Laplace transform: Review
Laplace transform: Review
Use
Definition of Laplace transform
Existence
Laplace transforms of some functions of time
Exponential
Impulse
Ramp
Sine, Cosine
.
Properties of Laplace Transforms
Translation:
[pic]
Multiplication by exponential:
[pic]
Differentiation:
[pic]
[pic]
Integration
[pic]
Laplace transforms (Section 2.3):
Final value theorem
Use: Can find f(() even if you do not know f(t).
[pic]
Conditions: f, df/dt Laplace transformable and poles of sF(s) have negative real parts.
Initial value theorem:
[pic]
Conditions: f, df/dt Laplace transformable
• Example: Mass suspended by a spring and a damper it hit by a hammer. Find initial displacement, velocity and final value of displacement.
Inverse Laplace Transform
Idea for finding inverse Laplace transforms. Break down F(s) into components of which you know the inverse Laplace transforms. Add the inverse Laplace transforms of these components.
[pic]
Cases: a) Distinct real poles
b) Complex poles
c) Multiple poles
a) [pic]
[pic]
b) Complex poles
Use: [pic]
[pic]
c) Multiple poles
[pic]
[pic]
Why not use:
[pic]
Try to find a and b:
[pic]
3 equations, 2 unknowns. Need to introduce another fraction so that you have 3 unknowns.
Solving differential equations using Laplace transforms
Example: mass-spring system
Importance of ultra simple models in real life
Problem:
Designed aircraft wing for low vibration.
Large safety margin.
The boss wants to redesign the wing to reduce weight. Recommends reducing plate thickness by 10%. Need to find if vibration level will still be acceptable. Need a quick answer.
.
Simplified model 2
• Find x(t) using simplified model 2
• Check if it still acceptable.
• Example: vibration of aircraft wing if engine detaches from wing.
[pic]
[pic]
[pic]
Linearize non linear systems
Non linear systems – very difficult to analyze
Linearize using Taylor series expansion about static equilibrium position
Example:
[pic]
Nonlinear because of the sin(.
Taylor expansion:
[pic]
-----------------------
Laplace or s domain
time domain
f(t-()
f(t)
(
t
static equilibrium position (no engine)
(0
x(t)
m
k
(
Differential
Equation
Solution
Algebraic
Equation
Laplace
Transform of
Solution
L
L-1
period= [pic]
(
t
x(t)
[pic]
[pic]
Simplification
Beam, same I and A as average I and A of wing section
F(t)=sin(t
F(t)=sin(t
F(t)=sin(t
x(t)
Equivalent static stiffness, k
c
m
Actual structure
Simplified
model 1
mass of wing/2
x(t)
x(t)
[pic]
linear approximation
sin(((
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