THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ANALYSIS OF POWER SPINNING OF CONES cY\ B. Avitzur C. T. Yang November, 1959 IP-402

\ifvMto'Th sje^^~ \)(V\IOI^-Y

ACKNOWLEDGMENT Assistance from Mr. Serope Kalpakcioglu of the Cincinnati Milling Machine Company and staffs in Spincraft, Inc. in obtaining the experimental data is deeply appreciated. ii

NOMENCLATURE in-lb w = rate of work or energy of deformation per unit volume in. m. in. min. W = total energy of deformation V =volume under the roller T-= effective stress J = effective strain rate R, 6, Z = polar coordinate directions of the spun cone 6 = strain rate ERR,' Sf3, zz = normal strain rates in R, 9 and Z directions RS,' E Z' gZR = shear strain rates UR, US, UZ = velocities in R, 8, and Z directions N = rotating speed of the cone, in revolutions per minute = strain deviator tensor Sij - stress deviator tensor A = modulus of plasticity e= hydrostatic strain rate eii = summation of three principal strain rates = ell e22 +33 X, Y, Z = rectangular coordinate directions of the spun cone k = yield limit in shear - O = yield limit in tension SRZ, Sze, S6R = deviator shear stresses TRZ, I Z@, BR = total shear stresses i j = total stress tensor S = hydrostatic stress tensor J - unit tensor s = surface area so = initial thickness of blank iii

X0 = 1/2 included angle of the cone n = number of revolutions r = radius of the round-off of the roller o x, y, z = coordinate directions for roller, z in the direction of feed, F; x in the direction normal to the cone surface; y is normal to xz plane F = feed of roller (in/rev) R = instantaneous radius at which the roller touches the cone po = radius of the torus on the roller T = time passed from the starting point t = tangential force WT = tangential power WF = feed power = shear strain = shear deformation *o W = power consumption to deform the cone in experimental tests, assumed equal to W T = starting time o W W = weighted power CosoN t = weighted tangential force = "so o iv

INTRODUCTION Spinning is generally a cold working process in which a rotating disk of sheet metal is deformed into a cup or cone by applying a localized pressure to the outside of the disk with a simple round-ended wood or metal stick. In common spinning practice the disk is deformed against a pattern having the final shape of the desired product. A very skillful machinist could, however, spin-form a cup or cone with no pattern-support at all. In these conventional types of spinning the thickness of the sheet metal is practically unchanged. In the last few years a new type of spinning called "shear spinning" was developed. It has also other names, such as "power spinning," "flow-turning" or'roll-forming." Essentially they mean the same operation. In this process the work is done on a machine having the pattern and blank attached onto the spindle as usual but the metal was deformed by a roller which is mechanically operated (see Fig. 1). Moreover, in this operation the radial distance at any point on the cone remains the same before and after spinning and the thickness is therefore bound to be changed. This process has been widely used in making television picture-tube cones, etc. Previous researchersi have done some work on the analysis of power, forces and stresses involved in this problem. But they are all limited to a certain extent. The authors attempted to study the problem in more detail — first with analytical approach and then the results were checked by experimental tests. -1

-2_L_ D, S THE FINISHED CONE THE CONE MANDREL.II THE BLANK THE OPERATION I j ROLLER F& r ro F1 A FEED { r| UNDEFORMED\ I ~~~~I \, |REGION ~~Rf02 R | Rfol.. T THE DIS- - -- — l - PLACEMENTS A- DLJ F Figure 1. The Cone,

-3THE NATURE OF DEFORMATION OF METAL IN THE PROCESS A sheet metal blank in power spinning could be deformed by a process of bending, shearing or a combination of both (see Fig. 2). In bending the radial distance of any point at mid-thickness of the cone (point o or s in Fig. 2) remains unchanged. The linesAB and CD in Fig. 2 remain straight and normal to the surfaces. Therefore, the thickness before and after spinning can be related by the following equation: Thickness A B = Thickness AB'sin^C In shearing, however, not only points at mid-thickness but any material point have the same radial distance before and after deformation. The straight lines AB and CD in lower half of Fig. 2 change to AB' and C'D' but remain parallel to the Z axis. Besides, AB = A'Bt and CD = C'D'; thus thickness after deformation =(thickness before deformation)'sino. From the experimental tests described later it was found that the actual deformation pattern is a combination of bending and shearing. However, for the case of small included angles of a spun cone, the deformation is close to pure shear. Also for the sake of simplicity the pure shear deformation is assumed in the following analysis. i THE ANALYSIS In the analysis the following assumptions were made: 1. Mises material is used, which implies that: (a) the material is homogeneous and isotropic, (b) there is no elastic deformation, and consequently no volumetric change, (c) there is no strain hardening. 2. The thickness of the blank is much smaller than the minimum radius of the cone. 3. The metal deforms under the roller in pure shear.

-4Q)BEND b) SHEAR/ / By B A D' C D Figure 2. The Shear and Bending Strains.

-54. The frictional force under the roller is neglected. This is justified, because the relative velocity between the cane and the roller is very small. Also this was substantiated by a previous publication by 2 Reichel which states that lubricants do not affect the force. It was observed in the actual process and thence followed in the analysis that metal deforms instantaneously underneath the roller. The rest of the cone is strain free at the instant. From a variety of roller shapes, as shown in Fig. 3, shape I was chosen to be analyzed. The roller is assumed to have its axis parallel to the side of the cone. The choice of the coordinate axes XYZ with the origin at 0 is indicated in Fig. 4. Another set of cylindrical polar coordinates (R,8,Z) with the same origin and Z axis as in XYZ coordinates was introduced. The assumed shear strain field is also illustrated in the lower part of Fig. 4. In the analysis the "Deformation Energy" theory was used which is indicated as follows: W = /g0r dV (1) where T = total energy of deformation V = volume worked underneath the roller Cs effective stress $ = effective strain rate o will be assumed to be a constant and % is to be derived from the velocity field. V is the product of the contact area between the cone and the roller and the thickness of the cone. The angles and dimensions of the cone and the shape and dimensions of the roller are shown in Figs. 1 and 3 respectively. Strain Rates The strain rates in cylindrical polar coordinates assume the form:

-6~~~~~LA~~ND~~v LAND X LAND^ V.l' FRONT SHAPE I SHAPE II SHAPE TI Figure 3. The Roller.

-7z I\ IX F u X ^ The i s Figure 4. The Deformations.

-8-) UR laU g Up Uz RRH -' ee R)' zz ~z R? ~ R- (2) E _l UR aUz) Rz: 2 \ * aR r = 1 (^Q F 1 Lz Where E.RR..8', ZZ' ERG, RZ' and ECZ are the components of the strain rate tensor field; UR, Ug, and Uz are the components of the velocity vector field. In order to solve our particular deformation field, the velocity field under the roller was computed first. Because the angular velocity of the spinning cone is constant, one gets the circumferential velocity at any point on the cane as follows: Ug s 21TRN (3) Where U9 is the circumferential velocity R is the radius of the point considered and N is the velocity of rotation, r.p.m. Also, because there is no change in the radius R at any point on the cone diring spinning, the radial velocity (UR) will be: U =0 O(4) It is apparent that the cone under the area of contact during deformation takes the shape of the roller. Let the geometry of the roller be described as follows: Z = H (R,G,n) (5) Thus, dZ dR + * d +; * dn (6) a_.R ~ de+ 6'an - (6) dand u~ZdHZ 8Z dG JZd (7) and dT JR dT e dT on dT Where T is the time passed from time T = To to the instant n is the number of revolutions passed from time T = T to the instant, and UZ is the velocity component in Z direction.

-9One now continues to compute the derivatives with respect to time in the fo llowing: n = N(T - To) (8) J —:N (9) d8_ U@ = 2- N d=f UR 21'N (10) dR - UR = O (11) dT Substituting the above derivatives into Equation (7) yields: uz 2N S+ - N = N 2r + az l (12) Therefore the complete velocity field can be summarized as: UR = o U8 = 2TWRN (13) Uz = N [2TJ' + where the velocity field is a function of R, and 8 only, and the derivatives and - are yet to be defined from the equation of the roller. Inserting the values of UR, Ug and UZ from Equation (13) into Equation (2) one gets the following strain rate field: i = )Uz RZ- rZr - 1 &.UZ = (14) ezz' 2R ae all other gij - 0 Stress-Strain Relation Levy-Mises stress-strain rate law of plasticity (incremental theory) was used, that is, <ij =YSij (15) where Si and fij are stress deviator and strain-rate deviator tensors respectively; 4 is a modulus of plasticity and a function of &ij. Owing to volume constancy in plastic region, the plastic strain rate deviator is equal to the plastic strain rate itself, because

-10e = ei,(16) where e = hydrostatic strain rate and eii = ell + e22 33 = 0 (17) Let a term "I" be defined as 1 t2.2 *2 t 2 2'2 12 1 k I 2' z) + fx + Z ) = i ij (tensor) (18) It has been shown by Prager and Hodge4 that: S =k ctij (19) ij t' kl' kl Where k = yield limit in simple shear =. The stress deviator field can therefore be written as follows: cat -7 ii~~ = k <tRZ RZ RZ - and Sez =Z k gZ (20) -and 0 =, = 2 / RZ + ez all other Sij - 0. The Power Let the rate of work per unit volume = %w ~i f i (21) Now Gj Sij + Sj (22) where 5ij = deviator stress S = hydrostatic tensor and ij = unit tensor, equals 1 when i=j equals 0 when i/j Again because of volume constancy, Cii =, thus w = Sij G iJ (23) Therefore * = SiJ' i iJ EiJ () (24) 2 21 (24) Now that the yielding condition J2 = S i k2 (25) One gets: w = 2k cij'ij 4= 6o'iJ iJ 2 ~c fI/ U (1z\2 = /i 6o URZ 2sjv +i/ (lU =2(26) =i,: 1(~j) +( g (26)

-11 - The rate of total work done on metal under the contact area, it = JidV, that is, To- + 1(^ \ r1 Zf dV if= (i (* )...: a )2 J surfaceLC I ) d (27) area Where ds = infinitesimal surface area (ds = Rd 8 dR). Because the terms under the square root are independent of Z, one gets: W=, f3.R kUZ + -) dR d. (28) Where the integration is to be done along the boundaries of the area of contact between the roller and the cone.:1 Let W be designated as weighted power, and defined as W'. / (29) ~soN or NW = + aR dR da (30) Also let if [1 UZ/Uz2 (31) R a/ RJ The following equation yields W'f J ~iR IR' ds (32) N J1 + a The function of the cylindrical portion of the roller is to smooth out the feed mark and the work done by it is very small compared with that done by the torus. Thereby the work done by the cylindrical portion of the roller is neglected. The actual and simplified areas of contact are shown in Figs. 5 and 6. Assuming c is independent of R and 8, one has: W I= id fJ JR (P\) dR d9 (33) NU7 Since -z changes appreciably for slight change of R, R itself can be considered as constant, (RRpo). Thereforet

-12LL..C u0 /' *, + 0C-0 < / +c O + I.. I, O / 0 0 <+ J / CJi ~,J + r 0 C N / + 0 0o U.) CD+o c \ o / " U) q O M U U) II U) + y -1 +\ r io < 0 L____j ~I I

-13g l1 41+J'I R = upper limit -- d WN R0 Jo[UZ R = lower limit d N =a Z d (34) Again since U N 2 +, one has L 89 8'nn' one has W- = Ro 4 2 ) de.+ f daa (35) Now that (A dB = F cosC 0 (36) W = 21Ro I -1TT F cosc(o + R( A t) dB (37) The value of aZ can be evaluated from the equation of a torus, the geometry of an which can be approximated by the following equation (see FiLg. 7): (a - Z)2 + (b- X)2 = r2 (38) 0 From Fig. 7 also one has 2 -R. 1o2 e Z Fn cosC( * + / sinco + ir b -Rcos (39) RF[ cos 9 - (Fn sinOCo -^o cosC. F coC( zn Fcs 2 - CR cos (Fn * sinOC - coso) 0.... FucosO% ~(40) )ZZ- -_ (41) dn _n _ upper limit n/R s lower limit Thus the equation W' can be evaluated except for the value of c- the strain ratio, which will be dealt with in the following section. Evaluation of the Sttain Ratio From the approximate equation of the torus, Equation (38), one can derive U sZ. dZ FcosOC [ cos -(Fn.sin - coso )]. -Rsin[ -2TrFsino: N N dT- /r2- [Rcos 4- (nsin<O- pcosO)2. (42) 1 [UZ 01 P ro2.cos 6 (2mT * sin e + F sincC ) + 2 rsin 6e (R cosG -R] N aR (43) and1 ri Z l rJ a 7'sin 8 2lWRosin S + F sin5C^J * 2wcos 98(R cos S -/) (44)

-14LINE DE RA Figure 6. Approximated Area of Contact. / x", //Po cos a'o ~~~ro~~ ~ ~X b= Fn sin ao -po cos o Fn sin a z Fn cos po sin ao a = Fn cos ao + Pos in abo Figure 7. Approximating the Torus by a Cylinder.

-15Thus d-; UR 2j 5 Jncos e(R cos 9 -,6) - r2 sin 9 (2ZM sin 9 + F sino) (2 L aUZ ai2in G (R cos8 + 3) + r 2 cos 8 (21TR sin Q + F sinda0) (45) where = Fn sinoC0 -/o cosOCo, and = r2 -[ cos 8 - (Fn-sinoCo -to cosoO)] (46) Evaluation of the Power It now remains to evaluate numerically the spinning power from the following equation: Wt -, {2fR F cosoCo J2 a) dB (47) where J is defined in Equation (45), Combining Equations (40) and (41), one has Z _ Rupper cos 8 -/3 Rlower cos 8 -/ (48) Cos2 R upper cos (ower cs - 2 and, J is solved as average for the area of contact. eav ge The boundaries of the approximated area of contact (Fig. 6) will now be computed. These boundaries are derived from the intersection of the cone with the roller. The equation of the toric portion of the roller (see Appendix) is: G [/(R cos cosC - Z sino)2 C R2 sin2 8 - pO i [R cos si Fn - 2 = Fi. (49) * [+ cos8 skrne+Z coso%0 Fn J r20 (49) and the equation of the portion of the cone which is in the process of being deformed (zone 2 of Fig. 8) is: Z = b * b2 - c where b /~ sinoCo F (n - cosoC(,O sincCo F (n-l)cos C c = (R cosC * pO)2 [R sinc (- - (n - -r2 ) s (R,COSO: +o)2 4 [R sinoCO - F (n -l)] 2 - r2J

-16Area of Contact The area of contact between the roller and the cone is shown in Fig. 5 orne and an approximated/is shown in Fig. 6. The boundaries of the approximate area of contact can now be described line by line as follows: Line AB ((R cosoo - Z sincC +,o) + (R sincO( + Z cos n) - ) - r 9 0 (51) Line BD R = RB = F (n - ) sinoo -f coso F(n-l) sinoC -0 cosOC (52) Line DE Line DE is computed from the intersection of the roller, G = O,with zone 2 of the cone. G and zone 2 of the cone were defined previously by Equations (49) and (50) respectively. Line AE R R=A RO F (n 2-r) sino - (, *+ r) cosk _ - F(n-l)'sinO<o - (o 4+ r) cosoC (53) Line EE' = 8 and 8E is found from the equation of the line DE for the value of E E R RA. The boundary line DE as well as the values of BE and i were computed on the IBM 650 digital computer. The numerical method used is described in Avitzur's thesis. The actual boundaries of the area of contact (Fig. 5) are fully defined in the same reference. In order to calculate the total energy of deformation under the torus, it is necessary to integrate within the boundaries of R and 8. Since the boundaries of integration have to be solved numerically, the IBM 650 digital computer was also used for the numerical evaluation of the integrals. Even though a computer was used, it is necessary to simplify the boundaries as shown by the dotted lines in Fig. 6. The programming of the computer is not shown in the paper for obvious reasons.

-17ZONE 3 ZONE 2 /-ZONE I THE BEND ^ /^ ^ SPIRAL B SPIRAL A Figure 8. Zones of the Cone. Figure 9Deformation of a Cube by Pure Sh —-— ear. I: /,A// Fig-7e 9'-oainfaCeyPu Shr Figure 9. Deformation of a Cube by Pure Shear.

-18Forces The total spinning power can be divided into the feed power and the tangential power, WF and WT respectively. Since WF is small compared with WT as observed from the experiment, the former is neglected here. Thus WT t-ue (54) where t = tangential force. Numerical calculations of the tangential forces versus feed, included angle and round-off radius were plotted and shown in Figs.13, 14, and 15, respectively. Solving the Power by a Simplified but Approximate Method (Deformation Theory) It is seemingly difficult to calculate the spinning power and the tangential force. What is more, the necessity of using a digital computer to evaluate the integrals handicaps one in getting a quick numerical answer. Therefore, the following simple method is suggested to obtain an approximate power and tangential force with comparatively little effort. A cubic block of unit volume as shown in Fig. 9 is subjected to pure shear lb. with load, k lqbi X 1 sq. in. = k lbs., sq-in. shear strain =' displacement? = 1'tanY work per unit volume = w = 3 tan (55) in3 total volume worked on the cone = V = 21RN so F-sirno i (56) 0rmm (min and Y = 900~ - (57) 0 W = total power = 21TRN sinoC so F 7 tany' s ( soNFR cosoC0 (58) Numerical calculations for a few examples of the tangential forces using this simplified method were also plotted in Figs. 13, 14 and 15 in dotted lines.

EXPERIMENTS AND RESULTS Two types of experiments were conducted. One was designed to study the nature of the deformation in spinning. The other was designed to measure forces between the tool and the work, and also the spinning power. 1. Investigation of the Deformation Pattern In the original disk.0125 inch holes were drilled and plugged with "sculp" metal (al. alloy) as shown in Fig. 10. After the canes were spun, the metal was carefully cut and filed until the holes were revealed. From the direction of the plugs, a three-dimensional deformation picture was constructed. A typical picture of the holes is shown in Fig. 11. This cone was spun and checked by the Cincinnati Milling Machine Company as part of a study conducted there. Detailed comparisons between the model cone and actual cones is given in Ref. 5. In Fig. 12 a top view or the radial line of holes shows the shear ERG of the cone. The angle OC- indicates the extent to which the outer surface slipped over the inner surface of the cone. Numerical measurements of the distortion of the plugged holes can be found in Ref. 5. In the analytical approach it was assumed thatoC = C 0. It was observed experimentally that~C< andoCR do not exceed a few degrees. In standard practice oCo is not over 75~, which means that the minimum shear angle (Y) is over 15~. Since og and oc are much smaller than 15~, it is justified to assume that =R 0 In the analytical approach -RZ was further assumed as pure shear, which means that cr 0. This assumption does not actually hold. For large oC, it 0' seems that the deformation is closer to pure bending. For smaller oo, the deformation is closer to pure shear rather than bending. For simplicity of the computation pure shear has been assumed throughout the study. (For the difference in power consumption between pure shear and bending, see Appendix 2 of Ref. 5.) -19

-20z rCUJ ~~~Z c~Z -- Z ( I LJ 0^~~~~~~~~~~~ z~ C I < — ~ ~ 0,, Va0O8 Af 1o o V I(,!081 ^ -- --- - - - ~ ~~~~~ vi 0e' ------

21TE PLUGGED HOLES: ~~~~~~~~~~~iii.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...... CROSS SECTION; AXIAL KADIAL CROSS SECTIO: ~!"............ii ~~~~~~iiiiii,;iiiiia~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........................t of a s un on...............cnntiMllngMahneCo)

-22_J 0 0~c~ &cv~rl I t3~~~~~~~~~CM Ha) H oj \ \K (I) \ ^ Y^ ^k C~ZK i x ~

-232o Measuring the Tangential Force and the Power Two sets of experiments were conducted separately at the Cincinnati Milling Machine Company and Spincraft, Inc. In the Cincinnati Milling Machine Company tests were made to measure the forces. On their power spinning machine the roller is mounted on a three-dimensional dynamometer to measure the forces between the roller and the cone, the feed force, tangential force and the axial thrust. In this study only the tangential force is of interest to us. The experimental results are given in Table 1. The effect of the parameters, feed, included angle and the round-off radius on the tangential force was plotted and shown in Figs. 13, 14, and 15. In Spincraft, Inc. the power consumption of a d-c motor which drives the main spindle was measured. The set-up is shown in Fig. 16. The tare horsepower and the horsepower for spinning were separately measured. The power consuption for deforming the cone (W~) is calculated as follows: ~ _ H.P. with load - tare H. P. efficiency Since it is not easy to get the efficiency, it was assumed to be 100% for the first approximation. The experimental results are given in Table 2 and plots shown in Figs. 13, 14, and 15. The true stress and strain curve for aluminun 1100 - H was obtained and shown in Fig. 17. The yield strength is taken as an average of the true stresses in the region after yielding. CONCLUSION Results from theoretical analysis and experimental tests agree fairly well regardless of difficulties involved in the force and power measurement. The forces calculated from the simplified deformation theory is the lowest among all the forces computed from the lengthy analysis using incremental theory. The effect of the variables on the tangential forces is observed as follows:

-241. As the radius Ro gets bigger, the tangential force is decreased, but it will never get to a value less than that predicted by the simplified method as shown in Figs. 13, 14, and 15. 2. The tangential force is linearly proportional to the yield limit 0-, and the blank thickness so 3. Increasing the feed (F) will increase the tangential force. 4. The larger the included angle (2o0) of.the cone, the smaller will be the force. 5. As the roller round-off radius ro approaches zero, the efficiency approaches its best value (i.e., smallest force). As r0 gets bigger than about 1/4 inch in practical conditions, a further increase of ro does not affect the force appreciably. 6. As the roller radius lo approaches zero, the force is least. When o ranges between 6" and 10" (6" co ( 10"), the tangential force changes very little.

-25K\,4 V.nTpve. O \ r -h p la 1 |u? _<8 "^' S l c t> r l - ^ c @4E-) 0 0 0 0 0 ) o -Fl oU )0 | _ Ip N ~, XE-1 |o o g:n 0 8 r.N S iS I snT p e 4 Cuu i a u I o P4 Cu | 4| oo H o 4 C) 04 - 4 CM o _npugj -4 _ u s |o8TI8S ~IV 4* @4 -4 sTpa o- m I. $4 onTpeH - 44p 0 ~',,-.RU! o o o o o o o o o o o o o o *nuTH zt.d PoApo.. -4 T auoo pa~ai(On An nn ~888 g — g ---—. os JO U4ThI0I P__T_ _ _. ~_____________g _ _ ________ - *JaT r# a g Q _ l | _ t- - n- -- t4 t t - -- -4- - c- 3 sfan iH H HH H H H H H H H H H H -u TaquUTS;. CJ Cu^ Cu Cu Cu Cu Cu Cu MN l C Cu Cu ( oTpUlOa or a nrlou. k cu4 -4 H M -~,,. 0 4 - w% w% oJ 0J 0u

-26— 4 | 4p Id fS P ~ | 9 A -- 2 I ~ cu. y T I 4J 4 snjgg In | g _ O r6 00 8 d leSwr # _ sIo o o o ct o o S o C 4J-' I 0 O..... _ h *D -, I _ 10 n m 0 - e 0 0 8 N M1 S pe c. Q. ~. o, to 0 - II In in d N N 0 on S j~ Pk S g3 nlpv'to I n __ N O,,nTPo. o o~m&_ o "0" O 2 | N A; F o n -a J g * oit s y~. W "2'~0 0 0 C.~,~P.< O_ o oo oooo oo? [ ____ __1 4 N N N N'4 rtO 8 r o |. o [.,=. d p | = ^; 5gg:z; C L n @ L _ go o.. ~I o Z t0 0 0 t0 0 0 a)j w H e a a'$e;;; X - ^ t7 g IieS' V ~ 31 ~le ~d ~le__ o _. o. S -... -. u JnpH I ed 2 _ N if S 2 N 2 2 2 -t I_ d paj _ _ N,-4 fToI4 4 4 4 uoj!q~N ntn-Io I~ PaH 0 ~In r(~~~~~~~~~~6D0 c

-27The Effect of the Feed (F) on the Weighted Tangential Force (t') - - - Predicted by the Simplified Method (Deformation Theory) Predicted by the Analysis (Incremental Theory) The Tangential Force t = ao'so't' (lb.) The Power W = 2nRoNt = 2t.ao-SQ-RoNtt' (lb-in) Spincraft Experimen:tal Data Spincraft Experimental Data.08 o- Ro - 2-19/52 in..08 0 — Ro = 3-5/8 in. D-_-Ro = (.4 in. /E Ro = 8 in. o.07 rg= 4.5 in. ro = 3/ in./8 in..0 - 00 r C n.w O07 - PO = 4.5 in. ro = 3/8 in. o I 5.05,.05.040.04 q /'p 0 a.036.03 z / /^/ z.02 - 02 ^ / // TEST NO. _ _^ A t * I ^ /^- TEST NO. //.02 / 112 113 114 115 12 133 134 135 0.02.04.06.08.10 0.02.04.06.08.10 FEED, IPR FEED, IPR Spincraft Experimental Data U.06.03 - - Cincinnati Experimental 0 0 4 / w- R = 7-/8 in.. = ( in., r.250 in. z I -03 0 Z03 0 W z.042 z./02. TEST NO., - TEST NOO S. I- 11.03.~ MI 3.02 " 7I h124 125 7. I I,0I, 0.02.04.06.08.10 0.01.04.06.08.0 FEED, IPR FEED, IPR SpFigure 15. Curves of Tangnaimental Forces Versus Feeds.ata Figure 13. Curves of Tangential Forces Versus Feeds.

-28The Effect of the Included Angle 2UO on the Weighted Tangential Force t' - - - Predicted by the Simplified Method (Deformation Theory) - Predicted by the Analysis (Incremental Theory) The Tan6ential Force = aoSot' (l1.) The \ Power W 2 = 2tRoNt 2AaosoRoNt' ( in ).08 nan. Spincraft Spincraft.07 Experimental Data 07 Experimental Data 0 \ O- Ro n'5 in. 0 C.) 6.06' Ros" o \ Po = 4.5 in. o 06 \ Po = 4.3 in..06 Ro0"- - R o - D-R in. 0-R0 8in. 4 r 3/8 in. r =1 5/8 in. W - o.05 t VE \ \\ w T N0. \ \ Ro ~ z 1S8.04 -.04 0 Rs ~IO\ r\ 0 0 0t t 6 0 1.04- 1 110.03 115 135 125.o 114 134 124C. 0 0 20 40 60 80 90 0 20 40 60 80 90 HALF THE INCLUDED ANGLE Cot DEG. HALF THE INCLUDED ANGLE CIO,DEG. 0 =.025 -- Cincinnati Experimental a TEST NO. Dta at RNO. 1 in. OMS 0.020 -2 Po in. 0 4 0 ro 1/4 in. HA L*10 F 0.028 ipr 0 20 1.O85 0 HALF;HE INLUDE TGET INCLUDED aA Ro = 1 n E. Id "~vV d "igure 14.025rves of TaeP 0- Cin innix me.ntl.0 lo- 0M 2.005 Mp - 7 n 0 20 40 60 1/40 in. HAL I0 F -.028 ipr Figure 14. Curves of Tangential Forces Versus Included Angles of Cones.

-2908r The Effect of the "Round-Off" ro on the Weiignte~ Ta., -.ntial Force t - - - Predicted by the Simplified Method (Deformation Theory. Predicted by tne Anlalysis (Inlcremental Theory).07 The Tangential Force t = aosot' (l-.) The Power W = 2bot = riaRo Nt(i- ).06 - Spincraft Experimental Data 0- Ro 3 in. 0- Ro 8 in. 0 PO =4.3 in. o.05 = F -. QGO i 0.0 ^.04 - / ^-~ —---- W 32 ~ 45 148 134 z TEST NO..02 - M9A M I I M 10 RF 2" -—. — R 10".01 a — Cincinnati Experimental Data at R = 1 in. Po ( illn. a0 = 35~ F =.050 ipr 0 125.250 375.500.625.750 ROUND OFF RADII, r, Figure 15. Curves of Tangential Forces Versus Round-Off Radii.

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REFERENCES 1. a. Colding, B. N., Shear Spinning,ASME Paper No. 59-PROD-2, 1959. b. Feola, J. N., Experimental Analysis of Shear Deformation, M. S. Thesis, Syracuse University, January, 1955. c. Siebel, E., and Droge, K. A., "Forces and Material Flow in Spinning," Werkstattstechnik and Maschinenbau 45 No. 1, S-9 (January, 1955). 2. Reichel, H., "Roll Spinning of Cone Shaped Aluminum Parts," Fertigungstechnik, Q, Part 1, No. 5, 181-184 (April, 1958); Part 2, No. 6, 252-260 (June, 1958). 3. Rouse, H. (Editor), Advanced Mechanics of Fluids, John Wiley and Sons, New York, 1959, p. 204. 4. Prager, W. and Hodge, P. H., Jr., Theory of Perfectly Plastic Solids. John Wiley and Sons, New York, 1951, p. 31. 5. Avitzur, B., Analysis of Power Spinning of Cones, Ph.D. Thesis, University of Michigan, June, 1959.

APPENDIX TRANSFORMATION OF THE COORDINATE SYSTEMS AND MATHEMATICAL DESCRIPTION OF THE TORIC PART OF THE ROLLER A. The Sets of Axis and Their Transformation Consider three coordinate systems (see Fig. 4): (1) (x,y,z) cartesian coordinate system, with the origin 0. The axis z is the axis of cylindrical symmetry for the roller, and the origin is the center of the toric portion of the roller. (2) (X,Y,Z) cartesian coordinate system, with the origin 0. The axis Z is the axis of cylindrical symmetry for the cone. (3) (R,e,Z) cylindrical polar coordinates with the same origin 0 and Z axis as the second cartesian system. The directional cosines for transformation from (x,y,z) system to (X,Y,Z) system and vice versa can be represented in the following way. Table AI - Directional Cosines x y z X cosoCo 0 sinOc Y o 1 I 0 Z -sin o0 0 1 cosoc The transformation scheme is according to the following equation. xi ai * aijX (Al) = bj + aijxi where: i = 1,2,3 denote the column number in Table AI j = 1,2,3 denote the row number in Table AI ai = denote the coordinates of the origin 0 in the (x,y,z) coordinate system, a1 a2 =0 -33

-34a3 =-Fn, b denote the coordinates of the origin 0' in the (X,Y,Z) coordinate system, bl = Fn SinO6, b = 0, 2 b3 = Fn cosQ o 2 = y, x3 -x X1 = X, X2 Y, and X3 Z. The transformation is now getting this shape: x: a1 * a 11X1 + a122 + al3X3 = a11X1 al3X3 = X coso50 - z sinoC -' 11 1 12~2 3 11 1 13X3 0 0 x2 = a2 + a21Xl + a22X2 + a23X3 = a22X2 Y= x3 = z = *33 x 4 + 322 * a33X3 =a3 3 a31 * a33X3 -Fn Xsin% + Zc.osO( And applying the second of Equation(A3], one gets: X1 I = x cosO o + z sin OC Fn sinoCo 0 0 0 X2 =Y=y X = Z = - x sinX + z cosOC + Fn cosoC 3 o o o The transformation from R, e Z system to (X,Y,Z) system is to be performed by: X = R cos 8 Y = R sin @ Z Z and from (X,Y,Z) to (R,e,Z) is performed through

-352 2 R XI +Y 8 cos'1 X = sin1, Y = tn'1 Y J 2+y2' ~fX2+y2' X Z=Z The transformation from either system to any other system of the three is now given. x = X cosK.~o - Z sinoCo = R cos 9 cosQo - Z sinOC y - Y = R sin 8 tz = X sinOo + Z cosOC( - Fn = R cos 8 sinOCo + Z cosc,0 -Fn X = R cos 8 = x cosCo + z sinOC + Fn sinaCo (A2) Y R sin 8 = y Z Z Z = -x sinOC + z cosOC + Fn cos OC o 0 0 R X2 +Y2 = (x cosCo+ z cosOo + Fn sini co)2 y co1 X -1 y cos1 x cosr O+ z sin'o +4 Fn sinc\o 1 8cos 1sin,o' c V X2+y2 2+' cos'x0+z sinOC Fn sinO0)2+y2| z -x sin X + z cOSoo + Fn cosOC~ J B. The Roller The roller is composed of a half torus and a cylinder. The half torus exists for z >0. Its equations are: +z2 o y2 ]-r2 = O 0 jZ 2 2 y2 -. ^ j ^ 22 - r^ 0(A3) G = (R cos 8 cosO(o - Z sino C)2 + R2 sin2 8 -/o, (A3)'+ R cos 8 sine'; + Z coso( - Fn - r=

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