Analysing regular nonlinear vibrations of nano/micro plates based on the nonlocal theory and combination of reduced order modelling and multiple scale method

Awrejcewicz, Jan; Sypniewska–Kamińska, Grażyna; Mazur, Olga

doi:10.1016/j.ymssp.2021.108132

Cited by 22 publications

(9 citation statements)

References 46 publications

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“…In this paper, the homoclinic phenomena and chaotic motions of the buckled double layered nanoplates for each case of buckling modes are studied using the extended Melnikov method. Besides, the existence of transverse homoclinic orbits and homoclinic bifurcation are investigated because of significant importance since they can imply the occurrence of chaotic motion by the symbolic dynamics and Smale-Birkhoff theorem [23][24][25][26][27].…”

Section: Hamilton Formulation By Coupled Single-modes Solution For Bu...mentioning

confidence: 99%

“…The developed approach based on applying the Bubnov-Galerkin method has been used to reduce partial differential equations (PDEs) to an ordinary differential equation (ODE). It is to be pointed out that there are numerous important works in the field of chaotic dynamics of MEMS/NEMS and micro-/nanostructures [23][24][25][26], chaos prediction using Melnikov method [27,28], and utilizing the Winkler-Pasternak elastic foundation for structures [29] in which briefly are reviewed in continue. In ref.…”

Section: Introductionmentioning

confidence: 99%

“…In ref. [26], the analysis for geometrically nonlinear vibrations of rectangular micro/nanoplates with an account of small‐scale effects based on the nonlocal theory and combination of reduced order modeling (ROM) and multiple scale method (MSM) has been performed. Hamilton's principle was applied to derive the governing PDEs based on the Kirchhoff–Love hypotheses and geometric nonlinearity introduced through the von Karman theory.…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

Mamandi

2022

Z Angew Math Mech

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In this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

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Section: Hamilton Formulation By Coupled Single-modes Solution For Bu...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

Mamandi

2022

Z Angew Math Mech

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“…Currently, the nonlocal continuum theory has been applied in micro-mechanics more and more widely [6][7][8][9][10][11][12][13][14][15]. For example, recently, Awrejcewicz et al [15] investigated the vibration of rectangular micro/nanoplates using the nonlocal theory.…”

Section: Introductionmentioning

confidence: 99%

Lateral Free Vibration of Micro-Rods Using a Nonlocal Continuum Approach

Xie

Zhang²,

Chen

et al. 2022

J. Adv. App. Comput. Math.

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The lateral free vibration of micro-rods initially subjected to axial loads based on a nonlocal continuum theory is considered. The effects of nonlocal long-range interaction fields on the natural frequencies and vibration modes are examined. A simply supported micro-rod is taken as an example; the linear vibration responses are observed by two different methods, including the separation of variables and multiple scales analysis. The relations between the vibration mode and dimensionless coordinate and the relations between natural frequencies and nonlocal parameters are analyzed and discussed in detail. The numerical comparison shows that the theoretical results by two different approaches have a good agreement, which validates the present micro-rod model that can be used as a component of the micro-electromechanical system.

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“…Nonlinear modes of vibration of single-layered graphene sheets are studied by Ribeiro and Chuaqui [26] by Bubnov-Galerkin approach as well as harmonic balance method. Awrejcewicz et al [27] analyse the forced nonlinear vibrations of nano/micro plates based on multiple scale method Furthermore, when the plate is subjected to periodic in-plane load the parametric resonance may occur [28,29] and this matter should be investigated thoroughly.…”

Section: Introductionmentioning

confidence: 99%

Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory

2021

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Parametric vibrations of the single-layered graphene sheet (SLGS) are studied in the presented work. The equations of motion govern geometrically nonlinear oscillations. The appearance of small effects is analysed due to the application of the nonlocal elasticity theory. The approach is developed for rectangular simply supported small-scale plate and it employs the Bubnov–Galerkin method with a double mode model, which reduces the problem to investigation of the system of the second-order ordinary differential equations (ODEs). The dynamic behaviour of the micro/nanoplate with varying excitation parameter is analysed to determine the chaotic regimes. As well the influence of small-scale effects to change the nature of vibrations is studied. The bifurcation diagrams, phase plots, Poincaré sections and the largest Lyapunov exponent are constructed and analysed. It is established that the use of nonlocal equations in the dynamic analysis of graphene sheets leads to a significant alteration in the character of oscillations, including the appearance of chaotic attractors.

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Analysing regular nonlinear vibrations of nano/micro plates based on the nonlocal theory and combination of reduced order modelling and multiple scale method

Cited by 22 publications

References 46 publications

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

Lateral Free Vibration of Micro-Rods Using a Nonlocal Continuum Approach

Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory

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