A spectral-Tchebychev technique for solving linear and nonlinear beam equations

Yagci, Baris; Filiz, Sinan; Romero, Louis L.; Özdoğanlar, O. Burak

doi:10.1016/j.jsv.2008.09.040

Cited by 93 publications

(38 citation statements)

References 50 publications

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“…Use of the so-called Gauss-Lobatto points as the sampling points brings significant simplification to determination of Tchebychevexpansion polynomial coefficients and associated transformations [25]. Sampling of a function/(x, y, z) in a 3D domain produces a third-rank tensor form as…”

Section: G-lobatto Sampling and Transformationmentioning

confidence: 99%

“…However, displacement boundary conditions have to be imposed to the solution. The linear displacement boundary conditions for the 3D problem can be expressed as ßq = qb (25) where ß is the (linear) boundary condition operator and qi, includes the nonhomogeneous portion of the displacement boundary condition, which, in general, could be a function of time. It should be noted here that, due to the nature of the 3D problem, the boundary matrix ß includes only constant values.…”

Section: (24)mentioning

confidence: 99%

“…Their recursive nature allows robust formulation of different boundary-value problems [20]. Using the Tchebychev polynomials as basis functions, Yagci et al [25] have developed the spectral-Tchebychev approach for solving one-dimensional vibrations of linear and nonlinear beams for different boundary conditions. It was shown that the convergence was indeed exponential, and the solution accuracy was comparable to that from a 3D finite-elements method.…”

Section: Introductionmentioning

confidence: 99%

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A Spectral-Tchebychev Solution for Three-Dimensional Vibrations of Parallelepipeds Under Mixed Boundary Conditions

Filiz

Bediz

Romero

et al. 2012

Journal of Applied Mechanics

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Vibration behavior of structures with parallelepiped shape-including beams, plates, and solids-are critical for a broad range of practical applications. In this paper we describe a new approach, referred to here as the three-dimensional spectral-Tchebychev (3D-ST) technique, for solution of three-dimensional vibrations of parallelepipeds with different boundary conditions. An integral form of the boundary-value problem is derived using the extended Hamilton's principle. The unknown displacements are then expressed using a triple expansion of scaled Tchebychev polynomials, and analytical integration and differentiation operators are replaced by matrix operators. The boundary conditions are incorporated into the solution through basis recombination, allowing the use of the same set of Tchebychev functions as the basis functions for problems with different boundary conditions. As a result, the discretized equations of motion are obtained in terms of mass and stiffness matrices. To analyze the numerical convergence and precision of the 3D-ST solution, a number of case studies on beams, plates, and solids with different boundary conditions have been conducted. Overall, the calculated natural frequencies were shown to converge exponentially with the number of polynomials used in the Tchebychev expansion. Furthermore, the natural frequencies and mode shapes were in excellent agreement with those from a finite-element solution. It is concluded that the 3D-ST technique can be used for accurate and numerically efficient solution of three-dimensional parallelepiped vibrations under mixed boundary conditions.

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Section: G-lobatto Sampling and Transformationmentioning

confidence: 99%

Section: (24)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Spectral-Tchebychev Solution for Three-Dimensional Vibrations of Parallelepipeds Under Mixed Boundary Conditions

Filiz

Bediz

Romero

et al. 2012

Journal of Applied Mechanics

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“…Yagci et al [15] used a spectral Chebyshev technique for solving linear and nonlinear beam equations. Sari and Butcher [16] used the PS method for the free vibration analyses of non-rotating and rotating Timoshenko beams with damaged boundaries.…”

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of tapered columns under self-weight using pseudospectral method

Sudheer¹,

Harikrishna²,

Rao³

2016

J. vibroeng.

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Abstract. This paper deals with the vibration of tapered column which is affected by gravity using a pseudospectral formulation. The formulation is simple and easy-to-implement and is capable of dealing with different end conditions. Numerical examples of the effects of taper, cross section shapes and gravity on the vibration of columns are illustrated. The effectiveness of the pseudospectral method for vibration analysis of tapered heavy columns is validated by comparing the results with numerical techniques such as the numerical initial value method and differential quadrature method.

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“…Therefore, non-uniform beams have been the subject of research of many authors. The typically non-uniform beams have been circumscribed according to the Bernoulli--Euler [1][2][3] or Timoshenko [3][4][5][6][7][8] theory. The Timoshenko theory [9,10] is adequate for vibrations of higher modes or for short beams.…”

Section: Introductionmentioning

confidence: 99%