Fuzzy-Genetic Optimization for Engineering Problems ...

Fuzzy-Genetic Optimization for Engineering Problems Part II: Aeronautical and Mechanical Applications Gianni Caligiana, Franco Persiani, Gian Marco Saggiani DIEM – Faculty of Engineering Viale Risorgimento, 2 40136 Bologna, Italy

Abstract Aim of the paper is to show how a combined fuzzy-genetic algorithm (FGA) can be valuable in optimization problems involving many design parameters and several optimization criteria or imposed constraints. The new approach is described in detail in a companion paper [1] and a test case (with a known solution) is considered there, to validate the soundness of the proposed methodology. In this paper several applications in the aeronautical and mechanical fields are reported. A known comparative solution can be evaluated for one of these examples, while, for the others, optimal solutions are not available and only some answers can be found in literature for particular conditions. Keywords: Fuzzy Logic; Genetic algorithms; Fuzzy-genetic algorithms; Optimization; Design; Structural analisys; Wing tailoring; Plane slide bearing; Reduction gear train.

1. Introduction Genetic algorithm has proved to be a valuable tool in mechanical optimization. Usually it is driven by a fitness derived by a weighted global objective function, accounting for suitable optimization criteria and for imposed constraints. The choice of proper values for the weights is the most critical point in the whole procedure. Fuzzy algorithms are well suited to handle, in a rigorous analytical way, indeterminate information typical of human experience like the subjective assumptions of the designer in optimization. The proposed approach exploits the properties of fuzzy logic. The codified design variables, proper for every optimization problem, are elaborated by the finite element analysis or by the evaluation modules interfaced to genetic algorithm to derive the corresponding crisp values for the design functions and the constraints (which are globally considered as target functions, see the detailed description in the companion paper, [1]). The crisp values of these target functions are managed by a fuzzy engine (fuzzification) to obtain the linguistic values (expressed through fuzzy numbers) of the corresponding linguistic target functions. Linguistic target functions are handled according to a heuristic rulebase (inference phase) to obtain linguistic values of a linguistic fitness. Linguistic fitness is returned again to a crisp fitness (defuzzification phase) to be utilized by the genetic algorithm to improve the population of the solutions and to derive the near optimum suitable one. A test case is considered elsewhere [1]. Three optimization problems, in the aeronautical and mechanical fields, are here considered. a) First of all, the typical theoretical sizing of a plane slide bearing under hydrodynamic lubrication is examined. Aim of this search is to find the configuration corresponding to a minimum friction

coefficient, while maintaining a prefixed value for the minimum film thickness. The combined fuzzy-genetic algorithm (FGA) seems to be effective and reliable. b) A reduction gear train optimization is than reported. The optimization objective is to find a couple of gear with the minimal centre distance but satisfying gear-tooth surface and bending fatigue strength requirements. A known solution can be evaluated: some tests are performed to assess the influence of the number of distinct linguistic values (assumed for the target functions) on the FGA convergence speed and on the reliability of the obtained results. c) The last example of optimization is an application in the aeronautical field. The search is for a minimum weight composite wing while ensuring the attainment of a proper ratio between the divergence speed and the never exceeded speed. 2. Plane slide bearing The classic slide bearing under hydrodynamic lubrication [2, 3, 4] is shown schematically in Fig. 1. Applied load (N = 30 kN) and slide speed (V = 1.5 m/s) are assumed as data. The design parameters are: length (a) and width (b) of the slide, lubricant viscosity (µ), slide slope [m = (t2 - t1) / t1]. Design parameters “a” and “b” are constrained in the range [200-400] mm; “µ“ can be varied in the range [0.085-0.7] N s/m2, and “m” in the range [0.1-10]. Other constraints are introduced to avoid unfeasible solutions. The target functions, Ti, (with the meaning specified in § 2 of [1]) are the friction factor (f) and the minimum film thickness (t1) between the slide and the guide:

T1 = f = λ ( m )

µ V (a + b ) N

T2 = t1 =

6 µ V a2 b2 ψ ( m) N a+b

(1)

with the already mentioned meaning of symbols and with λ(m) and ψ(m) given by:

4 ln(1 + m) −6 + 2+m m λ (m) = ln(1 + m) −2 6 + 2 2m + m m2

ψ ( m) =

ln(1 + m) −2 + 2 2m + m m2

(2)

Details on how the above expressions can be derived are reported elsewhere [5]. Aim of the search is to find the configuration corresponding to the minimum available value for the friction coefficient (f), while maintaining the minimum film thickness (t1) greater or equal to to = 0.05 mm. The membership functions assumed for T1, T2 and for fitness are similar to those assumed in the test case reported in the companion paper [1]. In Fig. 2 and 3 the values of “f” and “t1/to”, corresponding to the fittest string of design parameters inside each population, are reported against the number of iterations of the FGA. As a control of the slide bearing configuration, the resulting eccentricity (Fig. 1) of the applied load is recorded in Fig. 4. In Table 1, all the final values of the design parameters for a typical run of the FGA, are shown. Particular solutions, found in literature [6, 7] and obtained assuming “ad hoc” crisp values for some of the design parameters, have been tested to confirm the soundness of the algorithm [5]. Table 1 slide bearing: results

a (mm) 202

b (mm) 200

µ (Pa s) 0.085

m 1.84

f 0.0024

t1 (mm) 0.051

e (mm) 20.7

3. Reduction gear train The sizing of a reduction gear train is examined. In particular, two spur gears are considered. For this case, a known theoretical solution [2, 3, 8] can be easily derived and compared with the numerical evaluation performed by the FGA. Assumed data for the problem are: input power (P = 5000 W), revolutions per minutes of the pinion (np = 1400 rpm) and velocity ratio (τ = 0.2). A 18NiCrMo5 steel is considered for the pinion and a 16CrNi4 for the gear. The design parameters are: module (m) and number of teeth (zp) of the pinion (those relative to the gear can be obtained as a consequence). They are coded to a 10 bit string, corresponding to 32 discrete values for each design parameter, accounting for commercial availability. Suitable constraints are considered to avoid teeth interference and to ensure a proper contact length. The target functions, Ti, (with the meaning specified in § 2 of [1]) are the centre distance (c) and the surface gear-tooth fatigue stress (σH):

T1 = c = d p

1 + (1 / τ ) 2

T2 = σ H = C p

Ft b dp I

Ko Kv Km

(3)

with the known meaning of the defined symbols and where [8]:

C p = 0.564

1 1 −ν Ep

2 p

+

Ep, Eg νp, νg

Ft = I=

60 P π np d p

sen ϕ cos ϕ R 2 R +1

ϕ

R=

is the elastic coefficient;

Eg are, respectively, the Young’s moduli of pinion and gear; are, respectively, the Poisson’s ratios for pinion and gear; is the tangential component of the load between teeth;

is the geometry factor; is the pressure angle;

dg dp

b=λm

dp Ko Kv Km

1 −ν g2

is the ratio of gear and pinion diameters; is the tooth width (λ = 10 is assumed); is the pitch diameter; is the velocity or dynamic factor; is the overload factor; is the mounting factor.

Aim of the search is to find the configuration corresponding to the minimum centre distance (c), while maintaining the gear-tooth surface stress for the pinion and for the gear (generally pinion is the more critical one) lower than the surface fatigue strength (SH), but as near as possible to SH itself:

S H = S fe C Li C R

(4)

where: Sfe CLi CR

is the surface fatigue strength (metallic spur gears, 107-cycle life, 99% reliability); is the life factor (= 1 at 107 cycles); is the reliability factor (= 1 for 99% reliability).

A safety factor, Fs = SH/σH = 1.4, is considered in reducer optimization so that an allowable surface fatigue strength Sall = SH/1.4 is actually utilized. A check is also performed relative to the effective fatigue gear-tooth-bending stress, which can be expressed as:

σ =

Ft bmJ

Ko Kv Km

(5)

with the known meaning of the defined symbols and where J is the geometry factor. For every feasible solution, effective bending stress must always be lower than the corresponding fatigue strength:

S n = S ' n C L CG C S k r k t k ms

(6)

where, [8]: S’n CL CG CS

is the standard endurance limit (in rotating bending; R.R. Moore testing machine); is the load factor (= 1 for bending loads); is the gradient factor; is the surface factor;

Actually, the same safety factor assumed for the surface fatigue stress (Fs = 1.4) is considered in comparing the effective fatigue stress to the corresponding fatigue strength. The membership functions assumed for T1, T2 and for fitness are similar to those assumed in the test case reported in the companion paper [1] for the cantilevered tube in bending and torsion. However, several tests were performed on the possible influence on the results of the number of linguistic values (or number of independent fuzzy numbers) defining linguistic target functions or linguistic fitness. In Table 2 the test named “m_n” means “m” linguistic values for the target functions and “n” linguistic values for the fitness (expressed, respectively, by “m” and “n” corresponding fuzzy numbers). Normalized run time is evaluated by dividing the time of each run by the lowest value obtained. At the end of 20 independent runs of the algorithm, the mean values for the normalized centre distance (better numerical estimate of the centre distance/known value of the calculated solution, c/co) and the mean value for the normalized surface stress (ratio between the surface stress and the allowable corresponding surface fatigue strength, σH/Sall) are recorded. Table 2 Mean values obtained on 20 independent runs of the FGA

series

σH/Sall

c/co

3_3 3_5 3_7 5_5 5_7 5_9

0.910 0.903 0.913 0.915 0.901 0.898

1.073 1.072 1.077 1.074 1.091 1.075

Normalized run time 1.108 1.000 1.013 3.687 5.633 6.709

Owing to the higher number of rules fired simultaneously in the inference phase [1], the increment of the number of linguistic values utilized increases hugely the relative run time. No other clear trend can be inferred, probably owing to the randomness, always present in the otherwise “fitness-driven” genetic algorithm. For example, when 5 fuzzy numbers are considered for the target functions (series 5_n), an increase of the fitness linguistic values “n” seems to lead to a worst performance relative to the normalized surface stress (values of σH/Sall are farther from the target value, equal to 1), while a not clear tendency appears for the normalized centre distance (c/co). Attention must be paid in examining these results. Any consideration derives from the examination of the mean value tendency and, therefore, a very unlucky run of the FGA can penalize several good results. Furthermore, owing to the low number of runs considered, no definitive conclusions can be drawn. More investigation is required. Optimization objective is nevertheless reached, because the results are always near to the theoretically evaluated solution. If a sufficient numbers of independent runs were performed, a solution very near to optimum can be derived. See, for example the histograms of Fig. 5 e 6 where 42 independent runs of the algorithm corresponding to a 3_7 series (3 fuzzy numbers for the target functions and 7 fuzzy numbers for the fitness) are considered. In Fig. 5 the values obtained by the FGA for the normalized surface stress (σH/Sall) are recorded as abscissas, while the number of times each value occurs is shown as ordinate. The greater part of the 42 values concentrates near the known solution (σH/Sall = 1). The error between the numerical data and the known solution can be less than 7%. For the normalized centre distance (c/co) a better trend can be observed (Fig. 6). The major part of the runs of the FGA gives results differing from the known solution for less than 5%. In this case, however, a particularly bad run of the FGA can be identified (Fig. 6). This outlines how important is, for a genetic algorithm (and, consequentely, also for the FGA) to repeat, for at least 20 times, the independent program runs, as suggested in literature [9], to avoid the attainment of bad results due to an unlucky event. The relatively high values of the errors between FGA’s results and known solution must not be misleading. In this example, discrete values for the design parameters are utilized, so that convergence to solution of the genetic engine of the FGA is hampered. Lower errors can be observed when continuous values for the design parameters can be allowed (see, for example, the test case in [1]).

4. Wing optimization A more complex example is presented. In this case, the final solution is not known and different design criteria must be considered together. The wing model consists of two “skins” symmetric with respect to the main surface, while the core is made of weightless webs, having infinite in-plane stiffness and zero out-of-plane stiffness (Fig 7). Each of the two skins consists of Nl laminae, and Nl is allowed to vary with the spanwise coordinate Y. Furthermore, a “lamina” consist of Ns sheets, or plies, each having thickness ts, so that the total thickness of the skin is given by: t = t(y) = Nl(y) Ns ts

(7)

In the followings, Nl is regarded as a continuous function of Y, although, as a matter of fact, is not; such approximation is justified by imposing: a) t