Chapter 4
Reliability of Chaotic Vibrations of
Euler-Bernoulli Beams with Clearance
4.1 Introduction
The methodology for detecting true chaos (in terms of nonlinear dynamics) developed
on the example of a structure composed of two beams with a small clearance is outlined. Euler-Bernoulli hypothesis is employed, and the contact interaction between
beams follows Kantor model. The complex nonlinearity results from von Kármán
geometric nonlinearity as well as the nonlinearity implied by the contact interaction. The governing PDEs are reduced to ODEs by the second-order finite difference
method (FDM). The obtained system of equations is solved by Runge-Kutta method
of different accuracies. To purify the signal from errors introduced by numerical
methods, the principal component analysis is employed and the sign of the first
Lyapunov exponent is estimated by Kantz, Wolf and Rosenstein methods [1].
The proposed methodology for the detection of true chaos is based on the complex
investigation of a signal/time history from the point of view of nonlinear dynamics, focusing on construction and analysis of the signals, power frequency spectra,
Poincaré pseudo-maps, wavelet spectra, Lyapunov exponents as well as 2D and 3D
phase portraits.
Owing to the introduced methodology, the convergence of FDM has been achieved
while solving the system of nonlinear PDEs/ODEs with an account of the geometric
von Kármán nonlinearity and the design nonlinearity for the mathematical model
established in the frame of the kinematic first-order hypotheses.
The employed methodology has proved that the chaotic vibrations of two-layer
beams can be detected based on the used methods and computational algorithms.
Although the magnitude of clearance between the beams is small, it has been found
that chaotic vibrations of beams appear even for small amplitudes. However, we
have illustrated that in spite of small amplitudes, there is a need to take the geometric
nonlinearity into account while constructing a feasible mathematical model. We
have detected, the occurrence of the phase synchronization of beam vibrations for
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_4
93
Reliability of Chaotic Vibrations of
Euler-Bernoulli Beams with Clearance
4.1 Introduction
The methodology for detecting true chaos (in terms of nonlinear dynamics) developed
on the example of a structure composed of two beams with a small clearance is outlined. Euler-Bernoulli hypothesis is employed, and the contact interaction between
beams follows Kantor model. The complex nonlinearity results from von Kármán
geometric nonlinearity as well as the nonlinearity implied by the contact interaction. The governing PDEs are reduced to ODEs by the second-order finite difference
method (FDM). The obtained system of equations is solved by Runge-Kutta method
of different accuracies. To purify the signal from errors introduced by numerical
methods, the principal component analysis is employed and the sign of the first
Lyapunov exponent is estimated by Kantz, Wolf and Rosenstein methods [1].
The proposed methodology for the detection of true chaos is based on the complex
investigation of a signal/time history from the point of view of nonlinear dynamics, focusing on construction and analysis of the signals, power frequency spectra,
Poincaré pseudo-maps, wavelet spectra, Lyapunov exponents as well as 2D and 3D
phase portraits.
Owing to the introduced methodology, the convergence of FDM has been achieved
while solving the system of nonlinear PDEs/ODEs with an account of the geometric
von Kármán nonlinearity and the design nonlinearity for the mathematical model
established in the frame of the kinematic first-order hypotheses.
The employed methodology has proved that the chaotic vibrations of two-layer
beams can be detected based on the used methods and computational algorithms.
Although the magnitude of clearance between the beams is small, it has been found
that chaotic vibrations of beams appear even for small amplitudes. However, we
have illustrated that in spite of small amplitudes, there is a need to take the geometric
nonlinearity into account while constructing a feasible mathematical model. We
have detected, the occurrence of the phase synchronization of beam vibrations for
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_4
93