128
11 The Beginning of the Theory of Stability of Equilibrium
then under these conditions, the critical value of the parameter coincides with the
smallest root of one of the two equations
p 2n = 0, , 2n−1 = 0,
(11.8)
or, which is the same thing, with the smallest root of the equation 2n = 0.
Keeping in mind that for p 2n = 0, Eq. (11.5) has a zero root (λ = 0), which
corresponds to arbitrary constant values of the generalized coordinates (11.4), we
have in this case a state of indifferent equilibrium. If the load is critical, we will say
that the elastic system loses Euler stability.
From the above, it follows that for 2n−1 = 0, at least two characteristic numbers
are purely imaginary (λ 1,2 = ±ωi). Since the real parts of the remaining roots of
Eq. (11.5) are negative, the system sets the periodic motion. This type of stability
loss is called s e l f - o s c i l l a t i n g.
11.10 Methods for Determining Critical Load
The smallest load at which the total potential energy of the elastic system under the
influence of forces loses a minimum in the state of equilibrium under study is called
e n e r g y - c r i t i c a l. The determination of this load is the e n e r g y m e t h o d
task of studying the stability of elastic systems.
The value of the load parameter, at which infinitely close forms of equilibrium are
possible, is called critical in Euler’s sense, and the corresponding load is Euler’s. 4
The value of the load parameter, at an arbitrarily small excess of which there is
no form of equilibrium infinitely close to the studied one, is called the ultimate, and
the corresponding load is called the u l t i m a t e d. Determining Euler and limit
loads is the task of a static method for studying the stability of elastic systems.
In the presence of non-conservative forces in an elastic system, the concept of
energy load, generally speaking, loses its meaning. Below (p. 131) we show that the
Euler load in such systems can be critical under certain conditions.
In some special cases, the Euler and energy-critical loads in conservative systems
may differ. For example, when the Euler value of the load parameter H = H e does
not change the sign of any of the coefficients a n (H ) in the expression (11.2), the
potential energy may not lose its minimum at H = H e , i.e. the Euler load may be
lower than the energy-critical one. The Euler load may not exist when the coefficient
a n (H ) changes the sign after going to infinity, and the energy-critical load exists.
It follows from the above, in particular, that the presence of the potential of forces
acting on an elastic system is neither necessary nor sufficient for the existence of an
Euler load or its coincidence with the critical load.
4 This load is also sometimes called the bifurcation or branching load of equilibrium forms.
11 The Beginning of the Theory of Stability of Equilibrium
then under these conditions, the critical value of the parameter coincides with the
smallest root of one of the two equations
p 2n = 0, , 2n−1 = 0,
(11.8)
or, which is the same thing, with the smallest root of the equation 2n = 0.
Keeping in mind that for p 2n = 0, Eq. (11.5) has a zero root (λ = 0), which
corresponds to arbitrary constant values of the generalized coordinates (11.4), we
have in this case a state of indifferent equilibrium. If the load is critical, we will say
that the elastic system loses Euler stability.
From the above, it follows that for 2n−1 = 0, at least two characteristic numbers
are purely imaginary (λ 1,2 = ±ωi). Since the real parts of the remaining roots of
Eq. (11.5) are negative, the system sets the periodic motion. This type of stability
loss is called s e l f - o s c i l l a t i n g.
11.10 Methods for Determining Critical Load
The smallest load at which the total potential energy of the elastic system under the
influence of forces loses a minimum in the state of equilibrium under study is called
e n e r g y - c r i t i c a l. The determination of this load is the e n e r g y m e t h o d
task of studying the stability of elastic systems.
The value of the load parameter, at which infinitely close forms of equilibrium are
possible, is called critical in Euler’s sense, and the corresponding load is Euler’s. 4
The value of the load parameter, at an arbitrarily small excess of which there is
no form of equilibrium infinitely close to the studied one, is called the ultimate, and
the corresponding load is called the u l t i m a t e d. Determining Euler and limit
loads is the task of a static method for studying the stability of elastic systems.
In the presence of non-conservative forces in an elastic system, the concept of
energy load, generally speaking, loses its meaning. Below (p. 131) we show that the
Euler load in such systems can be critical under certain conditions.
In some special cases, the Euler and energy-critical loads in conservative systems
may differ. For example, when the Euler value of the load parameter H = H e does
not change the sign of any of the coefficients a n (H ) in the expression (11.2), the
potential energy may not lose its minimum at H = H e , i.e. the Euler load may be
lower than the energy-critical one. The Euler load may not exist when the coefficient
a n (H ) changes the sign after going to infinity, and the energy-critical load exists.
It follows from the above, in particular, that the presence of the potential of forces
acting on an elastic system is neither necessary nor sufficient for the existence of an
Euler load or its coincidence with the critical load.
4 This load is also sometimes called the bifurcation or branching load of equilibrium forms.