Gear Tooth Profile Research Articles

Geometric synthesis in the profile of the involute tooth for improving the performance of bending in the material of the spur gear

Abstract Transferring power from one rotational axis to another in numbers of axes requires portable equipment such as gear. In operation, involuted teeth were reported as bending and pitting failures. Bending fatigue causes gear tooth cracks at the root, and pitting causes more wear. This bending failure was reduced with the help of material selection, geometric modifications like changes in profile shape, transmission error, module, tip relief, pressure angles, and the application of forces. A normal force acting on a gear tooth gets resolved into three components, like tangential, radial, and axial. When it comes to enhancing bending strength, the geometric forms of profiles have a significant impact. A variety of tools and techniques, including hobbing, shaping, grinding, etc., are available at the fingertips for altering gearing tooth. Changing the borders of gear tooth profiles with hobbing tools is becoming more popular among the authors as a means for altering gear tooth. There are four distinct geometric profiles that may be used to change gear geometries and enhance bending strength by reducing stress on the tooth. These profiles include trochoidal, circular, bezier, and cubic spines, and they are implemented using a hob cutter. It is noteworthy to note that these designs boost strength by integrating the shape of number of teeth on the spur gear. This study compares the number of teeth on the gear’s pinion with other geometric characteristics, including pressure angle, module, etc., in order to discover the optimal way for modifying the involute tooth of the gear in order to increase the bending resistance effectively of the spur gear. For the purpose of boosting the strength of gear material, a number of research endeavours have utilized circular profiles; nonetheless, this profile is older than the Bezier curve profile. Here, Bezier curves with a three-and five-point profile give better results than other profiles with significant reductions in bending stress for the same quantity of tooth on the gear.

Compensation modification of plastic gear tooth profile considering meshing deformation

Compensation modification of plastic gear tooth profile considering meshing deformation

Mathematical model and generation analysis for crossed helical gear system

Purpose Crossed helical gears have point contact and their surfaces are subject to high surface stress. Contact stress and root tooth stress are the most common sources of failure in crossed helical gear. This paper aims to study the load-carrying capacity and performance of crossed helical gear teeth with different gear tooth profiles. To overcome defects and reduce the sensitivity to small shaft angle changes. The combined tooth profile is designed to reduce the bending stresses, contact stresses and tooth deflection, and prevent pinion failure in the gearbox. Design/methodology/approach The principle of the method is a line contact is introduced instead of a point contact between two teeth in mesh with each other. The tooth surface of the helical gear is designed and cut by a modified tool. Higher normal pressure angles like 25° and 35° are used. The modified involute is accomplished to eliminate interference between the teeth. Engineering software packages have been applied to generate all crossed helical gears gear profiles. The modification is compounding curves consisting of an epicycloid-involute-hypocycloid gear teeth profile generated by the cutter modified. Findings The stresses in the crossed helical gear teeth profile were reduced by increasing the normal pressure angle values. Using a 35° pressure angle the enhancement percentage in contact stress and teeth fillet region will be about 29.345% and 15.421%, respectively. The best enhancement in a gear’s resistance is the epicycloid-involute-hypocycloid gear teeth profile. The enhancement was 32.610% and 18.588%. The skew in line of action in skewed helical gear will be sensitive when the crossing angles are small. Their teeth surface tends to be easily worn out; however, the wearing process will be reduced by using a proposed gear teeth profile. Practical implications The gear teeth to be modified are cut by a shaper process. The modifying rack cutter of this study can be used as a reference for creating a different helical gears sample. The helical teeth surface is modified to become an envelope of the other. This makes an original point contact change into a line contact. The epicycloid-involute-hypocycloid gear teeth profile is preferred for a higher contact ratio and a large load capacity. This work explicitly introduces a new method of kinematic consideration to improve the load capacity of crossed helical gear. Originality/value This paper showed some novel results by the unique shape of the rack-cutter designed to generate different helical angles and different gear positions. In the future, it will make valuable contributions to the further development of the dynamic performance of a crossed helical gear system through the study in the field of using asymmetric teeth profiles of helical gears with tip relief as the manner to enhance the crossed helical gear performance. Investigation of crossed helical gear by applying a predesigned parabolic function of transmission error enables the absorption of linear discontinuous functions caused by misalignment.

Analysis of involute-curve geometry using position gradients: Application to gear-tooth profile

This paper examines using non-rational polynomials to approximate the involute-curve geometry, which is fundamental in the design of gear systems. The parametric form of the involute curve in terms of an involute-angle parameter is first presented. To be able to compare the analytical geometry results and numerical results obtained using the finite-element (FE) method for describing the geometry, the relationship between the involute-angle parameter and the FE spatial coordinate is defined. A new relationship between the involute-curve arc length and the coordinate used in the FE parameterization is also developed. To this end, the involute-curve equation is used to determine the involute-angle parameter at the intersection with circles with arbitrary radii including the gear pitch, addendum, and base circles. The parameter relationship allows defining the nodal position-gradient vectors defined by differentiation with respect to the FE spatial coordinate from involute-curve tangent vector. Two procedures for determining the left involute curve from the right involute curve of the gear tooth are described. In the first procedure, a coordinate transformation based on Rodrigues transformation with axis of rotation defined by the tooth-center line is used; while in the second procedure, the analytical expression of the left involute curve is defined using the gear-tooth thickness at the base circle. A general procedure for determining the position gradients for a given geometry obtained from analytical curves or using imaging techniques is introduced. Such a procedure can be used in case of worn gear teeth that cannot be described by simple functions. A planar rectangular element based on the absolute nodal coordinate formulation (ANCF) is used to approximate the gear-tooth geometry. While the involute geometry cannot be described exactly using non-rational polynomials, the analysis and results of this investigation demonstrate that the FE solution leads to accurate geometry results, demonstrating the potential of using the position gradients in future strength-calculation studies.

Computerised design, simulation of meshing and stress analysis of bionic logarithmic spiral spur gear drives

Inspired by the inherent properties of logarithmic spiral in nature, a new cylindrical spur gear drives using logarithmic spiral as a gear tooth profile is proposed. The logarithmic spiral property is explored, and then the equation of meshing, conjugate tooth profile and contact path of the proposed gear pairs are derived. The meshing characteristics of the new pair are investigated, such as pressure angle, contact ratio and sliding ratio. Simulation analysis of the strength, transmission error and transmission efficiency are used to validate the performance of the gear pairs before production of the proposed gear, followed by gluing load capacity and bending fatigue strength testing of the fabricated gear pairs. The results show that the pressure angle at any point of the tooth profile on the proposed gear is constant and equal to the logarithmic spiral strike angle. The contact ratio of the gear decreases with the increase of the strike angle. Compared with that of involute gears, logarithmic spiral gears have better performance in terms of strength, transmission error, transmission efficiency and lifetime.

ОБОСНОВАНИЕ РАЦИОНАЛЬНОЙ КОНСТРУКЦИИ КРИВОШИПНОГО МЕХАНИЗМА КОЛЕСНО-ШАГАЮЩЕГО ДВИЖИТЕЛЯ

The article substantiates the rational design of the crank mechanism of a wheel-step mover. Kinematics analysis of a prototype of the wheel-step mover revealed the presence of oscillations in the vertical coordinate of the central axis during movement. To eliminate this drawback, differential geometry methods were used, thanks to which a method for calculating the non-circular profile of support shoes was created. Studies of the dynamics of the crank mechanism of the wheel-step mover showed that even in a steady state of motion, periodically acting inertial forces arise. To reduce their effect, non-circular gears should be used in the mover power drive. The dimensions of the crank mechanism links have been calculated, at which the cross-country ability of the running system, its kinematics and dynamics characteristics are improved, energy costs when laying a track are reduced, as well as the negative impact of the machine on the fertile soil layer. The polar coordinates of the involute points have been calculated. They are needed to construct the tooth profiles of non-circular gears. As the power drive of the crank mechanism, non-circular gears have symmetrical geometric parameters along two axes and provide a constant movement speed. Thus, the wheel-step running system can be used in agriculture, the timber industry, mining, and when working in emergency situations. The mover design being developed is intended for unmanned ground vehicles. This type of unmanned vehicles is at the initial stage of its development, inferior in frequency to air and marine unmanned vehicles. A targeted concentration of production and financial resources is necessary to ensure that domestic industry has a leading position in the production and sale of unmanned ground vehicles.

Method for a Comprehensive Analytical Study of Contact Pressures, Wear, and Durability of Metal-Polymer Helical and Spur Gears with Steel Pinion and Polyamide Gear

An analytical complex study of metal-polymer (MP) helical and spur gears by a new author's analytical method was carried out. The influence of the helix angle on the contact pressures in the engagement, the teeth wear, and the durability of the MP gears of different polymers has been established. MP gears with a pinion made of carbon steel (0.45% C) and gear made of basic polyamides PA6, PA66, and their composites with the following fillers: fiberglass (PA6+30GF), carbon fiber (PA6+30CF), molybdenum disulfide (PA6+MoS2), and oil filled cast polyamide (PA6+Oil) are studied. According to the developed procedure for determination of the angles of engagement parity change, it is shown that there is two-one-two-pair or three-two-three-pair engagement depending on the helix angle. Quantitative and qualitative regularities of changes in the initial maximum contact pressures and their reduction due to tooth wear over the time from the moment of the entrance of teeth to engagement up to their exit from it are established. In particular, at a helix angle of β = 10°, the contact pressures reduce to 33% compared to spur gears. The linear wear of the gear polymer tooth profile at selected points of engagement and different helix angles has been investigated, and the influence of the pressure angle on it is shown. It is established that its regularities of change during the engagement phase are the same for all polymeric materials studied. The durability of transmissions is calculated, and its comparative evaluation is given. At the helix angle β = 5°, a slight increase in durability is achieved compared to spur gears, and at β = 10° the durability of MP gears increases by about 1.5 times for all polymeric materials studied.

A novel harmonic movable tooth drive with almost surface contact conjugate tooth profiles: Parametric design, modification and analysis

This paper focuses on a novel harmonic movable tooth drive as an alternative of currently used harmonic gear drives. Different from the harmonic gear drives, the novel drive removes small module teeth from thin-walled flexible rim, instead, adopts several rigid movable teeth with logarithmic spiral tooth profile. Such tooth geometry makes almost surface contact possible under load condition, in comparison with the conventional involute tooth profile creating a linear contact. Aiming at the issue of cusp meshing between movable tooth and ring gear during the meshing process, this paper proposes a modification design method for the conjugate tooth profiles. First, the radial motion law of movable tooth is modified, in order to avoid rigid impacts at the exchange point of the rise travel and return travel. Then, the tooth profile of ring gear is generated by the envelope of the modified profile of movable tooth. Based on the formulated modification scheme, the equations of the conjugate tooth profiles are derived, and the influences of spiral angle and modification parameters on the meshing characteristics are systematically investigated. Finally, the wave generator cam profile and pivoted segment profile are established considering the strength consistency. The outcomes can provide significance guidance for the parametric design and optimization of this novel drive.

Tooth profile design and meshing characteristics analysis of internal meshing gears based on variable center meshing line

A design method of internal meshing gear based on variable center meshing line (VCML) is presented in this paper. The meshing line is an arc line connecting the intersection of the indexing circle connecting the inner and outer gears and the intersection of the tooth top circle, and the center of the line of engagement freely slides on the vertical bisection line of the line connecting the two intersection points. The top tooth profile equation of internal and external gears is derived by rack equation. According to the normal method of gear profile, the equation of gear root meshing line is derived by using the tooth top profile equation of the gear. According to the principle of plane meshing, the root profile equation of gear is derived by using the root meshing line equation of gear. The tooth profile of VCML gear can be deduced quickly by adjusting the center position of the meshing line. The stress, meshing stiffness, transmission error, contact ratio and sliding ratio of the VCML gears are analyzed by finite element method, and the area of significant performance improvement of the meshing line center of the VCML gears is found. The research results show that compared with involute gears, the maximum average stress of VCML gears is reduced by 64.99%, the meshing stiffness is increased by 193.67%, and the transmission error is reduced by 65.95%. This method improves the efficiency of new tooth profile design and transmission performance. By changing the center coordinates of the meshing line, the tooth profile of the VCML gear can be quickly redesigned, which greatly improves the efficiency of the new tooth profile design and transmission performance. This design method provides a new idea for the tooth profile design of internal meshing gear.

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

Geometric contact characteristics of variable hyperbolic circular-arc-tooth-trace cylindrical gear under different machine tool errors and sensitivity analysis for impact factors

Machine tool error is one of the most important research contents in gear transmission error, and contact characteristics are important factors affecting gear transmission performance. The establishment of tooth contact analysis (TCA) model of gear pair plays a guiding role in analyzing contact characteristics of gear pair. Based on the forming principle of variable hyperbolic circular-arc-tooth-trace cylindrical gear (VH-CATT gear), the structure model and coordinate system of VH-CATT gear machine tool are established. Based on the gear meshing conditions, the tooth surface mathematical model of the VH-CATT gear with machine tool error are established, and the TCA model with machine tool error is established. The influence of different machine tool errors on the geometric contact characteristics of the gear pair is analyzed. The results show that the pressure angle error, rotation error in y direction, and rotation error in z direction, translation error in y direction, and rotation error in z direction have a significant impact on the contact characteristics of tooth profile of VH-CATT gear.

Optimization of Tooth Profile Modification Amount and Manufacturing Tolerance Allocation for Rotate Vector Reducer Under Reliability Constraint

Abstract Rotate vector (RV) reducer is mainly applied in high-precision transmission. Its manufacturing errors strongly influence transmission accuracy. In order to select the optimal gear profile parameters and manufacturing accuracy to meet the transmission conditions in actual machining, a model of gear tooth contact analysis of involute and cycloid transmission considering tooth profile modification and manufacturing errors was established according to the principle of coordinate transformation and conjugate contact of gear. The sensitivity of different manufacturing errors to transmission errors was analyzed. At the same time, the optimization method of gear tooth profile modification parameters and manufacturing accuracy configuration under transmission reliability constraints was designed. The results showed that based on the optimization calculation of samples by the DIRECT algorithm, it was possible to obtain a set of RV reducer modification parameters and manufacturing tolerance bands with the lowest manufacturing cost under the condition of transmission accuracy. The selected parameters had high-reliability characteristics. Therefore, this method provides a theoretical optimization reference for manufacturing high-reliability RV reducer.

Design and transmission performance analysis of planthopper hip joint gear

To enhance the efficiency and load-carrying capacity of the jumping robot’s transmission system, the transmission characteristics of the planthopper hip gear were analyzed. The analysis revealed that the gear pair exhibited low transmission errors, high transmission efficiency, and significant torque capacity, making it suitable for high-speed, high-precision, and high-power density transmissions. A bionic design study of the jumping robot’s gear was conducted, drawing inspiration from the planthopper hip gear’s tooth shape and profile. A logarithmic spiral was assumed as the tooth profile based on its consistent pitch angle and curvature gradient characteristics. Parameters of the tooth profile of the bionic gear were calculated, and equations for the tooth surface were derived. A three-dimensional model of the bionic gear was constructed, and both its transmission efficiency and error were calculated. Using the finite element method, theoretical formulas for contact stress and bending stress in the bionic gear were deduced. The distribution characteristics of both bending and contact stress were analyzed, demonstrating that the bionic gear used in high-performance jumping robots possesses low transmission error, high transmission efficiency, and excellent contact strength, providing a theoretical foundation for enhancing their jumping capabilities.

Analysis of arbitrary tooth profiles of cylindrical gears using normal polar coordinates (Application to the generation of a gear tooth profile by a given tooth profile of rack cutter and its interference problem)

Analysis of arbitrary tooth profiles of cylindrical gears using normal polar coordinates (Application to the generation of a gear tooth profile by a given tooth profile of rack cutter and its interference problem)

Effects of Profile Shifting of a Cosine Gear

Cosine gears are a special type of gear drive, in which the tooth profile of the theoretical generating rack is described by the cosine function. According to recent research results, the cosine gear drive has a lower slip coefficient and the contact and bending stresses supported by the teeth during the load are smaller than those occurring in involute gears. The main advantage of the cosine drive pair lies in the apparent higher rigidity of the gear tooth: this can reduce flank wear and increase gear life, especially in heavy load and highspeed applications. In contrast to involute profile gears, cosine gears do not show undercut root fillet, even with small teeth number. This paper deals with cosine gear tooth profile design and the meshing of the cylindrical cosine gear pair conceived without, or with the application of the profile shifting. The mathematical model of the meshing was implemented in a Mathcad environment, showing that there exists a significant difference compared to the well-known involute gears. The simulation of the motion during the rolling of the gear pairs was realized with the Autolisp programming interface. Solid models were created and tested in the Autodesk Inventor environment.

Contact Analysis for Cycloid Pinwheel Mechanism by Isogeometric Finite Element

Cycloid drives are generally used in precision machinery requiring high-reduction ratios, such as robot joint (RV) reducers. The contact stress of cycloidal gears greatly affects lifetime and transmission performance. Traditional finite element method (FEM) has less computational efficiency for contact analysis of complex surface. Therefore, in this paper, isogeometric analysis (IGA) was employed to explore the multi-tooth contact problem of the cycloid pinwheel drive. Based on the nonuniform rational B spline (NURBS) curved surface generation method, the NURBS tooth profile of the cycloid gear was reconstructed. In addition, the NURBS surface of the cycloid gear–pin tooth–output pin was generated via the element splicing method. A geometrical analysis model of cycloid pinwheel drive was established to solve the contact force of the meshing pair under different input angles and compared with the finite element method in terms of convergence, resultant accuracy, and solving timeliness. The results show that isogeometric analysis has higher accuracy and efficiency than the finite element method in calculating the contact stress and contact force. The error of the IGA is only 8.8% for 10 × 10 elements in contact, while the error of the finite element method reaches about 40%. The method can improve the contact simulation accuracy of the cycloid drive and provides a reference for the design evaluation of RV reducer.

High-precision measurement of gear tooth profile using line spectral confocal method

The optical measurement of gear tooth profile has the advantages of high efficiency and non-contact, but its measurement accuracy has not reached the accuracy of the existing contact method. Therefore, a line spectral confocal high-precision gear tooth profile measurement method is proposed. First, the gear optical measurement coordinate system is established using a standard mandrel and the optimal measurement position model for the measured tooth profile not blocked by the adjacent tooth is established. Then, to solve the problem that the tooth profile information contained in the optical measurement results and contact measurement results is inconsistent and cannot be directly compared, a virtual probe envelope filtering method for the tooth profile optical measurement data is proposed. Next, to verify the accuracy of the line spectral confocal gear measurement system, the tooth profile deviation measurement experiments using a two-accuracy-level gear involute master are conducted. The results show that the proposed line spectral confocal method has high measurement repeatability and accuracy. The maximum absolute difference between the tooth profile deviation measurement results and the verification results is 0.16 µm, and that the tooth profile deviation curve maintains highly consistent with the verification curve. Finally, the tooth profile deviation measurement experiments using a two-accuracy-level spur gear are performed. The results indicate that the maximum absolute difference between the tooth profile deviation evaluation results measured with the proposed method and the gear measuring center is 0.28 µm. The maximum absolute difference between the two profile deviation curves by point-to-point is 0.52 μm, and the average absolute difference is 0.16 μm. The proposed line spectral confocal method can realize the high-precision measurement of gear tooth profile.

Tooth Profile Construction and Experimental Verification of Non-Circular Gear Based on Double Arc Active Design

The double arc pitch curve is a novel way for creating a non-circular gear tooth profile. Aiming at the problems of the rounding of non-integer teeth and the nonlinear equation system in the construction of double circular arc tooth profile, an active design method based on planetary gear transmission kinematics and gear-meshing theory is proposed. By studying the double arc pitch curve of the three engagement positions, combined with the uniform theory of the relative curvature of surfaces, the distribution characteristics of the solution of the nonlinear equation system are analyzed in depth, leading to the optimal solution of the characteristic parameter values of the pitch curve. Combined with UG modeling and ADAMS motion simulation, the gear engagement force and the angular velocity of the planetary gear around the center are compared and analyzed in the case of integer tooth ratio and non-integer tooth profile. The results show that the engagement force of the gear pair constructed by integer teeth is reduced by about 17%. The optimal combination of the numbers of teeth is determined by orthogonal experiments, on which trial production and hydraulic load tests are based. The results reveal that the torque of the product at a rated speed is about 2.6% higher than that of foreign prototypes, which verifies the effectiveness of the design method proposed in this paper.

A Hybrid Strategy for Profile Measurement of Micro Gear Teeth.

A hybrid strategy is proposed to meet the challenge of obtaining the profile of micro gear teeth with a small modulus. Firstly, the contact probe segmentally obtained the falling flank profiles with an auxiliary lifting mechanism to avoid interference when it climbs on the rising slope. Then, the noncontact chromatic confocal displacement sensor efficiently acquired the gear peak positions to carry out the two-point error separation with the gear peak positions from the probe measurement. Finally, actual experiments were carried out to obtain the profile of a harmonic drive flexspline. Compared with the commercial ultraprecise profiler, the proposed method provides measurement results with a deviation of less than 20 μm. In conclusion, the hybrid strategy is feasible and accurate for drawing the micro gear teeth profile without any collision between the measuring probes and the measured workpiece.

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