Statics 2e 343
Problem 3.66
In Prob. 3.65, imagine that the load-deflection data for the hanger is such that
FD.100 lb=in:/ ı C.200 lb=in:2/ ı2
and that three hangers are used to support
a straight rigid pipe from an uneven ceiling as shown. Assume the pipe may
undergo vertical motion only. Hanger 2 displaces by 0.50 in. more than hangers
1 and 3. Because of symmetry, equal forces are supported by hangers 1 and 3.
Further, the sum of the forces supported by all three hangers equals the weight
of the pipe and its contents, which is
600 lb
. Determine the deflection of the
pipe and the forces supported by each of the three hangers.
Problem 3.67
A model for the suspension of a vehicle is shown where the spring has stiffness
kD200 N=mm and an unstretched length of 360 mm.
(a)
Determine the value of
P
and the force supported by member
AB
so that
the suspension has the equilibrium position shown where member
AB
is
horizontal.
(b)
If
PD0
, determine the
x
and
y
coordinates of point
A
, namely
xA
and
yA.
Problem 3.68
Repeat Prob. 3.67 if the spring has stiffness
kD250 N=mm
and an unstretched
length of 340 mm.
Problem 3.69
If
kD5N=mm
and
WD100
N, determine
ı
. Springs are unstretched when
ıD0.
Problem 3.70
If
kD5N=mm
and
WD100
N, determine
ı
. Springs are unstretched when
ıD0.
Solution
Problem 3.71
The system shown can undergo vertical motion only, and both springs are unstretched when
WD0
. Determine the deflection
ı
in terms of
W
,
k1
, and
k2
. Show that your result is correct
for the following special cases: when
WD0
, when
k1D0
, when
k2D0
, when
k1! 1
,
and when k2! 1.
Problem 3.72
The system shown can undergo vertical motion only. The springs have stiffnesses
k1D240 lb=in:
and
k2D310 lb=in:
, and both springs have the same unstretched length
of
100 in
. The springs are installed such that the foundation where the springs are built-in
has hD8in.
(a) If WD0, determine the distance dand the force supported by each spring.
(b) If WD5000 lb, determine the distance dand the force supported by each spring.
Solution
dDWC.240 lb/in./.100 in./.310 lb/in/.8 in. 100 in./
550 lb/in. :(7)
Problem 3.73
The system shown can undergo vertical motion only and the springs are unstretched when
ıADıBD0
. Determine
ıA
,
ıB
, and the force supported by each spring if
mAD50 kg
,
mBD80 kg, k1D100 N=mm, k2D120 N=mm, and k3D140 N=mm.
Problem 3.74
The system shown can undergo vertical motion only and the springs are unstretched
when
ıADıBD0
. Determine
ıA
,
ıB
, and the force supported by each spring
if
mAD50 kg
,
mBD80 kg
,
k1D100 N=mm
,
k2D120 N=mm
, and
k3D
140 N=mm.
Problem 3.75
A fragile item
A
, with weight
W
, is to be shipped within a box
B
using the
vertical spring suspension shown. Springs 1 and 2 have stiffnesses
k1
and
k2
,
respectively, and unstretched lengths
L1
and
L2
, respectively. The springs are
installed by stretching them to the same length
h
and then attaching them to
A
.
The vertical deflection
ı
is measured such that
ıD0
is the position where
A
is
in the middle of box B.
(a) Show that Wand ıare related by ıDWk1.h L1/Ck2.h L2/
k1Ck2
.
(b) Explain why ı¤0when WD0.
(c)
Suggest some mathematical tests you can perform to verify the accuracy
of the expression in Part (a). For example, if
ıD0
and
k2D0
, show
that Whas the expected value.
Solution
Statics 2e 357
Problem 3.76
A model for the suspension of a truck is shown. Block
A
represents the chassis
of the truck, and it may undergo vertical motion only with no rotation. Force
F
represents the portion of the truck’s weight and payload that is supported by
this suspension. Due to
F
, block
A
deflects by amount
ı
. When
FD0
, both
springs are undeformed, and there is a gap
h
between the second spring and
the chassis. When
ı
is less than
h
, only one spring supports
F
; and when
ı
exceeds
h
, the second spring engages and helps support
F
. If
k1D1200 lb=in.
,
k2D600 lb=in., and hD1:3 in:,
(a) Determine ıwhen FD1400 lb.
(b) Determine ıwhen FD2800 lb.
Solution
Problem 3.77
The structure shown is a retractable tool holder that is used in a factory to support
a tool at point
D
. When the tool is to be used, the worker will grasp the tool and
apply a downward force to lower it to the position that is needed. When the tool
is not in use, the spring causes the tool to retract so that it is out of the way. The
spring has 2lb=in:stiffness and assume Wis vertical.
If
WD10 lb
and the structure is to have an equilibrium configuration where
D65ı, determine the unstretched length of the spring.
Problem 3.78
The structure shown is a retractable tool holder that is used in a factory to support
a tool at point
D
. When the tool is to be used, the worker will grasp the tool and
apply a downward force to lower it to the position that is needed. When the tool
is not in use, the spring causes the tool to retract so that it is out of the way. The
spring has 2lb=in:stiffness and assume Wis vertical.
If the unstretched length of the spring is
30 in:
, and
D˛D60ı
when
WD0, determine the value of Wneeded so that D80ı.
Problem 3.79
The structure shown is a retractable tool holder that is used in a factory to support
a tool at point
D
. When the tool is to be used, the worker will grasp the tool and
apply a downward force to lower it to the position that is needed. When the tool
is not in use, the spring causes the tool to retract so that it is out of the way. The
spring has 2lb=in:stiffness and assume Wis vertical.
If the unstretched length of the spring is
30 in:
, and
D˛D60ı
when
WD0, determine the value of Wneeded so that D100ı.
Problem 3.80
The structure shown is a retractable tool holder that is used in a factory to support
a tool at point
D
. When the tool is to be used, the worker will grasp the tool and
apply a downward force to lower it to the position that is needed. When the tool
is not in use, the spring causes the tool to retract so that it is out of the way. The
spring has 2lb=in:stiffness and assume Wis vertical.
If the unstretched length of the spring is
30 in:
, and
D˛D60ı
when
WD0
, determine the value of
that occurs when
WD15 lb
.Note: Unlike
Probs. 3.77–3.79, the system of equations for this problem is difficult to solve by
hand, and software such as Mathematica or Maple is very helpful.