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Here are some typical conclusions that can be drawn from this type of study, as shown on the plot. If we were
Fig. 3 Histogram of evaluation numbers required by the two epoch
algorithm to reach the weight threshold of 45.8 kg
Fig. 4 Minimum weight reducers designed for various service life
spans—note that the required service life has to be reduced to less than 2,000 h before any weight reductions will result aiming for, say, 2,000 h, we would need to sacrifice 75% of this service life for a 4 kg (roughly 10%) weight saving. Halving the required service life would only save us just over 1 kg. At the same time, if weight was our sole concern, there would be no point in making any service life sacrifices if we cannot allow it to drop below 2,000 h, as no weight
saving would result.
6 Conclusions and the way forward
Compared to many other industrial products, certain classes of mechanical transmissions (such as the reducers discussed here) offer the allure of a finite design space (as a result of the standardisation of their layout and most of their design variables), which should make global optimization considerably easier.
Alas, as a result of the ‘curse of dimensionality’, even these design spaces can be vast and can present a considerable challenge to the designer, especially when a multitude of inequalities constrain the design space.
In this paper we have shown how an evolutionary algorithm, augmented by a constraint handling heuristic based on a biological paradigm, can make solving such complex structural design problems a feasible proposition, even when detail design level constraints are taken into account.
We have considered a particular class of transmissions here, but in building the tool for solving the related design problem, we have not encountered any severe scalability issues (though clearly, a greater number of design variables would further increase the size of the design space and therefore longer GA run times may be required)—therefore, broader classes of transmissions could be considered for the Automated optimal design of a two-stage helical gear reducer 435 same treatment, leading ultimately to a generic transmission system design tool based on the evolutionary optimization concepts described here.
Taking a broader perspective, this type of heuristic might facilitate the solution of other heavily constrained design problems, which, as a result of the complexity of their constraint boundaries and relatively small size of their feasible regions, present unsurmountable obstacles to conventional problems. Complex engineering systems with multiple interactions between their subsystems are often the sources of such optimization problems. Consider, for instance, the conceptual design of airframes: an intensely multi-disciplinary problem typically yielding a variety of
constraints related to aerodynamic performance, structural design criteria, environmental impact, cabin design, payload positioning, etc. For similar reasons, the automotive industry encounters many similar problems too, adding crashworthiness to the above list as another typical source of highly restrictive constraints.
There is room for further development in terms of the fidelity of the analysis, which drives the optimization process described here—of course, given the relatively large number of objective function evaluations demanded by the sheer size of the design space, this would have to be achieved through a careful control of the computational cost of the analysis. In the same vein, further objective functions could also be considered—manufacturing cost is a potential example.
In parallel with such developments, there is also considerable
scope for the better understanding of a series of algorithm design questions related to the heuristic introduced here. For example, when is the best time to conclude an equilibrium epoch? A cursory study led to us using 40% as the ratio of feasible inpiduals being reached as an epoch termination criterion, but it is hard to tell at present