A global synthesis approach for optimizing the meshing performance of hypoid gears based on a swarm intelligence algorithm

PurposeThis paper aims to present a comparison of numerical methods for determining the contact pattern of Gleason-type bevel gears. The mathematical model of tooth contact analysis and the finite element method were taken into consideration. Conclusions have been drawn regarding the usefulness of the considered methods and the compatibility of results. The object of the analysis was a bevel gear characterised by an 18:43 gear ratio and arc tooth line, and manufactured according to the spiral generated modified-roll method.Design/methodology/approachThe mathematical model of tooth contact analysis consists of both the mathematical model of tooth generating and the mathematical model of operating gear set. The first model is used to generate tooth flanks of the pinion and the ring gear in the form of grids of points. Then, such tooth surfaces are used for the tooth contact analysis performed with the other model. It corresponds to the no-load gear meshing condition. The finite element method model was built on the basis of the same tooth flanks obtained with the former model. The commercial finite element method software Abaqus was used to perform two instances of the contact analysis: a very light load, corresponding to the former no-load condition, and the operating load condition. The results obtained using the two models, in the form of the contact pattern for no-load condition, were compared. The effect of heavy load on contact pattern position, shape and size was shown and discussed.FindingsThe mathematical models correctly reproduce the shape, position and size of the contact pattern; thus, they can be reliably used to assess the quality of the bevel gear at the early stage of its design.Practical implicationsDetermination of the correct geometry of the flank surfaces of the gear and pinion teeth through the observation of contact pattern is a fundamental step in designing of a new aircraft bevel gear.Originality/valueA possibility of the independent use of the mathematical analysis of the contact pattern has been shown, which, thanks to the compatibility of the results, does not have to be verified experimentally.

A new method of modified optimization of double helical gears is proposed based on reducing vibration and noise and raising machining efficiency. Firstly, the straight profile of rack-cutter edge is replaced by three segment parabolas, and the equation of the rack-cutter profile is ultimately represented in the rack-cutter surface. Secondly, the physical and mathematical model of tooth contact analysis and loaded tooth contact analysis of double helical gears are introduced and then the loaded transmission errors are obtained. The optimal modification parameters are achieved based on the minimum amplitude of loaded transmission error. Finally, a set of equipment for measuring loaded transmission error, the analysis software platform and the test-bed of vibration and noise are designed, respectively. The feasibility of the method of modified optimization is verified. Compared with vibration and noise before modification, those after modification averagely decrease 18% and 2.7 dB.

In order to reduce the sensitivity of straight bevel gear drives to misalignment, a new geometry of such gear drives is proposed in longitudinal direction. Point contact instead of line contact of tooth surfaces is achieved by longitudinal crowning of pinion tooth surface. The tooth surface modeling and tooth contact analysis (TCA) of straight bevel gear drives have been established. TCA program of a pair of straight bevel gears was performed in MATLAB and tooth bearing contact and transmission errors were obtained.

Value function approximation, such as Q-learning, is widely used in the discrete control rather than the continuous one because the optimal action in the discrete setting is more easily selected. Optimizing the action is a non-convex optimization problem with respect to the complex value function. Some notable studies simplify the non-convex optimization problem by assuming the value function as quadratic in the actions or by discretizing the action space. However, the performance of the output policy will decline if these studies’ premises do not hold. In order to address the problem, we propose a framework that combines swarm intelligence algorithms with value-based Reinforcement Learning, where the swarm intelligence algorithms are employed to search for the optimal action with respect to the state and the value function. To ensure the correctness of this framework, we conditionally claim the convergence rate of swarm intelligence algorithms with high probability. We then implement it by searching the batch optimal actions to various states on the GPU platform for the batch training. Furthermore, we employ the population-based atomic actions for the compatibility with the existing related work about solving discrete control problems. Four classical control models and four robot simulation environments are utilized in the comparisons. According to empirical results, our framework outputs a policy comparable with that of the policy-based algorithms by 10% timesteps in the continuous control. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the exploration-exploitation dilemma of Reinforcement Learning to solve continuous control tasks. To balance the exploration and exploitation, the stochastic exploration and the prioritized exploration are roughly two feasible ways, where the prioritized one is a better choice due to the higher data efficiency than the stochastic one, e.g. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon$</tex-math> </inline-formula> -greedy. Normally, the prioritized exploration works well in the value-based Reinforcement Learning algorithms rather than the policy-based ones; meanwhile, the policy-based algorithms are more suitable to continuous control tasks than the value-based ones. To tackle this conflict, we especially design a particle swarm optimization to maximize the Q-value of action in Q-learning. Our design can be hybridized by various swarm intelligence and value-based Reinforcement Learning algorithms. Also, it can be embedded in most intelligent control systems easily. The aim of this study is to solve the continuous control tasks by value-based algorithms as the first step of applying the prioritized exploration. The simulative results verify the effectiveness and efficiency of our design.

AGMA and ISO standards consider different types of deviation of gear geometry caused during manufacturing processes of cylindrical gears through a system of accuracy grades, in which each grade corresponds to deviations within a certain range. The gear drive behavior is affected by the level and type of deviation, causing transmission errors, shifts in the bearing contact, increase of contact and bending stresses, and vibration and noise. The main goal of this paper is the investigation of the influence of manufacturing errors on transmission errors and contact and bending stresses, and the determination of the corresponding flank modifications required for their compensation. The contact pattern and the unloaded function of transmission errors are obtained through application of tooth contact analysis (TCA). Application of finite element analysis (FEA) allows for contact and bending stresses to be determined along one or two cycles of meshing. Surface deformations are used for determination of the loaded function of transmission errors in which its peak-to-peak value is related to the vibration and noise response in operating conditions. Several numerical examples considering different levels and types of manufacturing error and their compensation through predefined flank modifications are presented.

Beveloid gears are widely applied in fields like ships, automobiles and industrial precision transmissions. In this paper, the formulas of the beveloid gear tooth surface used in marine transmissions were derived and a mesh model for the intersected beveloid gear pair was setup. Then loaded tooth contact analysis was performed using the finite element method considering the coupling of the assembly errors and the elastic deformation of tooth surface. Through the analysis, the influences of assembly errors on contact patterns, mesh force and tooth surface deformations were investigated. In a further step, the tooth profile modifications were performed to alleviate the edge contact and a subsequent major improvement of the mesh condition was obtained. Finally, loaded tooth contact experiments for marine gearboxes with small shaft angle were conducted. The tested results showed good correlation with the computed results. This work may provide some value for the practical design aiming at improved contact characteristics of the beveloid gears with intersected axes.

This article focuses on improving the meshing characteristics of a complementary curvilinear gear set generated by complementary rack cutters. Firstly, the mathematical model of a complementary curvilinear gear set was developed based on the theory of gearing. Tooth contact analysis (TCA) was conducted to explore the influences of assembly errors on the transmission error (TE) and contact pattern. A stable point contact was successfully introduced to replace the original line contact in the improved complementary curvilinear gear set, resulting in less sensitivity to assembly errors. Finally, finite element analysis (FEA) was also utilized to calculate contact stress of the original and improved complementary curvilinear gear set.

Undercutting and contact characteristics of longitudinal cycloidal spur gears generated by the dual face-hobbing method

Measurement and mathematical model of convexo-concave Novikov gear mesh

In this paper, the circular-arc curvilinear tooth gear drive is proposed. The gear and pinion tooth surfaces are generated by two complemented circular-arc rack cutters with curvilinear tooth-traces. According to the theory of gearing, the mathematical model of the proposed gear is developed. The tooth contact analysis technique is utilized to investigate the kinematical errors of circular-arc curvilinear tooth gear drives under different assembly errors. Contact patterns of the circular-arc curvilinear tooth gear drive are simulated by the developed computer-aided tooth contact analysis programs and surface topology method. Numerical examples are presented to show the kinematical errors of the circular-arc curvilinear tooth gear set under different assembly conditions. Relations among the circular-arc tooth profile, curvilinear tooth-trace, contact ratio, contact pattern, and kinematical error are also demonstrated by numerical examples.

Solid particle erosion prediction in elbows based on machine learning and swarm intelligence algorithm

A clustering algorithm is a solution for grouping a set of objects and for distribution centre location problems. But the common K-means clustering algorithm may give local optimal solutions. Swarm intelligent algorithms simulate the social behaviours of animals and avoid local optimal solutions. We employ three swarm intelligent algorithms to avoid these solutions. We propose a new algorithm for the clustering problem, the fruit-fly optimization K-means algorithm (FOA K-means). We designed a distribution centre location problem and three clustering indicators to evaluate the performance of algorithms. We compare the algorithms of K-means with the ant colony optimization algorithm (ACO K-means), particle swarm optimization algorithm (PSO K-means), and fruit-fly optimization algorithm. We find K-Means modified by the fruit-fly optimization algorithm (FOA K-means) has the best performance on convergence speed and three clustering indicators, compactness, separation, and integration. Thus, we can apply FOA K-means to improve the distribution centre location solution and the efficiency for distribution in the future.

Modified swarm intelligence algorithms for the pharmacy duty scheduling problem

This paper proposes a novel approach for distribution network reconfiguration (DNR) using swarm intelligence (SI) algorithms enhanced with shrinking encircling behavior to address non-convex optimization challenges in large-scale distribution networks. The methodology is applied to Taipower's distribution network, utilizing data from the Distribution Mapping Management System (DMMS) and Distribution Dispatch Control Center (DDCC) to create an OpenDSS-based simulation model. The proposed SI algorithms determine the optimal statuses of 4-way and 2-way feeder switches to minimize energy losses while adhering to constraints on bus voltage and feeder currents. Comparative evaluations of multiple SI algorithms identify those with shrinking encircling behavior as the most effective for the high-dimensional DNR problem. Additionally, a graph-theory-based rule is introduced to ensure radial network topology during reconfiguration. Numerical results demonstrate that the proposed approach reduces energy losses by 22.9%, achieves uniform feeder current distribution, and improves operational efficiency. These findings underscore the potential of this method to enhance modern distribution network management.

During the last three decades, the water resources engineering field has received a tremendous increase in the development and use of meta-heuristic algorithms like evolutionary algorithms (EA) and swarm intelligence (SI) algorithms for solving various kinds of optimization problems. The efficient design and operation of water resource systems is a challenging task and requires solutions through optimization. Further, real-life water resource management problems may involve several complexities like nonconvex, nonlinear and discontinuous functions, discrete variables, a large number of equality and inequality constraints, and often associated with multi-modal solutions. The objective function is not known analytically, and the conventional methods may face difficulties in finding optimal solutions. The issues lead to the development of various types of heuristic and meta-heuristic algorithms, which proved to be flexible and potential tools for solving several complex water resources problems. This paper provides a review of state-of-the-art methods and their use in planning and management of hydrological and water resources systems. It includes a brief overview of EAs (genetic algorithms, differential evolution, evolutionary strategies, etc.) and SI algorithms (particle swarm optimization, ant colony optimization, etc.), and applications in the areas of water distribution networks, water supply, and wastewater systems, reservoir operation and irrigation systems, watershed management, parameter estimation of hydrological models, urban drainage and sewer networks, and groundwater systems monitoring network design and groundwater remediation. This paper also provides insights, challenges, and need for algorithmic improvements and opportunities for future applications in the water resources field, in the face of rising problem complexities and uncertainties.