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Theory of Machines by Sadhu Singh

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A-4

LAPLACE TRANSFORMS

The Laplace transform F(s) of a function f(t) is defined as follows:

app04-ufig1

where ‘s’ is a complex variable called the Laplace operator and is equal to jw(s = σ + ) · e−st is called the kernel of the Laplace transformation.

1. Step input

f(t) = 0 for t < 0

= u0 for t ≥ 0

app04-ufig2

For a unit step,

app04-ufig3

2. Pulse input

f(t) = 0 for t < 0 and t > t0

= u0 for 0 < t < t0

3. Exponentially decaying function

f(t) = e−at

4. Ramp function

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