The determination of optimal design of the planetary gear train with a lightweight, a short center distance, and a high efficiency is an important issue in the preliminary design of power transmission systems. Conventional and traditional methods have been widely used in optimization. They are deterministic and limited to solve some mechanical problems with several variables and constraints. Therefore, some optimization methods have been developed, such as the nonconventional method, the genetic algorithm (GA). This paper describes a multiobjective optimization for the epicyclical gear train system using the GA. It is aimed to obtain the optimal dimensions for epicyclical gear components like a module, number of teeth, the tooth width, the shaft diameter of the gears, and a performed efficiency under the variation of operating mode of PGT system. The problem is formulated under the satisfaction of assembly and balance constraints, bending strength, contact strength of teeth, and other dimension conditions. The mathematical model and all steps of the GA are presented in detail.
Gear trains are used in most types of machinery and vehicles for power transmission. Epicyclical gear train takes a very significant place among the gear transmissions which are used in many branches of industry such as automobile, aerospace, machine tools, and turbines. It has several advantages like a smaller envelope size than a parallel shaft for the same power, low weight, coaxial shafts resulting in more compact installation, and high transmission ratio speed and efficiency [
The design of the planetary gear train is highly complicated and specific. This complexity leads to many design variables, mathematical formulations, constraints, and many influencing factors. Using conventional or traditional optimization techniques to solve a design problem, they could not provide an optimum result according to the complex shape and geometry of gears and many factors. Therefore, a stochastic approach genetic algorithm (GA) is applied to solve this problem and to obtain a satisfactory result as well [
Yan and Lai [
The objective of this study stems from the idea of providing an optimal design of the epicyclical gear train by the genetic algorithm approach. It would offer a global optimization of the geometrical parameters of the epicyclical gear train system as a module, number of teeth, tooth width, and shaft diameter, in order to obtain light weight, short center distance, and maximal efficiency. In addition, the specificity of the present work is that it takes into account the variation of the gear ratio or the operating modes of the PGT.
Originally developed by Holland, the genetic algorithm (GA) is a robust technique based on the natural selection and genetic production mechanism. This algorithm works with a group of possible solutions within a search space instead of a single solution, as described in gradient optimization methods [
Genetic algorithm is referred to as a search method of the optimal solution to simulate Darwin's genetic selection and biological evolution process. In fact, it is a series of random iterations and evolutionary computations which simulate the process of selection, crossover, and mutation occurred in natural selection and population genetic as described in Figure
Flowchart of GA for solving the problem.
The model of the application was a single stage of an epicyclical gear train type I with three planets as shown in Figure
Simple epicyclical gear drive type I [
Many factors and parameters like torque, material, tooth width, input speed, module, and others have been applied to design optimization as described in Table
Description of parameters and variables [
Parameters  Symbol  Coefficient value 

Tooth width 

[12 120] 
Module 

[2 20] 
Tooth number of sun gear 

[17 30] 
Tooth number of planet gear 

[17 50] 
Diameter of sun shaft 

[10 30] 
Diameter of planet shaft 

[10 40] 
Input power 

7.5 
Input speed 

1500 
Density material of standard steel 

7850 
Helix angle factor 

1 
Elastic coefficient 

189.98 
Contact ratio coefficient 

0.94 
Product of load factor 

1.75 
Tooth form factor 

2 
Stress concentration coefficient 

1.73 
Contact ratio coefficient 

1.08 
Allowable contact stress 

1100 
Allowable bending stress 

460 
Allowable shearing stress of shafts 

30.67 
The objective function is a quantity to be minimized or maximized by exploring a search space under the imposed constraints. In this paper, three objective functions have been defined as follows:
The total weight of the planetary gear train has been presented by the envelope of the three components: sun gear, planet gear, and ring gear. For the number of teeth, just the number of teeth of the sun and the planet will be considered as variables. The number of teeth
The minimization of the center distance determines smaller gears which would require less material and cost to make and less space to operate it. The following equation presents the center distance of the epicyclical gear train system [
The last objective function is the efficiency, which is more detailed in the part bellow and defined as follows:
This optimization has been formulated with the satisfaction of many constraints. The bending strength, the contact strength, and the dynamic factor
Dimensional constraint:
Balance and assembly condition for PGT:
Bending strength of teeth:
Contact strength of teeth: The expression of the contact stress in equation (
The condition of shaft diameters is defined in equation (
Therefore, the previous expression shows that eight constraints must be introduced in the objective functions. The syntax of the constraints in the genetic algorithm has been presented by the function
Amount of iterations and variables in a function are coded in bit patterns as shown in Table
Coding of variables.
Design variable vectors  Vectors  Randomized binary digits  String length 

Tooth width ( 
X(1)  111110101  9 
Module ( 
X(2)  1011011  7 
Number of teeth_sun ( 
X(3)  1101  4 
Number of teeth_planets ( 
X(4)  101011  6 
Shaft diameter_sun ( 
X(5)  11111  5 
Shaft diameter_planet ( 
X(6)  1110100  7 
A single chromosome  11111010110110111101101011111111110100  38 
Here,
The planetary gear train is a system with one configuration and many combinations. Indeed, it can have the case where at least one of the parts, planetary, ring, or the planet carrier, is fixed and the two others are either the input or the output. In this case, there are six different operating modes or transmission ratios and a different expression of transmission efficiency [
Basic circuit transmission relation diagram.
The transmission relation diagram of SPGT with one degree of freedom (1 DOF) has six basic cases, and their efficiency expression is given in Table
SPGT basic gear transmission diagram and efficiency.
Mode  Transmission diagram 

Transmission ratio  Efficiency 

1 




2 




3 




4 




5 




6 




Based on the kinematic equation of the gear system, the Willis formula, the torque equation, and the power balance equation have been obtained, as expressed in equations (
Firstly, assuming the ring gear
Table
Table
Optimization results of the epicyclical gear train design.
Function variables  Mode 1  Mode 2  Mode 3  Mode 4  Mode 5  Mode 6 

Number of teeth of the sun gear  21  30  21  27  21  21 
Number of teeth of the planet gear  18  18  18  18  18  18 
Number of teeth of the ring gear  57  66  57  63  57  57 
Module  2  2  2  2.75  2  2 
Shaft diameter (sun)  20  30  18  18  23  20 
Shaft diameter (planet)  17  17  21  21  20  11 
Width coefficient of tooth  12.00  12.17  12.50  12.14  13.70  13.01 
Tooth width  24.00  24.33  25.00  33.38  27.40  26.03 
Transmission ratio  −2.71  −0.45  0.26  3.34  0.73  1.37 
Weight of PGT  0.276  0.312  0.280  0.664  0.375  0.342 
Center distance  39  39  39  61.88  39  39 
For both operations of the PGT as speed reducer or speed multiplier, the center distance was the same 39 mm, but for the best weight, there is a margin of a difference of 66 g.
For the design parameters, the big size has been found in the second mode with tooth number of the sun gear of 30 and with a module of 2.75 mm in the 4th mode. For the rest of the modes, the value of the tooth number of the sun and planet gears was the same with a value of 21 and 18, respectively. The shaft diameter of the sun and planet gears has been found to be equal, respectively, 20 mm and 17 mm for the first mode and 20 mm and 11 mm for the last mode.
The average of the tooth width was 26.69 mm in different transmission ratio, but the best one was in the first mode with a value of 24 mm in the case of the PGT as a speed reducer and 26 mm in the 6th mode of PGT as a speed multiplier.
The figures below show the variation of the fitness functions during generations and transmission ratio variation. The first and the third mode converge rapidly to the final solution of the minimal weight than the rest configuration of the epicyclical gear train (see Figure
Variation of the optimum fitness for the weight with different transmission ratios.
Best solution of the center distance in six cases of the PGT system.
A Pareto front, presented as a plot of the weight and the center distance of the planetary gear train, gives a quantitative description of the compromise between weight and size in both cases of the operation mode of the epicyclical gear train system as a speed reducer and a speed multiplier (see Figures
Pareto front of the GA multiobjective optimization of the epicyclic gear train case of a speed multiplier.
Pareto front of the GA multiobjective optimization of the epicyclic gear train case of a speed reducer.
As mentioned beforehand in analysis efficiency, a planetary gear train is a system with one configuration and many combinations. Therefore, six results of the efficiency were obtained as shown in Table
Optimization results of the epicyclical gear train efficiency by the GA.
Types  1  2  3  4  5  6 

Range  −2.71  −2.2  −2.71  −2.33  −2.71  −2.71 
Transmission ratio  −2.71  −0.45  0.26  3.34  0.73  1.37 
Efficiency %  95.00  96.56  95.00  98.49  96.29  98.60 
Optimum results for efficiency in each type of PGT.
Gears are the most important elements of the power transmission system. This paper presents one of the powerful metaheuristic approaches to optimization of the epicyclical gear system with the spur gear by means of the genetic algorithm. The objective of this optimization has focused on the minimization of the hall weight of the PGT, minimization of center distance, and maximization of the efficiency. The latter has been accomplished using GA under dimensional conditions of gears, balance or assembly constraint of PGT, bending strength of teeth, contact strength of teeth, and torsional strength of the gear shaft. The design variables to be optimized were the module, the number of teeth for both the sun and planet gears, the tooth width, and the shaft diameter of the gears, especially sun and planet gears.
The objective functions were analyzed according to the six cases of the PGT operation. The fitness function of the weight was found to be equal to 0.276 kg in mode 1 of the PGT as a speed reducer and equal to 0.342 kg for the 6^{th} mode of the PGT as a speed multiplier. The value of the center distance was 39 mm for two operations of the PGT as a speed reducer and a speed multiplier. The module was found to be equal to 2 mm in several modes. The teeth number of the sun and planet gears was found to be equal to 21 and 18, respectively. The optimal tooth width values were found to be equal to 24.00 mm and 26.03 mm in the first and the last mode, respectively. The transmission efficiency was very important in the last type of planetary gear train system with the percentage of 98.60%, the case when the planet carrier was the output, the ring the input, and the sun gear fixed.
In addition, the change of the operating mode of the epicyclical gear train system has been a great impact on the variable of optimization and in the final solutions, which is the added value of this paper compared to the previous research quoted in the literature.
In light of this study, GA is an efficient approach for the optimization of complex problems with many variables, multiobjective functions, and constraints. It also provides a better result in a short time compared with other traditional optimization.
Center distance (mm)
Tooth width of gears (mm)
Primitive diameter of sun, planet, and ring gears (mm)
Shaft diameter of the sun and planet gears (mm)
The sun, the ring, and the planet carrier
Global gear ratio PGT
Coefficient of tooth length
Application factor
Transverse load factor (bending stress)
Face load factor (bending stress)
Transverse load factor (contact stress)
Face load factor (contact stress)
Dynamic factor
Module (mm)
Input speed (rpm)
Number of planets
Total weight of the epicyclical gear train (kg)
Constraint functions
Lower and upper boundaries of variables
Power of sun, planet carrier, and ring gear (kw)
Torque of sun, planet carrier, and ring gear (N·mm)
Number of teeth on sun, planet, and ring gears
Tooth form factor
Stress concentration coefficient
Contact ratio coefficient
Angular velocity of sun, planet carrier, and ring (rad·s^{−1})
Basic transmission ratio of PGT
Density material of the gear (kg m^{−3})
Tooth form factor
Gear material strength (N·mm^{−2})
Allowable shearing stress of shafts (Mpa)
Efficiency of planetary gear train
Internal and external gear pairs
Planetary gear train
One degree of freedom
Genetic algorithm.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.