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    An Asymptotic Method to Analyze the Vibrations of an Elastic Layer

    Source: Journal of Applied Mechanics:;1969:;volume( 036 ):;issue: 001::page 65
    Author:
    J. D. Achenbach
    DOI: 10.1115/1.3564587
    Publisher: The American Society of Mechanical Engineers (ASME)
    Abstract: The displacement components for both free and forced vibrations are sought as power series of the dimensionless wave number ε, where ε = 2π × layer thickness/wavelength. For the free vibration problem the object is to determine the frequencies, which are also sought as power series of the dimensionless wave number. The displacement and frequency expansions are substituted in the displacement equations of motion and in the boundary conditions. By collecting terms of the same order εn , a system of second-order, inhomogeneous, ordinary differential equations of the Helmholtz type is obtained, with the thickness variable as independent variable, and with associated boundary conditions. For free vibrations, subsequent integration yields the coefficients of εn for the displacements and the frequencies for all modes, and in the whole range of frequencies, but in a range of dimensionless wave numbers 0 < ε < ε* < 1, where ε* increases as more terms are retained in the expansions. For forced vibrations, the amplitudes are determined in an analogous manner if the external surface tractions are of sinusoidal dependence on the in-plane coordinates and on time. The response to surface tractions of more general spatial dependence is obtained by Fourier superposition.
    keyword(s): Vibration , Waves , Displacement , Frequency , Thickness , Free vibrations , Boundary-value problems , Equations of motion , Differential equations AND Wavelength ,
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      An Asymptotic Method to Analyze the Vibrations of an Elastic Layer

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    http://yetl.yabesh.ir/yetl1/handle/yetl/132901
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    contributor authorJ. D. Achenbach
    date accessioned2017-05-09T00:18:22Z
    date available2017-05-09T00:18:22Z
    date copyrightMarch, 1969
    date issued1969
    identifier issn0021-8936
    identifier otherJAMCAV-25885#65_1.pdf
    identifier urihttp://yetl.yabesh.ir/yetl/handle/yetl/132901
    description abstractThe displacement components for both free and forced vibrations are sought as power series of the dimensionless wave number ε, where ε = 2π × layer thickness/wavelength. For the free vibration problem the object is to determine the frequencies, which are also sought as power series of the dimensionless wave number. The displacement and frequency expansions are substituted in the displacement equations of motion and in the boundary conditions. By collecting terms of the same order εn , a system of second-order, inhomogeneous, ordinary differential equations of the Helmholtz type is obtained, with the thickness variable as independent variable, and with associated boundary conditions. For free vibrations, subsequent integration yields the coefficients of εn for the displacements and the frequencies for all modes, and in the whole range of frequencies, but in a range of dimensionless wave numbers 0 < ε < ε* < 1, where ε* increases as more terms are retained in the expansions. For forced vibrations, the amplitudes are determined in an analogous manner if the external surface tractions are of sinusoidal dependence on the in-plane coordinates and on time. The response to surface tractions of more general spatial dependence is obtained by Fourier superposition.
    publisherThe American Society of Mechanical Engineers (ASME)
    titleAn Asymptotic Method to Analyze the Vibrations of an Elastic Layer
    typeJournal Paper
    journal volume36
    journal issue1
    journal titleJournal of Applied Mechanics
    identifier doi10.1115/1.3564587
    journal fristpage65
    journal lastpage72
    identifier eissn1528-9036
    keywordsVibration
    keywordsWaves
    keywordsDisplacement
    keywordsFrequency
    keywordsThickness
    keywordsFree vibrations
    keywordsBoundary-value problems
    keywordsEquations of motion
    keywordsDifferential equations AND Wavelength
    treeJournal of Applied Mechanics:;1969:;volume( 036 ):;issue: 001
    contenttypeFulltext
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