Involute Gear Research Articles

Mathematical principle for calculating contacting curve length of involute Helicon gearing

Mathematical principle for calculating contacting curve length of involute Helicon gearing

Modelling Method and Meshing Characteristics of a Novel Curve-Surface Conjugate Internal Gear Drive

Space exploration has become a major focus in the field of technology, with gear transmissions in aerospace equipment playing a crucial role. In the extreme environment of space, gear transmissions face challenges like large temperature differentials, deformation and maintenance difficulties, which will severely impact transmission accuracy and service life. To meet the growing demands for high-performance gear transmissions with high transmission efficiency and error adaptability in the aerospace field, this paper proposes a novel curve-surface conjugate internal gear drive consisting of an involute internal gear and a curve-surface gear. The fundamental theory of curve-surface conjugation is introduced, and the construction method for curve-surface gear based on a selected contact path and meshing tube is presented. The analysis models including induced curvature, sliding ratio and tooth contact analysis with errors (ETCA) are simulated to evaluate the meshing characteristics. Additionally, prototypes are manufactured and experimental setups are established to validate the transmission performance. These results indicate that as the rotational speed increases, the transmission efficiency of the curve-surface conjugate internal gear drive improves, which is contrary to the trend observed in involute gear drives. And the transmission efficiency of the curve-surface conjugate internal gear drive surpasses that of the involute gear drive at higher rotational speeds. Moreover, this novel gear drive exhibits excellent error adaptability, maintaining intact contact paths and high transmission efficiency even in the presence of assembly errors. This study provides new ideas for the design and manufacture of high-performance gear transmissions from the perspective of spatial geometric elements.

Predictive Model for Scuffing Temperature Field Rise of Spiral Bevel Gears under Different Machining Conditions

Spiral bevel gears are extensively employed in mechanical transmissions; however, they are prone to adhesive wear when operating under high-speed and heavy-load conditions. Research indicates that the tooth surface roughness of gears significantly influences the friction and wear of the meshing gears. The present study delves into the origins of tooth surface roughness through the integration of the W-M function and fractal theory. Utilizing an involute helical gear with surface roughness for tooth cutting, a three-dimensional model is established with roughened tooth surfaces. This paper introduces an approach to developing three-dimensional gear models with roughness and applies the finite element method to perform thermodynamic analysis on gears exhibiting diverse levels of surface roughness. The thermal analysis of gears with varying degrees of roughness was conducted using the finite element method. Comparative analysis of the results under specific operating conditions elucidated the impact of roughness on tooth surface temperature rise. In order to validate the simulation model, an experimental test platform for spiral bevel gears of identical size was established. This model integrates tooth surface roughness with thermodynamic analysis, allowing for the rapid assessment of tooth surface temperature rise under different machining conditions, and reducing the cost of validating predicted tooth surface load-carrying capacity.

Lifespan Evaluation for a Standard RV Reducer based on Fatigue Strength Theory

Rotate vector (RV) reducers are the two-stage deceleration device comprising involute and cycloid-pin gear transmission mechanisms. As the typical deceleration elements, they are widely applied in industrial robots, digitally controlled machine tools and automation fields due to their compact structure and high precision. However, a reducer may lose precision after a long-term operation. Moreover, pitting and metal peeling of internal components may also occur, leading to a fatigue failure. Therefore, it is necessary to conduct investigations on lifespan evaluation for RV reducers. In this study, the CRV-80E reducer is taken as the research object. Firstly, according to its transmission characteristics, the lifespan evaluation model for a standard RV reducer is established based on fatigue strength theory and Palmgren-Miner linear cumulative damage law, with the internal crankshaft bearing is considered as the key component. Then, the simulations are carried out on crankshaft bearing by the ANSYS Workbench and SKF SimPro. The contact stress, deformation on bearing rollers, and the rated lifespan of the reducer are simulated. Finally, the accelerated life test is conducted on an RV reducer by increasing the external load with the positioning precision as an output and criterion. After the test, the reducer is disassembled, and metal peeling failure is observed on the crankshaft bearing while other parts are relatively intact. The test results validate the feasibility and accuracy of the service life evaluation model and simulation analysis. This study provides a new research approach for researchers and manufacturers and a design reference for engineers to improve the lifespan of RV reducers by optimizing crankshaft bearings, which have a certain academic value.

A novel forward performance-driven design method for gear parameters

Gear is one of the most crucial components of the transmission system, and the performance of gear directly affects the efficiency and reliability of the transmission system. Conventional methods for designing gear parameters involve several time-consuming and complex steps, which may not guarantee optimal performance. Therefore, we propose a new method for designing gear parameters that aims to improve efficiency and accuracy. First, the tooth surface equations of spur and helical involute gears suitable for symmetric and asymmetric teeth are deduced based on the gear-forming machining principle. Second, the performance evaluation models for load capacity, dynamic performance, efficiency, and power density of the gears are established based on the precise gear surface. The design objectives are standardized and evaluated comprehensively using a linear weighting method. Finally, a forward performance-driven design method of gear parameters is established. The proposed method is applied to a helical gear pair design case, and the results show that 90.7% of the individuals in the Pareto optimal front are asymmetric gears, with 9.3% being symmetric gears. This suggests that asymmetric gears have more opportunities to be optimal than symmetric gears. The highest-ranked gear designed using the proposed method is superior to the gear designed using conventional methods.

New method for strength analysis of involute beveloid gears using fractal theory

New method for strength analysis of involute beveloid gears using fractal theory

Quantifying the surface damage of involute gear flanks independent of the gear datum axis

Abstract Surface damage on involute gear tooth flanks can develop during the operation of a gearbox and affects their life, durability and efficiency. Understanding the extent and severity of this damage is critical, especially in long life applications such as wind turbines where gearboxes are a critical component that incur high down time and costs for replacement. Accessing gears in service is difficult and relating gear form measurements to the gear datum is often impossible without its removal. Additionally, damage measurements are typically not areal, which can miss the most severe areas, or require lengthy measurement times. This paper describes a method for characterising and quantifying localised damage relative to the undamaged tooth surface independent of the gear datum axis, and provides a method to estimate the damage location using involute co-ordinates. By taking soft replicas of the gear flank and measuring them with optical methods, the damage is characterised. This method allows for the areal evaluation of damage that relates to involute coordinates, which can be combined with nominal data or measurements after manufacture to create data sets for simulation in tooth contact analysis models, used to train condition monitoring models or for improved maintenance programming to improve reliability and reduce costly downtime.

High-ratio planetary gearbox with EC gearing for robot applications

AbstractThe drive system of robots and robot-like systems (RLS) is often designed with a combination of an e-motor and a gearbox with a high transmission ratio to optimize performance. The various types of possible robot gearboxes can be selected based on their characteristics, which strongly influence the performance of the entire robotic system. Planetary gear drives have advantages due to their high efficiency and low design complexity. Disadvantageous is the low transmission ratio per stage and the resulting large design space required with the currently predominant involute gearing. Using special tooth profile shapes, such as the eccentric cycloid (EC) gearing, enables a high transmission ratio per stage to be achieved, thus reducing the design space required. In order to evaluate the design, a description of the geometry and characteristics of the EC gearing is necessary. The application-optimized design can be made accessible on an interdisciplinary basis using a suitable description language for the product development of the complete robotic system. The paper shows the design and analysis of a planetary gearbox with a high transmission ratio for applications in robotics. The planetary gear stage is designed with the EC gearing, which offers advantages compared to the involute gearing. The performance of the selected gearing is evaluated based on various characteristics. This allows advantages to be identified compared to the established types of transmission for robots and RLS. Overall, the paper presents a new robot gearbox with a comprehensive description and analysis directly accessible for simulation or production using additive manufacturing.

A prediction method for the backlash error of robot precision reducers based on optimal assembly

Abstract The kinematic accuracy and service life of robot precision reducers are directly affected by their assembly accuracy, and the backlash error is one of the most important performance indicators for evaluating the assembly accuracy. A prediction method for the backlash error of robot precision reducers is proposed, which can quickly and accurately calculate the backlash error of the optimal assembly reducers. Based on the optimal assembly process, the sensitive relationship between the errors of the crucial components and the assembly accuracy of the reducer is analysed, and the crucial errors affecting the backlash error are identified. The cumulative backlash of the crucial error along the dimensional chain is clarified, and the corresponding relationship between the backlash and the backlash error is determined. On this basis, backlash prediction models are established for the involute planetary gear and cycloidal-pin gear transmission parts, respectively. The mapping relationship between dimensional errors and backlash is also derived. Furthermore, the backlash error of the reducer is obtained by calculating the backlash error caused by each error term of the high-speed stage and low-speed stage transmission parts. Finally, the effectiveness and correctness of the prediction method are verified by the optimal assembly of RV reducers and the measurement of the backlash error. The theory and method proposed in this paper can quickly and accurately predict the backlash error of the reducer for robots, effectively predict the assembly accuracy of the selective reducers, significantly improve the assembly efficiency of the reducers, and then provide theoretical guidance and technical support for enhancing the assembly quality and kinematic accuracy of robot reducers.

Theoretical and experimental study on a novel gear drive with high transmission efficiency

Abstract This paper introduces a novel gear drive characterized by high transmission efficiency. The gear incorporates cubic curves with corresponding tangents defining the normal section tooth profile. The generation principle of this new gear drive is thoroughly established. Utilizing gear meshing theory, key parameters such as the relative motion velocity, normal vector, and meshing equation are derived. Subsequently, the fundamental principle of the conjugate curve is formulated. The methodology for generating the tooth profile is also explored. An efficiency test was conducted on the gear, and the results demonstrate a superior transmission efficiency relative to that of traditional involute gears.

Analysis of the effect of circular spline tooth profile parameters on flexspline gear stress based on response surface methodology

Abstract This study presents a simulation model for the wave generator and flexspline assembly, examining the relationship between radial and main section deformations. Adjusting the wall thickness enables interference-free meshing of various sections. A response surface model for the effect of circular spline profile parameters on flexspline stress has been established. To assess how various parameters of the involute gear profile affect the meshing stress of flexspline. Thus providing a theoretical basis for optimizing gear profile modifications.

Optimizing Structural Steel Spur Gear Design for Reduced Stress Concentrations in Industry 4.0 Using Finite Element Analysis

Abstract: The advent of Industry 4.0 has highlighted the importance of spur gears in power transmission across the automotive, aerospace, and marine industries. However, these gears are susceptible to failure due to bending and contact stresses from fatigue. Designing an involute gear profile presents challenges, particularly in managing tooth root stresses and avoiding interference. This study tackles these issues through comprehensive stress analysis, using modeling and FEM assessment. The focus is on evaluating bending stresses at critical locations and identifying strategies to reduce stress concentrations at the gear tooth root. By enhancing gear robustness, this research could contribute to potential patents in gear technology. Background: Spur gears are crucial for power transmission but often fail due to fatigue-induced stresses and design challenges at the tooth root. Objective: This study aims to address spur gear design challenges through comprehensive stress analysis, using modeling and FEM to assess bending stresses and explore strategies to reduce stress concentrations at the tooth root. Methods: The research uses SolidWorks and ANSYS for FEM analysis, focusing on bending stresses and exploring modifications in gear design using various materials for optimization. Results: The analysis shows strategic modifications to the gear's fillet radius and addendum can reduce stress at the tooth root by 38.26%, with stainless steel being the optimal material for enhanced gear strength. Conclusion: The study emphasizes strategic adjustments in fillet radius and addendum to reduce stress in spur gears, highlighting stainless steel's superior performance for optimized design and enhanced strength.

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.

Assessment of the load capacity of Novikov gears and involute gears

An assessment is made of the load capacity of Novikov gears and involute gears of varying accuracy. The possibility of replacing high-precision involute gears with ground teeth with Novikov gears of medium precision is shown. Keywords:Novikov gear transmission, involute gear transmission, very hard teeth, accuracy, load capacity vkorotkin@sfedu.ru

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.

A Modified Approach to the Rack Generation of Beveloid Gears

The purpose of this paper is to present an easier and more efficient method for the determination of the geometry of a bevelled gear tooth. Based on a method that provides an easier way for the rack generation of involute helical gears, the mathematical model of a beveloid gear is studied. The mathematical procedure for developing two-dimensional cross-sections has been extended to three-dimensional gear models. A computer programme is developed to obtain generating and generated surfaces. The proposed algorithm is compared with the previous studies for verification and validation. The results demonstrate that the coordinates obtained from the given method are nearly the same on the start and end points of the main gear parts, such as the involute and root fillets regions. Also, between the limits, the values can be considered acceptable. A coordinate deviation of the gear profile has been observed in the mathematical model, because of the profile shift. Modifications have been developed in the equations to eliminate these cases. The main advantage of the proposed method is to obtain mathematical models without carrying out some of the calculation steps used in previous studies. Eventually, this feature will provide an easier and faster method to develop computer-aided models of the beveloid gear types.

Nonlinear Dynamic Analysis of a Spur Gear Pair System with Wear Considering the Meshing Position

In this paper, a nonlinear dynamic model of a parallel shaft gear system consisting of two involute spur gears is developed to investigate the coupling effect between the gradual surface wear of gear teeth over time and nonlinear dynamic characteristics. A uniform wear model that accounts for how the volumetric wear of the gear teeth affect their meshing position, backlash, and stiffness is proposed. Additionally, a nonlinear dynamic model with six degrees of freedom is described that considers friction, time-varying gear backlash, and time-varying meshing stiffness. The proposed model significantly changes the mesh stiffness, not only in terms of value but also in terms of contact ratio. Furthermore, the nonlinear dynamic characteristics of the gear system vary significantly. It is found that the gradual wear of gear teeth affects the meshing position and further has a significant impact on the nonlinear dynamic characteristics of the spur gear system. This paper provides a basis for studying the nonlinear dynamic characteristics of the spur gear system as it experiences the gradual wear of teeth over time.

Mathematical and computer modeling of transmission with non-traditional engagement for mining equipment drive

In modern gear mechanics, transmission mechanisms play a crucial role in converting the rotational motion of a driving shaft into the rotational motion of another shaft with varying angular speeds and torque. To achieve optimal designs for the next generation of transmission mechanisms, it is essential to develop mathematical models of their dynamic behavior, conduct computer simulations of the meshing geometry of key components, and visualize the operation of the mechanism. Despite the widespread use of involute gearing in mechanical transmissions, there is ongoing research into new types of gearing that offer advantages over traditional systems. The main challenges facing the industry include increasing the gear ratio in a single stage, enhancing load capacity, and improving efficiency compared to standard gear transmissions. This paper presents the results of mathematical and computer modeling, along with a comparative analysis of the eccentric-cycloid (EC) engagement with the involute gear transmission. Through analytical calculations, the energy-force parameters of the EC gearbox were determined, equivalent stresses and static deflections of transmission shafts were obtained. The paper includes the results of static analysis of elements of the new EC transmission, as well as an algorithm for computer modeling of contact stresses occurring in the engagement. Conclusively, by comparing contact stresses in traditional involute gear transmission, calculated using various analytical methods, with those in EC engagement determined through computer simulation, the advantages of the new transmission type and its potential application in mining equipment transmission mechanisms are highlighted.

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.

Algorithm for synthesis of plane gears according to the constant curvature criterion

This paper presents an algorithm for synthesizing plane gears based on the constant curvature criterion. The derived formulas for pinion gears can be applied in various mechanisms, including clockwork mechanisms, gun turret aiming systems, lifting and transport mechanisms, and planetary gearboxes with tracked propellers. These formulas enable effective analysis of transmission operation and optimal value selection, leading to more rational design and increased efficiency of the entire mechanism. Internal gearing technology allows for significant reductions in drive dimensions and metal consumption compared to traditional involute gears. Improving the technical characteristics of mechanical drives can be achieved by optimizing pinion gear design. This study focuses on the analysis of pinion gears, which reduce profile slippage and transmission wear, resulting in increased efficiency and convenience of operation. Additionally, pinion gears are technologically advanced and reliable. This work develops a mathematical model of a pinion transmission as a flat gearing with a constant gear ratio, providing a foundation for further research and practical applications.