2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin ωt (2.2) x = A sin ωt, ẋ = Aωcosωt (2.3) ζ = Aω = const. : (2.1) dζ dt iωζ = F (t)/m 2-2
D Dy(x) = f(x) y = g(x) y = g(x) + h(x) D(g + h) = Dg + Dh = f, Dh = 0 (2.4) h ζ = Ae iωt A t ζ = A(t)e iωt dζ dt iωζ A(t) = = Ȧeiωt + Aiωe iωt Aiωe iωt = Ȧeiωt = F (t)/m t 0 iωt F (t) dte m + A 0 ζ(t) = ẋ + iωx = e iωt t 0 iωt F (t) dte m + A 0e iωt A 0 = ζ(0) x ζ ω F (t) 2-3
2.2 2.2.1 ẍ + ω 2 x = F 0 m ( ẍ + ω 2 x F ) 0 = 0 mω 2 x = 0 x = F 0 /mω 2 2.1 : x(t) = F 0 ẋ(0) (1 cos ωt) + sin ωt + x(0) cos ωt mω2 ω x(t) F 0 mω 2 = ( x(0) F 0 mω 2 ) cos ωt + ẋ(0) sin ωt ω 2.2 F = ft : F (t) = ft ζ(t) = f m eiωt t 0 dtte iωt ω x(t) = f (ωt sin ωt) mω3 2-4
x t 2.2.2 F (t) = f cos ω t : : ẍ + ω 2 x = f m cos ω t ẍ + ω 2 x = f m eiω t 2-5
ω ω x = Ae iω t A( ω 2 + ω 2 )e iω t = f m eiω t A = f 1 m ω 2 ω 2 x = f 1 m ω 2 ω cos 2 ω t x = a cos(ωt + φ) + f 1 m ω 2 ω cos 2 ω t x(0) = 0, ẋ(0) = 0 (i) a cos φ + f 1 m ω 2 ω 2 = 0 (2.5) (ii) 0 = aω sin φ (2.6) (ii) sin φ = 0 φ = 0 φ = π φ = 0 : a = f 1 m ω 2 ω 2 φ = π a = f 1 m ω 2 ω 2 cos(ωt + π) = cos ωt 2-6
x = f 1 m ω 2 ω 2 (cos ω t cos ωt) (2.7) 2.3 : F (t) F (t) = f 2 ( ) e iω t + e iω t x(0) = 0, ẋ(0) = 0 ζ = f 2m eiωt t 0 ( ) dt e i(ω ω )t + e i(ω+ω )t ζ = f { i ( ) 2m eiωt e i(ω ω )t 1 ω ω = if 2m + i ( ) } e i(ω+ω )t 1 ω + ω { 1 ω ω ( e iω t e iωt ) + 1 ω + ω ( e iω t e iωt ) } ω : x = f m(ω 2 ω 2 ) (cos ω t cos ωt) (2.7) ω ω ω = ω + ω ω 0 f ωt cos(ωt + ωt) cos ωt lim x(t) = lim ω ω ω 0 m (2ω + ω)( ω) ωt f = t sin ωt (2.8) 2mω 2-7
x(t) : ω = ω ω ω (2.8) cos A cos B = 2 sin A + B 2 sin A B 2 (2.9) cos(x + Y ) = cos X cos Y sin X sin Y cos(x Y ) = cos X cos Y + sin X sin Y A + B = 2ω + ω, A B = ω, ω 2 ω 2 = ω(2ω + ω) x = 2f m 1 ω(2ω + ω) sin ( ω + ω 2 ) t sin ( ω 2 ) t (2.10) sin 2π/(ω + ( ω/2)) 2π/ω sin 4π/ ω 0 4π ω t 2-8
ω 2.2.3 ẍ + 2λẋ + ω 2 x = f t m eiω x = Ae iω t A A = f 1 m ω 2 ω 2 + 2iλω 1 A+iB = A ib A 2 +B 2 A = f ω 2 ω 2 2iλω (2.11) m (ω 2 ω 2 ) 2 + 4λ 2 ω 2 A ib = A 2 + B 2 e iφ, tan φ = B A ω 2 ω 2 2iλω = (ω 2 ω 2 ) 2 + 4λ 2 ω 2 e iφ (2.12) tan φ = 2λω ω 2 ω 2 (2.13) φ ω 2 ω 2 2λω (ω ω ) 2 + 4λ 2 ω 2 2-9
A = f 1 m (ω2 ω 2 ) 2 + 4λ 2 ω 2 e iφ (2.14) (ω 2 (ω 2 2λ 2 )) 2 + 4λ 2 (ω 2 λ 2 ) ω ω = ω 2 2λ 2 x f 1 m 2λ ω 2 λ 2 ω = ω 2 2λ 2 ω (2.13) ω = ω tan φ λ 0 φ (2.14) ω 2 ω 2 2iλω e iφ ω 2 ω 2 2iλω ω ω ω = 0 ω = ω 2-10
ω = 0 A φ = 0ω A φ ( e iφ ω = ω (i/2λω ) φ = π/2 ω A ω ω φ = π λ φ 0 π (2.14) λ = 0 ω = ω A : ω ω ( ) x = R(Ae iω t iφ ) t x = A cos(ω t φ). dt dx = A ω sin(ω t φ)dt. f cos ω t T = 2π/ω W = 1 T T 0 dtf cos ω t ( A ω sin(ω t φ)). cos α sin β = 1 (sin(α + β) sin(α β)), 2 sin(2ω t φ) W = 1 2 f A ω sin φ. φ ω 2 ω 2 2λω (ω ω ) 2 + 4λ 2 ω 2 2-11
2λω sin φ = (ω2 ω 2 ) 2 + 4λ 2 ω 2 W = f 2 4λ 2 ω 2 4mλ (ω 2 ω 2 ) 2 + 4λ 2 ω 2 A 2 ((2.14) ) ω ω 2 [ W = f ( ) ] 2 ω 2 ω 2 2 1 1 + 4mλ 2λω λ ω 0 = ω ( ω 2 ) = W 2λ ω ω ω δ ω ω ω [ ( ) ] 2 1 ωδ(2ω + ωδ) 1 + 2λ(ω + ωδ) 1 1 + ( ωδ λ 1/2 ω = ω ωδ = ±λ 2-12 ) 2.
Q 1 2 δ = ω 2λ Q (Quality factor) 2.2.4 C E R L: C: R: E: L L I Φ Φ = LI E induced = dφ dt = LdI dt 2-13
Φ I C Q Q = CV E RI + Q = E + E induced = E L di }{{} C dt V I = dq/dt or L d2 Q dt + RdQ 2 dt + 1 C Q = E (2.15) d 2 Q dt + R dq 2 L dt + 1 LC Q = E L (2.16) or m d2 x dt + αdx + kx = F 2 dt (2.17) d 2 x dt + 2λdx 2 dt + ω2 x = F m (2.18) Q x ( ) Φ L dq dt mv = p L m E F () R α 1 C k 2-14 ( )
: E = E 0 cos ω t E(t) = E 0 e iω t I(t) = I 0 e iω t I 0 ZI 0 e iω t = E 0 e iω t Z = iω L + R + 1 iω C = E 0/I 0 Z ( / ) ω = 1 LC, λ = R 2L I(t) = I 0 cos(ω t φ) I 0 = E 0 2λω R (ω2 ω 2 ) 2 + 4λ 2 ω 2 tan φ = 2λω ω 2 ω 2 I 0 2 W I 0 ω = ω φ = ω 2λ = 1 L R C 2-15