Genetic algorithms multi-objectives optimisation of nonstandard spur gears.

Chira, Flavia ; Banica, Mihai ; Butnar, Lucian 等

1. INTRODUCTION

The computational method of design developed and used by the paper authors has on the base of the calculus algorithms the "direct gears design". This permit to determine the geometrical parameters of any special gears without the constraints imposed by the standard classical generation of the involutes teeth profiles, considering as the first step of design the geometry of the gears and only as second step the geometry of the tool (Kapelevich, 2008). The most important geometrical parameter of asymmetric gears is the coefficient of asymmetry "k" defined as the ratio between the base circle diameter of the inactive flank and the base circle diameter of the active flank (Litvin et al., 2000). In relation with the coefficient of asymmetry there are two cases of asymmetrical gears having different advantages emphasised by different researchers: with k>1, better for reducing the contact stress and with k<1, better for reducing the bending stress (Karpat et al., 2006).

In order to design, analyse and compare, in the same time, the booth cases, this paper authors have been named the profile with bigger pressure angles "direct profile" and the profile with smaller pressure angles "inverted profile". Depending on the active flank the gears have been named "direct asymmetric gears" and "inverted asymmetric gears". The symmetric gear can be considered taking part, as a particular case with superior or inferior limit value of "k", of the booth categories. As initial data have been used the centre distance (a) and the numbers of teeth ([z.sub.1], [z.sub.2]).

[FIGURE 1 OMITTED]

2. ON THE GENETIC ALGORITHM

The objective of design optimisation is to find the set of parameters that minimize an objective function subject to a set of behavioural constraints. The group of parameters that can be varied to improve the design are called design variables, they are denoted by the vector X={[x.sub.1], [x.sub.2] [x.sub.N]}.

The optimisation process implies an objective function [PHI](X) that can be improved and that provides a basis for choice between alternative acceptable designs.

The constraints that impose lower and upper limits on the values of design variables are called side constraints: [A.sub.i]<[x.sub.i]<[B.sub.i]. Once the design variables, objective and constraint functions have been defined, the optimisation problem can be solve using Numerical Techniques or Evolutionary Algorithms.

Evolutionary algorithms (EAs) are stochastic search techniques based on computer implementations of some of the evolutionary mechanisms found in nature, such as fitness, selection, crossover and mutation, in order to solve the optimisation problems. EAs are a power and robust method of global optimisation because they explore very large solution spaces without being trapped by local minima.

A Genetic Algorithm is a machine learning technique modelled upon the natural process of evolution. It uses a stochastic, directed and highly parallel search based on principles of population genetics that artificially evolve solutions to a given problem.

3. GEAR DESIGN OPTIMISATION METHOD

3.1 The genes that represent a design solution

A Genetic Algorithm considers that a solution, a design vector X, is associated with one individual with only one chromosome having n genes. The genes represent the variables.

Designing asymmetric gears, with the designing program developed, as application in Matlab, it is possible to obtain many solutions. The initial data are the centre distance and the number of teeth. The resulting different performances are in relation with the choosing of the genes called "designing variable" that the designing engineer must introduce in the first line of the program.

The most important conditions for apply the genetic algorithms are: the possibility that the system can be described by a set of variable and the possibility of evaluation the system with an objective function.

An individual from the domain of possible solutions it is determined by the following genes:

[x.sub.1] = [[alpha].sub.wd]--the mesh angle on the direct profile "d";

[x.sub.2] = [[alpha].sub.wi]--the mesh angle on the inverted profile "i";

[x.sub.3] = f--the coefficient of modification the gear rack or racks profile angle [[alpha].sub.dc];

[x.sub.4] = cr--the number of generating gear rack;

[x.sub.5] = var--the variant of using the asymmetric gear.

This set of variables generates an asymmetric gear. The developed applications permit to determine the geometrical parameters and the model of the gears in meshing.

3.2 The evaluation and objective functions

For a rapid evaluation of the behaviour of the designed asymmetric gear under the load there were established the methods of calculus and there were developed, as Matlab applications, routines to carry out diagrams of variation of the following gear characteristics: the elasticity, implicitly the rigidity, of the asymmetric tooth and of the pairs of teeth in contact; the transmission error; the relative sliding speed and the specific sliding between flanks; the instantaneous and medium power loses; the bending stress at the bottom of the tooth, for the pinion and for the gear and the contact stress. For more accurate results, the frictional contact between teeth in meshing can be approached using generalized Jocobians and Newton method (Pop & Cioban, 2008).

Any of the significant values of those parameters, obtained during a meshing cycle, can be used as objective function for mono-objective optimisation having the aim to improve, to minimise, the respectively parameter (Chira & Banica, 2007).

For multi-objectives optimisation the objective function result from composing in different way, in relation with the beneficiary request, the evaluation function resulted from the mentioned functional parameters. To ensure the compatibility between the optimising program and the designing program that is the source for the evaluation functions it was necessary to realise the optimising program also as application in Matlab.

In this paper will be present some results obtained by using as evaluation function the bending stress to the pinion ([[sigma].sub.ech1]), the bending stress to the gear ([[sigma].sub.ech2]) and the contact stress ([[sigma].sub.H]).

Have been made optimisation with the following objective function resulting by weighted sum of the evaluation function:

[F.sub.obj] ([bar.X]) = P1 - [F.sub.1] ([bar.X]) + P2 - [F.sub.2] ([bar.X]) + P3 ([F.sub.3], ([bar.X])/10) (1)

4. EXAMPLES OF OPTIMIZATION

By using the developed software there has been performed optimal designing of the asymmetric gear using 3 different objective functions, for the same initial date: [z.sub.1] = 16, [z.sub.2] = 57 , a = 120 mm and a transmitted power P = 18kW at a rotation speed n = 1000rot / min. The results are presented in table 1 and figure 2. The domain of the possible solution it has been determined by choosing the extreme values of the variable: [[alpha].sub.wi] [member of] [30,40], [[alpha].sub.wi] [member of] [20,30], f [member of] [1,10], cr [member of] {1,2}, var [member of] {1,2}.

With these ranges for the variables, considering only the solutions corresponding to the natural values situated between the extreme values is analysed and compared a number of 4,840 possible solutions.

[FIGURE 2 OMITTED]

For emphasize the better performances that result by using optimal design the resulted values of the stresses can be compared with those corresponding to the symmetric gears generated with the gear rack with profile angle equals 20 degrees, presented in table 1, last column.

5. CONCLUSION

The necessary and enough conditions for using the optimising techniques that have on the base the genetic algorithms are: the possibility that the system that must be optimise can be describe by a set of variable; the quality of the system can be evaluated by one or more evaluation or objective function. Both are satisfied in the case of the method developed for the design of asymmetric gear, method that can be adapted to other special non-standard spur gears.

The optimisation method that use genetic algorithms have the advantage that can research in the same time many possible variants, use probabilistic methods for the transition from a generation to the next and can be used for complex objective function, that can result from other programs. It is important to know that the realised optimisation program can be used to solve many other problems for optimal design of different products. The only condition is to have the evaluation function provided by a program in the same language.

6. REFERENCES

Chira, F. & Banica, M. (2007). Studies on the Gears with Asymmetric Teeth, Risoprint, ISBN 978-973-751-495-0, Cluj-Napoca, Romania

Kapelevich, A.L. (2008). Direct Design Approach for High Performance Gear Transmissions, Gear Solution, January 2008, 22-31. Available from: http://www.akgears.com Accessed: 2009-02-05

Karpat, F.; Cavdar, K. & Babalic, C.,F. (2006). An Investigation on Analysis of Involutes Spur Gears with Asymmetric Teeth: Dynamic Load and Transmission Errors, The 2nd International Conference "Power Transmissions 2006" Proceedings, pp. 69-74, ISBN 8685211-78-6, 25-26 April, Novi-Sad, Serbia

Litvin F., L.; Lian, Q. & Kapelevich A., L. (2000). Asymmetric Modified Gear Drives: Reduction of Noise, Localization of Contact, Simulation of Meshing and Stress Analysis. Computer Methods in Applied Mechanics and Engineering, No.188, pp. 363-390, ISSN 0045-7825

Pop, N. & Cioban, H. (2008). Geneneralized Jocobians and Newton Method for Solving the Frictional Contact Problems, Annals of DAAAM for 2008&Proceedings of The 19th International DAAAM Symposium, Vol.19 No.1, October, 2008, pp. 1093-1094, ISSN 1726-9679

Tab. 1. Multi-objectives optimisation results

 [F.sub.obj1] [F.sub.obj2]

The weighted sum parameters in the objective function

P1 0.3 0.25
P2 0.3 0.35
P3 0.4 0.4

The values of the designing variables-genes resulting

[[alpha].sub.wd] 38 39
[[alpha].sub.wi] 20 20
f 10 9
cr 2 2
var 1 1

The values of the parameters used for evaluation (N/[mm.sup.2])

[MATHEMATICAL EXPRESSION
 NOT REPRODUCIBLE IN ASCII] 86.68 87.65
[MATHEMATICAL EXPRESSION
 NOT REPRODUCIBLE IN ASCII] 96.25 95.60
[[sigma].sub.H] 877.79 876.38

 [F.sub.obj3] Simet

The weighted sum parameters in the objective function

P1 0.15
P2 0.55
P3 0.3

The values of the designing variables-genes resulting

[[alpha].sub.wd] 40 20
[[alpha].sub.wi] 30 20
f 1 0
cr 2 1
var 2 1

The values of the parameters used for evaluation (N/[mm.sup.2])

[MATHEMATICAL EXPRESSION
 NOT REPRODUCIBLE IN ASCII] 91.11 86.12
[MATHEMATICAL EXPRESSION
 NOT REPRODUCIBLE IN ASCII] 72.93 151.0
[[sigma].sub.H] 1,052.1 1,117