
Stability analyses 223
The weighting coefcients from Equation (5.20) are computed by the
locally applied DQ method, and the characteristic polynomial (Appendix
A) is given as
φ=−+−α
zz
() 43
2
(5.21)
where γ = cos(θ) − 2,
r2[1cos() ] , and β = 2rsin(θ)γ. The complex
conjugate polynomial of Equation (5.21) is given as
φ=−+−α+βzz iz() 14 (3 )
(5.22)
The conditions
|(0)
*
and φ=−α−βi(0)3 are obtained from
Equations (5.21) and (5.22). For Equation (5.21) to be the VN polynomial,
φ−
(0)| |(0)|] 0
*2 2
results in (Miller, 1971)
−−α−β≥1(3) 0
22
(5.23)
It is difcult to analytically solve Equation (5.23). Therefore the roots of
ϕ