94
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
the investigated system, among others. In addition, all frequencies exhibited by beam
vibrations are in resonance relation with the excitation frequency.
In order to confirm the occurrence of true chaotic vibration, we have demonstrated
convergence of the numerical results regarding the signal, not only with respect to the
Fourier frequency power spectrum, which is recognized and often employed. Furthermore, we obtained convergence of the numerical results when the computation
step (with respect to the spatial coordinate) was increased twice.
We have detected the set of parameters and we have used the methods which guarantee reliability and truth of the obtained solutions. In particular, we have demonstrated that the increase in the number of beam partitions (nodes) implies regularization of the obtained vibrations. After comparing the results obtained within linear and
nonlinear problems and taking into account the contact between beams, we have also
illustrated that the account of the spatial coordinate x in u(x, t), i.e. the increase in
the number of equations, decreases the chaotic components in the analysed signals.
Furthermore, we have presented, illustrated and discussed that the continuous
mechanical system cannot be truncated to the system with a finite number of degrees
of freedom, but the problem is, indeed, of an infinite dimension.
Finally, we have shown how applications of the PCA for the purification of chaotic
signal of the beam allows localizing the fundamental frequency of vibrations of the
studied structure composed of two beams.
4.2 Literature Review
Beams and beam constructions are widely employed as elements of numerous devices
in today’s industry, machine construction, rocket industry and geology. In many
cases, the above-mentioned structural elements are subjected to complex external
dynamical excitations. Investigations of nonlinear dynamics and contact interactions
of the beam structures belong to important (but unsolved) challenging problems in
the field of fabricating various sensors and amplifiers.
Owing to the complexity of equations governing the nonlinear dynamics of two
geometrically nonlinear beams with a contact interaction, it is impossible to find
an exact analytical solution. In general, the problem can be solved using numerical
methods. However, in this case, the problem regarding reliability of the obtained
results is generated [2], in particular in the case of chaos detection and monitoring.
Chaotic vibrations are dangerous since they usually exhibit large amplitudes causing
large-scale reactions of the system, either leading directly to the system damage or
causing various harmful effects. On the other hand, in many cases, errors introduced
by numerical computations are identified with chaotic oscillations. In this work, we
define the truth of chaos by means of dealing with the problems of contact interaction
of two beams by the employed numerical method.
It is known that the fundamental characteristics of chaos is associated with sensitivity to the initial conditions. Definition of chaos introduced in 1989 by Devaney [3]
consists of three parts. In addition to the condition of existence of the dependence
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
the investigated system, among others. In addition, all frequencies exhibited by beam
vibrations are in resonance relation with the excitation frequency.
In order to confirm the occurrence of true chaotic vibration, we have demonstrated
convergence of the numerical results regarding the signal, not only with respect to the
Fourier frequency power spectrum, which is recognized and often employed. Furthermore, we obtained convergence of the numerical results when the computation
step (with respect to the spatial coordinate) was increased twice.
We have detected the set of parameters and we have used the methods which guarantee reliability and truth of the obtained solutions. In particular, we have demonstrated that the increase in the number of beam partitions (nodes) implies regularization of the obtained vibrations. After comparing the results obtained within linear and
nonlinear problems and taking into account the contact between beams, we have also
illustrated that the account of the spatial coordinate x in u(x, t), i.e. the increase in
the number of equations, decreases the chaotic components in the analysed signals.
Furthermore, we have presented, illustrated and discussed that the continuous
mechanical system cannot be truncated to the system with a finite number of degrees
of freedom, but the problem is, indeed, of an infinite dimension.
Finally, we have shown how applications of the PCA for the purification of chaotic
signal of the beam allows localizing the fundamental frequency of vibrations of the
studied structure composed of two beams.
4.2 Literature Review
Beams and beam constructions are widely employed as elements of numerous devices
in today’s industry, machine construction, rocket industry and geology. In many
cases, the above-mentioned structural elements are subjected to complex external
dynamical excitations. Investigations of nonlinear dynamics and contact interactions
of the beam structures belong to important (but unsolved) challenging problems in
the field of fabricating various sensors and amplifiers.
Owing to the complexity of equations governing the nonlinear dynamics of two
geometrically nonlinear beams with a contact interaction, it is impossible to find
an exact analytical solution. In general, the problem can be solved using numerical
methods. However, in this case, the problem regarding reliability of the obtained
results is generated [2], in particular in the case of chaos detection and monitoring.
Chaotic vibrations are dangerous since they usually exhibit large amplitudes causing
large-scale reactions of the system, either leading directly to the system damage or
causing various harmful effects. On the other hand, in many cases, errors introduced
by numerical computations are identified with chaotic oscillations. In this work, we
define the truth of chaos by means of dealing with the problems of contact interaction
of two beams by the employed numerical method.
It is known that the fundamental characteristics of chaos is associated with sensitivity to the initial conditions. Definition of chaos introduced in 1989 by Devaney [3]
consists of three parts. In addition to the condition of existence of the dependence