Comparing the direct normal form and multiple scales methods through frequency detuning

Elliott, A.J., Cammarano, A. , Neild, S.A., Hill, T.L. and Wagg, D.J. (2018) Comparing the direct normal form and multiple scales methods through frequency detuning. Nonlinear Dynamics, 94(4), pp. 2919-2935. (doi: 10.1007/s11071-018-4534-1)

Abstract

Approximate analytical methods, such as the multiple scales (MS) and direct normal form (DNF) techniques, have been used extensively for investigating nonlinear mechanical structures, due to their ability to offer insight into the system dynamics. A comparison of their accuracy has not previously been undertaken, so is addressed in this paper. This is achieved by computing the backbone curves of two systems: the single-degree-of-freedom Duffing oscillator and a non-symmetric, two-degree-of-freedom oscillator. The DNF method includes an inherent detuning, which can be physically interpreted as a series expansion about the natural frequencies of the underlying linear system and has previously been shown to increase its accuracy. In contrast, there is no such inbuilt detuning for MS, although one may be, and usually is, included. This paper investigates the use of the DNF detuning as the chosen detuning in the MS method as a way of equating the two techniques, demonstrating that the two can be made to give identical results up to ε2 order. For the examples considered here, the resulting predictions are more accurate than those provided by the standard MS technique. Wolfram Mathematica scripts implementing these methods have been provided to be used in conjunction with this paper to illustrate their practicality.

• Item Type: Articles
• Additional Information: S.A. Neild gratefully acknowledges support from the EPSRC via the fellowship EP/K005375/1.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Cammarano, Dr Andrea and Elliott, Mr Alexander
• Authors: Elliott, A.J., Cammarano, A., Neild, S.A., Hill, T.L., and Wagg, D.J.
• College/School: College of Science and Engineering > School of Engineering; College of Science and Engineering > School of Engineering > Systems Power and Energy
• Journal Name: Nonlinear Dynamics
• Publisher: Springer
• ISSN: 0924-090X
• ISSN (Online): 1573-269X
• Published Online: 14 September 2018
• Copyright Holders: Copyright © 2018 The Authors
• First Published: First published in Nonlinear Dynamics 94(4): 2919-2935
• Publisher Policy: Reproduced under a Creative Commons License