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## THREE DIMENSIONAL EQUILIBRIUM OF RIGID BODIES

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### Title: Slide 1 Author: Cinar Last modified by: Casper Created Date: 4/2/2012 9:13:44 AM Document presentation format: Ekran G sterisi (4:3) Company – PowerPoint PPT presentation

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Title: THREE DIMENSIONAL EQUILIBRIUM OF RIGID BODIES

1
THREE DIMENSIONAL EQUILIBRIUM OF RIGID BODIES
2
If forces acting on a rigid body are three
dimensional, six equations of equilibrium can be
used
3
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.
4
Representation of Support Reactions in Three
Dimensional Problems
5
String, Rope, Chain, Cord, Belt
6
Contact with Smooth Surface or Ball Support
7
Roller Support or Wheel on Rail (Makarali Mesnet
veya Ray Üzerinde Tekerlek)
8
Contact with Rough Surface
9
Ball-and-Socket Joint (Küresel Mafsal)
10
Sliding Support (Kayar mesnet)
11
Built-in / Fixed / Cantilever Support (Ankastre
Mesnet)
12
Double Hinge (Çift Mentese)
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!)
13
Single Hinge (Tek Mentese)
Couple reactions perpendicular to hinge axis due
to imbalance, depending on design, may also exert
force along the z axis.
14
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15
Couple reactions perpendicular to bearing axis
due to imbalance!
16
Double Radial / Journal / Ball Bearings Çift
No moment reaction occurs in double radial
bearings!
17
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18
Single Bearing (Thrust-Bearing) Tek Basma Yatagi
(Konik Yatak)
Force reactions in three directions moment
reactions in two dimensions (perpendicular to the
bearing axis)
19
Thrust Bearing (Basma Yatagi) and Radial Bearing
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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
(pulley) at B and is secured to the vertical x-y
plane at F. Calculate the magnitude of the
tension T1.
23
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.
24
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.
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5. Under the action of the 40 Nm 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.)
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29
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
q0and the bearing at C does not exert any axial
force. Knowing that q120 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.
30
50 cm
50 cm
40 cm
15 cm
15 cm
25 cm
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.
31
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.
32
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.
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