Gear Tooth Profile Research Articles

Study on the design principle of the LogiX gear tooth profile and the selection of its inherent basic parameters

The development of scientific technology and productivity has called for increasingly higher requirements of gear transmission performance. The key factor influencing dynamic gear performance is the form of the meshed gear tooth profile. To improve a gear’s transmission performance, a new type of gear called the LogiX gear was developed in the early 1990s. However, for this special kind of gear there remain many unknown theoretical and practical problems to be solved. In this paper, the design principle of this new type of gear is further studied and the mathematical module of its tooth profile deduced. The influence on the form of this type of tooth profile and its mesh performance by its inherent basic parameters is discussed, and reasonable selections for LogiX gear parameters are provided. Thus the theoretical system information about the LogiX gear are developed and enriched. This study impacts most significantly the improvement of load capacity, miniaturisation and durability of modern kinetic transmission products.

Algorithms for Involute and Octoidal Bevel-Gear Generation

A suitable formulation and the implementing algorithms for involute and octoidal bevel-gear generation are proposed in this paper. In particular, the exact spherical involute tooth profile of bevel gears and their crown rack is obtained through the pure-rolling motion of a great circle of the fundamental sphere on the base cone. Moreover, the tooth flank surface of octoidal bevel gears is obtained as the envelope of the tooth flat flank of the octoidal crown rack during the pure-rolling motion of its flat pitch (surface) on the pitch cone. The proposed algorithms have been implemented in MATLAB; several examples are included to illustrate their applicability.

Synthesis of the Base Curves For N-Lobed Elliptical Gears

Motivated by the need to synthesize the tooth profiles of noncircular gears, we approach the synthesis of the tooth profile of circular spur gears using their pitch circle, rather than their base circle. We do this by means of envelope theory. The proposed formulation gives the involute tooth profile and its well-known base circle for any pitch radius and profile angle of the rack cutter, which coincides with the pressure angle for circular gears. Then, the foregoing approach applies to the synthesis of the base curves of noncircular gears with involute tooth profiles and of their rack. We do this by resorting to basic differential geometry using the Euler–Savary Theorem, rather than to envelope theory. In particular, the formulation of both base curves for the right and left involute tooth profiles is obtained, for the first time, for N-lobed elliptical gears and their rack through the formulation of the pitch curves and their evolutes. The proposed formulation is illustrated with numerical results.

Computerized tooth profile generation and analysis of characteristics of elliptical gears with circular-arc teeth

This study describes rack cutters with circular-arc profile teeth to generate elliptical gears which rotate about one of their foci. A mathematical model for elliptical gears with circular-arc teeth is developed according to gear theory. The influence of the design parameters, such as the number of teeth, gear module, pressure angle at the pitch point and the major-axis, are investigated. The effects of the circular-arc radius of the rack cutter on both the undercutting of teeth and on pointed teeth of the generated circular-arc elliptical gear are also studied. Three numerical examples are presented to elucidate the generation of gear tooth profiles using the proposed mathematical model, and to investigate the phenomena of tooth undercutting and pointed teeth.

Optimum Design of the Involute-Cycloid Composite Tooth Profile Helical Gear

This paper deals with the optimum design of the involute-cycloid composite tooth profile helical gear, which was developed as a non-involute tooth profile gear based on cycloid tooth profile. The involute-cycloid composite tooth profile helical gear has some design parameters such as pressure angle and rolling circle, which define the involute-cycloid composite tooth profile curve. In this study, we developed a method to calculate the tooth root stress and the tooth contact stress of the involute-cycloid composite tooth profile helical gear. Then we compared the results with the experimentally measured data, and the validity of the present calculation method was certified by the good agreement of the calculated values with the measured data. Furthermore, we discussed the effects of the design parameters on the tooth bending and tooth contact strengths of the involute-cycloid composite tooth profile helical gear.

CNC flexible generating specific gear tooth profile based on standard involute gear hob

CNC flexible generating specific gear tooth profile based on standard involute gear hob

Computer aided design of elliptical gears with circular-arc teeth

An elliptical gear drive, whose rotation axis coincides with its geometric center, can be used as an oil pump. Rack cutters with circular-arc teeth generate the tooth profiles of circular-arc elliptical gears, which have a convex–concave tooth profile contact. The mathematical model of a circular-arc elliptical gear based on the theory of gearing and gear generation mechanism is also developed. Owing to its complex geometry, the profile of circular-arc elliptical gears may have tooth undercutting and pointed teeth. Design parameters such as pressure angle at pitch point, number of teeth, module, major axis, and circular-arc radius affect the tooth undercutting and pointed teeth of circular-arc elliptical gears. This paper proposes a systematic method to generate the tooth profile of circular-arc elliptical gears without tooth undercutting and pointed teeth. Some numerical examples illustrate the effectiveness of the computerized design process.

Nonlinear Dynamic Modeling of Gear-Shaft-Disk-Bearing Systems Using Finite Elements and Describing Functions

This study presents a new nonlinear dynamic model for a gear-shaft-disk-bearing system. A nonlinear dynamic model of a spur gear pair is coupled with linear finite element models of shafts carrying them, and with discrete models of bearings and disks. The nonlinear elasticity term resulting from backlash is expressed by a describing function, and a method developed in previous studies to determine multi harmonic responses of nonlinear multi-degree-of-freedom systems is employed for the solution. The excitations considered in the model are external static torque and internal excitation caused by mesh stiffness variation, gear errors and gear tooth profile modifications. The model suggested and the solution method presented combine the versatility of modeling a shaft-bearing-disk system that can have any configuration without a limitation to the total degree of freedom, with the accuracy of a nonlinear gear mesh interface model that allows to predict jumps and double solutions in frequency response. Thus any single stage gear mesh configuration can be modeled easily and accurately. With the model developed it is possible to calculate dynamic gear loads, dynamic bearing forces, dynamic transmission error and bearing displacements. Theoretical results obtained by using the method suggested are compared with the experimental data available in literature, as well as with the theoretical values calculated by employing a previously developed nonlinear single degree of freedom model.

207 歯元曲線形状最適化による浸炭歯車の強度向上

207 歯元曲線形状最適化による浸炭歯車の強度向上

114 インボリュート・サイクロイド合成歯形歯車の中心距離誤差と運転性能

114 インボリュート・サイクロイド合成歯形歯車の中心距離誤差と運転性能

A Study on Gear Honing Using Grinding Wheel Shaped in Internal Gear. Experimental Verification on Effectiveness of Proposed Honing Method.

A new honing method for extending the life of the grinding wheel is proposed in the previous paper. The method is that a setting angle between work shaft and grinding wheel shaft is changed according to the setting distance between the both axes in both process of dressing and honing. In this paper, the experimental verification on the availability of new method is reported. In the experiments, the negligible small change of the honed gear tooth profile is confirmed after the grinding wheel dressing in the amount of 10 mm.

Computer Aided Design of Elliptical Gears

An elliptical gear drive, whose rotation axis coincides with its geometric center, is simulated using generation with a rack cutter. The mathematical model of an elliptical gear based on the theory of gearing and gear generation mechanism is also developed. Owing to its complex geometry, the profile of elliptical gears may exhibit tooth undercutting, and pointed teeth. In addition, tooth undercutting of an elliptical gear based on the developed mathematical model of the elliptical gear and the theory of gearing is also investigated. Pointed teeth usually appear on the major axis of the gear elliptical pitch curve. Moreover, a geometric relation is developed and applied to prevent the pointed teeth appearing on elliptical gears. Furthermore, a computer simulation program is developed to generate the tooth profile of elliptical gears without tooth undercutting and pointed teeth. Various numerical examples illustrate the effectiveness of the computerized design process.

Mathematical model for parametric tooth profile of spur gear using line of action

A mathematical model for parametric tooth profile of spur gear is proposed by using a given equation of line of action. The line of action usually comprises only simple curves. It will be easier to manipulate the line of action when multisegment tooth profile curves are used. The proposed mathematical model will enhance the freedom of tooth profile design by combining the simple curves into the line of action. The curvature, sliding velocity, contact ratio, and the limitation of undercutting can be derived directly from the equation of line of action. Based on the proposed mathematical model, we can represent both mating tooth profiles by the single parameter of line of action. Therefore, it is easier to study the kinematical characteristics of mating gears with nonstandard tooth profiles by applying the proposed mathematical model.

Knowledge-based Geometry Generation for Spur and Helical Gears

Numerical analysis and true 3-D solid models of gear pairs are highly desirable to simulate real working conditions and reduce long and expensive test phases. This article presents methods, within a knowledge-based Automated Concurrent Engineering Software (ACES) for gear design and manufacturing system, for defining the gear tooth profile analytically from basic geometric parameters (diametral pitch, pressure and helix angles, number of teeth and tooth thickness) and determining the coordinates of all points on the working and the fillet portions of the tooth for standard and nonstandard gears, in order to generate FEM models for analysis, through automatic mesh generation and boundary condition assignment and produce a suitable gear construction (3-D model). The involute and trochoid curves can then be plotted on a Cartesian coordinate system emanating at the center of the gear. An accurate description of the tooth geometry and the associated mating-gears model for FEM analysis provide an early estimation of the full stress fields during the design phase, where there is still time to make significant changes. This 3-D solid model provides maximal value to a designer for machining the gear on a standard CNC milling machine without form cutters; far defining tool paths for Electric Discharge Machining (EDM); for rapid prototyping systems and for viewing VRML models. The system has a direct interface with a FEM processor (ANSYS) and a CAD tool (Pro/E). A case study is presented to demonstrate the system functionality with spur and helical gears.

A new method for seeking the optimum gear tooth profiles––the theoretical basis of Wildhaber–Novikov gearing

The object of this paper is to try and develop a new method for seeking the optimum gear tooth profiles. It is illustrated by an example for the line-contact helical gearing. The method developed may be summarized as follows: (1) Determine the essential requirements in accordance with the nature of the service that the gears are to render; (2) Let the paths of contact meet these requirements respectively as well as obey the general law of meshing; (3) Solve the resulting differential equations so that we may obtain the theoretical paths of contact with optimum performances; (4) Finally modify these theoretical paths of contact in order that an optimum basic rack tooth profile of the system may be obtained for practical application. The concrete content involved in the example may be regarded as the theoretical basis of Wildhaber–Novikov gearing. The method embodied in this paper may be regarded as a new field belonging to the theory of gearing.

3D Mesh Generation for Static Stress Determination in Spiral Noncircular Gears Used for Torque Balancing

Standard procedures available for evaluating gear fillet stresses are targeted to circular gears where little information is available about noncircular gears. In this paper, a mesh generation algorithm is presented for the discretization of gear tooth profiles of noncircular gear elements for static stress evaluation using FEM (Finite Element Method). The procedure assumes that the noncircular gear tooth profile is known. Delimiting a tooth domain from the gear structure and subdividing it into sectors, the gear tooth mesh is generated by mapping pregenerated patterns into the sectors. The discretized tooth domain is used in conjunction with the FEM to obtain stresses within the tooth fillet region. For comparison, this procedure was applied to determine the stress distribution in circular gears and rectangular beams. The results were in complete agreement with existing analytical solutions. An illustrating example is presented where a noncircular gear pair is introduced to eliminate the speed and torque fluctuation that exists in an electrical generator when an I.C. engine is used as the power source for the generator. The maximum fillet Von Mises stress is calculated for each angular position of the noncircular gear pair.

Computer solid modeling technologies applied to develop and form mathematical parametric tooth profiles of bevel gear and skew gear sets

This study is based on the parametric conjugate tooth profiles of bevel gear sets having the properties of invariant speed ratios, fixed skew axes (pitch line), zero length of common perpendicular line, two rotating axes intersecting at a point, etc. The geometric characteristics of the tooth profiles of bevel gears can be studied by using the kinematics parametric polar angular (pressure angle) function, the meshing equation and the constraints of continuity conditions. In this paper, the author derives the general profile equations, meshing constraint equations and non-undercut condition equations. By using the above-mentioned equations, gear geometric solid models are constructed. In addition, some gearing examples are presented to verify the developed theories. The non-undercut conditions and curvature properties are discussed in the paper. These developed theories have demonstrated bevel gear tooth profiles that can be parameterized by specifying certain relative parameters and are studied by a series of developed procedures. For the machining of these gears, the CAD/CAM software package from PRO/Engineering is used to transfer the solid models into NC programs. The method of transferring the manufacturing cutting NC codes to protect the tools and avoid impacting fixtures is demonstrated. This study is important for engineering design and for manufacturing factories involved in actual machining.

Computerized tooth profile generation of elliptical gears manufactured by shaper cutters

This work simulates an elliptical gear drive, the axis of rotation of which is coincident with its geometric center, manufactured by shaper cutters. The mathematical model of an elliptical gear is developed based on the theory of gearing and gear generation mechanisms. In addition, the tooth undercutting of the gear is also investigated based on the developed mathematical model of the elliptical gear, its unit normal vectors and a numerical method. A geometric relationship is developed and applied to prevent the occurrence of pointed teeth on elliptical gears. Further, this study also develops computer simulation programs to generate the tooth profile of elliptical gears without tooth undercutting and pointed teeth. Comparison of the angular velocity variations of the elliptical gear drives is also made. The results show that the developed elliptical gear drive can be utilized as an oil pump with a larger pumping volume and less angular velocity variation.

2310 特殊歯形歯車の歯面摩擦損失

We measured the friction losses of involute- cycloid tooth profile gear, modified cycloid tooth profile gear and involute gear using a power circulating-type gear testing machine, by the oil immersion formula, whose method is based on the oil temperature rise due to the power loss in the gear drives. It was found that the involute- cycloid tooth profile gear has the least friction loss in the three gears. Furthermore, we calculated the gear friction losses of these gears. The calculated value agreed approximately with the experimental data.

Modeling of X-Ray Fabrication of Macromechanical Structures

This paper evaluates the application of Lithographie, Galvanoformung, Abformung (LIGA) as a technique for fabricating millimeter to centimeter-scale mechanical components. The main focus of the work is an investigation of modeling approaches to estimate dimensional accuracy of complex profile features. Numerical models for X-ray exposure and an empirical model for polymethyl-methacrylate (PMMA) development are presented. Optical coordinate measurements are performed on two case studies associated with fabricating a planetary gear assembly: (1) a spider frame and (2) involute gear tooth profiles.