Nonlinear dynamic modeling and vibration analysis for early fault evolution of rolling bearings | Scientific Reports

Scientific Reports volume 14, Article number: 23687 (2024) Cite this article

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In rotating machinery, the condition of rolling bearings is paramount, directly influencing operational integrity. However, the literature on the fault evolution of rolling bearings in their nascent stages is notably limited. Addressing this gap, our study establishes an innovative nonlinear dynamic model for early fault evolution of rolling bearings based on collision impact. Firstly, considering the fault evolution characteristics, the influence of the rolling element and fault structure, the dynamic model of early fault evolution between the rolling element and the local fault is established. Secondly, according to the Hertzian contact deformation theory, a nonlinear dynamic model of rolling bearings expressed as mass-spring is established. Thirdly, the energy contribution method is used to integrate the fault evolution model and the nonlinear dynamic model of the rolling bearing. A nonlinear dynamic model of early fault evolution of the rolling bearing is proposed by using the Lagrangian equation. Comparing the simulation results of the nonlinear dynamic with the experimental results, it can be seen that the numerical model can effectively predict the evolution process and vibration characteristics of the fault evolution of rolling bearings in the early stage.

As a key component of rotating machinery, rolling bearings are widely used in the ship’s propulsion system, power generation system, sewage treatment system and so on. Rolling bearings are prone to fatigue, cracks, pitting corrosion, sliding, and other faults, which are likely to shorten the life of equipment, produce vibration and noise beyond standards, reduce the vitality of rotating equipment, and even cause significant economic losses1,2,3,4. The vibration characteristics of faulty rolling bearings at an early stage are not obvious and difficult to detect in time. Therefore, rolling bearing fault diagnosis plays a very important role in the maintenance and repair of equipment, in order to effectively monitor bearing faults, it is necessary to analyze the fault mechanism of rolling bearings. The nonlinear dynamic model of rolling bearings early fault evolution is established, by analyzing the dynamic response of the nonlinear dynamic model, the vibration response of rolling bearings in early fault evolution is revealed, which provides the theoretical basis and technical support for bearing fault monitoring and diagnosis5,6.

Many researchers have studied dynamic models of rolling bearings. For example, Niu et al7. established the dynamic model of rolling bearings and analyzed the vibration characteristics of the rolling bearing in case of local surface fault. Liu et al8,9,10. proposed an analytical model for analyzing the vibration response of rolling bearings. Based on the model, it is found that both surface dents and inner ring offsets will cause impact collisions between rolling elements and raceways. Yang et al11. established the dynamic analytical model of rolling bearings using the centralized mass method. Single and hybrid faults of rolling bearings are studied. Zheng et al12. established a nonlinear dynamic model of rolling bearings based on collision impact. The comparison shows that the nonlinear dynamic model can effectively predict the vibration characteristics of rolling bearing faults. Tian et al13. proposed a nonlinear collision model with time-varying contact stiffness/damping to describe the collision behavior of bearings, which verifies the correctness of the model through the comparison of test results and experimental results. Deng et al14. established a nonlinear dynamic model of high-speed ball bearings based on elastic fluid lubrication. The correctness of the nonlinear dynamic model is verified by experiments. Wang et al15. proposed a dynamic model of a five-degree-of-freedom rolling bearing coupled to the rotor system and studied the influence of the rotor on the bearing system. Jang et al16. established a dynamic model of faulty rolling bearings considering the contact and friction forces between each component in the rolling bearing. The stability index of the tank cage is studied, quantified and analyzed. Wang et al17. proposed an improved rolling bearing dynamic model considering bearing deformation and ball-inner raceway separation and studied the influence of appropriate preload on the use characteristics of rolling bearings. Luo et al18. studied the incentive mechanism of the inlet and outlet responses caused by REB outer ring layer cracks and proposed a contact model for rolling element layer fracture interactions, which is experimentally verified by the effectiveness of the model.

Many scholars have studied the change in the fault size of rolling bearings. Aoyu et al19. proposed a fault size pre-estimation method based on fault inlet and fault outlet vibration signals, and experiments verified the effectiveness of the proposed method. Thalji et al20. proposed a descriptive model of wear evolution over the life cycle of rolling bearings, which describes the stages and influencing factors of wear evolution. Thalji et al21. combined a rolling bearing dynamics model with a signal processing method to extract the vibration signatures during the wear of rolling bearings. Thalji et al22. established a dynamic model that can simulate the evolution of bearing rail wear over the service life of rolling bearings in a continuous manner. Kogan et al23. proposed a multi-body nonlinear dynamic model of the interaction between the rolling element and the peeling outer ring and studied the sensitivity between influencing parameters such as fault size and radial load. Liu et al24. established a dynamic model of the faulty rolling bearing. Based on the dynamic model, the contact load distribution and stiffness of rolling bearings under different fault sizes and operating conditions are studied. Li et al25. established a dynamic model of the angular contact ball bearing with the local fault in the outer ring. The results show that with the increase in running speed, the load distribution between the ball and the outer ring is much greater than the load distribution between the ball and the inner ring. Utpat et al26. established a three-dimensional finite element model of rolling bearings expressed as spring mass. The vibration amplitude and frequency characteristics of rolling bearings at different speeds, loads and fault sizes are studied. Nathan et al27. proposed an elastoplastic finite element model of rolling bearings, and the calculated results showed that the spherical density has the greatest effect on the damage at the fractured edge of the impact layer.

With the rapid development of machine learning, intelligent fault diagnosis methods with high diagnostic accuracy and good generalization have attracted more and more attention from scholars. Chen et al28. proposed an explainable artificial intelligence rolling bearing intelligent fault diagnosis algorithm. Rafiee et al29. established an intelligent transmission condition monitoring based on an artificial neural network. Vannucci et al30. proposed a monitoring algorithm using artificial intelligence to process data. However, the intelligent fault diagnosis algorithm needs enough prior data sets to identify the fault of the rolling bearing effectively, so it is necessary to establish the dynamic model of the rolling bearing to provide an effective vibration data set for the intelligent fault diagnosis algorithm.

In summary, The above articles have conducted in-depth research on the dynamic model of rolling bearings, but a large number of scholars have only studied the bearing fault at a specific time or calculated the fault vibration at multiple moments to simulate the evolution process of the fault. These methods cannot express the randomness and irreversibility of the fault evolution in detail, so it is necessary to study the dynamic model of the fault evolution of rolling bearings.

The study focuses on the evolution of rolling bearing fault dimensions more extensively than before. Firstly, the fault evolution law and characteristics of rolling bearings are analyzed, and the fault evolution model between rolling elements and fault areas is established. Secondly, according to the Hertzian contact deformation theory, the energy contribution expression of each component in rolling bearings and the energy expression of the fault evolution model are derived. Thirdly, the Lagrange equation is used to integrate the fault evolution model and the dynamic model of rolling bearings. Then, the nonlinear dynamic model of early fault evolution of rolling bearings is established.

The occurrence and change of faults in rolling bearings is a slow evolutionary process, and a typical collision impact phenomenon will be generated when the rolling elements pass through the defect area. The collision impact phenomenon caused by rolling bearing faults aggravates the instability of the motion system. With the continuous evolution of the fault, the unstable vibration of the system will become stronger and stronger. In the early stage, the vibration characteristics of rolling bearing faults are not obvious and difficult to detect in time, so the early fault evolution dynamic model between rolling elements and fault areas is established according to the evolution law and characteristics of rolling bearing faults. Figure 1 shows the evolution of faults on the outer raceway of rolling bearings. As shown in Fig. 1, there is a fault at 6 o’clock in the outer raceway of the rolling bearing. With the continuous development of faults, the sinking area of the rolling elements in the fault area is increasing, and the distance of the rolling elements moving in the fault area is also increasing. The collision impact between the rolling element and the end sidewall of the localized fault is the main vibration source of the early rolling bearing fault.

Diagram of the early fault evolution-outer race fault.

According to the contact characteristics between the rolling elements and the fault area, the dynamic model of the early fault evolution system can be divided into two parts: a three-dimensional vibration system and a collision impact system. The three-dimensional vibration system consists of an inner raceway, a rolling element and an outer raceway, which is generated as the rolling element initially contacts the beginning sidewall and disappeared as the rolling element contacts the end sidewall. The collision impact system consists of an outer raceway and a rolling element, which exists at the moment of collision between the rolling element and the end sidewall. After the rolling element leaves the fault area, it moves with a new initial value and collides with the fault area again, so as to reciprocate.

Due to the small size of the early fault, the distance between the beginning sidewall and the end sidewall is selected to express the deterioration degree of the fault. The collision force between the rolling element and the fault area will vary with \(\Delta l\). The equation of the outer raceway early fault evolution is as follows:

Where \({m_{i+r}}\), \({m_j}\)and \({m_{{\text{out}}}}\) represent the mass of the inner ring-rotor, the masses of the rolling ball and the masses of the outer ring, respectively; \(P_i(i=1,2)\)is the harmonic force; \({C_1}\), \({C_2}\)and \({K_1}\), \({K_2}\) represent the damping and stiffness, respectively; \(\alpha\)and \(\tau\)represent the angular velocity and initial phase, respectively; \({\dot {x}_{2+}}\)and \({\dot {x}_{2 - }}\) represent the instantaneous velocities of the rolling element before and after the collision, respectively; R is the collision coefficient; \(\Delta {l_1}\) and \(\Delta {l_2}\) indicate the degree of evolution of early fault, respectively; a and b are constant thresholds, respectively.

Diagram of the early fault evolution-inner race fault.

Similarly, when there is a fault in rolling bearing inner raceway (Fig. 2), the contact process between the rolling element and the fault area is similar to before. Because the inner raceway rotates with the rotating shaft, the speed of the inner raceway needs to be considered in establishing the dynamic model of the inner ring fault evolution. The equation of the inner raceway early fault evolution is as follows:

Where \({\dot {x}_{1+}}\), \({\dot {x}_{1 - }}\), \({\dot {x}_{2+}}\)and \({\dot {x}_{2 - }}\)represent the instantaneous velocities of the inner raceway and the rolling element before and after the collision, respectively.

The Newmark-β method is used to solve the equation of the early fault evolution system to obtain the acceleration response of the inner raceway and the rolling element. The vibration generated by the early fault evolution system is decomposed at the collision point, which will be used to establish a nonlinear dynamic model of the early fault evolution of rolling bearings. The movement of rolling bearings is cyclical. Therefore, the motion of the early fault evolution system is also periodic. A cycle of the early fault evolution system starts from the beginning sidewall and ends with the end sidewall. The cycle conditions are as follows:

Where \(\omega\) represent the no-measure angular velocity.

In rotating machinery, the inner raceway of the rolling bearing rotates with the shaft and the outer raceway of the rolling bearing is fixed to the housing. After loading, the mass center of the inner raceway will be offset compared to before loading, and the contact deformation will occur between the rolling element and the raceway, as shown in Fig. 3. The Lagrange equation (Eq. (6)) is used to couple the early fault evolution system with the nonlinear dynamic model of the rolling bearing. Therefore, the energy contribution of the fault evolution model and each component of the rolling bearing need to be calculated.

Where T and V are the kinetic energy and potential energy, respectively. \(\rho\) and f are vector with generalized DOF coordinates and vector with generalized contact forces, respectively.

Geometry relationship between rolling elements and raceways.

As shown in Fig. 3, simplify complex rolling bearing systems to mass-spring models, in practice, the mass center of the inner raceway and the rolling element will be shifted, \({x_{in}}\)and \({y_{in}}\) represent the offset of the inner center on the x-axis and y-axis, respectively. The mass center of the out raceway is fixed.

In rolling bearings, there is a spring connection between the rolling element and the raceway, which conforms to the Hertzian contact deformation theory. The mass center of the outer raceway of the rolling bearing is used as the reference point, and the expression equations of kinetic energy and potential energy of the rolling element are derived, respectively. The kinetic energy of a rolling element can be expressed as follows:

Where \({I_j}\) represents the moment of inertia of the jth rolling element; \({\vec {\rho }_j}\)represents the displacement vector and can be expressed as follows:

Where \({\rho _j}\) represents the distance between the rolling element center and the outer raceway center. \({\alpha _j}\) is used to describe the angular position of the j th rolling element. Under ideal conditions, the angular velocity of the rolling elements is as follows:

Where R and \({r_{ball}}\) represent the radius of the outer raceway and rolling element, respectively.

The kinetic energy of the jth rolling element and the total kinetic energy of the rolling element in the bearing can be expressed as follows:

Where z represents the number of rolling elements in the rolling bearing.

The total potential energy of rolling elements in the rolling bearings relative to the center of the outer raceway is as follows:

The inner raceway is fixed to the rotor and rotates with the rotor, relative to the mass center of the outer raceway, the inner raceway and rotor have the same offset distance (\({x_{in}}\) and \({y_{in}}\)), which is represented by vectors as follows:

According to Eq. (13), the kinetic energy formulas of the inner raceway (Eq. (14)) and the rotor (Eq. (15)) can be derived, respectively:

Where \({I_{in}}\) and \({I_{rotor}}\) is the moment of inertia of the inner raceway and rotor, respectively; The \({m_{rotor}}\) and \({m_{in}}\) are the quality of the rotor and inner raceway, respectively.

The potential energy of the inner raceway (Eq. (16)) and the rotor (Eq. (17)) relative to the center of the outer raceway is calculated as follows:

Assuming that the outer raceway is stationary, the displacement velocity \({\dot {r}_{out}}\) and angular velocity \({\dot {\phi }_{out}}\) of the outer raceway are zero. Hence, the kinetic and potential energy contributions of the outer raceway are zero.

In rolling bearings, the contact deformation between the rolling element and the raceway conforms to the Hertzian contact theory. According to the Hertzian contact theory, an expression of the contact potential energy between the rolling element and the raceway can be obtained:

Where \({K_i}\) is the contact stiffness between the rolling elements and the inner raceway, which can be obtained by Eq. (19); \({K_o}\) is the contact stiffness between the rolling elements and the outer raceway, which can be obtained by Eq. (20); \({\delta _{in}}\) and \({\delta _{out}}\) are the deformations of the inner raceway and outer raceway, which can be expressed by Eq. (24) and Eq. (25), respectively.

Where \({\rho _i}\) and \({\rho _o}\) are the sum curvatures of the inner and outer raceway, \({\delta _i}^{*}\) and \({\delta _o}^{*}\) represent functions of \(F({\rho _i})\) and \(F({\rho _o})\)31. The effective modulus of elasticity Eq. (21) and estimated damping Eq. (22) of rolling bearings can be calculated as follows:

Where \({\delta _{i/o}}\) is the contact deformation of the inner and outer raceways.

In practical applications, there is a gap \({\gamma _0}\) between the rolling element and the raceways. The expression of the contact deformation between the rolling element and the raceways is as follows:

Where r is the radius of the inner raceway; \({r_{ball}}\) is the radius of the rolling element; \({\chi _j}\) is the center distance between the rolling element and the inner raceway, which can be obtained by Eq. (26) and Eq. (27):

The early fault evolution system starts with the beginning sidewall and ends with the end sidewall, as shown in Figs. 1 and 2. The energy contribution of the early fault evolution system is mainly provided by the rolling element and the inner ring. The kinetic energy contribution of the early fault evolution system is much greater than the potential energy contribution of the early fault evolution system, so only the kinetic energy contribution of the early fault evolution system is considered. Depending on the angular position of the fault area in the rolling bearing, the velocity of the rolling elements and the inner raceway can be divided into x-axis and y-axis components, respectively. According to the kinetic energy theorem and Eqs. (1)–(5), the energy contribution of the early fault evolution system can be obtained:

Where \({\dot {x}_{ci}}\) and \({\dot {y}_{ci}}\) represent the velocity of the inner raceway on the x-axis and y-axis when the inner ring fault; \({\dot {x}_{co}}\) and \({\dot {y}_{co}}\) represent the velocity of the inner raceway on the x-axis and y-axis when the outer ring fault, respectively; \({\dot {x}_{cb}}\) and \({\dot {y}_{cb}}\)represent the velocity of the rolling element on the x-axis and y-axis, respectively.

The total kinetic energy and the total potential energy of the rolling bearing can be obtained by Eq. (30) and Eq. (31).

Where \({T_{i.r.}}\),\({T_{o.r.}}\), \({T_{r.e.}}\), \({T_{rotor}}\) and \({T_{c.i}}_{.}\) are the kinetic energy of the inner raceway, the outer raceway, rolling elements, the rotor and the early fault evolution system, respectively. \({V_{i.r.}}\),\({V_{o.r.}}\), \({V_{r.e.}}\), \({V_{rotor}}\)and \({V_{spring}}\) are the potential energy of the inner raceway, the outer raceway, rolling elements, the rotor and springs, respectively.

The obtained kinetic energy and potential energy are brought into Eq. 6 to obtain the nonlinear dynamic model of the faulty rolling bearing with the early fault evolution.

Where \({\ddot {x}_{ci}}\), \({\ddot {y}_{ci}}\), \({\ddot {x}_{co}}\), \({\ddot {y}_{co}}\), \({\ddot {x}_{cb}}\)and \({\ddot {y}_{cb}}\) represent the acceleration of the early fault evolution system, which have been decomposed into the Cartesian Coordinate System; \({\lambda _{in}}\)and\({\lambda _{out}}\)are the control factors for whether the early fault evolution system is triggered.

Rolling bearings consist of an inner raceway, an outer raceway, rolling elements and a cage, and each component of the rolling bearing has its characteristic frequency, which can be obtained by the following equation:

The characteristic frequency of the inner raceway:

The characteristic frequency of the outer raceway:

Where \(\psi\)represents the contact angle; \({f_r}\) is the rotation frequency.

The nonlinear dynamic equation system of rolling bearings with the early fault evolution are solved by the Newmark-β method, and the radial acceleration of rolling bearings during fault evolution is obtained. In this paper, the measured life cycle datasets of rolling bearings are provided by the Institute of Design Science and Basic Component at Xi’an Jiaotong University32. Figure 4 shows the main components of the rolling bearing testbed. As shown in Fig. 4, the rolling bearing testbed consists of an alternating current induction motor, support shaft, motor speed controller, a digital force display, a hydraulic loading system and so on. This testbed can be used to acquire life cycle vibration signals of rolling bearings under different speeds and radial forces. The rolling bearing designation used for numerical simulation and test verification is LDK UER204. The detailed parameters of LDK UER204 are shown in Table 1. The whole life cycle vibration signals of the rolling bearing (LDK UER204) under different speeds (2100, 2250 and 2400 rpm), different fault types and different loads are collected. The sampling frequency is 12 kHz.

Testbed of rolling element bearings.

Considering the preload force, lubrication and other conditions of the rolling bearing in practical applications, the initial conditions for setting the dynamic model are as follows. The internal radial clearance between the rolling element and the raceway is 1 µm. The initial distance between the rolling element and the mass center of the outer raceway is 18 mm. The calculation step of the Newmark-β method is 0.0001 s. The inner raceway and the rotor have the same parameters, where the displacement and velocity are set as follows: \({x_0}={10^{ - 1}}\), \({y_0}={10^{ - 1}}\), \({\dot {x}_0}=0\) and \({\dot {y}_0}=0\). Other parameters of the rolling bearing such as speed, loading force and so on are shown in Table 1. Figure5 shows photographs of rolling bearings with inner and outer raceway faults, respectively.

Figure 6 shows the measured vibration response of the rolling bearing during the whole operating life with an outer raceway fault. It also shows the local magnification of the vibration signal surge phase during early fault evolution. The start time is taken from the beginning of the bearing vibration signal, and the cut-off time is at the first significant stage jump of the vibration signal. The phase from the start time to the cut-off time is the early fault evolution stage of the rolling bearing, and Eq. (32) is used to calculate this stage, the calculation results are compared with the measured results. Regardless of the degree of the fault evolution, every rolling element will collide with the fault and provoke a corresponding characteristic frequency when passing through the defect area, which can be obtained by Eq. (33) and Eq. (34).

Photographs of the fault bearings: (a) inner race defect; (b) outer race defect .

Vibration response over the whole operating life of the rolling bearing (outer raceway defect, 2100 rpm).

Fig.7 shows the measured and simulated vertical acceleration response and its spectrum of the rolling bearing with an early outer raceway fault evolution at the speed of 2100 rpm, respectively. As can be seen from Fig. 7(a) and Fig. 7(c), significant periodic shock vibrations are shown in both the measured and calculated time-domain diagram. The measurement results and calculation results have similar vibration response amplitudes and development trends. The vibration amplitude of the measurement results and the calculation results in the initial stage of the fault (fault size is 0) is about 0.02 mm/s2. With the continuous evolution of the early outer raceway fault, the vibration amplitude of both results gradually increased from 0.0 mm/s2, and there is a pronounced jump increase in both results at a certain time. As can be seen from Fig. 7(b) and Fig. 7(d), the measured and calculated rotation frequencies are 34.28 Hz and 35 Hz, respectively, which are very close. The calculated outer raceway defect frequency is 107.5 Hz, which is very close to the measured outer raceway defect frequency of 108.1 Hz. Higher harmonics of the outer raceway defect frequencies appeared in both the measured and calculated spectrum. Further, according to the uncertainty and irreversibility of the fault evolution characteristics, the early fault evolution degree of the rolling bearing in practical applications can be estimated. The early fault evolution degree of the rolling bearing in practical applications can be estimated. The defect size range of the early fault generation stage is about 0 mm~0.3 mm. The defect size range of the early fault slow evolution stage is about: 0.3 mm~0.5 mm. The size range of the early fault accelerated evolution stage is about: 0.5 mm~0.8mm.

Vibration response to early fault evolution of the rolling bearing with outer raceway defect (2100 rpm): (a) experimental signal, (b) experimental FFT, (c) calculation signal and (d) calculation FFT.

Vibration response over the whole operating life of the rolling bearing (outer raceway defect, 2500 rpm).

Vibration response to early fault evolution of the rolling bearing with outer raceway defect (2500 rpm): (a) experimental signal , (b) experimental FFT , (c) calculation signal and (d) calculation FFT .

Figure 8 shows the measured vibration response of the rolling bearing over the entire operating life with an outer raceway defect at 2500 rpm, as well as important evolutionary points for early vibration signals. The early fault evolution phase of the rolling bearing begins at the beginning of the bearing vibration signal and ends at the first significant jump of the vibration signal. When the rotational speed is 2500 rpm, the calculation results are compared with the measurement results, as shown in Fig. 9.

As shown in Fig. 9(a) and Fig. 9(c), significant periodic shock vibrations, similar vibration response amplitudes and development trends are shown in the measured and calculated time-domain diagram. In the initial stage of the fault (fault size is 0), the amplitude of the measured and calculated results is about 0.025 mm/s2. With the continuous evolution of the early outer raceway fault, the amplitude of both results gradually increased from 0.025 mm/s2, and the vibration amplitude of the measured and calculated results increased significantly at a certain time. As shown in Fig. 9(b) and Fig. 9(d), the measured and calculated rotational frequencies are very close, which are 36.85 Hz and 37.99 Hz, respectively. The calculated outer raceway defect frequency is 116 Hz, which is very close to the measured outer raceway defect frequency of 114.5 Hz. Higher harmonics of the outer raceway defect frequency appear in the measured and calculated spectrum. Based on the uncertainty and irreversibility of fault evolution and Fig. 9, the degree of the rolling bearing early fault evolution in practical applications is estimated. The defect size range of the early fault generation stage is about 0 mm ~ 0.05 mm. The defect size range of the early fault slow evolution stage is about 0.05 mm ~ 0.1 mm. The size range of the early fault accelerated evolution stage is about 0.1 mm ~ 0.67 mm.

As shown in Figs. 5, 6, 7,8, 9; Table 2, both time-domain diagrams of the calculated and measured have significant periodic shock vibration, similar vibration development trend and vibration response amplitude. The rotational frequency, outer raceway fault frequency, and its multiple harmonics in the calculated and measured spectrum are consistent with each other, respectively. It can be seen that the nonlinear dynamic modeling for early fault evolution of rolling bearing based on collision impact proposed in this paper can effectively simulate the time-frequency characteristics of outer raceway faults. Furthermore, the model can be used to estimate the degree of fault evolution in the actual operation of rolling bearings.

Figure 10 shows the measured vibration response of rolling bearings over the entire operating life with inner raceway defects at 2250 rpm, as well as important evolution nodes of early vibration signals. It can be seen from Fig. 10 that in the early stage of the inner raceway fault, there are no significant changes in the acceleration vibration response of the rolling bearing, so it is necessary to study the evolution of the inner raceway early fault.

Vibration response over the whole operating life of the rolling bearing (inner raceway defect, 2250 rpm).

Vibration response to early fault evolution of the rolling bearing with inner raceway defect (2250 rpm): (a) experimental signal, (b) experimental FFT, (c) calculation signal and (d) calculation FFT.

As shown in Fig. 11(a) and Fig. 11(c), periodic shock vibrations, similar vibration response amplitudes and development trends are shown in the measured and calculated time-domain diagram. It can be seen from Fig. 11(a) that although there is no obvious step fluctuation in the time-domain diagram of the early inner raceway fault, the rolling bearing already has a hidden fault hazard with insignificant vibration amplitude. The nonlinear dynamic modeling for early fault evolution of rolling bearings in this paper can be used to infer the hidden fault hazards of inner raceway faults in time. As shown in Fig. 11 (b) and Fig. 11 (d), the measured and calculated rotational frequencies are very close to each other, which are 37.22 Hz and 37.49 Hz, respectively. The calculated inner raceway defect frequency is 185 Hz, which is very close to the measured inner raceway defect frequency of 185.1 Hz. In addition, higher harmonics of the inner raceway defect frequency appear in the measured and calculated spectrum. The above comparison results prove the correctness of the nonlinear dynamic modeling for early fault evolution of rolling bearings proposed in this paper. According to the characteristics of fault evolution and Fig. 11, it is possible to estimate the degree of early fault evolution in practical applications. The defect size range of the early fault generation stage is about 0 mm ~ 0.15 mm, the vibration response of the rolling bearing is stable during this stage, and it is difficult to detect hidden faults in time. The defect size range of the early fault slow evolution stage is about 0.15 mm ~ 0.45 mm, and the vibration response of the rolling bearing begins to fluctuate in this stage. The size range of the early fracture acceleration evolution stage is about 0.45 mm ~ 0.67 mm, the change of vibration response in this stage is gradually strong.

Figure 12 shows the measured vibration response of rolling bearings over the entire operating life with inner raceway defects at 2400 rpm, as well as important evolution nodes of early vibration signals. It can be seen from Fig. 12 that in the early stage of the inner raceway fault, there are no significant shock changes in the acceleration vibration response of the rolling bearing, so it is difficult to find hidden faults of the rolling bearing in time.

Vibration response over the whole operating life of the rolling bearing (inner raceway defect, 2400 rpm).

Vibration response to early fault evolution of the rolling bearing with inner raceway defect (2400 rpm): (a) experimental signal, (b) experimental FFT, (c) calculation signal and (d) calculation FFT.

As shown in Fig. 13(a), although there are no significant step fluctuations in the time domain diagram of the early inner raceway fault, the rolling bearing already has a hidden fault hazard with a small size. The nonlinear dynamic modeling for early fault evolution of rolling bearings proposed in this paper is used to calculate this stage, and the calculation results are shown in Fig. 13(c). As can be seen from Fig. 13 (a) and Fig. 13 (c), there are periodic shock vibrations and similar vibration trends in the measured and calculated time-domain diagram. As shown in Fig. 13 (b) and Fig. 13 (d), the measured and calculated rotational frequencies are very close to each other, which are 39.62 Hz and 39.99 Hz, respectively. The calculated inner raceway defect frequency is 196.6 Hz, which is very close to the measured inner raceway defect frequency of 192.3 Hz. In addition, higher-order harmonics of the frequency of inner raceway defects appear in the measured and calculated spectrum. According to the characteristics of fault evolution and Fig. 13, the nonlinear dynamic modeling for early fault evolution of rolling bearings is used to estimate the degree of early fault evolution in practical applications. The defect size range of the early fault generation stage is about 0 mm ~ 0.14 mm. During this stage, the vibration response of the rolling bearing is relatively stable, and it is difficult to detect hidden faults in time. The defect size range of the early fault slow evolution stage is about 0.14 mm ~ 0.43 mm, the vibration response of the rolling bearing begins to fluctuate in this stage. The size range of the early fracture acceleration evolution stage is about 0.45 mm ~ 0.71 mm, the change of vibration response in this stage is gradually stronger.

As shown in Figs. 10, 11, 12 and 13; Table 3, when the rolling bearing has an early inner raceway fault, the vibration characteristics of the rolling bearing are not obvious, which increases the difficulty of rolling bearings fault diagnosis. However, the calculated results of the proposed nonlinear dynamic model in this paper are consistent with the measured results. The comparative results show the nonlinear dynamic modeling for early fault evolution of rolling bearings can effectively simulate the evolution degree and characteristics of the inner raceway fault. This model can provide theoretical support for fault diagnosis of rolling bearings.

The nonlinear dynamic modeling for early fault evolution of rolling bearings is an ideal calculation model, which is not affected by external factors such as temperature changes, lubrication conditions and so on, thus the calculated results are relatively pure. These factors may be the main reasons for the discrepancy between the measured results and the calculated results.

A novel nonlinear dynamic modeling for early fault evolution of rolling bearings based on collision impact is developed to predict the fault evolution process and vibration characteristics of rolling bearings. The performance of the nonlinear dynamic model proposed in this paper is numerically studied, and the following major conclusions are summarized.

1. The new system of dynamical equations for early fault evolution can show the fault evolution process and collision impact characteristics between the rolling element and the fault area in detail, which is a more realistic dynamic model.

2. The energy contribution method is used to integrate the early fault evolution system into the rolling bearing to establish the nonlinear dynamic model for early fault evolution of rolling bearings. Comparing the calculated and measured results show that the model can effectively predict the evolution and vibration characteristics of the early fault of rolling bearings.

3. When the early fault of the inner or outer raceway occurs in the rolling bearing, the vibration fluctuation of the rolling bearing is not obvious, which increases the difficulty of rolling bearing fault diagnosis. The nonlinear dynamic model proposed in this paper can effectively simulate the evolution trend and vibration characteristics of rolling bearing’s early faults, which provide an important theoretical basis and technical support for rolling bearing fault diagnosis.

However, with the continuous development of rolling bearing faults, impact vibration between the rolling element and the fault area will become increasingly complex. This phenomenon must continue to be studied.

All data generated during this study are included in this published article.

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This work was supported by the National Natural Science Foundation of China (Grant No. 52241102).

College of Oceanography, Yantai University, Yantai, 264000, China

Longkui Zheng & Ning Luo

School of Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan, 400063, China

Yang Xiang

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Z.L. wrote the main manuscript text, conceptualization, methodology and software.Y.X. edited the main manuscript text and data analysis.L.N. prepared figures. All authors reviewed the manuscript.

Correspondence to Longkui Zheng or Yang Xiang.

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Zheng, L., Xiang, Y. & Luo, N. Nonlinear dynamic modeling and vibration analysis for early fault evolution of rolling bearings. Sci Rep 14, 23687 (2024). https://doi.org/10.1038/s41598-024-75126-5

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Received: 04 May 2024

Accepted: 01 October 2024

Published: 10 October 2024

DOI: https://doi.org/10.1038/s41598-024-75126-5

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