11.12 Stability Under Non-conservative Load (Example)
135
Fig. 11.4 Research on the stability of a non-conservative system
when γ exceeds 2π . Note that in the interval
0,
1
2
for η, the value p 4 passes from
positive to negative values through zero (indifferent equilibrium), and p 4 reaches
zero without changing the sign, at γ = π , when η =
1
2
; for η
1
2
, the coefficient
p 4 changes the sign, turning to infinity.
11.12.3 Investigation of the Value μ, (Formula (11.31))
Using formulas (11.29), μ can be converted to the form:
μ = mh
2
2 (μρ
2
+ 1)L(f, ρ
2 , c ij ),
(11.34)
where
L(f, ρ
2 , c ij ) = (ρ
2 c 11 − c 22 )f (μ) + c 12 c 21 ,
(11.35)
f (μ) =
μ
(μρ 2 + 1)
2
.
(11.36)
Substituting in (11.35) formula (11.24), we get
L(f, ρ
2 , c ij ) =≡
(G + H ) 2
2
[f S
2
+ (cos γ − 1)(cos γ − 1 − ηη)],
(11.37)
135
Fig. 11.4 Research on the stability of a non-conservative system
when γ exceeds 2π . Note that in the interval
0,
1
2
for η, the value p 4 passes from
positive to negative values through zero (indifferent equilibrium), and p 4 reaches
zero without changing the sign, at γ = π , when η =
1
2
; for η
1
2
, the coefficient
p 4 changes the sign, turning to infinity.
11.12.3 Investigation of the Value μ, (Formula (11.31))
Using formulas (11.29), μ can be converted to the form:
μ = mh
2
2 (μρ
2
+ 1)L(f, ρ
2 , c ij ),
(11.34)
where
L(f, ρ
2 , c ij ) = (ρ
2 c 11 − c 22 )f (μ) + c 12 c 21 ,
(11.35)
f (μ) =
μ
(μρ 2 + 1)
2
.
(11.36)
Substituting in (11.35) formula (11.24), we get
L(f, ρ
2 , c ij ) =≡
(G + H ) 2
2
[f S
2
+ (cos γ − 1)(cos γ − 1 − ηη)],
(11.37)