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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