Involute Helicoid Research Articles

A method for point-contact conjugate tooth surfaces with a constant instantaneous transmission ratio generated by point-contact conjugate involute helicoids

Point-contact conjugate tooth surfaces are insensitive to manufacturing errors and assembly errors and have no transmission errors. However, design conditions for the current method of generating point-contact conjugate tooth surfaces are strict or lacking. This study proposed a method for point-contact conjugate tooth surfaces with a constant instantaneous transmission ratio generated by point-contact conjugate involute helicoids. According to relative motions of single- and two-degree-of-freedom in generating, design criteria were given and a mathematical model of the method was established. The mechanism of point-contact conjugate tooth surfaces was obtained by investigating the relationship between the engagement lines/surfaces, and kinematics was studied, in which the mechanism with a constant instantaneous transmission ratio was geometrically analysed. Based on this method, four types of point-contact conjugate worm drives were studied, in which two novel types of worm drives were developed, i.e., a point-contact conjugate toroidal involute (TI) worm drive, and a point-contact conjugate involute cylindrical (ZI) worm drive. The method was validated for correctness using numerical simulations and meshing tests and the influence of adjustable parameters on the meshing performance was outlined.

Manufacturing Method for Large Cylindrical Worm Gear Set of ISO Type I on Universal CNC Machine Tools

Large cylindrical worm gear set of ISO type I are manufactured using endmill tools on universal CNC machine tools. This manufacturing method requires neither special gear-generating machines nor special tools. The tooth flank forms of ISO type I cylindrical worm gears are involute helicoids as a standard. The targeted theoretical tooth flanks of the worm and the mating worm wheel are determined based on a tooth contact analysis (TCA) of such worm gear set. The cutting conditions of the worm are determined after the offset distance between the worm axis, and the central axis of the endmill tool is calculated. Afterward, the worm is manufactured by controlling only two axes on machine tools using a swarf milling method by use of the side of the endmill tool under the determined conditions. Meanwhile, the targeted theoretical tooth flanks of the mating worm wheel are modeled in 3-dimensional computer-aided design software, and the worm wheel is manufactured by a swarf milling method in a computer-aided manufacturing process. The comparison of experimental and analytical tooth contact patterns indicates almost no difference between the two tooth contact patterns.

Meshing theory of involute worm drive

The subject of this paper is the establishment of a new meshing theory applicable to the involute worm drive. The method is proposed for determining the meshing characteristic parameters of the involute worm drive using the principal frame on the involute helicoid. A rigorous proof is presented to prove that a helicoid with constant helix parameter ground by a planar grinding wheel is an involute helicoid employing the uniqueness theorem for surfaces in differential geometry. The relationship among the helix angle of base circle, normal pressure angle and helix angle for the involute worm is proved. Using the proposed meshing theory, the numerical simulation of the involute worm drive is carried out to investigate the meshing performance of the involute worm drive. The numerical results reflect that the conjugate zone covers almost the entire worm gear tooth surface. The contact frequency is higher on one side of the worm gear tooth surface. The contact stress is higher near the end face of worm gear. An area of poor lubrication exists in the middle of the worm gear tooth surface.

Теоретические положения синтеза сопряженных зацеплений

The conditions for the synthesis of conjugate couplings are systematized on the basis of a single geometric-kinematic criterion, which consists in the equality of the velocity projections of the points of the active surfaces that make up the higher pair on the common contact-normal to these surfaces. The technology of formation of a family of conjugate couplings by a two — parameter envelope by an involute helicoid or its particular varieties-a circular cone or plane-is theoretically justified

Automatic modeling of the surfaces of the equal slope in AutoCad system through language AutoLisp

Graphical computer modeling is a real possibility of solving problems on the formation of surfaces by a kinematic method. It is due to the presence of automated design systems, such as MathCad, Mathematika, AutoCad, etc. The advantage of the AutoCad system is the presence of the built-in language of functional AutoLisp. Having a wide range of mathematical functions and functions for drawing graphic objects, the AutoLisp language allows you to write a program for drawing shells of any complexity, in a particular case - shells of the same ramp on the elliptical plane, the surface of the involute helicoid, and then transferring the results to the AutoCAD environment. Playing objects in slow motion makes it possible to compose mini-films about the formation of surfaces. The image of graphical constructions in three-dimensional space using the AutoCAD color palette enhances the expressiveness of the image of surfaces and their elements. The aim of the study is the possibility of modeling the surface of the equal slope ramp on an elliptical plane, carried out by its gradual drawing in a slow dynamic mode with the creation of a mini-film that allows the surface to be displayed on the monitor screen

Revisiting Generation and Meshing Properties of Beveloid Gears

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known. Beveloid gears are often regarded as a generalization of involute cylindrical gears involving one additional degree-of-freedom, in that the midplane of their (virtual) generating rack is inclined with respect to the axis of the gear being generated. A peculiarity of their generation process is that the motion of the generating planar surface, seen from the fixed space, is a rectilinear translation (while the gear blank is rotated about a fixed axis); the component of such translation that is orthogonal to the generating plane is the one that ultimately dictates the shape of the generated, envelope surface. Starting from this basic fact, we set out to revisit this type of generation-by-envelope process and to profitably use it to explore peculiar design layouts, in particular for the case of motion transmission between skew axes (and intersecting axes as a special case). Analytical derivations demonstrate the possibility of involute helicoid profiles (beveloids) transmitting motion between skew axes through line contact and, perhaps more importantly, they lead to the derivation of designs featuring insensitivity of the transmission ratio to all misalignments within relatively large limits. The theoretical developments are confirmed by various numerical examples.

Mechanism Study on Gear Tooth Surface Texture in Power Honing Process

Gear driving occupies an indispensable position in the field of mechanical transmission of the modern industry, better texture and mechanics characteristics of gear surface contribute to better meshing quality. Gear honing has been wildly used as gear finishing process because of its better gear surface texture and higher surface residual stress than gear grinding in recent years, especially for the development of power honing with internal gearing. In this paper, the gear surface model of the bull-gear in gear box was built based on the involute helicoid theory, and then the points of gear surface were converted to the honing wheel by using the space coordinate transform theory and matrix resolution method, the tooth surface model of the honing wheel was calculated based on the honing parameters and the conjugate contact principle of internal meshing. Through analysing the relative motion velocity vector of contact lines, the formation mechanism of gear surface texture was reflected clearly with different axis-crossing angles. The results of this research have important theoretical and instructional significance for the honing process. [Received 22 July 2016; Accepted 16 November 2016]

Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion

The curvature theories for envelope curve of a straight line in planar motion and envelope ruled surface of a plane in spatial motion are systematically presented in differential geometry language. Based on adjoint curve and adjoint surface methods as well as quasi-fixed line and quasi-fixed plane conditions, the centrode and axode are taken as two logical starting-points to study kinematic and geometric properties of the envelope curve of a line in two-dimensional motion and the envelope surface of a plane in three-dimensional motion. The analogical Euler–Savary equation of the line and the analogous infinitesimal Burmester theories of the plane are thoroughly revealed. The contact conditions of the plane-envelope and some common surfaces, such as circular and noncircular cylindrical surface, circular conical surface, and involute helicoid are also examined, and then the positions and dimensions of different osculating ruled surfaces are given. Two numerical examples are presented to demonstrate the curvature theories.

Study on ZI worm and helical gear drive with large transmission ratio

A ZI worm and helical gear drive is composed of an involute helicoid worm and a helical gear. Its application is limited due to less studies on the transmission performance with a large transmission ratio. In this paper, the design method of the drive is developed as: the parameters of the ZI worm should be the same as those of the helical gear; the helix angle β1 and the reference diameter factor d1∗ of the ZI worm should be greater than the minimum values proposed. The TCA of the drive shows that: the contact areas on the tooth surfaces are narrow; the effective face width of the ZI worm is larger than that of the helical gear; the contact ratio is less than 2; when the number of threads of the ZI worm changes from 1 to 5, the contact ratio decreases slightly, the effective face width of the worm decreases and that of the helical gear increases correspondingly. The instantaneous transmission ratio is proved theoretically to be constant and the assembly errors cause small shifts of the paths of contact.

Calculation of helical gear’s basic parameters using COP-data acquired by optical 3D digitizer

Gears are industrial components with a precise geometry. Identification of their basic parameters plays an important role in their reverse design and quality control process. This article describes a new approach for the calculation of helical gear’s basic parameters using optical data acquired by 3D digitizer. This approach is implemented by acquiring cloud-of-points data (COP-data) from the bearing seats and gear tooth surface. Cylinder surface fitting through COP-data acquired from bearing seats is performed for the determination of gear axis of rotation. In a final step, involute helicoid surface fitting through COP-data acquired from gear tooth surface determines the helical gear’s primary features. Particle swarm optimization algorithm as an efficient method is applied to perform the surface fitting process in this article.

Spiral involutes and their application in gear transmission

Involute helical gears mesh based on the intersections of involute helicoids. However, spiral involutes on the tooth surface do not participate in meshing directly. A new type of gear drive, the spiral involute gear drive, is proposed that works on the contact of spiral involutes. The generation of tooth profile is introduced in detail. Through relative-stagnation method, spiral involutes prove to have conjugation characteristics. To testify whether the transmission ratio of cylindrical spiral involute gears is constant, simulation is implemented in commercial codes ADAMS based on solid models of a pair of spiral involute gears. The computed results show that this novel gear drive can achieve a constant transmission ratio. Due to transmission with uniform velocity, cylindrical spiral involute gears can be used in transmission between intersecting axes. Milling and grinding apply to manufacturing of spiral involute gears.

Spiral Involutes and its Application in Gear Transmission

Involute helical gears mesh based on the intersections of involute helicoids. However, spiral involutes on the tooth surface do not participate in meshing directly. A new type of gear drive, the spiral involute gear drive is proposed that works on contact of spiral involutes. The generation of tooth profile is introduced in detail. Through relative-stagnation method spiral involutes prove to have conjugation characteristics. To testify whether the transmission ratio of cylindrical spiral involute gears is constant, simulation is implemented in commercial codes ADAMS based on solid models of a pair of spiral involute gears. The computed results show that this novel gear drive can achieve a constant transmission ratio. Due to transmission with uniform velocity, cylindrical spiral involute gears can be used in transmission between intersecting axes, so that generating milling and generating grinding can be achieved.

An Principle and Technology of On-Machine Measurement for Large Gear Hopping

For large gear with the characteristic of large structure, quality and modulus, its profile precision is usually measured by in-site measurement. Based on large gear on-machine measuring described, involute helicoid and its characteristic are studied. The definition of medium rack in the framework of gear meshing transmission is put forward, the scheme of on-machine measurement based on rack-shaped-edge probe is formed, which can be divided into rolled angle (phase angle) and pressure angle (meshing angle) measurement according to the rotation of gear and prob. Measurement grid is determined by measuring straight generatrix deviation of gear profile, furthermore total tooth profile deviation and total helix deviation are separated.

S1101-3-4 歯車の歯当たりとかみ合い評価法について : 可展円筒歯車歯面の図式展開について(伝動装置の基礎と応用セッション3)

Tooth bearing patterns give us very useful information about gear engagement situations. However, the detail analysis of gear engagement using actual contact patterns is not enough up to now, because the actual patterns are normally captured by a still camera as projected images. In the case of the involute helicoid surface, it is possible to develop it onto the plane geometrically. It is more convenient to compare the actual patterns with simulated ones on the plane. In this paper, geometrical developable surfaces are classified in the first place, and the involute helicoid surface is developed onto the plane geometrically with the help of the geodesic curvatures of tooth profiles and tooth leads. It is found that the lead circular helices are developed as the concentric circles on a plane, and the involute tooth profile is developed as an involute curve which has a certain diameter of a base circle.

GI-08 EVALUATION OF GEAR MEASURING INSTRUMENT USING MULTIPLE-TOOTH INVOLUTE HELICOID ARTIFACT(INSPECTION OF GEARS)

The reduction of the vibration and noise of gears is an important issue in mechanical devices such as vehicles and wind turbines. The characteristics of the vibration and noise of gears are markedly affected by deviations of the tooth flank form of micrometer order; therefore, strict quality control of the tooth flank form is required. The gear measuring instrument is generally calibrated by an involute helicoid artefact. In the tooth flank form measurement of the involute helicoid artefact, the setting error of the involute helicoid artefact largely affects the measurement result. The laborious setting operation is required for the calibration of the gear measuring instrument. Therefore frequent calibration of the gear measuring instrument has not been realized. In this study, evaluation method of the gear measuring instrument using multiple-tooth involute helicoid artifact is proposed to reduce the effect of the setting error. The efficiency of this method is confirmed by the measurement simulation using a computer program.

In-suit integrated measurement of profile precision of large gears based on rack-shaped-edge probe

Aiming at the characteristic of large gears, from line-group features of involute helicoid, straight generatrix is the instantaneous contact line between a pair of conjugated involute helicoid and it is the token of contact precision and reflects synthetically the total deviation of tooth profile and spiral line. In-suit integrated measurement based on rack-shaped-edge probe is put forward and the mathematic model is built. Two kinds of measuring principle and technology, varying phase-angle measurement and varying engagement angle measurement, can be obtained and mathematic models separating total deviation of tooth profile and spiral line are built.

Profile of grinding wheel for grinding TI worm

TI worm drive is a novel kind of hourglass worm, which consists of involute helical gear and enveloping hourglass worm. Accurate grinding for TI worm is the key manufacture technology for TI worm drive being popularized and applied, which means to obtain the profile of grinding wheel corresponding to the current grinding method. With the analysis of the movement relationship and the characteristic of the generating surface, the problem of accurate grinding for TI worm is converted into grinding for involute helical by shaped wheel. According to the theory of gear mesh, mesh equation for grinding involute helicoid by grinding wheel is deduced, the equation of the axial section profile of grinding wheel which can accurately grind TI worm is gained.

Simulation of whirling process and tool profiling for machining of worms

In comparison with machining of worm on a lathe or on a milling machine, whirling process uses a rotating tool ring carrying tools and encompassing the work blank. Due to nature of contact between tool and work blank, simulation of whirling process is necessary to understand the geometry of worm surface generated. In this paper, an attempt is made to simulate the whirling process and analyze the effect of tilting of tool head. In some applications worm geometry is specified and tool geometry has to be arrived at. An approach for profiling of tool is explained in this paper and validation of tool profile is carried out by simulation. Two worms, namely straight-sided in axial section and involute helicoid are taken as case studies and results are discussed.

Design, generation and stress analysis of face-gear drive with helical pinion

Two versions of face-gear drive geometry with a helical pinion are considered. One version is based on a screw involute helicoid. The second version is a new geometry developed as envelopes of two mismatched parabolic racks of the pinion and the shaper. A face-gear drive with a spur pinion is considered as a particular case of the developed theory. A new method of grinding or cutting of face-gears by a worm of special shape has been developed. The following problems are considered: (i) generation of a grinding worm with surfaces free of singularities and (ii) generation of face-gear by a shaper with surfaces free of undercutting and pointing. Tooth contact analysis and stress analysis are applied combined with investigation of tooth bearing contact on both sides. The following advantages have been achieved for the geometry proposed: (a) existence of a longitudinal bearing contact, (b) avoidance of edge contact, and (c) reduction of contact stresses. A phenomenon of asymmetry of tooth bearing contact for the driving and coast sides has been discovered. A computerized design, generation and stress analysis for the new types of face-gear drives have been developed.

Measuring Method of Transmission Error of Final Reduction Gearboxes and Gear Noise Occurrence Mechanism

The innovative measuring method for transmission error of final reduction gearboxes has been developed. Because laser speed sensors measure the rotary movement of the flanges connected rigidly to a gear pair, obtained data express the transmission error precisely. It has been confirmed that the transmission error measured by the method are in the relation of direct proportion to the vibration level of final reduction gearboxes. This has proven that the proposal of hypoid gears with involute helicoid tooth form is nonsense to reduce gear noise of final reduction gearboxes.