140
11 The Beginning of the Theory of Stability of Equilibrium
for example, for η = 1 and small values of the parameter a, the worst-case friction
reduces the critical load by almost 4 times.
Since the ratio μ of small friction coefficients is uncertain, the critical parameters
have to be determined from the assumption that the friction is the worst. In this
way, the area of reliable stability was determined above (see p. 134), where the
equilibrium of the system is stable under arbitrary small friction.
11.12.5 The influence of the spacing of the End Masses
Let us show that the effect of friction on the value of the critical load depends
significantly on the separation of the end masses, set by the parameter a. For this
purpose, let us consider the case when η = 1. At the same time (see Fig. 11.5),
the straight line γ ∗ = 2.33 touches the curve γ 0
∗ (a, 1) at the point a 0 = 0.77.
Obviously, for the values a = 0.77, the critical load is minimal (γ 0 ) and its value
does not depend on the relation (μ) of low friction coefficients.
Let γ 0 = 1. It can be shown that when η is continuously reduced from 1 to
1
2
,
the point a 0 (η) will move along the numeric axis from the value a 0 (1) = 0.77
to a 0
1
2
= ∞, respectively. Each point a 0 (η 0 ) gives the minimum of the
corresponding curve γ 0
∗ (a, η 0 ) and has similar properties as a 0 (1) = 0.77. At such
points of the minimum, there is an equality
ρ
2 c 11 = c 22 .
(11.52)
In this case, the conditions (11.49) and (11.50) obviously coincide for any μ; the
critical load is minimal and there is no destabilization.
In the area of reliable stability (Fig. 11.4), the equilibrium of the system is stable
for any values of the parameter a. Obviously, if the separation of the end masses is
unfavorable (a = a 0 ), then the loss of stability can occur directly above the curve
γ ∗ (η).
The significant effect of low friction on critical parameters in self-oscillating loss
of stability becomes apparent if we consider the general condition (11.31) in more
detail.
In fact, let the loss of stability occur due to a violation of the condition (11.31),
which can be represented as
p
2
1
p 3
p 1
p 2 −
p 3
p 1
2
p 0 − p 4
> 0.
(11.53)
11 The Beginning of the Theory of Stability of Equilibrium
for example, for η = 1 and small values of the parameter a, the worst-case friction
reduces the critical load by almost 4 times.
Since the ratio μ of small friction coefficients is uncertain, the critical parameters
have to be determined from the assumption that the friction is the worst. In this
way, the area of reliable stability was determined above (see p. 134), where the
equilibrium of the system is stable under arbitrary small friction.
11.12.5 The influence of the spacing of the End Masses
Let us show that the effect of friction on the value of the critical load depends
significantly on the separation of the end masses, set by the parameter a. For this
purpose, let us consider the case when η = 1. At the same time (see Fig. 11.5),
the straight line γ ∗ = 2.33 touches the curve γ 0
∗ (a, 1) at the point a 0 = 0.77.
Obviously, for the values a = 0.77, the critical load is minimal (γ 0 ) and its value
does not depend on the relation (μ) of low friction coefficients.
Let γ 0 = 1. It can be shown that when η is continuously reduced from 1 to
1
2
,
the point a 0 (η) will move along the numeric axis from the value a 0 (1) = 0.77
to a 0
1
2
= ∞, respectively. Each point a 0 (η 0 ) gives the minimum of the
corresponding curve γ 0
∗ (a, η 0 ) and has similar properties as a 0 (1) = 0.77. At such
points of the minimum, there is an equality
ρ
2 c 11 = c 22 .
(11.52)
In this case, the conditions (11.49) and (11.50) obviously coincide for any μ; the
critical load is minimal and there is no destabilization.
In the area of reliable stability (Fig. 11.4), the equilibrium of the system is stable
for any values of the parameter a. Obviously, if the separation of the end masses is
unfavorable (a = a 0 ), then the loss of stability can occur directly above the curve
γ ∗ (η).
The significant effect of low friction on critical parameters in self-oscillating loss
of stability becomes apparent if we consider the general condition (11.31) in more
detail.
In fact, let the loss of stability occur due to a violation of the condition (11.31),
which can be represented as
p
2
1
p 3
p 1
p 2 −
p 3
p 1
2
p 0 − p 4
> 0.
(11.53)