Simulation of an active vibration control system in a centerless grinding machine using a reduced updated FE model M.H. Fernandes a , I. Garitaonandia a, à , J. Albizuri b , J.M. Herna ´ ndez a , D. Barrenetxea c a Faculty of Mining Engineering, Department of Mechanical Engineering, University of the Basque Country, Colina de Beurko s/n, 48902 Barakaldo, Spain b Faculty of Engineering, Department of Mechanical Engineering, University of the Basque Country, Alameda de Urquijo s/n 48013 Bilbao, Spain c IDEKO Pol. Industrial de Arriaga, 2. 20870 Elgoibar, Spain article info Article history: Received 22 August 2008 Received in revised form 28 October 2008 Accepted 13 November 2008 Available online 27 November 2008 Keywords: Centerless grinding Active control Modelling Vibration simulation abstract In this paper, a novel and complete process to simulate an active vibration control system in a centerless grinding machine is presented. Based on the updated finite element (FE) model of the machine, the structural modifications performed to incorporate active elements are detailed, as well as the subsequent reduction procedure to obtain a low-order state space model. This reduced structural model was integrated in the cutting process model giving a tool adapted for the purpose of simulating different control laws. Using the developed model, a control algorithm, which previously had been implemented in the centerless grinding machine under study, was checked. The simulation results were in agreement with the experimentally obtained ones, showing that the designed model is able to reproduce machine behaviour with the control activated. This model constitutes a powerful tool to evaluate the effectiveness of different approaches to that of the described one, making it possible to tackle an optimisation process of the control system by means of simulations and, thus, avoiding the costs that would involve the practical implementation of each one. & 2008 Elsevier Ltd. All rights reserved. 1. Introduction One of the most important problems limiting the circularity of round parts in centerless grinding is the occurrence of self-excited vibrations caused by regenerative chatter. Due to the fact that this process is usually used to machine cylindrical elements with high surface finish requirements, the appearance of roundness errors is undesirable, and the study of procedures to minimize this negative effect has aroused the interest of several researchers. Traditionally, the developed solutions consist of avoiding the unstable operating regions through an adequate selection of set- up conditions, combining properly the geometric configuration of the machine with workpiece rotational speed [1–5]. However, from a productivity point of view, these solutions are not necessarily the recommended ones because the optimum cutting conditions can differ from the chatter-free configurations. Other procedures have been based on structural modifications to stiffen the most flexible components in the force transmission loop [6,7]. This stiffening produces an increase in the first resonant frequency, widening the low workpiece rotational speed stability zone. Nevertheless, such an alternative involves a redesign of the machine structure, giving rise to solutions that can be economically unfeasible. Taking into account the above-mentioned limitations, Albizuri et al. [8] proposed a novel approach based on the application of active vibration control. Using commercial piezoelectric actuators (A) in a feedback loop, they reduced the structural vibration level and, consequently, the roundness errors of the workpieces. Although the results they obtained were promising, they were presented without a previous mathematical development, which would allow them to optimise the active vibration system by means of simulations, concerning both the programmed con- trollers and the used sensors and actuators. Later, Garitaonandia et al. [9] characterized the dynamic properties of the machine combining finite element (FE) model updating and model order reduction techniques. The proposed approach was effective for the estimation of machine dynamic behaviour under usual cutting conditions, but the study was not extended to the simulation of any active vibration control scheme. The availability of a theoretical model describing both the structural and the controller characteristics is of major interest, as it permits one to predict the effectiveness of different control alternatives, giving insight into their behaviour before their practical implementation and providing valuable information to select the most suitable one. Therefore, the high costs (both economic and time costs), involving the experimental evaluation of each possible solution to make this selection, can be avoided. ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijmactool International Journal of Machine Tools & Manufacture 0890-6955/$ -see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2008.11.002 à Corresponding author. Tel.: +34946017815; fax: +34946017800. E-mail address: iker.garitaonandia@ehu.es (I. Garitaonandia). International Journal of Machine Tools & Manufacture 49 (2009) 239–245 Motivated by the benefits that can be obtained with a model having the described characteristics, in this paper a theoretical model derived from the FE formulation is developed to simulate the active vibration control system that had been previously implemented in [8]. Practical results obtained in the same reference will be used to evaluate the accuracy of the simulated predictions, thus validating the model. 2. FE model The simulation procedure presented in this paper is based on the updated FE model developed in [9]. Fig. 1 shows this model, where for greater clarity only solid models of the components have been shown. It consists of 53,200 nodes and 37,807 elements. As is reported in [9], this model predicted accurately the chatter behaviour observed in different machining tests. In these tests, the most violent vibrations appeared when the main chatter mode was excited at frequencies close to 55 Hz, where an important displacement of the lower slide relative to the swivel plate was presented in the longitudinal Z direction (see Fig. 1). Other less-severe vibrations were detected at about 130 Hz when the secondary chatter mode was excited, dominated by a local bending deformation of the workblade. 3. Integration of the actuators in the FE model To simulate the active control system, it is necessary to modify the FE model in order to incorporate the piezoelectric actuators. With this purpose, the design presented in [8] was followed, where two PI-247.30 piezoelectric actuators had been placed in the upper spindle support (D) area to obtain a strong control authority on the main chatter mode. This design is depicted in Fig. 2, where it can be seen that the integration of the actuators requires a design modification of the upper spindle support in such a way that two holes are made in it. Each actuator, which is located collinear to a force transducer (B), exerts the control force on its left side (according to Fig. 2) over support D and on its right side over an auxiliary component (E) that connects the two actuator-force transducer groups. This component transfers the two control forces to the lower spindle (C). Therefore, both actuators are placed in a parallel configuration in the loading path from the lower slide to the swivel plate, and they transmit the force to the lower spindle in a series configuration. Piezoelectric actuators like the ones used in this application consist of n-stacked ceramic layers of PZT material (lead–zirco- nate–titanate) that change in length when electrically charged. An important aspect to consider is the modelling of the force they exert. The relation among the externally applied voltage (V pzt ), the axial deformation of an actuator ( D L) and the actuator-force (F pzt ) is [10], F pzt ¼ K pzt ðd 33 nV pzt À D LÞ, (1) where K pzt represents the actuator stiffness and d 33 is the piezoelectric constant. Defining z as the vector containing the displacements of the different nodes in the FE model, the axial deformation of an actuator can be expressed by the following relation: D L ¼ b T z, (2) where b is the influence vector of the axial end displacements of the actuator. Substituting Eq. (2) into Eq. (1) and considering the centerless grinding machine structure formed by its mass matrix M, its damping matrix C and its stiffness matrix K, the incorporation of the forces exerted by the actuators in the equation of motion governing the dynamic equilibrium leads to [11], ARTICLE IN PRESS Fig. 1. Updated FE model of the centerless grinding machine. Fig. 2. Detail of piezoelectric actuators location area. M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245240 M € z þ C _ z þðK þ K pzt b 1 b T 1 þ K pzt b 2 b T 2 Þz ¼ L u F n þ b 1 K pzt  d 33 nV pzt_1 þ b 2 K pzt d 33 nV pzt_2 , (3) where F n is the normal grinding force and L u is its influence vector. Subscripts ‘‘1’’ and ‘‘2’’ have been used in b and V pzt to make a distinction between the two actuators. Eq. (3) shows that the effect of each actuator in the structure can be properly modelled through an axial stiffness K pzt between its ends (modelling the passive behaviour) and a pair of opposite forces of value K pzt d 33 nV pzt applied axially in the same ends (modelling the active behaviour). Following the design illustrated in Fig. 2, the lower spindle area of the FE model (Fig. 3a) was modified in order to incorporate the actuators, as shown in Fig. 3b. Each actuator was modelled with an element having the same stiffness as the piezoelectric stack and two lumped masses of half the total mass of the actuators were placed at their ends. The actuators were connected with the upper spindle support and with two elements modelling the auxiliary component. These elements were idealized as undeformable because it was considered that the displacement of the lower slide relative to the swivel plate was much bigger than the deformation of the auxiliary element. Finally, the rigid elements were connected to the lower spindle. It was verified that these structural modifications did not alter appreciably the natural frequencies corresponding to the main and the secondary chatter modes; hence, the passive behaviour of the structure remained basically unchanged. 4. Definition of the control strategy The active damping control strategy, which had been im- plemented by Albizuri et al. [8], consisted of measuring the machine vibrations with an accelerometer located in the upper spindle support and passing this acceleration signal through a controller based on a second-order filter (SOF) to generate a voltage feedback to the actuators proportional to the output of the filter [12]. Fig. 4 shows the control loop sketch in the modified FE model. The SOF controller is described by the general law given by HðsÞ¼ Àg o 2 f s 2 þ 2 x f o f s þ o 2 f . (4) The controller poles, which are defined by the filter properties ( o f , x f ), are located in the complex plane to produce an adequate migration of closed loop poles as the feedback gain g is increased. The minus sign in Eq. (4) is included to produce a negative feedback of the acceleration. As is shown in Fig. 4, the accelerometer is located very close to the actuators. The control configuration, where the measurement is done in the same degree of freedom (dof) as the excitation, is called the collocated control, and it is very demanding in practical applications because it enjoys the very attractive property of unconditional stability [12,13]. Physical constraints avoid exact collocation in this application. 5. Reduction procedure Once the FE model has been modified incorporating the active elements and the control strategy has been described, it is necessary to apply a reduction procedure to complete the required simulations at a reasonable computer cost. For this purpose, the modal truncation method was used [14], which is based on characterizing machine dynamics by the dominating vibration modes within the frequency range of interest. To cover the main and the secondary chatter modes, this frequency range was established between 0 and 160 Hz, where 15 vibration modes were calculated. For these modes, both natural frequencies and modal displacements corresponding to dof’s, where application of forces or acquisition of responses was required, were selected. Following the procedure described in [9], the extracted modal parameters, together with modal damping factors obtained experimentally, were used to obtain a state space model of order 30 designed for the purpose of predicting displacements, velocities and accelerations of selected dof’s. The integration of the reduced order structural model in the cutting process model demanded the consideration as inputs of the normal grinding force and the voltages V pzt_1 and V pzt_2 supplied to the piezoelectric actuators. The controller H(s) ARTICLE IN PRESS Fig. 3. Detail of the FE model in the upper spindle support area: (a) before modifications and (b) after modifications. Fig. 4. Control strategy. M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245 241 supplies the same voltage signals to both the actuators (V pzt_1 ¼ V pzt_2 ), so these two voltages were considered as one unique input V pzt . The variable describing machine deformation (y m ) and the acceleration of the dof corresponding to the accelerometer location (a) were selected as model outputs. After defining the necessary inputs and outputs, the state space model order was reduced, removing vibration modes with unimportant contribution in the input–output behaviour using the balanced truncation method [15]. As a result, a reduced state space representation was obtained, described by the three most controllable and observable modes (6 states), defined by _ x ¼ A r x þ B r F n K pzt d 33 nV pzt () , (5) y m a  ¼ C r x þ D r F n K pzt d 33 nV pzt () , (6) where A r , B r , C r and D r are the reduced matrices of the system. This reduced order state space model was validated comparing several frequency response functions (FRFs) obtained both experimen- tally and using the model. A good agreement was obtained between the responses, mainly at low frequencies and in the vicinity of the resonance peaks corresponding to the main and the secondary chatter modes. 6. Control system evaluation Once a state space model describing the structural behaviour of the modified centerless grinding machine has been developed, the following step in the simulation process consists of closing the loop between the acceleration and the voltages through the control law described in Eq. (4), as illustrated in Fig. 5. The controller design requires adjusting the frequency of the filter poles to the natural frequency of the mode that is intended to actively damp [12]. This application is focused on the reduction of vibrations related to the main chatter mode because this is the one responsible for the appearance and evolution of the most important roundness errors in workpieces, so o f was adjusted to 55 Hz. Filter damping was fixed in x f ¼ 0.5 [12]. 6.1. Evolution of structural roots A very important feature to consider in the simulation process is the choice of the feedback gain, as it modifies the structural dynamic behaviour, changing closed loop roots location. The evolution of these roots for increasing values of feedback gain is shown in Fig. 6, where the trajectories of compensator poles and structural roots selected in the balanced truncation process can be distinguished. The structural roots correspond to the following vibration modes:  a suspension mode at 209 rad/s (33.3 Hz),  the main chatter mode at 363 rad/s (57.8 Hz), ARTICLE IN PRESS Fig. 5. Feedback loop. Fig. 6. Root locus for increasing feedback gain. Fig. 7. (a) Evolution of the root corresponding to the main chatter mode and (b) comparison of receptance FRFs. M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245242  the secondary chatter mode at 797 rad/s (126.8 Hz). None of the trajectories shown in Fig. 6 crosses the imaginary axis and thus all roots remain in the left part of the complex plane. It is interesting to go deep into the evolution of the structural root associated with the main chatter mode, shown in Fig. 6 in a box and detailed in Fig. 7a. It can be seen that the root reaches a maximum damping of 22.5% for an optimum value of feedback gain. Damping of the root in open loop was 3.6%, and so the active damping capability of the proposed control strategy is demon- strated. Fig. 7b shows the receptance FRFs between the variable representing the structural deformation and the normal grinding force, both in open loop configuration and in closed loop with optimal gain. Resonant peaks corresponding to the main chatter mode and the suspension mode are substantially reduced. On the other hand, the peak corresponding to the secondary chatter mode remains practically unaltered, showing that actuators have no control authority over this mode. 6.2. Time domain simulations: experimental verification The procedure to obtain the maximum damping in the main chatter mode, described in the previous section, does not take into account the practical limitations that can be presented when the control system is implemented. These limitations arise because force generation capability of the actuators is limited and, in practice, the required active forces to counteract self-excited vibrations can exceed the admissible limits before feedback gain reaches its optimum value. The above-mentioned restriction results in an upper limit of the voltage that can be applied to the actuators. Taking into account that PI-247.30 piezoelectric actuators require an input voltage between 0 and 1000 V, it is very interesting to develop a methodology to predict the voltages demanded by the control system for different feedback gain values to assure that admissible limits are not exceeded. For this purpose, time domain simulation of the process was programmed, which is well suited to obtain such quantitative values. This procedure implies integrating the control loop shown in Fig. 5 in the chatter loop of the centerless grinding process, previously presented by several authors [6,9]. Fig. 8 shows the integrated model, where e 0 ,(1À e ), j 1 and j 2 are variables depending on the geometric configuration of the machine, s is the Laplace operator, o p is the angular velocity of the workpiece, K is the cutting stiffness and k eq is the equivalent contact stiffness. The control system influence over the structure ARTICLE IN PRESS Fig. 8. Centerless grinding process chatter loop with control algorithm integrated. Fig. 9. Theoretical evolution of accelerations: (a) at the grinding wheel head, (b) at the regulating wheel head and (c) at the workblade. Fig. 10. Experimental evolution of accelerations: (a) at the grinding wheel head, (b) at the regulating wheel head and (c) at the workblade. M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245 243 can be seen as follows: when a feedback gain value is selected, the relation between structural deformations and normal grinding force is expressed by the damped FRF, thus originating a reduction in vibration amplitudes. Several grinding cycles were simulated, increasing sequentially the feedback gain and, consequently, the voltage applied to the actuators. For each gain value, the workpiece was divided into 360 equal radial segments and its rotation was simulated segment-by- segment using the experimental cutting conditions programmed previously in the frame of the work performed in [8]. Roundness errors of workpieces were calculated following the procedure described in [9] and, additionally, the evolution of both normal grinding forces and voltages applied to the actuators was obtained. The last simulated grinding cycle corresponded to the one where the voltage applied to the actuators reached its limit. This situation happened for a smaller feedback gain value than the one with which maximum active damping had been obtained in Fig. 7. A subsequent increase in the feedback gain would saturate the actuators, meaning that their physical limits are an important constraint in this application. This last study was used to check the effectiveness of the theoretical model comparing two important simulated results to the corresponding experimental measurements. The first test was based on the comparison of vibration amplitudes in different components of the centerless grinding machine, whereas the second one was based on the comparison of final workpiece roundness errors. The results are detailed below. 6.3. Comparison of vibration amplitudes This study was undertaken using the normal grinding force evolution as input in the developed state space model to obtain some acceleration predictions as outputs. Figs. 9a–c show the theoretical evolution of accelerations in three dof’s located at the grinding wheel head, the regulating wheel head and the work- blade, respectively, both before and after the application of control law. Experimental measurements of the same variables are shown in Fig. 10 . Theoretical results agree with experimental measurements quite well, as they predict adequately the quantitative values of accelerations in different components of the machine. Further- more, the maximum reachable vibration reductions in these components are predicted correctly. Additionally, it can be seen that no vibration reduction is obtained in the workblade, as it could be expected taking into account that vibration of this component is dominated by the secondary chatter mode, which cannot be actively damped (Fig. 7b). In this case, simulation results show even an increase in acceleration amplitudes when control is applied. 6.4. Comparison of final workpiece roundness errors The final profile of the workpiece gives a quantitative measurement of the maximum error reduction that can be achieved with the SOF controller. Fig. 11a shows the theoretical shape simulated before the application of the control law whereas Fig. 11b illustrates the profile after its application. Figs. 12a and b show the profiles obtained experimentally. The theoretical model predicted a roundness error reduction of 41.7%, whereas the experimental error reduction had been of 32%. This result shows that computer calculations were highly realistic, which is a statement that is reinforced comparing theoretical and experimental workpiece profiles. 7. Conclusions The experimental testing of active vibration control systems in machine tools is a very costly and time- consuming task and, therefore, solutions based on virtual ARTICLE IN PRESS Fig. 11. Final theoretical workpiece profile: (a) without control and (b) with control. Fig. 12. Final experimental workpiece profile: (a) without control and (b) with control. M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245244 prototyping by numerical simulations are increasingly demanded. The work presented in this paper satisfies the demand existing in the centerless grinding sector, providing an inexpensive tool adapted for the purpose of evaluating the performance of different control alternatives before their practical implementation. The developed model integrates the description of both the mechanical structure and the control system following a mecha- tronic approach. It was validated comparing the experimental results obtained from a previously implemented second-order filter controller with the ones estimated numerically. Quantitative predictions concerning vibration reduction capability of the controller were in good agreement with experimentally obtained results. Additionally, the model has the ability to predict correctly the roundness error improvement in workpieces once the control has been activated. 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Nordmann, Comparison of control strategies for active vibration control of flexible structures, Archives of Control Science 13 (3) (2003) 303–312. [14] Z Q. Qu, Model Order Reduction Techniques, with Applications in Finite Element Analysis, Springer, New York, 2004. [15] B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Transactions on Automatic Control AC 26 (1) (1981) 17–32. ARTICLE IN PRESS M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245 245 . active system of reduction of vibrations in a centerless grinding machine using piezoelectric actuators, International Journal of Machine Tools and Manufacture. of the centerless grinding machine. Fig. 2. Detail of piezoelectric actuators location area. M.H. Fernandes et al. / International Journal of Machine Tools