Root Bending Stress Research Articles

Performance reliability evaluation of high‐pressure internal gear pump

AbstractWith the development of the high‐end equipment technology, the performance requirements of the internal gear pump (IGP) under high pressure are also increasing. However, the increase of working pressure will lead to the instability of gear pump performance in terms of volumetric efficiency, noise, reliability and so on, it is necessary to reasonably evaluate the reliability level of high‐pressure IGP. The reliability analysis of the high‐pressure IGP is carried out from the aspects of flow, noise, and gear strength in this paper. First, the output flow rate and far‐field flow‐induced noise of the high‐pressure IGP were obtained through fluid numerical simulation, and experimental verification was conducted. Then, based on the time‐varying meshing stiffness, backlash function and static transmission error of the gear pair, a nonlinear dynamic model of the internal meshing gear pair was established. The time‐varying meshing force was obtained through the dynamic model of the gear pair, and then the tooth contact stress and tooth root bending stress were obtained. Finally, considering the uncertain factors affecting the performance of the high‐pressure IGP, Latin hypercube sampling (LHS) combined with dendrite network (DD) was used for random response modeling. The performance reliability of the high‐pressure IGP, including output flow rate, far‐field flow‐induced noise, and the strength of gear pair, were estimated based on the fourth moment‐based saddlepoint approximation (FMSA). The reliability analysis results can provide a theoretical basis for the structural optimization design of the high‐pressure IGP.

Optimization of polyoxymethylene spur gear pair using meta-heuristic algorithms: A comparative study

Design-optimization of power transmitted elements such as gears is a complicated process to accomplish as the mathematical model acquires to be in compliance with the optimization algorithm used. Utilization of technical standards and handbooks assisted engineers and designers with designing the gears, however the results were inefficient, thus meta-heuristic algorithms were approved for optimization, for example, genetic algorithm. In this paper, optimization of single stage spur gear pair made of polyoxymethylene material was carried out using genetic algorithm (GA), artificial bee colony (ABC), JAYA, grey wolf optimizer (GWO), and whale optimization algorithm (WOA) in MATLAB. Three single functions were used as objectives, those are, weight, power loss, and center distance. Module ( m), face width ( b), number of pinion teeth ( z1), and finally profile shifts ( x1, x2) served as design variables. Unlike steel gears which were constrained by handling root bending stress and flank pressure, more constraints were added to the problem with the increase of complexity such as root and flank temperature, tooth deformation, and wear abrasion. The results showed a significant decrease in all three objectives when optimizing each objective solely with variation of the variables. It is also observed that JAYA has the superiority in decreasing the three objectives in comparison to GA and ABC nevertheless, in agreement with GWO, and WOA.

Theoretical and experimental research on tooth root bending stress of face gear

A tooth root bending stress analysis method is proposed to study the tooth surface load distribution and tooth root bending stress during the meshing process of orthogonal face gear. Firstly, the tooth surface equation, transition surface equation and tooth width constraints are derived. Then, Through the tooth contact analysis (TCA), the contact path and spots of the face gear pair are obtained. What’s more, the proposed method based on Load tooth contact analysis (LTCA) is utilized to calculate the tooth surface load distribution and root bending stress of the face gear along the mesh cycle and compared with the finite element method. Meanwhile, root bending stress tests were carried out to verify the proposed method. The comparisons indicate that the proposed method has higher computational efficiency compared with the finite element method (e.g. the finite element method takes about 5 h for a mesh cycle and the proposed method costs 40 s). The results show that the tooth surface load distribution and root bending stress of the proposed method are consistent with those obtained by the finite element method and gear root stress experiment.

Effect of tooth root fillet on root bending stress

The tooth root fillet exerts a significant effect on the root bending stress of gear teeth. However, the current study predominantly focuses on theoretical analysis and numerical methods, with insufficient emphasis on effective experimental verification. Therefore, this paper completes root bending stress tests on gear teeth and conducts an analysis of how the tooth root fillet impacts root bending stress. Firstly, the paper provides a summary of the generation theory of the tooth root fillet using rack cutters/hobs, which serves as the foundational machining principle for tooth root fillet. Secondly, using three typical tooth root fillet as examples, the study conducts root bending stress tests. The R-S-N curves corresponding to these three tooth root fillet profiles are derived using both conventional group methods and a variable load approach, involving lifting and lowering. Finally, the paper analyses the influence of these three types of tooth root fillet on the root bending stress of gear teeth. Through a comparison of S-N curves at the same reliability level, it identifies the optimal tooth root fillet among the three.

A Novel Tooth Modification Methodology for Improving the Load-Bearing Capacity of Non-Orthogonal Helical Face Gears

This study proposes a double-crown tooth surface modification technology that improves the load-carrying capacity of non-orthogonal helical tooth surface gears. The tooth modification is determined by a modified rack-cutter, and its feed motion is related to an intentionally designed transmission error. The novelty of the tooth modification design is that the transmission error can be pre-designed. First, changing the tooth profile of the tool enables preliminary modification along the tooth profile direction; second, by modifying the interaction between the tool and the machined gear, subsequent fine adjustments are made to the contact path. This two-stage tooth modification strategy not only retains the advantages of the traditional method but also significantly improves the balance of the load distribution on the tooth surface through an original contact path modification strategy. Through systematic tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA), it was verified that the new method reduces contact stress and tooth root bending stress and improves the gear’s resistance to misalignment errors. This research provides the basis and motivation for further exploring and improving this tooth profile modification technology to solve the challenges faced by more complex gear systems.

The Design, Analysis, and Optimization of a New Pitch Mechanism for Small Wind Turbines

This article proposes and designs a novel variable pitch adjustment device for small wind turbines. The generator spindle is designed to be hollow so that the drive rod passes through it and connects the pitch drive mechanism to the pitch actuator. The article introduces the basic structure and working principle of the pitch mechanism and verifies the feasibility of the pitch device by using 3D printing technology to produce a small-scale model. The stress analysis of the wind turbine was carried out using the unidirectional fluid–structure coupling method. The results show that the maximum equivalent stress of the pitch mechanism is 27.42 MPa, the maximum tooth surface contact stress of the gear is 38.40 MPa, and the maximum tooth root bending stress is 18.13 MPa. The rack synchronous disk, blade handle, and gear rack mechanism were designed with light weight using various optimization schemes. The results of the optimization showed that the overall mass of the pitch mechanism was reduced by 33.2%, improving the applicability of the new pitch mechanism.

Research on the Strength Calculation Method and Effects of Gear Parameters for High-Coincidence High-Tooth Gears

This article studies the calculation method for the tooth root bending stress of a high-tooth gear pair with a high contact ratio. The boundary point of the double-tooth meshing zone of the high-tooth gear pair is used as the loading point for the load, and the calculation formula for the bending stress at the dangerous section of the tooth root is obtained. By using ANSYS finite element simulation, the effect of the addendum coefficient, pressure angle, and other gear parameters on the bending stress of the tooth root is studied. The analysis shows that increasing the pressure angle will reduce the bending strength of the tooth root. Increasing the coefficient of a tooth’s top height will lead to an increase in the bending strength of the tooth root. Comparing the finite element analysis (FEA) results with the theoretical calculation results, the analysis shows that under low loads, the maximum error of the theoretical calculation values of the driving toothed gear and driven gear shall not exceed 13.53% and 15.42%, respectively. Under high loads, the maximum theoretical errors of the driving toothed gear and driven gear shall not exceed 8.78% and 10.91%, respectively. This verifies the correctness of the calculation method, which is of great significance for improving the load-bearing capacity of high-tooth gears and for guiding tooth shape design.

Design, Stress Analysis, and FEA Prediction of Nutation Drive with Face Gear

Background: Nutation face gear transmission is a patent about a new type of transmission based on meshing between two face gears. It replaces spur gear with internal face gear to form a “faceface” meshing gear pair and actualizes its deceleration function by combining the nutation principle, which has the advantages of both face gear and nutation drive. Objective: The purpose of this paper is to study the designing and manufacturing of a nutation face gear reducer to derive the force calculation formulas of the face gear, input shaft and bearings, calculate the contact stress of tooth surface and bending stress of tooth root, determine the basic parameters of the prototype, and analyze the processing points for the core parts of the prototype. Methods: Using the dynamic meshing force analysis method by combining it with modern digital design and manufacturing methods, the designing and manufacturing of a nutation face gear reducer are studied. Results: In the theoretical calculation, the maximum contact stress of the fixed side tooth surface is about 433MPa, and the average contact stress is about 345MPa; the maximum contact stress of the tooth surface on the rotating side is about 579MPa, with an average of about 502MPa; the comparison of which with the finite element analysis results verifies the theoretical calculation results. Conclusion: The dynamic meshing force analysis method of nutation face gear transmission is deduced and given. Based on this, gear tooth surface transient contact stress and tooth root bending stress are calculated.

Influence of gear–shaft interference fit assembly on the meshing characteristics of cylindrical gears considering comprehensive modifications

The interference fit assembly of shaft and gear has an important influence on the gear drive performance. In this work, the cylindrical gear active surfaces and fillet will be provided with comprehensive modifications including front and back end reliefs, tip and root reliefs, as well as a low-value lead crowning. The mathematical model of the gear flanks surfaces with the mentioned modifications has been developed. A finite element model of the gear set has been established to study the effect of the interference fit assembly of the gears with their shafts on the meshing characteristics of cylindrical gears with modified tooth surfaces. With the increment of the interference fit amount, the contact pattern is shifted towards the top edge of the gear flank. The increment of the interference fit amount also contributes to the shift of root stress distribution towards the root land of the tooth. The maximum value of root bending stress keeps a positive linear relationship with the amount of shaft–gear interference fit being considered.

소형 수문 권양기용 피니언의 강도설계

This paper presents an approach for the strength design of the pin jack-type gear system, that is, roller rack pinion (RRP) system, typically employed in small to mid-sized sluice gate devices. First, the load distribution of the pinion in the section of double tooth contact (DTC) was presented by the introduction of the load sharing factor considering the gear precision. Notably, the higher the gear precision, the lower was the contact force, contact stress, and tooth root bending stress (TRBS) in the section of DTC. Meanwhile, as the mounting distance (MD) increased, the pressure angle and geometry factor for the TRBS increased, and the tooth equivalent curvature decreased. Consequently, the change in contact force according to the increase of the MD was insignificant. However, the contact stress and TRBS significantly decreased, and the fatigue lifetime of the pinion increased.

Data-driven bending fatigue life forecasting and optimization via grinding Top-Rem tool parameters for spiral bevel gears

To establish a bridge between grinding tool parameters and loaded tooth fatigue life, an innovative data-driven root flank bending fatigue life forecasting and optimization via Top-Rem tool parameters was proposed for grinding spiral bevel gears. The recent machine settings modification is extended into grinding Top-Rem tool parameters modification in case that geometric accuracy and root bending fatigue life are integrated into a collaborative optimization. The proposed Top-Rem modification includes three key steps: (I) arc-shaped blade, (II) top part, and (III) top fillet part. Then, while root bending stress is determined by using finite element method (FEM)-based simulated loaded tooth contact analysis (SLTCA), data-driven fatigue life forecasting is developed by correlating with the multiaxial fatigue damage model based assessment. Moreover, data-driven bending fatigue life optimization model is established by using Top-Rem tool parameters modification, where the important constraints in target flank determination includes: (i) root overcutting, (ii) geometric accuracy, and, (iii) fatigue life. For high accuracy and efficiency, two different strategies are proposed: (i) the different parameters modification types; and, (ii) sensitivity analysis of grinding Top-Rem tool parameters. Finally, proposed method can verify that bending fatigue life can be significantly improved by modifying the key Top-Rem tool parameters in early stage of the whole life product development for spiral bevel gears.

기어 정밀도를 고려한 CRP 시스템 랙의 굽힘 강도설계

본 논문에서는 기존 랙-피니언의 인벌류트 피니언 치를 롤러로 대체한 CRP(cam rack pinion) 시스템 랙의 기어 정밀도(gear accuracy grade)를 고려하지 않았을 때와 고려하였을 때의 이뿌리 굽힘응력(TRBS: tooth root bending stress) 거동을 비교 고찰하였다. 하중전담구간(STC: single tooth contact)에서의 TRBS 거동은 기어 정밀도와 무관하게 거의 일치하였으나 STC 최고위치인 HPSTC(highest point of STC)를 지난 이후의 하중분담구간(DTC: double tooth contact)에서의 TRBS는 기어 정밀도에 따라 상당한 차이를 보였다. 또 설계제원 중 롤러 피니언 모듈이 증가하면 압력각의 감소로 최대 접촉력은 약간 감소하나 형상계수(bending strength geometry factor)의 감소로 인해 TRBS가 증가함을 확인하였다.

Effects of gear structural parameters on unsteady-state temperature field of large transmission ratio gear and tooth root thermal stress

A calculation method of tooth root stress for large transmission ratio gear and rack is proposed using a broken-line section model and thermal-mechanical coupling analysis. To describe the transient meshing dynamics between the gear and the rack, a contact deformation model between the gear and the rack is fitted by using the least square method, and the meshing temperature field is calculated using the Blok flash temperature theory. The method of the broken-line section is modified based on the fatigue crack growth path at the tooth root. A new broken-line section model to calculate the tooth root stress is established, and a stress permeability factor is introduced to identify the width of the broken line. Using the meshing temperature field of gear and rack, the calculation formulas of root bending stress and root compression stress are derived by the integral-iterative method under thermal-mechanical coupling, and the correctness of the maximum tooth root stress model is validated by the finite element method (FEM). Compared with other methods for analyzing the thermal strength of gear roots, the proposed novel broken-line section method has simpler calculations and higher accuracy, and it can systematically analyze the effect of structural parameters on the dynamic strength of gear transmission mechanisms under actual working conditions.

Investigation of Load Capacity of High-Contact-Ratio Internal Spur Gear Drive with Arc Path of Contact

Tooth root bending stress and surface contact stress are two major determinants of the load carrying capacity of gear drives. Generally, a high contact ratio has the potential to enhance the gear strength. In this paper, the basic procedures and methods for constructing a high-contact-ratio (HCR) internal spur gear pair with the arc path of the contact are presented. The effects of design and modification parameters such as deflection angle, modification angle, and addendum coefficient on gear drive are analyzed in detail based on 2D finite-element models (FEMs). A comparison of experimental and finite-element analyses (FEA) of the bending stress, contact stress, and contact ratio between HCR and involute internal spur gear drives was conducted to demonstrate the advantages of the HCR gear drive in terms of load capacity. The results show that appropriately increasing the deflection angle and addendum coefficient is beneficial for reducing bending and contact stresses. It was also observed that the bending and contact stresses of a HCR pinion are lower than those of an involute one. Moreover, the contact ratio increases with input torque.

Geometrical Analysis of Tooth Form Factor and Stress Correction Factor In Calculating Tooth Root Bending Stress on Internal Gear

One of the most critical disadvantages of plastics must be much lower strength than steel, which prevents plastic gears from having high load capacity. Internal gears have larger tooth root thickness and smaller tooth flank sliding than external ones. In addition, they have a concave tooth profile. These facts indicate that even plastic gears could realize relatively high load capacity. Although tooth root bending stresses greatly depend on fillet profile, failure of internal gears is generally not crucial in steel gear pairs, which has provoked few discussions on fillet profile. The present paper describes fillet profiles of internal gears, which the top land of the pinion cutter generates. Geometrical analyses defined the change in the profiles associated with the number of teeth and profile shift coefficient of pinion cutter with the possible maximum radius of top land. According to ISO 6336-3, which is the basic standard on the load capacity of gear pair, calculations of tooth form factors and stress correction factors determined the positions of the critical section for evaluating tooth root bending stress.

Design and finite element analysis of AISI 4340 alloy steel helical gear

The present study aims at design, modeling, modal analysis, static analysis and harmonic analysis of helical gear of AISI 4340 material. The dimensions of helical gear are obtained through comprehensive theoretical design procedure. Three dimensional model of helical gear was created by three dimensional modeling software. Ansys workbench finite element tool was used for harmonic analysis, static structural analysis and modal analysis of designed helical gear. Helical gears are commonly used components in power transmission system. Helical gears play a vital role in transmitting motion and power. Gear teeth are usually subjected to root bending stresses and tooth contact stresses. Study of these stresses during designing helical gear is very important. Hence numerical analysis of AISI 4340 helical gear is done using numerical analysis software tool to check the designed gear met the application requirement or not.

Precise mathematical model for the ratchet tooth root bending stress

Abstract. A ratchet is an essential component of the ratchet pawl mechanism. But the traditional ratchet strength check method has certain limitations in the design process. In this paper, the stress analysis of the ratchet is discussed and a precision mathematical model for the ratchet tooth root bending stress is proposed for the first time. This model was established by the folded section and defined by the incision effect theory. To test the prediction ability of the proposed mathematical model, the maximum stress of three standard ratchets and one non-standard ratchet were analyzed by the FEA (finite element analysis) method. The non-standard ratchet was adapted in the ratchet experiment to analyze its maximum stress. The analysis results presented in this paper show that the proposed mathematical model has a good predictability, regardless of whether it is a standard or non-standard ratchet. It is recommended that this model can be used to predict the ratchet tooth root bending stress in the ratchet design process.

Bending fatigue life oriented tooth flank dry-grinding tool modification for cleaner manufacturing of spiral bevel gear product

Gear tooth flank dry-cutting was beget with emergent and disruptive NC technology. This new technology is enabling ever-higher levels of production efficiency and sustainability. It also has the potential to dramatically influence sustainable development of spiral bevel gear product. Sufficient guidance, in this respect, is lacking in the scholarly or practitioner literature. In this work, focusing on spiral bevel gear production, we further examine dry-cutting technology in terms of application and sustainability implications. We introduce bending fatigue life design into gear dry-grinding by considering various economic, environmental and social attributes. We also develop bending fatigue life oriented tooth flank dry-grinding tool modification by incorporating a high life cycle assessment (LCA) and low life cycle cost (LCC). At first, data-driven tool modification design relating to dry-grinding tool parameters is developed. Then, finite element method (FEM) based root bending fatigue assessment is performed by using the multiaxial fatigue damage model. Here, with data-driven finite element modeling, simulated loaded tooth contact analysis (SLTCA) is employed to root bending stress determination. They were introduced into the multiaxial fatigue damage model based fatigue life assessment. Moreover, focusing on the minimum fatigue at dangerous point namely fatigue limit, bending fatigue life oriented dry-grinding tool modification model considering geometric accuracy is established. Then, data-driven adaptive feedback and optimization is performed by using tool modification. The final output tool modification parameters can be used to obtain the cleaner manufacturing of gear product both flank geometric accuracy and fatigue life. Finally, the numerical instance was given to verify the proposed method. By building a bridge between the tool parameters in early design and the fatigue life in actual transmission, the proposed method can get a significantly important access to cleaner manufacturing of spiral bevel gears product.

Analytical procedure for the optimization of plastic gear tooth root

The increasing diffusion of plastic gears in many industrial sectors makes it necessary to have a better knowledge of the bending strength properties of teeth in order to ensure an adequate design process. However, technological development has not gone hand in hand with the standard development necessary to carry out an optimized design process. To date, the specific standards for plastic gears refer mainly to design and calculation methods for the metal gears. So, the possible optimization strategy for the tooth root geometrical profile is also driven by the typical metal production processes. This paper presents an innovative approach, supported by an analytical procedure, to design optimized plastic gear wheels. The results obtained by the analytical approach is compared to the standardized one to highlight similarities and discrepancies. The goodness of the proposed results is verified also through finite element numerical analysis. The goal of the paper is to provide the designer with an analytical approach useful to optimize the root geometry design of plastic gears for lower root bending stress.

캠 랙 피니언 시스템의 강도설계

The cam rack pinion (CRP) system takes the traditional rack and pinion concept and advances it by replacing the pinion teeth with bearing supported rollers that engage a trochoid rack tooth profile. The bearing supported rollers eliminate the sliding friction of the conventional rack and pinion with smooth rolling friction that yields a 99% efficient rotary to the linear motion conversion. In this study, the fatigue failure life of a CRP system was investigated. First, we investigated the difference in rack tooth profiles by changing the profile shift coefficient of CRP systems. Next, the Hertz contact stresses and tooth root bending stresses were analyzed by varying the profile shift coefficient to determine the tooth fatigue characteristics. It was found that the variation in the contact force with increasing the profile shift coefficient was insignificant, but the Hertz contact and tooth root bending stresses decreased significantly.