Finite element models

As discussed in §2.3.2, finite element models are ubiquitous in understanding the structural dynamics of physical structures. While the section above discussed this modelling technique in terms of software-based methods, it must be noted that finite element models were origi- nally derived analytically. A thorough guide to the technical implementation of these is given in [139]. As this reference shows, there are a number of different implementations of the fi- nite element method. The primary focus of this thesis is the study of beam-like structures, which can be accurately modelled using the Galerkin method [2, 140]. As such, this will be the only finite element methodology investigated in this section.

Accurate beam models are of great importance to this thesis, due to their historic use in modelling macroscale structures and their more recent prominence in M/NEMS devices. As such, it is necessary to understand the development of various beam theories and the appli- cability of each. Note that these are continuous models for beams, which will then be dis- cretised using the Galerkin method. There are two leading linear methodologies employed, each with its own set of assumptions and associated advantages and disadvantages; namely, these are the theories of Euler-Bernoulli, as applied in [2], and Timoshenko, as defined in [140]. The principal difference between the two methodologies is that the Euler-Bernoulli theory assumes that the cross section of the beam remains perpendicular to the neutral axis,

whereas Timoshenko theory allows a shear deformation, so that these are no longer perpen- dicular. As a result of their prevalence, the two methods are regularly compared, for instance in [141], concluding that there are a number of cases in which the difference in accuracy resulting from their assumptions is negligible. An example of this can be observed in the modelling of rotating wind turbine and helicopter blades in [142]. However, the inclusion of shear deformations and the associated rotation allows a more accurate solution at high deflections. Note that a number of nonlinear beam methodologies have been developed (see [143], for a comparison of these), but these are beyond the scope of this thesis, which will focus on weakly nonlinear structures.

As with the software-based finite element methods above, the analytical structures must be discretised and appropriately treated. Across the literature, this is commonly achieved through the use of the Ritz-Galerkin method, henceforth referred to simply as the Galerkin method [2, 140]. The underlying assumption of this technique is that a continuous function in terms of two variables can be discretised as an infinite series of products of projection functions, each of which is then defined in terms of only one variable. For beams, the transverse displacement, w, is typically expressed as a function of the position along the beam, x, and time, t. Applying the Galerkin approximation, this displacement is expressed as w(x, t) = ∞ X k=1 φk(x)qk(t),

where the φk and qk terms are functions defining the shape and magnitude of the deflec-

tion, respectively. By truncating this expansion2_{and utilising the orthogonality of the linear}

mode shapes, it becomes possible to express the equations of motion as a finite system of second-order differential equations. This approach was used to model coupled-bending tor- sion vibrations of an asymmetric aerofoil in [144].

The Galerkin approximation is particularly useful for modelling simple continuous sys- tems, such as beams, plates, shells, and pipes. Extensive consideration has been given to developing Galerkin models for these structures with a number of classical BCs, an ex- pression that typically refers to clamped, pinned, free, and sliding beam tips. Derivations

2_{As an interesting aside, the HB technique discussed above is actually a specific application of the Galerkin}

method. By writing y(t) = N X k=1 Ake+jkωnt+ ¯Ake−jkωnt,

the displacement x is expressed in terms of Fourier coefficients, Ak, and time varying components, e±jkωnt.

and discussions of these cases can be readily found, for example, in [2, 140, 145]. These BCs have been widely used in the literature. For instance, circular plates have been con- sidered with free BCs in [146] and with clamped conditions in [39]. This variation can be seen to noticeably alter the mode shapes and natural frequencies of the plate. Further, the Galerkin method has been developed to include fluid-structure interactions, largely building upon these classical BCs [147–152]. While this provides a relatively uncomplicated model, this technique does not take into account the influence that the pressure induced by the fluid would have on the accuracy of these BCs.

In spite of this, there are a number of cases in which it is inaccurate to use these classical BCs, as doing so does not truly reflect the resonant behaviour of the system. The develop- ment of this approach has been somewhat incremental and has primarily been achieved by adapting or adding to the traditional cases. For instance, in [153], a cantilever bar model is expanded with a horizontal spring at its tip, leading to more pronounced hysteretic be- haviour. In [154], a cantilever beam is considered with a rotational spring at some point along its length, and this concept is expanded, in [155], through the addition of a linear compressive spring. These papers derive the mode shapes and frequencies for these con- figurations, though an investigation into the influence this has on the system dynamics was prohibited by the lack of sufficient computing power at the time they were written. Sim- ilar statements can be made regarding the system considered in [156, 157], in which both beam tips are supported by two springs, one rotational and the other translational. In [158], a compressive axial load is added to a number of classical beam configurations, revealing that the relative critical buckling load is consistent across a large number of these combinations, despite the differences in their resonant behaviour. Furthermore, systems such as these have been shown to exhibit interactions between the modes [159].

While undoubtedly more complex, the non-classical BCs discussed thus far have a rel- ative simplicity due to the fact that any additional components have been defined linearly. Of course, this is not guaranteed to be the case and the inclusion of nonlinear terms leads to solvability conditions that are markedly more complicated. This has been addressed in the work of [160], which considers the normal modes of a number of beams with non-classical, nonlinear BCs. However, the inclusion of a nonlinear term requires the updating of the stan- dard procedure for finding the mode shapes, instead applying Hamilton’s principle to find an approximate solution for the mode shapes. This use of an approximation adds some un- certainty to the model, which could be avoided if the exact mathematical definitions of the

BCs were used. An important aspect of this work is the conclusion that these nonlinear BCs influence the mode shapes and natural frequencies of the system. This concept could be developed further, as will be shown in this thesis.

The methodology of [160] has been expanded to allow the considerations of thin plates [161, 162], thick plates [163], and complex shell structures [164], as well as to consider the implications of non-local effects in the free vibrations of M/NEMS in [165]. These com- plex expansions increase the uncertainty related to the use of an approximation for the mode shape. In each of these cases, as opposed to assuming that the solution to the fourth-order equations of motion will comprise some combination of cos, sin, cosh, and sinh functions, it is proposed that this could be approximated through a truncated Fourier series. The valida- tion of this proposition would need to be done either numerically or experimentally, which increases the workload of the user.

Alternatively, it has been demonstrated that, should the cubic term be assumed to be small, then the analytical approximation methods of the previous section can be used to calculate the system behaviour. For example the MS method has been used to predict the behaviour of a beam supported by a nonlinear cubic spring in [166] and a spring with a magnetic interaction at the tip in [167]. In fact, the application of Hamilton’s principle in [160] is originally presented as an implementation of the HB method. As has already been discussed, these methods are themselves approximations based on a series of assumptions, so it is possible that excessive testing would be required to validate the model.

Numerical solutions for a beam with one nonlinear BC are provided in [168], which applies an iterative method that converges to a numerically correct mode shapes. This ap- proach is also applied in [169], which assumes that both BCs can be nonlinear. This iterative method initially considers the problem from a purely mathematical standpoint, similar to that in [170]. However, the latter study focuses on the existence of solutions, defining criteria to ensure that it is possible to obtain a number of mode shapes. This work is expanded in [171] to demonstrate that infinitely many solutions exist, as is expected from the more traditional engineering approach. These mathematical studies offer comfort to the engineering user of these techniques, but do not offer any great insight for their successful application.