Range-optimal design of an extended-expansion engine linkage

Abstract

Linkage design problems are conventionally posed as the maximisation of a kinematic quality measure subject to bounds on envelope dimensions, with performance requirements imposed as equality constraints. Applied to an extended-expansion (Atkinson) engine mechanism, that formulation is shown to be inadequate in three respects: it establishes no exchange rate between its competing objectives and so admits no solution, only a front; its objective is identically equal to the work done by the gas under the virtual-work identity and therefore quantifies no loss; and it determines no cross-section, so the inertia loads that govern the design at operating speed cannot be represented within it.

An alternative formulation is developed in which the objective is the application-level quantity of merit – distance travelled per unit of fuel consumed – and in which the competing geometric quantities are priced by the physical chain connecting them to it. The resulting problem is simultaneously multidisciplinary, mixed-integer, and, as specified, possessed of a feasible set of Lebesgue measure zero. Each property is shown to determine an element of the admissible solution method rather than merely to complicate it. The measure-zero feasible set, in particular, obliges the relaxation of the equality requirements into tolerance bands; those bands are then shown to be comparable in width with the manufacturing scatter of the components, so that reliability-based optimization arises as a consequence of the relaxation rather than as an addition to the problem.

Five quantitative results follow. The quasi-statically optimal geometry coincides with the transmission-angle singularity at which the inertia loads are largest and admits no feasible structure above 2000 rev/min, whereas a geometry displaced from it attains lower mass and greater range simultaneously. A tolerance study conducted against the specification rather than against a design identifies which of the stated bounds govern reliability and prices their relaxation at 0.47 % of the objective in exchange for a fall in the probability of violating a dimensional requirement from 0.664 to 1.9e-5. Power is taken from the shaft that turns twice per cycle, as on any four-stroke, so the mechanism’s advantage over a conventional slider-crank – sized by identical structural and tribological models, and optimised over its own degrees of freedom rather than proportioned by hand – is measured at equal speed and equal firing rate and amounts to 17.6 %; it decomposes into an indicated efficiency of 0.480 against 0.457, a mechanical efficiency of 0.865 against 0.787 and an engine mass of 12.9 kg against 16.9, so that extended expansion, the feature the topology exists for, is the smallest of the three. Imposing the coupled and structural constraints throughout the search rather than verifying them on its result is worth 4.9 % of the objective under an identical algorithm, the deterministic optima of both mechanisms being shown to be dominated by designs standing slightly off their active constraints. Finally, the first-order reliability model is shown to fail in two independent ways that no output of it signals: it prices only what its uncertain vector contains, so that the constraint governing the design – a gear face width at a reliability index of zero – lies outside the model entirely; and it linearises the expansion stroke at a tie between the two top dead centres that the optimizer itself has closed to 0.107 um, giving a failure probability seven times too small against 150 000 sampled builds. Both corrections are free.

The topology is finally removed from the assumptions altogether: a spring-connected synthesis, in which every candidate member is a spring whose stiffness is a design variable and is penalised towards rigid or absent, is run on the same requirements with no mechanism given. It returns discrete linkages and no alternative. The one start reaching a four-stroke motion returns an Otto engine whose two strokes are equal to six figures, for a reason that is an identity rather than an accident – a piston reached from one shaft alone is periodic in that shaft’s angle, so the asymmetry, which lives entirely in the first harmonic, is exactly zero. Extended expansion therefore requires the piston train to be reached from both shafts, which is what the three-cornered link of the studied mechanism does.

The methodological contribution is a demonstration, on a problem small enough to be verified throughout, that the choice of objective and the geometry of the feasible set determine both the design obtained and the class of algorithms that may legitimately be applied to obtain it.

Keywords

Multidisciplinary design optimization; mixed-integer nonlinear programming; reliability-based design optimization; mechanism synthesis; analytic sensitivities; extended-expansion engine.

Contents

The study

• 1. Introduction
  • 1.1 Four routes to the objective, and they conflict
  • 1.2 Two structural properties decide what may be applied
  • 1.3 Contribution
  • 1.4 Structure
• 2. Related work
  • 2.1 Extended-expansion engines and the mechanisms that realise them
  • 2.2 Formulating a mechanism-synthesis problem
  • 2.3 Architectures, derivatives, and thin feasible sets
  • 2.4 Discrete catalogue choices
  • 2.5 Design under tolerance
  • 2.6 Position of this work
• 3. Problem formulation
  • 3.1 The mechanism and its specification
  • 3.2 Design variables and objective
  • 3.3 Evaluating the objective
  • 3.4 Settings, and the baseline
  • 3.5 The specification cannot be met as written
• 4. Method
  • 4.1 The feasible set is thin, and that selects the optimizer
  • 4.2 Finite differences are wrong, not merely inaccurate
  • 4.3 The coupling is a field, not a handful of scalars
  • 4.4 The gear choice is discrete and pins a design variable
  • 4.5 The relaxed bands are comparable with the scatter
  • 4.6 The problem is nonconvex
  • 4.7 The problem solved
  • 4.8 Implementation
  • 4.9 Verification
• 5. Results
  • 5.1 The objective decides the design, and the quasi-static optimum is the worst place to be
  • 5.2 The relaxation is a promise about tolerance
  • 5.3 The reliability model failed twice, in different places
  • 5.4 The topology is worth 17.6 %, and the dimension count is not the reason
  • 5.5 The announced problem, solved, and where the search must be helped
  • 5.6 Asking for the topology instead of supplying one
  • 5.7 Giving the synthesis more freedom, and what it exposes
  • 5.8 Enumerating the topologies, and what the ranking was really measuring
  • 5.9 Alternatives, once the fit is posed as root-finding
  • 5.10 Pricing the alternative: what the five-bar is actually worth
• 6. Conclusions
  • 6.1 What the study establishes
  • 6.2 Limitations
  • 6.3 Further work
  • 6.4 Headline numbers
• References

Appendices

• Appendix A. Theory
  • A.1 Parametrisation
  • A.2 Kinematics
  • A.3 The Atkinson cycle
  • A.4 Quasi-static loads
  • A.5 The conventional objectives and constraints
  • A.6 Verification
  • A.7 Sizing the parts, and the coupling that creates
  • A.8 Failure modes and cross-sections
  • A.9 Convergence of the sizing loop
  • A.10 Derivatives
• Appendix B. Running the code
  • Install
  • Command line and library
  • Module map
  • Reproducing the results
  • Tests
  • Provenance
• Appendix C. Supporting measurements
  • C.1 How strongly the disciplines couple
  • C.2 What the analytic derivatives buy
  • C.3 The mixed-integer decomposition
  • C.4 Local optima
  • C.5 Reference designs
  • C.6 The range optimization, and why the projection matters
  • C.7 What the prescribed motion established
• API reference
  • Physics
  • Sizing and dynamics
  • Derivatives
  • The vehicle-level problem
  • Study of the result
  • Mixed-integer design
  • Optimization
  • Visualisation
  • Command line
  • Reference designs

At a glance

from exlink import COUPLED_DESIGN, evaluate
from exlink.robustness import failure_probability, format_reliability

# speed_rpm is the half-speed shaft's; the crankshaft turns twice per cycle
outcome = evaluate(COUPLED_DESIGN, speed_rpm=1000.0)
print(outcome.output_speed_rpm)      # 2000.0, the speed the study quotes
print(outcome.km_per_litre)          # 3338.3
print(outcome.budget.kilograms())    # where the mass actually is

print(format_reliability(failure_probability(COUPLED_DESIGN)))

Indices

• Index

• Module Index