###### Abstract  
This paper presents the process of the optimization of linear ball bearings with four-point connection using a genetic algorithm and the finite element method. Currently, modern steering systems without an intermediate shaft--steer-by-wire systems--are being developed. The focus of this paper was on the optimization of linear ball bearings with four-point connection, embedded between the outer and inner columns tube in terms of the number of balls in the bearing and the clearance between balls. The aim of the research was to maximize the first two natural frequencies in the steering system, which is crucial for improving the stability and efficiency of the system. Various factors influencing natural vibration such as bearing geometry, raceway and ball materials, and operating conditions (preload) were taken into account in the research. Preload is a major factor affecting not only linear motion but also natural frequency. In order to speed up the calculations, the author's simplified model of a linear bearing with the use of a system of springs was proposed. The nonlinear properties of the spring were determined on the basis of [PERSON]'s theory. A genetic optimization process resulted in a linear bearing structure that meets the natural frequency criteria. In addition, the full reference model was numerically compared with the simplified one, which showed convergent results of natural frequencies. linear ball bearings; natural frequency; preload; finite element method; optimization; genetic algorithm; steer-by-wire steering column +
Footnote †: journal: 8 September 2024  
ensures high precision, low friction and minimal wear (Figure 1). A significant difference between conventional ball bearings and linear bearings is linear axial movement instead of rotary movement. Unlike traditional plain bearings, these bearings use rolling elements, typically balls, to minimize friction and improve overall performance [2; 3; 4]. They therefore provide controlled linear motion with low friction, as well as efficiency and durability. The use of rolling elements not only reduces friction, but also contributes to better load distribution, making linear ball bearings suitable for applications with varying loads and speeds. Precision is a hallmark of linear ball bearings, making them ideal for applications where accuracy and repeatability are paramount. The design of these bearings allows for smooth and quiet operation, minimizing wear and extending the life of the system. The seating of four-point connection ball bearings in the steer-by-wire steering column under consideration is shown in Figure 2. As you can see, the outer and inner columns in the system act as both the outer and inner raceways of the ball bearing, respectively. The bearing balls are kept at a distance by a plastic cage (Figure 3). This enclosure is not included in the numerical model. Optimization is a key process in science, engineering, and management that involves finding the best solutions to problems by maximizing or minimizing the function of the goal while meeting certain constraints [5]. A flowchart of the optimization process is shown in Figure 4.  
Many methods have been developed to solve various types of optimization problems. Among them, two main optimization methods can be distinguished: deterministic and stochastic [5]. Deterministic algorithms, such as the gradient method or the Newtonian method, use information about derivatives of the objective function to systematically approach the optimal solution [6; 7]. Stochastic algorithms, on the other hand, such as genetic algorithms or simulated annealing, are based on random processes, which allows them to avoid local minimums and maxima. Particular attention is paid to genetic algorithms, which are inspired by the processes of natural evolution. These algorithms use selection, crossover, and mutation mechanisms to iteratively improve the solution population.