Asymptotic analysis of geometrically nonlinear beam vibrations: Kirchhoff and Bolotin equations

The paper analyzes various approximate models of geometrically nonlinear vibrations of a beam. In practice, simplified equations are often based on the quasi‐static Kirchhoff hypothesis—neglecting axial inertia. This hypothesis is justified with the prescribed end‐displacements of the beam in the ax...

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Bibliographic Details
Published in:Zeitschrift für angewandte Mathematik und Mechanik 2024-10, Vol.104 (10), p.n/a
Main Authors: Andrianov, Igor V., Koblik, Steve G.
Format: Article
Language:English
Subjects:
Asymptotic methods
Asymptotic properties
Hypotheses
Inertia
Linear equations
Stiffness
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Summary:The paper analyzes various approximate models of geometrically nonlinear vibrations of a beam. In practice, simplified equations are often based on the quasi‐static Kirchhoff hypothesis—neglecting axial inertia. This hypothesis is justified with the prescribed end‐displacements of the beam in the axial direction. Under dead loading, quasi‐static Kirchhoff hypothesis results in a linear equation. The corresponding approximate equations obtained in this paper are based on the asymptotic procedure. The ratio of bending stiffness to reduced tensile/compressive stiffness is taken as a small parameter. Axial inertia is taken into account in the equation of the first approximation. Introduced by V.V. Bolotin concept “nonlinear inertia” is discussed. The most common errors in using the quasi‐static Kirchhoff hypothesis are analyzed.
ISSN:0044-2267
1521-4001
DOI:10.1002/zamm.202400341