134
11 The Beginning of the Theory of Stability of Equilibrium
only in the arrangement of the coefficients c 12 and c 21 . These values are included in
the stability conditions only as a product of (c 12 c 21 ). It follows that the systems in
Fig. 11.3a, b are equivalent with respect to the stability of their equilibrium.
11.12.2 Area of Valid Stability
Below, we will investigate the system of Eq. (11.26). To determine the critical load
(see p. 125) it is enough to examine the coefficient p 4 and the value
μ ≡ p 3 (p 1 p 2 − p 0 p 3 ) − p
2
1 p 4 .
(11.31)
It is easy to show that the function defined by formula (11.25) on the
interval (0, 2π) has no zeros. It follows that the functions (11.24), as well as
p 2 (kl), p 3 (kl), p 4 (kl) and μ(kl), are continuous when 0 < kl < 2π . Let us
limit ourselves to the specified interval for now.
Using formulas (11.24) and (11.25), the coefficient p 4 can be converted as
follows:
p 4 =
(G + H ) 2
(γ )
[η + (1 − η) cos γ ] (γ = kl).
(11.32)
This shows that p 4 can change the sign for those values of γ that nullify either the
expression in the square brackets or the denominator of the right side of the equality
(11.32). The denominator ) of formula (11.32) changes the sign for γ > 0 for
the first time when γ 1 = 2π . Equating the numerator to zero, we get
cos γ = −
η
1 − η
.
(11.33)
The equality (11.33) is possible if and only if
−
η
1 − η
1,
i. e. for η
1
2
. Setting the values η
0 < η <
1
2
from Eq. (11.33), we find the
smallest values of γ 0 (η), which changes the sign of the expression in the square
brackets of formula (11.32). The visualization of the calculation results is shown in
Fig. 11.4.
The figure shows that the value p 4 for the values η that satisfy the condition
0 η <
1
2
becomes negative when γ > γ 0 (η) and the inequality γ 0 (0) < γ 0 (β) <
γ 0
1
2
is right; for the values η
1
2
, the coefficient p 4 goes to negative values
11 The Beginning of the Theory of Stability of Equilibrium
only in the arrangement of the coefficients c 12 and c 21 . These values are included in
the stability conditions only as a product of (c 12 c 21 ). It follows that the systems in
Fig. 11.3a, b are equivalent with respect to the stability of their equilibrium.
11.12.2 Area of Valid Stability
Below, we will investigate the system of Eq. (11.26). To determine the critical load
(see p. 125) it is enough to examine the coefficient p 4 and the value
μ ≡ p 3 (p 1 p 2 − p 0 p 3 ) − p
2
1 p 4 .
(11.31)
It is easy to show that the function defined by formula (11.25) on the
interval (0, 2π) has no zeros. It follows that the functions (11.24), as well as
p 2 (kl), p 3 (kl), p 4 (kl) and μ(kl), are continuous when 0 < kl < 2π . Let us
limit ourselves to the specified interval for now.
Using formulas (11.24) and (11.25), the coefficient p 4 can be converted as
follows:
p 4 =
(G + H ) 2
(γ )
[η + (1 − η) cos γ ] (γ = kl).
(11.32)
This shows that p 4 can change the sign for those values of γ that nullify either the
expression in the square brackets or the denominator of the right side of the equality
(11.32). The denominator ) of formula (11.32) changes the sign for γ > 0 for
the first time when γ 1 = 2π . Equating the numerator to zero, we get
cos γ = −
η
1 − η
.
(11.33)
The equality (11.33) is possible if and only if
−
η
1 − η
1,
i. e. for η
1
2
. Setting the values η
0 < η <
1
2
from Eq. (11.33), we find the
smallest values of γ 0 (η), which changes the sign of the expression in the square
brackets of formula (11.32). The visualization of the calculation results is shown in
Fig. 11.4.
The figure shows that the value p 4 for the values η that satisfy the condition
0 η <
1
2
becomes negative when γ > γ 0 (η) and the inequality γ 0 (0) < γ 0 (β) <
γ 0
1
2
is right; for the values η
1
2
, the coefficient p 4 goes to negative values