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Digital twins are required to process a large amount of data during operation, in order to achieve specific tasks, over the lifetime of the physical twin that they relate to. One important feature of processing data is the identification of trust in both the underlying data and processed information that arises from the data. Trust, as it is defined here, will typically be built from several contributory sources. While there are both quantitative and qualitative sources of trust, this paper focuses on the qualitative aspects of trust via the transparency of the algorithmic process that is available in the crystal-box modelling. The crystal-box idea is also extended to include the concept of a ‘crystal-box workflow’. The key idea is that in order to assist the user of the digital twin to interpret the results they are presented with, via the digital twin interface, the information needs to be contextualised. This work shows an example of how this can be done for a vibration testing (specifically modal testing) example on a scaled three-storey structure. The information is contextualised for the user via ‘profiles’, which collate and augment the processed information together. In particular, synthetic results are generated in order to augment a limited set of physically recorded data, and these synthetic results are then used to assist the user in contextualising the physically recorded data. Implementation results are shown using an open-source digital twin code called DTOP-Cristallo.

In this paper a new design for a small scale inerter-based dynamic vibration absorber (IDVA) is presented, based upon a pivoted-bar mechanism. There are several new innovations in this study. The first is to design, build and test an inerter-based device that does not need to be grounded, or placed between different parts of the structure in order to create relative motion. Instead, the relative motion is created between an auxiliary mass and the host structure. Secondly, the pivoted-bar mechanism is designed to act as a pure inerter, and avoids unbalanced inertance effects such as those that occur in the dynamic antiresonant vibration isolator (DAVI). Thirdly, the effects of parasitic mass are minimised by using (i) the appropriate device arrangements, (ii) numerical optimisation, and (iii) fine tuning of the device by adding small additional masses. Fourth, the optimum device damping values are obtained by using a gel damper that can be modelled as a hysteretic damper. In addition, the design uses frictionless flexure hinges that have a small amount of stiffness that can affect the device performance. It is shown how this can also be compensated for using the design optimisation and fine tuning strategies. Detailed design and analysis methodologies are provided, in order to extend the existing work on inerters towards practical design and implementation. In terms of applications, it is anticipated that the design would be suitable for small-size restricted-space applications. A prototype of the design for a small-scale and relatively high frequency application is manufactured. Experimental and numerical results show that the device provides a 18% improvement in performance compared to a classical tuned mass damper (TMD).

Computational models are widely used to describe engineering systems and to predict their behaviour. However, in many applications, these computational models do not capture the complete physics of the real system, leading to model discrepancy. Model discrepancy causes bias in the inferred model parameters when it is not properly accounted for. This paper proposes a novel approach that seeks to capture the functional form of the model discrepancy and reduce bias in the estimated model parameters through and equation discovery procedure. A sparse Bayesian model is proposed, where sparsity is introduced through a hierarchical prior structure, providing a mechanism for removing erroneous candidate model terms from a series of potential equations as a part of an equation discovery procedure. At the same time, a Gaussian process model is used to account for model discrepancy. These two modelling assumptions are combined in a Bayesian formulation, allowing the system parameters and model discrepancy to be inferred in a probabilistic manner with associated uncertainties quantified based on their posterior distributions. The resulting method is capable of simultaneously providing physical insights into the system behaviour, by selecting the appropriate candidate model components and their respective system parameters, whilst compensating for model discrepancy that may occur due to an incomplete set of candidate terms. In order to efficiently solve the statistical model, an expectation maximisation algorithm is proposed for performing inference, and illustrative examples are presented to validate the proposed method. It is shown that compared to using a conventional sparse Bayesian approach for performing equation discovery, such as the Relevance Vector Machine, the proposed method provides better equation selection and parameter estimation, with less bias in the parameter estimates.

The aim of this paper is to provide an introduction to using normal form transformations for linear and nonlinear structural dynamics examples. Starting with linear single-degree-of-freedom systems, a series of examples are presented that eventually lead to the analysis of a system of two coupled nonlinear oscillators. A key part of normal form transformations are the associated coordinate transformations. This review includes topics such as Jordan normal form and modal transformations for linear systems, while for nonlinear systems, near-identity transformations are discussed in detail. For nonlinear oscillators, the classical methods of Poincaré and Birkhoff are covered, alongside more recent approaches to normal form transformations. Other important topics such as nonlinear resonance, bifurcations, frequency detuning and the inclusion of damping are demonstrated using examples. Furthermore, the connection between normal form transformations and Lie series is described for both first and second-order differential equations. The use of normal form transformations to compute backbone curves is described along with an explanation of the relationship to nonlinear normal modes. Lastly, conclusions and possible future directions for research are given.

One of the requirements of the population-based approach to Structural Health Monitoring (SHM) proposed in the earlier papers in this sequence, is that structures be represented by points in an abstract space. Furthermore, these spaces should be metric spaces in a loose sense; i.e. there should be some measure of distance applicable to pairs of points; similar structures should then be ‘close’ in the metric. However, this geometrical construction is not enough for the framing of problems in data-based SHM, as it leaves undefined the notion of feature spaces. Interpreting the feature values on a structure-by-structure basis as a type of field over the space of structures, it seems sensible to borrow an idea from modern theoretical physics, and define feature assignments as sections in a vector bundle over the structure space. With this idea in place, one can interpret the effect of environmental and operational variations as gauge degrees of freedom, as in modern gauge field theories. One can then regard data normalisation procedures like cointegration as gauge-fixing operations. This paper will discuss the various geometrical structures required for an abstract theory of feature spaces in SHM, and will draw analogies with how these structures have shown their power in modern physics.\ud \ud \ud \ud Having motivated a number of problems in Population-Based SHM (PBSHM) in geometrical terms, it remains to show how these problems might be solved. In the second part of the paper, the problem of determining the normal condition cross section of a feature bundle is addressed. The solution is provided by the application of Graph Neural Networks (GNN), a versatile non-Euclidean machine learning algorithm which is not restricted to inputs and outputs from vector spaces. In particular, the algorithm is well suited to operating directly on the sort of graph structures which are an important part of the proposed framework for PBSHM. The solution of the normal section problem is demonstrated for a heterogeneous population of truss structures for which the feature of interest is the first natural frequency. The GNN approach is shown to not only solve the normal section problem, but also to accommodate varying temperatures across the population; it thus provides a means of data normalisation.

In this paper, a review of the nonlinear aspects of the mechanical inerter will be presented. The historical context goes back to the development of isolators and absorbers in the first half of the twentieth century. Both mechanical and fluid-based nonlinear inerter devices were developed in the mid- and late twentieth century. However, interest in the inerter really accelerated in the early 2000s following the work of Smith [87], who coined the term ‘inerter’ in the context of a force–current analogy between electrical and mechanical networks. Following the historical context, both fluid and mechanical inerter devices will be reviewed. Then, the application of nonlinear inerter-based isolators and absorbers is discussed. These include different types of nonlinear energy sinks, nonlinear inerter isolators and geometrically nonlinear inerter devices, many relying on concepts such as quasi-zero-stiffness springs. Finally, rocking structures with inerters attached are considered, before conclusions and some future directions for research are presented.

This paper explores the influence of linear hysteretic damping on the performance of passive tunedinerter devices. An inerter is a device that produces a force proportional to the relative acceleration across its two terminals; devices incorporating inerters have received widespread attention in the earthquake engineering community, because they offer the ability to improve the seismic response of structures. However, the majority of this research has assumed that the damping components within the tuned-inerter device exhibit viscous, rather than hysteretic, damping. This restriction imposes an essential question on how the hysteretic damping model will change the performance of the device compared to the viscous damping model. It is shown that the response of viscous\ud and hysteretic inerter systems have significant differences in displacement amplitude due to the frequency-dependency of the damping. Therefore, a new formulation for obtaining the optimum loss factor of the hysteretic damping in the inerter system is proposed. Next, the challenges associated with accurately predicting the time-response of a hysteretically damped system are discussed. A numerical time-integration method is extended to address these challenges, using\ud a new formulation that has the benefit of being broadly applicable to multi-degree-of-freedom hysteretic linear systems and nonstationary random signals. The results show that the earthquake responses from the hysteretic damping model can differ significantly from the ones obtained via the viscous model.

In this paper, experimental data from tests of a helical fluid inerter are used to model the observed hysteretic behavior. The novel idea is to test the feasibility of employing mem-models, which are time-invariant herein, to capture the observed phenomena by using physically meaningful state variables. Firstly we use a Masing model concept, identified with a multilayer feedforward neural network to capture the physical characteristics of the hysteresis functions. Following this, a more refined problem formulation based on the concept of a multi-element model including a mem-inerter is developed. This is compared with previous definitions in the literature and shown to be a more general model. Through-out this paper, numerical simulations are used to demonstrate the type of dynamic responses anticipated using the proposed time-invariant mem-models. Corresponding experimental measurements are processed to demonstrate and justify the new mem-modeling concepts. Focusing on identifying the unknown function forms in the proposed problem formulations, the results show that it is possible to formulate a unified model constructed using both the damper and inerter from the mem-model family. This model captures many of the more subtle features of the underlying physics, not captured by other forms of existing model.

This study proposes a novel approach to increase the chatter stability in machining operations. It shows the potential performance improvement when an inerter-based dynamic vibration absorber is employed in a machining operation. Tuned inerter based devices have been employed to decrease the magnitude of the vibrations in applications such as civil engineering structures and vehicle suspension systems but the nature of chatter in machining is different from these applications. Therefore, it requires a different tuning methodology to obtain the optimal design parameters. In this study, the machining operation is modelled as an undamped single degree of freedom system and different configurations of an inerter, a damper and two springs are used to ensure a stable region of operation. Strategies for the tuning parameters are developed both analytically and numerically. Using these techniques the performance improvement in the chatter stability provided by using inerter based devices instead of a traditional dynamic vibration absorber is demonstrated.

The use of viscous damping terms to simplify the damping of a vibrating system has been well established for decades. For solid materials whose energy dissipated per cycle is frequency-independent, an equivalent viscous damping has often been used. However, this may give inaccurate results, especially at higher excitation frequencies. Alternatively, a complex stiffness term can be used. In this case, a challenge arises for the time domain analysis due to the unstable poles in the resulting model. Several methods have been proposed to deal with this issue. The use of an analytic signal along with Hilbert transform and a time reversal technique is one of the first introduced methods. In this paper, we extend the method so that it can be used for solving the system equations of motion using the numerical integration algorithm solvers that are available in MATLAB. We also present the application of this extended method to simulate a multi-degree-of-freedom (MDOF) structure with supplemental passive vibration suppression systems using linear hysteretic damping in the time domain.

The current work introduces a novel combination of two Bayesian tools, Gaussian\ud Processes (GPs), and the use of the Approximate Bayesian Computation (ABC) algorithm\ud for kernel selection and parameter estimation for machine learning applications. The\ud combined methodology that this research article proposes and investigates offers the\ud possibility to use different metrics and summary statistics of the kernels used for Bayesian\ud regression. The presented work moves a step toward online, robust, consistent, and\ud automated mechanism to formulate optimal kernels (or even mean functions) and their\ud hyperparameters simultaneously offering confidence evaluation when these tools are\ud used for mathematical or engineering problems such as structural health monitoring\ud (SHM) and system identification (SI).

A short report is provided on the 6th European Conference on Structural Control which took place in\ud Sheffield from 11–13 July 2016

In this paper the (Formula presented.) nonlinear modal interactions that occur in a nonlinear three-degree-of-freedom lumped mass system, where (Formula presented.), are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. (Formula presented.)) is considered. However, due to the symmetries of the system under consideration, only (Formula presented.) modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in the frequency domain. The forced responses are computed using the continuation software AUTO-07p. A comparison of the results gives insights into the multi-modal interactions and shows how the frequency response of the system is related to those branches of the backbone curves that represent such interactions.

A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.

Resonance is an important phenomenon in vibrating systems and, in systems of nonlinear coupled oscillators, resonant interactions can occur between constituent parts of the system. In this paper, out-of-unison resonance is defined as a solution in which components of the response are 90° out-of-phase, in contrast to the in-unison responses that are normally considered. A well-known physical example of this is whirling, which can occur in a taut cable. Here, we use a normal form technique to obtain time-independent functions known as backbone curves. Considering a model of a cable, this approach is used to identify out-of-unison resonance and it is demonstrated that this corresponds to whirling. We then show how out-of-unison resonance can occur in other two degree-of-freedom nonlinear oscillators. Specifically, an in-line oscillator consisting of two masses connected by nonlinear springs—a type of system where out-of-unison resonance has not previously been identified—is shown to have specific parameter regions where out-of-unison resonance can occur. Finally, we demonstrate how the backbone curve analysis can be used to predict the responses of forced systems.

In this paper, we derive a frequency detuning method for multi-degree-of-freedom oscillators with nonlinear stiffness. This approach includes a matrix of detuning parameters, which are used to model the amplitude dependent variation in resonant frequencies for the system. As a result, we compare three different approximations for modeling the affect of the nonlinear stiffness on the linearized frequency of the system. In each case, the response of the primary resonances can be captured with the same level of accuracy. However, harmonic and subharmonic responses away from the primary response are captured with significant differences in accuracy. The detuning analysis is carried out using a normal form technique, and the analytical results are compared with numerical simulations of the response. Two examples are considered, the second of which is a two degree-of-freedom oscillator with cubic stiffnesses.

The problem of an Euler-Bernoulli cantilever beam whose free end impacts with a point constraint is revisited from the point of view of modal analysis. It is shown that there is non-uniqueness of consistent impact laws for a given modal truncation. Moreover, taking an N-mode compliant, bilinear formulation and passing to the rigid limit leads to a sequence of impact models that does not converge as N--> ∞. The dynamics of such truncated models are studied numerically and found to give rise to quite different dynamics depending on the number of degrees of freedom taken. The simulations are compared with results from simple experiments that show a propensity for multiple-tap dynamics, in which higher-order modes lead to rapidly cycling intermittent contact. The conclusion reached is that, to derive an accurate model, one needs to avoid the impact limit altogether, and take sufficiently many modes in the formulation to match the actual stiffness of the constraining stop.

In this article, a hybrid control framework is used to design semi-active controllers for vibration reduction. It is shown that the semi-active skyhook damper, typically used for vibration reduction, can be recast in the framework of an event-driven intermittent controller. By doing this, we can then exploit the well-developed techniques associated with hybrid control theory to design the semi-active control system. Illustrative simulation examples are based on a 2 degree-of-freedom system, often used to model the dynamics of a quarter car body model. The simulation results demonstrate how hybrid control design techniques can improve the overall performance of the semi-active control system.

Measurement of the vibrations of plates can offer significant challenges to the experimentalist, particularly when the plates are lightweight, exhibit large amplitude deflections, nonlinear responses or are initially curved. The use of accelerometers adds masses which can change the dynamics of lightweight plates. Large amplitude oscillations and initial curvatures cause complications when using a laser vibrometer, as they make it difficult to get consistent reflections back to the receiver. Furthermore, large or nonlinear oscillations challenge inherent assumptions on which the vibrometer’s algorithms depend. A high speed video camera avoids these issues, but makes it hard to extract numerical data. This paper describes a method that extends the capabilities of a high speed video camera by using a mirror, allowing post-processing software to stereoscopically resolve an array of points on the plate surface to 3D coordinates, capturing the complete shape and position of the plate throughout vibration. This method avoids all the problems mentioned above and gives very clear insight into plate vibration. Some example results of this method are presented, using thermally bistable carbon laminate plates filmed at a 1000 frames per second. These plates pose the challenges described, and also exhibit an unusual oscillatory motion where the plates ‘snap’ between two statically stable states. The method is shown to provide clear insight into the rich dynamics of these plates.

Quasi-active damping is a method of coupled mechanical and control system design using multiple semi-active dampers. By designing the systems such that the desired control force may always be achieved using a combination of the dampers, quasi-active damping seeks to approach levels of vibration isolation achievable through active damping, whilst retaining the desirable attributes of semi-active systems. In this article a design is proposed for a quasi-active, base-isolating suspension system. Control laws are firstly defined in a generalised form, where semi-active dampers are considered as idealised variable viscous dampers. This system is used to demonstrate in detail the principles of quasi-active damping, in particular the necessary interaction between mechanical and control systems. It is shown how such a system can produce a tunable, quasi-active region in the frequency response of very low displacement transmissibility. Quasi-active control laws are then proposed which are specific for use with magnetorheological dampers. These are validated in simulation using a realistic model of the damper dynamics, again producing a quasi-active region in the frequency response. Finally, the robustness of the magnetorheological, quasi-active suspension system is demonstrated. (C) 2010 Elsevier Ltd. All rights reserved.

In this paper we examine the use of dynamic friction within a bolted structure to improve damping properties of the structure. The structure considered for this paper consists of two steel beam-columns bolted together allowing dynamic friction to occur at the interface. This paper presents an analysis of the behaviour of the structure and the effect of friction on its dynamics. It also presents an analysis of the energy dissipation in the structure by means of friction and the optimization of the bolt tension in order to dissipate the maximum vibration energy. We define analytical expressions for the vibration behaviour before and after slip occurs as well as the condition at which the slip-stick transition occurs. An experiment, in which the measurements of the bolt tension, the slip within the structure and the bending velocity are made, is used to validate the model. The theoretical analysis gave very close agreement with the experimental results and the effective damping of the structure was increased by a factor of approximately 10 through the use of dynamic friction. (C) 2010 Elsevier Ltd. All rights reserved.

In this articles the authors present a review of non-linear structural control techniques. This is an area of growing importance in a range of engineering applications, where non-linear behaviour is encountered. Structural control is usually divided into three main areas: (a) passive (b) semi-active, and (c) active control. This article follows this convention, and highlights in each section the relevant state of the art for non-linear systems, with additional references to related linear approaches.

This paper presents a study of how different vibration modes contribute to the dynamics of an inclined cable that is parametrically excited close to a 2 : 1 internal resonance. The behaviour of inclined cables is important for the design and analysis of cable-stay bridges. In this work the cable vibrations are modelled by a four-mode model. This type of model has been used previously to study the onset of cable sway motion caused by internal resonances which occur due to the nonlinear modal coupling terms. A bifurcation study is carried out with numerical continuation techniques applied to the scaled and averaged modal equations. As part of this analysis, the amplitudes of the cable vibration response to support inputs is computed. These theoretical results are compared with experimental measurements taken from a 5.4 m long inclined cable with a vertical support input at the lower end. In general this comparison shows a very high level of agreement.

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first- and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

Causality, in the bond graph sense, is shown to provide a conceptual framework for the design of real-time dynamic substructure testing experiments. In particular, known stability problems with split-inertia substructured systems are reinterpreted as causality issues within the new conceptual framework.\ud \ud As an example, causality analysis is used to provide a practical solution to a split-inertia substructuring problem and the solution is experimentally verified.