MIE404 - Lecture 3 - Laplace Transforms and Intro to Block Diagrams.pdf
-
Upload
jashandeep-singh-kochhar -
Category
Documents
-
view
229 -
download
5
Transcript of MIE404 - Lecture 3 - Laplace Transforms and Intro to Block Diagrams.pdf
Lecture Overview
• Last Time• Modelling Mechanical and Electrical Systems
• This Lecture• Review of Laplace Transforms• Transfer Functions• Block Diagrams
9/16/2015 2MIE404 – Fall 2015
Laplace Transforms - Motivation
• Take and ODE (of any order)
• To an algebraic equation
• Benefits
9/16/2015 3MIE404 – Fall 2015
ܯ ሷݔ ݐ + ሶݔ ݐ + ݔ ݐ = (ݐ)
ଶݏܯ ݏ + ݏ ݏ + = (ݏ)ܨ
Laplace Transform - Definition
Given a function f(t) in the time domain, we define its Laplace transform, F(s), as
F(s) is the frequency domain representation of f(t).
9/16/2015 4MIE404 – Fall 2015
Finding the Laplace Transform• Example 1• Heaviside Step Function
9/17/2015 5MIE404 – Fall 2015
ݑ ݐ = ቊ0, ݐ < 01, ݐ 0
Finding the Laplace Transform
9/17/2015 6MIE404 – Fall 2015
• Example 1• Decaying Exponential
ݐ = ቊ 0, ݐ < 0௧ , ݐ 0
Properties of Laplace Transforms• Laplace transforms have many properties we will
exploit in the course• Linearity
• Differentiation
• Integration
9/17/2015 9MIE404 – Fall 2015
Converting Models to Laplace Domain• Consider our Mass-Spring Damper System
9/17/2015 10MIE404 – Fall 2015
f(t)
x(t)
M
b
k
(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ
Block Diagrams• Examples
9/17/2015 15MIE404 – Fall 2015
f(t)
x(t)
M
b
k
(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ
1ଶݏܯ + ݏ +
(ݏ)ܨ (ݏ)
Block Diagrams• Examples
9/17/2015 16MIE404 – Fall 2015
f(t)
x(t)
M
b
k
(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ
1ݏ
ሷݔ ሶݔ +ݔ
ܯ
ܯ
1ݏ
1ܯ
Building Up Block Diagram• Example – Coupled System
9/17/2015 18MIE404 – Fall 2015
x1(t)
M1
b
k1
f(t)M2k2
x2(t)