3.9 Self-excited Vibrations
85
Fig. 3.15 Example 3.10
The deflection being small, Eq. (b) becomes
F = 2ρ A 0 v
2
α
(c)
The slope at the free end of the cantilever due to a uniformly load is
α =
wl
3
6E I 0
=
4
3L
wL
4
8E I 0
=
4x
3L
(d)
The equation of motion of the equivalent mass is
m eq ¨
x + kx = F
(e)
Substituting proper values, Eq. (e) becomes
m
2
¨
x +
8E I 0
L 3 − 2ρ A 0 v
2 4
3L
x = 0
The system is unstable when x is negative, that is, when
v >
3E I 0
ρ A 0 L 2
(f)
85
Fig. 3.15 Example 3.10
The deflection being small, Eq. (b) becomes
F = 2ρ A 0 v
2
α
(c)
The slope at the free end of the cantilever due to a uniformly load is
α =
wl
3
6E I 0
=
4
3L
wL
4
8E I 0
=
4x
3L
(d)
The equation of motion of the equivalent mass is
m eq ¨
x + kx = F
(e)
Substituting proper values, Eq. (e) becomes
m
2
¨
x +
8E I 0
L 3 − 2ρ A 0 v
2 4
3L
x = 0
The system is unstable when x is negative, that is, when
v >
3E I 0
ρ A 0 L 2
(f)