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1 Stacking sequence optimization of composite wing using fractal branch and bound method Orbiting Plane : HOPE-X (JAXA)
2 Fractal Branch and Bound Method (FBBM) Fractal structure of design space
3 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
4 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 = 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
5 Frequency/o (/rad) 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
6 D D Candidate points and selected points c [ K ] + λ[ K ] µ [ M ] = A (, ) V: Candidate points and selected points 6
7 λ c (, ) y = β + + β + β + β + β β5 E adj 7
8 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 [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
10 N N Fractal branch structure of stacking-sequence 9 + β + β + β + β β5 f = β +
11 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
12 D Optimization flow [ K ] + λ[ K ] µ [ M ] = A Candidate points and selected points c Optimization flow FEM y Finite element assemblages for delta wing
13 λ c (, ) Optimization flow f + β + β + β + β β5 = β + λ = c Optimization flow Contour plot of response surface a/b=, = Contour plot of flutter limit a/b=, = R adj =.99 6
14 (FBBM) [(5/-5) ] s (, )=(., -. ) c =79.8 Optimization flow [5/-5/5/9/-5/9/5/-5 ] s (, )=(-.68, ) c = Contour plot of flutter limit a/b=, = 8
15 Optimization flow D opt -. Design space for zoomed response surface 9 Optimization flow Candidates Selected [ K ] + λ[ K ] µ [ M ] = A -. Candidate points and selected points for zoomed RS c 8 5
16 c λ = Optimization flow 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, ) c =
17 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
18 5, 5 9 =,,,,, 5 : =5, 5, 6, 7, 8, 9 c 5 D FEM 6 8
19 6.5% =.5 c tmp c n i (i=~6) 7 Optimal flutter limit parameter : c Base fiber angle : Response surface c opt Response surface of optimal flutter limit parameter c 5,
20 o = 6 [9//9/ /9 ] s, [9/ /9 / /9] s (, )=(-.7,. ) c opt = % =.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 )= (, )=(.,.) c =69.95 (, )=(-.8, -.6) c =8.8. (, )=(.,.) c =69.95 () (, )=(-.5,.) c =
21 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 )
22 SST Contour plot of response surface h/a=. Contour plot of flutter limit h/a=. R adj =.98 λ c = SST (FBBM) [(5/-5) ] s (, )=(., -. ) c =. 9 c c(fbbm)
23 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
24 SST Contour plot of response surface h/a=. Contour plot of flutter limit h/a=. R adj =.987 λ = c SST (FBBM) [9 8 ] s (, )=(-.,. ) c =5.9 7 c c(fbbm)
25 .5 h/a=. h/a=. h/a= h/a= h/a=
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