Precision, Stability and Productivity Increase in Throughfeed Centerless Grinding I. Gallego 1 (3), R. Lizarralde 2 , D. Barrenetxea 2 , P. J. Arrazola 1 1 Manufacturing Department, Faculty of Engineering – Mondragon University, Mondragon, Spain 2 Ideko Technological Center, Elgoibar, Spain Submitted by R. Bueno (1), San Sebastian, Spain Abstract Centerless grinding is a high precision manufacturing process commonly applied to the mass production of many industrial components. However, workpiece roundness is critically affected by geometric lobing and no practical tool has been developed to solve the problem in throughfeed working mode. Based on simulation methods previously applied to plunge grinding, a new software tool has been developed in this work. The software determines the optimal working configuration and can be used to reduce set-up time and improve three important features: 1) Precision, as the roundness error is rapidly corrected at the optimal configuration. 2) Productivity, since the workpiece stock can be significantly reduced. 3) Stability, because the process is less sensitive to the original roundness error of the workpiece. Keywords: Centerless grinding, Productivity, Simulation 1 INTRODUCTION Centerless grinding is a manufacturing process widely used as a high-productivity finishing technology. The main advantage of this process comes from the fact that clamping and centering operations are eliminated, reducing operation times and allowing automation via workpiece chargers and manipulators. However, workpiece roundness is critically influenced by two types of instabilities: chatter and geometric lobing. Chatter appears as a result of the coupling between the cutting process and the main frequencies of the machine. Simulation of dynamic instabilities and chatter avoidance will be treated in an upcoming paper. On the other hand, geometric lobing may appear when the machine configuration allows a number of lobes close to an integer to be present in the workpiece surface (see Figure 1), coinciding a lobe maximum in the regulating wheel with two minima in the grinding wheel and in the blade respectively. Under these conditions, together with its rotation, the workpiece oscillates between the grinding and regulating wheels during the process, leading to lobe regeneration. On the contrary, under stable conditions, the workpiece self-centers and high precision can be rapidly achieved. Figure 1: Geometric lobing in centerless grinding (size of lobes is exaggerated). In a previous work [1], a new practical tool developed to facilitate the set up of plunge centerless grinding was presented. Throughfeed grinding is much more complex to simulate. In this operation mode, the regulating wheel is swivelled to give the workpiece an additional axial movement. As a result, the grinder can be continuously fed with workpieces and productivity is significantly increased. Nevertheless, the process may be more difficult to set up. This paper presents the basis of a new software tool called SUA (Set-up Assistant), which is already commercially available, developed to increase precision and productivity in centerless grinding. 2 THROUGHFEED STABILITY SIMULATION Several models have been successfully developed in the bibliography [2-11], specially concerning plunge grinding. In mid sixties, Rowe et al. [2] derived an equation to obtain the roundness error as a function of the error at the contact points, which was applied to the grinder set up problem [3]. In 1967, Reeka [4] presented geometric stability charts, which showed the number of lobes generated as a function of the blade angle and the angle between the two tangents of the workpiece with the wheels. In the seventies and eighties, the existing models were refined by Miyashita, Hashimoto, Rowe and other authors [5-7], and dynamic effects were introduced in the equations. Both Bueno et al. [8] and Zhou et al. [9] showed that the contact stiffness is an essential parameter to predict geometric lobing correctly. More recently, Harrison and Pierce [10] applied the model of Zhou to plot stability maps in plunge grinding. Finally, in 2004 Hashimoto et al. [11] presented guidelines for determining proper set-up conditions to avoid spinners, chatter vibration and roundness problems. Following Furukawa et al. notation [12], let r ω ( ϕ ) be the radius defect in a point in the surface defined by the angle ϕ . In plunge operation (Figure 2), the equation that describes the process is: ( ) ( ) ( )( ) ( ) ϕϕϕεϕϕεϕ ωωωω Hrrr +−−−− ′ = 21 1 (1) Support blade Grindin g wheel 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 Re g ulatin g wheel Annals of the CIRP Vol. 55/1/2006 Figure 2: Geometric configuration in plunge grinding. Figure 3: Geometric configuration in throughfeed grinding (workpieces are machined in a continuous run). This equation simply describes the displacement of the contact point between the workpiece and the grinding wheel due to the presence of defects on the blade and on the regulating wheel. ε ′ and (1- ε ) are coefficients that relate the movement of the workpiece with respect to the grinding wheel originated by an unitary defect on the blade and on the regulating wheel respectively. H ω ( ϕ ) represents the distortion in the cutting depth due to the static deflection of the grinder, the deformation at the contact point between the workpiece and the regulating wheel and other minor effects. In throughfeed grinding (Figure 3), α angle remains constant, but β varies as the workpiece goes through the path between both wheels and the blade. Consequently, ϕ 2 in equation (1) does not remain constant anymore and, as a result, process simulation becomes more complex. The solution adopted in this work has been to transform the plunge model, which is known to be very reliable for this type of operation [1], into a more generalised simulation scheme for throughfeed grinding. The first step is to calculate the actual value of ϕ 2 as a function of the workpiece position in its path: () β λλ λ πϕ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅⋅+ ⋅− −= 100 0 2 sincos sin arctan L L L zx zh z (2) λ 0 and λ 1 are the feed and penetration angles of the regulating wheel after truing and positioning it (Figure 4). The next stage is to analyse the process stability degree all along the workpiece path in the grinder. For that purpose, the path is divided in a number of segments (typically near one hundred) and equation (1) with the actual value of ϕ 2 in each position is used. Every equation is then analysed as described in previous works [1, 13]. Laplace transformation is applied to it with the aim of studying unstable frequency components of the defect function. As it is well known, the poles of the function in the frequency domain will define whether the process is stable or unstable. The function has a very complex analytical form and all the significant poles should be found by means of a highly efficient algorithm. Levenberg-Marquardt is the most appropriate optimisation method, as it takes advantage from the safe local convergence given by first-order methods and the faster convergence of second-order methods. This algorithm is much faster than Simplex method used by Harrison and Pearce [10], being more suitable for throughfeed grinding due to the number of calculations that must be done. Using the output of these calculations, stability maps can be plotted, as is described in next section. 3 RESULTS 3.1 Stability diagram of a specific configuration In general, ϕ 2 angle changes considerably in throughfeed grinding. Therefore, it is very difficult to prevent the workpiece from passing through one geometrically unstable segment at least. Stability diagrams provide information about the less stable lobe component and its stability degree all along the workpiece path. A positive value of stability degree ( ωξ ) means that roundness error is corrected, while a negative value indicates that lobe regeneration is going to take place [1]. In Figure 5, three stability diagrams are plotted for different workpiece heights: 0.75 mm, 3.5 mm and 9.5 mm. Other configuration parameters are: blade angle 30º, wheels and workpiece diameters 632 mm, 310 mm and 33 mm respectively, λ 0 3º, ω r 49 min -1 , wheel width 500 mm, feed 2500 mm/min. The less stable component values are shown at the bottom of the graphs. It is important to point out that the presence of a high order unstable component in a short section of the stability diagram (Figure 5c) is not critical for the process, because the amplitude of these type of components is rapidly corrected in the rest of the path. The important configurations to avoid are those that have: 1) Low order components at the entry (typically 3 or 5), even in short segments, since lobe correction is much slower in this case. 2) High order components in a relatively long segment (more than 20% of the path). Figure 4: Some important parameters in throughfeed grinding. Grinding wheel Regulating wheel Workpiece exit Workpiece entry Feed λ 0 α α β ma x β min λ 1 λ 0 λ z h h d z L x 0 Diamond path Workpiece axis Regulating wheel axis (x 0, y 0 ,z 0 ) Y X θ α β h ϕ 1 ϕ 2 γ =α+ β Grinding wheel Regulating wheelBlade Workpiece (a) (b) (c) Figure 5: Stability diagrams, blade angle: 30º, workpiece height: (a) h=0.75 mm, (b) h=3.5 mm and (c) h=9.5 mm. Due to the computational efficiency of Levenberg- Marquardt algorithm, stability diagrams are obtained in a few seconds with an average computer. In the absence of vibrations, these plots may facilitate to a large extent process setting up. As wheel diameter changes after repeated dressing, stability diagrams may also be used to correct workpiece height and remain at stable configurations from the beginning to the end of wheels life. Furthermore, roundness error decreases very fast at optimal configuration, so productivity of manufacturing lines may be enhanced by reducing the stock to grind. In addition, if high process capability (cpk) is required, the use of stability diagrams may be very helpful, because the final result is less sensitive to the initial profile of workpiece, which is usually a matter of concern for grinding companies. 3.2 Stability maps By repeating the previous calculation for different working configurations, stability maps can be plotted (Figure 6). Figure 6a shows the less stable lobe components and Figure 6b represents the global stability of the process. The latter may be used at the first stages of the setting up, as it clearly displays the most stable areas (in blue) and the working configurations to avoid (dark red). Mathematically, global stability of each lobe component should be defined as the mean value of stability along the path. However, to further ensure that unstable segments are avoided, a penalty weight is applied to negative stability values. The weight value selected in our simulation is 2.5 for wheels of 500 mm or wider (as in Figure 6b), which decreases linearly to 1 for wheels of 100 mm or less. The reason for this change is that, from a stability point of view, throughfeed grinding with narrow wheels is not very dissimilar to plunge grinding, as ϕ 2 angle does not change very much. (a) (b) Figure 6: Stability maps, wheel with: 500 mm. (a) Number of lobes (b) Global stability degree (relative units). In blue: stable areas. In dark red: unstable areas. 0 5 10 15 20 30 35 40 45 Workpiece height (mm) Blade angle (º) -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 3 5 12 14 16 18 18 20 22 23 24 24 25 25 26 26 27 27 27 28 29 29 30 31 32 Workpiece height (mm) Blade angle (º) 0 5 10 15 20 30 35 40 45 0 50 100 150 200 250 300 350 400 450 500 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 STABLE INSTABLE Workpiece position (mm) Stability degree 3 13 7 5 30 0 50 100 150 200 250 300 350 400 450 500 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0 . 08 STABLE INSTABLE Workpiece position (mm) Stability degree 7 5 32 30 24 18 0 50 100 150 200 250 300 350 400 450 500 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 STABLE INSTABLE Workpiece position (mm) Stability degree 3 7 5 (a) (b) 0%0%0% -75% -30% -32% -90% -80% -70% -60% -50% -40% -30% -20% -10% 0% A mp li tu d e (%) n=3 n=5 n=27 Before After grinding 0%0%0% -74% -71% -77% n=3 n=5 n=27 Before After grinding Figure 7: Experimental evolution of 3, 5 and 27 lobe components, workpiece height: (a) 3.5 mm (b) 9.5 mm. Figure 8: Global stability degree, wheel width: 200 mm. Experimental tests were performed at the same working conditions of Figures 5 and 6 (workpiece stock: 0.2 mm). An amount of 36 workpieces with 3, 5 and 27 lobes were prepared. At low workpiece height, it is very well known that 3 lobes are formed. At h=3.5 mm (Figure 7a), the process is stable, but the amplitude of 3 and 5 components decreases very slowly. On the contrary, closer to the optimal working configuration, all the lobes decrease very rapidly (Figure 7b). 3.3 Influence of wheel width on stability Preceding maps have been obtained for a wheel width of 500 mm. For shorter wheels, there is a bigger risk of finding a high order unstable component in a relatively long segment. For the purposes of comparison, Figure 8 displays the global stability degree when the wheel width is 200 mm. As the process is now more unstable, the information that this graph provides is even more useful than for wider wheels. 4 CONCLUSIONS A new software tool has been developed to predict geometric lobing in throughfeed grinding. Stability determination of a specific working configuration requires only a few seconds in an average computer. Stability maps can be plotted by repeating the same calculation for different workpiece heights and blade angles. In the absence of vibrations and work rotation instabilities, these maps are very helpful to select the ideal working configuration and improve precision and productivity. Stability models provide the basis for a new simulation tool which has been practically implemented in industry by a software program (SUA, Set-up Assistant), which can be incorporated into the CNC control of centerless grinders. 5 ACKNOWLEDGMENTS This work has been carried out with the financial support of the Basque Country Government (projects UE 2005-4 and PI 2004-9) and the Spanish Government (projects DPI2003-09676-C02-01 and FIT-020200-2003-72). The authors wish to thank J. I. Marquinez for his contribution to this work. 6 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] Rowe W.B., Koeningsberger F., 1965, The Work Regenerative Effect in Centerless Grinding, Int. J. Mach. Tool Des. Res., 4:175-187. [3] Rowe W.B., Richards R.L., 1972, Geometric Stability Charts for the Centreless Grinding Process, J. Mech. Eng. Science, 14/2:155-158. [4] Reeka D., 1967, On the Relationship Between the Geometry of the Grinding Gap and the Roundness Error in Centerless Grinding , PhD. Diss., Tech. Hochschule, Aachen. [5] Miyashita, M., 1972, Unstable Vibration Analysis of Centerless Grinding System and Remedies for its Stabilisation, Annals of the CIRP, 21/1:103-104. [6] Miyashita M., Hashimoto F., and Kanai A., 1982, Diagram for Selecting Chatter Free Conditions of Centerless Grinding, Annals of the CIRP, 33/1:221- 223. [7] Rowe, W. B., Miyashita, M., Koenig, W., 1989, Centerless Grinding Research and Its Application, Annals of the CIRP, 38/2:617-624 [8] Bueno R., Zaratain M., Aguinagalde J. M., 1990, Geometric and Dynamic Stability in Centerless Grinding, Annals of the CIRP, 39:395-398. [9] Zhou S.S., Gartner J.R., Howes T.D., 1996, On the Relatioship between Setup Parameters & Lobing behavior in Centerless Grinding, Annals CIRP, 45: 341-346. [10] 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. [11] 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. [12] Furukawa Y., Miyashita M., Shiozaki S., 1972, Chatter Vibration in Centerless Grinding, Bull. of JSME, 15, 82:544-553. [13] 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. 0 5 10 15 20 30 35 40 45 Workpiece height (mm) Blade angle (º) -0.1 0 0.1 0.2 0.3 0.4 0.5 . Precision, Stability and Productivity Increase in Throughfeed Centerless Grinding I. Gallego 1 (3), R. Lizarralde 2 ,. to increase precision and productivity in centerless grinding. 2 THROUGHFEED STABILITY SIMULATION Several models have been successfully developed in