
Application of the Universal Mathematical Model of the Free Profile Forming to Determine the Contact Form and Area in the Grooves
V. S. Solod, Candidate of Engineering Science, Assistant Professor, Head of Department, It is known [1] that the errors of the calculation of the energypower parameters of pass rolling in the grooves by the method of the reduced or corresponding strip make up to 45  60%. Most errors are due to inaccurate determination of the projection of the metalroll contact surface. To improve the calculation accuracy the paper [1] presents empirical dependencies that, according to the authors, work only for the conditions under which they were obtained.
G. Tsoukhar [2] also presents engineering formulas to calculate the area of the projection of the striproll contact surface (contact surface), based on an approximate description of the form of the deformation zone by rectangles and trapeziums, but they also work only for the studied systems. To improve the accuracy of the calculation of the striproll contact form and area we present a universal mathematical model of metal forming in the grooves. The mathematical model is based on two local rules of profile transformation, namely: the rule of transverse «stickiness» and the rule of spread distribution along the height, the mathematical model is described in detail in Ref. [3]. Using the said mathematical model we can build the profiles of the deformation zone step by step taking into account the rules of the metal spread development on the free profile, based on the developed rules, as well as the dependence which defines the spread profile equation as (1)
b_{0} и b are the initial and the final strip width, respectively; L_{d} is the length of the deformation zone. To prove the validity of the present mathematical model for the description of the form and calculation of the area of the horizontal projection of the striproll contact line we used experimental data obtained by G. Tsoukhar, [2], for pass sequences «ovalsquare», «ovalround», «diamondsquare», «diamonddiamond». Modeling of the metal forming in the passes showed that the above equation of the spread profile is to be corrected due to different conditions of the spread development along the length of the deformation zone for the passes with grooves of different forms. The parameters conventional signs and the form of the passes used for modeling are shown in Fig. 1 and their geometry  in Table 14, Thus, it is suggested to use the equation of the spread profile along the horizontal plane of the stock symmetry in the following form (2) n is the exponent, characterizing the groove form (n=1...2) tha depends on the mean integral characteristic tgq_{ср} of the pass wall slope to the horizontal line q_{ср} on the strippass contact area , (3)
n is determined by the formula n = 3  2tgq_{ср} when tgq_{ср}<1 , and when tgq_{ср}≥1 it is taken equal to 1. It is to be mentioned that such a correction scarcely influences the stock crosssection form and area at the exit from the rolls. Fig. 2 shows the stock cross sections calculated by the present model taking into account the groove form coefficient n. The contact surface form and sizes calculated by the model, taking into account the groove form coefficient n are shown in Fig. 3 and in Table 14. The shaded area in the figure corresponds to the striproll contact area, the thick line  to the profile of the maximum width (spread) of the stock. The contact surface and crosssection forms of the stock obtained by means of the present model visually are in good agreement with the experimental results (see Ref. [2]). The obtained results prove that the developed method of the stock free profile description can be used for the development of computer modeling algorithms for the bar rolling in the grooves. References 1. Smirnov V.K., Shilov V.A., Inatovich Y.V. Roll Pass Design.  M.: Metallurgiya, 1987.  368 p. 2. Tsoukhar G. Power Influences for Pass Rolling, translated from German by V.G. Drozd, edited by Y.S. Rokotyan. M.: Metallurgiya, 1963.  103 p. 3. Solod V.S., Kulagin R.Y., Beygelzimer Y.Y. Universal Mathematical Model of the Metal Forming in the Grooves // Stal'.  2006.  № 8.

