The helical gear system is an important form of transmission in high-speed trains, and helical gear failure has a great impact on the transmission performance. To investigate the influence of wear parameters on the nonlinear dynamics of gear systems, the wear fault parameters of a bending-torsion-shaft coupling mode with six degrees of freedom are established. Using the variable step fourth-order Runge-Kutta numerical integration method, the gear dynamics model with fault parameters is analyzed to get the dynamic response of the helical gear system. The system excitation frequency, evolution of system periodic motion, quasi-periodic motion, and chaotic motion with variable fault parameters are analyzed qualitatively based on the results, including the system status judgment criteria of phase plane graph, Poincaré cross-section graph, bifurcation diagram, and RMS. The results show that wear fault affects the system differently at different frequencies. Finally, the correctness of the conclusion is verified through experiments, and the impact on the actual application process is analyzed.

Helical gears are not only a common transmission mecha-nism but also an important form of transmission in high-speed trains. Their reliability is thus important for the overall industrial processes (Wang et al., 2018). The high-speed train gear-box-driven gear is directly press-fitted on the axle, and the driving gear is connected with the traction motor through coupling and hangs on the cross beam through the gear box (Zhang, 2011). Therefore, the vibration characteristics of the gear system will directly affect the running performance of the high-speed train and the drive transmission system.

In recent years, many scholars have studied the non-linear dynamics of gear systems. Spitas and Spitas (2015) analyzed a coupled multi-DOF dynamic contact model with intermittent gear tooth contacts. Wang et al. (2011) made a preliminary analysis of a fault gear simulation method. Ma and Chen (2012) established a torsional gear pair model and analyzed the singularity of cracked gear and the dynamic response process of crack evolution. Zhang et al. (2011) took into account the dynamic distribution of load in the meshing area and the use of gear test standards and tolerances to determine the transmission error of gear pairs in the study of gear transmission system with tooth root crack failure. Parey and Tandon (2003) and Parey et al. (2006) established a variety of fault gear dynamics models and response signal analyses in the study of gear system dynamics with defects. Alshyyab and Kahraman et al. (2005) studied a kind of discrete Fourier transform of nonlinear time-varying kinetics models using the multiple harmonic balance method and found that stable subharmonic waves exist mainly in the form of softening. Chaari et al. (2009) quantified the reduction of meshing stiffness caused by partial failure of gears and proposed a finite element model to verify the analytical formula. Kahraman and Singh (1991) studied the frequency-response characteristics of nonlinear gear rotor-bearing systems with time-varying mesh stiffness. Shen et al. (2006) extended the incremental harmonic balance method to the nonlinear dynamic analysis of gear pairs. Rao et al. (2014) studied the torsional vibration characteristics of two-stage spur gear systems. The above studies mainly focused on the crack evolution and resonance response of faulty gears. Although there are many studies of normally operating spur gear transmission systems, there are fewer studies on the failure of helical gear transmissions system. In addition, there are few studies on the dynamic characteristics of helical gear systems with high-speed train fault parameters.

In this paper, the faulty helical gear of a high-speed train is taken as the research object, and time-varying pa-rameters, time-varying mesh stiffness, and time-varying mesh error are fully considered. The bending-torsion-axis coupling system model of the faulty helical gear is established. Through numerical analysis of the dynamic model combined with the phase diagram, Poincaré section map, bifurcation diagram, and spectrogram of the response of the nonlinear gear system, the dynamic response of the gear transmission system under wear fault is qualitatively analyzed.

The gear transmission systems of high-speed trains adopt helical gear
transmission. In the process of transmission, the meshing of gears generates
axial dynamic meshing force, so the transmission system has torsional
vibration, lateral vibration, and axial vibration, thus forming a meshing
type of bending-torsion-shaft coupling system dynamics mode (Li and Wang,
1999). Accordingly, the establishment of a high-speed train helical gear
transmission dynamics model is shown in Fig. 1. The system is a
three-dimensional vibration model. To simplify calculation, this model does
not consider friction of tooth surface and contains six degrees of freedom.
In Fig. 1,

Three-dimensional vibration model of gear system.

According to Newton's second law, the analytical model of the gear system in
Fig. 1 can be deduced as follows:

After that, make the Eq. (

Combine the last two torsional dynamics equations, and define the
non-dimensional internal excitation frequency

The basic parameters of gear pairs used in this paper are shown in Table 1.

A certain type of high-speed train helical gear basic parameters.

The stiffness excitation is a kind of dynamic excitation, which is formed due to the change of integrated stiffness with time in the gear meshing process. With the constant alternating of double teeth and multiple teeth meshing, the bearing load and elastic deformation between gears will alternate. Due to the tilt angle, the helical gears experience a reduced amount of abrupt change in the alternating changes of tooth pairs. However, considering the irregularity of contact line and interaction of multiple teeth, as well as the presence of manufacturing errors, there are also periodic changes in elastic deformation and meshing stiffness. Using the accumulated integral potential energy method (Wan et al., 2015), the curve of time-varying mesh stiffness is shown in Fig. 2.

Time-varying mesh stiffness curve.

The gear meshing stiffness varies with meshing position. When a pair of
gears meshes at the meshing point

The teeth are divided into

The stiffness corresponding to the Hertzian contact deformation of a pair of
teeth on meshing line is constant, and it is expressed as (Yang and Sun,
1985):

It is assumed that the tooth surface error is small enough relative to the
macroscopic structure of the gear, the actual contact point coincides with
the position of the theoretical contact point, and the normal direction of
each point after the contact does not change, regardless of the frictional
force and the lubricating oil effect when the tooth is engaged. Under the
action of the meshing force

SMITH uses an iterative method to calculate the transfer error. The iterative steps are described in (Han, 2003).

Due to the existence of gear transmission error and the influences of wear
and lubrication in the meshing process, the tooth clearance between meshed
gears and backlash function can be expressed by a piecewise function.
Assuming the tooth gap is

Lateral backlash function.

Axial backlash function.

The meshing damping is mainly related to the gear meshing damping ratio,
average meshing stiffness, and gear quality, and it can be calculated by Eq. (

Uniform wear of the tooth surface changes the tooth thickness as well as the
parameters

Effect of uniform wear of tooth surface on meshing stiffness.

Moreover, when gear teeth produce wear fault, the gear clearances will be
larger than in a normal gear (Wang et al., 2011), and the backlash
function is:

Bifurcation diagram normal conditions.

System response of

Figure 6 shows the bifurcation diagram of the gear system with the change of
shaft frequency (

System response of

System response changing with wear loss at

When

System response changing with wear loss at

Spectrogram.

When

Figure 11a, b shows frequency spectrum for fault-free gear systems,
and the amount of wear is

Figure 12 is a test bench customized to reduce the center distance of the
transmission ratio of a certain high-speed train gearbox. This test bench
can effectively simulate the motion of the gearbox under various operating
conditions. The input motor can input different speeds, the output motor can
apply different sized loads, and the gears interact with one-step meshing.
Because no obvious changes can be seen in the simulated spectrogram at low
frequencies, this experiment only verifies the effects of wear failure on
helical gear systems at high frequencies. The operating conditions were
shaft speed

Gearbox test bench.

Gears used in experiment.

Figures 14 and 15 show the time-domain signal and spectrogram of the normal
gear and wear fault gears under the same conditions. The calculated rotation
frequency of the driving gear was 73 Hertz, and the rotation frequency of the
driven gear was 29.2 Hertz. The meshing frequency was 2628 Hertz. From Fig. 14,
when the normal gear was at

Time-domain signal and spectrogram of the normal gear at

Time-domain signal and spectrogram of tooth surface wear gear at

Qualitative and quantitative methods were used to study the influence of
wear fault on the bifurcation and chaos characteristics of a helical gear
system. The specific conclusions are as follows:

With the increase of shaft frequency, the helical gear system of a high-speed train undergoes five stages of change: single-period motiondouble-period motionsingle-period motiondouble-period motionchaos motion. Wear failure will make the gear system move from single period motion to double period motion at low frequencies. At high frequencies, however, the gear system will gradually evolve from chaotic motion to double-period motion, and the sidebands of fundamental frequency and frequency doubling will decrease. Wear failures have the greatest impact on RMS, kurtosis, and peak-to-peak values at the initial failure stage; then all of them show a fluctuating state, and the fluctuation amplitude is small.

The influence of wear failure was verified experimentally by a gear test bench. The spectrum signal obtained at high frequencies was basically consistent with the simulation results.

In fault signal diagnosis of high-speed trains, detected signals can be used to identify and diagnose the types of faults and provide guidance for fault diagnosis of high-speed train gearboxes.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This research has been supported by the National Key Research and Development Program of China (grant no. 2016YFB1200402).

This paper was edited by Jinguo Liu and reviewed by two anonymous referees.