Problem 5.1
Given: A third spring of constant k
3
and a weight W
3
are added to the system of Example 5.3.
Find:
Displacements q
1
, q
2
, and q
3
Solution:
We will use a somewhat different approach than example 5.3
For a sigle spring, the strain energy is:
U
1
2
k
⋅
δ
2
⋅
=
Eq. 5.12
where
δ
is the stretch of the spring.
The force in the spring is:
Nk
δ
⋅
=
or
δ
2
N
k
⎛
⎝
⎞
⎠
2
=
Thus:
U
1
2
N
2
k
⋅
=
For the three springs, the strain energy is:
U
1
2
N
1
2
k
1
⋅
1
2
N
2
2
k
2
⋅
+
1
2
N
3
2
k
3
⋅
+
=
N
1
W
1
W
2
+
W
3
+
=
N
2
W
2
W
3
+
=
N
3
W
3
=
U
1
2
W
1
W
2
+
W
3
+
()
2
k
1
⋅
1
2
W
2
W
3
+
2
k
2
⋅
+
1
2
W
3
2
k
3
⋅
+
=
q
1
W
1
1
2
W
1
W
2
+
W
3
+
2
k
1
⋅
1
2
W
2
W
3
+
2
k
2
⋅
+
1
2
W
3
2
k
3
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
d
d
=
q
1
2W
1
⋅
2
⋅
+
3
⋅
+
2k
1
⋅
=
→
q
2
W
2
1
2
W
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 Spring '10
 MechanicsofSolids
 Energy, Force, Strain, Johannesburg, South African National Roads Agency, The Spring

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