A discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions

Rahmouni, Abdellatif; Beidouri, Zitouni; Benamar, Rhali

doi:10.1016/j.jsv.2013.04.011

Cited by 8 publications

(41 citation statements)

References 20 publications

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“…Using the Pythagorean Theorem, we express the nonlinear energy versus the characteristics of the beam [1], see Figure3:…”

Section: Results In the Linear Case For A Beam With Two Concentrated mentioning

confidence: 99%

“…The discrete model developed and validated in the case of a continuous beam presented in [1,2,3], was applied to beams with two or three concentrated masses.…”

Section: Discussionmentioning

confidence: 99%

“…The studied model of a beam with three concentrate dmasses M s , M t and M q is shown in Figure 1: respect to the neutral fiber (in the case of a rectangular section, r is equal to the thickness of the beam divided by √12). This study shows that the developed model may be used to study successfully non-linear vibrations of beams carrying many concentrated masses simply by changing the mass matrix but without any change in the linear and nonlinear stiffness tensors, corresponding to a uniform beam defined in [1].…”

Section: Presentation and Nomenclaturementioning

confidence: 98%

“…Application of the discrete model developed in the work [1,2,3] is made here to a Bernoulli beam carrying two or three concentrated masses at various locations and subject to geometrical nonlinear vibration due to large transverse displacements. This model focuses on the known physical phenomenon of the dynamic behavior: the stretching of the beam induces nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations

Rahmouni

Benamar

2016

MATEC Web Conf.

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Abstract:In a recent work, a discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions has been developed and validated via comparison with known results corresponding to nonlinear vibration of clamped beams carrying a concentrated mass. It is extended here to continuous beams carrying two or three concentrated masses at various locations and subjected to large vibration amplitudes. The discrete model used is an N-dof (N-Degrees of Freedom) system made of N masses placed at the ends of solid bars connected by springs, presenting the beam flexural rigidity. The large placed at the ends of solid bars connected by springs, presenting the beam flexural rigidity. The large transverse displacements of the bar ends induce a variation in their lengths giving rise to axial forces modelled by longitudinal springs causing nonlinearity. The calculations made allowed application of the semi -analytical model developed previously for nonlinear structural vibration involving three tensors, namely the mass tensor m ij , the linear rigidity tensor k ij and the nonlinearity tensor b ijkl presenting the effect of the change in the bar lengths. The addition of three concentrated masses studied here induces a change in the mass matrix. By application of Hamilton's principle and spectral analysis in the modal basis, the nonlinear vibration problem is reduced to a nonlinear algebraic system, using an explicit method, developed previously for non-linear structural vibration. This study shows that concentrated masses may be used for practical purposes to shift the resonant frequency; if the three masses locations are appropriately chosen.

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“…Using the Pythagorean Theorem, we express the nonlinear energy versus the characteristics of the beam [1], see Figure3:…”

Section: Results In the Linear Case For A Beam With Two Concentrated mentioning

confidence: 99%

“…The discrete model developed and validated in the case of a continuous beam presented in [1,2,3], was applied to beams with two or three concentrated masses.…”

Section: Discussionmentioning

confidence: 99%

Section: Presentation and Nomenclaturementioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations

Rahmouni

Benamar

2016

MATEC Web Conf.

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“…The aim of this work is the development of a flexible general discrete model for linear and nonlinear vibrations of beams resting on elastic foundations, via the adaptation and the extension of the approach presented in [9]. Accordingly, the process of discretization carried out is intended to allow efficient variation of beam and soil geometrical and mechanical characteristics, in order to make it easy to perform multiple parametric studies.…”

Section: Introductionmentioning

confidence: 99%

A Discrete Model for Nonlinear Vibration of Beams Resting on Various Types of Elastic Foundations

Khnaijar

Benamar

2017

Advances in Acoustics and Vibration

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This paper presents a discrete physical model to approach the problem of nonlinear vibrations of beams resting on elastic foundations. The model consists of a beam made of several small bars, evenly spaced. The bending stiffness is modeled by spiral springs, and the Winkler soil stiffness is modeled using linear vertical springs. Concentrated masses, presenting the inertia of the beam, are located at the bar ends. Finally, the nonlinear effect is presented by the axial forces in the bars, assumed to behave as longitudinal springs, due to the change in their length induced by the Pythagorean Theorem. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing any modification in the mass matrix, the stiffness matrix, and the nonlinearity tensor to be made separately. Therefore, once the model is established, various practical applications may be performed without the need of going through all the formulation again. The study of the nonlinear behavior makes the solution of the movement equation rise in complexity. By considering this discrete model and using the linearization method, one can achieve an idealized approach to this nonlinear problem and obtain quite easily approximate solutions.

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A new model for beam crack detection and localization using a discrete model

Khnaijar

Benamar

2017

Engineering Structures

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A discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions

Cited by 8 publications

References 20 publications

Nonlinear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations

Nonlinear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations

A Discrete Model for Nonlinear Vibration of Beams Resting on Various Types of Elastic Foundations

A new model for beam crack detection and localization using a discrete model

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