if forces acting on a rigid body are three dimensional, · pdf file · 2014-08-07if...
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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
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