Free vibrations of tapered beams with flexible ends

Rosa, M.A. De; Auciello, N.M.

doi:10.1016/0045-7949(95)00397-5

Cited by 60 publications

(33 citation statements)

References 9 publications

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“…In this paper we ask the question whether the moving mass has a significant effect on the tip deflection of the beam. Vibrations of tapered beams have for instance been studied by Goel [19], Mabie and Rogers [20] and De Rosa and Auciello [21], who all considered linearly varying cross-sectional dimensions, and Zhou [22] who considered more general polynomial tapering. We choose the tapering to be parabolic, in which case we can obtain the mode shapes of the free beam, without a mass, exactly by using a transformation of the (spatial) independent variable similar to that used in [22].…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of a tapered cantilever beam under a travelling mass

Zhao

Heijden

2015

Meccanica

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We study the vibration of a tapered cantilever (Euler-Bernoulli) beam carrying a moving mass. The tapering is assumed to be parabolic. Using the Galerkin method we find approximate solutions in an energy formulation that takes into account dynamic mass-beam coupling due to inertial, Coriolis and centrifugal effects. The approximate solutions are expanded in terms of the mode shapes of the free tapered beam, which can be obtained analytically. We then study the effect the tapering as well as the magnitude and velocity of the mass have on the tip deflections of the beam. We consider two different initial conditions, one where the mass starts moving from a statically deformed beam and one where the beam is initially triggered to vibrate. We find that tip deflections are more irregular for strongly tapered beams. Our results are of interest for barreled launch systems where tip deflections may adversely affect projectile motion.

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Section: Introductionmentioning

confidence: 99%

Dynamic analysis of a tapered cantilever beam under a travelling mass

Zhao

Heijden

2015

Meccanica

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“…Although any type of non-uniform beam can be approximated by a stepped beam with a suitable number of uniform sections, for comparison purposes a linearly tapered beam has been chosen from the published literature [10,13]. This has made a direct comparison of results obtained from the present method with those available in the literature.…”

Section: Example 4: Approximate Solution For Of a Tapered Beammentioning

confidence: 96%

Free vibration analysis of multiple-stepped beams by using Adomian decomposition method

Mao

2011

Mathematical and Computer Modelling

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a b s t r a c tThe Adomian decomposition method (ADM) is employed in this paper to investigate the free vibrations of the Euler-Bernoulli beams with multiple cross-section steps. The proposed ADM method can be used to analyze the vibration of beams consisting of an arbitrary number of steps through a recursive way. The solution can be obtained by solving a set of algebraic equations with only three unknown parameters. Furthermore, the method can be extended to obtain an approximate solution to vibration problems of any type of non-uniform beams. Several numerical examples are presented and compared to those given in the paper. It is shown that the ADM offers an accurate and effective method of free vibration analysis of multiple-stepped beams with arbitrary boundary conditions.

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“…Those authors showed that trigonometric functions work slightly better than the static deflections and highlight the accuracy of Rayleigh's quotient to the true natural frequencies. References [3][4][5][6] present exact solutions for the frequency equation of a Bernoulli-Euler beam restricting the stiffness coefficients, in order to reproduce some particular cases, and accounting for the rotation inertia of attached discs and their eccentricity. Similar problems are treated in Refs.…”

Section: Introductionmentioning

confidence: 99%

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

Silva

Maia

2011

Acta Mech Sin

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Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending, the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length. Based on Rayleigh's quotient, an iterative strategy is developed to find the approximated torsional stiffness coefficients, which allows the reconciliation between the theoretical model results and the experimental ones, obtained through impact tests. The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh's quotient but also on the mode shapes, considering the shape functions defined in branches. Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.

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Free vibrations of tapered beams with flexible ends

Cited by 60 publications

References 9 publications

Dynamic analysis of a tapered cantilever beam under a travelling mass

Dynamic analysis of a tapered cantilever beam under a travelling mass

Free vibration analysis of multiple-stepped beams by using Adomian decomposition method

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

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