CALCULATION METHODOLOGY FOR GEOMETRICAL CHARACTERISTICS OF THE FORMING TOOL FOR RIB COLD ROLLING

Calculation of the forming tool for rib cold rolling by the enveloping method is usually complex and very time-consuming. The cold rolling process is not always optimized, resulting in a poor quality of the rolled rib profi le and premature failure of the forming tool. Improvement in techniques of rib rolling on metal heat-exchange tubes is a challenging technological task, which has defi ned the purpose of the present work – the study of the effect of the forming tool for cold rolling of ribs by enveloping method on the accuracy of geometrical dimensions of ribbed surfaces. Absence of a well-developed calculation methodology for this type of tool now adversely affects the rate of manufacturing application of rib cold rolling processes. Since the main task when introducing the rib rolling process is to achieve the required accuracy of the work-piece, it seems expedient to improve calculation methodologies allowing designing a forming tool for rib cold rolling taking into account the required accuracy of geometrical dimensions of the ribbed surfaces. The present work proposes a methodology that allows performing calculations of parameters required for manufacturing and supervision over various rolling tools. Experimental approbation data of the outlined methodology results are presented. Because of deformation processing without additional technological operations, the effect of surface hardening has been achieved, and the roughness of rolled ribbed surfaces Ra was 0.8 μm, which is comparable in quality only with fi nishing machining. It was demonstrated that when calculating the coordinates of profi le points, the geometry of the lead-in, as well as rolls with a screw rib, it is particularly important to determine the angle of crush forming rolls and the y-coordinate of rolling bars as the main characteristics of the profi le.


INTRODUCTION
Improving the quality and competitiveness of a manufacturing range, the introduction of highly productive and resource-saving technologies, and continuous scientifi c and technological modernization with optimal use of resources are important conditions for the economic development, as shown in works [1][2][3]. The technological paradigm [4] of the modern production sector is characterized by increased stock utilization ratio and the average number of operations per workplace per month according to works [5,6]. In many respects, it is connected with a reduction of the relative role of subtractive technologies of materials processing by cutting due to the development of additive and hybrid technologies of geometry generation as well as methods of non-cutting shaping, as is proven in papers [7][8][9]. The use of plastic deformation methods reduces labor costs and the amount of processing waste, as demonstrated in papers [10][11][12]. In comparison with analogues made by cutting, products manufactured with the use of plastic deformation methods have added mechanical characteristics of surfaces, as is proven in publications [13][14][15][16] that is their competitive advantage. Common methods of metal processing nowadays are rolling, drawing, compression molding and hot and cold stamping. One of the most promising methods of plastic deformation for the formation of the ribbed surfaces is rolling with the help of the forming tool in the form of heads with a radial arrangement of rolls driven into rotation by way of simultaneous longitudinal cold pushing of the initial workpiece [34,35]. Possibility of the ribbed surfaces formation for heat-exchange devices can be achieved due to application of radically different technologies, as is proven in paper [17], however, the effi ciency of their operation will signifi cantly depend on the quality of surfaces. In particular, heat-exchange characteristics are signifi cantly reduced because of clogging of the ribbed surfaces in case of their low quality, as demonstrated in publications [18][19][20]. The roughness of ribbed heat exchange surfaces has a signifi cant impact on the effi ciency of their operation. With its high productivity, the rolling technique ensures high quality of the ribbed surface due to improving its smoothness and simultaneous hardening, as shown in paper [21]. Nevertheless, the industrial introduction of the cold rolling method is fraught with certain diffi culties. Since the cold rolling process is not always optimized according to papers [22][23][24][25][26], in practice premature destruction of the rolling tool in case of the low accuracy of the rolled rib profi le is often observed, as is proven by the authors of papers [27,28]. As can be seen from the above, the improvement of the rib rolling forming technology on metal Effective production volume, pcs. More than 500 More than 500 More than 1 More than 1

LITERATURE REVIEW
For the formation of the ribbed surfaces, various technological methods can be applied; however, the choice of one or another technology is defi ned not only by design features of a manufactured product but also by economic feasibility and productivity. In most cases, the manufacturing cost of technological operations of ribbed surfaces forming by cold rolling method turns out to be much lower in comparison with cutting processing, as shown in paper [10]. The reason for this is higher productivity of the method, a higher utilization factor of the processed material, lower costs for depreciation of equipment and replacement of machining tools as disclosed in work [29]. Besides, cold plastic processing has a number of technical advantages in comparison with other forming methods, the main of which are low roughness and hardening of the part surface (Table 1). Under pressure, which exceeds the value of the fl ow stress of the processed material as a condition shown in paper [10], plastic deformation occurs in the material at room temperature, as demonstrated in work [13], the result of which is not only the formation but also a signifi cant change in material characterization of the processed part, as is proved in paper [15]. Rolling is accompanied by an intensive increase in the surface hardness of the part and hardening of its material, as shown in paper [16]. Wear resistance of products manufactured by plastic deformation is 30-40 % over than of milled ones, and strength characteristics of rolled ribs are up by 10-20 %. The change in properties of the strained material is connected with changes in its structure, occurring at various scale levels under the action of technological processing conditions according to the Curie principle, as stated in papers [30][31][32]. Besides the change of the processed material properties under the action of the smooth and practically non-deformable tool, the roughness of the processed surface decreases according to work [21]. In addition to it, an important advantage of the rib rolling method is a high dimensional stability of the machining tool as indicated by the authors of paper [28], which is maintained at the level of the original one during the whole running time of the tool. Although the listed forming processes and changes in the structure and properties of the material happen simultaneously, they have different intensity and, therefore, the resulting cold rolling effect can be signifi cantly different. Depending on the required effect of processing, it is necessary to choose the appropriate technological modes for each particular pair of the forming tool and processed product taking into account the applied equipment and the type of the technological operation. Usually, the main task when introducing the rib rolling process is to achieve the required accuracy of the workpiece. Besides the experimental approach for the solution of this problem, it is expedient to apply the calculation methodologies allowing designing a forming tool for cold rolling of ribs taking into account the required accuracy of the ribbed surfaces. Since the accuracy is characterized not only by the accuracy of geometric dimensions but also by the accuracy of the shape and mutual location as well as by roughness of surfaces, special attention has been given to these particular parameters in the article.

METHODOLOGY
The deformation processing method is one of the most signifi cant factors determining the design of the machining tool. The kinematic process of rolling of the line profi le ribs by enveloping method is similar to the process of rib manufacturing by slotting or milling [33]. The difference consists only in a few working motions typical for each machining method. The basis for obtaining a theoretically correct tool profi le is the principle of mutual enveloping of the triangular rib profi le and the tool when rolling the centrode of the triangular rib profi le over the centrode of the tool without sliding. The following theoretical provisions must be implemented for cold rolling of ribs by enveloping method: • Conjugated profi les must have a common contact tangent and normal; • The normal drawn through the contact point of the conjugated profi les must pass through the profi ling pole, which is the tangential point of the tool and the centrodes of the triangular rib profi le. • Depending on the nature of the working motions during rolling, the forming tool can be classifi ed as follows: • A tool with a radius of the fi nite quantity centrode that performs a rotational motion during rolling (rollers, roller heads, etc.); • A tool with infi nite radius centrode, having a straightline direction of the main motion (fl at bars, round cross-section bars). The calculation methodology of the forming tool for rolling of the line profi le ribs presented in this paper is composed of three independent stages. At the fi rst stage, the calculations related to the manufacturing of the rolling tool are performed. These include: • Initial parameters selection of the reference profi le of the triangular ribs; • Calculation of the pitch circle radius of the triangular rib profi le; • Calculation of the workpiece diameter; • Calculation of basic geometrical parameters of the tool. 1. To achieve the required accuracy of the ribbed surface, the calculation of the rolling tool should be based on the reference rib profi le with adjusted dimensions. The adjustment shall be carried out, taking into account the elastic recovery of the workpiece metal by expression (1): where ∆d ср is the value of the absolute change of the rib profi le average diameter, σв is the ultimate tensile strength of the workpiece material, E is the modulus of elasticity of the workpiece material, d ср is the average diameter of the rib profi le. The choice of geometrical elements of the profi le ( Figure 1) is carried out according to the formulas: in which de is the outer average diameter of the manufactured product; deн is the outer average diameter of the crush forming roll; δe is the tolerance range of the average diameter; d 1 is the diameter of the manufactured product rib in its top point; d 1н is the diameter of the crush Figure 1: Tolerance range distribution of the crush forming roll and triangular ribs profi le forming roll in its lower point; δ 1 is the diameter tolerance range in the rib top of the crush forming roll; d 2 is the diameter of the manufactured product rib in its lower point; d 2н is the diameter of the manufactured product rib in its lower point; d 2н is the diameter of the crush forming roll in its top point; δ 2 is the diameter tolerance range in the lower point of the rib. After that, the constant parameter of the triangular ribs profi le е is calculated. All notations are shown graphically in Figures 2 and 3. For straight-sided ribs, the profi le parameter е corresponds to the expression (3a): where b is the width of the straight-sided rib. For the ribs of the triangular profi le, parameter e is found from the expression (3b): in which β is the pressure angle at the top of the rib; r 1 is the radius of the addendum circle along the full profi le; r is the radius of the pitch circle of the profi le. Tool wear during the rolling process and thermal deformations are not taken into account in the calculation, because they are so small that practically have no impact on the accuracy of the rolled products. 2. When calculating the minimum possible radius of the pitch circle of the triangular rib profi le, the formula (4a) is used for the crush forming rolls and formula (4b) is used for the rolling bars. The following notations are used in the formulas: y is the pressure angle of the triangular ribs; re is the radius of the addendum circle. These formulas defi ne the possibility of processing the tool profi le to its full height. The radius of the pitch circle of the rib profi le r is taken as close as possible to the calculated value of the workpiece radius in the range from r min to re, which ensures the correct division of the workpiece by the required number of ribs because the circular pitches of the conjugated profi les have the same value only on the pitch circles.
mula (5a), in which z is the number of rolling ribs: • The determination of the gearing ratio i by the formula (5b), in which z p is the number of ribs on the roll: • The determination of the centre-to-centre distance A by the formula (5c), in which R is the radius of the pitch circle of the rolling tool: 5. Basic geometrical parameters of the rolling tool (Figures 2 and 3) are calculated for the crush forming rolls according to the following formulas: • The number of ribs on the crush-forming roll z p is rounded to an integer value based on the approximate radius of the crush forming roll R СР : • The radius of the pitch circle of the rolling tool R: • The radius of the addendum circle of the rolling tool R e : • The radius of the dedendum circle of the rolling tool R i : • The rib height h based on the radius of the lower point of the rib r i when rolling: Then the calculation of additional parameters necessary for the complete control of the tool is performed, more specifi cally: 1) calculation of coordinates of profi le points (x, y); 2) calculation the diameter dp of the control crush forming roll. The curve of the working tool profi le, obtained because of profi ling by the enveloping method, is complex and therefore is usually described by equations in parametric form. One of the following geometric elements may serve as a parameter in the derivation of these equations: the current rotation angle, triangular rib profi le φ; variable radius of the triangular rib profi le ρ шл or tool profi le ρ.
In this case, the variable parameter for the crush forming rolls is the current tool radius ρ, and for the rolling barsthe x-coordinate. The procedure for the calculation of the profi le point coordinates is as follows.
1. For the crush forming rolls: 1) the current rotational angle φ is calculated; 2) the auxiliary angle δ (the vertical contact angle when rolling by the crush forming rolls) is calculated; 3) the pressure angle θ is determined; 4) coordinates х, у of the profi le points are calculated. 2. For the rolling bars: 1) the determination of the current rotational angle φ; 2) the calculation of the y-coordinate of the profi le. Formulas for calculating the point coordinates of the crush forming roll profi le are derived in a rectangular coordinate system associated with the roll profi le (see Fig.  2) and originating at the point O 1 , which is the center of the roll. By taking an arbitrary radius value ρ within the range from R i to R e , which corresponds to some rotational angle of the triangular ribs profi le φ and the tool profi le φ 1 = φ i , equality is written from the rectangular triangle BO 1 C: in which each of the summands is defi ned by the following expressions: where РC is the common normal line of the conjugate profi les: Substituting the found values into the equality (7) and solving the resulting square equation for sin (y±φ) , we obtain the expression (10): in which the ratio ρ/R is expressed by the coeffi cient k. From the same triangle BO 1 C let us defi ne sin(δ): The pressure angle represents the following difference: The coordinates of the profi le points are determined by the system of equations (13): After calculating the pressure angle θ, it is necessary to check the overlap of the angle θmax. Because of the wrong choice of the pitch circle radius of the crush forming roll R = r/i, the pressure angle θ may take a value higher than the maximum permissible θmax, which may lead to underestimating of the profi le height during the machining process. Therefore, the condition of θ̊ ≤ θmax must be followed, which, taking into account that θmax= (δt -δs)/2 will take the following form (14): where δ t is the angular pitch of the roll; δ s is the central angle corresponding to the width of the space along the arc of the pitch line. If this condition is not met, the value of the radius R should be increased by reducing the radius of the pitch circle of the triangular rib profi le r. In deriving the formulas for calculating the coordinates of the profi le points (x, y) of the rolling bars, the variable parameter is the x-coordinate of the point C, which belongs to the rib profi le and the bar profi le simultaneously, rotated relative to the initial position, shifted from the point P by a distance PO = rφ (see Figure 3). In this case, the rectangular coordinate system О ху with the beginning at the intersection of the bar profi le with the initial line is built so that the О х axis passes through the profi le centre of the triangular ribs and the О у axis coincides with the initial line of the bar. From the triangle РCB we will obtain the expression (15): By substituting РC from the equality (9), as well as by transforming and moving the entire numerical expression to the left side, we can obtain a square equation (16): The solution of the equation (16) is the expression (17) defi ning the angle φ as a function of the x-coordinate: where k 1 is a variable equal to x/r. The y-coordinate is defi ned by the following expression: From the presented analytical dependencies, it follows that the most important in terms of the required accuracy of the calculation are formulas for determining the minimum radius of the pitch circle of the profi le of the triangular ribs r min and the pressure angle of the crush forming rolls (or y-coordinates of the rolling bars). At the third stage, the calculation of geometrical parameters of the lead-in of the crush forming rolls, working according to the scheme with the axial feed of the workpiece, is performed. The calculation reduces to determining the length of the lead-in m 1 and the vertex angle α1 depending on the given values of the cone angle α.
The ribs of the crush forming rolls can have a straightline or curved profi le, which geometry is determined by the geometry of the cutting tool and the method of profi ling. The most expedient for economical energy consumption in metal deformation in the process of rib formation during rolling should be considered a profi le limited by a curve, which by nature close to the profi le curve of the calibration part of the rolls. Under this condition, at the transition from the lead-in to the calibrating part, the minimum energy is spent for the secondary metal deformation associated with the fi nal formation of the profi le of the triangular ribs with the help of the calibrating part of the rolls. This profi le can be obtained, for example, by machining with a shaping cutter using the enveloping method. Let the process of profi le cutting of the calibration and lead-in parts be performed with a tool whose profi le corresponds to the reference rib profi le. As the cone serves as a poloidal surface, when cutting the ribs of the leadin, with a constant gear ratio i and a constant angular pitch δi the circular pitch t0 has a variable value along the length of the lead-in t0 ≠ t0, and the width of the space along the arc of the pitch circle has the same values at any point of the poloidal cone. Therefore, the circular pitch t0 is reduced by reducing the thickness of the roll rib. At the same time, the profi le height is reduced, and at the point C (C') the profi le height is the same as the height of its foot. Then let us take the normal section I-I, passing through the point A, and the section II-II, crossing the polhode in the point B. We will project the rib profi les obtained in these sections onto the common plane parallel to the two secants. The circular steps t0 and t0` will be equal to their values taken on the elevation, and the radii of the pitch circles in these sections, representing the distances R ПР and Rʹ ПР from the considered points A and B to the actual axis of rotation of the roll, will have a variable value, which is determined by the formulas: The reduced gearing ratio is also different from the actual ratio (20): The pressure angle θ is equal to the difference of angles (21): Knowing the pressure angle of the triangular ribs Υ, the pressure angle of the roll from θ 1 and θ 2 as well as the reduced values of the gearing ratios i пр and i`п р , we will fi nd the values of radii of the addendum circles along the full profi le in these sections R 1 and R`1.
The taper lead-in actual angle α 1 is determined by the expression (22): We will fi nd the taper lead-in projection angle β 1 by expression (23) from the ACA 1 triangle: A segment of CA is defi ned by the expression (24) from the ACD triangle: The full profi le height AA' in the section I─I is defi ned by expression (25): By substituting dependencies (23)-(25) into the expression (22), we will get the dependence (26): We will defi ne the length of the lead-in m 1 from the triangle A 1 ЕК by the expression (27), and the segment BE will be found by the formula (28): By substituting the value of BE from the formula (28) into the expression (27) and identically transforming the formula (27), we will determine m 1 by the formula (29): It should be noted that the rib profi les in normal I─I and end III─III sections have different geometrical parameters. In particular, the curvature of their side surface depends on the pressure angle Υ of the rolling tool. The correlation between the actual angle Υ in the normal cross-section and its projection Υ 1 in the end section is determined by the equality (30): As can be seen from the above, it was demonstrated that it is possible to change the geometry of the lead-in profi le and bring it closer to the profi le geometry of the calibration part by changing the pressure angle of the tool making the lead-in profi le.

RESULTS AND DISCUSSION
Experimental testing of the calculation results gained according to the presented methodology was carried out.
Rolling was performed on a vertical milling machine in a three-rolls rolling device with feed-out of the workpiece. Test results of the rolled products demonstrated that the un straightness of the rib profi le in the normal section did not exceed 0.008 mm and the variation in a dimension of the rolls did not exceed 0.04 mm; the tangential pitch error was ± 0.015 mm. Because of deformation processing without additional technological operations, the effect of surface hardening has been achieved, and the roughness of the rolled ribbed surfaces Ra was 0.8 μm, which is comparable in quality only with fi nishing machining. Besides, it was found that the use of machining tools with the pitch or gradient of working coils error leads to the formation of surface defects such as "fold" regardless of the contour fi lling degree. As well, surface defects such as folds, layers and burrs ( Figure 4) as well as defects in the form of metal pitting and cracking may also take place.
The main reason for the formation of defects during the rolling process is related to the conditions of rib profi ling. Swelling of the profi le takes place due to the redistribution of elementary volumes of metal, displaced by the working coils of the forming tool. In the process of the profi le swelling, it appears that the emergence of both symmetrical and asymmetrical deformation is possible. The nature of the deformation depends on the way that working windings of the rolling tool pass over the surface of the forming rib, i.e. the windings either coincide or not. The formation of defects at the tip of the full rib profi le is possible due to symmetrical deformation. However, the defects in the rib profi le may have a different arrangement depending on the displacement value of the tool profi le in the deformation cycle. Adjustment of the rolling machine when the butt beat tool installation also leads to asymmetry of the metal deformation, with the conse-quence that the paths of the tool windings do not match and defects take place in the form of folds, layers and rolling laps, etc. Breaking the symmetry of metal deformation during cold rolling of the ribs mainly takes place in the following cases: • because of the wrong displacement of the forming tool; • because of the poor quality of the machining tools; • because of the complete fi lling of the winding profi le during rolling; • because of the low rigidity of the rolling machine used. The main types of destruction of the rolling tool are irreversible deformations and fi re cracking. Until the present time, there has not been formed a simple and unifi ed theoretical apparatus describing the infl uence of aggregate factors on the destruction of the tool during its manufacture and operation. Nevertheless, the most important factors that determine the durability of a forming tool can be divided into design and operational ones. The design factors are the geometry of the tool, chemical composition of its material, type of heat treatment, hereditary technological features of the production of the tool itself, etc. Operational factors include the deformation method, temperature and degree of deformation as well as the type of lubrication. Furthermore, subjective factors should also be mentioned, such as skills of workers and their compliance with technological instructions, and production culture in general. The variety of factors acting in the process of the forming tool operation and the lack of suffi cient amount of accumulated experimental data on the working conditions of the tool in many respects complicate the possibility of the theoretical analysis and development of new solutions to improve the quality of the forming tool.

CONCLUSIONS
Manufacturing application of cold rolling of the rib profi les leads to the signifi cant increase in productivity and quantity reduction in processing waste. The present work proposes a methodology that allows you to perform calculations of parameters required for manufacturing and supervision over various rolling tools. Experimental approbation data of the outlined methodology results are presented. It was demonstrated that determining the angle of the crush forming rolls and y-coordinates of the rolling bars as the main characteristics of the profi le is of particular importance for the calculation of the rolling tool profi le. The worked-out technological process has high productivity, and the obtained products at low roughness have high mechanical properties of the formed surfaces. Because of deformation processing without additional technological operations, the effect of surface hardening has been achieved, and the roughness of the rolled ribbed surfaces Ra was 0.8 μm, which is comparable in quality only with fi nishing machining.