## 2 Formulation of the Optimization Problem  
The criterion function in the optimization task, which depends on a set of design variables, is a mathematical representation of the goal that the algorithm is aiming for in the optimization process. The aim of the optimization is to define the optimum number of  
Figure 4: A block diagram of a physical problem posed using an optimization process [4].  
Figure 3: Geometric model of linear ball bearing with cage.  
balls in the linear bearing and the optimum ball spacing (Figure 5) by means of a criterion depending on the first two natural frequencies that occur in the steering column. In the optimization process, a simplified model of ball bearings with four-point connection was used. The number of balls in a linear bearing (z\({}_{i}\)) is defined in the range from 5 to 12. Ball spacing (c) is defined in the range from 5 to 20 mm. In the optimization process, an unequal limitation was introduced, described by Equation (1). The value of the clearance between the balls depends on the number of balls in the bearing.  
\[(\mathrm{z_{i}}-1)c\leq 82.5 \tag{1}\]  
where  
z\({}_{i}\)--number of balls in a linear bearing,  
c--ball spacing.  
The set of design variables in a genetic algorithm is stored in the form of a chromosome \(\mathrm{C_{h}}\) (Equation (2)). In the optimization process, the algorithm strives to find the global optimum of the target function (Equation (3)).  
\[\mathrm{C_{h}}=[\mathrm{x_{1}},\mathrm{x_{2}},\ldots,\mathrm{x_{i}},\ldots, \mathrm{x_{n}}] \tag{2}\]  
\[\max_{\mathrm{x}}\mathrm{J_{C_{h}}}(\mathrm{x}) \tag{3}\]  
where  
n--number of design variables,  
x\({}_{i}\)--value of the i-th variable from the acceptable range of the design variable \(\left[\mathrm{x_{i}}\right]_{\mathrm{min}}<\mathrm{x_{i}}<\left[\mathrm{x_{i} }\right]_{\mathrm{max}}\),  
\(\left[\mathrm{x_{i}}\right]_{\mathrm{min}}\)--minimum value of the design variable,  
\(\left[\mathrm{x_{i}}\right]_{\mathrm{max}}\)--maximum value of the design variable,  
\(\mathrm{J_{C_{h}}}\)--objective function.  
In the case of the natural frequency analysis of the steering system, both the vertical and horizontal forms of the frequency are taken into account. Therefore, the task becomes a multi-criteria task, and the objective function depends on two component functions. Our goal is to obtain the first natural frequency (vertical mod) above 40 Hz and the second natural frequency (lateral mod) above 45 Hz (Equation (4)).  
\[\max_{\mathrm{x}}\mathrm{J_{C_{h}}}(\mathrm{x_{1}},\mathrm{x_{2}})\to\left\{ \max_{\mathrm{x}}\mathrm{J_{1}}(\mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{f_{1}} =\frac{\omega_{1}}{2\pi}\geq 40\;\mathrm{Hz},\;\max_{\mathrm{x}}\mathrm{J_{2}}( \mathrm{x_{1}},\mathrm{x_{2}})=\mathrm{f_{2}}=\frac{\omega_{2}}{2\pi}\geq 45 \,\mathrm{Hz}\right\} \tag{4}\]  
where  
J\({}_{1}\),J\({}_{2}\)--component functions of the function J, for the first and second natural frequencies, respectively,  
x\({}_{1}\)--design variable related to the number of balls in the linear bearing,  
x\({}_{2}\)--design variable related to the spacing between balls in a linear bearing,  
f\({}_{1}\)--the first natural frequency associated with the vertical mod of the steering column,  
f\({}_{2}\)--the second natural frequency associated with the lateral mod of the steering column,  
Figure 5: Cross-section of a linear ball bearing with four-point connection describing the design variables of the optimization task.  
\(\omega_{1}\)--first circular natural frequency associated with the vertical mod of the steering column, \(\omega_{2}\)--the second circular natural frequency associated with the lateral mod of the steering column.