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Problem 3.50
Repeat Part (b) of Example 3.4 on p. 140, using optimization methods of calculus. Hint: Solve Eqs. (4) and
(5) in Example 3.4 for
FBD
as a function of
˛
. Then solve for the value of
˛
that makes
dFBD=d˛ D0
(this equation is difficult to solve analytically, and you may need to solve it graphically or by other
approximate means). This problem can also be effectively solved using computer mathematics programs
such as Mathematica or Maple.
Problem 3.51
A device for tensioning recording tape in a video cassette recorder is shown.
The tape wraps around small pins at
A
,
B
, and
C
. The pins at
A
and
C
are fixed,
and the pin at
B
is supported by a spring and can undergo vertical motion in the
frictionless slot. Friction between the tape and pins is negligible. The spring
has stiffness
kD0:5 N=mm
and is unstretched when
hD25 mm
. Neglecting
the size of the pins, determine the tension in the tape when
(a) hD18 mm.
(b) hD10 mm.
Problem 3.52
The brake linkage for a vehicle is actuated by pneumatic cylinder
AB
.
Cylinder
AB
, springs
CD
,
EF
, and
GH
, and the slotted tracks at
A
,
C
,
E
, and
G
are all parallel. If the slotted tracks are frictionless, and cylinder
AB
produces a tensile force of
12 kip
, determine the deflection
ı
and forces
in members AC ,AE, and AG. All springs are unstretched when ıD0.
Problem 3.53
The system shown may undergo vertical motion only. For the values of
mA
,
mB
,
k1
, and
k2
given below, determine the displacements ıAand ıB.
mAD100 kg, mBD80 kg, k1D200 N=mm, and k2D300 N=mm.
Problem 3.54
The system shown may undergo vertical motion only. For the values of
mA
,
mB
,
k1
, and
k2
given below, determine the displacements ıAand ıB.
mAD5kg, mBD4kg, k1D6N=mm, and k2D3N=mm.
Problem 3.55
In Fig. P3.55(a), four identical crates weighing
2000 lb
each are stacked one
on top of another, and in Fig. P3.55(b), a simple model for determining the
deformation of the stack of crates is shown. In this model, each spring has
the same stiffness
kD4500 lb=in:
and the forces are determined using the
following idealization. Half the weight of the top crate is applied to point
A
and
half to point
B
. Similarly, half the weight of the next crate is applied to point
B
and half to point C, and so on. Determine the deflections ıA,ıB,ıC, and ıD.
Problem 3.56
In Prob. 3.55, let the top two crates weigh
3000 lb
each, and the bottom two crates weigh
4000 lb
each, and
all springs have the same stiffness
kD5000 lb=in
. Use the scheme described in Prob. 3.55 to determine
the forces FA–FDand determine the deflections ıA,ıB,ıC, and ıD.
Problem 3.57
Collar
A
has negligible weight and slides without friction on the vertical bar
CD. Determine the vertical force Fthat will produce D30ıif
(a) Spring AB is unstretched when D0ı.
(b) Spring AB has an unstretched length of 2ft.
(c) Spring AB has an unstretched length of 4ft.
Problem 3.58
Point
A
is supported by springs
BC
and
DE
and cable segments
AB
and
AD
.
The springs and cables have negligible weight, and the springs have identical
stiffness
kD10 N=cm
and
20 cm
unstretched length. The structure has the
geometry shown when FD225 N.
(a) Determine the lengths of cables AB and AD.
(b) Determine the coordinates of point Awhen FD0.
334 Solutions Manual
Problem 3.59
Spring
AB
is supported by a frictionless roller at
B
so that it is always vertical.
If the spring is unstretched when
D0ı
, determine
and the forces in spring
AB
and bar
AC
when
F
has the value indicated. Hint: The force supported by
bar AC is zero when < 90ı, and may be nonzero when D90ı.
(a) FD25 lb.
(b) FD50 lb.
Problem 3.60
A collar with a pulley slides on a frictionless vertical bar
GH
. A string
ABCD
is wrapped around the pulley, where portion
AB
of the string is horizontal. A
spring with
2lb=in:
stiffness is placed between the collar and point
H
. The
spring has
8in:
unstretched length and
6in:
final length. The collar and pulley
have a combined weight of
5lb
, and the weights of all other components are
negligible. Determine the value of
P
required for equilibrium, and the reaction
between the collar and bar GH .
Problem 3.61
A collar with a pulley slides on a frictionless vertical bar
GH
. A string
ABCD
is wrapped around the pulley, where portion
AB
of the string is horizontal. A
spring with
2lb=in:
stiffness is placed between the collar and point
H
. The
spring has
8in:
unstretched length and
6in:
final length. The collar and pulley
have a combined weight of
5lb
, and the weights of all other components are
negligible. Determine the value of
P
required for equilibrium, and the reaction
between the collar and bar GH .
Problem 3.62
The machine shown is used for compacting powder. Collar
C
slides on plunger
AB
and is driven by a motor (not shown in the sketch) so that it has the oscillatory vertical
motion
ıD.10 mm/.sin t/
where
t
is time in seconds. Plunger
AB
weighs
4
N
and is pressed against the powder by the spring whose end is driven by the motion of
collar
C
. The spring has stiffness
kD0:1 N=mm
and
100 mm
unstretched length.
The
50 mm
dimension shown is the spring length when the machine is started (i.e.,
ıD0
and the powder column is at its initial height
hD110 mm
); this dimension
changes with the oscillation of collar
C
and as the plunger
A
moves down. Assume
there is no friction in the system (other than friction between individual grains of
powder), and assume the motion of
C
is slow enough that there are no dynamic
effects. Determine the largest and smallest forces the plunger applies to the powder
over a full cycle of motion of Cif
(a) The powder column is at its initial height hD110 mm.
(b) The powder column is at its compacted height hD80 mm.
Problem 3.63
A fuel pump is driven by a motor-powered flywheel. The pump behaves as a
spring with stiffness
2N=mm
that is unstretched when
˛D ˙60ı
. Neglect any
possible dynamic effects.
(a)
Determine the largest tensile and compressive forces spring
AB
expe-
riences during one revolution of the flywheel, and state the positions
˛
where these occur.
(b)
Without further analysis, is it certain that the largest tensile and compres-
sive forces in crank arm
BC
occur at the same positions
˛
determined in
Part (a)? Explain.
Problem 3.64
The suspension for the landing gear of an aircraft is shown. The wheel is attached
to bar
BC
, which slides vertically without friction in housing
A
, which is fixed to
the frame of the aircraft. The spring is precompressed so that it does not undergo
additional deflection until the force supported by the landing gear is sufficiently large.
Further, if the force supported by the landing gear exceeds a limit, the suspension
“bottoms out” and deflects no more. Specify the spring stiffness and initial length if
hD18 in:for F500 lb and hD12 in:for F1500 lb.
Problem 3.65
A hanger is made of cord-reinforced rubber. It is used as a spring support
with limited travel for a wide variety of applications. If prototype samples
are available, an effective means to characterize its nonlinear stiffness is by
testing in a laboratory, where forces of known magnitude are applied and the
deflections that result are measured. Imagine this produces the load-deflection
data provided in the table of Fig. P3.65.
(a)
Determine the constants
a
,
b
, and
c
that will fit the general quadratic
equation FDaCbı Ccı2to the load-deflection data for this hanger.
(b) Plot the load-deflection relation determined in Part (a).
(c)
Speculate on the range of values for
F
for which the relation obtained in
Part (a) will be reasonably accurate.
