three dimensional equilibrium of rigid bodies

THREE DIMENSIONAL

EQUILIBRIUM OF RIGID BODIES

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.

Representation of Support Reactions

in Three Dimensional Problems

String, Rope, Chain, Cord, Belt

T

Contact with Smooth Surface orBall 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 BearingsÇift Radyal Yatak (Bilyeli Yatak)

No moment reaction occurs in double radial bearings!

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) andRadial Bearing (Radyal Yatak)

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

GD

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. Under the action of the 40 N·m torque

applied to the vertical shaft, the

restraining cable AC limits the rotation of

the arm OA and attached shaft to an angle

of 60° measured from the y axis. The

collar D fastened to the shaft prevents

downward motion of the shaft in its

bearing. Calculate the bending moment

M, the compression P and the shear force

V in the shaft at section B. (Bending

moment, expressed as a vector, is normal

to the shaft axis and shear force is also

normal to the shaft axis.)

Bx Bz

By

My

Mx

TB

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 =0°and the bearing at C

does not exert any axial force. Knowing that

=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. Two rods are welded to form a T-shaped structure. The end D of the structure rests

against a frictionless vertical wall, while ends A and B are supported by radial bearings.

When a 600 N magnitude vertical force P is applied to the midpoint E of the part DC of the

structure, determine the reactions at D.

25 cm15 cm

15 cm

40 cm

50 cm

50 cm

8. 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.

z

y

x

9. 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

AzAy Bx

Bz

By

FCD

G

W=25(9.81) N