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Meccanica

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06-10-2023

An analytical investigation of nonlinear response and stability characteristics of beam with three-dimensional tip mass

Author: Pravesh Kumar

Published in: Meccanica | Issue 10/2023

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Abstract

This article presents the mathematical modelling and nonlinear dynamics of beam with three-dimensional tip mass undergoing large deformation in lateral and transverse directions. The beam is modelled as Euler–Bernoulli having large radius of curvature considering longitudinal, lateral, transverse as well as torsional deformations. The governing equations of motion and linearly coupled boundary conditions assimilating the moments and shear forces dues to three-dimensional tip mass are derived using Hamilton’s principle. The transcendental eigen-frequency equation and subsequently, the eigen-functions are obtained by free vibration analysis for beam with and without torsional deformation. In addition, the investigation of nonlinear behaviors of the system under combined external excitation and inherently present internal resonance conditions between lateral and transverse motions are explored. The inextensibility condition is used to define longitudinal deformation in terms of lateral and transverse deformations leading to highly nonlinear governing equations of motion. The Galerkin’s method is exploited for discretization using the obtained modal parameters and method of multiple scale is used to obtain the steady state solutions under combined resonance conditions. The spring softening behavior due to cubic nonlinearities exhibiting multi-valued solutions and multi-jump phenomenon are investigated. The system exhibits peculiar combination of pitchfork and saddle-node bifurcations in frequency response of system due to parametric variation of system attributes which have not been visible in previous studies. The obtained linear as well as nonlinear results have been compared with the existing results as well as numerical simulations. The outcomes of the present work shall enhance the ability of prediction and control of catastrophic failure of the base excited beam with three-dimensional tip mass.

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Title: An analytical investigation of nonlinear response and stability characteristics of beam with three-dimensional tip mass
Author: Pravesh Kumar
Publication date: 06-10-2023
Publisher: Springer Netherlands
Published in: Meccanica / Issue 10/2023
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-023-01710-0

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