Equações Diferenciais Ressonância - ICMC · 2021. 5. 19. · O fenonemo de resonancia ocorre...

Equações Diferenciais

RessonânciaTiago Pereira

[email protected]

ICMC

mailto:[email protected]?subject=

EDO’s com forçamento

Considere a EDO

https://sites.icmc.usp.br/tiago/index.html/edo.html

Ressonancia

Tiago Pereira

March 18, 2020

O fenonemo de resonancia ocorre quando um sistema que exibe oscilacoes amortecidasforcado periodicamente. A resposta do sistema pelo forcamente depende a frequencia doforcamente. Aqui temos duas possibilidades interessantes. O sistema ”ignora” o forcamentose desenvolve oscilacoes com amplitudes desprezivel. Ou absorve a energia do forcamentode maneira otima. Neste caso dizemos que o sistema esta em ressonancia.

Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t

Relembre que a equacao caracteristicaO truque utilizar a formula da Euler

e

i!t = cos!t+ i sin!t

e resolver a EDO na forma complexa

x

00 + �x

0 + !

20x = A0e

i!t

Chutamos que a solucao da particular

xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t

Substitutindo e ansatz na EDO obtemos

B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�

A melhor estrategia escrever B na forma polar

B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1



Considere a EDO


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Forçamento periódico



Considere a EDO


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

R L C



Considere a EDO


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1



Considere a EDO


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Qual é a amplitude de oscilação da resposta?


Truque: Utilizar a forma complexa


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Pela formula de Euler


Forma complexa


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1


Determinando a particular


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Chutando


Determinando a particular


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Obtemos


Voltando para a EDO


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Originalmente

Substituindo xp


Resolvendo


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1


Passando para polares


Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1

Ressonancia

Tiago Pereira

March 18, 2020


Considere a EDO

x

00 + �x

0 + !

20x = A0 sin!t


e



x

00 + �x

0 + !

20x = A0e

i!t


xp(t) = Be

i!t ) x

0p(t) = i!Be

i!t ) x

00p(t) = �!

2Be

i!t


B(�!

2 + i!� + !

20) = A0 ) B =

A0

(!20 � !

2) + i!�


B = re

i✓

E relembrar que

r

2 = BB̄ & ✓ = tan�1 ImB

ReB

1


Caso não homogêneo


Determinando o r

r

2 =A0

(!20 � !

2) + i!�

A0

(!20 � !

2)� i!�

(1)

=A

20

(!20 � !

2)2 + �

2!

2(2)

Portanto

r =A0p

(!20 � !

2)2 + �

2!

2

Donde temos

xp(t) = re

i(!t+✓)

Como a solucao da EDO a parte imaginaria temos

xp(t) = r sin(!t+ ✓)

O ganho em amplitude

g =r

A0

Nos diz quanta energia o sistema consegue obter do foramentoInspecionando obtemos

g(!) =1p

(!20 � !

2)2 + �

2!

2

Vamos analisar o maximo valor de g em funcao de !. Para isso analisamos

dg

d!

=1

2

�4!(!20 � !

2) + 2!�2

[(!20 � !

2)2 + �

2!

2]3/2

Portanto o maximo atingido em

�4!(!20 � !

2) + 2!�2 = 0

Logo

! =

p!

20 + �

2

2

2


Finalmente


Determinando o r

r

2 =A0

(!20 � !

2) + i!�

A0

(!20 � !

2)� i!�

(1)

=A

20

(!20 � !

2)2 + �

2!

2(2)

Portanto

r =A0p

(!20 � !

2)2 + �

2!

2

Donde temos

xp(t) = re

i(!t+✓)




g =r

A0


g(!) =1p

(!20 � !

2)2 + �

2!

2


dg

d!

=1

2

�4!(!20 � !

2) + 2!�2

[(!20 � !

2)2 + �

2!

2]3/2


�4!(!20 � !

2) + 2!�2 = 0

Logo

! =

p!

20 + �

2

2

2


Tomando apenas a parte Im


Determinando o r

r

2 =A0

(!20 � !

2) + i!�

A0

(!20 � !

2)� i!�

(1)

=A

20

(!20 � !

2)2 + �

2!

2(2)

Portanto

r =A0p

(!20 � !

2)2 + �

2!

2

Donde temos

xp(t) = re

i(!t+✓)




g =r

A0


g(!) =1p

(!20 � !

2)2 + �

2!

2


dg

d!

=1

2

�4!(!20 � !

2) + 2!�2

[(!20 � !

2)2 + �

2!

2]3/2


�4!(!20 � !

2) + 2!�2 = 0

Logo

! =

p!

20 + �

2

2

2


Comparando forçamento e Solução





defasagem




Nova Amplitude


Ganho em amplitude


g(!) =r(!)

A0

<latexit sha1_base64="TjO77YGD0fbLVuqHrOwzmyY7JYg=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0Wom5JIQTdC1Y3LCvYBTQiT6SQdOpOEmYlQQtZu/BU3LhRx6xe482+cthG09cCFwzn3cu89fsKoVJb1ZZSWlldW18rrlY3Nre0dc3evI+NUYNLGMYtFz0eSMBqRtqKKkV4iCOI+I11/dD3xu/dESBpHd2qcEJejMKIBxUhpyTMPw5oTcxKiE3gBnUAgnIkfJc8uPSv3zKpVt6aAi8QuSBUUaHnmpzOIccpJpDBDUvZtK1FuhoSimJG84qSSJAiPUEj6mkaIE+lm01dyeKyVAQxioStScKr+nsgQl3LMfd3JkRrKeW8i/uf1UxWcuxmNklSRCM8WBSmDKoaTXOCACoIVG2uCsKD6VoiHSOehdHoVHYI9//Ii6ZzW7Ua9cduoNq+KOMrgAByBGrDBGWiCG9ACbYDBA3gCL+DVeDSejTfjfdZaMoqZffAHxsc3ZfWZdg==</latexit>



0

5

10

1 2 3

!0 = 1

<latexit sha1_base64="uiuAaPKSssA5W5o+ltdm94LmqSQ=">AAAB83icbVDLSgNBEJz1GeMr6tHLYBA8hV0J6EUIevEYwTwgu4TZSW8yZB7LzKwQlvyGFw+KePVnvPk3TpI9aGJBQ1HVTXdXnHJmrO9/e2vrG5tb26Wd8u7e/sFh5ei4bVSmKbSo4kp3Y2KAMwktyyyHbqqBiJhDJx7fzfzOE2jDlHy0kxQiQYaSJYwS66QwVAKGpO/jGxz0K1W/5s+BV0lQkCoq0OxXvsKBopkAaSknxvQCP7VRTrRllMO0HGYGUkLHZAg9RyURYKJ8fvMUnztlgBOlXUmL5+rviZwIYyYidp2C2JFZ9mbif14vs8l1lDOZZhYkXSxKMo6twrMA8IBpoJZPHCFUM3crpiOiCbUuprILIVh+eZW0L2tBvVZ/qFcbt0UcJXSKztAFCtAVaqB71EQtRFGKntErevMy78V79z4WrWteMXOC/sD7/AFivZCb</latexit>

� = 0.1

<latexit sha1_base64="sfvrpSL44xZvJ5S1SWazk9HLhkE=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KokU9CIUvXisYD+gCWWy3bRLd5OwuxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwlRwbVz321lb39jc2i7tlHf39g8OK0fHbZ1kirIWTUSiuiFqJnjMWoYbwbqpYihDwTrh+G7md56Y0jyJH80kZYHEYcwjTtFYyfeHKCWSG+LWvH6l6tbcOcgq8QpShQLNfuXLHyQ0kyw2VKDWPc9NTZCjMpwKNi37mWYp0jEOWc/SGCXTQT6/eUrOrTIgUaJsxYbM1d8TOUqtJzK0nRLNSC97M/E/r5eZ6DrIeZxmhsV0sSjKBDEJmQVABlwxasTEEqSK21sJHaFCamxMZRuCt/zyKmlf1rx6rf5QrzZuizhKcApncAEeXEED7qEJLaCQwjO8wpuTOS/Ou/OxaF1zipkT+APn8wcN/JBk</latexit>

Ganho em amplitude

g

<latexit sha1_base64="SE/C+UXb2dfRF1y6jJlR+hjF/H0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORqUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6tWqtWavUb/M4inAG53AJHlxDHe6hAS1ggPAMr/DmPDovzrvzsWwtOPnMKfyB8/kDzruM8g==</latexit>

!

<latexit sha1_base64="d7KZX8l+HUff4JrmKwFLMt5hKnk=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0J6DHoxWME84BkCbOTTjJmdmaZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03jUo1wwZTQul2RA0KLrFhuRXYTjTSOBLYisa3M7/1hNpwJR/sJMEwpkPJB5xR66RmV8U4pL1S2a/4c5BVEuSkDDnqvdJXt69YGqO0TFBjOoGf2DCj2nImcFrspgYTysZ0iB1HJY3RhNn82ik5d0qfDJR2JS2Zq78nMhobM4kj1xlTOzLL3kz8z+ukdnAdZlwmqUXJFosGqSBWkdnrpM81MismjlCmubuVsBHVlFkXUNGFECy/vEqal5WgWqneV8u1mzyOApzCGVxAAFdQgzuoQwMYPMIzvMKbp7wX7937WLSuefnMCfyB9/kDknePIg==</latexit>



0

5

10

1 2 3

!0 = 1


� = 0.1


Ganho em amplitude

!res

<latexit sha1_base64="dLKQYDWAG3zqVf65jB1EtCJFuSM=">AAAB+XicbVBNS8NAEN34WetX1KOXxSJ4KokU9Fj04rGC/YAmhM120i7dzYbdTaGE/hMvHhTx6j/x5r9x2+agrQ8GHu/NMDMvzjjTxvO+nY3Nre2d3cpedf/g8OjYPTntaJkrCm0quVS9mGjgLIW2YYZDL1NARMyhG4/v5353AkozmT6ZaQahIMOUJYwSY6XIdQMpYEiiIlACK9CzyK15dW8BvE78ktRQiVbkfgUDSXMBqaGcaN33vcyEBVGGUQ6zapBryAgdkyH0LU2JAB0Wi8tn+NIqA5xIZSs1eKH+niiI0HoqYtspiBnpVW8u/uf1c5PchgVLs9xASpeLkpxjI/E8BjxgCqjhU0sIVczeiumIKEKNDatqQ/BXX14nneu636g3Hhu15l0ZRwWdowt0hXx0g5roAbVQG1E0Qc/oFb05hfPivDsfy9YNp5w5Q3/gfP4Ar96Tsw==</latexit>

g(!res)

<latexit sha1_base64="OczkxVEiBeTDdN1IkLgNAqBjn8w=">AAAB/HicbVBNS8NAEN3Ur1q/qj16WSxCvZRECnosevFYwX5AE8JmO0mX7iZhdyOUUP+KFw+KePWHePPfuG1z0NYHA4/3ZpiZF6ScKW3b31ZpY3Nre6e8W9nbPzg8qh6f9FSSSQpdmvBEDgKigLMYupppDoNUAhEBh34wuZ37/UeQiiXxg56m4AkSxSxklGgj+dVa1HATARHxc1cKLEHNLvxq3W7aC+B14hSkjgp0/OqXO0poJiDWlBOlho6dai8nUjPKYVZxMwUpoRMSwdDQmAhQXr44fobPjTLCYSJNxRov1N8TORFKTUVgOgXRY7XqzcX/vGGmw2svZ3GaaYjpclGYcawTPE8Cj5gEqvnUEEIlM7diOiaSUG3yqpgQnNWX10nvsum0mq37Vr19U8RRRqfoDDWQg65QG92hDuoiiqboGb2iN+vJerHerY9la8kqZmroD6zPH0aMlIk=</latexit>

g

<latexit sha1_base64="SE/C+UXb2dfRF1y6jJlR+hjF/H0=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipORqUK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasIbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp6tWqtWavUb/M4inAG53AJHlxDHe6hAS1ggPAMr/DmPDovzrvzsWwtOPnMKfyB8/kDzruM8g==</latexit>

!

<latexit sha1_base64="d7KZX8l+HUff4JrmKwFLMt5hKnk=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0J6DHoxWME84BkCbOTTjJmdmaZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEsGN9f1vb219Y3Nru7BT3N3bPzgsHR03jUo1wwZTQul2RA0KLrFhuRXYTjTSOBLYisa3M7/1hNpwJR/sJMEwpkPJB5xR66RmV8U4pL1S2a/4c5BVEuSkDDnqvdJXt69YGqO0TFBjOoGf2DCj2nImcFrspgYTysZ0iB1HJY3RhNn82ik5d0qfDJR2JS2Zq78nMhobM4kj1xlTOzLL3kz8z+ukdnAdZlwmqUXJFosGqSBWkdnrpM81MismjlCmubuVsBHVlFkXUNGFECy/vEqal5WgWqneV8u1mzyOApzCGVxAAFdQgzuoQwMYPMIzvMKbp7wX7937WLSuefnMCfyB9/kDknePIg==</latexit>



0

5

10

1 2 3

Ganho em amplitude

Determinando o r

r

2 =A0

(!20 � !

2) + i!�

A0

(!20 � !

2)� i!�

(1)

=A

20

(!20 � !

2)2 + �

2!

2(2)

Portanto

r =A0p

(!20 � !

2)2 + �

2!

2

Donde temos

xp(t) = re

i(!t+✓)




g =r

A0


g(!) =1p

(!20 � !

2)2 + �

2!

2


dg

d!

=1

2

�4!(!20 � !

2) + 2!�2

[(!20 � !

2)2 + �

2!

2]3/2


�4!(!20 � !

2) + 2!�2 = 0

Logo

! =

p!

20 + �

2

2

2

Determinando o r

r

2 =A0

(!20 � !

2) + i!�

A0

(!20 � !

2)� i!�

(1)

=A

20

(!20 � !

2)2 + �

2!

2(2)

Portanto

r =A0p

(!20 � !

2)2 + �

2!

2

Donde temos

xp(t) = re

i(!t+✓)




g =r

A0


g(!) =1p

(!20 � !

2)2 + �

2!

2


dg

d!

=1

2

�4!(!20 � !

2) + 2!�2

[(!20 � !

2)2 + �

2!

2]3/2


�4!(!20 � !

2) + 2!�2 = 0

Logo

! =

p!

20 + �

2

2

2



0

5

10

1 2 3

Ganho em amplitude

!res =

r!20 +

�2

2

<latexit sha1_base64="Qqxp4N9s8LjSr0+pMq9drn361Bc=">AAACInicbVBNS8NAEN34WetX1aOXxSIIQklKQT0IRS8eFawWmlom20ld3E3i7kYoIb/Fi3/FiwdFPQn+GLc1B7U+GHi8N8PMvCARXBvX/XCmpmdm5+ZLC+XFpeWV1cra+oWOU8WwxWIRq3YAGgWPsGW4EdhOFIIMBF4GN8cj//IOleZxdG6GCXYlDCIecgbGSr3KgR9LHEAv85WkCnVOD6mvb5XJCsO9qtNd6ocKWOYPQEq4qudZPc97lapbc8egk8QrSJUUOO1V3vx+zFKJkWECtO54bmK6GSjDmcC87KcaE2A3MMCOpRFI1N1s/GJOt63Sp2GsbEWGjtWfExlIrYcysJ0SzLX+643E/7xOasL9bsajJDUYse9FYSqoiekoL9rnCpkRQ0uAKW5vpewabBrGplq2IXh/X54kF/Wa16g1zhrV5lERR4lski2yQzyyR5rkhJySFmHknjySZ/LiPDhPzqvz/t065RQzG+QXnM8v3syj4A==</latexit>

-



0

5

10

1 2 3

Ganho em amplitudeg(!res) =

1q�

2!

20 +

�4

2

g(!res) =1

�!0

1q1 + �2

2!0

3

!0 = 1


� = 0.1


No nosso caso


Aplicações da Ressonância





T




Forçamento na ressonância




Pequenas amplitudesgasto de energia




! ⇡ !0

<latexit sha1_base64="RevTbaQEQ+hO6eTuI1SiRwBmmmw=">AAACAHicbVC7SgNBFJ31GeNr1cLCZjAIVmFXAloGbSwjmAdkl2V2MpsMmRczs2IIafwVGwtFbP0MO//GSbKFJh64cOace5l7T6oYNTYIvr2V1bX1jc3SVnl7Z3dv3z84bBmZa0yaWDKpOykyhFFBmpZaRjpKE8RTRtrp8Gbqtx+INlSKeztSJOaoL2hGMbJOSvzjSHLSRzBCSmn5COfPJEj8SlANZoDLJCxIBRRoJP5X1JM450RYzJAx3TBQNh4jbSlmZFKOckMUwkPUJ11HBeLExOPZARN45pQezKR2JSycqb8nxogbM+Kp6+TIDsyiNxX/87q5za7iMRUqt0Tg+UdZzqCVcJoG7FFNsGUjRxDW1O0K8QBphK3LrOxCCBdPXiati2pYq9buapX6dRFHCZyAU3AOQnAJ6uAWNEATYDABz+AVvHlP3ov37n3MW1e8YuYI/IH3+QMyYZYn</latexit>


Equações Diferenciais Ressonância - ICMC · 2021. 5. 19. · O fenonemo de resonancia ocorre...

Documents

Transcript of Equações Diferenciais Ressonância - ICMC · 2021. 5. 19. · O fenonemo de resonancia ocorre...