A Dynamical Basis for Computing the Modes of Euler–bernoulli and Timoshenko Beams

Claeyssen, Julio Cesar Ruiz; Soder, R.A.

doi:10.1006/jsvi.2002.5232

Cited by 12 publications

(5 citation statements)

References 5 publications

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“…In this work, this is accomplished by choosing in each segment a fundamental basis that is a translation of a fixed basis that is generated by an initial-value solution in the first segment. This later solution can be found in the work of Timoshenko et al [17] literature without the systematic treatment considered in [2,3,18]. We will consider the basis for the first segment that is constituted by the solution h(x) of the initial value problem…”

Section: The Fundamental Basismentioning

confidence: 99%

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Modal Formulation of Segmented Euler-Bernoulli Beams

Copetti

Claeyssen

Tsukazan

2007

Mathematical Problems in Engineering

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We consider the obtention of modes and frequencies of segmented Euler-Bernoulli beams with internal damping and external viscous damping at the discontinuities of the sections. This is done by following a Newtonian approach in terms of a fundamental response of stationary beams subject to both types of damping. The use of a basis generated by the fundamental solution of a differential equation of fourth-order allows to formulate the eigenvalue problem and to write the modes shapes in a compact manner. For this, we consider a block matrix that carries the boundary conditions and intermediate conditions at the beams and values of the fundamental matrix at the ends and intermediate points of the beam. For each segment, the elements of the basis have the same shape since they are chosen as a convenient translation of the elements of the basis for the first segment. Our method avoids the use of the first-order state formulation also to rely on the Euler basis of a differential equation of fourth-order and it allows to envision how conditions will influence a chosen basis.

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Section: The Fundamental Basismentioning

confidence: 99%

“…The methodology introduced by Tsukazan [1] in terms of a fundamental response [2,3] is applied here to a triple-span Euler-Bernoulli beam with internal damping of the type Kelvin-Voight and viscous external damping at the discontinuities of the sections.…”

Section: Introductionmentioning

confidence: 99%

Modal Formulation of Segmented Euler-Bernoulli Beams

Copetti

Claeyssen

Tsukazan

2007

Mathematical Problems in Engineering

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“…The computing of the roots of the characteristic equation can be, in principle, simplified by choosing an appropriate basis, that is, to make Θ the most sparse possible. This is accomplished by working with the dynamical basis [1].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Modal Analysis of a Beam with a Tip Rotor by Using a Fundamental Response

Claeyssen

Copetti

Balthazar

2003

MSF

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The shape modes of a clamped-free beam model with a tip rotor are determined by using a dynamical basis that is generated by a fundamental spatial free response. This is a non-classical distributed model for the displacements in the transverse directions of the beam which turns out to be coupled through boundary conditions due to rotation. Numerical calculations are performed by using the Ritz-Rayleigh method with several approximating basis.

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“…The shape of the waves is determined by a generating scalar function that appears in the closed form of the plane and modal matrix basis. The generating scalar function behaves quite well, regardless of varying parameters, that is, through limit procedures we can go from a dynamic situation to a static one [8]. Moreover, the modal-generating function is oscillatory above a critical frequency and evanescent below it.…”

Section: Introductionmentioning

confidence: 99%

Matrix basis for plane and modal waves in a Timoshenko beam

2016

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Plane waves and modal waves of the Timoshenko beam model are characterized in closed form by introducing robust matrix basis that behave according to the nature of frequency and wave or modal numbers. These new characterizations are given in terms of a finite number of coupling matrices and closed form generating scalar functions. Through Liouville’s technique, these latter are well behaved at critical or static situations. Eigenanalysis is formulated for exponential and modal waves. Modal waves are superposition of four plane waves, but there are plane waves that cannot be modal waves. Reflected and transmitted waves at an interface point are formulated in matrix terms, regardless of having a conservative or a dissipative situation. The matrix representation of modal waves is used in a crack problem for determining the reflected and transmitted matrices. Their euclidean norms are seen to be dominated by certain components at low and high frequencies. The matrix basis technique is also used with a non-local Timoshenko model and with the wave interaction with a boundary. The matrix basis allows to characterize reflected and transmitted waves in spectral and non-spectral form.

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A Dynamical Basis for Computing the Modes of Euler–bernoulli and Timoshenko Beams

Cited by 12 publications

References 5 publications

Modal Formulation of Segmented Euler-Bernoulli Beams

Modal Formulation of Segmented Euler-Bernoulli Beams

Modal Analysis of a Beam with a Tip Rotor by Using a Fundamental Response

Matrix basis for plane and modal waves in a Timoshenko beam

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