Mechanism Study of Symmetric Mechanism-Rolling Bearing Dynamic System Considering the Influence of Temperature Factor

Abstract

Due to the symmetry of their structure and the way they rotate, rolling bearings are often used at high temperatures and high speeds. When the temperature changes, the material properties of the rolling bearing change, which in turn causes the dynamic model equation of the rolling bearing to change. Given the problem in that the conventional dynamic model equation does not consider temperature changes, but because temperature changes cause changes in material performance parameters resulting in differences between the dynamic simulation signals and the actual signals, this paper fully considers three factors, namely, the thermal expansion coefficient, the hardness value, and the friction coefficient. With the influence of temperature changes, a study on the mechanism of the symmetric mechanism-rolling bearing dynamic system considering the influence of temperature factors is proposed. First, combined with the material properties of the rolling bearing, the changes in thermal expansion coefficient, hardness value, and friction coefficient caused by temperature changes were analyzed. Second, the functional relationship expressions obtained above were substituted into the existing conventional dynamic model. Finally, the comprehensive functional relationship formula was substituted into the conventional dynamic model to obtain the rolling bearing dynamic model under temperature difference changes. By studying the mechanism of the symmetric mechanism-rolling bearing dynamic system that considers the influence of temperature factors, this method can undertake a more comprehensive consideration of the dynamic analysis research into rolling bearing fault diagnosis, thereby verifying the effectiveness of the method.

1. Introduction

Due to the symmetry and rotational working mode of rolling bearing structures, rolling bearings play an important role in mechanical rotating equipment [1,2]. At the same time, the operating conditions of rolling bearings directly affect the operating status of the equipment [3,4]. Only by accurately performing condition monitoring and fault diagnosis of the rotating machinery, such as the rolling bearings, can we better guide the maintenance of equipment during operation [5,6]. The rolling bearing dynamic system takes each bearing element as the basic unit, establishes a simplified dynamic equation set based on tribology and dynamics principles, explores the vibration response characteristics under different working conditions, and provides a theoretical basis for status monitoring and fault diagnosis of rotating machinery [7,8].

In 2015, Sidra Khanam and J.K. Dutt [9] established and verified a bearing failure vibration model including collision force but did not consider temperature. In 2016, Wu [10] considered the changes in the clearance and load-bearing area of rolling bearings and verified the multi-body dynamic simulation and experimental model of fault bearings. However, the research on clearance and load-bearing areas still has limitations. In 2020, Li [11] studied the rolling bearing dynamic model of rolling elements with local pitting failure but did not consider the effect of temperature on pitting corrosion. In 2021, Gupta [12] conducted real-time dynamic modelling of lower-temperature bearings with thermal coupling but did not conduct in depth research on the operating conditions at lower temperatures. In 2022, Lei [13] studied the dynamic characteristics of a ball bearing with partial spalling of the outer ring and established a ball bearing dynamic model considering sliding and thermal expansion, but only considered a single factor of thermal expansion, without considering other factors. In 2023, Kim [14] studied the impact of heat on fatigue life by analyzing the thermomechanical behavior of double-row tapered roller bearings (DTRB) and established a fatigue life prediction model, but did not study the impact of heat on changes in the fault area. Likewise, in recent years, various studies have been conducted on bearing failure modes. In 2018, Francescod [15] studied white etching cracks but did not explore the thermal effects caused by temperature. In 2020, Milan [16] studied the damage of roller bearings of belt conveyor rollers and concluded that plastic deformation is the main cause of damage but did not study the relationship between temperature and plastic deformation. In 2022, Liu [17] conducted a study on motor bearings and concluded that the wear of the inner ring of the motor bearing is uniform, but the impact of thermal expansion on the bearing clearance was not considered.

The above research status concerns various models and the research on the dynamic characteristics of rolling bearing dynamics while also exploring the related forms of bearing failure. However, in actual operating conditions, as the operating time and load increase, the temperature of the rolling bearing will increase. Changes occur, which will cause changes in various characteristic parameters of the bearing material GCr15 bearing steel, including the thermal expansion coefficient, hardness value, and friction coefficient, which will affect the stiffness and damping of the rolling bearing dynamic system. Therefore, it is important to consider the influence of temperature factors and necessary to study the mechanism of the symmetric mechanism-rolling bearing dynamic system. By studying the mechanism of the symmetric mechanism-rolling bearing dynamic system that considers the influence of temperature factors, we can provide more reference and basis for bearing dynamics in actual operating conditions. This kind of research can optimize bearing design, improve performance and life, predict life, optimize working conditions, improve material selection, increase energy efficiency, and conduct reliability analysis. By understanding in depth the impact of temperature on the bearing system, we can achieve better results in practical applications, improve the reliability and efficiency of the mechanical system and prevent irreversible damage to the mechanical system caused by changes in the bearing dynamic model when the temperature rises.

Although the research system on dynamics has become increasingly mature, research on dynamic systems that considers temperature is still relatively scarce, and the impact on rolling bearing dynamic systems when temperature rises is still unclear. Therefore, this article proposes a study on the mechanism of the symmetric mechanism-rolling bearing dynamic system that considers the influence of temperature factors. The main contents are as follows: Section 2 analyzes the traditional dynamic model of rolling bearings; Section 3 analyzes and studies the bearing dynamic model that considers the influence of temperature factors and compares the results while Section 4 summarizes the conclusions.

2. Traditional Dynamic Model of Rolling Bearings

Traditional dynamics of rolling bearings mainly include the following seven important aspects of research and analysis: namely, load analysis, friction and wear analysis, bearing type and structure analysis, kinematic analysis, fatigue analysis, vibration and noise analysis, and lubrication analysis. The traditional dynamic equations of rolling bearings are equations used to describe the motion behavior of internal components of the bearing under the action of external loads and operating torques. These equations usually involve aspects of mechanics, friction, and kinematics within the bearing, including basic load equations, friction equations, bearing internal load distribution equations, rolling bearing radial and axial stiffness equations, and bearing fatigue life equations.

In this article, dynamic analysis is mainly conducted on the load force equation and motion relationship equation. The contact between the ball and raceway is simplified as a spring damping model, with contact stiffness and damping between the inner ring and shaft being $K_{r-in}$ and $C_{r-in}$, horizontal contact stiffness and damping between the outer ring and bearing seat being $K_1$ and $C_1$, vertical contact stiffness and damping being $K_2$ and $C_2$, and contact stiffness between the inner ring and outer ring being $K_{in-out}$. The simplified schematic diagram of rolling bearings is shown in Figure 1. The relative displacement in the horizontal direction of the inner and outer rings is $x_{in-out}=x_{in}-x_{out}$, and the relative displacement in the vertical direction is $y_{in-out}=y_{in}-y_{out}$. The schematic diagram of the bearing motion relationship is shown in Figure 2 [18].

In the dynamic system study in this article, the 6307 deep groove ball-bearing dynamic system is used. This dynamic system has great advantages and wide applicability and is suitable for dynamic system simulation research. It has mainly the following modules and advantages:

(1)

Multi-degree-of-freedom system modelling: This dynamic system allows the modelling of multiple degrees of freedom, which is very important for describing complex mechanical or structural systems. It can efficiently handle multiple vibration modes and multiple interacting components.

(2)

Rigid and non-rigid systems: This dynamic system can handle both rigid and non-rigid systems, which is very useful for various practical engineering and physics applications. It uses the ode15s solver for rigid and non-stiff ODEs.

(3)

Phase angle control: By introducing beta conditions, the dynamic system can introduce or exclude friction-damping effects under specific phase angle conditions. This capability allows application of different friction models to different parts of the system in different situations, making the simulation more flexible.

(4)

Spectrum analysis: This dynamic system also includes spectrum analysis of the system response, which is very helpful for analyzing the frequency characteristics and vibration modes of the system and can reveal the resonant frequency and frequency response of the system.

(5)

Hilbert envelope demodulation: The dynamic system also implements Hilbert envelope demodulation, which is a method for extracting the envelope of the vibration signal. This is valuable for analyzing useful information in vibration signals, for example, for fault diagnosis or structural health monitoring.

In the 6307 deep groove ball-bearing dynamic system in this article, the angular velocity of the bearing cage [19] is the following:

$${\omega }_c=\frac{{\omega }_s}{2}(1-\frac{d_b}{D})$$

(1)

In the formula, the angular velocity of the shaft is ${\omega }_s$, the diameter of the rolling element is $d_b$, and the diameter of the rolling bearing is $D$.

The angular position ${\theta }_i$ [19] of the $i-th$ ball at any time $t$ is as follows:

$${\theta }_i=\frac{2\pi (i-1)}{z}+{\omega }_{c}t+{\phi }_0$$

(2)

In the formula, when the moment of impact between the ball and the fault point is $t_0=0$, the initial angular position is ${\phi }_0$, the number of rolling elements is $z$, and the angular position of the ball is $\theta$.

The relative displacement in the horizontal $X$-axis direction between the inner and outer rings is as below:

$$x_{in-out}=x_{in}-x_{out}$$

(3)

The relative displacement in the vertical $Y$-axis direction is as below:

$$y_{in-out}=y_{in}-y_{out}$$

(4)

The contact deformation $\delta$, the radial clearance $c_r$, and the total contact deformation at the $i-th$ ball [19] are the following:

$${\delta }_i=(x_{in}-x_{out})\sin {\theta }_i+(y_{in}-y_{out})\cos {\theta }_i-c_r$$

(5)

There is no Hertzian contact force in the non-loading area of the rolling bearing during rotation. The Hertzian contact force formula in the load-bearing area is as follows:

$$Fi=Kin-{}_{out}{\delta }^{1.5}=Kin-{}_{out}{[(x_{in}-x_{out})\sin {\theta }_i+(y_{in}-y_{out})\cos {\theta }_i-c_r]}^{1.5}$$

(6)

The total contact force acting on the inner and outer rings = the sum of the contact forces generated by all balls, and the vector decomposition of the $X$-axis and $Y$-axis is performed. The component forces on the $X$-axis and $Y$-axis are, respectively, as follows [20]:

$$\left\{\begin{array}{c}{F}_X={\displaystyle \sum _{i=1}^z{F}_i}\sin {\theta }_i={\displaystyle \sum _{i=1}^z{K}_{in-out}}{[(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r]}^{1.5}\sin {\theta }_i\\ F_Y={\displaystyle \sum _{i=1}^z{F}_i}\cos {\theta }_i={\displaystyle \sum _{i=1}^z{K}_{in-out}}{[(x_{in}-x_{out})\sin {\theta }_i+(y_{in}-y_{out})\cos {\theta }_i-c_r]}^{1.5}\cos {\theta }_i\end{array}\right.$$

(7)

The specific equation diagram is shown in Figure 3.

On introducing spring stiffness $K_b$, friction damping $C_d$, and $b$ as variable constants, which take 0 or 1, the system of equations is as follows:

$$\left\{\begin{array}{c}{F}_X={\displaystyle \sum _{i=1}^z{K}_b}{[(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r-b{C}_d]}^{1.5}\cos {\theta }_i\\ F_Y={\displaystyle \sum _{i=1}^z{K}_b}{[(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r-b{C}_d]}^{1.5}\sin {\theta }_i\end{array}\right.$$

(8)

When simulating the dynamic system of the 6307 deep groove ball bearing, this article sets a proportional relationship between temperature and time. The temperature selection range is 20–110 °C, and for every 10 °C there is a node, with a total of ten temperature points being set. In the time domain waveform diagram, this article selects the interval 0–0.7 s for analysis. The following fault characteristic frequency calculation formula is used which is a common formula employed in rolling bearing fault diagnosis. On the one hand, it is to determine whether a fault occurs in the rolling bearing dynamic model. On the other hand, if a fault occurs, it is to find out the location of the fault in the rolling bearing dynamic model.

In the dynamic system of this article, the outer ring is fixed and the inner ring rotates. The following fault characteristic frequency calculation formula is used:

Inner ring fault frequency:

$$f_{in}=\frac{f_r}{2}{N}_b\left[1+\frac{d}{D}\cos \alpha \right]$$

(9)

Outer ring failure frequency:

$$f_{out}=\frac{f_r}{2}{N}_b\left[1-\frac{d}{D}\cos \alpha \right]$$

(10)

Rolling element rotation failure frequency:

$$f_b=\frac{f_{r}D}{2d}\left[1-{\left(\frac{d\cos \alpha }{D}\right)}^2\right]$$

(11)

Characteristic frequency of cage failure:

$$f_c=\frac{f_r}{2}\left(1-\frac{d}{D}\cos \alpha \right)$$

(12)

In the formula, $f_r$ is the bearing speed, $N_b$ is the number of rolling elements, $\alpha$ is the contact angle, where $\cos \alpha =1$ and $d$ is the diameter of the inner ring of the rolling bearing, while $D$ is the diameter of the outer ring of the rolling bearing.

In this dynamic system, $f_r=25.4r/s$, $N_b=8$, $d=14$ mm, $D=57.5$ mm. The dynamic modelling of the bearing inner ring fault obtains the time domain waveform diagram and spectrum diagram without considering the temperature, as shown in Figure 4. Through comparative analysis of the calculation results, it is determined that the rolling bearing fault characteristic frequency is the inner ring fault characteristic frequency, $f_{in}=126.463$ Hz.

3. Bearing Dynamic Model Considering the Influence of Temperature Factors

When studying the influence of temperature on various parameters, the state of spring stiffness $K_b$ and friction damping $C_d$ in the dynamic equation is mainly updated. Spring stiffness is the elastic characteristic of the bearing, which can reflect the bearing’s ability to resist deformation. Friction damping $K_b$ is the stability of the bearing when subjected to external impact and vibration. The initial spring stiffness is $K_{b_0}$ and the initial friction damping is $C_{d_0}$. In the bearing dynamics model that considers the influence of temperature factors, the changes and properties of rolling bearing materials in this article are mainly based on the thermal expansion coefficient, hardness value, and friction coefficient. The hardness value is presented in a decreasing form, and the thermal expansion coefficient and friction coefficient are presented in an increasing form.

3.1. Thermal Expansion Coefficient

The thermal expansion coefficient of GCr15 bearing steel is an important material physical property, which describes the degree of linear expansion of the material when the temperature changes. This coefficient is usually expressed as the change in length per degree Celsius (°C) (often expressed in microns/meter °C or inches/ft °F), and it changes with temperature [21]. In this article, to facilitate the study of the thermal expansion coefficient, the linear expansion coefficient is used.

When the temperature of a solid substance changes by 1 °C, the ratio of its change in length to its length at the original temperature (not necessarily 0 °C) is called the “linear expansion coefficient”. The symbol is $\partial$. Its definition is as follows:

$$\partial =\frac{\Delta L}{L_0\Delta T}1/°C$$

(13)

In the formula, $\partial$ is the linear expansion coefficient, $\Delta L$ represents the length change caused by temperature change, $L_0$ is the initial length of the object, and $\Delta T$ is the temperature change.

The linear expansion coefficient is further divided into the average linear expansion coefficient and the instantaneous linear expansion coefficient. The average linear expansion coefficient is the average expansion rate within a certain temperature range, while the instantaneous linear expansion coefficient is the expansion rate at a specific time and temperature point. In this article, the instantaneous thermal expansion coefficient is used because the thermal expansion coefficient every 10 °C from 20 °C to 110 °C is considered [22,23]. By reasonable selection of values and verification of bearing materials using relevant testing systems, we first obtain a data table of changes of instantaneous thermal expansion coefficient with temperature, as shown in

Table 1. Data table of instantaneous thermal expansion coefficient changes with temperature.

Temperature (°C)	20	30	40	50	60	70	80	90	100	110
Instantaneous Thermal Expansion Coefficient (10−5·K−1)	5	6	7	9	9.5	10	12	13	15	16

. The change of instantaneous thermal expansion coefficient with temperature is shown in Figure 5, showing an overall upward trend.

The change of instantaneous thermal expansion coefficient with temperature is shown in Figure 5, showing an overall upward trend.

In the dynamic system equation considering the thermal expansion coefficient,

$$\left\{\begin{array}{c}{K}_b=K_{b_0}\times [1+\partial (T-T_0)]\\ C_d=C_{d_0}\times [1+\partial (T-T_0)]\end{array}\right.$$

(14)

Among them, 1 is the real-time temperature, 2 is the initial temperature, and is substituted into the Equation set (8), to obtain the following system of equations:

$$\left\{\begin{array}{c}\begin{array}{c}{F}_X={\displaystyle \sum _{i=1}^z{K}_{b_0}\times [1+\partial (T-T_0)]}\{(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r\\ -b[C_{d_0}\times [1+\partial (T-T_0)]]{\}}^{1.5}\cos {\theta }_i\end{array}\\ \begin{array}{c}{F}_Y={\displaystyle \sum _{i=1}^z{K}_{b_0}\times [1+\partial (T-T_0)]}\{(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r\\ -b[C_{d_0}\times [1+\partial (T-T_0)]]{\}}^{1.5}\sin {\theta }_i\end{array}\end{array}\right.$$

(15)

To obtain more intuitively the change in the acceleration amplitude of the ordinate of the time domain diagram caused by the temperature change, the stiffness $K_b$ and the damping $C_d$ are individually changed. When considered alone, the functional relationship except $K_b$ in Equation (15) is all variables except $C_d$ are set to constants. Similarly, when considering $C_d$ alone, all variables in Equation (15) except the functional relationship are set to constants. At the same time, to simplify the comparison, $\cos {\theta }_i=\sin {\theta }_i$ is taken. Figure 6 is an $F_x$ trend chart after a single stiffness change caused by temperature, and Figure 7 is an $F_x$ trend chart after a single damping change caused by temperature.

In the same way, the trend chart after a single stiffness change caused by temperature and the trend chart after a single damping change caused by temperature are also shown in Figure 6 and Figure 7. After vector superposition of $F_x$ and $F_y$, where

$$\overrightarrow{F}=\overrightarrow{F_x}+\overrightarrow{F_y}$$

(16)

$$F=\sqrt{{\left(F_x\right)}^2+{\left(F_y\right)}^2+2F_x{F}_y\cos \partial }$$

(17)

$$\tan \alpha =\frac{\left|F_x\right|}{\left|F_y\right|}$$

(18)

$$a=\frac{F}{m}$$

(19)

In the formula, the system mass, $a$ is the system acceleration, the angle between $F_x$ and $F_y$, and different values are used for calculation in the superposition calculation.

The single stiffness and single damping changes caused by temperature are combined and calculated to obtain the trend chart of the resultant force $F$ changing with temperature, as shown in Figure 8. After that, the time domain waveform diagram of each temperature is summarized. The $X$-axis is time $t$, the $Y$-axis is amplitude $V$, and the $Z$-axis is temperature $T$. The time-domain waterfall diagram is shown in Figure 9.

The peak points in the waterfall chart are compared in the $Z$-axis temperature direction, the representative and significant time point of 0–0.7 s selected, and the corresponding ten points of the four coordinates of 0.079 s, 0.277 s, 0.435 s, and 0.633 s are also selected. By comparing the ordinates of the peak values of each temperature point, we can obtain a trend diagram of the peak point amplitude acceleration changing with temperature, as shown in Figure 10. Similarly, when analyzing other data points, the trends are the same, so they are not stated here.

It can be seen from Figure 10 that the peak point amplitude acceleration shows a decreasing trend with temperature, which is consistent with the theoretical calculation results in Figure 8, verifying the validity of the analysis of the impact of thermal expansion on the dynamic model. The thermal expansion coefficient is mainly reflected in dimensional changes, bearing clearance changes, and material selection in rolling bearing dynamics. At the same time, under different conditions, changes in the thermal expansion coefficient may have different effects on the structural integrity and performance of the rolling bearings. In high-temperature environments and with rapid temperature changes, the thermal expansion coefficient of rolling bearings changes more significantly. The materials of the bearing parts fail to match well the operating temperature range, which may lead to large dimensional changes, thus affecting structural integrity and performance.

The increase in temperature causes the thermal expansion coefficient $\alpha$ of the rolling bearing to increase, which in turn causes the stiffness $K_b$ and damping $C_d$ in the rolling bearing dynamic model to change, which causes the dynamic equations of the $X$-axis direction force $F_x$ and $Y$-axis direction force $F_y$ to change, and finally gives the total. When joining force $F$ changes, the acceleration also changes. The specific relationship is shown in Figure 11.

To sum up, it can be seen that the amplitude acceleration output by the rolling bearing becomes smaller because the temperature rises, which causes the thermal expansion coefficient of the GCr15 bearing steel, the material of the inner and outer rings of the bearing, to increase, the clearance becomes smaller, and the impact of the rolling bearing becomes smaller, which is more consistent with the actual working conditions.

3.2. Hardness Value

Hardness value is a physical property used to describe the hardness of a material and is usually used to measure the material’s resistance to external forces [24]. Common hardnesses include Rockwell hardness, Brinell hardness, Vickers hardness, etc. Rockwell hardness is used for measurement in this paper. Rockwell hardness is known as a widely used hardness testing method [25] and is commonly used to measure the hardness of metals and alloys. This test method uses the change in material under indentation at different depths to determine its hardness value.

When considering the impact of hardness value on the dynamic system, a reasonable value is selected based on the hardness value at different temperatures and the relevant testing system is used to verify the bearing material. First, a data table of the change of hardness value with temperature is obtained, as shown in

Table 2. The hardness value changes with the temperature data table.

Temperature (°C)	20	30	40	50	60	70	80	90	100	110
Hardness Value (HRC)	67.1	65.2	63.6	63.6	63.7	63.8	63.2	63	62.8	62.4

[26,27]. The trend chart of hardness value changing with temperature is shown in Figure 12, showing an overall downward trend.

The trend chart of hardness value changing with temperature is shown in Figure 12, showing an overall downward trend.

In the dynamic system equation considering hardness,

$$\left\{\begin{array}{c}{K}_b=K_{b_0}\times [1+\Delta h(H-H_0)/H_0]\\ C_d=C_{d_0}\times [1+\Delta h(H-H_0)/H_0]\end{array}\right.$$

(20)

Among them, are the hardness change coefficient, the real-time hardness, and the initial hardness. Considering that the bearing surface needs to withstand continuous friction and wear, we choose the initial hardness to be 63.6 HRC. After substituting into the parameter Equation set (8), the following equation set is obtained:

$$\left\{\begin{array}{c}{F}_X={\displaystyle \sum _{i=1}^z{K}_{b_0}\times [1+\Delta h(H-H_0)/H_0]}\{(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r\\ -b[C_{d_0}\times [1+\Delta h(H-H_0)/H_0]]{\}}^{1.5}\cos {\theta }_i\\ F_Y={\displaystyle \sum _{i=1}^z{K}_{b_0}\times [1+\Delta h(H-H_0)/H_0]}\{(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r\\ -b[C_{d_0}\times [1+\Delta h(H-H_0)/H_0]]{\}}^{1.5}\sin {\theta }_i\end{array}\right.$$

(21)

Figure 13 and Figure 14 show the $F_x$ trend graphs after the temperature causes changes in single stiffness and single damping respectively.

According to Equations (16)–(19), the single stiffness and single damping change caused by the temperature are vector combined and calculated to obtain the trend chart of the join forces $F$ changing with temperature, as shown in Figure 15. After that, the time domain waveform diagram of each temperature is summarized. The $X$-axis is time $t$, the $Y$-axis is amplitude $V$, and the $Z$-axis is temperature $T$. The time domain waterfall chart is shown in Figure 16.

Similarly, by selecting the corresponding ten temperature point peak-axis directions of the four coordinates of 0.079 s, 0.277 s, 0.435 s, and 0.633 s for comparison, and you can obtain the trend chart as shown in Figure 17.

It can be seen from Figure 17 that the peak point amplitude acceleration shows an increasing trend with temperature, which is consistent with the theoretical calculation results in Figure 15, verifying the validity of the analysis of the impact of hardness on the dynamic model. At the same time, changes in hardness values will also affect the distribution of bearing loads. In areas with high hardness values, bearings are more likely to withstand high loads, and in areas with low hardness values, they are more likely to deform.

The increase in temperature causes the hardness value $H$ of the rolling bearing to increase, which in turn causes the stiffness $K_b$ and damping $C_d$ in the rolling bearing dynamic model to change, which causes the dynamic equations of the $X$-axis direction force $F_x$ and the $Y$-axis direction force $F_y$ to change, and finally causes the resultant force $F$ changes; the acceleration also changes accordingly. The specific relationship is shown in Figure 18.

From the above analysis, it can be concluded that the amplitude acceleration output by the rolling bearing becomes larger because the increase in temperature will lead to changes in the heat treatment effect of the bearing. The reduced hardness value will weaken the strength of the bearing material, making the rolling elements more susceptible to deformation and stress under load, resulting in an increase in force. The dynamic model will experience greater force, which is more consistent with the actual working conditions.

3.3. Friction Coefficient

The friction coefficient is divided into average friction coefficient and instantaneous friction coefficient. The average friction coefficient and the instantaneous friction coefficient are two different ways to describe the nature and behavior of friction phenomena [28]. They are respectively applicable to different time scales and engineering scenarios. First of all, the average friction coefficient is a concept that represents the average value of the friction coefficient measured over some time or under certain conditions. This average takes into account changes and fluctuations in friction and is therefore often used to describe friction behavior over longer periods [29]. On the other hand, the instantaneous friction coefficient is a concept that represents the friction coefficient measured in a short time or instant. It is often used to describe instantaneous friction behavior, such as the initial friction experienced by an object when it overcomes static friction and initiates motion at the beginning of relative motion between two objects. The instantaneous coefficient of friction may be relatively high because it takes into account this momentary additional drag, while the subsequent friction may settle to a lower level [30,31]. In this study, on considering the universality of research on GCr15 bearing steel, the average friction coefficient is used for analysis.

When considering the impact of the friction coefficient on the dynamic system, a reasonable value is taken for the average friction coefficient at different temperatures and the relevant testing system is used to verify the bearing material. First, a data table of the change of hardness value with temperature is obtained, as shown in

Table 3. The average friction coefficient changes with temperature data table.

Temperature (°C)	20	30	40	50	60	70	80	90	100	110
Average Friction Coefficient	0.0150	0.0155	0.0157	0.0203	0.0205	0.0263	0.0812	0.0942	0.0986	0.1030

[32,33]. The trend chart of hardness value changing with temperature is shown in Figure 18, showing an overall upward trend.

The trend chart of hardness value changing with temperature is shown in Figure 19, showing an overall upward trend.

In the dynamic equation considering the friction coefficient,

$$\left\{\begin{array}{c}{K}_b=K_{b_0}\times [1+h(\mu -{\mu }_0)/{\mu }_0]\\ C_d=C_{d_0}\times [1+h(\mu -{\mu }_0)/{\mu }_0]\end{array}\right.$$

(22)

substituting into the parametric system of Equation (8), we obtain the following system of equations:

$$\left\{\begin{array}{c}{F}_X={\displaystyle \sum _{i=1}^z{K}_{b_0}\times [1+h(\mu -{\mu }_0)/{\mu }_0]}\{(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r\\ -b[C_{d_0}\times [1+\Delta h(H-H_0)/H_0]]{\}}^{1.5}\cos {\theta }_i\\ F_Y={\displaystyle \sum _{i=1}^z{K}_{b_0}\times [1+h(\mu -{\mu }_0)/{\mu }_0]}\{(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r\\ -b[C_{d_0}\times [1+\Delta h(H-H_0)/H_0]]{\}}^{1.5}\sin {\theta }_i\end{array}\right.$$

(23)

Figure 20 and Figure 21 are respectively the trend diagrams of $F_x$ after changes in single stiffness and single damping caused by temperature.

According to Equations (16)–(19), the $F_x$ and $F_y$ after the single stiffness and single damping change caused by temperature are vector combined are calculated to obtain the trend chart of the resultant force $F$ with temperature, as shown in Figure 22. After that, the time domain waveform diagram of each temperature is summarized. The $X$-axis is time $t$, the $Y$-axis is amplitude $V$, and the $Z$-axis is temperature $T$. The time domain waterfall chart is shown in Figure 23.

Similarly, by selecting the ten temperature point peaks corresponding to the four coordinates of 0.079 s, 0.277 s, 0.435 s, and 0.633 s for comparison in the $Z$-axis direction, the trend chart in Figure 24 can be obtained.

It can be seen from Figure 24 that the peak point amplitude acceleration shows a downward trend with temperature, which is consistent with the theoretical calculation results in Figure 22, verifying the effectiveness of the analysis of the impact of the friction coefficient on the dynamic model.

The increase in temperature causes the hardness value $H$ of the rolling bearing to increase, which in turn causes the stiffness $K_b$ and damping $C_d$ in the rolling bearing dynamic model to change, which causes the dynamic equations of the $X$-axis direction force $F_x$ and the $Y$-axis direction force $F_y$ to change, and finally gives the total. When force $F$ changes, the acceleration also changes. The specific relationship is shown in Figure 25.

From the above analysis, it can be concluded that the amplitude acceleration output by the rolling bearing becomes smaller because the increase in temperature usually causes changes in the lubricant in the bearing. For example, the viscosity of oil or grease decreases, thereby reducing the lubrication performance. When the lubrication performance decreases, the friction coefficient will increase, causing the rolling elements to experience greater resistance in the bearing, and then the acceleration in the dynamic model will become smaller, which is more consistent with the actual working conditions. In a real rolling bearing dynamic system, the friction caused by the friction coefficient is nonlinear, and as the temperature increases, the increase in friction will cause the vibration frequency of the system to decrease, and the system’s response to external excitation will also become slower.

3.4. Combining the Three Factors

By comparing the changes in the acceleration of the dynamic system model caused by the thermal expansion coefficient, hardness value, and friction coefficient, we can obtain the relationship diagram of the changes in rolling bearing parameters caused by the increase in temperature, as shown in Figure 26.

In the dynamic equation that comprehensively considers the three factors,

$$\left\{\begin{array}{c}{K}_b=K_{b_0}\times {\partial }_0\\ C_d=C_{d_0}\times {\partial }_0\\ {\partial }_0=1+\partial (T-T_0)+\Delta h(H-H_0)/H_0+h(\mu -{\mu }_0)/{\mu }_0\end{array}\right.$$

(24)

Combining the above equations and substituting them into the system of Equation (8), the following system of equations can be obtained:

$$\left\{\begin{array}{c}{F}_X={\displaystyle \sum _{i=1}^z{K}_{b_0}[1+\partial (T-T_0)+\Delta h(H-H_0)/H_0+h(\mu -{\mu }_0)/{\mu }_0]}\\ \{[(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r-b{C}_{d_0}\times \\ [1+\partial (T-T_0)+\Delta h(H-H_0)/H_0+h(\mu -{\mu }_0)/{\mu }_0]{\}}^{1.5}\cos {\theta }_i\\ F_Y={\displaystyle \sum _{i=1}^z{K}_{b_0}[1+\partial (T-T_0)+\Delta h(H-H_0)/H_0+h(\mu -{\mu }_0)/{\mu }_0]}\\ \{[(x_{in}-x_{out})\cos {\theta }_i+(y_{in}-y_{out})\sin {\theta }_i-c_r-b{C}_{d_0}\times \\ [1+\partial (T-T_0)+\Delta h(H-H_0)/H_0+h(\mu -{\mu }_0)/{\mu }_0]{\}}^{1.5}\sin {\theta }_i\end{array}\right.$$

(25)

The change of the resultant force $F$ with temperature after comprehensively considering the three factors is shown in Figure 27. The time domain waveform diagram of each temperature is also summarized. The $X$-axis is time $t$, the $Y$-axis is amplitude $V$, and the $Z$-axis is temperature $T$. The time domain waterfall chart is shown in Figure 28.

Similarly, by selecting the corresponding ten temperature point peaks at the four coordinates of 0.079 s, 0.277 s, 0.435 s, and 0.633 s for comparison in the $Z$-axis direction, the trend chart in Figure 29 can be obtained.

It can be seen from Figure 29 that the peak point amplitude acceleration shows a decreasing trend with temperature, which is consistent with the theoretical calculation results in Figure 27, verifying the effectiveness of the analysis of the impact of the dynamic model when considering the three factors. At the same time, to facilitate the analysis of the influence of temperature on the fault frequency of the dynamic model, this paper combines the initial simulation signal spectrum diagram of the dynamic model, as shown in Figure 30, and the experimental signal spectrum diagram of the 6307 bearing inner ring fault, as shown in Figure 31 with the temperature being 100 °C. The simulated signal spectrum diagram of °C is shown in Figure 32 for comparison.

By comparing Figure 30, Figure 31 and Figure 32, it can be seen that the fault characteristic frequencies of the three are $f_{in}=126.423$ Hz, $f_{in}=127$ Hz, and $f_{in}=126.823$ Hz, respectively. Through comparison, it can be seen that when temperature is properly considered, the fault characteristic frequency in the spectrum diagram of the simulation signal is closer to the fault characteristic frequency of the experimental signal, the error is reduced, and it is more consistent with reality.

To sum up, it can be seen from the above analysis that when the temperature increases, the increase in thermal expansion coefficient and friction coefficient will cause the amplitude acceleration to decrease, and the decrease in hardness value will cause the amplitude acceleration to increase. When the three factors are considered together, the overall amplitude acceleration shows a downward trend. This is more consistent with the actual working conditions. The characteristic frequency of rolling bearing faults has increased, which is also close to the actual working conditions. The difference between the kinetic model that considers temperature and the kinetic model that does not consider temperature affects the change of the overall kinetic model through the difference in initial condition parameters. Through the sensitivity analysis of the dynamic model considering temperature, comparing the change amount of the resultant force $F$ and the change amplitude amount of the acceleration, the friction coefficient becomes more significant.

The application of bearing dynamic models that consider temperature in bearing fault diagnosis can greatly improve the efficiency and accuracy of fault detection. This model can more comprehensively explain the operating conditions of the bearing system by extensively considering the influence of temperature, and can thus provide accurate bearing fault detection in the system bringing practical guidance to the industry. In the main, there are the following major industries:

(1)

High-speed mechanical systems: In high-speed rotating mechanical systems, the impact of temperature on bearing performance is particularly significant. Bearing dynamic models that account for temperature can help detect factors such as increased friction and thermal expansion, increasing sensitivity to potential failures in high-speed mechanical systems.

(2)

Industrial production lines: In production lines that require high precision and high efficiency and stability, bearing failure may cause production interruption and equipment damage. Through temperature-aware bearing dynamic models, potential signs of failure can be detected earlier and production line downtime can be reduced.

(3)

Aerospace field: In aircraft, spacecraft, and other equipment, the reliability of bearings is crucial for safety and performance. Through accurate fault detection, bearing faults can be identified in advance to avoid equipment failure at critical moments.

In these fields, accurate fault detection in bearing systems using bearing dynamic models that consider temperature is of great significance to improving system reliability, reducing maintenance costs, improving work efficiency, and ensuring safety.

4. Conclusions

By studying the mechanism of symmetric mechanism-rolling bearing dynamic systems considering the effect of the temperature factor, the main conclusions of this paper are as follows:

• Temperature changes cause the thermal expansion coefficient, hardness value and friction coefficient of the 6307 deep groove ball rolling bearing to change, respectively, which cause the spring stiffness $K_b$ and friction damping $C_d$ to change in the conventional dynamic model of the rolling bearing. Compared with the traditional dynamic equation, it has a more comprehensive approach.
• When the temperature increases, the increase in thermal expansion coefficient and friction coefficient cause the amplitude acceleration to decrease, and the decrease in hardness value causes the amplitude acceleration to increase. When considering each factor individually, the friction coefficient has a more significant impact. When the three factors are considered together, the overall amplitude acceleration shows a downward trend, which is more consistent with the actual working conditions.
• By comparing the fault characteristic frequencies of the inner ring of the 6307 deep groove ball rolling bearing in the three spectrum diagrams, it can be concluded that the fault characteristic frequency of the inner ring of the rolling bearing considering the temperature is closer to the experimental signal. Compared with the research on ordinary rolling bearing dynamic systems, temperature-based research on rolling bearing dynamic systems undertakes more comprehensive considerations, is more effective, and has practical guidance significance.