Sistemas Binários Moment of a Couple¡tica.pdfthe moments of both forces about a point lying on the line of action of one of the forces. For example, if moments are taken about point

Sistemas Binários

148 CHAPTER 4 FORCE SYSTEM RESULTANTS

4

4.6 Moment of a Couple

A couple is defined as two parallel forces that have the same magnitude,but opposite directions, and are separated by a perpendicular distance d,Fig. 4–25. Since the resultant force is zero, the only effect of a couple is toproduce a rotation or tendency of rotation in a specified direction. Forexample, imagine that you are driving a car with both hands on the steeringwheel and you are making a turn. One hand will push up on the wheelwhile the other hand pulls down, which causes the steering wheel to rotate.

The moment produced by a couple is called a couple moment. We candetermine its value by finding the sum of the moments of both coupleforces about any arbitrary point. For example, in Fig. 4–26, positionvectors and are directed from point O to points A and B lying onthe line of action of and F. The couple moment determined about Ois therefore

However or , so that

(4–13)

This result indicates that a couple moment is a free vector, i.e., it can actat any point since M depends only upon the position vector r directedbetween the forces and not the position vectors and directed fromthe arbitrary point O to the forces. This concept is unlike the moment ofa force, which requires a definite point (or axis) about which momentsare determined.

Scalar Formulation. The moment of a couple, M, Fig. 4–27, isdefined as having a magnitude of

(4–14)

where F is the magnitude of one of the forces and d is the perpendiculardistance or moment arm between the forces. The direction and sense ofthe couple moment are determined by the right-hand rule, where thethumb indicates this direction when the fingers are curled with the senseof rotation caused by the couple forces. In all cases, M will actperpendicular to the plane containing these forces.

Vector Formulation. The moment of a couple can also beexpressed by the vector cross product using Eq. 4–13, i.e.,

(4–15)

Application of this equation is easily remembered if one thinks of takingthe moments of both forces about a point lying on the line of action ofone of the forces. For example, if moments are taken about point A inFig. 4–26, the moment of is zero about this point, and the moment ofF is defined from Eq. 4–15.Therefore, in the formulation r is crossed withthe force F to which it is directed.

-F

M = r * F

M = Fd

rB,rA

M = r * F

r = rB - rArB = rA + r

M = rB * F + rA * -F = (rB - rA) * F

-FrBrA

F

!F

d

Fig. 4–25

O

BA

F

!F

rArB

r

Fig. 4–26

F

!Fd

M

Fig. 4–27

Um Binário é definido como sendo um par de forças paralelas de mesma intensidade e sentidos opostos, separadas pela distância perpendicular às suas linhas de ação d.

Como a resultante das forças é nula o único efeito do binário é produzir uma rotação ou tendência de rotação em uma dada direção.

Momento do binário


4





(4–13)



(4–14)



(4–15)


-F

M = r * F

M = Fd

rB,rA

M = r * F


M = rB * F + rA * -F = (rB - rA) * F

-FrBrA

F

!F

d

Fig. 4–25

O

BA

F

!F

rArB

r

Fig. 4–26

F

!Fd

M

Fig. 4–27

O momento do binário é dado pela soma dos momentos das componentes deste sistema em volta de qualquer ponto.

~M = ~rB ⇥ ~F + ~rA ⇥�~F

~M = ( ~rB � ~rA)⇥ ~F

~M = ~r ⇥ ~F

Importante: Este resultado indica que o momento do binário é um vetor livre, ou seja, ele pode atuar em qualquer ponto pois só depende do vetor posição r entre as forças e não dos vetores rA e rB dirigidos do ponto O até as forças.

Formulação Escalar:


4





(4–13)



(4–14)



(4–15)


-F

M = r * F

M = Fd

rB,rA

M = r * F


M = rB * F + rA * -F = (rB - rA) * F

-FrBrA

F

!F

d

Fig. 4–25

O

BA

F

!F

rArB

r

Fig. 4–26

F

!Fd

M

Fig. 4–27


4





(4–13)



(4–14)



(4–15)


-F

M = r * F

M = Fd

rB,rA

M = r * F


M = rB * F + rA * -F = (rB - rA) * F

-FrBrA

F

!F

d

Fig. 4–25

O

BA

F

!F

rArB

r

Fig. 4–26

F

!Fd

M

Fig. 4–27

Formulação Vetorial:


4





(4–13)



(4–14)



(4–15)


-F

M = r * F

M = Fd

rB,rA

M = r * F


M = rB * F + rA * -F = (rB - rA) * F

-FrBrA

F

!F

d

Fig. 4–25

O

BA

F

!F

rArB

r

Fig. 4–26

F

!Fd

M

Fig. 4–27


4





(4–13)



(4–14)



(4–15)


-F

M = r * F

M = Fd

rB,rA

M = r * F


M = rB * F + rA * -F = (rB - rA) * F

-FrBrA

F

!F

d

Fig. 4–25

O

BA

F

!F

rArB

r

Fig. 4–26

F

!Fd

M

Fig. 4–27

Binários Equivalentes

M2

M1

(a)

4.6 MOMENT OF A COUPLE 149

4

Equivalent Couples. If two couples produce a moment with the samemagnitude and direction, then these two couples are equivalent.For example,the two couples shown in Fig. 4–28 are equivalent because each couplemoment has a magnitude of , and each is directed into the plane of the page. Notice that larger forcesare required in the second case to create the same turning effectbecause the hands are placed closer together. Also, if the wheel wasconnected to the shaft at a point other than at its center, then the wheelwould still turn when each couple is applied since the couple isa free vector.

Resultant Couple Moment. Since couple moments are vectors,their resultant can be determined by vector addition. For example,consider the couple moments acting on the pipe in Fig. 4–29a.Since each couple moment is a free vector, we can join their tails at anyarbitrary point and find the resultant couple moment,as shown in Fig. 4–29b.

If more than two couple moments act on the body, we may generalizethis concept and write the vector resultant as

(4–16)

These concepts are illustrated numerically in the examples that follow.In general, problems projected in two dimensions should be solved usinga scalar analysis since the moment arms and force components are easyto determine.

MR = ©1r * F2

MR = M1 + M2

M1 and M2

12 N # m

M = 30 N(0.4 m) = 40 N(0.3 m) = 12 N # m

0.3 m0.4 m

30 N

40 N

40 N

30 N

Fig. 4–28

MR(b)

M2 M1

Fig. 4–29

Se dois binários produzem momentos com a mesma direção e intensidade eles são ditos equivalentes.

Nas duas figuras os momentos produzidos pelos dois binários são iguais: M=30.0,4=40.0,3=12N.m e cada um está dirigido para o centro da página.

• Neste exemplo, note que:

• Na segunda figura as forças precisam ser maiores para criar o mesmo efeito de rotação que na primeira figura pois as mãos estão mais próximas (menor a distância)

• Se a roda estivesse ligada ao eixo por outro ponto que não seu centro, ela ainda giraria desde que o momento é um vetor livre.

M2

M1

(a)


4




(4–16)


MR = ©1r * F2

MR = M1 + M2

M1 and M2

12 N # m

M = 30 N(0.4 m) = 40 N(0.3 m) = 12 N # m

0.3 m0.4 m

30 N

40 N

40 N

30 N

Fig. 4–28

MR(b)

M2 M1

Fig. 4–29

Momento resultante em um sistema binário

M2

M1

(a)


4




(4–16)


MR = ©1r * F2

MR = M1 + M2

M1 and M2

12 N # m

M = 30 N(0.4 m) = 40 N(0.3 m) = 12 N # m

0.3 m0.4 m

30 N

40 N

40 N

30 N

Fig. 4–28

MR(b)

M2 M1

Fig. 4–29

M2

M1

(a)


4




(4–16)


MR = ©1r * F2

MR = M1 + M2

M1 and M2

12 N # m

M = 30 N(0.4 m) = 40 N(0.3 m) = 12 N # m

0.3 m0.4 m

30 N

40 N

40 N

30 N

Fig. 4–28

MR(b)

M2 M1

Fig. 4–29

~MR =X

i

~ri ⇥ ~Fi


4

Steering wheels on vehicles have been madesmaller than on older vehicles becausepower steering does not require the driverto apply a large couple moment to the rim ofthe wheel.

F F

Important Points

● A couple moment is produced by two noncollinear forces thatare equal in magnitude but opposite in direction. Its effect is toproduce pure rotation, or tendency for rotation in a specifieddirection.

● A couple moment is a free vector, and as a result it causes thesame rotational effect on a body regardless of where the couplemoment is applied to the body.

● The moment of the two couple forces can be determined aboutany point. For convenience, this point is often chosen on the lineof action of one of the forces in order to eliminate the moment ofthis force about the point.

● In three dimensions the couple moment is often determinedusing the vector formulation, where r is directedfrom any point on the line of action of one of the forces to anypoint on the line of action of the other force F.

● A resultant couple moment is simply the vector sum of all thecouple moments of the system.

M = r * F,

EXAMPLE 4.10

Determine the resultant couple moment of the three couples actingon the plate in Fig. 4–30.

SOLUTIONAs shown the perpendicular distances between each pair of couple forcesare and . Considering counterclockwisecouple moments as positive, we have

a

b Ans.

The negative sign indicates that has a clockwise rotational sense.MR

= -950 lb # ft = 950 lb # ft= (-200 lb)(4 ft) + (450 lb)(3 ft) - (300 lb)(5 ft)

+MR = ©M; MR = -F1d1 + F2d2 - F3d3

d3 = 5 ftd2 = 3 ft,d1 = 4 ft,

F2 ! 450 lb

F1 ! 200 lbF3 ! 300 lb

F3 ! 300 lb

F2 ! 450 lb

d3 ! 5 ft

F1 ! 200 lb

A

B

d2 ! 3 ft

d1 ! 4 ft

Fig. 4–30

Exemplo 1: Determine o momento resultante devido aos 3 binários atuando no sistema abaixo

Exemplo 2: Determine a intensidade e a direção do binário atuando nas engrenagens abaixo.


4

EXAMPLE 4.11

Determine the magnitude and direction of the couple moment actingon the gear in Fig. 4–31a.

SOLUTIONThe easiest solution requires resolving each force into its componentsas shown in Fig. 4–31b. The couple moment can be determined bysumming the moments of these force components about any point, forexample, the center O of the gear or point A. If we considercounterclockwise moments as positive, we have

a

d Ans.ora

d Ans.= 43.9 N #m+M = ©MA; M = (600 cos 30° N)(0.2 m) - (600 sin 30° N)(0.2 m)

= 43.9 N #m+M = ©MO; M = (600 cos 30° N)(0.2 m) - (600 sin 30° N)(0.2 m)

This positive result indicates that M has a counterclockwise rotationalsense, so it is directed outward, perpendicular to the page.

NOTE: The same result can also be obtained using , where dis the perpendicular distance between the lines of action of the coupleforces, Fig. 4–31c. However, the computation for d is more involved.Realize that the couple moment is a free vector and can act at anypoint on the gear and produce the same turning effect about point O.

M = Fd

(b)

30!

F " 600 N600 sin 30! N

600 cos 30! N

30!

F " 600 N 600 sin 30! N

600 cos 30! N

0.2 m

O

A

30!

30!

(c)

F " 600 N

F " 600 N

O

d

Fig. 4–31

30!

30!

(a)

F " 600 N

F " 600 N

0.2 m

O


4

EXAMPLE 4.11

Determine the magnitude and direction of the couple moment actingon the gear in Fig. 4–31a.

SOLUTIONThe easiest solution requires resolving each force into its componentsas shown in Fig. 4–31b. The couple moment can be determined bysumming the moments of these force components about any point, forexample, the center O of the gear or point A. If we considercounterclockwise moments as positive, we have

a

d Ans.ora

d Ans.= 43.9 N #m+M = ©MA; M = (600 cos 30° N)(0.2 m) - (600 sin 30° N)(0.2 m)

= 43.9 N #m+M = ©MO; M = (600 cos 30° N)(0.2 m) - (600 sin 30° N)(0.2 m)

This positive result indicates that M has a counterclockwise rotationalsense, so it is directed outward, perpendicular to the page.

NOTE: The same result can also be obtained using , where dis the perpendicular distance between the lines of action of the coupleforces, Fig. 4–31c. However, the computation for d is more involved.Realize that the couple moment is a free vector and can act at anypoint on the gear and produce the same turning effect about point O.

M = Fd

(b)

30!

F " 600 N600 sin 30! N

600 cos 30! N

30!

F " 600 N 600 sin 30! N

600 cos 30! N

0.2 m

O

A

30!

30!

(c)

F " 600 N

F " 600 N

O

d

Fig. 4–31

30!

30!

(a)

F " 600 N

F " 600 N

0.2 m

O

Exemplo 3: Determine o momento do binário atuando no encanamento da figura. a) Vetorialmente


4

EXAMPLE 4.12

Determine the couple moment acting on the pipe shown in Fig. 4–32a.Segment AB is directed 30° below the x–y plane.

SOLUTION I (VECTOR ANALYSIS)The moment of the two couple forces can be found about any point. Ifpoint O is considered, Fig. 4–32b, we have

Ans.It is easier to take moments of the couple forces about a point lying onthe line of action of one of the forces, e.g., point A, Fig. 4–32c. In thiscase the moment of the force at A is zero, so that

Ans.

SOLUTION II (SCALAR ANALYSIS)Although this problem is shown in three dimensions, the geometry issimple enough to use the scalar equation The perpendiculardistance between the lines of action of the couple forces is

Fig. 4–32d. Hence, taking moments of theforces about either point A or point B yields

Applying the right-hand rule, M acts in the direction. Thus,

Ans.M = 5-130j6 lb # in.

- j

M = Fd = 25 lb 15.196 in.2 = 129.9 lb # in.

d = 6 cos 30° = 5.196 in.,

M = Fd.

= 5-130j6 lb # in.= 16 cos 30°i - 6 sin 30°k2 * 125k2M = rAB * 125k2

= 5-130j6 lb # in.= -200i - 129.9j + 200i= 18j2 * 1-25k2 + 16 cos 30°i + 8j - 6 sin 30°k2 * 125k2M = rA * 1-25k2 + rB * 125k2

O

z

30!

xy

25 lb

A

25 lb

B

8 in.

6 in.

(a)

Fig. 4–32

z

x25 lb

A

25 lb

B

(b)

yrB

rA

O

z

x

y

25 lb

A

25 lb

B(c)

rAB

O

6 in.

z

x

y

25 lb

A

25 lb

B(d)

30!

d

O


4

EXAMPLE 4.12




Ans.




Ans.M = 5-130j6 lb # in.

- j

M = Fd = 25 lb 15.196 in.2 = 129.9 lb # in.

d = 6 cos 30° = 5.196 in.,

M = Fd.



O

z

30!

xy

25 lb

A

25 lb

B

8 in.

6 in.

(a)

Fig. 4–32

z

x25 lb

A

25 lb

B

(b)

yrB

rA

O

z

x

y

25 lb

A

25 lb

B(c)

rAB

O

6 in.

z

x

y

25 lb

A

25 lb

B(d)

30!

d

O


4

EXAMPLE 4.12




Ans.




Ans.M = 5-130j6 lb # in.

- j

M = Fd = 25 lb 15.196 in.2 = 129.9 lb # in.

d = 6 cos 30° = 5.196 in.,

M = Fd.



O

z

30!

xy

25 lb

A

25 lb

B

8 in.

6 in.

(a)

Fig. 4–32

z

x25 lb

A

25 lb

B

(b)

yrB

rA

O

z

x

y

25 lb

A

25 lb

B(c)

rAB

O

6 in.

z

x

y

25 lb

A

25 lb

B(d)

30!

d

O

b) Escalarmente


4

EXAMPLE 4.12




Ans.




Ans.M = 5-130j6 lb # in.

- j

M = Fd = 25 lb 15.196 in.2 = 129.9 lb # in.

d = 6 cos 30° = 5.196 in.,

M = Fd.



O

z

30!

xy

25 lb

A

25 lb

B

8 in.

6 in.

(a)

Fig. 4–32

z

x25 lb

A

25 lb

B

(b)

yrB

rA

O

z

x

y

25 lb

A

25 lb

B(c)

rAB

O

6 in.

z

x

y

25 lb

A

25 lb

B(d)

30!

d

O

Documents

Sistemas Binários Moment of a Couple¡tica.pdfthe moments of both forces about a point lying on the line of action of one of the forces. For example, if moments are taken about point