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According to the mechanism and the meshing characteristics of a pair of meshing gears, at any of the meshing points, the pressure angle of the other gear at the meshing point can be obtained from the pressure angle of either gear at the meshing point, that is, αi2=arctan[z1 z2tanα0 -z1tanαi1z2] where α'0―——the pitch angle of the pitch is z1,z2―- the number of teeth of the active and driven gears, just know the pressure angle of any gear at the meshing point, you can find the point The single tooth pair meshing stiffness.
The single-tooth meshing stiffness of the standard assembly gear pair is substituted into r, rb, ra, rF, rx, rf into Sf, hr, h, hi, hx formula [7, 8], and the parameters required for the deformation compliance formula are obtained. Then, these parameters are substituted into the rectangular bending deformation flexibility, the trapezoidal bending deformation flexibility, the total shear deformation flexibility, and the deformation flexibility formula [8] caused by the root base tilt, then the modulus in each formula of the gear deformation flexibility m can be eliminated.
Gear pair meshing diagram For the driving gear, the engagement pressure angle αE1 is its tip pressure angle αa1, ie αE1=αa1=arccosz1cosα0z1 2h3a10. For the driven gear, the engagement pressure angle αA2 is its tip pressure angle αa2, ie αA2=αa2 =arccosz2cosα0z2 2h3a11 In, there is CE=Rb1tanαa1-tanα0CA=Rb2tanαa2-tanα012 For the driving gear, the distance from the point of the point to the center of the gear is RA1=CA2 R12-2CAR1cos90°-α013 For the driven gear, the point of the gear is at the center of the gear The distance is RE2=CE2 R2-2CER2cos90°-α014αA1=arccosRb1RA1αE2=arccosRb2RE215 Substituting Rb1, Rb2, R1, R2 and Equations 12-14 into Equation 15, αA1 can be expressed as αA1=y5z1, z2, α0, αa216αE2 can be expressed as αE2 =y6z1,z2,α0,αa117 are determined by the flexibility formula of contact deformation [8] and the formulas 6～11,16,17: 1 For the gear pair of standard assembly, the influence factors of the single tooth on the meshing stiffness are the number of gear teeth and points. Degree of pressure angle, displacement coefficient, load pressure angle, tip height coefficient, head clearance coefficient, tip pressure angle, elastic modulus and Poisson's ratio.
For pairs of different straight-tooth involute cylindrical gears with the same number of teeth, elastic modulus, Poisson's ratio and displacement coefficient, different modulus and standard assembly, the single-tooth meshing stiffness is exactly the same. Single-tooth meshing stiffness of non-standard assembly gear pairs For non-standard assembled gear pairs, the same can be obtained αA1=y7z1,z2,α0,α0,αa218αE2=y8z1,z2,α0,α0,αa119 from Equations 18 and 19 It can be seen that the influence of the single tooth on the meshing stiffness of the non-standard assembly gear pair increases the pressure angle α0' at the pitch circle.
Similarly, when the number of teeth is z>33, the same conclusion can be obtained. The example is verified in the following example, E=206×105N/mm2, v=013.
Comparison of single tooth pair meshing stiffness The parameters of the first pair of gear pairs are: z1=58, z2=85, α0=α0'=20°, m=3; the parameters of the second pair of gear pairs are: z1=58, z2 =85, α0=α0'=20°, m=10. The single tooth pair meshing stiffness of the two pairs of gear pairs calculated according to the Ishikawa formula adopts the above comparison criterion, the same below.
Figure 2 According to the Ishikawa formula, the pair of gear pairs have a single tooth pair meshing stiffness. It can be seen that the paired gear pairs have the same meshing stiffness value at the corresponding single tooth pair, indicating that the meshing stiffness of the spur gear is independent of the modulus m. .
Exploring the formula for calculating the meshing stiffness according to the new equivalent tooth shape. The new equivalent tooth shape is shown as shown in the figure, which uses a superposition of a small rectangle and two trapezoids. The upper base of the first trapezoid is the chord length at the top of the tooth tip, the lower base is the chord length at the index circle; the upper base of the second trapezoid is the chord length at the index circle, and the lower base is the effective root circle. The length of the string. The deformation and contact deformation compliance formula caused by the elastic inclination of the root base is the same as the Ishikawa formula. When the load acts on the second trapezoidal zone, the deformation compliance formula is the same as the Ishikawa formula.
The parameters of the calculation formula of the meshing stiffness according to the new equivalent tooth shape are as follows: s=mzsinπ2zh'=rcosπ2z-r2f-sf22hi=h'sf-hrsf-sh'i=hs-h'sas-sa The remaining parameters are the same as the Ishikawa formula.
Example Verification 1 Comparison of single tooth pair meshing stiffness The parameters of the first gear pair are: z1=58, z2=85, α0=α0'=20°, m=3; the parameters of the second gear pair are: z1=58 , z2=85, α0=α0'=20°, m=10. The single tooth pair meshing stiffness value of the two pairs of gear pairs calculated according to the new equivalent tooth shape obtained by the meshing stiffness formula. Similarly, the pair of gear pairs have the same meshing stiffness value at the corresponding single tooth pair, which further indicates that the meshing stiffness of the spur gear is independent of the modulus m. Fig.4 The single tooth pair meshing stiffness of the two gear pairs calculated according to the new equivalent tooth shape formula. The stiffness formula obtained by the new equivalent tooth shape is compared with the Ishikawa formula. The parameters of the gear pair are: z1=58, z2=85, α0=α0 '=20°, m=3. The meshing stiffness of the single tooth pair of the gear pair is calculated by the Ishikawa formula and the stiffness formula obtained by the new equivalent tooth shape, respectively. Figure 5 is a comparison of the meshing stiffness of a single tooth calculated by two formulas. The calculation result of the stiffness formula obtained by the new equivalent tooth shape is about 5 larger than the calculation result of the Ishikawa formula, and the γ is close to 1 It is about 3, and it is about 2 in the middle of γ. This reflects that the calculation result of the Ishikawa formula is smaller than the actual value.
Conclusion The factors affecting the meshing stiffness of the standard assembled spur gears are gear tooth number, index circle pressure angle, displacement coefficient, load pressure angle, tip height coefficient, head clearance coefficient and tip pressure angle. , modulus of elasticity and Poisson's ratio, independent of modulus. In the case of non-standard assembly, the influencing factors should also increase the pitch angle.
A comparison criterion of single tooth pair meshing stiffness is proposed. Under this premise, the meshing stiffness of the single-tooth pair of different straight-tooth involute cylindrical gear pairs with different modulus and other parameters is the same, that is, the meshing stiffness is independent of the modulus. 3 The comparison between the stiffness formula obtained by the new equivalent tooth shape and the calculation result of the Ishikawa formula shows that the calculation result of the Ishikawa formula is smaller than the actual value, and there is a certain error.
Single tooth gear bite hardness comparison principle
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