An Improved Coupled Dynamic Modeling for Exploring Gearbox Vibrations Considering Local Defects

Gearbox is a key part in machinery, in which gear, shaft and bearing operate together to transmit motion and power. The wide usage and high failure rate of gearbox make it attract much attention on its health monitoring and fault diagnosis. Dynamic modeling can study the mechanism under different faults and provide theoretical foundation for fault detection. However, current commonly used gear dynamic model usually neglects the influence of bearing and shaft, resulting in incomplete understanding on gearbox fault diagnosis especially under the effect of local defects on gear and shaft. To address this problem, an improved gear-shaft-bearing-housing dynamic model is proposed to reveal the vibration mechanism and responses considering shaft whirling and gear local defects. Firstly, an eighteen degree-of-freedom gearbox dynamic model is proposed, taking into account the interaction among gear, bearing and shaft. Secondly, the dynamic model is iteratively solved. Then, vibration responses are expounded and analyzed considering gear spalling and shaft crack. Numerical results show that the gear mesh frequency and its harmonics have higher amplitude through the spectrum. Vibration RMS and the shaft rotating frequency increase with the spalling size and shaft crack angle in general. An experiment is designed to verify the rationality of the proposed gearbox model. Lastly, comprehensive analysis under different spalling size and shaft crack angle are analyzed. Results show that when spalling size and crack angle is lager, RMS and the amplitude of shaft rotating frequency will not increase linearly. The dynamic model can accurately simulate the vibration of gear transmission system, which is helpful to gearbox fault diagnosis.


Introduction
Gear transmission system is a key component in rotating equipment, in which gear and bearing operate together to transmit motion and power.
Gearbox have been widely used in CNC machine, wind turbine, and other fields [1][2] [3]. With the development of mechanical equipment towards complexity and high-power density, the working condition of gearbox has become more complex and tougher. It will lead to local defects on gear or bearing with higher possibility, even the failure of gearbox or machine. If a fault is not detected in time, maintenance activity cannot be organized, which may cause a major accident and economic loss. Hence, gearbox fault diagnosis plays a key role in real production and application.
Vibration signal is usually taken as the preferred source data for fault diagnosis [4]. Especially, in a gearbox system, the acquired vibration data possesses rich information of each component, including the motion and coupled effect. Therefore, figuring out the mechanism of the interaction of gear, shaft and bearing in vibration signals is of great significance for gearbox fault diagnosis.
To investigate the dynamic performance of gear transmission systems, Ma et. al [5][6] studied the calculation of time-varying mesh stiffness for cracked spur gears. Chen et al. [7][8] [9] performed a series of investigations on the dynamic responses of spur gear on tooth root crack, deformation and errors. Wang et al. [10] explored the fault mode of gear tooth crack during the process of the propagation for detection through a simulation study. However, in these works, only the vibration responses of gear have been investigated, while the bearing and shaft were neglected. For exploring the dynamic behaviours of the gear transmission systems considering the couple effect of gear and bearing, Sawalhi and Randall [11] [12] presented a vibration model to analyse the interaction between the gear and the bearing with 34 degree of freedom (DOF), and vibration responses of bearing local defect was also investigated.
However, the interaction mechanism of gear meshing impact on bearing is not clear. Zeng et al. [13] studied the vibration response of different positions on gearbox system through finite element method and investigated the relation between the gear meshing stiffness (GMF) and vibration. Hu et al. [14] utilized a dynamic model under finite element node to investigate responses of taking a gear-rotor-bearing system at high speed into consideration the time-varying mesh stiffness (TVMS) and other factors. Zhou et al proposed a 16-DOF gear-rotor-bearing dynamic model considering the effect of gear transmission error, bearing clearances and eccentricity. Xiao et al. [16] investigated a gear-shaft-bearing-housing dynamic model and analysed impulsive gear mesh force on the vibration characteristics and energy dissipation considering 8 DOF. However, the dynamic gear meshing and local defect were ignored. Fernandez et al. [17] investigated the dynamic responses of gear transmission system under the influence of bearing clearance. Chen et al. [18] established a geared rotor-bearing system under the effect of gear meshing, shaft and oil-film and analysed the vibration characteristics response. Xu et al. [19] proposed a coupled nonlinear dynamic model considering gear, shaft and bearing and vibration characteristics under the influence of bearing clearances were studied. However, the shaft was regarded as rigid in these works.
Overall, it finds that the interaction among gear, shaft and bearing is not explained clearly especially under the effect of gear local defects and shaft deflection. Fatigue and local defects, such as spalling, will appear inevitably on gear surface after long-time operation. The vibration responses of the gear transmission system will show obvious impulses when enter the fault zone. Therefore, the health condition of the gearbox can be detected according to the impulses. However, the diagnostics features of local defects will be affected by distributed faults, such as shaft deflection, resulting in misjudgement on the degree of failure. The coupled effect of gear local defects and shaft deflection on vibration for fault diagnosis is still unclear.
Thus, to fill the gap, this study proposes an improved coupled gear-shaft-bearing-housing dynamic model considering the interaction mechanism among gear, shaft and bearing. Vibration responses when crack appear on gears are investigated. The influence of shaft whirling on vibration are analysed for monitoring gear spalling. The rest part of the paper is organized as follows. Section I expounds the proposed coupled dynamic model. Section II introduces the implementation of the numerical simulation.
Section III is the result and analysis. Section IV Figure 1 Schematic diagram of the gear-shaftbearing-housing dynamic model ,   [20]. b N means the number of rolling elements, and ij  denotes the angle position of the i-th ball.
The dynamic gear mesh force and the moment among the gear pair can be calculated through TVMS and damping [19]. (21) where, m K denotes the TVMS, pg  is dynamic transmission error and pg v is its differential coefficient, respectively, m C stands for the mesh damping, bp r and bg r the base radius of pinion and gear separately. Note that the improved model is constructed based on following considerations and assumptions.
(1) 18 DOF is considered in the dynamic model, which contains the translational motion of inner race, outer race and sensor on vertical (X) and horizontal (Y) direction, the translational motion of pinion and gear on vertical (X) and horizontal (Y) direction and the rotational motion of pinion and gear on axial direction. Therefore, for each bearing, it has 6 DOF. For pinion and gear, they have 6 DOF totally. Other DOF are neglected in this model. (2) The structure of the model is assumed to be symmetrical. It means that bearing 1 and bearing 2 are assumed to possess the same vibration waveform. Similarly, bearing 3 and bearing 4 are the same.

Contact Stiffness of Normal Gear
Under the influence of meshing force and torque, gear teeth will be deformed during operation, which mainly includes bending deformation, shear deformation, axial compression deformation, contact deformation and gear filletfoundation deformation [5][6][7][8]. The tooth force diagram of normal gear is shown in Figure 2. The bending stiffness, shear stiffness, axial compression stiffness, Hertz contact stiffness and stiffness caused by gear fillet-foundation deflection can be written as [8][19] where E is the elastic modulus and G is shearing modulus of gear tooth, and Ix and Ax are the moment of inertia and the area of the section at the distance x from the fixed end of the gear tooth. δf represents the fillet-foundation deflection.
According to the geometric relationship between involute tooth profile and rotation angle, Eq. (22) - (24) can be transformed to [21]        

Contact Stiffness of Faulty Gear
In gearbox, the percentage of gear fault accounts for about 60%, such as spalling and pitting on gear surface. When the tooth surface has spalling fault, the schematic diagram and force analysis of gear tooth meshing is shown in Figure 3. The bending stiffness, shearing stiffness and the axial compression stiffness in Figure 3 can be written as [22] Finally, for both normal gear and spalling gear, the integrated meshing stiffness of the pinion (subscript 1) and gear (subscript 2) can be written as  The deflection of the beam can be written as [23] 

Shaft Deflection and Shaft Stiffness
According to the principle of material mechanics, the deflection along the axis under F can be calculated as According to definition of stiffness, the bending stiffness of shaft at the position C for gear installation can be obtained as [24] When the gear is fixed at the middle of the shaft, the stiffness can be written as

Shaft Crack and Shaft Whirling
Manufacturing errors or assembly errors will lead to reciprocating impact of meshing between gear teeth. Shaft cracks are easy to appear at the installation position of gear under reciprocating impact condition, as shown in Figure 5. The magnitude and direction of the stress on the crack surface of the shaft will change with the operation of the shaft. Therefore, the crack on the shaft will present a breathing characteristic, which results in the change of the support stiffness. It is noted that transverse crack is considered in this study. The shaft crack will cause the shaft unbalance and then lead to shaft whirling. The schematic diagram of shaft whirling by shaft crack is shown in Figure 6.

Path of whirling
Bearing Bearing ω Figure 6 Diagram of shaft whirling The switching function   f  and the cosine wave model are used to simulate the opening and closing process of the shaft crack.
where  is the angle between the fixed coordinate system and rotor rotation coordinate system,  is the angular velocity of the pinion,  is the initial position angle of the shaft crack.
When the shaft crack occurs, the gravity of the gear and the meshing force of the gear teeth will have a transverse impact on one side of the shaft. Therefore, the stiffness of the supporting shaft on both x and y direction will decrease, which can be written as [25]   sx s sx where , sx y K  means change of the stiffness due to the crack, which is written by  [25].
The shaft whirling Rw can be calculated form the deflection caused by shaft crack.
The displacement excitation and its derivative caused by the shaft whirling along the line of action can be given by where Rw is the shaft whirling vector,   is the whirling frequency, which is equal to rotating frequency in this study.
Under the influences of shaft whirling, the dynamic transmission error pg  and its differential coefficient pg v can be given as where h is a coefficient to judge whether contact occurs between the two-meshing tooth.

Implementation of the Numerical Simulation Assumptions and Considerations
It is noted that the model is established on an ideal working condition, i.e., the influence of speed and load fluctuation are neglected. Besides, manufacturing errors, bearing skidding and temperature are also ignored in this model.

Main Parameters in Numerical Simulation
The main parameters adopted in numerical simulation are recorded in Table 1 and Table 2. Besides, the shaft rotating speed is 1500 rpm, while the torque provides by the brake is 20Nm. Table 3 displayed the characteristic frequencies of gear and bearing are shown in, containing rotating frequency, GMF, ball pass frequency on inner race (BPFI), ball pass frequency on outer race (BPFO) and cage frequency (CF).

Force analysis
In the dynamic model, the time-varying meshing stiffness of the gear pair can be calculated during the model solving by Eq. (19). The stiffness and the meshing force are shown in Figure 7. Note that when there is no external local defect on gear, bearing and shaft, the working condition is named "Baseline". In this model, gear and bearing operate together to transmit motion and power. Therefore, the response of bearing is of great concern. In this part, the load distribution of bearing is analyzed. The dynamic resultant force on each ball is shown in Figure 8.  Figure 8, it finds that the dynamic force on each ball represents periodical waveform, and the nine balls change in turn. The means that with the rotating of the bearing, the dynamic force change with the loaded region and unloaded region, which is consistent with the theoretical load distribution.

Vibration Responses under Gear Fault
In this section, vibration responses are analyzed considering baseline and spalling pinion. The parameter of the spalling is set as ls=2mm, ws=20mm, L=1.8mm, and ds=1mm. Vibration signal of bearing 1 in Figure 1 are extracted. Analysis form time domain and frequency domain are illustrated in Figure 9. As can be seen form Figure 9 (a) and (b), when spalling appears on gear surface, the waveform shows an obvious increase comparing with the baseline, and periodic impulses can be recognized form the waveform. As can be seen, periodic impulses with the period of 1/� �1 can be clearly detected when spalling occurs. After enlargement, periodic impulses with the period of 1/� 푚 is dominant. Through FFT spectrum in Figure 9 (c), the energy of GMF and its harmonics has higher amplitude. After the enlargement of the first order GMF, the sidebands are located at the both side of GMF, as shown in in Figure 9 (d). The spectrum amplitude of spalling gear shows an obvious increase comparing with the baseline. Figure 9 (e) depicts the envelope spectrum. From Figure  9 (e), the amplitude of � �1 of spalling gear is much higher than baseline, which verified the modulation phenomenon derived from gear spalling. It means that when gear spalling occurs, the modulation between the gear fault and GMF plays a vital effect.

Vibration Analysis under the Influence of Shaft Crack
To investigate the vibration features of gear system under the influences of shaft whirling, simulations under baseline and cracked shaft are carried out first. Vibration waveform and spectra of two cases is shown in Figure 10. As can be seen form Figure 10 (a) and (b), when crack appears on the shaft, the vibration waveform fluctuates more obvious than baseline. From the FFT spectrum in Figure 10 (c), GMF and its harmonics is dominant. After the enlargement of the first order GMF, the more sidebands are located at the both side of GMF when crack occurs, as shown in Figure 10 (d). Through envelope spectrum in Figure 10 (e), the amplitude of fr1 of cracked shaft is much higher than baseline, which verified the modulation phenomenon derived from shaft crack.

Experiment Verification Experiment Settings
A spur-gear gearbox test rig is designed to validate the proposed model. Vibration of the bearing housing was measured through vibration accelerometer. The test rig and measure system are shown in Figure 11. The gearbox consists of three shafts and two pairs of gear. It is noteworthy that the second pair of gear is utilized to verify the proposed model. The supported bearings are deep grove ball bearing 6205. The tooth number of studied gear pair is 25 and 53, respectively, which is consistent with the dynamic model. During the test, vibration data is acquired under normal gear, gear with spalling and shaft crack. The input rotational speed by the motor is 900rpm. The load is supplied by the magnetic powder brake and set as 60Nm.  Figure 11 Test facilities: (a) spur-gear gearbox test rig and (b) the structure of the gearbox In order to validate the dynamic model and vibration responses under gear spalling and shaft crack, spalling and crack were machined on gear and shaft, as shown in Figure 12. As shown in Figure 12, the spalling is located on the gear surface. The length of the spalling is 2mm the width of it is 10mm and the depth of it is 1mm. In the experiment, the crack circumference angle of the shaft is set as 60°. Besides, the shaft rotating speed is 1200rpm and the torque provided by the magnetic powder brake is 20Nm.  Table 7 exhibits the characteristic frequencies of the gearbox under speed of 1200rpm, which contains rotational frequency of shaft and GMF.  Figure 13 exhibits the analysis result of the vibration signal. As can be seen form Figure 13(a) and (b), the amplitude of gear spalling and shaft crack is higher than baseline and the impulsive wave is more obvious, which can be attributed to the excitation of the spalling and crack. The amplitude and RMS value are consistent with the simulation.
Form the FFT spectrum in Figure 13(c), spectrum concentrates on around the GMF fm and its harmonics. It means that GMF extract more energy during the operations. Sideband can be detected at the both side GMF. It can also find that the amplitude of sideband of cracked gear is higher than normal gear.
The spectrum based on envelope demodulation is exhibited in Figure 13(d). From Figure 13(d), the rotating frequency of the second shaft fr and its harmonics can be clearly detected form the envelope spectrum under three cases. The amplitude of fr and its harmonics of spalling gear is higher than baseline. It means that when spalling occurs on gear surface, shaft frequency fr and its harmonics will modulate on the GMF. Note that the fluctuation amplitude on fr can be attributed to the background noise, or assembly error. Even so, the findings of experiment in Figure 13 and simulation analysis is consistent, which verifies the accuracy of the proposed dynamic model.

Comprehensive Analysis Vibration Responses under the Influences of Spalling Length and Width
In this section, force and vibration responses under the influences of different spalling size is exhibited and compared. Figure 14 depicts the comparison on stiffness and dynamic force under different spalling length and width. As can be seen form Figure 14, the stiffness decreases with increase the length and width of the spalling, while the dynamic gear meshing force and vibration signals increase with them. It means that when crack appears on gear tooth root, the stiffness will decrease compared with baseline, which results in the increase of vibration.
RMS of stiffness and vibration and amplitude of rotating frequency of shaft 1, fr1, are calculated under different spalling length and width, as shown in Figure 15. As can be seen, the RMS of stiffness show downtrend as the increase of spalling length and width, while the RMS and the amplitude of fr1 of vibration increase by means of the increase of spalling length and width in general. It means that the increase of spalling will result in the increase of vibration. However, the fluctuations can be found in Figure  15 (b) and (c) when spalling length and width change. It means that when spalling size changes, there is a nonlinear relationship between the spalling size and vibration. As can be seen from Figure 16, the vibration of different shaft cracks shows impulsive waveform. RMS and the amplitude of fr1 of vibration increase with the increase of shaft crack angle in general. Similarly, fluctuations can be found in Figure 16 when the angle is around 30°. However, the total uptrend with the shaft crack angle is obvious.

Vibration Analysis under the Influence of Shaft Crack and Gear Fault
Vibration under different shaft crack and gear spalling are simulated and analyzed. The comparison results are shown in Figure 17. As can be seen from Figure 17, the vibration RMS of different shaft crack and gear spalling show uptrend. However, the increase of RMS is small. The amplitude of fr1 of the vibration acceleration show different results. As shown in Figure 17(c), when the spalling size is small, the amplitude of fr1 shows uptrend with the shaft crack angles. When the spalling width is 40mm, amplitude of fr1 fluctuates. It means that, when larger spalling appears with lager width, the amplitude of characteristic frequency fr1 will not show obvious increase with shaft crack angles, which is different with the result in Figure 16. Similarly, when the shaft crack is larger, including 50°, 60°, 70°, 80°and 90°, the amplitude of frequency fr1 decrease with the spalling width. It means that when gear spalling and shaft crack appear, the vibration is complex and nonlinear.

Discussion
Through the proposed dynamic model, vibration characteristics considering local defects on gear and shaft are investigated. From the result, RMS and the amplitude of shaft rotating frequency will fluctuate or increase nonlinearly when gear spalling size and shaft crack angle is larger. Therefore, it is not perfectly reasonable to monitor it form the single modulation frequency. It leads to difficulty for the fault diagnosis of gearbox and it should take care when gear spalling and shaft crack deteriorate. It is necessary to find new indicators for detection. Even so, the establishment of such a model can provides theoretical foundation for gearbox fault diagnosis. The model can be used to simulate vibration of gearbox system to study the vibration characteristics under local defects. Besides, the model has some limitations on applications. The structure of the model is assumed to be symmetrical. Geometry and assembly errors, such as misalignment, are ignored in this model. Besides, lubrication, temperature and bearing skidding are also ignored in this model. Therefore, there are several potential avenues for improving the proposed model. The influence of shaft whirling and bearing clearances on the time-varying stiffness of gear meshing is one of the future works, especially when the system is not symmetrical.
Study on the dynamic characteristics of shaft and bearing on time-vary gear mesh stiffness is one of the future works.

Conclusions
This study investigates a gear-shaft-bearinghousing vibration model considering the interaction between gear and bearing, as well as shaft whirling. Vibration responses analysis are carried out based on the model under the effect of gear spalling and shaft crack. Through numerical and experimental analysis, the key findings and the conclusions can be drawn as 1. The proposed coupled dynamic model exhibits better performance on revealing the mechanism in the gear-shaft-bearing system.
2. Based on the spectrum, GMF is dominant in the gearbox system. When local defects appear on gear surfaces or shaft, sideband s can be detected on both side of GMF and its harmonics.
3. With the increase of spalling size, RMS of stiffness show downtrend, resulting in the increase of vibration level, including the vibration RMS and the amplitude of first three order rotating frequency. It means that the growth of spalling will result in the increase of vibration and modulation.
4. An experiment was designed to certify the proposed dynamic model based on spurgear gearbox test rig. Experiment analysis and simulation result is consistent on vibration responses under the effect of gear spalling and shaft crack, which demonstrated the rationality of the enriched gear-shaft-bearing dynamic model.

Comprehensive vibration analysis under
different gear spalling and shaft crack shows that when gear spalling size and shaft crack angle is small, RMS and the amplitude of shaft rotating frequency has better linear growth, while fluctuations or downtrend will occur when gear spalling size and shaft crack angle is larger.