11.12 Stability Under Non-conservative Load (Example)
137
Hence, for η =
1
2
, we find that γ ∗
1
2
= π. Let η =
1
2;
then Eq. (11.42) can be
converted:
tg
γ
2
=
γ
2
1 −
1
2η
.
(11.43)
Setting η, one can find the corresponding value of γ 0 (η) from formula (11.43).
These values are shown graphically in Fig. 11.4.
From the above arguments, it follows that the equilibrium of the system under
consideration is always stable in the area that is covered with shading in Fig. 11.4.
Let us call it the reliable stability area. In addition, one can say the following:
1. when the parameter η is in the range
0,
1
2
or, equivalently, H < G (H
0, G > 0), the loss of stability occurs statically, and the critical load coincides
with the Euler load. The critical parameter γ 0 (η) increases with an increase in the
non-conservative component of the load, which indicates the stabilizing influence
of H ;
2. for values η that satisfy the condition
1
2
η 1, in other words, H G (H >
0, G 0), the loss of stability is self-oscillating; for η >
1
2
, the Euler load
does not exist. The critical parameter γ ∗ (η) decreases with an increase in the
non-conservative component of the load. This means that the tracking force can
have a destabilizing effect.
11.12.4 Investigation of the Effect of Friction
Now we will show that in this problem, taking into account small friction can
sometimes have a significant impact on the stability of the system equilibrium.
Let us first assume that there is no friction (b 1 = b 2 = 0). Then instead of
Eqs. (11.26), we have the system
m
d 2 v
dt 2 + c 11 v + c 12 ϕ = 0,
I
d 2 ϕ
dt 2 + c 21 v + c 22 ϕ = 0.
(11.44)
The characteristic equation of the system (11.44) has the form
p 0 ω
4
+ p 2 ω
2
+ p 4 = 0,
(11.45)
137
Hence, for η =
1
2
, we find that γ ∗
1
2
= π. Let η =
1
2;
then Eq. (11.42) can be
converted:
tg
γ
2
=
γ
2
1 −
1
2η
.
(11.43)
Setting η, one can find the corresponding value of γ 0 (η) from formula (11.43).
These values are shown graphically in Fig. 11.4.
From the above arguments, it follows that the equilibrium of the system under
consideration is always stable in the area that is covered with shading in Fig. 11.4.
Let us call it the reliable stability area. In addition, one can say the following:
1. when the parameter η is in the range
0,
1
2
or, equivalently, H < G (H
0, G > 0), the loss of stability occurs statically, and the critical load coincides
with the Euler load. The critical parameter γ 0 (η) increases with an increase in the
non-conservative component of the load, which indicates the stabilizing influence
of H ;
2. for values η that satisfy the condition
1
2
η 1, in other words, H G (H >
0, G 0), the loss of stability is self-oscillating; for η >
1
2
, the Euler load
does not exist. The critical parameter γ ∗ (η) decreases with an increase in the
non-conservative component of the load. This means that the tracking force can
have a destabilizing effect.
11.12.4 Investigation of the Effect of Friction
Now we will show that in this problem, taking into account small friction can
sometimes have a significant impact on the stability of the system equilibrium.
Let us first assume that there is no friction (b 1 = b 2 = 0). Then instead of
Eqs. (11.26), we have the system
m
d 2 v
dt 2 + c 11 v + c 12 ϕ = 0,
I
d 2 ϕ
dt 2 + c 21 v + c 22 ϕ = 0.
(11.44)
The characteristic equation of the system (11.44) has the form
p 0 ω
4
+ p 2 ω
2
+ p 4 = 0,
(11.45)