142
11 The Beginning of the Theory of Stability of Equilibrium
p 3
p 1
=
p 2
2p 0
.
(11.61)
In fact, if the equality (11.61) is fulfilled, then, as can be seen from formula
(11.55), stability occurs as long as the value (11.56) is positive, so that the
boundaries of the stability (( = 0) and instability (( 0 = 0) regions in this case
coincide. On the contrary, when there is no destabilization, the system parameters
must obviously meet the condition (11.61).
Note that small friction can stabilize an unstable equilibrium, while taking into
account the quantities of the second order of smallness (ε 2 ) is significant. Indeed,
the presence of the second term in the right part of formula (11.60) may slightly
expand the stability area (11.55). The latter becomes important in cases when the
system without taking into account friction is unstable or is located on the boundary
of the instability region (( 0 0).
References
1. N. Chetaev, Ustoichivost’ dvizheniya (Dynamic stability). (Nauka Publ., Moscow, 1965)
2. M. Il’in, K. Kolesnikov, S. Yu.S., Teoriya kolebanii: ucheb. dlya vuzov (Theory of vibrations:
textbook. for universities). (MGTU im. N.Eh. Baumana Publ., Moscow, 2003)
3. M. Leonov, Osnovy mekhaniki uprugogo tela (Fundamentals of elastic body mechanics). (Izd-vo
AS Kirg. SSR, Frunze, 1963)
4. V. Molotnikov, Osnovy teoreticheskoi mekhaniki (Fundamentals of theoretical mechanics).
(Feniks Publ., Rostov on Don, 2004)
11 The Beginning of the Theory of Stability of Equilibrium
p 3
p 1
=
p 2
2p 0
.
(11.61)
In fact, if the equality (11.61) is fulfilled, then, as can be seen from formula
(11.55), stability occurs as long as the value (11.56) is positive, so that the
boundaries of the stability (( = 0) and instability (( 0 = 0) regions in this case
coincide. On the contrary, when there is no destabilization, the system parameters
must obviously meet the condition (11.61).
Note that small friction can stabilize an unstable equilibrium, while taking into
account the quantities of the second order of smallness (ε 2 ) is significant. Indeed,
the presence of the second term in the right part of formula (11.60) may slightly
expand the stability area (11.55). The latter becomes important in cases when the
system without taking into account friction is unstable or is located on the boundary
of the instability region (( 0 0).
References
1. N. Chetaev, Ustoichivost’ dvizheniya (Dynamic stability). (Nauka Publ., Moscow, 1965)
2. M. Il’in, K. Kolesnikov, S. Yu.S., Teoriya kolebanii: ucheb. dlya vuzov (Theory of vibrations:
textbook. for universities). (MGTU im. N.Eh. Baumana Publ., Moscow, 2003)
3. M. Leonov, Osnovy mekhaniki uprugogo tela (Fundamentals of elastic body mechanics). (Izd-vo
AS Kirg. SSR, Frunze, 1963)
4. V. Molotnikov, Osnovy teoreticheskoi mekhaniki (Fundamentals of theoretical mechanics).
(Feniks Publ., Rostov on Don, 2004)