6 Movimiento Oscilatorio Forzado Amortiguado

7/24/2019 6 Movimiento Oscilatorio Forzado Amortiguado

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Vibracin forzada amortiguada

)cos( tQkxxcxm f=++ &&&

)cos(2 2 tqxxnx fn =++ &&&

)cos()( 43 tCtsenCx ff +=

)()cos( 43 tsenCtCx fff =&

( ))cos()( 432 tCtsenCx fff +=&&

( ) ( )( ) )cos()cos()(

)()cos(2)cos()(

43

2

4343

2

tqtCtsenC

tsenCtCntCtsenC

fffn

ffffff

=+

+++


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( ) ( )( ) )cos()cos()(

)()cos(2)cos()(

43

2

43432

tqtCtsenC

tsenCtCntCtsenC

fffn

ffffff

=++++

)cos()cos()cos(2)cos()()(2)(

4

2

34

2

3

2

43

2

tqtCtCntCtsenCtsenCntsenC

ffnffff

fnffff

=+++

0)cos(2)(24

2

34

2

3

2

43

2 =++++ tqCCnCtsenCCnCfnfffnff

( )

( ) 02

02

4

2

34

2

3

2

43

2

=++

=+

qCCnC

CCnC

nff

nff

( )22

4

3

2

fn

fCnC

= ( )

( )22222

22

4

4 fnf

fn

n

qC

+

=


3/24



( )222223

4

2

fnf

f

n

nqC

+=

( )( )22222

22

4

4 fnf

fn

n

qC

+

=

( )

( )

( )

)cos(

4

)(

4

222222

22

22222t

n

qtsen

n

nqx f

fnf

fn

f

fnf

f

+

+

+=

)cos( = tAx f

C4 C3


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( )( )

( ) )cos(

4)(

4

222222

22

22222t

n

qtsenn

nqx ffnf

fn

f

fnf

f

++

+=

)cos( = tAx f

( )

( )

( )

2

22222

222

22222

2

44

2

+

+

+=

fnf

fn

fnf

f

n

q

n

nqA

( )222222

2

4 fnfn

qA

+=

( ) ( ) ( ) ( )( )2222

2

2222

2

22222

14144

1

+

=

+

=

+

= n

n

f

n

f

n

n

fnf

q

n

q

n

qA


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)cos( =

tAx f

( )

=

=

=22

2

22 1

2

1

22

arctg

n

arctg

n

arctg

n

f

nf

fn

f

( ) ( ) ( )( )2222max

14

1

+=k

QA

( )

( ) ( ) ( )( )

( )

( ) ( ) ( )( )2

222

2

2222

2

2

14

1

2cos

14

1

2cos

+

=

+

=

arctgt

k

Qarctgt

qx

ff

n


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0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2

Wf/Wn

Beta

n/Wn=2n/Wn=1

n/Wn=0.5

n/Wn=0.25

n/Wn=0.15

n/Wn=0.01

Movimiento oscilatorio forzado amortiguado

Factor de amplificacin

( ) ( ) ( )( )2222 141

+=


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0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2

Wf/Wn

Angulod

efase

n/Wn=2

n/Wn=1

n/Wn=0.5

n/Wn=0.25

n/Wn=0.15

n/Wn=0.01

n/Wn=100

ngulo de fase de la respuesta

( )

=2

1

2

arctg


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Amplitud de la fuerza, Q = 10 Frecuencia, f

= 1

-15

-10

-5

0

5

10

15

0 5 10 15 20

tiempo

X

Fuerza, Wf=1

n/Wn=2

n/Wn=1

n/Wn=0.5n/Wn=0.25

n/Wn=0.15

n/Wn=0.001

n/Wn=100

Constante de restitucin k=2, Frecuencia natural n

= (g/Xest

)1/2 = 1.4


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= 2


= (g/Xest

)1/2 = 1.4

-15

-10

-5

0

5

10

15

0 5 10 15 20

tiempo

X

Fuerza, Wf=2

n/Wn=2

n/Wn=1

n/Wn=0.5

n/Wn=0.25n/Wn=0.15

n/Wn=0.001

n/Wn=100


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= 1.4


= (g/Xest

)1/2 = 1.4

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20

tiempo

X

Fuerza, Wf=1.4

n/Wn=2

n/Wn=1

n/Wn=0.5

n/Wn=0.25

n/Wn=0.15

n/Wn=100


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= 1.4


= (g/Xest

)1/2 = 1.4

-300

-200

-100

0

100

200

300

0 5 10 15 20

tiempo

X

Fuerza, Wf=1.4

n/Wn=2

n/Wn=1

n/Wn=0.5

n/Wn=0.25

n/Wn=0.15

n/Wn=0.001

n/Wn=100


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)cos( = tAx f

)2

cos()( +== tAtAsenx ffff&

xtAx fff22 )cos( ==&&

referencia

kA

cwfA

mwf2A

Q


13/24

Vibracin forzada amortiguada: anlisis transitorio

ph xxx +=

( )222223

4

2

fnf

f

n

nqC

+= ( )

( )2222222

4

4 fnf

fn

n

qC

+

=

)cos()()( 43 tCtsenCtx ffp +=

( ) )()cos( 21 tsenCtCetx ddtn

h +=

( ) 4100 CCxtx +===

( ) ( )fd CCnCxtx 32100 ++=== &&

( )

pd

CnC

d

tn

d

d

do

tn

xtsentCe

tsennxx

txetx

d

f +

+

++=

+

)()cos(

)()cos(

34

4

00

&


14/24

Vibracin forzada amortiguada, aplicacin: DESBALANCE ROTATORIO

( ) ( )[ ] xckxtext

mxmM &&& =++ cos

2

2

( ) xckxtemxM &&& = cos2


15/24



( )( )

( ) )cos(

4)(

4

222222

22

22222t

n

qtsen

n

nqx f

fnf

fn

f

fnf

f

+

+

+=

)cos( = tAx f

C4 C3

( ) ( ) ( ) ( )( )22222

22222 144

1

+=

+= n

fnf

q

n

qA

( )

=

=

2221

22

arctgnarctg

fn

f

( ) ( ) ( ) ( )( )22222

2

22222

2

144

1

+=+= n

fnfne

A

m

M


16/24

( ) ( ) ( )( )2

222

2

2

14

+

= n

e

A

m

M

( )

=2

1

2

arctg


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)cos( = tAx f

)2

cos()( +== tAtAsenx ffff&

xtAx fff22 )cos( ==&&

referencia

kA

cwfA

mwf2A

Q

Ft

( ) ( ) ( )( )2

222 14

1

+

=

k

QA

( ) ( ) ( )( )222222

22222

1441

41

+

+

=+=+=

Q

kAAckF ft

( )

=

21

2

arctg

( ) ( ) ( )( )222222

14

41

+

+=Q

Ft


18/24

( ) ( ) ( )( )2222

22

14

41

+

+=

Q

Ft


19/24

Fuerza pulsante peridica generalizada: Introduccin


20/24

Fuerza pulsante peridica generalizada: Aplicacin de Series de Fourier

++=11

0 )()cos()( tnsenbtmaatF nm

mndttnsentmsen

= )()(

mn

dttntm

=

)cos()cos(

0)()cos( = dttnsentm


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F(t) = Pwt/

para t perteneciente al intervalo [0, /w]

F(t) = Pwt/

2P

para t perteneciente al intervalo (/w,2/w]

Para describir la fuerza pulsante en funcin del tiempo se utiliza series deFourier.

Fuerza pulsante peridica generalizada: Aplicacin de Series de Fourier

/w

2/w


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ao

= w/2

0/w

Pwt/

dt

+ w/2

/w2/w

(Pwt/

2P) dt

ao

= P w2/(22) 02/w

t dt

-

P w/

/w2/w

dt

= P w2/(22 ) (t2/2)02/w

-

P w/

(t)/w2/w

= P

P = 0

an

= w/

0/w

Pwt/

cos(nwt) dt

+ w/

/w2/w

(Pwt/

2P) cos(nwt) dt

= w/

0

2/w

Pwt/

cos(nwt) dt

-

2P w/

/w

2/w

cos(nwt)dt

an

= Pw/(n2 ) 02/w

nwt

cos(nwt) dt

-

2P w/

/(nw) /w2/w

nw

cos(nwt)dt

Si u = nwt

, du

= nw

dt

, lmites de integracin: u

= nwt

an

= Pw/(n2)/(nw) 02n

u cos(u) du

-

2P w/

/(nw) n2n

cos(u) du

= Pw/(n2)/(nw) |cos(u) + u sen(u)| 02n

- 2P w/

/(nw) |sen(u)| n2n

= 0


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bn

= w/

0/w

Pwt/

sen(nwt) dt

+ w/

/w2/w

(Pwt/

2P) sen(nwt) dt

= w/

02/w

Pwt/

sen(nwt) dt

2P w/

/w2/w

sen(nwt) dt

= Pw/(n2)/(nw) 02n

u sen(u) du

-

2P w/

/(nw) n2n

sen(u) du

= Pw/(n2)/(nw) |sen(u) -

u cos(u)| 02n

+ 2P w/

/(nw) |cos(u)| n2n

= -

Pw/(n2)/(nw) 2n

+ 2P/(nw) w/

(1

(-1)n) =

bn

= -

2 P/(n) + 2P/(n) (1

(-1)n) = 2P(-1)n+1/(n )

Por lo tanto:

F(t) = 2P/

(-1)n+1

sen(nwt)/n

con: n = 1,2,3,

F(t) = 2P/

(sen(wt)

1/2sen(2wt) + ....)

Resultado idntico al reportado en Timoshenko

ejercicio 1.11-4.


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6 Movimiento Oscilatorio Forzado Amortiguado

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