Zadatak

Sile F1 i F2 nastaju djelovanjem ruku radnika na ruke elektrine builice .Zamijeniti ovaj sistem sila glavnim vektorom i glavnim momentom za taku O. Rezultate prikazati u vektorskoj obliku.

B

A

Glavni vektor jednak je vektorskom zbiru oba vektora:

N)1416(

421036

kjiF

kjkjiF

FF

O

O

iO

Najea komponenta sile je -14 N i njom se vri pritisak builice o dasku.

Glavni moment jednak je vektorskom zbiru momenata pojedinanih sila za taku O:

21

21

FrFrM

FOBFOAM

MM

BAO

O

OiO

Glavni vektor jednak je vektorskom zbiru momenata pojedinanih sila za taku O:

21

21

FrFrM

FOBFOAM

MM

BAO

O

OiO

4203,025,00

10363,0015,0

kjikjiMO

Nm)45,03,33,1(

4,045,03,39,0

kjiM

ikjiM

O

O

Komponente momenta MOx = 1,3 Nm i MOy = 3,3 Nm utiu na savijanje builice oko tih osa, dok komponenta MOz = - 0,45 Nm utie na uvijanje builice oko ose z.

Zadatak: Skup sila koji djeluje na kvadar prema slici redukovati na taku O (odrediti rezultantnu silu i moment). Zadano je:

1 2 3 4 120 NF F F F

0,3 ma

0,5 mb

0,7 mc

Rjeenje:

Geometrija kvadra

arctg cd

2 2d a b

2 2arctg c

a b

2 2

0,7arctg0,3 0,5

50,2

arctg ab

0,3arctg0,5

30,96

Komponente sile F1

1 1 cosxyF F

1 1 sinzF F

1 1

1 1

sin

cos sinx xy

x

F FF F

1 1

1 1

cos

cos cosy xy

y

F FF F

Glavni vektor rezultantna sila

1 4 1 4cos sinR xX F F F F

120 cos50,2 sin30,96 120RX

1 3 1 3cos cosR yY F F F F

120 cos50,2 cos30,96 120RY

1 2 1 2sinR zZ F F F F

120 sin50,2 120RZ

185,8615

99,516 N

N212, 4 N19

R

R

R

XYZ

159,516 185,869 212,194R R R RF X i Y j Z k i j k Y j Z k i j k159 516 185 869 212 194Y j Z kY j Z k 159 516 185 8696 185 869F X iX iii

2 2 2 2 2 2159,516 185,869 212,194324,066 N

R R R R

R

F X Y ZF

Ravan Oyz moment za osu x

2 1Ox yM F b F c

2 1 cos cosOxM F b F c

120 0,5 120 cos50,2 cos30,96 0,7106,109 Nm

Ox

Ox

MM

Ravan Oxz moment za osu y

0OyM

Ravan Oxy moment za osu z

4 1 3Oz yM F b F a F a

4 1 3cos cosOzM F b F a F a

120 0,5 120 cos50,2 cos30,96 0,3 120 0,3115 N,76 m1

Oz

Oz

MM

106,109 115,761O Ox Oy OzM M i M j M k i k i M j M k i k106 109 115 761i M j M kj M k 106 109

2 2 2 2 2106,109 115,76

157,034 Nm

1O Ox Oy Oz

O

M M M M

M

Zadatak Homogena ploa, oblika i dimenzija kao na slici, vezana je za vertikalnu osovinu u taki A pomou sfernog oslonca, a u taki B pomou cilindrinog oslonca. U taki D vezano je ue, iji je drugi kraj vezan za podlogu u taki E. Teina ploe je G=50 kN, a na plou djeluje vjetar w=10 kN/m2. Odrediti sve reakcije veza. Zadano je: BC=AD=2R, AB=3R, EA=R, R=1m.

R60

2R

1

,

5

R

1

,

5

R

R

x

y

z

A

E

D

C B

GW

T

Reakcije veza ploe Ploa e se osloboditi veza sa okolinom i njihov uticaj zamijenit e se reakcijama.

R60

x

y

z

A D

C B

GW

S

zT

yT

XA

YAZA

XBYB

T

Teite ploe Da bi se postavila aktivna sila teine ploe i sila vjetra, potrebno je odrediti koordinate teita:

m5,1

i

iiT A

zAz

m797,0

232

342

232

2

2

RRR

RRRRRR

AyAyi

iiT

T

y

z

D

C B

zT

yT R

2R

1

,

5

R

1

,

5

R

y

z

A

T

1

2

T1 T2 yT1

yT24R 3

Sila vjetra:

kN3,442

62

2

wRRwAW

Ugao sa slike:

6,2621

2

RR

ADEAtg

R60

x

y

z

D

C B

GW

S

zT

yT

XA

YAZA

XBYB

R60

2R

1

,

5

R

1

,

5

R

R

x

y

z

A

E

GW

T

Statiki uslovi ravnotee ploe su: 1. 060sinsin WSXXX BA 2. 060coscos WSYYY BA 3. 0 GZZ A 4. 0360cos RYzWyGM BTTx 5. 0360sin RXzWM BTy 6. 02sin60sin RSyWM Tz

3R

Iz prethodnih 6 jednaina dobivaju se nepoznate reakcije oslonaca ploe:

Iz 3. kN50GZA

Iz 4. kN36,2460cos31

TTB zWyGRY

Iz 5. kN18,1960sin31

TB zWRX

Iz 6. kN19,3460sinsin21

TyWRS

Iz 2. kN79,3260coscos WSYY BA

Iz 1. kN85,360sinsin WSXX BA

Reakcije oslonaca A i B imaju intenzitete:

kN31

kN92,5922

222

BBB

AAAA

YXF

ZYXF

Homogena ploa ABCD teine G = 6 kN optereena je silama F = 9 kN u takama A i B prema slici. Na plou djeluje spreg sila u ravni ploe momenta M = 5 kNm. Ploa je oslonjena sa 6 tapova zanemarljivih teina prema slici. Odrediti reakcije tapova ako su dimenzije: a = 5 m, b = 5 m, c = 4 m, d = 2 m.

Rjeenje:

Ploa ABCD osloboena veza

2 2 2 2

5arctg arctg 51,3444arctg arctg 38,6655arctg arctg 455

5 5arctg arctg 74,212

accbba

a bd

= = =

= = =

= = = + += = =

i

F ( M

i

)

F

( A )

F

( B )

G M

1

S

2

S

3

S

4

S

5

S

6

S

i

X

F 0 0 /

1

s i nS 0 0

4

s i nS 0

6

s i n s i nS ( 1 )

i

Y

0 0 0 / 0 0

3

c o sS 0

6

s i n c o sS ( 2 )

i

Z

0 - F - G /

1

c o sS 2

S

3

s i nS 4

c o sS 5

S

6

c o sS ( 3 )

i

x

a a a / 2 / 0 0 0 0 0 a

i

y

0 b b / 2 / 0 0 b b b b

i

z

0 0 / 0 0 0 0 0 0

x

i i i i

M

y Z z Y

0

F b

2

b

G

0 0 0

3

s i nb S 4

c o sb S 5

S b

6

c o sb S ( 4 )

y

i i i i

M

z X x Z

0

F a

2

a

G

0 0 0 0 0 0

6

c o sa S ( 5 )

z

i i i i

M

x Y y X

0 0 0

M

0 0 0

4

s i nb S 0

6

6

s i n c o s

s i n s i n

a S

b S

( 6 )

J e d n a i n e r a v n o t e e n a o s n o v u t a b e l e :

F

1

s i nS 4

s i nS 6

s i n s i n 0S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

3

c o sS 6

s i n c o s 0S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

F G +

1

c o sS +2

S +

3

s i nS +4

c o sS +5

S +

6

c o sS = 0 . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 )

3 4 5 6

s i n c o s c o s 0

2

b

F b G b S b S S b b S . . . . . . . . . . . . . . . . . . . . . ( 4 )

6

c o s 0

2

a

F a G a S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 5 )

4 6 6

s i n s i n c o s s i n s i n 0M b S a S b S . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 6 )

i z ( 5 ) :

6

2

4 4 , 0 9 1 k N

c o s

a

F a G

S

a

i z ( 2 ) :

6

3

s i n c o s

3 8 , 4 1 9 k N

c o s

S

S

i z ( 6 ) :

6 6

4

s i n c o s s i n s i n

1 , 2 8 1 k N

s i n

M a S b S

S

b

i z ( 1 ) :

4 6

1

s i n s i n s i n

2 8 , 1 7 4 k N

s i n

F S S

S

i z ( 4 ) :

3 4 6

5

s i n c o s c o s

2

2 4 , 8 k N

b

F b G b S b S b S

S

b

i z ( 3 ) :

2

S = F + G -

1

c o sS -3

s i nS -4

c o sS -5

S -

6

c o sS = 2 0 , 6 k N

Homogena ploa ABCD se odrava u ravnotei pomou sfernog zgloba, cilindrinog zgloba, te tapa zanemarljive teine, prema slici. Odrediti reakcije veza. Potrebni podaci su: G = 10 kN, F = 10 kN, a = 4 m, b = 4 m, c = 6 m

Rjeenje:

Ploa ABCD osloboena veza

6arctg arctg 56,3144arctg arctg 454

90 90 45 45

cbab

= = =

= = = = = =

iF F G AX AY AZ DX DY S

iX sinF 0 AX 0 0 DX 0 0 (1) iY cosF 0 0 AY 0 0 DY cosS (2) iZ 0 G 0 0 AZ 0 0 sinS (3) ix a a/2 0 0 0 0 0 a

iy b b/2 0 0 0 0 0 b

iz 0 c/2 0 0 0 c c c

x

i i i i

My Z z Y

= 0 2

b G 0 0 0 0 Dc Y sincosb Sc S

+ (4)

y

i i i i

Mz X x Z

= 0 2

a G 0 0 0 Dc X 0 sina S (5)

z

i i i i

Mx Y y X

=

cossin

a Fb F

+ 0 0 0 0 0 0 cosa S (6)

Jednaine ravnotee na osnovu tabele:

sinF + AX + DX = 0 ................................................................................... (1)

cosF + AY + DY cosS = 0 ........................................................................ (2)

G + AZ sinS =0 ...................................................................................... (3)

2b G Dc Y sin cosb S c S + =0 ......................................................... (4)

2a G + Dc X + sina S =0 ............................................................................ (5)

cos sina F b F + cosa S =0 ............................................................ (6)

iz(6): cos sincos

a F b FSa

+ = 25,495kNS =

iz (5): sin

2D

aa S GX

c

= 17,475 kNDX =

iz (4): sin cos

2D

b G b S c SY

c

+ = 3,333 kNDY =

iz (3): AZ =G + sinS 31,213 kNAZ = iz (2): AY = cosF DY cosS + 10,404kNAY = iz (1): AX = sinF DX 24,547 kNAX =

563. The cart supports the uniform crate having a mass of85 kg. Determine the vertical reactions on the three castersat A, B, and C. The caster at B is not shown. Neglect themass of the cart.

BA

C0.2 m 0.5 m

0.6 m 0.35 m

0.1 m

0.4 m0.2 m

0.35 m

565. If and , determinethe tension developed in cables AB, CD, and EF. Neglectthe weight of the plate.

y = 1 mP = 6 kN, x = 0.75 m z

F

B

D

A

y

x

x y

E

C

P

2 m2 m

566. Determine the location x and y of the point ofapplication of force P so that the tension developed incables AB, CD, and EF is the same. Neglect the weight ofthe plate.

z

F

B

D

A

y

x

x y

E

C

P

2 m2 m

*568. Determine the magnitude of force F that must beexerted on the handle at C to hold the 75-kg crate in theposition shown. Also, determine the components of reactionat the thrust bearing A and smooth journal bearing B.

F0.1 m

0.2 m

0.5 m

0.6 m

0.1 m

z

xy

A

B

C

569. The shaft is supported by three smooth journalbearings at A, B, and C. Determine the components ofreaction at these bearings.

0.6 mx B

CA

z

0.9 m

0.6 m

0.9 m y0.9 m

0.9 m

0.9 m

900 N

500 N

450 N600 N

570. Determine the tension in cables BD and CD andthe x, y, z components of reaction at the ball-and-socketjoint at A.

z

y

x

C

B A

3 m

300 N

D

1 m

0.5 m

1.5 m

*572. Determine the components of reaction acting at thesmooth journal bearings A, B, and C.

0.6 m45

x y

C

z

B

A0.4 m

0.8 m

0.4 m

450 N

300 N m

573. Determine the force components acting on the ball-and-socket at A, the reaction at the roller B and the tensionon the cord CD needed for equilibrium of the quartercircular plate.

z

x

350 N

1 m

2 m

60

3 m200 N

200 N

yB

A

C

D

577. The plate has a weight of W with center of gravity atG. Determine the distance d along line GH where thevertical force P = 0.75W will cause the tension in wire CD tobecome zero.

z

F

B

D

A

H

y

xG

d

E

C

P

L2

L2

L2

L2

578. The plate has a weight of W with center of gravity atG. Determine the tension developed in wires AB, CD, andEF if the force P = 0.75W is applied at d = L/2.

z

F

B

D

A

H

y

xG

d

E

C

P

L2

L2

L2

L2

579. The boom is supported by a ball-and-socket joint at Aand a guy wire at B. If the 5-kN loads lie in a plane which isparallel to the xy plane, determine the x, y, z components ofreaction at A and the tension in the cable at B.

z

5 kN

5 kN

y

x

3 m

2 m

1.5 m

30

30

BA

585. The circular plate has a weight W and center ofgravity at its center. If it is supported by three vertical cordstied to its edge, determine the largest distance d from thecenter to where any vertical force P can be applied so as notto cause the force in any one of the cables to become zero.

A

d120

120120

Cr

P

B

586. Solve Prob. 585 if the plates weight W is neglected.

A

d120

120120

Cr

P

B

587. A uniform square table having a weight W and sidesa is supported by three vertical legs. Determine the smallestvertical force P that can be applied to its top that will causeit to tip over.

a/2a/2

a

*592. The shaft assembly is supported by two smoothjournal bearings A and B and a short link DC. If a couplemoment is applied to the shaft as shown, determine thecomponents of force reaction at the journal bearings and theforce in the link. The link lies in a plane parallel to the yzplane and the bearings are properly aligned on the shaft.

250 mm300 mm

400 mm250 N m

y

Ax

20

120 mm

30D

B

z

C

123456