F (x) = kx x k F = F (x) U(0) U(x) = x F = kx 0 F (x )dx U(x) = U(0) + 1 2 kx2
x U(0) = 0 U(x) = 1 2 kx2 U(x) x 0 = 0 x 1 U(x) U(0) + U (0) x + 1 2 U (0) x 2 U (0) = 0 U(x) U(0) + 1 2 U (0) x 2 U(0) = a, U (0) = k U(x) = a + 1 2 kx2 F (x) = d U(x) = kx dx y F=-kx O x y F=-kx O x
ω = k m mẍ = kx ẍ + ω 2 x = 0, x e λt λ 2 + ω 2 = 0 λ ± = ±iω x(t) = Ae iωt + Be iωt e ±iωt = ωt ± i ωt x(t) = C ωt + D ωt δ = D/C x(t) = Acos(ωt δ) v(t) = ẋ(t) = Aω (ωt δ) E = K + U = 1 2 ka2 x(t), v(t) x(0) = x 0, v(0) = 0 x(0) = A δ = x 0, v(0) = Aω δ = 0 x 0 = A, δ = 0 x(0) = 0, v(0) = v 0 x(t) = x 0 (ωt), v(t) = x 0 ω( ωt) x(0) = A δ = 0, v(0) = Aω δ = v 0 δ = 0, A = v 0 /ω x(t) = v 0 ω (ωt), v(t) = v 0 (ωt)
f = b v cv 2ˆv ˆv x f = bv = bẋ mẍ + bẋ + kx = 0 b m = 2α, k m = ω2 0 ẍ + 2αẋ + ω 2 0x = 0 x e λt λ 2 + 2αλ + ω 2 0 = 0 λ 1,2 = α ± α 2 ω0 2 ( x(t) = e Ae αt α 2 ω0 2t t + Be α 2 ω0 2t)
b = 0 α = 0 x(t) = Ae iω 0t t + Be iω 0t x(t) = x 0 (ω 0 t δ) α < ω 0 α 2 ω 20 = i ω 2 0 α2 = iω 1 ω 1 = ω 2 0 α2 x(t) = e αt ( Ae iω 1t t + Be iω 1t ) x(t) = Ae αt (ω 1 t δ) A(t) = Ae αt ω 1 < ω 0 x(t) α [α 1 ] = [T ] τ = m b = 1 2α
K 0, U 0 K, U t K + U = K 0 + U 0 + W µδ E(t) = E 0 + W µδ W µδ f = bv t t W µδ = fdr = fvdt = bv 2 dt < 0 0 W µδ < 0 E(t) α ω 1 v = ẋ = d ( Ae αt (ω 1 t δ) ) dt = ω 1 Ae ( (ω αt 1 t δ) + α ) (ω 1 t δ) ω 1 0 α ω 1 v = ω 1 Ae αt (ω 1 t) x t 1.0 0.5 1 2 3 4 5 6 t 0.5 1.0
K(t) = 1 2 mω2 1A 2 e 2αt 2 (ω 1 t) U(t) = 1 2 kx2 = 1 2 ka2 e 2αt 2 (ω 1 t δ) E(t) = 1 2 ka2 e 2αt ( mω 2 1 2 (ω 1 t δ) + k 2 (ω 1 t δ) ) k = mω 2 0 ω 1 ω 0 E(t) = 1 2 ka2 e 2αt mω0 2 ( 2 (ω 0 t δ) + 2 (ω 0 t δ) ) E(t) = k 1 2 A2 e 2αt = E 0 e 2αt τ = 1/(2α) E(t) = e t τ E0 1/e α > ω 0 α 2 ω0 2 = β > 0 ( x(t) = e αt Ae βt + Be βt) β < α x(0) = 0, v(0) = v 0
x(t)ce αt (βt) v(t) = e αt (β (βt) α (βt)) α = ω 0 α λ 1, λ 2 λ 1 = λ 2 x(t) = Ae αt te αt x(t) = (A + Bt)e αt x t v t 0.3 0.04 0.02 0.2 1 2 3 4 5 6 7 t 0.1 0.02 1 2 3 4 5 6 7 t 0.04 x t 1.0 0.5 1 2 3 4 5 6 7 t 0.5 1.0
F (t) mẍ + bẋ + kx = F (t) b k m = 2α, ω F (t) 0 =, f(t) = m m ẍ + 2αẋ + x = f(t) L = d2 dt 2 + 2α d dt + ω2 0 Lx(t) = f(t) f(t) x µ (t) x 1 (t), x 2 (t) x(t) = x µ (t) + Ax 1 (t) + Bx 2 (t) Lx(t) = Lxµ(t) + ALx 1 (t) + BLx 2 (t) = f(t) + A 0 + B 0 = f(t)
f(t) = f 0 (ωt) Lx(t) = ẍ + 2αẋ + x = f 0 (ωt) y(t) f 0 Ly(t) = ÿ + 2αẏ + y = f 0 (ωt) z(t) = x(t) + iy(t) Lx(t) + ily(t) ẍ + iÿ + 2α(ẋ + iẏ) + x + iy = f 0 (ωt) + if 0 (ωt) ẍ + iÿ = z (ωt) + i (ωt) = e iωt z(t) Lz(t) = z(t) + 2αż(t) + ω 2 0z(t) = f 0 e iωt z(t) = Ce iωt ( ω 2 + 2αω i + ω 2 0) z(t) = f 0 e iωt C = ω 2 0 f 0 + 2αω i ω2
C f 0 C C = C e iδ δ C C CC = C 2 = f 2 0 (ω 2 0 ω2 ) 2 + 4α 2 ω 2 f 0 e iδ = C (ω 2 0 + 2αω i ω 2 ) f 0 ( δ + i δ) = C (ω 2 0 + 2αω i ω 2 ) δ = 2αω ω 2 0 ω2 C z(t) = Ce iωt = C e i(ωt δ) x(t) + iy(t) = C ( (ωt δ) + i (ωt δ)) x(t) = C (ωt δ) x oµ (t) = Ae λ +t + Be λ t, λ ± = α ± α 2 ω0 2 x(t) = C (ωt δ) + Ae λ +t + Be λ t
t (Aeλ +t + Be λ t ) 0 t x(t) = C (ωt δ), για t α < ω 0 λ ± = α ± i ω0 2 α2 = α ± iω 1 x oµ = Ae αt (ω 1 t δ 1 ) x(t) = C (ωt δ) + Ae αt (ω 1 t δ 1 ) C 2 = f 2 0 (ω 2 0 ω2 ) 2 + 4α 2 ω 2
C f 0 ω 0 ω α α ω 0 (2αω 0 ) 2 ω ω 0 C ω ω 0 ω = ω 0 C max = f 0 2αω 0 D = (ω0 2 ω2 ) 2 + 4α 2 ω 2 dd dω = 0 ω = ω0 2 2α2 α ω 0 C max f 0 /(2αω 0 ) x t 2 1 2 4 6 8 t 1 t
α ω = ω 0 C 2 C( ω) 2 = 1 2 C 2 max 4αω 2 0 (ω 2 ω 2 0 )2 + 4α 2 ω 2 0 = 1 2 x = ω 2 ω0 2 x 2 4α 2 x 4α 2 ω 2 0 = 0 ω 0 α ω ω 0 ± α δ ( ) 2αω δ = arctan ω0 2 ω2 ω ω 0 δ = arctan ( ) = π 2 C 2 100 80 60 40 C2 w, 1 2, 1 6 C2 w, 1 2, 1 8 C2 w, 1 2, 1 10 20 0.2 0.4 0.6 0.8 1.0 1.2 Ω C 2 ω = ω 0
ω ω ω 0 ω ω 0 δ π 2 π 2 ω δ = π/2 ω = ω 0 δ = π/2 α ω ω 0 δ π Ω 3.0 2.5 2.0 1.5 1.0 0.5 0.5 1.0 1.5 2.0 Ω Ω 0 δ ω = ω 0 π 2
m k l ω m ω R l k(r l) = m v2 R = mω2 R ω 0 = k m ω 2 0(R l) = ω 2 R R = l 1 ω2 ω 2 0 ω ω 0 R ω < ω 0 ϵ R r = R + ϵ F = (F r, F ϕ ) F ϕ = 0 F r = m( r r ϕ 2 ) = k(r l) F ϕ = m(r ϕ + 2ṙ ϕ) r rf ϕ = m(r 2 ϕ + 2rṙ ϕ) = 0
d dt (r2 ϕ) = 0 r 2 ϕ = C R 2 ω = r 2 ϕ ϕ = R 2 r 2 ω ϕ F r ) m ( r R4 r 3 ω2 = k(r l) R = 0 r = ϵ + R r = ϵ m ϵ R4 (ϵ + R) 3 ω2 = ω 2 0(ϵ + R l) ϵ/r 1 ( ε R + 1) 3 1 3ϵ R + 6ϵ2 R 2 + O ( ϵ 3) ϵ ϵ + 3ω 2 ϵ ω 2 R = ω 2 0(ϵ + R l) ω 2 0 (R l) = ω2 R ϵ + (3ω 2 + ω 2 0)ϵ = 0 ϵ = A (ω 1 t δ) A, δ ω 1 ω 1 = 3ω 2 + ω 20 = 3ω 2 + k m
m P k Q t = 0 v 0 ŷ R = OP = OQ = P Q/2 T = (0, R) k m r = k r, r = (x, y) ω 2 = k m x(t) = A (ωt δ 1 ) y(t) = B (ωt δ 2 ) T = 2π/ω A, B, δ 1,2 δ 2 = 0, δ 1 = δ x(t) = A (ωt δ) y(t) = B (ω) y T 0 P O Q x
δ δ = 0, π 2 δ = 0, ϕ = y x = A y A x δ = π 2 A, B ( x A ) 2 + ( y B ) 2 = 1 A, B x, y δ 0, π 2 x, x, y 1.0 0.5 0.0 0.5 1.0 2 1 0 1 2 δ = 0, π 4, π 2
δ 1 = 0 ( x a y ) 2 ( x ) 2 + = 1 b c a, b, c y(x) δ 2 = 0, π/4, π/2 δ 2 π/2 ( x A 2 y ) 2 ( x ) 2 + = 1 B A 4 4 4 2 2 2 0 0 0 2 2 2 4 4 2 0 2 4 4 4 2 0 2 4 4 4 2 0 2 4 ω 1 = 2ω 2 δ = 0, π 4, π 6
k 1 k 2 ω 2 1 = k 1 m, ω2 2 = k 2 m T 1 = 2π, T 2 = 2π ω 1 ω 2 m(ẍî + ÿĵ) = (k 1 xî + k 2 yĵ) x(t) = A (ω 1 t δ 1 ) y(t) = B (ω 2 t δ 2 ) n 1, n 2 T = n 1 T 1 = n 2 T 2 ω 1 ω 2 = n 1 n 2 [ A, A] [ B, B] k 1 = n 1 ω1 = n 1 k 2 n 2 ω 2 n 2
x 1 x 2 x 1 (0) = x 2 (0) = 0, ẋ 1 (0) = υ 1, ẋ 2 (0) = 0 x 1 x 2 mẍ 1 + mg x 1 + k(x 1 x 2 ) = 0 l mẍ 2 + mg x 2 + k(x 2 x 1 ) = 0 l Laplace L[ẍ 1 (t)] = s 2 ( ) ( ) = ( ) υ L[ẍ 2 (t)] = s 2 ( ) 1.0 0.5 2 1 1 2 0.5 1.0 1.0 0.5 2 1 1 2 0.5 1.0 ω 1 = 2, 5 ω2 = 2 δ = π 6
Laplace { m(s 2 υ) = mg } l + ( ) ms 2 = mg l + ( ) F 1, F 2 υ(s 2 + g l ( ) = + k m ) (s 2 + g l ) (2 k m + g l + s2 ) = υ { 1 2 s 2 + g l + 2 k + 1 } s 2 + g m l = υ { 1 2 s 2 + A 2 + 1 } s 2 + B 2 F 2 ω 2 1 = g l, ω2 2 = k m x n (t) = υ [ω 1 t] + ( ) n 1 [( ω1 2 + 2ω2 2 )t] 2 g ω 2 l 1 + 2ω2 2 n = 1, 2 x 1, x 2 ω 1 = 1, ω1 2 + 2ω2 2 = 3 1.0 0.5 5 10 15 20 25 0.5 1.0
m r U(r) = A( r R + a2 R r ) 0 < r < a, A, R r 0 k f(x, y) = 1 x(t) = A (2ωt) y(t) = B (ωt δ) δ = 0, π 4 k k 1, k 2 x 1, x 2 mẍ 1 = k 1 x 1 k 2 (x 1 x 2 ) mẍ 2 = k 1 x 2 k 2 (x 2 x 1 ) x = ( x1 x 2 ) ( 1 A = 1 ) k 1 M k 2 M k 1
r = ω 2 1 /ω2 2 d 2 dt 2 x + ω2 2A x = 0 R R 1 AR = I d 2 dt 2 R 1 x + ω 2 2R 1 ARR 1 x = 0 y = R 1 x a 1, a 2 ÿ 1 + ω 1 y 1 = 0, ÿ 2 + ω1 2 + ω2 2 y 2 = 0 a 1, a 2 ẍ + 2aẋ + ω 2 0x = f(t) f(t) h T h
l m ϕ x y F ϕ = m(r ϕ + 2ṙ ϕ), F r = m( r r ϕ 2 ) r = l = ṙ = r = 0 F ϕ = ml ϕ, F r = ml ϕ 2 mg mg ϕ, mg ϕ T mg ϕ mg ϕ ml ϕ = mg ϕ ml ϕ 2 = T mg ϕ mg ϕ ϕ T ϕ + g l ϕ = 0 ϕ ϕ ϕ + g l ϕ = 0 ϕ(t) = C (ωt + δ), ω = g l
ϕ ϕ d dt ϕ ϕ + ω 2 ϕ ϕ = 0 ( ) 1 2 ϕ 2 ω 2 ϕ = 0 1 2 ϕ 2 ω 2 ϕ = C ml 2 ml 2 C = E 1 2 m(l ϕ) 2 mgl ϕ = E s = lϕ v = ṡ = l ϕ K = 1 2 mv2 U = mgl ϕ K + U = E ϕ max < π v ϕmax = 0 ϕ max = 0 C = ω 2 ϕ max ϕ 1 2 ϕ 2 = ω 2 ϕ ω 2 ϕ max ϕ = ± 2ω ϕ ϕ max
ϕ = dϕ dt dt = ± 1 dϕ 2ω ϕ ϕmax ϕ = [0, ±ϕ max ] l ϕmax dϕ T = 4 2g ϕ ϕmax 0 ( D = ϕ ϕ max = 2 2 ϕ max 2 ϕ ) 2 2 k 2 = 2 ϕ max 2 D = 2(k 2 2 ϕ 2 ) θ ϕ 2 = k θ D D = 2k 2 2 θ θ = [0, π 2 ] dϕ = 2k θdθ ϕ/2 = 2k θdθ 1 2 ϕ/2 dϕ = 2k θdθ 1 k 2 2 θ
D ϕ max < π l T = 4 g π/2 0 dθ 1 k 2 2 θ ϕ 2 = k θ, k = ϕ max 2 T O y mg x O y T e e r mgsin mgcos x mg
k K( k2 k 2 1 ) π/2 dθ 1 k 2 = 0 1 k 2 2 θ k k k = 0 T = 2π l/g τ = T 4 l/g k 1 k 1 + 1 2 k2 2 (θ) + 3 8 k4 4 (θ) + O ( k 5) ϕ 0 = π ϕ0 0 dϕ dt = ± ϕ 2 0 + 2ω2 (1 + ϕ) v = l ϕ v = ± 2gl ϕ ϕ max 2.4 Τ 2.2 2.0 1.8 1.6 1.4 0.2 0.4 0.6 0.8 1.0 k τ
ϕ max ϕ max = π v = ± 2gl ϕ + 1 ϕ = ±nπ ϕ = 0 v = ±2 gl x ϕ/π v ϕ < π m k l 0 T 0 x x 1 (t) = A (5πt), y 1 (t) = B (3πt δ) φ 1.5 1.0 0.5 2 1 1 2 0.5 φ Π 1.0 1.5
x 2 (t) = A (πt), y 2 (t) = B ( 2πt) m v M L R C V I(t) Kirchoff v 1 AB, DC l l 0 m l 0 T T 0 0 l l m m m M
R + - L C l AB 3l/2 CD h A d h > d > d 0 F (r) t m d2 r dt 2 = L2 mr 3 + F (r) ϕ d 2 u + κu + λf(u) = 0 dϕ2 C D A B
κ f(u) F (r) ( 1 F (r) = k r 2 l ) r 3 k, l r(ϕ) A d d 0