Global coupled equations for dynamic analysis of planishing mill

Vol.13 No. 1Trans. Nonferrous Met. Soc. ChinaFeb. 2003Article ID: 1003 - 6326(2003)01 - 0014 - 06Global coupled equations for dynamic analysisof planishing millCAI Gan-wei(蔡敢为)', ZHONG Jue(钟掘)*(1. College of Mechanical Science & Engineering,Huazhong University of Science and Technology , W uhan 430074, China;2. College of Mechanical & Electrical Engineering,Central South University, Changsha 410083, China)Abstract: The dynamic properies of rlling mill are significantly influenced by many coupling fectors. According10 the coupled mechanical and electric dynamics theory, the global coupled equations for the dynamic analysis ofplanishing mill CM04 of Shanghai Baosteel Group Corporation were derived, by using finite element methods. Theseelastu-dynamic equations establish the coupling relations among the stand vibration system, torsional vibration sys-tetn, driving motors, etc. It provides theoretical basis to a certain extent for globally dynamic simulation, analysis ofstability of motion, prediction of abnormal operating mode, globally optimum design and control, etc.Key words: rllig mill; dynanie analysis; electromechanical couplingCLC number: TG333; TH113Document code: A1 INTRODUCTIONThe main constituents of planishing millCM04 of Shanghai Baosteel Group CorporationModern high-speed rolling mill is a complex e-(CBGS) are shown in Fig. 1. Its mechanical mainlectromechanical system in which many couplingblock is composed of stand vibration system andfactors exist. These factors have significant influ-torsional vibration system, and both of them areence on the dynamic properties of the rolling mill,coupled with rolling interface. The torsional vibra-which can't be explained only with separate cou-tion system is a multi-shaft system with each shaft-pling or their superposition. For example, the in-ing separately driven by direct-current motors, andthey are connected by the strip steel. The electro-lectromechanical coupling and interface couplingmagnetic fields among the air gap of electromotorscan hardly be explained by the superposition of ecouple shaftings with electrical system.lectromechanical coupled forced vibration and self-There has been a“ghostly vibration" withoutexcitation vibration of rolling interface. During theany obvious law in the working process of the plan-last few years, a lot of works on the worldwideishing mill. This often results in the light anddifficult problem of the vibration of rolling millshade alternating stripes on the plate being flat-have been done by many specialists from differenttened, which significantly influences the quality ofcountries, but few of them had attached enoughproducts. In this paper, the global coupled equa-importance to the many coupling factors which dotions, which indicate the relations among the standexist in the rolling mill, and the dynamic model ofvibration system, torsional vibration system, driv-rolling mill established by them can be classifieding motors of the planishing mill are established,into two kinds: stand vibration system and torsion-so as to study more deeply the multi-unit andal vibration system of main drive shafting[*1. Al-multi-dimensional relations among all kinds othough great success have been made on the studyprocesses and parameters, the law of feedback be-of such stand and torsional vibration subsyster,tween the operation mode and the function of ma-there are still many vibration behaviors can't be ex-chine set.plained reasonably. Therefore, it is quite necessaryto take all kinds of coupling factors into account2 VIBRATION EQUATIONS OF STAND VIBRA-and research the vibration principle of rolling millTION SYSTEMfrom the viewpoint of the whole system more deep-ly°!l.The stand vibration system of the planishingD Foundntion item: Projects( 50175031. 5085170) supported by the Na中国煤化工ina; PrleeOJY204)supported by the Natural Science Foundation of Hunan Province, Chinas:YHC N M H Gion of Education ofrofHunan Province, ChinaReceived date; 2002 - 06 -241 Accepted date; 2002- 10 -27Correspondence:CAl Can-wei, Profesor, + 86-27-87543872, caiganwei@163. comVol.13 No.1Global coupled equations for dynamic analysis of planishing mill。15。Stand vibrationShim block,systemoil cylinder, etc.Rolling interphase| DC motorStrip seMill housingCoiler shafting| TorsionalvibrationRolingysco/ funcoler satingMain drive shafing| Electrical drive & control systemFig.1 Sketch map of planishing mill CM04 systemmill is composed of mill housing， backing roll ,quations of the stand vibration system from the fi-working roll, oil cylinder, wedging block, bear-nite element model:ings and bearing brackets, etc. Previous studiesM.U +CU +KU.=Q .(1)demonstrate that the finite element method withwhere M， C and Ki respectively represent theappropriate discrete scale is suitable for the dynam-mass matrix, damping matrix, and stiffness matrixic analysis of rolling mil[lo]. Therefore, the vibra-of the stand vibration system; U, u and 0 aretion equations for the stand vibration system ofthe column matrix of the generalized coordinates offour-high mill are established with the finite ele-the system and its first and second derivatives; Qment model.is the column matrix of generalized force whose el-The model of the stand vibration system isements are composed of the draught pressure p.shown in Fig. 2. The mill housing is divided into18 beam elements. The oil cylinder and wedging3 DYNAMIC EQUATIONS OF TORSIONAL VI-BRATION SYSTEMblock are regarded as one pole element. The rigidi-ty of the oil film in the oil cylinder, the rigidity ofThe torsional vibration system of the planish-bearings and bearing brackets of upper backing rolling mill is composed of uncoiling shafting, mainand the flexural rigidity of upper backing roll aredrive shafting and coiling shafting. Each shaftingtaken into account in the computation of elementis independently driven by DC motor, and then isstiffness matrix. Each roll set is regarded as ajointed together through the strip. The dynamic a-lumped mass element, and these equivalent massesnalysis of torsional vibration system is as follows.are calculated according to the vibration modecurves of the rollers with conservation of energy3.1 Dynamic equations of stripprinciplelol. The interface between the backingThe thickness and elongation of the strip be-roll and the working roll is respectively simulateding planished are both very small(O. 3 - 3. 5mm,as elastic elements whose stiffness is the elastic0.5%- 2.1%), so the variation of the thickness ofcontact stiffness between the rollers. The constitu-the strip is neglected in the global model of plan-ent between the lower backing roll and the bottomhe strip there are thebeam is represented by a pole element. Accordingtens中国煤化工the uncoing andto the real condition of the planishing mill, twocoili-tral point there isconstraints are placed on node I and node 15.theTYH.CNMHG"by prie wbrereEach node and generalized coordinate is shownrolls. The strip is simulated with finite pole ele-in Fig.2. Therefore, we can obtain the dynamic e-ments, as shown in Fig. 3. The dynamic equation●16●Trans. Nonferrous Met. Soc. ChinaFeb. 2003ψU20[623L Uz1U24⑧七U19+ U25U17⑥ψU29A Xus|6十U16 .@②0H30U28 .③Uis七UI3@|0||Un4 UssU12→-U1o82236V34|2③|ius四↓438,之Us63|i3[②❷4 Uss |U4° Us12 L4014s U2④||4U44ouL十U回45 Us3U47↓U53IUso分HHt Usi⑥U48!➊Us2-UsqU468Fig.2 Calculation model of stand vibration system of planishing mill2:24262三王二三王.r° Usg2V60四U61U63团U64 五Fig.3 Finite element model of stripof the strip being composed of certain number ofid- body translational acceleration of every elementpole elements is:of the strip, which can be regarded as zero in sta-M2Dz +C:Uz+K:U2 =Qx -M2Uz(2)ble rolling; Qz is the generalized force column ma-where M2, Cz and K2 are mass matrix, damping中国煤化工foree T，T2 andmatrix and stiffness matrix of the strip systemn;fric. IU2，Uz and Uz are column matrix of elastic coordi-TYH.CNMH Gnate displacements of the system and its first and3.2" uynamnc equations OI main drive shaftingsecond derivatives; Ua is the column matrix of rig-The main drive shafting is composed of severalVol. 13 No. 1Global coupled equations for dynamic analysis of planishing millinertia components ( including electromotor,component j; w is the angular frequency of funda-clutch, working roll, etc) and elastic componentsmental component; 0 is the phase angle of har-(including coupling spindle, etc)， which can bemonic component j.simplified as shaft and board torsional vibrationFrom this“mass-spring system”, dynamic e-system, as shown in Fig. 4. When establishing thequations of main drive shafting is established :dynamic equation, the thick and short componentsMsUs +GUs +KgUs =Qs - MsUs .(5)are simplified as inertia, and their torsional rigiditywhere Mg, Cg， K: are respectively mass matrix,is neglected. The slender components are simpli-damping matrix and stiffness matrix of the mainfied as a torsional spring, and their rotary inertia isdrive shafting; U,s is the column matrix of rigid-neglected. Mscl, Ms are electromagnetic torque ofbody rotational acceleration of the main driveelectromotor, N; is working torque:shafting; Us is generalized coordinate, which re-N:= fRp(3)presents the elastic torsion motion that overlaps onwhere R is the radius of working roll, f is thethe rigid- body rotation of the main drive shafting;friction coefficient, p is the draught pressure.Qs is the generalized force column matrix beingAccording to Ref. [11],electromagneticcomposed of the torque of working roll, electricaltorque Mse. ; is made up of the torque resulted fromtorque of the electromotor, etc.DC component Is and the torque resulted from har-monic component of commutating currente11] :3.3Dynamic equations of uncoiler and coilerMe. = Cu.. ..+C.m.o.1shaftingcos(jut- 8,)(4)Uncoiler and coiler shafting is also a torsionalwhereCm.t is the constant of electromagneticvibration system being composed of inertia ele-torque of electromotor i,中is the magnetic flux ofments and elastic elements, as shown in Fig. 5 andelectromotor i; Im; is the amplitude of harmonicFig. 6. Ms, . is the electromagnetic torque that the1JU7o iMae,12930323334__43Nj⑧四\|四\|@\|②|国|Fig.4 Calculation model of main drive shafting(J,一Rotational inerti of the first electromotor, eletromotor shaft couplig, the second etromotor, shaft couplingof output shaft, intermediate shaft, spindle, working roll, 0 - Correponding torsionel elastic element;Mai，Ma - Electromagnetic torque of eletromotor; N2 - - -Torque of working rol)1gNsU73U7s1dc3637383940Mdc.4国\|国\面③33\●中国煤化工Fig.5 Calculation model oCNMHG