Among the wide diversity of nonlinear mechanical systems, it is possible
to distinguish a representative class of the systems which may be
characterised by the presence of threshold nonlinear positional forces.
Under particular configurations, such systems demonstrate a sudden
change in the behaviour of elastic and dissipative forces. Mathematical
study of such systems involves an analysis of equations of motion
containing large-factored nonlinear terms which are associated with the
above threshold nonlinearity. Due to this, we distinguish such
discontinuous systems from the much wider class of essentially nonlinear
systems, and define them as strongly nonlinear systems'. The vibration
occurring in strongly nonlinear systems may be characterised by a sudden
and abrupt change of the velocity at particular time instants. Such a
vibration is said to be non-smooth. The systems most studied from this
class are those with relaxation (Van Der Pol, Andronov, Vitt, Khaikhin,
Teodorchik, etc. [5,65,70,71,98,171,181]), where the non-smooth
vibration usually appears due to the presence of large nonconservative
nonlinear forces. Equations of motion describing the vibration with
relaxation may be written in such a manner that the highest derivative
is accompanied by a small parameter. The methods of integration of these
equations have been developed by Vasilieva and Butuzov [182], Volosov
and Morgunov [190], Dorodnitsin [38], Zheleztsov [201], Mischenko
and Rozov [115], Pontriagin [137], Tichonov [174,175], etc. In a
system with threshold nonlinearity, the non-smooth vibration occurs due
to the action of large conservative forces. This is distinct from a
system with relaxation.
