Ch 5 Laplace Transform Company LOGO 1 Company
- Slides: 67
Ch # 5 Laplace Transform Company LOGO 1
Company LOGO Topics 1. Get to know: Laplace Transform 2. Laplace Theory and Properties 3. Applications 2
Company LOGO Get to know: Laplace Transform § Widely used to solve Linear Ordinary Differential Equation )สมการอนพนธสามญเชงเสน ( § With known initial value problems § Laplace Transform: 3 Fundamental steps Step #1: The given ODE is transformed into an algebraic equation called “subsidiary equation” Step #2: The “subsidiary equation” is solved by purely algebraic operations Step #3: The solution in Step 2 is transformed back, resulting in the solution of the given problem 3
Company How to apply Laplace Transform to LOGO Differential Equation Time domain (t) Laplace domain (s) 1 Subsidiary eq (Laplace space) Differential eq (Input space) 2 Solve problem (Laplace space) Solutions (output space) 3 4
Company LOGO Step #1: Forward Laplace Transform �������� integrate ���� 5
Company LOGO Inverse Laplace Transform 6
Company LOGO Laplace Transform (Example#1) The interval of integration in (1) is infinite, such an integral is called improper integral § Laplace transform of a unit function is 1/s L(1) = 1/s 7
Company LOGO Laplace Transform (Example#2) Integration by part 8
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Company LOGO Laplace Transform Properties § Linearity คณสมบตเชงเสนของการแปลงลาปลาซ § First Shifting Theorem คณสมบตการเลอนใน S space (laplace space( § Laplace of Differential Equation § Laplace of Integral Equation § Second Shifting Theorem คณสมบตการเลอนใน t space (time space( § Differentiation of Laplace Transform § Integration of Laplace Transform 10
Company LOGO Proof: 11
Company LOGO Proof: 12
Company LOGO (2) Proof: 13
Company LOGO Non-existence of Laplace Transform Existence of Laplace Transform Non-existence of Laplace Transform 14
Company LOGO Integration by part 15
Company LOGO Helpful of Laplace Transform of Derivative 1. Solving Laplace Transform of complicated f(t) using Laplace Derivative of f(t) no actual integration of Laplace Transform 2. Solving Linear ODE System 16
Company LOGO Solve Laplace Transform of complicated f(t) What is Direct Method Indirect Method using Relation to Laplace of Derivative initial condition Derivative of f(t) Relation to Laplace of Derivative no actual integration of Laplace Transform
Company LOGO Solving Linear ODE System using Laplace Differential Equations: Initial Value Problems Continued
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Company LOGO Slide 9 Continued 21
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Company LOGO Second Shifting Theorem: time shifting (t-shifting) (2) 26
Company LOGO Second Shifting Theorem: time shifting (t-shifting) 27
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Company LOGO If the conversion of f(t) to f(t-a) is inconvenient, replace it by (4**) ใชสตร 1, 2, 3 จากตาราง 6. 9 32
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Company LOGO Kirchoff Voltage Law 34
Company LOGO Take Laplace Transform of Both sides 35
Company LOGO Slide 22 Laplace transform of integral Formula (2), slide 25 Formula (7), table 6. 9 (4*), slide 27 36
Company LOGO Short Impulses : Dirac’s Delta function Continued 37
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Company LOGO Pages 241 -243 c Continued 39
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Example 2: Hammerblow response of a Mass-spring system Company Find the response of the system in example 1 with the square wave replaced by a unit impulse at time t =1 LOGO Page 244 131 1 41
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Company LOGO L{h(t)} 43
Company LOGO Pages 251 -252 44
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Company LOGO Differentiation of Laplace Transform 46
Company LOGO Differentiation of transforms. Formula 21 -23 in Table 6. 9 47
Company LOGO Differentiation of Laplace Transform 48
Company LOGO Differentiation of Laplace Transform 49
Company LOGO Integration of Laplace Transform 50
Company LOGO Systems of ODEs Continued 51
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Company LOGO Page 264 a Continued 56
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Company LOGO Pages 265 -267 a Continued 59
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Company LOGO Pages 265 -267 d Continued 62
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