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journal homepage: www.elsevier.com/locate/jmatprotec

Analysis of splitting spinning force by the principal

stress method

Liang Huang, He Yang∗, Mei Zhan, Yuli Liu

College of Materials Science and Engineering, Northwestern Polytechnical University, P.O. Box 542,

Xi’an 710072, PR China

a r t i c l ei n f o

Keywords:

Splitting spinning force

Forming parameters

Principal stress method

Three-dimensional projected areas

a b s t r a c t

The splitting spinning which is designed to split a rotational disk blank into two flanges,

is one of newly rising, green flexible forming technologies, and it can be widely applied

to manufacture a whole pulley or wheel in fields of aerospace, automobile and train. The

investigation of forming parameters influencing on splitting spinning force can provide

the foundation for the choice of equipments, the design of dies and the determination

of processing parameters. This paper aims at developing a reasonable formula between

splitting spinning force and forming parameters by the principal stress method, and then

the reliability of the formula is verified by the comparisons with experimental data. Mean-

while, both a reasonable method of calculating the three-dimensional projected areas and a

more effective method of solving the average angle in the deformation zone are presented.

Furthermore, based on the formula, the laws of initial thickness and initial diameter of

workpiece, diameter and splitting angle of splitting roller and feed ratio of splitting spin-

ning influencing on splitting spinning force are investigated. The achievements may serve

as an important guide for the determination and optimization of forming parameters of

splitting spinning.

© 2007 Elsevier B.V. All rights reserved.

1.Introduction

The splitting spinning is designed to split a rotational disk

blank from the outer rectangular edge into two flanges using a

rollercalledsplittingrollerwithasharpcorner,andthenshap-

ing spinning is done by other two or three rollers (Hauk et al.,

2000; Schmoeckel and Hauk, 2000). The splitting spinning pro-

cess is a continuous and local plastic forming process with the

feeding of rollers, companying with many advantages, such

as, high quality, high efficiency, low cost, etc. Therefore, it can

be applied to manufacture a whole wheel or pulley with alu-

minum alloy or soft steel (in heating condition) in fields of

aerospace, automobile and train.

∗Corresponding author. Tel.: +86 29 8849 5632; fax: +86 29 8849 5632.

E-mail address: yanghe@nwpu.edu.cn (H. Yang).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmatprotec.2007.11.181

Until now, three methods solving the plastic forming are

theoretical analysis, experiments and numerical simulation.

Theoretical analysis is often adopted to develop the func-

tional relationship among forming parameters directly based

on some assumptions. In the past, much work about the the-

oretical analysis of spinning force aiming at cans and cones

have been made (Kalpakciaglu, 1961; Avitzur and Yang, 1960;

Chen et al., 2005), but the theoretical analysis of splitting spin-

ning force is scarce.

This paper aims at developing the theoretical analysis of

splitting spinning force by the principal stress method and

establishing a practical formula between spinning force and

forming parameters, and the reliability of the formula is veri-

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journal of materials processing technology 2 0 1 ( 2 0 0 8 ) 267–272

fiedbycomparisonswithexperimentaldata.Furthermore,the

influences of different forming parameters on spinning force

are investigated.

2.Analysis of splitting spinning force

2.1.Basic principles

The processes for the principal stress method are as follows.

(1) The stress balancing differential equations based on the

element body picked from deformable body are simplified

to ordinary differential equations. (2) The normal stress is

assumed to be the principal stress, removing the effects of

tangential stress. So, yield equation can be simplified to linear

equation. (3). Due to the solutions of stress balancing differen-

tialequationsandyieldequationbasedonthestressboundary

conditions, the stress distributions and the forming force are

both solved.

2.2. Assumptions and simplifications

In the theoretical analysis, the following assumptions and

simplifications are adopted: (1) the workpiece is homo-

geneous, isotropic elastic–plastic body, without volumetric

changeandno-strainhardening.(2)MaterialfollowsvonMises

yield criterion. (3) The frictional mode follows the classical

Coulomb friction rule, and the effects of strain rate and tem-

perature are neglected. (4) The workpiece deforms locally and

other parts are rigid.

2.3.Development of the functional relationship

Based on the global cylindrical coordinates of (R, ?, Z) and the

local plane rectangular coordinates of (X, Y), the force analysis

ofdeformationzoneofsplittingspinningisillustratedinFig.1.

Though the force analysis of the element body picking from

the quadrilateral called ABCD in the deformation zone, the

stress balancing differential equation is followed:

due to

?

Fx= 0, (?x+ d?x)h + 2?utan˛dx

= 2?dx + ?x(h + 2tan˛dx).

(˛ > 0)(1)

Fig. 1 – Force analysis of deformation zone of splitting

spinning. (a) Geometrical parameters analysis of

deformation zone and (b) Force analysis of element body.

The static balancing relationship is written as

?u+ ? tan˛ = ?y, namely,?u= ?y− ? tan˛,

(2)

According to the classical Coulomb friction mode, the shear

stress is expressed by ? =??u, and then Eq. (2) becomes

?u= ?y− ??utan˛, namely,?u=

?y

1 + ?tan˛.

(3)

According to von Mises yield criterion, we obtain

?x= ?y+ ˇKεn.

(4)

Here, ˇ is an additional coefficient for plane strain deforma-

tion approximation.Since deformation levels are various, the

strain ε may be approximately considered as the equivalent

strain ¯ ε. Thus, Eq. (4) becomes

?x= ?y+ ˇK¯ εn.

(5)

Consequently, Eq. (1) can be simplified, i.e.

d?x

2? + 2?xtan˛ − 2?utan˛=dx

h.

(6)

After replacing Eq. (3) into Eq. (6), it becomes

d?x

2? × ?y/(1 + ?tan˛) + 2?xtan˛ − 2tan˛ × ?y(1 + ?tan˛)

=dx

h.

(7)

And then, after replacing Eq. (4) into Eq. (7), it becomes

d?y

2?×?y/(1+?tan˛)+2(?y+ˇK¯ εn)tan˛−2tan˛×?y/(1 + ?tan˛)

=dx

h.

(8)

After rearranging Eq. (8), it becomes

d?y

M?y+ N=

dx

−2xtan˛ + h0

,

(9)

noting that M=(2?+2?tan2˛)/(1+?tan2˛), N = 2ˇK¯ εntan˛.

According to the integral rules, Eq. (9) becomes

?y=C

M(−2xtan˛ + h0)−M/2tan˛−N

M.

(10)

According to the stress boundary conditions, when x=xe,

?y=?ye, where ?yeis minor and can be negligible. Thus, the

constant value of C is given by

C = (N + M?ye)(−2xetan˛ + h0)M/2tan˛.

(11)

From Fig. 1, we obtain

xe=h0− t0

2tan˛.

(12)

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269

Fig. 2 – Details of disk blank/splitting roller contact area on the circumferential view.

Substitution of Eq. (12) into Eq. (11) leads to

C = (N + M?ye)t0M/2tan˛.

(13)

Hence, the compressive stress on contact surface of the work-

piece can be calculated by

?y=

?N

M+ ?ye

??−2xtan˛ + h0

t0

?−M/2tan˛

−N

M,

(14)

where h0=(t0+2Cssin˛)/cos˛, where Csis the reduction depth

per revolution.

The splitting spinning force can be calculated by

p =1

x

?x

0

?ydx

=

t0

−2xtan˛

??−2xtan˛+h0

?N

M+ ?ye

?

×

t0

?−M/2tan˛

−

?h0

t0

?−M/2tan˛?

−N

M.

(15)

Hence,

expressed by

three-dimensionalsplitting spinningforce is

Ft= pAt, Fr= pArandFz= pAz,

(16)

where At, Arand Azis the projected deformation areas of the

contact curved surface between disk blank and splitting roller

inthecircumferential,radialandaxialdirections,respectively.

Consequently, the total splitting spinning force can be

expressed by

F =

?

F2

t+ F2

r+ F2

z.

(17)

2.4.

deformation areas

Analysis of three-dimensional projected

2.4.1.

In this study, the splitting spinning force is divided into three

orthogonal force components, so the projected deformation

areas in the circumferential, radial and axial directions can be

calculated respectively (Chen et al., 2005).

Fig. 2 depicts the circumferential deformation area of disk

blank/splitting roller. The distance A1A2is computed as L1=

A1A2= (f + Cstan˛)cos˛, and the distance L2is computed as

Three-dimensional projected deformation areas

Fig. 3 – Details of disk blank/splitting roller contact area on the radial view.

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journal of materials processing technology 2 0 1 ( 2 0 0 8 ) 267–272

Fig. 4 – Details of disk blank/splitting roller contact area on the axial view.

L2=?cos˛+fsin˛cos˛. So, the circumferential contact areas

indicated by dotted hatch on the left side of Fig. 2 can be

approximately computed as follows:

At∼= (f + Cstan˛)cos˛(?cos˛ + f sin˛cos˛).

(18)

Fig.

blank/splitting roller. On the left side of the drawing is a

magnified detail of the contact area. So, the radial contact

areas can be calculated approximately as follows:

3 depictstheradial deformationarea ofdisk

Ar∼= 2(f + Cstan˛)cos˛Dr?0

2

.

(19)

Similarly, Fig. 4 depicts the axial deformation area of

disk blank/splitting roller. The distance L3 is computed as

L3=(f+Cstan˛)sin˛. So, the axial contact areas can be calcu-

lated approximately as follows:

Az∼=(f + Cstan˛)sin˛

2

Dr?0

2

.

(20)

2.4.2.

Fortheangleofdeformationzone-?0,amoreefficientcalculat-

ingmethodisgiven,thatistoreplace?0withtheaverageangle

of ?0. The details are followed that the contact pair between

disk blank and splitting roller is approximately simplified as

twoplanetangentcirclesinFig.5.InFig.5,O1A = Dr/2.Accord-

ing to the analysis, we obtain CD = f and CE = (h0− t0)/2tan˛.

R0is the instantaneous radius of disk blank, and the relation-

ship between R0and R is as follows:

Average angle of deformation zone

R = R0+ (CE − CD) = R0+

h0− t0

2tan˛ − f.

(21a)

In the right-angle triangle of O1AB, we obtain

AB =Dr

2

sin?0andO1B =Dr

2

cos?0.

(21b)

In the right-angle triangle of OAB, we obtain

AB2+ OB

2= OA

2.

(21c)

Fig. 5 – Illustration of two plane tangent circles between disk blank and splitting roller.

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271

Fig. 6 – Comparisons of computational results and

experiment data.

Meanwhile, we obtain

OO1= R +Dr

2

= O1B + OB.

(21d)

Therefore, ?0is obtained from the Eq. (21a) to the Eq. (21d) in

the following form:

?0= cos−1

?

D2

r/2 − f2+ R(Dr− 2f)

Dr(R + Dr/2)

?

.

(21)

2.5. Verification of theoretical analytic equations

The computational results are compared with experiment

data (Schmoeckel and Hauk, 2000) to verify the above equa-

tionsofsplittingspinningforce.Thecomputationalconditions

are comprised of processing parameters and material prop-

erties of splitting spinning. The processing parameters of

splitting spinning include the following: (1) initial diameter

of disk blank, 100.0mm; (2) initial thickness of disk blank,

10.0mm; (3) diameter of splitting roller, 99.17mm; (4) splitting

angle of splitting roller, 45◦; (5) radius of corner of splitting

roller, 1.0mm; (6) feed rate of splitting roller, 1.0mm/s; (7)

rotate speed of mandrels 150rpm; (8) feed ratio of splitting

spinning, 0.4mmrev−1; (9) reduction depth per revolution,

0.0mm; (10) Friction coefficient between disk blank and split-

ting roller, 0.05; (11) Friction coefficient between disk blank

and mandrels, 0.05. The Material properties of splitting spin-

ning include the following: (1) material type (China), LF2M; (2)

density, 2680kg/m3; (3) Young’s modulus, 70.0GPa; (4) yield

strength, 90.0MPa; (5) Poisson’s ratio, 0.3; (6) harden expo-

nent, 0.16; (7) constitutive equations, ? =Kεn(Shichun and

Miaoquan, 2002).

Based on the characteristics of splitting spinning, an addi-

tional coefficient of plane deformation ˇ=1.8 is adopted. The

comparisons of splitting spinning force between calculating

results and experiment data are shown in Fig. 6. In Fig. 6, there

is a discrepancy between the computational results and the

experimental data; however, its relative values are less than

Fig. 7 – Spinning force variations in different initial

thicknesses of disk blank.

17.02%. So, computational results are in good agreement with

experimental data, and the discrepancy is in control. Hence,

the proposed equations of splitting spinning force are vali-

dated.

3.Results and discussion

The equations of splitting spinning force may describe the

quantitative relationships between forming conditions and

forming results. Therefore, this study investigates the influ-

ences of different forming parameters on splitting spinning

force, and obtains the corresponding forming laws. And other

forming parameters are the same as the ones in Section 2.5.

The spinning force variations in different initial thick-

nesses of disk blank are shown in Fig. 7. In Fig. 7, the splitting

spinning force increases with the increase of feed amount in

thestateofdifferentinitialthicknessesofdiskblank.Butwhen

Fig. 8 – Spinning force variations in different feed ratios of

splitting spinning.

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thefeedamountislessthan4mm,thechangesofinitialthick-

nesses of disk blank have no obvious effect on the spinning

force.

Similarly, the spinning force increases with the increase of

initial diameter of disk blank, diameter of splitting roller and

the feed amount in the state of different angles of the splitting

rolleraccordingtoEq.(17).Butwhenthesplittingangleismore

than 30◦, the changes of splitting angle of splitting roller have

no obvious effect on the spinning force.

The spinning force variations in different feed ratios of

splitting spinning are shown in Fig. 8. When the value of feed

ratio of splitting spinning is smaller, the influence of feed ratio

onsplittingspinningforceisnotobvious;butwhenthevalueis

larger, the splitting spinning force increases with the increase

of feed amount obviously.

4. Conclusions

In this study of the theoretical analysis of splitting spinning

force, the following conclusions have been drawn.

1. A reliable theoretical model established by the principal

stress method is proposed for the calculation of the split-

ting spinning force. Meanwhile, both a reasonable method

of calculating the three-dimensional projected areas in

circumferential, radial and axial directions and a more

effective method of solving the average angle in the defor-

mation zone are presented.

2. On the basis of the above equations of the splitting spin-

ning force, the influencing laws of five important forming

parametersonthesplittingspinningprocessarediscussed,

including initial thickness of disk blank, initial diameter of

disk blank, diameter of splitting roller, splitting angle of

splitting roller and feed ratio of splitting spinning.

3. The achievements of this study may provide an important

guideline for the determination and optimization of form-

ing parameters of splitting spinning.

Acknowledgements

The authors would like to thank the National Natural Sci-

ence Foundation of China (Nos. 50405039 and 50575186), the

National Science Found of China for Distinguished Young

Scholars (No. 50225518) and the Foundation of NWPU for the

support given to this research.

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