Thermal characterization of cylindrical roller bearings based on thermal-structural coupling approach

Abstract

High-speed running bearings generate a large amount of heat, which has a significant impact on the accuracy of machine tool feeding system. In this paper, by applying the bearing proposed statics and local heat source method, considering the combined surface contact thermal resistance and lubricant viscous temperature characteristics of the influence of the thermal network method to determine the main thermal characteristics of the shaft system parameters, the establishment of the transient thermal equilibrium equations, in the iterative process of the shaft system temperature field and deformation of the coupling; through the solution, the bearing temperature field and the transient change of the thermal parameters of the characteristics of the bearing, analyze the working conditions and structural parameters of the bearing transient thermal characteristics curve; using the finite element method coupled field simulation and test bed test, to verify the theoretical model is reasonable. The influence of working conditions and structural parameters on the transient thermal characteristic curve of the bearing is analyzed. The reasonableness of the theoretical model is verified by using the finite element method coupled field simulation and test bench test. After in-depth analysis: no matter the initial clearance or the change of rotational speed and load, it will lead to the change of thermal equilibrium temperature of the bearings, and will produce heat-induced load. Therefore, by choosing the appropriate clearance, improving the viscosity of lubricant, and enhancing the air convection, the heat generation and thermal deformation can be effectively reduced, which provides an important method to support the optimization of the bearing structure and the design of thermal balance.

Keywords

High-speed roller bearing coupling analysis model thermal network method local heat source method thermally induced loads

Introduction

Rolling bearings are widely used in aero engines, machine tool spindles and other equipment requiring high speed rotation because of their low friction, high transmission efficiency and reliable operating characteristics, and their performance will greatly affect the stability and reliability of the equipment, thus affecting the normal operation of the equipment. When the equipment is in a high speed working environment, the friction between the contact parts of the bearing and the loss of churning oil of the roller will generate tremendous heat. If not consumed in time, this heat will cause thermal deformation and contact state change of the bearing. It may even appear to be damaged.^1–4 It is therefore crucial to explore the thermal stability of bearings in depth.

In the literature,⁵ a failure analysis model of a double row tapered roller bearing was constructed by combining finite element analysis with Fortran simulation technology to elucidate its failure’s causes better. In the literature,⁶ a temperature analysis model of a shaft-linked bearing was constructed using Ansys software to explore the causes of heat generation, heat transfer and the development of its characteristics over time. In the literature,⁷ an effective model for the thermal coupling analysis of machine tool spindles was constructed using the finite element method to explore the temperature variations and dynamic stiffness characteristics in depth. The finite element method allows for faster meshing and parameterization, thus better meeting the needs of the thermal analysis of the shaft system. In the literature,^8,9 a steady-state thermal analysis model of a ball bearing was constructed by means of thermal network analysis, and the effects of load, speed and oil supply on the thermal characteristics of the bearing were investigated in depth. In the literature,¹⁰ a thermodynamic coupling model of the bearing system was constructed by combining the thermal displacement of the bearing and its related mechanism, and its stiffness and vibration behavior in the rotor system was analyzed. The literature takes heat transfer and rolling bearing tribology as the theoretical basis, takes a type of electric spindle as an example, considers the influence of frictional heat and preload method, establishes a spindle-bearing analysis model, and studies the influence of the thermal characteristics of the bearing and application parameters on the kinetic performance.¹¹ The literature considered the deformation equation of the spindle-bearing system under the combined effect of assembly stress and thermal stress, and based on the thermal network method, the characteristic parameters such as the heat source and thermal boundary conditions of the shaft system were modified in real time to realize the thermal coupling analysis.¹² A set of thermal nodal equilibrium equations was established in the literature.¹³ The Newton-Raphson method was used to solve the equations, and the effects of bearing preload, spindle speed and lubricant viscosity on the nodal temperature of the shaft system temperature field were numerically analyzed. In the literature,¹⁴ thermal resistance, power loss and convective heat transfer calculation models were established; MATLAB was used to solve the heat balance equation and the steady-state temperature field distribution of the shaft system; in the literature,¹⁵ in order to predict and control the thermal performance of the electric spindle during operation and its influence on the dynamic characteristics of the electric spindle, a thermal-machine coupled dynamic model of the bearing considering the influence of the system thermal response and preload method was established. The spindle bearing’s friction loss and dynamic support stiffness during the operation were analyzed. A coupled analytical model for the thermal dynamic characteristics of high-speed electric spindles was established in the literature.¹⁶ The change in support stiffness under the combined effect of centrifugal force softening effect and thermal preload hardening effect and its influence on the dynamic performance of the spindle system were analyzed. In the literature,¹⁷ the radial elastic deformation of the spindle rotor and bearing inner ring under the action of centrifugal force was calculated for a high-speed spindle system, and the trend of the spindle rotor-bearing connection state with increasing speed was analyzed. In order to study the temperature distribution in the axle box during the operation of high-speed rolling stock, literature calculated the frictional power consumption of each contact part in the bearing based on the theory of frictional heat generation.¹⁸ A thermal mesh model of the axle box unit was established and the temperature distribution of each node in the axle box was solved. In order to reduce the friction power consumption of air-conditioning sliding vane compressor, the friction power consumption characteristics of cylindrical roller bearings used in the compressor were studied in the literature.¹⁹ Based on the theory of rolling bearing dynamics, a mathematical model of friction power consumption of cylindrical roller for compressor was established by considering the cyclic load factor, on the basis of which the influence of bearing working condition parameters and structural parameters on the friction power consumption characteristics of cylindrical roller bearings was analyzed. Literature established a finite element model of shaft-cylindrical roller bearing-housing system considering temperature effect and clearance change, and systematically studied the displacement of inner and outer rings, raceway stress and bearing stiffness characteristics.²⁰ Literature studied the friction performance and temperature distribution of high speed rail axle box bearings, established a theoretical analysis model and a thermal network analysis model of axle box bearing friction performance; and verified the accuracy of the theoretical model by using the high speed rail bearing thermal test; finally, based on these two models and actual engineering examples, the friction performance and temperature distribution of the axle box bearings are analyzed by the influencing factors.²¹

The above literature has systematically discussed the heat generation mechanism and heat transfer mechanism of the rotor bearing system but primarily focused on the temperature growth characteristics of the electric spindle, and the empirical formula method is still heavily used in the calculation of bearing heat generation. A more comprehensive study is needed for the transient characteristics of high-speed rolling bearings.

In this paper, the stress-deformation equation of the bearing system under thermal stress is deduced, and on this basis, the main thermal characteristic parameters of the shaft system are determined based on the thermal network method, and the transient thermal equilibrium equation is established, and the temperature field of the shaft system is coupled with the deformation in the iterative process to construct the transient thermal-structural coupling model. The transient variation characteristics of the bearing temperature field and thermal parameters are obtained through the solution, the influence of the operating conditions on the transient temperature rise curve of the bearing is analyzed, and the accuracy of the transient model is verified by combining it with tests. The study of the temperature field of a high-speed cylindrical roller bearing is of great significance for improving the rotational accuracy of the bearing and the machining accuracy of the machine tool. Also, it provides an essential basis for optimizing the bearing structure and thermal balance design. In this paper, the temperature field analysis and thermal-structural coupling deformation study are carried out for a cylindrical roller bearing of type NJ204 used in a machine tool spindle.

Model construction and analysis

Power loss calculation model

Rumbargei argues that cylindrical objects rotating at angular velocity incur power losses when the object is in a viscous fluid medium.

N_{f} = \frac{1}{8} ρ fA ω^{3} r^{3}

(1)

f = {\begin{matrix} \frac{16}{R e R e <} \\ 3 (\frac{R e}{2500} {()}^{0.856} (\frac{16}{R e} () R e \geq) 1.3 {(\frac{T a}{41})}^{0.539} (\frac{16}{R e} ())) \end{matrix}}

(2)

Re = ω rC / ν

(3)

Ta = Re \sqrt{C / r}

(4)

Where: $ρ$ is the lubricant density; $f$ is the friction factor; $A$ is the effective area of the cylindrical surface; $ω$ is the angular velocity of cylinder rotation; $r$ is the radius of the cylinder; $Re$ is the Reynolds number; $Ta$ is the Taylor number; $ν$ is the lubricant kinematic viscosity.

By applying the model to the inner ring, roller and cage surfaces of a rolling bearing system, the power losses on the surface of each component can be accurately estimated.

Rumbarger method heat transfer calculation model for bearings

The heat dissipation in rolling bearing system is in the form of heat conduction, heat convection and heat radiation, mainly considering convective heat exchange. Most high-speed rolling bearings use oil spray lubrication, bearing system convective heat transfer is mainly for the lubricant and the inner and outer ring raceway surface convective heat transfer, lubricant and rolling body surface and cage convective heat transfer, bearing seat outer surface and air convective heat transfer,¹⁹ its convective heat transfer coefficient is calculated as follows:

(1) For the bearing outer ring fixed, inner ring rotation, inner and outer ring and cage cylindrical surface between the fluid convection heat transfer coefficient is:

α_{1} = 0.175 \frac{k}{R} \sqrt{\frac{R ω C}{ν} \sqrt{\frac{C}{R}}} / \ln (1 + \frac{C}{R})

(5)

Where: $k$ is the lubricant’s thermal conductivity; $R$ is the inner raceway’s radius.

(2) The natural convection heat transfer coefficient between the outer surface of the bearing housing and the fluid (air) is:

α_{2} = 0.53 \frac{k}{D_{h}} {(G_{r} P_{r})}^{0.25}

(6)

Where: $D_{h}$ is the diameter of the cylindrical surface of the bearing housing; $G_{r}$ is the Grashof criterion; $P_{r}$ is the Pmndtl criterion.

(3) The forced convective heat transfer coefficient of the lubricant and the rolling body is:

α_{3} = 0.0986 {\frac{n}{ν} [1 \pm \frac{D_{w}}{D_{pw}}]}^{0.5} k {P_{r}}^{1 / 3}

(7)

Where: $D_{w}$ for the rolling body diameter; $D_{pw}$ for the roller section circle diameter.

(4) The forced convective heat transfer coefficient between the lubricant and the outer surface of the rotating shaft is:

α_{4} = 0.11 \frac{k}{d} {[0.5 {Re}^{2} P_{r}]}^{0.35}

(8)

Re = π ω_{1} d^{2} / ν

(9)

where: $ω, d$ are the angular velocity and diameter of the shaft respectively.

Determination of the friction factor based on dynamics simulation

Through the cylindrical roller bearing NJ204 force analysis can be seen, the bearing bearing area for the lower half of the circle, and contact load is also in the radial force line of action of the bottom one rolling body maximum, and away from the role of the line of each rolling body, the load gradually decreases. Because of the symmetrical characteristics of the bearing, we only consider the contact stress between the three different parts, see Figure 1 for details.

Figure 1.

Geometric model of cylindrical roller bearing.

The simulation was carried out by using the kinetic software Adams, inputting the structural parameters of the bearing and the operating parameters in turn, setting the simulation time and the solution step, setting the speed of the bearing to 6000 r/min, the radial load to 1.5 kN, the simulation time to 0.175 s, and the acceleration of the cage as shown in Figure 2. It can be seen from the figure that the cage operation is relatively chaotic when the bearing first runs. After a period of operation it tends to stabilize, with the root mean square value stabilizing at 0.7 m/s².

Figure 2.

Cage acceleration versus time.

Using Adams software, we were able to study the mechanical behavior of cylindrical roller bearings in order to gain a better understanding of their operation. We have simulated the changes in positive pressure and friction as the rollers move from a certain angle and present the results, as detailed in Figures 3 and 4.

Figure 3.

Positive pressure change of roller and collar.

Figure 4.

Roller and collar friction change.

The positive pressure and frictional force of the roller at different positions can be obtained by combining the data, as shown in Table 1.

Table 1.

Roller and raceway in different positions interaction forces.

Position	Position 1 (0°)	Position 2 (36°)	Position 3 (324°)
Positive pressure $F_{N} / N$	1133.6	732.1	649.2
Friction $f / N$	56.68	36.62	32.41

After careful kinetic studies, we have found that as the bearing rollers rotate, their friction factors with other components change accordingly. The friction factor taken in this paper is the average value. Based on the kinetic analysis, the friction factor between the roller and the inner ring is 0.05, the friction factor between the roller and the outer ring is 0.02, and the friction factor between the cage and the roller is 0.01.

Generally speaking, metals are in an elasto-plastic state of contact and the contact area of the contact surface becomes more significant when the load is increased. However, as the actual contact area changes at a slower rate than the load, the friction will change at a slower rate than the load. Therefore, as the load increases, the friction coefficient will gradually decrease. In a state of plastic deformation, the coefficient of friction remains relatively constant, independent of external forces. The coefficient of friction in the kinetic analysis is obtained by simulations, which better reflect the actual contact state.

Comparison of heat generation calculation models

Now take NJ204 type outer ring double block edge, inner ring single block edge cylindrical roller bearing as an example, select the lubricant motion viscosity for 8 mm²/s, respectively use different methods to calculate the bearing heat production,^22–26 including the approximate method, Palmgren method, Astridge method, Rumbarger method, Harris method and B M Jemidovich method. The structural parameters of the cylindrical roller bearing NJ204 are shown in Table 2. The inner ring can be rotated, while the outer ring is fixed. The relationship between the frictional power loss of the bearing and the rotational speed for different calculation models is shown in Figure 8 (Fr = 1.5 kN, Fa = 0).

Table 2.

Bearing structure parameters.

Parameter	Numerical value
Nominal internal diameter d/mm	20
Nominal outside diameter D/mm	47
Width B/mm	14
Number of rollers	10
Modulus of elasticity/(E₁/GPa)	207
Poisson ratio $ν_{1}$	0.3

According to Figure 5, we found that: the approximate method calculation formula has nothing to do with the speed, only with the load, so under the same load, its friction power and heat loss is unchanged. This method is suitable for the rough calculation of bearing heat under low speed working conditions but not suitable for high speed rolling bearings; under the same speed, Astridge (overall method) calculates the bearing friction power loss heat is the largest, and with the speed of heat production increase rate is also the largest; under the same radial load, with the increase of bearing speed, Rumbarger method and Harris method calculates the frictional power loss of the rolling bearing is relatively close; the Palmgren (integral method) and B M Jemidovich method (integral method) power loss calculations are roughly in the same order of magnitude. In order to determine the method, the theoretical method of the literature was used for the test backpropagation,²⁷ and it can be seen from the graph that the Rumbarger heat generation calculation model fits the test backpropagation curve better.

Figure 5.

Heat production for each heat generation model at different speeds.

According to Figure 6, the frictional heat generated by the bearing rises sharply when its speed rises and becomes more intense when it is faster. The sliding between the rollers and the inner and outer raceways leads to relatively low frictional heat; this is because the bearing is subjected to a higher external load and slides less, and the frictional heat it generates is reduced accordingly. The heat caused by elastic hysteresis has increased because the bearing speed increases will make the centrifugal force suffered by the roller increases, the outer raceway contact load value increases, and then the elastic hysteresis moment increases, the frictional heat increases; roller viscous frictional heat, roller and cage pocket hole and cage and raceway guide surface frictional heat have increased, mainly because with the increase in bearing speed, cage and The main reason is that with the increase of the bearing speed, the cage and roller rotation speed, the roller, cage, raceway guide surface between the joint force increased, the frictional heat and rise.

Figure 6.

Local heat production by the Rumbarger method.

Construction and calculation of a thermal network model considering thermal expansion

Thermal network diagram of the bearing system.

To simplify the solution of the temperature field problem, the following assumptions are made:

(1) The heat source of the bearing mainly originates from the raceway surfaces of its inner and outer rings;

(2) The frictional heat between the rolling body and the inner and outer rings is evenly distributed between the rollers and the inner and outer rings in a ratio of 1:1.

By simplifying the cylindrical roller bearing system, we can refer to Figure 7 to obtain the relevant information. Table 3 shows the specific significance of each thermal node. $T_{\infty}$ is the air temperature around the end cap’s outside and $T_{l \infty}$ is the lubricant’s inlet oil temperature. It is assumed that the starting and air temperatures of the bearing system are both 20°C, while the inlet oil temperature is maintained at 60°C.

Figure 7.

Hot node locations.

Table 3.

Hot nodes.

Hot nodes	Nodes indicate temperature	Hot nodes	Nodes indicate temperature
$T 1$	End face surface temperature	$T 7$	Inner ring raceway temperature
$T 2$	Bearing housing end face temperature	$T 8$	Average temperature of the inner ring
$T 3$	Average bearing housing temperature	$T 9$	Average shaft temperature
$T 4$	Average temperature of the outer ring	$T 10$	Average shaft end temperature
$T 5$	Outer ring raceway temperature	$T_{\infty}$	Ambient temperature
$T 6$	Average roller temperature	$T_{l \infty}$	Lubricant temperature

Thermal resistance calculation model for bearing thermal networks

Here the influence of thermal radiation is relatively small, based on how heat is transferred, so only heat conduction and convection are considered. The expressions for the heat conduction and convection heat resistance for the heat transfer process in the one-dimensional model based on the thermal network method are:

R_{c} = δ / kA

(10)

R_{c} = 1 / hA

(11)

In the formula: $k$ —Thermal conductivity of materials; $δ$ —Thermal conductivity characteristic length; $A$ —Heat exchange area; $h$ —Convective heat transfer coefficient.

The inner and outer ring of the bearing and bearing seat is simplified as a thin-walled cylinder, using the calculation formula of thermal resistance and the relevant parameters of the parts, which can be solved to get the axial thermal resistance and radial thermal resistance between the nodes, Table 4 is the thermal resistance of each part of the bearing system, $R_{c 1},$ $R_{c 2}$ for the contact thermal resistance between the cylindrical roller and the inner and outer ring.

Table 4.

Thermal resistance of the components of the bearing system.

Thermal resistance
Roller	$R_{b} = \frac{1}{π k_{b} L_{b}}, R_{b} = \frac{\ln (d_{b 1} / d_{b 2})}{2 π k_{b} L_{b}}, R_{bh} = \frac{1}{π h_{b} BD}$
Inner ring	$R_{i} = \frac{\ln (D_{ie} / D_{i})}{2 π k_{i} B}, R_{ih} = \frac{1}{π k_{i} d_{i} B}$
Outer ring	$R_{o} = \frac{\ln (D_{o} / D_{oe})}{2 π k_{o} B}, R_{oh} = \frac{1}{π k_{o} d_{0} B}$
Shaft	$\begin{array}{l} R_{o h} = \frac{1}{π k_{0} d_{0} B}, R_{s a} = \frac{4 L_{s}}{π k_{s} {d_{s}}^{2}}, R_{s r} = \frac{1}{π k_{s} B}, \\ R_{s h} = \frac{4}{π h_{s} {d_{s}}^{2}} \end{array}$
Bearing housing	$R_{hr} = \frac{\ln (D_{h} / D_{hc})}{2 π k_{h} C}, R_{ha} = \frac{4 L_{h}}{π k_{h} ({D_{h}}^{2} - {D_{he}}^{2})}$
End cap	$R_{da} = \frac{4 L_{d}}{π k_{d} ({D_{1}}^{2} - {D_{2}}^{2})}, R_{dh} = \frac{4}{π h_{d} {D_{1}}^{2}}$

From the theory of heat transfer, the convective heat transfer coefficient is:

h = N_{u} λ_{a} / L

(12)

In the formula: $λ_{a}$ —Air thermal conductivity; $L$ —Convection heat transfer characteristic length. For a cylinder take the diameter of the outer surface; $N_{u}$ —Nusselt number.

Bearing thermal expansion calculation model.

As the temperature rises, the individual components of the bearing will experience varying degrees of thermal expansion displacement, which will change the contact condition within the bearing. The mathematical formula can express the radial thermal expansion displacement of a high-speed roller bearing:

u_{b} = β_{b} T_{b} D_{b}

(13)

u_{i} = β_{i} T_{i} D_{i}

(14)

u_{0} = \frac{β_{0} (1 + ν) D_{0}}{3 (D_{0} + D_{h})} [T_{0} (2 D_{0} + D_{h}) + T_{h} (2 D_{0} + D_{h})]

(15)

In the formula: The subscripts $i$ , $o$ , $b$ and $h$ denotes the inner ring, outer ring, ball and housing respectively; $u$ , β , $T$ and $D$ are the radial thermal displacement, coefficient of thermal expansion, temperature and diameter respectively.

Thermal displacement of bearing components can lead to thermally induced loads on the contact surfaces of the rollers and raceways, which can be expressed by:

u_{t} = u_{b} + (\frac{u_{i} - u_{0}}{2}) \cos (\frac{α_{i} + α_{0}}{2})

(16)

F_{t} = k_{t} {u_{t}}^{1.5}

(17)

The formula: $α_{i},$ $α_{0}$ are the internal and external contact angles respectively; $k_{t}$ is the load deflection coefficient; $u_{t}$ is the total thermal deformation in the standard contact direction.

Establishment of the heat balance equation

By analyzing the cylindrical roller bearing system, we can link the thermal nodes in it by their thermal resistance and create a complete thermal network diagram using their corresponding structural dimensional parameters, thermal conduction coefficients, convective heat transfer coefficients and other factors, as shown in Figure 8.

Figure 8.

Thermal network diagram.

In the thermal network of a bearing system, the heat balance equation for each node can be described by the following equation:

\sum_{j = 1}^{m} \frac{T_{j} - T_{p}}{R_{jp}} + Q_{p} = m_{p} C_{p} \frac{\partial T_{p}}{\partial t}

(18)

In the formula: $T_{p}$ is the temperature of the node to be sought; $T_{j}$ is the temperature of each node associated with the node to be sought; $R_{jp}$ is the thermal resistance between nodes j and p; $Q_{p}$ is the amount of heat generated at the node to be sought; $m_{p}$ is the mass associated with the node; and $C_{p}$ C is the specific heat capacity of the material.

The model has 10 nodes, and according to the transient heat balance equation, the differential equations of 10 nodes can be established. The fourth-order Longacurta method was used to solve the set of differential equations to obtain the transient thermal characteristics of the bearing, the process is shown in Figure 9. We found that the viscosity of the lubricant had to be adjusted to ensure that it conformed to the desired parameters when carrying out the iterations, specifically by:

η = η_{0} e^{- γ (T_{oil} - T_{0})}

(19)

Figure 9.

Flow chart of the thermal-force coupling calculation.

Convergence verification of the theoretical model

The bearing was analyzed using a Matlab program written for the following conditions: speed 6000 r/min, radial load Fr 1500 N, kinematic viscosity of the lubricant 8, initial clearance of the bearing 20 μm.

The change of heat generation rate during the thermal balance of the bearing at 6000 r/min is shown in Figure 10. It can be seen that the heat generation rate of the bearing decreases rapidly at the initial stage of operation. Then the heat generation rate shows a slow decreasing trend and remains stable. From the start-up to the thermal equilibrium state, the heat generation rate decreases by 18.88%. The trend of the heat generation rate during the thermal equilibrium process is mainly related to the viscosity of the lubricant in the bearing cavity, as shown in Figure 11. The temperature of the lubricant, which absorbs heat in the bearing cavity, also rises, causing its viscosity to drop.

Figure 10.

Heat generation rate versus time.

Figure 11.

Lubricant kinematic viscosity versus time.

Experimentation and analysis

Test machine construction

In order to verify the feasibility of the theoretical calculations, a temperature measurement experiment was designed to verify indirectly.

According to the structure of the spindle system used in this study, a real-time monitoring scheme for the temperature of the end face of the inner ring of the bearing based on fiber optic sensing and fiber optic rotary connector (FORJ) is designed, that is, the left shaft end gland is redesigned, the fiber optic grating sensor (FBG) is arranged in its end face, and the FBG is welded to the FORJ and then led out through the newly-designed hollow end cap connector, which connects with the optical signal collector, which demodulates the FBG wavelength signal and sends it to the computer after processing to get the transient temperature at the measurement point. FBG wavelength signal demodulation instrument will be converted and sent to the computer, after processing can be obtained at the measurement point location of the transient temperature. Figure 12 shows the schematic diagram of the bearing inner ring temperature testing system based on FBG.

Figure 12.

Schematic diagram of FBG-based bearing inner ring temperature testing system.

The arrangement of the test bench and sensors is shown in Figure 13, and the basic information of the shaft system and bearings as well as the parameters of the test conditions are shown in Table 5.

Figure 13.

Test device.

Table 5.

Main technical parameters of the test bench.

Spindle type	Mechanical spindle
Driving method	Electric spindle drive
Lubrication method	Oil spraying (ambient temperature to +200°C)
Test bearing type	NJ204
Sensor	Outer ring: thermocouple temperature sensor. Inner ring: fiber optic grating sensor
Measurement and control method	Outer ring: PC automatic monitoring and recording. Inner ring: optical signal collector and PC

Analysis of results and discussion

Analysis of the influence law of structural parameters

Analysis of lubricant flow rate for 10 mL/s, inner ring speed for 15,000 r/min, Fr for 1.5 kN, different clearance under the bearing thermal equilibrium state inner and outer ring stress distribution characteristics. As can be seen from Figure 14, the thermal expansion of the inner ring is greater than the thermal expansion of the outer ring, for the running before the clearance is zero bearing; it’s running due to the difference between the expansion of the inner and outer ring so that the bearing in the negative clearance state, thermal expansion so that all the rollers are in the bearing state. With running before clearance increase, we can see that the bearing area roller number gradually reduces, subject to load roller stress increased.

Figure 14.

Stress variation with play at different collar angle positions at oil temperature 60°C.

The characteristics of the stress and displacement distribution in the inner and outer rings of the bearing in thermal equilibrium, when the lubricant temperature rises from 0°C to 80°C, are analyzed. The lubricant flow rate is 10 mL/s; the inner ring speed is 15,000 r/min, Fr is 1.5 kN. Figure 15 shows that the lubricant temperature has no significant effect on the inner and outer ring raceway stresses under the condition of large bearing clearance.

Figure 15.

Stress variation with oil temperature at different collar angular positions with a clearance of 20 μm.

As shown in Figure 16, it can be seen that the temperature of the lubricant has a significant effect on the displacement of the collar. As the oil temperature rises, the inner and outer ring displacements show a trend of decreasing and then increasing, and the outer ring displacement changes are more significant than the inner ring. When the temperature of the lubricating oil drops to a certain level, the viscosity will increase, instead of improving the friction loss of the bearing, which in turn makes the displacement caused by thermal expansion of the inner and outer rings increase. Zero clearance and clearance case inside and outside the circle displacement change amplitude difference is not significant, that the lubricant temperature on the bearing ring displacement influence than the influence of clearance.

Figure 16.

Variation of displacement with oil temperature for different angular positions of the collar at a clearance of 20 μm.

Analysis of the influence law of working parameters

The influence of the inner ring speed on the temperature rise of the bearing was analyzed, as shown in Figure 17. The initial clearance of the bearing is 20 μm; the speed increases from 6000 to 36,000 r/min, the temperature rise of each part of the bearing increases rapidly, among which the roller temperature rise is the largest about 150°C, the inner raceway temperature is the second, the following is the outer raceway surface temperature, the cage surface temperature, the above temperature distribution characteristics are closely related to the frictional power consumption and convection boundary conditions between the components of the bearing.

Figure 17.

Effect of speed on temperature rise at Fr = 1.5 kN.

Analysis of radial load’s influence on the bearing’s temperature rise, as shown in Figure 18. As can be seen from the figure, when the load is 250 N, the temperature of each part of the bearing is the highest, the reason is that the load is too light, resulting in serious slippage between the roller and the raceway, plus the convection heat transfer coefficient of the raceway surface is small, and then the roller and raceway sliding friction power consumption increases. However, as the load continues to increase, the temperature of the bearing will appear a sharp fall, followed by a gradual rise in temperature. It can be seen that the appropriate radial load will make the bearing avoid the slipping phenomenon, which has an essential significance to the bearing long life cycle.

Figure 18.

Radial load at n = 15,000 r/min effect on temperature rise.

The effect of lubricant viscosity on the temperature rise of the bearing is analyzed as shown in Figure 19. It can be seen from the figure that the temperature of each part of the bearing increases with the viscosity of the lubricant in a nearly straight line, which shows that the kinematic viscosity of the lubricant has a relatively significant effect on the bearing temperature; at different lubricant viscosities are the highest roller surface temperature, when the viscosity of the lubricant increases from 3 to 11 mm²/s, the roller temperature of the bearing increases from 81.5°C to 106.3°C. There are two main reasons for this phenomenon: (1) the frictional power consumption of the various parts of the bearing increases with the increase in lubricant viscosity; (2) the increase in lubricant viscosity leads to a deterioration in its fluidity, which, turn reduces the convective heat transfer performance of the lubricant.

Figure 19.

Fr = 1.5 kN, n = 15,000 r/min at effect of lubricant viscosity on temperature rise.

Simulation analysis of thermal-force coupled finite element model

In order to simplify the FEA model and to bring the analytical results close to the actual situation, the following assumptions are made:

(1) Bearing friction factor remains unchanged; ignore the impact of material wear, that the roller and raceway uniformly absorb the heat generated by roller and raceway friction;

(2) As the temperature changes, the thermal conductivity, coefficient of thermal expansion, and specific heat capacity of the material change accordingly, but its density remains the same;

(3) Ignoring the calculation of the flow and radiation fields, by using different heat transfer coefficients for different parts of the bearing, in order to simulate the flow and radiation fields and their effects on the structure.

Still, take NJ204 cylindrical roller bearing as an example; Table 6 gives the main structural parameters of the bearing; bearing material selection GCrl5 bearing steel, when the temperature is 100°C, thermal conductivity $k^{'}$ = 45 W/m.°C; given bearing speed $n$ = 15,000 r/min, radial load $Fr$ = 1.5 kN, axial load $Fa$ = 0, clearance is 0, the thermal conductivity of lubricating oil $k$ = 0.14 W/m.°C, inlet oil temperature 60°C. The Relevance Center of Sizing is set to Fine in Ansys Workbench, and the rest is set by default.

Table 6.

Simulation parameters.

Parameters Name	Calculation results
Contact parameters for GCrl5/ $(W / ° C)$	109
Heat transfer coefficient between inner/outer ring and rollers of bearings/ $(W / m^{2} \cdot ° C)$	25
Heat transfer coefficient between the inner/outer ring of the bearing and the outside world/ $(W / m^{2} \cdot ° C)$	150
Thermal conductivity/ $(W / m \cdot k)$	35

The Rumbarger regional analysis allows us to accurately understand the heat flow distribution and convective heat transfer coefficient of the NJ204 bearing. After inputting the bearing material parameters, heat flow density, and heat transfer boundary conditions into the Ansys Workbench interface, the Rumbarger mechanics module enables a better investigation of the thermal properties of the bearing.

A study was carried out on the contact load distribution of the raceway, taking the largest loaded roller and obtaining the stress and load distribution along the straight line of the roller respectively as shown in Figure 20. The contact stress in the inner ring is greater than the contact stress in the outer ring. As the rollers are shaped, no “edge effect” occurs, which helps to improve the bearing performance.

Figure 20.

Roller contact stresses along the roller busbar direction.

As can be seen from Figure 21, the inner and outer ring raceways are stress peaks, indicating that the bearing has more than one roller in the bearing state. As the load increases, the stress value gradually increases, the inner ring raceway stress is slightly more significant than the outer ring raceway, and the maximum equivalent force is 303 Mpa.

Figure 21.

Force of equivalent effect on a collar.

According to Figure 22, the thermal expansion of the inner ring increases significantly as the temperature of the inner ring is significantly higher than that of the outer ring during the operation of the bearing.

Figure 22.

Thermal expansion of the collar.

As can be seen from Figure 23, the bearing roller temperature rise is the highest, the inner ring raceway, the outer ring raceway next, and the lowest cage. The highest roller temperature rises in the roller, and the inner raceway junction, roller temperature rises along the radial direction, from the inner ring contact to the outer ring contact in descending order. For the inner ring, the highest temperature is distributed at the contact with the roller; as the blocked side is mainly for heat conduction, compared with the non-blocked side convection heat transfer, heat dissipation is faster, so the temperature rise on the blocked side is lower than that on the non-blocked side. This trend is consistent with the literature,²⁸ which has been verified experimentally and indirectly proves the feasibility of the theoretical approach.

Figure 23.

Temperature field under coupled thermal-force simulation: (a) Overall, (b) Outer ring, (c) Inter ring, (d) Cage and (e) Roller.

Comparison and analysis

Firstly, three repetitions of the test were carried out and the data were averaged to reduce the error in the test data. Then, the experimental data were compared with the theoretical and simulation results for the corresponding points, as shown in Figure 24. The deviation of the measured data from the theoretical model and the coupled thermal-force simulation is within 10%, which shows that the method in the paper is feasible; the transient thermal simulation does not take into account the influence of thermal deformation and thermally induced load on the temperature rise, so the deviation is more considerable.

Figure 24.

Comparison of temperature rise results.

Conclusion

In this paper, the transient heat balance equation is established by applying the bearing proposed statics and local heat source method, and the temperature field of the shaft system is coupled with the deformation in the iterative process; the following conclusions are drawn through the solution:

(1) The lubricant temperature has a significant effect on the collar displacement. With the increase in oil temperature, the displacement of both inner and outer rings shows a trend of decreasing and then increasing, and the displacement of the outer ring changes more than the inner ring. When the temperature of the lubricant drops to a certain level, the viscosity will increase, but improve the friction loss of the bearing, which in turn makes the inner and outer rings due to thermal expansion caused by the displacement increases. Zero clearance and clearance in the case of displacement of the inner and outer ring changes in the magnitude of the difference is not large, indicating that the lubricant temperature on the bearing collar displacement is greater than the impact of clearance.

(2) Appropriate radial load will make the bearing to avoid slipping phenomenon, reduce the heat and thermal deformation; bearing parts of the temperature with the increase in lubricant viscosity nearly linear increase, which indicates: bearing parts of the friction power consumption with the increase in lubricant viscosity and increase; lubricant viscosity increases will lead to its liquidity deterioration, reduces the lubricant’s convective heat transfer performance.

(3) The accuracy of the theoretical model is verified with the experiment and compared with the heat-force coupling model, and the results show that the use of the theoretical model proposed in this paper to predict the temperature field of the axial system can significantly reduce the error, and provide an important method for the optimization of the bearing structure and the design of heat balance.

Footnotes

Handling Editor: Sharmili Pandian

Author contributions

Writing—original draft preparation: C.M.Y.; Writing—review and editing: M.Y.Z.; Visualization: Y.S.C.; Supervision: M.Q., Y.F.D.; Project administration: F.F.C.; All authors have read and agreed to the published version of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The project was supported by the Open Foud of State Key Laboratory of Tribology in Advanced Equipment (SKLTKF20A03), the Henan Province Key R&D and Promotion Project, grant number 232102220043.

Institutional review board statement

Not applicable.

Informed consent statement

Not applicable.

ORCID iDs

Mengyu Zhu

Yanfang Dong

Data availability statement

Data are reported within the article.

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