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Lucian Tudose · Ovidiu Buiga · Cornel ¸Stefanache ·András Sóbester Abstract The design space of multi-stage transmissions is usually very large and heavily constrained. This places significant demands on the algorithm employed to search it, but successful optimization has the potential to yield considerably better designs than conventional heuristics, at the same time enabling a better understanding of the trade-offs between various objectives (such as service life and overall weight). Here we tackle a two-stage helical gear transmission design problem (complete with the sizing and selection of shafts, bearings, housing, etc.) using a two-phase evolutionary algorithm in a formulation that can be extended to 48808
1 Introduction
The complexity of the design of multi-stage reducers lies in the strong and often intractable connections between the design variables defining its sub-systems. In other words, an optimal reducer is generally not an assembly of components optimized in isolation, a fact overlooked by many conventional design heuristics. For instance, the impact of a certain choice of gear width and center distance may yield a minimum mass gearing, but the selection of this gearing may cascade through subsequent steps of the design process (sizing of shafts, further stages, bearings, housing, etc.) to ultimately lead to a heavier reducer than if a slight compromise had been made on the choice of that first gearing.
A typical example might be that selecting a smaller than optimal gear diameter (and a correspondingly greater contact width) could yield a somewhat heavier gearing, but a more compact layout and therefore a much lighter housing; it is worth mentioning though that in reality the impact on the overall objective tends to be much
less direct and therefore much more obscure than in this example.
Of course, in all but a few trivial cases, it is impossible to tell what that first compromise should have been, let alone what any subsequent choices should have been made with the overall goal in mind, instead of concentrating on the subsystem in hand. The chief reasons impeding a truly ‘holistic’ reasoning at every step of the design heuristic are the sheer number and the highly non-linear nature of the constraints and the objectives, the large number of design variables and the complexity of the interactions between them. Additionally, analytical models may not be available for these interactions and constraints, precluding higher level analytical
calculations that could predict the global effect of local design decisions.
The last two decades have seen an increasing awareness amongst the power transmission design community of the shortfalls of simple trial and error type methods conventionally used to tackle this highly constrained class of design problems and potential replacements have begun to emerge in the shape of expert systems (Ferguson et al.
1999; Abersek et al. 1996), synthesis tools based on spatial grammars (see the simulated annealing-driven, grammar based topological gearbox design tool described by Lin et al. 2009), particle swarm searches (Ray and Saini 2001), algorithms based on the modeling of civilizations and societies (Ray and Liew 2003), constrained quasi-Newton local searches (see the study by Thompson et al. 2000) into the fatigue life versus gearing volume trade-off) and evolutionary algorithms (the work of Li et al. (2008) on the application of a fuzzy-controlled genetic search to the optimization of a simple reducer model and the study by Gologlu and Zeyveli (2009) for recent examples). In fact, the latter category—headlined by genetic algorithms (GAs)—appears to be the direction of choice at present and there are two key reasons for this.
Firstly, GAs can handle the highly discretised design spaces of transmission systems. Standardisation and the favouring of off-the-shelf (as opposed to purpose-designed) subcomponents are the main reasons for most design variables only being permitted a pre-determined set of discrete values (as we shall see, this is the case with our own application too). Secondly, the full description of a class of reducers (say, that of the two-stage, helical gear family) generally requires a large number of design variables—typically, well over ten—and GAs have a fine track record in the global