An Influence Analysis of the Bearing Waviness on the Vibrations of a Flexible Gear

Abstract

Roller bearing manufacturing errors have been proven to be critical factors affecting the vibrations of gear systems. Waviness is one main form of manufacturing error affecting the operational performance and life of bearings. However, most previous studies did not completely incorporate the effects of the uneven bearing waviness on the flexible gear system vibrations. To characterize the contribution of the uneven bearing waviness on the vibrations of the gear system, a gear transmission system dynamics model considering shaft flexibility was established. The evenness sinusoidal waviness model (SWM) and uneven sinusoidal waviness model considering the time-varying contact (SWMS) were compared. The influences of the time-varying gear meshing stiffness excitations and flexibilities of shafts on the vibrations of the gear system were considered. A dynamic model was established, and the vibrations of the flexible gear system with the SWM and SWMS were compared. The vibrations induced by different amplitudes and orders of bearing waviness were analyzed. Note that the waviness of the bearing has a great influence on the system vibrations. The vibrations of the flexible gear system intensified with the increase in the bearing waviness order and amplitude. The vibrations from the gear system with the SWMS were bigger than those of the SWM. This paper introduces an alternative dynamic modeling model enabling the vibration analysis of the flexible gear system with evenness and uneven bearing waviness.

1. Introduction

As an important part, roller bearings are widely used in different industrial gear systems. Waviness should exist in the roller bearings. The vibration characteristics of the gear system with bearing waviness should provide critical insights for the condition monitoring and vibration control of practical applications.

The bearing and gear systems’ vibrations with waviness errors have been widely studied by previous works. Cao et al. [1] established a shaft–bearing-system dynamic model, ignoring shaft flexibility while incorporating the influences of bearing waviness and nonlinear stiffness. Bai et al. [2] studied the influence of the waviness on the cage speed based on the established shaft–bearing dynamic model. Choudhury et al. [3] proposed two simulation approaches for the solution of vibrations caused by the bearing waviness excitations. Liu et al. [4] proposed a dynamic modeling method of the ball bearing raceway waviness. Liu et al. [5] improved the time-dependent excitation model due to ball bearing raceway waviness. Xi et al. [6,7] developed a new vibration modeling method on the basis of the Hertzian contact theory. They studied the time and frequency vibration characteristics of a bearing spindle system incorporated into waviness. Shi et al. [8] presented a dynamic modeling method of a planetary gear set, where the waviness errors are incorporated into the needle roller bearing. Yang et al. [9] developed a rigid herringbone gear–bearing-system dynamic model by incorporating the influences of bearing waviness. Dai et al. [10] studied the rigid gear shaft–bearing system vibrations with local faults by proposing a dynamic model. Dikmen et al. [11] proposed a flexible shaft-system dynamic model based on finite element (FE) theory. They studied the change tendency of the system’s natural frequencies due to bearing stiffness variation. Neriyal et al. [12] further investigated the bending-torsional vibration of a gear system by using the FE dynamic modeling method. Liu et al. [13] also developed an FE model for studying the vibrations of a flexible rotor system. Yang et al. [14] incorporated the effects of the nonlinear support ball bearing stiffness. They refined the dynamic model of a gear set. Peng et al. [15] studied the load-sharing characteristic of the gear system under different space motions. Li et al. [16,17] proposed an improved dynamic model of a spline gear–bearing–shaft system. They incorporated the effects of the internal spline filet-foundation deflection on the meshing stiffness. Šafář et al. [18] investigated the influences of the surface waviness on the bearing vibrations by a comprehensive prediction model, which combines a fundamental trend expressed by a power function with periodic oscillations. Ebrahimi et al. [19] studied the tribo-dynamic effects introduced by surface waviness of an elastohydrodynamic lubrication line contact. Venner et al. [20] studied the deformation of bearing waviness under starvation lubrication conditions. Jurko et al. [21] investigated the effects of cone roller waviness on the bearing internal resistance. In the above literature, most recent works focused on the rigid gear–bearing system with and without local faults and waviness. Some works also studied the flexible gear–bearing system vibrations without incorporating the waviness. However, few works investigated the vibrations of flexible gear–bearings with uneven bearing waviness.

To overcome this gap, the influences of the bearing waviness on a flexible shaft–gear–bearing system vibration were studied. By utilizing the FE theory, a flexible shaft–gear–bearing system considering the bearing waviness was established. The bearing waviness was considered as the uneven sinusoidal waviness model considering the time-varying contact (SWMS). The gear system vibrations with the evenness sinusoidal waviness model (SWM) and SWNS were analyzed. The Newmark-β method was used for numerical integration of the governing dynamic equations. The acceleration waveforms and spectra results were compared, and the effectiveness of the bearing waviness amplitude and order on the flexible shaft–gear–bearing system vibrations was discussed. This study may provide a new approach and theoretical foundation for the vibration analysis of a flexible shaft–gear–bearing system incorporating the bearing waviness.

2. Dynamic Model of the Bearing with Uneven Waviness

The desirable smooth raceway surface should be a wavey one, as waviness errors exist on the bearing raceway surfaces. The raceway surface’s curvature radius is time-varying with the waviness dimension. Here, the influence of the surface roughness was not considered. The waviness of the bearing’s raceway was approximately regarded as the sinusoidal function waveforms, as given in Figure 1.

For the inner or outer raceway with waviness error, the raceway surface can be expressed as a continuous sinusoidal function. When the jth roller contacts the relative raceway, the waviness cases p_ij or p_oj with an amplitude A_ij or A_oj were as follows [3]:

$$p_{ij}={\displaystyle \sum _{i=1}^{l=WO}{A}_{il}\cos \left[-l\left({\omega }_i-{\omega }_c\right)t+2\pi l\left(j-1\right)/N_b+{\alpha }_{il}\right]}$$

(1)

$$p_{oj}={\displaystyle \sum _{i=1}^{l=WO}{A}_{ol}\cos \left[-l\left({\omega }_o-{\omega }_c\right)t+2\pi l\left(j-1\right)/N_b+{\alpha }_{ol}\right]}$$

(2)

where the subscripts i and o represented the inner and outer raceway; α_il and α_ol were the initial phase angles of radial waviness for the two raceways; β_il and β_ol were the initial axial ripple angles for two raceways; N_b was the number of bearing rollers; and ω_c was the cage rotation frequency, which was as follows:

$${\omega }_c=0.5\left[{\omega }_i\left(1-{\displaystyle \frac{d\cos \alpha }{d_m}}\right)+{\omega }_o\left(1+{\displaystyle \frac{d\cos \alpha }{d_m}}\right)\right]$$

(3)

In the past works, the bearing waviness was always considered as the SWM. In practice, the bearing waviness amplitude was inconstant, and the contact stiffness of the raceway and rollers was time-varying accordingly. In this paper, the SWMS was proposed, which incorporates both the influences of the time-variant contact stiffness and time-variant displacement. In Figure 2, the curves of waviness between the SWM and SWMS were compared.

At the arbitrary location (N_w) of SWMS in Figure 2, the waviness was as follows:

$$A_w={\displaystyle \sum _{i=1}^{l_w}\left(A_0+A_{wi}\sin \left(\frac{2\pi N_w}{n_{wi}}\right)\right)}$$

(4)

where A₀ represented the initial waviness; A_wi represented the amplitude of ith waviness, i = 1, 2, 3... l_w; l_w was the order of waviness; and the wavelength of ith waviness n_wi was as follows:

$$n_{wi}=\left\{\begin{array}{cc}{\phi }_{wi}{R}_{in}\hfill & \mathrm{inner}\mathrm{race}\mathrm{waviness}\mathrm{error}\\ {\phi }_{wi}{R}_{out}\hfill & \mathrm{outer}\mathrm{race}\mathrm{waviness}\mathrm{error}\end{array}\right.$$

(5)

where R_in and R_out represented the inner and outer raceway radii and φ_wi was the angle of the ith waviness peak. The sum of all the waviness peak angles in the same raceway was 2π. For the jth roller, N_w was as follows:

$$N_w=\left\{\begin{array}{cc}{R}_i({\displaystyle \frac{2\pi }{N_b}}\left(j-1\right)+{\omega }_{\mathrm{ca}}t+{\theta }_{in})\hfill & \mathrm{inner}\mathrm{race}\mathrm{waviness}\mathrm{error}\\ R_o({\displaystyle \frac{2\pi }{N_b}}\left(j-1\right)+({\omega }_{ca}-{\omega }_{sh})t+{\theta }_{in})\hfill & \mathrm{outer}\mathrm{race}\mathrm{waviness}\mathrm{error}\end{array}\right.$$

(6)

where j denoted the jth roller; θ_in was the initial angle of the roller; ω_ca was the cage angular velocity; and ω_sh was the angular velocity of the inner raceway. For the N_wth wave, the curvature of waviness was as follows:

$${\rho }_w={\displaystyle \frac{\left|A_{wi}{\left(\frac{2\pi }{n_{wi}}\right)}^2\sin \left(\frac{2\pi N_w}{n_{wi}}\right)\right|}{{\left(1+A_{wi}^2{\left(\frac{2\pi }{n_{wi}}\right)}^2{\cos }^2\left(\frac{2\pi N_w}{n_{wi}}\right)\right)}^{1.5}}}$$

(7)

When the waviness of the inner raceway was considered, the main curvatures of the roller and inner raceway in contact were as follows:

$${{\displaystyle \rho }}_{a1}={\displaystyle \frac{2}{d}},\hspace{1em}{{\displaystyle \rho }}_{a2}={\displaystyle \frac{2}{d}},\hspace{1em}{{\displaystyle \rho }}_{b1w}={{\displaystyle \rho }}_w,{{\displaystyle \rho }}_{b2}=-{\displaystyle \frac{1}{r_i}}$$

(8)

When the waviness of the outer raceway was considered, the main curvatures of the roller and inner raceway in contact were as follows:

$${{\displaystyle \rho }}_{a1}={\displaystyle \frac{2}{d}},\hspace{1em}{{\displaystyle \rho }}_{a2}={\displaystyle \frac{2}{d}},\hspace{1em}{{\displaystyle \rho }}_{b1w}=-{{\displaystyle \rho }}_w,{{\displaystyle \rho }}_{b2}=-{\displaystyle \frac{1}{r_o}}$$

(9)

$${\displaystyle \sum \rho }={\rho }_{a1}+{\rho }_{b1}+{\rho }_{a2}+{\rho }_{b2}$$

(10)

$${\displaystyle \sum {{\displaystyle \rho }}_1}={{\displaystyle \rho }}_{a1}+{{\displaystyle \rho }}_{b1},{\displaystyle \sum {{\displaystyle \rho }}_2}={{\displaystyle \rho }}_{a2}+{{\displaystyle \rho }}_{b2}$$

(11)

The stiffness of the roller-inner raceway contact or roller-outer raceway contact is as follows:

$$K_{in}\begin{array}{c}\end{array}or\begin{array}{c}\end{array}{K}_{out}={\left({\displaystyle \frac{{\pi }^2{c}^2{\overline{E}}^{2}U}{4.5T^3{\displaystyle \sum \rho }}}\right)}^{1/2}$$

(12)

where $\overline{E}$ is the equivalent elastic modulus of roller and raceway; c, T, and U are three parameters of the ellipse, which are as follows [22]:

$$c=1.0339\ln {\left({\displaystyle \frac{{\displaystyle \sum {{\displaystyle \rho }}_1}}{{\displaystyle \sum {{\displaystyle \rho }}_2}}}\right)}^{0.6360}$$

(13)

$$T=1.5277+0.6023\ln \left({\displaystyle \frac{{\displaystyle \sum {{\displaystyle \rho }}_1}}{{\displaystyle \sum {{\displaystyle \rho }}_2}}}\right)$$

(14)

$$U=1.0003+0.5968\left({\displaystyle \frac{{\displaystyle \sum {{\displaystyle \rho }}_2}}{{\displaystyle \sum {{\displaystyle \rho }}_1}}}\right)$$

(15)

The global contact stiffness between the roller and raceways was as follows:

$$K={\displaystyle \frac{1}{{\left[\frac{1}{{{\displaystyle K}}_{in}^n}+\frac{1}{{{\displaystyle K}}_{out}^n}\right]}^n}}$$

(16)

where the load-deformation index n of ball bearings was 1.5.

In the SWMS, for the two raceways, l_w was 12, 18, and 24. The maximum values of A_ws were 8 μm, 10 μm, and 12 μm. Figure 3 gives the waveform of contact stiffness from the SWM and SWMS, which should vary periodically with the contact angle. In the SWM, the contact stiffness was constant. In the SWMS, the contact stiffness between each roller and raceway changes periodically with the rotation angle, and the amplitude of contact stiffness increases with the increment of the waviness wave number. Compared with the SWM, the SWMS can more accurately describe the time-variant contact stiffness of the bearing when there is unevenly distributed waviness on the raceways.

3. Dynamic Model of a Flexible Gear System with Waviness

To reveal the mechanism of bearing waviness on the flexible gear system vibrations, a gear model with six degrees of freedom was proposed in Figure 4. The shaft flexibility was considered. The impact of waviness on the gear system was simulated by the excitation force acting on the bearings. Based on this model, the flexible gear system vibrations with the SWM and SWMS were analyzed.

The dynamic equation matrix of the flexible gear system model was as follows [23,24]:

$$M\ddot{v}(t)+C\dot{v}(t)+Kv(t)=P_0$$

(17)

For the discretization procedure of the flexible gear system, each shaft was divided into several segments with multiple nodes [25]. The gears and bearings’ geometrical centers are located on the corresponding nodes of the flexible shaft. Each shaft segment element contains two nodes positioned at both ends, with adjacent shaft segment elements connected sequentially through shared nodes.

In Formula (17), P₀ denoted the external and internal excitations applied to the system, and M, C, and K were global mass, damping, and stiffness matrices, represented as follows:

$$M=\left[\begin{array}{ccc}{M}_1& 0& 0\\ 0& M_2& 0\\ 0& 0& M_3\end{array}\right]$$

(18)

$$K=\left[\begin{array}{ccc}{K}_1+K_{\mathrm{m}11}& K_{\mathrm{m}12}& 0\\ K_{\mathrm{m}21}& K_2+K_{\mathrm{m}22}+K_{\mathrm{m}33}& K_{\mathrm{m}34}\\ 0& K_{\mathrm{m}43}& K_3+K_{\mathrm{m}44}\end{array}\right]$$

(19)

$$C=\left[\begin{array}{ccc}{C}_1+C_{\mathrm{m}11}& C_{\mathrm{m}12}& 0\\ C_{\mathrm{m}21}& C_2+C_{\mathrm{m}22}+C_{\mathrm{m}33}& C_{\mathrm{m}34}\\ 0& C_{\mathrm{m}43}& C_3+C_{\mathrm{m}44}\end{array}\right]$$

(20)

where M₁, M₂, and M₃ were the mass matrices of the three shafts; K₁, K₂, and K₃ were the stiffness matrices of the three shafts; and K₁, K₂, and K₃ were the damping matrices of the three shafts. Those matrices can be acquired by Equations (21)–(23), respectively, which are as follows:

$$M_{\mathrm{i}}={\left[\begin{array}{cccccccc}{M}_{\mathrm{s}11}^{\mathrm{i}-1}& M_{\mathrm{s}12}^{\mathrm{i}-1}& & & & & & \\ M_{\mathrm{s}21}^{\mathrm{i}-1}& M_{\mathrm{s}22}^{\mathrm{i}-1}+M_{\mathrm{s}11}^{\mathrm{i}-2}& M_{\mathrm{s}12}^{\mathrm{i}-2}& & & & & \\ & M_{\mathrm{s}21}^{\mathrm{i}-2}& M_{\mathrm{s}22}^{\mathrm{i}-2}+M_{\mathrm{s}11}^{\mathrm{i}-3}& M_{\mathrm{s}12}^{\mathrm{i}-3}& & & & \\ & & M_{\mathrm{s}21}^{\mathrm{i}-3}& M_{\mathrm{s}22}^{\mathrm{i}-3}+M_{\mathrm{s}11}^{\mathrm{i}-4}& M_{\mathrm{s}12}^{\mathrm{i}-4}& & & \\ & & & M_{\mathrm{s}21}^{\mathrm{i}-4}& \dots & \dots & & \\ & & & & \dots & \dots & M_{\mathrm{s}12}^{\mathrm{i}-(N_{\mathrm{i}}-2)}& \\ & & & & & M_{\mathrm{s}21}^{\mathrm{i}-(N_{\mathrm{i}}-2)}& M_{\mathrm{s}22}^{\mathrm{i}-(N_{\mathrm{i}}-2)}+M_{\mathrm{s}11}^{\mathrm{i}-(N_{\mathrm{i}}-1)}& M_{\mathrm{s}12}^{\mathrm{i}-(N_{\mathrm{i}}-1)}\\ & & & & & & M_{\mathrm{s}21}^{1\mathrm{i}-(N_{\mathrm{i}}-1)}& M_{\mathrm{s}22}^{\mathrm{i}-(N_{\mathrm{i}}-1)}\end{array}\right]}_{\mathrm{i}=1,2,3}$$

(21)

$$K_{\mathrm{i}}={\left[\begin{array}{cccccccc}{K}_{\mathrm{s}11}^{\mathrm{i}-1}& K_{\mathrm{s}12}^{\mathrm{i}-1}& & & & & & \\ K_{\mathrm{s}21}^{\mathrm{i}-1}& K_{\mathrm{s}22}^{\mathrm{i}-1}+K_{\mathrm{s}11}^{\mathrm{i}-2}& K_{\mathrm{s}12}^{\mathrm{i}-2}& & & & & \\ & K_{\mathrm{s}21}^{\mathrm{i}-2}& K_{\mathrm{s}22}^{\mathrm{i}-2}+K_{\mathrm{s}11}^{\mathrm{i}-3}& K_{\mathrm{s}12}^{\mathrm{i}-3}& & & & \\ & & K_{\mathrm{s}21}^{\mathrm{i}-3}& K_{\mathrm{s}22}^{\mathrm{i}-3}+K_{\mathrm{s}11}^{\mathrm{i}-4}& K_{\mathrm{s}12}^{\mathrm{i}-4}& & & \\ & & & K_{\mathrm{s}21}^{\mathrm{i}-4}& \dots & \dots & & \\ & & & & \dots & \dots & K_{\mathrm{s}12}^{\mathrm{i}-(N_{\mathrm{i}}-2)}& \\ & & & & & K_{\mathrm{s}21}^{\mathrm{i}-(N_{\mathrm{i}}-2)}& K_{\mathrm{s}22}^{\mathrm{i}-(N_{\mathrm{i}}-2)}+K_{\mathrm{s}11}^{\mathrm{i}-(N_{\mathrm{i}}-1)}& K_{\mathrm{s}12}^{\mathrm{i}-(N_{\mathrm{i}}-1)}\\ & & & & & & K_{\mathrm{s}21}^{\mathrm{i}-(N_{\mathrm{i}}-1)}& K_{\mathrm{s}22}^{\mathrm{i}-(N_{\mathrm{i}}-1)}\end{array}\right]}_{\mathrm{i}=1,2,3}$$

(22)

$${\mathbf{C}}_{\mathrm{i}}=a_1{M}_{\mathrm{i}}+a_2{K}_{\mathrm{i}},\mathrm{i}=1,2,3$$

(23)

K_m11, K_m12, K_m13, and K_m14 are the submatrices of the stiffness matrix of the following gear meshing unit:

$$K_{\mathrm{m}}=k_{\mathrm{m}}{V}^{\mathrm{T}}V=\left[\begin{array}{cc}{K}_{\mathrm{m}11}& K_{\mathrm{m}12}\\ K_{\mathrm{m}21}& K_{\mathrm{m}22}\end{array}\right]$$

(24)

$$C_{\mathrm{m}}=c_{\mathrm{m}}{V}^{\mathrm{T}}V=\left[\begin{array}{cc}{C}_{\mathrm{m}11}& C_{\mathrm{m}12}\\ C_{\mathrm{m}21}& C_{\mathrm{m}22}\end{array}\right]$$

(25)

where k_m is the time-varying meshing stiffness of gears. The time-varying meshing stiffness calculation method is based on Ref. [26]. V represents the projection vector of transformation from the displacements in all directions at the gear node to the meshing line direction, which can be expressed by [27,28,29].

In the modeling process, firstly, the structural dimensions and working conditions of the gear system components were defined, including the shaft dimension, gear parameters, bearing parameters, rotating speed, and load torque. Therefore, the global stiffness, mass, damping matrices, and load vector of the system were assembled according to Newton’s second law. Based on the mapping relationship between the local node number per shaft segment element and the global node number of the system, element matrix submatrices are superimposed sequentially into corresponding positions of the global matrix.

4. Results and Discussions

Here, the Newmark-β integral method was utilized to solve the dynamic equations of the gear system. By using the FFT method, the acceleration spectra from the SWM and SWMS were obtained and compared. The parameters of the gears and bearings in the gear system are given in

Table 1. Gear parameters.

Parameters	#1	#2	#3	#4
Material	40 Cr	40 Cr	40 Cr	40 Cr
Young’s modulus/E	209 Gpa	209 Gpa	209 Gpa	209 Gpa
Poisson’s ratio/v	0.30	0.30	0.30	0.30
Width/B	40 mm	40 mm	40 mm	40 mm
Modulus	4.5 mm	4.5 mm	4 mm	4 mm
Number of teeth	39	117	44	132
Pressure angle	20°	20°	20°	20°
Spiral angle	13.5°	13.5°	13.5°	13.5°
Rotation direction	right	left	Right	left
Height coefficient	1
Clearance coefficient	0.25

and

Table 2. Bearing parameters.

Parameters	Value
Inner diameter/mm	20
Outer diameter/mm	52
Width/mm	23
Roller diameter/mm	10.53
Number of rollers	6
Radial internal clearance/μm	10

. Due to the Y-direction aligning with the gear’s gravity it can effectively capture the dominant radial vibration components induced by the gear teeth interaction. Therefore, to maintain clarity and conciseness in the graphical presentation, only the relevant results of Y-direction acceleration are presented in the following sections.

Firstly, the system accelerations with different waviness orders were analyzed. Figure 5 demonstrates the comparisons of the Y-direction acceleration spectra of each bearing’s inner raceway from the gear system with the SWM and SWMS under an input speed of 3000 r/min. In Figure 5, the peak frequencies of accelerations at bearing #1 were the inner raceway roller passing frequency f_i1 and its harmonics. The comparisons of peak frequencies of the gear system with the SWMS and SWM for different bearing waviness orders were given in

Table 3. Peak frequencies of bearing #1 from the SWM and SWMS for different inner raceway waviness order cases.

lw	Peak Frequency (Hz)		Characteristic Frequency	Amplitude (m/s2)
	SWM	SWMS		SWM	SWMS	Difference
12	1551.0	1551.0	8 fi1	10.84	10.60	2.26%
14	1357.1	1357.1	7 fi1	7.26	6.88	5.52%
16	1551.0	1551.0	8 fi1	11.46	10.59	8.22%
18	2326.5	2326.5	12 fi1	11.97	11.81	1.35%
20	1938.8	1938.8	10 fi1	11.18	10.45	6.99%
22	2132.6	2132.6	11 fi1	7.59	7.62	0.39%
24	1551.0	1551.0	8 fi1	17.15	16.59	3.38%
26	2520.4	2520.4	13 fi1	7.42	7.62	2.62%

. In Figure 5, the peak frequencies were 8 f_i1 and 12 f_i1; and the difference between the peak frequencies was l_w/n times f_i1 when the waviness was established as the SWM, and the waviness order is n times the number of rollers. If the waviness order was not equal to a multiple of the number of rollers, the peak frequency was n/2 f_i1. The frequency amplitude increases with the order of inner raceway waviness. Compared with the SWM, the SWMS has more harmonics of f_i1. Therefore, the SWMS can be more accurate and reflect the authenticity of bearing waviness.

Figure 6 demonstrates the Y-direction accelerations RMS (Root Mean Square) values of bearings #1, #3, and #5 when the bearing waviness orders change from 0 to 26. When the SWMS and SWM were used to model the ripple degree of the bearing inner raceway of the input shaft, the RMS values of radial accelerations of bearings #1, #3, and #5 on the three shafts increased with the increment of ripple degree of the inner raceway. The RMS values of SWMS and SWM have similar increment trends; the radial vibration of SWMS was slightly higher than that of SWM. The results indicate that the influence of the uneven waviness on the bearing vibration can be better reflected based on the SWMS. When the waviness order was an integer multiple of the number of the bearing rollers, the RMS value was larger. Note that the order of bearing inner raceway waviness could affect the vibration level of the system. When the inner raceway waviness order was an integer multiple of the number of rollers, the vibration was most obvious.

Figure 7 demonstrates the Y-direction acceleration spectra of each bearing’s outer raceway from the SWM and SWMS under an input speed of 3000 r/min. In Figure 7, the peak frequency of accelerations at bearing #1 was the outer raceway roller passing frequency f_o1. The comparisons of peak frequencies from the SWMS and SWM with different bearing waviness orders are given in

Table 4. Peak frequencies of bearing #1 from the SWM and SWMS for different outer raceway waviness orders.

lw	Peak Frequency (Hz)		Characteristic Frequency	Amplitude (m/s2)
	SWM	SWMS		SWM	SWMS	Difference
12	1273.5	1273.5	12 fo1	3.21	3.22	0.31%
14	1485.7	1485.7	14 fo1	4.62	4.55	4.51%
16	849	849	8 fo1	5.21	5.20	0.19%
18	1273.5	1273.5	12 fo1	8.36	8.31	0.60%
20	1061.2	1061.2	10 fo1	7.84	7.77	0.90%
22	1167.4	1167.4	11 fo1	9.57	9.64	0.73%
24	1273.5	1273.5	12 fo1	12.33	12.20	1.06%
26	2520.4	2520.4	13 fi1	7.42	7.62	2.62%

. In Figure 7, the peak frequency is 12 f_o1, and the difference between the peak frequencies is l_w/n times f_o1 when the waviness is established as the SWM, and the waviness order is n times the roller number. If the waviness order was not equal to a multiple of the number of bearing rollers, the peak frequency was n/2 f_o1. The frequency amplitude increased with the order of the outer raceway waviness. Compared with the SWM, the SWMS has more harmonics of f_o1. Therefore, the SWMS was more accurate and could reflect the authenticity of bearing waviness.

Figure 8 demonstrates the Y-direction accelerations RMS values of bearings #1, #3, and #5 when the bearing outer raceway waviness orders change from 0 to 26. The RMS values of radial accelerations of bearings #1, #3, and #5 on the three shafts increased with the increment of ripple degree of the outer raceway. The RMS values of SWMS and SWM have similar increment trends; the radial vibration of SWMS was slightly higher than that of SWM. The results indicate that the influence of the uneven waviness on the bearing vibration can be better reflected based on the SWMS. When the waviness order was an integer multiple of the number of the bearing rollers, the RMS value was larger. Note that the order of bearing outer raceway waviness could affect the vibration level of the system. When the outer raceway waviness order was an integer multiple of the number of rollers, the vibration was most obvious.

5. Experimental Validation

For verification of the proposed flexible shaft–gear–bearing-system dynamic model, a test bench as depicted in Figure 9 is established. The test gearbox employed ISO accuracy class 7 gears with a module of 1.5 mm and the bearing 6304. The lubrication oil CKC220 was used. The rotating speeds were set as 2500 r/min and 3000 r/min, while a load torque of 50 N·m was applied by the brake. Allowing the gear system to operate steadily for 30 s, the acceleration time waveform of bearing #1 along gravity was acquired by the data acquisition system. The data acquisition system includes a single-axis accelerometer of sensitivity 9.91 mV/g, an LMS vibration acquisition instrument, and a computer. The sampling frequency and the sampling time of the signals were 20 kHz and 10 s, respectively. The parameters of the simulation model are consistent with those of the test bench. The time-domain waveforms of the simulation and test vibration accelerations obtained are shown in Figure 10a,c, and the spectra of the simulation and test vibration accelerations are obtained by using the FFT, and are shown in Figure 10b,d.

In Figure 10a,c, the amplitudes of the simulation and test acceleration waveforms are comparable. Moreover, the comparisons of the spectra results indicate that ball pass frequencies of the inner raceway f_i and its harmonics 3 f_i, 5 f_i, 6 f_i, 8 f_i, and 9 f_i are obvious in both spectra at a rotation speed of 2500 r/min. Meanwhile, the ball pass frequencies of the inner raceway and its harmonics 2 f_i, 3 f_i, 4 f_i, 6 f_i, and 7 f_i are obvious in both spectra at a rotation speed of 3000 r/min. The deviations of these characteristic frequencies from simulation and test results were less than 1%. Then the proposed dynamic modeling method can be verified to some extent.

6. Conclusions

In this paper a dynamic model of a flexible shaft–gear–bearing system considering the waviness of support bearings was proposed. The time-invariant and the time-varying excitations were considered. The effects of the waviness on the contact stiffness and vibrations were calculated and analyzed. The following conclusions are reached.

(1)

During the modeling process of bearing waviness, it was necessary to consider the unevenness of waviness on the raceway. The SWMS considers the time-variant contact stiffness and displacements could effectively obtain more accurate results, which should be closer to the actual results.

(2)

For the SWMS, the contact stiffness of the bearing changed periodically with the rotation angle, and its period was related to the order of waviness. The higher order could obtain a shorter period. The amplitude of contact stiffness was positively correlated with that of waviness.

(3)

Compared with the SWMS, the contact stiffness between the roller and raceway in the SWM was constant.