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Hydrodynamic lubrication analysis for tube spinning process
Wear 220 Ž1998. 145–153
Hydrodynamic lubrication analysis for tube spinning process Ismail Nawi a , S.M. Mahdavian b
b,)
a The Agency for Assessment and Application of Technology (BPPT), Jakarta, Indonesia Department of Mechanical and Manufacturing Engineering, RMIT, Melbourne, VIC 3083, Australia
Received 14 October 1997; accepted 12 June 1998
Abstract A theoretical analysis based on the two dimensional isothermal Reynolds equation was developed for the hydrodynamic lubrication of the tube spinning process. The linear velocity of the forming tool and rotational velocity of the mandrel both influence the establishment of a hydrodynamic lubricant film thickness at the inlet zone. Formation of a hydrodynamic lubricant film thickness at the inside of the tube is ruled by the eccentricity of the mandrel and tube. The theoretical and experimental estimates of film thickness were compared and are in agreement. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Reynolds equation; Tube spinning process; Lubrication
1. Introduction Tube spinning is a metal forming process, which is used to form pre-formed blanks either to be stretched further or modify shapes. The pre-formed cup shape product is placed over the mandrel and held firmly to the mandrel. During the forming process, the mandrel with the pre-formed cup rotates and the forming tool, with one or two small rollers used to apply localized pressure and moves forward over the mandrel length with constant velocity. This action stretches the cup and decreases its thickness. The presence of an effective lubricant film between contact surfaces in tube spinning process will increase the reduction in thickness, reduce tool wear, prevent cracking and wave forming build-up, and effect the surface roughness of the product. Nawi and Mahdavian w4,5x reported the evidence of forming a hydrodynamic lubrication film under controlled spinning process conditions. They studied the formation of the hydrodynamic lubrication in tube and cone spinning through comparison of changes in surface finish of lubricated and unlubricated spun products. Similar changes in forming forces and thickness of product were also observed. They also reported that an increase in lubricant viscosity effects the rate of metal build-up at the completion of the forming process, which is known the wave forming effect. Most of the available theoretical work for )
Corresponding author. Tel.: q61-399256010; Fax: q61-399256003; E-mail: [email protected]
the spinning process is only concerned with the deformation mechanics of the process. The results or analyses of plasto-hydrodynamic lubrication in other metal forming processes such as wire drawing given by Dowson et al. w1x, extrusion by Wilson and Walowit w6x, Mahdavian w2x, and deep drawing by Mahdavian and Shao w3x cannot be either implemented or modified for the spinning process to estimate the lubricant film thickness. The major difficulty in using these models is the additional relative movement between the tool and workpiece in the spinning process, which makes it different from the other processes. This paper is concerned with the development, using the two dimensional Reynolds equation, of a realistic steady isoviscous hydrodynamic lubrication model for the tube spinning process. This analysis includes both the tool and mandrel velocities. The analysis produces an estimate of the lubricant film thickness between the inside and outside surfaces of the workpiece, mandrel, and forming tool. The results of the analysis are compared with the previous experimental measurements of the authors.
2. Process analysis The lubrication of workpiece and tooling surfaces in the tube spinning process is shown in Fig. 1. The tube is clamped to the mandrel and is rotating with the same speed as the mandrel. The forming roller, while free to
0043-1648r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 Ž 9 8 . 0 0 2 4 8 - 8
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I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
rotate, is also advancing forward, parallel to the mandrel axis towards the trailing edge of the tube. The lubricant is drawn between the outside surface of the tube and the roller into the converging wedge spaces between these surfaces. The lubricant film between the mandrel and the inside tube surface, as shown, in Fig. 2 is formed due to both stretching of the tube wall in the feed direction over the surface of mandrel and also by squeezing the lubricant between the inside tube and mandrel surfaces. The spinning action, which is caused by continuous changes in the clearance between the tube and mandrel, is due to the eccentricity of the mandrel while rotating against the roller. Theoretical analyses are carried out to estimate the lubricant film thickness at the inside and outside surfaces of the tube in two stages: 1. Tube–Roller ŽOutside. lubricant film analysis: Ža. Inlet Zone Analysis Žb. Work Zone Analysis 2. Tube–Mandrel ŽInside. lubricant film analysis In addition to assumptions used for two-dimensional isoviscous Reynolds equation further assumptions are made to assist the modeling of the process. These additional assumptions are: Ž1. The interfaces between contacts of tooling are smooth and parallel. Ž2. Tube and mandrel are clamped together and the surface velocity of tube is equal to V1. Ž3. The surface velocity of the roller is constant and equal to V2 . Ž4. The roller leading section is similar to a truncated cone with angle a . The divergent wedge angle for position where the mandrel and roller axes are parallel is still equal to a and the relative position of these axes changes the wedge angle in the spinning process. The angle a is so called the angle of attack. Ž5. The flow stress
Fig. 2. Side view of the lubricated tube spinning.
sy of material is constant. Ž6. The contact between the roller and tube is theoretically a line contact but the width is estimated by Hertz theory. Assumptions 1 to 4 deal with representing the tooling conditions and their surface. Assumption 5 is common in plastic deformation analysis of workpiece. In most mechanical working processes elastic strains are negligible in relation to plastic strain, and it is therefore justifiable to neglect the elastic deformation. The rigid-perfectly-plastic model also assists to simplify the mathematics. In assumption 6 the Hertz theory is used only to estimate the size of surface contact between the tube and the roller. The support of solid mandrel behind the tube still meets the geometrical conditions required by Hertz theory for two parallel solid cylinder contacts. The assumption of rigidplastic behavior of the tube metal with constant flow stress for maximum compressive stress in Hertz equation is a sound estimate of calculating the surface contact width.
3. Tube–roller (outside) lubricant film analysis
Fig. 1. Lubricated tube spinning.
In order to model the lubrication process, the deformation of the tube by the roller is divided into zones as follows: 1. Inlet Zone is the zone where the lubricant enters between the roller and tube wall. In this area, the tube wall is rigid. 2. Work Zone is the zone where plastic deformation of the tube wall occurs and according to the assumption 5 it is perfectly plastic deformation. 3. Outlet Zone is the zone where the film thickness is assumed to be constant and the workpiece is rigid. These zones are shown in Fig. 3. In the inlet zone, both
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
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The film thickness hŽ y . in the plane y–z is a function of y, which can be found from the geometry of disc to disc system as shown in Fig. 4.Introducing 1rR s 1rR1 q 1 R 2
Ž 2.
h Ž y . s h i q Ž 1 q y 2r2 Rh i .
Ž 3.
The general form of the inlet film thickness h in direction of z-axis as a function of x and y from the geometry of the tube and roller is: h Ž x , y . s h i q Ž y 2r2 R . q Ž x y L . tan a
Fig. 3. The inlet zones of film thickness on the x – z plane.
the workpiece and mandrel surfaces remain rigid. Once the hydrodynamic lubricant film is created due to the inlet wedge action the interfaces between the workpiece and roller are completely separated. The boundaries of this zone are the points A and B, which are located at distances of Xa and L on the x-axis, which is a small portion of the roller width. The roller diameter is much greater than the width of the roller. At the inlet zone, the lubricant is subjected to two surface velocities. The relative rotation of the mandrel and roller is the surface velocity in y direction, and the roller movement in the direction of feed rate is the other surface velocity in x direction. The lubricant pressure is increased from ambient pressure to the level where the tube starts to yield at the boundary of the inlet and work zones. In the work zone, the plastic deformation of the tube proceeds from its inlet zone boundary with the constant flow stress over the work zone until it is completed at the boundary between the work and outlet zones. The local lubricant film varies in this zone and is influenced by the inlet film thickness. The lubricant pressure gradient in this zone is insignificant. The boundaries of the work zone are point’s B and C. In the outlet zone both the work piece and roller surfaces is rigid and separated by a constant film thickness. The analysis of this zone is similar to the other forming process such as extrusion and wire drawing which have already been carried out by numerous researches and does not significantly effect the estimating of the lubricant film thickness.
where Eqs. Ž1. and Ž3. are special cases of the Eq. Ž4.. The pressure gradient in the x direction is much greater than the pressure gradient in the y direction. This is justified through comparison of the pressure variations across the small length of inlet zone with the large radius of the roller. This means the steady Reynolds equation for the inlet zone for steady state becomes,
ErE x Ž h3 E prE x . s 6h Ž UE hrE x q V E hrE y .
h3 Ž d prd x . s y6h Ž Uh q Vx E hrE y . q C1
Ž 6.
Substitute for EhrE y and hŽ x,y . in Eq. Ž6. gives,
Ž d prd x . s y Ž 6h U . r h i q Ž y 2r2 R . 2
q Ž x y L . tan a 4 y Ž 6h Vxy . rR h i q Ž y 2r2 R . q Ž x y L . tan a 4
3
q Cir h i q Ž y 2r2 R . q Ž x y L . tan 4
3
Ž 7. For boundary conditions, d prd x s 0 at x s L and let
The geometry in Fig. 3 shows that the film thickness in the inlet zone along the x-axis in the plane x–z is given by:
Ž 1.
Ž 5.
Introducing, U1 q U2 s yU and V1 q V2 s yV in Eq. Ž5. then integration gives,
3.1. The inlet zone
h Ž x . s h i q Ž x y L . tan a
Ž 4.
Fig. 4. Inlet film thickness on the y – z plane.
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
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Simplifying the above equation, the inlet film thickness becomes, h i s Ž 3h V . Ur Ž V tan a . q drR . r Ž sy tan2a .
Ž 11 .
For the case d s 0, Eq. Ž11. becomes, h i s 3h Ur Ž sy tan a .
Ž 12 .
Eq. Ž12., which becomes independent of velocity V, is similar to the analysis of the inlet zone of extrusion given by Wilson w7x. Introducing the nondimensional parameters: for Hi s h irL, Ui s UrV , d i s drR , Fi s 3h Vrs L Eq. Ž12. becomes, h i s Fi Ž Ui tan a q d i . rtan2a
Fig. 5. Non dimensional inlet zone film thickness vs. the angle of attack for the various value of contact width d i .
y s d where d is very small so, y 2r2 R f 0.Hence Eq. Ž7. becomes, C1 s 6h Ž Uh i q VL drR . Ž 8. Ž . Ž . Substitute C1 from Eq. 8 into Eq. 7 then after integrating yields, p s Ž 6h U . rtan a h i q Ž y 2r2 R . q Ž x y L . tan a 4 q Ž 6h Vy . rR tan2 a h i q Ž y 2r2 R . q Ž x y L . tan a 4 q Ž 6h Vy . Ž Ltan a y h i y y 2r2 R . r2 R tan2 a h i q Ž y 2r2 R . q Ž x y L . tan a 4
2
y Ž 6h Uh i q 6h Vd LrR . r2tan a h i q Ž y 2r2 R . 2
q Ž x y L . tan a 4 C2
Ž 13 .
Eq. Ž13. is plotted in Fig. 5. The effect of the angle a to the lubricant film thickness for the various of non-dimensional half width of the Hertzian contact d i is shown in Fig. 5. The film thickness decreases as the roller angle is increased. Non dimensional value of d i effects the film thickness. A high value of d i generates a higher film thickness than a low value of d i . Wilson’s analysis of the extrusion process may be considered as a special case of the two dimensional analysis where d i is equal to zero. In another words in the extrusion process the die is not rotating in y-axis direction hence V1 and V2 are zero and the influence of d i is diminished. The surface velocity of mandrel Žmandrel rotation. causes to drag more viscous fluid between the diverging surfaces. This will change the pressure distribution and increase the inlet film thickness. The importance of the term Ž V P d . in increasing the inlet film thickness is noticeable in the inlet film thickness equation. The mandrel rotation influences the establishment of hydrodynamic lubricant film even a small contact area Žfraction of square millimeters. is recognized between the roller and the tube surfaces.
Ž 9.
Calculating the constant C2 from the boundary conditions, p s 0 at x s ` and y s `, substituting in Eq. Ž9. gives C2 s 0. Assume that the material starts to yield at value of p s sy Žyield stress. at x s L and, substitute, y s d for contact width from the following Eq. Ž10.. d is estimated from the half width of Hertz contact. In a real situation the contact between the roller and work piece is a Hertz contact. It is assumed that the pressure is maximum at the edge of the contact and remains constant to the position of y s 0. sy s Ž 6h Urh i tan a . q Ž 6h Vdrh i Rtan2a . q Ž 3h Vd Ž L tan a y h i . rh2i Rtan2a . y Ž Ž 3h Uh i q 3h Vd LrR . rh2i tan a .
Ž 10 .
Fig. 6. Non dimensional inlet zone film thickness vs. the angle of attack for the various value of feed rate.
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
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Fig. 6 shows the variation of nondimensional film thickness with respect to the Roller’s angle for different values of nondimensional velocities, Fi s 0.01 and d i s 0.01. The variation is non-linear and as the value of a is increased the film thickness is decreased. Any increase in the feed rate velocity increases the film thickness. In addition, it shows that if the feed rate velocity is increased, the film thickness also increases particularly at the value of a smaller than approximately 308. The rotational motion of the workpiece in addition to its linear motion assists the wedge action to enhance the formation of a thick lubricant film thickness. 3.2. The work zone analysis In the work zone, where the tube material is deformed at the constant pressure p, hence there is no rate of change of the pressure either with x or y so E prE x s 0 and E prE y s 0. Therefore, because d hrdt is also zero, the Reynolds equation may be written as,
Fig. 8. Non dimensional work zone film thickness vs. feed rate for a s 45 degree.
U E hrE x q V E hrE y s 0 or E hrE x q Ž VrU . E hrE y s 0 Ž 14 .
x ) L, hŽ x . s h i q Ž x y L . tan a
Hence, the initial condition of the case of characteristic equation is,
L0 F x F L, hŽ x . s Ž x Ž h i y Ž L0 y l h . tan b .
This equation can be solved by Langrange decomposition. The differential equation of the characteristic family is, d x s d yr Ž VrU . or d yrd x s VrU
Ž 15.
From the geometry of the work zone in Fig. 7, the initial conditions of the film thickness h can be found as,
qL Ž L 0 y l . tan b y h i l h . r Ž L y l h . If hŽ x, 0. is f Ž x . then the solution along this characteristic is hŽ x , y. s f Ž x R . s f Ž x y Uy r V ..Hence, the solution along this characteristic is, h x , y s Ž x y UyrV . Ž h i y Ž L0 y l h . tan b .
h x s x Ž h i y Ž Lo y l h . tan b . qL Ž L o y l h . tan b y h i l h 4 r Ž L y l h .
q L Ž L o y l h . tan b y h i l h . r Ž L y l h .
Ž 16 .
where, l h s L tan a y h i y L tan b q l o tan b 4 rtan a
Ž 17 .
Ž 18 .
Eq. Ž18. may be written in nondimensianal form, Hw s Ž X w y Uw d w R w . Ž Hi y Low y l w . tan b . q Ž Low y l w . tan b y Hi l w 4 r Ž l h y l w .
Ž 19 .
where,
d ) s y, d w s d )rR , Hw s hrL, Uw s UrV , Hi s h irL, l w s l hrL, Low s LorL, R w s RrL, X w s xrL
Ž 20 .
and, Hi is the nondimensional inlet film thickness. The non-dimensional work zone film thickness is plotted against the non-dimensional feed rate Ui for various value of b . The high feed rate increases the film thickness at the work zone as at the inlet zone. Large value of the angle b also contribute to the high value of lubricant film in this zone. This case is applied for a s 45 degree, and if a s b , the film thickness at the work zone is equal to the film thickness at the inlet zone ŽFig. 8..
4. Tube–mandrel (inside) film thickness analysis
Fig. 7. Work zone of the tube.
It was described previously that the tube is initially clamped to the mandrel and during the spinning process
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I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
Fig. 9. The eccentricity of the mandrel and tube.
both mandrel and workpiece rotate at the same speed. There is no slip between the tube and mandrel. The stretching effect in the direction of feed may result from the forming of an external surface of the tube’s wall by the roller but its sliding velocity over the mandrel is insignificant. Experimental studies show that a film is formed between the mandrel and the inside surface of the tube. The formation of film thickness is as a result of the squeezing action that is caused by the initial clearance between the mandrel and workpiece. When the roller is rotating with surface velocity V2 against the workpiece over the surface contact d ŽHertz contact width. the clearance and the slight eccentricity of mandrel rotation as shown in Fig. 9 causes the lubricant to be squeezed between the mandrel and internal surface of the tube. The lubricant pressure, which is built-up that consequently, separates the two surfaces from each other.
Additional assumptions for the inside wall analysis of the lubricant film thickness are: 1. The variation of pressure is considered in y direction, the direction of motion. 2. The contact between the roller and the tube is Hertz contact. 3. The eccentricity of the mandrel and the tube varies similar to a cam motion. 4. The inside surface of the tube is assumed to be stationary without any slip relative to the mandrel’s surface. In the y direction, the pressure gradient E prE y is in the order of p r d where d is the half of the Herztian contact, and in the x direction the pressure gradient E prE x is the in the order of prL where L is the tube length. Since L 4 d therefore, prd 4 prL so, E prE x - E prE y. Assumption number 4 gives, U s 0 and V s 0.Therefore, the two dimensional Reynolds equation becomes, drd y Ž h3 d prd y . s y3h d hrdt
Ž 21 .
Double integrating gives, p s y3h y 2r2 h 3 d hrdt q yC1 q C2
Ž 22 .
The boundary conditions where, p is maximum at y s 0, gives d prd y s 0, at y s 0, and C1 s 0.Hence, Eq. Ž22. becomes, p s y3h y 2r2 h 3 d hrdt q C2
Ž 23 .
C2 is obtained from the boundary conditions p s 0 at y s d , gives C2 s 3hd 2r2 h 3 d hrdt
Ž 24 .
where d is half width of the Hertz contact area. Substituting for C2 from Eq. Ž24. in Eq. Ž23. yields, p s y3h Ž d hrdt . r2 h3 Ž y 2 y d 2 .
Ž 25 .
The maximum pressure pmax occurs at the centerline of the
Fig. 10. Non dimensional inside of the tube wall film thickness with respect to the viscosity factor Fi for various value of non dimensional half contact area d i .
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
contact where y s 0. This pressure pmax does not exceed the yield strength of workpiece, pmax s sy . For pmax and y s 0 in Eq. Ž25. gives,
sy s 3hd 2 Ž d hrdt . r2 h3 X
ously, high lubricant viscosity produces a thick lubricant film. The influence of d i is also significant. The higher the value of d i the thicker the lubricant film will be.
Ž 26 . X
where, Õ s d hrdt, and h s h , hence, hX s 3hd 2 ÕXr2 sy 4
1r3
Ž 27 .
X
Õ is the approach speed of tube to the mandrel which is similar to cam motion. The squeezing velocity can be obtained from the cam velocity given by, ÕX s e v cos u
Ž 28 .
where, e is the eccentricity of the mandrel and the tube, v is the radial velocity of the roller, u is the angle of the roller position. Here u is 0, Let, H X s hXrR , dX s d ))rR
Ž 29 .
hence, H X 3 s 3h ÕX dX 2r2 sy R
ž
/
Ž 30 .
Introducing F X s 3h ÕXr2 sy R
Ž 31 .
X
H simplifies to: H X s Ž F Xd X 2 .
151
1r3
Ž 32 .
Fig. 10 shows the variation of non-dimensional film thickness with respect to the non-dimensional viscosity factor for different value of non-dimensional half width contact area d i . The variation is non-linear and the value of film thickness increases if the value of Fi is increased. Obvi-
5. Theoretical and experiment comparison of film thickness. 5.1. Workpiece and roller (outside of the tube) The theoretical film thickness is calculated from the Eq. Ž12. for the following values for the tool geometry, tool dimensions, material property of blank, and spinning process parameters: R s 0.034 m, U s 0.00525 mrs, s y s 5 MPa, V s 10 mrs, a s 458, d s 0.001 m. Lubricant viscosities used in experiment are: Castor Oil, Drawing Oil, and DK 1172 with the viscosity of 0.082 Pa s, 0.164 Pa s, and 1.472 Pa s, respectively. The variation of film thickness with respect to different lubricant viscosity from Eq. Ž12. for the above values are plotted in Fig. 11 for comparison with experimental estimates of the film thickness taken from Mahdavian and Nawi w4x. The experimental estimate of lubricant film thickness values increases as the lubricant viscosity is increased. The theoretical curve follows the same trend as the experimental results. The theoretical film thickness is less than the experimental results. This is due to the neglect of the pressure dependency of viscosity in the analysis. The trend of increasing film thickness with viscosity for both theory and experiment results are similar, and still the theoretical model may be used to predict the formation of a lubricant film in the process.
Fig. 11. Comparison of the outside lubricant film thickness between experimental and theoretical results of the tube spinning Ž x-axis not in scale..
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
152
Fig. 12. Comparison of the inside lubricant film thickness between experimental and theoretical results of the tube spinning Ž x-axis not in scale..
5.2. Theoretical and experimental comparison of the film thickness between workpiece and mandrel (inside of the tube) The film thickness between the workpiece and the mandrel is determined by Eq. Ž27.. The roller velocity in the z direction Ž ÕX . where only the mandrel surface is lubricated is defined by using Eq. Ž28.. The clearance between the mandrel and the tube is 0.25 mm, hence the eccentricity is 0.5 mm and for the position u s 0 the velocity becomes ÕX s 13 mmrs. These values are used to calculate the inside film thickness. Fig. 12 shows the comparison between theory and the experimental results. Lubricant film thickness between the workpiece–mandrel theoretically and practically produce similar trend as shown in Fig. 12. The experimental results are higher than the theoretical results beyond 0.164 Pa s. This is because of the factors or variables that have not been considered in the formulation of the equation such as the influence of temperature and pressure changes in viscosity. Another factor that can be influenced to this phenomena is the estimation of the eccentricity between the mandrel and the tube. Since the eccentricity can affect the speed of the lubricant on z direction, the eccentricity has to be estimated accurately.
extrusion process. The variation of film thickness was plotted for the various parameters and it was concluded that the formation of high lubricant thickness is only achieved under certain conditions. It was also shown that the angle of attack a Žthe roller profile. influences the magnitude of the film thickness; large value of a reduces the thickness more than a small value of the angle. Increasing the feed rate andror mandrel rotation results in a higher film thickness. The work zone film thickness is mainly related to the inlet zone film thickness, the deformation angle and the angle of attack. A comparison between the theory and experimental film thickness measurements was carried out. The theory was used to estimate the film thickness both for inside and outside of the workpiece with the parameters used in experiments. The comparison between theoretical and experimental film thickness indicated that the inside film thickness from theory was higher than the experimental measurement but beyond the viscosity of 0.164 Pa s the experimental data produce higher film thickness. However, the differences are insignificant for the inside film thickness. The outside film thickness for the theory is less than the result from experimental data and the viscosity pressure variation and the possibility of occurrence of a mixed lubrication regime as the result of asperity contact should be considered in future analysis.
6. Conclusions 7. Biographies Using the Reynolds equation derived a theoretical analysis for the film thickness in tube spinning. Theoretical models were developed for the inlet zone and work zone at the outside of the workpiece wall. The film thickness at the inside of the wall, between the workpiece and mandrel, was also analyzed. The result of the theoretical model for the inlet zone film thickness was obtained in a closed form equation. The result of the model for the case where the Hertz contact is neglected is identical to the Wilson theory w7x that was developed for the inlet zone analysis of
M. Mahdavian received his BS Eng. from Tehran Polytechnic Institute in 1969, MS Northwestern University, and PhD from University of Massachusetts in 1975. He then joined the faculty of Engineering, of Shiraz University in Iran for 5 years. He is currently a senior lecturer at the Department of Mechanical and Manufacturing Engineering of Royal Melbourne Institute of Technology. His research interest is in areas of process tribology, manufacturing processes, and automation.
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
I. Nawi received his MS from Bandung Institute of Technology, Indonesia in 1989 and his PhD from Royal Melbourne Institute of Technology in 1996. He is currently with the Indonesian Agency for Assessment and Application of Technology. His research interests are in Flexible Manufacturing Systems and Manufacturing Processes. Appendix A. Nomenclature
a1 a b u v h d d) d )) sy d i s drR d w s d )rR dX s d ))rR F X s 3h ÕXr2 sy R Fi s 3h Vrs L h hX HX Hi s h irL Ho hi Hw s h r L lh
lo L Low s LorL
Angle of the roller profile for h o Angle of attack Workpiece inlet angle at the work zone Angle of contact area in y direction, angle of the roller position Radial velocity of the roller Viscosity Half width of Hertz contact area Half width of Hertz contact area of the work zone Half width of Hertz contact area of the mandrel–tube Yield stress Non dimensional contact width of inlet Non dimensional contact width of the work zone Non dimensional contact width of the mandrel–workpiece mandrel–tube eccentricity Non dimensional viscosity factor of the workpiece–mandrel Non dimensional viscosity factor of the tube Film thickness Film thickness between the mandrel and workpiece of the tube Non dimensional mandrel–workpiece film thickness Non dimensional inlet zone film thickness of the tube Work zone film thickness at distance x Inlet film thickness of tube Non dimensional work zone film thickness of the tube Length of contact area in Hertz contact, viscous thermal parameter, width of Hertz contact Distance on x-axis to the end of work zone Distance on x-axis to the end of inlet zone Non dimensional length at the end of work zone
L w s l hrl n p r r1 r2 R R1 R2 R w s RrL U1 U2 Ui s UrV Uw s UrV V ÕX V1 V2 x X w s xrL Yw y
153
Non dimensional width of the contact of roller and workpiece Revolutionrminute Pressure Radius of round ended roller profile Radius of roller profile on the work zone Radius of the deformation on the work zone Resultant of R 1 and R 2 Radius of the roller Radius of the mandrel Non dimensional resultant radius of the work zone Speed of the tool Speed of the mandrel Non dimensional feed rate Non dimensional feed rate of the work zone analysis Total surface velocity of mandrel and roller Lubricant velocity in z direction Surface velocity of mandrel Surface velocity of roller Distance on the x-axis Non dimensional distance on the xaxis Non dimensional distance on the yaxis Distance variable on y-axis
References w1x D. Dowson, B. Parson, P.J. Lidgitt, An elastoplastohydrodynamic lubrication analysis of wire drawing process, Elastohydodynamic Lubrication Symposium, Inst. Mech. Eng., p. 97, 1972. w2x S.M. Mahdavian, A thermal hydrodynamic lubrication analysis for hydrodynamic extrusion of a work hardening metal, ASME J. Tribology 108 Ž1986. 368. w3x S.M. Mahdavian, Z.M. Shao, Isoviscous hydrodynamic lubrication of deep drawing and it is comparison with experiment, ASME J. Tribology 115 Ž1993. 111–118. w4x S.M. Mahdavian, I. Nawi, Hydrodynamic Lubrication in Tube Spinning Process, Conference on Seamless Tube Technology, Tube and Pipe—Hong Kong, Organized by International Tube Association, pp. 184–189, Nov. 1993. w5x I. Nawi, S.M. Mahdavian, Hydrodynamic Lubrication in Metal Spinning, ASME Proceeding of Symposium on Tribology in Manufacturing Processes, Chicago, TRIBO—Vol.5rPED—Vol. 69, pp. 119– 123, Nov. 1994. w6x W.R.D. Wilson, J.A. Walowit, An isothermal hydrodynamic lubrication theory for hydrostatic extrusion and drawing process with conical dies, J. Lubrication Technology, Trans. ASME 93 Ž1. Ž1971. 69–74, Ser. F. w7x W.R.D. Wilson, The temporary breakdown of hydrodynamic lubrication during the initiation of extrusion, Int. J. Mech. Sci. 13 Ž1971. 17–28.
Hydrodynamic lubrication analysis for tube spinning process Ismail Nawi a , S.M. Mahdavian b
b,)
a The Agency for Assessment and Application of Technology (BPPT), Jakarta, Indonesia Department of Mechanical and Manufacturing Engineering, RMIT, Melbourne, VIC 3083, Australia
Received 14 October 1997; accepted 12 June 1998
Abstract A theoretical analysis based on the two dimensional isothermal Reynolds equation was developed for the hydrodynamic lubrication of the tube spinning process. The linear velocity of the forming tool and rotational velocity of the mandrel both influence the establishment of a hydrodynamic lubricant film thickness at the inlet zone. Formation of a hydrodynamic lubricant film thickness at the inside of the tube is ruled by the eccentricity of the mandrel and tube. The theoretical and experimental estimates of film thickness were compared and are in agreement. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Reynolds equation; Tube spinning process; Lubrication
1. Introduction Tube spinning is a metal forming process, which is used to form pre-formed blanks either to be stretched further or modify shapes. The pre-formed cup shape product is placed over the mandrel and held firmly to the mandrel. During the forming process, the mandrel with the pre-formed cup rotates and the forming tool, with one or two small rollers used to apply localized pressure and moves forward over the mandrel length with constant velocity. This action stretches the cup and decreases its thickness. The presence of an effective lubricant film between contact surfaces in tube spinning process will increase the reduction in thickness, reduce tool wear, prevent cracking and wave forming build-up, and effect the surface roughness of the product. Nawi and Mahdavian w4,5x reported the evidence of forming a hydrodynamic lubrication film under controlled spinning process conditions. They studied the formation of the hydrodynamic lubrication in tube and cone spinning through comparison of changes in surface finish of lubricated and unlubricated spun products. Similar changes in forming forces and thickness of product were also observed. They also reported that an increase in lubricant viscosity effects the rate of metal build-up at the completion of the forming process, which is known the wave forming effect. Most of the available theoretical work for )
Corresponding author. Tel.: q61-399256010; Fax: q61-399256003; E-mail: [email protected]
the spinning process is only concerned with the deformation mechanics of the process. The results or analyses of plasto-hydrodynamic lubrication in other metal forming processes such as wire drawing given by Dowson et al. w1x, extrusion by Wilson and Walowit w6x, Mahdavian w2x, and deep drawing by Mahdavian and Shao w3x cannot be either implemented or modified for the spinning process to estimate the lubricant film thickness. The major difficulty in using these models is the additional relative movement between the tool and workpiece in the spinning process, which makes it different from the other processes. This paper is concerned with the development, using the two dimensional Reynolds equation, of a realistic steady isoviscous hydrodynamic lubrication model for the tube spinning process. This analysis includes both the tool and mandrel velocities. The analysis produces an estimate of the lubricant film thickness between the inside and outside surfaces of the workpiece, mandrel, and forming tool. The results of the analysis are compared with the previous experimental measurements of the authors.
2. Process analysis The lubrication of workpiece and tooling surfaces in the tube spinning process is shown in Fig. 1. The tube is clamped to the mandrel and is rotating with the same speed as the mandrel. The forming roller, while free to
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rotate, is also advancing forward, parallel to the mandrel axis towards the trailing edge of the tube. The lubricant is drawn between the outside surface of the tube and the roller into the converging wedge spaces between these surfaces. The lubricant film between the mandrel and the inside tube surface, as shown, in Fig. 2 is formed due to both stretching of the tube wall in the feed direction over the surface of mandrel and also by squeezing the lubricant between the inside tube and mandrel surfaces. The spinning action, which is caused by continuous changes in the clearance between the tube and mandrel, is due to the eccentricity of the mandrel while rotating against the roller. Theoretical analyses are carried out to estimate the lubricant film thickness at the inside and outside surfaces of the tube in two stages: 1. Tube–Roller ŽOutside. lubricant film analysis: Ža. Inlet Zone Analysis Žb. Work Zone Analysis 2. Tube–Mandrel ŽInside. lubricant film analysis In addition to assumptions used for two-dimensional isoviscous Reynolds equation further assumptions are made to assist the modeling of the process. These additional assumptions are: Ž1. The interfaces between contacts of tooling are smooth and parallel. Ž2. Tube and mandrel are clamped together and the surface velocity of tube is equal to V1. Ž3. The surface velocity of the roller is constant and equal to V2 . Ž4. The roller leading section is similar to a truncated cone with angle a . The divergent wedge angle for position where the mandrel and roller axes are parallel is still equal to a and the relative position of these axes changes the wedge angle in the spinning process. The angle a is so called the angle of attack. Ž5. The flow stress
Fig. 2. Side view of the lubricated tube spinning.
sy of material is constant. Ž6. The contact between the roller and tube is theoretically a line contact but the width is estimated by Hertz theory. Assumptions 1 to 4 deal with representing the tooling conditions and their surface. Assumption 5 is common in plastic deformation analysis of workpiece. In most mechanical working processes elastic strains are negligible in relation to plastic strain, and it is therefore justifiable to neglect the elastic deformation. The rigid-perfectly-plastic model also assists to simplify the mathematics. In assumption 6 the Hertz theory is used only to estimate the size of surface contact between the tube and the roller. The support of solid mandrel behind the tube still meets the geometrical conditions required by Hertz theory for two parallel solid cylinder contacts. The assumption of rigidplastic behavior of the tube metal with constant flow stress for maximum compressive stress in Hertz equation is a sound estimate of calculating the surface contact width.
3. Tube–roller (outside) lubricant film analysis
Fig. 1. Lubricated tube spinning.
In order to model the lubrication process, the deformation of the tube by the roller is divided into zones as follows: 1. Inlet Zone is the zone where the lubricant enters between the roller and tube wall. In this area, the tube wall is rigid. 2. Work Zone is the zone where plastic deformation of the tube wall occurs and according to the assumption 5 it is perfectly plastic deformation. 3. Outlet Zone is the zone where the film thickness is assumed to be constant and the workpiece is rigid. These zones are shown in Fig. 3. In the inlet zone, both
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The film thickness hŽ y . in the plane y–z is a function of y, which can be found from the geometry of disc to disc system as shown in Fig. 4.Introducing 1rR s 1rR1 q 1 R 2
Ž 2.
h Ž y . s h i q Ž 1 q y 2r2 Rh i .
Ž 3.
The general form of the inlet film thickness h in direction of z-axis as a function of x and y from the geometry of the tube and roller is: h Ž x , y . s h i q Ž y 2r2 R . q Ž x y L . tan a
Fig. 3. The inlet zones of film thickness on the x – z plane.
the workpiece and mandrel surfaces remain rigid. Once the hydrodynamic lubricant film is created due to the inlet wedge action the interfaces between the workpiece and roller are completely separated. The boundaries of this zone are the points A and B, which are located at distances of Xa and L on the x-axis, which is a small portion of the roller width. The roller diameter is much greater than the width of the roller. At the inlet zone, the lubricant is subjected to two surface velocities. The relative rotation of the mandrel and roller is the surface velocity in y direction, and the roller movement in the direction of feed rate is the other surface velocity in x direction. The lubricant pressure is increased from ambient pressure to the level where the tube starts to yield at the boundary of the inlet and work zones. In the work zone, the plastic deformation of the tube proceeds from its inlet zone boundary with the constant flow stress over the work zone until it is completed at the boundary between the work and outlet zones. The local lubricant film varies in this zone and is influenced by the inlet film thickness. The lubricant pressure gradient in this zone is insignificant. The boundaries of the work zone are point’s B and C. In the outlet zone both the work piece and roller surfaces is rigid and separated by a constant film thickness. The analysis of this zone is similar to the other forming process such as extrusion and wire drawing which have already been carried out by numerous researches and does not significantly effect the estimating of the lubricant film thickness.
where Eqs. Ž1. and Ž3. are special cases of the Eq. Ž4.. The pressure gradient in the x direction is much greater than the pressure gradient in the y direction. This is justified through comparison of the pressure variations across the small length of inlet zone with the large radius of the roller. This means the steady Reynolds equation for the inlet zone for steady state becomes,
ErE x Ž h3 E prE x . s 6h Ž UE hrE x q V E hrE y .
h3 Ž d prd x . s y6h Ž Uh q Vx E hrE y . q C1
Ž 6.
Substitute for EhrE y and hŽ x,y . in Eq. Ž6. gives,
Ž d prd x . s y Ž 6h U . r h i q Ž y 2r2 R . 2
q Ž x y L . tan a 4 y Ž 6h Vxy . rR h i q Ž y 2r2 R . q Ž x y L . tan a 4
3
q Cir h i q Ž y 2r2 R . q Ž x y L . tan 4
3
Ž 7. For boundary conditions, d prd x s 0 at x s L and let
The geometry in Fig. 3 shows that the film thickness in the inlet zone along the x-axis in the plane x–z is given by:
Ž 1.
Ž 5.
Introducing, U1 q U2 s yU and V1 q V2 s yV in Eq. Ž5. then integration gives,
3.1. The inlet zone
h Ž x . s h i q Ž x y L . tan a
Ž 4.
Fig. 4. Inlet film thickness on the y – z plane.
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
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Simplifying the above equation, the inlet film thickness becomes, h i s Ž 3h V . Ur Ž V tan a . q drR . r Ž sy tan2a .
Ž 11 .
For the case d s 0, Eq. Ž11. becomes, h i s 3h Ur Ž sy tan a .
Ž 12 .
Eq. Ž12., which becomes independent of velocity V, is similar to the analysis of the inlet zone of extrusion given by Wilson w7x. Introducing the nondimensional parameters: for Hi s h irL, Ui s UrV , d i s drR , Fi s 3h Vrs L Eq. Ž12. becomes, h i s Fi Ž Ui tan a q d i . rtan2a
Fig. 5. Non dimensional inlet zone film thickness vs. the angle of attack for the various value of contact width d i .
y s d where d is very small so, y 2r2 R f 0.Hence Eq. Ž7. becomes, C1 s 6h Ž Uh i q VL drR . Ž 8. Ž . Ž . Substitute C1 from Eq. 8 into Eq. 7 then after integrating yields, p s Ž 6h U . rtan a h i q Ž y 2r2 R . q Ž x y L . tan a 4 q Ž 6h Vy . rR tan2 a h i q Ž y 2r2 R . q Ž x y L . tan a 4 q Ž 6h Vy . Ž Ltan a y h i y y 2r2 R . r2 R tan2 a h i q Ž y 2r2 R . q Ž x y L . tan a 4
2
y Ž 6h Uh i q 6h Vd LrR . r2tan a h i q Ž y 2r2 R . 2
q Ž x y L . tan a 4 C2
Ž 13 .
Eq. Ž13. is plotted in Fig. 5. The effect of the angle a to the lubricant film thickness for the various of non-dimensional half width of the Hertzian contact d i is shown in Fig. 5. The film thickness decreases as the roller angle is increased. Non dimensional value of d i effects the film thickness. A high value of d i generates a higher film thickness than a low value of d i . Wilson’s analysis of the extrusion process may be considered as a special case of the two dimensional analysis where d i is equal to zero. In another words in the extrusion process the die is not rotating in y-axis direction hence V1 and V2 are zero and the influence of d i is diminished. The surface velocity of mandrel Žmandrel rotation. causes to drag more viscous fluid between the diverging surfaces. This will change the pressure distribution and increase the inlet film thickness. The importance of the term Ž V P d . in increasing the inlet film thickness is noticeable in the inlet film thickness equation. The mandrel rotation influences the establishment of hydrodynamic lubricant film even a small contact area Žfraction of square millimeters. is recognized between the roller and the tube surfaces.
Ž 9.
Calculating the constant C2 from the boundary conditions, p s 0 at x s ` and y s `, substituting in Eq. Ž9. gives C2 s 0. Assume that the material starts to yield at value of p s sy Žyield stress. at x s L and, substitute, y s d for contact width from the following Eq. Ž10.. d is estimated from the half width of Hertz contact. In a real situation the contact between the roller and work piece is a Hertz contact. It is assumed that the pressure is maximum at the edge of the contact and remains constant to the position of y s 0. sy s Ž 6h Urh i tan a . q Ž 6h Vdrh i Rtan2a . q Ž 3h Vd Ž L tan a y h i . rh2i Rtan2a . y Ž Ž 3h Uh i q 3h Vd LrR . rh2i tan a .
Ž 10 .
Fig. 6. Non dimensional inlet zone film thickness vs. the angle of attack for the various value of feed rate.
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
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Fig. 6 shows the variation of nondimensional film thickness with respect to the Roller’s angle for different values of nondimensional velocities, Fi s 0.01 and d i s 0.01. The variation is non-linear and as the value of a is increased the film thickness is decreased. Any increase in the feed rate velocity increases the film thickness. In addition, it shows that if the feed rate velocity is increased, the film thickness also increases particularly at the value of a smaller than approximately 308. The rotational motion of the workpiece in addition to its linear motion assists the wedge action to enhance the formation of a thick lubricant film thickness. 3.2. The work zone analysis In the work zone, where the tube material is deformed at the constant pressure p, hence there is no rate of change of the pressure either with x or y so E prE x s 0 and E prE y s 0. Therefore, because d hrdt is also zero, the Reynolds equation may be written as,
Fig. 8. Non dimensional work zone film thickness vs. feed rate for a s 45 degree.
U E hrE x q V E hrE y s 0 or E hrE x q Ž VrU . E hrE y s 0 Ž 14 .
x ) L, hŽ x . s h i q Ž x y L . tan a
Hence, the initial condition of the case of characteristic equation is,
L0 F x F L, hŽ x . s Ž x Ž h i y Ž L0 y l h . tan b .
This equation can be solved by Langrange decomposition. The differential equation of the characteristic family is, d x s d yr Ž VrU . or d yrd x s VrU
Ž 15.
From the geometry of the work zone in Fig. 7, the initial conditions of the film thickness h can be found as,
qL Ž L 0 y l . tan b y h i l h . r Ž L y l h . If hŽ x, 0. is f Ž x . then the solution along this characteristic is hŽ x , y. s f Ž x R . s f Ž x y Uy r V ..Hence, the solution along this characteristic is, h x , y s Ž x y UyrV . Ž h i y Ž L0 y l h . tan b .
h x s x Ž h i y Ž Lo y l h . tan b . qL Ž L o y l h . tan b y h i l h 4 r Ž L y l h .
q L Ž L o y l h . tan b y h i l h . r Ž L y l h .
Ž 16 .
where, l h s L tan a y h i y L tan b q l o tan b 4 rtan a
Ž 17 .
Ž 18 .
Eq. Ž18. may be written in nondimensianal form, Hw s Ž X w y Uw d w R w . Ž Hi y Low y l w . tan b . q Ž Low y l w . tan b y Hi l w 4 r Ž l h y l w .
Ž 19 .
where,
d ) s y, d w s d )rR , Hw s hrL, Uw s UrV , Hi s h irL, l w s l hrL, Low s LorL, R w s RrL, X w s xrL
Ž 20 .
and, Hi is the nondimensional inlet film thickness. The non-dimensional work zone film thickness is plotted against the non-dimensional feed rate Ui for various value of b . The high feed rate increases the film thickness at the work zone as at the inlet zone. Large value of the angle b also contribute to the high value of lubricant film in this zone. This case is applied for a s 45 degree, and if a s b , the film thickness at the work zone is equal to the film thickness at the inlet zone ŽFig. 8..
4. Tube–mandrel (inside) film thickness analysis
Fig. 7. Work zone of the tube.
It was described previously that the tube is initially clamped to the mandrel and during the spinning process
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Fig. 9. The eccentricity of the mandrel and tube.
both mandrel and workpiece rotate at the same speed. There is no slip between the tube and mandrel. The stretching effect in the direction of feed may result from the forming of an external surface of the tube’s wall by the roller but its sliding velocity over the mandrel is insignificant. Experimental studies show that a film is formed between the mandrel and the inside surface of the tube. The formation of film thickness is as a result of the squeezing action that is caused by the initial clearance between the mandrel and workpiece. When the roller is rotating with surface velocity V2 against the workpiece over the surface contact d ŽHertz contact width. the clearance and the slight eccentricity of mandrel rotation as shown in Fig. 9 causes the lubricant to be squeezed between the mandrel and internal surface of the tube. The lubricant pressure, which is built-up that consequently, separates the two surfaces from each other.
Additional assumptions for the inside wall analysis of the lubricant film thickness are: 1. The variation of pressure is considered in y direction, the direction of motion. 2. The contact between the roller and the tube is Hertz contact. 3. The eccentricity of the mandrel and the tube varies similar to a cam motion. 4. The inside surface of the tube is assumed to be stationary without any slip relative to the mandrel’s surface. In the y direction, the pressure gradient E prE y is in the order of p r d where d is the half of the Herztian contact, and in the x direction the pressure gradient E prE x is the in the order of prL where L is the tube length. Since L 4 d therefore, prd 4 prL so, E prE x - E prE y. Assumption number 4 gives, U s 0 and V s 0.Therefore, the two dimensional Reynolds equation becomes, drd y Ž h3 d prd y . s y3h d hrdt
Ž 21 .
Double integrating gives, p s y3h y 2r2 h 3 d hrdt q yC1 q C2
Ž 22 .
The boundary conditions where, p is maximum at y s 0, gives d prd y s 0, at y s 0, and C1 s 0.Hence, Eq. Ž22. becomes, p s y3h y 2r2 h 3 d hrdt q C2
Ž 23 .
C2 is obtained from the boundary conditions p s 0 at y s d , gives C2 s 3hd 2r2 h 3 d hrdt
Ž 24 .
where d is half width of the Hertz contact area. Substituting for C2 from Eq. Ž24. in Eq. Ž23. yields, p s y3h Ž d hrdt . r2 h3 Ž y 2 y d 2 .
Ž 25 .
The maximum pressure pmax occurs at the centerline of the
Fig. 10. Non dimensional inside of the tube wall film thickness with respect to the viscosity factor Fi for various value of non dimensional half contact area d i .
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
contact where y s 0. This pressure pmax does not exceed the yield strength of workpiece, pmax s sy . For pmax and y s 0 in Eq. Ž25. gives,
sy s 3hd 2 Ž d hrdt . r2 h3 X
ously, high lubricant viscosity produces a thick lubricant film. The influence of d i is also significant. The higher the value of d i the thicker the lubricant film will be.
Ž 26 . X
where, Õ s d hrdt, and h s h , hence, hX s 3hd 2 ÕXr2 sy 4
1r3
Ž 27 .
X
Õ is the approach speed of tube to the mandrel which is similar to cam motion. The squeezing velocity can be obtained from the cam velocity given by, ÕX s e v cos u
Ž 28 .
where, e is the eccentricity of the mandrel and the tube, v is the radial velocity of the roller, u is the angle of the roller position. Here u is 0, Let, H X s hXrR , dX s d ))rR
Ž 29 .
hence, H X 3 s 3h ÕX dX 2r2 sy R
ž
/
Ž 30 .
Introducing F X s 3h ÕXr2 sy R
Ž 31 .
X
H simplifies to: H X s Ž F Xd X 2 .
151
1r3
Ž 32 .
Fig. 10 shows the variation of non-dimensional film thickness with respect to the non-dimensional viscosity factor for different value of non-dimensional half width contact area d i . The variation is non-linear and the value of film thickness increases if the value of Fi is increased. Obvi-
5. Theoretical and experiment comparison of film thickness. 5.1. Workpiece and roller (outside of the tube) The theoretical film thickness is calculated from the Eq. Ž12. for the following values for the tool geometry, tool dimensions, material property of blank, and spinning process parameters: R s 0.034 m, U s 0.00525 mrs, s y s 5 MPa, V s 10 mrs, a s 458, d s 0.001 m. Lubricant viscosities used in experiment are: Castor Oil, Drawing Oil, and DK 1172 with the viscosity of 0.082 Pa s, 0.164 Pa s, and 1.472 Pa s, respectively. The variation of film thickness with respect to different lubricant viscosity from Eq. Ž12. for the above values are plotted in Fig. 11 for comparison with experimental estimates of the film thickness taken from Mahdavian and Nawi w4x. The experimental estimate of lubricant film thickness values increases as the lubricant viscosity is increased. The theoretical curve follows the same trend as the experimental results. The theoretical film thickness is less than the experimental results. This is due to the neglect of the pressure dependency of viscosity in the analysis. The trend of increasing film thickness with viscosity for both theory and experiment results are similar, and still the theoretical model may be used to predict the formation of a lubricant film in the process.
Fig. 11. Comparison of the outside lubricant film thickness between experimental and theoretical results of the tube spinning Ž x-axis not in scale..
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152
Fig. 12. Comparison of the inside lubricant film thickness between experimental and theoretical results of the tube spinning Ž x-axis not in scale..
5.2. Theoretical and experimental comparison of the film thickness between workpiece and mandrel (inside of the tube) The film thickness between the workpiece and the mandrel is determined by Eq. Ž27.. The roller velocity in the z direction Ž ÕX . where only the mandrel surface is lubricated is defined by using Eq. Ž28.. The clearance between the mandrel and the tube is 0.25 mm, hence the eccentricity is 0.5 mm and for the position u s 0 the velocity becomes ÕX s 13 mmrs. These values are used to calculate the inside film thickness. Fig. 12 shows the comparison between theory and the experimental results. Lubricant film thickness between the workpiece–mandrel theoretically and practically produce similar trend as shown in Fig. 12. The experimental results are higher than the theoretical results beyond 0.164 Pa s. This is because of the factors or variables that have not been considered in the formulation of the equation such as the influence of temperature and pressure changes in viscosity. Another factor that can be influenced to this phenomena is the estimation of the eccentricity between the mandrel and the tube. Since the eccentricity can affect the speed of the lubricant on z direction, the eccentricity has to be estimated accurately.
extrusion process. The variation of film thickness was plotted for the various parameters and it was concluded that the formation of high lubricant thickness is only achieved under certain conditions. It was also shown that the angle of attack a Žthe roller profile. influences the magnitude of the film thickness; large value of a reduces the thickness more than a small value of the angle. Increasing the feed rate andror mandrel rotation results in a higher film thickness. The work zone film thickness is mainly related to the inlet zone film thickness, the deformation angle and the angle of attack. A comparison between the theory and experimental film thickness measurements was carried out. The theory was used to estimate the film thickness both for inside and outside of the workpiece with the parameters used in experiments. The comparison between theoretical and experimental film thickness indicated that the inside film thickness from theory was higher than the experimental measurement but beyond the viscosity of 0.164 Pa s the experimental data produce higher film thickness. However, the differences are insignificant for the inside film thickness. The outside film thickness for the theory is less than the result from experimental data and the viscosity pressure variation and the possibility of occurrence of a mixed lubrication regime as the result of asperity contact should be considered in future analysis.
6. Conclusions 7. Biographies Using the Reynolds equation derived a theoretical analysis for the film thickness in tube spinning. Theoretical models were developed for the inlet zone and work zone at the outside of the workpiece wall. The film thickness at the inside of the wall, between the workpiece and mandrel, was also analyzed. The result of the theoretical model for the inlet zone film thickness was obtained in a closed form equation. The result of the model for the case where the Hertz contact is neglected is identical to the Wilson theory w7x that was developed for the inlet zone analysis of
M. Mahdavian received his BS Eng. from Tehran Polytechnic Institute in 1969, MS Northwestern University, and PhD from University of Massachusetts in 1975. He then joined the faculty of Engineering, of Shiraz University in Iran for 5 years. He is currently a senior lecturer at the Department of Mechanical and Manufacturing Engineering of Royal Melbourne Institute of Technology. His research interest is in areas of process tribology, manufacturing processes, and automation.
I. Nawi, S.M. MahdaÕianr Wear 220 (1998) 145–153
I. Nawi received his MS from Bandung Institute of Technology, Indonesia in 1989 and his PhD from Royal Melbourne Institute of Technology in 1996. He is currently with the Indonesian Agency for Assessment and Application of Technology. His research interests are in Flexible Manufacturing Systems and Manufacturing Processes. Appendix A. Nomenclature
a1 a b u v h d d) d )) sy d i s drR d w s d )rR dX s d ))rR F X s 3h ÕXr2 sy R Fi s 3h Vrs L h hX HX Hi s h irL Ho hi Hw s h r L lh
lo L Low s LorL
Angle of the roller profile for h o Angle of attack Workpiece inlet angle at the work zone Angle of contact area in y direction, angle of the roller position Radial velocity of the roller Viscosity Half width of Hertz contact area Half width of Hertz contact area of the work zone Half width of Hertz contact area of the mandrel–tube Yield stress Non dimensional contact width of inlet Non dimensional contact width of the work zone Non dimensional contact width of the mandrel–workpiece mandrel–tube eccentricity Non dimensional viscosity factor of the workpiece–mandrel Non dimensional viscosity factor of the tube Film thickness Film thickness between the mandrel and workpiece of the tube Non dimensional mandrel–workpiece film thickness Non dimensional inlet zone film thickness of the tube Work zone film thickness at distance x Inlet film thickness of tube Non dimensional work zone film thickness of the tube Length of contact area in Hertz contact, viscous thermal parameter, width of Hertz contact Distance on x-axis to the end of work zone Distance on x-axis to the end of inlet zone Non dimensional length at the end of work zone
L w s l hrl n p r r1 r2 R R1 R2 R w s RrL U1 U2 Ui s UrV Uw s UrV V ÕX V1 V2 x X w s xrL Yw y
153
Non dimensional width of the contact of roller and workpiece Revolutionrminute Pressure Radius of round ended roller profile Radius of roller profile on the work zone Radius of the deformation on the work zone Resultant of R 1 and R 2 Radius of the roller Radius of the mandrel Non dimensional resultant radius of the work zone Speed of the tool Speed of the mandrel Non dimensional feed rate Non dimensional feed rate of the work zone analysis Total surface velocity of mandrel and roller Lubricant velocity in z direction Surface velocity of mandrel Surface velocity of roller Distance on the x-axis Non dimensional distance on the xaxis Non dimensional distance on the yaxis Distance variable on y-axis
References w1x D. Dowson, B. Parson, P.J. Lidgitt, An elastoplastohydrodynamic lubrication analysis of wire drawing process, Elastohydodynamic Lubrication Symposium, Inst. Mech. Eng., p. 97, 1972. w2x S.M. Mahdavian, A thermal hydrodynamic lubrication analysis for hydrodynamic extrusion of a work hardening metal, ASME J. Tribology 108 Ž1986. 368. w3x S.M. Mahdavian, Z.M. Shao, Isoviscous hydrodynamic lubrication of deep drawing and it is comparison with experiment, ASME J. Tribology 115 Ž1993. 111–118. w4x S.M. Mahdavian, I. Nawi, Hydrodynamic Lubrication in Tube Spinning Process, Conference on Seamless Tube Technology, Tube and Pipe—Hong Kong, Organized by International Tube Association, pp. 184–189, Nov. 1993. w5x I. Nawi, S.M. Mahdavian, Hydrodynamic Lubrication in Metal Spinning, ASME Proceeding of Symposium on Tribology in Manufacturing Processes, Chicago, TRIBO—Vol.5rPED—Vol. 69, pp. 119– 123, Nov. 1994. w6x W.R.D. Wilson, J.A. Walowit, An isothermal hydrodynamic lubrication theory for hydrostatic extrusion and drawing process with conical dies, J. Lubrication Technology, Trans. ASME 93 Ž1. Ž1971. 69–74, Ser. F. w7x W.R.D. Wilson, The temporary breakdown of hydrodynamic lubrication during the initiation of extrusion, Int. J. Mech. Sci. 13 Ž1971. 17–28.