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The effect of variation in the curvature of part on the formability in incremental forming: An experimental investigation
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181 www.elsevier.com/locate/ijmactool
Short communication
The effect of variation in the curvature of part on the formability in incremental forming: An experimental investigation G. Hussain, L. Gao, N. Hayat, L. Qijian College of Mechanical Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 14 December 2006; received in revised form 2 May 2007; accepted 8 May 2007 Available online 24 May 2007
Abstract The maximum wall angle or thinning limit can be used as formability parameter in negative incremental forming. In the present study, the effect of the curvature of a part’s generatrix on the formability of an aluminum sheet has been investigated systematically. It was found that the formability increases as the radius of curvature decreases. r 2007 Elsevier Ltd. All rights reserved. Keywords: Negative incremental forming; Maximum wall angle; Generatrix; Radius of curvature
1. Introduction Due to changes in the pattern of customer’s demand, the small-batch production is getting increased attention in the manufacturing industry. Because of the low-cost tooling and flexibility, single-point incremental forming (SPIF) offers a promising alternative to produce customized small batches. In this process, the final component shape is formed by the relative movement of the tool with respect to the blank rather than the die shape. Though SPIF is being investigated extensively [1–4], yet the process is not mature enough to be employed for industrial applications. In order to use it at a large scale, some aspects of the process need to be further clarified. The present work is an extension of the previous work carried out by Hussain and Gao [5]. Before moving forward, a brief account of the two tests conducted in the aforementioned work is summarized as follows: In the first test, a conical frustum having varying wall angle was formed to fracture. While, in the second test, cones and pyramids, a set of each, with a variety of wall angles were formed.
Corresponding author. Tel.: +86 13675161625.
E-mail address: [email protected] (G. Hussain). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.05.001
According to the results, the first test showed higher formability than the second test indicating that the curvature of the test specimen affects the formability [5]. Strano et al. [6] have partially investigated the effect of curvature of directrix, and reported that the formability remains unaffected. However, no work has been reported regarding the effect of curvature of generatrix on the formability. The present investigation is an attempt in this direction. In order to quantify the formability, it has been defined as the maximum wall angle (ymax) that a sheet would endure without fracturing. The current work was carried out by employing varying wall angle conical frustum (VWACF) test as reported in Hussain and Gao [5]. 2. Design of test specimens In order to investigate the effect of curvature on ymax, the radius of curvature (R) was varied from 115 to 1000 mm in order to generate part geometry from generatrix, as shown in Table 1. The included angles of the generatices were varied so as to keep the blank size constant. Therefore, the major diameter (D1) of the generated surfaces remained unchanged, while the minor diameter (D2) and depth (h) of each surface underwent change. Fig. 1 presents an example of a surface generated by an arbitrary generatrix. It is obvious from the figure that the
ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
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Table 1 Design data of the test specimens having various radii of curvature of generatrix Generatrix
Test specimen
R (mm)a
P1 ðx1 ; y1 Þ (mm)b
P2 ðx2 ; y2 Þ (mm)c
y1 (deg.)d
y2 (deg.)e
h (mm)f
D1 (mm)g
D2 (mm)h
115 200 300 500 750 1000
73.94, 88.0788 159.83, 120.2263 257.27, 154.3119 445.6, 226.8053 677.25, 322.2304 906.8, 421.5611
113.26, 19.9292 193.23, 51.5962 286.97, 87.4541 475.6, 154.2875 707, 250.3018 936.9, 349.5975
40 53 59 63 64.5 65
80 75 73 72 70.5 69.5
68.15 68.63 66.85 72.51 71.92 72
110 110 110 110 110 110
32 43.7 51 50.5 51 50.3
a
Radius of curvature of generatrix. Initial point of generatrix. c End point of generatrix. d Slope angle on the initial point of generatrix. e Slope angle on the end point of generatrix. f Designed depth of test specimen. g Major diameter of the test specimen. h Minor diameter of the test specimen. b
Fig. 1. Illustration of generatrix, generated surface and terminology.
wall angle continuously varies along the generatrix from point P1 ðx1 ; y1 Þ to P2 ðx2 ; y2 Þ, thus inducing a corresponding change in the wall thickness of the specimen (see Ref. [5] for details). Due to these variations in thickness, a fracture on a point Dðxd ; yd Þ could occur whenever the thinning limit is surpassed. The instantaneous wall angle (yp) and thickness (tp) on an arbitrary point Pðxp ; yp Þ can be computed as derived in [5] and presented below: y p 1 1 y1 hp yp ¼ cos ¼ cos , (1) R R tp ¼ t0 cos y ¼
t0 ðy hp Þ. R 1
(2)
3. Experimental procedure Under the same forming conditions as employed in [5], six test specimens (conical frustums) generated by gener-
atrices having various radii of curvature were formed to fracture. A few representatives of these specimens are shown in Fig. 2. In order to check the variation in results, four specimens with each generatrix were formed. Using a depth gauge (0.01 mm least count), a number of points on each specimen were marked close to the fracture point. While, the actual thickness on each marked point was measured with a dial gauge indicator having least count of 0.001 mm (the dial gauge was calibrated before it was used). Fig. 3 depicts the thickness profile of the specimen having 115 mm radius of curvature of generatrix. The analytical thickness, predicted by Eq. (2), and the actual thickness measured directly from the specimen are in accordance on a transition point, i.e., Cðxc ; yc Þ. The wall angle on this transition point has been regarded as ymax. The detailed data (average values) of the critical points, i.e., transition and fracture points, for each specimen have been presented in Table 2. The maximum dispersion in the data was found to be 70.15%. 4. Results and discussion Fig. 4 shows that the value of ymax(varying) decreases as the radius (R) of curvature of generatrix increases. The rate of decrease in ymax(varying) is higher from 115 to 500 mm. From 500 mm onward, the curve shows a steady decrease with an increase in the radius. It means that as the radius approaches to infinity, ymax(varying) will show a trend similar to that as shown by ymax(constant) obtained from the specimens having constant wall angles formed in the previous work [5]. At the minimum radius of curvature (115 mm), the formability, i.e., ymax(varying), is about 4% higher than ymax(constant). Moreover, the variation in ymax(varying) is about 2% within the range of investigated radii of curvature. It can be said that the radius of curvature of generatrix affects the formability.
ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
Fig. 2. Some of the test specimens formed for the present investigation.
Fig. 3. Thickness profile of the test specimen having 115 mm radius of curvature of generatrix.
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ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
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Table 2 Average data of the transition and fracture points of the test specimens having various radii of curvature of generatrix R (mm) Depth of specimen measured to Theoretical and experimental thickness values of a critical point specimen on fracture point Dðxd ; yd Þ
115 200 300 500 750 1000
Theoretical and experimental thickness values of specimen on transition point Cðxc ; yc Þ
hd (mm)a
hc (mm)b
td(th) (mm)c
td(exp) (mm)d
tc(th) (mm)e
tc(exp) (mm)f
50.29 53.07 49.95 50.06 51.29 51.81
46.29 47.08 42.46 38.8 36.2 36.36
0.299 0.305 0.316 0.322 0.328 0.336
0.261 0.272 0.285 0.283 0.299 0.303
0.334 0.336 0.342 0.346 0.350 0.353
0.334 0.336 0.342 0.345 0.350 0.354
a
Depth of specimen measured to fracture point Dðxd ; yd Þ. Depth of specimen measured to transition point Cðxc ; yc Þ. c Theoretical thickness of specimen on Dðxd ; yd Þ. d Experimental thickness of specimen on Dðxd ; yd Þ. e Theoretical thickness of specimen on Cðxc ; yc Þ. f Experimental thickness of specimen on Cðxc ; yc Þ. b
defined as the limiting angle curve (LAC), as indicated in Fig. 4. The general form of the empirical model may be written as ymax ðvaryingÞ ¼ cRl
ðfor 0oRo8Þ,
where c is a constant and l the rate. 5. Conclusions The effect of radius (R) of curvature of the generatrix of the test specimen was investigated in the present study. An empirical model describing the effect of change in R was developed and validated as well. The current investigation has laid down a direction following which the limiting angle curves for other materials of interest can also be determined. The following important conclusions can be drawn from the study:
Fig. 4. Effect of the radius of curvature of generatrix on the formability: empirical model and limiting angle curve.
For 115pRp1000, ymax(varying) is governed by the negative-exponential empirical model, which has been presented in Fig. 4. The validity of the model was cross checked by forming two specimens based on any two radii, say 150 and 850 mm. For the selected radii, ymax(varying) was predicted with the help of the empirical model. The design data of the specimens formed for the aforesaid purpose is given in Table 3. In the Table, 68.41 and 67.31 are the predicted values of ymax(varying), which were selected as the slopes on the end points of the generatrices, i.e., circular arcs. Both of these specimens were successfully formed without fracture. This result validates the empirical model. Therefore, the curve of the empirical model may be
1) The formability, i.e., ymax, of a sheet metal shows a slight dependence on the R value of the test specimen. 2) For the small values of R (up to 500 mm), the formability increases sharply as the value of R decreases. However, when the value of R approaches towards infinity, its effect on the formability diminishes accordingly. 3) At the minimum value of R, i.e., 115 mm, the formability shown by the test specimen having varying wall angle was found to be 4% higher than that shown by a set of specimens formed with various constant wall angles. While, at the maximum value of R, i.e., 1000 mm, 1.7% increase in the formability was found. 4) The empirical model developed in the current work was validated. Based on this model, the limiting angle curve was defined. The curve followed a negative-exponential trend. Using this curve, the formability of the experimental material can be predicted without conducting test in the investigated range of R.
ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
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Table 3 Design data of the specimens formed to validate the empirical model Generatrix
Specimen
R (mm)
P1 ðx1 ; y1 Þ (mm)
P2 ðx2 ; y2 Þ (mm)
y1 (deg.)
y2 ¼ ymax ðvaryingÞ (deg.)a
h (mm)
D1 (mm)
D2 (mm)
150 850
96.5, 114.8379 748, 403.7276
139.48, 55.1857 784.5, 327.199
40 61.6
68.4 67.3
59.7 68.63
110 110
24 37.6
a Slope angle on the end point of generatrix. This is the maxiumum wall angle, predicted by the empirical model, that the specimen was expected to endure without fracturing.
References [1] T. Maki, Sheet Fluid Forming and Sheet Die-less NC Forming, Amino Corporation, Japan. [2] S. Matsubara, Incremental backward bulge forming of a sheet metal with a hemispherical head tool, Journal of Society For Technology of Plasticity 35 (1994) 1311. [3] J. Jeswiet, Incremental single point forming, in: The Proceedings of Conference of the North American Manufacturing Research Institution (NAMRI), Gainesville, Florida, 2001, p. MF01-246.
[4] J.-J. Park, Y.H. Kim, Fundamental studies on the incremental sheet metal forming technique, Journal of Materials Processing Technology 140 (2003) 447–453. [5] G. Hussain, L. Gao, A novel method to test the thinning limits of sheet-metals in negative incremental forming, International Journal of Machine Tools and Manufacture 47 (2007) 419–435. [6] M. Strano, L. Carrino, M. Ruggiero, Representation of forming limits for negative incremental forming of thin sheet metals, in: The Proceedings of IDDRG, Siendelfingen, Maggio, 2004, pp. 198–207.
International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181 www.elsevier.com/locate/ijmactool
Short communication
The effect of variation in the curvature of part on the formability in incremental forming: An experimental investigation G. Hussain, L. Gao, N. Hayat, L. Qijian College of Mechanical Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 14 December 2006; received in revised form 2 May 2007; accepted 8 May 2007 Available online 24 May 2007
Abstract The maximum wall angle or thinning limit can be used as formability parameter in negative incremental forming. In the present study, the effect of the curvature of a part’s generatrix on the formability of an aluminum sheet has been investigated systematically. It was found that the formability increases as the radius of curvature decreases. r 2007 Elsevier Ltd. All rights reserved. Keywords: Negative incremental forming; Maximum wall angle; Generatrix; Radius of curvature
1. Introduction Due to changes in the pattern of customer’s demand, the small-batch production is getting increased attention in the manufacturing industry. Because of the low-cost tooling and flexibility, single-point incremental forming (SPIF) offers a promising alternative to produce customized small batches. In this process, the final component shape is formed by the relative movement of the tool with respect to the blank rather than the die shape. Though SPIF is being investigated extensively [1–4], yet the process is not mature enough to be employed for industrial applications. In order to use it at a large scale, some aspects of the process need to be further clarified. The present work is an extension of the previous work carried out by Hussain and Gao [5]. Before moving forward, a brief account of the two tests conducted in the aforementioned work is summarized as follows: In the first test, a conical frustum having varying wall angle was formed to fracture. While, in the second test, cones and pyramids, a set of each, with a variety of wall angles were formed.
Corresponding author. Tel.: +86 13675161625.
E-mail address: [email protected] (G. Hussain). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.05.001
According to the results, the first test showed higher formability than the second test indicating that the curvature of the test specimen affects the formability [5]. Strano et al. [6] have partially investigated the effect of curvature of directrix, and reported that the formability remains unaffected. However, no work has been reported regarding the effect of curvature of generatrix on the formability. The present investigation is an attempt in this direction. In order to quantify the formability, it has been defined as the maximum wall angle (ymax) that a sheet would endure without fracturing. The current work was carried out by employing varying wall angle conical frustum (VWACF) test as reported in Hussain and Gao [5]. 2. Design of test specimens In order to investigate the effect of curvature on ymax, the radius of curvature (R) was varied from 115 to 1000 mm in order to generate part geometry from generatrix, as shown in Table 1. The included angles of the generatices were varied so as to keep the blank size constant. Therefore, the major diameter (D1) of the generated surfaces remained unchanged, while the minor diameter (D2) and depth (h) of each surface underwent change. Fig. 1 presents an example of a surface generated by an arbitrary generatrix. It is obvious from the figure that the
ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
2178
Table 1 Design data of the test specimens having various radii of curvature of generatrix Generatrix
Test specimen
R (mm)a
P1 ðx1 ; y1 Þ (mm)b
P2 ðx2 ; y2 Þ (mm)c
y1 (deg.)d
y2 (deg.)e
h (mm)f
D1 (mm)g
D2 (mm)h
115 200 300 500 750 1000
73.94, 88.0788 159.83, 120.2263 257.27, 154.3119 445.6, 226.8053 677.25, 322.2304 906.8, 421.5611
113.26, 19.9292 193.23, 51.5962 286.97, 87.4541 475.6, 154.2875 707, 250.3018 936.9, 349.5975
40 53 59 63 64.5 65
80 75 73 72 70.5 69.5
68.15 68.63 66.85 72.51 71.92 72
110 110 110 110 110 110
32 43.7 51 50.5 51 50.3
a
Radius of curvature of generatrix. Initial point of generatrix. c End point of generatrix. d Slope angle on the initial point of generatrix. e Slope angle on the end point of generatrix. f Designed depth of test specimen. g Major diameter of the test specimen. h Minor diameter of the test specimen. b
Fig. 1. Illustration of generatrix, generated surface and terminology.
wall angle continuously varies along the generatrix from point P1 ðx1 ; y1 Þ to P2 ðx2 ; y2 Þ, thus inducing a corresponding change in the wall thickness of the specimen (see Ref. [5] for details). Due to these variations in thickness, a fracture on a point Dðxd ; yd Þ could occur whenever the thinning limit is surpassed. The instantaneous wall angle (yp) and thickness (tp) on an arbitrary point Pðxp ; yp Þ can be computed as derived in [5] and presented below: y p 1 1 y1 hp yp ¼ cos ¼ cos , (1) R R tp ¼ t0 cos y ¼
t0 ðy hp Þ. R 1
(2)
3. Experimental procedure Under the same forming conditions as employed in [5], six test specimens (conical frustums) generated by gener-
atrices having various radii of curvature were formed to fracture. A few representatives of these specimens are shown in Fig. 2. In order to check the variation in results, four specimens with each generatrix were formed. Using a depth gauge (0.01 mm least count), a number of points on each specimen were marked close to the fracture point. While, the actual thickness on each marked point was measured with a dial gauge indicator having least count of 0.001 mm (the dial gauge was calibrated before it was used). Fig. 3 depicts the thickness profile of the specimen having 115 mm radius of curvature of generatrix. The analytical thickness, predicted by Eq. (2), and the actual thickness measured directly from the specimen are in accordance on a transition point, i.e., Cðxc ; yc Þ. The wall angle on this transition point has been regarded as ymax. The detailed data (average values) of the critical points, i.e., transition and fracture points, for each specimen have been presented in Table 2. The maximum dispersion in the data was found to be 70.15%. 4. Results and discussion Fig. 4 shows that the value of ymax(varying) decreases as the radius (R) of curvature of generatrix increases. The rate of decrease in ymax(varying) is higher from 115 to 500 mm. From 500 mm onward, the curve shows a steady decrease with an increase in the radius. It means that as the radius approaches to infinity, ymax(varying) will show a trend similar to that as shown by ymax(constant) obtained from the specimens having constant wall angles formed in the previous work [5]. At the minimum radius of curvature (115 mm), the formability, i.e., ymax(varying), is about 4% higher than ymax(constant). Moreover, the variation in ymax(varying) is about 2% within the range of investigated radii of curvature. It can be said that the radius of curvature of generatrix affects the formability.
ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
Fig. 2. Some of the test specimens formed for the present investigation.
Fig. 3. Thickness profile of the test specimen having 115 mm radius of curvature of generatrix.
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ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
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Table 2 Average data of the transition and fracture points of the test specimens having various radii of curvature of generatrix R (mm) Depth of specimen measured to Theoretical and experimental thickness values of a critical point specimen on fracture point Dðxd ; yd Þ
115 200 300 500 750 1000
Theoretical and experimental thickness values of specimen on transition point Cðxc ; yc Þ
hd (mm)a
hc (mm)b
td(th) (mm)c
td(exp) (mm)d
tc(th) (mm)e
tc(exp) (mm)f
50.29 53.07 49.95 50.06 51.29 51.81
46.29 47.08 42.46 38.8 36.2 36.36
0.299 0.305 0.316 0.322 0.328 0.336
0.261 0.272 0.285 0.283 0.299 0.303
0.334 0.336 0.342 0.346 0.350 0.353
0.334 0.336 0.342 0.345 0.350 0.354
a
Depth of specimen measured to fracture point Dðxd ; yd Þ. Depth of specimen measured to transition point Cðxc ; yc Þ. c Theoretical thickness of specimen on Dðxd ; yd Þ. d Experimental thickness of specimen on Dðxd ; yd Þ. e Theoretical thickness of specimen on Cðxc ; yc Þ. f Experimental thickness of specimen on Cðxc ; yc Þ. b
defined as the limiting angle curve (LAC), as indicated in Fig. 4. The general form of the empirical model may be written as ymax ðvaryingÞ ¼ cRl
ðfor 0oRo8Þ,
where c is a constant and l the rate. 5. Conclusions The effect of radius (R) of curvature of the generatrix of the test specimen was investigated in the present study. An empirical model describing the effect of change in R was developed and validated as well. The current investigation has laid down a direction following which the limiting angle curves for other materials of interest can also be determined. The following important conclusions can be drawn from the study:
Fig. 4. Effect of the radius of curvature of generatrix on the formability: empirical model and limiting angle curve.
For 115pRp1000, ymax(varying) is governed by the negative-exponential empirical model, which has been presented in Fig. 4. The validity of the model was cross checked by forming two specimens based on any two radii, say 150 and 850 mm. For the selected radii, ymax(varying) was predicted with the help of the empirical model. The design data of the specimens formed for the aforesaid purpose is given in Table 3. In the Table, 68.41 and 67.31 are the predicted values of ymax(varying), which were selected as the slopes on the end points of the generatrices, i.e., circular arcs. Both of these specimens were successfully formed without fracture. This result validates the empirical model. Therefore, the curve of the empirical model may be
1) The formability, i.e., ymax, of a sheet metal shows a slight dependence on the R value of the test specimen. 2) For the small values of R (up to 500 mm), the formability increases sharply as the value of R decreases. However, when the value of R approaches towards infinity, its effect on the formability diminishes accordingly. 3) At the minimum value of R, i.e., 115 mm, the formability shown by the test specimen having varying wall angle was found to be 4% higher than that shown by a set of specimens formed with various constant wall angles. While, at the maximum value of R, i.e., 1000 mm, 1.7% increase in the formability was found. 4) The empirical model developed in the current work was validated. Based on this model, the limiting angle curve was defined. The curve followed a negative-exponential trend. Using this curve, the formability of the experimental material can be predicted without conducting test in the investigated range of R.
ARTICLE IN PRESS G. Hussain et al. / International Journal of Machine Tools & Manufacture 47 (2007) 2177–2181
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Table 3 Design data of the specimens formed to validate the empirical model Generatrix
Specimen
R (mm)
P1 ðx1 ; y1 Þ (mm)
P2 ðx2 ; y2 Þ (mm)
y1 (deg.)
y2 ¼ ymax ðvaryingÞ (deg.)a
h (mm)
D1 (mm)
D2 (mm)
150 850
96.5, 114.8379 748, 403.7276
139.48, 55.1857 784.5, 327.199
40 61.6
68.4 67.3
59.7 68.63
110 110
24 37.6
a Slope angle on the end point of generatrix. This is the maxiumum wall angle, predicted by the empirical model, that the specimen was expected to endure without fracturing.
References [1] T. Maki, Sheet Fluid Forming and Sheet Die-less NC Forming, Amino Corporation, Japan. [2] S. Matsubara, Incremental backward bulge forming of a sheet metal with a hemispherical head tool, Journal of Society For Technology of Plasticity 35 (1994) 1311. [3] J. Jeswiet, Incremental single point forming, in: The Proceedings of Conference of the North American Manufacturing Research Institution (NAMRI), Gainesville, Florida, 2001, p. MF01-246.
[4] J.-J. Park, Y.H. Kim, Fundamental studies on the incremental sheet metal forming technique, Journal of Materials Processing Technology 140 (2003) 447–453. [5] G. Hussain, L. Gao, A novel method to test the thinning limits of sheet-metals in negative incremental forming, International Journal of Machine Tools and Manufacture 47 (2007) 419–435. [6] M. Strano, L. Carrino, M. Ruggiero, Representation of forming limits for negative incremental forming of thin sheet metals, in: The Proceedings of IDDRG, Siendelfingen, Maggio, 2004, pp. 198–207.