Jean Le Rond d Alembert, ( ) 2005

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1 Jean Le Rond d Alembert, ( ) 2005

3 lumped mass modeling) x x A l ρ 5.4

4 x 10 x y(x, t) y M(x, t) Q(x, t) E I 2 y/ 2 M = EI 2 y 2 (5.1) x z Q Q = M = EI 3 y 3 (5.2) Q(x + dx) {Q(x) + Q(x)/ dx} ( Q(x + dx) = Q(x) + Q(x) ) dx = EI 3 y 3 EI 4 y dx (5.3) 4 y ρadx

5 ρadx 2 y t 2 = EI 3 y 3 + ρadx 2 y t 2 ( EI 3 y 3 EI 4 y 4 dx = EI 4 y 4 dx ) (5.4) 2 y t 2 + EI 4 y ρa 4 = 0 (5.5) 2 = EI/ρA (5.5) y(x, t) = φ(x)e jωt (5.6) 2 d4 φ(x) dx 4 ω 2 φ(x) = 0 d 4 φ(x) ( ω ) 2 dx 4 φ(x) = 0 (5.7) x φ 5.7 φ(x) = De sx s ( ω ) 2 s 4 = 0 ω ω s = ±, ±j (5.8) φ(x) ω φ(x) = C 1 e x + C 2 e ω x + C 3 e j ω x + C 4 e j ω x (5.9)

6 C 1 C x φ(x) l ρ E A I x = 0 x = l φ(0) = 0 (5.10) d 2 φ(x) dx 2 = 0 (5.11) x=0 φ(l) = 0 (5.12) d 2 φ(x) dx 2 = 0 (5.13) x=l 5.9 C 1 C 4 C 1 e l ω C 1 + C 2 + C 3 + C 4 = 0 (5.14) C 1 + C 2 C 3 C 4 = 0 (5.15) + C2 e l ω + C3 e jl ω + C4 e jl ω = 0 (5.16)

7 C 1 e l ω + C2 e l ω C3 e jl ω C4 e jl ω = 0 (5.17) C 1 = C 2 C 3 = C 4 C 2 e l ω + C2 e l ω C4 e jl ω + C4 e jl ω = 0 (5.18) C 2 e l ω + C2 e l ω + C4 e jl ω C4 e jl ω = 0 (5.19) ( C 2 e l ω ) ω + e l = 0 (5.20) ω = 0 C 2 = 0 C ( C 4 e jl ω ) ω e jl = 0 (5.21) C 4 = 0 C 4 = 0 ( ) ω 2sin l = 0 (5.22), Ω s

8 ω l = π, 2π, 3π, 4π ( π ) 2 EI Ω s = l ρa, ( 3π l ) 2 EI ρa, ( ) 2 2π EI l ρa, ( 4π l ) 2 EI ρa (5.23) 5.9 ( C 4 e jl Ω s e jl Ω ) s Csin sin ( ( l l Ωs Ωs ) ) ( ) iπx sin l (i = 1, 2, 3 ) (5.24) C 5.4 x = 0

9 φ(0) = 0 (5.25) dφ(x) dx = 0 (5.26) x=0 x = l d 2 φ(x) dx 2 d 3 φ(x) dx 3 = 0 x=l = 0 x=l (5.27) C 1 + C 2 + C 3 + C 4 = 0 C 1 e l ω C 1 e l ω + C2 e l ω C2 e l ω (C 1 C 2 ) + j(c 3 C 4 ) = 0 C3 e jl ω C3 e jl ω C4 e jl ω = 0 + C4 e jl ω = 0 (5.28) C j j C 2 0 e l ω e l ω e jl ω e jl ω = e l ω e l ω je jl ω je jl C 3 0 ω 0 C 4

10 [ 1 2 (1 + j)el ω ( 1 + j)e l ω ω e jl 1 2 ( 1 + j)el ω 1 2 (1 + j)e l ω ω e jl 1 2 (1 + j)el ω 1 2 ( 1 + j)e l ω ] [ ] 0 [ C3 C 4 = 0 [ ] [ ] [ ] a11 a 12 C3 0 = a 21 a 22 C 4 0 ω je jl 1 2 ( 1 + j)el ω (1 + j)e l ω + je jl ω ] a 11 a 22 a 12 a 21 = 0 (5.29) T l 2 T x l x dx t y(x, t) y(x, t) y

11 ρ ρdx y 2 y/ t 2 x T y T y (5.30) x + dx y y x y/ x dx ( ) x y dx y + ( ) y dx (5.31) T (x + dx) y ( T + T ) { y dx + ( ) } { y y dx T + ( ) } y dx (5.32) ρdx 2 y t 2 ρ 2 y t 2 ρ 2 y t 2 = T = T = T 2 y { y + ( ) y ( ) } y dx T t (5.33)

12 ρ 2 y t 2 2 y t 2 2 y t 2 = T 2 y 2 = T ρ 2 y 2 = 2 2 y 2 (5.34) 5.34 (wave equation) = T/ρ y(x, t) y(x, t) = φ(x)e jωt (5.35) φ(x) x ω ω 2 φ(x)e jωt = 2 d2 φ(x) dx 2 e jωt d 2 φ(x) ( ω ) 2 dx 2 + φ(x) = 0 (5.36) 5.36 ( ω ) ( ω ) φ(x) = A os x + B sin x (5.37) A B 2 osine sine

13 φ(0) = 0 φ(l) = 0 (5.38) 5.37 B sin A = 0 ( ) ωl = 0 (5.39) A B A = B sin ( ) ωl = 0 (5.40) ωl = π, 2π, 3π (5.41) ω 1 Ω i (i = 1, 2, 3, ) Ω i = iπ l = iπ l T (i = 1, 2, 3 ) (5.42) ρ i 5.42 Ω i 5.37 ω A = 0 ( ) iπ φ(x) = B sin l x (i = 1, 2, 3 ) (5.43)

14 } T D S P Q L [ h : (a) 1 (b) (5.37)

15 a1 440Hz. E = 206GPa ρ = 7860kg/m 3 l = 1m φ = 1.175mm T [N] 1) 5.7 E = 206GPa ρ = 7860kg/m 3 h = 0.6mm 440Hz 1) JIS SWRS82A SWRS87A SWRS92A /2 0.2mm 90

16 a1 5.7

1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2

1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2 1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2