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Abstract Approximate linearised models can be important for preliminary design of floating wind turbines, but their value depends on how well they approximate the real-world non-linear behaviour. This paper focuses on the non-linear inertial coupling between motion of the floating platform and the blade dynamics, using a simplified model to demonstrate how the inertial coupling works, and systematically studying the linearity of the dynamic blade response to different directions, amplitudes and frequencies of motion. Simplified equations of motion are derived and approximately solved analytically, showing that the blade response contains harmonics at a range of frequencies, some linear and some non-linear in the amplitude of the platform motion. Comparison to numerical simulations shows that the analytical results were qualitatively useful but inaccurate for large platform motions. Because of the multiple harmonics in the response, there are more combinations of rotor speeds and platform motions leading to large resonant blade responses and non-linear behaviour than might be expected. Overall, for realistically low rotor speeds and platform frequencies (below 20 rpm and 0.2 Hz), non-linear inertial loading due to platform motion should be negligible. The implications of this work for the use of linearised structural models and the relevance of scale model testing are discussed. Highlights • Blade response has harmonics at combinations of rotor speed and platform frequency. • Some response harmonics are linear, some non-linear, in amplitude of platform motion. • Multiple harmonics could lead to unexpected non-linear & resonant blade responses. • Structural non-linearity due to platform motion should not be an issue in practice. [ABSTRACT FROM AUTHOR]

Statistical energy analysis (SEA) is a prominent method for predicting the high frequency response of complex structures under steady loading where the structure is split into subsystems and the subsystem energies are calculated. Since at high frequencies, the dynamic response of nominally identical structures can differ greatly, methods have been developed to predict both the mean and variance of the energy in the subsystems of a system across an ensemble of systems. SEA can be extended to predict the transient response of a system, either to shock or time-varying inputs and is known as transient SEA, although this formulation has so far only been interested in the mean response. In this paper, a method for predicting the variance of the transient response is derived by considering how an individual realisation can deviate from the mean. A matrix differential equation for the covariance of the subsystem energies is derived which is driven by terms representing the variability in the system. These variance terms are provided by assuming that the natural frequencies in each subsystem conform to the Gaussian orthogonal ensemble. The accuracy of the method is investigated both numerically and experimentally using systems involving coupled plates and its limitations are discussed. [ABSTRACT FROM AUTHOR]

The hybrid deterministic-statistical energy analysis method has proven to be a versatile framework for modeling built-up vibro-acoustic systems. The stiff system components are modeled deterministically, e.g., using the finite element method, while the wave fields in the flexible components are modeled as diffuse. In the present paper, the hybrid method is extended such that not only the ensemble mean and variance of the harmonic system response can be computed, but also of the band-averaged system response. This variance represents the uncertainty that is due to the assumption of a diffuse field in the flexible components of the hybrid system. The developments start with a cross-frequency generalization of the reciprocity relationship between the total energy in a diffuse field and the cross spectrum of the blocked reverberant loading at the boundaries of that field. By making extensive use of this generalization in a first-order perturbation analysis, explicit expressions are derived for the cross-frequency and band-averaged variance of the vibrational energies in the diffuse components and for the cross-frequency and band-averaged variance of the cross spectrum of the vibro-acoustic field response of the deterministic components. These expressions are extensively validated against detailed Monte Carlo analyses of coupled plate systems in which diffuse fields are simulated by randomly distributing small point masses across the flexible components, and good agreement is found. [ABSTRACT FROM AUTHOR]

Abstract When information about a distribution consists of statistical moments only, a self-consistent approach to deriving a subjective probability density function (pdf) is Maximum Entropy. Nonetheless, the available information may have uncertainty, and statistical moments maybe known only to lie in a certain domain. If Maximum Entropy is used to find the distribution with the largest entropy whose statistical moments lie within the domain, the information at only a single point in the domain would be used and other information would be discarded. In this paper, the bounded information on statistical moments is used to construct a family of Maximum Entropy distributions, leading to an uncertain probability function. This uncertainty description enables the investigation of how the uncertainty in the probabilistic assignment affects the predicted performance of an engineering system with respect to safety, quality and design constraints. It is shown that the pdf which maximizes (or equivalently minimizes) an engineering metric is potentially different from the pdf which maximizes the entropy. The feasibility of the proposed uncertainty model is shown through its application to: (i) fatigue failure analysis of a structural joint; (ii) evaluation of the probability that a response variable of an engineering system exceeds a critical level, and (iii) random vibration. [ABSTRACT FROM AUTHOR]

This work is concerned with the statistical properties of the frequency response function of the energy of a random system. Earlier studies have considered the statistical distribution of the function at a single frequency, or alternatively the statistics of a band-average of the function. In contrast the present analysis considers the statistical fluctuations over a frequency band, and results are obtained for the mean rate at which the function crosses a specified level (or equivalently, the average number of times the level is crossed within the band). Results are also obtained for the probability of crossing a specified level at least once, the mean rate of occurrence of peaks, and the mean trough-to-peak height. The analysis is based on the assumption that the natural frequencies and mode shapes of the system have statistical properties that are governed by the Gaussian Orthogonal Ensemble (GOE), and the validity of this assumption is demonstrated by comparison with numerical simulations for a random plate. The work has application to the assessment of the performance of dynamic systems that are sensitive to random imperfections. [ABSTRACT FROM AUTHOR]

If the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the Gaussian Orthogonal Ensemble (GOE) of random matrices. Previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation. The derivation of these expressions employed random point process theory, and in the theoretical development it was assumed that the modal generalised forces were uncorrelated. Although this assumption is often valid, there are cases in which correlations between the generalised forces can significantly affect the response variance, and in the present work the existing theory is extended to include correlations of this type. The extended theory is applicable to both single frequency responses and to band average responses, and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure. [ABSTRACT FROM AUTHOR]

• Non-intrusive sensitivity toolbox for digital twin to support design decisions. • Efficient probabilistic gradient estimation using likelihood ratio method. • Metrics independent of design requirements using entropy and Fisher information. • Consistent identification of high value data throughout the design process. Successful application of digital twins in the design process requires a tailored approach to identify high value information from the uncertain data. We propose a non-intrusive sensitivity metric toolbox that integrates black-box digital twins in the design and decision process under uncertainties. The toolbox captures the evolving nature of the key design performance indicators (KPI) and provide both KPI-free and KPI-based metrics. The KPI-free metrics, which are based on entropy and Fisher information but independent of design KPIs, is shown to give good indication of the most influential data for KPI-based metrics. This suggests a consistent identification of high value data throughout the design process. [ABSTRACT FROM AUTHOR]

A hybrid Finite Element-Statistical Energy Analysis (FE-SEA) method has been developed and improved over the past fifteen years to analyse complex vibro-acoustic built-up systems, comprised by both deterministic and statistical components, and the FE-SEA equations have been derived considering that the power input to the structure is due to prescribed forces applied to the degrees of freedom of either the deterministic and/or the statistical components. In this paper, it is presented a generalised set of FE-SEA equations to estimate the response of a system when prescribed displacements contribute to the power input to the complex system. It was found that the equations are not necessarily reversible, meaning that if a set of displacements estimated from the hybrid approach with prescribed forces are used as input to the same randomised structure, the response will be altered due to its statistical nature. The generalised set of FE-SEA equations is validated against numerical data of FE Monte Carlo simulations of two complex built-up structures, where the input is given to the statistical system and to a deterministic resonant structure, respectively. • Hybrid FE-SEA equations derived for prescribed displacements. • Input forces differ to the reacting forces when the same displacements are imposed. • Computational time significantly reduced in vibro-acoustic analysis. [ABSTRACT FROM AUTHOR]

Hybrid techniques, such as a Finite Element-Statistical Energy Analysis (FE-SEA), have been widely employed to model vibro-acoustic systems in order to estimate the ensemble averaged response due to external inputs, assuming that the governing equations of motion of the comprising structures and connections are linear. However, nonlinear behaviour can be found in the interfaces between the point where the input is applied and the system itself, resulting in the force being transmitted through a nonlinear path, that might affect the structural or acoustic response of the system depending on the amplitude of the input. With the aim of improving the estimation of the mean squared response of the system subjected to random inputs through a nonlinear interface, an equivalent linearisation approach has been adopted to linearise the equations of an existing FE-SEA method that supports prescribed displacements as inputs rather than forces. Results from numerical simulations of a nonlinear single degree of freedom system demonstrate the ability of the equivalent linearisation approach to estimate the mean square response with a minimum error, even for systems with strong nonlinearities. The method here developed has been validated against experimental data collected from a vibro-acoustic system excited through a nonlinear massless and undamped spring. • Equivalent linearisation applied to a hybrid FE-SEA. • Structural and acoustic mean square responses are accurately estimated • Computational time significantly reduced in nonlinear vibro-acoustic analysis. [ABSTRACT FROM AUTHOR]

Abstract: When considering the sound transmission through a wall in between two rooms, in an important part of the audio frequency range, the local response of the rooms is highly sensitive to uncertainty in spatial variations in geometry, material properties and boundary conditions, which have a wave scattering effect, while the local response of the wall is rather insensitive to such uncertainty. For this mid-frequency range, a computationally efficient modeling strategy is adopted that accounts for this uncertainty. The partitioning wall is modeled deterministically, e.g. with finite elements. The rooms are modeled in a very efficient, nonparametric stochastic way, as in statistical energy analysis. All components are coupled by means of a rigorous power balance. This hybrid strategy is extended so that the mean and variance of the sound transmission loss can be computed as well as the transition frequency that loosely marks the boundary between low- and high-frequency behavior of a vibro-acoustic component. The method is first validated in a simulation study, and then applied for predicting the airborne sound insulation of a series of partition walls of increasing complexity: a thin plastic plate, a wall consisting of gypsum blocks, a thicker masonry wall and a double glazing. It is found that the uncertainty caused by random scattering is important except at very high frequencies, where the modal overlap of the rooms is very high. The results are compared with laboratory measurements, and both are found to agree within the prediction uncertainty in the considered frequency range. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the development of efficient algorithms for propagating parametric uncertainty within the context of the hybrid Finite Element/Statistical Energy Analysis (FE/SEA) approach to the analysis of complex vibro-acoustic systems. This approach models the system as a combination of SEA subsystems and FE components; it is assumed that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. The method has been recently generalised by allowing the FE components to possess parametric uncertainty, leading to two ensembles of uncertainty: a non-parametric one (SEA subsystems) and a parametric one (FE components). The SEA subsystems ensemble is dealt with analytically, while the effect of the additional FE components ensemble can be dealt with by Monte Carlo Simulations. However, this approach can be computationally intensive when applied to complex engineering systems having many uncertain parameters. Two different strategies are proposed: (i) the combination of the hybrid FE/SEA method with the First Order Reliability Method which allows the probability of the non-parametric ensemble average of a response variable exceeding a barrier to be calculated and (ii) the combination of the hybrid FE/SEA method with Laplace's method which allows the evaluation of the probability of a response variable exceeding a limit value. The proposed approaches are illustrated using two built-up plate systems with uncertain properties and the results are validated against direct integration, Monte Carlo simulations of the FE and of the hybrid FE/SEA models. [Copyright &y& Elsevier]

Abstract: A method is presented to predict the transient response of a structure at the driving point following an impact or a shock loading. The displacement and the contact force are calculated solving the discrete convolution between the impulse response and the contact force itself, expressed in terms of a nonlinear Hertzian contact stiffness. Application of random point process theory allows the calculation of the impulse response function from knowledge of the modal density and the geometric characteristics of the structure only. The theory is applied to a wide range of structures and results are experimentally verified for the case of a rigid object hitting a beam, a plate, a thin and a thick cylinder and for the impact between two cylinders. The modal density of the flexural modes for a thick slender cylinder is derived analytically. Good agreement is found between experimental, simulated and published results, showing the reliability of the method for a wide range of situations including impacts and pyroshock applications. [Copyright &y& Elsevier]

Two existing methods of probabilistic inference are based on variational principles: maximum entropy and minimum Fisher information. In each case, a probability density function is inferred by setting the first variation of a functional to zero, subject to information constraints. This study considers whether other functionals could be used for this purpose, and by starting with requirements for self-consistency and invariance, it is shown that the most general admissible functional is just a linear combination of entropy and Fisher information, with the proviso that the normal definition of Fisher information is modified by the inclusion of a prior. This amounts to an axiomatic derivation of entropy and Fisher information. The concern is with continuous random variables and both the single- and multivariable cases are considered. A number of examples are considered to compare inference based on entropy with that based on Fisher information, and to highlight the role of boundary conditions for inference based on Fisher information. [ABSTRACT FROM AUTHOR]

Abstract: An existing hybrid finite element (FE)/statistical energy analysis (SEA) approach to the analysis of the mid- and high frequency vibrations of a complex built-up system is extended here to a wider class of uncertainty modeling. In the original approach, the constituent parts of the system are considered to be either deterministic, and modeled using FE, or highly random, and modeled using SEA. A non-parametric model of randomness is employed in the SEA components, based on diffuse wave theory and the Gaussian Orthogonal Ensemble (GOE), and this enables the mean and variance of second order quantities such as vibrational energy and response cross-spectra to be predicted. In the present work the assumption that the FE components are deterministic is relaxed by the introduction of a parametric model of uncertainty in these components. The parametric uncertainty may be modeled either probabilistically, or by using a non-probabilistic approach such as interval analysis, and it is shown how these descriptions can be combined with the non-parametric uncertainty in the SEA subsystems to yield an overall assessment of the performance of the system. The method is illustrated by application to an example built-up plate system which has random properties, and benchmark comparisons are made with full Monte Carlo simulations. [Copyright &y& Elsevier]

Abstract: This paper considers the estimation of statistics of displacement of a vibrating rectangular plate with random wave scatterers. The influence of uncertainty is investigated using point impedance theory. Coherent boundary effects are seen, which decrease when the number of scatterers increases. The boundary effect is investigated using images and the first side and corner reflections are found to be a minimum requirement to estimate the spatial correlation. Statistics for point driven response are investigated under the assumption that the statistics of the natural frequencies follow those of the Gaussian Orthogonal Ensemble (GOE). The estimates are compared with Monte Carlo simulation results, and they show good agreement. [Copyright &y& Elsevier]

Abstract: This work presents active control of high-frequency vibration using skyhook dampers. The choice of the damper gain and its optimal location is crucial for the effective implementation of active vibration control. In vibration control, certain sensor/actuator locations are preferable for reducing structural vibration while using minimum control effort. In order to perform optimisation on a general built-up structure to control vibration, it is necessary to have a good modelling technique to predict the performance of the controller. The present work exploits the hybrid modelling approach, which combines the finite element method (FEM) and statistical energy analysis (SEA) to provide efficient response predictions at medium to high frequencies. The hybrid method is implemented here for a general network of plates, coupled via springs, to allow study of a variety of generic control design problems. By combining the hybrid method with numerical optimisation using a genetic algorithm, optimal skyhook damper gains and locations are obtained. The optimal controller gain and location found from the hybrid method are compared with results from a deterministic modelling method. Good agreement between the results is observed, whereas results from the hybrid method are found in a significantly reduced amount of time. [Copyright &y& Elsevier]

The vibro-acoustic response of built-up structures, consisting of stiff components with low modal density and flexible components with high modal density, is sensitive to small imperfections in the flexible components. In this paper, the uncertainty of the response is considered by modeling the low modal density master system as deterministic and the high modal density subsystems in a nonparametric stochastic way, i.e., carrying a diffuse wave field, and by subsequently computing the response probability density function. The master system's mean squared response amplitude follows a singular noncentral complex Wishart distribution conditional on the subsystem energies. For a single degree of freedom, this is equivalent to a chi-square or an exponential distribution, depending on the loading conditions. The subsystem energies follow approximately a chi-square distribution when their relative variance is smaller than unity. The results are validated by application to plate structures, and good agreement with Monte Carlo simulations is found. [ABSTRACT FROM AUTHOR]

Large sections of many types of engineering construction can be considered to constitute a two-dimensional periodic structure, with examples ranging from an orthogonally stiffened shell to a honeycomb sandwich panel. In this paper, a method is presented for computing the boundary (or edge) impedance of a semi-infinite two-dimensional periodic structure, a quantity which is referred to as the direct field boundary impedance matrix. This terminology arises from the fact that none of the waves generated at the boundary (the direct field) are reflected back to the boundary in a semi-infinite system. The direct field impedance matrix can be used to calculate elastic wave transmission coefficients, and also to calculate the coupling loss factors (CLFs), which are required by the statistical energy analysis (SEA) approach to predicting high frequency vibration levels in built-up systems. The calculation of the relevant CLFs enables a two-dimensional periodic region of a structure to be modeled very efficiently as a single subsystem within SEA, and also within related methods, such as a recently developed hybrid approach, which couples the finite element method with SEA. The analysis is illustrated by various numerical examples involving stiffened plate structures. [ABSTRACT FROM AUTHOR]

Abstract: Recently a hybrid analysis method has been developed in which the various components of a complex vibro-acoustic system can be modeled either deterministically or statistically. The coupling between these two types of component is effected by using a diffuse field reciprocity relation, which relates the cross-spectrum of the forces at the boundary of a statistical component to the vibrational energy level of the component. In practical applications, components may be coupled over a domain, rather than just at the component boundaries—for example, the coupling between a plate and an acoustic volume involves the surface of the plate. It is shown in the present work that the reciprocity relation and the hybrid method can be extended to this situation, so that, for example, it is possible to couple a statistical model of a plate to a finite element model of an acoustical cavity, or to a statistical acoustical cavity. The coupling of statistical acoustical and structural components is not new, but rather is done routinely within statistical energy analysis (SEA); this part of the present work demonstrates that the relevant coupling loss factors can be found by using the hybrid method. The analysis is illustrated by various numerical examples. [Copyright &y& Elsevier]

In the statistical energy analysis (SEA) of high frequency noise and vibration, a complex engineering structure is represented as an assembly of subsystems. The response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem, and these energies are found by employing the principle of power balance. Strictly the computed energy is an average taken over an ensemble of random structures, and for many years there has been interest in extending the SEA prediction to the variance of the energy. A variance prediction method for a general built-up structure is presented here. Closed form expressions for the variance are obtained in terms of the standard SEA parameters and an additional set of parameters αk that describe the nature of the power input to each subsystem k, and αks that describe the nature of the coupling between subsystems k and s. The theory is validated by comparison with Monte Carlo simulations of plate networks and structural-acoustic systems [ABSTRACT FROM AUTHOR]

The global dynamic response of a deep water floating production system needs to be predicted with coupled analysis methods to ensure accuracy and reliability. Two types of coupling can be identified: one is between the floating platform and the mooring lines/risers, while the other is between the mean offset, the wave frequency, and the low frequency motions of the system. At present, it is unfeasible to employ fully coupled time domain analysis on a routine basis due to the prohibitive computational time. This has spurred the development of more efficient methods, including frequency domain approaches. A good understanding of the intricate coupling mechanisms is paramount for making appropriate approximations in an efficient method. To this end, a simplified two degree-of-freedom system representing the surge motion of a vessel and the fundamental vibration mode of the lines is studied for physical insight. Within this framework, the frequency domain equations are rigorously formulated, and the nonlinearities in the restoring forces and drag are statistically linearized. The model allows key coupling effects to be understood; among other things, the equations demonstrate how the wave frequency dynamics of the mooring lines are coupled to the low frequency motions of the vessel. Subsequently, the effects of making certain simplifications are investigated through a series of frequency domain analyses, and comparisons are made to simulations in the time domain. The work highlights the effect of some common approximations, and recommendations are made regarding the development of efficient modeling techniques. [ABSTRACT FROM AUTHOR]

Sensitivity analysis identifies the influential parameters and it is a key step in the design process. Focusing on reliability based design, the probability of a failure mode and its sensitivity to input uncertainties are obtained in a single Monte Carlo simulation using the Likelihood Ratio method. For correlated multiple failure modes, a sensitivity matrix is proposed. The dominant singular vectors of the sensitivity matrix are found to indicate the most important parameters for combined sensitivity of multiple failure modes. Therefore, the sensitivity matrix can guide resources allocation so that most uncertainties can be reduced for system reliabilities. • Sensitivity of system reliability with unknown failure configurations. • Efficient simulation for the most influential design parameters. • Failure probability and its first derivative in a single simulation run. [ABSTRACT FROM AUTHOR]

Abstract: Sound radiation into open space from a vibrating structure has been investigated since Rayleigh. On the other hand the sound power transferring into neighboring reverberant subsystems has also been rigorously studied using statistical energy analysis, particularly for the high frequency range. Falling between the two well-known problems, pressure and intensity fields from the sound radiation have not yet been widely studied using statistical methods. In this paper, the sound radiation from a vibrating thin plate having uncertain dynamic properties is investigated. Estimates are developed for the reverberant vibration field in the uncertain plate subjected to a point-excitation, and for the ensemble average of pressure from the direct field and from the reverberant field, leading to an estimate of the average sound intensity. The power radiated from the plate and the radiation efficiency is also derived. Monte Carlo simulations are conducted with an ensemble of plates with randomly-distributed point masses, and the simulation results compare well with the estimates. [Copyright &y& Elsevier]

Vibro-acoustic analysis of complex systems at higher frequencies faces two challenges: how to compute the response without using an excessive number of degrees of freedom (DOFs), and how to quantify the uncertainty of the response due to small spatial variations in geometry, material properties, and boundary conditions, which have a wave scattering effect? In this study, a general method of analysis is presented that provides an answer to both questions while overcoming most limitations of statistical energy analysis. The fundamental idea is to numerically compute an artificial ensemble of realizations for the components of the built-up system that are highly sensitive to small random wave scatterers. This can be efficiently performed because their eigenvalue spacings and mode shapes conform to Gaussian orthogonal ensemble spacings and Gaussian random fields, respectively. The DOFs of the overall system are therefore limited to those of the deterministic components and the interface DOFs of the random components. The method is extensively validated by application to plate structures. Good agreement between the predicted response probability distributions and the results of detailed parametric probabilistic models is obtained, also for cases of low modal overlap, single point loading, and strong subsystem coupling. [ABSTRACT FROM AUTHOR]

Abstract: Acoustic radiation from a structure can be expressed in terms of “modal radiation” and “modal coefficients”. This paper investigates the contributions of these two modal properties to radiation excited by a point force. Sound radiation from two basic structures is considered: a baffled rectangular plate and a closed spherical shell. The plate behaviour is familiar, and governed by the relation between the natural frequency of a mode and its coincidence frequency. For a closed spherical shell, there are either zero or two “critical frequencies”, depending on the radius and thickness. When there are two the shell radiates well both above and below the two frequencies, and poorly in the frequency range between them. [Copyright &y& Elsevier]

Focuses on a study concerning a method based on reduced basis approximation concepts for improved first-order approximation of eigenvalues and eigenvectors of modified structural dynamic systems. Problem statement and conventional methods; Improved first-order approximation; Demonstration examples.