A CALCULATION FOR COMPENSATING THE ERRORS DUE TO SPRINGBACK WHEN FORMING METAL SHEET BY SINGLE POINT INCREMENTAL FORMING ( SPIF )

The question of compensating for the error of dimension due to springback phenomenon when forming metal sheet by SPIF method is being one of the challenges that the researchers of SPIF in the world trying to solve. This paper is only a recommendation that is based on the macro analysis of a sheet metal forming model when machining by SPIF method for calculating a reasonable recompensated feeding that almost all researchers have not been interested in yet: Considering the metal sheet workpiece is elasto-plastic and the sphere tool tip is elastic, the authors attempt to calculate for compensating the error of dimension due to elastic deforming of the tool tip. The metal sheet is clamped by a cantilever joint that has an evident sinking at the machining area that is also calculated to add to the compensating feeding value. The paper also studies the limited force for ensuring the elastic deforming at these working area of the sheet to eliminate all the unexpected plastic deforming of the sheet. With two small but novel contributions, this study can help to take theoretical model for elastic forming of metal sheet closer to real situation.


INTRODUCTION
The deformation of manufacturing installations is an unavoided phenomenon in almost all pressing machines.In this technology, on one hand, we attempt to progress the plastic deformation of the workpiece as much as possible.On the other hand we have to restrict one of the manufacturing installations such as machine, spindle, tools, clamping installations… to the minimum with in the purpose of increasing the accuracy of the products.

Especially in the Single Point Incremental
Forming method, a recent technology of metal sheet forming, the unexpected deformation of the product after forming (The Springback phenomenon) is a critical question that the researchers in SPIF field are interesting.
The goal of this paper is to describe the analyzing calculation for providing the TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 13, SỐ K4 -2010 The compensative values are composed: -Elastic deformations of the tangent surface of the punch and the metal sheet.
-Elastic deformations of the volume of the cantilever part of the punch.
-Elastic deformations of the clamping installation.
-Elastic deformations due to the elastic sinking of the sheet.

Elastic deformations of the punch when machining
In figure 1, we can see the sphere tip punch that is mounted in the spindle of a CNC milling machine.To consider the absolute rigidity of the spindle and the carriage machine, their deformations, if exist, are infinitesimal, the deformation of the punch can be divided in 3 sections: -Section 1: the deformation of the sphere surface of the tangent area (y 1 ) is equal to the depth t of feeding rate.
-Section 2: a part of phere area (y 2 ) of the length of D/2-t that has a variable section.
-Section 3: the tail of the punch to the clamping area of length (y 3 ) when its initial value is l 0 =AB.
Since the elastic deformation is calculated by (1) we can apply Ludwid 's formula for calculating the elastic stress at an arbitrary tangent angle ϕ on the sphere section of the sheet.
Formula (2) describes the elastic stress at an arbitrary point in arbitrary tangent area of sheet and punch.It has the same direction of strain.This means it has tangent direction with the sphere at an arbitrary line that makes an angle ϕ (Figure 2) with the axe of the punch.
We can consider it the normal elastic stress in The stress of the circumference direction σ T =0 due to the non deformation on circumference.
Let's consider an infinitesimal cube volume in the tangent area in figure 2.
According to Von Mise critical, we write down 3 main orthogonal stresses of the cube.From [7] we can find out the relationship among the main stresses: With the condition of the positive of σ R , we can eliminate the negative value: Replace (3) into (4) we have the normal stress on the sheet surface and with the law of Newton III it is also the normal stress on the spheral surface of the punch.
Select "+" sign and interest in the worst case that is the maximum stress: it appears at the top C' of the punch (ϕ=0) The tangent strain is ε= σ R /E P , where E P is Young's modulus of the punch From (6) we can calculate the maximum strain at the top of the punch (at ϕ=0) The tangent depth is t (Figure 2), we can calculate the displacement of the shorted dimension at tangent area y 1 =t.εMax :

Elastic deformation of the volume of the cantilever part of the punch y 3 :
By the cantilever clamped section, this part of the punch is also pressed.
With its diameter D and the length L of the punch the pressed deformation is calculated as: TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 13, SỐ K4 -2010 Trang 19 Axial force P Z is calculated in the downward feeding rate : Calculate its maximum value when σ R reaches its critical value in ( 6) The shorted pressed displacement y 1 in Z direction [7] is: Replace (8) into we have the dispacement of spheral area y 2 is :

Total strain due by the elastic of the punch y p = y 1+ y 2+ y 3
From ( 7), ( 9) and ( 10) we can calculate the total strain of the punch: Trang 21

Deformation generated by the sinking of the sheet when forming:
The maximum axial resultant P Zmax can cause the sinking of the sheet.Let's observe figure 3 with the simple clamping plate (round in general case) but the shape of the sheet is more complex.L Max is the maximum distance from the gutter of the clamping plate to the minimum radius of the sheet.

Total compensation:
Addition all the values in ( 11), ( 12), and (13) we get the total compensation: for remedying the damaging effects of the deformations of workpiece (metal sheet) and increasing the accuracy of the dimensions of the products.In an acceptable hypothesis of the absolute rigidity of the spindle, carriage, the paper only concentrates in the calculation for compensation the deformation of the secondary installations for CNC milling machine when forming metal sheet in SPIF technology.