Asymptotic Analysis of the Free in-Plane Vibrations of Beams With Arbitrarily Varying Curvature and Cross-Section

Tarnopolskaya, Tanya; Hoog, Frank de; Fletcher, Neville H.; Thwaites, S.

doi:10.1006/jsvi.1996.0507

Cited by 39 publications

(27 citation statements)

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“…Figure 6 shows the mode transition of mode 2 from dominant 1IB to dominant 2IB as the initial curvature k 2 increases for the pinned-pinned case. This phenomenon, which is often referred to as veering [Chen and Ginsberg 1992] or frequency mode transition [Tarnopolskaya et al 1996;Tarnopolskaya et al 1999], results from the nature of the eigenvalues of a general self-adjoint Table 5. Vibration frequencies versus initial curvature k 2 for free-free, clampedclamped, and pinned-pinned boundary conditions.…”

Section: Coupled Free-vibration Frequencies For Initially Curved Beamsmentioning

confidence: 99%

“…for in-plane [Chidamparam and Leissa 1995;Tarnopolskaya et al 1996;Fung 2004] and out-of-plane vibration [Irie et al 1982;Howson and Jemah 1999] will be compared with the results from the present work. Results for coupled vibration are also presented as part of an investigation of low-frequency mode transition [Tarnopolskaya et al 1999], also referred to as veering phenomena [Chen and Ginsberg 1992].…”

Section: Introductionmentioning

confidence: 96%

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Vibration characteristics of curved beams

Chang

Hodges

2009

J. Mech. Mater. Struct.

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The paper presents a concise framework studying the coupled vibration of curved beams, whether the curvature is built-in or is caused by loading. The governing equations used are both geometrically exact and fully intrinsic, with a maximum degree of nonlinearity equal to two. For beams with initial curvature, the equations of motion are linearized about the reference state. For beams that are curved because of the loading, the equations of motion are linearized about the equilibrium state. A central difference spatial discretization scheme is applied, and the resulting linearized ordinary differential equations are cast as an eigenvalue problem. Numerical examples are presented, including: (1) validation of the analysis for both in-plane and out-of-plane vibration by comparison with published results, and (2) presentation of results for vibration of curved beams with free-free, clamped-clamped, and pinned-pinned boundary conditions. For coupled vibration, the numerical results also exhibit the low-frequency mode transition or veering phenomenon. Substantial differences are also shown between the natural frequencies of curved beams and straight beams, and between initially curved and bent beams with the same geometry.

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Section: Coupled Free-vibration Frequencies For Initially Curved Beamsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Vibration characteristics of curved beams

Chang

Hodges

2009

J. Mech. Mater. Struct.

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“…However, the revealed effect for higher eigenfrequencies is noted, apparently, for the first time, since it is for waves whose period is of the same order as the period of modulation, and the existing analytical methods [14][15][16][17][18][19] are not applicable for analyzing such eigenproperties. Note however, that for parameters (26) only the first eigenmode has a period which is much higher than the period of modulation (since period of the second eigenmode is l ).…”

Section: Case: Zero Mean Cross-section Modulationmentioning

confidence: 99%

“…In particular, the exact solutions have been found for some specific cases of tapering [9][10][11][12]. On the other hand, several approximate analytical methods for determining the eigenfrequencies of non-uniform beams and rods were developed in [4,13,14]. In particular, the WKB method [15] was proposed for the analysis of spatially slowly varying beams and rods [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

“…So, either the amplitude of modulation should be small [4], or the variation of structural parameters should be slow [16][17][18] or rapid [19], or a certain parameter (e.g. the thickness) of the corresponding homogeneous structure should be small [14]. In the present paper none of these restrictions are imposed, meaning that in particular the "mid-frequency" eigenmodes, with a period of the same order as the period of modulation, are considered.…”

Section: Introductionmentioning

confidence: 99%

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Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties

Sorokin

Thomsen

2015

Journal of Sound and Vibration

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A beam with periodically continuously varying spatial properties is analyzed. This structure is a generic model for various systems widely used in industry, e.g. risers, rotor blades, and similar. The aim is to reveal effects of periodic spatial modulation both on the beam eigenfrequencies and eigenmodes. Special attention is given to "mid-frequency" eigenmodes having period of the same order as the period of modulation, which cannot be captured by the conventional analytical methods. In particular, the paper addresses prediction of bandgaps and their influence on the distribution of eigenfrequencies. For analyzing the problem considered, the method of varying amplitudes is employed. A connection of this method with the classical Hill's infinite determinant method and the method of space-harmonics is noted. A dispersion relation of the considered non-uniform periodic structure is obtained, and values of the modulation amplitudes at which frequency bandgaps arise are determined. It is shown that eigenfrequencies of the beam can lie within the bandgaps, and that such eigenfrequencies can be considerably affected by modulation. It is revealed that there is an abrupt shift in the effect of modulation on eigenfrequencies, and that modulations of the beam mass per unit length and of the beam stiffness affect them oppositely. It is shown that eigenmodes having a period close to the period of modulation comprise a long-wave component; this illustrates the capacity of non-uniform structures to sustain long-wave oscillations on comparatively high frequencies.

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Phononic Band Gaps in Periodic Cellular Materials

Liebold‐Ribeiro

Körner

2013

Adv Eng Mater

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The dynamic mechanical properties of finite two-dimensional periodic cellular materials are investigated by finite element eigenmode analysis for different architectures of the unit cell. Frequency band gaps are examined in quadratic and hexagonal lattice topologies with regular, inverted, and chiral architecture. Pronounced band gaps develop for chiral lattices. The formation of band gaps can be traced back to the resonance behavior of the elementary building blocks of the cellular structure for different boundary conditions (mode transition). Based on the findings of this work periodic lattice materials with specific band gaps can be designed. Fig. 11. Eigenmode analysis of a single curved strut (L ¼ 5.0 mm, t ¼ 0.2 mm) performed using hinged-hinged and free-free boundary conditions. The plot (left) shows the frequency as a function of the strut amplitude A, whereas the shapes of the different vibration modes of bands I-IV are depicted on the right.

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Asymptotic Analysis of the Free in-Plane Vibrations of Beams With Arbitrarily Varying Curvature and Cross-Section

Cited by 39 publications

References 0 publications

Vibration characteristics of curved beams

Vibration characteristics of curved beams

Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties

Phononic Band Gaps in Periodic Cellular Materials

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