Dinamik -Sistemet Me Një Shkallë Lirie

Punoi: Burim ZEQIRI Faqja 1

( )P t

0P

/ 2 0

m

Detyra 1:

Nga tabelat e mposhtme jan dhne karakteristikat gjeometrike dhe fiziko-mekanike t skems s

treguar n figur:

Ec(GPa) Es(GPa) m(ton) a(m) b(cm) h(cm) d(cm)

3

29 200 15 4 4 25 5*

7

Rastet e ngarkimit

I P(t)=Po sin( )t Po=40kN 45rad

II P(t) grafikisht Po=40kN 0.2s


36.79

73.57

36.79

73.57

147.15Q KN

1) T bhet analiza e lkundjeve t lira t strukturs t gjendet karakteristikat dinamike (

, ,T f ) pr tre rastet e ndryshme t variabls s dhna.

229 29000000c

KNE GPa

m

2200 200000000s

KNE GPa

m

3 340.25 0.5 0.0026041

12 12c

bhI m

275520.83c cE I KNm

Pr :

1 0.03d m 2 2

2

1

0.030.0007068

4 4

dA m

1 141360sE A KN

2 0.05d m 2 2

2

2

0.050.0019631

4 4

dA m

2 392620sE A KN

3 0.07d m 2 2

2

3

0.070.003847

4 4

dA m

3 769545sE A KN

15000 9.81 147150 147.15Q m g N KN

a) Aplikojm ngarkesn 147.15Q KN


294.30

52.03

-36.79

-36.79

-52.03

0.25

0.25

0.5

0.5

1.0P

M

N

b) Aplikojm ngarkesn njsi 1.0P


2.0

0.353

-0.25

-0.25

-0.353

M

N

Llogarisim zhvendosjen statike

st

MM NNU dx dx

EI EA

1

1 1 1( 4 2 2 294.3) (0.3535 52.025 5.656 2 36.78 0.25 2 4)3

stUEI EA

a) Pr rastin kur d=3cm

,1

1 1 1( 4 2 2 294.3) (0.3535 52.025 5.656 2 36.78 0.25 2 4)

75520.83 3 141360stU

,1 0.022756stU m


1

,1

9.8120.7538

0.0227756st

g rad

U s

1

1

2 2 3.1410.30269

20.7538T s

1

1

1 13.306

0.30269f Hz

T

b) Pr rastin kur d=5cm

,2

1 1 1( 4 2 2 294.3) (0.3535 52.025 5.656 2 36.78 0.25 2 4)

75520.83 3 392620stU

,2 0.02150stU m

2

,1

9.8121.3605

0.02105st

g rad

U s

2

2

2 2 3.1410.29414

21.3605T s

2

2

1 13.399

0.29414f Hz

T

c) Pr rastin kur d=7cm

,3

1 1 1( 4 2 2 294.3) (0.3535 52.025 5.656 2 36.78 0.25 2 4)

75520.83 3 769545stU

,3 0.021149stU m

3

,3

9.8121.536

0.021149st

g rad

U s

3

3

2 2 3.1410.2916

21.536T s

3

3

1 13.428

0.2916f Hz

T

Duke i krahasuar t tre rastet kur kemi diameter t ndryshm t shufrs (d=3cm, d=5cm, d=7cm),

vrejm se me rritjen e diametrit zvoglohet zhvendosja statike e strukturs, me rast me rritjen e

diametrit rritet edhe shtangsia e strukturs q nnkupton zvoglimin e periods s lkundjeve t lira.


( )P t

m

2) Pr rastin I t ngarkimit t prcaktohet reagimi dinamik maksimal dhe t ndrtohet

diagramat e forcave t brendshme maksimale (M dhe N).

I P(t)=Po sin( )t Po=40kN 45rad

Pr d=5cm karakteristikat dinamike t fituar mbas kalkulimit jan:

2 21.3605rad

s

2 0.29414T s

2 3.399f Hz

Caktimi i koficentit dinamik rasti pa shuarje

2 2

1 1= = =0.290836

451- 1-

21.360

d 0P =P =0.290840=11.633KN

Caktimi i koficentit dinamik rasti me shuarje


0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.106

0.290

1.0

10%

2 2* 1 21.3605 1 0.1 21.146rad

s

*

2 2

0.2940.2969

1 1 0.1

TT s

- Karakteristikat dinamike t strukturs n rastin me shuarje 10% dhe pa shuarje sht i

neglizhushm.


dP

m

317.56

maxM

a) Caktimi i momenteve maksimale

158.783dQ P KN


56.317

-56.317

-39.69

-39.69

dP

m

maxN

b) Caktimi i momenteve minimale

135.517dQ P KN


271.032

47.906

-33.87

-33.87

-47.906

minM

minN


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

1

2

3

4

5

6

7

8

9

10

11 12 13 14 15 16 17 18 19 20

10

20

30

40

( )tP

( )s

3) Pr rastin II nprmjet zgjedhjes numerike t Integralit t Dyhamelit, t ndrtohen diagrama

e zhvendosjeve dhe shpejtsive t sistemit n funksion t kohs. T krahasohet kjo diagram me

diagramn e zhvendosjeve statike.

II P(t) grafikisht Po=40kN 0.2s

- Integrali i Dyhamelit sht si m posht:

( )0

1sin

t

t tU P t d

m

- Zgjedhja e prgjithshme pr prcaktimin e zhvendosjeve dhe shpejtsis jepen si m posht:

1 11

cos( ) sin (1 cos )ii i iP

U U t V t tk

1 1sin cos( ) sini

i i i

PV U t V t t

k

Prcaktojm forcat vepruese pr gjdo interval t kohs s veprimit t forcs.

0.20.01

20 20t s


0.01 0.02 0.03

2

3

t=0.01s

Nga ngjashmria e trekndshave caktojm vlern e forcs pr gjdo intercal t

kohs t. Caktojm vlern e forcs n ordinatn 2.

Pr kohn t=0.015s dhe

0

0.5 0.5

i

i

P P

t t

02 240

0.5 0.1 0.01 0.005 340.5 0.1

PP t t kN

Duke i marrur nga llogaria paraprake karakteristikat dinamike t strukturs themi se:

21.3605rad

s

0.01t s

- Pr kohn t=0

00 0 1 0 11

cos( ) sin (1 cos )P

U U t V t tk

0

400 0 (1 cos21.3605 0.05) 0.01304

6990.499U cm

00 0 1 0 1sin cos( ) sinP

V U t V t tk

040

0 0 21.3605 sin 21.3605 0.05 2.5906990.499

cmV

s

- Pr kohn t=0.015s

11 1 1 1 11

cos( ) sin (1 cos )P

U U t V t tk

11

0.01304 cos(21.3605 0.05) 2.59sin 21.3605 0.0521.3605

38(1 cos21.3605 0.05) 0.051

6990.499

U

cm

11 1 1 1 1sin cos( ) sinP

V U t V t tk

1 21.3605 0.01304 sin 21.3605 0.05 2.590cos(21.3605 0.05)

3821.3605 sin 21.3605 0.05 4.933

6990.499

V

cm

s


Hapi t

t

tP kN

( )stU cm

( )iU cm

( / )iV cm s

0 0.0 0.01 40

0.572205

0.01304 2.590

1 0.01 0.01 38 0.543595

0.051 4.933

2 0.02 0.01 34 0.486374

0.11 6.792

3 0.03 0.01 30 0.429154

0.182 7.821

4 0.04 0.01 26 0.371933

0.264 8.503

5 0.05 0.01 22 0.314713

0.35 8.539

6 0.06 0.01 18 0.257492

0.429 7.544

7 0.07 0.01 14 0.200272

0.499 6.336

8 0.08 0.01 10 0.143051

0.554 4.58

9 0.09 0.01 6 0.085831

0.589 2.356

10 0.1 0.01 2 0.02861

0.6 -0.235

11 0.11 0.01 40 0.572205

0.597 -0.355

12 0.12 0.01 40 0.572205

0.593 -0.459

13 0.13 0.01 40 0.572205

0.588 -0.543

14 0.14 0.01 40 0.572205

0.582 -0.602

15 0.15 0.01 40 0.572205

0.576 -0.634

16 0.16 0.01 40 0.572205

0.570 -0.637

17 0.17 0.01 40 0.572205

0.564 -0.612

18 0.18 0.01 40 0.572205

0.558 -0.585

19 0.19 0.01 40 0.572205

0.553 -0.481

20 0.20 0.01 40 0.572205

0.549 -0.383


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

SHPEJTSIA (cm/s)

ZHVENDOSJA (cm)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

ZHVENDOSJA STATIKE (cm)

ZHVENDOSJA DINAMIKE (cm)

( )t s

( )t s

( )

( / )

U cm

V cm s

( )U cm

a) Krahasimi i zhvendosjeve dinamike dhe shpejtsis

b) Krahasimi i zhvendosjeve statike dhe atyre dinamike.

Dinamik -Sistemet Me Një Shkallë Lirie

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