One-row planetary gear optimal design via a new two-phase evolutionary algorithm.

Tudose, Lucian ; Buiga, Olvidiu ; Jucan, Daniela 等

1. INTRODUCTION

Gears optimization problem induces a number of challenges especially when the design problem involves gear geometry, kinematics and strength. The resulting optimization problem involves design variables which can be integer (number of teeth), discrete (normal module), and real (gear width). Many researchers have reported solutions to this problem. A design synthesis of a nine-speed gear drive was presented (Osman et al., 1978). The objective of the synthesis was to minimize the size of all gears from the mesh and speed ratio so that the size of the largest gear is kept to a minimum. An optimized design of helical gear reducer is performed (Li & Symmons 1996) in order to minimize the centre distance. Other researchers have developed several applications using different design and calculation methods (Deb, Pratap, & Moitra, 2000; Ray & Saini, 2003). An optimal design problem of a two-stage speed reducer is presented (Rui, L. et al., 2008). The purpose of this research is to obtain the multi-objective optimization design scheme of a gear reducer with a Fuzzy Genetic Algorithm. A study of the optimization and regression techniques for optimum determination of partial ratios of two-step, helical gearboxes in order to obtain the minimal length of the reducer case was proposed (Vu, 2008). The design problems above mentioned are too simply formulate related to the design realities.

In our particularly case we deal with a planetary transmission. The planetary gears are gears that have gear wheels with moving axes (Grote & Antonsson, 2009). The most commonly used ordinary single-row planetary gear (Fig. 1) consists of a central wheel [z.sub.a] with external teeth, a stationary central wheel [z.sub.b] with internal teeth, planetary pinions [z.sub.g], wheels with external teeth interlocking simultaneously with [z.sub.a] and [z.sub.b], and carriers h, to which the axes of the planetary pinions are locked. The carrier is joined to the low-speed shaft.

2. TWO PHASES ENHANCED EVOLUTIONARY ALGORITHM

An optimization problem consists of an objective function accompanied by certain number of constraints. Optimization problems with a very large number of constraints can be very difficult to solve. In order to remove this shortcoming, a two-phase enhanced evolutionary algorithm (2PhEA) inspired from the evolutionary concept of punctuated equilibrium is presented in this paper. The main idea behind this algorithm is its operation in two phases. In each phase, the individual's fitness is determined by another factor. In Phase 1, the individual's fitness depends only on the way in which an individual is more suitable (or not) in terms of constraints (population "fight for survival" and there is no interest for the best individual). This phase is a kind of "feasible individual generator". The algorithm moves into the second phase when the number of feasible individuals of the population exceeds a preset threshold. Phase 2 is a common evolutionary algorithm (sometimes a simple GA).

3. DESIGN PROBLEM FORMULATION

The aim of our optimal design is to obtain an one-row planetary gear as compact as possible. The input data are:

Electrical engine horsepower: P=2.9 kW; Overall transmission ratio: [i.sub.T]=7.6; Rotational speed of input shaft: [n.sub.1]=925 rpm;

[FIGURE 1 OMITTED]

4. OPTIMIZATION OF PLANETARY GEAR

4.1 Genes

The design problem genes are presented in Table 1.

In order to determine the tooth number [z.sub.b] of the stationary central wheel b the authors of this paper introduced two new variables Rdir and [DELTA]u. By introducing the variable [DELTA]u, we increased the possible teeth number of the stationary central wheel [z.sub.b], from 53 possible values to 68.

4.2 Objective function

The volume of the circumscribed cylinder of the stationary central wheel b can be written (using standard nomenclature):

V=[pi] x [b.sub.b] x [([d.sub.fb]/2+S).sup.2][right arrow]min (1)

where: S=2.2 x [m.sub.n]+0.05 x [b.sub.b]

4.3 Constraints

The solutions of the optimization program have to satisfy a set of 32 constraints, which could be defined as follow: the Hertz stress should be less or equal to the allowable Hertz stress for both external and internal toothing (2 constraints), the bending stress ([[sigma].sub.Fa,g,b]) at the tooth base has to be less or equal to the allowable bending stress ([[sigma].sub.FPa,g,b]) for all planetary gears (3 constraints), the normal addendum modification coefficient should be in such range that the undercutting of the central wheel and of the planetary pinion teeth does not get worse (2 constraints), the normal addendum modification coefficient of the planetary pinions and stationary central wheel should be in the range of [-0.5 ... 1] (2 constraints), the profile shift should be in such a range that the tooth thickness at the top of all planetary gears does not decrease under a certain value (3 constraints), radial contact ration should be greater than a certain imposed value (2 constraints), for the span measurement several conditions should be satisfied (11 constraints), the number of both external and internal gearings teeth should be co-prime numbers (2 constraints), three constraints about avoiding tooth interference and two constraints about planetary pinions mounting conditions.

4.4 Results

In solving this optimization problem, our own software Cambrian v.3.2 was used. Written in Java, Cambrian is a platform that allows the assembling of all sort of evolutionary algorithms (including 2PhEA) in an original manner. The optimal values found for all above considered genes are presented in Table 2.

4.5 Conclusions

The comparative study between the classical design solution (obtained using Grote & Antonsson, 2009) and the optimal design solution leads to the following conclusions:

* The volume of the circumscribed cylinder of the stationary central wheel b calculated with the classical method is 2.06 [10.sup.-3] [m.sup.3] while the optimal design solution offers a smaller volume, equals to 1.47 x [10.sup.-3] [m.sup.3], i.e. a reduction by 29.33%;

* After optimization the working centre distance decreased from 80 mm to 56 mm;

* In the optimal design solution the gears are slightly wider than those obtained in classical design manner;

* In figure 2 an overlay image representing the optimal (1) and classical (2) solutions of the single-row planetary gear are presented. Note that we used the same input data in both cases.

[FIGURE 2 OMITTED]

As a next step we will increase the complexity of our optimization problem by introducing the shafts and the transmission case. The objective function could be the mass of the entire single-row planetary gear.

This work has been supported by the Grant of the Romanian Government PN II CNCSIS ID_1077 (2007-2010).

5. REFERENCES

Deb, K.; Pratap, A. & Moitra, S. (2000). Mechanical component design for multiple objectives using elitist non-dominated sorting GA, In: Parallel Problem Solving from Nature--PPSN VI, Schoenauer, M. et al. (Eds.) pp 859-868, Springer, ISBN 978-3540410560, Berlin

Grote, K.-H.; Antonsson, E.K. (2009). Springer Handbook of Mechanical Engineering, Springer, ISBN 298-3540491316, Berlin

Li, X. & Symmons G.R. (1996). Optimal design of involute profile helical gears, Mechanism and machine theory Vol. 31, Issue 6, (August 1996) pp 717-728, ISSN 0094-11XXXX

Osman, M.O.M.; Sankar, S. & Dukkipati R.V. (1978). Design synthesis of a nine-speed machine tool gear transmission using multiparameter optimization, Journal of Mechanical Design, Transactions of ASME, Vol. 100, pp 303-310, ISSN 1050-0472

Ray, T.; Saini, P. (2003). Engineering design optimization using a swarm with intelligent information sharing among individuals, IEEE Transactions on Evolutionary Computation Vol. 7, No.4, pp 391-392, ISSN 1089-778X

Rui, L. et al. (2008). Multi-objective optimization design of gear reducer based on adaptive genetic algorithm, Proceedings of the 12th Int. Conf. on Computer Supported Cooperative Work in Design, pp 229-233, ISBN 978-14244-1650-9, April 2008, China

Vu Ngoc Pi (2008). A new study on optimal calculation of partial transmission ratios of two-step helical gearboxes, Proceedings of the 2nd WSEAS Int. Conf on Computer Engineering and Applications Stevens Point (Ed.), pp 162-165, ISBN 978-960-6766-47-3, January 2008, Mexico

Tab. 1. Genes of the optimization problem

 Genes Range

Tooth number of the central pinion, [z.sub.a] 21 ... 29
Number of planetary pinions, [n.sub.w] 2 ... 5
Working centre distance, [a.sub.w] 56 ... 315
Normal addendum modification
 coefficient, [x.sub.na] -0.5 ... 1
Length width coefficient, [[psi].sub.a] 0.2 ... 0.8
Standard pitch cylinder helix angle, [beta] 5 ... 21.75[degrees]
Transmission ratio variation, [DELTA]u -0.04 ... +0.04
Round direction, Rdir 0 and 1

Tab. 2. Gene values obtained after optimization

 Genes Values

 Optimal Classical

Tooth number of the central
 pinion, [z.sub.a] 22 23
Number of planetary pinions,
 [n.sub.w] 2 2
Working centre distance, [a.sub.w] 56 80
Normal addendum modification
 coefficient, [x.sub.na] 0.988 0.7
Length width coefficient,
 [[psi].sub.a] 0.785 0.39
Standard pitch cylinder helix
 angle, [beta] 16.5[degrees] 15[degrees]
Transmission ratio variation,
 [DELTA]u 0.01669 --
Round direction, Rdir 1 --