if forces acting on a rigid body are three dimensional,...

If forces acting on a rigid body are three dimensional, six

equations of equilibrium can be used:

0

0

0

0

z

y

x

F

F

F

FR

0

0

0

0

z

y

x

iR

M

M

M

CFrMM

At most six unknowns can be determined in three

dimensional rigid body equilibrium problems.

These unknowns will generally involve support/

bearing reactions and tension forces in ropes, wires,

etc. In general, employing the moment equation first

may facilitate the solution of the problem.

Moment can either be taken about a point where

the number of unknown values is maximum, or it

can be taken about an axis / line, where the lines

of action of unknown forces intersect.

String, Rope, Chain, Cord, Belt

T

Contact with Smooth Surface or Ball Support

Roller Support or Wheel on Rail

(Makaralı Mesnet veya Ray Üzerinde Tekerlek)

Contact with Rough Surface

Ball-and-Socket Joint

(Küresel Mafsal)

Sliding Support

(Kayar mesnet)

Built-in / Fixed / Cantilever Support

(Ankastre Mesnet)

If hinges are supporting forces along the hinge axis, force reactions

will occur along hinge axis

(In case of double hinge no moment reactions occur!)

Double Hinge (Çift Menteşe)

Couple reactions perpendicular to hinge axis due to imbalance,

depending on design, may also exert force along the z axis.

Single Hinge

(Tek Menteşe)

Couple reactions perpendicular to bearing axis due to imbalance!

Radial / Journal /Ball Bearing

(Tek Radyal Yatak)

Double Radial / Journal / Ball Bearings Çift Radyal Yatak (Bilyeli Yatak)

No moment reaction occurs in double radial bearings!

Ball Bearing (Radyal Yatak)

Ball Bearing (Radyal Yatak)

Single Bearing (Thrust-Bearing)

Tek Basma Yatağı (Konik Yatak)

Force reactions in three directions + moment reactions in two

dimensions (perpendicular to the bearing axis)

Thrust Bearing (Basma Yatağı) and

Radial Bearing (Radyal Yatak)

Ball Bearing (Radyal Yatak)

Thrust Bearing (BasmaYatağı)

Collar on Shaft with Pin (Mil Üzerinde

Bilezik, Pimli Bağlantı)

1. The 9 m steel boom has a mass of 600 kg with center of mass at midlength. It is

supported by a ball and socket joint at A and the two cables under tensions T1 and T2.

The cable which supports the 2000 kg load leads through a sheave (pulley) at B and

is secured to the vertical x-y plane at F. Calculate the magnitude of the tension T1.

2. The shaft, lever and handle are welded together and constitute a single rigid

body. Their combined mass is 28 kg with mass center at G. The assembly is

mounted in bearings A and B, and rotation is prevented by link CD. Determine the

forces exerted on the shaft by bearings A and B while the 30 N.m couple is applied

to the handle as shown.

3. The lever AB is welded to the bent rod BCD which is supported by bearing E and

cable DG. Assuming that the bearing can exert an axial thrust and couples about axes

parallel to the x and z axes, determine the tension in cable DG and the reaction at E

under the action of the 220 N force. The mass of ABCD is neglected.

220 N

240 mm

60 mm

250 mm

225 mm

160 mm

120 mm

G D

C

E

B

y

z

x

A

FF

4. The shaft assembly (consisting of welded pieces AB, ED and CD) is supported

by a thrust bearing at A and a radial bearing at B. The assembly is subjected to a

force at C and a couple . If it is known that the y component of the reaction

at bearing B is (N), determine the vector expressions of the force , couple

and the bearing reactions at A and B. Link ED lies in the yz plane. ED=250 mm.

F

M

j104

F

M

Ax

Az

Ay

Bz

By

5. The assembly shown is welded to collar A which fits on the vertical pin shown.

The collar is free to move about or along the y axis. For the loading shown,

determine the tension in each cable and the reaction at A.

Mx

Mz

Ax Az

TCF

TDE

6. For the portion of a machine shown, the

100 mm diameter pulley A and wheel B are

fixed to a shaft supported by bearings at C

and D. The spring of constant 360 N/m is

unstretched when q=0°and the bearing at C

does not exert any axial force. Knowing that

q=120° and the machine is at rest and in

equilibrium, determine the tension T and the

reactions at C and D. Neglect the weights of

the shaft, pulley and wheel.

150 mm

100 mm

50 mm

25 mm

90 mm

300 mm

25 mm

150 N

x

z

y

T

7. A 450 mm long uniform rod

AB has a weight of 304 N and is

attached to a ball-and-socket joint

at A. The end B of the rod rests

against an inclined frictionless

surface and is held in the

equilibrium position shown by

cord BC. Knowing that cord BC is

450 mm long, determine the

tension in the cord and the

reactions at A and B.

255 mm

W

75 mm

450 mm

450 mm

660 mm

B

A

y

z

x

4

3

G

z

y

x

8. The 25 kg rectangular access door is held in the 90° open position by the single prop CD.

Determine the force F in the prop and the magnitude of the force normal to the hinge axis AB in

each of the small hinges A and B.

z

y

x

Ax

Az Ay Bx

Bz

By

FCD

G

W=25(9.81) N