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- けいざぶろう あさま
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3 1. 10/ / /23 ( ) 4. 10/ / 6 ( ) 6. 11/ /20 ( ) 8. 11/ / 4 ( ) / /18 skyline 12. 1/ 8 ALE 13. 1/15 ALE 14. 1/22 ALE 3
4 t + t M t+ t ü + C t+ t u + t+ t Q = t+ t F M C t t + t Newmark-β t M, C Q Newton-Raphson M t+ t ü (k) + C t+ t u (k) + t+ t K (k 1) u (k) = t+ t F t+ t Q (k 1) t+ t u (k) = t u (k 1) + u (k) t+ t u (k) = t u (k 1) + u (k) t+ tü (k) = t ü (k 1) + ü (k) u (k), u (k), ü (k) 4
5 t t + t t+τü = t ü + τ t (t+τ ü t ü) τ τ = t t+ t u = t u + t 2 ( t+ tü + t ü ) t+ t u = t u + t t u + t2 3 tü + t2 t+ tü 6 u (k) = t 2 ü(k) u (k) = t2 6 ü(k) 5
6 Newmark -β 1 Newmark-β 1959 Newmark,, t.,, t + t M t+ t ü + C t+ t u + t+ t Q = t+ t F Newmark-β, t + t t,. t+ t u = t u + t [ γ t+ t ü +(1 γ) t ü ] {( ) } 1 t+ t u = t u + t t u + t 2 2 β tü + β t+ t ü 6
7 Newmark -β 2 Newmark-β, t + t t,. t+ t u = t u + t [ γ t+ t ü +(1 γ) t ü ] {( ) 1 t+ t u = t u + t t u + t 2 2 β tü } + β t+ t ü, γ = 1 2, β = 1 6, γ = 1 2, β = 1 4, t+ t u t+ t u. t+ t u = t u + t 2 (t+ t ü + t ü) t+ t u = t u + t t u + t2 4 (t ü + t+ t ü) t t + t t+τü = 1 2 (t ü + t+ t ü) (0 τ t) (1),. γ = 1 2, β = 1 4 (trapezoidal rule) 7
8 Newmark -β 3 Newmark-β, t + t t,,, t + t,. t+ tü = 1 u t u) 1 ( ) 1 t u β t 2(t+ t β t 2β 1 tü { [ t+ t 1 u = t u + (1 γ) t ü + γ u t u) 1 t u ( 1 ]} β t 2(t+ t β t 2β 1)t ü t = t u + t(1 γ) t ü + γ β t (t+ t u t u) γ ( ) 1 t u γ β 2β 1 t t ü = γ ( β t (t+ t u t u)+ 1 γ ) ( t u + t 1 γ ) tü 2β 2β, K Q t+ t t+ t Q = K t+ t u Newmark-β ( 1 β t 2M + γ ) β t C + K t+ t u = t+ t F + M [ 1 β t 2 t u C ( ) 1 t u + β t 2β 1 [ ( ) γ γ t u + β t β 1 tü ] ( ) ] γ t u + 2β 1 t t ü (2) 8
9 Newmark-β ( 1 β t 2M + γ ) β t C + K Newmark -β 4 t+ t u = t+ t F + M [ 1 β t 2 t u + 1 ( ) 1 t u + β t 2β 1 [ ( ) γ γ + C t u + β t β 1 ] tü ( ) ] γ t u + 2β 1 t t ü ( ) 1 M + γ β t 2 β t C + K, γ = 1 2, β = 1 4, t., γ = 1 2, β = 1 t,, 6., γ>
10 Newmark -β 5,, Newton-Raphson (k =2, 3, ). M t+ t ü (k) + C t+ t u (k) + t+ t K (k 1) u (k) = t+ t F t+ t Q (k 1) t+ t u (k) = t+ t u (k 1) + u (k) t+ t u (k) = t+ t u (k 1) + u (k) t+ tü (k) = t+ t ü (k 1) + ü (k), (k) k., Newmark β,,, k k 1 u (k) = γ β t u(k) (3) u (k) ( 1 β t 2M + γ ) β t C + K u (k) ü (k) = 1 β t 2 u(k) (4) = t+ t F t+ t Q (k 1) M t+ t ü (k 1) C t+ t u (k 1) (5) 10
11 Newmark -β 6 Newmark-β. 1. (a) M C. (b). a 0 = 1 a β t 2 1 = γ a β t 2 = 1 β t a 3 = 1 2β 1 a 4 = γ β 1 a 5 = γ 2β 1 a 6 = γ t a 7 =(1 γ) t 2. (a) t + t. K eff = a 0 M + a 1 C + K (b) t + t. t+ t F eff = t+ t F t Q + M(a 2t u + a 3tü)+C(a 4t u + a5ü) (c) t + t t+ t u. K eff u = t F t+ t eff u = t u + u (d),. i. U (0) = U, i =0. ii. i = i +1 iii. i 1. 11
12 t+ t F (i 1) eff = t+ t F M t+ t ü (i 1) C t+ t u (i 1) t+ t Q (i 1) iv. i. K eff u (i) = t+ t F (i 1) eff v.. u = u + u (i) vi., t + t,,. t+ t ü = a 3 u + a 4t u + a 5tü t+ t u = t u + a 6tü + a t+ tü 7 t+ t u = t u + u 12
13 1 t t + t t t + t,,,,. [t, t + t], u t+ t/2 u, u ü. t u t u t+ t/2 = ut+ t u t t ü t = ut+ t/2 u t t/2 t u t = ut+ t/2 + u t t/2 t (6) 13
14 2 M tü + C t u + t Q = t F, u t+ t/2. ( M u t+ t/2 = t + C ) 1 [ ( M (F t Q t (u t )) + 2 t C 2 u t+ t = u t + t u t+ t/2 ) u t t/2 ] (7) t + t u t+ t t ü t. (7), M t + C 2,, u t+ t/2.,, M (lumped mass matrix),.,,,., 1. 14
15 t. t., u t/2, t =0 u 0. u 0 = 0 u t/2 = u t/2 (8) u t/2 = tm 1 (F t Q t (u t ))/2 (9) 15
16 3. 1. (a) M C. (b) t. (c). M eff = M t + C 2 (d) 0 0 u, 0 u 0 ü, t. 2. (a) t. F t eff = F t Q t (u t )+ ( M t C 2 ) u t t/2 (b) t + t/2 u t+ t/2. u t+ t/2 = M 1 eff F t eff (c) t + t u t+ t. t ü t.,, t. 16
17 1, consistant (lumped).. consistant, ρ 0, ρ t 0,t δu ρ 0 üdv = δu ρ t üd t v = δu t MÜ (10) V t v. (10),, consistant. (lumped),, consistant,, lumped.,,, lumped. 17
18 1 lumped,, V e N i dv e. lumped 1. V e N (i) dv i 2. V e N (i) N (i) dv i serendipity 5. consistent 18
19 1 MK Ax = λx Kx = λmx MK L M = LL T Kx = λmx Kx = λll T x L 1 Kx = λl T x L T x = y x = L T y L 1 KL T y = λy MK 19
20 2 R = xt Ax x T x = xt λx x T x = λ 20
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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