138
11 The Beginning of the Theory of Stability of Equilibrium
and the coefficients p 0 , p 2 , p 4 are still determined by formulas (11.29).
The zero solution of the system (11.44) is stable by the first approximation if and
only if all the coefficients of Eq. (11.45) and its discriminant
0 = p
2
2 − 4p 0 p 4
(11.46)
are positive because all the roots of Eq. (11.45) are purely imaginary.
Reasoning similarly, as in the presence of friction, it is not difficult to establish
that a trivial solution of the system (11.44) can become unstable due to violation of
one of the inequalities
p 4 > 0, , 0 > 0.
(11.47)
If the first one is violated, the system under study loses stability statically, passing
through a state of indifferent equilibrium at p 4 = 0; if the second one is violated,
the loss of stability is self-oscillating.
As shown above (see p. 134), in the presence of friction, the necessary and
sufficient stability conditions for the zero solution of the system (11.26) had the
form:
p 4 > 0, , μ > 0.
(11.48)
Comparing the conditions (11.47) and (11.48), we see that in the case of s t a t i c
stability loss, small friction does not affect the critical parameters.
Let the stability loss be self-oscillating
1
2
η 1
. Compare the conditions
0 > 0 and μ > 0. Substituting the values of the coefficients (11.29) and (11.46),
in the absence of friction, we will have the following stability condition
1
4ρ 2 (ρ
2 c 11 − c 22 )
2
+ c 12 c 21 > 0.
(11.49)
In the presence of friction, the stability condition according to formula (11.35) has
the form
(ρ
2 c 11 − c 22 )
2
· f (μ) + c 12 c 21 > 0.
(11.50)
Since the left side of the latter inequality depends on the ratio of small friction
coefficients μ, at b 1 → 0 and b 2 → 0, the condition (11.50) does not go,
generally speaking, to the condition (11.49). In other words, the critical parameters
determined with low friction are usually different from the critical parameters found
without it.
11 The Beginning of the Theory of Stability of Equilibrium
and the coefficients p 0 , p 2 , p 4 are still determined by formulas (11.29).
The zero solution of the system (11.44) is stable by the first approximation if and
only if all the coefficients of Eq. (11.45) and its discriminant
0 = p
2
2 − 4p 0 p 4
(11.46)
are positive because all the roots of Eq. (11.45) are purely imaginary.
Reasoning similarly, as in the presence of friction, it is not difficult to establish
that a trivial solution of the system (11.44) can become unstable due to violation of
one of the inequalities
p 4 > 0, , 0 > 0.
(11.47)
If the first one is violated, the system under study loses stability statically, passing
through a state of indifferent equilibrium at p 4 = 0; if the second one is violated,
the loss of stability is self-oscillating.
As shown above (see p. 134), in the presence of friction, the necessary and
sufficient stability conditions for the zero solution of the system (11.26) had the
form:
p 4 > 0, , μ > 0.
(11.48)
Comparing the conditions (11.47) and (11.48), we see that in the case of s t a t i c
stability loss, small friction does not affect the critical parameters.
Let the stability loss be self-oscillating
1
2
η 1
. Compare the conditions
0 > 0 and μ > 0. Substituting the values of the coefficients (11.29) and (11.46),
in the absence of friction, we will have the following stability condition
1
4ρ 2 (ρ
2 c 11 − c 22 )
2
+ c 12 c 21 > 0.
(11.49)
In the presence of friction, the stability condition according to formula (11.35) has
the form
(ρ
2 c 11 − c 22 )
2
· f (μ) + c 12 c 21 > 0.
(11.50)
Since the left side of the latter inequality depends on the ratio of small friction
coefficients μ, at b 1 → 0 and b 2 → 0, the condition (11.50) does not go,
generally speaking, to the condition (11.49). In other words, the critical parameters
determined with low friction are usually different from the critical parameters found
without it.