Nonlinear Resonance Analysis of Slender Portal Frames under Base Excitation

Luis Fernando Paullo Muñoz,Andréa Silva,Paulo B. Gonçalves,Ricardo A. M. Silveira

doi:10.1155/2017/5281237

Abstract

The dynamic nonlinear response and stability of slender structures in the main resonance regions are a topic of importance in structural analysis. In complex problems, the determination of the response in the frequency domain indirectly obtained through analyses in time domain can lead to huge computational effort in large systems. In nonlinear cases, the response in the frequency domain becomes even more cumbersome because of the possibility of multiple solutions for certain forcing frequencies. Those solutions can be stable and unstable, in particular saddle-node bifurcation at the turning points along the resonance curves. In this work, an incremental technique for direct calculation of the nonlinear response in frequency domain of plane frames subjected to base excitation is proposed. The transformation of equations of motion to the frequency domain is made through the harmonic balance method in conjunction with the Galerkin method. The resulting system of nonlinear equations in terms of the modal amplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain the nonlinear resonance curves. Suitable examples are presented, and the influence of the frame geometric parameters and base motion on the nonlinear resonance curves is investigated.

Highlights

Lightweight framed structures are extensively used in engineering applications such as buildings, industrial constructions, off-shore platforms, and aerospace structures
The resulting system of nonlinear equations in terms of the modal amplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain the nonlinear resonance curves
An incremental technique for the direct calculation of the nonlinear resonance curves of plane frames discretized by the finite element method and subjected to a base excitation is proposed

Summary

Introduction

Lightweight framed structures are extensively used in engineering applications such as buildings, industrial constructions, off-shore platforms, and aerospace structures. The determination of a structure’s response in the frequency domain, characterized by the resonance curves and bifurcation diagrams, is, in some cases, indirectly obtained through a series of analyses in time domain, varying step by step a selected control parameter (see Parker and Chua [21]) In this brute force approach, for each control parameter value, the equations of motion must be integrated long enough to reach the steady-state regime. An incremental technique for the direct calculation of the nonlinear dynamic response in frequency domain of nonlinear plane frames discretized by the finite element method and subjected to a base excitation is proposed. The resulting system of nonlinear equations in terms of the modal amplitudes and forcing frequency is solved by the Newton-Raphson method together with an arc-length procedure to obtain the nonlinear resonance curves. Examples of commonly used frame geometries are presented and the influence of vertical and horizontal base motions on the frame nonlinear response as a function of the frame geometric parameters is analyzed

Formulation

Numerical Examples

Frequency Domain Analysis of Slender Frames

Conclusions