82
3 Forced Vibration of Single Degree of Freedom System
The damping force is expressed as
F d = ± a · ˙
x
2
where negative sign is considered with positive ˙
x and vice versa.
The energy dissipated per cycle is
E d =
F D du =
2π
ω
0
F d ˙
x dt
= 2
π
ω
0
(a ˙
x
2
) ˙
x dt
= 2 a ω
2 A
3
π/ω
0
sin
3
ω t dt
=
8
3
a ω
2 A
3
The equivalent viscous damping is given by
π C eq ω A
2
=
8
3
aω
2 A
3
or
C eq =
8
3π
aω A
From Eq. (3.52) substituting c = C eq , we get
A =
F 0
cp
=
3π F 0
8a p Aa
or
A =
3π F 0
8a p 2
1
2
3 Forced Vibration of Single Degree of Freedom System
The damping force is expressed as
F d = ± a · ˙
x
2
where negative sign is considered with positive ˙
x and vice versa.
The energy dissipated per cycle is
E d =
F D du =
2π
ω
0
F d ˙
x dt
= 2
π
ω
0
(a ˙
x
2
) ˙
x dt
= 2 a ω
2 A
3
π/ω
0
sin
3
ω t dt
=
8
3
a ω
2 A
3
The equivalent viscous damping is given by
π C eq ω A
2
=
8
3
aω
2 A
3
or
C eq =
8
3π
aω A
From Eq. (3.52) substituting c = C eq , we get
A =
F 0
cp
=
3π F 0
8a p Aa
or
A =
3π F 0
8a p 2
1
2