As an important branch of mechanical transmission, worm drives are extensively used owing to their advantages of compact structure, large gear ratio, stable transmission, and self-locking. Among them, hourglass worm drive is commonly used in aerospace, ship and submarine, weapons and equipment, mining metallurgy, manufacturing equipment, and other important fields of the national economy and defense due to its unique advantages of low noise, multi-tooth instantaneous contact, smooth transmission, high-load capacity and long service life , , .
For the research of toroidal worm drive, the straight profile toroidal worm transmission pair was first proposed by Hindley, and S. Cone successfully developed it and specialized large-scale production . Meanwhile, E. Wildhaber developed a positive plane primary envelope hourglass worm drive based on a helical tooth plane envelope hourglass worm drive . Y. Zhao proposed a dual tori double-enveloping hourglass worm drive and analyzed the meshing traits, contact line distribution, correction processing theory, and so on of the transmission pair in detail , , . X. Deng etal. researched the meshing characteristics, temperature field, and optimum process of machining complex surfaces on zero clearance roller enveloping toroidal worm drive , , . In addition, the Technical University of Dresden proposed the toroidal involute (TI) worm drive  and proved through prototype tests that the transmission pair has apposite load allocation, multiple tooth instantaneous contact, and great dependability. Y. Chen etal. proposed the wedge tooth involute gear enveloping toroidal worm drive, discussed the backlash adjustment method and the principle of tooth flank wear and compensation, developed the transmission sub-sample, and carried out physical tests , , . At the same time, H. Li etal. have also performed numerous researches works on the cutting performance and the processing exactness of the toroidal worm wheel hob , , . Although hourglass worm drives offer many advantages, they are extremely sensitive to errors in industrial applications. The reason is that the hourglass worm drive adopts the dual model processing method, and the worm is a toroidal rotary body and forms multiple tooth instantaneous line contact, which makes the conjugate tooth surface more constrained, resulting in the difficulty of high finishing, and finally resulting in its extreme sensitivity to processing errors and assembly errors. With the improvement of the performance requirements of transmission devices, it is urgent to explore new principles and methods for the development of drive and transmission, and how to decrease the susceptibility of the hourglass worm drive to various errors is a long-term research hotspot of scholars.
To diminish the effect of processing errors and assemblage errors on the worm drive and reduce the error sensitivity, F. Litvin etal. researched the impact of errors on the contact zone of the cylindrical worm drive, proposed to form a mismatch point meshing cylindrical worm drive with a small transmission error by increasing the diameter of the worm wheel hob, and systematically analyzed the change law of its contact trajectory . M. Shunmugam etal. optimized the parameters of the hob tooth shape with a horizontal plane of gear, carried out scientific and controllable and reasonable modification of the tooth flank of the toroidal worm wheel, and obtained the instant tooth contact state of the straight profile hourglass worm drive . V. Simon etal. analyzed the contact region of the tooth surface when the cylindrical worm drive was loaded under the condition of tooth shape error and assembly error and offered a method to improve the contact area by modifying the worm tooth surface . A. Falah etal. researched the impact of assembly error on the change of contact area between the tooth flanks of the worm and the gear of cylindrical worm transmission, as well as analyzed the change of interdental load . To diminish the sensitivity of worm drive to error, Y. Zhao etal. designed a toroidal worm drive with instantaneous point contact obtained by rationally modifying the worm gear tooth surface, studied the kinematics of the point-conjugate tooth surface couple, and explained the philosophy of non-specific mismatched drive , , . Y. Chen etal. proposed a mismatched TI worm drive with multiple tooth point contact based on the medium gear given the problem of the TI worm transmission pair being sensitive to an assembly error, to verify the bearing capacity and error compliance through TCA . X. Dong systematically and in detail summarized and analyzed the mismatch modified method and contact analysis method of toroidal worm pair in view of the sensitivity of toroidal worm pair to processing errors and assemblage errors  ; through the analysis of planar secondary enveloping hourglass worm pair, cone secondary enveloping hourglass worm pair, double cone secondary enveloped hourglass worm pair, and straight hourglass worm pair, the sensitivity of toroidal worm pair to errors under different modification methods was explored, the characteristics of two mismatch contact analysis methods, tooth surface tangent contact analysis method and tooth surface zero gap contact analysis method, were compared, and the purpose of reducing the sensitivity of toroidal worm pair to errors could be achieved by modifying the tooth surface of the worm gear.
Based on breaking through the traditional TI worm drive full conjugate principle and dual forming method, our research group used the variable tooth thickness involute gear and plane envelope hourglass worm that can be non-dual precision grinding as meshing components, constructed a precision point contact transmission pair with the local conjugate principle, proposed a unique hourglass worm drive with point meshing , combined with the changeable tooth thickness involute gear envelope toroidal worm drive , then developed an innovative point-contact toroidal worm drive with the variable gap between the tooth flanks of the worm and gear , and the evolution mechanism of the tooth surface point-line in its whole life cycle was investigated . On this basis, we further analyzed the local conjugate mechanism, used the standard involute gear as the medium tooth surface, combined the TI worm and modified involute gear to form local conjugate point meshing, and proposed a novel configuration method of toroidal worm pair with point-contact. Specifically, based on analyzing the standard involute forming theory, a linear shrinkage modification method was proposed, and the modification involute was obtained by defining the three modification criteria and modification function; by establishing four representative involute modification functions, cosinoidal modification function, parabolic modification function, hyperbolic modification function, and elliptic modification function, the mathematical model of this novel transmission pair are derived; the tooth flanks of the standard involute gear and modified involute gear are tangent to a helical line along the tooth width direction, the standard involute gear tooth flank is in complete conjugate meshing touch with the worm tooth flank, and the standard involute gear tooth flank is used as the medium tooth surface and removed, so it can be determined that the tooth flanks of the modified involute gear and hourglass worm form a local conjugate meshing point contact.
It is worth noting that both components of the novel hourglass worm drive can be produced by high-precision grinding , and this novel transmission pair has low sensitivity to errors. At the same time, in view of the fact that the target modification point and modification amount of the modification function of the modified involute gear can be selected according to the needs of specific working conditions, the novel transmission pair also has the advantages of arbitrary contact point position and better lubrication performance.
In this paper, a modification criterion of the modification involute and a novel point-contact hourglass worm drive based on modified involute are proposed, and the mathematical model of the worm pair is derived; the contact point between the worm and gear is solved by TCA, and the contact ellipse is solved by Hertzian point contact theory; the meshing correctness is verified by numerical analysis and FEA, and the impact of the modification parameters on the contact stress is investigated; the accurate 3D error model is established, and the error sensitivity to center distance error, shaft angle error, and comprehensive error was studied.
Considering that the parameters of the tooth surfaces of the left and right flack of the novel transmission pair, including helical angle, pressure angle, and modulus, are the same, only the left flack is analyzed and discussed here.
Tooth contact analysis
A virtual medium gear is constructed, and its tooth flank is exactly the same as the standard involute gear. The surface ƩIII can be tangent to the surface ƩI in linear contact, which is a helical, Lii, along the tooth width direction, as reflected in Fig.7 (a). At the same time, the surface ƩII is a spatial spiral surface, which is generated by the surface ƩI as the mother surface, the surfaces ƩI and ƩII are also in linear contact, and the contact line is the line Li, as reflected in Fig.7
Numerical analysis parameters
So as to confirm the exactness of the meshing transmission and the consistency of the theoretical contact point and the actual contact area of this novel transmission pair, the key design parameters are given, four different modification positions of the modified involute gear are considered, and the specific parameters are detailed in Table2.
The lines Li () and Lii () and the point P between the local conjugate surfaces of this novel transmission pair of the examples in Table2
Error sensitivity analysis
The purpose of proposing the novel point-contact hourglass worm drive is to reduce the sensitivity of the hourglass worm drive with multi-tooth line contact to errors and at the same time improve the tooth surface contact state (the contact area and contact stress). Before analyzing the error sensitivity of the novel worm drive, the tooth surface contact state in the zero error state is first studied to obtain the transmission performance of the transmission pair under different torques. Take
In this study, a modification method of the involute gear tooth surface and a novel transmission pair with point-contact are proposed, and the conclusions can be recapitulated as follows:
The involute modification method and criterion of the standard involute are proposed, five representative modification functions are established, and the mathematical model of this novel transmission pair is derived.
TCA of this novel transmission pair is presented, the feature of multi-tooth point-contact is
Declaration of Competing Interest
We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
This work wassupported by the National Natural Science Foundation of China (Grant No. 52075054) and the Innovation Group Science Fund of Chongqing Natural Science Foundation (No. cstc2019jcyj-cxttX0003).
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