Spiral bevel gears occupy several advantages such as high contact ratio, strong carrying capacity, and smooth operation, which become one of the most widely used components in highspeed stage of the aeronautical transmission system. Its dynamic characteristics are addressed by many scholars. However, spiral bevel gears, especially tooth fracture occurrence and monitoring, are not to be investigated, according to the limited published issues. Therefore, this paper establishes a threedimensional model and finite element model of the Gleason spiral bevel gear pair. The model considers the effect of tooth root fracture on the system due to fatigue. Finite element method is used to compute the mesh generation, set the boundary condition, and carry out the dynamic load. The harmonic response spectra of the base under tooth fracture are calculated and the influence of main parameters on monitoring failure is investigated as well. The results show that the change of torque affects insignificantly the determination of whether or not the system has tooth fracture. The intermediate frequency interval (200 Hz–1000 Hz) is the best interval to judge tooth fracture occurrence. The best fault test region is located in the working area where the system is going through meshing. The simulation calculation provides a theoretical reference for spiral bevel gear system test and fault diagnosis.
Spiral bevel gears are one of the most important components of the aeronautical transmission system. Due to the harsh working environment, failure occurs sometimes, and the most common situation is resonance failure. The failure of the spiral bevel gear transmission system often leads to serious accidents; therefore, accurate detection, the positioning of the fault, and eliminating hidden danger have very important significance in improving the operating efficiency of the gear system.
Most studies were related to spiral bevel gear modeling and tooth contact analysis (TCA); Tsai and Chin [
About spiral bevel gear fault detection, Zakrajsek et al. [
In the area of spiral gear dynamics analysis, Li and Hu [
Therefore, through system vibration signal to determine the occurrence of tooth fracture, thus eliminating the fault, is a meaningful work. In this paper, a pair of Gleason spiral bevel gears is mathematically modeled and a threedimensional solid model is generated as well. Finite element method is applied to analyze the harmonic response of the system; deformation amplitude and phase of the base are calculated. Therefore some tooth fracture features and main parameters’ influences are analyzed to assist the fault recognition.
Typical spiral bevel gears applied in aeronautical transmission system are manufactured by Gleason face hobbing process [
In the spherical coordinate system, the spherical involute equation on one side of outer space width is
The spherical involute equation on the other side of outer space width is
When spiral bevel gears are in cutting processing, tooth pitch surface unfolded drawing is shown in Figure
Pitch surface unfolded map.
The distance between cutting tool circle
In circle
In circle
According to the spherical geometrical relationship, spherical angle
Thus,
The spherical involute equation on one side of inner space width could be derived by substituting
In order to improve the accuracy of the threedimensional model, in this paper, a number of equally spaced auxiliary spherical involute lines are inserted in the direction of the tooth line. Substitute (RB) to (R0.1nB) in (
Based on the abovementioned involute equation, key points of the curve are established by “law command” in CATIA. Then “spline command” is applied to connect these key points for the tooth profile involute curve, and “multisection surface command” is used to generate the tooth surface. Moreover, tooth space is modeled by “split command,” which is based on these tooth surfaces. Finally, with the Boolean subtraction calculation of the solids, the parametric modeling of the spiral bevel gear could be successfully achieved, which verifies the correctness of the design method [
In addition, due to longterm operation in the highspeed circumstance, tooth root is prone to fatigue fracture. This paper simulates the fractured tooth and completes the healthy and fracture spiral bevel gear assembly model, as shown in Figure
System parameters.
Active gear  Driven gear  

Modulus  6  
Pressure angle (°)  20  
Tooth number  15  46 
Tooth width (m)  0.44  0.44 
Shaft angle 
90  
Mean spiral angle (°)  35 
Spiral bevel gear assembly model.
Healthy model
Tooth fracture model
The geometric model in CATIA is introduced into ANSYS workbench 18.1, so finite element analysis model is obtained. As friction contact analysis is a nonlinear problem, in order to save the computing resources and improve calculation accuracy, hexahedral meshing method is applied for calculation, and then the grid is refined in contact areas. The grid size of the noncontact region of this model is defined as 5 mm, and the contact area is defined as 1 mm, which ensures the rationality of the mesh and the maximum efficiency. At the same time, in order to ensure the convergence of the calculation results, the meshing model is firstly calculated, and then the mesh size of the contact area is gradually reduced. If the difference between calculation results is small after multiple attempts, the result is convergent and acceptable. Figure
Spiral bevel gear meshing.
Healthy model
Tooth fracture model
Contact area of healthy model
Contact area of tooth fracture model
After mesh is complete, the model is processing multiple settings, that is, pretreatment process:
Material definition: the model is set according to steel properties, the active and driven gears belong to the same material, Young’s modulus is 2.1 × 10^{11} Pa, Poisson’s ratio is 0.3, and the density is 7850 kg/m^{3}.
Contact definition: the calculation of the spiral bevel gear pair is surfacetosurface contact problem between the elastomers. It is of great importance to select contact surface and target surface; the inappropriate selection will lead to excessive penetration, affecting the accuracy of the solution. In general, when the convex and concave surfaces contact, the concave surface should be set as the target surface, so the spiral bevel gear outer tooth surface is defined as the target surface. In addition, the amount of penetration between the two contact surfaces depends on the normal contact stiffness. If the normal contact stiffness is too large, it will increase solution iteration number, which may lead to nonconvergence; if the normal contact stiffness is too small, the penetration between nodes could be too large, resulting in model instability. Based on the above analysis, this paper firstly uses the small normal contact stiffness coefficient and then gradually increases until analysis results deviation is so small. The results show that the best normal contact stiffness coefficient is 1.0 when the normal contact stiffness coefficient is set from 0.001, 0.01, 0.1, 1.0, and 1.1.
Rotational pair definition: the inner surfaces of the gears are set as reference surfaces, and then the rotational centers of reference surfaces are created. Finally, a pair of rotational gears is defined.
Solver definition: the general equation for harmonic response analysis is
where [
The timevarying meshing stiffness, meshing line displacement, and dynamic meshing force can be regarded as periodic format and can be expanded in Fourier series under fundamental meshing frequency. In this paper, dynamic meshing force is simulated as the excitation load, the torque is loaded in the sinusoidal form, the sweep frequency is set from 0 to 2000 Hz, initial phase angle is 0°, and solution intervals are 100.
Boundary conditions and load setting: degree of freedom (DOF) is released only in rotational direction, and the value of torque is 1200 N·m, which is gradually applied on the active gear.
The system is solved by the augmented Lagrangian method in ANSYS and the first six natural frequencies of the system are obtained, as shown in Table
Natural frequency.
Order  Healthy tooth model frequency (Hz)  Fracture tooth model frequency (Hz)  Deviation (Hz) 

























The base of the driven gear is the optimum position for deformation vibration detection sensor and acceleration vibration detection sensor so that the base is also set in response to the output in finite element method. Figure
Harmonic response spectrum of spiral bevel gear.
Deformation amplitude
Deformation phase angle
Acceleration amplitude
Acceleration phase angle
Here, in order to separate ideal monitoring frequency, three frequency intervals are defined to describe the sweep frequency, that is, low frequency interval (below than 200 Hz), intermediate frequency interval (200 Hz to 1000 Hz), and high frequency interval (greater than 1000 Hz) [
In low frequency interval, the tooth fracture model has a significant peak in the deformation amplitude spectrum, while the healthy tooth model has a valley in this interval. However, in the acceleration amplitude spectrum, the peaks of two models are equivalent.
In intermediate frequency interval, the tooth fracture model does not show an obvious peak in deformation and acceleration amplitude spectrum, while the peak value of healthy tooth model is very obvious. And there is no other interference signal in this interval and adjacent interval, so it can be effective in monitoring frequency interval for tooth fracture detection.
In high frequency interval, the acceleration amplitude spectrum is peaked at about 1050 Hz in fracture model, and the peak of healthy tooth model is four times that of the fracture model.
Compared with Figures
Compared with Figures
In brief, low frequency, intermediate frequency, and high frequency interval sections all have tooth fracture characteristic signal; however, the intermediate frequency interval (200 Hz–1000 Hz) is the best interval for detection among others.
In order to investigate the effect of torque (dynamic meshing force) and other excitations on the harmonic response of the spiral bevel gear system, tooth fracture model is target object, and the torque of different sizes is loaded without changing the other pretreatment process, so excitation’s influence on tooth fracture detection could be obtained. The harmonic response of each load is calculated, respectively, and the result is shown in Figure
Influence of torque on harmonic response of spiral bevel gear.
Deformation amplitude
Deformation phase angle
In order to improve the accuracy of the test, the influence of the measuring point position on the harmonic response of the spiral bevel gear system is studied. The responses of the different measuring points are calculated without changing the other pretreatment, so the best monitoring area could be found. The position of the test points is shown in Figure
Measuring points position.
Harmonic responses of spiral bevel gear under different measuring points.
Deformation amplitude
Deformation phase angle
Based on the meshing principle and gear cutting process, the tooth involute equation of the bevel gear is obtained, and the parametric spiral bevel gear threedimensional model and the finite element model are established.
The analysis results draw the following conclusions:
The authors declare that they have no conflicts of interest.
The work described in this paper was fully supported by the National Natural Science Foundation of PRC (Grant nos. 51375226 and 51475226); Postgraduate Research and Practice Innovation Program of Jiangsu Province; China Scholarship Council’s support for joint research with Professor Yeping Xiong in University of Southampton.