Laplace Transform

변환의 정의, 역변환

F (s) = ℒ (f) = ∫_0^∞^(-st) f (t) t

f (t) = ℒ^(-1) (F)

선형성

ℒ {af (t) + bg (t)} = aℒ {f (t)} + bℒ {g (t)}

s 이동 (제 1이동 정리)

ℒ {^atf (t)} = F (s - a)

ℒ^(-1) {F (s - a)} = ^atf (t)

함수의 미분

ℒ (f^′) = sℒ (f) - f (0)

ℒ (f^″) = s^2ℒ (f) - sf (0) - f^′ (0)

ℒ (f^(n)) = s^nℒ (f) - s^(n - 1) f (0) - … - f^(n - 1) (0)

함수의 적분

ℒ {∫_0^tf (τ) τ} = 1/sℒ (f)

t 이동 (제 2 이동 정리)

ℒ {f (t - a) u (t - a)} = ^(-as) F (s)

ℒ^(-1) {^(-as) F (s)} = f (t - a) u (t - a)

변환의 미분

ℒ {tf (t)} = -F^′ (s)

변환의 적분

ℒ {(f (t))/t} = ∫_s^∞F (Overscript[s, ~]) Overscript[s, ~]

합성곱

(f * g) (t) = ∫_0^tf (τ) g (t - τ) τ

= ∫_0^tf (t - τ) g (τ) τ

ℒ (f * g) = ℒ (f) ℒ (g)

f는 주기 p인 주기함수

ℒ (f) = 1/(1 - ^(-ps)) ∫_0^p^(-st) f (t) t

라플라스 변환표

F(s)=L{f(t)} f(t)
1/s 1
1/s^2 t
1/s^n t^(n - 1)/(n - 1) !
1/s^(1/2) 1/πt^(1/2)
1/s^(3/2) 2 (t/π)^(1/2)
1/s^a t^(a - 1)/Γ (a)
1/(s - a) ^at
1/(s - a)^2 t^at
1/(s - a)^n 1/(n - 1) ! t^(n - 1) ^at
1/(s - a)^k 1/(Γ (k)) t^(k - 1) ^at
1/((s - a) (s - b)) 1/(a - b) (^at - ^bt)
s/((s - a) (s - b)) 1/(a - b) (a^at - b^bt)
1/(s^2 + ω^2) 1/ωSin[ωt]
s/(s^2 + ω^2) Cos[ωt]
1/(s^2 - a^2) 1/aSinh[at]
s/(s^2 - a^2) Cosh[at]
1/((s - a)^2 + ω^2) 1/ω^atSin[ωt]
(s - a)/((s - a)^2 + ω^2) ^atCos[ωt]
1/(s (s^2 + ω^2)) 1/ω^2 (1 - Cos[ωt])
1/(s^2 (s^2 + ω^2)) 1/ω^3 (ωt - Sin[ωt])
1/(s^2 + ω^2)^2 1/(2ω^3) (Sin[ωt] - ωt Cos[ωt])
s/(s^2 + ω^2)^2 t/(2ω) Sin[ωt]
s^2/(s^2 + ω^2)^2 1/(2ω) (Sin[ωt] + ωt Cos[ωt])
s/((s^2 + a^2) (s^2 + b^2)) 1/(b^2 - a^2) (Cos[at] - Cos[bt])
1/(s^4 + 4k^4) 1/(4k^3) (Sin[kt] Cos[kt] - Cos[kt] Sinh[kt])
s/(s^4 + 4k^4) 1/(2k^2) Sin[kt] Sin[kt]
1/(s^4 - k^4) 1/(2k^3) (Sinh[kt] - Sin[kt])
s/(s^4 - k^4) 1/sk^2 (Cosh[kt] - Cos[kt])
(s - a)^(1/2) - (s - b)^(1/2) 1/(2 (π t^3)^(1/2)) (^bt - ^at)
1/((s + a)^(1/2) (s + b)^(1/2)) ^(-(a + b) t/2) I_0 ((a - b)/2t)
1/(s^2 + a^2)^(1/2) J_0 (at)
s/(s - a)^(3/2) 1/πt^(1/2) ^at (1 + 2at)
1/(s^2 - a^2)^k π^(1/2)/(Γ (k)) (t/(2a))^(t - 1/2) I_ (k - 1/2) (at)
^(-as)/s u(t-a)
^(-as) δ(t-a)
1/s^(-k/s) J_0 (2kt^(1/2))
1/s^(1/2) ^(-k/s) 1/πt^(1/2) Cos[2kt^(1/2)]
1/s^(3/2) ^(k/s) 1/πt^(1/2) Sinh[2kt^(1/2)]
^(-ks^(1/2)) k/(2 (π t^3)^(1/2)) ^(-k^2/4t)
1/sIn[s] -In[t]-Υ (Υ≈0.5772)
In[(s - a)/(s - b)] 1/t (^bt - ^at)
In[(s^2 + ω^2)/s^2] 2/t (1 - Cos[ωt])
In[(s^2 - a^2)/s^2] 2/t (1 - Cosh[at])
arc Tan[ω/s] 1/tSin[ωt]
1/sarc Cot[s] Si(t)

자유운동 (free motion)

제차미분방정식

m y^″ + c y^′ + k y = 0

m - 질량

m y^″ - 관성력

c y^′ - 감쇠력

k y - 용수철의 힘

강제운동 (forced Motion)

m y^″ + c y^′ + k y = r (t)

r (t) - 외부로부터의 힘

입력 (input)

구동력 (driving force)

라플라스 변환

u (t - a) = {{{0}, {1}} {{t<a 일때}, {t>a 일때}}

ℒ {u (t - a)} = ^(-as)/s

ℒ {u (t - a)} = ∫_0^∞^(-st) u (t - a) t

= ∫_0^a^(-st) 0t + ∫_a^∞^(-st) 1t = -1/s^(-st) | _0^∞ = ^(-as)/s

ℒ {δ (t - a)} = ^(-as)

p285

y^″ + 3y^′ + 2y = r (t), y (0) = 0, y^′ (0) = 0

구형파

r (t) = u (t - 1) - u (t - 2)

t = 1에서의 단위충격

r (t) = δ (t - 1)

s^2Y + 3sY + 2Y = 1/s (^(-s) - ^(-2))

Y (s) = F (S) (^(-s) - ^(-2))

F (s) = 1/(s (s + 1) (s + 2))

F (s) = (1/2)/s - 1/(s + 1) + (1/2)/(s + 2)

f (t) = ℒ^(-1) (F) = 1/2 - ^(-t) + 1/2^(-2t)

y = ℒ^(-1) (F (s) ^(-s) - F (s) ^(-2s))

= f (t - 1) u (t - 1) - f (t - 2) u (t - 2)

s0 = Plot[0, {t, 0, 1}, DisplayFunction→Identity] ; s1 = Plot[{1/2 - ^(-(t - 1)) + 1/2^(-2 (t - 1))}, {t, 1, 2}, DisplayFunction→Identity] ;

s2 = Plot[{-^(-(t - 1)) + ^(-(t - 2)) + 1/2^(-2 (t - 1)) - 1/2^(-2 (t - 2))}, {t, 2, 10}, DisplayFunction→Identity] ;

Show[{s0, s1, s2}, {PlotRange→All, DisplayFunction→$DisplayFunction}] ;

[Graphics:HTMLFiles/index_130.gif]

s^2Y + 3sY + 2Y = ^(-s)

Y = F (s) ^(-s)

F (s) = 1/((s + 1) (s + 2)) = 1/(s + 1) - 1/(s + 2)

f (t) = ℒ^(-1) (F) = ^(-t) - ^(-2t)

y (t) = ℒ^(-1) (^(-s) F (s)) = f (t - 1) u (t - 1) = {{{0, (0≤t<1)}, {^(-(t - 1)) - ^(-2 (t - 1)), (t>1)}}

Plot[{^(-(t - 1)) - ^(-2 (t - 1))}, {t, 1, 10}] ;

[Graphics:HTMLFiles/index_137.gif]

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