Asymptotic study of linear vibrations of a stretched beam with concentrated masses and discrete elastic supports
The asymptotic analysis following  and based on the homogenization technique [46,49,166] in frameworks of linear dynamics for arbitrary ranges of frequencies is applied in this section to the infinite 1D system which consists of elastically supported discrete masses, linked by beams. Three scale regions of eigenfrequencies are found. The first one corresponds to the continuum approach, when the system studied can be described as an effectively continuum homogeneous beam and corrections are of a higher order of magnitude. The second region corresponds to the antiphase mode where neighboring masses vibrate with slowly varying amplitudes. The highest range of frequencies reflects the short beam vibrations between neighboring masses which are immobile in the first term approach. The completeness of the spectrum analysis is shown. Dispersion relations and the peculiarities of the corresponding eigenmodes are discussed. The system studied admits generalizations and may itself serve as an adequate model for various technical applications: civil engineering, ship building, etc.
KeywordsAsymptotic Analysis Main Term Asymptotic Series Main Frequency Linear Vibration
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