As has been observed in §2.1, the potential scope for structures exhibiting nonlinear charac- teristics is extremely broad, both in terms of the variety and the scale of these mechanical systems. These range from large, lightweight wind turbines [13, 23] to increasingly small and intricate micro- and nanoscale devices [22]. Whether this behaviour is desired or not, the accurate modelling of the associated phenomena presents a significant challenge to engi- neers.
In linear dynamic structures, there is an underlying assumption that each linear normal mode has a natural frequency for its vibration, and that the value of this is invariant to the behaviour of the system [24]. It can be further observed that these linear modes are orthogo- nal to one another. Mathematically speaking, this means that it is not possible for one mode to be written as a linear combination of the others, but the practical implication of this is that the contribution of one mode to the response has no influence on that of the others. As such, linear systems follow the principle of superposition, by which the full system response can simply be found through the summation of the contributions of the linear modes. These modelling techniques and assumptions undoubtedly provide a useful framework, which has been used to great effect in the modelling of classical engineering structures. However, the assumptions of uncoupled modes and of invariant natural frequencies do not hold for nonlinear structures, so this approach must be developed or replaced to accommodate more complicated behaviour.
Nonlinearity can be introduced to mechanical structures in a number of ways. The fo- cus of this thesis will be on geometric nonlinearity, in which the geometry of the structure changes in deformation, leading to changes in the structural properties and, hence, the system response. These systems are typically characterised using polynomial expressions, though the approach differs depending on the size of the deformation, as summarised in [9]. It must be noted that, although not discussed in this thesis, it is possible for the material of a structure, or any contact it may have with other bodies (such as impact or friction), may also lead to nonlinear behaviour. However, in contrast with geometric nonlinearities, it may not be possible to use a polynomial expression in terms of the displacement to model these nonlinear effects, particularly in the case of contact nonlinearities. Further details of such phenomena may be found, for example, in [25]. From this point forward, any references to ‘nonlinear’ behaviour will be taken to refer to geometric nonlinearity.
In nonlinear structures, the natural frequencies have been observed to shift as the re- sponse amplitude changes [26]. Therefore, the resonance peak will appear at different fre- quencies for different excitation levels. Although this difference may appear simple, it can add a great deal of complexity to both the response itself and the mathematical tools re- quired to capture it. For a linear model, the peak of the response will be exactly at the linear natural frequency, regardless of the amplitude. Therefore, the locus defined by the peak of the curve at different displacements is simply a straight, vertical line. In the nonlin- ear case, the aforementioned shifting of the natural frequency at higher amplitudes creates
a locus (or backbone curve) that can move away from the linear natural frequency at higher displacement levels. This can lead to the phenomenon of multistability, in which a single forcing frequency may induce a stable periodic orbit for more than one amplitude, leading to systems that exhibit hysteretic behaviour [26]. An example of this has been observed in the ‘jumping’ phenomenon in aircraft dynamics, as addressed in [27]. Furthermore, if the modes are not uncoupled, it is possible for an internal resonance to occur between them, in which energy may transfer from one to another, as remarked in [28].
These behavioural phenomena correspond to nonlinear terms in the equations of motion. These terms typically result in analytical approximations that are extremely complicated and, hence, require a great deal of further manipulation to allow the system to be understood. As such, the typical approach is to approximate the response in some way, either using an analytical method or by employing a numerical strategy. In either case, the solutions are no longer exact, and their accuracy is dependent on the assumptions made in the creation of the approximation. There are a number of methods that may be applied, and the user must decide which assumptions will lead to an accurate approximation of the system behaviour. The most suitable and applicable method, and the implications of its assumptions, are not necessarily known a priori.
For the methods discussed in this thesis to become as reliable as their linear counterparts, it is important to remove the uncertainty associated with their accuracy and usability. As summarised in Chapter 1, a key aim of this thesis is to extend the understanding of a number of nonlinear techniques. As such, the current chapter will address the historical and recent developments in the field, as well as highlighting those areas in which further research may be beneficial to the practitioner.