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23 1 11 A 2 A.1..................................... 2 A.2.................................. 4 A.3............................... 5 A.4.................................... 6 A.5....................... 8 A.5.1........................ 8 A.5.2.............................. 9 B - 10 B.1 RLC with................................ 10 B.2 -.................................. 10 B.3 RLC.............................. 11 B.4......................... 11 B.4.1................................. 11 B.4.2.............................. 13 B.4.3...................... 14 C 16 C.1......................................... 16 C.2................................. 17 C.3.......................................... 18 C.4.......................................... 21 C.5..................................... 22 C.6................................. 23 1

A URL http://www.kozosoft.co.jp/gijyutu/s28.html http://www7b.biglobe.ne.jp/oldman-galil library/page102.html http://www.jtccm.or.jp/library/jtccm/public/b-number/0512 kikou.pdf http://www.jtccm.or.jp/library/jtccm/public/mokuji06/0612 kikou.pdf http://pedpa.co.jp/library/tower.html http://ja.wikipedia.org/wiki/ A.1 ( 1) ( 7) 2

1: http://www.kozosoft.co.jp/gijyutu/s28.html ( 2) ( 8) [ ] 2: http://www.kozosoft.co.jp/gijyutu/s28.html 3

A.2 n k n, n m n n x n (t) m 1 ẍ 1 (t) = k 1 x 1 (t) + k 2 {x 2 (t) x 1 (t)} (A.1) m 2 ẍ 2 (t) = k 2 {x 2 (t) x 1 (t)} k 3 m 1 = m 2 M k 1 = k 2 K K M ω2 0 ẍ 1 (t) = 2ω0x 2 1 (t) + ω0x 2 2 (t) ẍ 2 (t) = ω0x 2 1 (t) ω0x 2 2 (t) (A.2) (A.3) (A.4)! x 1 (t) x(t) = x 2 (t)! 2 1 H = 1 1 (A.5) ẍ(t) = ω 2 0 H x(t) (A.6) H H s det (H s I) = 0 (A.7) I s = 3 5, 3 + 5 2 2 s 3 5 ω 1 = ω 0 = 0.618ω 0 2 s 3 + 5 ω 2 = ω 0 = 1.618ω 0 2 (A.8) (A.9) 3 4

3: M = 70[t] = 70, 000[kg], K = 20, 000[kN/m] ω 0 = 16.9[rad/s] (A.10) ω 1 = 10.444[rad/s], ω 2 = 27.344[rad/s] (A.11) T 1 = 0.601[s], T 2 = 0.229[s] (A.12) T 1 3 A.3 p80-83 {m n, k n } T 1 = 0.341 [s], T 1 = 1.083 [s] 5

4: A.4 1 k n s h T = 2π g (A.13) h = 20 [m] T = 8.8 [s] 1.2 [s] h N N 1 6

5: 7

A.5 A.5.1 N ẍ(t) = ω0 2 H x(t) ẍ 1 (t) 2 1 x 1 (t) ẍ 2 (t) = ω0 2 1 2 1 x 2 (t)..... ẍ N (t) 1 1 x N (t) N x 0 (t) = 0, x N+1 (t) = x N (t) x 0 (t) ẍ 1 (t) 1 2 1 x 1 (t) ẍ 2 (t) = ω0 2 1 2 1 x 2 (t)..... ẍ N (t) 1 2 1 x N (t) x N+1 (t) (A.14) (A.15) n (n = 1, 2,..., N) ẍ n (t) = ω 2 0{x n+1 (t) 2x n (t) + x n 1 (t)} (A.16) x n (t) = ψ(nd, t) d 2 ψ(nd, t) 2 ψ(nd + d, t) 2ψ(nd, t) + ψ(nd d, t) t 2 = (ω 0 d) d 2 (A.17) N h = Nd d d 0 nd ξ 2 ψ(ξ, t) t 2 = (ω 0 d) 2 2 ψ(ξ, t) ξ 2 (A.18) K M G 2 ρ K = G S d ω 0 = M = ρsd r s K M = G ρ d 2 s G ρ v 2 http://ja.wikipedia.org/wiki/ 2 ψ(ξ, t) t 2 = G ρ 2 ψ(ξ, t) ξ 2 (A.19) (A.20) (A.21) 8

A.5.2 λ 1 h λ 1 = 4h (A.22) λ 1 = vt 1 T 1 T 1 = 4L v = 4 r ρ G h (A.23) T 1 = 0.02h 9

B - W.T.Thomson http://www2.city.suginami.tokyo.jp/news/detail/6257/kagakunotane06 2.pdf B.1 RLC with R L C RLC v(t) = V cos ωt (B.1) i(t) q(t) KVL i(t) = dq dt q(t) (B.2) Ri(t) + L di dt + q(t) = v(t) (B.3) C L d2 q dt 2 + R dq dt + q(t) = v(t) (B.4) C B.2 - M, k, Γ, f(t), q(t) M d2 q dt 2 + Γdq + kq(t) = f(t) dt (B.5) 10

q v f M C = 1 k Γ 1 2 Mv2 q i v L C R 1 2 Li2 1 2C q2 1 2C q2 1: - B.3 RLC q ω k M ω = r k M (B.6) RLC - RLC 1 LC ω = 1 LC (B.7) B.4 B.4.1 RLC L d2 q dt 2 + R dq dt + q(t) = V cos ωt (B.8) C 11

ω cos ωt sin ωt R A cos ωt + B sin ωt (B.9) ansatz cos sin exchange (B.8) L d2 q dt 2 + R d q dt + q C = V ejωt (B.10) (1) (B.10) q(t) (2) (B.8) q(t) q(t) = < q(t) (B.8) (1) (B.8) (B.8), (B.10) L di dt + Ri(t) + 1 C Z L dĩ dt + Rĩ(t) + 1 Z C ĩ(t) = Ie jωt i(t)dt = V cos ωt ĩ(t)dt = V e jωt µ jωl + R + 1 I = V jωc (B.11) (B.12) (B.13) (B.14) 12

( ) Z jωl + R + 1 jωc I V ZI = V (B.15) (B.16) ĩ(t) = V Z ejωt = V Z) ej(ωt arg (B.17) Z (2) i(t) = <ĩ(t) = V cos(ωt arg Z) Z (B.18) arg Z phase-delay Z Z ωl 1 ωc = 0 ω = 1 LC (B.19) B.4.2 RLC KVL Z jωl (jωc) 1 13

Z I V V = ZI (B.20) ω (1) jωl (jωc) 1 Z (2) i(t) = V Z cos(ωt arg Z) B.4.3 M d2 q dt 2 + Γdq dt + 1 q(t) = F cos ωt C (B.21) C = k 1 v i = dq dt f v = dq dt M dv dt + Γv(t) + 1 Z v(t)dt = F cos ωt C M dṽ dt + Γṽ(t) + 1 Z C ṽ(t)dt = F e jωt (B.22) (B.23) ( ) ṽ(t) = V e jωt µ jωm + Γ + 1 Q = F jωc µ Z Γ + j ωm 1 ωc F V ZV = F ṽ(t) = F Z ejωt = F Z) ej(ωt arg (B.28) Z v(t) = <ṽ(t) = F cos(ωt arg Z) Z (B.29) (B.24) (B.25) (B.26) (B.27) f v = f Γ 14

arg Z phase-delay Z Z http://ew3.ee.uec.ac.jp/etc/acoustics 1.pdf http://www.logos.ic.i.u-tokyo.ac.jp/ chik/infotech/murota-1.pdf M jωm ωm 1 ωc ω = 1 MC (B.30) C (jωc) 1 Z F Q ZV = F (B.31) v(t) = F Z cos(ωt arg Z) 15

C H T = H --, C.1! 2 1 H = 1 2 (C.1) H 6 6: H!! x, y i = 1 0, j = 0 1 H H = a b (C.2)! 2 a 1! 1, b 2 (C.3) 16

H i, j a, b H i = a H j = b (C.4) c i +d j c a +d b 7: H i, j c i +d j c a +d b i, j H i = a, H j = b 1 det(h) = 3 [ ] H H 1 H H 1 = H 1 H = I (C.5) I H 6 H C.2 H u u 6 H u = s u (C.6) s det(h s I) = 0 (C.7) s 1 = 1, s 2 = 3 17

s 1 = 1, s 2 = 3!! u 1 = 1 1, u 2 = 1 1 2 1 2 1 H u 1 = u 1 H u 2 = 3 u 2 (C.8) (C.9) 8 i, j u 1, u 2 H H u 1 s 1 = 1 u 2 s 2 = 3 C.3 H i, j u 1, u 2 i, j u 1, u 2 U u 1 u 2 (C.10) i, j u 1, u 2 U i = u 1 (C.11) U j = u 2 u 1, u 2 i, j U U 1 U! U = 1 1 1 = 2 1 1 cos 45 o sin 45 o! sin 45 o cos 45 o 45 o!! U 1 = 1 1 1 cos 45 o sin 45 o = 2 1 1 sin 45 o cos 45 o (C.12) (C.13) 45 o 18

8: H c u 1 +d u 2 cs 1 u 1 +ds 2 u 2 u 1, u 2 H u 1 = s 1 u 1, H u 2 = s 2 u 2 H U 1, S, U 19

9: H U 1, S, U H (i) 45 o (ii) s 1 = 1 s 2 = 3 (iii)45 o 20

U, U 1 H! U 1 s 1 0 H U = S 0 s 2 (C.14) U 1 H U = S H = U S U 1 (C.15) H 8 U 1 : 45 o i, j u 1, u 2 S: s 1 = 1 s 2 = 3 S U: 45 o i, j C.4 H H U 1, S, U U 1 U S s 1 = 1 s 2 = 3 s 1 s 2 S s 1 0 det(s) = Ø 0 s 2 Ø = s 1s 2 (C.16) [ ] 2 2! a 11 a 12 A = a 21 a 22 (C.17) det(a) = Ø a 11 a 12 a 21 a 22 Ø = a 11a 22 a 12 a 21 (C.18) H 2 1 det(h) = Ø 1 2 Ø = 3 (C.19) s 1 s 2 = 3 21

[ ] 2 2 det(a s I) = 0! a 11 a 12 A = a 21 a 22 (C.20) (a 11 s)(a 22 s) a 12 a 21 = 0 (C.21) s 1, s 2 (a 11 s)(a 22 s) a 12 a 21 = (s 1 s)(s 2 s) (C.22) s = 0 a 11 a 22 a 12 a 21 = s 1 s 2 (C.23) Ø ØØ Ø a 11 a 12 a 21 a 22 Ø = Ø s 1 0 0 s 2 Ø (C.24) (Q.E.D.) H det(h) = det(s) H [ ] 3 3 3 3 C.5 H det(h) H 1 det(h 1 ) H det(h 1 ) = 1 det(h) = 1 3 det(h 1 ) det(h) = 1 (C.25) (C.26) 22

[ 1] det(h) = 0 det(h 1 ) = det(h) = 0 H 1 [ 2] A, B det(a) det(b) = det(a B) (C.27) (C.26) H 1 H = I C.6 H = a b! b a (C.28) a b (1) H s 1, s 2 (2) H u 1, u 2 u i 1 (3) U = u 1 u 2 (C.29) H H = U S U 1? U 1 :? S U :?? :? (4) (a) U 1 (b) S U 1 (c) U S U 1 H * 23