11.12 Stability Under Non-conservative Load (Example)
133
m
d 2 v
dt 2 + h 1
dv
dt
+ c 11 v + c 12 ϕ = 0,
I
d 2 ϕ
dt 2 + h 2
dϕ
dt
+ c 21 v + c 22 ϕ = 0.
(11.26)
We look for solutions to Eqs. (11.26) in the form
v = A exp λt, ϕ = B exp λt,
(11.27)
where A, B are constants, and λ is a characteristic indicator.
Substituting the solutions (11.27) into Eqs. (11.26) gives a system of homogeneous algebraic equations with respect to A and B. The condition for the existence
of a non-zero solution of this system is that its determinant is equal to zero
mλ 2 + b 1 λ + c 11
c 12
c 21
lλ 2 + h 2 λ + c 22
= 0.
By revealing this determinant, we obtain the characteristic equation:
p 0 λ
4
+ p 1 λ
3
+ p 2 λ
2
+ p 3 λ + p 4 = 0,
(11.28)
where
p 0 = lm, p 1 = h 2 m(μρ 2 + 1),
p 2 = m(ρ 2 c 11 + c 22 ) + b 1 b 2 , p 3 = h 2 (μc 22 + c 11 ),
p 4 = c 11 c 22 − c 12 c 21 , μ = h 1 /h 2 .
(11.29)
In this case, in the coefficient p 2 , the product of small quantities b 1 , b 2 will be
ignored in the future (unless otherwise specified).
In conclusion of this point, we note the following. The oscillations of the system
shown in Fig. 11.3b are described under similar assumptions by the following
equations
m
d 2 v 1
dt 2 + b 1
dv 1
dt
+ c 11 v 1 + c 21 ϕ 1 = 0,
I
d 2 ϕ 1
dt 2 + b 2
dϕ 1
dt
+ c 12 v 1 + c 22 ϕ = 0,
(11.30)
moreover, the coefficients c ij (i, j = 1, 2) are determined by formulas
(11.24, 11.25). The systems of Eqs. (11.26) and (11.30) differ from each other
133
m
d 2 v
dt 2 + h 1
dv
dt
+ c 11 v + c 12 ϕ = 0,
I
d 2 ϕ
dt 2 + h 2
dϕ
dt
+ c 21 v + c 22 ϕ = 0.
(11.26)
We look for solutions to Eqs. (11.26) in the form
v = A exp λt, ϕ = B exp λt,
(11.27)
where A, B are constants, and λ is a characteristic indicator.
Substituting the solutions (11.27) into Eqs. (11.26) gives a system of homogeneous algebraic equations with respect to A and B. The condition for the existence
of a non-zero solution of this system is that its determinant is equal to zero
mλ 2 + b 1 λ + c 11
c 12
c 21
lλ 2 + h 2 λ + c 22
= 0.
By revealing this determinant, we obtain the characteristic equation:
p 0 λ
4
+ p 1 λ
3
+ p 2 λ
2
+ p 3 λ + p 4 = 0,
(11.28)
where
p 0 = lm, p 1 = h 2 m(μρ 2 + 1),
p 2 = m(ρ 2 c 11 + c 22 ) + b 1 b 2 , p 3 = h 2 (μc 22 + c 11 ),
p 4 = c 11 c 22 − c 12 c 21 , μ = h 1 /h 2 .
(11.29)
In this case, in the coefficient p 2 , the product of small quantities b 1 , b 2 will be
ignored in the future (unless otherwise specified).
In conclusion of this point, we note the following. The oscillations of the system
shown in Fig. 11.3b are described under similar assumptions by the following
equations
m
d 2 v 1
dt 2 + b 1
dv 1
dt
+ c 11 v 1 + c 21 ϕ 1 = 0,
I
d 2 ϕ 1
dt 2 + b 2
dϕ 1
dt
+ c 12 v 1 + c 22 ϕ = 0,
(11.30)
moreover, the coefficients c ij (i, j = 1, 2) are determined by formulas
(11.24, 11.25). The systems of Eqs. (11.26) and (11.30) differ from each other