11.12 Stability Under Non-conservative Load (Example)
131
Euler force, then Eq. (11.16) for λ = λ 1 will have a solution that grows indefinitely
over time according to the potential law. For the case under consideration, the Euler
force is critical.
When λ = 0, Eq. (11.15) becomes independent of the nature of the mass
distribution and passes into the static equation, from which the Euler forces are
determined. Thus, the Euler load can be defined as the smallest load, at which the
eigenvalue of the corresponding dynamic problem vanishes.
11.12 Stability Under Non-conservative Load (Example)
The study of equilibrium stability in the general case of non-conservative loads is
a complex mathematical problem. Therefore, examples of this type of problem in
publications are quite rare. This section only provides an illustration of the effect
of non-conservative loads on the oscillation and stability of the simplest model of a
cantilever rod compressed by spaced masses.
11.12.1 Equations of Perturbed Motion
We will consider an elastic rod of length l with two separated masses at the free
end, compressed by the gravity G of the end masses and the tracking force H
(Fig. 11.3a). Ignoring the mass of the rod, we consider here a system with two
degrees of freedom.
a
b
Fig. 11.3 The simplest non-conservative system
131
Euler force, then Eq. (11.16) for λ = λ 1 will have a solution that grows indefinitely
over time according to the potential law. For the case under consideration, the Euler
force is critical.
When λ = 0, Eq. (11.15) becomes independent of the nature of the mass
distribution and passes into the static equation, from which the Euler forces are
determined. Thus, the Euler load can be defined as the smallest load, at which the
eigenvalue of the corresponding dynamic problem vanishes.
11.12 Stability Under Non-conservative Load (Example)
The study of equilibrium stability in the general case of non-conservative loads is
a complex mathematical problem. Therefore, examples of this type of problem in
publications are quite rare. This section only provides an illustration of the effect
of non-conservative loads on the oscillation and stability of the simplest model of a
cantilever rod compressed by spaced masses.
11.12.1 Equations of Perturbed Motion
We will consider an elastic rod of length l with two separated masses at the free
end, compressed by the gravity G of the end masses and the tracking force H
(Fig. 11.3a). Ignoring the mass of the rod, we consider here a system with two
degrees of freedom.
a
b
Fig. 11.3 The simplest non-conservative system