高速铣削颤振系统稳定性及分岔的Chebyshev多项式数值分析

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Mar.2013机 床 与 液 压Hydromechatronics Engineering Vo1.41 No.6DOI：10.3969/j.issn.1001-3881.2013.06.004Prediction Chatter Stability and Bifurcation inM illing M achineZHAO Demin ，ZHANG Qichang. Department ofChina University2.Department ofEngineering Mechanics，Colege of StorageofPetroleum，Qingdao 266555，China；& Transportation and Architectural Engineering。

Mechanics，Colege ofMechanical Engineering，Tianfin University，Tianfin 300072，China1.IntroductionAbstract：The shifted Chebyshev polynomials and Floquet theory are adopted for the predictionchaer stability and bifurcation in miling.The stability Iobes diagram is obtained.The stability inmiling can wel be predicted by the lobes diagram.The muliti-periodic and Hopf bifurcations aredetected by the Eigen-values analysis.The results showed that the stability solution of the systemtransform from the stable equilibrium point to the Iimf cycle oscilatory after multiple cycle bifurca-tion.and it transforms to the quasi-periodic oscilation after Hopf bifurcation.The numerical re-suits of the Poincar6 section prove that the occurence of the quasi-periodic oscilation。

Key words：miling，chaer stability，Chebyshev polynomials，bifurcationHigh-speed milling is in aerospace，ship，andmany other industries due to its advantages such ashigh materia1 remove rates，beter surface finish andlow cost.However，the chatter vibration of the ma-chine tool·workpiece system is not only one of themain limitations for poor workpiece surface qualitybut also promotes wear of the machine tools. Thebasic and comprehensive mechanism of the machinechater was presented by Tobias[1]and Ahintas[2]。

Because the cuting force is time-varying，it canbe approximated by the zero-order or one·-order Fou-tier series. Based on this principle，the analyticalstability prediction method in frequency domain wasReceived：2012-12-09Project supposed by the Fundamental Research Funds forthe Central Universities(1 1 CX04049A)，National Natu-ral Science Foundation of China(10872141)ZHAO Demin，Doctor.E-mail：zhaodemin### upc.edu.cnintroduced for the stability lobes in milling by Altintas[2-3].Altintas[4]and Tang[5]summarized ana。

1yrical stability prediction method in equeney do-main and semi.discretization method in time domainfor the two-or muhi-degree-of-freedom (MDOF)sys-tem moda1.The stability analysis on an uncertain dy-namics milling model was performed and probabilisticinstead of determ inistic stability lobes were obtainedin Reference[6].Faassen[7]and Quintana[8]presented an experimental method to identify stabilitylobes diagram in miling operation.Gradisek[9]re。

vealed periodic and quasi-periodic chatter by usingthe semi-discretization method. The quasi-periodicsolutions of the time-periodic delay diferential equa·-tions in high speed milling system were also identifiedby some miling experiments[10]。

The time·varying periodic cutting force approxi·-mated by the zero-order or one·order Fourier series isnot accurate for high-speed milling.Chebyshev poly·nomial[1 1-14]is an eficient computational schemefor the analysis of the periodic system. Therefore，this paper presented a stability theory which predictschatter stability and bifurcation based on Chebyshevpolynomial rather than Fourier series。

ZHAO Demin，et al：Prediction Chatter Stability and Bifurcation in Milling Machine 232.Dynamics model of millingThe cross sectional figure of the 2-degree-0ffreedom(2-DOF)high-speed miling tool-workpiecesystem is shown in Fig.1.The too]with the diameterD.and teeth number z rotates at an angular speed(rad/s).The radial immersion angle of the ith toothvaries with time as： (t)Ot2.r(i-1)/z.aand a describes the axial and radial depth of immer-sion，respectively.The dynamics model of this ma-chine tool-workpiece system is given by)c )K )圳W here，M ，C and K are the mode mass，dampingand stifness matrix，respectively，F(t)is the cutingforce。

Fig.1 General sketch of the milling dynamicmodel with 2.D0F2.1.Cutting forces modelThe machine tool chatter vibrations occur due toa self excitation mechanism in generation of chipthickness during machining operations.An oscillatorysurface finish left by one of the tooth is removed bythe succeeding oscillatory tooth due to the structuralvibrations.According to Altintas[4]，the resultedchip thickness becomes also oscilatory，which couldbe expressed byhi(t)(zXx(t)sin (t)Ay(t)coskj(t))g( (t)) (2)where Ax( ) (t)- (t-T)，Ay(t)Y(t)-Y(t-T)，g(咖 )describes a unit step function deter-mining whether or not the J tooth is in cuting。

g( ) (3)Where， and咖 are the start and exit angles of thecutter to and from the cutting，respectively。

The tangential F and radial F cutting forcesacting on the tooth j are proportional to the axialdepth of cut ap and chip thickness ( )，F k，aPhi(t)，F krF (4)Where，the cutting coefficients kf and k are con-stant.Resolve the cutting forces in the and Y direc-tionF ：-F日cosbj-F sin由iF”F sin J-F口cosckiThe total cutting forces on the cuttersted by all the teeth are given byN-1 N-lF ∑F ，F Rearanging Eq.(6)in matrix )知 ；where：(5)contribu。

∑ (6)form yieldsi) n (f)儿△yJ(7)0 ( )E -gj(t)[sin2bj(t) (1-cos2cbi(t))]axy(f)J0Ⅳ-1∑JoⅣ-1- gj(t)[(1cos2gb(t))krsin2dpj(t)]( )∑gj(t)[(1-cos2，i(t))-ksin2gbi(t)]JoⅣ-1∑gj(t)[sin2d(t)- ，(1COS2bj(t))]02.2.Governing structure dynamics modelSubstituting Eq.(7)into Eq.(1)yields the fol-lowing coupled delayed different equations with peri-odic coeficient：0(8)In order to normalize the delay period to T 1，we apply the folowing transform ation： ，f ，d (9)Let X ，Y， ，Y )T 1， 2， 3， 4)T.Eq。

(8)can be expressed asX B(t) G( ) -1) (10)(t) (t)，-1≤t≤0Where，B(-t)and c(-t)are al time periodic matrixwith T1，and their expressions are given in Appen-dix A。

3.Shifted Chebyshev polynomials analysisThe shift Chebyshev polynomials can be genera-1J o -p 口l -1， rJ y 2 肼2 甜 ∞ . .y ”y ，J L ，l Lr J 、 、 ) r - -，L ， y - -、 、/ /y r，(L 1J 、 、， ， /、 、 /L /.........L

The coefficients bj can be derived by，( ( i)击J1，2 3-(13)6。 ( )击 0，Let (i) (i)， (i)，， - (j) isan m x l column vector of the shift Chebyshev poly-nomials and(i)，4 (i) (14)Where，厶is 4-order identity and denotes as Kro-necker product.Based on the theory of the ordinarydiferential equation，the solution of the Eq.(10)isgiven by- Xl(-t)X1(0)(B(s)X(s)C(s) (s-1))ds (15)Suppose that X (i)，B( )，C(j)， (i-1)，x(o)can be expressed by the folowing equations：X (i)f0) m。，日(i) (i) Fc(i)于(i) D， ( -1) (i) mo(O)于(i) T(1 mo(16)Here，m1，m0 are the nm x 1 Chevbyshev coefficientsvectors of the solution vector X1(i)，the initial func-tion (i-1)，respectively.F，D are the nm x nChebyshev coeficients matrices of (i)，C(i)，re-spectively.T(1)is defined as厂1 1])：，n I .I.-I l l： ： ：：L 0 0 j(17)Substitute Eqs.(16)and(17)into Eq.(15)，ityields：(i) m (i) T(1)m0c( (i) F (i) ，n (j) D于(i) mo)6 (18)Apply Chebyshev operation matrices G，Q andQ expresed in Appendix B，it yields the Eq。

(19).The process of the derivation can reference toReference[11-14]for details。

(i) Dmo (19)By simplify Eq.(19)，we obtain：[，-e ，]m [7(1)e ]m。(20)Similarly，in the interval[i-1，i]，the ithChevbyshev coefficient vector relates to the counter-part of the previous interval as[，- OF m [ (1) D m (21)W can be defined as an approximately monodro-my operator [，- ，] [ (1) 。] (22)Based on the Floquet theory ，the Eigen-valuesof monodromy operator W can predict the asymptoticstability of system.The cycle solution of the non -smooth dynamical system is stable if al the Eigen-values lie within the unit circle.The multiple cyclebifurcation occurs if one Eigen-value go through unitcircle at point--1 and the saddle·note bifurcation oc·-curs if the one Eigen-value go through unit circle atpoint1.The Hopf bifurcation will take place if onepair Eigen-values go through unit circle at complexnumber。

4.Simulation and discussThe system parameters are chosen according toAltintas[4].Throughout of the paper，the value ofthe parameters are chose as： 900(N/ram )，k 0.3，∞ 510 Hz，(to 802 Hz， 0.04， 0.05，k 96.2 x 10～N/m，k 47.5 x 10 N/m. 4.The cutter applied has four flutes with zerohelix and the cuting condition is half immersiondown milling.The 12-order shift Chebyshev polyno-mial is used in the simulation。

4.1.Chatter stability analysisFig.2 gives the lobes diagram about spindlespeed versus axial depth of cut a .The curve ofthe lobes demonstrates that the system with the pa。

rameters in the region of below the curve is stable asshown in Fig.3 and contrast to that the system is un。

stable if the parameters are in the region of above thecurve.If the system parameters are on the curve，thesystem iS in critical stability。

ZHAO Demin，et al：Prediction Chatter Stability and Bifurcation in Milling Machine 250 40 30 20.10· 0 1-0 2- 0 3. 0 425201 510504Spindle Speed/(r·min )Fig.2 Stability lobes4 。

'1r 0.60 40 20.0 2.0 40 50 100 15O 0 50 l0O l50fFig.3 The time history plots of l and 2，when力：2.0×10 (r/min)，Up：18 mm4.2.Chatter bifurcation analysisThe bifurcation analysis is only discussed whenthe milling system changes from stability to criticalstability.When the parameters of the system are cho-sen on the critical stability curve，the real and imagi-nary part of one pairs of the Eigen-values of monodro·-my operator W， whose modules is max among allEigen-values.varies as the spindle speed as shownin Fig.4.The results indicate that when the spindlespeed is in the region approximately 1.755 ×10 ≤力≤2.238×10 .the multiple cycle bifurcation takesplace.When the spindle speed is in the region 0.5×10 ≤ <1.755 x 10 or 2.238×10 < ≤5×10 。

the Hopf bifurcation takes place.No Eigen-values gothrough unit circle at1，thus the saddle-note bifur-cation never occurs。

Fig.5 and Fig.6 give the phase plane plots ofand Y direction when 2.0×10 (r/min)，a。

20.2 mm and 3.5×10 (r/min)，a 19.8mm.Fig.5 demonstrates that the system converge tolimit cycle oscilation(LCO)after multiple cycle bi-furcation.The quasi-periodic oscillation occurs afterHopf bifurcation as shown in Fig.6。

看言 0写§ 0量2 3 4Spindle Speed/(r·min )2 3 4Fig.4 Th e real and imaginary parts of the Eigen-valuesVeTSUS spindle speed.0 10 .0 05 0 0 O5 0 10l O0 50. O 5. 1 Ol0 .0 05 O 0 O5 0 10Fig.5 The phase plane plots of and y directions，with/22.0×10 (r/min)，Ⅱ 20.2 mm0，3O 20 10- 0 1. 0 2. 0 2 .0 1 0 0 l 0 2 .1 0 .0 5 0 0 5 1 0Fig。6 The phase plane plots of and Y directionswith力3.5×10 (r/min)，Up19.8 mmThe Poincar6 section figures as shown in Fig.7(a)，(b)and Fig.7(c)，(d)are obtained by per-forming 1.5 ×10 and 21×10 iterative times.re-spectively.When 3.5×10 (r/min)，a。19.8mm，the Poincar6 sections of and Y directions areapproximately elipse， which also confirm occur-rences of the quasi-periodic motions.Frequency con-ponents ration of the response，demonstrated in Fig.7(a)，(b)，is approximately 1：3。

昌吕、≈ ；u 0专4 0Hydromechatronics EngineeringFig.7 The Poincar6 section of in and Y directionswhich are obtained by performing diferentiterative times，when力3.5×10 (r/min)，o 19.8 mm，(a)，(b)：1.5×10 times；(c)，(d)：21×10 times5.ConclusionsThe paper investigated the machine tool-work-piece system chatter vibrations in high speed milling。

The shift Chebyshev polynomial and Floquet theoryare efficiently adopted for this type of time-varyingperiodic delayed system.The stability and bifurcationare analyzed in the paper and the primary resuhs ofthe present investigation can be summarized as fol-lows：The stability lobes have been obtained，whichcan give stability information about tool spindle speedand axial depth of cutting.Chatter bifurcation is ana-lyzed by the Eigen-values of monodromy operator andthe results confirm the multiple cycle bifurcation andHopf bifurcation have onset in the miling system。

After multiple cycle bifurcation，the stability solutionof the system transforms from the stable equilibriumpoint to the LCO.After Hopf bifurcation，the stabili。

ty solution transforms to the quasi·periodic oscila·-tion.Th e Poincar6 sections obtained also prove theoccurrence of the quasi-periodic oscillation and givethe frequency components ration of the response.Ourachievement in this paper can provide important in-form ation for design of the 5-axial milling machine。

Th is method is could be used to study the time，-peri·-odic delay-differential dynamics system。

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ZHAO Demin，et al：Prediction Chatter Stability and Bifurcation in Milling Machine 27Appendix AAppendix BC ：O0- 2i)-l apktT22i)0 O 0O 0 0- 如 鲁 。

631- ∞ 丢 譬n (i)，632丢 鲁n (i)，63-2 ∞64l 鲁 ( )，6让- 丢apk，T2鲁 (i)，64-2G Q。

121816116- - 1/301201402 m 2 0 fm -n 1 a22n320 m- 12口。 % 0，2 1(。 。。

1(。：口 ) 。： 1(口。。，) 口。等 1(。 。 )。， 1(口 。 1(口。。 ) 警 芋口m- 10 m- 220m- 32r Q lQoa m- 42(Continued from 41 page)0 0 1 0 0O 0 0 0 -............。.......L 8 。 。 。 ； 。 。-～。

l 0 O - O ； ～0 。-8 o -8 0 ； 0QU Baojun，et al：Research and Design of Embedded SefiM DeviceServer on the DNC System 41[2][3]XIONG Bin.Theoretical discussion of the agile DNCsyem[J].Compmer Integrated Manufacturing System，1999，5(6)：1-6，25。

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[4][5]DNC系统嵌入式串口服务器的研究与开发曲宝军 ，郭 超山东理工大学 机械工程学院，山东 淄博 255049YAN Weiguo，WANG Minjie，WANG Minrui.Researchof the Ethemet communication technology based on Eth-ernet and TCP/IP[J].Journal of Da Lian University ofTechnology，2003，43(1)：77-81。

Jean J.Labrosse.IC/OS-II-Open source real-timeembedded operating system[M].Bei Jing：China electricpower press，2001。

摘要：针对当前使用的通用串口服务器在数控加工中出现的-些问题，提出了适用于DNC系统的数据传输机制。并论述了-种专用于DNC系统的嵌入式串口服务器的软、硬件设计方法。研究了串口服务器的体系结构以及 内核设计，提出了多任务内核的具体实现方法。

关键词：串口服务器；DNC系统；C/OS-II；以太网中图分类号：TP23(Continued on 27 page)高速铣削颤振系统稳定性及分岔的 Chebyshev多项式数值分析赵德敏h，张琪昌1.中国石油大学(华东)储运与建筑工程学院 工程力学系，山东 青岛 266580；2.天津大学 机械工程学院 力学系，天津 300072摘要：采用 Cheb)rshev多项式法和 Floquet理论相结合来预测铣床运行中的颤振和分岔。得到 了稳定性极限形图，可以准确地预示机床的稳定性。通过系统的特征值分析得到此系统发生了倍周期分岔和 Hopf分岔。系统由稳定的平衡点通过倍周期分岔收敛到稳定的极限环运动 ，由Hopf分岔转化到概周期运动。庞加莱截面的数值结果也证实了概周期运动的发生。

关键词 ：铣削；颤振稳定性；Chebyshev多项式法；分岔中图分类号：TH17