Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

Abstract

Reduced-order models are essential to study nonlinear vibrations of structures and structural components. The natural mode discretization is based on a twostep analysis. In the first step, the natural modes of the structure are obtained. Because this is a linear analysis, the structure can be discretized with a very large number of degrees of freedom. Then, in the second step, a small number of these natural modes are used to discretize the nonlinear vibration problem with a huge reduction in the number of degrees of freedom. This study finds a recipe to select the natural modes that must be retained to study nonlinear vibrations of an angle-ply laminated circular cylindrical shell that the author has previously studied by using admissible functions defined on the whole structure, so that an accuracy analysis is performed. The higher-order shear deformation theory developed by Amabili and Reddy is used to model the shell.

Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic manifold method with modal derivatives

“…Moreover, the error on Γ will be smaller than the error on the correction factor C, being Γ composed of other terms that are not affected by the reduction method. The conclusion is that ρ ∈ [3,4] can be understood as a transition region, and converged results thanks to slow/fast assumption can be faithfully obtained over 4, but below 3 caution has to be exercised. Figure 1 shows also other interesting features on the behaviour of the type of nonlinearity.…”

“…When referring to linear methods, one can also distinguish techniques where the best orthogonal basis is computed once and from all. Modal basis [3,12,34,56], Ritz vectors [20], dual modes [25] and proper orthogonal decomposition (POD) [4,23,27] fall into that family. The proper generalised decomposition (PGD) [10,14] under its progressive variant (pPGD) as defined in [32] also belongs to that case since additional vectors are added when the dynamics is becoming more complex.…”

Non-linear vibrations of shells: A literature review from 2003 to 2013

“…He found an increased non linearity for a 0/30 lay up shell with respect to an identical symmetric cross ply 0/90/0 laminated shell. Moreover, Amabili [219] developed a new technique for obtaining reduced order models of angle ply circular cylindrical shells by using natural modes of vibration. In Refs.…”

Intelligent computing for Duffing-Harmonic oscillator equation via the bio-evolutionary optimization algorithm

“…Problems of nonlinear vibration in conservative systems, bearing a wide history, have been interpreted to a great extent, in order to study various parametric behaviors of nonlinear vibrations of several mechanical objects. [1][2][3] In this regard, several methods have been established to analyze these problems, analytically as well as numerically. [4][5][6] Particularly, among many well-known oscillatory equations, the Duffing equation is found to be the most widely studied equation for its eminent applications in different fields of science, engineering, and biology.…”