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Abstract—Chatter vibration in machining operations is a limiting factor in enhancing metal cutting efficiency. This paper presents a novel semi-active intelligent control technique for chatter vibration supersession using tunable magnetorheological (MR) dampers. Structural dynamic characteristics of the machine tool (e.g., real and imaginary parts of the frequency response function) are the main parameters which specify the stability lobes diagram (SLD), i.e., the border between stable and unstable cutting conditions. In the present work, chatter reduction is achieved by altering these factors semi-actively, by means of a MR damper. First, a lumped model for MR damper is presented using modified Bouc-Wen model. Subsequently, integrated simulation software is developed for studying the vibration of a lathe machine equipped with a MR damper. An innovative real-time criterion is presented to recognize the chatter occurrence. A fuzzy controller is designed to calculate the voltage to be sent to MR damper at each instant to prevent chatter occurrence. The obtained results show that the proposed method has been successful in reducing the chatter conditions and improving the stability of turning operation with very low energy consumption.

I. INTRODUCTION ACHINING of industrial parts with high speed and

high material removal rate is a major requirement for manufacturing companies to be able to produce their products with minimum cost and compete with other manufacturers. In this context, chatter vibration is a limiting factor for increasing material removal rate. Chatter is an undesirable self-excited vibration that is generated mainly due to regenerative effect. It causes harmful effects such as poor surface finish, premature tool wear, potential damage to the machine or the tool, and load noises.

Application of smart materials and instruments as sensors and actuators for vibration suppression has attracted significant attention in recent years. In this research, an integrated model of a lathe and a smart magnetorheological (MR) damper along with a fuzzy control algorithm is presented to avoid chatter occurrence in a lathe machine.

Due to importance of chatter reduction, a lot of researches have been conducted in this field. Available literature can be divided in four categories. The first category includes researches which treat this problem in passive mode. These researches attempt to prevent the chatter vibration through proper design of the structure of the machine tool, by different means such as increasing the rigidity of the system, changing the cutter geometrical shape [1] and insertion of vibration

Manuscript received March 1, 2010. The authors are with Mechanical Engineering Department, Isfahan University of Technology (IUT), Isfahan, Iran; (phone: (+98)311-3915226; fax: (+98)311-3912628; e-mail: behbahani@cc.iut.ac.ir).

absorbers [2]. Lack of intelligence and flexibility, and not optimized design for various machining operations and conditions are the main disadvantages of these approaches. Also possible changes in the workpiece-machine structural dynamics cannot be considered. The second category refers to those researches which apply active control techniques for chatter suppression, such as using piezoelectric actuators [3] and magnetostrictive actuators [4]. These systems provide high flexibility to compensate the potential vibration under different operations and conditions. This approach is more expensive than other approaches, due to the cost of the required actuator and its drive and also due to high energy consumption. Moreover, these systems have safety concerns, because if a problem occurs in the energy source, the performance of the system can be considerably degraded. Possible destabilizing has also been reported due to active control solutions [5]. The third category includes those researches which present approaches to avoid chatter by changing the machining conditions, such as changing axial depth of cut or rotational spindle speed [6]. Obviously these approaches make some limitations in machining operation. Sometimes the spindle speed or depth of cut cannot be varied to lie in the stable zone due to machining constraints; hence the feed must be halted during the process. The other category which is the subject of this study is chatter suppression by semi-active methods. In this approach, dynamic properties of some parts of the machine can be changed. Insertion of variable dampers in the system is a common approach. MR dampers are a good example of variable dampers, where the damping characteristics of the damper are varied by the input voltage to the damper. MR dampers have been used for vibration reduction in many mechanical systems such as vehicle suspension system [7], washing machine base [8], and shock absorbers [9].

There is very little literature on semi-active chatter control by means of MR damper. Application of electrorheological (ER) fluids for chatter suppression has been reported in paper [10]. In 2009, a research activity has been conducted on using MR fluid instead of ER fluid in a boring process. The MR fluid showed better performance on controlling the stiffness of the system by means of a magnetic field [11]. It was shown that MR fluid has positive effect on the dominant frequencies of the system to improve the stability of the system in low spindle speeds. However, in high rotational speeds, a large increase in stiffness was needed which could not be provided by the MR fluid. This research did not investigate the influence of MR damper device on the damping of the system. Recently, possible positive effect on stability of a machining

Mechatronic Modeling and Control of a Lathe Machine Equipped with a MR Damper for Chatter Suppression

D. Sajedipour, S. Behbahani*, and S.M.K. Tabatabaei

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2010 8th IEEE International Conference onControl and AutomationXiamen, China, June 9-11, 2010

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process by means of MR dampers was reported through several experiments [5]. In these experiments, fixed voltages were given to the MR damper and stability improvements were depicted. However, no analytical study, modeling and simulation, or controller design was presented. Obviously, these are major requirements in effective application of MR dampers for chatter control, which have been overlooked in literature.

In present work, an integrated model for chatter analysis of a lathe machine equipped with a MR damper is presented. The influence of MR damper excited with constant voltage on the stability of a turning operation is investigated to show that the presented approach is able to predict the obtained results in previous work. Obviously, exciting with constant voltage can hardly be considered as the optimum solution to the problem. Next, by analysis of several effective parameters, a fuzzy controller is designed to calculate the voltage to be supplied to the MR damper. The proposed controller is a semi-active controller which requires much less power in comparison with other control approaches. Also, this system provides safer solution in comparison with active controllers, because if a problem occurs in the energy source or control system, MR damper converts to a passive damper and it can still be effective in chatter reduction. A new approach for on-line chatter detection based on a new defined chatter detection index (CDI) is also presented. It is based on the analysis of vibration signals of the system.

The paper is organized as follows. First, an integrated modeling of the chatter vibration in machining along with a lumped model for MR damper is presented. Both time domain and frequency domain approaches for chatter study are presented. Subsequently, a fuzzy control approach is presented.

II. MODELING Chatter study of a machine tool is commonly conducted by drawing its stability lobes. These are diagrams which depict

the border between stable and unstable machining conditions, where the limited depth of cut is plotted versus spindle speed. Reviewing the formulation of the procedure, it may be concluded that chatter is a result of dynamic interactions between machining conditions and structural dynamic specifications of the machine tool [12]. Structural dynamics of the machine tool is presented in the form of the frequency response function (FRF) between the force and relative displacement of the tool tip. In particular, real and imaginary parts and the phase of FRF are the main parameters used in the generation of stability lobes. Obviously, altering the structural characteristics of the machine by the use of an external device can affect the stability lobes. The idea of the present work is to use a MR damper for this purpose. The schematic diagram of the proposed system is shown in fig. 1. In this section, a lumped model for MR damper and a model for chatter study in turning operation are presented, and then an integrated model for the study of influence of MR damper on chatter is presented.

A. MR damper modeling

Schematic view of a MR damper is presented in fig. 2(a). The oil used in these dampers contains micron-sized, magnetically polarizable particles; hence its viscosity changes when a magnetic field is applied. In a MR damper, a coil exists around the damper to generate magnetic field. The damping of the damper can be varied by means of the voltage supplied to this coil.

Several dynamic models have been proposed in literature to illustrate the behavior of a MR damper [13]–[15]. In this research, the modified Bouc-Wen model presented in [13] was used, because of proper match between simulation results of this model and experimental results as reported in [13]. Considering the fact that chatter occurs in high frequencies and the range of machine tool vibrations is high, MR damper model needs to have adequate precision in all conditions such as various types of excitation, under different amplitude and frequencies. These requirements are best satisfied with modified Bouc-Wen model in comparison with other proposed models. Fig. 2(b) shows the equivalent mechanical system

Fig. 1. Schematic of proposed system model.

a b Fig. 2: Schematic of MR damper (a), and modified Bouc-Wen mechanical model of MR damper (b).

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used in modified Bouc-Wen model. The governing equations used in this model are expressed as follows:

� � � � � �0 0 1 0F z c x y k x y k x x�� � � � � � �� � � �1

� � � �1n nz x y z z x y z x y ��� � � � � � �� � � � � � � �2

� � � �0 1 0 01y c c z c x k x y�� � � � � � �� � � �� � � �3

In this model, F is the total force generated by the system, z is the evolutionary hysteretic variable that accounts for the time history of the response, k1 is the accumulator stiffness, c0

is the viscous damping observed at larger velocities, c1 represents a dashpot which is included in the model to produce the roll-off at low velocities, k1 indicates a parameter to control the stiffness at large velocities, and x0 is the initial displacement of spring k1, associated with the nominal damper force due to the accumulator [13]. In the model, functional dependence of the parameters on the applied voltage can be represented by following equations:

1 1 1 0 0 0, ,a b a b a bu c c c u c c c u� � �� � � � � � � �4

� �u u v�� � �� � �5

v is the voltage applied to the MR damper. In [13], the optimal values of 14 inherent parameters of this model for a prototype were identified experimentally. This prototype was 21.5 cm long in its extended position, and had a ±2.5 cm stroke. A similar damper with similar parameters is considered for this research. A model was developed and its response under same excitation applied in [13] was simulated to be compared with simulated and experimental results reported in [13]. Fig. 3 shows the results under 2.5 Hz sinusoidal excitation with amplitude of 1.5 cm.

B. Linear chatter modeling

1) Frequency domain Analysis: Chatter in the machine tool is a harmful vibration which is the result of self-excitation mechanism that occurs during generation of chip thickness in machining operations [12]. Two main sources of chatter are regenerative effect and mode coupling. In most machining processes, regenerative chatter occurs earlier than mode coupling chatter [12]. It results from phase difference that is generated between two consecutive vibration waves, as shown in fig. 4. This phase difference causes a varying removed chip thickness. Sometimes the chip thickness can grow exponentially and lead to major damages to tool or workpiece [12]. Various models for chatter in turning have presented by

researchers that most of them are proposed linear models [12], [16], and some have preferred to use nonlinear models [17].

Despite the real nonlinearity of chatter, linear stability theory explains chatter in best form [12]. The stability expression used in this research is thus a reliable linear theory, first obtained by Tlusty [16]. In this theory, it is assumed that cutting force that causes relative vibration is exerted only in radial direction, thus relative displacement between tool and workpiece is in this direction (fig. 4). The final dynamic chip thickness can be expressed as follows:

� � � �0( ) [ ]h t h y t y t T� � � � � �6

h0 is the purposed chip thickness, i.e., feed rate of the machine tool, y(t) is the relative displacement of tool-workpiece at current time along assumed exerted machining force i.e., perpendicular to workpiece surface, and y(t-T) is the relative displacement during previous revolution of cut. T indicates the spindle revolution period that is the main cause of regenerative chatter occurrence. According to this theory, system equation of motion can be expressed as:

� � � � � � � �fmy t cy t ky t F t� � ��� � � �7

Where m, c, and k are equivalent mass, damping and stiffness of the systems, and Ff (t) is the machining force in current time, which is expressed as follows:

� � � � � � � �0[ ]f f fF t K ah t K a h y t T y t� � � � � � �8

Where Kf is the cutting constant of machining force, and a indicates the axial depth of cut. The dynamic chip thickness can be expressed in Laplace domain as follows:

� � � �� � � �

0 0( ) ( ) ( 1) ,

( ) ( ) ( )

sT sT

f f

h s h y s e y s h e y s

y s F s s K ah s s

� �� � � � � �

� � � � � �9

Where, �(s) is the transfer function of the system. In this theory, the stability lobes diagram (SLD), that indicates the maximum allowable axial depth of cut for each spindle speed can be generated. The procedure of generation of SLDs has been explained in [12]. Note that final equations that are used to obtain SLD are as follows:

Fig. 3: Simulation Results of MR damper with modified Bouc-Wen model under 2.5 Hz sinusoidal excitation with amplitude of 1.5 cm.

Fig. 4: Assumed 2-DOF cutting process and schematic of dynamic chip thickness generation system.

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� �1tan , 3 2H G� � � ��� � � � �10

� � � �2 2 60T k f n� � �� � � � �11

� �lim 1 2 ( )f ca K G �� � � �12

� is the phase angle of the structure, H and G are imaginary and real parts of transfer function, � is the phase shift between the current and previous waves, k is the integer number of waves, fc is the chatter frequency in Hertz, and n is the spindle speed in rev/min. alim Indicates the maximum axial depth of cut for stable machining, and G(�c) is the real part of transfer function at chatter frequency [12]. According to equation (12), chatter can be generated in frequencies where G(�c) is negative. To plot the SLDs, a chatter frequency around the natural frequency is chosen so that G(�c) is negative. The phase angle �, the phase shift �, and finally the critical depth of cut are calculated according to equations (10) and (12). After that the corresponding spindle speeds for obtained critical depth of cut are calculated using equation (11), by substituting k = 0, 1, 2, …, 10. The case study presented here is for a lathe machine with two dominant degrees of freedom, similar to the case study presented in [12]. Note that this lathe machine has two dominant natural frequencies in 150 Hz and 250 Hz. 2) Time domain chatter analysis: when a nonlinear element such as a MR damper or a nonlinear controller is added to the machine tool, total system equations will become nonlinear. In this situation, chatter analysis of the machine cannot be performed in the Laplace domain. It means that, the linear chatter theory and analysis presented in section B. 1 cannot be applied anymore. A time-domain chatter analysis and SLD generation algorithm are presented here. A block diagram model of the system was generated and the delay term (e-sT) was modeled by a delay block. Time domain simulation of a machining condition (i.e., spindle speed and axial depth of cut) indicates whether the corresponding point is in the stable region or in unstable region. In the stable conditions, it is observed that vibration signals of relative displacement (or machining forces) converge to a small value. In the unstable conditions, vibration signals diverge to large values. Software was developed to detect the convergence or divergence of obtained relative displacement. At any spindle speed, the depth of cut is increased little by little until the border of stable and unstable conditions is detected. The procedure is repeated for different spindle speeds and the SLDs are plotted. The main difficulty in this algorithm is in automatically (not visually) recognizing the stability or instability of vibration signals. In this context, some researchers have tried to develop a threshold value for chatter detection, by investigation of spectral density of process signals [18], and some others have used the audio signals for chatter detection [6].

Here, a novel algorithm is presented for chatter detection which is based on a new defined concept, called chatter detection index (CDI). It is applicable for on-line chatter detection, and also for chatter detection in time domain generation of stability lobes. The flow chart of the proposed approach is shown in fig. 5. By processing and analysis of

displacement/force signals in a 0.3 second interval, a chatter detection index is computed, which indicates whether the amplitude of the signals is increasing or not. Also, it can indicate how far a machining condition is from the stability border. First, the variance of signal is computed over time. Then the signal data in two time intervals are separated, first between 0.26s and 0.28s, and second between 0.28s and 0.3s. Next, these two branches of signals are entered into a cumulative sum block to compute sum of all magnitudes during these two ranges. CDI is then computed by dividing these values to each other. A comparison between SLDs for a linear system obtained from illustrated time domain analysis, and SLDs from frequency domain analysis is shown in fig. 6. Comparison between these two generated SLDs verifies the correctness of the proposed time domain SLDs generation, chatter detection algorithm, and proposed CDI concept.

III. INTEGRATED LATHE-MR DAMPER MODELING In order to study the MR damper influence on the stability

of a lathe machine, the governing equations have to be solved which are coupled together. Fig. 1 shows the schematic diagram of the proposed system. �MRD indicates the angle between axis of MR damper and direction of vibration, which is assumed to be zero in this study. According to the equations of MR damper model and equations of vibration of turning process, i.e., equations (1), (2), (3), (7), and (8), total equations of motion of the system

Fig. 6: Comparison between proposed time domain approach ( ) with analytical frequency domain approach ( ).

Fig. 5: Flow chart of the proposed chatter detection approach.

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can be expressed as:

� � � � � �� � � � � �0[ ]f M R D

my t cy t ky t

K a h y t T y t F t

� � �

� � � �

�� � (13)

Where FMRD is the force applied by MR damper against relative movement, which is added due to the presence of the MR damper. Block diagram of the new system is presented in fig. 7. Fig. 8 shows force-time and force-velocity diagrams of MR damper, respectively for V=0v and V=2.25v voltages, for a stable machining conditions, i.e., a=0.011 and T=60/n=60/7000. It is obvious that the maximum magnitude of MR damper force that is applied to the system in “on-state” is more than its equivalent force in “off-state”. The simulation results under a constant voltage indicated that SLDs would shift upward and a bit rightward. As expected and reported in available literature [5], MR damper causes improvement in chatter stability specifications of the machine. The reason of upward movement of SLDs is a significant increase in equivalent damping of the system. The amount of equivalent damping ratio (�) increases, hence G(�c) decreases which leads to an increased alim and SLDs shifted upward. Low rightward movement of SLDs is due to small increase in the natural frequency of the machine, due to low stiffness added by the MR damper. Since chatter frequency is close but a bit larger from the dominant structural mode of the system [12], fc

increases too which is the reason of small rightward shift in SLDs. When the applied voltage to MR damper is increased, the equivalent damping of the whole system is increased. This leads to a more stable system.

IV. CONTROL Designing a control scheme to identify the chatter

occurrence and apply appropriate voltage to MR damper is presented in this section. The objective is to control the vibration in a semi-active way by changing the characteristics of the structure. A fuzzy controller is designed for this purpose. The procedure of computation of the voltage is as follows (see fig. 9):

For each specific machining condition (i.e., axial depth of cut, a, and rotational spindle speed, n), CDI is computed according to the flowchart presented in section B. 2. The inputs to the fuzzy controller include calculated CDI, along with other parameters such as machining condition, current MR damper voltage, and previous CDI. The output of the controller is the voltage to be applied to MR damper.

A set of fuzzy rules is used to calculate the output voltage. In case that chatter continues to exist, it is detected in next time step (0.3 sec) and a necessary change in voltage is applied to get out from chatter vibration. The voltage modification is repeated until chatter vibration is overcome. Obviously, it cannot be claimed that chatter can be avoided for all cutting conditions, due to MR damper capacity limitations. MFs of the fuzzy controller inputs and output have been shown in fig. 10. Through gathering intuitive knowledge about the system dynamics and controller effect, a fuzzy rule base was proposed that contains seven rules as follows: 1) If (CDI is stable) and (previous-CDI is stable) and

(current-voltage is low), then (next-voltage is low). 2) If (CDI is stable) and (previous-CDI is unstable) and

(depth-proportion is low), then (next-voltage is low). 3) If (CDI is stable) and (previous-CDI is unstable) and

(depth-proportion is high), then (next-voltage is high). 4) If (CDI is unstable), then (next-voltage is high). 5) If (CDI is stable) and (previous-CDI is stable) and (depth-

proportion is low) and (current-voltage is high), then (next-voltage is low).

6) If (CDI is stable) and (previous-CDI is stable) and (depth-proportion is high) and (current-voltage is high), then (next-voltage is high).

Using these rules with weight factor of 1 for all of them, controller was designed and simulated. Stability lobes diagram was then generated. Fig. 11 depicts a comparison between

Fig. 7: Block diagram of integrated lathe-MR damper system.

Fig. 8: Damper force vs. time, velocity and displacement, for V=0 v and V=2.25 v (left and right columns), in a same stable machining conditions.

Fig. 9: Schematic block diagram of control system.

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SLDs obtained by controller with the SLDs of non controlled lathe machine. Fig. 12 shows time domain vibration of the particular cutting condition that has been unstable before using the controller. Obviously, the controller has been successful in stabilizing the system.

It is concluded that MR damper and the proposed controller has a significant positive effect on the stability specifications of the system and on stability lobes diagrams.

V. CONCLUSION An integrated mechatronic model for a lathe machine

equipped with a MR damper along with a fuzzy controller was presented. Modeling and analytical tools for analysis and controller design were developed. Simulation results confirmed the experimental results obtained in a recent research, for the basic condition of imposing constant voltage to the damper. A fuzzy controller was designed to compute the voltage to be imposed to MR damper, based on vibration behavior of the system. The controller significantly improved the system’s stability characteristics, and the obtained results showed quite successful performance of the proposed ideas in enhancing the stability of turning process.

REFERENCES [1] Y. Altintas, S. Engin, and E. Budak, “Analytical Stability Prediction and

Design of Variable Pitch Cutters,” ASME Int. Mech. Eng. Congress & Exposition, Anaheim, California, MED 8 pp. 141–148, 1998.

[2] Y. S. Tarng, J. Y. Kao, and E. C. Lee, “Chatter Suppression in Turning Operations with Tuned Vibration Absorber,” Int. J. Mach. Tools & Manuf., 2000, vol. 43, pp. 55–60.

[3] W. H. Zhu, , M. B. Jun, , and Y. Altintas, “A fast tool servo design for precision turning of shafts on conventional CNC lathes,” Int. J. Mach. Tools & Manuf., Vol. 41, pp. 953–965, 2001.

[4] P. Pagliarulo, H. Kuhnen, C. May, and H. Janocha, “Tunable Magnetostrictive Dynamic Vibration Absorber,” Int. J. Mach. Tools & Manuf., vol. 59, pp. 105–127, 2003.

[5] D. Sathianarayanan, L. Karunamoorthy, J. Srinivasan, G. S. Kandasami, and K. Palanikumar, “Chatter Suppression in Boring Operation Using Magnetorheological Fluid Damper,” Materials & Manufacturing Processes, vol. 23, pp. 329–335, 2008.

[6] Y. Altintas, and P. K. Chan, “In Process Detection and Suppression of Chatter in Milling,”. J. Mach. Tools Manuf., vol. 32, pp. 329–347, 1992.

[7] M. Yu, X. M. Dong, S. B. Choi, and C. R. Liao, “Human Simulated Intelligent Control of Vehicle Suspension System with MR Dampers,” Journal of Sound and Vibration, 2008.

[8] F. Previdi, and C. Spelta, “Vibration Control in a Washing Machine by Using Magnetorheological Dampers,” working paper, series Information Technology, no. 10, 2007.

[9] B. Sapinski, and M. Rosol, “MR Damper Performance for Shock Isolation,” J. Theor.& Applied Mech, vol. 45, no. 1, pp. 133–145, 2007.

[10] M. Wang, and R. Y. Fei,” On-line Chatter Detection and Control in Boring Based on an Electrorheological Fluid,” J. Mechatronics, vol. 11, no. 7, pp. 779–792, 2001.

[11] D. Mei, T. Kong, A. J. Shih, and Z. Chen, “Magnetorheological Fluid-controlled Boring Bar for Chatter Suppression,” J. materials processing technology. Vol. 209, pp. 1861–1870, 2009.

[12] Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge, UK: Cambridge University Press, 2000, pp. 97–104.

[13] B. F. Spencer, Jr., S. J. Dyke, M. K. Sain, and J. D. Carlson, “Phenomenological Model for Magnetorheological Dampers,” J. Eng. Mech., vol. 123, no. 3, pp. 230–238, march 1997.

[14] X. Song, M. Ahmadian, and S. C. Southward, “Modeling Magneto-Rheological Dampers with Application of Non-parametric Approach,” J. Intell. Mater. Syst. Struct., vol. 16, no. 5, pp. 421–432, 2005.

[15] A. Dominguez, R. Sedaghati, and I. Stiharu, “A New Dynamic Hysteresis Model for Magnetorheological Dampers,” J. Smart Mater. Struct. vol. 15, pp. 1179–1189, 2006.

[16] J. Tlusty, and M. Polacek, “The Stability of Machine Tools Against Self-Excited Vibrations in Machining,” Proc. ASME Int. Research in Production Eng., 1963, pp. 465–474.

[17] J. Tlusty, and F. Ismail, “Basic Nonlinearity in Machining Chatter,” Annals of the CIRP, 1981, vol. 30, pp. 21–25.

[18] S. Y. Liang, R. L. Hecker, and R. G. Landers, “Machining Process Monitoring and Control: The State-of-the-Art,” J. Manuf. Sci. Eng., vol. 126, pp. 297–310, may. 2004.

Fig. 11: Comparison of SLDs for basic turning operation ( ), with controlled by MR damper using fuzzy controller ( ).

Figure 10: Membership functions of the fuzzy controller.

Fig. 12: Vibration signals of an unstable condition, i.e., a =0.012 mm, n =4000 rpm, controlled by proposed semi-active approach.

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