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Stacking sequence optimization of composite wing using fractal branch and bound method Orbiting Plane : HOPE-X (JAXA)

Fractal Branch and Bound Method (FBBM) Fractal structure of design space

5 y V a 9º º b Coordinates of a delta wing and supersonic flow a/b=. h/a=. - (T8/6) E L = GPa E T =.8 GPa G LT =5.9 Gpa ν L =. 6

7 ( ) ),, ( 66 t y p t w h y w D y w D D w D L + = + + + ρ t w w y p + = µ λ ), ( [ ] [ ] [ ] = + M K K A µ λ Piston Theory = M q λ = M M V λ µ { } t i e w ω = 8 ( ) ), ( 66 y p t w h y w D y w D D w D L + = + + + ρ = 5 6 6 66 U U U U U U h D D D D D D Sym. k (number of laminate) N h N = Z = Z Z k- Z k Sym. k (number of laminate) N h N = Z = Z Z k- Z k ( ) = = = N k k k k k a k a k θ θ θ θ sin sin cos cos = N k N a n Sectional view of a laminate

Frequency/o (/rad) 5 5 5 5 55 6 65 7 c -5 Flutter Parameter Coalescence of st-nd eigenvalues of a delta wing a λ c = D λ c D (, )=(,) D 9 : λ c (, ) 5

D D Candidate points and selected points c [ K ] + λ[ K ] µ [ M ] = A (, ) V: Candidate points and selected points 6

λ c (, ) y = β + + β + β + β + β β5 E adj 7

E adj E n n = n ( k ) ( y y ) ˆ i i adj S yy y : yˆ : S : y yy k : n : E adj.95 E adj 5 = = N ( a k a k ) k = a n cosθ k cosθ k sin θ k sin θ k N k = N θ cosθ cosθ = o ± 5 9 o o :,5,9, = fractal structure of a design space 6 8

[9// /] [5// /] [// /] s The case of the outermost ply is 9º 5º layer Fractal Structure of Design Space 7 [5/9//]s The case of 5º-ply in the outermost ply and 9º-ply in the second ply 8 9

N N Fractal branch structure of stacking-sequence 9 + β + β + β + β β5 f = β +

a/b= h/a=. - (T8/6) E L = GPa E T =.8 GPa G LT =5.9 Gpa ν L =. y V a 9º º b Coordinates of a delta wing and supersonic flow Optimization flow D Candidate points and selected points

D Optimization flow [ K ] + λ[ K ] µ [ M ] = A Candidate points and selected points c Optimization flow FEM y 66 65 Finite element assemblages for delta wing

λ c (, ) Optimization flow f + β + β + β + β β5 = β + λ = c 68.5 +. 9.5 7.5.85 5 Optimization flow.5 5.5 5 6 6 7 7 -.5 8 -.5 8 - - -.5.5 Contour plot of response surface a/b=, = - - -.5.5 Contour plot of flutter limit a/b=, = R adj =.99 6

(FBBM) [(5/-5) ] s (, )=(., -. ) c =79.8 Optimization flow [5/-5/5/9/-5/9/5/-5 ] s (, )=(-.68, -.656 ) c =8.89 6. 7.5 -.5-5 6 7 8 - -.5.5 Contour plot of flutter limit a/b=, = 8

Optimization flow..5 -. -.5. -.5.5. D opt -. Design space for zoomed response surface 9 Optimization flow -. 5 -.5.. -.5.5.5 Candidates Selected -.5 -.75 [ K ] + λ[ K ] µ [ M ] = A -. Candidate points and selected points for zoomed RS c 8 5

c λ = 69.78 +.6 7.9 5. + 7. 8.5 7 7 Optimization flow -.5 75 -.5 75 8 8 -.5 -.5 -.75 -.75 - -.5 -.5.5.5 - -.5 -.5.5.5 Contour plot of zoomed RS a/b=, = Contour plot of flutter limit (-.5.5,.) a/b=, = R adj =.99 (FBBM) [5/-5/9/(5/-5) /9] s (, )=(-.8, -.6 ) c =8.8 Optimization flow [5/-5/5/9/-5/9/5/-5 ] s (, )=(-.68, -.656 ) c =8.89.8 6

y V y 9º b Coordinates of a delta wing and supersonic flow º a a/b= h/a=. - (T8/6) E L = GPa E T =.8 GPa G LT =5.9 Gpa ν L =. c = 9 c c 7

5, 5 9 =,,,,, 5 : =5, 5, 6, 7, 8, 9 c 5 D FEM 6 8

6.5% =.5 c tmp c n i (i=~6) 7 Optimal flutter limit parameter : c 8 6 8 6 6 8 Base fiber angle : Response surface c opt Response surface of optimal flutter limit parameter c 5, 5 9 8 9

o = 6 [9//9/ /9 ] s, [9/ /9 / /9] s (, )=(-.7,. ) c opt =8.99 5.5% =.5 c opt ( o ) c opt ( o -5 ) c opt ( o ) c opt ( o +5 ) c opt ( o )=85.85 c opt ( o -5 )=.8, c opt ( o +5 )=69.96 9 (, )=(.,.) c =69.95 (, )=(-.8, -.6) c =8.8. (, )=(.,.) c =69.95 () (, )=(-.5,.) c =8.99.7.

SST y.76 a/b=/.76 V 9º º. h/a=. - Coordinates of a SST wing and supersonic flow (T8/6) E L = GPa E T =.8 GPa G LT =5.9 Gpa ν L =. SST D FEM )

SST.5.5 5 5 -.5 5 - - -.5.5 -.5 5 - - -.5.5 Contour plot of response surface h/a=. Contour plot of flutter limit h/a=. R adj =.98 λ c =.7 +. 5..87.97 + 8.78 SST (FBBM) [(5/-5) ] s (, )=(., -. ) c =. 9 c.79 -.96875 7.99 -.79 -.96875 8.6.565 -.96875 7.7 c(fbbm) -.565. -.96875 -. 8.65.6

SST y.76 a/b=/.76 V 9º º. h/a=. - Coordinates of a SST wing and supersonic flow (T8/6) E L = GPa E T =.8 GPa G LT =5.9 Gpa ν L =. 5 SST D FEM 6

SST.5 5 5 8.5 5 5 8 6 6 -.5 -.5 - - -.5.5 - - -.5.5 Contour plot of response surface h/a=. Contour plot of flutter limit h/a=. R adj =.987 λ = c 6.5.85.57 +..66.8 7 SST (FBBM) [9 8 ] s (, )=(-.,. ) c =5.9 7 c.9969. 5..9766. 5.955.96875. 5.895 c(fbbm).988..96875 -. 5.6 5.669 8

.5 h/a=. h/a=. h/a=.. 5 -.5 - h/a=. - -.5.5 h/a=. 9 9 5 5

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