Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials

Sari, Ma’en S.; Butcher, Eric A.

doi:10.1016/j.ijengsci.2010.05.008

Cited by 38 publications

(12 citation statements)

References 18 publications

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“…In [48], the effects of non-ideal boundary conditions on the natural frequencies of beams and columns with variable cross-section subject to follower forces were investigated. Moreover, Ruta [49] dealt with linear vibrations of the Timoshenko beam with variable strength and geometric parameters.…”

Section: Introductionmentioning

confidence: 99%

Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials

Filippi

Pagani

Petrolo

et al. 2015

Composite Structures

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Please cite this article as: Filippi, M., Pagani, A., Petrolo, M., Colonna, G., Carrera, E., Static and free vibration analysis of laminated beams by refined theory based on Chebyshev Polynomials, Composite Structures (2015), doi: http://dx.Abstract This paper presents a new class of refined beam theories for static and dynamic analysis of composite structures. These beam models are obtained by implementing higher-order expansions of Chebyshev polynomials for the three components of the displacement field over the beam cross-section. The Carrera Unified Formulation (CUF) is adopted to obtain higher-order beam models. The governing equations are written in terms of fundamental nuclei, which are independent of the choice of the expansion order and the interpolating polynomials. Static and free vibration analysis of laminated beams and thin walled boxes has been carried out. Results obtained with the novel Chebyshev Expansion (CE) model have been compared with those available in the literature. For comparison, Taylor-like Expansion (TE) and Lagrange Expansion (LE) CUF models, commercial codes, analytical and experimental data are exploited. The performances of refined beam models in terms of computational cost and accuracy in comparison to the reference solutions have been assessed. The analysis performed has pointed out the high level of accuracy reached by the refined beam models with lower computational costs than 2D and 3D Finite Elements.

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Section: Introductionmentioning

confidence: 99%

Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials

Filippi

Pagani

Petrolo

et al. 2015

Composite Structures

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“…The accuracy of the method was verified by comparing the solutions with available results in the literature. Chebyshev polynomials are used extensively in the solution of engineering problems due to their fast convergence and accuracy as compared to other orthogonal functions as noted in Sari and Butcher (2010), Filippi et al (2015). Moreover they are easy to programme in symbolic form and the required accuracy can be attained by the number of polynomials (Sari and Butcher, 2010).…”

Section: Discussionmentioning

confidence: 99%

Buckling of nonuniform carbon nanotubes under concentrated and distributed axial loads

Robinson

Adali

2017

Mech. Sci.

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Abstract. Buckling of nonuniform carbon nanotubes are studied with the axial load taken as a combination of concentrated and axially distributed loads. Constitutive modelling of the nanotubes is implemented via nonlocal continuum mechanics. Problem solutions are obtained by employing a weak formulation of the problem and the Rayleigh-Ritz method which is implemented by using orthogonal Chebyshev polynomials. The accuracy of the method of solution is verified against available results. Solutions are obtained for the cases of uniformly distributed and triangularly distributed axial loads. Contour plots are given to assess the effect of nonuniform crosssections and the small-scale parameter on the buckling load for a combination of simply supported, clamped and free boundary conditions.

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“…After these have been applied, a single boundary condition at = 1 is applied. In the current work, the third boundary condition listed in (14) to (16) is imposed. Only a single right hand boundary condition needs to be applied since it follows that, for the value of 1 , the fourth and final boundary condition will automatically be satisfied.…”

Section: Problem Formulationmentioning

confidence: 99%

“…With regard to a system with a step discontinuity in the mass per unit length and bending stiffness, [10,11] demonstrated that an analytical solution can be found for the transverse vibrations by applying continuity conditions at the location of the step. Other times, the solution to a nonuniform, Euler-Bernoulli beam structure can be expressed in terms of special functions such as Bessel functions and Chebyshev polynomials [12][13][14]. In [12,13] it is shown that tapered beams admit a solution that can be expressed using Bessel functions.…”

Section: Introductionmentioning

confidence: 99%

“…In [12,13] it is shown that tapered beams admit a solution that can be expressed using Bessel functions. Chebyshev polynomials are used in [14] for the solution to a beam of exponentially varying width clamped at one end with the other end attached to a translational or torsional spring. Unfortunately, it is not always possible to determine an exact analytical solution or express the solution in terms of these special functions with exception of very specific types of nonuniformities some of which are discussed in [7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

Martin

Salehian

2018

Shock and Vibration

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The Lindstedt-Poincaré method is applied to a nonuniform Euler-Bernoulli beam model for the free transverse vibrations of the system. The nonuniformities in the system include spatially varying and piecewise continuous bending stiffness and mass per unit length. The expression for the natural frequencies is obtained up to second-order and the expression for the mode shapes is obtained up to first-order. The explicit dependence of the natural frequencies and mode shapes on reference values for the bending stiffness and the mass per unit length of the system is determined. Multiple methods for choosing these reference values are presented and are compared using numerical examples.

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Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials

Cited by 38 publications

References 18 publications

Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials

Static and free vibration analysis of laminated beams by refined theory based on Chebyshev polynomials

Buckling of nonuniform carbon nanotubes under concentrated and distributed axial loads

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

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