4.2 Literature Review
95
on the initial conditions, it includes the intermittency/transitivity condition as well
as the regularity condition understood as the density exhibited by periodic points or
periodicity. In 1992, Banks et al. [4] proved that sensitivity to the initial conditions
can be removed, which means that transitivity and periodicity are sufficient for the
occurrence of chaos.
One should also mention a definition of chaos proposed by Kundsen [5], according
to which a function defined on the bounded metric space can be understood as a
chaotic one if it is characterized by an essential dependence on initial conditions.
Owing to the definition of chaos given by Gulick [6], chaotic orbits exist if there
is either essential sensitivity to the initial conditions or at least one of the Lyapunov
exponents is positive in each point of the considered chaotic domain. In other words,
the studied orbit tends finally neither to a periodic nor a chaotic one (we refer to
Gulick definition while checking/validating the existence of true chaos).
The further obtained solutions depend on a chosen kinematic hypothesis, boundary
and initial conditions, number of employed intervals of integration regarding beams
with respect to FDM and methods used for solving Cauchy problem.
We are aimed at studying nonlinear dynamical features of a structural system
composed of two beams with a small clearance. The beams are modelled using the
first-order kinematic hypotheses. The lower beam (beam 2) can be viewed as an
elastic foundation for the upper beam (beam 1), while the latter beam is subjected to
the transverse uniformly distributed harmonic load.
In contrast to numerous other works dealing with truncation of the problem of an
infinite dimension to that of a few degrees of freedom (usually two or three) or those
employing the so-called reduced-order approach, we are aimed at studying the so far
defined problem as that having an infinite number of degrees of freedom.
In general, the solution to the problem can essentially depend on the employed
method of reduction from PDEs to ODEs, the method used to solve Cauchy problem
as well as on boundary and initial conditions. The solution given by FDM of the
approximation O(c
2
) will depend on the number of partitions of the integration
interval for solving Cauchy problem with respect to time. The analysis of the obtained
results is carried out by the methods of nonlinear dynamics and the theory regarding
the geometrically nonlinear beams with the account of the contact interaction.
4.3 Mathematical Model
The following main steps of investigations are carried out to numerically solve nonlinear problems of structures composed of two beams with a clearance and subjected
to the harmonic load, and thus to detect chaotic vibrations:
1. Since nonlinear PDEs are reduced to ODEs due to the employed FDM of the
second-order accuracy, the obtained solutions essentially depend on the number
of partitions of the beam. In other words, we need to find the number of n beam
partitions, for which the solutions obtained for 2n and n partitions coincide. We
95
on the initial conditions, it includes the intermittency/transitivity condition as well
as the regularity condition understood as the density exhibited by periodic points or
periodicity. In 1992, Banks et al. [4] proved that sensitivity to the initial conditions
can be removed, which means that transitivity and periodicity are sufficient for the
occurrence of chaos.
One should also mention a definition of chaos proposed by Kundsen [5], according
to which a function defined on the bounded metric space can be understood as a
chaotic one if it is characterized by an essential dependence on initial conditions.
Owing to the definition of chaos given by Gulick [6], chaotic orbits exist if there
is either essential sensitivity to the initial conditions or at least one of the Lyapunov
exponents is positive in each point of the considered chaotic domain. In other words,
the studied orbit tends finally neither to a periodic nor a chaotic one (we refer to
Gulick definition while checking/validating the existence of true chaos).
The further obtained solutions depend on a chosen kinematic hypothesis, boundary
and initial conditions, number of employed intervals of integration regarding beams
with respect to FDM and methods used for solving Cauchy problem.
We are aimed at studying nonlinear dynamical features of a structural system
composed of two beams with a small clearance. The beams are modelled using the
first-order kinematic hypotheses. The lower beam (beam 2) can be viewed as an
elastic foundation for the upper beam (beam 1), while the latter beam is subjected to
the transverse uniformly distributed harmonic load.
In contrast to numerous other works dealing with truncation of the problem of an
infinite dimension to that of a few degrees of freedom (usually two or three) or those
employing the so-called reduced-order approach, we are aimed at studying the so far
defined problem as that having an infinite number of degrees of freedom.
In general, the solution to the problem can essentially depend on the employed
method of reduction from PDEs to ODEs, the method used to solve Cauchy problem
as well as on boundary and initial conditions. The solution given by FDM of the
approximation O(c
2
) will depend on the number of partitions of the integration
interval for solving Cauchy problem with respect to time. The analysis of the obtained
results is carried out by the methods of nonlinear dynamics and the theory regarding
the geometrically nonlinear beams with the account of the contact interaction.
4.3 Mathematical Model
The following main steps of investigations are carried out to numerically solve nonlinear problems of structures composed of two beams with a clearance and subjected
to the harmonic load, and thus to detect chaotic vibrations:
1. Since nonlinear PDEs are reduced to ODEs due to the employed FDM of the
second-order accuracy, the obtained solutions essentially depend on the number
of partitions of the beam. In other words, we need to find the number of n beam
partitions, for which the solutions obtained for 2n and n partitions coincide. We