Non-Gaussian additive and multiplicative noise-induced chaos in the lateral vibration of a viscoelastic plate: A fully analytic approach

Abstract

In this study, the chaotic behavior of a viscoelastic plate under integrated non-Gaussian additive and multiplicative bounded noise is investigated with an analytical approach. First, the governing equation of motion of the system was derived by introducing a set of dimensionless parameters. After that, the modified version of Melinkov’s function in terms of statistical indices was obtained, and then, for a spectrum of bounded noises, the boarder curves of chaotic areas were obtained. The model of sine/cosine Wiener was chosen for the bounded noise which enables the researcher to produce a range of wide and narrowband noises. For the case that the excitation is only additive, or only multiplicative, and that both excitations exist simultaneously, the effect of variations in structural properties and noise characteristics on the chaos area were investigated. It was shown that at frequencies close to the natural frequency of the corresponding linear system, narrowband excitations affected the chaotic behavior more than the wideband ones and vice versa. To validate the results, a numerical simulation was also made.

Keywords

Non-Gaussian excitation bounded noise Kelvin–Voigt viscoelastic plate Melnikov’s method noise-induced chaos structural stability

Get full access to this article

View all access options for this article.

References

Abedi

Asnafi

(2015) To reduce the instability region in the nonlinear transverse vibration of randomly excited plates using orthotropic P-FG material. Nonlinear Dynamics 80: 1413–1430.

Abedi

Asnafi

Karami

(2014) To obtain approximate probability density functions for a class of axially moving viscoelastic plates under external and parametric white noise excitation. Nonlinear Dynamics 78: 1717–1727.

Alijani

Amabili

(2013) Nonlinear vibrations of laminated and sandwich rectangular plates with free edges. Part 1: theory and numerical simulations. Composite Structures 105: 422–436.

Amabili

(2008) Nonlinear Vibrations and Stability of Shells and Plates. New York: Cambridge University Press.

Amabili

(2016) Nonlinear vibrations of viscoelastic rectangular plates. Journal of Sound and Vibration 362: 142–156.

Amabili

(2018a) Nonlinear damping in large-amplitude vibrations: modelling and experiments. Nonlinear Dynamics: 1–14.

Amabili

(2018b) Nonlinear damping in nonlinear vibrations of rectangular plates: derivation from viscoelasticity and experimental validation. Journal of the Mechanics and Physics of Solids 118: 275–292.

Amabili

(2018c) Nonlinear Mechanics of Shells and Plates in Composite, Soft and Biological Materials. New York: Cambridge University Press.

Asnafi

(2019) Chaotic analysis of Kelvin–Voigt viscoelastic plates under combined transverse periodic and white noise excitation: an analytic approach. Acta Mechanica: 1–16.

10.

Balasubramanian

Ferrari

Amabili

, et al. (2017) Experimental and theoretical study on large amplitude vibrations of clamped rubber plates. International Journal of Non-Linear Mechanics 94: 36–45.

11.

D'onofrio

(2013) Bounded Noises in Physics, Biology, and Engineering. New York: Springer.

12.

Deng

Zhong

Liu

(2019) Stochastic stability of viscoelastic plates under bounded noise excitation. European Journal of Mechanics-A/Solids 78: 103849.

13.

Farokhi

Ghayesh

(2017) Nonlinear resonant response of imperfect extensible Timoshenko microbeams. International Journal of Mechanics and Materials in Design 13: 43–55.

14.

Farokhi

Ghayesh

Gholipour

(2017) Dynamics of functionally graded micro-cantilevers. International Journal of Engineering Science 115: 117–130.

15.

Ghayesh

(2012a) Nonlinear dynamic response of a simply-supported Kelvin–Voigt viscoelastic beam, additionally supported by a nonlinear spring. Nonlinear Analysis: Real World Applications 13: 1319–1333.

16.

Ghayesh

(2012b) Subharmonic dynamics of an axially accelerating beam. Archive of Applied Mechanics 82: 1169–1181.

17.

Ghayesh

(2018a) Dynamics of functionally graded viscoelastic microbeams. International Journal of Engineering Science 124: 115–131.

18.

Ghayesh

(2018b) Functionally graded microbeams: simultaneous presence of imperfection and viscoelasticity. International Journal of Mechanical Sciences 140: 339–350.

19.

Ghayesh

(2018c) Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Applied Mathematical Modelling 59: 583–596.

20.

Ghayesh

(2019a) Asymmetric viscoelastic nonlinear vibrations of imperfect AFG beams. Applied Acoustics 154: 121–128.

21.

Ghayesh

(2019b) Dynamical analysis of multilayered cantilevers. Communications in Nonlinear Science and Numerical Simulation 71: 244–253.

22.

Ghayesh

(2019c) Mechanics of viscoelastic functionally graded microcantilevers. European Journal of Mechanics-A/Solids 73: 492–499.

23.

Ghayesh

(2019d) Nonlinear oscillations of FG cantilevers. Applied Acoustics 145: 393–398.

24.

Ghayesh

(2019e) Resonant vibrations of FG viscoelastic imperfect Timoshenko beams. Journal of Vibration and Control 25: 1823–1832.

25.

Ghayesh

(2019f) Viscoelastic dynamics of axially FG microbeams. International Journal of Engineering Science 135: 75–85.

26.

Ghayesh

(2019g) Viscoelastic mechanics of Timoshenko functionally graded imperfect microbeams. Composite Structures 225: 110974.

27.

Ghayesh

(2019h) Viscoelastic nonlinear dynamic behaviour of Timoshenko FG beams. The European Physical Journal Plus 134: 401.

28.

Ghayesh

Amabili

(2013a) Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance. Nonlinear Dynamics 73: 39–52.

29.

Ghayesh

Amabili

(2013b) Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support. Mechanism and Machine Theory 67: 1–16.

30.

Ghayesh

Yourdkhani

Balar

, et al. (2010) Vibrations and stability of axially traveling laminated beams. Applied Mathematics and Computation 217: 545–556.

31.

Ghayesh

Kazemirad

Darabi

, et al. (2012) Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system. Archive of Applied Mechanics 82: 317–331.

32.

Ghayesh

Farokhi

Alici

(2015) Subcritical parametric dynamics of microbeams. International Journal of Engineering Science 95: 36–48.

33.

Ghayesh

Farokhi

Gholipour

, et al. (2017) Nonlinear bending and forced vibrations of axially functionally graded tapered microbeams. International Journal of Engineering Science 120: 51–62.

34.

Kazemirad

Ghayesh

Amabili

(2013) Thermo-mechanical nonlinear dynamics of a buckled axially moving beam. Archive of Applied Mechanics 83: 25–42.

35.

Lakes

(2009) Viscoelastic Materials. New York: Cambridge University Press.

36.

Lin

Y-K

Cai

(1995) Probabilistic Structural Dynamics. Advanced Theory and Applications. New York: McGraw-Hill.

37.

Mott

Roland

(2009) Limits to Poisson’s ratio in isotropic materials. Physical Review B 80: 132104.

38.

Potapov

(2011) Nonlinear vibrations and stability of elastic and viscoelastic systems under random stationary loads. Mechanics of Solids 46: 444.

39.

Timoshenko

Woinowsky-Krieger

(1959) Theory of Plates and Shells. New York: McGraw-Hill.

40.

Wang

C-J

Lin

Q-F

Yao

Y-G

, et al. (2019) Dynamics of a stochastic system driven by cross-correlated sine-Wiener bounded noises. Nonlinear Dynamics 95: 1941–1956.

41.

Wiggins

(2013) Global Bifurcations and Chaos: Analytical Methods. New York: Springer Science & Business Media.

42.

Younesian

Norouzi

(2016) Chaos prediction in nonlinear viscoelastic plates subjected to subsonic flow and external load using extended Melnikov's method. Nonlinear Dynamics 84: 1163–1179.