常用函数的拉氏变换表
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常用函数的拉氏变换表
拉氏变换
L[f(t)]=F(s)=∫0∞f(t)e−stdt;(s+σ+jω为复变量) {L}[f(t)]=F(s)=\int_0^{\infty}f(t)e^{-st}dt ;(s+\sigma+j\omega为复变量) L[f(t)]=F(s)=∫0∞f(t)e−stdt;(s+σ+jω为复变量)
| 序号 | 原函数f(t) | 象函数F(s) |
|---|---|---|
| 1 | δ(t)\delta (t) δ(t) | 1 |
| 2 | ε(t)\varepsilon (t) ε(t) | 1s\frac{1}{s} s1 |
| 3 | t | 1s2\frac{1}{s^2}s21 |
| 4 | tn−1(n−1)!,n=1,2,...\frac{t^{n-1}}{(n-1)!},n=1,2,...(n−1)!tn−1,n=1,2,... | 1sn\frac{1}{s^n} sn1 |
| 5 | e−ate^{-at} e−at | 1s+a\frac{1}{s+a} s+a1 |
| 6 | sinωtsin\omega t sinωt | ωs2+ω2\frac{\omega}{s^2+\omega ^2} s2+ω2ω |
| 7 | cosωtcos\omega t cosωt | ss2+ω2\frac{s}{s^2+\omega ^2} s2+ω2s |
| 8 | 1−e−at1-e^{-at}1−e−at | as(s+a)\frac{a}{s(s+a)}s(s+a)a |
| 9 | e−atsinωte^{-at}sin\omega t e−atsinωt | ω(s+a)2+ω2 \frac{\omega}{(s+a)^2+\omega^2}(s+a)2+ω2ω |
| 10 | e−atcosωte^{-at}cos\omega t e−atcosωt | ω+a(s+a)2+ω2 \frac{\omega+a}{(s+a)^2+\omega^2}(s+a)2+ω2ω+a |
| 11 | 1−11−ξ2e−ξωntsin(1−ξ2ωnt+θ)θ=arctan(1−ξ2/ξ)1-\frac{1}{\sqrt{1-\xi^2}}e^{-\xi\omega_nt}sin(\sqrt{1-\xi^2}\omega_nt+\theta) \\ \theta=arctan(\sqrt{1-\xi^2}/\xi)1−1−ξ21e−ξωntsin(1−ξ2ωnt+θ)θ=arctan(1−ξ2/ξ) | ωn2s(s2+2ξωns+ωn2)\frac{\omega_n^2}{s(s^2+2\xi\omega_ns+\omega_n^2)}s(s2+2ξωns+ωn2)ωn2 |
| 12 | ωn1−ξ2e−ξωntsin(1−ξ2ωnt)\frac{\omega_n}{\sqrt{1-\xi^2}}e^{-\xi\omega_nt}sin(\sqrt{1-\xi^2}\omega_nt)1−ξ2ωne−ξωntsin(1−ξ2ωnt) | ωn2s2+2ξωns+ωn2\frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}s2+2ξωns+ωn2ωn2 |
| 13 | 1β−a(e−at−e−βt)\frac{1}{\beta-a}(e^{-at}-e^{-\beta t})β−a1(e−at−e−βt) | 1(s+a)(s+β)\frac{1}{(s+a)(s+\beta)}(s+a)(s+β)1 |
| 14 | 1a2(e−at+at−1)\frac{1}{a^2}(e^{-at}+at-1)a21(e−at+at−1) | 1s2(s+a)\frac{1}{s^2(s+a)}s2(s+a)1 |
| 15 | 1(n−1)!tn−1e−at,n=1,2,...\frac{1}{(n-1)!}t^{n-1}e^{-at},n=1,2,...(n−1)!1tn−1e−at,n=1,2,... | 1(s+a)n\frac{1}{(s+a)^n}(s+a)n1 |
| 16 | 1a2[1−(1+at)e−at]\frac{1}{a^2}[1-(1+at)e^{-at}]a21[1−(1+at)e−at] | 1s(s+a)2\frac{1}{s(s+a)^2}s(s+a)21 |
| 17 | 1ω2[1−cos(wt)]\frac{1}{\omega^2}[1-cos(wt)]ω21[1−cos(wt)] | 1s(s2+ω2)\frac{1}{s(s^2+\omega^2)}s(s2+ω2)1 |
| 18 | a0(ω0−a02+ω2)1/2ω2cos(ωt+ψ)ψ=arctan(ω/a0)\frac{a_0}{(\omega_0}-\frac{a_0^2+\omega^2)^{1/2}}{\omega^2}cos(\omega t+\psi)\\ \psi=arctan(\omega/a_0)(ω0a0−ω2a02+ω2)1/2cos(ωt+ψ)ψ=arctan(ω/a0) | s+a0s(s2+ω2)\frac{s+a_0}{s(s^2+\omega^2)}s(s2+ω2)s+a0 |
| 19 | 1ab+1ab(a−b)(be−at−ae−bt)\frac{1}{ab}+\frac{1}{ab(a-b)}(be^{-at}-ae^{-bt})ab1+ab(a−b)1(be−at−ae−bt) | 1s(s+a)(s+b)\frac{1}{s(s+a)(s+b)}s(s+a)(s+b)1 |
| 20 | a0ab+a0−aa(a−b)e−at+a0−bb(b−a)e−bt)\frac{a_0}{ab}+\frac{a_0-a}{a(a-b)}e^{-at}+\frac{a_0-b}{b(b-a)}e^{-bt})aba0+a(a−b)a0−ae−at+b(b−a)a0−be−bt) | s+a0s(s+a)(s+b)\frac{s+a_0}{s(s+a)(s+b)}s(s+a)(s+b)s+a0 |
| 21 | a0ab+a2−a1−a0a(a−b)e−at+b2−a1b+a0b(a−b)e−bt)\frac{a_0}{ab}+\frac{a^2-a_1-a_0}{a(a-b)}e^{-at}+\frac{b^2-a_1 b+a_0}{b(a-b)}e^{-bt})aba0+a(a−b)a2−a1−a0e−at+b(a−b)b2−a1b+a0e−bt) | s+a1s+a0s(s+a)(s+b)\frac{s+a_1s+a_0}{s(s+a)(s+b)}s(s+a)(s+b)s+a1s+a0 |
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