Page 33 - June

METALWORKING EQUIPMENT AND TOOLS

and the main focus is on low-frequency vibration modes [1, 5].
However, self-oscillations at different frequencies and with different intensities are
observed at very different cutting conditions and in frictional contact. Moreover, self-oscillations
of various amplitudes almost always accompany the separation of chips in various technological
processes [4]. The very formation of articular shavings can be attributed to a self-oscillating
process, while the used dynamic models of the cutting process do not take into account this
phenomenon [4, 5].
More useful to understand the vibration resistance of cutting in more a wide frequency
range can be considered the stability of the equilibrium position of the cutting tool during the
process. In accordance with the theorems of Lagrange and Kelvin [6], a sufficient condition
for the stability of the equilibrium position of a mechanical system is the presence of a local
minimum of potential energy in this position. This position is called energetically favorable
state (EFS), and a system with an excess of potential energy is called nonequilibrium, excited.
The potential energy of a system with s degrees of freedom is a function of the generalized
coordinates of the system and is related to them by a homogeneous quadratic form [7]:

where at q1= 0, … qs = 0 (1).

In expression (1), symbols сij represent generalized coefficients rigidity.
In accordance with a sufficient condition for the stability of the equilibrium of mechanical
systems, it is possible to consider the position of the cutter without load and under load. Without
calculations, it can be concluded that the initial position of the cutting tool has a minimum of
potential energy and is in the EFS position, since any deviation from this position is associated
with elastic deformation, a corresponding increase in potential energy and a transition to an
excited state. Since at cutting in any modes, a load acts on the tool, causing a deviation from
the EFS, elastic deformations and a corresponding increase in potential energy, then the new
position will be an excited state that cannot be considered stable. It has a potential deformation
energy, which is greater than its value when the tool is in the EFS. In this interpretation, it is
already necessary to pose the question not about the stability of the cutting process, but about
the stabilization of the system in a nonequilibrium state, about those restraining conditions
that do not allow the elastic system to arbitrarily reduce the potential energy for the transition
to EFS. This includes methods for reducing the potential energy present during cutting. If in

Stanochniy park 33