Spherical Involute Research Articles

Constructing geometrical models of spherical analogs of the involute of a circle and cycloid

The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid

Contact Path Method of Generating Spherical Conjugated Curves as Bevel Gearing Tooth Profiles

Abstract This paper proposes a general method to generate spherical conjugated curves based on their contact path. The main objective of generating conjugated curves is to use them as gearing tooth profiles. The derivation is based on the prescribed spherical contact path, the equation of meshing, and the rigid body motion between two bodies. This paper demonstrates the method by assigning great circle and small circle contact paths in circular pitch pair transmission to generate conjugated curves. With the great circle contact path, the conjugated curves are exact spherical involute, and with the small circle contact path, the conjugated curves are spherical cycloid. The contact properties of the conjugated curves, such as induced curvature and the sliding ratio, are also provided with mathematical equations. Based on the results, the procedures and the geometries between the planar and spherical contact path methods have corresponding relationships. In contrast to the Camus theorem, the exact spherical involute could be generated from a similar procedure of planar involute cases. The ultimate goal of this research topic is to propose a general approach to generating conjugated profiles in planar, spherical, and spatial cases. The successful derivation of the spherical cases in this paper paves the way for spatial cases in the future.

Theoretical model analysis and case simulation of spherical involute surface

Theoretical model analysis and case simulation of spherical involute surface

Theoretical model analysis and case simulation of spherical involute surface

The theoretical model of a spherical involute surface is the premise and foundation of the 3D modelling design technology of spiral bevel gear. The forming process of the surface is also the forming process of its mathematical model. Based on the generating principle of spherical involute on cone and surface analysis, the system model is gradually divided into blocks from curve to surface, and the surface of the bevel gear is obtained by parameter coupling. The modelling process of curve, surface or initial spiral can be independently referenced. It can realise the precise modelling of complex spiral profile surfaces such as variable spiral angles. In this paper, the mathematical model of spiral bevel gear was simulated and analysed on the MATLAB2019b platform. The simulation results verified the correctness of the algorithm of the spiral bevel gear model and the practicability of design optimisation.

Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Tooth Surface Modeling and Stress Analysis of 3-DOF Spherical Gear

A new type of spherical gear is proposed, which is formed by combining the tooth profile of the plane involute annular tooth surface and spherical involute spherical cone tooth surface. 3-DOF of pitch, deflection and rotation can be realized without principle error transmission. Based on the forming principle of plane involute and spherical involute, the equations of annular tooth surface and spherical cone tooth surface are derived by coordinate transformation theory; Combined with MATLAB and Pro/E, the 3D accurate model of 3-DOF spherical gear was obtained, and the contact stress of tooth surface was analyzed based on the finite element method. The simulation results show that the stress concentration always occurs at the tooth root and the tooth top of the spherical and conical tooth surface under the condition of constant spherical distance and the same torque but different coaxial intersection angle, which can provide reference for the tooth surface modification and optimization of spherical gear.

Kinematic analysis and experimental verification of an oval noncircular bevel gears with rotational and axial translational motions

This paper proposes a new type of gear transmission based on the space engagement theory, which can achieve not only simple motion transmission between intersecting axes, but also the compound motion of the rotation and axial translation between them. An oval noncircular bevel gear engagement coordinate system was first established with parameter equations of the pitch cone described in spherical coordinates. Then, the spherical involute equations and formula for calculating displacement and axial translational velocity of the gears were derived. Experimental tests were carried out to validate the kinematic characteristics of oval noncircular bevel gear transmission calculated from theoretical models. The influences of eccentricity on the indicators, such as axial displacement, rotational angle, angular velocity and acceleration, periodicity of the line of contact and periodicity of each motion characteristic parameter, have been analyzed and discussed in last. Experimental verification and kinematic analysis indicate that the proposed gear design method is appropriate with great potential for industrial applications.

Parametric Modeling and Contact Analysis of Skew Spiral Bevel Gears

According to the principle of spherical involute generation, the mathematical model of skew spiral bevel gear tooth flank is derived, according to the mathematical model of the tooth flank, a solid model of the bevel gear is constructed, and then the meshing equation is derived by regulation that the relative motion velocity vector is perpendicular to the common normal line at the meshing point for a pair of gears. Finally, the contact line of skew spiral bevel gears meshing is analyzed, and the relative relationship between the meshing contact line and the tooth flank generation line are derived to provide a theoretical model for design, machining and assembly of gear.

Investigation on the semi-conjugate tooth model and disc milling process of logarithmic spiral bevel gears of pure-rolling contact

Conventional logarithmic spiral bevel gears (LSBGs) introduce full-conjugate surfaces (line contact) to generate teeth of the pinion and the gear. However, in general, these full-conjugate surfaces are constructed by spherical involute curves and it is difficult to manufacture these surfaces. Therefore, this article tries to use semi-conjugate surfaces (point contact) as the tooth profiles to make they can be manufactured by the disc milling process and of pure-rolling contact. The design method, manufacture kinematics and stress distribution situations of the LSBGs with semi-conjugate surfaces are investigated. Conjugate surface theory and spatial conjugate curve meshing theory are both introduced to complete the analytical arguments. Finite element analysis (FEA) is introduced to evaluate the contact mechanical characteristics of the LSBGs under loads. From the analytical and simulated results, it is concluded that, through the disc milling process, the LSBGs of continuous pure-rolling contact can be manufactured and mesh correctly. Besides, the manufactured LSBGs maintain pure-rolling contact approximately when they are under loads.

Wavelet application research on tooth surface of spherical involute spiral bevel gear

In order to improve the machining accuracy of spiral bevel gears, a scheme based on the cubic B-spline wavelet was proposed. Based on the principle of spherical involute formation, a program was made in MATLAB to obtain the coordinate data of discrete points of the tooth surface. Furthermore, the high frequency information of discrete point data was eliminated, low frequency information was reserved based on the wavelet theory, and a set of new discrete points on the tooth surface was reconstructed. When these new discrete points were accurately fitted through the cubic NURBS surface, a tooth surface with lower curvature and lower complexity was obtained. The digital model of the tooth surface obtained through this method was machined by CNC, and the finishing tooth surfaces were measured using a three-coordinate measuring instrument. These experiments have proven that tooth surfaces processed by the wavelet had higher machining accuracy, when compared with the original one. A new method was provided to improve the machining accuracy of the spiral bevel gear by reconstructing its digital model.

A method for three-dimensional precise modelling of spiral bevel gear

A method of three-dimensional precise modelling is given based on the basic meshing principle of spiral bevel gear. With the help of the spherical involute equation and the base conic helix equation, a series of data points of tooth profile and tooth trace are calculated by programming with MATLAB, according to the known basic parameters of spiral bevel gear. Then the exact curve is generated, and it is introduced into the UG environment for modelling. Taking the automobile rear axle driven gear as an example, the concrete method of the 3D solid modelling is illustrated. It is proved that the 3D accurate modelling method is feasible and effective by its application in the measurement and evaluation of the tooth surface error of the spiral bevel gear.

Method for the geometric modeling and rapid prototyping of involute bevel gears

The growth of additive manufacturing technology allows the fabrication of complete functional devices with complex geometries with ease and low cost. This technology allows the fabrication of pieces that could not be made in the past with traditional manufacturing techniques, in this case, bevel gears with exact spherical involute (ESI). Focusing on this issue, this paper presents a method for the geometric design and fabrication of practical industrial-like ESI bevel gears with variable surface detail. The definition of the tooth profiles on back cones is proposed introducing a projection procedure for straight and spiral toothing. An in-house-developed software package was developed to test the method, and a pair of plastic examples were fabricated on a generic 3D printer. A comparison between the fabricated pinion and its STL model was conducted for validation purposes. Mean deviations of $0.22$ and $0.15~mm$ were obtained for the whole model and for the contact surfaces comparison, respectively. Thus, the fabrication of bevel gears using AMT is achieved, and can be implemented to a specific application using a particular additive manufacturing technique.

Derivation of design formula for a shifted spherical involute bevel gear pair

The development of shifted spherical involute bevel gear will lead to increased flexibility and convenience in the design of the cross-axis gear system. The shifted spherical involute bevel gear is identified by the axis angle change between the crown rack and work piece. The new design formula for the shifted spherical involute bevel gear pair was derived mathematically from the condition that the arc length sum of the pinion and gear base cones is same as that of the disk of action. Using the formula, it was shown that a bevel gear pair can fit the design requirements with great flexibility. The gear transmission error of the designed and CNC processed gear pair was directly measured to experimentally verify its validity.

Mathematical definition and computerized modeling of spherical involute and octoidal bevel gears generated by crown gear

The involute profile is the most commonly used tooth profile for cylindrical gears. However, its counterpart for bevel gears is not commonly used because its generation is not compatible with the use of easy-to-manufacture cutting tools and market-oriented generation methods. Nowadays, with additive manufacturing technology growing across industries to manufacture parts and full-scale products, the spherical involute profile deserves to be given the possibility to become the reference profile for bevel gears generated by additive manufacturing, forging or plastic molding. In this work, both the spherical involute and the octoidal form systems for bevel gears are mathematically defined. The geometry of their respective crown-gears for generation is derived and the computerized modeling of the geometries of the generated bevel gears presented. The comparison between the obtained gear tooth geometries is performed from the point of view of the obtained geometric deviations, their behavior to the change of the shaft angle and the evolution of contact and bending stresses for different values of torque applied to the pinion. The results showed that the spherical involute profile can handle better with the changes in the shaft angle, although the octoidal profile presents lower contact and bending stresses.

Simulation and optimization of computer numerical control-milling model for machining a spiral bevel gear with new tooth flank

Distinguished with the traditional tooth flank of spiral bevel gears, an accurate spherical involute tooth profile is of obvious advantages for the design and manufacture. For obtaining the gear model with enough tooth flank accuracy, simulation and optimization methodologies of computer numerical control-milling model are proposed. Initially, it gives an identification of the spherical involute tooth flank based on an improved formation theory. A universal simulation machining of the universal multi-axis computer numerical control-milling center is proposed to obtain an initial model. Then, in order to get/for getting improved tooth flank accuracy of the initial gear model, it presents some optimization methods included in the applications which are (1) non-uniform rational B-spline reconstruction by the Skinning method and (2) an overall interpolation based on Energy method. Finally, some given numerical examples are utilized to verify the form error of tooth flank. In addition, a closed-loop experiment scheme is provided to verify the validity of proposed methods.

Formational principle and accurate fitting methodology for a new tooth surface of the spiral bevel gear

Distinguished with some basic theory and approaches of traditional tooth surface design, the spiral bevel gear with new tooth surface, namely spherical involute tooth surface, is investigated. Firstly, it improved its formational principle based on the spherical involute theory. As a key part, the parametric equation of the generating line was made a detailed derivation. To this end, fast and accurate solutions of the boundary curves and the tooth profile curve family are accomplished, respectively. Then, taking advantages of modelling techniques of cubic non-uniform rational B-spline (NURBS) curve and surface in CAD/CAM, a reconstruction method for the spherical involute tooth surface was proposed to enhance the precision. At last, related optimisation schemes associated with constructed NURBS tooth surface are utilised to obtain the parameterisation of tooth surface data and higher-accuracy fitting. Given numerical examples indicate that the accuracy of the tooth surface is obviously enhanced and is enough to provide gear finite element (FEM) analysis the data and the basic model in its digitised design and manufacturing.

A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem

Understanding the geometry of gears with skew axes is a highly demanding task, which can be eased by invoking Study's Principle of Transference. By means of this principle, spherical geometry can be readily ported into its spatial counterpart using dual algebra. This paper is based on Martin Disteli's work and on the authors' previous results, where Camus' auxiliary curve is extended to the case of skew gears. We focus on the spatial analog of one particular case of cycloid bevel gears: When the auxiliary curve is specified as a pole tangent, we obtain “pathologic” spherical involute gears; the profiles are always interpenetrating at the meshing point because of G2-contact. The spatial analog of the pole tangent, a skew orthogonal helicoid, leads to G2-contact at a single point combined with an interpenetration of the flanks. However, when instead of a line a plane is attached to the right helicoid, the envelopes of this plane under the roll-sliding of the auxiliary surface (AS) along the axodes are developable ruled surfaces. These serve as conjugate tooth flanks with a permanent line contact. Our results show that these flanks are geometrically sound, which should lead to a generalization of octoidal bevel gears, or even of bevel gears carrying teeth designed with the exact spherical involute, to the spatial case, i.e., for gears with skew axes.

Actual Tooth Contact Analysis of Straight Bevel Gears

Theoretically, spherical involutes are used as one of the base topographies for straight bevel gears. Actual bevel gears, however, have deviations from their intended topographies due to manufacturing errors, heat treatment deviations, and finishing processes. Measuring the physical parts with coordinate measuring machines (CMMs), this study proposes a new approach to capture such deviations. The measured deviations from spherical involute are expressed in form of a third-order two-dimensional (2D) polynomial function and added to the base topography to duplicate the geometry of the actual part; tooth thickness deviation is also accounted for and corrected through changing the theoretical tooth thickness. The resultant surfaces are then used to construct ease-off and surface of roll angle topographies and to perform tooth contact analysis (TCA) and calculate motion transmission error (TE). At the end a sample straight bevel gear set is measured and utilizing the proposed approach its predicted TCA is compared to the experimental TCA obtained from roll tester. The results show very good correlation between the predicted and actual TCA of the parts. Utilizing the proposed methodology, the other bevel gear base profile geometries (such as octoids) can also be analyzed. In the proposed approach, the difference between other base geometries and spherical involutes can be treated as deviations from spherical involutes and can be taken into account to perform TCA.

Spiral bevel gear true tooth surface precise modeling and experiments studies based on machining adjustment parameters

The true tooth surface of a spiral bevel gear is not the standard spherical involute surface, since the microscopic tooth surface varies according to machining adjustment parameters, with different tooth contact forces, stress distributions, and other intrinsic properties. Therefore, it is necessary to propose the method whose advancement and feasibility can be verified by gear cutting and contact pattern experiments, to obtain a precisely digitized true tooth surface of spiral bevel gear based on machining adjustment parameters, which will lay a solid foundation for subsequent true tooth contact analysis, transmission error analysis, gear cutting, etc.

Three-dimensional optical measuring method for the tooth surface of high-order oval bevel gears

In view of the complexity of geometry design theory for high-order oval bevel gears’ tooth profile and the difficulties in detecting the tooth surface error, the researches into tooth surface error detection and analysis methods of high-order oval bevel gears were launched. In the view of the feature of three-dimensional (3D) optical measuring method, a 3D optical scanner was used. Meanwhile, the tooth profile parametric equations of a bevel gear cutter with standard spherical involute and non-circular bevel gear were deduced based on the spatial meshing theory and the theory of spherical triangle. Then the 3D optical scanning detection experiment of oval bevel gears was designed and accomplished. After that, the reverse engineering was performed according to the point cloud which was obtained before, and a tooth surface which within a certain tolerance range can be obtained from the reverse engineering. The normal deviation nephogram of theoretical tooth surface and processed tooth surface can be obtained by comparing the tooth surface which is gotten from reverse engineering with theoretical tooth surface. Finally, the measurement and visualisation for processing tooth surface of high-order oval bevel gears were achieved. Meanwhile, the machining accuracy class of five-axis precision milling method was obtained.