The tooth of the double arc gear is a space spiral. The mathematical model of the double arc gear described by the equidistant line has an error when the helix angle is not 0, so the analysis result is also biased, and the double arc of any corner position The gear tooth profile mathematical model can accurately and realistically describe any point on the double arc gear tooth profile curve. Therefore, based on the mathematical model, the bending stress of double-arc gear is studied, and the influence of the parameters of gear teeth on the bending stress of the gear is revealed.
It provides a simple and fast new theoretical basis for the digital design, parameter optimization and manufacturing of double arc gears.
Mathematical model of the tooth profile of a double circular arc gear 1.1 The tooth profile mathematical model of a double circular arc gear sorts the equations of the tooth profile curves in each segment and is unified into the following form: x=rcos (r-Si)sin(i)cosiy =rsin-(r-Si)cos(i)cosi(1) where: is the involute angle of the tooth profile of each segment (rad); Si is the angle of the envelope point of the segment on each tooth profile curve The distance of the line, Si={Shg, Shf, Shj, Sha}, i is the angle between the CT line and the tool pitch line, i={gt, ft, jt, at}. The meaning of other parameters in equation (1) And calculation details.
1.2 Model Accuracy Analysis The accuracy of the tooth profile model was numerically verified. According to the mathematical model formula (1), the regenerative gear model is repeated in the Pro/E environment with different gear configuration parameters. Each time a gear model is generated, the generated tooth profile curve is compared with the envelope formed by the fan-forming method, and The distance between the two is measured, and the measurement error is less than 10-6 mm (the default accuracy of Pro/E).
Therefore, the gear model used in this paper is sufficient to meet the accuracy requirements of engineering analysis.
Finite Element Analysis of Bending Stress of Double Arc Gears Based on the powerful function of Pro/E parametric modeling, a finite element analysis of a series of accurate models of double arc gears with different parameters was carried out. Due to the limitation of computer capacity and speed, only three teeth of the gear are taken as research objects. In addition, the common failure modes of the double circular arc gears in actual use show that the root, the concave teeth and the tooth waist of the double circular arc gear are its weak parts, so the bending stress of the three tooth profiles is only described in this paper. the study.
2.1 Distribution of bending stress in each segment of double arc gears A double arc gear model with a number of teeth z=16, modulus m=3 and helix angle of 20 is constructed in Pro/E to rigidly constrain the inner hole of the gear model. The degree of freedom, ignoring the influence of the keyway on the gear stress, the load is directly applied to the tooth surface of one side of the gear with a load of 10 MPa. The automatic mesh is used, and Multi-PassAdaptive is used. ), set the convergence precision to 5.
Finally, the finite element analysis model of the gear is obtained.
Using the Mechaniea module of Pro/E software to carry out finite element analysis on the bending stress of the model, the bending stress cloud diagram of the double circular arc gear is obtained. The stress values ​​on both sides of the gear teeth are output to Excel, and then the data is used to draw the chart. , the abscissa is the arc length of the tooth profile, and the ordinate is the stress value.
The stress value on the pressure side of the same tooth is larger than the stress value on the tension side. It can also be concluded that the compressive stress and the tensile stress are consistent along the whole tooth profile curve; the maximum stress of the entire tooth profile occurs at the root of the tooth, and its position is close to the intersection of the root and the concave tooth; the maximum stress of the concave tooth occurs at At the intersection with the root; although the maximum stress of the tooth waist is smaller than the maximum stress of the concave tooth and the root, the stress is abrupt, and the sudden change point is about 2/3 of the length of the tooth curve (along the tooth waist to the root). Direction); the stress at the intersection of the concave tooth and the tooth waist is the smallest. The above analysis results verify the common failure modes of double arc gears: double arc gears are usually broken from the root of the tooth during engineering use, and sometimes breakage occurs at the waist of the tooth. Therefore, when designing and processing gears, the thickness of the root and the maximum stress of the tooth waist should be increased to reduce the occurrence of the tooth breakage.
2.2 Influence of different component parameters of the gear on the bending stress of the gear 2.2.1 Influence of the helix angle on the bending stress of the gear In Pro/E, the number of teeth is z=16, the modulus is m=3, and the helix angle is from 035 to 36 double arcs. The gear model (loading and restraining method is the same as above), one point is set on the tooth profile of the tooth waist, the concave tooth and the root of the tooth on the tension side of the tooth, and then the Mechaniea module of Pro/E software is used for the 3 The bending stress of a fixed point is subjected to finite element analysis, so that the bending stress of the gear changes with the helix angle. The results of the analysis are shown in Figure 6.
Fig. 6 The bending stress of the double circular arc gear varies with the helix angle. As can be seen from Fig. 6, as the helix angle increases, the bending stress of the gear decreases. According to the results of finite element analysis, the relationship between the bending stress of the gear and the helix angle is fitted into a quadratic polynomial: the relationship between the bending stress of the root and the helix angle: g=-0.00272 0.0193 3.9208 concave bending stress and helix angle Relationship between: f=-0.00142 0.0027 2.4708 The relationship between the bending stress of the tooth waist and the helix angle: j=-0.00182 0.019 1.84522.2.2 The influence of the modulus on the bending stress of the gear. The number of teeth is z=16, the spiral is constructed in Pro/E. Angle = 0, modulus m from 317 a total of 15 double arc gear models (loading, restraint and the selected research objects are the same as above), using the Pro/E software Mechaniea module to carry out the bending stress of the above three fixed points Finite element analysis shows the variation of the bending stress of the gear with the modulus. The analysis results are shown in Fig. 7. As the modulus increases, the bending stress of the gear decreases gradually. According to the finite element analysis results, the relationship between the bending stress of the gear and the modulus is fitted to the power relationship.
The relationship between the bending stress of the root and the modulus: g=11.018m-0.9162 The relationship between the bending stress of the concave tooth and the modulus: = 16.62m-1.6715 The relationship between the bending stress of the tooth and the modulus: j=5.5166 M-1.05312.2.3 Effect of the number of teeth on the bending stress of the gear In the Pro/E, a model of 16 double arc gears with modulus m=3, helix angle=0, and number of teeth z from 1530 was constructed (loading, restraint and extraction) The object point is still the same as above. Using the Mechaniea module of Pro/E software to carry out the finite element analysis of the bending stress of the three fixed points mentioned above, the variation law of the bending stress of the gear with the number of teeth is obtained. The influence of the number of teeth on the bending stress of the gear teeth It is not large, but it still maintains a certain change rule: as the number of teeth increases, the bending stress of the tooth root and the concave tooth gradually decreases, and the bending stress of the tooth waist gradually increases. Similarly, according to the results of finite element analysis, the relationship between the bending stress of the gear and the number of teeth can be fitted to the power relationship: the relationship between the bending stress of the root and the number of teeth: g=9.7489z-0.3293 concave bending stress and number of teeth The relationship between: f=6.3962z-0.1928 The relationship between the bending stress of the tooth and the number of teeth: j=0.638z-0.3993 Conclusion (1) The compressive stress and tensile stress of the double circular arc gear teeth change the same, the compressive stress ratio The tensile stress is large.
(2) The maximum bending stress of the double arc gear occurs at the root of the tooth. Although the bending stress value of the tooth waist is not particularly large, there is a sudden change, so the tooth fracture usually occurs at the root or the tooth waist.
(3) The bending stress of the double circular arc gear teeth will gradually decrease with the increase of the helix angle and the modulus; the bending stress of the tooth root and the concave teeth will gradually decrease with the increase of the number of teeth, and The bending stress of the tooth waist gradually increases as the number of teeth increases. Therefore, in the design, manufacture and selection of double-arc gears, the optimal helix angle, modulus and number of teeth can be reasonably determined by referring to the various relations obtained by fitting, so as to reduce the influence of bending stress as much as possible.
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