An accurate model for nonlinear forced vibrations of higher-order and Timoshenko cantilever beams

作者：Marco Amabili · 发表于：Composite Structures · 年份：2026 · DOI：10.1016/j.compstruct.2025.120035 · 被引用次数：10 · 研究领域：Bladed Disk Vibration Dynamics、Composite Structure Analysis and Optimization、Aeroelasticity and Vibration Control

An accurate higher-order model for planar nonlinear vibrations of cantilever beams is developed, taking into account nonlinear terms in (i) transverse displacement, (ii) longitudinal displacement, and (iii) rotation, as well as geometric imperfections, shear deformation, and rotary inertia. The model can be reduced to the Timoshenko beam theory, which is a first-order model, when neglecting higher-order terms. It is quite significant that the discretization is obtained by series expansions that satisfy both geometric (essential) and natural boundary conditions. A validation of the present model and a detailed convergence analysis are presented. It is crucial to adopt a proper expansion of the longitudinal displacement, which also satisfies the natural boundary condition of zero longitudinal force at the beam’s tip. Expansions that do not satisfy this boundary condition do not converge and result in over-predicting the system nonlinearity. Results show a very weak hardening-type nonlinearity for cantilever beams. In the case of geometric imperfections in the shape of the fundamental mode, an increase in the natural frequency and in the nonlinearity is observed, as well as differences in the time domain response of the generalized coordinates.