Parametric model of a three-roll unit of radial-shear rolling mini-mill

Introduction

Modern radial shear rolling (RSR) mini-mills are characterized by a simple design, utilizing optimal feed angles (18 – 22°) and rolling angles (not exceeding 10 – 12°) [1]. Typically, these mills are specifically assembled for the production of particular products, determined by the mill’s dimensions and structural features. Despite their compactness and high mobility, these mills offer a wide range of full-scale capabilities. Helical rolling mini-mills align with the requirements of lean production [2] and facilitate the application of innovative RSR methods, which enhance the material properties of long products through intensive plastic deformation [3; 4]. For example, pure magnesium has shown significant improvement in its mechanical properties through three roller helical rolling, attributed to changes in its microstructure [5]. Researchers in [6; 7] have documented the formation of a gradient structure and the improved properties of austenite stainless steel AISI 321. The technological capabilities of RSR mini-mills have been successfully tested using titanium rods in accordance with specific requirements [8].

For example, the «30-80» three-roller piercing mini-mill is specifically designed for hot piercing of ingots into shell with a bottom [9 – 11]. It utilized a wedge lock device as a screw-down mechanism, and the rollers are characterized by pre-feed and rolling angles according to the design. The mini-mill stands out for its mobility, compactness, and the absence of the need for assembly on a power basement.

Furthermore, numerous industrial and research entities have successfully implemented the «14-40» three-roller helical mini-mills [1; 12]. A comprehensive overview of the main designs and specifications of RSR mini-mills can be found in [1]. These new-generation mini-mills exhibit lower metal intensity, reduced energy consumption, and offer simplified operation compared to small section mills using longitudinal rolling techniques. This makes them particularly attractive to small and medium-sized companies operating in the metallurgy and mechanical engineering sectors.

Previously, researchers [13 – 15] developed a systematic approach to the design of mill stand based on the unified “metal deformation–mill design” system, enabling the production of items with predetermined properties.

The new mill stand designs are based on several key principles: the use of a close-top roll housing, positioning the working rollers at consistent angles of feed and rolling, and separation the actuating mechanism of roller positioning from direct action of the rolling force. These design principles have not only simplified the mini-mill design but also improved its operational conditions.

Typically, the design process for such mills begins with the deformation zone and the predetermined main geometrical profiles of the rollers and rolled ingots. Given the wide range of dimensions for RSR mills and the need to design custom equipment to meet specific customer demands, it is advisable to utilize the parametrization capabilities available in most modern CAD systems during the design process. These capabilities significantly reduce the time required for developing computer models and facilitate subsequent refinement and error elimination.

Modern CAD systems offer parameterization capabilities, allowing for the creation of a wide range of para­meters. These parameters can be static (representing a fixed value), calculated (expressed through mathematical equations based on static parameters), or reference (referring to the actual value of existing geometry for use in other geometrical units). CAD systems enable the establishment of semantic connections between constructed geometry and developed parameters of any type, thereby creating associative bonds between different components of the designed equipment.

The choice and implementation of an appropriate design methodology play a crucial role in the design process. A well-defined methodology allows for the development of a unified approach to constructing associative bonds and establishing parametric dependencies between individual units [16]. This formalization of parametric simulation methods brings the design process closer to the original intent of the engineer [17], promotes standardization and internal consistency within the simulated model, reduces development time, and maximizes the potential for reusing previously developed parts and units. Employing an appropriate and well-designed methodology enables faster modifications of models compared to an informal approach [18; 19]. The development of design methods is closely tied to the advancement of CAD systems and their application in engineering design tasks. Ultimately, these methods represent the accumulated engineering experience of researchers aimed at enhancing the flexibility and validity of the designed models [20]. An incorrect or non-formalized methodology can have a negative impact on the overall quality of the model and the time required for the design process [21]. There are documented cases where the correct selection of a methodology, particularly in the context of MDO (Multidisciplinary design optimization) tasks, has led to significant achievements in aircraft production [22].

There are three formalized methodologies known for developing computer models, aimed at enhancing model flexibility, reduction development and modification time, and improving overall model stability.

Horizontal Modeling methodology: This simulation strategy, patented by Delphi Technologies [23], focuses on reducing critical errors that may arise during model modification by elimination of dependences in the existing geometry and parameters. Traditional CAD systems typically employ a vertical tree structure for elements, which can lead to errors when modifications are made. The Horizontal Modeling Methodology proposes minimizing such dependencies and instead referring to unaffected initial elements, such as basic planes and the center of coordinates. This transforms the vertical tree structure into a horizontal one, reducing the occurrence of errors.

Explicit Reference Modelling methodology (ERM): Proposed in [18], ERM aims to minimize the number of dependencies and restrictions related to the current geometry in the model, as it tends to vary with changes in the model. Instead, references are used, which can refer to auxiliary geometric objects. Unlike the Horizontal Modeling Methodology, ERM is focused on maintaining the global concept of parametrical approach.

Resilient Modeling methodology [24] was introduced in [25]. This methodology emphasizes the creation of robust models with extensive possibilities for reusing units and elements. It involves optimizing parent-child dependencies and organizing the structure of elements in a more comprehensible manner.

Each of these methodologies has its advantages and disadvantages depending on the pattern of the developed model. The key criterion for the operability of a parametric model is its ability to adapt to changes made to its elements. The operability of these methodologies has been demonstrated in various optimization scenarios, including the automatic generation of numerous geometrical variations and units [26].

This work focuses on the parametric simulation of a roller unit using generalized algorithm for designing of deformation zone of a three-roller RSR mill in Autodesk Inventor software. The model is developed following the recommendations of the Explicit Reference Modelling.

Formulation of the problem and experimental methods

Geometry of the spatial position of longitudinal profile rolls

In order to construct and calculate the geometry of the spatial position of longitudinal profile rollers, a procedure known as virtual squeezes is utilized, which was developed and proposed for calculations of RSR mini-mills [13]. This procedure is based on invariant geometry relations of RSR mills, taking into account changes in virtual angles of feed and rolling along the axis of the deformation zone. It allows for efficient parametrization of roller dimensions and the deformation zone [14]. The procedure is based on the layout depicted in Fig. 1, a, where:

– З represents the considered cross section of the ingot with radius r_P and center at О;

– В represents the cross section of the roller with radius R_P and center at О_В, in contact with the ingot cross section З at point F;

– Ol and CL are the rolling axes for the ingot and the deformation area, respectively;

– GG_B is the common perpendicular (eccentricate) with length Е to the roller and rolling axes, meaning \(\angle \)OGG_B = \(\angle \)CG_BG = 90°;

– ОС is the line perpendicular to the rolling axis with length Р, intersecting the roller axis, denoted as \(\angle \)СOG = 90°. In drum type mills, it is positioned on the drum axis;

– β, δ, γ represent the angles of feed, rolling and crossing, respectively.

The set of notions used, such as technological squeeze, geometrical squeeze, and design layout, were introduces in [27; 28]. According to the procedure, the determination of the roller profile is simplified to calculating the radius of the roller’s cross section that will come into contact with the ingot’s cross section of a known radius r_i. This contact point is located at a known distance Δl_i along the rolling axis from the technological squeeze (from point О) (Fig. 1, b).

Calculation of roll calibration using the deformation center

Now, let us illustrate the calculations based on the procedure by determining the roller radius in extreme cross section of the deformation zone using the initial data for «30-70» mill (Table 1). The roller diameter in the squeeze is chosen to ensure the minimum pass without the roller surface crossing. The maximum feasible roller diameter in the squeeze, taking into account the feed and rolling angles, as well as the gap between rollers, is determined by the equation [29]

where \(D_{\rm{в}}^{\max }\) is the maximum possible diameter in the squeeze, mm; \(d_{\rm{к}}^{\min }\) is the minimum roller pass diameter in the squeeze, mm; Δ is the gap between adjacent rollers, mm; φ is the inclination angle of the generator of the input roller cone, deg; k is the coefficient that relates the roller diameter to the pass diameter.

The coefficient k incorporates parameters such as the feed angle β, rolling angle δ and the number of operating rollers, and can be determined using the equation:

These conditions are subsequently verified by analyzing interference in the CAD system.

The calculated data obtained are sufficient for plotting the roller calibration for three cross sections The calculated results are summarized in Table 2.

Similar calculations for roller unit calibration can be based on a higher number of cross sections to achieve greater precision. The calculation process for each cross section is similar to the aforementioned procedure. By using the described algorithm, it is possible to create a table of roller calibration with required step for production using a numerically controlled machine.

Key experimental findings

The Explicit Reference Modeling methodology was chosen as the foundation for this work. All calculated parameters from Tables 1 and 2 were incorporated as user parameters into the Autodesk Inventor models. These parameters are directly calculated and determined within the Autodesk Inventor software.

To simplify the calculations within Autodesk Inventor, all the parameters are encoded as two-symbol variables (consisting of a Latin letter and a digit). The parameters are categorized as follows:

– parameters X: these serve as the main initial data for the model calculation;

– parameters A: these are used to determine the parameters of technological and geometrical squeezes;

– parameters B: these are used to determine the parameters of the virtual squeeze in cross section 1 of the deformation zone input;

– parameters C: these are used to determine the parameters of the virtual squeeze in cross section 2 of the deformation zone output.

A comprehensive list of the parameters is provided in Table 3.

The accuracy of the constructed model is verified using the parameters X1, X2, A2, B5, B6, C5 and C7. The following relationship holds true for these parameters:

The deformation zone in the model is created using a 3D sketch. The explicit parameters of this sketch are linked to the user-defined parameters. When the dependencies and initial data (X) are correctly established, the 3D sketch of the model will have zero degrees of freedom. This means that the constructed geometry variant is the only possible one given the preset parameters. Removing any initial parameter (X) will introduce degrees of freedom to the deformation zone model.

The layout of the mutual position of the roller and rolling axes is depicted in Fig. 2.

Fig. 3 illustrates the parameters A, B, C, which are used to construct the geometrical and technological roller squeezes.

The deformation zone is constructed starting from the technological squeeze on the rolling axis, which serves as the common coordinate center of the model O. The rolling axis is plotted passing through the coordinate center. The distance l between technological and geometrical squeezes along the rolling axis (parameter A5) is determined to obtain segment OG. An auxiliary line is drawn from the point O, perpendicular to segment OG. Along this line, the rod radius in the squeeze r_P and the roller radius in the squeeze R_P are consecutively plotted (parameters X4 and X3, respectively) to obtain the point C. The segment CG_B (А4) coincides with the roller axis. Perpendiculars from segment OG are used to construct the feed angle β (parameter X1). Another auxiliary line is drawn from the top of angle β, parallel to segment ОС. Point G_B is obtained by intersecting this auxiliary line with the perpendicular. The eccentricate E (parameter A3) is plotted from point G to intersect the auxiliary line. The construction of the other virtual squeezes follows a similar procedure.

Figures 4–6 depict the parametric models of constructed roller–body assemblies for various mill dimensions.

These models can be used to simulate the deformation zone and design roller parts for three-roller mini-mills. The developed parametric model is suitable for three-roller mills with rolling angles of working rollers δ ranging from 5 to 15° and feed angles δ ranging from 18 to 22°.

Notable features of these models include fast construction and reconstruction of roller joints using parametrization. By employing universal modular joints between individual parameters of each model and mini-mill units, the design time for mills can be significantly reduced. Additionally, the constructed model ensures the production of high-quality items based on the specified initial model parameters (X).

Fig. 7 illustrates the deformation zone consisting of three rollers. The deformation zone can be obtained by utilizing a circular array or creating an assembly by positioning the rollers around the sketch at a 120° angle, in accordance with the preset rolling and feed angles.

The parametrization method greatly facilitates the design process of roller joints and rolling mills, reducing the manual effort required for adjusting parts and angle ratios.

Conclusions

By applying a parameterizable design algorithm, engineers can efficiently develop a computer model. Specifically, the method for plotting the deformation zone, based on the radii of the rolled ingot, roller, feed angle, and rolling angle, enables the rapid creation of a 3D model for roller calibration and the deformation zone. This model serves as a foundation for selecting other structural units of the mill.

The algorithm for designing the deformation zone was developed as a function of adjustable parameters, which are utilized for parametrization within a CAD system. The algorithm is based on the concept of virtual squeezes, which leverage the invariant relationships of RSR roller geometry in relation to the desired parameters of the technological squeeze. These parameters include the diameters of the roller and rod, as well as the feed and rolling angles along the rolling axis and/or roller axis. In the development of the computer model for the roll joint of the “30-70” mill, an explicit reference simulation methodology was employed, incorporating parameterizable properties that can be varied based on the mill’s objectives and specific design requirements. The resulting model is adaptive, allowing for the easy determination of dimensions and spatial positions of rollers and the deformation zone in the mini-mill.

Further research efforts will focus on the development of an adaptive parametric model, encompassing the assembly of component units such as the shaft and body of working roller, bearing supports, pads, screw-down mechanism, and stands.