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

Sorokin, Vladislav; Thomsen, Jon Juel

doi:10.1016/j.jsv.2015.03.002

Cited by 11 publications

(8 citation statements)

References 30 publications

(75 reference statements)

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“…Similar equations have been considered, e.g. in the papers [18,28], for problems associated with the non-moving structures and materials. In the following section, its solution by the Method of Varying Amplitudes will be briefly described.…”

Section: Governing Equationsmentioning

confidence: 99%

“…In the following section, its solution by the Method of Varying Amplitudes will be briefly described. A comparison of the MVA with the classical methods is given in the papers [18,19,28,29]. It is noted, in particular, that for linear equations with periodic coefficients the method gives the same results as the approaches based on the classical Floquet theory.…”

Section: Governing Equationsmentioning

confidence: 99%

“…As it was noted in Section 3, the compound wave traveling in the moving periodic string comprises components with different dimensional frequencies, cf. (28), (31):…”

Section: Moving Periodic Stringmentioning

confidence: 99%

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Wave propagation in axially moving periodic strings

Sorokin

Thomsen

2017

Journal of Sound and Vibration

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Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

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Wave propagation in axially moving periodic strings

Sorokin

Thomsen

2017

Journal of Sound and Vibration

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“…Perturbation theory is also a powerful tool for obtaining approximate solutions for systems with complex geometries for which closed form analytical solutions are not readily available. With respect to a nonuniform EulerBernoulli beam, the recently developed method of varying amplitudes, a perturbation theory approach, was applied first in [19] to a system with continuous, and periodically varying, mass, and stiffness properties. While quite a powerful method, this approach primarily builds on the assumption of periodic variations in the system parameters over the length of the structure.…”

Section: Introductionmentioning

confidence: 99%

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

Martin

Salehian

2018

Shock and Vibration

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The Lindstedt-Poincaré method is applied to a nonuniform Euler-Bernoulli beam model for the free transverse vibrations of the system. The nonuniformities in the system include spatially varying and piecewise continuous bending stiffness and mass per unit length. The expression for the natural frequencies is obtained up to second-order and the expression for the mode shapes is obtained up to first-order. The explicit dependence of the natural frequencies and mode shapes on reference values for the bending stiffness and the mass per unit length of the system is determined. Multiple methods for choosing these reference values are presented and are compared using numerical examples.

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“…Periodic structures with internal cutoff frequencies are of interest for numerous applications: As an example, we mention elastically supported periodic strings and beams [1][2][3], composite materials [4], phononic crystals [5][6][7], and vibration absorbers in fluid carrying pipes [8]. It is well renowned that a string supported by a Winkler foundation exhibits a cutoff frequency [9, §1.5.2] !…”

Section: Introductionmentioning

confidence: 99%

Multi‐parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies

Kaplunov

Nobili

2016

Math Methods in App Sciences

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In this paper, we consider periodic waveguides in the shape of a inhomogeneous string or beam partially supported by an uniform elastic Winkler foundation. A multi-parametric analysis is developed to take into account the presence of internal cutoff frequencies and strong contrast of the problem parameters. This leads to asymptotic conditions supporting non-typical quasi-static uniform or, possibly, linear micro-scale displacement variations over the high-frequency domain. Macro-scale governing equations are derived within the framework of the Floquet-Bloch theory as well as using a high-frequency-type homogenization procedure adjusted to a string with variable parameters. It is found that, for the string problem, the associated macroscale equation is the same as that applying to a string resting on a Winkler foundation. Remarkably, for the beam problem, the macro-scale behavior is governed by the same equation as for a beam supported by a two-parameter Pasternak foundation. Copyright c 0000 John Wiley & Sons, Ltd.Keywords: Periodic waveguide; cutoff frequency; homogenization; contrast; high-frequency IntroductionPeriodic structures with internal cutoff frequencies are of interest for numerous applications: as an example, we mention elastically supported periodic strings and beams [1,2], composite materials [3], phononic crystals [4,5,6] and vibration absorbers in fluid carrying pipes [7]. It is well renowned that a string supported by a Winkler foundation exhibits a cutoffwhere β is Winkler foundation modulus and ρ is the string linear mass density. As it is shown in [9], a two-phase piecewise periodic string supported by a uniform Winkler foundation cannot be treated by the conventional "low-frequency" homogenization method [10,11]. For the latter, the sought for macro-scale homogenized equation is of the same form as the original equation governing the behavior of the periodic system. Besides, a quasi-static uniform variation of the displacement field is retrieved at the micro-scale (see also [12]). On the other hand, the periodically supported string problem can be efficiently treated through a high-frequency asymptotic homogenization procedure, as established in [9,13,14,15]. Within this approach and contrarily to the classical setup, the homogenized macro-scale equation takes, as a rule, a different form than the original equation. Moreover, sinusoidal variations at the micro-scale are found which correspond to the eigenforms of the unit cell. Dynamic homogenization has been the subject of a number of remarkable contributions among which we mention [16,17,18,19,20,21,22] and also [23,24], dealing with the important case of periodic waveguides with contrast properties.In this paper, we show that quasi-static uniform (or, possibly, linear) micro-scale variation over the high-frequency range is still possible for strongly inhomogeneous periodic structures with internal cutoffs. As an example, a two-phase periodic waveguide in the shape of a string or a beam supported by an elastic Winkler foundation is studi...

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

Cited by 11 publications

References 30 publications

Wave propagation in axially moving periodic strings

Wave propagation in axially moving periodic strings

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

Multi‐parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies

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