 1 st crtcal speed nd crtcal speed z D(mm) n(r/mn) Hz (r/mn) Hz (r/mn) Hz I M O O O From Table 7, t can be seen that each of the output shafts satsfes the requrement of dynamc smlarty when each rotor rotates up to the workng speed. 4 Conclusons Ths paper s proposed a smlarty desgn method for the geared rotor system n ntegrally centrfugal compressor based on smlarty desgn prncples. Accordng to smlarty desgn method of the geometrc dstorton, the dynamc characterstcs of the scaled test rg below the workng speed are same wth the prototype. By comparng the vbraton l and Campbell dagram between the prototype rotor system and test rg rotor system, t shows that the rotor system obtaned by the smlarty desgn method has smlar dynamc characterstcs at the workng speed, whch meets the basc requrements of the ntegrally centrfugal compressor s dynamcs experments. 5 Acknowledgements Ths proect s supported by Natonal Natural Scence Foundaton of Chna (Grant No ). References 1. Km K H, Fleeter S, Compressor unsteady aerodynamc response to rotatng stall and surge exctatons, Journal of Propulson and Power, 1(5)(1994) Wehrman J G, Walder T E, Haryett N J, The Use of Integrally Geared Compressors Based on Two Industral Gas Companes Experence, Proceedngs of the thrty second Turbomachnery Symposum, Texas A&M Unversty, 3. 9

11 3. Gruntfest P M, Androns L, Marscher W D, Rotor nstablty problems n an ntegrally geared compressor supported by tltng pad bearngs, Turbomachnery Symposum Proceedngs, Texas A&M Unversty, (1) Morn M, Pnell M, Venturn M, Analyss of bogas compresson system dynamcs. Appled Energy, 86(11)(9) Sabns G M, Harrs H G, Whte R N, et al., Structural Modelng and Expermental Technques, Boca Raton: CRC Press, (1999) Cho U, Dutson A J, Wood K L, et al., An Advanced Method to Correlate Scale Models Wth Dstorted Confguratons, Journal of Mechancal Desgn, 17(1) (5) Hu P, Research on Smlarty Law of Rotor Torsonal Vbraton Test, Mechancal Engneer, S1(1996) Luo Z, Sh H, Chen X, Yu Q, Dynamc Smlarty Desgn Method of Dstorton Expermental Model for Multaxal Rotor System, Journal of Northeastern Unversty Nature Scence, 36(4) (15) Luo Z, Chen G, L J, Wang F, Desgn of Dynamc Smlarty Model of Rotor System Consderng the Bearng Stffness, Journal Of Northeastern Unversty Nature Scence, 36(3) (15) Wang C, Study on Integrated Control Method for Centrfugal Compressor, Dalan Unversty of Technology, Luo Z, Guo J, Tang R, Han Q, Wang D, Smlarty Research on the Nonlnear Dynamc Characterstcs of Rollng Bearng n Rotor-Bearng System, Journal of Dynamcs and Control, 14(3) (16)