3.9 Self-excited Vibrations
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3.9 Self-excited Vibrations
We have seen that the external dynamic loading acting on the system is independent
of the response of the system, such as the displacement, velocity and acceleration. But
there are certain systems where the external dynamic load is a function of the motion
itself. Such systems are called self-excited vibrating systems. Typical examples of
this category are the flutter of aeroplanes, aerodynamic vibration of bridges, the
shimming of automobile wheels and flow-induced vibrations.
Self-excited vibrations may be linear or nonlinear. If the vibration causes an
increase in energy of the system, the amplitude will go on increasing and the system
will become dynamically unstable. Only a convergence of the displacement of the
motion or its steady behaviour with time will result in a dynamically stable system.
As an example, let us consider a viscously damped single d.o.f. linear system
subjected to an internal force which is function of velocity. The equation of motion
with usual notation is given by
m ¨
x + c ˙
x + kx = F 0 ˙
x
(3.53)
Equation (3.53) when rearranged is given by
˙
x +
c − F 0
m
˙
x +
k
m
x = 0
(3.54)
Substituting an assumed solution x = A e
λ t where A is a constant in Eq. (3.54),
we get,
λ
2
+
(c − F 0 )
m
x +
k
m
= 0
(3.55)
which leads to the following solution
λ 1, 2 = −
c − F 0
2m
±
c − F 0
2m
2
−
k
m
(3.56)
For positive damping, c > F 0 . This will result in a vibrating system similar to
damped free vibrations and the system will remain stable.
Let us investigate the possibility of negative damping. In this case, c > F 0 . Three
cases arise.
Case I:
c − F 0
2m
2
>
k
m
(3.57)
In this case, both λ 1 and λ 2 are real and positive.