Response approximation techniques

The ability to accurately capture and predict the response of nonlinear structures is an impor- tant challenge in theoretical and modelling studies. This has been illustrated by the example structures in §2.1, the success of which can not be guaranteed without a sufficient under- standing of the system dynamics.

The atypical physical characteristics observed in these structures translate to nonlinear terms in the equations of motion. It is the inclusion of such terms that leads to overly com-

plex analytical solutions that provide little insight without significant extra algebraic ma- nipulation. This leaves the practitioner with two options to predict the behaviour of these structures. On one hand, an analytical method could be used to approximate the behaviour, as will be discussed in §2.4.1. Alternatively, the periodic response across a certain frequency range can be created numerically by selecting an initial periodic orbit in the linear regime and applying a continuation method. The latter of these options is discussed here.

Numerical continuation programmes, such as the AUTO-07p package1 [29, 30], Mat- Cont[31, 32], and Continuation Core (COCO) [33, 34], have been widely applied to non- linear structures. In particular, the work of [18] utilises numerical continuation to assess the nonlinear dynamics of an autogyro beyond the parameter window that can be achieved experimentally. A similar methodology was applied in [35], in which experimental results were supplemented with numerical responses. The lack of detailed knowledge of the basins of attraction led to some stable solutions being experimentally inaccessible.

Due to the multitude of experimentally-observed nonlinear behaviour in M/NEMS struc- tures, as addressed above, the use of numerical continuation methods for such systems has become increasingly common.

A series of studies by Caruntu et al. has utilised AUTO-07p to predict bifurcation be- haviour in both MEMS cantilever beams [36–38] and circular plates [39, 40]. However, the continuation had to be applied to a reduced-order model of the structures, since the pro- cess can be extremely computationally expensive for larger systems. A similar microarch structure, modelled as a pinned-pinned beam and reduced using the Galerkin method, is investigated using the same software in [41], which observed both internal resonance and softening-to-hardening behaviour in the structure. The work of [42] utilised the MatCont package to analyse the lumped-mass model of a MEMS oscillator; in this case, the multiple scales method was required to reduce the order of the system. The examples of this para- graph highlight the fact that, while it is possible to apply numerical continuation techniques to a variety of real-world structures, it is typically necessary to first approximate the system dynamics using a small number of equations. This is a result of the rapid increase in com- putational expense associated with the continuation of large systems of nonlinear equations and provides motivation both for the pursuit of reduced-order modelling techniques and of analytical approximation methods that do not have such a steep increase in computation time.

1_{AUTO-07p is the current version of this software, but the programme has been in development since the}

The effectiveness of these general numerical continuation software packages in nonlinear vibration applications has inspired the development of a number of methodologies that are aimed specifically at predicting nonlinear behaviour in mechanical structures. The NNMcont programme is one such example of these [43]; its methodology is described in [44]. The pro- gramme employs a shooting strategy that allows the continuation of NNMs to be achieved. Recently, this method has been used to investigate the NNMs of a gong [45], in which the nonlinear model was shown to match well with experimental results, though only when a large number of modes was included.

One potential drawback of the NNMcont programme is its dependence on extensive nu- merical solutions that would introduce a large computational expense for complex structures, a point which is addressed by Peeters et al. in [44]. To address this, in [46, 47], Kuether et al. expanded the algorithm through the use of a predictor-corrector step, so that only a single period was required to be calculated in the finite element software. The methodology begins by using a single linear mode of the system to produce the required period and expands the modal basis once the error becomes too great. The results generated using this technique showed good agreement with those found using the original NNMcont method, but were found in a matter of minutes, whereas the full model required over four days of calculation to create.

It should be noted that the continuation concept has also been pursued outside of numeri- cal and analytical systems, with an experimental strategy being applied in the development of control-based continuation methods. The theoretical framework for this technique was ini- tially outlined in [48], which provided a strategy for the continuation of periodic orbits and bifurcations. The methodology was then successfully applied to a nonlinear energy harvester in [49] and a cantilever beam with magnetic interaction at the tip in [50]. The application and understanding of this technique continues to be expanded. In [51], the optimisation of an impact oscillator was investigated through the introduction of a tuning scheme. The same system was used to investigate the possibility of calculating the stability of each periodic os- cillation in [52], with further analysis of both stability and bifurcation for a SDOF oscillator given in [53]. This discrete system has also been used to calculate the backbone curve of a physical structure in [54].