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

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 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. Electronic supplementary material The online version of this article (10.1007/s11071-018-4534-1) contains supplementary material, which is available to authorized users.

Using frequency detuning to compare analytical approximations for forced responses

“…The results presented in this paper, along with those of [3], have demonstrated that the MS and DNF are able to provide identical results. Given that the implementation of the two techniques requires a similar number of steps, this leaves the user to make a decision based on the nuances of the particular system they are considering.…”

“…These resonances have also 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]. This paper expands the recent discussion by the authors in [3], which considered the calculation of the free response of two nonlinear mass-spring systems using the MS and DNF techniques. The authors applied the frequency detuning from the DNF method, as explored in [33,34], in the MS technique to show that it is possible for both methods to give identical solutions.…”

“…Here, it can be seen that this remains true in the forced case. Thus, the MS solution can be considered as an approximation to the DNF and HB expressions, a point which was demonstrated to lead to divergent backbone curves in [3].…”

“…While the complexity and variation in such structures are constantly increasing, the development of mathematical methods for assessing the equations of motion of these systems continues to be an important branch of research. As has been addressed in the literature [1][2][3][4][5][6], there are two main ways in which the free and harmonically forced response of a nonlinear system can be approximated. Numerical continuation methods [2] can be used to follow the entire set of periodic responses over a region of the parameter space.…”

“…These methods were chosen due to their wide application in the literature and, therefore, the results provide the user with the make a more informed decision regarding the accuracy and applicability of the methods. It should be noted that these techniques do not represent the entire field of analytical methods for approximating nonlinear systems; for example, the more mathematically rigorous spectral manifolds approach may also be applied (see [12]) A more complete review of the associated literature is presented in [3]. However, as motivation for their comparison, we present some prominent studies including each of the considered techniques.…”