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

2. 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

3. 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

4. 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

5. 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

6. 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

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

8. 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:

9. 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

10. 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

11. 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

12. 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.

13. 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

14. 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.

15. 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

16. 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:

17. 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

18. 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