Impact of Modal Parameters on Milling Process Chatter Stability Lobes

Impact of Modal Parameters on Milling Process Chatter Stability Lobes'LI Zhongqun LIU Qiang( School of Mechanical Engineering and Automation , BeiHang University Beijing 100083 , China ,E-mail zhqunli@ 163. com )Abstract : Modals of the machine/ tool and machine/ part system are the principal factors affecting the stablity of a millingprocess. Based on the modeling of chatter stability of milling process，the influence of modal parameters on chatter stability lobesindependently or jointly has been analyzed by simulation . Peak to- vally specific zalue , lobe coefficient and the corresponding calcu-lation formula have been put forcard. General lares and steps of modal simplification for multimodality system have been summa-rized.Key words: milling process ; chatter stability lobes ; dsynamic stiffness ; modal parameler1 IntroductionNowadays , milling is widely used in fabricating of components of autos , aircrafts , moulds and dies. Withthe rapid development of numerical control techniques , the production efficiency and product quality are in thelimelight. However , as milling is an intermittent process , a periodic force interacts between the cutting tool andworkpiece which makes the chatter vibration developed at specific conditions. This phenomenon may result in areduced productivity , increased cost and inconsistent product quality. Thus ,it is useful to predict the chatter bymodeling the milling process and further regard it as a constraint in cutting parameters optimization.For a long time ,many researchers were devoted to the chatter mechanism ; its modeling and a series of pre-diction methods were developed. The first acurate model about self- excited vibrations in orthogonal cutting wasperformed by Tlusty 1 and Tobias 21. They identified the most powerful source of self-excitation , regeneration ,was associated with the dynamics of the machine tool and the feedback between the subsequent cuts on the samecutting surface. Minis etc. 31 applied the periodic differential equation to two dimensional milling process andnumerically solved it by Nyquist criterion. Budak and Altintas 41 developed a stabilty method which leads to ananalytical determination of stability limits by expanding the periodic function into the Fourier series and keepingonly the average item. The critical axial cutting depths were obtained by frequency sweeping around the system's natural frequencies.Based on the linear stability theory developed by Budak and Altintas , cutting force cofficients and modalparameters of machining system were used to obtain the chatter stability lobes diagram，which can be used as aninstruction for chatter-free cutting and a constraint for further cutting parameters optimization. The method hasbeen validated by a large amount of cutting experiments 4-71. According to its modeling , the chatter stabilitylobes are mainly affected by system' s modal parameters. However , the quantitative contribution of certain pa-rameter can not be obtained explicitly. Therefore , analyzing the effect of modal parameters on chatter stabilitylobes systematically is significant for fixing an appropriate cutting condition.2 Modeling of Chatter Stability for MillingA milling system can be simplified as an elastic damping system with two orthogonal modal directions asshown in Fig. 1. The time-varying dynamic milling force can be expressed as{F }eiot =一aK[1 -e-inT ][ AoIG( icoe )]KF }eiot(1)Where {F }represents the amplitude of the dynamic milling force {F( t )},wc is the chatter frequency ,a isthe axial cutting depth ,K, is the empirical cutting force coefficient [ Ao ] is the average term in the Fourier se-ries expansions of the directional coefficients matrix[ A, ][ G( iw. )]is the transfer function matrix of machine/tool and machine/ workpiece system. Eq. ( 1 ) has a non- trivia中国煤化工rminant is zero and thecorresponding eigenvalue is given as( A )MHCNMHGA =-aK(1-e ioT)=--(a1士V ap2- 4ao )(2)* Supported by the Fundamental Research Project of COSTIND( K1203020507 )- 19ao=2( iw, )3..( iwc Xaxasy-axjQsx)Where :(3)a1=a.$( iwc )+ax.,( iwe )The transfer functions are complex A is a complex number and the axial cutting depth( a )is a real num-ber. Thus , when substituting e w。! =cos w.T+ isinwfT and A= Ap+ iA| in Eq. (2 ), the imaginary part ofthe equation has to vanish yieldingA_sinwe. TK = tanψ(4)ARA relation between the chatter frequency and the spindle speed can be obtained :w。T= ε + 2kπε = π-2ψ(5)n=60/(NT)Where ε is the phase difference between the inner and the outer modulations k is the vibration numberwithin a tooth period n is the spindle speed( rpm )and N is the tooth number. The stability limits is obtainedas follows :alim =-2πAr{1 + K2 YNK,(6)Therefore , for a given cutting geometry , cutting force coefficients , transfer functions and chatter frequen-cy aA1 and AR can be determined from Eq. (2 ),and can be used in Eqs. ( 5 )and( 6 ) to determine the corre-sponding spindle speed and stability limit. When the procedure is repeated for a range of chatter frequencies andnumber of vibration waves k , the stability lobe diagram for a milling system is obtained.3 Chatter Stability Lobes ChartA typical chatter stability lobes with only the first-order modal in both x and y directios was shown irFig.2. The abscissa is the spindle speed ; the longitudinal coordinate is the coressponding critical axial cuttingdepth. Milling below the boundaries is stable ; otherwise , the chatter will occur. The chart is made up of a seriesof stability lobes. For the benefit of the further analyses , several terms were be defined first : The lobe matchingthe system dominant modal is called the first stablity lobe , the proximate one from right to left is called the sec-ond and the third lobe. The lobe with the highest peak within the chart' s scope is called the valid dominant sta-bility lobe. The precondition for the first lobe to be the valid dominant one is :w,S( nmax' N )60(7)Where wn is the natural frequency of the dominant modal m max is the upper limit of the spindle speed used insimulation and N is the tooth number.0卜unstable4Stable主F113161922Spindle speedUrpmFig.1 Chatter modeling for millingFig.2 Chatter stability lobesEach lobe' s peak to valley is defined as the peak-to-valley specific value λ; ,the valid dominant lobe' s one iscalled maximal peak-to-valley specific value λmax , and each lobe' s peak-to-valley specific value to the maximalpeak- to- valley specific value is called the lobe coefficient C; and can be expressed as :λ;= C; λmex(8)4 Impact of Modal Parameters on the Chatter Stability Lobes4.1 Impact of the Stiffness on the Chatter Stability Lobes f中国煤化工directions are identicalIt is assumed that the natural frequency and the damping(o,=1 002 Hz ,5 =0.021 2). Material' s tangential and radisTYHCNMHG are Ke = 586 Nm2”,Kr= 115 N/m2. Slot milling with a 2-flutte cutter of 16 mm' s diameter. The simulation result is depicted inFig.3 , the valley and the maximal peak to- value specific value was listed in Table 2.一191-Table 1 Stiffness used in the chatter stability lobes simulationSimulation No.simulsimu2simu3simu4simu52.17e64. 34e68. 68e64.34e6Stiffness( N/m)2. 17e6Fig. 3 shows that lobes' peaks and valleys increase with the stiffness whereas their horizontal positions areleft unchanged. The effect of stiffness on lobes are reflected quantitatively in Table 2 , a increase of nearly 50%of valley is obtained with a double stiffness in single direction and a increase of nearly 100% of valley is obtainedwith a double stiffness in both directions. No effect was found of the stiffness on the maximal peak- to-valley spe-cific value.2.5-simu_ 1--- simu_ 22.0simu 3--- simu_ 3e 2.5-_ siou 5f 2.0t0.50.40.60.8101.2141.61.82.022 2.400.40.60.81.012141.61.82.022 2.4Spindle ped(r/pmn)Spindle speed(r/prm)Fig3 Impact of siffness on stability lobes of single mode systemFig4 Impact natural frequency on stability lobe for single modal systemTable 2 Relation of valley , maximal peak-to-value specific value vs. stiffnessValley( mm)0.1560.2210.314 .0.3120. 623Max. peak to valley specific value8.278.268.294.2 Impact of Natural Frequency on Chatter Stability Lobes of Single Modal SystemSimilarly ,it is assumed that modal parameters are identical in x and y directions and the stiffness and thedamping ratio are left unchanged( ζ=0.021 ,k =2.17 N/m ), three simulations were conducted by changingthe natural frequency( 802 Hz ,902 Hz and 1 002 Hz). The result is depicted in Fig. 4.In Fig.4，the corresponding peaks and valleys shift rightwards with the increase of natural frequencywhereas the amount of peaks and valleys are kept unchanged.In order to reveal the influence of the natural frequency on the maximal peak- to-valley specific value ,a lotof simulations were performed. The relation of maximal peak- to- valley specific value and the natural frequency isshown in Fig.5.1.8r! 16-。damping ratin=0.02--- simu_simu_4-◆damping ratio= 0.04; 1.21.00f0.8-.6-.2400 600 800 1000 1200 1400 1600180000.40.60.81.01.2141.61.820222.4Natural frequency/HzSpindle speed/rpmFig.5 Impact of natural frequency on maximal peak-to-valleyFig.6 Impact of damping ratio on stability lobesspecific value for single modal Bystem3 distinct maximal peak- to-valley specific values were shown in Fig.5 with a constant damping ratio whilethe natural frequency is changed from 300 Hz to 1900 Hz. T中国煤化工near 800 Hz and 1 600Hz which is an integral multiple of the simulation spindle speeYY HCNMHG) With the increase ofnatural frequency , the chatter stability lobes move rightwards(seerig.4). w nen Tne natural frequency is lowerthan 800 Hz , the maximal peak- to valley specific value corresponds to the first lobe ; when within 800-1 600 Hz ,it corresponds to the second lobe ; when higher than 1 600 Hz ,it corresponds to the third lobe. As the dampingratio( ζ=0. 02 ,the first , second and third lobe' s peak-to-valley specific value are λ1= 17.0入2=8.7小3=6.1_ 19and the lobe cofficientsare Ci=1 ,C2=0.52 C3=0. 36 ias the damping ratiois ζ=0.04 ,入1=8.82 λ2=4.71小3=3.37 ,C1=1 ,C2=0.53 C3=0.38.With an increase of natural frequency ,the first ,second and third lobe moves out of the simulation scope inturn , leading to a change of maximal peak and an unchanged maximal peak- to-valley specific value. The damp-ing ratio has some effect on the maximal peak- to-valley specific value and has no effect on lobe coefficient.4.3 Impact of the Aamping Ratio on the Chatter Stability of Single Modal SystemUnder the condition of constant natural frequency and stiffness( wn=1 002 Hz k =2.17e6 N/m ),3 simu-lations were conducted with different damping ratios( ζ1=0.021 52=0.042 1ζ3= 0.084 ) and the results wereshown in Fig. 6.Fig. 6 shows that both the lobes' peak and valley rise with the damping ratio , but the peak has less increas-ing than the valley. No shift of the peak' s level position was found.To ilustrate the impact of the damping ratio on the maximal peak to- valley specific value , several simula-tions were done and the relationship between the damping ratio and the maximal peak to-valley specific value wasshown in Table 3.Table 3 Relation between damping ratio and maximal peak-to-valley speific value5ζ0.0050. 0080.0100.0200. 0400.0600. 0800. 1000.1200. 140λmex65.6041.5033. 3017.008.826.104.753.933.393.00Table 3 shows that the maximal peak-to-valley specific value appears a reciprocal relationship with thedamping ratio which can be represented by Eq.( 9 ). The comparison between the estimated and the simulatedone is shown in Fig. 7.λmx= 0.36/ζ(9)一. simu 130◆simulatedsimu 2。estimatedsimu 35害05|0f言1kA0.02 0.04 0.06 0.08 0.10 0.12 0.140.40.60.81.01.2141.61.82.0222.4Darmping/ratioSpindle speed/pmFig.7 Impact of damping ratio on max.peak to-valley specific valueFig.8 Inpact of modal superposition on stability lobes4.4 Impact of Modal Superposition on the Stability LobesIn active system , besides the dominant one other modals will have some effect on the stability lobes. Agroup of simulations were performed to reveal the essence. simulation 1 : single model ,Wn1 = 741 Hz , 51=0.021 ,K1= 5e6 N/m ;Simulation 2 :single modal (0n2=1 002 Hz ζ2=0.021 ,K2= 2. 17 N/m Simulation3 :double modals( the above two modals ). The results were shown in Fig. 8.Fig.8 shows that the lobes of co- existing modals can be obtained from the minimal critical axial cuttingdepth of each spindle speed when single modal exists alone. In short , the lobe' s valley of superposed modals isthat of the dominant modal existing alone except its partial peaks would be cut off. If the stability boundariesgenerated by a non-dominant modal alone does not intersect with that of the dominant one , the effect of thisnon-dominant modal can be left out of account.4.5 Impact of Single Modal Parameter Alternation on the Stability Lobe of Multimodality SystemIt is supposed that there are two modals in a system and the modal parameters are wn1= 741 Hz ζ1 =0.021 ,K1= 1.07e7 N/m ,(wn2=1 002 Hz, 52=0.021 , K2=2.17e6 N/m. To reveal the impact of certainmodal parameter alternation on the lobes of the whole system , six simulations were done by changing the stiff-ness of the dominant modal ,see Table 1 ,( the sixth simulation中国煤化工the presence of only thedominant modal ). The result was shown in Fig. 9.In Fig. 9 , the impact of the stiffness alternation on lobes iTYHc N M H Galmost the same as thatin single mode system except for some peaks would be cut off by lobes generated by non-dominant modals i.e.the axial critical cutting depths for each spindle speed increase with the modal stiffness. Similarity is the impactof natural frequency and damping ratio on multimodality system..一193一4.0p一simu 13.5simu 3; 3....... simu 3.03.0.-. simu 6.2.5 t2.0|.5E 1.0o.50.50.40.6081012141.61.82022240040.60.810121416182024Spindle speed/pmFig9 Impact of siffness on stability lobes of multimodality systemFig.10 Impact impact of siffness of stability anddamping ratio on stability lobes4.6 Integrated Impact of Stiffness and Damping Ratio. simu 1----- simu 2on Chatter Stability LobesThree simulations were conducted by keeping a con-2.0..*... simu 3- simu 4stant product of the stiffness and the damping ratio and 9.55. simu.一simu6changing the individual stiffness and damping ratio( ζ3:52:ζ1=4:2:1 K3:K2:Kq= 1:2:4) The result was ..0-depicted in Fig. 10.In Fig. 10 , when the product of the stiffness and thedamping ratio , is kept constant , the lobes' valley are fixedand the peaks have a slightly increase.00.40.60.81.01.2141.61.8202224.7 The Impact of Unsymmetrical Modals on the Stabili-Spindle speed(r/pm)ty LobesFig11 Impact of unsynmetrical modal on stability lobesFor simplicity，the modals used in the above simulation are all assumed to be symmetrical in both x and ydirections. However , they are usually unsymmetrical in active system due to the structure of spindle system. Toreveal the effect , 8 simulations were conducted. The simulation parameters refer to Table 4 , the comparativelobes were shown in Fig.11 and the maximal peak- to-valley specific value was depicted in Table 5.Table 4 Unsymmetrical modal s effect on stability lobes simulation parametersSimulationModal parameters in x directionModal parameters in y directionNo.K( N/m)w,( Hz)simul5000.022E6simu24E6simu30.04simu40.04.simu50.02 .E6simu6600simu70. 04Fig. 11 shows that the chatter stability lobes have the highest peak and the chart has the most regular shapewhen the modal parameters are identicalin x and y directions. It may result from the resonance in these two di-rections. Furthermore , Table 5 also shows that it has the highest peak- to- valley specific value when the modalparameters are identical in x and y directions.Table 5 Maximal peak- to-valley specific values of various unsymmetrical modalSimulation No.imulsimu8入max .17.112.312.44.014.05 .4.55.4.635 The Modal Simplification of Actual Systerf中国煤化工Generally , the machine/ tool or the machine/ workpiece s:MHCN M H Gtem. When conductingmodal hammer test and modal parameters identification , whether certain modal should be kept has to be con-fronted with. Preserving overmuch modal may result in a complicated process for chatter stability simulation.Otherwise ,it may lead to a rough simulation precision. The following laws and steps can be used in modal sim-plification according to the above analysis.- 19Of all the three modal parameters , the product of the stiffness and the damping ratio( K ,ζ )i.e. the ma-chining system dynamic sifness determines the stability degree of a machining system. Of all modals in a sys-tem , the dominant modal has the minimal dynamic stiffness and governs the basic shape of system' s chatter sta-bility lobes. Therefore , it should always be kept.The peak- to valley specific value of the dominant modal was calculated based on its damping ratio and Eq.( 9 ); then the rightmost lobe in the chart is determined to be the first , the second or the third lobe by its naturalfrequency and Eq. ( 7); lastly，the true maximal peak- to-valley specific value λmx is calculated by Eq. (8 ).If the ratio of certain modal' s dynamic stiffness to the dominant modal' s dynamic stiffness is larger than themaximal peak- to-valley specific value of the dominant modal λmax i.e. the stability lobe diagram resulting fromthis modal alone does not intersect with the diagram generated by the dominant modal alone. This modal can bediscarded.6 ConclusionsAlthough modal parameters such as the natural frequency，the damping ratio and the stiffness all have directeffects on the chatter stability lobes , there are different among their effects. Natural frequency makes the stabili-ty lobes shift horizontally , the damping ratio and the stiffness make the lobes' peak and valley move vertically.The synthetic action caused by the multimodality is that the modal with the least dynamic stiffness becomes thedominant modal. Its contribution makes up the main body of the stability lobes , and some of their peaks may becut off by other modals with relative stiffer modals. When the stability lobe' s chart of certain modal does not in-tersect with the one caused by the dominant modal , the modal can be neglected. The above research findingsmay be used as a guide for modal parameters identification , machining process plan and machine tool design.References[1] J. Tlusty. Manufacturing Process and Equipment [ M] Prentice Hal , New Jersey , 2002.[2] S. A. Tobias. Machine Tool Vibration[ M ] Blackie London , 1965.[3] Minis , R. Yanushevsky. A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling. Journal of Engi-neering for Industry , 1993( 115):1-8.[4] Y. Altintas , E. Budak. Analytical Prediction of Stability Lobes in Milling. Annals of the CIRP 1995 A4( 1 )357-362.[5] E. Budak, Y. Altintas. Analytical Prediction of Chatter Stability in Milling Part I : General formulation. Journal of DynamicSystens , Measurement and Control ,1998( 120 ) 22-30.[6] E. Budak , Y. Altintas. Analytical Prediction of Chatter Stability in Milling _Part II :Application of the General Formulation toCommon Milling Systems. Journal of Dynamic Systems , Measurement and Control 1998( 120 )31-36.[7] R.P.H. Faassen ,N. van de Wouw ,J.AJ. Oosterling , H. Nijmeijer. Prediction of Regenerative Chatter by Modeling andAnalysis of High- speed Milling. International Journal of Machine Tools & Manufacture , 2003( 43 ) :1437-1446.中国煤化工MHCNMHG一195一