ME375 Handouts - Fall 2002 MESB 374 System Modeling and Analysis Laplace Transform and Its Applications

Laplace Transform Motivation Laplace Transform ME375 Handouts - Fall 2002 Laplace Transform Motivation Laplace Transform Review of Complex Numbers Definition Time Domain vs s-Domain Important Properties Inverse Laplace Transform Solving ODEs with Laplace Transform

Motivation Time domain Time domain model solution ME375 Handouts - Fall 2002 Motivation Time domain model Time domain solution Classic calculus techniques Integration & Convolution Frequency domain model Frequency domain solution Algebraic techniques A quick way to solve for the solution of a linear-time-invariant (LTI) ODE for various inputs with either zero or non-zero ICs

Review of Complex Numbers ME375 Handouts - Fall 2002 Review of Complex Numbers Common Forms of a Complex Number: Coordinate Form: Phasor (Euler) Form: Moving Between Representations Phasor (Euler) Form ® Coordinate Form Coordinate Form ® Phasor (Euler) Form Real Img. z x A y e j = + cos sin f

Definition of Laplace Transform ME375 Handouts - Fall 2002 Definition of Laplace Transform Laplace Transform One Sided Laplace Transform where s is a complex variable that can be represented by s=s +jw f (t) is a function of time that equals to 0 when t < 0. Inverse Laplace Transform A function of complex variable s A function of time t A function of time t A function of complex variable s

Laplace Transforms of Common Functions ME375 Handouts - Fall 2002 Laplace Transforms of Common Functions Some simple examples Steps Exponentials Ramps Trigonometric Impulses y t y Unit Step Unit Ramp t y t Exponential

Laplace Transforms of Common Functions ME375 Handouts - Fall 2002 Laplace Transforms of Common Functions t y Trigonometric t y Unit Impulse

Important Properties Differentiation Linearity Given Given a and b are arbitrary constants, then Q: If u(t) = u1(t) + 4 u2(t) what is the Laplace transform of u(t) ? Differentiation Given The Laplace transform of the derivative of f (t) is: For zero initial condition:

Important Properties Integration Given The Laplace transform of the definite integral of f (t) is: Conclusion: Q : Given that the Laplace transforms of a unit step function u(t) = 1 and f(t) = sin(2t) are What is the Laplace transform of

Obtaining Time Information from Frequency Domain ME375 Handouts - Fall 2002 Obtaining Time Information from Frequency Domain Initial Value Theorem Ex: Final Value Theorem y t Note: FVT applies only when f (∞) exists ! y t

Inverse Laplace Transform ME375 Handouts - Fall 2002 Inverse Laplace Transform Given an s-domain function F(s), the inverse Laplace transform is used to obtain the corresponding time domain function f (t). Procedure: Write F(s) as a rational function of s. Use long division to write F(s) as the sum of a strictly proper rational function and a quotient part. Use Partial-Fraction Expansion (PFE) to break up the strictly proper rational function as a series of components, whose inverse Laplace transforms are known. Apply inverse Laplace transform to individual components. Read the slide to emphasize the procedure. Do not forget the linearity

One Example Find inverse Laplace transform of Residual Formula ME375 Handouts - Fall 2002 One Example Find inverse Laplace transform of Residual Formula Conjugate Complex pole results in vibration.

Use Laplace Transform to Solve ODEs ME375 Handouts - Fall 2002 Use Laplace Transform to Solve ODEs Differential Equations (ODEs) + Initial Conditions (ICs) (Time Domain) Solve ODE y(t): Solution in Time Domain L [ · ] L -1 [ · ] Solve Algebraic Equation Algebraic Equations ( s-domain ) Y(s): Solution in Laplace Domain

Examples Q: Use LT to solve the free response of a 1st Order System. ME375 Handouts - Fall 2002 Examples Q: Use LT to solve the free response of a 1st Order System. Q: Use LT to find the step response of a 1st Order System. Zero input response Zero initial condition response Combination of them Q: What is the step response when the initial condition is not zero, say y(0) = 5.

Use LT and ILT to Solve for Responses ME375 Handouts - Fall 2002 Use LT and ILT to Solve for Responses Find the free response of a 2nd order system with two distinct real characteristic roots: Dominant pole

Use LT and ILT to Solve for Responses ME375 Handouts - Fall 2002 Use LT and ILT to Solve for Responses Find the free response of a 2nd order system with two identical real characteristic roots:

Use LT and ILT to Solve for Responses ME375 Handouts - Fall 2002 Use LT and ILT to Solve for Responses Find the free response of a 2nd order system with complex characteristic roots: Conjugate complex poles result in vibration

Use LT and ILT to Solve for Responses ME375 Handouts - Fall 2002 Use LT and ILT to Solve for Responses Find the unit step response of a 2nd order system: 2nd system Zero initial condition response. Forced response