Page 490 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 14.7 Self-Excited Oscillations 477

and the phase can be determined from
^0
tan ^
By assuming the first approximation to be
'
A,) = A cos cot
its substitution into the differential equation results in
[{(ol - (o^)A + ^¡jlA^] cos cot - C(o a sin cot + \/jiA^ cos Scot
(14.6-9)

= /In COS cot —Bn sin cot
We again ignore the cosScot term and equate coefficients of cos cot and sin cot to
obtain

[col ~ o)^)A + - /!()
(14.6-10)
ccoA = Bn
By squaring and adding these results, the relationship between the frequency,
amplitude, and force becomes

— \[(ol —co^^ A + + \ccoA] (14.6-11)
By fixing c, and F, the frequency ratio co/co,^ can be computed for assigned
values of A.

14.7 SELF-EXCITED OSCILLATIONS

Oscillations that depend on the motion itself are called self-excited. The shimmy of
automobile wheels, the flutter of airplane wings, and the oscillations of the van der
Pol equation are some examples.
Self-excited oscillations may occur in a linear or a nonlinear system. The
motion is induced by an excitation that is some function of the velocity or of the
velocity and the displacement. If the motion of the system tends to increase the
energy of the system, the amplitude will increase, and the system may become
unstable.
As an example, consider a viscously damped single-DOF linear system
excited by a force that is some function of the velocity. Its equation of motion is
mx ^ cx ^ kx = F{x) (14.7-1)
Rearranging the equation to the form
mx + [cx — F(x)] F kx = 0 (14.7-2)
we can recognize the possibility of negative damping if F(x) becomes greater
than cx.