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Vibration characteristics of railway locomotive induced by gear tooth root crack fault under transient conditions
Journal Pre-proofs Vibration Characteristics of Railway Locomotive Induced by Gear Tooth Root Crack Fault under Transient Conditions Jianzheng Jiang, Zaigang Chen, Wanming Zhai, Tao Zhang, Yifan Li PII: DOI: Reference:
S1350-6307(19)30510-2 https://doi.org/10.1016/j.engfailanal.2019.104285 EFA 104285
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Engineering Failure Analysis
Received Date: Revised Date: Accepted Date:
12 April 2019 29 August 2019 4 November 2019
Please cite this article as: Jiang, J., Chen, Z., Zhai, W., Zhang, T., Li, Y., Vibration Characteristics of Railway Locomotive Induced by Gear Tooth Root Crack Fault under Transient Conditions, Engineering Failure Analysis (2019), doi: https://doi.org/10.1016/j.engfailanal.2019.104285
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Vibration Characteristics of Railway Locomotive Induced by Gear Tooth Root Crack Fault under Transient Conditions Jianzheng Jiang1,2, Zaigang Chen1,3*, Wanming Zhai1, Tao Zhang1, Yifan Li4 1
State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, People's Republic of China 2
3
China Railway Eryuan Engineering Group Co., Ltd. Chengdu 610031, People's Republic of China
The State Key Laboratory of Heavy Duty AC Drive Electric Locomotive Systems Integration, CRRC Zhuzhou Locomotive Co., LTD, Zhuzhou 412001, People's Republic of China 4
School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, People's Republic of China
*Corresponding author: Associate Prof. Zaigang CHEN Tel: +86-(0)28-86466504; Fax: +86-(0)28-87601843; E-mail address: [email protected] Abstract: Dynamic interactions between wheel and rail are becoming more intensified under the development of locomotive towards high-speed, heavy-haul and high-power directions, which are more likely to cause failures in gear transmission system of high-power heavy-haul electric locomotive. The dynamic impact of the locomotive gear transmission system becomes more drastic, especially for the extreme operation conditions, such as traction/braking process. Gear fault diagnosis under unsteady conditions has always been a hot and difficult research spot, and revealing of the gear fault vibration characteristics under the transient conditions is the premise and basis for effective fault detection and diagnosis. In order to reveal the gear fault vibration characteristics of railway locomotive dynamics system, a spatial dynamic model of a heavy-haul electric locomotive considering the dynamic coupling effect of gear transmission system is proposed based on the multibody dynamics theory in this paper. Then the dynamic responses of the locomotive under transient condition are obtained by considering the complicated excitations induced by the wheel-rail nonlinear contact, gear mesh and tooth root crack. The time-frequency analysis and angular synchronous average method are adopted to investigate the fault vibration feature of the tooth root crack. Finally, the distribution and variation of vibration characteristics of the gear tooth root crack fault evolution are revealed by condition indicators such as the Crest Factor (CF), Kurtosis (K), Fourth Order Figure of Merit (FM4), M6A, and M8A. The results indicate that: (1) the time-frequency analysis results of vibration acceleration of wheelset and bogie frame and dynamic mesh force could reflect the fundamental mesh frequency and its harmonics, however, only the vertical and longitudinal vibration acceleration of the wheelset and the gear dynamic mesh force could reflect the fault vibration characteristics in time-frequency analysis results; (2) the FM4, M6A and M8A values of locomotive system vibration signal could effectively reflect the crack propagation on the dynamic features of the locomotive system after being processed by angular synchronous averaging method.
Keywords: heavy-haul locomotive, multi-rigid body dynamics, tooth root crack, time-frequency analysis, angular synchronous average
1 Introduction The prompt growth of the world’s freight transportation has accelerated the development of heavy-haul railway transportation. For example, by 2017, the operating mileage of the Chinese railways reached 127,000 kilometers, and the total volume of railway freight transports reached 3.689 billion tons with an increase of 10.7% over the previous year [1]. Developing high-power heavy haul electric locomotives so as to meet the requirements towards the long marshalling and large axle-load train is imperative, which could improve the transportation capacity of the heavy-haul railway. However, operating the long-marshalling and large axle-load train will bring severe challenge to the service safety of the vehicles and the track structure. As a vital device in locomotive structure, gear transmission system endures the combined action of both the internal and external excitations such as the time-varying gear mesh stiffness, the wheel-rail nonlinear contact, and the traction/braking moment of motor, which is likely to produce failures to the gear and other components. Compared with healthy gears, the faulty gears will change the effective tooth thickness and contact area during meshing process, and further alter the stiffness and the error excitation of gears, which may change the dynamic responses of the gear transmission system. For example, a detailed review paper [2] further shows that the time-varying mesh stiffness (TVMS) of faulty gears is usually introduced into the dynamic model of gear transmission system to study the gear fault vibration features, like Ref [3]. Accordingly, how to accurately and efficiently obtain the time-varying mesh stiffness of gears with or without faults is significant to accurately investigate the fault features of the gear transmission system. For example, Ma et al. [4] proposed an improved mesh stiffness by considering the accurate tooth root transition curve and a parabolic curve to simulate the crack propagation path. Liang et al. [5] reviewed four methods for calculating the time-varying mesh stiffness of gears with or without faults based on existed literature, namely, square wave method, potential energy principle method, finite element method and experimental method. Chen et al. [6] established an analytical-finite element model to calculate the TVMS of spur gears based on finite element theory and loaded tooth contact analysis, where the complex gear foundation and crack propagation path were considered. Chen et al. [7] proposed an improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack which enables more accurate calculation of gear mesh stiffness. Liu et al. [8] investigated the gear mesh behavior by considering the work holding equipment errors based on the three-face cutter. Various forms of gear faults will lead to abnormal vibration of gear transmission system and shorten the service life of the system. At the same time, condition monitoring of gear transmission system can effectively avoid further deterioration of gear failure [9]. Based on published literature, Sharma and Parey [10] summarized the statistical indicators that can be used for condition monitoring and fault diagnose of gear transmission system in time domain and frequency domain. In classical railway vehicle dynamics research, vehicle system and track system are regarded as two independent subsystems. Scholars neglected the influence of track system on vehicle system when they study vehicle system dynamics [11,12]. In fact, the vibration of vehicle system and track system will affect each other through wheel-rail contact interface, thus, it is essential to build a large-scale system coupled by wheel-rail contact interface [13]. In recent years, the increasing operation speed and axle load of train has further aggravated the dynamic interaction between vehicle system and track system, which attracted the attention of scholars, and a series of detailed
and complex dynamic models were established. Zhai and Sun [14] proposed the vehicle-track coupled dynamic model considering the vehicle system and track system as a coupled large-scale system, and the wheel-rail dynamic interaction was studied from the overall point of view of vehicle-track integrated system. Besides, the research results have been widely used in engineering field. Zhai [15] also proposed an efficient and fast explicit numerical integration method for solving large multi-degree-of-freedom vehicle-track coupled system. Further development of computer technology has provided the better conditions for researchers to solve such large-scale dynamics systems. Researchers could establish a more detailed and comprehensive dynamic model to study the dynamic characteristics of vehicle and track structures, so as to ensure the safe service of vehicle and track structures. Based on the vehicle-track coupled dynamics model [14], Chen et al. [16-19] established a locomotive-track vertical-longitudinal coupled dynamic model considering the coupling effect of gear transmission system under complex wheel-rail excitation, based on the established model, they studied the dynamic response of the system under complex wheel-rail excitation and gear dynamic mesh force [16], the dynamic characteristics of the locomotive system under traction condition [17], and the vibration characteristics of the locomotive system when tooth root crack appeared in the gear transmission system [18,19]. Yao et al. [20] studied the vibration characteristic of components of traction drive system and the dynamic characteristics of locomotive traction drive system under saturated wheel-rail adhesion. Zhang et al. [21] based on the field test data of vibration acceleration and dynamic stress of high-speed train gearbox and did research on gearbox fault diagnose. Chen et al. [22] studied the method of extracting vibration characteristics of locomotive gear transmission system under complex excitations such as wheel-rail nonlinear contact. Based on vehicle-track coupled dynamics, Zhang et al. [23,24] established a 3-D locomotive-track coupled dynamics model considering the coupling effect of gear transmission system, and studied the dynamic effects of gear transmission system on vehicle system. Generally speaking, very few literatures have studied the effect of tooth root crack on the dynamic performance of locomotive dynamics systems, especially in spatial dynamics systems. To make up this gap, a spatial dynamics model of a high-power heavy-haul electric locomotive with gear transmission system considering tooth root crack is also established in this paper based on the multibody dynamics theory, where all the six degrees of freedom are included for each component in the locomotive. This dynamics model could supply more realistic vibration environment for investigation of gear tooth fault feature. And then, the fault vibration characteristics induced by the gear tooth root crack are studied in the vibration environment of the entire locomotive dynamics system, and effect of the crack propagation along the depth on the dynamic performance of the system is quantitatively analyzed. The angular synchronous average technique is used to extract the feature of weak fault signal from the strong vibration environment. And finally, the statistical indicators, namely the CF, K, FM4, M6A, and M8A, are used to assess the gear tooth root crack fault severity. The research results could be expected to offer the theoretical guidance to ensure the safe and efficient transportation of the heavy haul railway. The remaining part of this paper is organized as follows: Section 2 gives a description of modeling of heavy-haul locomotive considering gear transmission with tooth root crack. Section 3 investigates the influence of tooth root crack on the dynamic performance of locomotive system by time-frequency analysis. Section 4 analyzes the effect of crack size on the dynamic performance of locomotive system combing the statistical indicators and angular synchronous
averaging method. Section 5 gives the conclusions.
2 Spatial dynamics model of heavy-haul locomotive with gear tooth root crack fault 2.1 Gear time-varying mesh stiffness excitation with tooth root crack fault After some years of research and development since they were imported to China, the HX series of high-power heavy-haul electric locomotives have been popularly used. As the object of this study, the spur gear transmission systems are employed in the typical heavy-haul electric locomotives, and its main design parameters are shown in Table 1. Table 1
Main design parameters of the gear transmissions
Specificatin
Pinion
Gear
Material
Steel
Pressure angle(degree)
20
Number of teeth
23
120
Tooth width (mm)
175
136
Radius of the inner hub (mm)
23
70
Addendum coefficient
1
Tip clearance coefficient
0.25
The gear transmission system of a high-power heavy-haul locomotive is prone to failure under the combined action of emergency traction/braking conditions and poor track geometric irregularity excitations. Actually, there have been many faults found in the gear transmission system during the operation process. According to reference [25], a tooth root crack is seeded into one tooth of the pinion of the locomotive gear transmission system where the tooth root crack depth has a uniform distribution along the tooth width in this paper and shown in Fig. 1a). Different crack depth values, namely, 0 mm, 2 mm, 5 mm and 9 mm, are assumed respectively. The corresponding time-varying gear mesh stiffness of the locomotive is obtained by using the more accurate analytical calculation model proposed by Chen and Shao [9, 26] based on the principle of potential energy. The time-varying mesh stiffness results are shown in Fig. 1.
Mesh stiffness/ (MN/mm)
2.7
2.0 1.6 1.2 0.0
a)
0mm 2mm 5mm 9mm
2.4
0.4
0.8
b)
1.2
Mesh cycle
1.6
2.0
Fig. 1 Sketch of cracked gears and TVMS of gear pair with different crack sizes: a) cracked gears, b) calculated TVMS
Then, the dynamic model of the gear transmission system of the heavy-haul electric locomotive considering torsional vibration is established by using the Moved Marker technique and function expression provided by the software of SIMPACK. It should be noted that the -82 congruent marker in Simpack is adopted in this manuscript, this moved marker allows the position and orientation of a marker which is defined on one body to be determined by the position and orientation of another marker on a different body. Afterwards, the obtained time-varying gear mesh stiffness of the locomotive is introduced into the gear transmission dynamics model as an internal excitation.
2.2 Dynamics model of the heavy-haul electric locomotive The heavy-haul electric locomotive mainly consists of one car-body, two bogie frames, four wheelsets, and four gearboxes. In addition, there are primary and secondary suspension units which play a role in connecting major components and improving locomotive dynamic performance. The main design parameters of the heavy-haul electric locomotive are shown in Table 2. The established dynamics model of the gear transmission system with tooth root crack is then incorporated into the dynamics model of the traditional locomotive dynamics model where the coupling effect between the gear transmission system and the vehicle system is considered. This dynamics model enables the more realistic simulation of the gear fault induced vibrations in the whole locomotive dynamics system. The established locomotive dynamics model is shown in Fig. 2, which has totally 43 degrees of freedom. It is assumed that the tooth root crack appears in the gear transmission system mounted on the first wheelset. It should be noted that the dynamics model of the heavy-haul electric locomotive without tooth root crack had been verified by the field test data, and the relevant contents can be referred to our previously published paper [27]. Compared to our previously published paper, the fault vibration characteristic by the transmission system with the appearance of tooth root crack is further studied by using the time-frequency analysis and the statistical indicators in this paper, which are going to be introduced in Section 4. Based on this work, different influencing law of the tooth root crack on the spatial vibrations (e.g. vertical, longitudinal, lateral vibrations) of different components could be obtained so as to supply more guidance to the gear fault diagnosis and condition monitoring of railway locomotives. Table 2
Main design parameters of the locomotive dynamics system
Specification
Value
Car body mass
6.26 ´ 104 kg
Bogie mass
6.275 ´ 103 kg
Wheelset mass
2.77 ´ 103 kg
Motor mass
2.66 ´ 103 kg
Mass moment of inertia of car body (roll)
2.76 ´ 105 kg m2
Mass moment of inertia of car body (pitch)
1.434 ´ 106 kg m2
Mass moment of inertia of car body (yaw)
1.22 ´ 106 kg m2
Mass moment of inertia of bogie (roll)
5.39 ´ 103 kg m2
Mass moment of inertia of bogie (pitch)
1.311 ´ 104 kg m2
Mass moment of inertia of bogie (yaw)
1.68 ´ 104 kg m2
Mass moment of inertia of wheelset (roll)
2.48 ´ 103 kg m2
Mass moment of inertia of wheelset (pitch)
1.081 ´ 103 kg m2
Mass moment of inertia of wheelset (yaw)
2.96 ´ 103 kg m2
Mass moment of inertia of motor (roll)
6.04 ´ 102 kg m2
Mass moment of inertia of motor (pitch)
4 ´ 102 kg m2
Mass moment of inertia of motor (yaw)
8.01 ´ 102 kg m2
Vertical stiffness of primary suspension
2.1 MN/m
Lateral stiffness of primary suspension
5.7 MN/m
Longitudinal stiffness of primary suspension
1.44 MN/m
Vertical damping coefficient of primary suspension
25000 N · s/m
Vertical stiffness of second suspension
1.07 MN/m
Lateral stiffness of second suspension
0.332 MN/m
Longitudinal stiffness of secondary suspension
0.332 MN/m
Vertical damping coefficient of secondary suspension
45000 N · s/m
Lateral damping coefficient of secondary suspension
79000 N · s/m
Stiffness of secondary lateral stop
1.575 MN/m
Longitudinal stiffness of axle box link
164.5 MN/m
Lateral stiffness of axle box link
57 MN/m
Wheel radius
0.625 m
Fig 2. Dynamics model of the heavy-haul electrical locomotive
3 Effect of tooth root crack on locomotive dynamics in transient conditions The locomotive traction characteristic curve is as shown in Fig. 3. This traction torque is applied to the rotor of the electric motor, and then delivered to the pinion of the gear transmission system, and finally transmitted to the wheel-rail contact interface to generate the longitudinal forces so as to overcome the operation resistance of the entire train. Thus, the practical simulation on the transient operation of the locomotive and investigation on the gear tooth root crack fault vibration characteristics are becoming possible with using this locomotive dynamics model. In this investigation, the fourth class geometric irregularities of the Association of American Railroads is
adopted. The following dynamic responses in the locomotive system are extracted for the following analysis: the vertical vibration acceleration (VVA), the lateral vibration acceleration (LVA), and the longitudinal vibration acceleration (LOVA) of the first wheelset; and the dynamic mesh force (DMF) of the gear transmission system on first wheelset; as well as the vertical, lateral and longitudinal vibration acceleration of the front bogie frame. Tractive torque of motor/(kN.m)
12 10 8 6 4 2 0
20
40 60 80 Running speed/(km/h)
100
120
Fig.3 Tractive characteristic curve of the locomotive motor
In the driving process where the locomotive is accelerated, the calculated dynamic responses of the locomotive dynamics system is of nonstationary signals. So, the short-time Fourier transform technique is adopted to analyze the variation trend of the vibration frequency structure versus time. The obtained time-frequency results with using the short-time Fourier transform are shown in Figs. 4 to 9, respectively. The time-frequency distribution of the dynamic mesh force, the lateral, vertical and longitudinal vibration accelerations of the first wheelset are respectively given in Figs. 4 to 7. It should be noted that Fig. 4a), Fig. 5a), Fig. 6a) and Fig. 7a) correspond respectively to the calculated results without tooth root crack, while Fig. 4b), Fig. 5b), Fig. 6b), Fig. 7b) show respectively the calculated results in the case that a 9 mm tooth root crack is present. According to Figs. 4 to 7, it can be found that the dynamic mesh force, the lateral, vertical and longitudinal vibration acceleration of the first wheelset could clearly reflect the fundamental gear mesh frequency and its harmonics, besides, there appear also some sidebands that symmetrically distributed about the mesh frequency and its harmonics regardless of whether there is a tooth root crack or not during the operation process of the locomotive. The difference lies at: (1) the fault characteristic frequency induced by the tooth root crack appears in the high frequency band (600-1200Hz), as shown in Fig. 4b), Fig. 6b) and Fig.7b), which indicates the vibration information of the tooth root crack is embodied in the dynamic mesh force, the vertical and longitudinal vibration acceleration of the wheelset; (2) the phenomenon of sideband symmetrically distributed around the fundamental mesh frequency and its harmonics in the lateral and longitudinal vibration acceleration is more obvious than others, as shown in Figs. 5 and 7; (3) the lateral vibration acceleration of the gear pair is difficult to reflect the fault vibration characteristics caused by tooth root crack, as shown in Fig. 5, which indicates that the tooth root crack has no obvious influence on the locomotive lateral dynamics; (4) the vertical and the longitudinal vibration acceleration of wheelset contain more gear fault-related information which could reflect the sidebands more clearly.
a) healthy
b) with crack
Fig. 4 Time-frequency distribution of gear dynamic mesh force
a) healthy
b) with crack
Fig. 5 Time-frequency distribution of wheeelset lateral vibrations
a) healthy
b) with crack
Fig. 6 Time-frequency distribution of wheeelset vertical vibrations
a) healthy
b) with crack
Fig. 7 Time-frequency distribution of wheeelset longitudinal vibrations
Figures 8 and 9 display the time-frequency distribution results of the lateral and the vertical vibration accelerations, respectively. Here, Figs. 8a) and 9a) correspond to the results without tooth root crack, and Figs. 8b) and 9b) illustrate the results corresponding to the case with a tooth root crack of 9 mm depth, respectively. According to Fig 9, it can be found that the vertical vibration acceleration of the bogie frame could reflect the fundamental gear mesh frequency and its harmonics, and the sidebands are also symmetrically distributed around them. However, it is very difficult to find the characteristic frequencies induced by gear mesh and tooth root crack in the lateral vibration acceleration of the bogie frame, as shown in Fig 8. Besides the longitudinal vibration acceleration of the bogie frame are similar to those of the lateral vibration acceleration which is not displayed here.
a) healthy
b) with crack
Fig. 8 Time-frequency distribution of bogie frame lateral vibrations
a) healthy
b) with crack
Fig. 9 Time-frequency distribution of bogie frame vertical vibrations
Comparsions of these results presented above, such as the dynamic mesh force (in Fig. 4), the vibration acceleration of wheelset (in Figs. 5-7), and the vibration acceleration of bogie frame (in Figs. 8-9), it can be found that: (1) attenuation effect of primary suspension to high frequency excitation and the weakness of the fault vibration characteristics (especially for the incipient fault stage) caused by tooth root crack make it very difficult to transmit these vibration signals to the bogie frame. Therefore, the fault vibration characteristic frequencies caused by tooth root crack are hardly found in the vibration acceleration of the bogie frame, and very little proportion of the gear mesh frequency and its harmonic components could be transmitted to the bogie frame. (2) There is no characteristic frequency components induced by tooth root crack in the lateral vibration acceleration of the wheelset and the bogie frame due to that the spur gear transmission system will not generate the excitation along the axis of the gear shaft.
4 Fault feature of gear tooth root crack propagation based on angular synchronous average method Condition Monitoring and fault diagnosis of the locomotive gear transmission system are the final purpose of studying the vibration characteristics of gear tooth root crack. While cross-term may occur if the response signal is processed by short-time Fourier transform, which is a disadvantage in condition monitoring and fault diagnosis for gear tooth root crack fault. Thus angular synchronous averaging method (ASA) is used to enhance the fault feature from the transient signals since it has been proven that the ASA method is an effective way for the transient signals. The formula for the angular synchronous average is shown as follows,
y ( nDq ) =
1 N
N -1
å
x( nDq - r MDq )
(1)
r =0
where, Δθ denotes the angular displacement interval after resampling the original data, equals 0.0001π, M denotes the number of data points in each period. N denotes the number of data segments used for weighted average. y(nΔθ) represents the processed data. In this paper, the ASA method is used to process the original data. The whole original signal is divided into small segments with angular displacement of 2π. Figure 10 is the original response signal of the dynamic mesh force for healthy gears and faulty gear with tooth root crack in 9 mm depth. It can be seen that it’s very difficult to find the fault
vibration characteristics. In addition, Fig. 11 is the dynamic mesh force of gears with different crack depths processed by ASA method. According to Fig. 11, the fault vibration characteristics caused by the tooth root crack can be clearly found. When the cracked tooth is entering the meshing zone, the amplitude of the dynamic mesh force increases, while after entered, the amplitude of the dynamic mesh force decreases relative to the healthy condition.
-40 -60
DMF/kN
-80 -100 -120 -140
Crack-0mm Crack-9mm
-160 -180 0
2000
4000
6000
8000
Angular displacement of pinion/rad
Fig .10 Original signal of dynamic meshing force
-60
Failure zone
DMF/kN
-70
Crack-0mm Crack-5mm
Crack-2mm Crack-9mm
-80 -90
-100 -110 -120 0.5
1.0
1.5
2.0
Angular displacement of pinion/rad
2.5
3.0
Fig .11 Dynamic mesh force after ASA
The energy of vibration signal will change when the gears have faults, luckily the condition indicators could reflect the energy change of the vibration signal caused by gear fault, which could be used for condition monitoring and fault diagnose [10]. In this paper, effects of the crack depth on the locomotive dynamics system are also revealed by using some statistical indicators, such as the Crest Factor (CF), Kurtosis (K), Fourth Order Figure of Merit (FM4), M6A, M8A [10]. These condition indicators mentioned above are introduced briefly by the following Eqs. (2) to (6), respectively. Crest Factor, which is defined as the ratio of the maximum peak value of the time domain signal to the root mean square value, is calculated as,
CF =
x0- p k RMS x
(2)
Kurtosis could highlight the high amplitude signal while suppressed the low amplitude signal
due to the amplitude of the signal is processed by the fourth order. It is given by, N
K=
N å i =1 ( xi - x)4
(3)
N
(å i =1 ( xi - x) 2 ) 2
Zero Order Figure of Merit (FM0) was developed as a robust indicator of major faults in a gear mesh, while the Fourth Order Figure of Merit (FM4) was further developed to detect faults isolated to only a finite number of teeth, which is shown as, N
FM 4 =
N å i =1 (di - d )4
(4)
N
(å i =1 (di - d ) 2 ) 2
Another indicator, M6A, uses 6 moments, which is believed to be more sensitive to the peaks in difference signals. The calculation is shown in Eq. (5). Similarly, the calculation of the indicator, M8A, is shown in Eq. (6). N
M 6A =
N 2 å i =1 (di - d )6
M 8A =
N 2 å i =1 (di - d )8
(5)
N
(å i =1 (di - d ) 2 )3 N
(6)
N
(å i =1 (di - d ) 2 ) 4
In addition, the significance of various parameters in Eqs. (2) to (6) can be referred to Ref. [10]. Relative to the healthy condition, the change percentage of CF, K, FM4, M6A and M8A of dynamic mesh force and vertical vibration acceleration of wheelset which calculated from the locomotive system vibration signal that processed by angular synchronous averaging method under different crack depth are presented in Fig. 12. According to Fig. 12a), it can be found that the crack propagation along crack depth will decrease the FM4, M6A and M8A of gear dynamic mesh force and these 3 condition indicators almost keep the same trend of change, as well increase the K value of the dynamic mesh force. While there is no significant change in CF. It can also be found that the FM4, M6A and M8A values of vertical vibration acceleration of wheel set will increase with the crack propagation along crack depth from Fig. 12b), besides, there is no significant change in the CF and K values of vertical vibration acceleration of wheelset. 5%
100% CF FM4 K M6A M8A
0%
-10%
Change percentage
Change percentage
80% CF FM4 K M6A M8A
-20%
60%
40%
20% -30%
0%
-35% 0
2
4 6 Crack depth /mm
8
10
-10% 0
2
4 6 Crack depth /mm
8
10
Fig. 12 Change percentage of various statistical indicators versus crack depth: a) dynamic mesh force, b) Vertical vibration acceleration of wheelset.
Figure 13 shows the change percentage of CF, K, FM4, M6A and M8A of longitudinal and lateral vibration acceleration of wheelset which calculated from the locomotive system vibration signal that processed by angular synchronous averaging method under different crack depth. Similar to the change of condition indicators FM4, M6A and M7A of vertical vibration acceleration of wheelset, the FM4, M6A and M8A values of longitudinal vibration acceleration of wheelset will increase with the crack propagation along crack depth, while they are smaller than those of vertical vibration acceleration of wheelset as it shows in Fig.13 a). Besides, the CF will decrease at first and then increase with crack propagation along crack depth, however the variation of K is not obvious according to Fig. 13a). In addition, the trend in condition indicators of lateral vibration acceleration of wheelset is not obvious as it shows in Fig.13 b). 11%
70% CF FM4 K M6A M8A
Change percentage
8%
60%
CF FM4 K M6A M8A
40%
Change percentage
10%
6% 4% 2%
20% 0%
-20%
0% -40%
-2%
-50%
0
2
4 6 Crack depth /mm
8
10
0
2
4 6 Crack depth /mm
8
10
Fig 13 Change percentage of various statistical indicators versus crack depth: a) longitudinal vibration acceleration of wheelset, b) lateral vibration acceleration of wheelset
It can be seen from Figs. 12 and 13 that, the change of FM4, M6A, M8A of the vertical vibration acceleration of the wheelset is larger than that of the dynamic mesh force and the longitudinal vibration acceleration of wheelset, which could effectively reflect the crack propagation. The M8A is more sensitive to crack depth than M6A and FM4. And again, the lateral vibrations of the wheelset are not sensitive to the gear tooth root crack fault.
5 Conclusions A detailed three dimensional locomotive dynamics model is established to investigate the dynamic features of gear tooth root crack in a heavy-haul locomotive system. The fault features are analyzed by using the time-frequency analysis technology and represented by the statistical indicators. The main conclusions of this paper can be drawn as follows: 1) The time-frequency analysis results of dynamic meshing force, vertical and longitudinal vibration acceleration of wheelset can reflect the characteristic frequency of gear meshing and fault vibration characteristic frequency caused by tooth root crack. 2) The tooth root crack has little effect on the lateral dynamic performance of locomotive. Besides the characteristic frequency components of the fault vibration caused by tooth root crack is difficult to transfer to the bogie frame cause of the attenuation effect by the primary suspension system. 3) The FM4, M6A and M8A of dynamic mesh force and vertical vibration acceleration could
reflect the influence of crack depth on the dynamic characteristics of the system. The effect law of crack depth on FM4, M6A and M8A values of the vertical and longitudinal vibration accelerations of wheelset are similar with each other.
Acknowledgement The authors are grateful for the financial support provided by the National Natural Science Foundation of China (Grant Nos. 51775453, 51735012), the Fundamental Research Funds for the State Key Laboratory of Traction Power of Southwest Jiaotong University (Grant No. 2019TPL-T09), and the Fundamental Research Funds for the Central Universities (Grant No. 2682019YQ04).
Reference [1] China Railway General Company 2017 statistical bulletin. Ministry of development and reform of China Railway: http://www.china-railway.com.cn/cpyfw/tjxx/201803/t20180328_70388.html [2] Ma Hui, Zeng Jin, Feng Ranjiao. et al. Review on dynamics of cracked gear systems. Eng Fail Anal 2015; 55: 224–245. [3] Ma Hui, Pang Xu, Zeng Jin, et al. Effects of gear crack propagation paths on vibration responses of the perforated gear system. Mechanical System and Signal Processing 2015; 62, 113-128. [4] Ma Hui, Song Rongze, Pang Xu. Time-varying mesh stiffness calculation of cracked spur gears. Eng Fail Anal 2014; 44: 179-194. [5] Liang Xihui, Zuo Ming J, Feng Zhipeng. Dynamic modeling of gearbox faults: A review. Mech Syst Signal Process 2018; 98: 852-876. [6] Chen Kangkang, Huangfu Yifan, Ma Hui, et al.Calculation of mesh stiffness of spur gears considering complexfoundation types and crack propagation paths. Mech Syst Signal Process2019; 130: 273-292. [7] Chen Zaigang, Zhang Jie, Zhai Wanming, Wang Yawen, Liu Jianxin. Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack. Engineering Failure Analysis, 2017, 82, 72-81. [8] Liu Siyuan, Song Chaosheng, Zhu Caichao, Liang Chengcheng, Yang Xingyu. Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear. Mechanism and Machine Theory, 2019, 138, 95-111. [9] Chen Zaigang, Shao Yimin. Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth. Eng Fail Anal 2011; 18: 2149-2164. [10] Sharma Vikas, Parey Anand. A review of gear fault diagnoise using various condition indicators. Procedia Engineering 2016; 144: 253–263. [11] Garg VK, Dukkipati RV. Dynamics of railway vehicle systems. Toronto: Academic Press. 1984. [12] Wickens AH. Fundamentals of rail vehicle dynamics: guidance and stability. Lisse: Swets &Zeitlinger. 2003. [13] Zhai Wanming, Wang Kaiyun, Cai Chengbiao. Fundamentals of vehicle–track coupled dynamics. Vehicle System Dynamics 2009; 47: 1349–1376. [14] Zhai Wanming, Sun Xiang. A detailed model for investigating vertical interaction between railway vehicle and track. Vehicle System Dynamics 1994. [15] Zhai Wanming. Two simple fast integration methods for large-scale dynamic problems in engineering. International Journal for Numerical Methods in Engineering 1996; 39(24): 4199–4214. [16] Chen Zaigang, Zhai Wanming, Wang Kaiyun. Dynamic investigation of a locomotive with effect of gear transmissions under tractive conditions. Journal of Sound and Vibration 2017; 408: 220–223.
[17] Chen Zaigang, Zhai Wanming, Wang Kaiyun. A locomotive–track coupled vertical dynamics model with gear transmissions. Vehicle System Dynamics 2016; 55(2): 244-267. [18] Chen Zaigang, Zhai Wanming, Wang Kaiyun. Locomotive dynamic performance under traction/braking conditions considering effect of gear transmissions. Vehicle System Dynamics 2017: 1–21. [19] Chen Zaigang, Zhang Jie, Wang Zichao, et al. Study on vibration characteristics of a locomotive with gear tooth root crack by simulation. 2016 Prognostics and System Health Management Conference; 2017: 1–5. [20] Yao Yuan, Zhang Hongjun, Luo Shihui. An analysis of resonance effects in locomotive drive systems experiencing wheel/rail saturation adhesion. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 2014; 228(1): 4-15. [21] Zhang Bing, Andy C.C. Tan, Lin Jian-hui. Gearbox fault diagnosis of high-speed railway train. Eng Fail Anal 2016; 66: 407-420. [22] Chen Zaigang, Zhai Wanming, Wang Kaiyun. Vibration feature evolution of locomotive with tooth root crack propagation of gear transmission system. Mech Syst Signal Process 2019; 115: 29-44. [23] Zhang Tao, Chen Zaigang, Zhai Wanming, et al. Establishment and validation of a locomotive–track coupled spatial dynamics model considering dynamic effect of gear transmissions. Mech Syst Signal Process 2019; 119: 328-345. [24] Zhang Tao, Chen Zaigang, Zhai Wanming, et al. Effect of the drive system on locomotive dynamic characteristics using different dynamics models. SCIENCE CHINA -Technological Sciences 2019; 62(2): 308-320. [25] Jiang Jianzheng, Chen Zaigang, He Chunyan, et al. Fault vibration features of heavy-haul locomotive with tooth root crack in gear transmissions [C]. 2018 International Conference on Sensing, Diagnostics, Prognostics, and Control. IEEE, 2018, DOI: 10.1109/SDPC.2018.8664929. [26] Chen Zaigang, Shao Yimin. Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack. Mechanism and Machine Theory, 2013, 62, 63-74. [27] He Chunyan, Chen Zaigang, Zhai Wanming, et al. A spatial dynamics model for heavy-haul electric locomotive considering the dynamic coupling effect of gear transmissions. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 2019, 233(9), 961–973.
Highlights
l
A locomotive spatial dynamics model considering gear transmission system with gear tooth root crack is established.
l
The dynamic responses of the locomotive under transient condition are obtained by considering the complicated multiple excitations.
l
Time-frequency analysis technique is used to reveal the fault vibration features of different components under transient condition.
l
Combination of angular synchronous average method and statistical indicators is proposed for extraction of tooth root crack fault feature.
l
The studied results could supply some guidance for locomotive gear tooth fault diagnosis.
Conflict of interest The authors declared that they have no conflicts of interest to this work.
S1350-6307(19)30510-2 https://doi.org/10.1016/j.engfailanal.2019.104285 EFA 104285
To appear in:
Engineering Failure Analysis
Received Date: Revised Date: Accepted Date:
12 April 2019 29 August 2019 4 November 2019
Please cite this article as: Jiang, J., Chen, Z., Zhai, W., Zhang, T., Li, Y., Vibration Characteristics of Railway Locomotive Induced by Gear Tooth Root Crack Fault under Transient Conditions, Engineering Failure Analysis (2019), doi: https://doi.org/10.1016/j.engfailanal.2019.104285
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© 2019 Published by Elsevier Ltd.
Vibration Characteristics of Railway Locomotive Induced by Gear Tooth Root Crack Fault under Transient Conditions Jianzheng Jiang1,2, Zaigang Chen1,3*, Wanming Zhai1, Tao Zhang1, Yifan Li4 1
State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, People's Republic of China 2
3
China Railway Eryuan Engineering Group Co., Ltd. Chengdu 610031, People's Republic of China
The State Key Laboratory of Heavy Duty AC Drive Electric Locomotive Systems Integration, CRRC Zhuzhou Locomotive Co., LTD, Zhuzhou 412001, People's Republic of China 4
School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, People's Republic of China
*Corresponding author: Associate Prof. Zaigang CHEN Tel: +86-(0)28-86466504; Fax: +86-(0)28-87601843; E-mail address: [email protected] Abstract: Dynamic interactions between wheel and rail are becoming more intensified under the development of locomotive towards high-speed, heavy-haul and high-power directions, which are more likely to cause failures in gear transmission system of high-power heavy-haul electric locomotive. The dynamic impact of the locomotive gear transmission system becomes more drastic, especially for the extreme operation conditions, such as traction/braking process. Gear fault diagnosis under unsteady conditions has always been a hot and difficult research spot, and revealing of the gear fault vibration characteristics under the transient conditions is the premise and basis for effective fault detection and diagnosis. In order to reveal the gear fault vibration characteristics of railway locomotive dynamics system, a spatial dynamic model of a heavy-haul electric locomotive considering the dynamic coupling effect of gear transmission system is proposed based on the multibody dynamics theory in this paper. Then the dynamic responses of the locomotive under transient condition are obtained by considering the complicated excitations induced by the wheel-rail nonlinear contact, gear mesh and tooth root crack. The time-frequency analysis and angular synchronous average method are adopted to investigate the fault vibration feature of the tooth root crack. Finally, the distribution and variation of vibration characteristics of the gear tooth root crack fault evolution are revealed by condition indicators such as the Crest Factor (CF), Kurtosis (K), Fourth Order Figure of Merit (FM4), M6A, and M8A. The results indicate that: (1) the time-frequency analysis results of vibration acceleration of wheelset and bogie frame and dynamic mesh force could reflect the fundamental mesh frequency and its harmonics, however, only the vertical and longitudinal vibration acceleration of the wheelset and the gear dynamic mesh force could reflect the fault vibration characteristics in time-frequency analysis results; (2) the FM4, M6A and M8A values of locomotive system vibration signal could effectively reflect the crack propagation on the dynamic features of the locomotive system after being processed by angular synchronous averaging method.
Keywords: heavy-haul locomotive, multi-rigid body dynamics, tooth root crack, time-frequency analysis, angular synchronous average
1 Introduction The prompt growth of the world’s freight transportation has accelerated the development of heavy-haul railway transportation. For example, by 2017, the operating mileage of the Chinese railways reached 127,000 kilometers, and the total volume of railway freight transports reached 3.689 billion tons with an increase of 10.7% over the previous year [1]. Developing high-power heavy haul electric locomotives so as to meet the requirements towards the long marshalling and large axle-load train is imperative, which could improve the transportation capacity of the heavy-haul railway. However, operating the long-marshalling and large axle-load train will bring severe challenge to the service safety of the vehicles and the track structure. As a vital device in locomotive structure, gear transmission system endures the combined action of both the internal and external excitations such as the time-varying gear mesh stiffness, the wheel-rail nonlinear contact, and the traction/braking moment of motor, which is likely to produce failures to the gear and other components. Compared with healthy gears, the faulty gears will change the effective tooth thickness and contact area during meshing process, and further alter the stiffness and the error excitation of gears, which may change the dynamic responses of the gear transmission system. For example, a detailed review paper [2] further shows that the time-varying mesh stiffness (TVMS) of faulty gears is usually introduced into the dynamic model of gear transmission system to study the gear fault vibration features, like Ref [3]. Accordingly, how to accurately and efficiently obtain the time-varying mesh stiffness of gears with or without faults is significant to accurately investigate the fault features of the gear transmission system. For example, Ma et al. [4] proposed an improved mesh stiffness by considering the accurate tooth root transition curve and a parabolic curve to simulate the crack propagation path. Liang et al. [5] reviewed four methods for calculating the time-varying mesh stiffness of gears with or without faults based on existed literature, namely, square wave method, potential energy principle method, finite element method and experimental method. Chen et al. [6] established an analytical-finite element model to calculate the TVMS of spur gears based on finite element theory and loaded tooth contact analysis, where the complex gear foundation and crack propagation path were considered. Chen et al. [7] proposed an improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack which enables more accurate calculation of gear mesh stiffness. Liu et al. [8] investigated the gear mesh behavior by considering the work holding equipment errors based on the three-face cutter. Various forms of gear faults will lead to abnormal vibration of gear transmission system and shorten the service life of the system. At the same time, condition monitoring of gear transmission system can effectively avoid further deterioration of gear failure [9]. Based on published literature, Sharma and Parey [10] summarized the statistical indicators that can be used for condition monitoring and fault diagnose of gear transmission system in time domain and frequency domain. In classical railway vehicle dynamics research, vehicle system and track system are regarded as two independent subsystems. Scholars neglected the influence of track system on vehicle system when they study vehicle system dynamics [11,12]. In fact, the vibration of vehicle system and track system will affect each other through wheel-rail contact interface, thus, it is essential to build a large-scale system coupled by wheel-rail contact interface [13]. In recent years, the increasing operation speed and axle load of train has further aggravated the dynamic interaction between vehicle system and track system, which attracted the attention of scholars, and a series of detailed
and complex dynamic models were established. Zhai and Sun [14] proposed the vehicle-track coupled dynamic model considering the vehicle system and track system as a coupled large-scale system, and the wheel-rail dynamic interaction was studied from the overall point of view of vehicle-track integrated system. Besides, the research results have been widely used in engineering field. Zhai [15] also proposed an efficient and fast explicit numerical integration method for solving large multi-degree-of-freedom vehicle-track coupled system. Further development of computer technology has provided the better conditions for researchers to solve such large-scale dynamics systems. Researchers could establish a more detailed and comprehensive dynamic model to study the dynamic characteristics of vehicle and track structures, so as to ensure the safe service of vehicle and track structures. Based on the vehicle-track coupled dynamics model [14], Chen et al. [16-19] established a locomotive-track vertical-longitudinal coupled dynamic model considering the coupling effect of gear transmission system under complex wheel-rail excitation, based on the established model, they studied the dynamic response of the system under complex wheel-rail excitation and gear dynamic mesh force [16], the dynamic characteristics of the locomotive system under traction condition [17], and the vibration characteristics of the locomotive system when tooth root crack appeared in the gear transmission system [18,19]. Yao et al. [20] studied the vibration characteristic of components of traction drive system and the dynamic characteristics of locomotive traction drive system under saturated wheel-rail adhesion. Zhang et al. [21] based on the field test data of vibration acceleration and dynamic stress of high-speed train gearbox and did research on gearbox fault diagnose. Chen et al. [22] studied the method of extracting vibration characteristics of locomotive gear transmission system under complex excitations such as wheel-rail nonlinear contact. Based on vehicle-track coupled dynamics, Zhang et al. [23,24] established a 3-D locomotive-track coupled dynamics model considering the coupling effect of gear transmission system, and studied the dynamic effects of gear transmission system on vehicle system. Generally speaking, very few literatures have studied the effect of tooth root crack on the dynamic performance of locomotive dynamics systems, especially in spatial dynamics systems. To make up this gap, a spatial dynamics model of a high-power heavy-haul electric locomotive with gear transmission system considering tooth root crack is also established in this paper based on the multibody dynamics theory, where all the six degrees of freedom are included for each component in the locomotive. This dynamics model could supply more realistic vibration environment for investigation of gear tooth fault feature. And then, the fault vibration characteristics induced by the gear tooth root crack are studied in the vibration environment of the entire locomotive dynamics system, and effect of the crack propagation along the depth on the dynamic performance of the system is quantitatively analyzed. The angular synchronous average technique is used to extract the feature of weak fault signal from the strong vibration environment. And finally, the statistical indicators, namely the CF, K, FM4, M6A, and M8A, are used to assess the gear tooth root crack fault severity. The research results could be expected to offer the theoretical guidance to ensure the safe and efficient transportation of the heavy haul railway. The remaining part of this paper is organized as follows: Section 2 gives a description of modeling of heavy-haul locomotive considering gear transmission with tooth root crack. Section 3 investigates the influence of tooth root crack on the dynamic performance of locomotive system by time-frequency analysis. Section 4 analyzes the effect of crack size on the dynamic performance of locomotive system combing the statistical indicators and angular synchronous
averaging method. Section 5 gives the conclusions.
2 Spatial dynamics model of heavy-haul locomotive with gear tooth root crack fault 2.1 Gear time-varying mesh stiffness excitation with tooth root crack fault After some years of research and development since they were imported to China, the HX series of high-power heavy-haul electric locomotives have been popularly used. As the object of this study, the spur gear transmission systems are employed in the typical heavy-haul electric locomotives, and its main design parameters are shown in Table 1. Table 1
Main design parameters of the gear transmissions
Specificatin
Pinion
Gear
Material
Steel
Pressure angle(degree)
20
Number of teeth
23
120
Tooth width (mm)
175
136
Radius of the inner hub (mm)
23
70
Addendum coefficient
1
Tip clearance coefficient
0.25
The gear transmission system of a high-power heavy-haul locomotive is prone to failure under the combined action of emergency traction/braking conditions and poor track geometric irregularity excitations. Actually, there have been many faults found in the gear transmission system during the operation process. According to reference [25], a tooth root crack is seeded into one tooth of the pinion of the locomotive gear transmission system where the tooth root crack depth has a uniform distribution along the tooth width in this paper and shown in Fig. 1a). Different crack depth values, namely, 0 mm, 2 mm, 5 mm and 9 mm, are assumed respectively. The corresponding time-varying gear mesh stiffness of the locomotive is obtained by using the more accurate analytical calculation model proposed by Chen and Shao [9, 26] based on the principle of potential energy. The time-varying mesh stiffness results are shown in Fig. 1.
Mesh stiffness/ (MN/mm)
2.7
2.0 1.6 1.2 0.0
a)
0mm 2mm 5mm 9mm
2.4
0.4
0.8
b)
1.2
Mesh cycle
1.6
2.0
Fig. 1 Sketch of cracked gears and TVMS of gear pair with different crack sizes: a) cracked gears, b) calculated TVMS
Then, the dynamic model of the gear transmission system of the heavy-haul electric locomotive considering torsional vibration is established by using the Moved Marker technique and function expression provided by the software of SIMPACK. It should be noted that the -82 congruent marker in Simpack is adopted in this manuscript, this moved marker allows the position and orientation of a marker which is defined on one body to be determined by the position and orientation of another marker on a different body. Afterwards, the obtained time-varying gear mesh stiffness of the locomotive is introduced into the gear transmission dynamics model as an internal excitation.
2.2 Dynamics model of the heavy-haul electric locomotive The heavy-haul electric locomotive mainly consists of one car-body, two bogie frames, four wheelsets, and four gearboxes. In addition, there are primary and secondary suspension units which play a role in connecting major components and improving locomotive dynamic performance. The main design parameters of the heavy-haul electric locomotive are shown in Table 2. The established dynamics model of the gear transmission system with tooth root crack is then incorporated into the dynamics model of the traditional locomotive dynamics model where the coupling effect between the gear transmission system and the vehicle system is considered. This dynamics model enables the more realistic simulation of the gear fault induced vibrations in the whole locomotive dynamics system. The established locomotive dynamics model is shown in Fig. 2, which has totally 43 degrees of freedom. It is assumed that the tooth root crack appears in the gear transmission system mounted on the first wheelset. It should be noted that the dynamics model of the heavy-haul electric locomotive without tooth root crack had been verified by the field test data, and the relevant contents can be referred to our previously published paper [27]. Compared to our previously published paper, the fault vibration characteristic by the transmission system with the appearance of tooth root crack is further studied by using the time-frequency analysis and the statistical indicators in this paper, which are going to be introduced in Section 4. Based on this work, different influencing law of the tooth root crack on the spatial vibrations (e.g. vertical, longitudinal, lateral vibrations) of different components could be obtained so as to supply more guidance to the gear fault diagnosis and condition monitoring of railway locomotives. Table 2
Main design parameters of the locomotive dynamics system
Specification
Value
Car body mass
6.26 ´ 104 kg
Bogie mass
6.275 ´ 103 kg
Wheelset mass
2.77 ´ 103 kg
Motor mass
2.66 ´ 103 kg
Mass moment of inertia of car body (roll)
2.76 ´ 105 kg m2
Mass moment of inertia of car body (pitch)
1.434 ´ 106 kg m2
Mass moment of inertia of car body (yaw)
1.22 ´ 106 kg m2
Mass moment of inertia of bogie (roll)
5.39 ´ 103 kg m2
Mass moment of inertia of bogie (pitch)
1.311 ´ 104 kg m2
Mass moment of inertia of bogie (yaw)
1.68 ´ 104 kg m2
Mass moment of inertia of wheelset (roll)
2.48 ´ 103 kg m2
Mass moment of inertia of wheelset (pitch)
1.081 ´ 103 kg m2
Mass moment of inertia of wheelset (yaw)
2.96 ´ 103 kg m2
Mass moment of inertia of motor (roll)
6.04 ´ 102 kg m2
Mass moment of inertia of motor (pitch)
4 ´ 102 kg m2
Mass moment of inertia of motor (yaw)
8.01 ´ 102 kg m2
Vertical stiffness of primary suspension
2.1 MN/m
Lateral stiffness of primary suspension
5.7 MN/m
Longitudinal stiffness of primary suspension
1.44 MN/m
Vertical damping coefficient of primary suspension
25000 N · s/m
Vertical stiffness of second suspension
1.07 MN/m
Lateral stiffness of second suspension
0.332 MN/m
Longitudinal stiffness of secondary suspension
0.332 MN/m
Vertical damping coefficient of secondary suspension
45000 N · s/m
Lateral damping coefficient of secondary suspension
79000 N · s/m
Stiffness of secondary lateral stop
1.575 MN/m
Longitudinal stiffness of axle box link
164.5 MN/m
Lateral stiffness of axle box link
57 MN/m
Wheel radius
0.625 m
Fig 2. Dynamics model of the heavy-haul electrical locomotive
3 Effect of tooth root crack on locomotive dynamics in transient conditions The locomotive traction characteristic curve is as shown in Fig. 3. This traction torque is applied to the rotor of the electric motor, and then delivered to the pinion of the gear transmission system, and finally transmitted to the wheel-rail contact interface to generate the longitudinal forces so as to overcome the operation resistance of the entire train. Thus, the practical simulation on the transient operation of the locomotive and investigation on the gear tooth root crack fault vibration characteristics are becoming possible with using this locomotive dynamics model. In this investigation, the fourth class geometric irregularities of the Association of American Railroads is
adopted. The following dynamic responses in the locomotive system are extracted for the following analysis: the vertical vibration acceleration (VVA), the lateral vibration acceleration (LVA), and the longitudinal vibration acceleration (LOVA) of the first wheelset; and the dynamic mesh force (DMF) of the gear transmission system on first wheelset; as well as the vertical, lateral and longitudinal vibration acceleration of the front bogie frame. Tractive torque of motor/(kN.m)
12 10 8 6 4 2 0
20
40 60 80 Running speed/(km/h)
100
120
Fig.3 Tractive characteristic curve of the locomotive motor
In the driving process where the locomotive is accelerated, the calculated dynamic responses of the locomotive dynamics system is of nonstationary signals. So, the short-time Fourier transform technique is adopted to analyze the variation trend of the vibration frequency structure versus time. The obtained time-frequency results with using the short-time Fourier transform are shown in Figs. 4 to 9, respectively. The time-frequency distribution of the dynamic mesh force, the lateral, vertical and longitudinal vibration accelerations of the first wheelset are respectively given in Figs. 4 to 7. It should be noted that Fig. 4a), Fig. 5a), Fig. 6a) and Fig. 7a) correspond respectively to the calculated results without tooth root crack, while Fig. 4b), Fig. 5b), Fig. 6b), Fig. 7b) show respectively the calculated results in the case that a 9 mm tooth root crack is present. According to Figs. 4 to 7, it can be found that the dynamic mesh force, the lateral, vertical and longitudinal vibration acceleration of the first wheelset could clearly reflect the fundamental gear mesh frequency and its harmonics, besides, there appear also some sidebands that symmetrically distributed about the mesh frequency and its harmonics regardless of whether there is a tooth root crack or not during the operation process of the locomotive. The difference lies at: (1) the fault characteristic frequency induced by the tooth root crack appears in the high frequency band (600-1200Hz), as shown in Fig. 4b), Fig. 6b) and Fig.7b), which indicates the vibration information of the tooth root crack is embodied in the dynamic mesh force, the vertical and longitudinal vibration acceleration of the wheelset; (2) the phenomenon of sideband symmetrically distributed around the fundamental mesh frequency and its harmonics in the lateral and longitudinal vibration acceleration is more obvious than others, as shown in Figs. 5 and 7; (3) the lateral vibration acceleration of the gear pair is difficult to reflect the fault vibration characteristics caused by tooth root crack, as shown in Fig. 5, which indicates that the tooth root crack has no obvious influence on the locomotive lateral dynamics; (4) the vertical and the longitudinal vibration acceleration of wheelset contain more gear fault-related information which could reflect the sidebands more clearly.
a) healthy
b) with crack
Fig. 4 Time-frequency distribution of gear dynamic mesh force
a) healthy
b) with crack
Fig. 5 Time-frequency distribution of wheeelset lateral vibrations
a) healthy
b) with crack
Fig. 6 Time-frequency distribution of wheeelset vertical vibrations
a) healthy
b) with crack
Fig. 7 Time-frequency distribution of wheeelset longitudinal vibrations
Figures 8 and 9 display the time-frequency distribution results of the lateral and the vertical vibration accelerations, respectively. Here, Figs. 8a) and 9a) correspond to the results without tooth root crack, and Figs. 8b) and 9b) illustrate the results corresponding to the case with a tooth root crack of 9 mm depth, respectively. According to Fig 9, it can be found that the vertical vibration acceleration of the bogie frame could reflect the fundamental gear mesh frequency and its harmonics, and the sidebands are also symmetrically distributed around them. However, it is very difficult to find the characteristic frequencies induced by gear mesh and tooth root crack in the lateral vibration acceleration of the bogie frame, as shown in Fig 8. Besides the longitudinal vibration acceleration of the bogie frame are similar to those of the lateral vibration acceleration which is not displayed here.
a) healthy
b) with crack
Fig. 8 Time-frequency distribution of bogie frame lateral vibrations
a) healthy
b) with crack
Fig. 9 Time-frequency distribution of bogie frame vertical vibrations
Comparsions of these results presented above, such as the dynamic mesh force (in Fig. 4), the vibration acceleration of wheelset (in Figs. 5-7), and the vibration acceleration of bogie frame (in Figs. 8-9), it can be found that: (1) attenuation effect of primary suspension to high frequency excitation and the weakness of the fault vibration characteristics (especially for the incipient fault stage) caused by tooth root crack make it very difficult to transmit these vibration signals to the bogie frame. Therefore, the fault vibration characteristic frequencies caused by tooth root crack are hardly found in the vibration acceleration of the bogie frame, and very little proportion of the gear mesh frequency and its harmonic components could be transmitted to the bogie frame. (2) There is no characteristic frequency components induced by tooth root crack in the lateral vibration acceleration of the wheelset and the bogie frame due to that the spur gear transmission system will not generate the excitation along the axis of the gear shaft.
4 Fault feature of gear tooth root crack propagation based on angular synchronous average method Condition Monitoring and fault diagnosis of the locomotive gear transmission system are the final purpose of studying the vibration characteristics of gear tooth root crack. While cross-term may occur if the response signal is processed by short-time Fourier transform, which is a disadvantage in condition monitoring and fault diagnosis for gear tooth root crack fault. Thus angular synchronous averaging method (ASA) is used to enhance the fault feature from the transient signals since it has been proven that the ASA method is an effective way for the transient signals. The formula for the angular synchronous average is shown as follows,
y ( nDq ) =
1 N
N -1
å
x( nDq - r MDq )
(1)
r =0
where, Δθ denotes the angular displacement interval after resampling the original data, equals 0.0001π, M denotes the number of data points in each period. N denotes the number of data segments used for weighted average. y(nΔθ) represents the processed data. In this paper, the ASA method is used to process the original data. The whole original signal is divided into small segments with angular displacement of 2π. Figure 10 is the original response signal of the dynamic mesh force for healthy gears and faulty gear with tooth root crack in 9 mm depth. It can be seen that it’s very difficult to find the fault
vibration characteristics. In addition, Fig. 11 is the dynamic mesh force of gears with different crack depths processed by ASA method. According to Fig. 11, the fault vibration characteristics caused by the tooth root crack can be clearly found. When the cracked tooth is entering the meshing zone, the amplitude of the dynamic mesh force increases, while after entered, the amplitude of the dynamic mesh force decreases relative to the healthy condition.
-40 -60
DMF/kN
-80 -100 -120 -140
Crack-0mm Crack-9mm
-160 -180 0
2000
4000
6000
8000
Angular displacement of pinion/rad
Fig .10 Original signal of dynamic meshing force
-60
Failure zone
DMF/kN
-70
Crack-0mm Crack-5mm
Crack-2mm Crack-9mm
-80 -90
-100 -110 -120 0.5
1.0
1.5
2.0
Angular displacement of pinion/rad
2.5
3.0
Fig .11 Dynamic mesh force after ASA
The energy of vibration signal will change when the gears have faults, luckily the condition indicators could reflect the energy change of the vibration signal caused by gear fault, which could be used for condition monitoring and fault diagnose [10]. In this paper, effects of the crack depth on the locomotive dynamics system are also revealed by using some statistical indicators, such as the Crest Factor (CF), Kurtosis (K), Fourth Order Figure of Merit (FM4), M6A, M8A [10]. These condition indicators mentioned above are introduced briefly by the following Eqs. (2) to (6), respectively. Crest Factor, which is defined as the ratio of the maximum peak value of the time domain signal to the root mean square value, is calculated as,
CF =
x0- p k RMS x
(2)
Kurtosis could highlight the high amplitude signal while suppressed the low amplitude signal
due to the amplitude of the signal is processed by the fourth order. It is given by, N
K=
N å i =1 ( xi - x)4
(3)
N
(å i =1 ( xi - x) 2 ) 2
Zero Order Figure of Merit (FM0) was developed as a robust indicator of major faults in a gear mesh, while the Fourth Order Figure of Merit (FM4) was further developed to detect faults isolated to only a finite number of teeth, which is shown as, N
FM 4 =
N å i =1 (di - d )4
(4)
N
(å i =1 (di - d ) 2 ) 2
Another indicator, M6A, uses 6 moments, which is believed to be more sensitive to the peaks in difference signals. The calculation is shown in Eq. (5). Similarly, the calculation of the indicator, M8A, is shown in Eq. (6). N
M 6A =
N 2 å i =1 (di - d )6
M 8A =
N 2 å i =1 (di - d )8
(5)
N
(å i =1 (di - d ) 2 )3 N
(6)
N
(å i =1 (di - d ) 2 ) 4
In addition, the significance of various parameters in Eqs. (2) to (6) can be referred to Ref. [10]. Relative to the healthy condition, the change percentage of CF, K, FM4, M6A and M8A of dynamic mesh force and vertical vibration acceleration of wheelset which calculated from the locomotive system vibration signal that processed by angular synchronous averaging method under different crack depth are presented in Fig. 12. According to Fig. 12a), it can be found that the crack propagation along crack depth will decrease the FM4, M6A and M8A of gear dynamic mesh force and these 3 condition indicators almost keep the same trend of change, as well increase the K value of the dynamic mesh force. While there is no significant change in CF. It can also be found that the FM4, M6A and M8A values of vertical vibration acceleration of wheel set will increase with the crack propagation along crack depth from Fig. 12b), besides, there is no significant change in the CF and K values of vertical vibration acceleration of wheelset. 5%
100% CF FM4 K M6A M8A
0%
-10%
Change percentage
Change percentage
80% CF FM4 K M6A M8A
-20%
60%
40%
20% -30%
0%
-35% 0
2
4 6 Crack depth /mm
8
10
-10% 0
2
4 6 Crack depth /mm
8
10
Fig. 12 Change percentage of various statistical indicators versus crack depth: a) dynamic mesh force, b) Vertical vibration acceleration of wheelset.
Figure 13 shows the change percentage of CF, K, FM4, M6A and M8A of longitudinal and lateral vibration acceleration of wheelset which calculated from the locomotive system vibration signal that processed by angular synchronous averaging method under different crack depth. Similar to the change of condition indicators FM4, M6A and M7A of vertical vibration acceleration of wheelset, the FM4, M6A and M8A values of longitudinal vibration acceleration of wheelset will increase with the crack propagation along crack depth, while they are smaller than those of vertical vibration acceleration of wheelset as it shows in Fig.13 a). Besides, the CF will decrease at first and then increase with crack propagation along crack depth, however the variation of K is not obvious according to Fig. 13a). In addition, the trend in condition indicators of lateral vibration acceleration of wheelset is not obvious as it shows in Fig.13 b). 11%
70% CF FM4 K M6A M8A
Change percentage
8%
60%
CF FM4 K M6A M8A
40%
Change percentage
10%
6% 4% 2%
20% 0%
-20%
0% -40%
-2%
-50%
0
2
4 6 Crack depth /mm
8
10
0
2
4 6 Crack depth /mm
8
10
Fig 13 Change percentage of various statistical indicators versus crack depth: a) longitudinal vibration acceleration of wheelset, b) lateral vibration acceleration of wheelset
It can be seen from Figs. 12 and 13 that, the change of FM4, M6A, M8A of the vertical vibration acceleration of the wheelset is larger than that of the dynamic mesh force and the longitudinal vibration acceleration of wheelset, which could effectively reflect the crack propagation. The M8A is more sensitive to crack depth than M6A and FM4. And again, the lateral vibrations of the wheelset are not sensitive to the gear tooth root crack fault.
5 Conclusions A detailed three dimensional locomotive dynamics model is established to investigate the dynamic features of gear tooth root crack in a heavy-haul locomotive system. The fault features are analyzed by using the time-frequency analysis technology and represented by the statistical indicators. The main conclusions of this paper can be drawn as follows: 1) The time-frequency analysis results of dynamic meshing force, vertical and longitudinal vibration acceleration of wheelset can reflect the characteristic frequency of gear meshing and fault vibration characteristic frequency caused by tooth root crack. 2) The tooth root crack has little effect on the lateral dynamic performance of locomotive. Besides the characteristic frequency components of the fault vibration caused by tooth root crack is difficult to transfer to the bogie frame cause of the attenuation effect by the primary suspension system. 3) The FM4, M6A and M8A of dynamic mesh force and vertical vibration acceleration could
reflect the influence of crack depth on the dynamic characteristics of the system. The effect law of crack depth on FM4, M6A and M8A values of the vertical and longitudinal vibration accelerations of wheelset are similar with each other.
Acknowledgement The authors are grateful for the financial support provided by the National Natural Science Foundation of China (Grant Nos. 51775453, 51735012), the Fundamental Research Funds for the State Key Laboratory of Traction Power of Southwest Jiaotong University (Grant No. 2019TPL-T09), and the Fundamental Research Funds for the Central Universities (Grant No. 2682019YQ04).
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Highlights
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A locomotive spatial dynamics model considering gear transmission system with gear tooth root crack is established.
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The dynamic responses of the locomotive under transient condition are obtained by considering the complicated multiple excitations.
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Time-frequency analysis technique is used to reveal the fault vibration features of different components under transient condition.
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Combination of angular synchronous average method and statistical indicators is proposed for extraction of tooth root crack fault feature.
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The studied results could supply some guidance for locomotive gear tooth fault diagnosis.
Conflict of interest The authors declared that they have no conflicts of interest to this work.