Multi-Objective Optimization of Planetary Gearbox with Adaptive Hybrid Particle Swarm Differential Evolution Algorithm

This paper considers the problem of constrained multi-objective non-linear optimization of planetary gearbox based on hybrid metaheuristic algorithm. Optimal design of planetary gear trains requires simultaneous minimization of multiple conflicting objectives, such as gearbox volume, center distance, contact ratio, power loss, etc. In this regard, the theoretical formulation and numerical procedure for the calculation of the planetary gearbox power efficiency has been developed. To successfully solve the stated constrained multi-objective optimization problem, in this paper a hybrid algorithm between particle swarm optimization and differential evolution algorithms has been proposed and applied to considered problem. Here, the mutation operators from the differential evolution algorithm have been incorporated into the velocity update equation of the particle swarm optimization algorithm, with the adaptive population spacing parameter employed to select the appropriate mutation operator for the current optimization condition. It has been shown that the proposed algorithm successfully obtains the solutions of the non-convex Pareto set, and reveals key insights in reducing the weight, improving efficiency and preventing premature failure of gears. Compared to other well-known algorithms, the numerical simulation results indicate that the proposed algorithm shows improved optimization performance in terms of the quality of the obtained Pareto solutions

Multi-Objective Optimization of Planetary Gearbox with Adaptive Hybrid Particle Swarm Differential Evolution Algorithm

This paper considers the problem of constrained multi-objective non-linear optimization of planetary gearbox based on hybrid metaheuristic algorithm. Optimal design of planetary gear trains requires simultaneous minimization of multiple conflicting objectives, such as gearbox volume, center distance, contact ratio, power loss, etc. In this regard, the theoretical formulation and numerical procedure for the calculation of the planetary gearbox power efficiency has been developed. To successfully solve the stated constrained multi-objective optimization problem, in this paper a hybrid algorithm between particle swarm optimization and differential evolution algorithms has been proposed and applied to considered problem. Here, the mutation operators from the differential evolution algorithm have been incorporated into the velocity update equation of the particle swarm optimization algorithm, with the adaptive population spacing parameter employed to select the appropriate mutation operator for the current optimization condition. It has been shown that the proposed algorithm successfully obtains the solutions of the non-convex Pareto set, and reveals key insights in reducing the weight, improving efficiency and preventing premature failure of gears. Compared to other well-known algorithms, the numerical simulation results indicate that the proposed algorithm shows improved optimization performance in terms of the quality of the obtained Pareto solutions

Multi-objective optimization of planetary gearboxes based on adaptive hybrid metaheuristic algorithms

Планетарни преносници спадају у групу механичких зупчастих преносника, који су широко заступљени за трансформацију и пренос снаге првенствено због компактности конструкције, високе поузданости и степена искоришћења. Полазећи од функције, коју треба да испуни у оквиру неке конструкције, као и све строжих захтева у погледу перформанси преносника, предмет истраживања ове докторске дисертације је развој вишекритеријумског оптимизационог модела планетарног преносника. Развијени модели који задовољавају низ строгих захтева у погледу: компактности конструкције, минимизације губитака у преносу снаге, равномерности расподеле оптерећења, као и поузданости у различитим експлоатационим условима. За постављени проблем вишекритеријумске оптимизације планетарног преносника формиране су одговарајуће критеријумске функције и дефинисан је низ функционалних ограничења, као и одговарајући домени примене свих релевантних величина зупчастих парова са спољашњим и унутрашњим озубљењем и планетарног преносника као сложеног система, у циљу несметане монтаже, поузданог рада и спрезања зупчастих парова. У оквиру докторске дисертације, развијен је одговарајући механички модел за одређивање степена искоришћења истовремено спрегнутих зупчастих парова, у зависности од њихових геометријских параметара, као и од услова подмазивања. Критеријумске функције формираног вишекритеријумског оптимизационог проблема су нелинеарне и неконвексне па се глобално оптимално решење не може нумерички одредити конвенционалним методама оптимизације. Према томе, у циљу решавања овог комплексног оптимизационог проблема потребно је применити метахеуристичке оптимизационе алгоритме. Дакле, постоји стална потреба за побољшањем постојећих и развојем нових метахеуристичких алгоритама и то развојем адаптивних техника за подешавање управљачких параметара и хибридизацијом алгоритама. У складу са тим, у оквиру докторске дисертације детаљно су разматрани метахеуристички алгоритми, који припадају групи еволутивних алгоритама, као што су алгоритам диференцијалне еволуције (Differential Evolution, DE), генетски алгоритам (Genetic Algorithm, GA) као и алгоритми инспирисани биолошким системима у природи, и то алгоритам оптимизације ројем честица (Partical Swarm Optimization, PSO). У циљу отклањања недостатака метахеуристичких алгоритама који се јављају током оптимизационог процеса, алгоритми предложени у оквиру дисертације су модификовани кроз: развој адаптивних механизама за подешавање вредности управљачких параметара, и хибридизацију адекватних алгоритама. У циљу формирања ефикасног хибридног алгоритма, процесу хибридизације је претходила детаљна нумеричка симулација и статистичка анализа перформанси разматраних алгоритама. Према томе, хибридизацијом алгоритама постигнуто је адекватно искоришћење предности једног и истовремена елиминација недостатака другог алгоритма...Planetary gearboxes have a wide application in the field of transformation and transmission of power from the drive to the working machine, due to the compact structure, high reliability and efficiency. Due to the increasingly stringent performance requirements, which planetary gearboxes must satisfy, the research in this dissertation is focused on the problem of multi-objective non-linear optimization of planetary gearbox based on hybrid metaheuristic algorithm, which satisfies a number of strict requirements, such as: compact construction, minimization of power loss, load distribution and reliability in different operating conditions. In this work the formulations of the objective functions for the considered multiobjective optimization problem of planetary gearbox have been outlined along with the appropriate constraints. The formulated constraints have been analyzed and appropriate domains of practical applications of internal and external gear pairs have been formulated, with the aim to ensure proper working, mounting and meshing of considered gears. Furthermore, the theoretical formulation and numerical procedure for the calculation of the planetary gearbox power efficiency has been developed in this work. However, the objective functions and developed constraints of the considered Multiobjective planetary gearbox optimization problem are nonlinear and multimodal functions, and therefore the global optimal solution cannot be obtained using the conventional optimization methods. Therefore, in order to solve this multiobjective and complex optimization problem, the research in this dissertation is focused on metaheuristic optimization algorithms, which belong to the group of evolutionary algorithms, including: differential evolution algorithm and genetic algorithm, as well as the algorithms inspired by the biological systems, such as particle swarm optimization algorithm. To overcome difficulties in solving complex optimization problems, in this thesis the considered algorithms are modified with the development of adaptive techniques for setting the values of control parameters and hybridization of algorithms. In order to create an effective hybrid algorithm, the hybridization process has been preceded by extensive numerical simulations and statistical analysis of advantages and disadvantages of each algorithm. Therefore, the proposed hybridization of algorithms and introduction of adaptive control parameters can successfully combine the advantages and avoid disadvantages of each algorithm. In this way, the proposed modifications successfully combine the advantages of each algorithm and avoid their disadvantages, thus significantly expanding the scale of implementation of the proposed algorithms for complex optimization problems..

Efficiency analysis of planetary gears

By kinematic combinations of toothed pairs with external and internal contacts, we can obtain planetary gears with a considerably improved performance than the corresponding ones with fixed axes, as well as planetary gears with notably poor performance regarding the efficiency. In regard to that, the reference literature and papers almost regularly emphasize that planetary gears, under the same technical conditions, have a smaller mass and a higher degree of efficiency than the ones with fixed axes. The main aim of this paper is to examine the above statement and to determine the scope of the gear ratios in which the planetary gears are more suitable than the fixed axes gears

Typified machine parts series load capacity analysis from aspect of structural strength

Application of typization in the process of designing mechanical sub-assemblies and assemblies is one of the ways to reduce the cost of production. Therefore, nowadays, not only roller bearings, bolts, wedges, etc. are produced as standard machine elements but, by the usage of typization, a production of a large series of typified subassemblies and assemblies, such as electric motors, pumps, power transmissions, etc., is increasing. Increased application of typified parts, sub-assemblies and assemblies in mechanical systems requires an increase in their safety and reliability during operation. Accordingly, in this paper, the load capacity of the typified machine parts series from the aspect of their structural strength is analyzed. It has been shown that there is a scattering of calculated results of the safety factor of members of the typified series from the aspect of the structural strength. The paper presents a proposal for a calculation methodology by which the mentioned scattering of the results of load capacity of typified machine parts series can be significantly reduced

Multi-objective optimization of planetary gearboxes based on adaptive hybrid metaheuristic algorithms

Планетарни преносници спадају у групу механичких зупчастих преносника, који су широко заступљени за трансформацију и пренос снаге првенствено због компактности конструкције, високе поузданости и степена искоришћења. Полазећи од функције, коју треба да испуни у оквиру неке конструкције, као и све строжих захтева у погледу перформанси преносника, предмет истраживања ове докторске дисертације је развој вишекритеријумског оптимизационог модела планетарног преносника. Развијени модели који задовољавају низ строгих захтева у погледу: компактности конструкције, минимизације губитака у преносу снаге, равномерности расподеле оптерећења, као и поузданости у различитим експлоатационим условима. За постављени проблем вишекритеријумске оптимизације планетарног преносника формиране су одговарајуће критеријумске функције и дефинисан је низ функционалних ограничења, као и одговарајући домени примене свих релевантних величина зупчастих парова са спољашњим и унутрашњим озубљењем и планетарног преносника као сложеног система, у циљу несметане монтаже, поузданог рада и спрезања зупчастих парова. У оквиру докторске дисертације, развијен је одговарајући механички модел за одређивање степена искоришћења истовремено спрегнутих зупчастих парова, у зависности од њихових геометријских параметара, као и од услова подмазивања. Критеријумске функције формираног вишекритеријумског оптимизационог проблема су нелинеарне и неконвексне па се глобално оптимално решење не може нумерички одредити конвенционалним методама оптимизације. Према томе, у циљу решавања овог комплексног оптимизационог проблема потребно је применити метахеуристичке оптимизационе алгоритме. Дакле, постоји стална потреба за побољшањем постојећих и развојем нових метахеуристичких алгоритама и то развојем адаптивних техника за подешавање управљачких параметара и хибридизацијом алгоритама. У складу са тим, у оквиру докторске дисертације детаљно су разматрани метахеуристички алгоритми, који припадају групи еволутивних алгоритама, као што су алгоритам диференцијалне еволуције (Differential Evolution, DE), генетски алгоритам (Genetic Algorithm, GA) као и алгоритми инспирисани биолошким системима у природи, и то алгоритам оптимизације ројем честица (Partical Swarm Optimization, PSO). У циљу отклањања недостатака метахеуристичких алгоритама који се јављају током оптимизационог процеса, алгоритми предложени у оквиру дисертације су модификовани кроз: развој адаптивних механизама за подешавање вредности управљачких параметара, и хибридизацију адекватних алгоритама. У циљу формирања ефикасног хибридног алгоритма, процесу хибридизације је претходила детаљна нумеричка симулација и статистичка анализа перформанси разматраних алгоритама. Према томе, хибридизацијом алгоритама постигнуто је адекватно искоришћење предности једног и истовремена елиминација недостатака другог алгоритма...Planetary gearboxes have a wide application in the field of transformation and transmission of power from the drive to the working machine, due to the compact structure, high reliability and efficiency. Due to the increasingly stringent performance requirements, which planetary gearboxes must satisfy, the research in this dissertation is focused on the problem of multi-objective non-linear optimization of planetary gearbox based on hybrid metaheuristic algorithm, which satisfies a number of strict requirements, such as: compact construction, minimization of power loss, load distribution and reliability in different operating conditions. In this work the formulations of the objective functions for the considered multiobjective optimization problem of planetary gearbox have been outlined along with the appropriate constraints. The formulated constraints have been analyzed and appropriate domains of practical applications of internal and external gear pairs have been formulated, with the aim to ensure proper working, mounting and meshing of considered gears. Furthermore, the theoretical formulation and numerical procedure for the calculation of the planetary gearbox power efficiency has been developed in this work. However, the objective functions and developed constraints of the considered Multiobjective planetary gearbox optimization problem are nonlinear and multimodal functions, and therefore the global optimal solution cannot be obtained using the conventional optimization methods. Therefore, in order to solve this multiobjective and complex optimization problem, the research in this dissertation is focused on metaheuristic optimization algorithms, which belong to the group of evolutionary algorithms, including: differential evolution algorithm and genetic algorithm, as well as the algorithms inspired by the biological systems, such as particle swarm optimization algorithm. To overcome difficulties in solving complex optimization problems, in this thesis the considered algorithms are modified with the development of adaptive techniques for setting the values of control parameters and hybridization of algorithms. In order to create an effective hybrid algorithm, the hybridization process has been preceded by extensive numerical simulations and statistical analysis of advantages and disadvantages of each algorithm. Therefore, the proposed hybridization of algorithms and introduction of adaptive control parameters can successfully combine the advantages and avoid disadvantages of each algorithm. In this way, the proposed modifications successfully combine the advantages of each algorithm and avoid their disadvantages, thus significantly expanding the scale of implementation of the proposed algorithms for complex optimization problems..

Planetary gear pair design using metaheuristic algorithms

This paper considers the problem of formulating the non-linear optimization model for determining the optimal parameters of planetary gearbox, which is solved using a metaheuristic optimization algorithm. To determine the optimal parameters of the planetary gearbox it is necessary to formulate complex objective functions and minimize them, which is often a conflicting problem. To solve this complex optimization problem, in this paper we propose to employ metaheuristic algorithms, which characterize pseudo-randomness and the ability to find the global optimal solution to the multimodal optimization problems. The proposed metaheuristic method is based on the Genetic Algorithm (GA) which is hybridized with the local-search Nelder-Mead method. For the considered optimization problem, the appropriate software is implemented in the MATLAB software package, to verify the results. Inside the considered optimization module, we have defined the appropriate objective functions and constraints which determine the construction of the planetary gearbox. The optimal parameters of the planetary gearbox obtained using the proposed metaheuristic algorithm are compared with the results obtained using several well-known algorithms in the literature. The simulation results of the proposed optimization method indicate a significant improvement in planetary gearbox performance compared to the parameters obtained with well-known algorithms

TDOA based approach for accurate target localization based on hybrid genetic algorithm

Accurate localization of target based on time difference of arrival (TDOA) measurements is of crucial importance in a large number of different military and civil applications, especially in security systems, radars, sonars etc. This paper focuses on the determining the position of a target from a set of TDOA measurements obtained on several receivers whose positions are known. The considered target localization problem is formulated as the optimization problem, where the corresponding objective function is obtained based on least squares (LS) method. Due to the complexity of the considered problem, the resulting objective function is highly nonlinear and multimodal. Therefore, to solve this complex optimization problem this paper proposes the hybridization of Genetic Algorithm (GA) with well-known Gauss-Newton (GN) method. The performance of considered hybrid algorithm is investigated and compared to well-known conventional optimization algorithms in solving the considered TDOA based localization problem. The simulation results of the proposed optimization method indicate a significant improvement in localization accuracy compared to well-known algorithms

Optimization of planetary gears and effects of thin-rimed gear on fillet stress

Planetary gears take a very significant place among the gear transmissions, and they are widely used in military and civil industry applications such as marine vehicles, aircraft engines, helicopters and heavy machinery. Planetary gears are complex mechanisms which can be decomposed into external and internal gears with the corresponding interaction, which requires geometrical conditions in order to perform the mounting and an appropriate meshing of the gears during their work. Planetary gears have a number of advantages as compared to the transmission with fixed shafts such as a compact design, with co-axial shafts, high power density and higher efficiency, which is achieved by reducing gear weight using thin-rimed gears. The purpose of this paper is to present the optimization model for the planetary gears, where the objective function is the weight of gears, and functional constraints imposed upon their respective structural design. Hence, the objective is to minimize rim thickness of the gear in order to achieve high-performance power transmission and minimize weight. This paper presents the results of an investigation with finite element analysis (FEM) into the effects of thin-rimmed gear geometry on the root fillet stress distribution

Interference analysis of internal involute spur gear pair

In this paper, we have considered tip interference and radial interference of internal involute spur gears. Equations of tooth profiles are provided and operating constraints for internal gears are defined. The undercutting and interference of the teeth profiles are the main problems for the practical application of internal gears in planetary gear trains, which requires particular attention in the design of planetary gear trains. The problem of avoiding interference of internal involute spur gears is especially challenging. Therefore, it is necessary to create the corresponding geometric models and express the above requirements by the corresponding functional constraints to verify the engagement condition. The developed geometrical model of the tooth surfaces is most helpful in the analysis of meshing interference, contact, and stress analyzing, manufacturing, measuring, and optimizing internal gear sets. The numerical results are tested by computerized simulation of the meshing of internal involute spur gears