Abstract
In this paper, a more computationally convenient singularity condition of the enveloped surface is proposed using the theory of linear algebra. Its preconditions are only the tangential vector of the enveloping surface, the relative velocity vector, and the total differential of the meshing function. It avoids calculating the curvature parameters of the enveloping surface. It is proved that the singularity conditions of enveloped surface from different references are equivalent to each other and the relational equations among them are obtained. The curvature interference theory for the involute worm drive is established using the proposed singularity condition. The equation for the singularity trajectory is obtained. The calculation method for the singularity trajectory is proposed and its numerical result is obtained. The influence of the design parameters on the singularity trajectory is studied using the proposed curvature interference theory. The study results show that the risk of curvature interference is high when the transmission ratio is too small, especially in the case of the single-threaded worm and large modulus. The proposed singularity condition can also be applied to study the curvature interference mechanism in other types of the worm drive and to study the undercutting mechanism when machining the worm drive.
Highlights
The machining and meshing processes of the gears are essentially the enveloping process
It is due to this feature that the singularity trajectory on the enveloped surface is known as the curvature interference limit line or the first kind of limit line,[3,4,5]
By observing each of the examples ffi–þ in Figures 7(a) to (d), 8(a), (b) and 9 individually, with a constant modulus and a constant number of the worm threads, when the calculated modification coefficient of the worm gear is negative after selecting the transmission ratio, the lower singularity trajectory may appear on the worm gear tooth surface, which can lead to curvature interference
Summary
The machining and meshing processes of the gears are essentially the enveloping process. In practice, the singularity conditions, equations (14) and (16), inevitably require the calculation of the coefficients, E, F and G, of the first fundamental form for the enveloping surface S1.2–5,9,10 In particular, to simplify the calculation of the normal vector N~ of the contact line, it is usually necessary to calculate the curvature parameters of first in the main frameÈ fP; ~g1, ~g2, ~n1Ég or the unit orthogonal moving frame P; ~aj, ~ah, ~n1 according to Dong[4] and Zhao and Zhang[19,20].
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