126
11 The Beginning of the Theory of Stability of Equilibrium
so-called Euler forces. The latter method is familiar to the reader from the course of
resistance of materials.
The problem of determining the critical load is often replaced by a simpler problem of investigating first approximation equations or conditions for the existence of
extreme values of potential energy (in the presence of a force potential) or a state
of indifferent equilibrium, etc. Despite the fact that this substitution is not always
legal, these particular methods of determining the critical load are widely used in
many computer programs for analyzing structures.
11.7 The Theorem on Stability by the First Approximation
In the case of a system with a finite (n) number of degrees of freedom, the first
approximation equations of perturbed motion are a system of ordinary homogeneous
differential equations with constant coefficients depending on the load parameter.
Representing the generalized coordinates in this case as
q i = A i exp λt (i = 1, 2, . . . , n),
(11.4)
we obtain from the first approximation equations a known (characteristic) equation
for determining the c h a r a c t e r i s t i c i n d i c a t o r λ:
p 0 λ
2n
+ p 1 λ
2n−1
+ . . . + p 2n = 0 (p 0 > 0),
(11.5)
where p 0 , . . . , p 2n are coefficients that depend in a known way on the system and
the value of the load parameter.
For these systems, Lyapunov proved the following theorems.
Theorem 11.1 If the real parts of all the roots of Eq. (11.5) are negative, then the
considered equilibrium state will be stable (asymptotically).
Theorem 11.2 If among the roots of Eq. (11.5) there is at least one with a positive
real part, then the considered equilibrium state is unstable.
Note The study of (non)stability by the first approximation, generally speaking,
cannot give a solution to the problem of (non)stability only in the case when the real
parts of the roots of the characteristic equation have zero and negative values.
11.8 The Raus–Hurwitz criterion
In order for all the roots of Eq. (11.5) to have negative real parts, it is necessary and
sufficient to perform the inequalities
11 The Beginning of the Theory of Stability of Equilibrium
so-called Euler forces. The latter method is familiar to the reader from the course of
resistance of materials.
The problem of determining the critical load is often replaced by a simpler problem of investigating first approximation equations or conditions for the existence of
extreme values of potential energy (in the presence of a force potential) or a state
of indifferent equilibrium, etc. Despite the fact that this substitution is not always
legal, these particular methods of determining the critical load are widely used in
many computer programs for analyzing structures.
11.7 The Theorem on Stability by the First Approximation
In the case of a system with a finite (n) number of degrees of freedom, the first
approximation equations of perturbed motion are a system of ordinary homogeneous
differential equations with constant coefficients depending on the load parameter.
Representing the generalized coordinates in this case as
q i = A i exp λt (i = 1, 2, . . . , n),
(11.4)
we obtain from the first approximation equations a known (characteristic) equation
for determining the c h a r a c t e r i s t i c i n d i c a t o r λ:
p 0 λ
2n
+ p 1 λ
2n−1
+ . . . + p 2n = 0 (p 0 > 0),
(11.5)
where p 0 , . . . , p 2n are coefficients that depend in a known way on the system and
the value of the load parameter.
For these systems, Lyapunov proved the following theorems.
Theorem 11.1 If the real parts of all the roots of Eq. (11.5) are negative, then the
considered equilibrium state will be stable (asymptotically).
Theorem 11.2 If among the roots of Eq. (11.5) there is at least one with a positive
real part, then the considered equilibrium state is unstable.
Note The study of (non)stability by the first approximation, generally speaking,
cannot give a solution to the problem of (non)stability only in the case when the real
parts of the roots of the characteristic equation have zero and negative values.
11.8 The Raus–Hurwitz criterion
In order for all the roots of Eq. (11.5) to have negative real parts, it is necessary and
sufficient to perform the inequalities