This paper establishes a quasi-static elastic model of helical gear based on the change rule of contact line length, the parabolic model of single tooth mesh stiffness per unit of contact length, and ISO6336-1 standard, aiming at the accuracy issue of the approximate replacement method of mesh stiffness of helical gear. It conducts a comparative study with the approximate replacement method based on the quasi-static rigid model and ISO6336-1 standard. The influence laws of geometric parameters on the comprehensive mesh stiffness and load distribution coefficient of helical gear are analyzed, and the influence changes of different broken tooth forms on the mesh stiffness and load distribution coefficient are discussed, providing a basis for the study of fault dynamics of helical gear.
1. Introduction
During the gear transmission process, the time-varying mesh stiffness is one of the main internal excitation sources of the transmission system vibration, and it is also the key to determining the load distribution among teeth. Accurately and quickly determining the time-varying mesh stiffness is of great significance for the dynamic analysis and strength calculation of gears. The main calculation methods of gear mesh stiffness include the material mechanics method, the finite element method, and the approximate replacement method. The basic idea of the material mechanics potential energy method is to regard the gear tooth as a variable cross-section cantilever beam on the base circle, and calculate the mesh stiffness of the gear by deriving the bending potential energy, shear potential energy, compression potential energy, and Hertz contact potential energy stored in the gear tooth during the meshing process. Although it is faster and more efficient than the finite element method, due to its own limitations, it cannot be directly used to calculate the mesh stiffness of helical gears. Based on the differential idea, WANG Q B et al. sliced the helical gear along the tooth width and calculated the mesh stiffness of the thin spur gear by using the potential energy method, and then obtained the mesh stiffness of the helical gear by 并联 superposition of the thin gear stiffness. On the basis of the slice model and the potential energy method, WEI Jing et al. proposed a calculation model of the mesh stiffness of the tooth profile modified helical gear by improving the calculation method of the meshing line length and position of the helical gear, which can not only calculate the mesh stiffness of the helical gear, but also determine the tooth profile modification amount. The finite element method can consider the three-dimensional effect of the deformation of the helical gear, with high accuracy, but it generally requires the assistance of high-performance computers and takes a long time. BU Zhonghong et al. proposed an improved method for calculating the mesh stiffness based on the linear programming method on the basis of the finite element method, and studied the change rule of the mesh stiffness within one meshing period by calculating the mesh stiffness of the internal and external meshing helical gears under different helix angles. CHANG Lehao et al. proposed a method for calculating the mesh stiffness of helical gears by combining the finite element method and the elastic contact theory, and verified it by comparing with the calculation results of the aviation standard. Currently, in the study of the dynamics and fault characteristics of helical gears, in order to achieve the rapid calculation of the mesh stiffness, JIANG Hanjun et al. adopted the approximate replacement method, which replaced the change of the gear mesh stiffness with the change of the gear contact line length, and approximated the mesh stiffness as a harmonic-like form through the Fourier series expansion. However, the approximate replacement method does not consider the change of the unit length mesh stiffness during the gear meshing process, and its calculation accuracy needs to be further studied. In this paper, based on the change rule of the contact line length of the helical gear, combined with the parabolic model of the unit length mesh stiffness of a single tooth and the ISO6336-1 standard, a quasi-static elastic model of the helical gear is established, and a comparative study is conducted with the approximate replacement method based on the quasi-static rigid model. By comparing with the ISO standard results, the accuracy of the quasi-static elastic model is verified. The influence laws of geometric parameters and broken tooth faults on the mesh stiffness and load distribution coefficient of the helical gear are analyzed and discussed, providing a basis for the study of the fault dynamics of the helical gear.
2. Quasi-Static Rigid Model of Helical Gear
The approximate replacement method of the mesh stiffness of helical gear assumes that the load on the gear during the meshing process is uniformly distributed along the contact line, the gear deformation is also uniformly distributed along the contact line, the unit length mesh stiffness of the gear is constant, and the mesh stiffness of the gear is only related to the contact line length.
The meshing plane of the helical gear is expanded according to the base circle, as shown in Figure 1. In Figure 1, is the end face contact ratio, is the axial contact ratio, is the total contact ratio, is the base circle helix angle, b is the tooth width, and is the end face meshing coordinate, and its size is the distance from the intersection of the tooth contact line and the end face projected onto the entry end on the base circle.
During one meshing period, when the gear tooth starts to mesh from the front end face point and exits from the rear end face point , gradually increases from 0 to . The length of the contact line of a single pair of teeth l can be expressed as:
Let , and the dimensionless form of the equation is:
where , and is a periodic linear function, which is:
The length of the ith contact line at any moment can be expressed as:
During one meshing period, the total contact line length is:
where N is the number of simultaneously meshing tooth pairs, and floor is the floor function.
When the helical gear is in quasi-static rigid meshing, its comprehensive mesh stiffness is:
where is the total contact line length of the gear, and is the unit length mesh stiffness, , and its value can be obtained according to the standard ISO63361.
According to the total contact line length , the quasi-static load on the tooth surface of the ith pair of teeth can be expressed as:
Then the load distribution coefficient among teeth is:
where is the total normal quasi-static load, and R is the load distribution coefficient among teeth.
3. Quasi-Static Elastic Model of Helical Gear
Due to the helical effect, the meshing area of the helical gear usually has two or more pairs of teeth meshing simultaneously at different tooth height positions. Although the mesh stiffness of the gear is proportional to the contact line length of the gear tooth, the mesh stiffness at the contact points at different tooth heights is different. SANCHEZ M B et al. studied the mesh stiffness of spur gears and helical gears in detail, and derived the mesh stiffness formula of a single pair of teeth, and verified it by the numerical method. The research shows that the mesh stiffness of a single pair of teeth is approximately a semi-sinusoidal or parabolic distribution, and the unit length mesh stiffness of a single pair of teeth relative to the maximum mesh stiffness can be expressed as:
where , and is the ratio of the minimum stiffness to the maximum stiffness of a single pair of teeth. Figure 2 shows the result comparison of the two models when the gear pair in Table 1 has and . It can be seen that when the spur gear is transmitting, the results obtained by the two models are relatively consistent.
According to the definition of , the mesh stiffness of a single pair of helical gear teeth can be expressed as:
where is the maximum mesh stiffness of a single pair of teeth, which can be calculated according to the standard ISO6336-1.
During one meshing period, the comprehensive mesh stiffness of the helical gear is:
According to the single tooth stiffness and the comprehensive stiffness of the gear, the tooth surface load of the ith pair of teeth can be expressed as:
Then the load distribution coefficient is:
4. Comparative Analysis of Calculation Results
Take a pair of external meshing gear pairs as an example, and the parameters are shown in Table 1. In order to study the influence of , the calculations are carried out for the two cases of and . Figure 3 shows the comparison of the meshing stiffness results calculated by the two models. In the figure, , , , and are the single tooth mesh stiffness and the comprehensive mesh stiffness of the quasi-static rigid model and the elastic model, respectively.
It can be seen that the smaller the , the smaller the single tooth stiffness and the comprehensive mesh stiffness , and the greater the relative error of the calculation results of the two models. In terms of the comprehensive mesh stiffness, the relative error between and is larger in the multi-tooth meshing area. When , the relative error between the two is about 5%, and when , the difference between the two is about 5.5% in the multi-tooth meshing area. When , the relative error between the two is about 2% in the single-tooth meshing area, and when , it is about 5.5%.
In order to verify the accuracy of the model, the average values of the comprehensive mesh stiffness of the two models at different helix angles are calculated and compared with the results of the standard ISO6336-1, as shown in Table 2. It can be seen from the comparison that when , the average mesh stiffness obtained by the quasi-static elastic model (Quasi Static Elastic Model, QSEM) has the smallest error with the results of the standard ISO6336-1, and the maximum error does not exceed 4%; when , the relative error of the calculation results with the standard results is about 6% on average, and the calculation results of the quasi-static rigid model (Quasi Static Rigid Model, QSRM) have a large error with the results of the ISO standard, and the overall error is more than 10%.
Figure 4 shows the comparison of the load distribution coefficients of the two models when the helical gear is transmitted. It can be seen that the change trend of the load distribution coefficient of the helical gear is similar to the change of the single tooth mesh stiffness, which gradually increases to the peak and then gradually decreases, and the smaller the , the steeper the curve change.
5. Analysis of the Influence of Main Geometric Parameters
From the above analysis, it can be known that the results of the quasi-static elastic model when are in good agreement with the ISO standard results. The influence of geometric parameters is analyzed based on this model. Figure 5 shows the influence of the helix angle on the comprehensive mesh stiffness and the load distribution coefficient of the helical gear. It can be seen from Figure 5a that with the increase of the helix angle, the comprehensive mesh stiffness of the helical gear first decreases and then increases, and when , the fluctuation of the comprehensive mesh stiffness is the smallest. According to Figure 5b, it can be known that the larger the helix angle, the smaller the peak value of the load distribution coefficient, and the flatter the curve.
Figure 6 shows the influence of the tooth width on the comprehensive mesh stiffness and the load distribution coefficient. It can be seen from Figure 6a that with the increase of the tooth width, the comprehensive mesh stiffness of the helical gear continuously increases, and when , the comprehensive mesh stiffness of the helical gear is approximately a straight line, and the fluctuation is the smallest. In Figure 6b, with the increase of the tooth width, the peak value of the load distribution coefficient continuously decreases, and the larger the tooth width, the flatter the curve.
When in Figure 5a and in Figure 6a, the fluctuations of the comprehensive mesh stiffness are both small. In order to further analyze the fluctuation rule of the mesh stiffness, the comprehensive mesh stiffness fluctuation factor is defined. Figure 7 shows the change rule of the stiffness fluctuation factor with the geometric parameters. It can be seen from Figures 7a and 7b that with the increase of the helix angle and the tooth width, the mesh stiffness fluctuation factor shows a fluctuating change of first decreasing, then increasing, and then decreasing. When the helix angle is equal to a certain value and the tooth width b is equal to a certain value, the axial contact ratio is close to an integer, or the stiffness fluctuation factor is located in the trough, and the mesh stiffness fluctuation is the smallest.
6. Analysis of the Influence of Broken Tooth Fault
6.1 Typical Broken Tooth Fault Model of Helical Gear
Tooth breakage is a relatively serious gear fault, which will greatly change the internal stiffness excitation and meshing state of the gear. This paper mainly studies two typical forms of broken tooth faults. The first is the local broken tooth along the contact line direction, and the second is the partial or whole tooth damage along the tooth width direction, as shown in Figure 8.
When a gear has a broken tooth, the contact area of the meshing tooth surface will be reduced, the contact line length of the gear will be changed, and then the mesh stiffness and transmission quality of the gear will be affected. According to Figure 8a, when the tooth is broken along the contact line direction, the reduction amount of the contact line length of a single pair of teeth is as follows。
