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Raspunsul dinamic al sistemelor cu 1GDL

Note de curs: www.cosminchiorean.com

Sistem oscilant cu un singur grad de libertate

tumP

tucP

tkuP

in

a

e

Fortele care actioneaza pe directia gradului de libertate

)(tPPPP eain

Echilibrul dinamic: Principiul lui D’Alembert

)()( tPtkutuctum

m

c

m

k

2

2

)(1

)(2 2 tPm

tututu

Vibratii libere neamortizate

02

tutu

tCtCtu sincos)( 21

0;0 00

uuuu

0201 ;u

CuC

tu

tutu

sincos)( 00

Vibratii libere neamortizate

tu

tutu

sincos)( 00

cos

sin

0

0

Au

Au

tAtu sin)(

0

0

2

02

0 ;

u

uarctg

uuA

sT

2 11 s

Tf srad

T/

2

tAtu cos)(

)(sin)( 22 tutAtu

tAtu sin)(

Raspunsul dinamic la actiunea unui impuls finit

m

Hu

tu

tutu

sin)cos()( 00

m

Hu

u

0

0 0

tm

Htu sin)(

Raspunsul dinamic la actiunea unei forte perturbatoare oarecare

)(02 tfm

Ptutu

tdtfm

Ptdu ,sin)( 0

tt

dtfm

Ptdutu

0

0

0

sin)(

Raspunsul dinamic la actiunea unei forte perturbatoare oarecare

2

00

m

P

k

Pust

tudtfm

Ptu st

t

02

0 sin)(

dtftt

0

sin

stst utuu maxmax

Raspunsul dinamic la actiunea unei forte armonice

tPtP sin)( 0 t

m

Ptutu

sin02

)()()( 21 tututu tCtCtu sincos)( 211

tNtMtu sincos)(2

22

0

0

m

PN

M

t

m

PtCtCtu

sinsincos)(

22

021

022

0

0

0

0

uu

uu

22

002

01

m

PuC

uC

ttm

Pt

ututu

sinsinsincos)(

22

000

Raspunsul dinamic la actiunea unei forte armonice

ttm

Ptu

sinsin)(

22

0

tuttm

Ptu st

propriivibratii

fortatevibratii

sinsin

1

1)(

22

0

ttt

sinsin

1

12

2

2

1

1

sin

1

1

tt

tutu st

sin

1

12

Raspunsul dinamic la actiunea unei forte armonice

tttm

Pttt

m

Ptt

m

Ptu

cossin

2

1

2

sin1

cos

1

sinsin

lim)(2

0

2

0

2

22

0

sin2

1

cos2

1

Bt

B

tutBm

Ptu st

cos)(

2

0

1

2

2

ttm

Ptt

m

Ptu

2sin

2cos2

4sinsin

4)( 00

ttAttm

Ptu

cos)(cossin4

2)( 0

Vibratii libere amortizate

02 2

tuutu

m

k

m

c

2

2

02 22 rr

22

2,1 r

Vibratii libere amortizate: Amortizare critica

21

22 0

rr

cr

mkm

kmmmccr 2222

m

m

c

c

cr 2

2

tCCetu t

21)(

tuuuetu t

000)(

Vibratii libere amortizate: Amortizare supracritica

;1;crcc

22

2,1

22 0

r

ttt eCeCetu2222

21)(

Vibratii libere amortizate: Amortizare subcritica

;1;crcc

*2222

2,1 iir

2

2

22* 11

tCtCetu t *

2

*

1 sincos)(

tAetu t *sin)(2

1

2

2

2

1

C

Carctg

CCA

tu

ttuetu

uuC

uC

t *

*

0*

*

*

0

*

0

*

02

01

sinsincos)(

Vibratii libere amortizate: Amortizare subcritica

Vibratii libere amortizate: Amortizare subcritica

Vibratii libere amortizate: Amortizare subcritica

Vibratii fortate amortizate

tm

Ptuutu

sin2 02

)()()( 21 tututu tNtMtu sincos)(2

02

2

22

022

NM

m

PNM

22222

0

22222

22

0

4

2

4

m

PN

m

PM

cos

sin

AN

AM

22

22222

022

2

4

1

arctgN

Marctg

m

PNMA

tm

PtAtu sin

4

1sin)(

22222

02

Vibratii fortate amortizate

stationarfortatevibratii

liberevibratii

tt tm

PtAetAtAetu

sin

4

1sinsinsin)(

22222

0**

tutm

Ptutu st

*

2

22

2

2

22

02 sin

41

1)()(

tt sin

41

1

2

22

2

2

2

*

2

22

2

2

2

*

41

1

Vibratii fortate amortizate

Vibratii fortate amortizate

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