The contribution of Gustav R. Kirchhoff to the dynamics of tapered beams

DOI10.1002/zamm.201600250OpenAlexW2617635065MaRDI QIDQ6087230

Antonio Cazzani, Luciano Rosati, P. Ruge

Publication date: 11 December 2023

Published in: ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1002/zamm.201600250

zbMATH Keywords

structural dynamicsvibration analysistapered beamsEuler-Bernoulli beam theoryGustav R. Kirchhoff

Mathematics Subject Classification ID

Rods (beams, columns, shafts, arches, rings, etc.) (74K10) Vibrations in dynamical problems in solid mechanics (74H45) Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators (34L10) History of mathematics in the 19th century (01A55) History of mechanics of deformable solids (74-03) Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators (34L15) Free motions in linear vibration theory (70J30)

Cites Work

• A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model
• The complete works of Gabrio Piola: Volume I. Commented English translation of the original Italian works, published between 1825 and 1856
• Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials
• On generalized Cosserat-type theories of plates and shells: a short review and bibliography
• Geometric continuum mechanics and induced beam theories
• The special functions and their approximations. Vol. I, II
• Consistent tangent operator for an exact Kirchhoff rod model
• An unconstrained dynamic approach for the generalised beam theory
• Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates
• Reference frequencies for the validation of numerical solutions of transverse vibrations of non-uniform beams
• Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives
• The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results
• Generalized Hypergeometric Function Solutions on the Transverse Vibration of a Class of Nonuniform Beams
• At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola
• Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua”^** by E. Hellinger
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