A new approach for analyzing internal load and fatigue life of deep groove ball bearing in high-speed gearbox system

Abstract

The study of the bearing fatigue life in high-speed gearboxes is an integral part of high-speed gear transmission research. There is still a lack of research on the analysis and calculating the bearing internal dynamic load and fatigue life considering gear engagement and centrifugal force simultaneously. Therefore, the paper proposes a new approach to acquire the internal load and fatigue life of the deep groove ball bearing in a high-speed gearbox system. Firstly, the modeling method of the gear mesh, flexible shaft, high-speed deep groove ball bearing, and housing is established, constructing the gearbox system dynamic model, which has the ability to calculate the bearing internal load. Then, the calculation method for analyzing the fatigue life of bearings is given based on the linear damage theory. The results show that the gear engagement and centrifugal force have meaningful effects on bearing internal loads and fatigue life, especially at high speeds. Both of them should be considered when evaluating the bearing life. The calculation method and model proposed in this paper provide a new technical approach for service performance evaluation and reliability design of high-speed deep groove ball bearing.

Keywords

bearing internal load fatigue life high-speed deep groove ball bearing centrifugal force gear engagement

1. Introduction

Deep groove ball bearings with the advantages of simple structure, low coefficient of friction, and high limiting speed are ubiquitously employed in gear systems. As support parts of the gear system, the rolling bearing plays a crucial role in the vibration transmission process and has a direct impact on the dynamic characteristics of the gear system (Fernandez-Del-Rincon et al., 2017; Guo and Parker, 2010; Kahraman and Singh, 1991; Liu et al., 2019a; Yang and Lim, 2011). Thus, the bearing fatigue life is significant for the durability and reliability of gear systems. The normal failure form of rolling bearings is the pitting corrosion damage of internal and external raceways or rolling elements due to the periodic change of contact stress. And the contact stress is closely related to the bearing internal load. Nowadays, bearings in high-speed gear trains are operating at high speed, centrifugal force and gyroscopic effects will have a significant influence on the internal load distribution, which is essential for the bearing fatigue life. In addition, the effects of gear engagement on the bearing internal load distribution and fatigue life have not received sufficient attention. Unfortunately, there is still a lack of research on the analysis and calculating the bearing internal load distribution and fatigue life considering gear engagement and centrifugal force comprehensively.

Many researchers have investigated the bearing internal load and fatigue life during the past decades. Lundberg and Palmgren (1947) proposed an equation to calculate the bearing fatigue life. This method is widely adopted to estimate fatigue life. Similarly, the approach illustrates the relationship between internal bearing load and bearing life. Later, Nagatomo et al. (2012) studied the effect of internal bearing load distribution on bearing life by theory and experiment. In his research, the elastic deformation of the bearing ring was considered in calculating the internal load distribution. Arakere (2016) discussed the traditional empirical probabilistic approach to bearing life prediction and its limitations. Considering the limitations of the life model established by Lundberg and Palmgren (1947). Ioannides and Harris (1985) proposed a new fatigue life calculation method by considering the postulation of a statistical relationship between the probability of survival, the fatigue life, and a stress-related fatigue criterion level above a fatigue limit for an elementary volume of material. Oswald et al. (2012) gave the relationship between bearing life and internal clearance based on a static bearing model. Warda and Chudzik (2014) studied the method for calculating the life of cylindrical roller bearings considering radial clearance and roller profile. The studies by Oswald et al. (2012) and Warda and Chudzik (2014) illustrated the effect of clearance on bearing life. To further improve the accuracy of life calculation, a more accurate bearing model is essential to calculate the internal load. Zhang et al. (2017) established a quasi-dynamic model to study the load distribution of ball bearings under arbitrary preload. Fang et al. (2020) proposed a new modeling method of the ball bearing to calculate the internal load distribution. Liu et al. (2019b) presented a new analytical approach considering the axial preload force and contact angle to solve the internal load distribution of the bearing. Zheng et al. (2019) investigated the internal load and contact pressure distribution of a double-row tapered roller bearing based on a quasi-static bearing model. Li et al. (2018) developed a model of angular contact ball bearing considering local defects in the outer raceway and analyzed the internal loads of the bearing. In addition, Shi et al. (2016) presented a new calculation method of relative fatigue life considering surface texture on high-speed and heavy-load ball bearing. He et al. (2018) proposed an approach of testing the fatigue life using a small sample test. And experiments were conducted to determine the actual fatigue life of a small sample.

In summary, the time-varying bearing force is neglected when analyzing bearing internal loads and fatigue life. In gear systems, the dynamic meshing force generated by gear engagement will transfer to the bearings, leading to large fluctuations in bearing load and affecting the bearing internal load distribution and fatigue life (Kong et al., 2021). On the other hand, gearbox systems are moving towards high speeds. Bearings are operating at high speed, resulting in a large centrifugal force on the rolling elements. Therefore, when analyzing the internal bearing loads and life in high-speed gear systems, the dynamic bearing load and centrifugal force should be considered. However, the current bearing model for calculating internal loads treats the bearing load as a constant. And load factors are set empirically to account for bearing load fluctuations. Fortunately, with the development of gear dynamic models, it has been possible to calculate the dynamic bearing force due to the gear meshing behavior. Nevertheless, in the gear dynamic model, the bearing models are simplified to linear springs (Chen et la., 2015; Hu et al., 2020; Kahraman et al., 1990; Kang and Kahraman, 2012; Kong et al., 2020; Ouyang et al., 2019) or time-varying springs (Liu et al., 2019a; Yang and Lim, 2011) to consider the bearing flexibility without the possibility to calculate the rolling element load.

The first purpose of this paper is to propose a novel approach for calculating the bearing internal load and fatigue life by coupling the dynamic model of the gearbox system and the quasi-dynamic bearing model. Another purpose is to investigate the effect of gear engagement and centrifugal force on the internal load and fatigue life of deep groove ball bearings in high-speed gear systems. Section 2 gives the modeling method of the gear mesh, flexible shaft, high-speed deep groove ball bearing, and housing, constructing the gearbox system dynamic model. Section 3 shows the calculation process of internal load and fatigue life of bearing based on the gearbox system dynamic model. In Section 4, the effects of gear engagement and centrifugal force on the internal dynamic load and fatigue life are discussed in detail. Conclusions are given in Section 5.

2. Dynamic model of the gear-shaft-bearing-housing system

Figure 1 shows the physical diagram and three-dimensional model of the gearbox system. The gear parameters and the dimensional parameters of the gear shaft are listed in Tables 1 and 2, respectively. Table 3 gives the bearing parameters. The dynamic modeling method is as follows.

Table 1.

Gear parameters.

	Pinion	Gear
Number of teeth	29	49
Module (mm)	3	3
Transverse pressure angle (deg)	25	25
Face width (mm)	20	20
Load (Nm)	44.39	75

Table 2.

Dimension parameters of shafts.

	Pinion shaft			Gear shaft
	Length (mm)	Inner diameter (mm)	Outer diameter (mm)	Length (mm)	Inner diameter (mm)	Inner diameter (mm)
L1	34	6	16	34	6	16
L2	23	6	20	23	6	20
L3	20	6	24	20	6	24
L4	100	0	30	100	0	30
L5	25	0	40	25	0	40
L6	42	0	35	—	—	—
L7	45	0	38	—	—	—
L8	—	—	—	500	0	28
L9	—	—	—	63	0	30

Table 3.

Bearing parameters.

	Value
Inner diameter (mm)	30
Outer diameter (mm)	90
Ball diameter (mm)	17
Number of balls	6

2.1. Gear mesh element modeling

The schematic diagram of the meshing relationship of a spur gear pair established is shown in Figure 2. The gear body is simplified to a rigid disk. The angle between the line of action and the x-direction is

γ = φ + ζ (\frac{π}{2} - α)

(1)

where,

φ

is the angle of the vector

O_{1} O_{2}

with the X-axis.

ζ

represents the direction of rotation of the pinion. The pinion rotates counterclockwise with

ζ = 1

, while the pinion rotates clockwise with

ζ = - 1

α

is pressure angle.

The motion equation in matrix form can be written as (Chen et al., 2015)

M_{m} q_{m}^{″} + γ_{m 0} K_{m} q_{m} + γ_{m 0} C_{m} q_{m}^{'} = {(f^{m})}^{T} (γ_{m 0} k_{m} (t) e (t) + γ_{m 0} c_{m} e^{'} (t) - γ_{m 1} k_{m} (t) B)

(2)

where

q_{m} = {(x_{p}, y_{p}, z_{p}, θ_{p x}, θ_{p y}, θ_{p z}, x_{g}, y_{g}, z_{g}, θ_{g x}, θ_{g y}, θ_{g z})}^{T}

(3)

M_{m} = d i a g (m_{p}, m_{p}, m_{p}, J_{D p}, J_{D p}, J_{p}, m_{g}, m_{g}, m_{g}, J_{D g}, J_{D g}, J_{g})

(4)

K_{m} = k_{m} (t) {(f^{m})}^{T} f^{m}

(5)

C_{m} = c_{m} {(f^{m})}^{T} f^{m}

(6)

γ_{m 1} = {\begin{array}{c} - 1 & δ_{m} > B \\ 0 & e l s e \\ 1 & δ_{m} < - B \end{array}, γ_{m 0} = {\begin{array}{c} 1 & | δ_{m} | > B \\ 0 & e l s e \end{array}

(7)

δ_{m} = [(x_{p} - x_{g}) \sin (α) + (y_{p} - y_{g}) \cos (α) + (r_{b p} θ_{p} + r_{b g} θ_{g})] - e (t)

(8)

f^{m} = (c_{γ}, s_{γ}, 0,0,0, ζ R_{b p}, - c_{γ}, - s_{γ}, 0,0,0, ζ R_{b g})

(9)

c_{γ} = \cos γ; s_{γ} = \sin γ

(10)

here,

r_{b j} (j = p, g)

is the base radius of gear

j

, and

a_{j} (a = x, y, z, j = p, g)

θ_{a j}

are displacements of the gear.

m_{i} (i = p, g)

is the mass of the gear rotor,

J_{D i} (i = p, g)

is the diameter rotational inertia, and

J_{i} (i = p, g)

is the pole rotational inertia.

k_{m} (t)

is the time-varying meshing stiffness, and

c_{m}

is the meshing damping. B is half of the backlash.

e (t)

is the static transmission error.

2.2. Flexible shaft element modeling

In this paper, the Timoshenko beam element is used to establish the flexible shaft. As shown in Figure 3. A beam element consists of two nodes, each node has 6° of freedom. Then, the nodal displacement vector is given by

q_{s} = {[x_{i}, y_{i}, z_{i}, θ_{x i}, θ_{y i}, θ_{z i}, x_{(i + 1)}, y_{(i + 1)}, z_{(i + 1)}, θ_{x (i + 1)}, θ_{y (i + 1)}, θ_{z (i + 1)}]}^{T}

(11)

The motion equation in matrix form can be obtained as

M_{s} q_{s}^{″} + (C_{s} + Ω G_{s}) q_{s}^{'} + K_{s} q_{s} = 0

(12)

Here

M_{s}, C_{s}, G_{s}, K_{s}

are the mass matrix, damping matrix, gyroscopic matrix, and stiffness matrix, respectively. The expressions can be found in Appendix A of Ref. (Chen et al., 2015).

2.3. High-speed deep groove ball bearing modeling

Bearings are usually simplified as a spring-damper system and modeled as a stiffness matrix and damp matrix (Lim and Singh, 1990). It is a critical reason that the gear dynamic model established loses the ability to calculate the internal load of bearings. Therefore, in this paper, a nonlinear bearing force model is used instead of the static bearing stiffness model. Figure 4 displays the model of the high-speed deep groove ball bearing. The outer ring is fixed while the inner ring rotates with the shaft. Assuming the speed of the shaft is $Ω$ , without considering the case of rolling body slippage, the rolling element rotational speed is

Ω_{g} = \frac{Ω}{2} (1 - \frac{D}{d_{m}})

(13)

where

D

is the rolling body diameter,

d_{m}

is the bearing pitch circle diameter. The position of the jth rolling element with time can be expressed as

ψ_{j} (t) = \frac{2 π Ω_{g}}{60} t + \frac{2 π}{Z} (j - 1); (j = 1,2, . . ., Z)

(14)

Here,

Z

is the number of rolling elements.

As shown in Figure 5, load analysis is utilized to account for the effects of centrifugal force. After including centrifugal force, the equilibrium equation $Q_{i j} = Q_{o j}$ is destroyed by centrifugal force $F_{c j}$ . The new equilibrium equation can be written as (Hartis, 2006)

Q_{i j} + F_{c j} - Q_{o j} = 0

(15)

The load equilibrium equation of the rolling body in the non-load area is

Q_{i j} = 0; Q_{o j} = F_{c j}

(16)

where

Q_{o j} = {\begin{array}{c} K_{o} δ_{o j}^{1.5} & δ_{o j} > 0 \\ 0 & e l s e \end{array}

(17)

Q_{i j} = {\begin{array}{c} K_{i} δ_{i j}^{1.5} & δ_{i j} > 0 \\ 0 & e l s e \end{array}

(18)

F_{c j} = \frac{ρ π D^{3} ω_{g}^{2} d_{m}}{12}

(19)

Here,

ρ

is the material density.

K_{o}

and

K_{i}

are the contact stiffness coefficients.

δ_{o j}

and

δ_{i j}

denotes the contact deformation.

The total radial deformation δ_rj of the jth rolling body is

δ_{r j} = δ_{o j} + δ_{i j}

(20)

As shown in Figure 4, the total radial deformation of the jth rolling body can also be expressed in terms of $x (t)$ and $y (t)$

δ_{r j} = x (t) \cos (ψ_{j} (t)) + y (t) \sin (ψ_{j} (t)) - 0.5 P d

(21)

Here,

P d

is the radial clearance.

x (t)

and

y (t)

are expressed as the difference between the displacement of the inner and outer rings of the bearing, respectively, that is

\begin{array}{l} x (t) = x_{s} (t) - x_{h} (t) \\ y (t) = y_{s} (t) - y_{h} (t) \end{array}

(22)

where

x_{s}, y_{s}

is the nodal displacement of the connecting bearing on the support shaft.

x_{h}, y_{h}

is the nodal displacement of the connecting bearing on the housing.

The equilibrium equation on the inner ring of the bearing is

F_{x i} = \sum_{j = 1}^{Z} Q_{i j} \cos (ψ_{j} (t))

(23)

F_{y i} = \sum_{j = 1}^{Z} Q_{i j} \sin (ψ_{j} (t))

(24)

and the equilibrium equation on the outer ring of the bearing is

F_{x o} = \sum_{j = 1}^{Z} Q_{o j} \cos (ψ_{j} (t))

(25)

F_{y o} = \sum_{j = 1}^{Z} Q_{o j} \sin (ψ_{j} (t))

(26)

2.4. Housing element modeling

Researchers commonly use the finite element method to build the housing model (Guo et al., 2014; Liu and Yuan, 2020; Lu et al., 2020; Xu et al., 2019). However, the full finite element model contains a large number of degrees of freedom. And it is hard to be applied to the gear dynamic model directly. Abbes et al. (2007) utilized the modal synthesis method for reducing the degree of freedom, which vastly promotes the application of the finite element method in the modeling of housing.

As shown in Figure 6, an established reference node in the center of the bearing seat hole connects to the housing node through a rigid beam element. Subsequently, the reference node will be connected to the node of the flexible shaft through the bearing element. The detailed introduction can be found in Ref (Abbes et al., 2007; Kong et al., 2020)

2.5. Equation of the gearbox system

According to the modeling approach in sections 2.1 to 2.4, the total differential equation of motion of the gearbox system can be expressed as

M q^{″} + (C + Ω G) q^{'} + K f (q) = F (t)

(27)

In the formula, $M$ , $C$ , $G$ , and $K$ are the overall mass matrix, the damping matrix, the gyroscopic matrix, and the stiffness matrix of the system, respectively. Proportional damping is used in the differential equation of motion. $f (q)$ is the nonlinear displacement function due to gear backlash. $F (t)$ is the vibration excitation which includes external torque excitation, internal excitation and nonlinear bearing force.

3. Internal load and fatigue life calculation method

In a gear system, the bearing load is time-varying. The nonlinear bearing load can be written as the function of displacement variables, referring to equations (23)–(26). Thus, the displacement can be input directly to acquire the nonlinear bearing load. Figure 7 displays the specific analysis process. Firstly, the dynamic model of the gear system is established. After giving the initial value of the displacement vector, the $N e w m a r k - β$ method is employed to solve the motion equation (27). The nodal displacements $x_{s}, y_{s}, x_{h}, y_{h}$ are introduced into the high-speed deep groove ball bearing model to obtain the nonlinear bearing force, which is brought into the gear system dynamic model to calculate the displacement at the next moment. A stable dynamic response is obtained after a certain number of iterations. The bearing internal load is saved to facilitate the calculation of the bearing fatigue life.

According to Lundberg and Palmgren's theory [6], the fatigue life of the inner and outer ring raceway contact is estimated according to the following equations, respectively

L_{i} = {(\frac{Q_{c i}}{Q_{i}})}^{3}

(28)

L_{o} = {(\frac{Q_{c o}}{Q_{o}})}^{3}

(29)

Q_{c}

is the basic dynamic capacity at a raceway contact, and

Q

is the equivalent rolling element load acting on the raceway contact. The subscripts

i

and

o

refer the inner and outer ring raceway, respectively. Their expression are

Q_{c i} = 93.2 {(\frac{r_{i}}{r_{i} - \frac{D}{2}})}^{0.41} \frac{{(1 - γ)}^{1.39}}{{(1 + γ)}^{\frac{1}{3}}} γ^{0.3} D^{1.8} Z^{- \frac{1}{3}}

(30)

Q_{i} = {(\frac{1}{Z} \sum_{j = 1}^{j = Z} Q_{j}^{3})}^{\frac{1}{3}}

(31)

Q_{c o} = 93.2 {(\frac{r_{o}}{r_{o} - \frac{D}{2}})}^{0.41} \frac{{(1 + γ)}^{1.39}}{{(1 - γ)}^{\frac{1}{3}}} γ^{0.3} D^{1.8} Z^{- \frac{1}{3}}

(32)

Q_{o} = {(\frac{1}{Z} \sum_{j = 1}^{j = Z} Q_{j}^{\frac{10}{3}})}^{0.3}

(33)

Here,

γ = \frac{D}{d_{m}}

r

is the radius of curvature of the raceway,

D

is the rolling element diameter,

d_{m}

is the bearing pitch diameter,

Z

is the number of rolling elements.

Q_{j}

is the load of the j-th rolling body.

The total bearing life is

L = {(L_{i}^{- \frac{10}{9}} + L_{o}^{- \frac{10}{9}})}^{- 0.9}

(34)

During operation, the rolling element load $Q_{j}$ varies with time. To obtain the fatigue life of this situation, the linear cumulative damage theory is used to consider the effects of time-varying load on fatigue life. The rule can be written as

\sum_{i = 1}^{i = n} \frac{N_{i}}{L_{i}} = 1

(35)

where

N_{i}

represents the number of revolutions under condition

i

and

L_{i}

is the fatigue life under condition

i

n

represents the number of conditions.

The average equivalent rolling element load $Q_{m}$ should be obtained primarily.

Q_{m} = {(\frac{\sum_{k = 1}^{k = n} Q_{k}^{3} N_{k}}{N})}^{\frac{1}{3}}

(36)

where

N_{k}

is the number of revolutions under condition

k

. N is the number of revolutions in total.

Q_{m}

can be written as a function of time

Q_{m} = {(\frac{\sum_{k = 1}^{k = n} Q_{k}^{3} d t}{T})}^{\frac{1}{3}}

(37)

where

T

is the time period

d t

denotes time interval.

Q_{k}

is the equivalent rolling element load at a given moment

t

written as

Q_{k i} = {(\frac{1}{Z} \sum_{j = 1}^{j = Z} Q_{i j} {(t)}^{3})}^{\frac{1}{3}}

(38)

Q_{k o} = {(\frac{1}{Z} \sum_{j = 1}^{j = Z} Q_{o j} {(t)}^{\frac{10}{3}})}^{0.3}

(39)

Then, the bearing life can be written as

L = {(\frac{Q_{c}}{Q_{m}})}^{3}

(40)

4. Results and discussions

Figure 8 shows the finite element model of the gearbox system. The element type of the housing finite element model is the second-order tetrahedral element, the number of nodes is 4031, and elements are 1937. The flexible shaft finite element model adopts the Timoshenko beam element. The number of nodes of the pinion shaft is 15, and the gear shaft is 36.

The time-varying mesh stiffness and static transmission error curves of the gear pair in the simulation calculation are shown in Figure 9. The time-varying mesh stiffness was calculated by the finite element method. The calculation process can be deduced in our previous work (Hu et al., 2016). And the static transmission error obtained can be found in reference (Li et al., 2019).

4.1. Dynamic analysis of gearbox system and experimental verification

Firstly, the proposed dynamic model is analyzed and verified. Bearing dynamic loads are directly related to bearing life, so this paper analyzes the dynamic characteristics of bearing forces. Figure 10 shows the variation of bearing force with speed. It can be observed that three main resonant speeds appear accompanied by three harmonic resonant speeds. At these speeds, the dynamic bearing forces show large fluctuations which may lead to reduced bearing life. This also illustrates the importance of gear meshing in the analysis of bearing life.

For the validity of the subsequent analysis, the dynamic model proposed in this paper is experimentally verified. Vibration testing on the housing is conveniently performed and can be used to validate the model in this paper. The acceleration sensor is installed at the top position of the housing as shown in Figure 11. The sampling frequency is set to 10 kHz. The test range has a rotational speed of 500–5000 Rpm to ensure safety. The rotational speed varies with an increase of 50 Rpm. The acceleration values are recorded under steady state conditions. The damping values of the housing are derived by repeated comparisons with the experiments. For comparison, the vibration acceleration values at the same locations are extracted in the proposed dynamic model. The experimental and theoretical root-mean-square values of the vibration acceleration in the X-direction as a function of speed are shown in Figure 11. The trend shows that the measured values of acceleration are in good agreement with the predicted values.

4.2. Internal load analysis

The fatigue life of bearings is related to the maximum load of the rolling body (Harris, 2006). Therefore, the maximum load of the rolling element is studied. For the convenience of description, the maximum contact force between the rolling body and inner raceway is labeled as Qi-max. And the maximum contact force between the rolling body and outer raceway is marked as Qo-max. The variation of Qi-max and Qo-max with rotational speed is shown in Figure 12. It can be seen that the gear engagement and centrifugal force have a great influence on the load of the rolling body. Multiple peaks appear in the curves of Qi-max and Qo-max with rotational speed when the gear engagement is considered. The rotational speed corresponding to the wave peaks is the resonant speed of the gearbox system. In addition, the amplitudes of Qi-max and Qo-max are equally larger than the results of the static bearing model even when operating at non-resonant speeds. These phenomena illustrate the importance of dynamic bearing loads due to the gear meshing behavior in the analysis of bearing internal load. On the other hand, the greater centrifugal force will change the internal load distribution, referring to equation (15). Thus, the amplitude of Qi-max and Qo-max increases with the increase of speed. The graph allows us to visually determine whether to focus on the effect of gear engagement or centrifugal force at a certain rotational speed, or whether to consider the influence of both gear meshing and centrifugal force. For example, it is mainly the gear meshing that affects the amplitude of Qi-max with the rotational speed between 500 rpm∼17,450 rpm. It can also be noticed that the gear meshing affects Qi-max and Qo-max to the same extent. And the effect of centrifugal force on Qo-max is much greater than the effect on Qi-max.

Three rotational speeds are selected to further analyze the time-varying load of rolling element. The contact force between the rolling element and the inner race is $Q_{i}$ , and the contact force with the outer race is $Q_{o}$ . The time-varying load of $Q_{i}$ and $Q_{o}$ at 500 rpm, 11,000 rpm, and 20,000 rpm are given in Figures 13–15, respectively. Three speeds represent three different operating conditions. 500 rpm is the low-speed condition, 11,000 rpm is the resonant speed condition, and 20,000 rpm is the high-speed condition. Overall, the gear meshing behavior affects the internal load of the bearing at any rotational speed, while the centrifugal force plays a key role at higher rotational speeds.

4.3. Fatigue life analysis

After obtaining the internal load, the bearing fatigue life can be calculated, and the specific calculation method is given in Section 3. Figure 16 shows the comparison of bearing fatigue life for four cases to indicate the importance of the gear engagement and centrifugal force.

(i) Calculate the bearing fatigue life based on the static bearing model. As shown in “Life-static” in Figure 16, the fatigue life will not change with speed because of the non-consideration of the effects of gear meshing and centrifugal force.

(ii) Calculate the bearing fatigue life based on a quasi-dynamic bearing model. As shown in “Life-Fc” in Figure 16, the bearing model only considers the effect of centrifugal force and ignores gear meshing. The bearing life gradually decreases as the speed rises. At 20,000 rpm, the error reaches 35.5% compared with the life estimated by the static bearing model.

(iii) Calculate the bearing fatigue life based on the gearbox system dynamic model while neglecting centrifugal force. As shown in “Life-gear engagement” in Figure 16, the effect of gear meshing on bearing fatigue life is different at different speeds. It is noteworthy that the bearing life decreases sharply at the resonant rotational speed. The reason is the presence of large fluctuations in bearing rolling element loads (as shown in Figure 14), resulting in reduced bearing life. At high speed, gear meshing has little effect on bearing life. Overall, the bearing load fluctuation due to the gear meshing behavior will reduce the bearing life.

(iv) Calculate the bearing fatigue life based on the gearbox system dynamic model, taking into account the effects of gear engagement and centrifugal force. As shown in “Life-gear engagement-Fc” in Figure 16, the trend of bearing fatigue life with speed is similar to the superposition of cases (ii) and (iii). However, from the results at high speed, the bearing fatigue life is lower than the superimposed results of cases (ii) and (iii). In the light of the above analysis, the effects of both centrifugal force and gear engagement should be considered when evaluating the bearing life at high speed.

5. Conclusions

This paper studies the bearing internal dynamic load and fatigue life in the high-speed gear system. A nonlinear model of high-speed deep groove ball bearing is established and introduced into the finite element model of the gearbox system. The gearbox dynamic model has the ability to calculate the bearing internal dynamic load. And the paper gives the solution method of the internal bearing load and bearing fatigue life. The effects of gear engagement and centrifugal force on the internal load and fatigue life are investigated. The main conclusions are as follows.

1. The dynamic rolling element load due to gear meshing behavior is significant for the analysis of fatigue life. The rolling element load amplitude at arbitrary speeds is higher than the results acquired by the static bearing model. In the resonance region, the rolling element load amplitude increases significantly, leading to a sharp decrease in the bearing fatigue life. In the non-resonant region, the bearing life is lower than those calculated by the static model after considering gear meshing.

2. The effects of centrifugal force on the bearing internal load and life depend mainly on rotational speeds. At the lower rotational speed, the centrifugal force of the rolling element is small and has little effect on the internal load and life. As the rotational speed rises, the higher centrifugal force changes the internal load distribution, and the rolling element load amplitude increases with it. At 20,000 rpm, the maximum contact force between the rolling body and outer raceway increases by 99.52%, and the bearing life decreases by 35.5%.

In summary, the approach and dynamic model proposed in this paper effectively analyze the internal dynamic load of the bearing and its fatigue life in the high-speed gear transmission system. It has significance for guiding the service performance evaluation and structural optimization design of high-speed deep groove ball bearings. It also provides a new method for the bearing life evaluation in high-speed gear transmission systems.

Footnotes

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: This work is supported by the National Natural Science Foundation of China (NSFC) through Grant No.52005515 and the National Key R&D Program of China through Grant No. 2020YFB2008200. The authors also gratefully acknowledge the supports of the Project of State Key Laboratory of High-Performance Complex Manufacturing, Central South University, through Grant No. ZZYJKT2021-06, and the Hunan Provincial Natural Science Foundation of China No. 2021JJ40740.

ORCID iDs

Zehua Hu

Siyu Chen

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