International Journal of Machine Tools & Manufacture 48 (2008) 832–840 Dynamic model of a centerless grinding machine based on an updated FE model I. Garitaonandia a,Ã , M.H. Fernandes a , J. Albizuri b a Department of Mechanical Engineering, Faculty of Mining Engineering, University of the Basque Country, Colina de Beurko s/n 48902 Barakaldo, Spain b Department of Mechanical Engineering, Faculty of Engineering, University of the Basque Country, Alameda de Urquijo s/n 48013 Bilbao, Spain Received 20 August 2007; received in revised form 29 November 2007; accepted 4 December 2007 Available online 15 December 2007 Abstract Centerless grinding operations present some characteristic features that make the process especially susceptible to regenerative chatter instabilities. Theoretical study of these vibrations present some difficulties due to the large amount of parameters involved in the process and, in addition, such a study requires a precise determination of dynamic properties of the particular machine under study. Taking into account the important role of the dynamic characteristics in the process, in this paper both analytic and experimental approaches are used with the aim of studying precisely the vibration modes participating in the response. Using as reference results obtained from an experimental modal analysis (EMA) performed in the machine, the finite element (FE) model was validated and improved using correlation and updating techniques. This updated model was used to obtain a state space reduced order model with which several simulations were carried out. The simulations were compared with results obtained in machining tests and it was demonstrated that the model predicts accurately the dynamic behavior of the centerless grinding machine, especially concerning on chatter. r 2007 Elsevier Ltd. All rights reserved. Keywords: Centerless grinding; Finite element; Experimental modal analysis; Model reduction 1. Introduction Centerless grinding is a chip removal process in which the workpiece is not clamped, but it is just supported by the regulating wheel, the grinding wheel and the workblade (Fig. 1). This configuration allows a simple and easy way to load/ unload workpieces with minimal interruption of the process, providing high flexibility in the sense that high productivities together with precis e dimensional tolerances of the parts can be obtained. On the other hand, problems associated with roundness errors are very common in these machines because of the floating center of the workpiece. As a consequence, surface errors of the workpiece, after contacting the workblade and the regulating wheel, produce a displacement of its center that can lead to an error regeneration mechanism. Several researchers have studied the origin and evolution of roundness errors [1–4]. These researches have shown that instabilities are produced due to geometric and dynamic causes. Geometric instabilities are specific of centerless grinding and they are produced as a consequence of the geometric configuration of the machine, independent of the structural characteristics and the workpiece angular velocity. Dynamic instabilities have their origin in the interaction between the regenerative process and the dynamic of the structure. In this case, self-excited vibra- tions appear limiting the surface quality of the workpieces. The study of this last problem requires an adequate knowledge of the dynamic properties of the machine, so it is of great assistance to have numerical models including these properties in order to predict the dynamic response of the machine-process system both in its original design and in a design with modifications. For the purpose of obtaining an adequate model, in this paper finite element (FE) models correlation-updating techniques are used by means of experimental data. ARTICLE IN PRESS www.elsevier.com/locate/ijmactool 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.12.001 Ã Corresponding author. Tel.: +34 946 014967; fax: +34 946 017 800. E-mail address: iker.garitaonandia@ehu.es (I. Garitaonandia). 2. System modeling Previously to the developm ent of this work, Albizuri et al. [5] studied the vibratory response of the machine under study using a lumped mass model, which character- ized the moving components guided by the two ball screw s in the feed direction (see Fig. 1). These are the components with larger vibration amplitudes in chatter conditions. This model, which has the advantage of its simplicity, is somewhat limited because it supposes several infinitely rigid components of the machine as the bed, the grinding wheel and the grinding wheel head, so it does not model adequately the force transmission path from the cutting point between the workpiece and the grinding wheel to the contact point between the workpiece and the regulating wheel. Due to the mentioned limitation, in this work special attention has been paid to the development of an updated FE model that will predict the dynamic response of the machine under operation conditions, givin g an insight into the real behavior of different components. 2.1. FE model Dynamic characteristics of centerless grinding machine were studied by means of a FE model using ANSYS software. This model, which consists of 53,200 nodes and 37,807 elements, is depicted in Fig. 2. This figure shows the global coordinate system used in the model, where z-axis was defined as the longitudinal axis of the machine, x-axis as the transversal one and y-axis as the vertical one. ARTICLE IN PRESS Nomenclature f FEM mode shape obtained from FE model f EMA mode shape obtained from EMA U matrix of mode sh apes X diagonal matrix of natural frequencies n diagonal matrix of modal dampings A, B, C, D matrices defining state space model L u input forces influence matrix q modal coordinate vector u state space output vector y state space input vector x state space state vector j 1 , j 2 , g, y, h centerless grinding geometric parameters (see Fig. 8)  0 ¼ sin j 2 sinðj 2 Àj 1 Þ depth of cut increment due to unitary waviness error of the workpiece at workblade contact point 1 À  ¼ sin j 1 sinðj 2 Àj 1 Þ depth of cut increment due to unitary waviness error of the workpiece at regulating wheel contact point s Laplace operator o t , o p, o a angular velocity of grinding wheel, work- piece and regulating wheel, respectively (Hz) K cutting stiffness (N/mm) b workpiece width (mm) k 0 cr contact stiffness per unit width between grind- ing wheel and workpiece (N/mm/mm) k 0 cs contact stiffness per unit width between reg- ulating wheel and workpiece (N/mm/mm) G(s) machine transfer function Fig. 1. Scheme of the centerless grinding machine. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 833 A total of 15 mode shapes and natural frequencies were extracted in the 0–160 Hz frequency range. This range of interest was defined taking into account that for the grinding machine under study, in chatter conditions, the most severe vibrations were observed in the neighborhood of 60 Hz, while other less severe vibrations appeared at about 130 Hz. This model predicts adequately the elastic-inertial properties of different components of the machine. A major prob lem arises when introducing the stiffness and damping properties of joint elements into the model because when studying joints, there are a lot of uncertain- ties that make not possible to model them precisely [6].In order to overcome this obstacle, the development of a good FE model requires the use of experimental data. 2.2. Experimental analysis An impact testing experimental modal analysis (EMA) was performed in the machine. Responses were measured in 69 points using triaxial accelero meters, so aceleration/ force frequency response functions (FRF) corresponding to 207 degrees of freedom were obtained. Fig. 3 illustrates the geometry used in the analysis, in which the arrow shows the excitation point and direction. This excitation direction was selected in order to excite modes with high modal displacement components in z direction. The FRFs obtained in the analysis were studied in LMS Cada-X software, obtaining 10 natural frequencies, mode shapes and damping factors in the frequency range of interest. 2.3. Theoretical–experimental correlation Numerically obtained mode shapes were correlated with the experimental ones in FEMtools software using the modal assurance criterion (MAC) [7]: MAC FEM; EMAðÞ¼ ðf T FEM f EMA Þ 2 ðf T FEM f FEM Þðf T EMA f EMA Þ . (1) MAC values obtained between the first 15 numerical mode shapes and the first 10 experimental ones are shown in Table 1. In this table, four values of MAC corresponding to as many paired mode shapes have been highlighted due to their importance in the development of this work. The first three pairs show MAC values above 85%, while the last pair shows a lower value. Although these MAC values point out that the correlation between the corresponding numerical and experimental mode shapes is adequate, it can be seen that there are significant differences in the natural frequencies of these mode shapes, so it was necessary to impr ove the FE model by means of an updating process. In the updating process, the three numerical natural frequencies with higher MAC values were selected as responses to be improved. To select adequate parameters to be updated, a sensitivity analysis was performed and it was concluded that the parameters most influencing the estimation of the mentioned natural frequencies were the stiffness values of the joint elements connecting the bed to the foundation and the axial stiffness of the lower slide ball screw. These stiffness values were improved iteratively in order to match the numerical natural frequencies to the experi- mental ones. A Bayesian parameter estimation technique was used [7] for this purpo se. Fig. 4 shows the MAC matrix obtained after the updating process. In this figure, it can be seen that an adequate corre- lation remains between the previously paired mode shapes. Moreover, Table 2 shows that the difference between the updated natural frequencies and the experimental ones are ARTICLE IN PRESS Fig. 2. FE model of the machine. Z Y X Fig. 3. Geometry of the EMA. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840834 small. Numerical frequency 13 also was improved in the updating process. 3. Updated FE model characteristics Once an updated FE model has been obtained, in the development of a dynamic analysis it is important to study the model in order to identify the modes with higher contribution to the response in operation conditions. In this analysis, it is necessary to take in mind that in centerless grinding operations the normal force generated in the cutting point between the workpiece and the grinding wheel is produced mainly in z direction, so the relative excitability of the different modes in that direction was evaluated calculating their modal participation factors (MPF). The result is shown in Fig. 5, where the MPF have been normalized so that the largest value has unit magnitude. From this figure it can be concluded that there are three modes with highest contribution to the dynamic response. In the mode shape at 33.48 Hz all the components of the machine move in phase in the longitudinal direction in a suspension movement with respect to the supports of the bed. The mode shape corresponding to natural frequency of 58.59 Hz is the one which is excited normally when chatter vibrations appear in the centerless grinding machine, so it is called the main chatter mode. Fig. 6 shows the animation for this mode shape. It is seen that this mode corresponds to an out of phase movement between the heads of the two wheels. The mode shape at 127.41 Hz corresponds to a mode, which has been excited only in some tests performed in the machine, always leading less vibration amplitudes than the previous one. ARTICLE IN PRESS Table 1 MAC values before updating, in % FE mode shapes (Hz) EMA mode shapes (Hz) 12345678910 (33.41) (48.43) (58.91) (76.96) (90.52) (108.44) (122.03) (128.48) (129.86) (144.02) 1 (30.16) 0.618 0.155 0.182 0.922 3.69 0.776 0.0768 0.18 0.0025 1.15e-4 2 (31.84) 95.7 5.45 18 0.0242 0.713 11.8 1.84 0.204 0.736 0.937 3 (45.9) 0.391 88 0.289 10.9 2.03 1.32eÀ3 0.0741 7.04 0.529 0.158 4 (54.85) 2.78 0.307 94.5 17.2 1.2 0.0131 5.53 5.8eÀ3 3.99 1.29 5 (67.38) 0.388 7.26eÀ5 0.753 7.88 6.07 10.9 8.6 2.99eÀ3 2.55 1.32 6 (73.5) 0.176 0.00821 0.0938 17.9 1.02 8.79 0.0144 0.217 0.669 8.12eÀ3 7 (82.27) 2.56 0.836 0.163 36.2 22.2 4.56eÀ4 0.0427 0.793 1.21 2.35 8 (93.03) 0.463 0.333 0.0339 0.242 19.3 7.9 10.9 0.49 0.916 4.92 9 (96.84) 0.472 0.212 0.423 5.5 25.2 42.8 42.1 8.44 4.76 8.03 10 (102.02) 0.901 0.0206 0.114 0.829 7.89 50.9 14 0.467 36.8 0.116 11 (113.28) 1.08eÀ3 2.4 0.0251 0.125 2.23 0.587 0.0959 0.451 0.322 0.316 12 (122.13) 0.565 0.234 3.76 1.49 0.987 1.08 10.9 0.734 17.7 8.2 13 (126.82) 0.0334 0.0806 2.43 0.747 0.332 0.0645 3.88 64.2 20.5 8.87 14 (149.88) 4.32 0.688 6.23 0.0856 0.229 3.93 1.35 3.23eÀ3 38.4 2.24 15 (159.44) 1.26 0.2 3.67 9.15 19.1 3.12 10.4 4.84 8.41 1.6 Bold numbers correspond to important mode shape pairs. Fig. 4. MAC matrix after the updating process, in %. Table 2 Comparison between updated numerical values and experimental results Pair FE mode Hz EMA mode Hz Diff. (%) MAC (%) 1 2 33.48 1 33.41 0.20 96.2 2 3 48.43 2 48.43 0.00 87.5 3 4 58.59 3 58.91 À0.55 93.2 4 13 127.41 8 128.48 À0.83 64.6 I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 835 4. FE model order reduction The large number of degrees of freedom of the updated FE model implies computationally expensive calculations in order to simulate the vibration behavior of the machine. This restriction makes necessary the reduction of the size in such a way that the reduced model and the original model will have the same frequen cy response charact eristics in the frequency range of interest. The big size of the FE model was reduced using the modal coordinate reduction [8], which is based on the fact that the dynamic response of the centerless grinding machine in the frequency range of interest is dominated by the first 15 structural modes, so it is possible to simulate its dynamic behavior using these modes and neglecting the rest. Several representative degrees of freedom were selected defining the points in which application of forces or acquisition of responses was required. Using as reference the different lists obtained from the updated FE model, truncated U matrix was created retaining the modal contributions of the mentioned degrees of freedom for the first 15 mode shapes. The first 15 updated natural frequencies were used to create X matrix and the damping properties obtained experimentally were used to construct n matrix. These matrices were used in MATLAB environment to obtain a modal model of the structure in state space defined by _ x ¼ Ax þ Bu; y ¼ Cx þ Du: (2) The state vector was selected as follows [9]: x ¼ Xq _ q "# . (3) In Eq. (3), the size of the state vector (and thus the order of the modal model) is twice the modes included in the model, being this size much less than the order of the ARTICLE IN PRESS Fig. 5. Modal participation factors. Fig. 6. Main chatter mode animation. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840836 updated FE model used as reference. The resulting state space model A and B matrices are A ¼ 0 X ÀX À2nX "# ; B ¼ 0 U T L u "# . (4) C and D matrices of system (2) depend on the required outputs, so the described model can be used to simulate displacements, velocities and accelerations of selected degrees of freedom. FRFs acceleration/force obtained both experimentally and using the state space modal model between input degree of freedom j and output degree of freedom k (see Fig. 2) were compared. The result is displayed in Fig. 7 . This figure shows that the state space modal model reflects adequately the system dynamic below approxi- mately 70 Hz, whi le above this frequency the discrepancies with the experi mental results are higher. Likewise, it can be seen that both the FRFs show three important resonance peaks corresponding to the mode shapes with higher MPF obtained in Fig. 5. 5. Experimental verification In order to evaluate the effectiveness of the state space reduced model obtained in previous section, it was used to perform a theoretical study of chatter instabilities in the centerless grinding machine under study. Fig. 8 shows a detail of the configuration of the machine and Fig. 9 shows the block diagram used to study its stability, which is based on similar diagrams presented in previous works [3,4]. The term G(s) takes into account the dynamic flexibility of the machine and it was obtained considering the three modes with major dynamic contribution in feed direction (see Fig. 7). This idea was carried out using controllability and observability criteria [10]. The state space modal model defined by Eq. (2) was transformed in the balanced realization, in which the controllability and observability matrices are equal and strictly diagonal, being the terms of the diagonal a quantitative measure of the relative importance of the different states in the input–output behavior of the system. This realization was divided into a dominant subsystem formed by the six more controllable and observable states, and a weak one, formed by the rest of the states. This last subsystem was eliminated residualiz- ing the least controllable and observable states in order to include their static contribution in the response [11]. 5.1. Analysis of chatter frequencies The characteristic equation of the block diagram shown in Fig. 9 is 1 À e À2pðs=o p Þ  K ð1 À Þ bk 0 cr þ 1 bk 0 cs þ GðsÞ ! þ 1 À  0 e Àj 1 ðs=o p Þ þð1 À Þe Àj 2 ðs=o p Þ ¼ 0. ð5Þ To guarantee stabl e cutting conditions, all the roots of this equation must be in the left side of the complex plane. If one of the roots is located in the right side of the complex plane, the system is unstable and during the grinding process the response grows in time causing the regeneration of a roundness error in the workpiece. The complete resolution of the roots of the characteristic equation is not an easy task due to the transcendental nature of the equation to be solved, formed by three time delays, so there are infinitely many solutions satisfying it. In this application, these roots were solved using the root locus technique [12], obtaining the solutions of the characteristic equation (Eq. (5)) for increasing values of cutting stiffness in the 0- N range. This technique plots the evolution of the different roots, so it can be determined which one becomes unstable. The application of this method requires the resolution of the characteristic equation for a zero cutting stiffness value. These solutions are:  the poles of the transfer function G(s),  an infinite number of poles of ð1 À e À2pðs=o p Þ Þ located at minus infinity,  the roots of the geometric characteristic equation 1 À  0 e Àj 1 ðs=o p Þ þð1 À Þe Àj 2 ðs=o p Þ ¼ 0 ÀÁ . The initial esti- mations of these roots were obtained using an iter- ative graphical procedure, which consisted in modifying ARTICLE IN PRESS Fig. 7. FRFs between j–k degrees of freedom. Fig. 8. Geometry of centerless grinding. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 837 sequentially the values of the possible roots until the real and imaginary parts of the geometric characteristic equations were annulated. These initial estimations were used to obtain the final solutions using Newton– Raphson method [13]. The first set of roots obtained for K ¼ 0 were used as initial estimations for the next increment of the cutting stiffness using Newton–Raphson method, and so on until reaching the cutting stiffness value obtained experimentally for the particular geometric configuration of the machine under study. In order to compare the results obtained theoretically with those obtained experimen tally, several simulations were performed programming geometric conditions with which previously cutting tests had been done in the machine. These conditions are shown in Table 3. Contact stiffness values assumed in the simulations corresponded to typical values for centerless grinding of steel components using a vitrified grinding wheel together with a rubber-bonded regulating wheel [13]. For illustration purposes, Fig. 10 shows the evolution of the root loci for a workpiece angular velocity of 6.2 Hz. In this figure, structural pole on 58.59 Hz migrates towards the imaginary axis for increasing values of cutting stiffness ARTICLE IN PRESS Fig. 9. Block diagram of centerless grinding. Table 3 Cutting conditions Workblade angle, y 301 Center height angle, g 81 Regulating wheel diameter 340 mm Grinding wheel diameter 628 mm Workpiece diameter 46.2 mm Workpiece width 368 mm Fig. 10. Root locus for increasing cutting stiffness. o p ¼ 6.2 Hz. Fig. 11. Comparison between theoretical and experimental chatter frequencies. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840838 until the pole crosses it and thus, it is instabilized. Chatter frequency is determined by the imaginary part of the unstable root after concluding the loci. This procedure was repeated for different values of workpiece angular velocity in the 0–20 Hz range. Chatter frequencies obtained theoretically were compared with the experimentally measured ones, as it is shown in Fig. 11. This figure shows that theoretical predictions are in agreement with experimental results. 5.2. Time domain simulations The reduced order model validation was completed with various time domain simulations in order to qua ntify the number and the amplitudes of the undulations produced in the workpiece in chatter conditions. The workpiece was discretized in 360 equal segments and its rotation was simulated segment by segment. Taking as reference the process block diagram shown in Fig. 9, time evolution of the roundness errors of the workpiece was obtained and in each segment rotation, integrating numerically this evolu- tion using Runge–Kutta algorithm [14], the errors were calculated as a function of the dynamic response of the machine and the errors in the previous pass and in the contact points with the workblade and the regulating wheel. Nonlinear effects as contact loss between the workpiece and the grinding wheel and spark-out process were taken into account [15]. The simulations were done programming a regulating wheel infeed rate of 1 mm/min, a total feed of 0.2 mm and a spark-out time of 2 s. Fig. 12a shows the final theoretical profile obtained for a workpiece angular velocity of 6.2 Hz, while Fig. 12b shows the real profile obtained under the same conditions programmed in the simulations. It is shown that workpiece profiles obtained both theore- tically and experimentally are quite similar. Moreover, theoretically predicted roundness error is within the same order of magni tude of the experimentally measured one. 6. Conclusions In this work, a dynamic model of a centerless grinding machine has been developed performing a detailed study of mode shapes that are excited in chatter conditions. The combined use of numerical FE model updating techniques via experimental modal data and model reduc- tion techniques resulted in a state space model representing adequately the modes with major modal contribution in machine vibrations. The presented methodology advances the state-of-the-art in modeling procedures of centerless grinding machines. Simulations demonstrated that the model predicts accurately both the appearance of chatter vibrations for different machine configurations and the time evolution of workpiece roundness errors under unstable operation conditions. Thus, this model represents a powerful tool to define optimal set up conditions in order to increase the productivity in centerless grinding practice. Acknowledgments The authors are grateful to IDEKO Technological Center for the provision of numerical and experimental facilities to conduct this work. References [1] J.P. Gurney, An analysis of centerless grinding, ASME Journal of Engineering for Industry 87 (1964) 163–174. [2] W.B. Rowe, F. Koenigsberger, The ‘‘work-regenerative’’ effect in centreless grinding, International Journal of Machine Tool Design and Research 4 (1965) 175–187. ARTICLE IN PRESS Fig. 12. Final workpiece profile: (a) theoretical and (b) experimental. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 839 [3] Y. Furukawa, M. Miyashita, S. Shiozaki, Vibration analysis and work-rounding mechanism in centerless grinding, International Journal of Machine Tool Design and Research 11 (1971) 145–175. [4] M. Miyashita, F. Hashimoto, A. Kanai, Diagram for selecting chatter free conditions of centerless grinding, Annals of the CIRP 31 (1) (1982) 221–223. [5] J. Albizuri, M.H. Fernandes, I. Garitaonandia, X. Sabalza, R. Uribe- Etxeberria, J.M. Herna ´ ndez, An active system of reduction of vibrations in a centerless grinding machine using piezoelectric actuators, International Journal of Machine Tools and Manufacture 47 (10) (2007) 1607–1614. [6] H. Ahmadian, J.E. Mottershead, M.I. Friswell, Joint modelling for finite element model updating, in: International Modal Analysis Conference (IMAC), 14th, Dearborn, MI; United States, 12–15 February, 1996, pp. 591–596. [7] M.I. Friswell, J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. [8] Z Q. Qu, Model Order Reduction Techniques, with Applications in Finite Element Analysis, Springer, New York, 2004. [9] A. Preumont, Vibration Control of Active Structures. An Introduc- tion, second ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. [10] B.C. Moore, Principal component analysis in linear systems: controllability, observabillity and model reduction, IEEE Transac- tions on Automatic Control 26 (1) (1981) 17–32. [11] K.V. Fernando, H. Nicholson, Singular perturbational model reduction of balanced systems, IEEE Transactions on Automatic Control 27 (2) (1982) 466–468. [12] N. Olgac, M. Hosek, A new perspective and analysis for regenerative machine tool chatter, International Journal of Machine Tools and Manufacture 38 (7) (1998) 783–798. [13] S.S. Zhou, Centerless Grinding Process and Apparatus Therefor, US Patent, July 1, 1997. [14] S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, McGraw-Hill, New York, 1988. [15] F.J. Nieto, J.M. Etxabe, J.G. Gime ´ nez, Influence of contact loss between workpiece and grinding wheel on the roundness error in centreless grinding, International Journal of Machine Tools and Manufacture 38 (10–11) (1998) 1371–1398. ARTICLE IN PRESS I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840840 . model of the machine. Z Y X Fig. 3. Geometry of the EMA. I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–8408 34 small International Journal of Machine Tools & Manufacture 48 (2008) 832–840 Dynamic model of a centerless grinding machine based on an