《高等岩石力学》课程教学资源（PPT课件）4 岩石流变力学

4.1.4Laplace积分变换①定义[r(t)] = F(s) = (~ f(t)e-s$tdtt>0[r()] = 1 H(t) = 3(1）单位阶跃函数0t≤oS0+8(t) = [6(t)] = 1(2)单位脉冲函数100t+H(t) = 8(t)二者关系

4.1.4 Laplace积分变换 ① 定义  (t) F(s) f(t)e dt st 0 −   L f = =  ( ) s L H t 1 = Lδ (t) = 1 ( )      = 0 0 1 0 t t H t ( )     = = 0 0 1 0 t t  t H(t) =δ(t)  ⑴ 单位阶跃函数 ⑵ 单位脉冲函数 二者关系

m.t"(3)幂函数Sm+11一(4)指数函数a+s②逆变换f(t) = L'[F(s)] =2元iJV-Ia例1-1-s+

② 逆变换 ( )  ( ) F (s)e dτ πi f t L F s sτ γ iω γ-iω -  + = = 2 1 1 例 1 ( ) t - - - - e s L s L s s L s s L − = −       + −       =       + = −       + 1 1 1 1 1 1 1 1 1 1 1 1 1   +1 = m m s m! Lt   s L + = − a 1 e at ⑶ 幂函数 ⑷ 指数函数

例 2 Lll+e-t=+一2S2S+③Laplace变换的性质设L[r(t)] = F(s)1）线性性质L[af(t)+ bf,(t)± ..] = al[(t)± bL[(t)+ ..[af(t)± bf;(t)± ..] = a'[()]+ br'[()]±

例 2 ( ) - - t e s s s L s s L − = − +       + = − +       + t 1 1 1 1 1 1 1 2 1 2 1 ③ Laplace变换的性质 1）线性性质 Laf1 (t)  bf2 (t)   = aLf1 (t)  bLf2 (t)   设 Lf (t) = F (s) L -1 af1 (t)  bf2 (t)   = aL-1 f1 (t)  bL-1 f2 (t)  

2）相似性质[r(et)]-3）微分性质[f()] = s"F(s) - s"-f(o) - s"- f'(o) - ... - fa-(o)4）积分性质4r () . .. t → (s)5）位移性质[eat f(t) = F(s - a)

2）相似性质  ( )         = c s F c L f ct 1 3）微分性质  ( ) ( ) (0) (0) (0) (n) n n 1 n 2 (n 1) L f t s F s s f s f f − − − = − −  −  − 4）积分性质 ( ) F (s) s L f t dt dt n t t t 1 0 0 0 =          5）位移性质 e ( ) ( a) at L f t = F s −

6）延迟性质[r(t - a)] = eas F(s)卷积定理下列积分称作函数f和g的卷积，记作（t)*g(t)f(t) * g(t) = ['f(t - ≤);(5)ds卷积定理：两函数卷积的Laplace变换等于两函数Laplace变换的乘积L[r(t) * g(t) = F(s) . G(s)

④ 卷积定理 f(t) g(t) f(t -  )g( )d t 0  = 下列积分称作函数 f 和 g 的卷积，记作f(t)*g(t) 卷积定理：两函数卷积的Laplace变换等于两函 数Laplace变换的乘积。 Lf(t)  g(t) = F(s)  G(s) 6）延迟性质 Lf (t ) F (s) as a e − − =