Page 89 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

76 Harmonically Excited Vibration Chap. 3

Complex stiffness. In the calculation of the flutter speeds of airplane
wings and tail surfaces, the concept of complex stiffness is used. It is arrived at by
assuming the oscillations to be harmonic, which enables Eq. (3.9-3) to be written as

mx +

By factoring out the stiffness k and letting y = a/irk, the preceding equation
becomes
mx + k{l -h iy)x = (3.9-4)

The quantity k{\ + iy) is called the complex stiffness and y is the structural
damping factor.
Using the concept of complex stiffness for problems in structural vibrations is
advantageous in that one needs only to multiply the stiffness terms in the system by
(1 -f iy). The method is justified, however, only for harmonic oscillations. With the
solution X = Xe''^^ the steady-state amplitude from Eq. (3.9-4) becomes

- (3.9-5)

(/c —moff) + iyk
The amplitude at resonance is then

(3.9-6)

Comparing this with the resonant response of a system with viscous damping

lA'I = T l

we conclude that with equal amplitudes at resonance, the structural damping
factor is equal to twice the viscous damping factor.

Frequency response with structural damping. By starting with Eq.
(3.9-5), the complex frequency response for structural damping can be shown to be
a circle. Letting co/co^ = r and multiplying and dividing by its complex conjugate
give a complex frequency response of

1 1 - y
H{r) = -I- /- = X iy
~ r^) + iy + y2 (1 - C) + y'^
where
1 - r^

X = and
C - r ^ ) +y^