Wheel Regenerative Chatter of Surface Grinding

Transcription

1 Hongqi Li Graduate Research Assistant Yung C. Shin Professor School of Mechanical Engineering, Purdue University, West Lafayette, IN Wheel Regenerative Chatter of Surface Grinding In this paper we present a comprehensive dynamic model that simulates surface grinding processes and predict their regenerative chatter characteristics. The model considers special aspects in surface grinding processes, such as interrupted grinding on a series of surfaces and step-like wheel wear along the axial direction due to crossfeed. A new theory for the wheel regenerative chatter mechanism, which describes the regenerative force as a function of not only the instantaneous chip thickness but also the distributed uneven grit wear/dullness, is introduced and applied in the model. Using the model, explanations are provided for those unrevealed wheel regenerative chatter phenomena observed from the experimental results in literature. The model is validated by comparing the simulated chatter frequencies and thresholds with the experimental results. DOI: / Introduction Chatter is a critical problem for grinding processes because grinding processes are inherently unstable while accuracy and surface finish are the two major purposes of grinding processes. Many studies have been conducted on grinding chatter problems, as reviewed by Inasaki et al. 1. Especially there was a blossom of studies in this area during the period of Those studies were mostly for cylindrical grinding processes and rarely for surface grinding processes. Among those studies on surface grinding, Inasaki and Yonetsu 3 experimentally investigated the characteristics of the undulations generated on the wheel due to chatter and the influence of the grinding conditions on the chatter generation process. Later they carried out more experiments to explore the effect of attritious wear on the chatter generation process 4. Thompson proposed a model to consider the reverse motion 5 and performed some surface grinding tests under chatter conditions with different fixture setups for the workpiece 6. There are two types of chatter for cylindrical grinding: workpiece regenerative chatter and wheel regenerative chatter Workpiece regenerative chatter occurs at a relative high workpiece speed and grows very fast. This type of chatter can be ascribed to the force regeneration from the undulations of successively ground surfaces of the workpiece. For surface grinding, however, it is accepted by most researchers that workpiece regenerative chatter is unlikely to occur because the phase between the successively ground surfaces, which is critical for chatter generation, cannot be maintained due to the interruption caused by the overtravel of the wheel and crossfeed. Thus, the chatter of surface grinding is mainly the other type, wheel regenerative chatter. According to the experiments of Inasaki and Yonetsu 3,4 and Thompson 6, the chatter of surface grinding behaves as the wheel regenerative chatter in the following aspects: 1 Chatter occurs even at a very low workpiece speed, and the amplitude of the vibration increases very slowly during grinding after dressing. 2 At the early stage of chatter, it is difficult to distinguish chatter from forced vibration, or see chatter marks on the workpiece surface. 3 As chatter develops, the waves generated on the wheel surface can be observed or measured, and chatter marks can be seen in some cases. Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received June 1, 2005; final manuscript received September 8, Review conducted by A. J. Shih. The current theory of wheel regenerative chatter 1 explains that at a low workpiece speed, on one hand, the amplitude of the undulations generated on the workpiece is negligible because of the wave filtering effect; on the other hand, the undulations generated on the wheel surface vary the depth of cut as well as the grinding force. Hence, the corresponding delayed response may magnify the undulations on the wheel and consequently cause instability. Under the wheel regenerative chatter condition, however, undulations are generated on the surfaces of both the workpiece and grinding wheel, thus the two-degree-of-freedom geometrical interactions caused by both the wheel and workpiece surface undulations along the contact length must be modeled accurately for chatter prediction. Since the current theory of wheel regenerative chatter neglects the effects of workpiece undulations and uses only the variation of depth of cut to represent the geometrical interactions in surface grinding, it is too simplistic to describe the real wheel regenerative chatter generation process, and there is no strong proof that it is valid or dominant. In addition, the existing models based on the current wheel regenerative chatter theory cannot explain many experimental observations reported in literature: 1 There is no difference in chatter occurrence whether a single workpiece is continuously or interrupted ground, or a series of workpieces are ground 1. 2 All dominant frequencies under wheel regenerative chatter conditions are approximately the multiplicities of the wheel rotational speed. Figure 1 shows chatter occurrence with three conditions obtained by Thompson 6. 3 Phase delays of chatter vibration and wheel undulations are very small. This observation is actually a support to the previous one. For the test case shown in Fig. 1a with four undulations, a one-revolution delay was reported after 4.8 min at 3000 rpm by Thompson 6. After conversion, the actual phase delay is 0.025, and the chatter frequency is % smaller than four times the wheel speed of 200 Hz. Another experimental result for cylindrical grinding by Snoeys and Brown 7 also shows a small phase delay around As shown in Fig. 2 from Weck and Alldieck 13, more than one dominant frequency can appear under chatter conditions, each of which is approximately a multiplicity of the wheel rotational speed. 5 The high-order harmonics of the chatter frequencies can be observed as chatter grows, as shown in Fig. 3 by Hashimoto et al. 11. Journal of Manufacturing Science and Engineering MAY 2006, Vol. 128 / 393 Copyright 2006 by ASME

2 Fig. 1 Three tests by Thompson 6 with different fixture setups under the chatter condition, which show that dominant chatter frequencies are very close to multiplications of an integer n and wheel rotational speeds: The spikes on the top show the marks per wheel revolution, and the waves on the bottom are the vibration signal The objective of this work is to develop a dynamic model of surface grinding processes to accurately predict wheel regenerative chatter. A time domain simulation model is extended from the model for the cylindrical plunge grinding processes 14 by applying a new theory for the wheel regenerative chatter mechanism, which describes the regenerative force as a function of not only the instantaneous chip thickness, but also the distributed uneven grit wear/dullness. With the new theory, all the observed wheel regenerative chatter characteristics discussed above can be explained or predicted by the model. Since the new model is developed for surface grinding processes, special characteristics different from those of other grinding processes, such as interrupted grinding of a series of surfaces and step-like wheel wear along the axial direction of the wheel due to cross-feed, are also taken into consideration in the model. 2 Dynamic Simulation Model The dynamic simulation model of surface grinding processes is developed by modifying the model for cylindrical plunge grinding processes developed by the authors Grinding Kinematics. For the time domain simulation, the step size of discretized time is determined by the frequencies of interest. At each time step i, the angular position of the grinding wheel, g, and the linear position of the workpiece, l w, are represented by: g = i g Fig. 2 Multiple chatter frequencies that are approximately multiplicities of the wheel rotational speed 13 1 Fig. 4 Kinematics of surface grinding l w = g w 2 g where g and w are the rotational speed of the grinding wheel and the linear speed of the workpiece, respectively, and g is the angular increment of the wheel for each time step. For convenience, g is chosen such that the number of steps per wheel revolution, n g, is an integer. n g = 2 3 g In general, surface grinding chatter is assumed to be a problem of two and a half dimensions, i.e., a two-dimensional problem in normal and traverse directions with consideration of cross-feed in the axial direction. Figure 4 shows the two-dimensional kinematics of the surface grinding, in which scales are exaggerated to show the relative motion. Dynamic responses of displacement and velocity in both directions are represented as x g, y g, ẋ g, ẏ g and x w, y w, ẋ w, ẏ w for the grinding wheel and workpiece, respectively. R g g is the instantaneous radius of the grinding wheel, which also represents its surface profiles, and a n is the static or nominal depth of cut. 2.2 Uncut Chip Thickness and Dynamic Force. Since the grinding chatter is a macro-level phenomenon, it is straightforward to model the dynamic grinding force and grinding wheel surface based on an averaging concept: average number of grits per unit area, N a, and average force per grit, f, both of which are functions of uncut chip thickness, C t. The contact length along the wheel is discretized into a few segments and each segment corresponds to a wheel angular increment, usually g, shown in Fig. 5. It is also computationally efficient to set g =n seg, where n seg is an integer. In each segment, the uncut chip thickness is assumed to be uniform and equal to the average value. To discuss the uncut chip thickness, the grinding wheel and its grits are considered as a peripheral milling tool and its milling teeth, respectively. Based on the milling theory 15, the static uncut chip thickness, C stat, can be approximated as, Fig The chatter frequencies and their second harmonics C stat =2 w sin 4 g N r where N r is the number of active grits per revolution, which is analogous to the number of teeth of a milling cutter. N r is the term 394 / Vol. 128, MAY 2006 Transactions of the ASME

3 Fig. 5 Geometrical interaction between wheel and workpiece related to grinding wheel surface topography and is difficult to accurately measure during a dynamic process. To avoid measurement of grinding wheel surface topography, one can replace N r in 4 with a known number to obtain a scaled form of static uncut chip thickness, C s, C s =2 w sin g N 5 where N=n g n seg, which is the total number of discretized segments per wheel revolution. It can be seen later that using the scaled form of uncut chip thickness one can generate a mathematically equivalent form of grinding force. In Eq. 5, g is a constant for a given wheel speed without considering the rotational vibration. The corresponding dynamic chip thickness, C d, can be obtained by replacing the static velocity term in radial direction, w sin, with its dynamic velocity term, ẏ w ẏ g and ẋ w ẋ g, projected in radial direction. C d =2 ẏ w ẏ g g N sin +2ẋ w ẋ g g N cos 6 The geometric variation of the workpiece and wheel represented by the term C w also contributes to the instantaneous uncut chip thickness. C w is similar to the runout term in milling processes, but it is a dynamic term due to the continuous wheel wear rather than the static runout in milling. C w is obtained from the accumulated wheel wear in the radial direction, R g,atthe current and previous time steps. As a result, the instantaneous uncut chip thickness, C t, can be represented by the summation of the contributions from the three terms, C t = C s + C d + C w 7 Whether or not the wheel and workpiece are in contact at the segment location is determined by their surface profiles and relative positions between them. At each time step, the surface profiles of the wheel and workpiece are updated so as to describe the regenerative process as follows: the overlapped portion between the wheel and workpiece, which represents the removed material during the time step, is simply subtracted from the workpiece profile, while the volume to be subtracted from the wheel profile is obtained by the volume of removed workpiece material divided by the grinding ratio, and C w is updated for each segment. The force of a segment without wheel workpiece contact or with negative chip thickness is set to zero. The specific forces per unit wheel width of each contacting segment in the tangential and radial directions are represented as a function of the instantaneous chip thickness, C t, using the following forms similar to those of the cutting processes 16: f t f r = K t K rc t where K t and K r are the specific grinding force coefficients for the tangential and radial direction respectively. In Eq. 8, C t is known during the simulation, while K t and K r, dependent on workpiece material properties and wheel wear conditions, are to be determined as a function of C t. The segment force becomes zero when the uncut chip thickness of C t is below zero due to losing local contact of the wheel and workpiece. On the other hand, when C t is positive but very small, the specific grinding energy is relatively large due to plowing and sliding in the grinding process. This inverse relationship between the specific energy and the uncut chip thickness, often called the size effect, also exists in other machining processes besides grinding. In Eq. 8, the specific force coefficients, K t and K r, are analogous to the specific energy, and the size effect is considered by the inverse relationship between K t,k r and the uncut chip thickness, C t. The semiempirical relationship between the static grinding force and the equivalent chip thickness has been well studied and verified 17,20, and the corresponding static force model can capture the dominant force characteristics with proper calibration, which are inherited by the derived dynamic model in this work. In the next section, derivations of K t and K r from the existing empirical static force models are presented. To calculate the total specific grinding forces, F x and F y, the segment forces, f t and f r, are first transformed into the force components f x and f y in the Cartesian coordinate system, and then added up for all the segments where f x f y F x F y = T fr f t = is the transformation matrix. segments fx f y sin T =cos sin cos Specific Force Coefficients. Specific grinding force coefficients K t and K r can be derived from an existing static force model, such as the semiempirical force model introduced by Choi and Shin 17. The static force model right after dressing includes the effects of the grinding condition, dressing condition, and wheel size: F x F y = Cx f 1 f d s 2 f d D 3 f g h 4 eq C ya 11 where C x, C y, f 1, f 2, f 3, and f 4 are constant coefficients, which are obtained from typically simple static grinding tests, a d and s d are the dressing depth of cut and dressing lead, respectively, D g is the wheel diameter, and h eq is the equivalent chip thickness, represented by h eq = w a n 12 g where w and g are the surface speeds of the workpiece and grinding wheel, respectively, and a n is the depth of cut. To obtain specific force coefficients one needs to represent forces as a function of uncut chip thickness, C t. However, it is more convenient to represent forces as a function of immersion Journal of Manufacturing Science and Engineering MAY 2006, Vol. 128 / 395

4 angle, i, because a simple relationship between the depth of cut and immersion angle can be represented as follows: a n = R g 1 cos i 13 By substituting Eq. 12 and 13 into Eq. 11, the specific grinding forces are represented as a function of immersion angle, i F x F y = Cx C yc1 cos i f 4 14 where C=a d f 1 s d f 2 D g f 3 w R g / g f 4 is a constant for a given condition. Then the specific segment forces at any angular position can be established from the derivatives of the total specific forces in Eq. 14 f x f y i = d d i F x F y = Cx C ycf 4 1 cos i f 4 1 sin i 15 The specific segment forces can also be obtained by substituting f t and f r in Eq. 8 into Eq. 9. Note that the static uncut chip thickness, C s, is used for the calculation of the static forces. f x f y = T Kr s 16 K tc After substituting the static uncut chip thickness, C s, in Eq. 5 into Eq. 16 and setting = i, the segment forces at angular position i become f x f y i = T i Kr K t 2 w g N sin i 17 By comparing the right-hand side of Eqs. 15 and 17, the specific grinding force coefficients K t and K r are obtained as a function of angular position i 18 K r K t i = T i Cx C y N gcf 4 1 cos i f4 1 2 w At the last step of the derivation, the angular position, i, is mapped to the corresponding uncut chip thickness, C t, using static chip thickness in Eq. 5 K r K tc t = Kr K t i =sin 1 C t g N/2 w 19 Because K t and K r are derived from a static force model, the static chip thickness relationship in Eq. 5 is used to map i to an uncut chip thickness of C t. However, C t in Eq. 19 is actually the instantaneous uncut chip thickness, and K t and K r, dependent on C t, are the instantaneous specific grinding force coefficients to be used for dynamic analysis. 2.4 Wear of Grinding Wheels. In the previous section, the effect of wheel wear on the instantaneous chip thickness is considered, and the wheel wear is modeled as a function of removed material based on the measured G ratio. Another effect of wheel wear is the force variation due to the grit dullness change as time goes. Malkin and Cook 18 associated the dullness with wear flat that can be measured using optical sensors. Their experiments showed that the force variation is significant and has a reasonably good correlation with wear flat. Since wear is time dependent, the static force variation can be represented as a function of grinding time, t, or equivalently as a function of accumulated material removal per unit width, V m, Fig. 6 F x = F x0 W x t = F x0 W x V m 20 F y = F y0 W y t = F y0 W y V m 21 where F x0 and F y0 are the initial forces per unit width right after dressing, and W x and W y are the wear-force functions, which depend on the workpiece material, wheel type, and dressing condition. The accumulated material removal per unit width, V m, can be represented using the following form: V m = w a n t 22 where w is the surface speed of the workpiece and a n is the depth of cut. The wear-force functions are determined experimentally in empirical forms for a given workpiece/wheel pair and a certain dressing condition. A simple form has been used for the grinding wear-force functions in 17 as follows: W x =1+ x V x m 23 W y =1+ y V y m 24 where x, x, y, and y, are constants determined experimentally. During the chatter development, the accumulated wheel wear in the radial direction, R g, distributes unevenly along the wheel circumference. It will be shown in later sections that the uneven wear, which represents unevenly distributed dullness to some extent, can be a source of regenerative chatter. To model the uneven wear, the wear-force functions in Eqs. 23 and 24 should be represented as functions of R instead of V m. Using the definition of the G ratio, G, the following relationship can be obtained: V m G = 25 2R g R In Eq. 25 the accumulated material removal per unit width, V m, can be represented in the form: V m =2R g RG 26 By considering force variations due to wheel wear using the wearforce functions, specific grinding force coefficients K t and K r in Eq. 18 become C xw x R, K r = T i K t,r Wheel wear in the axial direction C y W y R N gcf 4 1 cos i f4 1 2 w 27 where all the parameters on the right-hand side are either the known process parameters or the experimentally obtained coefficients of the empirical static force model. Another issue associated with the wheel wear of surface grinding is the step-like wheel wear along the axial direction of the grinding wheel due to crossfeed. This type of wheel wear is considered in the model separately. Figure 6 shows an example of the step-like wheel surface after a certain number of passes. For the purpose of solution efficiency, the average step-shape wheel and workpiece surface for each stroke are generated statically using the static deflection. At each pass, the average force and equilib- 396 / Vol. 128, MAY 2006 Transactions of the ASME

5 rium position can be solved by iteratively updating the real depth of cut due to deflection and the force as a function of the depth of cut. 2.5 Dynamic Responses. A comprehensive model for dynamic responses is adopted from the model of the cylindrical plunge grinding process, which was originally used for the cutting process simulation in 16. The model is discussed briefly here for completeness. The frequency response functions at the center of grinding wheel or workpiece surface can be represented as: H = H xx H yx H xy H yy 28 Each frequency response function, H pq, in which subscript p and q can be substituted by x or y, is represented in the following form from an experimental modal analysis, m d e H pq = r=m i r 2 + Z pq 29 M 1 r + jc r + K r where r, m 1, and m d are the mode number and the lowest and highest modes of interest, respectively, and M r, C r, and K r are the normalized effective modal mass, damping, and stiffness, respectively. r, the so-called modal angle, represents the phase shift between modes due to nonlinearity and other effects. The residual term, Z pq, often called residue flexibility, is used to compensate for the error introduced by the modes of truncation, in practice, for the static deflection error. In grinding, the compensation can be for the contact deformation of the grinding wheel and workpiece. For the system described in Eq. 29, the response of displacement contributed by the residue flexibility to the force F q t can be easily calculated from their multiplication in time F q t Z pq, while the response contributed by the mode r is obtained using step by step integration methods, such as Duhamel s integral. 2.6 A New Wheel Regenerative Chatter Theory. The wheel regenerative chatter is characterized by the wear undulations on the wheel surface, where grits wear unevenly and generate distributed dullness. Based on the grit dulling process and its effects on grinding forces, the new chatter theory describes the mechanism of wheel regenerative chatter as follows. Since a wheel segment or grit is duller with deeper attritious radial wear, the undulations generated on the wheel surface, which are the distributed depths of radial wear for an initially round wheel, roughly represent the degree of the grit dullness. In general, a duller grit has a larger specific cutting energy, which is described by force coefficients, K r and K t. Thus, grinding with an undulated/dulled wheel, the forces in each segment can change due to the resultant variations of not only the chip thickness, C t, but also the K r and K t due to grit dullness according to the mechanistic force model in Eqs. 8 and 27. It must be noted that the effects of chip thickness and grit dullness on grinding forces are comparably significant 18,20. As a result, chatter occurs when the delayed responses of motion corresponding to the fluctuating forces magnify the undulations on the wheel surface as well as the undulated dullness of the wheel grits. By considering the interaction between the wheel dulling and chatter developing process, the wheel regenerative chatter is more accurately modeled in the new theory. The basic concept of the wheel regenerative chatter theory is further illustrated by a simplified block diagram in Fig. 7, which shows the flow and feedback of the entire system. The actual implementation of simulation of the surface grinding system, however, is more complicated than the diagram. The bottom portion of the diagram shows the basic feedback loop. The instantaneous uncut chip thickness, C t, has three contributions: the static chip thickness, C s, the dynamic chip thickness due to vibration, C d, and the wear of the wheel, C w. The instantaneous grinding force coefficients, K t and K r, and the segment forces, f r and f t, are calculated from the static force model and the Fig. 7 Block diagram for the new theory of wheel regenerative chatter instantaneous chip thickness. Then f r and f t are transformed into forces in Cartesian coordinates, f x and f y. The total grinding forces, F x and F y, are then obtained by adding up the forces of all the segments along the contact portion of the wheel. The contact length/angle is determined by the relative position between the wheel and workpiece and their surface profiles. The dynamic motions x and y between the grinding wheel and workpiece responding to the grinding forces are obtained by integration through time. The top portion of the diagram shows the wheel wear effects. The total wheel wear/radial variation of the wheel, R, is updated from the removed material governed by the G ratio. The total wear of the wheel, R, is used to calculate the wear-force function, WR, for the additional force variation due to wear. The wear, R, is also used to calculate the chip thickness variation contributed by the wheel surface undulations. 3 Experimental Validation 3.1 Test Setup. Grinding tests were carried out on a MAZAK CNC machining center shown in Fig. 8. An aluminum oxide grinding wheel 38A60KVBE and hardened 4140 steel workpieces were employed in the grinding tests. The wheel and workpiece had dimensions of 150 mm diameter20 mm width and 100 mm length50 mm width50 mm height, respectively. Since the workpiece width is larger than the wheel width, a crossfeed of 2.5 mm was used to grind the entire surface. A waterbased cutting fluid was used for both the grinding and dressing processes. A dressing lead of 0.12 mm per revolution and a dressing depth of 0.02 mm were used for wheel dressing at the beginning of each test, and the tests were kept running without dressing until the occurrence of significant chatter. The other grinding conditions of the test cases are listed in Table 1. The instrumentation for the experiments is shown in Fig. 9. As shown, a Kistler 9225B dynamometer was mounted between the workpiece and fixture, and a PCB 308B accelerometer was Fig. 8 Test setup on a MAZAK CNC machine tool Journal of Manufacturing Science and Engineering MAY 2006, Vol. 128 / 397

6 Table 1 Grinding test conditions Test # Spindle speed rpm Workpiece speed ipm/mm/s Depth of cut inch/mm / / / / / / / / / / / / / / / / / /0.03 mounted on the workpiece in the traverse direction. Using the setup, force and vibration could be measured on-line during the grinding tests. Off-line measurement of the wheel surface was also performed after a certain number of passes using a laser triangulation sensor or a capacitance probe combined with an optical tachometer for synchronization. Structure dynamics of both the grinding wheel and workpiece were obtained through the experimental modal analysis, and the extracted modal parameters are listed in Table Model Coefficients. According to the static force model with wheel wear consideration, the forces per unit width in normal and its orthogonal direction can be represented by F x = F x0 1+ x V x m 30 Fig. 10 Average grinding forces for three test cases. a Force in normal direction. b Force in traverse direction. Fig. 9 Table 2 Instrumentation of grinding experiments Axis Modal parameters of grinding system Modal mass kg Damping ratio Stiffness kn/m Modal angle deg Natural freq. Hz Wheel/spindle X Y Workpiece X Y F y = F y0 1+ y V y m 31 where F x0 and F y0 are initial forces per unit width right after dressing in normal x axis and its orthogonal y axis direction, respectively, and x, y and x, y are the coefficients for the wear-force functions, which represent force variations as a function of accumulated material removal per unit width, V m. Figure 10 plots the measured forces versus accumulated material removal under three grinding conditions before chatter becomes significant. As shown, the measured forces in both directions, F x and F y, have roughly linear relationships with material removal. Accordingly x and y in Eqs. 30 and 31 can be set to one, and the forces can be curve-fitted using linear functions, in which x and y actually represent the constant force growth rates associated with the accumulated material removal per unit width under a given grinding condition. Figure 11 shows the initial forces per unit width, F x0 and F y0, and the force growth rates, x and y, versus equivalent chip thickness, h eq, which were obtained by curve fitting the measured forces of each individual test case. As shown, both the initial forces per unit width and the force growth rates have reasonably good correlations with equivalent chip thickness, h eq. In addition, the force growth rates in the two directions agree well with each other x y and can be fitted using the same curve/equation. Using curve fitting data in Fig. 11a and the coefficients provided in 17, all the coefficients of the static force model in Eq. 11 have been obtained. And after combining the force variation model as a function of accumulated material removal per unit width, V m, the total force per unit width can be represented as: F x F y = 3.27 a d s d D g h eq h eq V m 32 where the units are N/mm for F x and F, y mm/rev for a d, mm 3 /mm for V m and mm for all other parameters. However, 398 / Vol. 128, MAY 2006 Transactions of the ASME

7 Fig. 11 The coefficients of the force in Eqs. 30 and 31. a Initial forces per unit width, F x0 and F y0. b Force growth rates, x and y. the dynamic force variation needs to be described as a function of wheel radial wear, R, instead of V m, due to the uneven wear along the wheel surface. Thus the model also requires the relationship between the workpiece material removal and wheel radial wear, which is governed by the G ratio. The G ratio used in the simulation was obtained in an empirical form by curve fitting the data of single grit wear tests by Lal and Shaw 19 shown in Fig. 12, G = 9674h eq g 33 where the units of the parameters are m for equivalent chip thickness, h eq, and m/s for wheel surface speed, g. 3.3 Chatter Frequencies. The frequency analysis of the measured forces and vibration has been performed and the results, which agree with the observations of wheel regenerative chatter discussed in Sec. 1, show the appearance of a bunch of chatter frequencies, which are approximately the multiplicities of the wheel spindle speed. As an example, the measured forces in different passes of test case #5 are shown in Fig. 13 in frequency domain during chatter development. The force of the 25th pass is Fig. 12 G ratio obtained from data of overcut fly milling tests for single grits in 19 Fig. 13 Measured forces during the chatter development of test case #5. Spindle speed=2000 rpm, workpiece speed =84.7 mm/s, depth of cut=30 m. magnified and displayed at the top of Fig. 13 with a second X axis in units of the multiplicities of the wheel spindle speed. As shown, about eight significant frequencies are observed during the chatter development, and they correspond to 19 to 26 times the spindle speed of 2000 rpm 33.3 Hz, respectively. Simulations have been carried out under various grinding conditions using the obtained input parameters of structure dynamics, static force, and G ratio. The simulation results also show that multiple frequencies appear in one chatter occurrence, which are approximately the multiplicities of the spindle speeds. As an example, Fig. 14 shows the FFT plots of the simulated forces and vibration under the same grinding condition as that of Fig. 13, in which chatter frequencies appear within a frequency range of 19 to 30 times the spindle speed. It can be seen that the lower upper bounds of the chatter frequencies shown in Fig. 13 agree with those in Fig. 14. In addition, those chatter frequencies for both Figs. 13 and 14 within the bounds are the multiplicities of 33.3 Hz 2000 rpm and the frequencies in the middle between the bounds have larger amplitudes than those close to the bounds. The corresponding wheel and workpiece surface profiles of case #5 have also been obtained during the simulation and are shown in Figs. 15 and 16, respectively. It can be seen that the final wheel surface profile in Fig. 15 has roughly 18 waves/undulations with nonuniform amplitudes and wavelengths, which can result in multiple chatter frequencies. The workpiece surface profile in Fig. 16 also exhibits undulations whose amplitudes are comparable to those of the corresponding wheel undulations, and hence the workpiece surface profiles are not negligible for the prediction of wheel regenerative chatter. The wheel surface profiles in Fig. 15 also show a small phase delay, a one-revolution delay after about 3 m wheel radial wear. Because of the small phase delay, the force and vibration signals are approximately periodic with frequencies corresponding to the base frequency, i.e., the wheel spindle speed. Unlike fast growing chatter, such as milling and tuning chatter, and workpiece regenerative chatter of grinding, in which phase delay is relatively large and chatter frequency can be any value for a given spindle speed, a slowly developing chatter like the wheel regenerative chatter has the unique characteristics of small phase delay and approximate periodicity with the wheel Journal of Manufacturing Science and Engineering MAY 2006, Vol. 128 / 399

8 Fig. 14 Simulated forces and vibration of case #5 in frequency domain under a chatter condition. a Force. b Spindle vibration. rotational speed. In order to compare the predicted chatter frequencies with the measured ones under various grinding conditions, the cases for comparison are grouped by varying only one of the three major grinding parameters: spindle speed, workpiece speed, and depth of cut. The comparisons of the three groups are shown in Figs. 17, 18, and 19, respectively. Both the predicted and measured chatter frequencies are obtained from the force signals of the traverse direction in the frequency domain after chatter develops to a certain level. Similar to the frequencies in the force signals shown in Figs. 13 and 14a, there appear a bunch of chatter frequencies for each case/condition, which are represented by a series of circles/ Fig. 15 Simulated wheel surface generation of case #5 under a chatter condition Fig. 16 Simulated workpiece surface profile of case #5 under a chatter condition Fig. 17 Chatter frequencies for different spindle speeds with workpiece speed=84.7 mm/s and depth of cut=30 m. a Predicted. b Measured. bubbles in Figs. 17, 18, and 19. The size of each circle represents the normalized force amplitude peak value of the corresponding frequency at that condition as illustrated in the legend, where a size of 1.0 corresponds to the largest peak value in the FFT plot of the force signals. For easy reading of the chatter frequencies and recognition of the corresponding multiplicities, grid lines corresponding to multiplicities of the spindle speeds are also plotted in the figures. Figure 17 shows both the predicted and measured chatter frequencies under various spindle speeds with other grinding parameters unchanged. As shown, the measured chatter frequencies fall in the area bounded by the predicted ones. In other words, the measured and predicted chatter frequencies have reasonably good agreement. Similar agreements between the measured and the predicted chatter frequencies under different conditions are also shown in Figs. 18 and 19 for the other two cases with respect to the workpiece speed and depth of cut. The comparison results between the predicted and measured chatter frequencies shown in Figs. 17, 18, and 19 are similar to those in Figs. 13 and 14. For example, Fig. 17 shows that predicted and measured chatter frequencies of case #5 at 2000 rpm are 19 to 30 and 19 to 26 times the speed of 2000 rpm 33.3 Hz, respectively. The same results are also seen from Figs. 13 and Chatter Growth. Besides chatter frequency, the chatter growth rate is another important characteristic of chatter. An accurately predicted chatter growth rate can help optimize the dressing intervals of the grinding process. Since this model simulates surface grinding processes in the time domain, the chatter growth rate is usually obtained from the predicted force or vibration history. Figure 20 shows a typical simulated force history when wheel regenerative chatter of surface grinding occurs. As shown, 400 / Vol. 128, MAY 2006 Transactions of the ASME

9 Fig. 20 Simulated time history of normal force under a chatter condition Fig. 18 Chatter frequencies for different workpiece speeds with spindle speed=2500 rpm and depth of cut=30 m. a Predicted. b Measured. the actual grinding force, which is around 25 N, is interrupted and divided into a number of force intervals separated by zero intervals. The force intervals represent the strokes of a surface grinding process, and the zero intervals represent air cutting considered by the model. Random lengths of zero intervals are used in the model to consider the uncertain wheel phase shift between strokes in a real surface grinding process. It can also be seen in Fig. 20 that peak-to-peak force of the force intervals, which is a measure of chatter, grows as time goes, and hence the chatter growth rate can be calculated from the growth rate of the peak-to-peak force of each stroke. In fact, the measured chatter growth rate can be obtained in the exactly same way from the measured force history. Peak-to-peak forces obtained from each simulated stroke are used for chatter growth analysis and their characteristics associated with G ratios are studied in the following. Figure 21a shows the simulated peak-to-peak forces versus accumulated material removal under the conditions of various G ratios. It can be seen that the peak-to-peak force signature has a good correlation with the G ratio such that a smaller G ratio results in a faster chatter growth. For further comparisons of the curves with different G ratios, Fig. 21b is transformed from Fig. 21a by scaling the values of removed workpiece material for the curves. The scaling factors for the curves, s, are the ratios of 500 to their G ratios. As shown in Fig. 21b, the peak-to-peak forces calculated using different G ratios agree well with each other. It is obvious that the G ratio has an approximate linear relationship with the volume of removed workpiece material. In practice, according to Fig. 19 Chatter frequencies for different depths of cut with spindle speed=2500 rpm and workpiece speed=84.7 mm/ s. a Predicted. b Measured. Fig. 21 Simulated peak-to-peak forces in the normal direction with various G ratios. a Before scaling. b After scaling with factor s. Journal of Manufacturing Science and Engineering MAY 2006, Vol. 128 / 401

10 Fig. 22 Chatter growth represented by peak-to-peak forces. Wheel speed=2000 rpm, workpiece speed=200 ipm 84.7 mm/s, depth of cut= in 30 m. this linear characteristic, the grinding process can be simulated using a relative small G ratio and a good estimation of the chatter growth can be obtained by scaling the volume of removed workpiece material, thereby saving a significant amount of computing time in simulation. Using the G ratios calculated from Eq. 33, peak-to-peak forces have been predicted as a function of material removal per unit width for the test conditions listed in Table 1. Figure 22 shows predicted peak-to-peak forces against measured ones in traverse and normal directions for an example grinding test case. For comparison purposes, the predicted and measured peak-topeak forces were shifted by subtracting a constant value. As shown in the example, the predicted peak-to-peak force graph agrees well with the measured ones, and both the predicted and measured peak-to-peak forces increase slowly at first and start to grow fast at a certain point of material removal, where the wheel should be redressed before further grinding. This redressing point, which is also the chatter threshold associated with accumulated workpiece material removal per unit width, can be determined by setting a peak-to-peak force limit above its initial value. For example, a one-newton-force limit is set in Fig. 22 marked by the horizontal line, and the chatter threshold from the predicted peakto-peak force is then determined by their intersection, i.e., 610 mm 3 /mm of material removal, which is marked by the vertical line. The chatter thresholds of the measured peak-to-peak forces can be estimated in a similar way after curve fitting. The chatter growth was validated by comparing the chatter thresholds obtained from the predicted peak-to-peak forces with those obtained from the measured ones. For comparison purposes, the test cases are grouped by varying only one of the three major grinding parameters: wheel spindle speed, depth of cut, and workpiece speed. The comparisons of the three groups are shown in Figs. 23a, 23b, and 23c, respectively. In Fig. 23 the square marks connected by a thick line represent the predicted chatter thresholds while the paired dash marks represent the chatter thresholds obtained from the measured peak-to-peak forces in both normal and traverse directions. The thin line segment bounded by the dash mark pair provides the possible range of measured chatter thresholds. It can be seen that most of the predicted chatter thresholds fall into the corresponding measured ranges of chatter thresholds, indicating reasonable agreement. In addition, both the measured and predicted chatter thresholds show the same trends. Since a chatter threshold is associated with accumulated material removal, or is a function of time, a smaller value of threshold indicates that chatter reaches a given limit earlier or chatter grows faster in the entire process. Figure 23a shows the predicted and measured chatter thresholds under various spindle speeds with other grinding parameters unchanged. As shown, the chatter threshold decreases when the wheel speed increases, except for the interval between 2000 and 2250 rpm. This trend can be explained from the variation of the G ratio and equivalent chip thickness. Based on Eqs. 12 and 33, a higher wheel speed, g, causes a smaller equivalent chip thickness, h eq, as well as a smaller G ratio, G, with other parameters remaining constant, and a smaller G ratio has a smaller chatter threshold based on the analysis shown in Fig. 21. In addition, for the same amount of material removal a smaller equivalent chip thickness, h eq, generates a longer sliding length, which helps the grit dulling and wheel regenerative chatter growing process and results in a smaller chatter threshold. The agreements of trends between the measured and predicted chatter thresholds are also shown in Figs. 23b and 23c for the cases in the other two groups. The trend explanation for Figs. 23b and 23c is similar to that for Fig. 23a: a lower workpiece speed and/or depth of cut causes a smaller equivalent chip thickness and G ratio based on Eqs. 12 and 33, thereby resulting in smaller chatter thresholds. According to the comparison in Secs. 3.3 and 3.4, the measured and predicted chatter frequencies and thresholds have reasonably good agreement. The discrepancies shown in the comparison are likely from the topography of the wheel surface, wheel unbalance, spindle nonlinearity, and other uncertainties. The effects of fracture wear on force variations and wheel surface generations, which are not considered in the model, may also be the reasons for some discrepancies. The existing models on grinding chatter problems 1 13 are mostly for cylindrical grinding processes and rarely for surface grinding processes. Compared with the exiting models, the new model is capable of describing the dynamic surface grinding process more accurately in several critical aspects. First, unlike the other models that consider only the motion in the normal direction, it considers the complete two-dimensional kinematics, dynamics, surface profiles, and the geometrical interactions between the wheel and workpiece. In addition the cross-feed motion as well as its effect on uneven wheel wear in the axial direction has also been modeled. Second, the concentrated grinding force mod- Fig. 23 Chatter thresholds for different test conditions. a Depth of cut= in. 30m, workpiece speed =200 rpm 84.7 mm/ s. b Wheel speed=2500 rpm, workpiece speed=200 ipm 84.7 mm/ s. c Wheel speed =2500 rpm, depth of cut= in. 30 m. 402 / Vol. 128, MAY 2006 Transactions of the ASME

11 els, dependent on the dept of cut 5 13, are improved to a distributed grinding force model along the contact length as a function of the normalized uncut chip thickness, which is a twodimensional force model in the radial and tangential directions. Third, multiple degrees of freedom DOFs in the normal and traverse directions, instead of a single DOF in 5 13, are used for both the grinding wheel and workpiece to compute the dynamic responses as well as their effects on uncut chip thickness. More importantly, it considers the time-variant effects of grinding wheel wear on the forces and surface profiles, and accurately describes the coupled force regeneration, wheel surface undulation, and grit dulling processes. Consequently, a new theory for the wheel regenerative chatter mechanism is introduced. In summary, the new model has the unique ability of accurately predicting the regenerative forces of a multi-dof and twodimensional surface grinding system as a function of not only the instantaneous chip thickness, but also the distributed uneven grit wear/dullness. However, the required static force and G ratio models are currently in empirical or semiempirical forms because no existing analytical models can provide enough accuracy. When accurate analytical force and G ratio models become available, they can be easily integrated into the current dynamic surface grinding model. 4 Conclusion In this paper we presented a comprehensive dynamic model that simulates surface grinding processes and predicts their regenerative chatter characteristics. The special aspects of a surface grinding process have been considered such as the linear traverse motion, interrupted grinding of a series of surfaces and step-like wheel wear along the axial direction due to the cross-feed. A new theory for the wheel regenerative chatter mechanism, which describes the regenerative force as a function of not only the instantaneous chip thickness but also the distributed uneven grit wear/dullness, has been introduced and applied in the model. Using the model, explanations were provided for those unrevealed wheel regenerative chatter phenomena observed from the experimental results in the literature. Using the new theory, the characteristics of wheel regenerative chatter, which is a regenerative chatter with relatively slow chatter growth, have been discussed. The dynamic model has been validated by comparing the simulated chatter results with experimental ones. The good predictions of chatter frequencies and thresholds show that the mechanism described in the new theory is indeed the dominant mechanism of the wheel regenerative chatter in surface grinding processes. Although the new theory has been introduced and implemented in the model for surface grinding processes, it should be valid for other grinding processes. Acknowledgment The authors gratefully wish to acknowledge the support of the National Science Foundation through Grant No DMI, and the NIST/ATP through Grant No. 70NANB3H3064. Nomenclature C t chip thickness m C s,c d static and dynamic chip thickness m C w chip thickness variation due to grinding wheel wear m K t,k r specific force coefficients N/m 2 F,F total, specific grinding force N, N/m f force of segments N G G ratio, grinding ratio R radius m velocity m/s V m material removal per unit width m 2 W wear-force function w grinding width m x,y displacements m ẋ,ẏ velocities m/s angular position or angle radian spindle rotating speed radian/s interval of angular position radian Subscripts g,w grinding wheel and workpiece x,y cartesian coordinates t,r tangential and radial direction s,d static and dynamic 0 initial References 1 Inasaki, I., Karpuschewski, B., and Lee, H. -S., 2001, Grinding Chatter Origin and Suppression, CIRP Ann. 50, pp Malkin, S., 1973, Review of Material Processing Literature , 2-Grinding, ASME J. Eng. Ind. 95, pp Inasaki, I., and Yonetsu, S., 1968, Surface Waves Generated on the Grinding Wheel, Trans. Jpn. Soc. Mech. Eng. 11, pp Inasaki, I., and Yonetsu, S., 1969, Surface Waves Generated on the Grinding Wheel Report 2, Trans. Jpn. Soc. Mech. Eng. 12, pp Thompson, R. A., 1971, The Dynamic Behavior of Surface Grinding, Part 1 A Mathematical Treatment of Surface Grinding, Trans. ASME 93, pp Thompson, R. A., 1971, The Dynamic Behavior of Surface Grinding, Part 2 Some Surface Grinding Tests, Trans. ASME 93, pp Snoeys, R., and Brown, D., 1969, Dominating Parameters in Grinding Wheel and Workpiece Regenerative Chatter, Proc. 10th Int. Machine Tool Design and Research Conf., pp Bartalucci, B., and Lisini, G. G., 1969, Grinding Process Instability, ASME J. Eng. Ind. 91, pp Kaliszer, H., 1970, Analysis of Chatter Vibration during Grinding, Proc. 11th Int. Machine Tool Design and Research Conf., pp Inasaki, I., and Yonetsu, S., 1977, Regenerative Chatter in Grinding, Proc. 18th Int. Machine Tool Design and Research Conf., pp Hashimoto, F., Kanai, A., and Miyashita, M., 1984, Growing Mechanism of Chatter Vibration in Grinding Processes and Chatter Stabilization Index of Grinding Wheel, CIRP Ann. 50, pp Matsubara, T., Mizumoto, H., and Yamamoto, H., 1987, Experimental Analysis of Work Regenerative Chatter in Plunge Grinding, Bull. Jpn. Soc. Precis. Eng. 21, pp Weck, M., and Alldieck, J., 1989, The Originating Mechanisms of Wheel Regenerative Grinding Vibration, CIRP Ann. 38, pp Li, H., and Shin, Y. C., 2006, A Time-domain Dynamic Model for Chatter Prediction of Cylindrical Plunge Grinding Processes, ASME J. Manuf. Sci. Eng., in press. 15 Li, H., and Shin, Y. C., 2006 A Comprehensive Dynamic End Milling Simulation Model, ASME J. Manuf. Sci. Eng., in press. 16 Shin, Y. C., and Waters, A. J., 1994, Face Milling Process Modeling With Structural Nonlinearity, Trans. North Am. Manuf. Res. Inst. SME 22, pp Choi, T., and Shin, Y. C., 2005, Generalized Intelligent Grinding Advisory System, submitted to Int. J. Prod. Res. 18 Malkin, S., and Cook, N. H., 1971, The Wear of Grinding Wheels, Part 1 Attritious Wear, ASME J. Eng. Ind. 93, pp Lal, G. K., and Shaw, M. C., 1972, Wear of Single Abrasive Grains in Fine Grinding, Proc. Int. Grinding Conf., Carnegie Press, Pittsburgh, PA, pp Snoeys, R., Peters, J., and Decneut, A., 1974, The Significance of Chip Thickness in Grinding, CIRP Ann. 23, pp Journal of Manufacturing Science and Engineering MAY 2006, Vol. 128 / 403