Computational experiment on determination of load on the rollers of continuous casting machine

The article is devoted to calculations of load on rollers of a curvilinear continuous casting machine during bending of the ingot in the levelling section. The calculation takes into account the elastic and plastic characteristics of metal and the flexibility of the roller apparatus. Determination of load on rollers together with meeting other requirements such as reducing the cooling water consumption, minimizing the energy consumption enables optimizing the technology of slab cooling in the secondary cooling zone (SCZ) and obtaining metal of the required quality.


Introduction
Currently, advanced continuous casting machines (CCM) are used for metal casting. In curvilinear-radial CCM, the most loaded section of the technological axis of machine is the levelling section, where the slab is bent. Here an ingot is straightened from a radial shape to a straight line and it is moved to a horizontal plane. Calculation of loads due to levelling is a complex and challenging procedure both from mathematical and technological points of view as many parameters of the machine and cast metal must be taken into account. These are the plastic and elastic properties of the slab metal itself, which also change both along the slab cross section and along the technological axis of the machine; compliance (displacement) of the roller apparatus itself due to the enormous forces and moments of forces acting during bending.
The mathematical model for calculating the load on rollers at a constant speed of slab withdrawal, i.e. when the load on slab during its straightening in each roller, the rigidity of the slab metal, and the thickness of the formed slab do not change in time, was considered in [1]. In this paper the differential equations with 4n boundary conditions were solved using the iterative method. At the first stage, the position of the bent slab axis is found from deflection of rollers. At the second stage, the deflection of rollers is found from the position of the bent axis. Since these differential equations are not solved directly, a whole series of techniques of both mathematical and computational nature was used to solve the problem.
In [1], the load on rollers was found by a modification of mathematical model, where the curvature of the elastic component of the technological path ) (x y   was adopted instead of the true curvature It is impossible or difficult to determine the load on rollers from field experiments and by construction a physical model based on them, since there are several tens of pairs of rollers on the levelling section. Therefore, it is more affordable to determine the load on CCM rollers from computational experiment. Firstly, it is based on creation of a mathematical model of object under study, and secondly, when conducting it, the initial mathematical model for calculating the load on rollers must undergo a series of transformations. These transformations, ultimately, should lead to such a computational algorithm that can be implemented on a computer. The ways of transforming mathematical models are directly related to mathematical methods that are used for the quantitative analysis of these models. Methods of transforming mathematical models of different levels (meta-level, micro-level) are given in [2]. Next, computer calculations, processing and analysis of the obtained results are carried out.
The purpose of this study is to determine the load on rollers at the levelling section based on the methods of computational experiment by creating a mathematical model of object using iteration method.

Materials and methods
Consider sequentially the main stages of computational experiment for determining the load on rollers at the levelling section of CCM with constant stationary casting of metal [3].
1. Construction of mathematical model for determining the load on rollers. Figure 1 shows a model of displacements (trajectories) along the technological axis in slab during straightening of the ingot at CCM in the presence (absence) of various casting parameters.
The trajectories are described as follows: and meet the levelling conditions at points i x : , ( ) (x z is the curvature corresponding to elastic deformation caused by external forces. The figure shows that the relations between the displacements are written by the following equations: In addition, we assume that functions   2. The algorithm for calculating the load on straightening section, taking into account the full curvature formula, will be as follows. To obtain the algorithm, the initial mathematical model must undergo a series of transformations. These transformations, ultimately, should lead to such an algorithm that can be implemented on computer. The ways of transforming mathematical models are directly related to mathematical methods that are used for the quantitative analysis of these models. Methods of transforming mathematical models of different levels (meta-level, micro-level) are given in [2].
In this paper, we consider two ways of converting a mathematical model for calculating the load on rollers at the CCM straightening section for the quasi-stationary mode of metal casting. The mathematical model of loading the CCM roller apparatus, which takes into account the elastic and plastic properties of the slab material, varying both in the cross section of the slab and along the technological axis of machine, as well as flexibility of the roller apparatus itself, which has the ability to redistribute the loads on these rollers, is given in [2].

Results and discussion
The procedure for solving differential equations with 4n boundary conditions in [2] was performed using the iterative method. The calculation algorithm was as follows: at the first stage, the position of the bent axis of slab  1 + n is the amount of rollers.
8. Thus, the iteration continues until sufficient computational accuracy is achieved.
By substituting the following data: slab width  The initial results are obtained after the k-th iteration, depending on the accuracy of the task. The initial approximations should be chosen as close as possible to the true solution, since if the initial approximations are poorly defined, it is impossible to solve the system of equations.

Conclusions
As a result of the studies, an algorithm was developed for determining the load on the rollers in the secondary cooling zone of CCM during ingot straightening based on a computational experiment using the iteration method. Using the created mathematical model of levelling and a computational experiment, the loads on the rollers are determined when the slab is straightened in a continuous casting machine.