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, Op and Og represent the centroids of the driving and driven gears, respectively; mp and mg represent mass; Rp and Rg represent the radius of the base circle; Ip and Ig represent the moment of inertia; kpy and kgy represent stiffness in the tangential direction; kpz and kgz represent stiffness in the axial direction; cpy and cgy represent damping in the tangential direction; km and cm represent gear meshing stiffness and damping factors, respectively; fh represents the gear meshing displacement function; Tp represents the external load torque on the driving wheel; Tg represents the output torque; and β is the tilt angle of gears.

According to Newton's second law, the analytical model of the gear system in Fig. 1 can be deduced as follows: 1mpyp¨+cpyyp˙+kpyfy(yp)=-Fymgyg¨+cgyyg˙+kgyfy(yg)=Fympzp¨+cpzzp˙+kpzfz(zp)=-Fzmgzg¨+cgzzg˙+kgzfz(zg)=FzIpθp¨=Tp-FyRPIgθg¨=-Tg+FgRP In Eq. (), Fy and Fz represent the tangential and axial dynamic engagement forces, respectively, and they can be expressed as follows: Fz=sin⁡b{kmfzzp-tgb(yp+qpRp)-zg+tgb(yg+qgRg)-em(t)sin⁡b+cmzp-tgb˙(yp˙+qp˙Rp)2-zg˙-tgb(yg˙+qg˙Rg)-em(t)˙sin⁡b} Fy=cos⁡βkmfyyp+θpRp-yg+θgRg-em(t)cos⁡β+cmyp˙+θp˙Rp-yg˙3+θg˙Rg-em(˙t)cos⁡β In Eqs. () and (), “.” represents taking the derivative for t; yp and zp represent the panning vibration displacement of the main driven gear in the y direction, yg and zg represent the panning vibration displacement of the main driven gear in the z direction; and θp and θg represent the corner vibration displacement of the main gear and driven gear.

After that, make the Eq. () dimensionless; add the gear pair gap length b; assume ζ1=yp/b, ζ2=yg/b, ζ3=zp/b, ζ4=zg/b and ζ5=(θpRp-θgRg)/b; define the tangential relative displacement of the gear system as y1 , and define the axial relative displacement as y2 (Wu et al., 2016). Then: 4y1=ζ1-ζ2+ζ5-em(t)/b,5y2=ζ3-ζ4-tgβ(ζ1-ζ2+ζ5)-em(t)/b,6τ=wnt,7wn=km/me,8me=I1I2I1R22+I2R12,9w=wp/wn, In the above equations, τ represents unit time after dimensionless processing; wn represents the gear's natural frequency; and me represents the gear's equivalent mass.

Combine the last two torsional dynamics equations, and define the non-dimensional internal excitation frequency w=wp/wn. The non-dimensional system dynamics equations are obtained from Eq. (): 10ζ1+¨2η11ζ1+˙K11ζ1=-cos⁡β[K10fy(y1)+2ε10y1]˙,ζ2+¨2η21ζ2+˙K21ζ2=cos⁡β[K20fy(y1)+2ε20y1]˙,ζ3+¨2η12ζ3+˙K12ζ3=-sin⁡β[K10fz(y2)+2ε10y2]˙,ζ4+¨2η22ζ4+˙K22ζ4=sin⁡β[K20fz(y2)+2ε20y2]˙,ζ5+¨cos⁡β[K0fy(y1)+2ε0y1]˙=FK30/b. In Eq. (10), the variables are defined as follows: η11=cpy2mpwnK11=kpympwn2ε10=cm2mpwnη21=cqy2mqwnK21=kqymqwn2ε20=cm2mqwnη12=cpz2mpwnK12=kpzmpwn2K10=kmmpwn2η22=cqz2mqwnK22=kqzmqwn2K20=kmmqwn2K0=kmmewn2ε0=cm2mewnK30=1mewn2F=TpRp=TqRq

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

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.

The gear meshing stiffness varies with meshing position. When a pair of gears meshes at the meshing point j, the gear meshing stiffness can be expressed as: 11kj=11kbj1+1kbj2+1kfj1+1kfj2+1khj, where the subscript numbers 1 and 2 represent the driving wheel and the driven wheel, respectively; kbj1 and kbj2 represent the stiffness corresponding to the bending, shearing, and compression deformation of the teeth; kfj1 and kfj2 represent the stiffness corresponding to the deformation of the gear; and khj is the stiffness corresponding to the Hertz contact deformation. During the involute gear meshing process, single and double teeth alternately appear, and a complete meshing period is divided into two parts: single tooth meshing and double tooth meshing. When the double teeth are engaged, the meshing stiffness is a superposition of the meshing stiffness of the two pairs of teeth. The gear meshing stiffness in a complete meshing cycle can be determined by ascertaining the single tooth meshing zone and the double tooth meshing zone from the gear geometry.

The teeth are divided into n micro-elements, and the portion connected to the micro-element close to the crest is regarded as a rigid body. The cross-section of each micro-element is rectangular. It is assumed that the thickness of each micro-element is Li, the cross-sectional area is Ai, the cross-sectional area moment is Ii , the distance from the micro-element to the meshing point is Sij, the pressure angle corresponding to the meshing point is αj, and the distance between the meshing point and the neutral layer of the tooth is Yj. According to the theory of Timoshinko beam, the stiffness corresponding to the bending, shearing, and compression deformation of the teeth is (Li and Wang, 1999): kbj=∑i=1n1EeLi3+3Li2Sij+3LiSij23Iicos⁡2αj-Li2+2LiYijSij2Iicos⁡2αj12+(12+11v)Li5Aicos⁡2αj+LiAisin⁡2αj-1, Sainsot (Sainsot et al., 2004) proposed that the stiffness corresponding to the deformation of the involute spur gear based on the Muskhelishvili elastic ring theory can be expressed as: kff=13cos⁡2αiBEeL∗uf2Sf2+M∗ufSf+P∗(1+Q∗tan⁡αi)-1, The parameters in Eq. () are shown in Sainsot et al. (2004).

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): 14khj=2πB1-v12E1+1-v22E2-1, where B is the actual contact tooth width, E1 and E2 are the elastic moduli of the two gears, and v1 and v2 are Poisson's ratios of the two gears.

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 P, if the transmission error generated by the gear pair is δ, there is a load balance relationship: 15P=ks∑i=1nui=ks∑i=1n(δ-εi), where ks is the stiffness of each contact point, and SMITH considers the stiffness of each slice to be the same, which is derived from the single tooth stiffness formula in ISO6336. Additionally, εi is the initial gap amount at contact point i, and ui is the deformation of each contact point. When δ<εi, it indicates that the point has been touched, so ui takes a positive value, and when δ>εi, it indicates that the point is not in contact, so ui is set equal to zero.

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 Y, the nonlinear backlash function is (Wang and Kang, 2015): 16f(Y)=Y-1,Y>10,|Y|≤1Y+1,Y<1, The corresponding lateral and axial backlash functions can be expressed as: 17f(Y)=Y-sin⁡β,Y>sin⁡β0,|Y|≤sin⁡βY+sin⁡β,Y<sin⁡β, 18f(Y)=Y-cos⁡β,Y>cos⁡β0,|Y|≤cos⁡βY+cos⁡β,Y<cos⁡β, The lateral and axial backlash functions are shown in Figs. 3 and 4.

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. () (Zhu, 2004): 19cm=2εKsr1r2I1I2r12I1+r22I2, where ε is the gear pair relative meshing damping coefficient, Ks is the average meshing stiffness, I1 and I2 are moments of inertia, and r1 and r2 are base circle radii.

Uniform wear of the tooth surface changes the tooth thickness as well as the parameters Ai and Ii of the micro-element, which reduces the meshing rigidity. The wear depth corresponding to micro-element i is hi, and the cross-sectional area (Ai′) and the cross-sectional area moment (Ii′) of the micro-element after the tooth surface is uniformly worn are 20Ai′=Ai-Bhi,21Ii′=Ii-(yi-hi)2hiB. where yi is the distance from the tooth surface at micro-element i to the neutral layer. Uniform wear of the flank results in a decrease in kbj, thereby reducing the meshing stiffness of the gear. Substituting Eqs. () and () into Eq. (), Eqs. ()–() are used to obtain the gear meshing stiffness after the tooth surface is uniformly worn. The effect of wear fault on gear meshing stiffness is shown in Fig. 5.

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: 22f(Y)=Y-(1+hi)0Y+(1+hi)Y>(1+hi)|Y|≤(1+hi)Y<-(1+hi), The corresponding lateral and axial backlash functions can be expressed as: 23fy(Y)=Y-(sin⁡β+hi)0Y+(sin⁡β+hi)Y>(sin⁡β+hi)|Y|≤(sin⁡β+hi)Y<-(sin⁡β+hi), 24fz(Y)=Y-(cos⁡β+hi)0Y+(cos⁡β+hi)Y>(cos⁡β+hi)|Y|≤(cos⁡β+hi)Y<-(cos⁡β+hi),

Figure 6 shows the bifurcation diagram of the gear system with the change of shaft frequency (wk) when the gear system did not fail, that is, the fault value was m=0. As can be seen from Fig. 6, with the increase of wk, the gear system underwent five stages of change: single-period motion → double-period motion → single-period motion → double-period motion → chaos motion. When 40 Hertz <wk< 55 Hertz, the gear system exhibited single period motion, and when wk>72 Hertz, the gear system exhibited chaos motion. Figures 7 and 8 display the phase plane and Poincaré cross-sections of the gear system, respectively. When the shaft frequency was wk=45 Hertz, the phase plane of the system was one closed curve, and the Poincaré cross-section was one point, indicating that their motion was single-periodic; when wk=75 Hertz, the phase plane of the system was composed of many closed curves, and the Poincaré section contained many discrete points, indicating that its motion was chaotic.

When wk=43 Hertz, the phase plane, Poincaré section, bifurcation diagram, and waterfall plot of the helical gear system changing with the amount of wear of gear are shown in Fig. 9. It can be seen that phase diagram changed from one closed curve to two closed curves with increasing wear (h); the Poincaré section changed from one point to two discrete points; the bifurcation diagram gradually changed from single-period motion to double-period motion; the spectrum diagram did not change significantly, which shows that gear system changed from single-cycle motion to double-cycle motion with the increase of fault parameters. Figure 9e shows the RMS, kurtosis, and peak-to-peak changes of the acceleration spectrum. During the initial stage of fault, the wear fault had a great influence on the RMS, kurtosis, and peak-to-peak values, which all increased rapidly, but the RMS, kurtosis, and peak-to-peak values all showed fluctuations in state afterwards, indicating that the impact was slight.

When wk=72.5 Hertz, the changes in the phase diagram, Poincaré section diagram, and bifurcation diagram of the helical gear system in response to the increase of wear are shown in Fig. 10. The phase plane changed from many closed curves to two closed curves with the increase of fault parameters; the Poincaré section changed from many discrete points to two points; and the bifurcation diagram gradually changed from a chaotic state to four-fold periodic motion. In summary, all tooth wear failure will change the movement of the gear system from chaos to double-cycle motion. Figure 10d shows the RMS, kurtosis, and peak-peak values of the acceleration spectrum. The RMS, kurtosis, and peak-peak values rapidly increased at the initial stage of fault, and then all of them showed a fluctuating state, and the fluctuation amplitude was small.

Figure 11a, b shows frequency spectrum for fault-free gear systems, and the amount of wear is h=0.4 mm, from which we can see that wear failure reduced the amplitude of the sideband frequency of fundamental frequency and frequency doubling, and the amplitude of frequency doubling increased when wk=72.5 Hertz.