THREE DIMENSIONAL EQUILIBRIUM OF RIGID BODIES
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Transcript of THREE DIMENSIONAL EQUILIBRIUM OF RIGID BODIES
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THREE DIMENSIONAL
EQUILIBRIUM OF RIGID BODIES
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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
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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.
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Representation of Support Reactions
in Three Dimensional Problems
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String, Rope, Chain, Cord, Belt
T
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Contact with Smooth Surface orBall Support
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Roller Support or Wheel on Rail(Makaralı Mesnet veya Ray Üzerinde Tekerlek)
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Contact with Rough Surface
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Ball-and-Socket Joint(Küresel Mafsal)
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Sliding Support(Kayar mesnet)
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Built-in / Fixed / Cantilever Support
(Ankastre Mesnet)
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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)
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Couple reactions perpendicular to hinge axis due to imbalance,
depending on design, may also exert force along the z axis.
Single Hinge (Tek Menteşe)
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Couple reactions perpendicular to bearing axis due to imbalance!
Radial / Journal /Ball Bearing(Tek Radyal Yatak)
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Double Radial / Journal / Ball BearingsÇift Radyal Yatak (Bilyeli Yatak)
No moment reaction occurs in double radial bearings!
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Single Bearing (Thrust-Bearing)Tek Basma Yatağı (Konik Yatak)
Force reactions in three directions + moment reactions in two dimensions (perpendicular to the bearing axis)
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Thrust Bearing (Basma Yatağı) andRadial Bearing (Radyal Yatak)
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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 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.
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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.
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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
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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
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Ax
Az
Ay
Bz
By
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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.)
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Bx Bz
By
My
Mx
TB
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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
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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
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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.
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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.
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z
y
x
Ax
AzAy Bx
Bz
By
FCD
G
W=25(9.81) N