On the solution of S(ω)x=0 by a Newtonian procedure

Simpson, A.

doi:10.1016/0022-460x(84)90476-0

Cited by 36 publications

(15 citation statements)

References 3 publications

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“…Alternatives to bisection have been developed to accelerate convergence when using the Wittrick-Williams algorithm (Simpson, 1984;Williams and Kennedy, 1988;Kennedy and Williams, 1991) but further work is needed in this area.…”

Section: Introductionmentioning

confidence: 99%

An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies

Kennedy

Williams

2004

International Journal of Solids and Structures

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Section: Introductionmentioning

confidence: 99%

An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies

Kennedy

Williams

2004

International Journal of Solids and Structures

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“…The dynamic-sti ness matrix is a frequencydependent complex matrix, hence Equation (4) is more di cult to solve. There are several methods to solve this non-linear eigenvalue problem, such as the inverse iteration method [10], Newton's method [11], and Jacobi-Davidson iterations method [12].…”

Section: Substituting Equation (3) Into Equation (1) Results Inmentioning

confidence: 99%

Free-vibration analysis of a three-dimensional soil-structure system

Wegner

Zhang

2000

Earthquake Engng. Struct. Dyn.

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A procedure which involves a non-linear eigenvalue problem and is based on the substructure method is proposed for the free-vibration analysis of a soil-structure system. In this procedure, the structure is modelled by the standard ÿnite element method, while the unbounded soil is modelled by the scaled boundary ÿnite element method. The fundamental frequency, and the corresponding radiation damping ratio as well as the modal shape are obtained by using inverse iteration. The free vibration of a dam-foundation system, a hemispherical cavity and a hemispherical deposit are analysed in detail. The numerical results are compared with available results and are also veriÿed by the Fourier transform of the impulsive response calculated in the time domain by the three-dimensional soil-structure-wave interaction analysis procedure proposed in our previous paper. The fundamental frequency obtained by the present procedure is very close to that obtained by Touhei and Ohmachi, but the damping ratio and the imaginary part of modal shape are signiÿcantly di erent due to the di erent deÿnition of damping ratio. This study shows that although the classical mode-superposition method is not applicable to a soil-structure system due to the frequency dependence of the radiation damping, it is still of interest in earthquake engineering to evaluate the fundamental frequency and the corresponding radiation damping ratio of the soil-structure system.

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“…l ; ! u ) after the solution of Equation (10) can be used as a means to judge whether ! a is su ciently close to !…”

Section: Newton's Methodsmentioning

confidence: 99%

“…Equation (2), has enabled many logical procedures to be developed which are guaranteed to converge on all required natural frequencies with certainty and to any desired accuracy [3,[9][10][11][12]. If the ÿrst required natural frequency is the J r1 th one, the simplest of these methods is bisection [3] in which the W-W algorithm is ÿrst used to check whether an initial, user speciÿed, trial value !…”

Section: Introductionmentioning

confidence: 99%

Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

Yuan

Williams

et al. 2003

Numerical Meth Engineering

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When exact dynamic sti ness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally e cient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is ÿrst reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identiÿed by the Wittrick-Williams algorithm. Then the generalized linear eigenvalue problem is e ectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick-Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large-scale structures.

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On the solution of S(ω)x=0 by a Newtonian procedure

Cited by 36 publications

References 3 publications

An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies

An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies

Free-vibration analysis of a three-dimensional soil-structure system

Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

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