Research on Axial Trail of High-Speed Gas Painting Automizor

Transactions of Tianjin UniversityVol 9 No. 1ar.2003Research on Axial Trail of High-SpeedGas Painting AutomizorYANG Zhi-yong(杨志永), ZHANG Da-we(张大卫), HUANG Tian(黄田)School of Mechanical Engineering Tianjin University, Tianjin 300072, ChinaAbstract The non-linear dynamic model is set up of one type of high-speed painting automizor with gas suppoing system. The stability of motion and dynamic response of the gas painting automizor system are studied overrelatively wide range of rotating speed by numerical analytic method the critical velocity under working condition is found, and rotate stability and critical condition are discussed in theory. Furthermore, the range of thecritical parameter of the system when Hopf bifurcation occurs and the law between axis trace and bearing clear-ce are acquired,tooKeywords gas bearing painting automizor stability of motion axial trail critical velocityArticle ID:006-4982(2003)1-0045-0The auto-static painting is one of the most important speed gas painting automizor. Suppose gas has noworking procedures in making the high- level car i stability pressibility and it is the 2-D constant flow. The functionof motion and axial trail in process are the key factors that of motion and continuity of gas in cylindrical coordinate areffect the quality of car surface and the performance, lifeduand reliability of painting automizor which is an important(1a)part. Up to now, the research on stability of motion hasbeen done well by many scholars1迎=a2(1b)sults have also been gotten. But for the gas bearing forced damp are usually solved with1=0dz(1c)linear method thus some important dynamic characters areignored inevitably and most of the objects they studied are0r dethe big rotor-bearing system of generator which is simpli- wherefied into beam or pad which has many concentrated mas-ofses. The dynamic response is analyzed with modal synthe-u, -Radial speed of gassis, impedance mapping method and other dynamic sub-6— Angle between randeaxIstructure methods. But the response of this type of rotorr--Radial distance between the shaft and bearing ibearing system with high speed light load and single bear-z-Coordinate in axis directioning has not been found by nowAccording to the above, this paper will aim at a typeu-Dynamic viscidity of gas and u=pu iof high-speed gas painting automizor system and settinDensity of gasup non-linear dynamic functien the basis ofp--Pressuremeric analysis it is necessary to analyze the stability of mo-Considering the axes disturbance and boundary condition and dynamic response in wide rotate range and getle pressuredistributing of gas bearing by solving thecritical value and the rule between the track of axes centerIng clearanceTH中国煤化工CNMHG1 Capacity force of gas bearing** Aceepted date 2002-09-02YANG Zhi-yong born in 1972 male Dr associate Prof.E-mail azy (@aFig 1 shows the mechanical structure drawing of high-Supported by Tianjin Key Projec( No. 993802611Transactions of Tianjin University Vol 9 No I 2003EP=Po=0+1Bud 2e+25(2+5 cos 0 )sin 0(2+E2)H0(2+62)(1+cos)EaH(1+5cos0)(1+)(3)L-Length of bearing iE=l+8 cos 0,udtcP--Angle between random point and x axisDifferential coefficient ofDifferential coefficient of eAccording to transformation between the capacity forcein Cartesian coordinates and polar coordinate thenF=FFFF sinFront sheltand for sin cos c=xe=x+y2 e=Back shellrx t v1bud.LHoHLetA=,22、,B=X2+2(2+2C=x1=ym2+2),D1-x2+2Capacity force value in x and y axis arδDy )6)I-Painting cup 2-Shaft 3--Gas bearing 4--Push trayF,=-n“n-24X4y+Bx)+(Gy+Dx)}(7Fig. 1 Mechanical structure drawing of high-speed2 Axial trail of shaftgas painting automizorIn order to test the performance of motion of shaft themass and centrifugal force are both ignored temporarily thePeso-Pressure when the angle 0=0axial trail will be analyzed when only the gravity of shaft isn-Rotating speed of automizorconsidered. The relation between the bearing parametersd-Diameter of shaftwith the load is shown in Fig. 2e-Offset of center between the shaft and bearingHo-Offset of radius between the shaft and bearingnd HoCapacity force of bearing is gotten by integral EqF,=01(n-2i)E1+2E2](4))E3+2E4](5)H中国煤化工Er(1-2X2+2)CNMH Gers with loadE2246YANG Zhi-yong et al Research on Axial Trail of High-Speed Gas Painting Automizor12udLthe dynamie differential equation of gas painting automizorF sin eB6。√1-(2+2)is given in cup center coordinateFt mew sIn (t(10)FAccording to the above equation, thenSuppose at= KT, and K is positive integer T is non-di-d sin 0) 26 ud Lmension time. Change the parameter into non-dimensionsin 002(1-FH02(1-2)format(8)KxIntegral sin 0 of Eq (8)HHow(1-52)6 LD…=ecausebove has not accurate resultthe axial trail can be given by integrating one by one in aHsmall range. The following is an integral in a small rangeHo(1-8eHK281(1-y4=sThen the differential function in non-dimension formatThe integral of Eq (8)isis given( the sign x y e in function is still expressed1-52by x y s)(9)5FH0E1-2i=f+ea sir KThere C, is decided by boundary condition E=5 0=0(11)f +g+eG co( KT)3 Dynamic model of system and analysis of wherestabilityFf =e=tK-2 x-Ax+ By )+(-Cx+Dy)jince it is impossible that the center of axes andpainting cup are in the same line when painting automizorftK-2 x Ay +Bx)+( cy + Dx)]it is necessary to analyze the force of theExpand the gas supporting force into Taylor series inpainting cup so as to analyze the influence of motion on the point( xo p yo p ), and then the force f and f, are exproduct qualitypressed asFig 3 shows the force analysis with cup in motion.f=f20+a1△x+a2△x+a3△y+a4△jf,=f,wherea(i=1 2 3 A)Taylor series coefficient of f,b i=l 2 3 4)Taylor series coefficient of f,d, dd df,dfo,;o,The differential equation may be changed into standI form in the critical pointX=AFig 3 Force analysis of gas bearing in motionH中国煤化工CNMHGw is rotating speed which is expressed in Fig 3, thenT D Pcos TTransactions of Tianjin University Vol 9 No I 20030100position is Ci>-09, the axial trail of shaft does not include center of bearing, and its shape shows as ellAwell. If the original position C) is selected properly its ax0001ial trail can also show as a circle or one pointb, b, b bFig 5 and Fig. 6 show that the axial trail of cup centerIt can be estimated whether the system is stable or notby solving the eigenvalue of matrix A,. At the same timethe axial trail can also be gotten by solving equation groupwhich changed through Eq (10)01x2-21y2Ax,+By)a1+2 ty1(-Cx,+Dy,1+ea,cos(Kr(13)-1-2y1x2-x1y2Ay +Bx)C=-0.6Cy,+ Dx,)+g+ea sin( KrC1=-084 SimulationsThe physical and geometrical parameters of gas painting automizor are given in Tab. 1. By solving the function(13 by four-rank Rounge-Ktsimulation result is shown in figs 4-7Tab 1 Physical and geometrical parametersig. 4 Axial trail of shaftof gas painting automizorHty,/(Min radius D/mm17.8Min radius d/Max radius D./mmMax radius d/mmm。-L51.30Rotating speed n/t Time r without dimensionTime r without dimensionDensity of cup P /(tm)Density of tray p, /(t. m-3)Equivalent thickness of cup 8,/mmEquivalent radius of cup r,/mmEquivalent thickness of tray 8,/mEquivalent radius of tray r2/mmFig. 4 shows the change of axial trail when the shaft中国煤化工forced by gravity gets into equilibrium state at workinCNMHGO005 0.010L…xakspeed of n =30 000 r/min. When the original position isC,s-0.9, the axial trail of shaft includes center of bearFig 5 Response of system in the conditionws as anWhen the ori(H0=8.5umn=9000r/min48YANG Zhi-yong et al Research on Axial Trail of High-Speed Gas Painting Automizorin this rotating speed rangeK日8Fig. 7 shows the relation between the failure rotatingspeed andthe bigger the bearing clearance is the smaller the failure rotating speed is in wholetrend. The minimum failure speed is 2 000 r/min in work505 ConclusionsTime r without dimensionTime r without dimensionThis paper is aimed at a type of high-speed gas paiing automizor system and setting up non-linear dynamicfunction. The simulation of the gas painting automizor andcup shows that1) The eccentricity angle of shaft has relation with the-00100.00500.0050010original position in order to overcome the gravit2) The time historyIt toFig. 6 Response of system in the conditionemanative by comparison when the rotating speed is ran-(H0=8.5μmn=10000r/mn)ging from n=9 000 r/min to 10 000 r/min, this shows gaspainting automizor has Hopf bifurcation in this speed8000range3) The minimum failure speed is 2 000 r/ min in work6state and the relation between the minimum failure rotaig spThe above offers an important foundation for the de-sign maintenance and service of gas painting automizor il200engineeringReferencesClearance H/μm1] Abdul-waged N. Stability and unbalance response of large turbineFig. 7 Relation between the minimum failure rotatingbearing[ J]. Transactions of the AsMe ,J of Lubrication Technologyspeed and clearance between shaft and bearing982104(1)56-762] Nikolakopoulos P G. Dynamic stability of linear misaligned journalhen the bearing clearance is Ho=8. 5 um and the rotorbearings via Lyapunov's direct method[ J ] Tribology Transactions1997A01)138-146rotates at different speeds. The time histories of x-axis andy-axis are from convergent to emanative by comparisoning[ J] Tribology Transactions, 1997, 40(1): 147-155This shows that gas painting automizor has Hopf bifurcation中国煤化工CNMHG