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A lower upper-bound solution for shear spinning of cones
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International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
A lower upper-bound solution for shear spinning of cones Chul Kima , S.Y. Junga , J.C. Choib;∗ a
b
Research Institute of Mechanical Technology at Pusan National University, Kumjeong-Gu, Pusan, South Korea Department of Mechanical Design Engineering, ERC for NSDM at Pusan National University, 30 Changjeon-dong, Kumjeong-Gu, Pusan, 609-735, South Korea Received 17 September 2002; received in revised form 4 November 2003; accepted 5 November 2003
Abstract The shear spinning process, where the plastic deformation zone is localized in a very small portion of the work piece, is introduced for the manufacturing of large conical shapes. This process seemingly shows a promise for increasingly broader application to the production of axially symmetric parts. This study is to gain a better understanding of the process of shear spinning and to propose a lower upper-bound solution for shear spinning cones. Velocity 5elds and strain rates are derived from considering the adequate deformation mode and the contact factor is introduced to obtain the lower upper-bound power. The theoretical values are compared with the experimental results of various materials. Al-1100-0, Al-1100-H14, and Al-6061-0. A comparison shows that theoretical prediction is reasonably in good agreement with experimental results. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Shear spinning; Upper bound solution; Contact factor; Dynamometer
1. Introduction The rolling process which uses a roller and a mandrel to deform the metal plates, plastically, into conical or other shapes of circular cross-section, such as hemispheres, may be technically described as spinning. Conventional spinning or hand-spinning requires a blank which has the same wall thickness as the 5nished part. However, shear spinning di=ers from conventional spinning in that the blank diameter is the same as that of the 5nished part and the thickness of the 5nished part is determined by the blank thickness multiplied by the sine of half the included angle of the 5nished part. This “sine law” expresses the conditions that any circular element retains its original diameter during spinning and that the total volume of metal in a certain diameter will not change. In contrast, the term of conventional spinning is also used for the spinning process when the radial position of an element in ∗
Corresponding author.
0020-7403/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2003.11.002
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Nomenclature f Ft Ff Fn N m r0 ra ; rc ; ri ; rb S S1 S2 t0 tf Vx ; Vy ; Vz Vr ; V ; Vz w˙ zD H m d H ˙ ˙rz ; ˙z ; ˙r H Hm m p ; H H
feed of roller tangential force component in shear spinning feed force component normal force component speed of rotation of mandrel contact factor corner radius of roller cone radii area BQFCRB area BMICRB area BQFIMB initial thickness of blank 5nal thickness of cone velocity in x; y; z direction velocity in r; ; z direction total energy of deformation z-value of the curve SMR half cone angle of mandrel mean e=ective stress total e=ective strain strain rate strain rate tensors in cylindrical coordinates 5nite e=ective strain mean e=ective strain in the deformation zone length of blank in its rotational direction assumed actual contact length ideal geometric contact length radii of curvature of deformation
the blank reduces appreciably while the blank is deformed. The spinning process is an example of a new trend, which exists in metal working operations. This trend is toward localizing the deformation zone to a small region of the work piece in order to reduce the forming forces and consequently make it possible to reduce the size of the machine required to carry out such processes. Because of the importance of the spinning process in the formation of aircraft and missile components, a number of investigations on the spinning process, both theoretical and experimental, have appeared in the previous literatures. For example, Colding [1] considered cone spinning as a combination of rolling and extrusion process, whereas Kalpakcioglu [2] assumed a simple shear mechanism for the analysis of the working forces. The complex straining e=ect was introduced into the solution of cone spinning by Avitzur and Yang [3], Kobayashi et al. [4] and Hayama et al. [5]. But their equations for predicting the tangential component force are of a complex nature and require long computation time
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
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for its solution. Hayama and Amano [6,7] studied the contact between blank and roller during shear spinning by experimental and theoretical procedure. Moreover, Hayama [8] estimated concretely the three components of working force using the contact area between blank and roller obtained from his previous work. A test method for determining the spinnability of cones was proposed by Kegg [9]. This consists of shear spinning a blank on an ellipsoidal mandrel. On the other hand, Kalpakcioglu proposed the stress system to get the maximum allowable reduction in shear spinning process. Sortais et al. [10] studied the cone wall thickness variation in conventional spinning of cones and derived the theoretical tangential force component by the deformation energy method. Kobayashi established the condition of Lange wrinkling in conventional spinning of cones by modifying the theory of instability in deep-drawing of cups. And Slater gives the approximate upper-bound estimates for tangential force during shear spinning of cones assuming the ideal axi-symmetric deformation. But this assumption is not good enough for the shear spinning process. As a result, it is concluded that each previous theory for the working forces agreed with the experimental data for only a limited range of process variables, and did not give good results for other working conditions. 2. Mechanics of shear spinning of cones 2.1. Deformation mode The deformation mechanism of shear spinning of cones is shown schematically in Fig. 1(a). The blank material is a disk of diameter Db and uniform thickness t0 . The disk is mounted on a circular conical mandrel, which is clamped to the head of the spinning machine and rotated. A forming roller is driven on tracks on the bed of the machine parallel to the side of the mandrel. The process is characterized by the fact that the radial position of an element in the blank remains constant during deformation and the angular velocity is constant through the whole work piece except the deformation zone. This demands that the initial disk thickness t0 and the 5nal thickness of the cone tf for the half cone angle are related by the equation: tf = t0 sin :
(1)
And, for simplicity, it is assumed that during one revolution of the mandrel, the roller holds the same position, and after one complete revolution of the mandrel, the roller feeds f sin to x- direction and f cos to z- direction. The form of deformation of the z– plane about a radial element of the blank is assumed to take the form of the solid line SQMR in Fig. 1(b), where z and denote, respectively, the height and the length of the blank in its rotational direction. Fig. 1(b) corresponds to Figs. 2 and 3(a). The curved line m shows where the roller begins to make contact with the blank, while the line is the one obtained as a result of the geometrical calculation. The line where the bending starts can be drawn as BQFH in Fig. 3(a) and the shaded area, BIC, is the actual contact area. Fig. 3(b) shows the radial region of deformation. The deformation region is bounded by rb and rc . Let us consider an element r in Fig. 1(b). The element starts to deform at point S where its radius of curvature tend to in5nity. After passing the point S, the radius of curvature decreases gradually and at point Q one half portion of an element begins to yield by uniaxial tension or compression by bending. After passing the point Q, the circumferential extension of an element r is assumed to
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 1. Schematic description of the deformation in shear spinning of cones: (a) R–Z plane; (b) Z– plane.
y
x
θ vx vθ vR vy
v
Fig. 2. Velocity components in X –Y plane.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
be constant until it reaches the point M while the shear strain Q − m zero to (cot "2 − cot "1 ) along QM, Q where Q and m are given by the following equations:
rz
1897
is increased monotonically from
m = m =r:
(2)
Q = Q =r:
(3)
The di=erence of ((@ rz =@r) dr) makes the radius of curvature of the element decrease along QM and the element reaches the point M. After passing the point M, the extension of an element changes its sign and assumed to be constant while shear strain rz is increased monotonically, starting from ((Q − m )=Q )(cot "2 − cot "1 ) to (cot "2 − cot "1 ) on the roller along MR. After reaching the point R, the circumferential extension of an element r will be unloaded elastically. In Fig. 3(a), the plastic deformation occurs only in region BFCB and elastic bending occurs in region FHLCF.
Fig. 3. Model of deformation: (a) deformation region in –r plane; (b) shear deformation.
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To get the contact length m , we assumed the following relations: 2 m2 = (r) H = 2(1 − m) 2NZ
(4)
and
m (r): H 1−m Noting that the ideal geometric contact length is given by H (r) =
(5)
= r0 ;
(6)
where (r) H and H (r) are the mean radius of curvature for (r; ) and (r; ), respectively, and 0 is the ideal geometric contact angle and is given by the following equation: √ 1 −D + D2 − CF −1 rc − ; (7) 0 = cos r C where A = 2(rc + R0 cos ); B = r 2 + (z − z02 )2 + 2(z − z02 )R0 sin − rc2 − r02 ; C = A2 + 4R20 sin2 ; D = AB − 2R20 (z − z02 ) sin 2; F = B2 − 4R20 {r02 − (z − z02 )2 cos2 }; z − z02 = r02 − (r01 − r)2 − f cos for rb 6 r 6 r01 ; = r0 − f cos
for r01 6 r 6 rc :
From Eq. (4), we obtain √ m = 1 − m
(8)
2.2. Velocity 7elds and strain rates To solve our particular deformation 5eld, the velocity 5eld in the deformation zone was computed 5rst. Because of the deformation mode, the Low line has to be of the form, (x; y; z) = x2 + y2 = C1
(9)
"(x; y; z) = z − zD = C2 ;
(10)
where zD is the z-value of the curve SMR in Fig. 1(b). RM is the contacted portion of the roller with the blank and MS is composed as in Fig. 1(b) to get the same shear strain, z , along both sides of point M.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
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When S and M approach N, the Low model becomes similar to the one proposed by Avitzur for shear spinning of cones. In this case a velocity discontinuity exists at N and the shear loss term, which is neglected by Avitzur, should be involved in the power. The components of the velocity 5eld, Vx ; Vy ; Vz , assume the form,
@ @" @ @" − ; Vx = ) @y @z @x @y
@ @" @ @" Vy = ) − ; (11) @z @x @x @z
@ @" @ @" Vz = ) − @x @y @y @x and from Fig. 2, one gets V = Vy cos − Vx sin = 2*rN; Vr = Vy sin + Vx cos :
(12)
Inserting Eqs. (9) and (10) into Eqs. (11) and (12), one gets ) = −*N:
(13)
Vr = 0; V = 2*rN;
@zD @zD −y Vz = −2) x @y @x
= −2)
@zD @zD = 2*N ; @ @
(14)
noting that x = @y=@ and −y = @x=@. From the velocity 5elds of Eq. (14), one gets the strain rates in cylindrical polar coordinates as follows: rz =
@ 2 ZD 1 @Vz = *N ; 2 @r @r@
1 @ 2 ZD 1 @Vz = *N ; 2 r@ r @ 2 all other ij = 0. Z =
(15)
2.3. The power The rate of total work done on metal under the deformation zone becomes
H @Vz 2 1 @Vz 2 √ + dV: w˙ = @r r @ 3 vol:
(16)
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Inserting Eq. (10) into (11), one gets 2 2
2 @ ZD 1 @2 ZD 2*Nt0 m + 2 ds w˙ = √ @r@ r @ 2 3 S 2 2 2 2 2 2 2 2 2*Nt0 m @ ZD @ ZD 1 @ ZD 1 @ ZD = √ + 2 ds1 + + 2 ds2 2 @r@ r @ @r@ r @ 3 S1 S2
2 2 2 2
2 2 @ ZD @ ZD r r 1−m 2*Nt0 m ds1 + + + ds2 : = √ @r@ H @r@ m H 3 S1 S2
(17)
To obtain a lower upper-bound on energy, one needs @w˙ = 0: @m
(18)
Di=erentiating Eq. (17) with respect to m and inserting it into Eq. (18), one obtains m=1
and S1 = 0; S2 = S
(19)
Inserting Eq. (19) into Eq. (17), one obtains 2 2*Nt0 H m RH @ ZD √ w˙ = ds 3 S @r@ @ZD r=rc 2*Nt0 H m RH √ d = 3 @ r=rb 2*Nt0 H m RH √ NZD = 3 r=rc =
2*Nt0 H m RH √ f cos ; 3
where RH =
rc +r01 ; @Z@D |r=rb 2
(20)
=0
NZD |r=rc = f cos : Eq. (20) was proposed by Avitzur and Yang. They derived it by a simpli5ed model of pure shear deformation. They neglected the shear loss term along - in Fig. 4. The shear loss along - is pretty high and cannot be neglected in the calculation of the power in usual upper-bound solution based on the kinematically admissible velocity 5elds. In this analysis, the frictional losses between blank and roller was neglected because of the small discontinuity of the tangential velocity on the roller. In Avitzur the power consumed by the feed force is less than 7% of the total power in a typical case. So we can neglect the feed power and the power consumed can be regarded as done entirely
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1901
Fig. 4. Deformation in process proposed by Avitzur.
by the tangential force. One now gets w˙ = Ft v ; H . where v = 2*RN And the tangential force component on the roller may be written in the form w˙ Ft = : H 2*RN From Eq. (20) one obtains the tangential component of forces as follows: H m Ft = √ t0 f cos : 3 Ft 1 = √ t0 f cos : H m 3
(21)
(22)
(23) (24)
3. Experimental procedure In order to test the validity of the foregoing theory and furnish the Low stress required for calculating the working forces, the spinning test and the plane strain compression test were conducted on aluminum alloys. 3.1. Spinning test The experimental work of shear spinning was carried out on a spinning machine, Auto-spin 5060, and working forces were measured by three components of tool dynamometer, which was specially designed and manufactured as shown in Fig. 5. It was manufactured to measure the three force components, named as tangential force, feed force, and normal force, which has load range, 150 kg, for tangential force and 1000 kg for the others. Alloy steel, SCM2, was used as a base metal in manufacturing the tool dynamometer. Fig. 6 shows the con5gurations of the tool dynamometer and the locations of the electric resistance strain gauge (EA-06-125BT-120, M-M co.) which are
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 5. Tool and dynamometer set-up and worked cones.
represented by circled numbers. And the circuits of the strain gauge are described in Fig. 7. The tool dynamometer was calibrated in the calibration stand(Model B-2, LTV Aerospace Co.) where tool dynamometer was loaded individually for the 3-directions to the range of dead weight, while keeping the balance. 20, 50, 100 lb weights were employed and the excitation voltage to the strain gauge circuits was 5 V. And the force components were calibrated at the reference point 292:5 mm from the datum line, therefore, the calibrated forces should be multiplied by 292.5/Real distance, where Real distance is 230 mm for tangential force and 292:5 mm for feed force. Let Ft ; Ff , and Fn be the calibration load and R1 ; R2 , and R3 be the corresponding strains (V), respectively. Then it can be written as R 1 = a 1 F t + b 1 F f + c 1 Fn ; R2 = a 2 F t + b 2 F f + c 2 F n ;
(25)
R 3 = a 3 F t + b 3 F f + c 3 Fn in which a1 ; b1 , and c1 are the transformation coeRcients of the strain corresponding to the main force components when Ft ; Ff , and Fn act individually.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
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Fig. 6. Tool dynamometer and roller.
For the other coeRcients, the same procedures can be adopted. The coeRcients ai ; bi , and ci can be solved by the linear least-squares method, and Eq. (25) can be written in the form, F t = A 1 R 1 + B 1 R2 + C 1 R 3 ; F f = A 2 R 1 + B 2 R2 + C 2 R 3 ;
(26)
F n = A 3 R1 + B 3 R2 + C 3 R 3 : The three force components can be calculated from Eq. (26). The results of computation are shown as follows: Ft (kg) = (A1 R1 + B1 R2 + C1 R3 )
292:5 1 ; 230 2:2
Ff (kg) = (A2 R1 + B2 R2 + C2 R3 )
1 ; 2:2
Fn (kg) = (A3 R1 + B3 R2 + C3 R3 )
1 ; 2:2
where A1 = 0:02661; A2 = 0:00373; A3 = 0:0001296; B1 = −0:0007503; B2 = −0:1274; B3 = −0:000804; C1 = −0:00009679; C2 = 0:01106; C3 = 0:2396.
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Fig. 7. Strain gauge circuits.
The experimental values were recorded on the data coder (7404, Hewlett-Packard Co.) and signal conditioner(Type I-080, Bell & Howell Co.). It is necessary to obtain the instantaneous cone radius, ra , corresponding to each force reading, since the force component as well as the cone radius change during a single spinning test. After the blank was spun into a cone, the 5nal thickness was checked according to Eq. (1) by using the ultrasonic pulse echo type thickness gauge (Model CL 204 K-B Co.). The test conditions employed in the spinning tests are given in Fig. 8 and the typical read-out of the three components is shown in Fig. 9. 3.2. Plane-strain compression test The compression test and shear test are most e=ective to obtain the deformation characteristics applicable to several metal working operations as a shear spinning process. In order to maintain equality between the test specimen and the blank for spinning, the plane strain compression tests are applied. It should, however, be emphasized that the yield stress measured is Y , the yield stress
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911 Condition Material Diameter of Roller DR(mm) Thickness to(mm)
Fig. 10 Al 1100-H14
Fig. 11 Al 1100-0
Fig. 12 Al 6061-0
Fig. 13 Al 6061-0
125
125
125
125
3.2
3.2
2.3
2.3
Feed f(mm/rev.)
0.25 0.85 0.45 1.05 0.65
Mean Flow Stress
14.91 ~ 15.00
0.25 0.85 0.45 1.05 0.65 11.65 ~11.88
0.25 0.85 0.45 1.05 0.65 15.52 ~15.67
0.25 0.85 0.45 1.05 0.65 15.59 ~15.81
26°50`
26°50`
26°50`
26°50`
7200
7200
7037
7037
5
5
9
5
100
100
100
100
100
100
100
100
σm (kg/mm2)
Half Cone Angle α (deg.)
Young’s Modulus E(kg/mm2) Round-off Radius ro
Revolution per Minutes N(rpm) Measuring Radius ra(mm)
1905
Fig. 8. Test conditions in shear spinning of cones.
Fig. 9. Typical read-out of working forces.
in plain strain, not y , the yield stress in uniaxial tension, as measured in the tensile test. These are related: Y = 1:155y
(27)
The measured strain in the thickness direction, h , and the e=ective strain, , H are related: H = 1:115h
(28)
The experiments were done for three kinds of specimens, AL 1100-0, Al 1100-H14 and Al 6061-0. And each specimen was compressed to get extension along three directions, 90◦ ; 45◦ , and 0◦ , to the rolling direction of the original sheet. The experimental values are taken, with the mean value of the three kinds of specimen for each tested material, in which the linear least-squares method was applied.
1906
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
4. Discussion The tangential force in spinning is a very important quantity and is of great interest both to the designers of power spinning machines and to the production engineers faced with the problems of suRcient power to carry out the spinning operations [6]. The experimental working forces and the theoretical values of the spinning processes are shown in Figs. 10–13. The measured working forces are in reasonable agreement with the theoretical evaluations for all working conditions. The e=ect of material properties can be shown in Figs. 10–11. Both the strain-hardening material, Al-1100-0 and the non-straining material, Al-1100-H14 give fairly good agreement with
Fig. 10. Relation between tangential forces and feed of roller f.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1907
Fig. 11. Relation between tangential forces and feed of roller f.
theoretical values. Reviewing from Figs. 12–13, it is found that the change of corner radius of roller r0 gives no large di=erences of working forces, that is, as the radius of the round-o= of the roller gets bigger, the working forces change slightly. In the calculation of theoretical values, the mean e=ective stress, H m is the Low stress of material at the mean e=ective strain, Hm , which is given by rc 1 cot "2 √ dr Hm = rb − r c r b 3 r0 − r02 − (rc − rb )2 √ = ; (29) 3(rc − rb ) where cot "2 =
r0 − r r02
− (rc − rb )2
for rb 6 r 6 rc :
1908
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 12. Relation between tangential forces and feed of roller of f.
Figs. 14–16 show the stress–strain relations of plane strain compression test for three Al alloys, the mean e=ective strain Hm in shear spinning process is given by Eq. (29) and the mean e=ective stresses for the range, 0.36 – 0.40, of Hm are evaluated as H m = 14:91–15:00 kg=mm2
for Al 1100 − H14;
= 11:65–11:88 kg=mm2
for Al 1100 − 0;
= 15:52–15:81 kg=mm2
for Al 6061 − 0:
The mean e=ective stress, H m , is the average value of H in the deformation region in Eq. (17). So the Low stress of material at the mean e=ective strain, Hm is the mean e=ective stress, H m . Kobayashi et al. [4] used the mean e=ective stress as follows H m =
H d H d H
(30)
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1909
Fig. 13. Relation between tangential forces and feed of roller f.
Fig. 14. Equivalent stress–strain curve from compression test (1100-0).
which is independent of the history of deformation in the deforming zone. The mean e=ective stress, H m , for the mean e=ective strain, Hm , makes the theoretical values of working of force approach to the measured values.
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Fig. 15. Equivalent stress–strain curve from compression test (1100-H14).
Fig. 16. Equivalent stress–strain curve from compression test (6061-0).
5. Conclusion In the study of the mechanics of the shear spinning of cones, the following conclusions have been reached: 1. A simpli5ed solution of shear spinning of cones by a pure shear model coincides with the lower upper-bound solution derived in the present paper. 2. The tangential force is linearly proportional to the yield limit, H m blank thickness, t0 feed of roller f and cosine of the included semi-cone angle, . 3. Theoretical values can be applied to strain-hardening materials as well as non-strain hardening materials.
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4. The present theory gives a better agreement with the experimental data than any other existing theories for a wide range of process variables. Acknowledgements This work was supported by Research Funds of Pusan National University through Brain Busan 21. References [1] Colding BN. Shear spinning. ASME No. 59-prod-2, May, 1959. [2] Kalpackcioglu S. On the mechanics of shear spinning. Journal of Engineering for Industry. Transactions of ASME, Ser B 1961;83:125–30. [3] Avitzur B, Yang CT. Analysis of power spinning of cones. Journal of Engineering for Industry. Transactions of ASME, Ser B 1960;82:231–45. [4] Kobayashi S, Hall IK, Thomsen EG. A theory of shear spinning of cones. Journal of Engineering for Industry. Transactions of ASME, Ser B 1961;83:485–95. [5] Hayama M, Murota T, Kudo H. Study of shear spinning. (A) 1st Report, Transactions of JSME 1964;30(220):1450–7. (B) 2nd Report, Transactions of JSME 1964;30(220):1458–65. (C) 3rd Report, Transactions of JSME 1965;31(228):1259–69. [6] Hayama M, Amano T. Analysis of contact form of roller on sheet blank in shear spinning of cones. Journal of JSTP 1975;16(174):559–65. [7] Hayama M, Amano T. Experiments on the mechanism of shear spinning of cones. Journal of JSTP 1975;16(172): 371–8. [8] Hayama M. Analysis of working forces in shear spinning of cones. Journal of JSTP 1975;16(175):627–35. [9] Kegg RL. A new test method for determination of spinnability of metals. Journal of Engineering for Industry. Transactions of ASME, Ser B 1961;119 –24:119–24. [10] Sortais HC, Kobayashi S, Thomsen EG. Mechanics of conventional spinning. Journal of Engineering for Industry. Transactions of ASME, Ser B 1963;85:44–8:346–50.
International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
A lower upper-bound solution for shear spinning of cones Chul Kima , S.Y. Junga , J.C. Choib;∗ a
b
Research Institute of Mechanical Technology at Pusan National University, Kumjeong-Gu, Pusan, South Korea Department of Mechanical Design Engineering, ERC for NSDM at Pusan National University, 30 Changjeon-dong, Kumjeong-Gu, Pusan, 609-735, South Korea Received 17 September 2002; received in revised form 4 November 2003; accepted 5 November 2003
Abstract The shear spinning process, where the plastic deformation zone is localized in a very small portion of the work piece, is introduced for the manufacturing of large conical shapes. This process seemingly shows a promise for increasingly broader application to the production of axially symmetric parts. This study is to gain a better understanding of the process of shear spinning and to propose a lower upper-bound solution for shear spinning cones. Velocity 5elds and strain rates are derived from considering the adequate deformation mode and the contact factor is introduced to obtain the lower upper-bound power. The theoretical values are compared with the experimental results of various materials. Al-1100-0, Al-1100-H14, and Al-6061-0. A comparison shows that theoretical prediction is reasonably in good agreement with experimental results. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Shear spinning; Upper bound solution; Contact factor; Dynamometer
1. Introduction The rolling process which uses a roller and a mandrel to deform the metal plates, plastically, into conical or other shapes of circular cross-section, such as hemispheres, may be technically described as spinning. Conventional spinning or hand-spinning requires a blank which has the same wall thickness as the 5nished part. However, shear spinning di=ers from conventional spinning in that the blank diameter is the same as that of the 5nished part and the thickness of the 5nished part is determined by the blank thickness multiplied by the sine of half the included angle of the 5nished part. This “sine law” expresses the conditions that any circular element retains its original diameter during spinning and that the total volume of metal in a certain diameter will not change. In contrast, the term of conventional spinning is also used for the spinning process when the radial position of an element in ∗
Corresponding author.
0020-7403/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2003.11.002
1894
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Nomenclature f Ft Ff Fn N m r0 ra ; rc ; ri ; rb S S1 S2 t0 tf Vx ; Vy ; Vz Vr ; V ; Vz w˙ zD H m d H ˙ ˙rz ; ˙z ; ˙r H Hm m p ; H H
feed of roller tangential force component in shear spinning feed force component normal force component speed of rotation of mandrel contact factor corner radius of roller cone radii area BQFCRB area BMICRB area BQFIMB initial thickness of blank 5nal thickness of cone velocity in x; y; z direction velocity in r; ; z direction total energy of deformation z-value of the curve SMR half cone angle of mandrel mean e=ective stress total e=ective strain strain rate strain rate tensors in cylindrical coordinates 5nite e=ective strain mean e=ective strain in the deformation zone length of blank in its rotational direction assumed actual contact length ideal geometric contact length radii of curvature of deformation
the blank reduces appreciably while the blank is deformed. The spinning process is an example of a new trend, which exists in metal working operations. This trend is toward localizing the deformation zone to a small region of the work piece in order to reduce the forming forces and consequently make it possible to reduce the size of the machine required to carry out such processes. Because of the importance of the spinning process in the formation of aircraft and missile components, a number of investigations on the spinning process, both theoretical and experimental, have appeared in the previous literatures. For example, Colding [1] considered cone spinning as a combination of rolling and extrusion process, whereas Kalpakcioglu [2] assumed a simple shear mechanism for the analysis of the working forces. The complex straining e=ect was introduced into the solution of cone spinning by Avitzur and Yang [3], Kobayashi et al. [4] and Hayama et al. [5]. But their equations for predicting the tangential component force are of a complex nature and require long computation time
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1895
for its solution. Hayama and Amano [6,7] studied the contact between blank and roller during shear spinning by experimental and theoretical procedure. Moreover, Hayama [8] estimated concretely the three components of working force using the contact area between blank and roller obtained from his previous work. A test method for determining the spinnability of cones was proposed by Kegg [9]. This consists of shear spinning a blank on an ellipsoidal mandrel. On the other hand, Kalpakcioglu proposed the stress system to get the maximum allowable reduction in shear spinning process. Sortais et al. [10] studied the cone wall thickness variation in conventional spinning of cones and derived the theoretical tangential force component by the deformation energy method. Kobayashi established the condition of Lange wrinkling in conventional spinning of cones by modifying the theory of instability in deep-drawing of cups. And Slater gives the approximate upper-bound estimates for tangential force during shear spinning of cones assuming the ideal axi-symmetric deformation. But this assumption is not good enough for the shear spinning process. As a result, it is concluded that each previous theory for the working forces agreed with the experimental data for only a limited range of process variables, and did not give good results for other working conditions. 2. Mechanics of shear spinning of cones 2.1. Deformation mode The deformation mechanism of shear spinning of cones is shown schematically in Fig. 1(a). The blank material is a disk of diameter Db and uniform thickness t0 . The disk is mounted on a circular conical mandrel, which is clamped to the head of the spinning machine and rotated. A forming roller is driven on tracks on the bed of the machine parallel to the side of the mandrel. The process is characterized by the fact that the radial position of an element in the blank remains constant during deformation and the angular velocity is constant through the whole work piece except the deformation zone. This demands that the initial disk thickness t0 and the 5nal thickness of the cone tf for the half cone angle are related by the equation: tf = t0 sin :
(1)
And, for simplicity, it is assumed that during one revolution of the mandrel, the roller holds the same position, and after one complete revolution of the mandrel, the roller feeds f sin to x- direction and f cos to z- direction. The form of deformation of the z– plane about a radial element of the blank is assumed to take the form of the solid line SQMR in Fig. 1(b), where z and denote, respectively, the height and the length of the blank in its rotational direction. Fig. 1(b) corresponds to Figs. 2 and 3(a). The curved line m shows where the roller begins to make contact with the blank, while the line is the one obtained as a result of the geometrical calculation. The line where the bending starts can be drawn as BQFH in Fig. 3(a) and the shaded area, BIC, is the actual contact area. Fig. 3(b) shows the radial region of deformation. The deformation region is bounded by rb and rc . Let us consider an element r in Fig. 1(b). The element starts to deform at point S where its radius of curvature tend to in5nity. After passing the point S, the radius of curvature decreases gradually and at point Q one half portion of an element begins to yield by uniaxial tension or compression by bending. After passing the point Q, the circumferential extension of an element r is assumed to
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 1. Schematic description of the deformation in shear spinning of cones: (a) R–Z plane; (b) Z– plane.
y
x
θ vx vθ vR vy
v
Fig. 2. Velocity components in X –Y plane.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
be constant until it reaches the point M while the shear strain Q − m zero to (cot "2 − cot "1 ) along QM, Q where Q and m are given by the following equations:
rz
1897
is increased monotonically from
m = m =r:
(2)
Q = Q =r:
(3)
The di=erence of ((@ rz =@r) dr) makes the radius of curvature of the element decrease along QM and the element reaches the point M. After passing the point M, the extension of an element changes its sign and assumed to be constant while shear strain rz is increased monotonically, starting from ((Q − m )=Q )(cot "2 − cot "1 ) to (cot "2 − cot "1 ) on the roller along MR. After reaching the point R, the circumferential extension of an element r will be unloaded elastically. In Fig. 3(a), the plastic deformation occurs only in region BFCB and elastic bending occurs in region FHLCF.
Fig. 3. Model of deformation: (a) deformation region in –r plane; (b) shear deformation.
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To get the contact length m , we assumed the following relations: 2 m2 = (r) H = 2(1 − m) 2NZ
(4)
and
m (r): H 1−m Noting that the ideal geometric contact length is given by H (r) =
(5)
= r0 ;
(6)
where (r) H and H (r) are the mean radius of curvature for (r; ) and (r; ), respectively, and 0 is the ideal geometric contact angle and is given by the following equation: √ 1 −D + D2 − CF −1 rc − ; (7) 0 = cos r C where A = 2(rc + R0 cos ); B = r 2 + (z − z02 )2 + 2(z − z02 )R0 sin − rc2 − r02 ; C = A2 + 4R20 sin2 ; D = AB − 2R20 (z − z02 ) sin 2; F = B2 − 4R20 {r02 − (z − z02 )2 cos2 }; z − z02 = r02 − (r01 − r)2 − f cos for rb 6 r 6 r01 ; = r0 − f cos
for r01 6 r 6 rc :
From Eq. (4), we obtain √ m = 1 − m
(8)
2.2. Velocity 7elds and strain rates To solve our particular deformation 5eld, the velocity 5eld in the deformation zone was computed 5rst. Because of the deformation mode, the Low line has to be of the form, (x; y; z) = x2 + y2 = C1
(9)
"(x; y; z) = z − zD = C2 ;
(10)
where zD is the z-value of the curve SMR in Fig. 1(b). RM is the contacted portion of the roller with the blank and MS is composed as in Fig. 1(b) to get the same shear strain, z , along both sides of point M.
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When S and M approach N, the Low model becomes similar to the one proposed by Avitzur for shear spinning of cones. In this case a velocity discontinuity exists at N and the shear loss term, which is neglected by Avitzur, should be involved in the power. The components of the velocity 5eld, Vx ; Vy ; Vz , assume the form,
@ @" @ @" − ; Vx = ) @y @z @x @y
@ @" @ @" Vy = ) − ; (11) @z @x @x @z
@ @" @ @" Vz = ) − @x @y @y @x and from Fig. 2, one gets V = Vy cos − Vx sin = 2*rN; Vr = Vy sin + Vx cos :
(12)
Inserting Eqs. (9) and (10) into Eqs. (11) and (12), one gets ) = −*N:
(13)
Vr = 0; V = 2*rN;
@zD @zD −y Vz = −2) x @y @x
= −2)
@zD @zD = 2*N ; @ @
(14)
noting that x = @y=@ and −y = @x=@. From the velocity 5elds of Eq. (14), one gets the strain rates in cylindrical polar coordinates as follows: rz =
@ 2 ZD 1 @Vz = *N ; 2 @r @r@
1 @ 2 ZD 1 @Vz = *N ; 2 r@ r @ 2 all other ij = 0. Z =
(15)
2.3. The power The rate of total work done on metal under the deformation zone becomes
H @Vz 2 1 @Vz 2 √ + dV: w˙ = @r r @ 3 vol:
(16)
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Inserting Eq. (10) into (11), one gets 2 2
2 @ ZD 1 @2 ZD 2*Nt0 m + 2 ds w˙ = √ @r@ r @ 2 3 S 2 2 2 2 2 2 2 2 2*Nt0 m @ ZD @ ZD 1 @ ZD 1 @ ZD = √ + 2 ds1 + + 2 ds2 2 @r@ r @ @r@ r @ 3 S1 S2
2 2 2 2
2 2 @ ZD @ ZD r r 1−m 2*Nt0 m ds1 + + + ds2 : = √ @r@ H @r@ m H 3 S1 S2
(17)
To obtain a lower upper-bound on energy, one needs @w˙ = 0: @m
(18)
Di=erentiating Eq. (17) with respect to m and inserting it into Eq. (18), one obtains m=1
and S1 = 0; S2 = S
(19)
Inserting Eq. (19) into Eq. (17), one obtains 2 2*Nt0 H m RH @ ZD √ w˙ = ds 3 S @r@ @ZD r=rc 2*Nt0 H m RH √ d = 3 @ r=rb 2*Nt0 H m RH √ NZD = 3 r=rc =
2*Nt0 H m RH √ f cos ; 3
where RH =
rc +r01 ; @Z@D |r=rb 2
(20)
=0
NZD |r=rc = f cos : Eq. (20) was proposed by Avitzur and Yang. They derived it by a simpli5ed model of pure shear deformation. They neglected the shear loss term along - in Fig. 4. The shear loss along - is pretty high and cannot be neglected in the calculation of the power in usual upper-bound solution based on the kinematically admissible velocity 5elds. In this analysis, the frictional losses between blank and roller was neglected because of the small discontinuity of the tangential velocity on the roller. In Avitzur the power consumed by the feed force is less than 7% of the total power in a typical case. So we can neglect the feed power and the power consumed can be regarded as done entirely
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1901
Fig. 4. Deformation in process proposed by Avitzur.
by the tangential force. One now gets w˙ = Ft v ; H . where v = 2*RN And the tangential force component on the roller may be written in the form w˙ Ft = : H 2*RN From Eq. (20) one obtains the tangential component of forces as follows: H m Ft = √ t0 f cos : 3 Ft 1 = √ t0 f cos : H m 3
(21)
(22)
(23) (24)
3. Experimental procedure In order to test the validity of the foregoing theory and furnish the Low stress required for calculating the working forces, the spinning test and the plane strain compression test were conducted on aluminum alloys. 3.1. Spinning test The experimental work of shear spinning was carried out on a spinning machine, Auto-spin 5060, and working forces were measured by three components of tool dynamometer, which was specially designed and manufactured as shown in Fig. 5. It was manufactured to measure the three force components, named as tangential force, feed force, and normal force, which has load range, 150 kg, for tangential force and 1000 kg for the others. Alloy steel, SCM2, was used as a base metal in manufacturing the tool dynamometer. Fig. 6 shows the con5gurations of the tool dynamometer and the locations of the electric resistance strain gauge (EA-06-125BT-120, M-M co.) which are
1902
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 5. Tool and dynamometer set-up and worked cones.
represented by circled numbers. And the circuits of the strain gauge are described in Fig. 7. The tool dynamometer was calibrated in the calibration stand(Model B-2, LTV Aerospace Co.) where tool dynamometer was loaded individually for the 3-directions to the range of dead weight, while keeping the balance. 20, 50, 100 lb weights were employed and the excitation voltage to the strain gauge circuits was 5 V. And the force components were calibrated at the reference point 292:5 mm from the datum line, therefore, the calibrated forces should be multiplied by 292.5/Real distance, where Real distance is 230 mm for tangential force and 292:5 mm for feed force. Let Ft ; Ff , and Fn be the calibration load and R1 ; R2 , and R3 be the corresponding strains (V), respectively. Then it can be written as R 1 = a 1 F t + b 1 F f + c 1 Fn ; R2 = a 2 F t + b 2 F f + c 2 F n ;
(25)
R 3 = a 3 F t + b 3 F f + c 3 Fn in which a1 ; b1 , and c1 are the transformation coeRcients of the strain corresponding to the main force components when Ft ; Ff , and Fn act individually.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1903
Fig. 6. Tool dynamometer and roller.
For the other coeRcients, the same procedures can be adopted. The coeRcients ai ; bi , and ci can be solved by the linear least-squares method, and Eq. (25) can be written in the form, F t = A 1 R 1 + B 1 R2 + C 1 R 3 ; F f = A 2 R 1 + B 2 R2 + C 2 R 3 ;
(26)
F n = A 3 R1 + B 3 R2 + C 3 R 3 : The three force components can be calculated from Eq. (26). The results of computation are shown as follows: Ft (kg) = (A1 R1 + B1 R2 + C1 R3 )
292:5 1 ; 230 2:2
Ff (kg) = (A2 R1 + B2 R2 + C2 R3 )
1 ; 2:2
Fn (kg) = (A3 R1 + B3 R2 + C3 R3 )
1 ; 2:2
where A1 = 0:02661; A2 = 0:00373; A3 = 0:0001296; B1 = −0:0007503; B2 = −0:1274; B3 = −0:000804; C1 = −0:00009679; C2 = 0:01106; C3 = 0:2396.
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 7. Strain gauge circuits.
The experimental values were recorded on the data coder (7404, Hewlett-Packard Co.) and signal conditioner(Type I-080, Bell & Howell Co.). It is necessary to obtain the instantaneous cone radius, ra , corresponding to each force reading, since the force component as well as the cone radius change during a single spinning test. After the blank was spun into a cone, the 5nal thickness was checked according to Eq. (1) by using the ultrasonic pulse echo type thickness gauge (Model CL 204 K-B Co.). The test conditions employed in the spinning tests are given in Fig. 8 and the typical read-out of the three components is shown in Fig. 9. 3.2. Plane-strain compression test The compression test and shear test are most e=ective to obtain the deformation characteristics applicable to several metal working operations as a shear spinning process. In order to maintain equality between the test specimen and the blank for spinning, the plane strain compression tests are applied. It should, however, be emphasized that the yield stress measured is Y , the yield stress
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911 Condition Material Diameter of Roller DR(mm) Thickness to(mm)
Fig. 10 Al 1100-H14
Fig. 11 Al 1100-0
Fig. 12 Al 6061-0
Fig. 13 Al 6061-0
125
125
125
125
3.2
3.2
2.3
2.3
Feed f(mm/rev.)
0.25 0.85 0.45 1.05 0.65
Mean Flow Stress
14.91 ~ 15.00
0.25 0.85 0.45 1.05 0.65 11.65 ~11.88
0.25 0.85 0.45 1.05 0.65 15.52 ~15.67
0.25 0.85 0.45 1.05 0.65 15.59 ~15.81
26°50`
26°50`
26°50`
26°50`
7200
7200
7037
7037
5
5
9
5
100
100
100
100
100
100
100
100
σm (kg/mm2)
Half Cone Angle α (deg.)
Young’s Modulus E(kg/mm2) Round-off Radius ro
Revolution per Minutes N(rpm) Measuring Radius ra(mm)
1905
Fig. 8. Test conditions in shear spinning of cones.
Fig. 9. Typical read-out of working forces.
in plain strain, not y , the yield stress in uniaxial tension, as measured in the tensile test. These are related: Y = 1:155y
(27)
The measured strain in the thickness direction, h , and the e=ective strain, , H are related: H = 1:115h
(28)
The experiments were done for three kinds of specimens, AL 1100-0, Al 1100-H14 and Al 6061-0. And each specimen was compressed to get extension along three directions, 90◦ ; 45◦ , and 0◦ , to the rolling direction of the original sheet. The experimental values are taken, with the mean value of the three kinds of specimen for each tested material, in which the linear least-squares method was applied.
1906
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
4. Discussion The tangential force in spinning is a very important quantity and is of great interest both to the designers of power spinning machines and to the production engineers faced with the problems of suRcient power to carry out the spinning operations [6]. The experimental working forces and the theoretical values of the spinning processes are shown in Figs. 10–13. The measured working forces are in reasonable agreement with the theoretical evaluations for all working conditions. The e=ect of material properties can be shown in Figs. 10–11. Both the strain-hardening material, Al-1100-0 and the non-straining material, Al-1100-H14 give fairly good agreement with
Fig. 10. Relation between tangential forces and feed of roller f.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1907
Fig. 11. Relation between tangential forces and feed of roller f.
theoretical values. Reviewing from Figs. 12–13, it is found that the change of corner radius of roller r0 gives no large di=erences of working forces, that is, as the radius of the round-o= of the roller gets bigger, the working forces change slightly. In the calculation of theoretical values, the mean e=ective stress, H m is the Low stress of material at the mean e=ective strain, Hm , which is given by rc 1 cot "2 √ dr Hm = rb − r c r b 3 r0 − r02 − (rc − rb )2 √ = ; (29) 3(rc − rb ) where cot "2 =
r0 − r r02
− (rc − rb )2
for rb 6 r 6 rc :
1908
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 12. Relation between tangential forces and feed of roller of f.
Figs. 14–16 show the stress–strain relations of plane strain compression test for three Al alloys, the mean e=ective strain Hm in shear spinning process is given by Eq. (29) and the mean e=ective stresses for the range, 0.36 – 0.40, of Hm are evaluated as H m = 14:91–15:00 kg=mm2
for Al 1100 − H14;
= 11:65–11:88 kg=mm2
for Al 1100 − 0;
= 15:52–15:81 kg=mm2
for Al 6061 − 0:
The mean e=ective stress, H m , is the average value of H in the deformation region in Eq. (17). So the Low stress of material at the mean e=ective strain, Hm is the mean e=ective stress, H m . Kobayashi et al. [4] used the mean e=ective stress as follows H m =
H d H d H
(30)
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1909
Fig. 13. Relation between tangential forces and feed of roller f.
Fig. 14. Equivalent stress–strain curve from compression test (1100-0).
which is independent of the history of deformation in the deforming zone. The mean e=ective stress, H m , for the mean e=ective strain, Hm , makes the theoretical values of working of force approach to the measured values.
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C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
Fig. 15. Equivalent stress–strain curve from compression test (1100-H14).
Fig. 16. Equivalent stress–strain curve from compression test (6061-0).
5. Conclusion In the study of the mechanics of the shear spinning of cones, the following conclusions have been reached: 1. A simpli5ed solution of shear spinning of cones by a pure shear model coincides with the lower upper-bound solution derived in the present paper. 2. The tangential force is linearly proportional to the yield limit, H m blank thickness, t0 feed of roller f and cosine of the included semi-cone angle, . 3. Theoretical values can be applied to strain-hardening materials as well as non-strain hardening materials.
C. Kim et al. / International Journal of Mechanical Sciences 45 (2003) 1893 – 1911
1911
4. The present theory gives a better agreement with the experimental data than any other existing theories for a wide range of process variables. Acknowledgements This work was supported by Research Funds of Pusan National University through Brain Busan 21. References [1] Colding BN. Shear spinning. ASME No. 59-prod-2, May, 1959. [2] Kalpackcioglu S. On the mechanics of shear spinning. Journal of Engineering for Industry. Transactions of ASME, Ser B 1961;83:125–30. [3] Avitzur B, Yang CT. Analysis of power spinning of cones. Journal of Engineering for Industry. Transactions of ASME, Ser B 1960;82:231–45. [4] Kobayashi S, Hall IK, Thomsen EG. A theory of shear spinning of cones. Journal of Engineering for Industry. Transactions of ASME, Ser B 1961;83:485–95. [5] Hayama M, Murota T, Kudo H. Study of shear spinning. (A) 1st Report, Transactions of JSME 1964;30(220):1450–7. (B) 2nd Report, Transactions of JSME 1964;30(220):1458–65. (C) 3rd Report, Transactions of JSME 1965;31(228):1259–69. [6] Hayama M, Amano T. Analysis of contact form of roller on sheet blank in shear spinning of cones. Journal of JSTP 1975;16(174):559–65. [7] Hayama M, Amano T. Experiments on the mechanism of shear spinning of cones. Journal of JSTP 1975;16(172): 371–8. [8] Hayama M. Analysis of working forces in shear spinning of cones. Journal of JSTP 1975;16(175):627–35. [9] Kegg RL. A new test method for determination of spinnability of metals. Journal of Engineering for Industry. Transactions of ASME, Ser B 1961;119 –24:119–24. [10] Sortais HC, Kobayashi S, Thomsen EG. Mechanics of conventional spinning. Journal of Engineering for Industry. Transactions of ASME, Ser B 1963;85:44–8:346–50.