Parametric Optimization of Linear Ball Bearing with Four-Point Connection in Steer-by-Wire Steering Column by Means of Genetic Algorithm

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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 Hertz’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.

1. Introduction

In the design of steering systems, the reliability of design solutions is important. Recently, engineers have been paying a lot of attention to the development of steering systems without an intermediate shaft, i.e., steer-by-wire systems [1]. The steer-by-wire steering system replaces the mechanical intermediate shaft in classic electric power steering (EKK) systems with a “wire” to enable the exchange of information between the steering actuator (hand wheel actuator—HWA) and the actuator on the front axle side (front axle actuator—FAA). In addition, the change in the axial or angular position of the column is performed by the actuators responsible for changing these positions, and the key element playing an important role in the structure of the steering column is the linear bearing. One of the available types of bearings is the linear ball bearing, which is used in the case of the analyzed steering column. Optimizing linear ball bearings is an important step in the engineering process of steering column construction. These bearings play a key role in ensuring smooth and precise steering column movement, which directly affects driving comfort and safety.

Seating linear ball bearings with four-point connection in the steering column is one of the proposals for connections between the columns. The design of linear ball bearings 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. Thanks to their flexibility and ability to search large solution spaces, genetic algorithms are widely used in the optimization of complex engineering problems [6,7]. This paper presents an example of the application of a bio-inspired genetically inspired algorithm in the optimization of the structure of a linear ball bearing in terms of the criterion depending on the first two natural frequencies in the steering column. The design variables of the task are the number of balls in the linear bearing and the spacing between them.

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 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.

Figure 5. Cross-section of a linear ball bearing with four-point connection describing the design variables of the optimization task.

Figure 5. Cross-section of a linear ball bearing with four-point connection describing the design variables of the optimization task.

$$\left({\mathrm{z}}_{\mathrm{i}}-1\right)\mathrm{c}\le 82.5$$

(1)

where

• ${\mathrm{z}}_{\mathrm{i}}$—number of balls in a linear bearing,
• $\mathrm{c}$—ball spacing.

The set of design variables in a genetic algorithm is stored in the form of a chromosome C_h (Equation (2)). In the optimization process, the algorithm strives to find the global optimum of the target function (Equation (3)).

$${\mathrm{C}}_{\mathrm{h}}=\left[{\mathrm{x}}_1,{\mathrm{x}}_2,\dots {,\mathrm{x}}_{\mathrm{i}},\dots {,\mathrm{x}}_{\mathrm{n}}\right]$$

(2)

$$\underset{\mathrm{x}}{\max }{\mathrm{J}}_{{\mathrm{C}}_{\mathrm{h}}}\left(\mathrm{x}\right)$$

(3)

where

• $\mathrm{n}$—number of design variables,
• ${\mathrm{x}}_{\mathrm{i}}$—value of the i-th variable from the acceptable range of the design variable ${\left[{\mathrm{x}}_{\mathrm{i}}\right]}_{\mathrm{m}\mathrm{i}\mathrm{n}}<{\mathrm{x}}_{\mathrm{i}}<{\left[{\mathrm{x}}_{\mathrm{i}}\right]}_{\mathrm{m}\mathrm{a}\mathrm{x}}$,
• ${\left[{\mathrm{x}}_{\mathrm{i}}\right]}_{\mathrm{m}\mathrm{i}\mathrm{n}}$—minimum value of the design variable,
• ${\left[{\mathrm{x}}_{\mathrm{i}}\right]}_{\mathrm{m}\mathrm{a}\mathrm{x}}$—maximum value of the design variable,
• ${\mathrm{J}}_{{\mathrm{C}}_{\mathrm{h}}}$—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)).

$$\underset{\mathrm{x}}{\max }{\mathrm{J}}_{{\mathrm{C}}_{\mathrm{h}}}\left({\mathrm{x}}_1,{\mathrm{x}}_2\right)\to \left\{\underset{\mathrm{x}}{\max }{\mathrm{J}}_1({\mathrm{x}}_1,{\mathrm{x}}_2)={\mathrm{f}}_1={\displaystyle \frac{{\mathsf{\omega }}_1}{2\mathsf{\pi }}}\ge 40 \mathrm{H}\mathrm{z}, \underset{\mathrm{x}}{\max }{\mathrm{J}}_2({\mathrm{x}}_1,{\mathrm{x}}_2)={\mathrm{f}}_2={\displaystyle \frac{{\mathsf{\omega }}_2}{2\mathsf{\pi }}}\ge 45 \mathrm{H}\mathrm{z}\right\}$$

(4)

where

• ${\mathrm{J}}_1,{\mathrm{J}}_2$—component functions of the function J, for the first and second natural frequencies, respectively,
• ${\mathrm{x}}_1$—design variable related to the number of balls in the linear bearing,
• ${\mathrm{x}}_2$—design variable related to the spacing between balls in a linear bearing,
• ${\mathrm{f}}_1$—the first natural frequency associated with the vertical mod of the steering column,
• ${\mathrm{f}}_2$—the second natural frequency associated with the lateral mod of the steering column,
• ${\mathsf{\omega }}_1$—first circular natural frequency associated with the vertical mod of the steering column,
• ${\mathsf{\omega }}_2$—the second circular natural frequency associated with the lateral mod of the steering column.

Genetic Algorithm

The genetic algorithm is one of the bio-inspired methods—its principle of operation is modeled on the processes of Darwin’s natural evolution [8,9]. Genetic algorithms are inspired by the processes of natural evolution and work by the selection, crossing and mutation of genes. The optimization process begins with a random population of initial solutions, which are then evaluated for their suitability to solve the problem. The best solutions are selected for further reproduction, during which genotypes are crossed and mutated, leading to a new generation of solutions. It is an artificial intelligence technique that uses mechanisms inspired by natural selection and genetics to find solutions to optimization problems [10,11,12]. The optimization algorithm begins its process by generating an initial population, which then evolves in subsequent iterations (Figure 6) [13,14].

Each population P(t) consists of a set of chromosomes of a certain number. Each chromosome ${\mathrm{C}}_{\mathrm{t}}^{\mathrm{j}}$, in turn, presents a geometric representation to a given structural form of the system.

$$\mathrm{P}\left(\mathrm{t}\right)=\left[{\mathrm{C}}_{\mathrm{t}}^1,{\mathrm{C}}_{\mathrm{t}}^1,\dots ,{\mathrm{C}}_{\mathrm{t}}^{\mathrm{j}},\dots ,{\mathrm{C}}_{\mathrm{t}}^{\mathrm{N}},\right]$$

(5)

where

• $\mathrm{t}$—population index (iterations),
• $\mathrm{N}$—the number of chromosomes in the population,
• $\mathrm{j}$—chromosome index,
• ${\mathrm{C}}_{\mathrm{t}}^{\mathrm{j}}$—J-th chromosome in the population t.

$${\mathrm{C}}_{\mathrm{t}}^{\mathrm{j}}=\left[{\mathrm{x}}_1^{\mathrm{j}},{\mathrm{x}}_2^{\mathrm{j}},\dots ,{\mathrm{x}}_{\mathrm{i}}^{\mathrm{j}},\dots ,{\mathrm{x}}_{\mathrm{n}}^{\mathrm{j}},\right]$$

(6)

where

• $\mathrm{n}$—the number of genes in a chromosome,
• ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{j}}$—I-th gene on the j-th chromosome.

A new population is created by crossing selected chromosomes and introducing mutations. Mutations lead to the creation of new solutions that can potentially be better than previous ones. The old population is replaced by a new population that was created by the process of crossing and mutation. The process is repeated iteratively until the final condition is reached. The condition for stopping an algorithm can be defined as, for example, achieving an optimal solution, a set number of generations, or no improvement in subsequent generations.

3. Modal Analysis

Modal analysis belongs to the broadly understood dynamic analysis, the final results of which are the eigenvalue and mode modes. Modal analysis [15] is an essential tool for understanding the structural dynamics of objects such as structures and vibrating objects and the resistance of these objects to specific loads [16]. Modal analysis enables the testing, optimization, and validation of machines and structures. The general equation of motion is of the form:

$$\mathrm{M}\ddot{\mathrm{q}}\left(\mathrm{t}\right)+\mathrm{C}\dot{\mathrm{q}}\left(\mathrm{t}\right)+\mathrm{K}\mathrm{q}\left(\mathrm{t}\right)=\mathrm{F}\left(\mathrm{t}\right)$$

(7)

where

• $\mathrm{M}\ddot{\mathrm{q}}\left(\mathrm{t}\right)$, $\mathrm{C}\dot{\mathrm{q}}\left(\mathrm{t}\right)$, $\mathrm{K}\mathrm{q}\left(\mathrm{t}\right)$—matrices, respectively: the mass, stiffness and damping of the system,
• $\ddot{\mathrm{q}}\left(\mathrm{t}\right),\dot{\mathrm{q}}\left(\mathrm{t}\right),\mathrm{q}\left(\mathrm{t}\right)$—vectors, respectively: accelerations, velocities and displacements,
• $\mathrm{F}\left(\mathrm{t}\right)$—external load vector.

When we consider that the damping and load are zero, we are dealing with harmonic motion (Equation (8)).

$$\mathrm{M}\ddot{\mathrm{q}}\left(\mathrm{t}\right)+\mathrm{K}\mathrm{q}\left(\mathrm{t}\right)=0$$

(8)

In further consideration, the displacement vector described by Equation (9) is assumed:

$$\mathrm{q}\left(\mathrm{t}\right)={\mathrm{q}}_{\mathrm{a}}\sin \left(\mathsf{\omega }\mathrm{t}\right)$$

(9)

where

• ${\mathrm{q}}_{\mathrm{a}}$—vibration amplitude vector,
• $\mathsf{\omega }$—circular frequency,
• $\mathrm{f}$—frequency,
• $\mathrm{T}$—vibration period.

After substituting (8) to (9), we obtain the expression (10):

$$\left(\mathrm{K}-{\mathsf{\omega }}^2\mathrm{M}\right){\mathrm{q}}_{\mathrm{a}}=0$$

(10)

Equation (10) is a homogeneous system of linear equations, where the obvious solution, the so-called trivial solution, is ${\mathrm{q}}_{\mathrm{a}}=0$. The system can have a different solution if the matrix $\mathrm{K}-{\mathsf{\omega }}^2\mathrm{M}$ is a singular matrix.

$$\mathrm{d}\mathrm{e}\mathrm{t}\left(\mathrm{K}-{\mathsf{\omega }}^2\mathrm{M}\right)=0$$

(11)

The above equation is a polynomial of nth degree due to the variable ${\mathsf{\omega }}^2$. Zero places are sought ${\mathsf{\omega }}_{\mathrm{i}}^2$ from this equation, called natural frequencies, and the corresponding vectors are called displacement vectors ${\mathrm{q}}_{\mathrm{a}\mathrm{i}}$. The general formula of natural frequencies is written by Equation (12).

$${\mathsf{\omega }}_0=\sqrt{{\displaystyle \frac{\mathrm{k}}{\mathrm{m}}}}$$

(12)

where

• ${\mathsf{\omega }}_0$—natural frequency,
• $\mathrm{k}$—system stiffness,
• $\mathrm{m}$—system Weight.

The eigenvalue depends on the stiffness and mass, where the stiffness is determined by the boundary conditions adopted for the system and the material stiffness described by Young’s modulus and the mass of the system material described by its density. In addition, the stiffness depends on the contact stiffness between the balls and the raceway.

4. Mathematical Description for Linear Bearings

Four-point connection linear ball bearings are advanced mechanical components that enable precise linear motion with minimal friction. Their design is based on the use of rolling balls, which are supported at four contact points, ensuring an even distribution of loads and increased stability. The design of a linear ball bearing consists of rolling balls, housings to maintain the appropriate distance between the balls and raceways. The four-point connection means that each ball has four contact points with the raceways, allowing the loads to be evenly distributed along the entire length of the bearing. This allows these bearings to accommodate both axial and radial loads, which increases their versatility. The study focused only on radial loading, which plays a key role in natural frequency but also wear.

4.1. Wear and Preload of Linear Ball Bearing with Four-Point Connection

Bearing wear is determined by the degradation of the bearing raceway surface layer after a certain number of cycles [17,18]. The parameter that describes the reliability of a bearing is efficiency. The three stages of wear progress of the bearing raceways shown in Figure 7 can be identified. The first stage is the bedding-in. The next stage is the main operation of the bearing (bearing life stage) and the final stage is accelerated wear. The final stage of bearing operation occurs when microcracks, adhesion or a lack of lubrication occur [19]. A key factor in bearing design is the selection of the appropriate preload value to increase bearing life. According to Lunberg and Palmgren [20,21], preload distribution plays a key role in bearing life. The load in the bearing is caused by preload, which is determined by external loads and rotational speed. In a linear bearing, preload is caused only by an external load.

The value of preload in linear bearings is obtained by the tolerances of the raceways and the accuracy of the balls in the bearing. The preload in a linear bearing affects the final value of the resistance to movement and the natural frequency. A high preload value increases the stiffness of the connection, which affects the natural frequency, increases the resistance of linear motion, but also reduces the durability. Linear ball bearings with four-point connection have a preload value that is twice as low as that of linear ball bearings with two-point connection, while maintaining the same tolerances for raceways and bearing balls.

The reaction force of the ball with the raceway is due to the preload, as shown in Figure 8. The sum of the reaction force of the ball with the raceways should correspond to the value of the preload (Equation (13)) [22,23,24]. For the sake of simplification, it can be assumed that the distribution of reaction forces in contact between the ball and individual raceways is the same [22]. In addition, the value of the reaction force of the ball with the raceway depends on the angle of contact (Equation (14)).

Figure 8. Force distribution for one ball in linear ball bearing with four-point connection.

Figure 8. Force distribution for one ball in linear ball bearing with four-point connection.

$${\mathrm{F}}_{\mathrm{N}}={\mathrm{F}}_{\mathrm{r}\mathrm{i}1}+{\mathrm{F}}_{\mathrm{r}\mathrm{o}2}={\mathrm{F}}_{\mathrm{r}\mathrm{i}2}+{\mathrm{F}}_{\mathrm{r}\mathrm{o}1}$$

(13)

$${\mathrm{F}}_{\mathrm{r}}·\cos \mathsf{\alpha }={\mathrm{F}}_{\mathrm{r}\mathrm{i}1}+{\mathrm{F}}_{\mathrm{r}\mathrm{o}2}+{\mathrm{F}}_{\mathrm{r}\mathrm{i}2}+{\mathrm{F}}_{\mathrm{r}\mathrm{o}1}$$

(14)

where

• ${\mathrm{F}}_{\mathrm{r}\mathrm{i}1},{\mathrm{F}}_{\mathrm{r}\mathrm{i}2},{\mathrm{F}}_{\mathrm{r}\mathrm{o}1},{\mathrm{F}}_{\mathrm{r}\mathrm{o}2}$—reaction forces on a specific contact surface, where ri is the inner raceway and ro is the outer raceway,
• ${\mathrm{F}}_{\mathrm{N}}$—preload,
• ${\mathrm{F}}_{\mathrm{r}}$—external load,
• $\mathsf{\alpha }$—contact angle between the bearing ball and the raceway.

4.2. Hertz Contact Theory

In the case of linear ball bearings, the balls roll between the inner and outer raceways. In this case, the Hertz pressure will depend on factors such as preload, contact surface geometry and the material properties of the bearing raceway. The Hertz contact equation for spherical contact between two elastic bodies is given by Equation (15) [25].

$$\mathrm{P}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{r}\mathrm{i}1}}{\mathrm{d}\sqrt{\mathrm{R}}}}$$

(15)

where

• $\mathrm{P}$—Hertz pressure,
• $\mathrm{d}$—diameter of the spherical contact zone,
• $\mathrm{R}$—effective radius of curvature $\mathrm{R}={\displaystyle \frac{1}{{\mathrm{R}}_1}}+{\displaystyle \frac{1}{{\mathrm{R}}_2}}$ for a ball bearing with two radii of curvature ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ (Figure 9).

The pressure for the linear ball bearing is elliptical in shape. For simplicity, a spherical shape is sometimes used [26,27]. The spherical shape of the contact is a simplified model of the contact of the balls with the raceways; therefore, in order to better reflect the actual contact of the ball with the raceways, an elliptical shape was assumed (Figure 10) [28]. In addition, it was assumed that there is no lubrication between the balls and the raceway, i.e., the minimum distance between the ball and the h_min raceway is equal to zero. The h_min > 0 parameter also takes into account the lubrication of the bearing. The effect of lubrication was omitted in the paper due to the fact that it affects linear motion and not natural frequency.

4.3. Mathematical Model of Linear Contact Ball Bearing

Pressure-induced stresses are important in the joints between elastic bodies in the design of bearings, gears or other components in contact. The field of science describing this phenomenon is known as contact mechanics. Hertz [25] described the basic mathematical relationships of surface stresses and deformations created during pressure. Hertz’s theory of pressure states that the stress on the contact surface usually significantly exceeds the yield point or even the strength limit, which can be explained by the occurrence of a state of triaxial compression [25]. Hertz’s equations have been supported by experimental studies carried out for gearboxes or ball and roller bearings [29].

We assume that the connection of the balls in the bearing with the raceways is so rigid that there is no slippage effect, so a ball bearing in a static state can be considered. The nonlinear properties of contact are based on the Hertz theory of pressures of two bodies in contact and loaded with a total force F_r [25], where F_r is the external load. The nonlinearity in contact between the linear bearing balls and raceways is caused by elastic deformation, friction or a situation in which there is a significant load causing yielding.

From Equation (14), it follows that the preload depends on the contact angle. The contact angle of the balls with the raceways of the linear bearing is defined by the shape of the outer and inner raceway surfaces modeled by the Gothic arch, shown in Figure 11 (in practice, a V-shaped model is also used).

Deep groove ball bearings have a high degree of contact, which gives them a relatively high load-carrying capacity. The tangential factor (t_f) describes where two curves or surfaces meet, or where they share a tangent. In linear bearings, the contact coefficient describes the ratio of the raceway radius to the ball radius. Bearing housing diameter (${\mathrm{C}}_{\mathrm{D}}$), for a linear bearing, specifies the distance from the shaft axis to the center of the ball. The adopted model assumes that the linear bearing is mounted in the axis, as a result of which the bearing installation diameter is zero and the contact between the ball and the raceway is shown in Figure 12.

For the adopted ball diameter and the adopted shape of the raceway, it is possible to select the general geometric properties of the connection, such as contact angle ($\mathrm{a})$, tangential factor (${\mathrm{t}}_{\mathrm{f}})$ and bearing cage diameter (${\mathrm{C}}_{\mathrm{D}})$. From the above parameters, it is possible to determine the radii of contact curvature for a ball bearing with four-point connection (Equations (16)–(18)).

$${\mathrm{R}}_1={\displaystyle \frac{\mathrm{D}}{2}}$$

(16)

$${\mathrm{R}}_2={\mathrm{R}}_1·2·{\mathrm{t}}_{\mathrm{f}}$$

(17)

$${\mathrm{R}}_2^{\prime }={\displaystyle \frac{{\mathrm{C}}_{\mathrm{D}}}{2}}\left({\displaystyle \frac{+}{-}}\right){\mathrm{R}}_2\cos \mathsf{\alpha }$$

(18)

where

• $\mathrm{a}$—angle of contact between the bearing ball and the raceway,
• $\mathrm{D}$—linear bearing ball diameter,
• ${\mathrm{R}}_1$—minimum radius of curvature of the balls in a linear bearing,
• ${\mathrm{R}}_2$—minimum radius of contact between the inner and outer raceways,
• ${\mathrm{R}}_2^{\prime }$—maximum radius of contact between the inner and outer raceways,
• ${\mathrm{C}}_{\mathrm{D}}$—bearing cage diameter,
• ${\mathrm{t}}_{\mathrm{f}}$—tangential factor.

After determining the values of the radii of the contact curvatures R₁ and R₂, the equivalent coefficient of the radii of contact curvature was determined (Equation (19)) [25].

$${\mathrm{K}}_{\mathrm{D}}={\displaystyle \frac{1.5}{\frac{1}{{\mathrm{R}}_1}+\frac{1}{{\mathrm{R}}_2}+\frac{1}{{\mathrm{R}}_1^{\prime }}+\frac{1}{{\mathrm{R}}_2^{\prime }}}}$$

(19)

$$\cos \mathsf{\theta }={\displaystyle \frac{{\mathrm{K}}_{\mathrm{D}}}{1.5}}\sqrt{{\left({\displaystyle \frac{1}{{\mathrm{R}}_1}}-{\displaystyle \frac{1}{{\mathrm{R}}_1^{\prime }}}\right)}^2+{\left({\displaystyle \frac{1}{{\mathrm{R}}_2}}-{\displaystyle \frac{1}{{\mathrm{R}}_2^{\prime }}}\right)}^2+2\left({\displaystyle \frac{1}{{\mathrm{R}}_1}}-{\displaystyle \frac{1}{{\mathrm{R}}_1^{\prime }}}\right)\left({\displaystyle \frac{1}{{\mathrm{R}}_2}}-{\displaystyle \frac{1}{{\mathrm{R}}_2^{\prime }}}\right)\cos 2\mathsf{\phi }}$$

(20)

where

• ${\mathrm{K}}_{\mathrm{D}}$—equivalent radius factor of contact curvatures,
• $\mathsf{\phi }$—the angle between the planes tangent to the bodies in contact at the point of contact (Figure 12),
• $\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }$—angular dependence on the radii of contact curvatures and the angle of contact occurrence.

Taking into account the defined material properties of balls and raceways, using the expression (21) [22], we determine the equivalent material coefficient in contact ${\mathrm{C}}_{\mathrm{E}}$.

$${\mathrm{C}}_{\mathrm{E}}={\displaystyle \frac{1-{\mathsf{\vartheta }}_1^2}{{\mathrm{E}}_1}}+{\displaystyle \frac{1-{\mathsf{\vartheta }}_2^2}{{\mathrm{E}}_2}}$$

(21)

where

• ${\mathrm{C}}_{\mathrm{E}}$—equivalent material factor in contact,
• ${\mathsf{\vartheta }}_1$—Poisson ratio for the ball,
• ${\mathsf{\vartheta }}_2$—Poisson’s ratio for treadmills,
• ${\mathrm{E}}_1$—Young’s modulus for the ball,
• ${\mathrm{E}}_1$—Young’s modulus for treadmills.

Taking into account the values set K_D and C_E and the values of the angular parameter $\mathsf{\lambda }$ [26], from Equations (22)–(25) [25] we can determine the contact parameters, such as the maximum and minimum semi-axis of elliptical contact, maximum stress in contact, and strain in contact caused by a given load, understood as relative motion in the direction of the load axis of two points of the bodies in contact [25].

$$\mathrm{a}=\mathsf{\mu }\sqrt[3]{{\mathrm{F}}_{\mathrm{r}}{\mathrm{K}}_{\mathrm{D}}{\mathrm{C}}_{\mathrm{E}}}$$

(22)

$$\mathrm{b}=\mathsf{\kappa }\sqrt[3]{{\mathrm{F}}_{\mathrm{r}}{\mathrm{K}}_{\mathrm{D}}{\mathrm{C}}_{\mathrm{E}}}$$

(23)

$${\mathsf{\sigma }}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}={\displaystyle \frac{1.5{\mathrm{F}}_{\mathrm{r}}}{\mathsf{\pi }\mathrm{a}\mathrm{b}}}$$

(24)

$$\mathrm{y}=\mathsf{\lambda }\sqrt[3]{{\displaystyle \frac{{{\mathrm{F}}_{\mathrm{r}}}^2{\mathrm{C}}_{\mathrm{E}}^2}{{\mathrm{K}}_{\mathrm{D}}}}}$$

(25)

where

• ${\mathsf{\sigma }}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}$—maximum Hertz stress in contact,
• $\mathrm{y}$—deformation in contact,
• a—maximum semi-axis of elliptical contact of the ball with the raceway,
• b—minimum half-shaft of elliptical contact between the ball and the raceway,
• κ, λ and μ—transcendental auxiliary functions (Figure 13), whose values (contact coefficients) are selected on the basis of the determined value $\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }$ described by Equation (21). The values of the function were determined experimentally [29].

Assuming that the contact force increases linearly, it is possible to estimate the contact strain for a linear ball bearing with four-point connection as the sum of the deformations in individual contacts of the balls with opposite raceways based on Equations (22)–(25). Taking into account the total contact strain for the linear bearing and the forces acting in contact, it is possible to determine the exponential contact load factor (n), which is described by Equation (26). The load factor of one ball (${\mathrm{C}}_1$) describes Equation (27). On the basis of Equations (28) and (29), the following are determined, respectively: the exponential stiffness factor (k) and the stiffness coefficient (${\mathrm{C}}_2$).

$$\mathrm{n}={\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{P}}_{\mathrm{x}2}-\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{P}}_{\mathrm{x}1}}{\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{y}}_{\mathrm{y}2}-\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{y}}_{\mathrm{y}1}}}$$

(26)

$${\mathrm{C}}_1={\displaystyle \frac{{\mathrm{F}}_{\mathrm{r}\mathrm{x}}}{{\mathrm{y}}^{\mathrm{n}}}}$$

(27)

$$\mathrm{k}=\mathrm{n}-1$$

(28)

$${\mathrm{C}}_2={\mathrm{C}}_1\ast \mathrm{n}$$

(29)

where

• n—exponential load factor,
• k—exponential stiffness factor,
• C₁—load factor of one ball,
• C₂—coefficient of stiffness of one ball.

In the finite element method, contact occurs when the nodes of field 1 penetrate the surface formed from the nodes of field 2. The depth of penetration of the nodes of body 1 caused by force determines the stiffness of the contact (Equation (30)). Equation (30) is the equation for the spring stiffness in a linear ball bearing with four-point connection (Figure 14). Based on Equations (26)–(29) and Equation (30) (spring stiffness), the nonlinear contact stiffness for the corresponding force is calculated as F_n (Equations (31) and (32)).

Figure 14. Contact stiffness in finite element method [30].

Figure 14. Contact stiffness in finite element method [30].

$${\mathrm{F}}_{\mathrm{n}}={-\mathrm{k}}_{\mathrm{n}}·\left({\mathrm{y}}_{\mathrm{n}}-{\mathrm{y}}_0\right)$$

(30)

$${\mathrm{F}}_{\mathrm{n}\mathrm{x}}={-\mathrm{C}}_1·{\mathrm{y}}_{\mathrm{n}\mathrm{x}}^{\mathrm{n}}$$

(31)

$${\mathrm{C}}_{\mathrm{F}}={\mathrm{C}}_2·{\mathrm{y}}_{\mathrm{n}\mathrm{x}}^{\mathrm{k}}$$

(32)

where

• C_F—contact stiffness corresponding to the deformation in contact,
• F_nx—preload for ball with treadmill-given load,
• y_nx—strain in contact induced by force-given load.

5. Numerical Problem Definition

The aim of the research is to define a simplified linear model of a ball bearing with four-point contact and to evaluate the results obtained using the proposed simplified model and the reference model. Both models use a 0.7 mm density of hexagonal first order mesh. Moreover, the quality of the mesh was achieved by a Jacobian above 0.6 and an aspect ratio around 1.2. The natural frequency results were compared for the reference and simplified models. The comparative simulation was carried out using the ABAQUS 2019 finite element method software. In the reference model, the modal analysis was preceded by a static analysis so that the nodes involved in the contact were included in the natural frequency analysis (taking into account the ellipsoidal contact). The simplified model assumes that there is a nonlinear contact and the characteristics of this contact are modeled by springs with nonlinear stiffness. The nodes of these springs are not in direct contact with the raceways, but are connected by beam elements of infinite stiffness to the corresponding raceways. The arrangement of the beam elements corresponds to an ellipsoidal contact shape and these elements are in contact with the raceways. The balls in the linear bearing are placed between the upper block and the lower block, and the raceways have a Gothic arc shape with an angle of 45°. Figure 15 shows a geometric model of a linear bearing. It is assumed that all elements are made of steel and their weights are given in

Table 1. Mass distribution in a system.

Components	Mass [kg]
Upper block	1.6
Bottom block	2.67

. Simulations were carried out for three cases with ball numbers of 8, 10 and 12.

5.1. Reference Model

The reference model consists of two massive blocks that represent the inner and outer raceways and the balls. The components have been discretized with cube solid elements of the first order. The upper surface of the outer block is restrained in three directions, and an evenly distributed load of 300 N is applied to the lower surface of the inner block (Figure 16). A nonlinear contact is assumed between the balls and raceways to reflect the ellipsoidal pressure distribution in contact (Figure 17). The ellipsoidal contact pressure distribution will also be considered in the modal analysis after the static analysis.

5.2. Simplified Model

The main difference between the reference and the simplified model is the absence of balls, which have been replaced by springs with nonlinear characteristics. These springs are connected by beam elements that are in linear contact with the raceways, as shown in Figure 18 and Figure 19. The connection with the beam elements allows for the ellipsoidal contact of the ball with the bearing raceway to be reproduced. Contact nonlinearity was considered in the spring.

5.3. Properties of Springs with Nonlinear Characteristics

For the adopted geometric parameters describing the model (

Table 2. Geometric parameters of linear ball bearing with four-point connection.

Geometrical Properties	-
Ball diameter	8	[mm]
Tangential factor	0.625	[-]
Cage diameter	8	[mm]
Contact angle	45	[°]

), on the basis of Equations (16)–(18), it is possible to determine the radii of contact curvature (

Table 3. Radii of curvature contact for four-point linear ball bearing.

	Ball	Inner Race	Outer Race
Min radius (R1, R2)	4	5	5	[mm]
Max radius (R1′, R2′)	-	1.172	6.828	[mm]

).

The model assumes that the raceways and balls are made of steel and the material properties of the system are shown in

Table 4. Material properties.

Material Properties	-
Young modulus for balls (E1)	200,000	[MPa]
Young modulus for bearing race (E2)	200,000	[MPa]
Poisson ratio ($\mathsf{\vartheta }$)	0.3	[-]

.

On the basis of Equations (19)–(21), the following were determined (

Table 5. Contact rate values.

	Inner Race	Outer Race
cosθ	0.913	0.349
Principal curvature 1/R2	−0.2	−0.2
Principal curvature 1/R2′	0.854	−0.146
KD	1.3	9.769
CE	9.1	9.1 × 10−6
μ	3.295	1.295
κ	0.446	0.796
λ	0.493	0.728

):

On the basis of the data presented in Table 5 and on the basis of Equations (21)–(25), the contact parameters were determined, such as: maximum contact stresses (${\mathsf{\sigma }}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}$), contact deformation (y) and maximum/minimum ellipse semi-axis (a, b) (

Table 6. Dependence of ball–raceway contact parameters on load [24].

O U T P U T
	Herz Contact Stresses		Total Rel. Motion	y: Rel. Motion of Approach		c: Max Semiaxis Ellipt. Contact		d: Min Semiaxis Ellipt. Contact
Input Ball Force	Inner Race	Outer Race	Total Rel. Motion	Inner Race	Outer Race	Inner Race	Outer Race	Inner Race	Outer Race
0	0	0	0	0	0	0	0	0	0
10	1349.0	500.849	0.00160	0.00091	0.00068	0.16174	0.12453	0.02188	0.07655
20	1699.7	631.030	0.00254	0.00144	0.00109	0.20379	0.15689	0.02756	0.09644
50	2306.8	856.440	0.00468	0.00267	0.00201	0.27658	0.21294	0.03741	0.13090
100	2906.4	1079.04	0.00743	0.00423	0.00319	0.34847	0.26829	0.04714	0.16492
200	3661.9	1359.51	0.01180	0.00672	0.00507	0.43905	0.33802	0.05939	0.20779
500	4969.9	1845.14	0.02174	0.01239	0.00934	0.59589	0.45877	0.08061	0.28201
1000	6261.7	2324.73	0.03451	0.01967	0.01483	0.75077	0.57802	0.10156	0.35532
2000	7889.3	2928.98	0.05478	0.03123	0.02355	0.94591	0.72826	0.12796	0.44767
5000	10,707	3975.24	0.10091	0.05753	0.04338	1.28380	0.98840	0.17366	0.60759
10 × 105	13,490	5008.49	0.16019	0.09132	0.06886	1.61749	1.24531	0.21880	0.76551
20 × 105	16,997	6310.31	0.25428	0.14496	0.10932	2.03791	1.56899	0.27568	0.96449
50 × 105	23,068	8564.41	0.46840	0.26703	0.20136	2.76587	2.12945	0.37415	1.30902
10 × 105	29,064.6	10,790.4	0.74354	0.42389	0.31965	3.48478	2.68293	0.47141	1.64926

).

On the basis of the data presented in Table 6 and expressions (26)–(29), the stiffness coefficients were determined (C₂), as well as the loads for one ball (C₁) and exponential coefficients (n, k)(

Table 7. Stiffness and load ratio of linear ball bearing.

C1	155,969
C2	233,954
n	1.5
k	0.5

).

In order to determine the parameters of the spring with nonlinear characteristics, it was assumed that the balls were subjected to an external load of 300 N. The value of the external load was converted to preload on the basis of Equations (13) and (14). The value of preload equal to 106 N was determined. On the basis of Equations (30) and (32), the nonlinear contact characteristics of the balls with the raceways were determined (Figure 20).

5.4. Results of Numerical Analyses

The modal analysis was performed for both models, taking into account the same boundary conditions and using the FEM calculation program for modal analysis—LANCZOS. The first twenty natural frequencies were included in the analysis. The reference and simplified models were compared, with 8, 10 and 12 balls in the rope bearing (the influence of temperature was omitted). For both models, a load of 300 N was applied. Figure 21 shows the results in the form of displacements obtained from the frequency analysis.

The differences in natural frequencies between the reference model and the simplified model for the same number of balls are shown in

Table 8. Percentage difference between the simplified model and the reference model for the first 10 natural frequencies.

No. Natural Frequencies	Differences in Natural Frequency [%]
	8 Balls in the Bearing	10 Balls in the Bearing	12 Balls in the Bearing
1	0.315	0.318	0.318
2	0.290	0.576	1.041
3	0.047	0.888	1.829
4	1.179	1.195	1.423
5	0.781	0.897	0.904
6	0.874	0.936	1.018
7	0.629	1.100	1.388
8	0.818	0.963	1.151
9	0.505	0.671	0.740
10	0.250	0.328	0.400

and in the form of a diagram in Figure 22.

6. Optimization Using a Genetic Algorithm

The tests were carried out using two different solutions in the field of steering system design. In the case of the first solution, a steel outer column tube was used, and in the case of the second solution—an aluminum outer column tube. The geometrical forms of the two variants of the outer columns are different due to the process of their production (Figure 23). Linear ball bearings with four-point connection are mounted between the outer and inner columns. The number of balls in a linear bearing varies between 5 and 12. The spacing between the balls must meet the condition of Equation (1) and must not be smaller than the diameter of the balls, which in the assumed model is 4 mm. The preload of the ball bearings is 300 N. The linear bearing housing is omitted from the numerical model because the balls in the linear bearing are assumed to work uniformly. The steering wheels in both cases have been modeled in accordance with the applicable standards in terms of weight and stiffness.

In the optimization process, the first two natural frequencies that occur in the column, i.e., the vertical and horizontal form of the natural frequency, were taken into account. A genetic algorithm was used in the optimization process. The aim is to obtain the highest possible first two natural frequencies. In addition, it was assumed that the first natural frequency must be above 40 Hz and the second above 45 Hz. If the natural frequency values do not meet the assumed criteria, the solutions that are closest to the assumed goal are selected from the population and passed on to obtain the optimal solution. The parameters of the genetic algorithm that were adopted in the optimization are listed in

Table 9. Genetic algorithm parameters.

Parameter Name	Value
Maximum iterations	50
Minimum iterations	20
Population size	54
Type code	real
Distribution index	5
Mutation rate	0.01
Elite population [%]	10

. An elite approach was assumed—the declared percentage of the best solutions passes unchanged to the offspring population. This is a very important strategy to ensure that the value gained in the solutions does not deteriorate. If the elite population is high, it reduces the chances of introducing a new gene. The genetic algorithm uses a distribution index, which controls whether the offspring are closer or further from the parental individuals, this coefficient ranges from 3 to 10.

In the process of optimizing the steering column with the steel outer column tube, convergence was achieved in 26 iterations (Figure 24). A linear dependence of the vertical frequency form on the number of balls and a parabolic relationship of the lateral frequency form were noted (Figure 25). A greater effect on the natural frequency of the number of balls was also noted compared to the spacing between them, which results from the relationship described in Equation (1) (Figure 26). The optimal solution was achieved for the case of using 12 balls and a spacing between the balls of 7.5 mm. The natural frequency values for this case are 40.2 Hz for the vertical vibration frequency and 46.4 Hz for the lateral vibration frequency (Figure 27 and Figure 32).

In the second case, the optimization of the steering column model, whose outer column tube is made of aluminum, showed convergence after 26 iterations (Figure 28). The influence of the extreme values of the number of balls and the spacing between the balls on the natural frequencies was noted (Figure 29 and Figure 30). The optimal solution was achieved for the case of 12 balls and a spacing between the balls of 7.5 mm, and the natural frequency values are 40.4 Hz for the vertical frequency and 46.4 Hz for the lateral frequency form (Figure 31 and Figure 33).

Figure 28. History of convergence for natural frequencies for aluminium outer column tube: (a) vertical mode, (b) lateral mode.

Figure 28. History of convergence for natural frequencies for aluminium outer column tube: (a) vertical mode, (b) lateral mode.

Figure 29. Influence of the number of balls for natural frequency: (a) vertical mode, (b) lateral mode.

Figure 29. Influence of the number of balls for natural frequency: (a) vertical mode, (b) lateral mode.

Figure 30. Influence of the spacing between the balls for the natural frequency: (a) vertical mode, (b) lateral mode.

Figure 30. Influence of the spacing between the balls for the natural frequency: (a) vertical mode, (b) lateral mode.

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 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 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.

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.

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 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.