Mathematical Formulas for Industrial and Mechanical Engineering

3. Allows development of an analytical model which permits use of a discontinuous

(piecewise continuous) forcing function and the use of an integral term in the forcing

function (important for control)

4. System analysis using Laplace transform

X(s)

G(s)

Dynamic system

Y(s)

YðsÞ5 GðsÞ XðsÞ

yðtÞ5 L

21

ðYðsÞÞ

! inverse Laplace transform

Definition Laplace transform of x(t)

L½xðtÞ5 XðsÞ5

ð

N

0

xðtÞe

2st

dt ðs 5 σ 1 jωÞ

5.39 Table of Laplace Transforms

Function, f(t) Laplace Transform, F(s)

11/s

T 1/s

2

t

2

2/s

3

t

n

n!/s

n11

e

2at

1/(s 1 a)

t

n

e

2at

n!/(s 1 a)

n11

sin(bt) b/(s

2

1 b

2

)

cos(bt) s/(s

2

1 b

2

)

e

2at

sin(bt) b/((s 1 a)

2

1 b

2

)

e

2at

cos(bt)(s 1 a)/((s 1 a)

2

1 b

2

)

sinh(bt) b/(s

2

2 b

2

)

cosh( ...