Annals of the CIRP Vol. 56/1/2007 -347- doi:10.1016/j.cirp.2007.05.080 Intelligent Centerless Grinding: Global Solution for Process Instabilities and Optimal Cycle Design I. Gallego (3) Manufacturing Department, Faculty of Engineering – Mondragon University, Mondragon, Spain Submitted by R. Bueno (1), San Sebastian, Spain Abstract Centerless grinding productivity is largely limited by three types of instabilities: chatter, geometric lobing and workpiece rotation problems. Regardless of its negative effect in manufacturing plants, no functional tool has been developed to set up the process, because it involves the simultaneous resolution of several coupled problems. In this paper, new simulation techniques are described to determine instability-free configurations, making it possible to guarantee that the final workpiece profile is round. With this information and taking into account other process restrictions, like system static stiffness and workpiece tolerance, the optimal grinding cycle is designed. These results have been implemented into an intelligent tool to assist the application of this research in industrial environments. Keywords: Centerless Grinding, Productivity, Simulation 1 INTRODUCTION In centerless grinding, the workpiece is not clamped, but simply supported between the grinding wheel, the blade and the regulating wheel (figure 1), reducing the machine idle time and avoiding the necessity of centring holes on the workpiece. Due to the enormous manipulating time and manufacturing cost saving that this implies, centerless grinding is extensively used in the mass production of components in automotive and bearing industries for example. Nevertheless, the process suffers from three kinds of instabilities that may limit its precision and productivity. 1) Chatter, whose growing is much more pronounced than in conventional grinding. 2) Geometric lobing, which appears when the workpiece does not self-centre and begins to oscillate between the wheels. 3) Work-holding instability, which appears when the regulating wheel is not able to make the workpiece spin at its peripheral velocity. In previous works, techniques to avoid geometric lobing in infeed [1] and throughfeed [2] have been shown. Nevertheless, as stated by Hashimoto, all the problems should be solved at once, because stable conditions for one instability may not be so proper for another one [3]. Figure 1: Geometric configuration in infeed grinding. In this work, a new technology has been developed to give a global solution to all instabilities at the same time and design the optimal grinding cycle, so as to obtain the required workpiece tolerance in the minimum time without thermal damage. The application of these results in grinding industry has potentially a great impact, reducing process set-up time and decreasing production stops motivated by out-of-roundness issues. 2 PROCESS SIMULATION 2.1 Process equations In this section, the basic equations governing the centerless grinding process are shown. A new and more coherent notation has been employed to simplify some of the expressions. Let r w (t) and G r w (t) be the radius and radius defect of the workpiece in an infeed process (figure 1). The radius defect varies due to strictly geometric reasons, because the cutting forces cause changes in the deflection of the machine, workpiece and wheels, and because vibrations may arise. This way, G r w (t) can be expressed as:         ttttr DKgw HHHG  (1) where H g (t) is the geometric displacement of the workpiece due to roundness errors passing through the contact points with the blade and regulating wheel. H K (t) represents the time variation of the system static deflection (along the cutting direction) and H D (t) is the term representing the vibrations generated in the process. The geometric term H g (t) was derived by Dall [4] and refined by Rowe et al. [5]. It can be expressed as:       rwrbwbg WGWGH  trgtrgt (2) being  bw WG tr and  rw WG tr the radius defect at the contact points with the blade and regulating wheel. g b and g r are two geometrical parameters [1]. Geometric lobing stability has been studied in the frequency domain [6,7]. Applying the Laplace transform to equation (2), it is obtained: T  J r h M   M   J J r  J s Grinding wheel Regulating wheelBlade Workpiece -348- r b rbwg W W H s s egegsRs (3) In a grinding process, the cutting force F c (t) and the instant depth of cut, a e (t) are related by a parameter called cutting stiffness (k w ): )( ewc taktF (4) The cutting force produces a deflection of the machine, wheels and workpiece, which makes the radius reduction to accumulate a delay in relation to the programmed feed. From a stability point of view, it is not a matter of interest to know the delay, but the variations in deflection generated by the radius defect evolution. Defining W as the rotation period of the workpiece, it is obtained: trtrktF wwwc GWGG (5) The second term of equation (1) can be expressed as: eq c K k tF t G H (6) where k eq is the equivalent stiffness of the system. k eq can be determined in two different ways: experimentally or theoretically with the help of a data- base. In the first case, several tests must be performed employing different feed levels. Based on the difference of diameters between tests with and without spark-out, the relation of the cutting force and the total deflection of the system can be obtained. In the second case, the value of k eq can be predicted analytically using this expression: 1 crr 1 cs 1 m 1 eq kgkkk (7) This equation is derived from a spring-model, which includes the machine and blade stiffness (k m ) and the wheels/work contact stiffness (k cs , k cr ). Determination of equivalent stiffness and its dependencies on feed rate, wheel type, etc. is essential to predict instabilities accurately [8,9]. Regarding cutting stiffness, its value may be estimated with analytical approximations, but a final experimental calibration is recommended to have chatter prediction maps and the optimal grinding cycle very close to reality. Introducing expression (5) in equation (6): trtrKtrtr k k t wwww eq w K GWGGWGH (8) K parameter represents the flexibility of the system and relates the amount of deformation of the system to different depths of cut. Centerless grinding chatter was extensively studied by Miyashita, Hashimoto et al. [10,11] and Rowe et al. [12,13,14]. These authors developed successful models that can be used to predict chatter-free configurations. Bueno et al. [8] and Nieto et al. [15] determined the threshold stability value of the cutting stiffness as a function of the workpiece rotation frequency and introduced non linear effects in the model, such as the spark-out and lobe filtering. Hashimoto and Zhou [16] proved that filtering effects have a big effect on high waviness stabilisation. According to these authors, contact stiffness is again an important factor on chatter growing, as it has been stated for other types of grinding processes too [17]. To introduce vibrations in the model, the dynamic flexibility of the machine H(s) may be used. Centerless grinding process is usually well defined with a two- dimensional modal analysis at the three points of contact between the workpiece, blade and wheels. Nevertheless, if H(s) is introduced in the model without any correction, it will add an extra contribution to the static deflection of the system, already considered in equation (8). Subtracting this contribution, the expression of H D in Laplace domain turns into: Ư á á ạ ã ă ă â Đ m 1 2 r r rr 22 r r wwD 2 1 N r s V ss V esRks ZZ[Z H W (9) N m : number of considered vibration modes. m r , Z r , [ r : modal mass, frequency and damping of r mode. V r may be expressed in this way: ^` ^`^` ^` PXXCV T rr T r (10) {X r }: vector containing the relative deformations at the contact points of r mode. {C}: vector quantifying the real displacement at the cutting point due to a displacement of the contact points. {P}: vector relating the forces at the contact points with the normal force at the cutting point. The last term in equation (9) is the referred correction to H(s). We consider that this is an essential enhancement of preceding models, as it has a significant influence on the dynamic stability maps shown in section 4. In addition, it should be pointed out that when vibrations are introduced in the model, it is very important not to disregard the static term H K (t), because it may be proved that, in that hypothetical case, geometric lobing could appear with lobe numbers very far from integer, which is not physically possible. Rearranging all the terms deduced in equation (1): 0 1 2 11 m r b 1 2 r r rr 22 r r w rb w ắ ẵ đ ằ ằ ẳ ô ô ơ ê Ư W W W W ZZ[Z s N r s s s e V ss V k eKegeg sR (11) As it is well known, the poles of ' R w (s) will define whether the process is stable or not. The poles can be expressed in terms of defect regeneration frequency ( Z ) and damping ( [ ): s=- Z[ +i Z (1- [ 2 ) 1/2 . Regeneration will be unstable for [ <0. For that reason, it is necessary to find the roots of the next function: W W W ZZ[Z s N r s s e V ss V kK egegs á á ạ ã ă ă â Đ ằ ằ ẳ ô ô ơ ê Ư 1 2 1f m r b 1 2 r r rr 22 r r w rb (12) In the next section, the methodology to get all the significant roots of f(s) is shown. All these equations have been deduced for plunge processes. The adaptation of the model for throughfeed processes was explained by Meis [18] and Gallego et al. [2]. Other remarkable approaches to improve productivity can be found in the bibliography, like the one recently proposed by Klocke et al. [19], which involves working below center so that higher feed rates can be employed, -349- but using a new type of functional blade to avoid geometric lobing. Other authors, like Harrison and Pearce [20], have proposed changing machine configuration in- process to allow a faster correction of the initial roundness error of the workpiece. Finally, it should be mentioned that the equations to establish the limits for work rotation instabilities were deduced by Hashimoto et al. [21]. 2.2 Instabilities determination In contrast to milling process, where it is just necessary to know the chatter limits, in centerless grinding it is also necessary to determine the absolute value of the stability degree of the process. This is because at the optimal configuration, where all the lobes are stable with the maximum possible stability degree, initial roundness error correction is faster. This way, the process is less sensitive to changes in the roundness of entering workpieces and a final round profile may be obtained. Consequently, the development of a completely reliable and efficient method is the key to determine the best working configuration in a reasonable time. This is one of the major contributions of this work. In the past, several types of techniques have been employed: graphical methods [11], numerical methods [22], Taylor’s series approximation [9] or the Simplex method [23]. Established that f(s) is a complex function of complex variable, those s values which verify f(s)=0 will also verify |f(s)| 2 =0. Naming D and E the real and imaginary parts of s, the function is graphically shown in figure 2. This way, pole-finding of ' R w (s) has been transformed into root finding of a real function of two real variables. Being |f(s)| 2 positive, the problem is transformed into finding local minima. |f(s)| 2 has many minima close to D =0 axis. Those configurations with positive real part roots (i.e. [ <0), will be unstable. Those configurations with negative real part in all the roots, with the highest possible absolute value, will be the optimal configurations. The solution to this problem is to use an appropriate mesh in the ( D , E ) plane and then, starting from each point of the mesh, apply the best possible optimisation algorithm to find the closest minimum as fast as possible. With regard to the mesh, it is easy to demonstrate that the characteristic function can not have two different minima for the same value of E near D =0. This way, the mesh in D can be avoided. The optimum mesh is a row of points at D =0 from E = Z w to E =n max Z w , where Z w is the rotation speed of the workpiece and n max is the maximum number of lobes that can appear over the workpiece, usually not higher than 40 or 50. The number of points recommended for the E grid is n max . The best optimisation algorithm for the |f(s)| 2 function is Levenberg-Marquardt [24]. This algorithm is a hybrid method that combines the safe local convergence given by the steepest descent algorithm with the Newton method for a faster convergence. The only problem is that it is necessary to determine the first and second derivatives of the function analytically, leading to quite complex expressions. Nevertheless, by rearranging terms it is possible to include as many modes as desired in the function without excessive complication of expressions. The advantage of this technique is that it is possible to determine whether a certain working configuration is stable or not considering all instabilities in less than 0.1 seconds in an average computer. Repeating the calculus for many configurations it is possible to plot stability maps like the ones shown in the next sections. Figure 2: |f(s)| 2 function. 3 GEOMETRIC LOBING SUPPRESSION The two previous works on geometric lobing in infeed and throughfeed [1,2] have led to the development of a commercial set-up software, called Estarta SUA (Set Up Assistant). As an example, in figure 3 a stability map of the process is shown as represented in Estarta SUA. Stability maps are 2D or 3D graphs that define stable and unstable areas for different set-up parameters. In the case of geometric lobing, stability maps are plotted as a function of the blade angle (ș) and workpiece height above centre (h), two variables that are easily controlled by machine operators. Figure 3 has been obtained for the next conditions: wheels and workpiece diameters D s = 630 mm, D r = 310 mm and D w = 36 mm; K = 2.9; Q’ = 1.13 mm 2 s -1 ; feed: 1.2 mm min -1 ; Z r = 15 min -1 . The reliability of geometric lobing simulation is guaranteed by the fact that it is daily used by Estarta manufacturer and its customers with optimal results. 4 CHATTER PREDICTION Centerless grinding is especially sensitive to chatter. The high value of cutting stiffness that arise when grinding long workpieces causes the excitation of the main vibration modes (opening and closing of wheelheads) and also modes associated to the workblade. In centerless grinding, chatter caused by regeneration of a lobed profile on grinding wheels is less common than in other processes and will not be considered in this work. Chatter presents a great dependence on workpiece rotation frequency. Because of this, it is interesting to obtain stability maps for any combination of blade angle (ș), workpiece height (h) and regulating wheel rotating frequency ( Z r ), including at the same time geometric and dynamic phenomena. Figure 3: Geometric lobing stability map. In blue: stable areas. In red: unstable configurations. 0 5 10 15 20 25 50 40 30 20 10 Height (mm) Blade Angle (º) 5 7 36 32 28 24 20 16 31 35 22 18 33 14 22 24 26 28 18 31 33 35 14 16 18 20 22 24 26 28 30 32 34 36 34 36 32 30 28 26 30 32 34 36 34 9 36 26 10 15 10 5 0 0 0.5 -0.5 10 15 20 D E -350- 0 5 10 15 Height (mm) Figure 4. Chatter and geometric lobing stability map. In dark blue: stable areas. In red: unstable areas. Star size is proportional to the experimental vibration amplitude, round points represent tests without chatter. In figure 4 a stability map is shown as a function of h and Z r for a blade angle of 30º. Red areas represent configurations susceptible to chatter, light blue zones correspond to geometric lobing, while dark blue areas are stable for both chatter and geometric lobing. Black lines separate stable and unstable areas. Several tests have been performed in order to check the simulations, using two grinders of different manufacturers in the next conditions: 1. Machine 1: small grinder, wheelhead power 8 KW, wheelhead opening frequency: 90.8 Hz; T = 30º; D s = = 325 mm, D r = 220 mm, D w = 24 mm; workpiece length: L w = 25 mm; K eq = 14.2 N/Pm; Q’ = = 0.75 mm 2 s -1 ; feed: 1.2 mm min -1 . 2. Machine 2: large grinder, wheelhead power 60 kW, wheelhead opening frequency: 58.3 Hz; T = 30º; D s = = 628 mm, D r = 340 mm; D w = 47 mm; L w = 368 mm; K eq = 69.7 N/Pm; Q’ = 1.23 mm 2 s -1 ; feed: 1 mm min -1 . In the theoretical maps obtained for these cases, chatter free and geometric lobing free areas can be observed when using low heights and workpiece rotation speeds, as well as some transient areas at higher rotation speeds. There are also stable areas elsewhere, but they are too small from a practical point of view to be used. For machine 1, the map is checked with experimental results in figure 4. The size of the stars is proportional to the experimental vibration amplitude, while the round dots mean that no dynamic instability is excited. A good correlation is observed between simulation and experimental results. For machine 2, the simulation has been likewise reliable, in spite of the fact that the machine and process are very different from the previous case. A remarkable property observed in the theoretical maps is that the size of unstable areas changes as a function of the equivalent stiffness. For a given machine, changes in contact stiffness caused by workpiece length, feed rate or wheel type have an impact on stability maps and explain dynamic variations observed in production plants. For example, a more flexible regulating wheel reduces the size of unstable areas significantly. It should be noted the importance of the existence of dynamically and geometrically stable zones on the maps at high rotating speeds. It has been proved that in these configurations, as the workpiece turns many revolutions during the process, better roundness and roughness qualities are achieved. Moreover, a high rotating speed also prevents the workpiece from thermal damage. On the other hand, as we presented for geometric lobing |1,2], it is possible to solve the equations in time domain. This possibility opens new ways for chatter suppression, such as variable workpiece rotation velocity. This point will be discussed in a future work. 5 WORK ROTATION INSTABILITY AVOIDANCE The regulating wheel is the element controlling the rotational movement of the workpiece, exerting a brake moment over it by friction. In certain conditions, for example when high feed velocities or dull wheels are employed, the workpiece dragging becomes unstable, generating shakes, irregular velocities, jumps and accelerations, with risks for machine operators. In the absence of other kind of instabilities, these phenomena are the limiting factor to productivity, because the process feed defines the cutting force exerted by the grinding wheel and the required brake moment to control the movement of the workpiece. On the other hand, another function of the regulating wheel is to rotate the workpiece before beginning the grinding process. If the workpiece does not rotate, a flat band may be generated in the periphery of the workpiece. As mentioned before, the model and equations describing these phenomena were fully developed by Hashimoto et al. [21]. The model is conditioned by how precisely the values of the friction coefficients in the contacts ( P b , P r ) are introduced. With this objective two methodologies have been developed: one in laboratory and another in situ in the process. In laboratory, tribometer measurements have been performed with disc-on-disc geometry, due to its similarity with the real process, using pressures at the contact area identical to real processes. To obtain the values in situ, two force sensors in the slides and a Kistler plate under the workblade have been employed. On tables 1 and 2 the results obtained for a specific wheel (rubber based A80 regulating wheel in contact with hardened F-522 steel) are displayed. The friction coefficient depends strongly on the wheel topography (related to the dressing and the wheel wear) and the lubricant used, but not so strongly on the exerted pressure and the grain mesh. The working configuration employed in table 2 belongs to a working regime in which the regulating wheel has no problem to hold back the workpiece, as the relationship between tangential and normal forces at the contact point ( n r t r FF ) is smaller than the limit friction coefficient. Based on these friction coefficient values, it is easy to plot stability maps as a function of feed, h, T , Z r or other parameters, showing spinning free areas (see figure 5). With the aid of these maps, it is possible to know the available margin to increase the feed without risks. 6 8 10 12 14 16 18 20 22 21 19 9 11 13 15 17 21 19 9 11 13 15 17 7 Regulating wheel speed (min - 1 ) -5 -351- Rough dressing Fine dressing Pressure (MPa) ȝ R Pressure (MPa) ȝ R 70 0,29 70 0,22 100 0,31 100 0,19 Table 1: Friction coefficients between regulating wheel and F-522 steel as obtained in tribometer for different dressing conditions and pressures. Rough dressing Fine dressing Feed mm/min Pressure MPa n r t r F F Feed mm/min Pressure MPa n r t r F F 1,6 45 0,25 1,6 52 0,20 3,2 59 0,25 3,2 73 0,20 5,7 72 0,25 5,7 89 0,20 8 86 0,24 - - - Table 2: Relationship between tangential and normal forces at regulating wheel-steel contact as obtained in a sensorised grinder for different dressing conditions and pressures. In a future paper, a complete study of friction coefficient values for different wheels and conditions will be shown. 6 OPTIMAL CYCLE DESIGN The main reference on optimal cycle calculation in grinding processes is the work developed by Malkin [25]. Based on this work, the author has successfully developed the GrindSim software to set-up and optimise cylindrical grinding processes. Two possible criteria can be used to design a centerless grinding cycle: 1) Minimise the process cycle time or 2) Adjust the cycle to a previously established process time, minimising wheel wear. This last option is very common in production lines. The process parameters to be optimised include the feed for each stage of the infeed cycle, the stock removal in each stage and the spark-out time. The restrictions to apply are given by the maximum power to employ in the roughing process free from burning problems [25] and the required tolerance and roughness of the final workpiece. To define the optimal cycle, the feeds to use are fixed first. The first feed will depend on the criteria chosen to optimise the cycle. In the first case, it is deduced from the maximum power that is possible to use. If production time is pre-established the strategy is similar, although it is Figure 5: Work rotation stability map for 5 mm/min feed and a dull regulating wheel (friction coefficient: 0.20). Figure 6. Grinding cycle evolution with 4 feeds and spark- out: (a) Deflection. (b) Real radius of workpiece (blue) and programmed position for the wheel (black). necessary to employ iterative methods. Once the roughing feed is known, the rest are obtained through a series of given relations (depending on the wheel in use). Machining time for each feed can be calculated by establishing a proportionality with the deflection ( G ) accumulated in the previous stage (figure 6a), although it would still be necessary to determine the time for the first stage and the spark-out. The referred proportionality can be adjusted depending on how aggressively we want to design the process. The spark-out time is calculated establishing that the exponential reduction of the radius defect fits the required radial tolerance ( G f < G tol ), ensuring also that, during a time interval before and after the process stops, the workpiece is within tolerances. It should be noted that the programmed final position of the wheel exceeds slightly the desired dimension of the workpiece (figure 6b), in the same quantity as the average accumulated radius defect at the end of the process. CONCLUSIONS The main conclusions of this work can be summarised as follows: 1. Both an enhanced model and computation algorithm for centerless grinding have been developed. 2. All the process instabilities can be predicted together. At the optimal configuration, roundness correction is faster, making the process less sensitive to changes in the quality of entering workpieces. 3. Grinding cycles have been designed to obtain the required workpiece tolerance at minimum production time or, alternatively, minimum wheel wear. Practical application of these results may increase significantly precision and productivity of many industrial processes, reducing set-up time and decreasing production stops motivated by chatter or out-of-roundness issues. 0 2 4 6 8 0 2 4 6 8 (a) (b) 1 3 2 4 0 x 10 -2 12.0 11.9 11.8 11.7 Time (s) Radius (mm) System deflection (mm) G 1 G 2 G 3 G 4 G f Tolerance, G tol Tolerance Spinning Free Zone Transition Zone Spinning Zone Blade angle (º) -5 0 5 10 15 10 15 20 25 30 35 40 45 50 Height (mm) -352- To assist the implementation of grinding simulation in industry, an intelligent software tool has been developed (Estarta SUA), which can be incorporated into the CNC control of centerless grinders. 7 ACKNOWLEDGMENTS This work has been carried out with the financial support of the Basque Country Government (projects UE 2005-4 and IT-2005/043) and the Spanish Government (projects FIT-020200-2003-72 and DPI2003-09676-C02-01). The author wishes to acknowledge his colleagues from Ideko Tecnological Centre (R. Lizarralde, D. Barrenetxea and G. Aguirre), Mondragon University (J. I. Marquínez, J. Madariaga and R. Fernández), Estarta (I. Muguerza) and Manhattan Abrasives (P. Cárdenas) for their contribution to this work. 8 REFERENCES [1] Lizarralde, R., Barrenetxea, D., Gallego, I., Marquinez, J.I., 2005, Practical Application of New Simulation Methods for the Elimination of Geometric Instabilities in Centerless Grinding, Annals of the CIRP, 54/1:273-276. [2] Gallego, I., Lizarralde, R., Barrenetxea, D., Arrazola, P.J.; 2006, Precision, Stability and Productivity Increase in Throughfeed Centerless Grinding, Annals of the CIRP, 55/1: 351-354. [3] Hashimoto, F., Lahoti, G.D., 2004, Optimization of Set-up Conditions for Stability of The Centerless Grinding Process, Annals of the CIRP, 53/1:271- 274. 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[22] Frost, M.; Fursdon, P.M.T., 1985, Towards optimum centerless grinding. ASME M.C. Shaw Grinding Symposium:313-328. [23] Harrison A. J. L., Pearce T. R. A., 2002, Prediction of lobe growth and decay in centreless grinding based on geometric considerations Proc. Instn. Mech. Engrs., Part B: J. Engineering Manufacture, 216:1201-1216. [24] Gallego, I., Barrenetxea, D., Rodríguez, A., Marquínez, J. I., Unanue, A, Zarate, E., 2003, Geometric lobing suppression in centerless grinding by new simulation techniques, The 36th CIRP- International Seminar on Manufacturing Systems:163-170. [25] Malkin, S., 1989, Grinding Technology: theory and applications of machining with abrasives, Society of Manufacturing Engineers, Dearborn, Michigan . doi:10.1016/j.cirp.2007.05.080 Intelligent Centerless Grinding: Global Solution for Process Instabilities and Optimal Cycle Design I. Gallego (3) Manufacturing. values for different wheels and conditions will be shown. 6 OPTIMAL CYCLE DESIGN The main reference on optimal cycle calculation in grinding processes