05 Oscillateur mécanique en régime forcé

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Chapitre 5 : Oscillateur mécanique en régime forcé. Mécanique. Page 1 sur 7. I Equation différentielle du mouvement de l'oscillateur harmonique amorti en ...
= − ! • "

•

$

•

$

=− # − %

&

&

= −µ ×

(#

'

= −µ ×

=

)

*

=−

⇔

=

+

=−

⇔ +

µ

#Ω

=

#Ω

+ -λ =

.

,

+

−µ +

µ

ω /

= -λ =

ω- =

ω- =

+ -λ + ω - =

#Ω

µ

*

λ ω

1λ 0 =

1ω 0 = 1 0=

2

&

#

#Ω ×

+ (* '

#

−

−

#

'

'

!

#Ω

3 #Ω = +

+ +

+

%

=

=

#Ω

9

(

4

5

&

+ -λ + ω - = 4 9 • 4 > =
⇔ −Ω - +

⇔ −Ω ⇔

+

=

⇔

=

ω

×

Ω

+ω-

-

Ω

=

ωΩ

ω - − Ω- +

(# ω

= −

# =

Ω +ω- =

-

=

%

+

(# ω -

=

Ω

−

-

ω

+

Ω (ω

Ω

ω

#Ω + (# ω -

=

(− )

- -

-

+

-

( −

− =−

# −

+

-

=

( −

π−

< # ≥

-

>

-

! * 9

#Ω

=

#

(# ω -

(− )

- -

=

#

→

-

+

+

∈

.

=

#

→∞

&

#

(− )

- -

+

&'

-

ω-

A

#Ω .

#

#

&

-

&

-

2

#

@#

=

⇔ -# − ⇔

=

-

# −-

=

+

-

−

-

⇔ #

-

-

− -# −

-

=

# ≥

?

•

−

-

-

> ⇔

> ( -

=

#

=

−

>> .

&

−

• =

≈

&

-

-

'

-

&

≈

#

ω-

< ⇔

< ( #

ω-

> ( -

ω-

< ( -

Ω −

-

-

≈

#Ω =

4

C

A

6 Ω+ − Ω−

'

=

4

=

D

=Ω

> =Ω

=

Ω+ − Ω− ≈ Ω

−

-

-

< Ω+

>>

#Ω + ϕ + π ( -

= Ω +π ( -

=

ω

=ω

2

# = #Ω + ϕ # = −Ω #Ω + ϕ = Ω Ω

&

-

Ω− < Ω

&

ω

>>

(− )

- -

+

-

B

! 9

# 2*

% *A

=

#

→

→∞

4

=

#

& = #

#

@#

-

(

# −

- -

⇔-

# −

- -

⇔- # − ⇔ = & &

-

-

+

−

-

+

=

#Ω (# ω

(

⇔

-

= 222 =

⇔

-

+

∆=

-

=

)-

ε×

# −

-

−

-

)=

=

-

Ω=ω # =

-

)-

(# ω - -

⇔

=

⇔

-

(− )

- -

−

-

-

+

=

-

-

-

=ε # −

-

ε =±

%

# (ε = ε

+B

=

Ω

ω

±

=

-

≥ Ω+ − Ω−

=

−

=

+

−

+B

-

−

ω

-

+- (

&

&

− =

−ε

>

#−-

-

-

-

=

-

-

-(

−

-

+

ω #Ω =

⇔

(− )

(-#

#Ω

= # =

-

-

+-

-

- -

+ -# −

-

# − =

=

)-

(# ω

= ⇔-

>

-

.

Ω−

ω

−

+

= =

-

#∀

+B +

=

Ω+

ω

+ =

-

-

+B

>>

&

=

ω&

-

F

Ω

ω

* =

⋅ = −µ × ⋅ = −µ ×

-

= −µ × ×

-

#Ω + ϕ + π ( -

#< !

= µ×

= −

-

×

-

=

-

µ×

ω

=

-

-

-

-

(− )

- -

+

%

-

=

Ω

ω

-

-

!

&

! =

! +

-

& &

# −

- -

=

! +

! #Ω =

= ! #Ω = +∞ =

! #Ω −

Ω+ =

-

!

Ω=ω

.

-

!

&

=

-

µ×

-

ω

&

# −

-

&

E

!

!

&

!

&

-

ω

Ω−

Ω

Ω+

! ! "

:

ε =ε +ε =

ε

-

-

Ω-

=

-

B :

"

ε

-

Ω- +

-

-

+

=

-

B

ε

= =

=

= !

-

B

#Ω + ϕ

Ω- +

=

-

B

(Ω

-

+ω-

) ,

'

-

B

+ !

(Ω

+ω-

-

)

=− !

!

# = +

+∞

µ

."

+

A

×

#"Ω + Ψ"

# . # =

"=

"

#

#

= ."

+∞

2

-π Ω

.

9

,

+

#

$

= !

&

,

"=

.

-

)

:

>

-

-

#

+

:

-

#Ω + ϕ +

9

"

-

=! +!

ε ε

=

.

=

#"Ω

"

# =

"

#"Ω + ϕ " + Ψ"