An analytical method for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach

Loghmani, Masih; Yazdi, Mohammad Reza Hairi

doi:10.1016/j.rinp.2018.08.046

Cited by 22 publications

(17 citation statements)

References 76 publications

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“…The dimensionless natural transverse frequencies Ω are calculated by solving the eigenvalue problem resulted from the discretization of Equations (25) and (26).…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The boundary conditions given in Equation 32are discretized using the CSCM as in Equation 28, except that each differentiation matrix D 1 is substituted with the differentiation matrix D 2 . Using the proposed method, Equations (25) and (26) are given as:…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Moreover, Zhang et al [13] applied Eringen's elasticity theory to present the scale parameter coefficients for the natural vibrations and buckling behavior of simply supported rectangular plates. Further studies about this topic were examined by other authors [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

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Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

2020

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The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

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“…The dimensionless natural transverse frequencies Ω are calculated by solving the eigenvalue problem resulted from the discretization of Equations (25) and (26).…”

Section: Mathematical Modelmentioning

confidence: 99%

Section: Solution Proceduresmentioning

confidence: 99%

See 1 more Smart Citation

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

2020

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“…The crack can make the structures more flexible and so are able to decrease the natural frequencies of the structures [12][13][14][15]. As a result, analysis of such structures may be applied for structural health monitoring.…”

Section: Introductionmentioning

confidence: 99%

Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Elkhalek¹,

Osman²,

Matbuly³

2019

ENGMATH

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This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler-Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multisegments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.

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“…Vibrations of nano-beams containing cracks and other defects are studied by Roostai and Haghpanahi [14], Loya [15], also by Lellep and Lenbaum [16], Loghmani and Yazdi [17], Hossain and Lellep [18,19]. In the paper [19] the effect of the temperature is taken into account.…”

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confidence: 99%

Natural vibrations of curved nano-beams and nano-arches

Lellep¹,

Shahid²

2021

Preprint

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The natural vibrations of curved nano-beams and nanoarches are studied. The nano-arches under consideration have piece wise constant thickness; these are weakened with stable cracks located at re-entrant corners of the steps. A method of determination of natural frequencies is developed making use of the method of weightless rotating spring. The aim of the paper is to assess the sensitivity of the eigenfrequencies on the geometrical and physical parameters of the nano-arch. The results of the calculations favourably compare with similar works of other researchers.

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An analytical method for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach

Cited by 22 publications

References 76 publications

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Natural vibrations of curved nano-beams and nano-arches

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