1 Probability finite element sampling method In the PDS module of ANSYS, the commonly used probabilistic analysis methods include Monte Carlo method and response surface method. Of course, users can customize other methods. Among them, Monte Carlo method includes direct sampling and Latin hypercube. Sampling, response surface method includes central composite sampling method and Box-Behnken matrix sampling method. This section briefly describes the basic principles of the above four sampling methods.
1.1 Monte Carlo method 1.1.1 Direct sampling Monte Carlo direct sampling method is the most common and traditional Monte Carlo analysis method, this method is very common, because it mimics the natural process, everyone can observe and imagine, so It is also very easy to understand. For this method, a simulated cycle represents a component under specific load and boundary conditions. However, this method is not always feasible, mainly because the sampling process is not memory. For example, if two random input variables are set, X1 and X2, which are subject to uniform distribution, and between 0 and 1, in the range of values, 15 subsamples are taken. In the sampling point, there may be two Or multiple sampling points overlap or are relatively close, as shown in 1(a), such sampling should be avoided as much as possible during the calculation process.
1.1.2 Latin hypercube sampling Latin hypercube sampling: Latin hypercube sampling is a high-level and more efficient Monte Carlo method. The difference between Latin hypercube sampling and direct sampling is that Latin hypercube sampling has memory. Avoid sampling points or sample points that are close, as shown in 1(b). Usually Latin hypercube sampling is 20 to 40 less than the direct sampling method, but for more complex problems, it also requires a large number of sampling times.
1.2 Response surface method 1.2.1 Center composite sampling method Center composite sampling, sampling points are the center point, N coordinate axis points, the sum of 2N-f points centered on the N-dimensional hypercube center, N is a random input The number of variables, f is the factorial parameter of the central compound method. When f=0, it is called full-order sampling. When f=1, it is called half-order sampling. With the number of random input variables, f values ​​are different. Generally speaking, as the number of random input variables increases, the value of f increases, which makes the calculated number of cycles more reasonable. The f value is automatically estimated according to the number of random input variables in the calculation.
The central composite sampling is shown in (a).
1.2.2 Box-Behnken Matrix Sampling Method The number of sampling points of the Box-Behnken matrix method is the center point plus the midpoint of each side of the N-dimensional hypercube. The sampling point of the Box-Behnken matrix method is shown in (b).
2 Probability finite element analysis of gear meshing process based on different sampling methods 2.1 Establishment of probabilistic finite element model Finite element analysis of the meshing process of a pair of tiny gears, the gears are processed by quasi-LIGA method, the finite element model diagram is as shown, tiny For the material parameters of the gear and the load to be applied, E1, 1, E2 and 2 are the corresponding symbols used in the calculation, assuming that the above parameters obey the Gaussian distribution.
1 Gear material parameters and load parameters Pinion material parameters Big gear Material parameters Elastic modulus E1/GPa Poisson's ratio 1 Elastic modulus E2/GPa Poisson's ratio 2 Nominal value 710.31710.31 Standard deviation 3.550.01554.260.0186
2.2 Verification of finite element model The gear contact analysis of this paper belongs to nonlinear analysis. The analysis results are mainly affected by the established contact relationship and the number of elements and the degree of unit density. Dividing the grid is an important part of establishing the finite element model. There are many problems to be considered, and the amount of work required is large. The divided grid form will have a direct impact on the calculation accuracy and calculation scale. In this paper, according to the Hertz contact theory, the contact half width is estimated, and 1/5 of the half width is used as the unit division length of the local region of the gear meshing position. When the material parameters and load of the gear are nominal values, the design parameters and theoretical calculation of the meshing gear are calculated. As shown in the results of the finite element calculation, it can be seen from the results in the table that the finite element model has a reasonable unit density at the gear meshing, the contact relationship is processed correctly, and the calculation result is accurate, which satisfies the engineering needs. The correct establishment of the finite element model not only improves the computational efficiency of the subsequent probability finite element analysis but also ensures the calculation accuracy.
2 design parameters of the meshing gear and theoretical calculations and finite element calculation results comparison modulus m / mm pressure angle / (°) number of teeth Z1 axis diameter d1/mm number of teeth Z2 axis diameter d2 / mm tooth width B / mm finite element results / MPa Hertz theory result / MPa error / 0.120120.4200.60.68146.04140.913.51
2.3 Analysis of calculation results 2.3.1 Monte Carlo calculation results analysis Using ANSYS PDS module for analysis, when Monte Carlo direct sampling and Latin hypercube sampling method are selected, the number of iterations is set to 100 times, and the above two methods are applied. The calculation results of the maximum equivalent stress at the meshing point in the probability sampling analysis are as shown, and the results of the sensitivity analysis are as shown. According to the sampling calculation results, the fitted response surface map is as shown. It can be seen that in the direct sampling analysis, the maximum equivalent stress at the gear meshing point is 296.8 MPa and the minimum is 270.61 MPa. In the Latin hypercube sampling analysis, the maximum equivalent stress at the gear meshing joint is 297.01 MPa and the minimum is 271.78. MPa, the maximum difference between the two sampling methods is about 0.4, but by looking at the sample file, it can be seen that some sampling points are close to overlap in direct sampling, mainly because the direct sampling method has no memory. It can be seen from 5 that the calculation results of the two sampling methods are basically the same. It can be seen that the elastic modulus of the material parameters of the large and small gears has the greatest influence on the maximum equivalent stress at the gear meshing, which is most affected by the elastic modulus of the large gear. To ensure the reliability of the gear system operation, we must first ensure the elastic modulus of the large and small gears, that is, to control the processing of large and small gears and heat treatment processes. It can be seen from 6 that the response surfaces fitted by the two sampling methods are basically the same.
2.3.2 Response surface method calculation result analysis The application response surface method calculation result is shown in Fig. 7. The ANSYS PDS module automatically calculates the iteration number 25. From Fig. 7, the center composite method sampling calculation result and the Box-Behnken matrix method sampling calculation The results are basically the same and the shape of the surface is basically similar. Based on the calculation results of the response surface, the extended calculation of Monte Carlo method can also be performed. The sampling point is 10000. Because the calculation is based on the fitting result, the calculation speed is very fast, and the calculation result is as shown in the figure. It can be seen that in the central composite sampling analysis, the maximum equivalent stress at the gear meshing point is 304.8 MPa and the minimum is 263.97 MPa. In the Box-Behnken matrix sampling analysis, the maximum equivalent stress at the gear meshing joint is 304.43 MPa. The minimum value is 263.97 MPa, and the maximum difference between the two sampling methods is about 0.2. Compared with the Monte Carlo calculation results, including the Monte Carlo method, it can be seen that the response surface method effectively reduces the number of iterations. And the reliability of the system is effectively predicted.
3 Ending the paper Based on the established nonlinear finite element model of gear meshing contact, Monte Carlo method (direct sampling and Latin hypercube sampling) and response surface method (central composite sampling and Box-Behnken matrix method) Sampling) The gear system was calculated. According to the comparison of results, on the basis of the calculation of the response surface method, the Monte Carlo expansion calculation can be performed to obtain more accurate analysis results, and the calculation efficiency is high. In addition, the reliability of the gear system operation is known by calculation. The elastic modulus of the two gear material parameters has a great influence, and the elastic modulus of the large gear has the greatest influence on the system. To ensure the reliability of the system operation, the above two parameters must be strictly controlled. Through the above research, the probabilistic finite element analysis method and reference basis are provided for the reliability design of the gear system, thus effectively ensuring the reliability of the gear system operation.
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