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Effects of process parameters on workpiece roundness in tangential-feed centerless grinding using a surface grinder
Journal of Materials Processing Technology 210 (2010) 759–766
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Effects of process parameters on workpiece roundness in tangential-feed centerless grinding using a surface grinder W. Xu a,∗ , Y. Wu b , T. Sato b , W. Lin b a b
Graduate School, Akita Prefectural University, 84-4 Tsuchiya-ebinokuchi, Yirihonjo, Akita 015-0055, Japan Dept. of Machine Intelligence and Systems Engineering, Akita Prefectural University, 84-4 Tsuchiya-ebinokuchi, Yurihonjo, Akita 015-0055, Japan
a r t i c l e
i n f o
Article history: Received 19 October 2009 Received in revised form 5 January 2010 Accepted 8 January 2010
Keywords: Centerless grinding Surface grinder Ultrasonic vibration Roundness Shoe
a b s t r a c t The present authors proposed a new centerless grinding method using a surface grinder in their previous study [Wu, Y., Kondo, T., Kato, M., 2005. A new centerless grinding technique using a surface grinder. J. Mater. Process. Technol. 162–163, 709–717]. In this method, a compact centerless grinding unit composed mainly of an ultrasonic elliptic-vibration shoe is installed onto the worktable of a multipurpose surface grinder to perform tangential-feed centerless grinding operations. However, for the complete establishment of the new method it is crucial to clarify the workpiece rounding process and the effects of process parameters such as the worktable feed rate, the stock removal and the workpiece rotational speed on the machining accuracy, i.e., workpiece roundness, so that the optimum grinding conditions can be determined. In this paper, the effects of the process parameters on workpiece roundness are investigated by simulation and experiments. For the simulation analysis, a grinding model taking into account the elastic deformation of the machine is created. Then, a practical way to determine the machiningelasticity parameter is developed. Further, simulation analysis is carried out to predict the variation of workpiece roundness during grinding and to discover how the process parameters affect the roundness. Finally, actual grinding operations are performed by installing the previously constructed unit onto a CNC surface grinder to confirm the simulation results. The obtained results indicate that: (1) a slower worktable feed rate and higher workpiece rotational speed give better roundness; (2) better roundness can be also obtained when the stock removal is set at a larger value; (3) the workpiece roundness was improved from an initial value of 23.9 m to a final value of 0.84 m after grinding. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In the manufacturing industry, for high accuracy and high productivity machining of cylindrical components, such as bearing raceways, silicon ingots, pin gauges, crankshafts, ferrules and catheters, centerless grinding operations have been extensively carried out on specialized centerless grinders (Serope, 2001). Two types of centerless grinders are available commercially, one that uses a regulating wheel and one that uses a shoe, and these grinders are different in how they support the workpiece and control the workpiece rotational speed. Especially since the invention of the regulating wheel type centerless grinder by Heim in 1915 (Yonetsu, 1966), much research has been devoted to clarify the rounding process by Rowe et al. (1964, 1965), optimizing the machining conditions by Rowe and Bell (1986) and Wu et al. (1996), achieving stability of the workpiece during grinding by Miyashita et al. (1982),
∗ Corresponding author. Tel.: +81 184 272157; fax: +81 184 27 2165. E-mail addresses: [email protected], [email protected] (W. Xu). 0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2010.01.003
Epureanu et al. (1997) and Albizuri et al. (2007), and achieving safe machining operation by Hashimoto et al. (1982) for the sake of enhancing machining accuracy and efficiency. Shoe type centerless grinding has also attracted attention from both industrial and academic researchers. Yang and Zhang (1998) designed a flat vacuum-hydrostatic shoe to increase load capacity and stiffness for high precision applications of shoe centerless grinding. Then Yang et al. (1999) and Zhang et al. (1999) analyzed the process stability in vacuum-hydrostatic shoe centerless grinding. In addition, Zhang et al. (2003) developed a geometry model to predict the lobing generation in shoe centerless grinding and used the model for the analysis of grinding processes. From the viewpoint of production cost, the two types of centerless grinders are highly suitable for small-variety and largevolume production, because the loading/unloading of workpieces is extremely easy and fast. However, the centerless grinder is a special-purpose machine and is relatively costly, putting it at a disadvantage for large-variety and small-volume production, the demand for which has increased rapidly in recent years. As a solution to this problem, we proposed a new centerless grinding technique that can be performed on a multipurpose machine such
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as a surface grinder (rather than a centerless grinder) in our previous work (Wu et al., 2005). This method is based on the concept of ultrasonic-shoe centerless grinding, which has been studied by Wu et al. (2003, 2004a,b). In the method, a compact unit consisting mainly of an ultrasonic elliptic-vibration shoe, a blade, and their respective holders is installed on the worktable of a multipurpose surface grinder to conduct tangential-feed centerless grinding. The function of the ultrasonic elliptic-vibration shoe is to hold the cylindrical workpiece in conjunction with the blade, and to control the workpiece rotational speed with the elliptic motion on its upper end-face. In the previous study (Wu et al., 2005), an actually constructed unit was installed on the table of a CNC surface grinder, and tangential-feed centerless grinding operations were performed involving pin-shaped workpieces. For the complete establishment of this new method, it is crucial to clarify the effects of process parameters such as the feed rate of worktable, the stock removal and the workpiece rotational speed on the machining accuracy, i.e., workpiece roundness, so that the optimum grinding conditions can be confirmed. For this purpose, this paper presents a simulation method to predict the variation of workpiece roundness during tangential-feed centerless grinding performed on a surface grinder. In order to evaluate and discuss the effects of process parameters on workpiece roundness, simulation analysis was carried out followed by experimental confirmation on the previously constructed experimental rig.
2. Operation principle of tangential-feed centerless grinding performed on a surface grinder Fig. 1 schematically illustrates the operation principle of the new centerless grinding technique and the detailed construction of the experimental rig. The rig is established by installing a grinding unit, mainly composed of an ultrasonic elliptic-vibration shoe and its holder, a blade and its holder, and a base plate, onto the worktable of a surface grinder via an electromagnetic chuck. A workpiece constrained between the unit and the grinding wheel is fed in the direction of worktable movement at a feed rate of Vf . Once the workpiece interferes with the clockwise rotating grinding wheel, actual grinding action begins, and the grinding is finished when the ground workpiece loses contact with the grinding wheel with forward movement of the worktable. During grinding, the workpiece rotational speed nw is controlled by the elliptic motion on the upper end-face of the shoe, and the stock removal ı is determined by adjusting the gap between the grinding wheel and the
Fig. 1. Schematic illustration of tangential-feed centerless grinding performed on a surface grinder.
worktable. This method provides an alternative centerless grinding technique especially suited to large-variety and small-volume production at low cost, achieved by simply installing the compact centerless grinding unit on a surface grinder. As shown in Fig. 1, the shoe is fixed on its holder via a spacer (electric isolation), and the blade is held directly on its holder by means of bolts. The shoe and blade are fastened on the metal base plate so that the unit can be mounted on the worktable by means of an electromagnetic chuck. The shoe is constructed by bonding a piezoelectric ceramic device (PZT) with two separated electrodes onto a metal elastic body (stainless steel, SUS304). When two alternating current (AC) signals (over 20 kHz) with a phase difference of to each other generated by a wave function generator are applied to the PZT after being amplified by means of power amplifiers, bending and longitudinal ultrasonic vibration are excited simultaneously. The synthesis of vibration displacements in the two directions creates an elliptic motion on the end-faces of the metal elastic body. Consequently, the workpiece rotation is controlled by the frictional force between workpiece and shoe, so that the peripheral speed of the workpiece is maintained at the same value as the bending vibration speed on the shoe end-face (Xu et al., 2009a,b). The workpiece rotational speed can be adjusted by changing the value of parameters such as the amplitude Vp–p and frequency f of the voltage applied to the PZT (Ueha and Tomikawa, 1991, Kenjo and Yubita, 1991), because the shoe bending vibration speed varies with the variation of the applied voltage. In addition, a pre-load is applied to the shoe at its lower end-face in its longitudinal direction using a coil spring to prevent the PZT from breaking due to resonance.
Fig. 2. Geometrical arrangements in tangential-feed centerless grinding using a surface grinder: before grinding (a) and during grinding (b).
W. Xu et al. / Journal of Materials Processing Technology 210 (2010) 759–766
3. Simulation Fig. 2(a) and (b) shows the geometrical arrangements of the unit, the workpiece and the grinding wheel before grinding and during grinding, respectively. The shoe and the blade with a tilt angle of are fixed on the L-shaped worktable, and thus the relative position of the blade to the shoe is kept constant during grinding. The workpiece with an initial radius of 0 , constrained between the shoe, the blade and the grinding wheel, is fed forward on the worktable at a feed rate of Vf , and is ground at point A by the grinding wheel rotating at a speed of ng . In the meantime, the workpiece rotational speed nw is controlled by the elliptic motion on the shoe end-face.
Solving Eqs. (5) and (6) simultaneously yields the XYcoordinates of the workpiece center Owt at time t, as follows:
⎧ ⎨ ⎩
XC0 = XOw0 YC0 = YOw0 − 0
Y − YB0 = tan (X − XB0 )
(1)
Y − YC0 = 0
(2)
Substituting coordinates of points B and C into Eqs. (1) and (2), respectively, gives PX + QY + R = 0
(3)
Y − YOw0 + 0 = 0
(4)
where P = tan , Q = −1, R = YOw0 + 0 cos − tan (XOw0 − 0 sin ). During grinding (see Fig. 2(b)), the coordinates of the workpiece center Owt and the grinding wheel center Ogt will vary as the material is removed. Let the instantaneous workpiece radius in the direction parallel to the X-axis after grinding for time t be (t), and the workpiece radius at points A, B and C can be expressed with (t − TA ), (t − TB ), and (t − TC ), respectively, where TA = [ − 2˛(t)]/4nw , TB = ( + 2)/4nw and TC = 3/4nw are the time delays for points A, B and C. Since the (t − TB ) and (t − TC ) are equal to the distances from the Owt to the blade end-face and to the shoe upper end-face, respectively, they can be obtained as follows from the geometrical arrangement in Fig. 2(b) by using Eqs. (3) and (4). |PXOw (t) + QYOw (t) + R|
P2 + Q 2
(t − TC ) = YOw (t) − YOw0 + 0
(7)
XOg (t) = XOg0 − Vf t = XOw0 +
2(0 + Rg )ı − ı2 − Vf t
(8)
In addition, the following relationships are established from the geometrical arrangement in Fig. 2(b).
[XA (t) − XOg (t)]2 + [YA (t) − YOg (t)]2 = Rg2 YA (t) − YOw (t) = cot ˛(t)[XA (t) − XOw (t)]
(9)
where [XOg (t) − XOw (t)] [YOg (t) − YOw (t)]
tan ˛(t) =
(10)
Therefore, the XY-coordinates of point A are decided as follows by using Eqs. (9) and (10):
⎧ ⎨
(−V −
V 2 − 4UW ) 2U ⎩ YA (t) = cot ˛(t)[XA (t) − XOw (t)] + YOw (t) XA (t) =
(11)
where U = 1 + cot2 ˛(t), V =2[cot ˛(t)YOw (t)−cot2 ˛(t)XOw (t)−cot 2 2 (t)+[cot ˛(t)X ˛(t)YOg (t)−XOg (t)], W =XOg Ow (t)−YOw (t) + YOg (t)] − Rg2 .
Then, the linear equations representing the blade end-face and shoe upper surface in this coordinate system can be written as
(t − TB ) =
P 2 + Q 2 (t − TB ) − QYOw (t) − R P YOw (t) = (t − TC ) + YOw0 − 0
YOg (t) = YOg0 = YOw0 + 0 + Rg − ı
In this model, several assumptions are made: (1) the workpiece is in constant contact with the blade and the shoe at points B and C, respectively, during grinding; (2) the vibration of the entire machine is too small to be regarded, and no chatter occurs on the machine due to the ultrasonic elliptic-vibration of the shoe; (3) the workpiece rotation is always stable, and no variation of rotational speed during grinding occurs; (4) the wear of the grinding wheel is too small to be recognized, and the grinding wheel radius is kept constant during grinding. Let a XY-coordinate system be located on the worktable. An optional point O on the worktable is the origin of the coordinate system. The X-axis is taken in the horizontal direction and the Yaxis in vertical direction. Before grinding (see Fig. 2(a)), the initial XY-coordinates of the grinding wheel center Og0 and the workpiece center Ow0 are (XOg0 , YOg0 ) and (XOw0 , YOw0 ), respectively. Thus, the XY-coordinates of the initial blade contact point B (XB0 , YB0 ) and the shoe contact point C (XC0 , YC0 ) can be obtained from the geometrical arrangement in Fig. 2(a), as follows: XB0 = XOw0 − 0 sin YB0 = YOw0 + 0 cos
XOw (t) =
In this moment, the XY-coordinates of the grinding wheel center Ogt are obtained from the geometrical arrangement in Fig. 2(b) as
3.1. Simulation model
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(5) (6)
Eventually, the workpiece radius (t − TA ) at point A after grinding for time t is obtained from the XY-coordinates of the workpiece center Ow and the grinding point A as follows: (t − TA ) =
[XA (t) − XOw (t)]2 + [YA (t) − YOw (t)]2
(12)
The apparent wheel depth of cut would be = (t − TA − T) − (t − TA ), where T is the time required for one revolution of the workpiece. If the grinding system has an ideal stiffness, the true wheel depth of cut would be equal to the apparent wheel depth of cut. However, our previous work (Xu et al., 2009c), in which the grinding model for simulation analysis was developed under the ideal stiffness, revealed that the simulation results do not agree well with the actual results obtained experimentally sometime. This is because the grinding system withstands the elastic deformation caused by the grinding force during actual grinding. Rowe et al. introduced a dimensionless parameter called machining-elasticity parameter k as a measure to indicate the elastic deformation of centerless grinding system which is defined as a quotient between the true depth of cut and the apparent depth of cut with Eq. (13) (Rowe and Barash, 1964; Marinescu et al., 2006). k=
true wheel depth of cut = apparent wheel depth of cut
(13)
Also Rowe et al. pointed out that the parameter k depends on the cutting stiffness Km and the whole grinding system static stiffness Ke in a relationship of k = Ke /(Km + Ke ). Following Rowe et al.’s consideration, the true wheel depth of cut can be calculated as = k in the current work, resulting in the true workpiece radius
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Fig. 4. A result of grinding force measurement.
Solving Eq. (16) yields
Fig. 3. Apparent radius and true radius after workpiece rotates for i revolutions.
i =
at point A is
i−1
= (t − TA − T ) − k{(t − TA − T )
[XA (t) − XOw (t)]2 + [YA (t) − YOw (t)]2 }
(14)
However, the wheel depth of cut calculated using these equations is less than zero occasionally. Obviously this phenomenon would not happen. Therefore, Eq. (14) should be modified as
(t − TA ) = (t − TA − T ) − k((t − TA − T ) −
− i = a − bi
1 ı 2
(17)
(18)
a=2
where
ı(2Rg + 20 − ı)Vf nw + Vf2 /2n2w (2Rg + 20 −
+ 20 − ı). ı), b = Therefore, according to the definition of the machiningelasticity parameter of k = (i−1 − i )/(i−1 − i ), the true radius Vf2 /n2w (2Rg
[XA (t) − XOw (t)]2 + [YA (t) − YOw (t)]2 )
(t − TA ) ≤ (t − TA − T )
(15)
(t − TA ) > (t − TA − T )
N of the workpiece after grinding for N revolutions can be obtained as follows:
3.2. Determination of the machining-elasticity parameter As described above, the machining-elasticity parameter k depends on the stiffness of the grinding system. If the simulation result is to be trusted, the value of k should be determined for the given grinding system. Rowe et al. proposed a method for the determination of machining-elasticity parameter k in conventional in-feed centerless grinding (Rowe et al., 1965), in which a parameter proportional to the true wheel depth of cut, i.e., the grinding power, during in-feed or spark-out was measured to obtain the parameter k. However, this method is available only for in-feed centerless grinding with a constant apparent depth of cut generated by the uniform in-feed movement of grinding wheel, and thus not suitable for the tangential-feed centerless grinding being dealt with in our current work because the apparent depth of cut always varies during grinding (see Xu et al., 2009d). Therefore, an alternative procedure to determine the value of k is proposed as below. Fig. 3 shows the apparent workpiece radius i and the true radius i after the workpiece is rotated for i revolutions, during which time the workpiece moves for distance Si . SN is the distance that the workpiece moved from its initial position to the position where its center is right below the grinding wheel center. In this duration, the workpiece rotates for N revolutions. Thus, based on the geometrical arrangement shown in Fig. 3, the following relationship is obtained: (Rg + i ) = (SN − Si )2 + (Rg + 20 − i − ı) 2
Si = Vf i/nw
+ 0 −
4Rg + 40 − 2ı
(t − TA ) = (t − TA − T )
where
2
Subsequently, using Eq. (17) yields
(t − TA ) = (t − TA − T ) −
−
ı(2Rg + 20 − ı) − Vf i/nw
2
(i = 1, 2, . . . , N) 2
(Rg + 0 )2 − (Rg + 0 − ı) =
(16) and
ı(2Rg + 20 − ı).
SN =
N
N−1 1
= (1 − k) 0 + k(1 − k)
N
N−i i
+ . . . + k(1 − k)
+ . . . + kN N
= (1 − k) (0 − 1 ) + (1 − k) + (1 − k)(N−1
) + − N N
N
= a[(1 − k) + (1 − k) + 2(1 − k)
N−1
N−1
N−1
(1 − 2 ) + . . .
(19)
(0 = 0 ) N
+ . . . + (1 − k)] − b[(1 − k)
+ . . . + N(1 − k)] + N
= Because of 0 < k < 1 and N 1 in Eq. (19), and the relation of N 0 − ı/2 is obtained by substituting N for i in Eq. (16), the value of N is approximately calculated with Eq. (20).
N = (a − Nb − 2b)
k2 − 1 1−k ı −b + 0 − 2 k k2
(20)
Further, solving Eq. (20) yields
k=
−F +
F 2 − 4EG
2E
(21)
where E = N − 0 + ı/2 + a − Nb − b, F = −a + Nb + 2b, G = −b. Consequently, the value of k can be obtained using Eq. (21), as long as the parameters Rg , ı, nw , Vf , 0 , N and N are known. In an actual grinding operation, the values of Rg , ı, nw , Vf and 0 are set up as the grinding conditions. In addition, N can be known by measuring the diameter of the workpiece ground for N revolutions. Since N is determined using the relation of N = nw TN where TN is the time that the workpiece moves for distance SN (see Fig. 3), its value can be obtained as long as TN is known. In this paper, TN is
W. Xu et al. / Journal of Materials Processing Technology 210 (2010) 759–766
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ing its peak, until the grinding has been performed for time TN , and eventually the grinding action finishes in time Tg after spark-out for time Tg − TN . Although the actual grinding action happened during the time Tg = 75.34 s, the time TN = 73.85 s was just that the workpiece moved for the distance SN . Hence, the value of N in the current conditions was N = 1.2 × 73.85 = 88.62. In addition, the workpiece radius was N = 2.450 mm after grinding. Subsequently, the value of k under the current conditions was k = 0.089. Changing the workpiece stock removal ı but keeping the other parameters constant yielded the results shown in Fig. 5. It is found that k increased gradually as the stock removal increased, and its mean value was around 0.084. Fig. 5. Measuring results of parameter k.
measured based on the variation of grinding force during grinding. For example, Fig. 4 shows a typical variation of grinding force obtained in an actual grinding operation conducted on the previously constructed experimental rig (see Wu et al., 2005) under the conditions of Rg = 90 mm, ı = 0.1 mm, nw = 1.2 s−1 , Vf = 0.05 mm/s and 0 = 2.499 mm. It can be seen that the grinding force increases rapidly in the beginning and then decreases gradually after reach-
3.3. Simulation procedure The initial workpiece with a diameter of 5 mm is shown in Fig. 6(a). A flat with depth of 25 m in radial direction is generated on its circumference to indicate the initial roundness. In simulation, the workpiece was divided equally into 360 sections along its circumference, as shown in Fig. 6(b), and thus the initial workpiece profile could be represented with 360 radius of (i) (i = 1–360). In any given time t during grinding, as long as (t − TB ) and (t − TC )
Fig. 6. Initial workpiece profile (a), division of workpiece (b), and calculation flowchart (c) for simulation analysis.
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Table 1 Grinding conditions for simulation. Workpiece initial radius 0 (mm) Workpiece rotational speed nw (s−1 ) Workpiece initial roundness (m) Worktable feed rate Vf (mm/s) Stock removal ı (mm) Blade angle (◦ ) Grinding wheel radius Rg (mm) Machining-elasticity parameter k
2.5 0.1–3.2 25 0.03–0.15 0.25–0.125 60 90 0.084
Fig. 9. The main portion of the grinding apparatus.
Fig. 7. Variation of roundness during grinding.
are known, the instantaneous radius of (t − TA ) can be calculated using the basic Eqs. (1)–(15). Following this procedure, every 360 of workpiece radius is obtained and used to draw the profile and calculate the roundness of the workpiece. The simulation flowchart is shown in Fig. 6(c).
ally, some new comparatively small convex and concave areas are formed at the initial stage, but the following grinding process decreases the height of these areas, which eventually improves the workpiece roundness. Fig. 8(a)–(c) shows the effects of the worktable feed rate Vf , the workpiece rotational speed nw and the stock removal ı on the final roundness, respectively. It is obvious that the workpiece roundness increases with the increasing worktable feed rate Vf , but decreases as the workpiece rotational speed nw and the stock removal ı increase. From the simulation results, it is clear that a better final roundness needs a higher workpiece rotational speed, a larger stock removal, and a lower worktable feed rate.
3.4. Simulation results 4. Experiments Table 1 shows the grinding conditions for the simulation analysis. Fig. 7 shows the variation of the workpiece roundness and some of the workpiece profiles during grinding obtained under Vf = 0.067 mm/s, ı = 0.075 mm, nw = 1.2 s−1 , k = 0.084. It is obvious that as the size of the initial flat indentation decreases gradu-
In order to confirm the simulation results, the centerless grinding unit constructed in the work (Wu et al., 2005) was installed on the worktable of a CNC surface grinder (SGT-315RPA by Nagase Integrex Co., Ltd.) (see Fig. 9). The experimental details to be
Fig. 8. Effects of worktable feed rate Vf (a), stock removal ı (b), and workpiece rotational speed nw (c) on the final workpiece roundness.
W. Xu et al. / Journal of Materials Processing Technology 210 (2010) 759–766 Table 2 Grinding conditions. Grinding wheel Workpiece (mm) Input voltage amplitude Vp–p (V) Input voltage frequency f (kHz) Input voltage phase difference (◦ ) Grinding wheel speed (m/s) Worktable feed rate Vf (mm/s) Stock removal ı (mm) Blade angle (◦ ) Pre-load of spring (N)
SDC400N180 × 15 × 75 TH10, 5 × L13 30–120 46.5 90 20 0.03–0.15 0.025–0.125 60 15
described below were designed based on our previous work that dealt with the workpiece rotation control characteristics and the effects of grinding parameters on workpiece roundness (Xu et al., 2009b). 4.1. Experimental conditions and procedure The grinding experiments were carried out involving the workpiece as shown in Fig. 6(a), and its dimension was the same as that used in simulation, and a flat indentation of 25 m in radial direction was generated by surface grinding to indicate the initial roundness. In order to maintain the stability of workpiece, the width of workpiece must be smaller than that of grinding wheel. The other grinding conditions are tabulated in Table 2. In addition, since the workpiece rotational speed depends on the AC voltage applied to the PZT, the relationship between the applied voltage and the workpiece rotational speed should be obtained during an actual grinding operation. For this purpose, the rotational motion
765
of the workpiece end-face was recorded by using a high-speed digital camera (VW-6000 by Keyence Co., Ltd.) in the experiments, and the workpiece rotational speed was obtained from the recorded workpiece end-face motion using an animated image-processing software (Movie Editor by Keyence Co., Ltd.). The grinding procedure was as follows: first, the unit constraining a workpiece was fastened on the worktable, and its position was adjusted carefully so that the unit was at the lower left side of the grinding wheel. Next, the stock removal was given by adjusting the gap between the grinding wheel and the worktable. The worktable was moved forward to carry the workpiece toward the grinding wheel. Once the workpiece passed through and reached right-side of grinding wheel, the grinding operation was completed. In experiments, the grinding operations under the same grinding conditions were carried out on more than three workpieces. After grinding, the workpiece roundness was measured with a roundness measurement instrument (Rondcom55A by Tokyo Seimitsu Co., Ltd.) at five different cross-sections, and the average value of the five measurements was used to indicate the workpiece roundness. Changing the worktable feed rate, workpiece rotational speed and stock removal, and repeating the above-mentioned procedures yielded the workpiece roundness for different grinding conditions. 4.2. Experimental results and discussion Table 3 shows the relationship between the workpiece rotational speed nw and the AC voltage Vp–p applied to the PZT under the conditions of f = 46.5 kHz, = 90◦ , ı = 0.075 mm, Vf = 0.067 mm/s. As can be seen, the workpiece rotational speed increases
Fig. 10. Photos and profiles of the workpiece before grinding (a) and after grinding (b) under the conditions of f = 46.5 kHz,
= 90◦ , Vp–p = 50 V, ı = 0.075 mm and Vf = 0.033 mm/s.
Fig. 11. Effects of worktable feed rate Vf (a), stock removal ı (b), and workpiece rotational speed nw (c) on the final roundness.
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Table 3 Relationship between the workpiece rotational speed nw and the applied volt= 90◦ , ı = 0.075 mm, age Vp–p under the grinding conditions of f = 46.5 kHz, Vf = 0.067 mm/s. Vp–p (V) nw (s−1 )
30 0.9
50 1.5
70 1.8
90 2.2
100 2.5
120 3.5
(4) Better roundness can be obtained when the stock removal is set at a larger value. (5) The workpiece roundness was improved from an initial value of 23.9 m to a final value of 0.84 m after grinding under the current conditions. Acknowledgements
monotonously with the increasing amplitude of applied voltage. This means that the required workpiece rotational speed can be given exactly by applying the corresponding voltage to the PZT. Fig. 10(a) and (b) shows the photographs and cross-section profiles of the workpiece before grinding and after grinding under the grinding conditions of f = 46.5 kHz, = 90◦ , Vp–p = 50 V, ı = 0.075 mm, Vf = 0.033 mm/s, respectively. The workpiece roundness was improved greatly from the initial value of 23.95 m to the final one of 0.84 m. The roundness results obtained for different worktable feed rates Vf , stock removals ı and amplitudes of voltage Vp–p applied to the PZT are shown in Fig. 11(a), (b) and (c), respectively. It can be seen from Fig. 11(a) that the worktable feed rate affects the workpiece roundness significantly. The roundness was improved greatly from 13.13 m to 0.84 m as long as the worktable feed rate was changed from 0.15 mm/s to 0.033 mm/s. Fig. 11(b) shows that the stock removal ı affects the roundness significantly. The workpiece roundness was 16.502 m when the stock removal was 0.025 mm, but it reached 2.55 m as the stock removal was increased to 0.125 mm. Finally, as shown in Fig. 12(c), the workpiece roundness improved from 6.052 m to 3.038 m as the workpiece rotational speed nw increased from 0.9 s−1 to 3.5 s−1 , meaning a better roundness can be obtained when a higher voltage is applied to the PZT (see Table 3). The experimental results shown in Fig. 11 indicate that a higher workpiece rotational speed (via a larger value of Vp–p ), a larger stock removal, and a lower worktable feed rate should be chosen to achieve a higher machining accuracy (i.e., better workpiece roundness). These findings are the same as those obtained in our previous work (Xu et al., 2009b) and agreed well with the simulation results. 5. Conclusions As a step for establishing the tangential-feed centerless grinding technique performed on a surface grinder, a simulation method, in which the elastic deformation of the grinding machine was taken into account with a machining-elasticity parameter, was developed to clarify workpiece rounding process and predict workpiece roundness during grinding. After proposing a practical way to determine the machining-elasticity parameter, the effects of the process parameters such as workpiece rotational speed, stock removal and workpiece feed rate on the workpiece roundness were investigated by means of a simulation method followed by experimental confirmation. The obtained simulation and experimental results can be summarized as follows. (1) The machining-elasticity parameter increases with the increasing stock removal, and its mean value in the current conditions was 0.084. (2) The workpiece rotational speed can be controlled exactly by the elliptic motion of the shoe, and the larger amplitudes of applied voltage could induce higher workpiece rotational speed. (3) A slower worktable feed rate and a faster workpiece rotational speed result in better roundness.
This research was financially supported in part by Grants-in-Aid for Science Research from the Japan Society for the Promotion of Science (Grant No. 17560100). The authors also gratefully acknowledge the financial support from the Osawa Scientific Studies Grants Foundation and the Machine Tool Engineering Foundation. References Albizuri, J., Fernandes, M.H., Garitaonandia, I., Sabalza, X., Uribe-Etxeberria, R., Hernández, J.M., 2007. An active system of reduction of vibrations in a centerless grinding machine using piezoelectric actuators. Int. J. Mach. Tools Manuf. 47, 1607–1614. Epureanu, B.I., Dowell, E.H., Montoya, F.M., 1997. Pattern formation and linear stability analysis in centerless grinding. Proc. Inst. Mech. Eng. B: J. Eng. Manuf. 211 (8), 619–626. Hashimoto, F., Suzuki, N., Kanai, A., Miyashita, M., 1982. Critical range of set-up conditions of centerless grinding and problem of safe machining operation. J. JSPE 48 (8), 996–1001 (in Japanese). Kenjo, N., Yubita, S., 1991. Primer of Ultrasonic Motor. Sougoudenshi Publication, Tokyo (in Japanese). Marinescu, I.D., Hitchiner, M., Uhlmann, E., Rowe, W.B., Inasaki, I., 2006. Handbook of Machining with Grinding Wheels. CRC, New York, pp. 521–522. Miyashita, M., Hashimoto, F., Kanai, A., 1982. Diagram for selecting chatter free conditions of centerless grinding. Ann. CIRP 31 (1), 221–223. Rowe, W.B., Barash, M.M., 1964. Computer method for investigating the inherent accuracy of centerless grinding. Int. J. Mach. Tool Des. Res. 4, 91–116. Rowe, W.B., Bell, W.F., 1986. Optimization studies in high removal rate centerless grinding. Ann. CIRP 35 (1), 235–238. Rowe, W.B., Barash, M.M., Koenigsberger, F., 1965. Some roundness characteristics of centerless grinding. Int. J. Mach. Tool Des. Res. 5, 203–215. Serope, K., 2001. Manufacturing Engineering and Technology, 4th ed. Prentice Hall, New Jersey, pp. 21–27. Ueha, S., Tomikawa, Y., 1991. New Ultrasonic Motor. Torikkepus Publication, Japan (in Japanese). Wu, Y., Syoji, K., Kuriyagawa, T., Tachibana, T., 1996. Studies on centerless grinding (2nd Report: optimum grinding conditions). J. JSPE 62 (3), 433–437 (in Japanese). Wu, Y., Fan, Y., Kato, M., Wang, J., Syoji, K., Kuriyagawa, T., 2003. A new centerless grinding technique without employing a regulating wheel. Key Eng. Mater. 238–239, 355–360. Wu, Y., Fan, Y., Kato, M., Kuriyagawa, T., Syoji, K., Tachibana, T., 2004a. Development of an ultrasonic elliptic vibration shoe centerless grinding technique. J. Mater. Process. Technol. 155–156, 1780–1787. Wu, Y., Fan, Y., Kato, M., Tachibana, T., Syoji, K., Kuriyagawa, T., 2004b. Determination of an optimum geometrical arrangement of workpiece in the ultrasonic ellipticvibration shoe centerless grinding. Key Eng. Mater. 257–258, 495–500. Wu, Y., Kondo, T., Kato, M., 2005. A new centerless grinding technique using a surface grinder. J. Mater. Process. Technol. 162–163, 709–717. Xu, W., Wu, Y., Sato, T., Lin, W., 2009a. Workpiece rotation motion in centerless grinding using surface grinder. In: Proceedings of MECJ-09, Morioka, Japan, pp. 259–260. Xu, W., Wu, Y., Sato, T., Liang, Z., Lin, W., 2009b. Experimental study of tangentialfeed centerless grinding process performed on surface grinder. Mater. Sci. Forum. 626–627, 17–22. Xu, W., Wu, Y., Sato, T., Lin, W., 2009c. Development of tangential-feed centerless grinding method using surface grinder. Trans. JSME-C 75 (757), 50–56 (in Japanese). Xu, W., Wu, Y., Sato, T., Lin, W., 2009d. A method to increase material removal rate in tangential-feed centerless grinding performed on surface grinder. In: Proceedings of the 5th International Conference on LEM21, Osaka, Japan, pp. 559–562. Yang, Y., Zhang, B., 1998. Design of a vacuum-hydrostatic shoe for centerless grinding. In: Technical Paper of the 26th NAMRC, pp. 113–118. Yang, Y., Zhang, B., Wang, J., 1999. Vacuum-preloaded hydrostatic shoe for centerless grinding. Ann. CIRP 48 (1), 269–272. Yonetsu, S., 1966. Centerless Grinding Technique. Nikkan kougyou shinbun-sha, Tokyo, pp. 3–5 (in Japanese). Zhang, B., Gan, Z., Yang, Y., Howes, T.D., 1999. Workholding stability in shoe centerless grinding. J. Manuf. Sci. Eng. T. ASME 121, 41–48. Zhang, H., Lieh, J., Yen, D., Song, X., Rui, X., 2003. Geometry analysis and simulation in shoe centerless grinding. J. Manuf. Sci. Eng. T. ASME 125, 304–309.
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Effects of process parameters on workpiece roundness in tangential-feed centerless grinding using a surface grinder W. Xu a,∗ , Y. Wu b , T. Sato b , W. Lin b a b
Graduate School, Akita Prefectural University, 84-4 Tsuchiya-ebinokuchi, Yirihonjo, Akita 015-0055, Japan Dept. of Machine Intelligence and Systems Engineering, Akita Prefectural University, 84-4 Tsuchiya-ebinokuchi, Yurihonjo, Akita 015-0055, Japan
a r t i c l e
i n f o
Article history: Received 19 October 2009 Received in revised form 5 January 2010 Accepted 8 January 2010
Keywords: Centerless grinding Surface grinder Ultrasonic vibration Roundness Shoe
a b s t r a c t The present authors proposed a new centerless grinding method using a surface grinder in their previous study [Wu, Y., Kondo, T., Kato, M., 2005. A new centerless grinding technique using a surface grinder. J. Mater. Process. Technol. 162–163, 709–717]. In this method, a compact centerless grinding unit composed mainly of an ultrasonic elliptic-vibration shoe is installed onto the worktable of a multipurpose surface grinder to perform tangential-feed centerless grinding operations. However, for the complete establishment of the new method it is crucial to clarify the workpiece rounding process and the effects of process parameters such as the worktable feed rate, the stock removal and the workpiece rotational speed on the machining accuracy, i.e., workpiece roundness, so that the optimum grinding conditions can be determined. In this paper, the effects of the process parameters on workpiece roundness are investigated by simulation and experiments. For the simulation analysis, a grinding model taking into account the elastic deformation of the machine is created. Then, a practical way to determine the machiningelasticity parameter is developed. Further, simulation analysis is carried out to predict the variation of workpiece roundness during grinding and to discover how the process parameters affect the roundness. Finally, actual grinding operations are performed by installing the previously constructed unit onto a CNC surface grinder to confirm the simulation results. The obtained results indicate that: (1) a slower worktable feed rate and higher workpiece rotational speed give better roundness; (2) better roundness can be also obtained when the stock removal is set at a larger value; (3) the workpiece roundness was improved from an initial value of 23.9 m to a final value of 0.84 m after grinding. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In the manufacturing industry, for high accuracy and high productivity machining of cylindrical components, such as bearing raceways, silicon ingots, pin gauges, crankshafts, ferrules and catheters, centerless grinding operations have been extensively carried out on specialized centerless grinders (Serope, 2001). Two types of centerless grinders are available commercially, one that uses a regulating wheel and one that uses a shoe, and these grinders are different in how they support the workpiece and control the workpiece rotational speed. Especially since the invention of the regulating wheel type centerless grinder by Heim in 1915 (Yonetsu, 1966), much research has been devoted to clarify the rounding process by Rowe et al. (1964, 1965), optimizing the machining conditions by Rowe and Bell (1986) and Wu et al. (1996), achieving stability of the workpiece during grinding by Miyashita et al. (1982),
∗ Corresponding author. Tel.: +81 184 272157; fax: +81 184 27 2165. E-mail addresses: [email protected], [email protected] (W. Xu). 0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2010.01.003
Epureanu et al. (1997) and Albizuri et al. (2007), and achieving safe machining operation by Hashimoto et al. (1982) for the sake of enhancing machining accuracy and efficiency. Shoe type centerless grinding has also attracted attention from both industrial and academic researchers. Yang and Zhang (1998) designed a flat vacuum-hydrostatic shoe to increase load capacity and stiffness for high precision applications of shoe centerless grinding. Then Yang et al. (1999) and Zhang et al. (1999) analyzed the process stability in vacuum-hydrostatic shoe centerless grinding. In addition, Zhang et al. (2003) developed a geometry model to predict the lobing generation in shoe centerless grinding and used the model for the analysis of grinding processes. From the viewpoint of production cost, the two types of centerless grinders are highly suitable for small-variety and largevolume production, because the loading/unloading of workpieces is extremely easy and fast. However, the centerless grinder is a special-purpose machine and is relatively costly, putting it at a disadvantage for large-variety and small-volume production, the demand for which has increased rapidly in recent years. As a solution to this problem, we proposed a new centerless grinding technique that can be performed on a multipurpose machine such
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as a surface grinder (rather than a centerless grinder) in our previous work (Wu et al., 2005). This method is based on the concept of ultrasonic-shoe centerless grinding, which has been studied by Wu et al. (2003, 2004a,b). In the method, a compact unit consisting mainly of an ultrasonic elliptic-vibration shoe, a blade, and their respective holders is installed on the worktable of a multipurpose surface grinder to conduct tangential-feed centerless grinding. The function of the ultrasonic elliptic-vibration shoe is to hold the cylindrical workpiece in conjunction with the blade, and to control the workpiece rotational speed with the elliptic motion on its upper end-face. In the previous study (Wu et al., 2005), an actually constructed unit was installed on the table of a CNC surface grinder, and tangential-feed centerless grinding operations were performed involving pin-shaped workpieces. For the complete establishment of this new method, it is crucial to clarify the effects of process parameters such as the feed rate of worktable, the stock removal and the workpiece rotational speed on the machining accuracy, i.e., workpiece roundness, so that the optimum grinding conditions can be confirmed. For this purpose, this paper presents a simulation method to predict the variation of workpiece roundness during tangential-feed centerless grinding performed on a surface grinder. In order to evaluate and discuss the effects of process parameters on workpiece roundness, simulation analysis was carried out followed by experimental confirmation on the previously constructed experimental rig.
2. Operation principle of tangential-feed centerless grinding performed on a surface grinder Fig. 1 schematically illustrates the operation principle of the new centerless grinding technique and the detailed construction of the experimental rig. The rig is established by installing a grinding unit, mainly composed of an ultrasonic elliptic-vibration shoe and its holder, a blade and its holder, and a base plate, onto the worktable of a surface grinder via an electromagnetic chuck. A workpiece constrained between the unit and the grinding wheel is fed in the direction of worktable movement at a feed rate of Vf . Once the workpiece interferes with the clockwise rotating grinding wheel, actual grinding action begins, and the grinding is finished when the ground workpiece loses contact with the grinding wheel with forward movement of the worktable. During grinding, the workpiece rotational speed nw is controlled by the elliptic motion on the upper end-face of the shoe, and the stock removal ı is determined by adjusting the gap between the grinding wheel and the
Fig. 1. Schematic illustration of tangential-feed centerless grinding performed on a surface grinder.
worktable. This method provides an alternative centerless grinding technique especially suited to large-variety and small-volume production at low cost, achieved by simply installing the compact centerless grinding unit on a surface grinder. As shown in Fig. 1, the shoe is fixed on its holder via a spacer (electric isolation), and the blade is held directly on its holder by means of bolts. The shoe and blade are fastened on the metal base plate so that the unit can be mounted on the worktable by means of an electromagnetic chuck. The shoe is constructed by bonding a piezoelectric ceramic device (PZT) with two separated electrodes onto a metal elastic body (stainless steel, SUS304). When two alternating current (AC) signals (over 20 kHz) with a phase difference of to each other generated by a wave function generator are applied to the PZT after being amplified by means of power amplifiers, bending and longitudinal ultrasonic vibration are excited simultaneously. The synthesis of vibration displacements in the two directions creates an elliptic motion on the end-faces of the metal elastic body. Consequently, the workpiece rotation is controlled by the frictional force between workpiece and shoe, so that the peripheral speed of the workpiece is maintained at the same value as the bending vibration speed on the shoe end-face (Xu et al., 2009a,b). The workpiece rotational speed can be adjusted by changing the value of parameters such as the amplitude Vp–p and frequency f of the voltage applied to the PZT (Ueha and Tomikawa, 1991, Kenjo and Yubita, 1991), because the shoe bending vibration speed varies with the variation of the applied voltage. In addition, a pre-load is applied to the shoe at its lower end-face in its longitudinal direction using a coil spring to prevent the PZT from breaking due to resonance.
Fig. 2. Geometrical arrangements in tangential-feed centerless grinding using a surface grinder: before grinding (a) and during grinding (b).
W. Xu et al. / Journal of Materials Processing Technology 210 (2010) 759–766
3. Simulation Fig. 2(a) and (b) shows the geometrical arrangements of the unit, the workpiece and the grinding wheel before grinding and during grinding, respectively. The shoe and the blade with a tilt angle of are fixed on the L-shaped worktable, and thus the relative position of the blade to the shoe is kept constant during grinding. The workpiece with an initial radius of 0 , constrained between the shoe, the blade and the grinding wheel, is fed forward on the worktable at a feed rate of Vf , and is ground at point A by the grinding wheel rotating at a speed of ng . In the meantime, the workpiece rotational speed nw is controlled by the elliptic motion on the shoe end-face.
Solving Eqs. (5) and (6) simultaneously yields the XYcoordinates of the workpiece center Owt at time t, as follows:
⎧ ⎨ ⎩
XC0 = XOw0 YC0 = YOw0 − 0
Y − YB0 = tan (X − XB0 )
(1)
Y − YC0 = 0
(2)
Substituting coordinates of points B and C into Eqs. (1) and (2), respectively, gives PX + QY + R = 0
(3)
Y − YOw0 + 0 = 0
(4)
where P = tan , Q = −1, R = YOw0 + 0 cos − tan (XOw0 − 0 sin ). During grinding (see Fig. 2(b)), the coordinates of the workpiece center Owt and the grinding wheel center Ogt will vary as the material is removed. Let the instantaneous workpiece radius in the direction parallel to the X-axis after grinding for time t be (t), and the workpiece radius at points A, B and C can be expressed with (t − TA ), (t − TB ), and (t − TC ), respectively, where TA = [ − 2˛(t)]/4nw , TB = ( + 2)/4nw and TC = 3/4nw are the time delays for points A, B and C. Since the (t − TB ) and (t − TC ) are equal to the distances from the Owt to the blade end-face and to the shoe upper end-face, respectively, they can be obtained as follows from the geometrical arrangement in Fig. 2(b) by using Eqs. (3) and (4). |PXOw (t) + QYOw (t) + R|
P2 + Q 2
(t − TC ) = YOw (t) − YOw0 + 0
(7)
XOg (t) = XOg0 − Vf t = XOw0 +
2(0 + Rg )ı − ı2 − Vf t
(8)
In addition, the following relationships are established from the geometrical arrangement in Fig. 2(b).
[XA (t) − XOg (t)]2 + [YA (t) − YOg (t)]2 = Rg2 YA (t) − YOw (t) = cot ˛(t)[XA (t) − XOw (t)]
(9)
where [XOg (t) − XOw (t)] [YOg (t) − YOw (t)]
tan ˛(t) =
(10)
Therefore, the XY-coordinates of point A are decided as follows by using Eqs. (9) and (10):
⎧ ⎨
(−V −
V 2 − 4UW ) 2U ⎩ YA (t) = cot ˛(t)[XA (t) − XOw (t)] + YOw (t) XA (t) =
(11)
where U = 1 + cot2 ˛(t), V =2[cot ˛(t)YOw (t)−cot2 ˛(t)XOw (t)−cot 2 2 (t)+[cot ˛(t)X ˛(t)YOg (t)−XOg (t)], W =XOg Ow (t)−YOw (t) + YOg (t)] − Rg2 .
Then, the linear equations representing the blade end-face and shoe upper surface in this coordinate system can be written as
(t − TB ) =
P 2 + Q 2 (t − TB ) − QYOw (t) − R P YOw (t) = (t − TC ) + YOw0 − 0
YOg (t) = YOg0 = YOw0 + 0 + Rg − ı
In this model, several assumptions are made: (1) the workpiece is in constant contact with the blade and the shoe at points B and C, respectively, during grinding; (2) the vibration of the entire machine is too small to be regarded, and no chatter occurs on the machine due to the ultrasonic elliptic-vibration of the shoe; (3) the workpiece rotation is always stable, and no variation of rotational speed during grinding occurs; (4) the wear of the grinding wheel is too small to be recognized, and the grinding wheel radius is kept constant during grinding. Let a XY-coordinate system be located on the worktable. An optional point O on the worktable is the origin of the coordinate system. The X-axis is taken in the horizontal direction and the Yaxis in vertical direction. Before grinding (see Fig. 2(a)), the initial XY-coordinates of the grinding wheel center Og0 and the workpiece center Ow0 are (XOg0 , YOg0 ) and (XOw0 , YOw0 ), respectively. Thus, the XY-coordinates of the initial blade contact point B (XB0 , YB0 ) and the shoe contact point C (XC0 , YC0 ) can be obtained from the geometrical arrangement in Fig. 2(a), as follows: XB0 = XOw0 − 0 sin YB0 = YOw0 + 0 cos
XOw (t) =
In this moment, the XY-coordinates of the grinding wheel center Ogt are obtained from the geometrical arrangement in Fig. 2(b) as
3.1. Simulation model
761
(5) (6)
Eventually, the workpiece radius (t − TA ) at point A after grinding for time t is obtained from the XY-coordinates of the workpiece center Ow and the grinding point A as follows: (t − TA ) =
[XA (t) − XOw (t)]2 + [YA (t) − YOw (t)]2
(12)
The apparent wheel depth of cut would be = (t − TA − T) − (t − TA ), where T is the time required for one revolution of the workpiece. If the grinding system has an ideal stiffness, the true wheel depth of cut would be equal to the apparent wheel depth of cut. However, our previous work (Xu et al., 2009c), in which the grinding model for simulation analysis was developed under the ideal stiffness, revealed that the simulation results do not agree well with the actual results obtained experimentally sometime. This is because the grinding system withstands the elastic deformation caused by the grinding force during actual grinding. Rowe et al. introduced a dimensionless parameter called machining-elasticity parameter k as a measure to indicate the elastic deformation of centerless grinding system which is defined as a quotient between the true depth of cut and the apparent depth of cut with Eq. (13) (Rowe and Barash, 1964; Marinescu et al., 2006). k=
true wheel depth of cut = apparent wheel depth of cut
(13)
Also Rowe et al. pointed out that the parameter k depends on the cutting stiffness Km and the whole grinding system static stiffness Ke in a relationship of k = Ke /(Km + Ke ). Following Rowe et al.’s consideration, the true wheel depth of cut can be calculated as = k in the current work, resulting in the true workpiece radius
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Fig. 4. A result of grinding force measurement.
Solving Eq. (16) yields
Fig. 3. Apparent radius and true radius after workpiece rotates for i revolutions.
i =
at point A is
i−1
= (t − TA − T ) − k{(t − TA − T )
[XA (t) − XOw (t)]2 + [YA (t) − YOw (t)]2 }
(14)
However, the wheel depth of cut calculated using these equations is less than zero occasionally. Obviously this phenomenon would not happen. Therefore, Eq. (14) should be modified as
(t − TA ) = (t − TA − T ) − k((t − TA − T ) −
− i = a − bi
1 ı 2
(17)
(18)
a=2
where
ı(2Rg + 20 − ı)Vf nw + Vf2 /2n2w (2Rg + 20 −
+ 20 − ı). ı), b = Therefore, according to the definition of the machiningelasticity parameter of k = (i−1 − i )/(i−1 − i ), the true radius Vf2 /n2w (2Rg
[XA (t) − XOw (t)]2 + [YA (t) − YOw (t)]2 )
(t − TA ) ≤ (t − TA − T )
(15)
(t − TA ) > (t − TA − T )
N of the workpiece after grinding for N revolutions can be obtained as follows:
3.2. Determination of the machining-elasticity parameter As described above, the machining-elasticity parameter k depends on the stiffness of the grinding system. If the simulation result is to be trusted, the value of k should be determined for the given grinding system. Rowe et al. proposed a method for the determination of machining-elasticity parameter k in conventional in-feed centerless grinding (Rowe et al., 1965), in which a parameter proportional to the true wheel depth of cut, i.e., the grinding power, during in-feed or spark-out was measured to obtain the parameter k. However, this method is available only for in-feed centerless grinding with a constant apparent depth of cut generated by the uniform in-feed movement of grinding wheel, and thus not suitable for the tangential-feed centerless grinding being dealt with in our current work because the apparent depth of cut always varies during grinding (see Xu et al., 2009d). Therefore, an alternative procedure to determine the value of k is proposed as below. Fig. 3 shows the apparent workpiece radius i and the true radius i after the workpiece is rotated for i revolutions, during which time the workpiece moves for distance Si . SN is the distance that the workpiece moved from its initial position to the position where its center is right below the grinding wheel center. In this duration, the workpiece rotates for N revolutions. Thus, based on the geometrical arrangement shown in Fig. 3, the following relationship is obtained: (Rg + i ) = (SN − Si )2 + (Rg + 20 − i − ı) 2
Si = Vf i/nw
+ 0 −
4Rg + 40 − 2ı
(t − TA ) = (t − TA − T )
where
2
Subsequently, using Eq. (17) yields
(t − TA ) = (t − TA − T ) −
−
ı(2Rg + 20 − ı) − Vf i/nw
2
(i = 1, 2, . . . , N) 2
(Rg + 0 )2 − (Rg + 0 − ı) =
(16) and
ı(2Rg + 20 − ı).
SN =
N
N−1 1
= (1 − k) 0 + k(1 − k)
N
N−i i
+ . . . + k(1 − k)
+ . . . + kN N
= (1 − k) (0 − 1 ) + (1 − k) + (1 − k)(N−1
) + − N N
N
= a[(1 − k) + (1 − k) + 2(1 − k)
N−1
N−1
N−1
(1 − 2 ) + . . .
(19)
(0 = 0 ) N
+ . . . + (1 − k)] − b[(1 − k)
+ . . . + N(1 − k)] + N
= Because of 0 < k < 1 and N 1 in Eq. (19), and the relation of N 0 − ı/2 is obtained by substituting N for i in Eq. (16), the value of N is approximately calculated with Eq. (20).
N = (a − Nb − 2b)
k2 − 1 1−k ı −b + 0 − 2 k k2
(20)
Further, solving Eq. (20) yields
k=
−F +
F 2 − 4EG
2E
(21)
where E = N − 0 + ı/2 + a − Nb − b, F = −a + Nb + 2b, G = −b. Consequently, the value of k can be obtained using Eq. (21), as long as the parameters Rg , ı, nw , Vf , 0 , N and N are known. In an actual grinding operation, the values of Rg , ı, nw , Vf and 0 are set up as the grinding conditions. In addition, N can be known by measuring the diameter of the workpiece ground for N revolutions. Since N is determined using the relation of N = nw TN where TN is the time that the workpiece moves for distance SN (see Fig. 3), its value can be obtained as long as TN is known. In this paper, TN is
W. Xu et al. / Journal of Materials Processing Technology 210 (2010) 759–766
763
ing its peak, until the grinding has been performed for time TN , and eventually the grinding action finishes in time Tg after spark-out for time Tg − TN . Although the actual grinding action happened during the time Tg = 75.34 s, the time TN = 73.85 s was just that the workpiece moved for the distance SN . Hence, the value of N in the current conditions was N = 1.2 × 73.85 = 88.62. In addition, the workpiece radius was N = 2.450 mm after grinding. Subsequently, the value of k under the current conditions was k = 0.089. Changing the workpiece stock removal ı but keeping the other parameters constant yielded the results shown in Fig. 5. It is found that k increased gradually as the stock removal increased, and its mean value was around 0.084. Fig. 5. Measuring results of parameter k.
measured based on the variation of grinding force during grinding. For example, Fig. 4 shows a typical variation of grinding force obtained in an actual grinding operation conducted on the previously constructed experimental rig (see Wu et al., 2005) under the conditions of Rg = 90 mm, ı = 0.1 mm, nw = 1.2 s−1 , Vf = 0.05 mm/s and 0 = 2.499 mm. It can be seen that the grinding force increases rapidly in the beginning and then decreases gradually after reach-
3.3. Simulation procedure The initial workpiece with a diameter of 5 mm is shown in Fig. 6(a). A flat with depth of 25 m in radial direction is generated on its circumference to indicate the initial roundness. In simulation, the workpiece was divided equally into 360 sections along its circumference, as shown in Fig. 6(b), and thus the initial workpiece profile could be represented with 360 radius of (i) (i = 1–360). In any given time t during grinding, as long as (t − TB ) and (t − TC )
Fig. 6. Initial workpiece profile (a), division of workpiece (b), and calculation flowchart (c) for simulation analysis.
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Table 1 Grinding conditions for simulation. Workpiece initial radius 0 (mm) Workpiece rotational speed nw (s−1 ) Workpiece initial roundness (m) Worktable feed rate Vf (mm/s) Stock removal ı (mm) Blade angle (◦ ) Grinding wheel radius Rg (mm) Machining-elasticity parameter k
2.5 0.1–3.2 25 0.03–0.15 0.25–0.125 60 90 0.084
Fig. 9. The main portion of the grinding apparatus.
Fig. 7. Variation of roundness during grinding.
are known, the instantaneous radius of (t − TA ) can be calculated using the basic Eqs. (1)–(15). Following this procedure, every 360 of workpiece radius is obtained and used to draw the profile and calculate the roundness of the workpiece. The simulation flowchart is shown in Fig. 6(c).
ally, some new comparatively small convex and concave areas are formed at the initial stage, but the following grinding process decreases the height of these areas, which eventually improves the workpiece roundness. Fig. 8(a)–(c) shows the effects of the worktable feed rate Vf , the workpiece rotational speed nw and the stock removal ı on the final roundness, respectively. It is obvious that the workpiece roundness increases with the increasing worktable feed rate Vf , but decreases as the workpiece rotational speed nw and the stock removal ı increase. From the simulation results, it is clear that a better final roundness needs a higher workpiece rotational speed, a larger stock removal, and a lower worktable feed rate.
3.4. Simulation results 4. Experiments Table 1 shows the grinding conditions for the simulation analysis. Fig. 7 shows the variation of the workpiece roundness and some of the workpiece profiles during grinding obtained under Vf = 0.067 mm/s, ı = 0.075 mm, nw = 1.2 s−1 , k = 0.084. It is obvious that as the size of the initial flat indentation decreases gradu-
In order to confirm the simulation results, the centerless grinding unit constructed in the work (Wu et al., 2005) was installed on the worktable of a CNC surface grinder (SGT-315RPA by Nagase Integrex Co., Ltd.) (see Fig. 9). The experimental details to be
Fig. 8. Effects of worktable feed rate Vf (a), stock removal ı (b), and workpiece rotational speed nw (c) on the final workpiece roundness.
W. Xu et al. / Journal of Materials Processing Technology 210 (2010) 759–766 Table 2 Grinding conditions. Grinding wheel Workpiece (mm) Input voltage amplitude Vp–p (V) Input voltage frequency f (kHz) Input voltage phase difference (◦ ) Grinding wheel speed (m/s) Worktable feed rate Vf (mm/s) Stock removal ı (mm) Blade angle (◦ ) Pre-load of spring (N)
SDC400N180 × 15 × 75 TH10, 5 × L13 30–120 46.5 90 20 0.03–0.15 0.025–0.125 60 15
described below were designed based on our previous work that dealt with the workpiece rotation control characteristics and the effects of grinding parameters on workpiece roundness (Xu et al., 2009b). 4.1. Experimental conditions and procedure The grinding experiments were carried out involving the workpiece as shown in Fig. 6(a), and its dimension was the same as that used in simulation, and a flat indentation of 25 m in radial direction was generated by surface grinding to indicate the initial roundness. In order to maintain the stability of workpiece, the width of workpiece must be smaller than that of grinding wheel. The other grinding conditions are tabulated in Table 2. In addition, since the workpiece rotational speed depends on the AC voltage applied to the PZT, the relationship between the applied voltage and the workpiece rotational speed should be obtained during an actual grinding operation. For this purpose, the rotational motion
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of the workpiece end-face was recorded by using a high-speed digital camera (VW-6000 by Keyence Co., Ltd.) in the experiments, and the workpiece rotational speed was obtained from the recorded workpiece end-face motion using an animated image-processing software (Movie Editor by Keyence Co., Ltd.). The grinding procedure was as follows: first, the unit constraining a workpiece was fastened on the worktable, and its position was adjusted carefully so that the unit was at the lower left side of the grinding wheel. Next, the stock removal was given by adjusting the gap between the grinding wheel and the worktable. The worktable was moved forward to carry the workpiece toward the grinding wheel. Once the workpiece passed through and reached right-side of grinding wheel, the grinding operation was completed. In experiments, the grinding operations under the same grinding conditions were carried out on more than three workpieces. After grinding, the workpiece roundness was measured with a roundness measurement instrument (Rondcom55A by Tokyo Seimitsu Co., Ltd.) at five different cross-sections, and the average value of the five measurements was used to indicate the workpiece roundness. Changing the worktable feed rate, workpiece rotational speed and stock removal, and repeating the above-mentioned procedures yielded the workpiece roundness for different grinding conditions. 4.2. Experimental results and discussion Table 3 shows the relationship between the workpiece rotational speed nw and the AC voltage Vp–p applied to the PZT under the conditions of f = 46.5 kHz, = 90◦ , ı = 0.075 mm, Vf = 0.067 mm/s. As can be seen, the workpiece rotational speed increases
Fig. 10. Photos and profiles of the workpiece before grinding (a) and after grinding (b) under the conditions of f = 46.5 kHz,
= 90◦ , Vp–p = 50 V, ı = 0.075 mm and Vf = 0.033 mm/s.
Fig. 11. Effects of worktable feed rate Vf (a), stock removal ı (b), and workpiece rotational speed nw (c) on the final roundness.
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Table 3 Relationship between the workpiece rotational speed nw and the applied volt= 90◦ , ı = 0.075 mm, age Vp–p under the grinding conditions of f = 46.5 kHz, Vf = 0.067 mm/s. Vp–p (V) nw (s−1 )
30 0.9
50 1.5
70 1.8
90 2.2
100 2.5
120 3.5
(4) Better roundness can be obtained when the stock removal is set at a larger value. (5) The workpiece roundness was improved from an initial value of 23.9 m to a final value of 0.84 m after grinding under the current conditions. Acknowledgements
monotonously with the increasing amplitude of applied voltage. This means that the required workpiece rotational speed can be given exactly by applying the corresponding voltage to the PZT. Fig. 10(a) and (b) shows the photographs and cross-section profiles of the workpiece before grinding and after grinding under the grinding conditions of f = 46.5 kHz, = 90◦ , Vp–p = 50 V, ı = 0.075 mm, Vf = 0.033 mm/s, respectively. The workpiece roundness was improved greatly from the initial value of 23.95 m to the final one of 0.84 m. The roundness results obtained for different worktable feed rates Vf , stock removals ı and amplitudes of voltage Vp–p applied to the PZT are shown in Fig. 11(a), (b) and (c), respectively. It can be seen from Fig. 11(a) that the worktable feed rate affects the workpiece roundness significantly. The roundness was improved greatly from 13.13 m to 0.84 m as long as the worktable feed rate was changed from 0.15 mm/s to 0.033 mm/s. Fig. 11(b) shows that the stock removal ı affects the roundness significantly. The workpiece roundness was 16.502 m when the stock removal was 0.025 mm, but it reached 2.55 m as the stock removal was increased to 0.125 mm. Finally, as shown in Fig. 12(c), the workpiece roundness improved from 6.052 m to 3.038 m as the workpiece rotational speed nw increased from 0.9 s−1 to 3.5 s−1 , meaning a better roundness can be obtained when a higher voltage is applied to the PZT (see Table 3). The experimental results shown in Fig. 11 indicate that a higher workpiece rotational speed (via a larger value of Vp–p ), a larger stock removal, and a lower worktable feed rate should be chosen to achieve a higher machining accuracy (i.e., better workpiece roundness). These findings are the same as those obtained in our previous work (Xu et al., 2009b) and agreed well with the simulation results. 5. Conclusions As a step for establishing the tangential-feed centerless grinding technique performed on a surface grinder, a simulation method, in which the elastic deformation of the grinding machine was taken into account with a machining-elasticity parameter, was developed to clarify workpiece rounding process and predict workpiece roundness during grinding. After proposing a practical way to determine the machining-elasticity parameter, the effects of the process parameters such as workpiece rotational speed, stock removal and workpiece feed rate on the workpiece roundness were investigated by means of a simulation method followed by experimental confirmation. The obtained simulation and experimental results can be summarized as follows. (1) The machining-elasticity parameter increases with the increasing stock removal, and its mean value in the current conditions was 0.084. (2) The workpiece rotational speed can be controlled exactly by the elliptic motion of the shoe, and the larger amplitudes of applied voltage could induce higher workpiece rotational speed. (3) A slower worktable feed rate and a faster workpiece rotational speed result in better roundness.
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