Abstract
It is possible to use numerical techniques to provide solutions to nonlinear dynamical systems that can be considered exact up to numerical tolerances. However, often, this does not provide the user with sufficient information to fully understand the behaviour of these systems. To address this issue, it is common practice to find an approximate solution using an analytical method, which can be used to develop a more thorough appreciation of how the parameters of a system influence its response. This paper considers three such techniques—the harmonic balance, multiple scales, and direct normal form methods—in their ability to accurately capture the forced response of nonlinear structures. Using frequency detuning as a method of comparison, it is shown that it is possible for all three methods to give identical solutions, should particular conditions be used.
Highlights
Modelling the dynamical behaviour of nonlinear structures remains an active area of research across the engineering disciplines, and one which becomes increasingly relevant as the push for more lightweight and efficient structures continues
This paper presents and compares three analytical approximation methods, both in terms of a general nonlinear dynamical system and the Duffing oscillator, which has been widely used in the development and study of such techniques
It has been shown that the harmonic balance (HB) and direct normal form (DNF) give identical solutions for the Duffing oscillator, the inclusion of a bookkeeping parameter in the latter provides the potential for a simpler management of weak terms in more complex structures
Summary
Modelling the dynamical behaviour of nonlinear structures remains an active area of research across the engineering disciplines, and one which becomes increasingly relevant as the push for more lightweight and efficient structures continues. The accuracy of three analytical methods in capturing the behaviour of a nonlinear system is assessed These are the harmonic balance (HB), the multiple scales (MS), and direct normal form (DNF) techniques. It has been widely used in the prediction of nonlinear dynamic behaviour, such as bifurcations in the frequency–amplitude relationship [19,20,21,22] and internal resonances [23,24,25] These resonances have been investigated using the DNF method [26,27,28,29], which has been further applied to explore the significance of nonlinear normal modes in relation to forced cases [30] and identify nonlinearity in structures [31,32].
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