Abstract

The crack failure is prone to happen for the gear teeth under alternating loads. The crack reduces the time-varying meshing stiffness, thus causing changes in the system vibration responses. Moreover, the tooth friction leads to the flash temperature on the tooth surface, and the flash temperature causes the deformation of the tooth profile. The flash temperature stiffness generated by this deformation is an important component of the resultant meshing stiffness, and inevitably influences the dynamic characteristics. Hence, this paper makes an effort to provide a clear understanding of the relationship between the tooth crack and nonlinear dynamics for a multistage gear transmission system considering the flash temperature in a mixed friction state. The time-varying meshing stiffness and flash temperature stiffness are separately calculated using the potential energy method and Hertz's theory, and the torsional dynamic model is established by the lumped mass method for the system with high-speed driving gear crack in the mixed friction state. The influences of the flash temperature and crack on the dynamic behaviors are studied through a bifurcation diagram, frequency spectrum, phase portrait, and Poincare map, and the frequency spectra of numerical simulation and experimental testing are compared. The results indicate that both flash temperature and tooth crack decrease the time-varying meshing stiffness, the flash temperature increases the displacement amplitude of chaotic motion, and the tooth crack enlarges the interval of chaotic motion. Meanwhile, the results provide a suitable meshing frequency, which puts the system in a stable working state. The research provides significant guidance for monitoring the crack fault of multistage gear transmission systems.

1 Introduction

Tooth crack often occurs during the process of gear drive since the teeth are subjected to fluctuating cycle stress. It damages the teeth's surface material, reduces the teeth's effective cross-sectional area, changes the gears' meshing stiffness, and ultimately affects the dynamic characteristics of the gear system [1]. Moreover, the large amount of heat generated by friction raises the local temperature near the meshing point of gear teeth, i.e., the phenomenon of flash temperature on the tooth surface, which leads to the local deformation of the tooth profile. This deformation decreases the time-varying meshing stiffness of the gear pair, thus changing the system's dynamic performance [2]. Hence, it is necessary to investigate the dynamic crack characteristics of the multistage gear transmission system considering the flash temperature in a mixed lubrication state.

The tooth crack influences the time-varying meshing stiffness of gears, and much effort is spent on the dynamic characteristics study of gear systems with cracks. Meng et al. [3] calculated the time-varying meshing stiffness of gears with different crack lengths using the potential energy method, and analyzed the dynamic response of a 6-degree-of-freedom gear system with different crack lengths. Liu et al. [4] gained the time-varying meshing stiffness of gears with different crack depths by the energy method and finite element method, and analyzed the vibration response of gear systems with different root crack depths. Wang et al. [5] proposed an improved model for the calculation of the mesh stiffness of a helical gear system with crack, and analyzed the vibration response of the system with the propagation of gear cracks. Ren et al. [6] established the time-varying meshing stiffness model of spur gears with crack by the potential energy method, and discussed the time-domain characteristics of vibration of gear different crack depth and crack angle. Qiao et al. [7] obtained the mesh stiffness under the single-stage and multistage gear cracks with the finite element method, and studied the dynamic characteristics and statistical indicators under different cracks. Wan et al. [8] established a time-varying meshing stiffness calculation model based on the energy method to solve tooth root cracks and tooth surface spalling, and analyzed the differences in vibration response characteristics between the two types of faults through dynamic simulation. Shen et al. [9] calculated the time-varying meshing stiffness of gears with cracks using the finite element method, and studied the influence of crack location on the vibration response of the two-stage gear transmission system. Xie et al. [10] obtained the time-varying meshing stiffness of gear pair with cracks adopting the improved potential energy method, and studied the dynamic response of planetary gear transmission systems with double-tooth cracks of the sun gear. Kong et al. [11] setup the time-varying meshing stiffness calculation model of gears under the action of cracks and wear using the potential energy method, and analyzed the dynamic characteristics of gears under different degrees of cracks and wear. Xiang et al. [12] analyzed the mesh stiffness of gears with root cracks by the potential energy method, and compared the nonlinear dynamic features of the cracked and normal system by bifurcation diagram, time series, phase trajectory, Poincare map, and spectrum diagram.

Most of the energy consumed by friction is converted into heat, resulting in an increase in the tooth surface temperature thus influencing the gear dynamic characteristics. Xu et al. [13] established a meshing stiffness model using flash temperature theory, and studied the influence of the dynamic wear model considering the tooth contact flash temperature on the dynamic characteristics of a gear-bearing system. Sun et al. [14] considered the effect of temperature on the material's elastic modulus and Poisson's ratio and related the temperature to the time-varying meshing stiffness, and analyzed the influence of temperature on the nonlinear dynamic of the gear system. Pan et al. [15] built a coupled nonlinear dynamic model with a 10-degree-of-freedom gear-shaft-bearing transmission system considering time-varying meshing stiffness (the stiffness caused by temperature and shock), and investigated the effect of the contact temperature on the dynamic characteristics. Liu et al. [16] setup the two-stage planetary gear system model considering the temperature of the system, and researched the influence of gear temperature on the vibration of the planetary gear system. Zhang et al. [17] presented a nonlinear dynamic model of a wind turbine gearbox considering the tooth contact temperature, and studied the dynamic wear characteristics of gear transmission systems considering the tooth contact temperature. Lu et al. [18] proposed a thermal time-varying stiffness calculation model of spur gear pairs, and revealed the influencing mechanism of temperature on the tooth profile error, mesh stiffness, total deformation, and backlash.

Taken together, the impacts of tooth crack and flash temperature on the dynamic characteristics have mostly been investigated separately, and the chaotic characteristics of the gear system considering crack fault and flash temperature in a mixed friction state have not been analyzed. The time-varying meshing stiffness and flash temperature stiffness are calculated separately using the potential energy method and Hertz's theory. The nonlinear dynamic model of a multistage gear transmission system with high-speed gear crack is built, and the dynamic equations of the model are derived and solved. The simulated and experimental frequency domain characteristics are compared, and the impacts of flashing temperature and tooth crack on the vibration characteristics of the system are studied.

2 Resultant Meshing Stiffness Calculation

During gear operation, the two contact surfaces undergo elastic deformation under the action of force, which causes time-varying meshing stiffness. Meanwhile, there is friction between the contact surfaces, and the friction can also undergo thermal deformation that causes time-varying flash temperature stiffness of gears. Therefore, the resultant meshing stiffness of gears, ksum(t), should include time-varying meshing stiffness, kt(t), and time-varying flash temperature stiffness, kflash(t). The expression of ksum(t) is
(1)

2.1 Time-Varying Meshing Stiffness Calculation

2.1.1 Time-Varying Meshing Stiffness of Health Gear.

The spur gear tooth is assumed as a cantilever beam starting from the roof circle, as shown in Fig. 1. The potential energy method has the advantages of higher calculation efficiency and accuracy. The shear stiffness ks, bending stiffness kb, axial compression stiffness ka, Hertzian contact stiffness kh, and fillet-foundation stiffness kf are calculated using the potential energy method from Refs. [19,20]
(2)
(3)
(4)
(5)
(6)

where χ is the Poisson's ratio, N is the number of gear teeth, E is the elastic modulus, bk is the tooth width, α0=20deg, α1 is the pressure angle at the meshing point, α2 is half of the tooth angle on the base circle, and α3 is half of the tooth angle on the dedendum circle, uF=rb,i[(α1+α2)sinα1+cosα1cosα2], sF=2rb,isinα2, rb,i is the base circle radius, and the coefficients L*,M*,P*,andQ* are the same as those in Ref. [20].

Fig. 1
Cantilever beam model of the health gear tooth
Fig. 1
Cantilever beam model of the health gear tooth
Close modal
For the single-tooth-pair meshing duration, the time-varying meshing stiffness kt(t) is expressed as
(7)
For the double-tooth-pair meshing duration, the time-varying meshing stiffness kt(t) is written as follows:
(8)

where g =1,2 denotes the driving and driven gears and y denotes the yth meshing gear tooth.

2.1.2 Time-Varying Meshing Stiffness of Gear With Crack.

The tooth crack propagation is a slow process, and the cantilever beam model of the spur gear tooth with crack is shown in Fig. 2. Assuming that the crack is a straight line starting from the root circle and having a length of q, and the angle between the crack line with the centerline of the tooth is defined ν. According to Ref. [10], when a crack exceeds the centerline of the tooth, the gear teeth will quickly fracture. Therefore, the main research focuses on the crack situation before the crack reaches the centerline.

Fig. 2
Cantilever beam model of the tooth with crack
Fig. 2
Cantilever beam model of the tooth with crack
Close modal

The crack causes changes in shear stiffness and bending stiffness. The shear stiffness kscrack and bending stiffness kbcrack of the gear with crack are obtained respectively according to Ref. [21]

  1. When hchr & α1> αc
    (9)
    (10)
    where hc is the distance from the crack end point A to the tooth central line, hr is half of the roof chordal tooth thickness, αc is the angle at the force action point K, and αr is half of the center angle corresponding to the tooth root circle's chord through point A.
  2. When hc<hr or when hahr & α1αc
    (11)
    (12)
    For the single-tooth-pair meshing duration, the time-varying meshing stiffness of the gears with crack, ktcrack(t), is expressed as
    (13)
For the double-tooth-pair meshing duration, the time-varying meshing stiffness of the gears with crack, ktcrack(t), is written as follows:
(14)

The time-varying meshing stiffness of the high-speed fixed-axis gears with crack is calculated, where the crack angle of the driving gear is ν=70 deg, and the crack lengths are q =0.5 mm, q =1.0 mm, and q =1.5 mm (when q =1.5 mm, the crack line exactly reaches the centerline of the gear teeth), as shown in Fig. 3. The time-varying meshing stiffness decreases after tooth crack, and gradually decreases with increases of the crack length. Moreover, there is a phenomenon of time-varying meshing stiffness decreasing significantly over time in both double and single tooth meshing regions.

Fig. 3
Time-varying meshing stiffness of high-speed fixed-axis gears under different crack conditions
Fig. 3
Time-varying meshing stiffness of high-speed fixed-axis gears under different crack conditions
Close modal

2.2 Time-Varying Flash Temperature Stiffness Calculation

2.2.1 Flash Temperature.

The tooth contact temperature, TC, is the sum of the flash temperature, TF, and the bulk temperature, TB. according to Blok's flash temperature theory [15], the expression of TF is
(15)

where μ(t) is the friction coefficient under mixed lubrication state, ϖ is the temperature rise coefficient, and takes 0.83, fe is the normal load on tooth surface per unit tooth width, vg is the tangential velocity of the tooth surface, B(t) is the half width of the tooth surface contact band, yg is the thermal conductivity of tooth surfaces, ρg is the material density, and σg is the specific heat capacity.

According to Ref. [22], μ(t) is expressed as
(16)

where Sa is the average tooth surface roughness, ηM is the dynamic viscosity coefficient of lubricating oil, vs(t) is the sliding velocity of meshing tooth pair, and vs(t)=|ν1(t)ν2(t)|; vc(t) is the entraining velocity of meshing tooth pair, and vc(t)=|ν1(t)+ν2(t)|/2.

The expression of vg is
(17)
where rb,g is the base circle radius of the driving and driven gears, ωg is the angular velocity, rg(t) is the distance from the meshing point to the center of the gear, and the expression of rg(t) is
(18)
(19)

where α is the pressure angle and ra,2 is the addendum circle radius of the driven gear.

The expression of B(t) is
(20)

where ψ is the calculation coefficient, and takes 1.128, fn is the normal load on the tooth surface, Rg is the curvature radius of gear tooth profile, and Rg(t)=rg(t)sin(arccos(rb,g/rg(t)).

2.2.2 Tooth Profile Deformation.

The increase in tooth surface temperature causes the thermal deformation of tooth surfaces, and the actual tooth profile of the gear cannot overlap with the theoretical tooth profile. According to Refs. [17,23,24], the tooth profile deformation caused by tooth surface contact temperature variation for the driving and driven gears, εg(t), is expressed as
(21)

where g is the linear expansion coefficient of the material, og is the thermal deformation of the base circle, and og=grb,gt(rz,g)+grb,g(1+χg)[rb,g2(12χg)rz,g2][t(rb,g)t(rz,g)]/[(1χg)(rb,g2rz,g2)]; t(rz,g) is the temperature of the gear shaft in steady-working state, rz,g is the radius of the gear shaft, and t(rb,g) is the temperature of the base circle in steady-working state; αa,g is the addendum pressure angle, and αa,g=arccos(rb,g/ra,g), ιg is the tooth thickness, and ιg(t)=lgrg(t)/rd,g2rg(t)(inv(arccos(rb,g/rg(t)))invα), lg is the tooth thickness of the reference circle, and rd,g is the reference circle radius, invα is the involute function, and invα=tanαα.

2.2.3 Time-Varying Flash Temperature Stiffness.

Based on Hertz's contact theory, the stiffness change caused by flash temperature on tooth surfaces of the driving and driven gears kflash,g(t) is calculated as follows:
(22)
The deformation of two tooth surfaces at the meshing point is in one line. The equivalent meshing stiffness caused by flash temperature on two tooth surfaces, i.e., the time-varying flash temperature stiffness kflash(t) is described as follows:
(23)

2.3 Resultant Meshing Stiffness Without and With the Crack.

The time-varying meshing stiffness is affected by flash temperature on the tooth surface, and the resultant meshing stiffness of the high-speed fixed-axis gears without and with the crack is calculated, as presented in Fig. 4, where ksumcrack(t) is the resultant meshing stiffness of the gears with crack. It is indicated that the time-varying meshing stiffness significantly decreases when the flash temperature is considered, and has a faster decrease in double-teeth meshing region and a slightly slower decrease in single-tooth meshing region. Moreover, the resultant meshing stiffness is reduced by the tooth crack.

Fig. 4
Resultant meshing stiffness of high-speed fixed-axis gears without and with the crack
Fig. 4
Resultant meshing stiffness of high-speed fixed-axis gears without and with the crack
Close modal

3 Dynamic Model for a Multi-Stage Gear Transmission System Considering the Flash Temperature

3.1 Dynamic Model of the System.

The lumped mass method not only calculates simply and accurately, but also reflects the dynamic characteristics of gears. According to Refs. [12,19,25,26], a lumped parameter model is established for the multistage gear transmission system composed of a two-stage fixed-axis gear system and a one-stage planetary gear system, as shown in Fig. 5. This model has been used to solve the dynamic responses of the multistage gear transmission system. The two-stage fixed-axis gear system includes high-speed driving and driven gears I1, I2, and low-speed driving and driven gears II1, II2. In the one-stage planetary gear system, s, pn(n = 1,2,3,4), r, and c represent the sun gear, nth planetary gears, inner ring gear, and planet carrier. The torque is input by the high-speed driving gear and output by the planet carrier, the input torque is Tin and the output torque is Tout. Besides, every gear is a spur gear.

Fig. 5
Dynamic model of the multistage gear transmission system
Fig. 5
Dynamic model of the multistage gear transmission system
Close modal

3.2 Dynamic Equations of the System.

The dynamic equations of the multistage gear transmission system shown in Fig. 5 are built using the Lagrange's equation from Ref. [17]
(24)
(25)
(26)
(27)
(28)
(29)
(30)
where Ji is the moment of inertia, and i = I1, I2, II1, II2, s,pn, r, c denotes the high-speed driving and driven gears, low-speed driving and driven gears, sun gear, planet gears, inner ring gear, and planet carrier; θi is the angular displacement; cj is the damping coefficient, and j = I, II, spn, rpn denotes the high-speed fixed-axis gear pair, low-speed fixed-axis gear pair, sun and planet gear pairs, and planet and inner ring gear pairs; xj is the relative displacement on the meshing line; ksum,j is the resultant meshing stiffness; f(xj) is the nonlinear function of the backlash; Ff,j is the tooth surface friction force; and Li is the tooth surface friction arm. According to Ref. [19], the expression of xj is
(31)
(32)
(33)
(34)
where ej(t) is the comprehensive transmission error. According to Ref. [12], ej(t) is expressed using the first harmonic form of the meshing function
(35)
where eamp,j and φj are the amplitude and initial phase of the comprehensive transmission error; ωj is the meshing frequency.
According to Refs. [17,19], cj is expressed as
(36)

where ξ is the damping ratio and takes 0.03–0.17, kavg,j is the average value of the resultant meshing stiffness, mj,g is the mass, and g =1,2 denotes the driving and driven gears.

According to Refs. [12,15,19,25], f(xj) is expressed as
(37)

where bj is half of the backlash.

To facilitate the solution of dynamic equations, ksum,j(t) is expressed using Fourier series expansion from Refs. [15,17,25,27], and the first harmonic term is taken
(38)

where kamp,j and ϕj is the amplitude and initial phase of the resultant meshing stiffness.

According to Coulomb's law from Ref. [17], Ff,j(t) is expressed as
(39)

where λj(t) is the direction coefficient.

According to Ref. [27], the friction arm of the driving and driven gears for the external meshing gear pair, Lg(t), is described as
(40)
(41)
According to Ref. [27], the friction arm of the driving and driven gears for the internal meshing gear pair, Lg(t), is described as
(42)
(43)
Combine Eqs. (24)(34), define the time nominal scale ωe, and order τ=ωet, where ωe=kavg,I(JI1rb,I22+(JI2+JII1)rb,I12)/JI1(JI2+JII1). The nominal dimension p is introduced, and takes bI. The dimensionless excitation frequency Ωj, dimensionless displacement uj, dimensionless velocity u˙j, and dimensionless acceleration u¨j are calculated by Ωj=ωj/ωe, uj=xj/p, u˙j=x˙j/pωe, and u¨j=x¨j/pωe2, respectively. The dimensionless equations of the system are expressed as follows:
(44)
(45)
(46)
(47)

where G(j)=cjωeu˙j+ksum,j(τ)ωe2f(uj); ma1=JI1(JI2+JII1)JI1rb,I22+(JI2+JII1)rb,I12; ma2=JI2+JII1rb,I1rb,I2; ma3=(JI2+JII1)JII2(JI2+JII1)rb,II22+JII2rb,II12; ma4=JII2+Jsrb,II2rb,s; ms=JII2+Jsrb,s2; mc=Jcrb,c2; mpn=Jpnrb,pn2; q1=JI1LI2rb,I2+(JI2+JII1)LI1rb,I1JI1rb,I22+(JI2+JII1)rb,I12; q2=LI2rb,I2; q3=LII1rb,II1; q4=(JI2+JII1)LII2rb,II2+JII2LII1rb,II1(JI2+JII1)rb,II22+JII2rb,II12; q5=Lsrb,s; q6=LII2rb,II2; q7=Lpnsrb,pn; q8=Lpnrrb,pn.

4 Nonlinear Dynamic Characteristics Analysis

The multistage gear transmission system dynamic model considering the flash temperature and high-speed driving gear crack under mixed friction state are solved using the 4–5 vary-step Runge–Kutta method with the advantages of simple, low computational demand, and high accuracy according to Refs. [17,19,28,29]. The 4–5 vary-step Runge–Kutta method provides candidate solutions by a fourth-order method and controls errors by a fifth-order method. The calculation accuracy is estimated at each step, and the time-step size is reduced to meet the specified accuracy. The solution in this paper goes very smoothly and does not get stuck in a long-running process, which means there are no discrete events. If the discrete event occurs, it can be addressed by reducing the default value of the minimum step or smoothing the backlash function. The impacts of flash temperature and crack on the dynamic characteristics are analyzed through the bifurcation diagram, frequency spectrum, phase portrait, and Poincare map of high-speed fixed-axis gears. To eliminate the transient response, the first 400 revolutions are omitted from the calculation results. The gear parameters of the system are shown in Table 1, and Tin = 6.5 N⋅m, Tout = 265 N⋅m.

Table 1

Gear parameters of the multistage gear transmission system

High-speed fixed-axis gearsLow-speed fixed-axis gearsplanetary gears
Gear parametersI1I2II1II2spnr
Tooth number2910036902836100
Modulus (mm)1.51.51.51.5111
Radius of base circle (mm)20.470.525.363.4131747
Gear mass (g)1251224.522411114134.6
Moment of inertia (kg m2)0.0560.1440.0070.01
Tooth width (mm)30303030202020
Poisson's ratio0.30.30.30.30.30.30.3
Pressure angle (deg)20202020202020
Thermal conductivity [J/(m s °C)]46.4746.4746.4746.4746.4746.4746.47
Material density (kg/m3)7850785078507850785078507850
Elastic modulus (GPa)206206206206206206206
Specific heat capacity [J/(kg°C)]481.5481.5481.5481.5481.5481.5481.5
Linear expansion coefficient (×10−5)1.161.161.161.161.161.161.16
Gear shaft radius (mm)510515555
Temperature of shaft (°C)20202020202020
Temperature of base circle surface (°C)100100100100100100100
Initial temperature of tooth surface (°C)20202020202020
Steady bulk temperature of tooth surface (°C)100100100100100100100
High-speed fixed-axis gearsLow-speed fixed-axis gearsplanetary gears
Gear parametersI1I2II1II2spnr
Tooth number2910036902836100
Modulus (mm)1.51.51.51.5111
Radius of base circle (mm)20.470.525.363.4131747
Gear mass (g)1251224.522411114134.6
Moment of inertia (kg m2)0.0560.1440.0070.01
Tooth width (mm)30303030202020
Poisson's ratio0.30.30.30.30.30.30.3
Pressure angle (deg)20202020202020
Thermal conductivity [J/(m s °C)]46.4746.4746.4746.4746.4746.4746.47
Material density (kg/m3)7850785078507850785078507850
Elastic modulus (GPa)206206206206206206206
Specific heat capacity [J/(kg°C)]481.5481.5481.5481.5481.5481.5481.5
Linear expansion coefficient (×10−5)1.161.161.161.161.161.161.16
Gear shaft radius (mm)510515555
Temperature of shaft (°C)20202020202020
Temperature of base circle surface (°C)100100100100100100100
Initial temperature of tooth surface (°C)20202020202020
Steady bulk temperature of tooth surface (°C)100100100100100100100

4.1 Influence of the Flash Temperature on the Dynamics.

The Eqs. (44)(47) are solved in the interval from 0 to 800 T, where T = 2*pi/ΩI and ΩI is the excitation frequency and cannot be zero. Moreover, the system is in quasi-periodic motion when ΩI is more than 2.5. Thus, the displacement bifurcation diagrams of high-speed fixed-axis gears for the multistage gear transmission system without and with flash temperature are shown in Fig. 6 when ΩI ∈ [0.01, 2.5], where ξ=0.07, eamp,j=2μm, bI=bII=5μm, and bspn=brpn=2μm. In Fig. 6(a), when the flash temperature is not considered, the high-speed fixed-axis gears are in periodic motion as ΩI ∈ [0.01, 0.56], enter into chaotic motion by cataclysm as ΩI ∈ [0.56, 0.7], transform into quasi-periodic motion as ΩI ∈ [0.7, 0.92], transit to 2T-quasi-periodic motion by doubling bifurcation route as ΩI ∈ [0.92, 1.09], still are in 2T-quasi-periodic motion after cataclysm as ΩI ∈ [1.09, 1.14], enter into chaotic motion and the displacement amplitude decreases gradually as ΩI ∈ [1.14, 1.77], transform into 2T-quasi-periodic motion by inverse-doubling bifurcation route as ΩI ∈ [1.77, 1.88], and are in quasi-periodic motion as ΩI ∈ [1.88, 2.5]. In Fig. 6(b), when considering the flash temperature, the high-speed fixed-axis gears are in periodic motion as ΩI ∈ [0.01, 0.57], enter into chaotic motion by cataclysm as ΩI ∈ [0.57, 0.7], transform into quasi-periodic motion as ΩI ∈ [0.7, 0.99], transit to 2T-quasi-periodic motion by doubling bifurcation route as ΩI ∈ [0.99, 1.11], still are in 2T-quasi-periodic motion after cataclysm as ΩI ∈ [1.11, 1.16], enter into chaotic motion and the displacement amplitude decreases gradually as ΩI ∈ [1.16, 1.74], transform into 2T-quasi-periodic motion by inverse-doubling bifurcation route as ΩI ∈ [1.74, 1.85], and are in quasi-periodic motion as ΩI ∈ [1.85, 2.5]. Taken together, the flash temperature enlarges the interval of quasi-periodic motion and reduces the interval of 2T-quasi-periodic motion in the low-frequency region, and increases the displacement amplitude and shortens the interval of chaotic motion in the high-frequency region.

Fig. 6
Displacement bifurcation diagrams of high-speed fixed-axis gears for the system in mixed friction states (a) without the flash temperature and (b) with the flash temperature
Fig. 6
Displacement bifurcation diagrams of high-speed fixed-axis gears for the system in mixed friction states (a) without the flash temperature and (b) with the flash temperature
Close modal

Due to the significant impact of flash temperature on the bifurcation characteristics, further analysis of the frequency spectrum, phase portrait, and Poincare map of high-speed fixed-axis gears are conducted when the system considering the flash temperature in a mixed friction state, as shown in Fig. 7. When ΩI = 0.62, the high-speed gears are in chaotic motion, the meshing frequency of high-speed fixed-axis gears, fI, the meshing frequency of low-speed fixed-axis gears, fII, and the sideband fIfII occur in the frequency spectrum, the phase portrait is a multicircle winding curve group, and the Poincare map is a set of discrete points. When ΩI = 0.8, the high-speed gears are in quasi-periodic motion, fI and fIfII occur in the frequency spectrum, the phase portrait is a closed curve band, and the Poincare map has a point group. When ΩI = 1.08, the high-speed gears are in 2T-quasi-periodic motion, fI and 1/2fI occur in the frequency spectrum, the phase portrait is a closed 2-circle curve band, and the Poincare map has two point groups. When ΩI = 1.13, the high-speed gears still are in 2T-quasi-periodic motion, the amplitude of 1/2fI increases in the frequency spectrum, the closed two-circle curve band on phase portrait gradually disappears, and the distance between two point groups on the Poincare map increases. When ΩI = 1.5, the high-speed gears are in chaotic motion, fI occurs in the frequency spectrum, the phase portrait is a complex winding curve group, and the Poincare map has a strange attractor. When ΩI = 1.78, the high-speed gears are in 2T-quasi-periodic motion, fI and 1/2fI occur in the frequency spectrum, the phase portrait is a closed 2-circle smooth curve band, and the Poincare map has two point groups. When ΩI = 2.1, the high-speed gears are in quasi-periodic motion, only fI occurs in the frequency spectrum and the amplitude of fI is the highest, the phase portrait is a closed smooth curve band, and the Poincare map has a point group.

Fig. 7
Dynamic responses of high-speed fixed-axis gears considering the flash temperature in mixed friction state (a) ΩI = 0.62, (b) ΩI = 0.8, (c) ΩI = 1.08, (d) ΩI = 1.13, (e) ΩI = 1.5, (f) ΩI = 1.78, and (g) ΩI = 2.1Dynamic responses of high-speed fixed-axis gears considering the flash temperature in mixed friction state (a) ΩI = 0.62, (b) ΩI = 0.8, (c) ΩI = 1.08, (d) ΩI = 1.13, (e) ΩI = 1.5, (f) ΩI = 1.78, and (g) ΩI = 2.1
Fig. 7
Dynamic responses of high-speed fixed-axis gears considering the flash temperature in mixed friction state (a) ΩI = 0.62, (b) ΩI = 0.8, (c) ΩI = 1.08, (d) ΩI = 1.13, (e) ΩI = 1.5, (f) ΩI = 1.78, and (g) ΩI = 2.1Dynamic responses of high-speed fixed-axis gears considering the flash temperature in mixed friction state (a) ΩI = 0.62, (b) ΩI = 0.8, (c) ΩI = 1.08, (d) ΩI = 1.13, (e) ΩI = 1.5, (f) ΩI = 1.78, and (g) ΩI = 2.1
Close modal

4.2 Influence of the Crack on the Dynamics of the System Considering the Flash Temperature

4.2.1 Experimental Validation of Gear Crack Characteristics.

Based on the multistage gear transmission system fault diagnosis platform shown in Fig. 8, the dynamics experiments are conducted for the system without and with crack, where, ν = 70 deg, q =1.0 mm. The acceleration vibration signals at the axial direction of the bearing block of the high-speed driving gear are collected using VB7 portable vibration analyzer, and the frequency spectrum is achieved by fast Fourier transform (FFT) and is normalized, as shown in Fig. 9. The motor rotation frequency is 40 Hz, the sampling frequency is 3000 Hz, and the number of sampling points is 2048. The main components of frequency spectra in health and crack states are fI, fII, the second harmonic frequency of low-speed fixed-axis gears 2fII, the input shaft rotation frequency fd, and the harmonics of fd. The amplitudes of fI, fII, and 2fII significantly increase after crack, and the fault sidebands fI±fd appear around fI.

Fig. 8
Test platform of the multistage gear transmission system (1—variable-frequency motor, 2—torque sensor and encoder, 3—radial load of bearing, 4—two-stage fixed-axis gearbox, 5—one-stage planetary gearbox, and 6—magnetic brake)
Fig. 8
Test platform of the multistage gear transmission system (1—variable-frequency motor, 2—torque sensor and encoder, 3—radial load of bearing, 4—two-stage fixed-axis gearbox, 5—one-stage planetary gearbox, and 6—magnetic brake)
Close modal
Fig. 9
Frequency spectrum of experiment (a) health state and (b) crack state
Fig. 9
Frequency spectrum of experiment (a) health state and (b) crack state
Close modal

Meanwhile, the numerical frequency domain results of high-speed driving gear without and with cracks are also presented in Fig. 10. The main frequency components remain consistent with those of experiments. When the system does not consider the flash temperature, the amplitude of fI increases after gear crack, and fI±fd occur near fI. When the system considers the flash temperature, the amplitudes of fI, fII, and 2fII all increase after gear crack, the fI±fd still occur near fI, and the changing trend of the numerical spectrum results, in this case, is consistent with the experimental data. However, the numerical calculation ignores the transverse and axial vibrations of the transmission shaft, and fd does not appear.

Fig. 10
Frequency spectrum of simulation (a) health state without the flash temperature, (b) crack state without the flash temperature, (c) health state with the flash temperature, and (d) crack state with the flash temperature
Fig. 10
Frequency spectrum of simulation (a) health state without the flash temperature, (b) crack state without the flash temperature, (c) health state with the flash temperature, and (d) crack state with the flash temperature
Close modal

4.2.2 Crack Characteristics Analysis of the System Considering the Flash Temperature.

The displacement bifurcation diagram of high-speed fixed-axis gears considering crack and flash temperature is presented in Fig. 11 when the system is in a mixed friction state and ΩI ∈ [0.01, 2.5], where, ν = 70 deg, q =1.0 mm, ξ=0.07, eamp,j=2μm, bI=bII=5μm, and bspn=brpn=2μm. The high-speed fixed-axis gears are in periodic motion as ΩI ∈ [0.01, 0.57], enter into chaotic motion by cataclysm as ΩI ∈ [0.57, 0.7], transform into quasi-periodic motion as ΩI ∈ [0.7, 0.97], transit to 2T-quasi-periodic motion by doubling bifurcation route as ΩI ∈ [0.97, 1.11], still are in 2T-quasi-periodic motion after cataclysm as ΩI ∈ [1.11, 1.16], enter into chaotic motion and the displacement amplitude decreases gradually as ΩI ∈ [1.16, 1.76], transform into 2T-quasi-periodic motion by inverse-doubling bifurcation route as ΩI ∈ [1.76, 1.86], and are in quasi-periodic motion as ΩI ∈ [1.86, 2.5]. Taken together, the high-speed fixed-axis gears enter 2T-quasi-periodic motion in advance, and the intervals of 2T-quasi-periodic motion and chaotic motion increase after the gear crack.

Fig. 11
Displacement bifurcation diagrams of high-speed fixed-axis gears with crack for the system considering the flash temperature
Fig. 11
Displacement bifurcation diagrams of high-speed fixed-axis gears with crack for the system considering the flash temperature
Close modal

The frequency spectrum, phase portrait, and Poincare map of high-speed fixed-axis gears for the system considering crack and flash temperature are presented in Fig. 12. when ΩI = 0.6, the high-speed gears are in chaotic motion, fI, fII, fIfII, and the fault sidebands fI±fd occur in the frequency spectrum, the phase portrait is a multicircle winding curve group, and the Poincare map is a set of discrete points. When ΩI = 0.9, the high-speed gears are in quasi-periodic motion, the amplitude of fI increases in the frequency spectrum, the phase portrait is a closed curve band, and the Poincare map has a point group. When ΩI = 1.0, the high-speed gears are in 2T-quasi-periodic motion, fI and 1/2fI occur in the frequency spectrum, the phase portrait is a closed two-circle curve band, and the Poincare map has two point groups. When ΩI = 1.7, the high-speed gears are in chaotic motion, the amplitude of fI increases and that of 1/2fI decreases in the frequency spectrum, the phase portrait is a complex winding curve group, and the Poincare map has a strange attractor. When ΩI = 1.85, the high-speed gears are in 2T-quasi-periodic motion, the amplitudes of fI and 1/2fI increase in the frequency spectrum, the phase portrait is a closed two-circle smooth curve band, and the Poincare map has two point groups. When ΩI = 2.0, the high-speed gears are in quasi-periodic motion, only fI occurs in the frequency spectrum and the amplitude of fI is the highest, the phase portrait is a closed smooth curve band, and the Poincare map has a point group.

Fig. 12
Dynamic responses of high-speed fixed-axis gears with crack for the system considering the flash temperature (a) ΩI = 0.6, (b) ΩI = 0.9, (c) ΩI = 1.0, (d) ΩI = 1.7, (e) ΩI = 1.85, and (f) ΩI = 2.0Dynamic responses of high-speed fixed-axis gears with crack for the system considering the flash temperature (a) ΩI = 0.6, (b) ΩI = 0.9, (c) ΩI = 1.0, (d) ΩI = 1.7, (e) ΩI = 1.85, and (f) ΩI = 2.0
Fig. 12
Dynamic responses of high-speed fixed-axis gears with crack for the system considering the flash temperature (a) ΩI = 0.6, (b) ΩI = 0.9, (c) ΩI = 1.0, (d) ΩI = 1.7, (e) ΩI = 1.85, and (f) ΩI = 2.0Dynamic responses of high-speed fixed-axis gears with crack for the system considering the flash temperature (a) ΩI = 0.6, (b) ΩI = 0.9, (c) ΩI = 1.0, (d) ΩI = 1.7, (e) ΩI = 1.85, and (f) ΩI = 2.0
Close modal

5 Conclusions

In the present work, a detailed study is implemented on the influences of the flash temperature and crack on the dynamics of a multistage gear transmission system. The torsional dynamic model is established for the system considering the flash temperature and high-speed driving gear crack, and the dynamic characteristics are analyzed through a bifurcation diagram, frequency spectrum, phase portrait, and Poincare map, and the frequency spectra of numerical simulation and experimental testing are compared. The main findings of the study are as follows:

  1. The flash temperature decreases the time-vary meshing stiffness, thus prolonging the quasi-periodic motion interval and shortening the 2T-quasi-periodic motion interval in the low-frequency region, and reducing the interval and increasing the displacement amplitude of chaotic motion in the high-frequency region.

  2. The changing trend of the numerical frequency spectrum results when considering the flash temperature is consistent with that of the experimental data for the high-speed fixed-axis gears without and with cracks.

  3. The crack makes the time-vary meshing stiffness decrease, and the time-vary meshing stiffness also decreases with the increase of the crack length.

  4. For the multi-stage gear transmission system considering the flash temperature, the 2T-quasi-periodic motion is in advance and the intervals of 2T-quasi-periodic motion and chaotic motion enlarge after the high-speed fixed-axis driving gear crack occurs.

These results provide a suitable meshing frequency for stable gear system operation, which makes the system avoid bifurcation points and chaotic intervals. Moreover, the established dynamic model considers the basic factors (i.e., time-varying meshing stiffness, comprehensive transmission error, damping, and backlash) and non-negligible nonlinear factors (i.e., friction and flash temperature caused by friction), which is closer to the fact. The spectrum responses obtained by this dynamic model are more accurate, greatly improving the reliability of fault monitoring.

Funding Data

  • Science and Technology Planning Project of Tianjin, Natural Science Foundation of Tianjin City (Grant No. 23JCZDJC00460; Funder ID: 10.13039/501100006606).

  • National Natural Science Foundation of China (Grant No. 51805369; Funder ID: 10.13039/501100001809).

Conflict of Interests

The authors declare that there is no conflict of interest regarding the publication of this article.

Data Availability Statement

There are no conflicts of interest.

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