Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation

Trabelssi, Mohamed; El-Borgi, Sami; Fernandes, Ralston; Ke, Liao-Liang

doi:10.1016/j.compositesb.2018.08.132

Cited by 33 publications

(35 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1Ã dx = lÃ . Further details about this simplification can be found in [36]. With further manipulations, it can be shown that the nonlocal stress resultants can be written entirely in terms of displacements…”

Section: Hamilton's Principlementioning

confidence: 99%

“…To address the shortcoming related to estimating higher-order derivatives, the problem was generally solved using high-order collocation methods like DQM [35,36] or the quadrature element method (QEM). This method is a high-order method used to solve FEM problems using a single or few high-order elements without the need to explicitly identify shape functions [37].…”

mentioning

confidence: 99%

“…The foundation models the interaction between the beam and the medium in which the beam is embedded such as a protein microtubule embedded in a matrix [1] or a carbon nanotube (CNT) embedded in a foundation [3]. To model [65] the forced vibration response, the authors propose two new numerical methods to estimate the frequency response curves of the nanobeam which are validated based on results obtained by the main authors using the differential quadrature method [36]. The closest study to this work is the paper published recently by Jin and Wang [16] who investigated the free vibration response of a linear and classical Timoshenko graded beam using WQEM.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

2020

Self Cite

View full text Add to dashboard Cite

The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen's nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first-or second-order interpolation functions. Hamilton's principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems. Keywords Functionally graded nanobeam • Nonlocal theory • Weak form quadrature element method (WQEM) • Free and forced vibration • Nonlinear von'Kármán strain • Frequency response curve 1 Introduction Nanobeams, nanoplates, nanoshells and other small-scale structural elements constitute the building blocks of micro-and nanoelectromechanical systems (MEMS and NEMS), actuators, sensors and atomic force micro

show abstract

Section: Hamilton's Principlementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…It has been reported that the single-mode Galerkin's method is suitable for analyzing nonlinear free vibration of nanobeams at low amplitudes. 76,77 The¯rst-mode approximation of Galerkin's technique is given as: where the linear¯rst mode shapes u ðxÞ, w ðxÞ and ðxÞ are calculated from linear vibration equations (Eq. (41)).…”

Section: Solution Proceduresmentioning

confidence: 99%

Surface Energy Effects on the Nonlinear Free Vibration of Functionally Graded Timoshenko Nanobeams Based on Modified Couple Stress Theory

Attia

Shanab

Mohamed

et al. 2019

Int. J. Str. Stab. Dyn.

View full text Add to dashboard Cite

An integrated nonlinear couple stress-surface energy continuum model is developed to study the nonlinear vibration characteristics of size-dependent functionally graded nanobeams for the first time. The nanobeam theory is formulated based on the Timoshenko kinematics, augmented by von Kármán’s geometric nonlinearity. The modified couple stress and Gurtin–Murdoch surface elasticity theories are incorporated to capture the long-range interaction and surface energy, respectively. Unlike existing Timoshenko nanobeam models, the effects of surface elasticity, residual surface stress, surface mass density and Poisson’s ratio, in addition to bending and axial deformations, are incorporated in the newly developed model. A power law function is used to model the material distribution through the thickness of the beam, considering the gradation of bulk and surface material parameters. A variational formulation of the nonlinear nonclassical governing equations and associated nonclassical boundary conditions is established by employing Hamilton’s principle. The generalized differential quadrature method is exploited in conjunction with either the Pseudo-arclength continuation or Runge–Kutta method to solve the problem with an exact implementation of the nonclassical boundary conditions. The formulation and solution procedure presented are verified by comparing the obtained results with available ones. Based on the parametric study, it is concluded that the nonclassical boundary conditions, material length scale parameter, residual surface stress, surface elasticity, bulk elasticity modulus, gradient index, nonlinear amplitude and thickness have important influences on the linear and nonlinear vibration responses of functionally graded Timoshenko nanobeams.

show abstract

“…Nonlinear analysis of free vibrations of beams is performed in [37]. Nonlocal free and forced vibrations of beams are investigated in [38]. Vibration system with nonlinear coupling is analysed in [39].…”

Section: Introductionmentioning

confidence: 99%

Vibroimpact mechanism in one separate case

Ragulskis¹,

Ragulskis²

2019

Math. models, eng.

View full text Add to dashboard Cite

It is known that in the vibroimpact system at the chosen values of parameters linear relationship between impact velocities and eigenfrequencies may exist. The purpose of this paper is to reveal the qualities of the systems of this type. Investigations are performed by analytical and numerical methods. It is determined that in the systems of this type nonlinear solutions with infinite series of harmonics exist. Multivalued stable and unstable regimes do not exist in the systems. The obtained analytical relationships enabled to reveal new qualities of the systems and to make useful conclusions. Keywords: characteristics of impact velocity and eigenfrequencies, free and decaying vibrations, phase trajectories of motions, harmonics of motions up to infinity.

show abstract

Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation

Cited by 33 publications

References 43 publications

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

Surface Energy Effects on the Nonlinear Free Vibration of Functionally Graded Timoshenko Nanobeams Based on Modified Couple Stress Theory

Vibroimpact mechanism in one separate case

Contact Info

Product

Resources

About