Jounal of Machine Engineeing, Vol. 11, No. 4, 211 Batosz POWALKA 1 Macin CHODZKO 1 Kzysztof JEMIELNIAK 2 milling, chatte, opeational modal analysis STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL DATA Pediction of pocess stability in milling is usually based on expeimental fequency esponse functions of machine tool and cutting foce model. Altenatively, opeational modal analysis may be applied fo the pediction of a stability diagam. It does not equie modal impact test but uses vibation signal acquied duing actual cutting to identify modal paametes. Such an appoach consides bounday conditions that may be diffeent duing cutting. On the othe hand synthesis of fequency esponse function is not feasible due to the lack of scaled modal esidues. Thus, in this pape modal mass obtained by means of impact test is used to calculate fequency esponse functions. Estimation of modal damping and natual fequency is caied out using only acceleation signals measued duing flexible wokpiece machining. 1. INTRODUCTION Milling of flexible pats is a challenging task due to the occuence of egeneative chatte. The mechanism of egeneative chatte was explained by Tlusty and Polacek [1], and by Tobias and Fishwick [2]. A typical appoach to avoid chatte vibation is the selection of spindle otation speed and depth of cut based on a stability lobes diagam [2]. Since its intoduction by Tobias, numeous methods of stability lobes constuction calculation have been developed. A compehensive eview of the methods has been given by Altintas and Weck [3]. Geneally, in ode to geneate a stability lobes diagam, it is necessay to have infomation egading the cutting pocess and machine tool dynamics. The cutting pocess is fequently descibed by the mechanistic foce model, which assumes the popotionality of the cutting foce to the chip coss-sectional aea. Usually, fo pactical applications, the dynamics of a machine tool ae epesented in tems of a fequency esponse function (FRF) matix. Typically, an FRF matix is identified though the impact test. This test not only equies tained pesonnel but is also a time-consuming task. Theefoe, it may not always be feasible in an industial envionment. Also, modal paametes, i.e. modal damping and natual fequencies obtained by means of impact test, might diffe fom those obtained duing cutting due to the speed-dependent dynamics of the 1 West Pomeania Univesity of Technology, Szczecin, Poland 2 Wasaw Univesity of Technology, Poland
71 Stability Analysis in Milling Based on Opeational Modal Data spindle and diffeent bounday conditions. Seveal eseaches have pefomed impact tests with a otating spindle to take into account its speed-dependent dynamics [4],[5],[6]. Such an appoach is, howeve, not ecommended due to injuy isk and the excessive machine tool immobilization that is equied to identify dynamics at vaious spindle speeds. This limitation was ovecome by Gagnol et al. [7], who constucted an accuate speed-dependent model of the spindle and used it fo calculating a stability diagam. Abele and Fiedle [8] pesented the method of calculating dynamic behaviou duing milling that also consideed the change in bounday conditions. Thei method equies a dynamomete to measue cutting foces and thus has limited industial applicability. Zaghbani and Songmene [9] poposed a diffeent appoach that equies no measuement of cutting foces. Instead, they employed the opeational modal analysis (OMA) to extact modal damping and natual fequencies fom the acceleation signal measued duing actual machining. They applied OMA to find dynamic paametes and then used them to pedict tool-spindle based chatte instability. In ode to eliminate the so-called vitual modes that coespond to the hamonic excitation, they poposed a complicated algoithm.thei algoithm was limited to the estimation of modal damping and natual fequency, wheeas modal esidues wee not identified, which made it impossible to synthesize an FRF matix using only output data. In this pape, a modified OMA which takes into account hamonic excitations is applied. This significantly simplifies the selection of eal stuctual modes. 2. METHOD OF GENERATION OF STABILITY LOBES DIAGRAM BASED ON OPERATIONAL DATA The poposed method consists of the following steps: Measuement of acceleation signals in the close vicinity of the tool-wokpiece inteface Extaction of modal paametes fom acceleation signal, i.e., modal damping, natual fequency Geneation of stability lobes diagam using synthesized FRF matix. Modal esidues ae eplaced by modal masses obtained fom the impact test. 2.1. ESTIMATION OF MODAL PARAMETERS In this pape, a modified Eigensystem Realization Algoithm (ERA) [1] method in the pesence of hamonic excitation was applied in ode to extact modal paametes fom the output-only data. The ERA method is based on the Natual Excitation Technique [11]: if the system is excited by stationay white noise, the auto- and coss-coelation functions between the esponse signals ae simila to the impulse esponse signal and can be expessed in a discete time domain as: R 2N ( k t ) = C exp( s k t ) = 1 s = ω ξ + iω 1 ξ 2 (1)
72 Batosz POWALKA, Macin CHODZKO, Kzysztof JEMIELNIAK o in fequency domain S xx = C 2N = 1iω s (2) whee R epesents matix built of auto and coss-coelation functions at kth discete time, N is the numbe of modes in the system, C is a matix of constants associated with the th mode, ω and ζ ae the th non-damped natual fequency and damping atio, Sxx is the powe spectal density matix of the esponse signal. The OMA vesion of ERA consists of building Hankel matices fom the matices given by equation (1) fo diffeent discete time intevals. This is used to compute a system matix. Eigenvalues of this matix ae used to extact damping atio and natual fequencies. In the pesence of hamonic excitation, in addition to the andom loads, the coelation functions will include non-decaying components. If the system is excited by mh hamonic components of fequencies ωj (j=1,2,,mh) then application of classical OMA algoithms will yield mh additional non-damped modes with poles sj=±iωj. Theoetically, in ode to identify vitual modes, a zeo-damping citeion may be used, but in pactice, damping atio is diffeent than zeo. To cope with this difficulty, Mohanty and Rixen [12] poposed a modified vesion of the ERA which foces the modal solution to have non-damped poles with natual fequencies equal to excitation fequencies. The fequency of hamonic excitation is assumed to be known a pioi as it is in the case of milling. In the pesent study, hamonic components wee assumed to be multiples of the spindle otation fequency. Consequently, the stabilization diagam does not include vitual modes but only potential stuctual modes (numeical modes might still be pesent). Stable poles ae then selected fom the stabilization diagam. The FRF matix which is equied fo stability analysis may be expessed in tems of modal paametes as: G ( iω) = A 2N = 1iω s Modal esidues A can be deived fom the elationship between powe spectal density, FRF matix and powe spectal density of the excitation foces: S =G xx ( iω) S G( iω) H When the system is excited by stationay white noise with vaiance σ2, the powe spectal density matix Sxx, which is a Fouie tansfom of the R matix takes fom: FF (3) (4) S xx 2 = 2σ G t ( iω) G( iω) H (5) Equations (5), (3) and (2) elate modal esidue matices A to matices of constants C as C = H AA 2 2 σ t 2ω ξ (6) which is valid when the damping is light.
73 Stability Analysis in Milling Based on Opeational Modal Data Since vaiance of the excitation foces is unknown, modal esidues ae not known exactly and, theefoe, an FRF matix cannot be accuately synthesized. The scaling facto may be found though the analytical model of the tested stuctue o fom the mass change effect. These appoaches equie eithe an accuate model of the stuctue o caying out an additional test afte intoducing mass changes at chosen measuement point. Both equiements ae not feasible in an industial envionment. In this study a modal mass identified in the impulse test was used to sytntesize OMA-based FRF. Cicle fit method was adopted fo estimation of modal masses independently in two othogonal diections X and Y. FRF matix was then constucted as whee G x G ( iω) G = 1/ m ω + iωζ ω + ω ( iω) x G ( iω) 1/ m x y ( iω) =, G ( ) 2 2 y iω = 2 2 ω + iωζ ω + ω (7) (8) 2.2. GENERATION OF STABILITY LOBES USING OPERATIONAL MODAL DATA Analytical pediction of stability in milling poposed by Altintas and Budak [13] was applied to geneate stability lobes. The chaacteistic equation fo finding limit depth of cut ap is: det ( I a G( iω ) G ( 1 exp( iω T ))) = (9) p c cp whee Gcp is the immesion dependent matix, which is a function of cutting coefficients, and T is tooth peiod. In ode to build this matix, specific cutting coefficients fo machined mateial cutting tool ae equied. FRF matix G(iωc) obtained fom an impact test expesses a esponse of the stuctue to a unit hamonic excitation. This equation is used to find limit stability conditions, i.e., depth of cut, spindle speed and chatte fequency. c 3. EXPERIMENTAL VERIFICATION Figue 1 shows an expeimental stand used fo the veification of OMA-based stability pediction. The stand imitates a flexible wokpiece that is esponsible fo the occuence of chatte vibation. The FRF matices in the X- and Y-axis diection equied fo conventional stability analysis wee fist identified though impact modal test. The most flexible mode of the wokpiece is at 1162 Hz (see Fig. 2). Modal damping of this mode is.491%. FRF measued at tool tip was also measued, and it tuned out that the tool is significantly stiffe than the wokpiece. Cutting foce coefficients wee estimated fom the calibation tests pefomed fo vaious feed ates. Cutting foces wee measued using
74 Batosz POWALKA, Macin CHODZKO, Kzysztof JEMIELNIAK igidly mounted Kistle dynamomete type 9752. The cutting mateial was aluminum PA6, and the cutte was 2 mm 2-fluted Isca HM9E9AD2-2-W2_XL with insets HM 9APCT132R-PDR made of IC28. The estimated specific cutting foce coefficients in adial and tangential diection ae K=41.9 N/mm2 and Kt=128.5 N/mm2, espectively. Fig. 1. View of the expeimental stand Expeimentally identified FRF matix and cutting foce model coefficients wee used to geneate stability lobe diagam fo full immesion cutting. Such geneated stability lobes diagam is compaed with diagams constucted using in-opeation acceleation signals. In ode to pedict stability limit based on opeational data two cutting tests wee pefomed. Feed diection duing both cutting tests coesponded to the X-axis which is maked on the machined wokpiece (Fig. 1). G x (iω) [mm/n 4 x 1-5 3 2 1 Impact test OMA 1 15 11 115 12 125 13 135 14 145 15 Fequency [Hz] G y (iω) [mm/n 1 x 1-3.8.6.4.2 Impact test OMA 1 15 11 115 12 125 13 135 14 145 15 Fequency [Hz] Fig. 2. Compaison of FRFs obtained though impact test and synthesized fom signals acquied duing full-immesion aluminum cutting at spindle speed 81 RPM, feed.1mm/tooth and depth of cut ap=2mm Depth of cut ap was set to 2 mm, and the spindle speed was set to 81 RPM to povide stable cutting. Duing cutting tests, the acceleation signals wee measued by the same set of acceleometes that was used in impact testing. Estimation of modal paametes
75 Stability Analysis in Milling Based on Opeational Modal Data was caied out using X- and Y-axes acceleation signals measued by a senso fixed diectly to the wokpiece. It must be noted that the tansient vibation signal, as the tool enteed the wokpiece, was not included in the analysis. Stabilization diagams constucted on the basis of opeational data ae shown in Fig. 3. In each analysis, 2 fist multiples of spindle otation fequency (135Hz) included in the modal solution wee emoved fom the stabilization diagam. The stabilization diagams contain only so-called stable poles, i.e., poles that do not change fequency moe than 1% and damping moe than 2% though consecutive model odes. OMA-based FRFs wee synthesized accoding to the fomula (8) and compaed with the impact test esults in Fig. 2. It tuned out that only the most flexible mode had an impact on the synthesized FRF. The othe identified poles contibuted insignificantly to the FRF. Modal damping and natual fequency obtained fom the opeational data wee.45% and 1172 Hz espectively. Model ode 3 2 Powe spectal density (db) 1 2 4 6 8 1 12 14 16 18 2 Fequency [Hz] Fig. 3. Stabilization diagams obtained fom the signals acquied duing full-immesion aluminum cutting at spindle speed 81 RPM, feed.1mm/tooth and depth of cut ap=2mm Figue 4 visualizes the diffeence between stability diagams fom impact test and fom actual cutting at depth 2 mm, espectively. It can be obseved that lobes fom the impact test and the depth of cut 2 mm ae vey simila. b) Depth of cut [mm] 3 4 5 6 7 8 9 1 1 5 Impact test OMA 3 4 5 6 7 8 9 1 Spindle speed [RPM] Fig. 4. Analytical stability lobe diagams fom impact test and fom signals acquied duing full-immesion aluminum cutting at spindle speed 81 RPM, feed.1mm/tooth and depth of cutap=2mm. Squae ( ) and coss (x) indicate stable and unstable cutting conditions duing expeimental test, espectively
76 Batosz POWALKA, Macin CHODZKO, Kzysztof JEMIELNIAK In ode to veify the stability diagam, additional cutting tests wee caied out. In this study, acceleation and the quality of the machined suface wee inspected visually to detemine chatte occuence. We obseved acceleation and inspect visually the quality of machined suface. Fig. 5 shows acceleation signal in the feed diection fo vaious depths of cut at 639 RPM. This speed coesponds to the bottom of the stability lobes. Visual analysis (Fig. 6) of the suface quality classified cutting tests with 2, 2.5 and 3 mm at 639 RPM as chatte, while the test with 1mm esulted in a chatte-fee suface. All cutting tests pefomed at 81 RPM (ap=3, 4 mm) wee stable. 15 1 mm 2 mm 2.5 mm 3 mm 1 X: 124 Y: 7.27 Acceleation [m/s 2 ] 5-5 -1-15 Fig. 5. Acceleation signal in the feed diection fo vaious depths of cut at 639 RPM Fig. 6. Suface photos fo vaious depths of cut at 639 RPM 4. CONCLUSION Stability analysis that utilizes modal paametes obtained fom opeational data is pesented. Results of the analyses agee with conventional chatte pediction. A discepancy in absolute limit depth of cut (bottom of lobes) is obseved. This discepancy is caused by diffeent values of modal damping obtained by means of impact test and opeational data that may be due to diffeent bounday conditions. Accuate finding of limit depth of cut exclusively (in this study modal mass fom the impact test was assumed) fom the OMA based stability analysis will be a subject of futhe eseach. It equies the development of a method fo finding scaled values of modal esidues. In ou futue wok, we will also
77 Stability Analysis in Milling Based on Opeational Modal Data focus on applying opeational data fo chatte pediction when it is impossible to place sensos in close vicinity of the tool-wokpiece inteface. ACKNOWLEDGMENTS Financial suppot of Stuctual Funds in the Opeational Pogamme - Innovative Economy (IE OP) financed fom the Euopean Regional Development Fund - Poject "Moden mateial technologies in aeospace industy", N POIG.1.1.2--15/8- is gatefully acknowledged REFERENCES [1] TLUSTY J., POLACEK M., 1963, The Stability of Machine Tools Against Self-excited Vibations in Machining, Poceedings of the ASME Intenational Reseach in Poduction Engineeing, 465-474. [2] TOBIAS S. A., FISHWICK W., 1958, Theoy of Regeneative Machine Tool Chatte, The Enginee, London, 25/199-23. [3] ALTINTAS Y., WECK M., 24, Chatte Stability in Metal Cutting and Ginding, Annals of the CIRP, Key Note Pape of STC-M, 53/2/619-642. [4] FAASSEN, R. P. H., VAN DE WOUW, N., OOSTERLING, J.A.J., NIJMEIJER, H., 23, Pediction of Regeneative Chatte by Modelling and Analysis of High-speed Milling, Intenational Jounal of Machine Tools and Manufactue, 43/1437 1446. [5] SCHMITZ, T., ZIEGERT, J., AND STANISLAUS, C., 24, A Method fo Pedicting Chatte Stability fo Systems with Speed-Dependent Spindle Dynamics, SME Technical Pape TP4PUB182, Tansactions of NAMRI/SME, 32/17-24. [6] CHEN C., WANG K., SHIN Y., 1994, An Integated Appoach Towad the Dynamic Analysis of High-Speed Spindles, Pat 1: System Model, ASME J. of Vibations and Acoustics, 116/56-513. [7] GAGNOL V., BOUZGARROU B. C., RAY P., BARRA B., 27, Model-based Chatte Stability Pediction fo High-speed Spindles, Intenational Jounal of Machine Tools and Manufactue, 47/1176 1186. [8] ABELE E., FIEDLER U., 24, Ceating Stability Lobe Diagams duing Milling, Annals of the CIRP, 53/1/39 312. [9] ZAGHBANI I., SONGMENE V., 29, Estimation of Machine-Tool Dynamic Paametes duing Machining Opeation though Opeational Modal Analysis, Intenational Jounal of Machine Tools and Manufactue, 49/12-13/947 957. [1] JUANG J. N., PAPPA R. S., 1985, An EigensystemRealization Algoithm fo Model Paamete Identification and Model Reduction, Jounal of Guidance Contol and Dynamics, 8/5/62 627. [11] JAMES G. H., CARNE T. G., LAUFFER,J. P., 1993, The Natual Excitation Technique fo Modal Paametes Extaction fom Opeating Wind Tubines, SAND92-1666, UC-261. [12] MOHANTY, P. RIXEN, D.J., 26, Modified ERA method fo Opeational Modal Analysis in the Pesence of Hamonic Excitations, Mechanical Systems and Signal Pocessing, 2/1/114-13 [13] ALTINTAS, Y., BUDAK, E., 1995, Analytical Pediction of Stability Lobes in Milling, Annals of the CIRP, 44/1/357 362.