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The eigenvalue solutions of this equation yield the modal frequencies,ω , and modal deflection
amplitudes, { q } . The free vibration mode shapes can be used to diagonalize the left-hand side of
Equation 2-4:
[ { }{ } ... ] = {} φ
q (2-6)
1 2 q
{} q []{} φ r
= (2-7)
This results in transforming the forced vibration equation into modal coordinates:
2
1 2
[ ω M + iωC
+ K]{}
r = ρV
[ Q]{}
r
− (2-8)
2
The classical flutter equation is obtained by converting the damping term into “structural”
damping:
⎡ 2
1 2 ⎤
⎢
− ω M + ( 1+
ig)
K − ρV
Q {} = 0
⎣
2 ⎥
r
(2-9)
⎦
The Strouhal number defines the reduced frequency as:
b
k
V
ω
= (2-10)
Rearranging the classical flutter equation from Equation 2-9 into eigenvalue form gives:
⎡ 2
⎛ 1 1 b 1 −1
⎞ +
⎢ ⎜ + ρ M Q(
k)
−
2
2 2
2
⎢⎣
2
⎟
⎝ ωi
k ωi
⎠ ω
( 1 ig)
⎤
[]{} I r = 0
Solving this equation yields the frequency-damping-velocity relationships:
( 1+
ig )
⎥
⎥⎦
(2-11)
λ f = λreal
+ λimag
=
(2-12)
2
ω
b
V = ,
k λ
real
λimag
g = ,
λ
real
1
ω = (2-13)
λ
real
22