A calculation standard of tooth root stress and to

2022-08-16
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Abstract: This paper uses a high-performance parallel computer and ANSYS software to calculate the three-dimensional root stress and deformation of a real gear, provides a calculation standard for root stress and tooth deformation, provides a new foundation for gear dynamic, optimization and reliability design and CAE, and also tests the finite element parallel calculation ability of ANSYS software. The calculation is carried out on the dawning 4000A supercomputer. The degree of freedom of the calculation model is 6.59 million, there are 5 working conditions, the calculation time is 2 hours, and 4 CPUs are used

key words: tooth root stress, tooth deformation, supercomputer, parallel calculation, calculation standard

1 preface

the calculation of tooth root stress and tooth deformation of more than 500 enterprises at home and abroad is the basic problem of gear design and research. In the past, due to the limitation of computers, many researchers and analysts used two-dimensional or three-dimensional equivalent gear tooth models to calculate the root stress and gear tooth deformation. There are errors in the calculation models, and the reliability of the calculation models and results is low. Therefore, the calculation results are difficult to compare, especially with the test results, resulting in many repeated tests. Therefore, it is necessary to establish a standard for the calculation of three-dimensional tooth root stress and tooth deformation, which can be used as a reference for gear design and researchers and CAE researchers. Using high-performance computing, it is now possible to simulate and analyze the gear three-dimensional whole wheel model, and accurately calculate the gear root stress and tooth deformation. In this paper, a shearer gear is calculated on the Shuguang 4000A supercomputer

2 calculation of gear root stress and tooth deformation

Table 1 shows the parameters for calculating gears and machining tools. The tools are convex hobs, the addendum height coefficient is 1, and the shaft diameter (gear inner circle diameter) of gear fitting is 90 mm. Table 1 Calculation of tooth root stress gear parameters

2.1 gear three-dimensional calculation model

the load is uniformly or linearly distributed on the top of the tooth, the acting load is set as F = 100n/mm, the total load under different working conditions is the same, and the gear inner circle is fixed and constrained. Gear elastic modulus E = 2.06e5 n/mm2, Poisson's ratio ν= 0.3。

2.1.1 lattice model

Figure 1 is the geometric model for calculating the gear entity. Due to the stress concentration at the root of the gear, the three-dimensional quadratic isoparametric element should be selected. The element solid95 is used in ANSYS software, and the whole gear lattice is shown in Figure 2. The whole wheel lattice is divided into three parts, the load acting on the tooth sector, the adjacent tooth sector (left and right) and the rest of the tooth sector, and each part is divided into two parts: the tooth and the rim (wheel width). Due to the high grid density of the tooth root of the gear sector under load (Fig. 3), the grid density of other parts is also increased correspondingly, and the circumferential grid density of the rim part is the highest (relative to other rim parts). Because the stress is mainly concentrated in the local range of the load acting gear sector, a relatively sparse lattice can be used with its spacer gear sector, and the lattice of the adjacent gear sector is the transition lattice between the load acting gear sector and the spacer gear sector. The grids in the other gear tooth sectors are the same and can be copied. The radial grid density of all wheel flange parts is the same, and the tooth direction density of the whole wheel grid is the same. The load distribution acts on the nodes at the top of the tooth. The grid and uniformly distributed load of the load acting on the tooth are shown in Figure 3. There are certain requirements for the length ratio of three sides of the cell

Figure 1 gear solid three-dimensional geometric model figure 2 gear three-dimensional whole gear lattice model

Figure 3 gear tooth three-dimensional lattice model

2.1.2 calculation time and parallel performance

gear calculation model has 518940 units, nodes and common degrees of freedom. The calculation conditions are 5. The calculation time is in Table 2, and the solution method uses the preconditioned iterative solver of ANSYS. The calculation time of different CPU numbers is shown in Figure 4. The acceleration ratio of two CPUs is 2.17, and the acceleration ratio of four CPUs is 3 22。 The efficiency of shared memory parallel solution of ANSYS software is very high, which is not inferior to distributed parallel solution. The acceleration ratio of four CPUs can reach more than 3. Table 2 calculation time of three-dimensional full tooth root stress finite element model under multiple working conditions on Dawning 4000A

Figure 4 Calculation and solution time (CPU and wall clock wall time)

2.2 gear root stress

calculation results of maximum tooth root stress under different working conditions and eccentric load modes are in Table 3, σ F is the maximum tooth root tensile stress (i.e. the maximum principal stress) on the tension side σ 1), σ C is the maximum root compressive stress on the compression side (i.e. the minimum principal stress σ 3), table 4 shows the five load cases acting on the tooth top. Figure 5 shows the tooth direction distribution of tooth root stress on the tension side of uniformly distributed load, σ I = σ 1 - σ 3= 2Tmax, the stress intensity is equal to 2 times the maximum shear stress Tmax, σ E is the equivalent stress. For plane stress or plane strain, theoretically σ 1= σ 1. In fact, because the calculation error is not absolutely equal [2]. For plane stress, theoretically equivalent stress σ E = σ 1. Plane strain, theoretically equivalent force:. According to the calculation results in Figure 5, most of the gears (60%) work in the plane strain state in the middle and in the plane stress state at the extreme end. The maximum stress is at the transition from the strain state to the plane stress, and the distance from the end is about one-fifth (23%) of the tooth length. The end stress is the smallest, only 65% of the maximum stress, and the Middle stress is slightly smaller (1.73% smaller) than the maximum stress. Under condition 2, the triple load is uniformly distributed and acts on the middle third of the gear tooth surface, but the root stress increases only 2.1 times. In working condition 3, the triple load is uniformly distributed and acts on one third of one end of the gear tooth surface, and the tooth root stress increases by 2.65 times. Under condition 4, the linear distribution of double load at one end acts on the gear tooth surface, and the root stress increases only 1.63 times. In working condition 5, three times the load at one end is linearly distributed on the gear tooth surface, and the root stress increases by 2.23 times

Figure 5 tooth direction distribution of tooth root stress under uniformly distributed load

σ 1 is the maximum principal stress, σ I = σ 1- σ three σ E is the equivalent stress

Table 3 root stress and partial load coefficient under different load conditions

Table 4 five load conditions

2.3 gear tooth deformation

gear tooth deformation includes tooth contact deformation, deflection deformation and body deformation. The tooth deformation literature [3] has been studied, but it is only a two-dimensional equivalent model. This paper studies and calculates the influence of the three-dimensional model of tooth deflection and body deformation, and also calculates the tooth deflection and body deformation under different load conditions and eccentric loads. Figure 6 shows the gear deformation under the uniformly distributed load of condition 1, U1 is the deformation of the load acting on the symmetrical point of the gear tooth, and U2 is the deformation of the symmetrical point of the gear tooth adjacent to the direction of the load acting line of the load acting on the gear tooth. Therefore, u 2 approximately represents the deformation of the gear body, and u 1- u 2 can be defined as the tooth deflection deformation. The calculation results show that the deformation of gear with uniform load changes a little along the tooth direction, and the change of U1 is consistent with the root stress, but the amplitude is very small. The maximum value of U1 is 5.461 μ m, and the minimum value is 5.321 μ M. U 2 changes in contrast to U 1, with a maximum of 1.460 microns and a minimum of 1.444 microns. Therefore, the deformation deflection of gear teeth is 3.939 microns, and the deformation of gear body is 1.452 microns

Figure 6 uniformly distributed load gear deformation tooth direction distribution

u1 is the deformation of the symmetrical point of the gear tooth under the load, U2 is the deformation of the symmetrical point of the adjacent gear tooth

2.4 error analysis of the calculation model

the discrete error of the calculation model from Figure 2 has been very small. According to the calculation results of Figure 5, according to the difference between the maximum principal stress and the stress intensity, the calculation error of the root stress is estimated to be 3 ‰. The local discretization error of tooth root stress calculated by ANSYS software is only 0.35 ‰, so the reliability of the calculation model is very high, and the calculation error must be less than 1%. On the other hand, the constraints of the gear calculation model assume that the inner circle matched with the shaft is a fixed constraint. In fact, there are many constraints, so different constraints must have an impact on the root stress and gear deformation. This effect on the tooth root stress and tooth deformation is certainly very small. For those with large flange thickness, it can be ignored (such as this paper), but whether it is less than 1% needs further study. The contact load of gears is assumed to be uniformly distributed or linearly distributed on the tooth surface. In fact, the contact load is not assumed to be ideally distributed. Therefore, it is an important topic to establish a pair of three-dimensional contact simulation and analysis models of gears

the discrete error of the gear deformation calculation model is smaller than the root stress, and the calculation error is certainly less than 1%, but the change of the gear deformation tooth direction distribution may be greater than 1%. In this paper, the change of tooth direction distribution of gear body deformation under uniform load is 1.1%, and the total deformation change is 2.6% (Fig. 6). Therefore, the possible error of calculating gear tooth deformation is about 1%. Gear deformation is regarded as a reliable expert, which is mainly affected by elastic modulus. The error and randomness of elastic modulus may make the error of gear deformation reach% and need further research. On the other hand, the gear deformation (especially the body deformation) is greatly affected by the gear body structure. The body structure may make the error of gear deformation reach more than 10% [4], but it has little influence on the gear tooth deformation, which may also reach 1%. Therefore, further research is also needed. Similarly, the deformation of gears with different load distributions varies along the tooth direction, so it is necessary to establish a three-dimensional contact simulation analysis model of gears

3 conclusion

by establishing the three-dimensional integral gear gbj118 (8) simulation analysis model of civil building sound insulation design code, using a high-performance computer, the tooth root stress and gear deformation can be accurately calculated, and the calculation error of the calculation model must be less than 1%. The distribution of root stress and tooth deformation under different load distributions can be accurately calculated. A new calculation standard of root stress and tooth deformation has been established. The influence of three-dimensional model on root stress and tooth deformation has been shown. The calculation of root stress and tooth deformation has been further developed, and the dynamic design, optimization design, reliability design and CAE of gears have a new foundation

ansys software's parallel computing has been applied and tested, supercomputing has been applied, 6.59 million degrees of freedom have been successfully run on Dawning 4000A, the advantages of parallel computing have been brought into play, and a new ANSYS parallel computing benchmark has been produced

[References]

[1] edited by Wu Jize and Wang Tong. Tooth root transition curve and tooth root stress. National Defense Industry Press, 1989

[2] Yang Shenghua, Wang qinkuo. Application and trend of new finite element method in tooth root stress calculation. Proceedings of "mechanics 2000" academic conference, 2000.8:

[3] Yang Shenghua. Gear contact finite element analysis Journal of computational mechanics, 2003, 20 (2):

[4] Yang Shenghua Application of substructure and sub model method in gear stress and deformation calculation Computational mechanics in engineering and science, Beijing: Peking University Press, 2001:

[5] Yang Shenghua, high performance computing and CAE, Proceedings of the second China CAE Engineering Analysis Technology Annual Conference (ccatac, 2006), 2006.7. (end)

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