44
2 Free Vibration of Single Degree of Freedom System
Fig. 2.24 Example 2.15
The dynamics of a helical spring is somewhat complicated. It has been simplified
as an elastic rod, assuming that it is uniformly stretched. If the velocity of the lumped
mass is ˙
x, then the velocity of the spring element at a distance ξ is
˙
x
ξ
L
If ρ is the mass per unit length of the spring, then
T E max =
1
2
m ˙
x
2
max +
L
0
1
2
ρ
ξ
L
˙
x max
2
dξ
=
1
2
m + ρ
L
3
˙
x
2
max
Let
x = A sin pt
then
˙
x max = A p
Therefore,
T E max =
1
2
m + ρ
L
3
(A p)
2
2 Free Vibration of Single Degree of Freedom System
Fig. 2.24 Example 2.15
The dynamics of a helical spring is somewhat complicated. It has been simplified
as an elastic rod, assuming that it is uniformly stretched. If the velocity of the lumped
mass is ˙
x, then the velocity of the spring element at a distance ξ is
˙
x
ξ
L
If ρ is the mass per unit length of the spring, then
T E max =
1
2
m ˙
x
2
max +
L
0
1
2
ρ
ξ
L
˙
x max
2
dξ
=
1
2
m + ρ
L
3
˙
x
2
max
Let
x = A sin pt
then
˙
x max = A p
Therefore,
T E max =
1
2
m + ρ
L
3
(A p)
2