**4.2. Radial feeding force calculation**

The feeding force is critical to the choice and design of the capacity of the spinning equipment. An analytical model for calculating the forming forces is very useful, especially when a quick prediction of forming force is required.

Stamping-Forging Processing of Sheet Metal Parts 49

θ *r s* σσσ

 θ

= − ( )/ *w ah a* . *h* is the width of

 τ−== , and

(3)

= , plasticity condition 2

along the cylinder surface, the mean feeding force *f* on the body of one unit

θθ

ρ

<sup>0</sup> 11 1 cos sin sin cos ) 2 2 <sup>2</sup> *r s*

= = + −−

the sector body in radial direction, *σθ* and *σr* are stresses in tangential direction and radial

Fig. 26 shows the compressed zone of the workpiece**,** the total feeding force *F* can be

θ

*m m <sup>m</sup> f d*

tan ( / ) *a b* <sup>−</sup> <sup>=</sup> , 2 2 *h ab* = + ,

Substituting boundary condition 1

θ

σ ρ

 , <sup>1</sup> 1 θ

integrating *<sup>r</sup>*

expressed as

σ

where, *m hh* =− + (2 ) / ρ

**Figure 26.** Sketch of compressed zone

can be expressed as:

θ θ = , 0 θ σ

<sup>1</sup> 1

direction, respectively. *σs* is the tensile strength, *τ* is the shear yield stress.

θθ σ ρ

As shown in Fig. 25, the final filled zone is the corner enclosed by roller, spindles and workpiece. A small sector body with thickness of one unit is analyzed to calculate the feeding force by slab method.

**Figure 25.** Sketch of principal stress on sector unit body

According to Fig. 25, the force equilibrium equation of the small sector body in *θ* direction is

$$
\pi (2\rho + h)d\theta \times 1 + (\sigma\_{\theta} + d\sigma\_{\theta})h \times 1 - \sigma\_{\theta}h \times 1 = 0 \tag{2}
$$

Substituting boundary condition 1 θ θ = , 0 θ σ = , plasticity condition 2 θ *r s* σσσ τ −== , and integrating *<sup>r</sup>* σ along the cylinder surface, the mean feeding force *f* on the body of one unit can be expressed as:

$$f = l\_0^{\theta\_1} \sigma\_r \rho \cos \theta d\theta = \sigma\_s \rho \left[ \frac{m}{2} + \frac{m\theta\_1}{2} \sin \theta\_1 - \sin \theta\_1 - \frac{m}{2} \cos \theta\_1 \right] \tag{3}$$

where, *m hh* =− + (2 ) / ρ , <sup>1</sup> 1 θ tan ( / ) *a b* <sup>−</sup> <sup>=</sup> , 2 2 *h ab* = + , ρ = − ( )/ *w ah a* . *h* is the width of the sector body in radial direction, *σθ* and *σr* are stresses in tangential direction and radial direction, respectively. *σs* is the tensile strength, *τ* is the shear yield stress.

Fig. 26 shows the compressed zone of the workpiece**,** the total feeding force *F* can be expressed as

**Figure 26.** Sketch of compressed zone

48 Metal Forming – Process, Tools, Design

force by slab method.

**4.2. Radial feeding force calculation** 

when a quick prediction of forming force is required.

**Figure 25.** Sketch of principal stress on sector unit body

τρ

(2 ) 1 ( ) 1 1 0 *hd d h h*

 θ

The feeding force is critical to the choice and design of the capacity of the spinning equipment. An analytical model for calculating the forming forces is very useful, especially

As shown in Fig. 25, the final filled zone is the corner enclosed by roller, spindles and workpiece. A small sector body with thickness of one unit is analyzed to calculate the feeding

According to Fig. 25, the force equilibrium equation of the small sector body in *θ* direction is

θθ

 σ

 σ

 θ

 σ+ ×+ + ×− ×= (2)

$$F = \left[ {}\_{0}^{l} p dt = \sigma\_{s} \rho \left[ \frac{m}{2} + \frac{m\theta\_{1}}{2} \sin \theta\_{1} - \sin \theta\_{1} - \frac{m}{2} \cos \theta\_{1} \right] \right] l \tag{4}$$

Stamping-Forging Processing of Sheet Metal Parts 51

The middle span *b1* of the groove was calculated by equation (4.1). The parameter values of

Step No. *a b1 c1 c2 r1 r2 d*  1 1.5 4.10 4 5 0.5 1.5 17.4 2 1.5 5.56 4 5 0.5 2.0 14.4 3 1.5 7.57 3 4 0.5 2.6 12.1 4 1.5 10.2 2 3 0.5 0 10

the tooling are given in Table 2.

**Table 2.** Values of die dimension

**Figure 29.** Shape of ring roller

where, *Rr* and *Rw* are the radius of roller groove and final workpiece, respectively. ∆ is the feeding distance of the roller in one circle of the work-piece. The radii of the workpiece before the last circle is *Rw* + ∆, according to Heron's Formula, the length of contacted zone is

$$l = \sqrt{(2r\_r + 2r\_w + \Delta)\Delta(2r\_r - \Delta)(2r\_w + \Delta)} \mid [\mathcal{Q}(r\_r + r\_w)]$$

The key to using the equation 4.4 is to obtain the values of *a* and *b*. In fact, it could be supposed that *a* equals to *b,* and set the value to be the allowable radii *<sup>c</sup> r* of the required parts. Then 1 θ π= / 4 , *<sup>c</sup> abr* = = .

#### **4.3. Application**

In this section, an application example of SFP to manufacture a disc-like part of car with thickened rim will be introduced.

Fig. 27 shows a typical part manufactured by rim thickening. The rim thickness is 3.33 times to that of the center portion. The material is 1045 steel, whose Young's modulus is 210 GPa. The relationship of true stress to true strain at room temperature is *<sup>n</sup> C*ε σ <sup>−</sup> = with *C*=1019.7 MPa and *n*=0.11, respectively. Firstly, a pre-formed part shown in Fig. 28 was made by stamping.

**Figure 27.** Sketch of part manufactured by rim thickening

**Figure 28.** Pre-formed parts by stamping

Because of <sup>3</sup> <sup>3</sup> <sup>0</sup> / 1.494 1.4 *<sup>N</sup>* λ = => *t t* and <sup>4</sup> <sup>4</sup> <sup>0</sup> / 1.351 1.4 *<sup>N</sup>* λ = =< *t t* , according to section 4.1, a four-step thickening process is required. The diameter of the spindle was 240 mm, which had the same value with the inner diameter of the thickened rim. The rollers' shape and parameters are shown in Fig. 29, two angular parameters *c*1 and *c*2 between the groove walls and middle plane, and a fillet with *r1* are designed to avoid scratch of the work-piece. The middle span *b1* of the groove was calculated by equation (4.1). The parameter values of the tooling are given in Table 2.


50 Metal Forming – Process, Tools, Design

parts. Then 1

stamping.

**4.3. Application** 

θ π

thickened rim will be introduced.

*l*

= / 4 , *<sup>c</sup> abr* = = .

**Figure 27.** Sketch of part manufactured by rim thickening

<sup>0</sup> / 1.494 1.4 *<sup>N</sup>*

= => *t t* and <sup>4</sup> <sup>4</sup>

**Figure 28.** Pre-formed parts by stamping

λ

Because of <sup>3</sup> <sup>3</sup>

*s*

supposed that *a* equals to *b,* and set the value to be the allowable radii *<sup>c</sup>*

The relationship of true stress to true strain at room temperature is *<sup>n</sup> C*

σ ρ

1 <sup>0</sup> 11 1 sin sin cos ) 2 2 <sup>2</sup>

*m m <sup>m</sup> F pdt <sup>l</sup>* θ

== + −−

where, *Rr* and *Rw* are the radius of roller groove and final workpiece, respectively. ∆ is the feeding distance of the roller in one circle of the work-piece. The radii of the workpiece before the last circle is *Rw* + ∆, according to Heron's Formula, the length of contacted zone is

(2 2 ) (2 )(2 ) / [2( )] *r w r w rw l r r r r rr* = + +Δ Δ −Δ +Δ +

The key to using the equation 4.4 is to obtain the values of *a* and *b*. In fact, it could be

In this section, an application example of SFP to manufacture a disc-like part of car with

Fig. 27 shows a typical part manufactured by rim thickening. The rim thickness is 3.33 times to that of the center portion. The material is 1045 steel, whose Young's modulus is 210 GPa.

MPa and *n*=0.11, respectively. Firstly, a pre-formed part shown in Fig. 28 was made by

λ

4.1, a four-step thickening process is required. The diameter of the spindle was 240 mm, which had the same value with the inner diameter of the thickened rim. The rollers' shape and parameters are shown in Fig. 29, two angular parameters *c*1 and *c*2 between the groove walls and middle plane, and a fillet with *r1* are designed to avoid scratch of the work-piece.

θθ

 θ (4)

*r* of the required

ε

<sup>−</sup> = with *C*=1019.7

σ

<sup>0</sup> / 1.351 1.4 *<sup>N</sup>*

= =< *t t* , according to section

**Figure 29.** Shape of ring roller

The spinning machine with clamping capacity of 1000 kN was employed, as shown in Fig. 30. The tooling action is controlled by a PLC unit. The groove of the roller was heat-treated to hardness of HRC 58-62, and polished to surface roughness of 0.4 μm. The clamping force was set to be 500 kN during the rotary forming process. Graphite emulsion was used for lubricant and cooling. Feed speed was 0.05 mm per circle.

Stamping-Forging Processing of Sheet Metal Parts 53

**Figure 31.** Parts with thickened rim made by multi-step spinning

Xin-Yun Wang, Jun-song Jin, Lei Deng and Qiu Zheng

*Science and Technology, Wuhan, China* 

*State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of* 

[1] P. Jiang. Combined technology of cold stamping and forging for sheet metal and its

[2] P. Jiang, X. He, Y. Wu, H. Xie. New compound plastic forming technologies and their

[3] C.J. Tan, K. Mori, Y. Abe. Forming of tailor blanks having local thickening for control of wall thickness of stamped products. Journal of materials processing technology.

[4] K. Mori, S.Nishijima, C.J.Tan. Two-stage cold stamping of magnesium alloy cups having small corner radius. International Journal of Machine Tools & Manufacture.

[5] K. Mori, Y. Abe, K. Osakada, S. Hiramatsu.Plate forging of tailored blanks having local thickening for deep drawing of square cups.Journal of Materials Processing

[6] X.Y. Wang, M.L. Guo, J.C. Luo, K. Ouyang, J.C. Xia. Stamping-forging hybrid forming of double layer cup with different wall thickness. Materials Research Innovations,

application. Automobile technology and material. 2000,(9):8-11.

applications. Forging and stamping.2000,(1):38-41.

**Author details** 

**5. References** 

2008,202:443–449.

2009,49 :767-772.

2011,15(S1):435-438.

Technology. 2011,211:1569-1574.

Fig. 31 shows the parts manufactured by multi-step spinning. According to section 4.2 and the shape of the final part, we can get *a* = *b* = *rc* = 1*mm*, and the final feeding force is 86 kN calculated by equation 4.4 and 90 kN measured in experiment. The value of experiment is 4.6% higher than that calculated by equation 4.4.

**Figure 30.** Multi-step spinning machine

**Figure 31.** Parts with thickened rim made by multi-step spinning
