Friction Coefficient s Numerical Determination for Hot Flat Steel Rolling at Low Strain Rate

Transcription

1 American ournal of Applied Matematics 05; 3(5: -8 Publised online September 6, 05 (ttp:// doi: 0.648/j.ajam ISS: (Print; ISS: X (Online riction Coefficient s umerical Determination for Hot lat Steel olling at Low Strain ate Peter Aiyedun, *, O. Ogunlade, A. O. Oni, Olayide Adetunji Department of Mecanical Engineering, University of Agriculture, P. M.., Abeokuta, igeria Department of Mecanical Engineering, University of Lagos, Lagos, igeria address: laiyedun_494@yaoo.com (P. Aiyedun, femday@yaoo.com (A.O. Oni To cite tis article: Peter Aiyedun, O. Ogunlade, A. O. Oni, Olayide Adetunji. riction Coefficient s umerical Determination for Hot lat Steel olling at Low Strain ate. American ournal of Applied Matematics. Vol. 3, o. 5, 05, pp. -8. doi: 0.648/j.ajam Abstract: In tis paper, a non-linear quadratic expression of te coefficient of friction at bot entry and exit sides ave been derived from te equations of rolling load in Hot olling land and ord s program (. Te study developed a numerical model for te estimation of coefficient of friction for steel (HC SS36 of different ticknesses on two ig reversing mills. Te equations for coefficient of friction on entry and exist sides of te mills were modelled from Hot olling land and ord s program (. Te equations were modelled suc tat te friction coefficient can be expressed as a function of process parameters measurable during rolling. Te capability of te model was verified by using a number of specimens of HC SS36 wit predetermined ot rolling experimental data. A good agreement was noted between te predicted friction coefficient and te measured one. eywords: olling Mill A and, Coefficient of riction, Strain ate. Introduction Te effects of friction in rolling operations ave always been an important topic of interest. Tis is due to te fact tat friction significantly affects te quality of materials and te processing parameters. or tis reason, frictional effects are considered to be a major variable tat must be controlled in order to optimize processing procedures to acieve economic goals witout impairing surface quality, te desired geometry or internal structure of te product. Terefore, an accurate evaluation of te friction conditions during rolling operations is required as a quality control measure. Te friction coefficient is regarded to be one of te most important parameters in rolling processes (Altınkaya et al., 04. Different metods for te predictions of coefficient of friction ave been widely reported. Te most common approac is to evaluate coefficient of friction from one or more rolling parameters. Tis is because te direct measurement of friction coefficient is a difficult procedure (Gudur et al., 008; Tan et al., 008. Te early works of Ekelund (933, Wusatowski (969 and Sparling (96 involved empirical models by examining rolling parameters from te observed experimental investigations data. Te notable teoretical models for te evaluation coefficient of friction include omogeneous and inomogeneous deformation equations. Te teory of omogeneous deformation equations was introduced by arman (95. Tis teoretical tecnique was establised from a consideration of equilibrium of forces acting on te elemental slab in te zone of deformation. Te principal assumption is tat te deformation of materials is omogenous trougout te wole deformation zone. However, te teoretical results generally sow poor agreement wit experimental results (ing, 00. Orowan (943 in is teory of inomogenous deformation assumed tat stress distribution in a vertical plane is not omogenous. He obtained a more accurate result. However, Sims (954 improved on Orowan s original metod but wit te assumption tat sticking friction exists along te arc of contact in ot rolling. A good agreement was sown between te results of experiment and calculation. urtermore, Cen et al. (04 also improved arman equation for ot-rolled strip. A new rolling pressure formula was deduced based on compreensive consideration of te slipping and sticking friction on te contact arc between ot-rolled strip and work rolls. Teir work also analysed te influence of friction condition, flow stress, roller distortion and oter factors on te results of rolling pressure and force per unit widt. Te computational results demonstrate tat te proposed new

2 American ournal of Applied Matematics 05; 3(5: -8 model improves te setting precision of roll force and can be applied to online control of ot rolled strip. Te introduction and application of oter teoretical metods suc as slip line analysis (Alexander, 955, te upper bound teory (Siebel, 95; onson and udo, 960 and finite element metod (Hwang and oun, 99; Moon and Lee, 008; Wang et al., 00; Zang and Cui, 0 ave been used for te analysis of coefficient of friction in metal rolling operations. However, tese teoretical metods as not been able to provide detailed beaviour of work piece tus coefficient of friction (Ginzburg, 989; obert, 983. Eac teoretical metod as its own merits and limitations. Altoug many investigators are working on rolling and many prints are available (Legrand et al., 0; Zang et al., 0; Parvizi and Abrinia, 04, some fields remain practically unexplored. Since tis is an expanding field, tere is still muc work tat needs to be done on rolling design. In tis work, Te land and ord s approac were slipping friction exist trougout te roll gap using Orowon s general equation of flat rolling, is also used to numerically determine te coefficient of friction for ot flat rolling of steel at low ( ( I P = 0.5 ( ( ( ( strain rate. In order to determine te coefficient of friction, a quadratic expression of te coefficient at bot entry and exit sides were generated from te Hot olling land and ord s program ( of Aiyedun (984 by making it te subject of formula from te equations of te rolling load troug matematical analysis. MATLA computer algoritm was ten used to develop a computer code for te coefficient of friction equation using sequential quadratic programming metod.. Matematical Model.. Determination of Coefficient of riction from Hot olling land and ord s ( Program Te matematical model used for te simulation is derived from te rolling load (P of Hot olling land and ord s ( program.... Coefficient of riction at te Entry Zone ( ( cos ( tan ( P ( 0.5 Cos = ( ( ( ( ( ( ( tan ( tan ( tan ( We multiply bot sides by and expand te expression, we ave P cos = ( ( ( ( ( ( ( tan ( tan ( tan (3

3 3 Peter Aiyedun et al.: riction Coefficient s umerical Determination for Hot lat Steel olling at Low Strain ate P 8 ( ( cos ( ( ( tan ( tan ( tan (4 = ( ( tan tan ( tan tan 4 P ( (5 8 cos ( 4 ( ( tan tan = 0 [ ] ( [ ( tan ( ( tan 4P (6 8 4 ( cos [( tan ] = 0 Tis expression for in te equation above is a complex non-linear quadratic equation wic can only be solved by a numerical metod. rom El-alay [6], te above expression can furter be expressed as: ( [ {( tan tan ( } {( tan tan } ' 4P r r 8 cos 4 (7 [( tan tan ] = 0... Coefficient of riction at te Exit Zone = P (8

4 American ournal of Applied Matematics 05; 3(5: -8 4 = ( ( ( 0.5 ( cos ( tan (9 We multiply troug by 4 and expand te expression, we ave = ( Log Cos ( ( ( ( ( ( ( ( tan ( tan ( tan ( [ 4 ( I ( ( ( tan ( tan = 8 cos ( ( tan [ ( ] ( tan = ( ( [ ( ( tan ( tan 8 cos ( 4 (

5 5 Peter Aiyedun et al.: riction Coefficient s umerical Determination for Hot lat Steel olling at Low Strain ate [ ] ( [ ( ( tan ( 4 ( tan tan 8 cos ( tan = 0 (3 but = P we ave:- ( in equation (8 and substituting into equation (3, [ ] ( [ ( ( tan ( 4 ( tan tan P ( 8 cos ( tan = 0 [ ] ( [ ( ( tan tan (4 ( tan tan 8 cos (5 4 [ P ( tan tan ] = 0 Also, from El-alay (966, te above expression can furter be expressed as; 4 ' ( [ {( ( tan tan } { ( tan tan } r r 8 cos 4 (6 [ P ( tan tan ] = 0.. Model Simulation and Validation Te equation of te rolling load (P was retrieved from Hot olling land and ord s program in wic a quadratic expression for te coefficient of friction was ten generated. Te computation of te coefficient of friction equation was integrated into te Sequential Quadratic Programming (SQP metod, wic is a numerical metod, and was run on MATLA computer software. Validation of te computer codes was ten carried out using a number of specimens of HC SS36 wit predetermined ot rolling experimental data from Tables 4. and 4. of Aiyedun (984. Tese specimens wit different initial tickness were ot rolled at varying low reductions, widts, furnace temperatures, rolling speed etc. on two-ig reversing rolling mill A (Specimens H0-H57 and

6 American ournal of Applied Matematics 05; 3(5: -8 6 (Specimens P30-P esults and Discussion Te computed results of te coefficient of friction ( at te entry and exit sides against te rolling conditions (reduction (% and equivalent strain rate are illustrated grapically in igs A comparison between Tamano and Yanagimoto (970 and te present work is presented in ig. 5. ig. 3 sows te effect of equivalent strain rate on te coefficient of friction for HC SS36 at low reduction on specimen rolled on mill A (H3-H35 and H36-H44 on te entry and exit sides. It was observed tat as te strain rate increased, te coefficient of friction on bot te entry and exit sides first decreased to a minimum value before increasing on te exit side. 3.. Effect of te olling Conditions on te Coefficient of riction ( on te Entry and Exit Sides ig. sows te effect of reduction (% on te coefficient of friction for HC SS36 on specimens rolled on mill A (H3-H35 and H36-H44 on te entry and exit sides. It was observed tat as te percentage reduction increased, te coefficient of friction decreases accordingly to a minimum value before increasing, tat of te exit side being more pronounced. ig. 3. Effect of Equivalent Strain rate on te Coefficient of friction for HC SS36 at low reduction, on Specimens rolled on Mill A on te entry and exit sides. ig. 4 sows te effect of equivalent strain rate on te coefficient of friction for HC SS36 at low reduction on specimen rolled on mill (P4-P44 and P47-P53 on te entry and exit side. It was observed tat as te strain rate increased, te coefficient of friction on bot te entry and exit side first decreased uniformly to a minimum value before increasing tat of te exit side being more pronounced. ig.. Effect of eduction, r (% on te Coefficient of friction for HC SS36 on Specimens rolled on Mill A on te entry and exit sides. ig. sows te effect of reduction (% on te coefficient of friction for HC SS36 on specimens rolled on mill (P4-P44 and P47-P53 on te entry and exit side. It was observed tat as te percentage reduction decreased before increasing, te coefficient of friction on bot te entry and exit side first decreases to a minimum value before increasing, tat of te exit side being more pronounced. ig. 4. Effect of Equivalent Strain rate on te Coefficient of friction for HC SS36 at low reduction, on Specimens rolled on Mill on te entry and exit sides. 3.. Comparison of te Present Work wit (Tamano and Yanagimoto 970 ig.. Effect of eduction, r (% on te Coefficient of friction for HC SS36 on Specimens rolled on Mill on te entry and exit sides. Te solution in tis work at te entry side is compared wit Tamano and Yanagimoto (970. It is natural tat tere sould be a great difference between te present solution and Tamano and Yanagimoto (970 because of te difference in te state of friction. Altoug, te present solution deals wit friction coefficient in te roll gap for various specimens wit varying and approximately te same percentage reduction

7 7 Peter Aiyedun et al.: riction Coefficient s umerical Determination for Hot lat Steel olling at Low Strain ate wereas Tamano and Yanagimoto (970 addresses te situation of rolling in te wole state of friction. Tis encompasses slipping friction, mixed friction or sticking friction wic depend on geometrical and pysical rolling factors (viz. radius of roll, tickness of strip before and after rolling, rolling temperature, yield stress of materials, and surface and lubricating condition of roll and material. ot te coefficient of friction in te two solutions tends to decrease as te percentage reduction increased, and as te strain rate increased. ig. 5. elation between te rolling conditions and te state of friction, after Tamano and Yanagimoto ( Conclusions In tis paper, a non-linear quadratic expression of te coefficient of friction at bot entry and exit sides ave been derived from te equations of rolling load in Hot olling land and ord s program ( of Aiyedun (984. rom te present teoretical investigation, te following conclusions were drawn: i. land and ord s approac gave a satisfactory result for te estimation of te coefficient of friction in ot flat rolling of steel (HC SS36 at low strain rate wen compared wit te solution found in Tamano and Yanagimoto (970. ii. or specimen rolled on mill A (H3-H35 and H36-H44 on te entry and exit sides, te effect of reduction (% on te coefficient of friction ( for HC SS36 sows tat, as te percentage reduction increased, te coefficient of friction ( decreases accordingly before increasing, tat of te exit side being more pronounced. iii. or specimen rolled on mill (P4-P44 and P47-P53 on te entry and exit sides, te effect of reduction (% on te coefficient of friction ( for HC SS36 sows tat, as te percentage reduction decreased before increasing, te coefficient of friction ( first decreases to a minimum value before increasing, tat of te exit side being more pronounced. iv. or specimen rolled on mill A (H3-H35 and H36- H44 on te entry and exit sides, te effect of equivalent strain rate on te coefficient of friction ( for HC SS36 at low reduction, sows tat, as te strain rate increased, te coefficient of friction ( first decreased to a minimum value before increasing, tat of te exit side being more pronounced. v. or specimen rolled on mill (P4-P44 and P47-P53 on te entry and exit sides, te effect of equivalent strain rate on te coefficient of friction ( for HC SS36 at low reduction, sows tat, as te strain rate increased, te coefficient of friction ( first decreased to a minimum value before increasing, tat of te exit side being more pronounced. vi. Wen te present work was compared wit te solutions in Tamano and Yanagimoto (970, tere was a great difference in te state of friction. Te present solution deals wit friction coefficient in te roll gap for various specimens wit varying and approximately te same percentage reduction wile Tamano and Yanagimoto (970 addresses te situation of rolling in te wole state of friction (slipping friction, mixed friction or sticking friction wic depend on geometrical and pysical rolling factors (radius of roll, tickness of strip before and after rolling, rolling temperature, yield stress of materials, and surface and lubricating condition of roll and material. ot te coefficient of friction in te two solutions tends to decrease as te percentage reduction increased, and as te strain rate increased. vii. At low strain rate ( s-, te coefficient of friction decreases uniformly at te wole roll gap for te range of specimens observed. omenclature = tickness before rolling (mm; = tickness after rolling (mm x = tickness of material at some point in te roll gap. S = specific normal pressure (/mm P = vertical roll pressure (/mm Q = orizontal compressive stress in te material at point of tickness ; at xdx, stress = σxdσx = roll radius (undeformed(mm = elastically deformed roll radius (mm = angle subtended by a point on roll surface at center of curvature of arc of contact wit respect to line joining roll centers φ = angle subtended by neutral plane wit respect to vertical α = angle of entry (i.e. maximum value of in rad = angle at wic sticking starts in ot rolling bland & ford s approac (radians = angle at wic sticking ends in ot rolling bland & ford s approac (radians = neutral angle (radians = equivalent strain rate(s- ε = angle of contact in ot rolling bland & ford s approac (radians

8 American ournal of Applied Matematics 05; 3(5: -8 8 H = land & ord s function P = rolling load ( τ = rictional stress acting on te roll surface, kg/mm = Coefficient of friction = widt of Strip (mm D = Diameter of oll cylinder specimen (mm = rolling draugt V = entry velocity (mm V = exit velocity (mm σ = flow stress of te material in plane compression (kg/mm α = geometrical factor r = reduction, expressed as a fraction C = integration constant Suffixes - conditions at exit plane refers to conditions at entry plane n refers to conditions at te neutral plane f refers to conditions at te frictional transition plane on te exit side b refers to conditions at te frictional transition plane on te entry side eferences [] Altınkaya, H. Orak, I. M. Esen. I. (04. Artificial neural network application for modeling te rail rolling process. Expert Systems wit Applications 4: [] Alexander,. M. (97. Proc.. Soc., A36, [3] Aiyedun, P. O. (984. A Study of Loads and Torque for Ligt eduction in Hot lat olling at Low strain rates. PD. Tesis. University of Seffield, U. [4] Cen, S. Li, W. Liu, X. (04. Calculation of rolling pressure distribution and force based on improved arman equation for ot strip mill. International ournal of Mecanical Sciences [5] Ekelund, S. (933. Steel, 93, Weekly issues for Aug. to Oct 933. [6] El-alay,. E. H. A (966. A study of te factors affecting friction and teir effects upon load, [7] Torque and spread in ot flat rolling. PD. Tesis, University of Seffield, U. [8] Ginzburg, V.. (989. Steel-rolling Tecnoy, Pittsburg, Pennsylvania. [9] Gudur, P. P. Salunke, M. A. Dixit, U. S. (008. A teoretical study on te application of asymmetric rolling for te estimation of friction. International ournal of Mecanical Sciences 50: [0] Hwang, S. M. oun, M. S. (99. Analysis of ot-strip rolling by a penalty rigid-viscoplastic finite element metod, Int.. Mec. Sci. 34: [] ing, L. (00. olling mill roll design, Duran tesis, Duram University. [] onson, W. and udo, H. (960. Int.. Mec. Sci.,, [3] arman, V (95. T., angew, Z. Mat. Mec., 5, [4] Legrand,. Lavalard, T. Martins, A. (0. ew concept of friction sensor for strip rolling: Teoretical analysis. Wear ( [5] Moon, C. H. Lee, Y. (008. Approximate model for predicting roll force and torque in plate rolling wit peening effect considered, ISI Int. 48: [6] Orowan, E. (943. Proc. Inst. Mec. Eng. 50, [7] Parvizi, A. Abrinia,. (04. A two dimensional upper bound analysis of te ring rolling process wit experimental and EM verifications. International ournal of Mecanical Sciences 79: [8] oberts, W. L. (988. lat Processing of Steel, Dekker, York. [9] Siebel, E. (95. Eisen, Stal. 45, [0] Sims,.. (954. Proc, Inst. Mec. Eng.68, [] Sparling, L. M. G. (96. ormula for spread in Hot lat olling, Proc. Int. Mec. Eng., Vol. 75, o.ll, pp [] Tan, X. Yan, X. uster,. P. agunatan S. Wang,. (008. Dynamic friction model and its application in flat rolling. ournal of materials processing tecnoy 07: 34. [3] Tamano, T. and Yanagimoto, S. (970. Teory of rolling for te ange of mixed riction, ulletin of te SME, 3: 63: [4] Wang, X. Peng, Y. Xu, L. Liu, H. (00. A 3-D differential metod for solving rolling force of PC ot strip mill,. Iron Steel es. Int. 7: [5] Wusatowski, Z. (969. undamentals of olling Gliwice, Poland. [6] Zang,. Cui, Z. (0. Continuous EM simulation of multi-pass plate ot rolling suitable for plate sape analysis,. Cent. Sout Univ. Tecnol. 8 (0 6. [7] Zang, S. H. Zao, D. W. Gao, C.. Wang G. D. (0. Analysis of asymmetrical seet rolling by slab metod. International ournal of Mecanical Sciences 65: