Engineering Nonlinearity

Lead Research Organisation: University of Bristol
Department Name: Mechanical Engineering


The aim of this proposal is to transform the design and manufacture of structural systems by relieving the bottleneck caused by the current practice of restricting designs to a linear dynamic regime. Our ambition is to not only address the challenge of dealing with nonlinearity, but to unlock the huge potential which can be gained from exploiting its positive attributes. The outputs will be a suite of novel modelling and control techniques which can be used directly in the design processes for structural systems, which we will demonstrate on a series of industry based experimental demonstrators. These design tools will enable a transformation in the performance of engineering structural systems which are under rapidly increasing demands from technological, economic and environmental pressures.

The performance of engineering structures and systems is governed by how well they behave in their operating environment. For a significant number of engineering sectors, such as wind power generation, automotive, medical robotics, aerospace and large civil infrastructure, dynamic effects dominate the operational regime. As a result, understanding structural dynamics is crucial for ensuring that we have safe, reliable and efficient structures. In fact, the related mathematical problems extend to other modelling problems encountered in other important research areas such as systems biology, physiological modelling and information technology.

So what exactly is the problem we are seeking to address in this proposal? Typically, when the behaviour of an engineering system is linear, computer simulations can be used to make very accurate predictions of its dynamic behaviour. The concept of end-to-end simulation and virtual prototyping, verification and testing has become a key paradigm across many sectors. The problem with this simulation based approach is that it is built on implicit assumptions of repeatability and linearity. For example, many structural analysis methods are based on the concept of a frequency domain charaterisation, which assumes that response of the system can be characterised by linear superposition of the response to each frequency seperately. But, the response of nonlinear systems is known to display amplitude dependence, sensitivity to transient effects in the forcing, and potential bistability or multiplicity of outcome for the same input frequency. As a result, when the system is nonlinear (which is nearly always the case for a large number of important industrial problems) it is almost impossible to make dynamic predictions without introducing very limiting approximations
and simplifications. For example, throughout recent history, there have been many examples of unwanted vibrations; Failure of the Tacoma Narrows bridge (1940); cable-deck coupled vibrations on the DongTing Lake Bridge (1999); human induced vibration on the Millennium Bridge (2000); NASA Helios failure (2003); Coupling between thrusters and natural frequencies of the flexible structure on the International Space Station (2009); Landing gear shimmy.

In many cases, the complexity of modern designs has outstripped our ability to understand their dynamic behaviour in detail. Even with the benefit of high power computing, which has enabled engineers to carry out detailed simulations, interpreting results from these simulations is a fundamental bottleneck, and it would seem that our ability to match experimental results is not improving, due primarily to the combination of random and uncertain effects and the failure of the linear superposition approach. As a result a new type of structural dynamics, which fully embraces nonlinearity, is urgently needed to enable the most efficient design and manufacture of the next generation of engineering structures.

Planned Impact

This highly ambitious, multi-disciplinary, and innovative programme is aimed at transforming the design and manufacture of structural systems. The goal is not only to improve performance and reduce cost, but to exploit new aspects of nonlinearity for positive effect. Relevant sectors include, energy, aerospace, offshore, renewables, and medical engineering, but will extend to conventional automotive and other ground transport and civil engineering. A range of stakeholders will benefit:

Companies with a high dependence on structural dynamic expertise for the design and manufacture of their products will gain access to new technology, either through directly licensing intellectual property developed during the programme, or through subsequent co-development projects. Technology transfer will be facilitated by specialist teams at the five partner institutions, coordinated by Research & Enterprise Development Department at Bristol, who have extensive experience in balancing the need to protect inventions while encouraging commercial exploitation. Industrial partners will thus gain new capabilities allowing them to develop new products, leading to a competitive advantage, and ultimately, UK wealth creation.

In addition, there are a variety of smaller, specialist manufacturing and consulting companies, for example, in civil/structural engineering and the automotive sectors, who also require structural dynamic expertise. These companies will benefit from new software products developed as a direct impact from this proposed programme. Faster, more efficient design tools and techniques will enable such companies to be more competitive in an era where economic and environmental pressures are rapidly increasing the demands on the performance of all civil, mechanical and aerospace systems.


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Agrawal DK (2013) Modeling nonlinearities in MEMS oscillators. in IEEE transactions on ultrasonics, ferroelectrics, and frequency control

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Arrieta A (2012) Dynamic control for morphing of bi-stable composites in Journal of Intelligent Material Systems and Structures

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Arrieta A (2012) Passive load alleviation bi-stable morphing concept in AIP Advances

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Becker W (2013) Bayesian sensitivity analysis of bifurcating nonlinear models in Mechanical Systems and Signal Processing

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Ben Abdessalem A (2018) Model selection and parameter estimation in structural dynamics using approximate Bayesian computation in Mechanical Systems and Signal Processing

Description There were three objectives in this project. Objective 1 Modelling: to develop the advanced modelling techniques required to achieve nonlinear analysis and testing of multi-degree-of-freedom systems.
Development of modelling techniques in ENL focused around three main topics. First modelling nonlinear multi-degree-of-freedom systems for modal testing. One of the key novelties developed here was the use of backbone curve models, which allowed both a reduced order model to be created and also parameter identification to be applied. This allowed a strong connection to other methods to be established, and these are described in more detail under Objectives 2 and 3. For example, to use the backbone curve procedure in the presence of uncertainty, a backbone curve identification method using control based continuation was developed . This fusion of identification methods with nonlinear modelling is key to creating a genuinely new nonlinear modal testing method with the potential for a high level of industrial impact. An example of this is that during the project ENL researchers compiled an industry handbook for modal testing structures with nonlinearities for Rolls-Royce.
The second main area of novel nonlinear modeling was the development of nonlinear statistical energy analysis (NSEA) . This is a major advance that will enable modeling of mid- to high-frequency nonlinear multi-modal systems that was previously not possible. This breakthrough was made possible by modelling the coupling of energy flow between different frequency bands in the system being considered. It is planned that this new methodology will be incorporated into commercial software code, and therefore will have direct impact on industrial practice.
The third major area of novel development has been the modeling of constitutive relationships for dynamic friction applications. Here the crucial insight has been in the use of rate-and-state models, combined with new experimental tests enabling comprehensive validation. Already, work with Stirling Dynamics has improved modelling in industrial friction problems using multibody dynamics software (SimMechanics) to exploit the new knowledge.
Objective 2 Identification: to derive novel data-based identification methods for multi-degree-of-freedom nonlinear systems.
One of the first novel achievements in identification was a new method for simultaneously identifying both models and parameters. This is a significant step forward for the identification of structural systems. It was achieved using Bayesian identification schemes for parametric white and grey box models appropriate to nonlinear structural dynamics. The result was a method for simultaneously identifying both models and parameters . In particular models were selected based on choosing the model with least complexity whilst retaining required fidelity. A key feature of the new technique is that it can choose between models with a different number of parameters. This allows for the identification method to seek the model that fits the data without overfitting, as well as estimating parameters simultaneously. The new algorithms have been applied, for the first time, in the context of structural dynamics and on large, industrially relevant, datasets. The work has already led to papers with industrial co-authors.
Another novel contribution in this area is a the development of a very fast method for Approximate Bayesian Computation (ABC) applied to system identification . This allows much more efficient use of the ABC algorithm, which due to its novelty and properties has the prospects to become the method of choice for a range of practical problems. For example, machine learning is gaining a significant interest in offshore wind industry .
Objective 3 Control: particularly to exploit nonlinearity in structural dynamics design via control.
The control work in ENL has focused on three areas. First an experimental energy harvester has been developed, which acts as one of the demonstrators in the project. It consists of a moving mass, ball screw and rotational generator. It is based on an analysis of the greater electromagnetic coupling coefficient of such a rotating generator system, over a linear moving coil system , allowing a stronger mechanical effect when electrically shunting the generator. Compared with that harvested using a linear electromagnetic damper, the power output of the nonlinear device is always larger when the excitation is below its maximum level. The experimental harvester is significantly larger than most designs, having a 10 kg inertial mass, and was designed to generate tens of watts, from sea motion on a sailing ship for example. Initial experimental tests have shown that the damping can be significantly increased by electrical shunting of the generator, but that the parasitic damping in the mechanical system is larger than expected and is also nonlinear. This provided a good case study for identification of this nonlinear damping, undertaken in collaboration with Objective 2 . Other novel aspects of energy harvesting were carried out, and led (amongst other things) to a generalized mass law which defines the maximum possible energy that can be harvested from a vibrating system .
The second area in the control domain has been related to understanding the dynamics of the cochlea. This has important applications related to hearing loss in humans, and particularly cochlear implant technology. The model has been developed at the microscopic level, using the finite element method. The individual motion of each component within the organ of Corti, due to acoustic and electrical excitation, is calculated and compared with recorded measurements, showing a similar internal behaviour to that seen experimentally. The fully active and nonlinear dynamic response can then be calculated, using the predicted linear responses of the basilar membrane motion and of the deflection of the hair bundles of the outer hair cells when driven by the fluid or the outer hair cells, and the feedback loop due to the nonlinear response of the outer hair cells. This provides a prediction of the fully active internal response of the organ of Corti, which currently awaits experimental verification.
Finally in the control domain there has been work on structural control using passive, semi-active and active control techniques. This included work on inerter systems , and hybrid semi-active with active control .
Exploitation Route The new scientific ideas developed in this project are taking hold in the wider research community - one example of this is the IMAC conference (Orlando, Florida, starting in Feb. 2015 and onwards), held dedicated sessions on the exploitation of nonlinearity in structural dynamics. This demonstrates just one way that the ENL project is having an influence beyond the summation of its parts.
Sectors Aerospace, Defence and Marine

Description Examples from the project include: -ENL researchers are worked with industry partners to improve the fidelity of multibody dynamics software (such as SimMechanics) for modelling industrial friction problems -A handbook for modal testing structures with nonlinearities was been developed for industry partners -ENL researchers worked with industry partners on improved control techniques for rotorcraft -Energy blog, The Carbon Brief listed ENL research as one of the "Five innovations that could cut the cost of offshore wind" -A special issue of the Philosophical Transactions of the Royal Society was produced that included papers with industrial co-authors.
First Year Of Impact 2012
Sector Aerospace, Defence and Marine
Impact Types Economic

Title Research data supporting Validation of a constitutive law for friction-induced vibration under different wear conditions 
Description The experimental data, simulation results and associated Matlab files are all included in this dataset. They are organised according to the figure numbers in the paper. 
Type Of Material Database/Collection of data 
Year Produced 2017 
Provided To Others? Yes