tu )( tu ≥∀= 0 t 1)( tu <∀= 0 t 0)( 1sin −= a ⋅+⋅−⋅⋅⋅−− tutaea )(1sin111 ⋅⋅−⋅⋅⋅− tutaea )(1sin1 ⋅+ p ⋅+⋅ pp apap +++ ⋅+⋅ pp ⋅⋅⋅ tute ap ++ ⋅ tut )( ⋅ tuA )(

Sciences de l’ingénieur

annexe1 Lycée Jacques Amyot

Auxerre 27/09/2005 Page 1 sur 1

ANNEXE 1

TRANSFORMEES DE LAPLACE COURANTES

Original (variable t)

(Fonction)

Image (variable p)

(Transformée)

Original (variable t)

(fonction)

Image (variable p)

(Transformée)

)

δ (t

¹

) ( 1 ⋅ u t

les fonctions linéaires sont proportionnelles donc :

) (t u A ⋅

p 1

p A

( ) ( ) sin ω ⋅ t ⋅ u t

2

2

ω

ω + p

( ) ( ) cos ω ⋅ t ⋅ u t

2 2

+ ω p

De la même manière :

p ) (t u t ⋅

) (t u t A ⋅ ⋅

et de façon plus générale :

) (t u t A ⋅

ⁿ

⋅

2

1 p p

2

A

1

!

⋅

_n+

p A n

( ) t u (t ) sh ω ⋅ ⋅

2

2

ω

ω

− p

) (t u e

^−a⋅t

⋅

a p +

1 ch ( ) ω ⋅ t ⋅ u (t )

2 2

− ω p

p

) (t u t e

^a⋅t

⋅

ⁿ

⋅

)

1

(

!

− a

ⁿ+

p

n ( ^e

^−a⋅t

^{⋅ sin} ( ) ^{ω ⋅ t} ) ^{⋅ u ( t )}

( )

²

ω

²

ω

+ + a p )

(

1 e u t

t

⎟⎟ ⋅

⎠

⎜⎜ ⎞

⎝

⎛ −

^−τ

^p ⋅ ( 1 + τ ⋅ ^p )

1 ( ^e

^−a⋅t

^{⋅ cos} ( ) ^{ω ⋅ t} ) ^{⋅ u ( t )}

( + )

²

+ ω

²

+ a p

a p

) (t u e t

t

⎟⎟ ⋅

⎠

⎞

⎜⎜ ⎝

⎛ − τ + τ ⋅

^−τ

p ⋅ ( 1 + τ ⋅ p )

1

2

)

2

e u ( t

t

^t

⎟⎟ ⋅

⎠

⎜⎜ ⎞

⎝

⎛ ⋅

^−τ

τ ( 1 )

²

1

⋅ p + τ

D’autre part :

Original (variable t) Image (variable p)

( ¹ ) ^{( )}

sin 1

2

2

e a t u t

a

t

a

⋅ ⋅ − ⋅ ⋅

− ⋅

⋅

⋅

−

ω

ω

_ω

2 2

1 1 2

1

p a ⋅ p + ⋅ + ⋅

ω ω

( ¹ ) ^{( )}

sin 1

1 1

²

2

e a t u t

a

t

a

⋅ ⋅ − ⋅ + ⋅

− ⋅

−

^{− ⋅ω⋅}

ω ϕ

avec :

1

2

sin ϕ = − a

et

cos ϕ = a

⎟ ⎠

⎜ ⎞

⎝

⎛ + ⋅ ⋅ + ⋅

⋅ 2 1

₂ ²

1

1

p a p

p ω ω

Nota :

u (t )

est appelée fonction de Heaviside, elle est telle que :

u ( t ) = 1 ∀ t ≥ 0

et

u ( t ) = 0 ∀ t < 0