Towards Deep and Simple Understanding of the Transcendental Eigenproblem of Structural Vibrations

Williams, F.W.; Yuan, Songmei; Ye, K.; Kennedy, David; Djoudi, M. S.

doi:10.1006/jsvi.2002.5016

Cited by 25 publications

(12 citation statements)

References 9 publications

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“…This work consolidates the possibility of using an iterative process for extracting eigenvalues, without the compulsory use of the Wittrick-Williams algorithm [6,10,11,14]. Ref.…”

Section: Discussionmentioning

confidence: 99%

Power secant method applied to natural frequency extraction of Timoshenko beam structures

Dias

2010

Lat. Am. j. solids struct.

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This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named "power deflation", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem.Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.

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“…This work consolidates the possibility of using an iterative process for extracting eigenvalues, without the compulsory use of the Wittrick-Williams algorithm [6,10,11,14]. Ref.…”

Section: Discussionmentioning

confidence: 99%

Power secant method applied to natural frequency extraction of Timoshenko beam structures

Dias

2010

Lat. Am. j. solids struct.

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“…Various algorithms have been proposed for dynamic analysis of frame structures, including those based on continuous mass method of Ovunc [6], the integral method of Antes and co-workers [7], and the solution of transcendental eigenvalue problem of Williams and co-workers [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Exact solution of vibration problems of frame structures

2008

Numer Methods Biomed Eng

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SUMMARYAlthough exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to get exact solutions for free vibration and harmonic analyses for nontrivial cases. This paper extends an earlier study on exact solutions of axial vibration problems of elastic bars to dynamic analyses of elastic frame structures. New shape functions for the transverse displacement field are constructed by using the homogeneous governing equations and then a novel beam element is formulated. Combining the new (bending) beam element and the one developed earlier for elastic bars yields a new element for general frame structures. The new frame element can be used to get exact solutions for both natural frequency and undamped harmonic analyses of frame structures. Illustrative examples are presented to demonstrate the effectiveness of the new element and the algorithm.

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“…Various methods have been proposed for accurately determining the dynamic behaviours of frame structures, including continuous mass method of Ovunc [6], the integral method of Antes and co-workers [7] and algorithms based on the solution of transcendental eigenvalue problem of Williams and co-workers [8][9][10][11]. Because special numerical algorithms are involved in these methods, it may not be simple to integrate them into conventional finite element analysis programs.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions of axial vibration problems of elastic bars

2007

Numerical Meth Engineering

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SUMMARYWhile exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to obtain exact solutions for free vibration and harmonic analyses for non-trivial cases. This paper presents a study on new finite element formulation and algorithms for exact solutions of undamped axial vibration problems of elastic bars. Appropriate shape functions are constructed by using the homogeneous governing equations, and based on the new shape functions, a novel element is formulated. An iterative procedure is proposed for determining both the exact natural frequency values and the corresponding vibration mode shapes. Exact solutions can also be obtained for undamped harmonic response analyses by using the new element, as its stiffness and mass matrices are exact for a specified frequency. Illustrative examples are presented to demonstrate the effectiveness of the proposed element and algorithm.

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Towards Deep and Simple Understanding of the Transcendental Eigenproblem of Structural Vibrations

Cited by 25 publications

References 9 publications

Power secant method applied to natural frequency extraction of Timoshenko beam structures

Power secant method applied to natural frequency extraction of Timoshenko beam structures

Exact solution of vibration problems of frame structures

Exact solutions of axial vibration problems of elastic bars

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