A toolpath strategy for improving geometric accuracy in double-sided incremental sheet forming

Sttr ULLAH, Xioqing LI, Peng XU, Ynle LI, Ki HAN,Dongsheng LI,b

a School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China

b Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education, School of Mechanical Engineering, Shandong University, Jinan 250061, China

KEYWORDS incremental sheet forming;double-sided incremental forming;geometric accuracy;springback;grey relational analyses

Abstract The double-sided incremental forming(DSIF)improved the process flexibility compared to other incremental sheet forming(ISF)processes.Despite the flexible nature,it faces the challenge of low geometric precision like ISF variants.In this work,two strategies are used to overcome this.First, a novel method is employed to determine the optimal support tool location for improving geometric precision. In this method, the toolpath oriented the tools to each other systematically in the circumferential direction. Besides, it squeezed the sheet by the same amount at the point of interest. The impacts of various support tool positions in the circumferential direction are evaluated for geometric precision. The results demonstrate that the support tool should support the master tool within 10°to its local normal in the circumferential direction to improve the geometric accuracy. Second, a two-stage process reduced the geometric error of the part by incrementally accommodating the springback error by artificially increasing the step size for the second stage.With the optimal support tool position and two-stage DSIF, the geometric precision of the part has improved significantly.The proposed method is compared to the best DSIF toolpath strategies for geometric accuracy,surface roughness,forming time,and sheet thickness fluctuations using grey relational analysis(GRA).It outperforms the other toolpath strategies including single-stage DSIF,accumulative double-sided incremental forming(ADSIF),and two-stage mixed double sided incremental forming(MDSIF).Our approach can improve geometric precision in complex parts by successfully employing the support tool and managing the springback incrementally.

1. Introduction and review

Ideas need to be converted quickly into products and analyzed to meet the requirements of the industrial revolution. It demands the manufacturing process with less changeover time and tooling cost. Decades-old conventional manufacturing processes have a long time and capital cost requirements even for small batch production and prototype development due to component-specific and expensive dies preparation. In recent years,ISF mounted to success due to its flexible nature for prototype and small-batch production. It is cost-effective, has a short lead time, and utilizes generic tooling. In ISF, flat metal sheets are incrementally deformed into complex threedimensional components using a computer numerically controlled(CNC)generic tool stylus.A sheet of material is peripherally clamped along all outer edges.1The salient characteristics of ISF are: higher formability, less forming forces, greater flexibility, reduced lead time, low cost for production of customized and low-volume components.2In addition to this, ISF can form a wide range of materials such as steel, aluminum, copper, polymers, titanium, etc.3It has considerable potential in the aerospace industry, prototyping in automotive, on-site repair for military applications, personalized products in the medical, architecture, etc.4The existing ISF process can be broadly classified into three categories:single-point incremental forming (SPIF), two-point incremental forming (TPIF), and DSIF.5

In SPIF, a peripherally clamped sheet is deformed incrementally by a single, small hemispherical-ended tool moving along a preprogrammed toolpath. The geometric accuracy obtained with the SPIF is not up to the mark. In TPIF, an extra full or partial die is used on the other side of the sheet to enhance the part accuracy.6Good accuracy in TPIF due to dies involvement compromises the process flexibility. DSIF has gained the researcher’s attention as it effectively addresses the shortcoming of SPIF and TPIF. The DSIF process requires the use of two tools, one on either side of the sheet(one for forming and the other for acting as a local moveable die). The coordinates of the master (forming) tool are usually obtained from the CAD/CAM software. The support tool coordinates are calculated with respect to the master tool coordinates. Both tools move in a synchronized manner. The sequence of the process is almost similar to the SPIF process except for additional support tooling and its relative movement with the master tool (Fig. 1).7

Fig. 1 Schematics of ISF processes7.

Researchers have utilized various techniques to improve the geometric accuracy with DSIF. Toolpath compensations,adaptations, multi-stage, and heat-assisted processes were among the notable strategies employed by researchers.Malhotra et al.8formed a cone using various sets of squeeze factors(1, 0.9, and 0.85). The geometric accuracy obtained at a part wall region with DSIF improved as compared to SPIF. Lingam et al.9improved the component precision by utilizing the toolpath compensation strategy to account for the tool and sheet compliance in smaller parts. Praveen et al.10extend the same work to the large-size components. Konka et al.11further improved the Praveen et al.10work by incorporating the machine error compliances in the toolpath to avoid the tool-sheet lost contact problem, which was thought to be the main cause of geometric precision degradation. Moser et al.12improved the toolpath to avoid the tool-sheet lost contact by modifying the sine law. Their toolpath was specific to shamrock part. Ren et al.13employed the contact force algorithm on the support tool to maintain a successful contact between the support tool and the sheet. Ren et al.14further combined the contact force algorithm with the in-situ springback strategy.The prior simulation required for the springback determination was time consuming. Component geometric accuracy improved with these toolpath compensation strategies,although these have some limitations.For high-wall-angle part formation, multi-stage DSIF processes were primarily used. It was used by Moser et al.15for cylindrical components and Zhang et al.16for the clover flange.In both cases,the part wall angle was 90°. Ultrasonic-assisted process can be helpful for reducing force requirements and geometric precision improvements17.

Aside from the toolpath compensation and multi-stage processes, the flexible support tool different positions, toolpath variation,and adjustment by pressure techniques is also trialed by researchers for accuracy improvement.The flexible support tool makes the DSIF process unique as other ISF variants have the limitation for such adjustments. Malhotra et al.18were the first to utilized this feature.They proposed an ADSIF toolpath, where the master and the support tool move in an outward direction after forming the inner annulus. It moves outward by an equal amount to step size in the same plane.The already processed inner material moves in the negative Z-direction by rigid body motion. ADSIF can process more virgin material.In a conventional DSIF toolpath,the required profile is obtained by the tools from the outer to the inner annulus. With ADSIF, the part geometric precision improved significantly.It faces some challenges(i)better geometric accuracy is obtainable only with a low step size(ii)the relationship between the toolpath parameters and desired shape demands trial-and-error experimentations, which makes the process time-consuming. Zhang et al.19utilized the combination of ADSIF and DSIF (MDSIF) with a relatively larger step size to reduce the long forming time. They obtained the same geometric accuracy earlier obtained with ADSIF and reduced the forming time too.Meier et al.20utilized a support tool capable of applying force on the master tool to avoid the tool-sheet lost contact problem. The support tool was also capable of orienting its position (angle) to the master tool. They reported geometric precision improvement when the master tool was supported locally. A detailed study on the defective skull reported the same conclusion for the local support both in aligned and normal tool configuration, which were the variation of the support tool position in the meridional direction21.Lu et al.22did a detailed study by changing the support tool orientation in a meridional direction. They focused on deformation mechanism in DSIF instead of geometric precision improvement. Wang et al.23utilized the support tool orientation to the master tool at various positions in the meridional direction by squeezing and reverse bending the sheet with independently control tools to mitigate the effect of springback.Reverse bending showed far better effects on geometric accuracy (as it reduced the springback effectively) improvement than squeezing. All these studies focused on the axisymmetric component,where local support by the support tool was effective.Shear spinning,a well-established method for axisymmetric structures, runs at several orders of magnitude faster than any ISF. The part geometry determines the speed limit for ISF: the further the part deviates from axi-symmetry, the slower the process must run.24Also, the geometric precision in asymmetric part is more challenging due to the transition from the convex to straight and convex to concave region.The limitation of slower speed and convex to straight region transition is focused for the asymmetric part as such components study are limited with DSIF.

From the literature review,it is evident that the main focus of the limited studies was on the support tool optimal position determination in the meridional direction.Due to the unavailability of any effective method, to date, the support tool position in the circumferential direction is not studied. In this article, a novel technique based on pyramid shape forming was proposed to find the support tool’s optimal location in the circumferential direction. The circumferential position of the support tool was fixed at a different location at the corner region in experiments,and its effects on geometric precision at the straight wall were observed. Based on these experiments,the support tool minimum deviation in the circumferential direction from the master tool local normal was determined.It will address the challenge of transition from convex to straight region in complex part. Secondly, the springback determination techniques in forming trialed to date are time expensive due to simulation involvements. An effective strategy to improve the geometric precision by countering the springback incrementally is proposed.This strategy is referred to as DDSIF(double-stage double sided incremental forming)in this work. With this strategy, the part is formed in two stages with relatively large step size to reduce forming time.In the first stage,the component is formed,with a normal step size.In the second stage,the geometric error in the component size due to the springback is rectified by adding an artificial step size enhancement (ASE) to the first stage step size. The best toolpath strategies (DSIF, ADSIF, and MDSIF) were compared with DDSIF on a single platform for a 1 mm thick sheet. Besides geometric precision, surface roughness, forming time, sheet thinning was also compared,and the best toolpath strategy was determined based on the GRA.

2. Experimental setup

In this section, the test machine parameters used for forming the part and measuring the formed part different qualities characteristics are presented. Also, the DSIF toolpath generation procedure is explained.

2.1. Machine specification

A custom-designed DSIF machine having a tool on both sides of the clamped sheet was utilized to form the parts (Fig. 2).The X-axis tools movement was adjusted by two servo motors on a gear-rack assembly, whereas the Y and Z-axis by servo motors on the leads screw. The velocity of the support tool was adjusted automatically by the control system to compensate for the master tool, thereby maintained a fully synchronized motion. Both have a hemispherical end with a diameter of 10 mm and are made of hard steel.The positioning accuracy of the machine was within 100 μm in all the three translational degrees of freedom: X, Y and Z-axis. The working space was limited to a flexible fixture size of 500 mm × 500 mm. A constant tool speed of 5 mm/sec was maintained for all the testing cases. Teflon grease was used to reduce the friction between the tools and a sheet.

The formed parts geometries were digitized using a portable free-scan X3 laser scanner. Different sections of the formed parts were recorded and compared in order to evaluate the performance of different strategies.The root means square values of the error along the target section were used as criteria to compare the deviation.19Mitutoyo Surftest surface roughness tester was used for surface roughness measurement on the inner side of the part. ISO 4288:1996 recommendations were followed for surface roughness measurement. Sheet thickness of the form part were determined by ultrasonic sheet metal thickness sensor.

2.2. Toolpath generation for DSIF process

The CAD/CAM software provides the master tool coordinates(Xm, Ymand Zm). Using Eqs. (1) and (2), the support tool is defined in DSIF and ADSIF via two parametersDandS(Fig. 3).

where D is the distance between the axis of the two tools in the XY plane; Sis the vertical distance between the bottom of the sheet and the tip of the support tool in the XZ plane; Zmand Zsare the master and support tool position in the XZ plane;Rmand Rsis the master and support tool position in the XY plane.In DSIF,parametersDandSwere determined by utilizing the sine law and the normal tool configuration.From Fig.3(b),it is evident thatD=d·sinθ, and S=rm+rs-d·cosθ+to,where rmand rsis the master and support tool radius, θ is the local wall angle, and tois original sheet thickness. The D and S changes continuously as part height increases due to the dependence of don the sheet thickness at the contact point(ti) (Fig. 3(b)).

In ADSIF, although the support tool coordinates can be obtained by the above-described method.However,better geometric accuracy could be obtained by adopting the coordinates based on trial-and-error of D and S combinations.While optimizing tools position in ADSIF, Ndip-Agbor et al.25reported that for a constant S value, increasing Dresults in forming a smaller angle as the squeezing of the sheet between the tools is reduced, and hence the geometric accuracy is compromised.The effect of increasing the parameterSwhile keepingDconstant leads to the reduced wall angle and accuracy. To get the desired precision, there should be a trade-off between the D and S values. For a 0.5 mm thick sheet with a 5 mm tool diameter, the best D and S combination trialed to date is 2.5 mm and 0.43 mm.13,19,26As 1 mm thick sheet with 10 mm tools diameter were processed in this work, therefore,before starting experiments on the pyramid, some preliminary trial-and-error experiments were performed for optimal parameters combination determination and were used wherever required.

Fig. 2 DSIF machine.

Fig. 3 DSIF and ADSIF toolpath generation strategy.

In ISF,the final sheet thickness is usually calculated by sine law. For the 40° wall angle part having an initial sheet thickness of 1 mm, the final sheet thickness should be 0.766 mm（tf=to·cosθ）. The change in sheet thickness from 1 mm to 0.766 mm does not occur abruptly,as the forming of part gets started.For the support tool position determination,the sheet thickness at the contact point (ti) was not considered as 0.766 mm from the beginning but incorporated by utilizing the straight-line equation. The slope (m) of the straight-line equation was calculated by Eq. (3).where Rois the part opening dimension at a specific section;Rfis the final dimension of the part on the same section(Fig.3(b)).Riis the part dimension on the same section(where Rfand Rois considered) at any arbitrary height; tois original sheet thickness, and tfis the final sheet thickness predicted by sine law. For pyramid of 45 × 45 × 15 mm size having 40° wall angles: Ro= 22.5 mm, Rf=2.87 mm, to=1 mm,and tf= 0.766 mm, the value of slope (m) and Y-intercept(c) are calculated as 0.012 and 0.732, respectively. The value of sheet thickness at contact point at different height for 45 × 45 × 15 mm pyramid are listed in Table 1.

Due to the incorporation of sine law via a straight-line equation, the sheet thickness value varies as the part height increases, which changes the parametersDandS, and is shown in Table 2. The D and S parameters calculated based on Eqs. (1) and (2) and the sine law are used for DSIF toolpath generations. For the ADSIF toolpath, the optimal Dand S parameters from the trial-and-error tryout experimentations were maintained during the entire experiment due to the process capability to process virgin material.

3. Methodology

In this section,first,the methodology and test plan will be discussed for the two strategies for geometric precision improvements. The first strategy is about finding the optimal support tool position with a novel method. The second is about the ASE calculation procedure for the second stage in two-stage forming for springback reduction. After that, the test plan and calculating procedure used to compare different toolpaths is discussed.

3.1. The support tool optimal position determination

To find the minimum possible deviation of the support tool in the circumferential direction from the local normal of the master tool to have the least effect on geometric precision degradation, the support tool positions were varied in the circumferential direction at the corner region from point 4＇to point 4＇＇while forming the pyramid in different experiments(Fig. 4). The sizes of the pyramids formed were 45 × 45 × 15 mm with 40° wall angles. The sheet materialwas AA7B04 having a 1 mm thickness. A step size of 0.5 mm was utilized in the first stage,and 0.603 mm in the second stage.Five circumferential positions for the support tool were utilized in five different experiments. In the first experiments, the support tool position was set by offsetting the master tool positions in linear directions to point 4＇＇(Fig. 4(a)). The angle between the local normal of the master tool and the support tool was 42.21°. In experiment no. 2, the support tool was adjusted at a 30°angle to the local normal of the master tool.The support tool adjustment was changed to 20°, 10° and finally to 0° in remaining experiments.The support tool adjustment of 0°represents its adjustment in line with the local normal of the master tool because the support tool maintains the same distance with the master tool when they travel to points 5-5＇.

Table 1 Sheet thickness at various levels of part height.

Table 2 Distance between tools at various height.

Fig. 4 Tools position in circumferential direction, and distance between axis of tools at different height.

The pyramid geometry was divided into 15 equal parts(contours)along the part height to find the master and support tool position to each other in a circumferential direction as the part height increases during the forming.For this purpose,the CAD/CAM drafting module was used. The master tool position at each control point was based on the NC code generated in the CAD/CAM software. The support tool was positioned based on the principle (aforementioned) at the required distanceDfrom the master tool position.To be succinct,two arbitrary heights are shown in Fig. 4. Along the part height, the circumferential position between the tools does not vary significantly and is almost the same as adjusted at the top contour.

To observe the circumferential position variation from the corner to the center region,the machine was programmed such that both tools arrived at the center region simultaneously from the corner region by following the shortest route.Different adjustments of support tool position in the circumferential position (except when φ=0°) made the support tool toolpath longer(4＇＇-5＇)when it was moving from the corner region to the center;to arrive at center region at the same time,they reached simultaneously at the mid region too.During the movement in the same contour,the distance between the tools was changing.It became minimum at the mid region represented by L2in Fig. 4. The distance between the tools were the same in all experiments at the corner region (L1) and at the center region(L3) in the same contour and was adjusted as Dfrom Table 2.The distance L2at the mid region was obtained from the CAD/CAM drafting tool and verified by running the machine without clamping the sheet on it.For different circumferential position, the value of L2was different and less than L1and L3except for φ= 0°, where the distance between the tools was same across the complete contour.

As the tools traveled onward from the mid region, the distance between them got started increasing (exceptφ= 0°). At points 5-5＇(center region of the straight wall), the tool gap became the same as it was adjusted at the corner. The circumferential angle between the tools (φ) decreased gradually to zero when it reaches center region. The circumferential angle(φM)at the mid region was almost the half of the initial adjustment at the corner regions. This toolpath helps in finding the circumferential deviation effect of the support tool from the local normal of the master tool (4-4＇) on geometric precision of the component.

3.2. ASE determination and incorporation

To determine and validate the ASE functionality,two pyramid from different material was formed. A pyramid having a size of 45 mm × 45 mm × 15 mm from AA7B04, and 100 mm × 100 mm × 30 mm with 45° wall angle from DC04 was formed. The support tool was positioned at the local normal while forming both the pyramid. A step size of 0.5 mm was maintained in the first stage.

To compensate for the geometric error of the components due to sheet springback, the part was formed in two-stages.In the first stage, with a step size of 0.5 mm, the target part height was not achieved due to sheet springback (Fig. 5). To achieve the target height, a second stage based on ASE was conducted. The step size was increased artificially by a factor to compensate for the height difference between the target and the formed part after the first stage. The factor value was obtained by dividing the part height difference (between the target and part height obtained in the first stage) by the total number of steps(contours)in which the part was formed.The difference between the target height and the height obtained after the first stage was 3 mm in the pyramid formed from AA7B04.Step size was artificially enhanced by 0.103 mm(3/29) as the part was formed in twenty-nine contours. This factor was added to each contour height in an incremental manner, which (for example) changed the first two contours to - 0.603 (- 0.5- 0.103), and -1.206 (-1-(2 × 0.103)) in the second stage. Similarly, the last contour height was changed to-17.487 mm from-14.50 mm.It increased the step size to 0.603 mm in the second stage. As evident from Fig. 5, the ASE has the capability to reduce the geometric error both in the radial and Z direction. The step size for the second stage is equal to the summation of step size in the first stage plus ASE.The second stage tries to deform the sheet to the required positions in the Z and radial(XY)planes.The same procedure was applied to the pyramid formed from the DC04 for ASE determination and incorporation in the second stage. In the second ASE stage, step-down size was adjusted by adding a constant factor of 0.12 mm (7.2/60) to every step height incrementally.

3.3. Toolpaths comparison

Different toolpaths was compared based on forming a pyramid having 45 mm×45 mm×15 mm size and 40°wall angles.The toolpaths considered are the single-stage (DSIF and ADSIF), and two-stage (MDSIF and DDSIF). The experimental plan is displayed in Table 3.

As an ADSIF displayed the best geometric precision with 0.025 mm step size,18therefore, the same step size was implemented for Tests No. 1 and 2. In ADSIF, the optimal D and S combination was obtained based on the experimental trialand-error tryout procedure for a 1 mm thick sheet. In the DSIF toolpath, the D and S values are utilized from Table 2.

Fig. 5 Principle for artificial step size enhancement.

A coarse step size of 0.5 mm was utilized in the first stage in Test No. 3 and 4, which led to part forming in twenty-nine contours. In DDSIF, the second stage step size due to an ASE accommodation was 0.603 mm. It aimed to reduce the geometric error incrementally. The D and Ssetting for each stage was kept the same as Test No.1.In Test No.4(MDSIF),the pyramid was first formed by ADSIF, and then with DSIF having a 0.5 mm step-down size. For the first A-stage, the D and Swere the same as was utilized in Test No.2.For the second D-stage, the D and S were similar to Test No. 1 setting.MDSIF was initially trialed by Zhang et al.19The step size,part profile,and sheet thickness were different from this study.

These four toolpaths are compared based on different responses such as geometric precision along different sections,part height, surface roughness, forming time, and sheet thickness variation.Obtained results from these tests have different units and magnitude.It was not possible to obtain conclusions based on variable information. Decision for the best toolpath was determined based on GRA. There are three main steps in GRA (i) data normalization (ii) grey relational co-efficient(GRC) determination (iii) grade calculation for different tests.Some researchers normalize the data based on responses,27while others uses S/N ratio.28In present work, data was normalized based on responses.

Responses were normalized based on different criteria.Eqs.(6), (7), and (8) are the equations for data normalization for‘‘larger-is-better”, ‘‘smaller-is-better”, and ‘‘target-is-better”,respectively. The ‘‘larger-is-better” option was utilized for sheet thickness since larger sheet thickness after forming leads to good mechanical properties of metal. The ‘‘smaller-isbetter” criterion was used for surface roughness, geometric error, and forming time. The criterion of ‘‘target-is-better”was utilized for forming height. Data normalization for multi-responses was conducted to mitigate the effects of different units and their variability.A suitable data in the range of 0 to 1 was obtained for all the responses.

where i and j = 1, 2, 3,...., n, and n is the number of experimental data items. Zijis the normalize response; yijis the observed response value obtained from experiments; min (yij)and max(yij）is the minimum and maximum response obtained from experiments,and T is the target value.After normalizing the data, the GRC was determined by using Eq. (9).

Table 3 Experimental plan for toolpath comparison.

●0 ≤ξ ≤1.ξis the distinguishing coefficient. Usually, it is taken as 0.527,29

All responses are graded into a single unit-less combined response obtained by using the Eq.(10)based on GRC.Test/-experiment which obtained the higher-grade was considered the best test among the DOE tests.

4. Results and discussion

4.1. Determination of optimal circumferential position

In this section,a discussion about geometric precision with different support tool circumferential positions and its mechanism is discussed.

The geometric error at the part opening (part height = 0 mm) is higher when the master tool is supported at 42.21° in the corner (Fig. 6).With this adjustment, the geometric error in the part is observed up to 7 mm part height.For support tool adjustment at 30°, the geometric error is reduced at the part opening, and also its effects are observed to lesser height of 5.5 mm. This trend continues with the reduction of the circumferential angle adjustment of the support tool. At the support tool circumferential position of 10° and 0° to the local normal of the master tool, the geometric error is below 1 and 0.5 mm along the complete forming height.The geometric precision improvement after a part height of 7 mm for φ=42.21° shows that if the support tool can readjust its circumferential position to 0° (which is obtained at point 5＇in all contour) within 9.15 mm (Fig. 7) distance from point 4＇＇to point 5＇,then its effects on geometric precision deterioration can be minimized. Similarly, for initial adjustment of support tool at φ=30°, the support tool should readjust itself to local normal within the distance of 11.53 mm, for φ = 20°,13.59 mm,andφ=10°,16.18 mm.From these results,it is evident that if the support tool supports the master tool by more than 10° to the local normal of the master tool, it can lead to geometric error maximization.If the support tool is capable to readjust its position at the local normal within a certain distance, then the effects of the wrong adjustment of the support tool can be minimized.It means that for complex parts transitioning from convex to straight, the master and support tool toolpaths should be extracted from more control points.It will help the support tool readjustment to the master tool if lagged or surpassed due to any transition. As the current work is based on the support tools moving circumferentially to a local-normal of the master tool, the size of the pyramid is immaterial.

Fig. 6 Effect of support tool circumferential position on geometric precision along section AA.

At the corner region, the support tool smaller region is in contact to the master tool for all the circumferential position trialed in this study, therefore, the precision along the entire part height is not affected as neither the master tool was lagged behind or exceeded by the support tool in this region (Fig. 8).

The distance and circumferential orientation between the tools were changing when they travel from the corner to the center region.At the mid region,the distance between the tools(L2)became the minimum(Fig.9).The sheet got squeezed by a factor of 0.93 for circumferential angle adjustment of 42.21°.The additional squeezing at the mid region decreases with the angle reduction and it became zero for 0° (Fig. 10). The higher circumferential angle adjustment resulted in the support tool lagging the master tool due to a relatively larger path,which deteriorates the geometric error due to the reverse bending and additional squeezing of the sheet. The smaller angle circumferential adjustments at the corner region leads to less lagging and squeezing. It helped in reducing the reverse bending negative effect on geometric precision. As tools cross the mid region, the support tool starts orienting its position into the master tool local normal,and transfer the error along with them to the center region.

Fig. 7 Distance between corner and center region along part height.

4.2. Geometric precision improvement by incrementally compensating the springback

In this section,the improvement in geometric accuracy with the incremental springback compensation approach is analyzed.

The geometric error of the pyramid formed with the twostage DSIF process enhances the geometric accuracy significantly. Each step was artificially enlarged in the second stage,resulting in a better part size in the radial and Z directions.Fig. 11 depicts the geometric error of the pyramid(45 mm × 45 mm × 15 mm) for the first and second stages.The geometric error is up to 2.97 mm in the first stage.The part height obtained in the first stage was 12 mm. With the second ASE stage, the geometric error is less than 0.45 mm along the complete part height. Fig. 12 display the geometric error of a relatively large pyramid(100 mm×100 mm×30 mm),formed from DC04.The maximum geometric error obtained in the first stage is 7.26 mm, and the part height acquired was 22.8 mm.The second ASE stage has reduced the maximum geometric error to 0.57 mm,and part height is increased to 29.6 mm.

Both pyramids are formed with a 0.5 mm step size in the first stage, which reduced the number of toolpath counters,and reduced the forming time.Due to its capacity to efficiently incrementally reduce the springback even for large step sizes,this method offers the potential to increase geometric precision in a short time. Also, the calculation procedure for an ASE determination is simple and less time-consuming.

In Section 4.1, the pyramid was formed with ASE while changing the support tool position. Even though the step size was increased in the second stage, geometric accuracy deteriorated when the support tool diverged from the local normal.In this section, the support tool was at the local normal, and the geometric accuracy improved when the second stage stepdown size was increased for springback reduction.

Fig. 8 Support tool circumferential position effect on geometric precision along section BB.

Fig. 9 Squeezing, over-bending and reverse-bending region at mid region.

Fig. 10 Sheet squeezing at mid region for different circumferential position adjustment at corner region.

Fig. 11 Geometric precision improvements with incremental springback accommodation in second stage (Material: AA7B04,Pyramid size: 45 mm × 45 mm × 15 mm).

Fig. 12 Geometric precision improvements with incremental springback accommodation in the second stage (Material: DC04,Pyramid size: 100 mm × 100 mm × 30 mm).

4.3. Toolpath comparison

Different responses of the four toolpaths are compared and then analyzed with GRA to determine the optimal toolpath.The responses compared are forming height, surface roughness, geometric precision, time required for part forming and sheet thinning.

The forming height and surface roughness comparison among the ADSIF, DSIF, DDSIF and MDSIF are depicted in Fig. 13. The error in part height is -1.46, 1.29, 0.08,-1.35 mm,respectively,and is obtained by subtracting the target part height from the obtained part height. The negative error represents the undersized part height, whereas the positive values indicate the oversized part. Among all toolpaths,the most precise part height of 15.08 mm is obtained with DDSIF. With ADSIF toolpath,the obtained precision in part height (although oversize having 16.29 mm height) is ranked second. As regard to surface roughness, DSIF and ADSIF outperformed the DDSIF. DSIF gives the lowest surface roughness of 0.804 μm, whereas the surface roughness obtained with ADSIF and DDSIF is 1.25 and 1.345 μm,respectively. The base metal surface roughness was 0.320 μm.The tool marks on the part surface are evident in the DDSIF due to additional stretching in the second stage due to larger step size.

DDSIF and MDSIF take about 30 min for forming the pyramid,whereas ADSIF and DSIF took 240 min due to small step size utilization. The geometric precision with ADSIF reduced substantially when the step size was increased to 0.5 mm in MDSIF. The A-stage in MDSIF displays the poor performance for forming the pyramid shape properly.The corner region radius becomes too large. Due to this reason, the second stage was not executed with the ASE. With the second D-stage, the part profile and part height get improved. The two-stage process with a large step size displays better potential for formability control than a single-stage process with even a fine step size.

ADSIF, DSIF and DDSIF exhibits low geometric error along the section AA (Fig. 14). The geometric error is consistently below 1 mm along the complete forming height.DDSIF precision is relatively better from the remaining two toolpath and its geometric error is below 0.5 mm. MDSIF with coarse step size gives the lowest geometric precision, and part error is above 1 mm along the complete part height.

DDSIF also shows the best geometric precision along the section BB with an error below 1 mm (Fig. 15). ADSIF and DSIF has not maintained the same performance along the section BB. It may be due to more increasing stiffness at the corner than any other region of the part. In DDSIF, the second stage has stretched the sheet in a more effective manner to improve the geometric accuracy at the corner.

The average of the geometric error(AGE)along the section AA and BB is lowest for the DDSIF having values of 0.29 and 0.38 mm,respectively(Fig. 16). The overall average geometric error(OAGE)is the average of AGE values along sections AA and BB. DDSIF has the smallest OAGE value among all the toolpaths. The absolute difference in AGE along section AA and BB is also lowest (0.036 mm) for DDSIF. Besides this,other toolpaths which involved DSIF at any stage performed better for the AGE difference. The increased difference in AGE between different sections of the same part can be troublesome,if the next iteration is planned for geometric accuracy improvement with ASE.

Fig. 13 Forming height and surface roughness comparison of different toolpath techniques.

Fig. 14 Geometric error comparison along section AA.

Fig. 15 Geometric error comparison along section BB.

The thickness reduction of components severely deteriorates the part performance and is of great concern in the industry. Thickness reduction is one of the limitations of incremental forming. The sheet thickness variations are primarily dependent on the tool trajectory, therefore, the sheet thickness obtained with toolpaths has also been compared.The thickness variation in two-stage forming is apparent than the single-stage DSIF and ADSIF (Fig. 17), and is due to the smaller step size utilization. The uniform and closer to the original sheet thickness is the target. It should not be at the cost of the other responses. The uniform sheet thickness obtained with DSIF is relatively closer to the original sheet thickness of 1 mm than the ADSIF.In DSIF, the smaller step size has utilized more material and made the material flow from outside to inside. Also, the second reason may be more forming height formation in ADSIF than the target and less forming height acquired in DSIF. In ADSIF, virgin material is always processed and is the reason for uniform sheet thickness across the part. DDSIF performance with respect to the sheet thickness variation is not as effective as was with geometric precision,forming height and time.The thickness obtained with it is on the higher side than the ADSIF for most of the part height.

Fig. 16 Average geometric error comparison along section AA and BB.

Fig.17 Thickness distribution comparison for different toolpath strategies.

Fig. 18 Grade obtained by different toolpath strategies.

From the comparison of toolpaths for different responses,any toolpaths have not displayed clear advantages over the other toolpaths.DDSIF toolpath gives the best geometric precision along the section AA and BB. The processing time required for the part forming is lower for DDSIF and MDSIF.DSIF and ADSIF perform better with respect to the surface roughness. Further, the geometric accuracy and forming height obtained with ADSIF is ranked second after the DDSIF. For thickness distribution along with the part height,DSIF and ADSIF give better results.

Fig. 18 depicts the grade values based on part height, surface roughness,error along section AA and BB,forming time,and a minimum sheet thickness of the part (Calculation at Appendix A).DDSIF has displayed the higher grade,followed by DSIF,and ADSIF,respectively.Works on other responses such as static and dynamic strength are underway, and they may change the overall grade. Further, MDSIF in previous studies gives the better results with relatively low step size of 0.1 mm. It will be trialed with smaller step size. The forming time will be more,however,it may have good impact on other responses, which can improve it grading.

5. Conclusions

In this work, a novel method is presented to find the support tool minimum deviation in the circumferential direction from the local normal of the master tool. Besides, to minimize the geometric error of the part due to the sheet springback, a two-stage DSIF was utilized with an ASE. The main focus of the research in DSIF was the geometric precision improvement of the part, although DSIF also poses the challenges of long forming time,sheet thickness thinning,and the rough surface finish. All the main limitations associated with the DSIF are, for the first time, simultaneously considered. The best toolpaths from the literature and the proposed toolpath in this work with an ASE are compared on a single platform with a multi-response GRA method. Based on this work, the following are concluded:

(1) The support tool should support the master tool by less than 10°in the circumferential direction to its local normal to ensure the component precision. A higher deviation of the support tool from the master tool local normal can increase the geometric error due to; (i) ineffective squeezing of the sheet at the required region (ii)reverse bending of the sheet due to the support tool lagging movement behind the master tool.

(2) The geometric error reduction from 7.26 mm to 0.57 mm for DC04, and from 2.97 to 0.45 mm in AA7B04 shows the effectiveness of the incremental springback accommodation techniques. Due to its capacity to efficiently incrementally reduce the springback even for large step sizes by using an ASE in the second stages, it has the potential to increase geometric precision in a short time.

(3) Based on the GRA, the optimal toolpath among the DSIF, ADSIF, DDSIF, and MDSIF is the two-stage DDSIF with an ASE. Geometric precision and forming height in two-stage DDSIF improved with less forming time.Sheet thickness variation in DDSIF along the part height was apparent,and on higher side than the ADSIF for most of the part. Surface roughness of 1.35 μm achieved with DDSIF is reasonable, although it is relatively on higher side than the DSIF and ADSIF.

The DSIF process faces the challenges of geometric precision in a component having the transition region from convex to straight and convex to concave.The first case regarding the convex to straight region transition is focused in this work,and suggestions are put forward. Detail study on the second challenge concerning convex to concave region transition is needed to be more focused in future. Also, the two-stage process can be conducted in a single stage by incorporating an ASE in the first stage, provided some reliable numerical or analytical solution for ASE prediction is established.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 52075025, 51975328), and Project funded by China Postdoctoral Science Foundation (No.2021T140418).

Appendix A.Different toolpath grade calculation based on GRA

Toolpaths Obtained responses from different toolpaths Normalization Height(mm)Ra(μm)Error-AA(mm)Error-BB(mm)Time(min)Thickness(mm)Height Ra Error-AA Error-BB Time Thickness DSIF 13.54 0.80 0.74 1.21 240.0 0.88 1.00 1.00 0.71 0.56 0.00 1.00 ADSIF 16.29 1.25 0.49 0.59 240.0 0.79 0.88 0.55 0.87 0.87 0.00 0.25 DDSIF 15.08 1.35 0.29 0.33 30.0 0.76 0.00 0.46 1.00 1.00 1.00 0.00 ADSIF 13.65 1.80 1.85 2.31 30.0 0.81 0.92 0.00 0.00 0.00 1.00 0.42 Toolpaths Deviation sequence Grey Relational Co-efficient (GRC) Combined Height(mm)Ra(μm)Error-AA(mm)Error-BB(mm)Time(min)Thickness(mm)Height(mm)Ra(μm)Error-AA(mm)Error-BB(mm)Time(min)Thickness(mm)Grade Order DSIF 1.00 1.00 0.63 0.53 0.33 1.00 1.00 1.00 0.63 0.53 0.33 1.00 0.87 2 ADSIF 0.80 0.53 0.80 0.79 0.33 0.40 0.80 0.53 0.80 0.79 0.33 0.40 0.71 3 DDSIF 0.33 0.48 1.00 1.00 1.00 0.33 0.33 0.48 1.00 1.00 1.00 0.33 0.95 1 ADSIF 0.86 0.33 0.33 0.33 1.00 0.46 0.86 0.33 0.33 0.33 1.00 0.46 0.62 4