Chapter 11
The Beginning of the Theory of Stability
of Equilibrium
11.1 Stability and Instability
Consider some mechanical system that is in equilibrium. Imagine that the points
of this system were given infinitesimal speeds. The system will start a movement
called perturbed. If with an unlimited increase in time, the deviations of the system
points from the equilibrium position remain small for all perturbed movements, then
the equilibrium of the system is called s t a b l e.
In the event that the deviations of the system from the equilibrium position
decrease indefinitely, the equilibrium is called a s y m p t o t i c a l l y s t a b l e.
Otherwise, the equilibrium is called u n s t a b l e.
11.2 Work and Classification of Forces
Let some constant force be applied at an arbitrary point on the body. Let, further, the
body make infinitely small movements. If a force moves with its point of application,
then they say that the work of a given force is equal to the product of the magnitude
of the force and the projection of its movement in the direction of the force. If the
displacements of the force and the points of its application do not coincide, the work
should be defined as the product of the modulus of force and the corresponding
projection of the displacement of the initial point of its application. With this in
mind, we consider the work of a force applied to a completely rigid body subject to
the following conditions:
(a) the force does not change the line of its action;
(b) the force is associated with the same point of application and when the body
moves, it is always perpendicular to some line invariably connected with the
body (tracking force).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_11
121
The Beginning of the Theory of Stability
of Equilibrium
11.1 Stability and Instability
Consider some mechanical system that is in equilibrium. Imagine that the points
of this system were given infinitesimal speeds. The system will start a movement
called perturbed. If with an unlimited increase in time, the deviations of the system
points from the equilibrium position remain small for all perturbed movements, then
the equilibrium of the system is called s t a b l e.
In the event that the deviations of the system from the equilibrium position
decrease indefinitely, the equilibrium is called a s y m p t o t i c a l l y s t a b l e.
Otherwise, the equilibrium is called u n s t a b l e.
11.2 Work and Classification of Forces
Let some constant force be applied at an arbitrary point on the body. Let, further, the
body make infinitely small movements. If a force moves with its point of application,
then they say that the work of a given force is equal to the product of the magnitude
of the force and the projection of its movement in the direction of the force. If the
displacements of the force and the points of its application do not coincide, the work
should be defined as the product of the modulus of force and the corresponding
projection of the displacement of the initial point of its application. With this in
mind, we consider the work of a force applied to a completely rigid body subject to
the following conditions:
(a) the force does not change the line of its action;
(b) the force is associated with the same point of application and when the body
moves, it is always perpendicular to some line invariably connected with the
body (tracking force).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_11
121