## 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

### 2.1 Time-Varying Meshing Stiffness Calculation

#### 2.1.1 Time-Varying Meshing Stiffness of Health Gear.

where $\chi $ is the Poisson's ratio, *N* is the number of gear teeth, *E* is the elastic modulus, *b*_{k} is the tooth width, $\alpha 0=20\u2009deg$, $\alpha 1$ is the pressure angle at the meshing point, $\alpha 2$ is half of the tooth angle on the base circle, and $\alpha 3$ is half of the tooth angle on the dedendum circle, $uF=rb,i[(\alpha 1+\alpha 2)sin\u2009\alpha 1+cos\u2009\alpha 1\u2212cos\u2009\alpha 2]$, $sF=2rb,i\u2009sin\u2009\alpha 2$, $rb,i$ is the base circle radius, and the coefficients $L*,M*,P*,and\u2009Q*$ are the same as those in Ref. [20].

where *g *=* *1,2 denotes the driving and driven gears and *y* denotes the *y*th 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 $\nu $. 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.

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]

- When $hc\u2265hr$ & $\alpha 1$> $\alpha c$$1kscrack=\u222b\alpha 3\alpha r2.4(1+\chi )\u2009cos2\alpha 1\u2009sin\u2009\alpha Ebk(sin\u2009\alpha 3\u2212qrbsin\u2009v+sin\u2009\alpha )3d\alpha +2.4(1+\chi )\u2009cos2\alpha 1(cos\u2009\alpha 2\u2212N\u22122.5N\u2009cos\u2009\alpha 0cos\u2009\alpha 3)Ebk(2\u2009sin\u2009\alpha 2\u2212qrbsin\u2009v)+\u222b\u2212\alpha c\alpha 22.4(1+\chi )(\alpha 2\u2212\alpha )cos\u2009\alpha \u2009cos2\alpha 1Ebk[sin\u2009\alpha +(\alpha 2\u2212\alpha )cos\u2009\alpha +sin\u2009\alpha 2\u2212qrbsin\u2009\nu ]d\alpha +\u222b\u2212\alpha 1\u2212\alpha c1.2(1+\chi )(\alpha 2\u2212\alpha )cos\u2009\alpha \u2009cos2\alpha 1Ebk[sin\u2009\alpha +(\alpha 2\u2212\alpha )cos\u2009\alpha ]d\alpha $(9)where $hc$ is the distance from the crack end point$1kbcrack=\u222b\alpha 3\alpha r12\u2009sin\u2009\alpha [N\u2009cos\u2009\alpha 0N\u22122.5\u2212(cos\u2009\alpha +cos\u2009\alpha 3\u2212cos\u2009\alpha r\u2212qrrcos\u2009v)cos\u2009\alpha 1]2Ebk(sin\u2009\alpha 3+sin\u2009\alpha \u2212qrrsin\u2009v)3d\alpha +4[1\u2212(N\u22122.5)cos\u2009\alpha 1\u2009cos\u2009\alpha 3N\u2009cos\u2009\alpha 0]3\u22124(1\u2212cos\u2009\alpha 1\u2009cos\u2009\alpha 2)3Ebk\u2009cos\u2009\alpha 1(2\u2009sin\u2009\alpha 2\u2212qrbsin\u2009v)3+\u222b\u2212\alpha c\alpha 212{1+cos\u2009\alpha 1[(\alpha 2\u2212\alpha )sin\u2009\alpha \u2212cos\u2009\alpha ]}2(\alpha 2\u2212\alpha )cos\u2009\alpha Ebk[sin\u2009\alpha +(\alpha 2\u2212\alpha )cos\u2009\alpha +sin\u2009\alpha 2\u2212qrbsin\u2009\nu ]3d\alpha +\u222b\u2212\alpha 1\u2212\alpha c3{1+cos\u2009\alpha 1[(\alpha 2\u2212\alpha )sin\u2009\alpha \u2212cos\u2009\alpha ]}2(\alpha 2\u2212\alpha )cos\u2009\alpha 2Ebk[sin\u2009\alpha +(\alpha 2\u2212\alpha )cos\u2009\alpha ]3d\alpha $(10)
*A*to the tooth central line, $hr$ is half of the roof chordal tooth thickness, $\alpha c$ is the angle at the force action point*K*, and $\alpha r$ is half of the center angle corresponding to the tooth root circle's chord through point*A*. - When $hc$<$hr$ or when $ha\u2265hr$ & $\alpha 1\u2264\alpha c$$1kscrack=\u222b\alpha 3\alpha r2.4(1+\chi )\u2009cos2\alpha 1\u2009sin\u2009\alpha Ebk(sin\u2009\alpha 3\u2212qrbsin\u2009v+sin\u2009\alpha )3d\alpha +2.4(1+\chi )\u2009cos2\alpha 1(cos\u2009\alpha 2\u2212N\u22122.5N\u2009cos\u2009\alpha 0cos\u2009\alpha 3)Ebk(2\u2009sin\u2009\alpha 2\u2212qrbsin\u2009v)+\u222b\u2212\alpha 1\alpha 22.4(1+\chi )(\alpha 2\u2212\alpha )cos\u2009\alpha \u2009cos2\alpha 1Ebk[sin\u2009\alpha +(\alpha 2\u2212\alpha )cos\u2009\alpha +sin\u2009\alpha 2\u2212qrbsin\u2009\nu ]d\alpha $(11)For the single-tooth-pair meshing duration, the time-varying meshing stiffness of the gears with crack, $ktcrack(t)$, is expressed as$1kbcrack=\u222b\alpha 3\alpha r12\u2009sin\u2009\alpha [N\u2009cos\u2009\alpha 0N\u22122.5\u2212(cos\u2009\alpha +cos\u2009\alpha 3\u2212cos\u2009\alpha r\u2212qrrcos\u2009v)cos\u2009\alpha 1]2Ebk(sin\u2009\alpha 3+sin\u2009\alpha \u2212qrrsin\u2009v)3d\alpha +4[1\u2212(N\u22122.5)cos\u2009\alpha 1\u2009cos\u2009\alpha 3N\u2009cos\u2009\alpha 0]3\u22124(1\u2212cos\u2009\alpha 1\u2009cos\u2009\alpha 2)3Ebk\u2009cos\u2009\alpha 1(2\u2009sin\u2009\alpha 2\u2212qrbsin\u2009v)3+\u222b\u2212\alpha 1\alpha 212{1+cos\u2009\alpha 1[(\alpha 2\u2212\alpha )sin\u2009\alpha \u2212cos\u2009\alpha ]}2(\alpha 2\u2212\alpha )cos\u2009\alpha Ebk[sin\u2009\alpha +(\alpha 2\u2212\alpha )cos\u2009\alpha +sin\u2009\alpha 2\u2212qrbsin\u2009\nu ]3d\alpha $(12)$ktcrack(t)=11kh+1kb1crack+1ks1crack+1ka1+1kf1+1kb2+1ks2+1ka2+1kf2$(13)

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 $\nu $=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.

### 2.2 Time-Varying Flash Temperature Stiffness Calculation

#### 2.2.1 Flash Temperature.

where $\mu (t)$ is the friction coefficient under mixed lubrication state, $\varpi $ 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, $\rho g$ is the material density, and $\sigma g$ is the specific heat capacity.

where $Sa$ is the average tooth surface roughness, $\eta M$ is the dynamic viscosity coefficient of lubricating oil, $vs(t)$ is the sliding velocity of meshing tooth pair, and $vs(t)=|\nu 1(t)\u2212\nu 2(t)|$; $vc(t)$ is the entraining velocity of meshing tooth pair, and $vc(t)=|\nu 1(t)+\nu 2(t)|/2$.

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

where $\psi $ 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.

where $\u2113g$ is the linear expansion coefficient of the material, $og$ is the thermal deformation of the base circle, and $og=\u2113grb,gt(rz,g)+\u2113grb,g(1+\chi g)[rb,g2(1\u22122\chi g)\u2212rz,g2][t(rb,g)\u2212t(rz,g)]/[(1\u2212\chi g)(rb,g2\u2212rz,g2)]$; $t(rz,g)$ 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; $\alpha a,g$ is the addendum pressure angle, and $\alpha a,g=arccos(rb,g/ra,g)$, $\iota g$ is the tooth thickness, and $\iota g(t)=lgrg(t)/rd,g\u22122rg(t)(inv(arccos(rb,g/rg(t)))\u2212inv\alpha )$, $lg$ is the tooth thickness of the reference circle, and $rd,g$ is the reference circle radius, $inv\alpha $ is the involute function, and $inv\alpha =tan\u2009\alpha \u2212\alpha $.

#### 2.2.3 Time-Varying Flash Temperature Stiffness.

### 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.

## 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 *I*1, *I*2, and low-speed driving and driven gears *II*1, *II*2. In the one-stage planetary gear system, *s*, *pn*(n = 1,2,3,4), *r*, and *c* represent the sun gear, *n*th 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 *T*_{in} and the output torque is *T*_{out}. Besides, every gear is a spur gear.

### 3.2 Dynamic Equations of the System.

*J*is the moment of inertia, and

_{i}*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; $\theta i$ 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

where $\xi $ 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.

where $bj$ is half of the backlash.

where $kamp,j$ and $\varphi j$ is the amplitude and initial phase of the resultant meshing stiffness.

where $\lambda j(t)$ is the direction coefficient.

*p*is introduced, and takes $bI$. The dimensionless excitation frequency $\Omega j$, dimensionless displacement $uj$, dimensionless velocity $u\u02d9j$, and dimensionless acceleration $u\xa8j$ are calculated by $\Omega j=\omega j/\omega e$, $uj=xj/p$, $u\u02d9j=x\u02d9j/p\omega e$, and $u\xa8j=x\xa8j/p\omega e2$, respectively. The dimensionless equations of the system are expressed as follows:

where $G(j)=cj\omega eu\u02d9j+ksum,j(\tau )\omega e2f(uj)$; $ma1=JI1(JI2+JII1)JI1rb,\u200aI22+(JI2+JII1)rb,\u200aI12$; $ma2=JI2+JII1rb,\u200aI1rb,\u200aI2$; $ma3=(JI2+JII1)JII2(JI2+JII1)rb,\u200aII22+JII2rb,\u200aII12$; $ma4=JII2+Jsrb,\u200aII2rb,\u200as$; $ms=JII2+Jsrb,\u200as2$; $mc=Jcrb,\u200ac2$; $mpn=Jpnrb,\u200apn2$; $q1=JI1LI2rb,\u200aI2+(JI2+JII1)LI1rb,\u200aI1JI1rb,\u200aI22+(JI2+JII1)rb,\u200aI12$; $q2=LI2rb,\u200aI2$; $q3=LII1rb,\u200aII1$; $q4=(JI2+JII1)LII2rb,\u200aII2+JII2LII1rb,\u200aII1(JI2+JII1)rb,\u200aII22+JII2rb,\u200aII12$; $q5=Lsrb,\u200as$; $q6=LII2rb,\u200aII2$; $q7=Lpnsrb,\u200apn$; $q8=Lpnrrb,\u200apn$.

## 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.

High-speed fixed-axis gears | Low-speed fixed-axis gears | planetary gears | |||||
---|---|---|---|---|---|---|---|

Gear parameters | I1 | I2 | II1 | II2 | s | pn | r |

Tooth number | 29 | 100 | 36 | 90 | 28 | 36 | 100 |

Modulus (mm) | 1.5 | 1.5 | 1.5 | 1.5 | 1 | 1 | 1 |

Radius of base circle (mm) | 20.4 | 70.5 | 25.3 | 63.4 | 13 | 17 | 47 |

Gear mass (g) | 125 | 1224.5 | 224 | 1111 | 41 | 34.6 | — |

Moment of inertia (kg m^{2}) | 0.05 | 6 | 0.14 | 4 | 0.007 | 0.01 | — |

Tooth width (mm) | 30 | 30 | 30 | 30 | 20 | 20 | 20 |

Poisson's ratio | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

Pressure angle (deg) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

Thermal conductivity [J/(m s °C)] | 46.47 | 46.47 | 46.47 | 46.47 | 46.47 | 46.47 | 46.47 |

Material density (kg/m^{3}) | 7850 | 7850 | 7850 | 7850 | 7850 | 7850 | 7850 |

Elastic modulus (GPa) | 206 | 206 | 206 | 206 | 206 | 206 | 206 |

Specific heat capacity [J/(kg °C)] | 481.5 | 481.5 | 481.5 | 481.5 | 481.5 | 481.5 | 481.5 |

Linear expansion coefficient (×10^{−5}) | 1.16 | 1.16 | 1.16 | 1.16 | 1.16 | 1.16 | 1.16 |

Gear shaft radius (mm) | 5 | 10 | 5 | 15 | 5 | 5 | 5 |

Temperature of shaft (°C) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

Temperature of base circle surface (°C) | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

Initial temperature of tooth surface (°C) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

Steady bulk temperature of tooth surface (°C) | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

High-speed fixed-axis gears | Low-speed fixed-axis gears | planetary gears | |||||
---|---|---|---|---|---|---|---|

Gear parameters | I1 | I2 | II1 | II2 | s | pn | r |

Tooth number | 29 | 100 | 36 | 90 | 28 | 36 | 100 |

Modulus (mm) | 1.5 | 1.5 | 1.5 | 1.5 | 1 | 1 | 1 |

Radius of base circle (mm) | 20.4 | 70.5 | 25.3 | 63.4 | 13 | 17 | 47 |

Gear mass (g) | 125 | 1224.5 | 224 | 1111 | 41 | 34.6 | — |

Moment of inertia (kg m^{2}) | 0.05 | 6 | 0.14 | 4 | 0.007 | 0.01 | — |

Tooth width (mm) | 30 | 30 | 30 | 30 | 20 | 20 | 20 |

Poisson's ratio | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

Pressure angle (deg) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

Thermal conductivity [J/(m s °C)] | 46.47 | 46.47 | 46.47 | 46.47 | 46.47 | 46.47 | 46.47 |

Material density (kg/m^{3}) | 7850 | 7850 | 7850 | 7850 | 7850 | 7850 | 7850 |

Elastic modulus (GPa) | 206 | 206 | 206 | 206 | 206 | 206 | 206 |

Specific heat capacity [J/(kg °C)] | 481.5 | 481.5 | 481.5 | 481.5 | 481.5 | 481.5 | 481.5 |

Linear expansion coefficient (×10^{−5}) | 1.16 | 1.16 | 1.16 | 1.16 | 1.16 | 1.16 | 1.16 |

Gear shaft radius (mm) | 5 | 10 | 5 | 15 | 5 | 5 | 5 |

Temperature of shaft (°C) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

Temperature of base circle surface (°C) | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

Initial temperature of tooth surface (°C) | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

Steady bulk temperature of tooth surface (°C) | 100 | 100 | 100 | 100 | 100 | 100 | 100 |

### 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/*Ω _{I}* and

*Ω*is the excitation frequency and cannot be zero. Moreover, the system is in quasi-periodic motion when

_{I}*Ω*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

_{I}*Ω*∈ [0.01, 2.5], where $\xi =0.07$, $eamp,j=2\u2009\mu m$, $bI=bII=5\u2009\mu m$, and $bspn=brpn=2\u2009\mu m$. In Fig. 6(a), when the flash temperature is not considered, the high-speed fixed-axis gears are in periodic motion as

_{I}*Ω*∈ [0.01, 0.56], enter into chaotic motion by cataclysm as

_{I}*Ω*∈ [0.56, 0.7], transform into quasi-periodic motion as

_{I}*Ω*∈ [0.7, 0.92], transit to 2T-quasi-periodic motion by doubling bifurcation route as

_{I}*Ω*∈ [0.92, 1.09], still are in 2T-quasi-periodic motion after cataclysm as

_{I}*Ω*∈ [1.09, 1.14], enter into chaotic motion and the displacement amplitude decreases gradually as

_{I}*Ω*∈ [1.14, 1.77], transform into 2T-quasi-periodic motion by inverse-doubling bifurcation route as

_{I}*Ω*∈ [1.77, 1.88], and are in quasi-periodic motion as

_{I}*Ω*∈ [1.88, 2.5]. In Fig. 6(b), when considering the flash temperature, the high-speed fixed-axis gears are in periodic motion as

_{I}*Ω*∈ [0.01, 0.57], enter into chaotic motion by cataclysm as

_{I}*Ω*∈ [0.57, 0.7], transform into quasi-periodic motion as

_{I}*Ω*∈ [0.7, 0.99], transit to 2T-quasi-periodic motion by doubling bifurcation route as

_{I}*Ω*∈ [0.99, 1.11], still are in 2T-quasi-periodic motion after cataclysm as

_{I}*Ω*∈ [1.11, 1.16], enter into chaotic motion and the displacement amplitude decreases gradually as

_{I}*Ω*∈ [1.16, 1.74], transform into 2T-quasi-periodic motion by inverse-doubling bifurcation route as

_{I}*Ω*∈ [1.74, 1.85], and are in quasi-periodic motion as

_{I}*Ω*∈ [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.

_{I}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 *Ω _{I}* = 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 $fI\u2212fII$ 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

*Ω*= 0.8, the high-speed gears are in quasi-periodic motion, $fI$ and $fI\u2212fII$ occur in the frequency spectrum, the phase portrait is a closed curve band, and the Poincare map has a point group. When

_{I}*Ω*= 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

_{I}*Ω*= 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

_{I}*Ω*= 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

_{I}*Ω*= 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

_{I}*Ω*= 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.

_{I}### 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, $\nu $ = 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 $fI\xb1fd$ appear around $fI$.

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 $fI\xb1fd$ occur near $fI$. When the system considers the flash temperature, the amplitudes of $fI$, $fII$, and $2fII$ all increase after gear crack, the $fI\xb1fd$ 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.

#### 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 *Ω _{I}* ∈ [0.01, 2.5], where, $\nu $ = 70 deg,

*q*=

*1.0 mm, $\xi =0.07$, $eamp,j=2\mu m$, $bI=bII=5\mu m$, and $bspn=brpn=2\mu m$. The high-speed fixed-axis gears are in periodic motion as*

*Ω*∈ [0.01, 0.57], enter into chaotic motion by cataclysm as

_{I}*Ω*∈ [0.57, 0.7], transform into quasi-periodic motion as

_{I}*Ω*∈ [0.7, 0.97], transit to 2T-quasi-periodic motion by doubling bifurcation route as

_{I}*Ω*∈ [0.97, 1.11], still are in 2T-quasi-periodic motion after cataclysm as

_{I}*Ω*∈ [1.11, 1.16], enter into chaotic motion and the displacement amplitude decreases gradually as

_{I}*Ω*∈ [1.16, 1.76], transform into 2T-quasi-periodic motion by inverse-doubling bifurcation route as

_{I}*Ω*∈ [1.76, 1.86], and are in quasi-periodic motion as

_{I}*Ω*∈ [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.

_{I}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 *Ω _{I}* = 0.6, the high-speed gears are in chaotic motion, $fI$, $fII$, $fI\u2212fII$, and the fault sidebands $fI\xb1fd$ 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

*Ω*= 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

_{I}*Ω*= 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

_{I}*Ω*= 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

_{I}*Ω*= 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

_{I}*Ω*= 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.

_{I}## 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:

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.

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.

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

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.