We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev–Pelekh–Reddy model). The von Kármán geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, as well as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the second order of accuracy, and next via the Runge–Kutta method of orders from two to eight, as well as the Newmark method. Investigations of the non-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based on the Sano–Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, and our results are authentic, reliable, and accurate.
Skip Nav Destination
,
,
,
Article navigation
February 2021
Research Article|
February 02 2021
On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness Available to Purchase
Special Collection:
Recent Advances in Modeling Complex Systems: Theory and Applications
T. V. Yakovleva
;
T. V. Yakovleva
a)
1
Department of Mathematics and Modeling, Saratov State Technical University
, 77 Politehnicheskaya St., 410054 Saratov, Russian Federation
Search for other works by this author on:
J. Awrejcewicz
;
J. Awrejcewicz
b)
2
Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology
, 1/15 Stefanowski St., 90-924 Lodz, Poland
b)Author to whom correspondence should be addressed: [email protected]
Search for other works by this author on:
V. S. Kruzhilin;
V. S. Kruzhilin
c)
1
Department of Mathematics and Modeling, Saratov State Technical University
, 77 Politehnicheskaya St., 410054 Saratov, Russian Federation
Search for other works by this author on:
V. A. Krysko
V. A. Krysko
d)
1
Department of Mathematics and Modeling, Saratov State Technical University
, 77 Politehnicheskaya St., 410054 Saratov, Russian Federation
Search for other works by this author on:
T. V. Yakovleva
1,a)
J. Awrejcewicz
2,b)
V. S. Kruzhilin
1,c)
V. A. Krysko
1,d)
1
Department of Mathematics and Modeling, Saratov State Technical University
, 77 Politehnicheskaya St., 410054 Saratov, Russian Federation
2
Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology
, 1/15 Stefanowski St., 90-924 Lodz, Poland
a)
Electronic mail: [email protected]
b)Author to whom correspondence should be addressed: [email protected]
c)
Electronic mail: [email protected]
d)
Electronic mail: [email protected]
Note: This paper belongs to the Focus Issue, Recent Advances in Modeling Complex Systems: Theory and Applications.
Chaos 31, 023107 (2021)
Article history
Received:
October 06 2020
Accepted:
January 06 2021
Citation
T. V. Yakovleva, J. Awrejcewicz, V. S. Kruzhilin, V. A. Krysko; On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness. Chaos 1 February 2021; 31 (2): 023107. https://doi.org/10.1063/5.0032069
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto–Sivashinsky test case
Rambod Mojgani, Ashesh Chattopadhyay, et al.
Recent achievements in nonlinear dynamics, synchronization, and networks
Dibakar Ghosh, Norbert Marwan, et al.
Enhancing reservoir predictions of chaotic time series by incorporating delayed values of input and reservoir variables
Luk Fleddermann, Sebastian Herzog, et al.
Related Content
Nonlinear dynamics of contact interaction of a size-dependent plate supported by a size-dependent beam
Chaos (May 2018)
Chaotic phenomena of charged particles in crystal lattices
Chaos (May 2006)
Chaotic vibrations of flexible functionally graded porous closed size-dependent cylindrical shells
AIP Conf. Proc. (June 2024)
Chaotic vibrations of size-dependent flexible rectangular plates
Chaos (April 2021)