## 7 Discussion  
The optimization process was carried out for two structural cases, taking into account the steel or aluminum outer column tube. The problem was the multiple criteria--two criteria related to the maximization of two natural forms of the system were taken into account. The optimization results showed that for both design cases, the optimal natural  
Figure 31: Three-dimensional diagram of the first two natural frequencies of aluminium outer column tube for the number of balls and the spacing between the balls.  
Figure 32: Results of displacements for the optimal solution for the vertical form of the natural frequency of the model: (**a**) outer column tube made of steel, (**b**) outer column tube made of aluminum.  
Figure 33: Results of displacements for the optimal solution for the horizontal form of the natural frequency of the model: (**a**) outer column tube made of steel, (**b**) outer column tube made of aluminum.  
frequencies were obtained for 12 balls with a spacing of 7.5 mm between them. In the first case, where the outer column tube is made of steel, the value of the first natural frequency of the vertical mode was 40.2 Hz and the second natural frequency of the lateral mod was 46.4 Hz. In the second case, in which the outer column was made of aluminum, the value of the first natural frequency of the vertical mode was 40.4 Hz and the second natural frequency of the lateral mod was 46.4 Hz. Using a genetic algorithm, for both column cases, a simplified modeling of ball bearings with four-point connection was used, using a system of nonlinear springs replacing the ball body. A comparison of the simplified numerical modeling method of linear ball bearings with four-point connection with the reference model showed a similar value of natural frequency with the use of different number of balls, and thus good accuracy of the proposed simplified model. The maximum difference in vibration frequency was less than 1.5%. The simplification of the model allowed us to significantly reduce the calculation time, reducing it by approximately three times as much. The presented research results are the first stage of a work aimed at developing a method for optimizing the joints occurring in the steer-by-wire steering system using an artificial intelligence algorithm [31; 32; 33]. Further work in the model will take into account additional parameters of the linear bearing, such as preload, contact angle, raceway radius and raceway material, and the implementation of a new optimization algorithm like grey wolf. Therefore, the method assumes the use of a population algorithm, in which case it is important to use a numerical model, leading to time savings and the simplification of the complexity of the problem, which is extremely important in industrial practice. The proposed optimization method, based on the bio-inspired algorithm and the finite element method with a simplified model of good accuracy, allows for optimization in the field of practical design problems of modern steering systems. It ensures an optimal solution with appropriate performance parameters, assuming minimized vibrations and thus achieving maximum comfort of use.  
Writing--original draft preparation, A.Z.; writing--review and editing, M.S. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Not applicable. Not applicable. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. Study completed with support of ZF Steering system Poland Sp. z o.o. (Konwojowa 94). Author [PERSON] was employed by the company ZF Active & Passive Safety Technology Steering Systems, ul. Konwojowa 94, 43-346 Bielsko-Biala, Poland. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.