In nonlinear dynamical systems, internal resonance is a well known phenomenon. Natural frequencies may change with vibration amplitude, becoming commensurable or close at certain energy levels. Nonlinear modes then interact based on their participating oscillators. Here, we investigate effects on the response curve of a system as neighboring oscillators are modified. This provides new insights into the relation of internal resonance and the frequency response of coupled nonlinear systems.We use methods of Harmonic Balance and a continuation algorithm to compute nonlinear frequency response curves and nonlinear normal modes for a model system. It is arranged as a chain of Duffing oscillators with separated natural frequencies. We demonstrate a kinking effect in the response curve once internal resonance emerges in the system's forced response. The kink is observed as a shift to higher response levels, stretching the response curve.We also show how this kinking effect is modified by adapting the nonlinearities of neighboring oscillators. The system's response branches kink vertically as an oscillator in the chain is made linear, stacking on top of each other. Multiple stable solutions at a certain frequency may then be observed. We show that these solutions correspond to varying degrees of vibration localization in the system. We expect our results to be valuable in system design. It should be possible to exploit the observed kinking effect, shaping the response curve by adding suitable oscillators of various characteristics. The engineer may thus be able to reach a target system response level at a certain frequency.

Detecting structural defects is one of the primary challenges engineers face. Consequently, the development of techniques and methods capable of detecting structural defects has always been critical. It should be emphasized that crack detection is only meaningful if it occurs before the final stages of structural failure. Accordingly, the early identification of structural defects has become a significant research challenge, motivating the development of techniques and diagnostic parameters that can effectively capture and reflect the structure’s nonlinearity or non-uniform behavior. This study aims to provide a more detailed examination of modulation phenomena observed in the measured response using the vibro-acoustic modulation (VAM) method, and propose a new model that simultaneously incorporates all three conventional modulation types (amplitude, frequency, and phase), which may offer a more accurate representation of the response signal behavior. Both theoretical and experimental results clearly confirm that the phase shifts of individual frequency components in the frequency domain vary throughout the lifetime of the tested specimen. This behavior, as anticipated by the proposed model, reveals a strong correlation between phase shifts and modulation indices (MIs). Furthermore, the relative sensitivity analysis indicates that the phase shift is more sensitive than the modulation index (MI), suggesting its strong potential as an indicator for early defect detection in structural components.

The accurate prediction of vessel responses in waves is crucial for decision-making and contribute to the operational safety and risk minimization. Short-term predictions can be carried out by estimating the vessel’s motions and loads based on incident waves. Existing model-based approaches either require computationally intensive simulations that compromise real-time capability or use simplified models affecting the accuracy of the prediction. Therefore, this study explores the feasibility of using neural networks for mapping time signals of surface elevation data and a set of corresponding ship responses, i.e. the heave and pitch motions as well as the vertical bending moment. The approach followed here is built on the assumption that the wave profile amidships is known. A synthetic dataset was generated using a time-domain strip theory solver with considerations of non-linear effects on motions and loads due to large amplitude waves in a variety of irregular, long-crested sea state conditions. We propose two different neural network models, a multi-layer perceptron (MLP) and a fully convolutional neural network (FCNN), and compare their performances on measurement data obtained from model tests in a seakeeping basin. The evaluations also include the freak wave reproduction of the ‘new year wave’. The proposed networks are able to estimate the motions and bending moment accurately for a wide range of sea state conditions, surpassing current state-of-the-art models on the given data sets.

In transducer arrays, symmetric grouping of identical elements is often employed to achieve uniform array performance. Such arrays can possess high coupling, preventing localized operation of individual transducers. This paper provides insight into how forced vibration localizes in a symmetric system of coupled oscillators. We use a simple lumped-parameter model of highly coupled oscillators derived from ultrasound transducer arrays. Forced vibration localization can be shown to be inversely related to the coupling strength between the oscillators. The results demonstrate how forced vibration in a coupled symmetric system may localize through modal superposition and how it may be tuned via the spacing of natural frequencies. Breaking the system’s symmetry leads to normal mode localization, which can be shown to affect the forced vibration response. The results reveal a variation in the system’s resonance frequency, attributed to curve veering effects.

The traditional measurement techniques to acquire the linear dynamic response of a single component are well established and have been in use for many decades to provide reliable input data for model updating. The measurement of assembled structures normally follows a very similar approach, although the presence of joints can introduce a nonlinear dynamic behaviour, which impacts the measurement results. Applying traditional linear test methods to a highly nonlinear structure, such as a dovetail joint in an aircraft engine blade-disk connection, does not necessarily take the special features of the nonlinear response into account and may lead to unreliable data. This is particularly true, if modal information such as damping are required. The influence of the measurement setup and the test procedures must be well understood for an accurate measurement. In this paper the influence of the different measurement components on a simple clamped beam and a compressor blade dove tail test rig will be investigated. A particular focus will be on the support of the test rig, the location and control of the excitation and the influence of the accelerometer on the response. Based on the findings an approach will be recommended that allows a reliable measurement of the dynamic behaviour of this heavily nonlinear structure.

Localized vibrations, arising from nonlinearities or symmetry breaking, pose a challenge in engineering, as the resulting high-amplitude vibrations may result in component failure due to fatigue. During operation, the emergence of localization is difficult to predict, partly because of changing parameters over the life cycle of a system. This work proposes a novel, network-based approach to detect an imminent localized vibration. Synthetic measurement data are used to generate a functional network, which captures the dynamic interplay of the machine parts, complementary to their geometric coupling. Analysis of these functional networks reveals an impending localized vibration and its location. The method is demonstrated using a model system for a bladed disk, a ring composed of coupled nonlinear Duffing oscillators. Results indicate that the proposed method is robust against small parameter uncertainties, added measurement noise, and the length of the measurement data samples. The source code for this work is available at C. Geier [(2024). "Code for paper Exploring localization in nonlinear oscillator systems through network-based predictions," Zenodo. https://doi.org/10.5281/zenodo.12611988].

The assimilation and prediction of phase-resolved surface gravity waves are critical challenges in ocean science and engineering. Potential flow theory (PFT) has been widely employed to develop wave models and numerical techniques for wave prediction. However, traditional wave prediction methods are often limited. For example, most simplified wave models have a limited ability to capture strong wave nonlinearity, while fully nonlinear PFT solvers often fail to meet the speed requirements of engineering applications. This computational inefficiency also hinders the development of effective data assimilation techniques, which are required to reconstruct spatial wave information from sparse measurements to initialize the wave prediction. To address these challenges, we propose a novel solver method that leverages physics-informed neural networks (PINNs) that parameterize PFT solutions as neural networks. This provides a computationally inexpensive way to assimilate and predict wave data. The proposed PINN framework is validated through comparisons with analytical linear PFT solutions and experimental data collected in a laboratory wave flume. The results demonstrate that our approach accurately captures and predicts irregular, nonlinear, and dispersive wave surface dynamics. Moreover, the PINN can infer the fully nonlinear velocity potential throughout the entire fluid volume solely from surface elevation measurements, enabling the calculation of fluid velocities that are difficult to measure experimentally.

Accurate real-time reconstruction of phase-resolved ocean wave fields remains a critical yet largely unsolved problem, primarily due to the absence of practical data assimilation methods for obtaining initial conditions from sparse or indirect wave measurements. While recent advances in supervised deep learning have shown potential for this purpose, they require large labeled datasets of ground truth wave data, which are infeasible to obtain in real-world scenarios. To overcome this limitation, we propose a physics-informed neural operator (PINO) framework for reconstructing spatially and temporally phase-resolved, nonlinear ocean wave fields from sparse measurements, without the need for ground truth data during training. This is achieved by embedding residuals of the free surface boundary conditions of ocean gravity waves into the loss function, constraining the solution space in a soft manner. In the current implementation, the framework is demonstrated for long-crested, unidirectional wave surfaces, where the wave propagation direction is aligned with the radar scanning direction. Within this setting, we validate our approach using highly realistic synthetic wave measurements by demonstrating the accurate reconstruction of nonlinear wave fields from both buoy time series and radar snapshots. Our results indicate that PINOs enable accurate, real-time reconstruction and generalize robustly across a wide range of wave conditions, thereby paving the way for future extensions of this framework toward multidirectional sea states and thus operational wave reconstruction in realistic marine environments.

Modeling the dynamics of systems with many interacting components, such as robots, wind turbines, and trusses, remains challenging today. These systems often display complex oscillatory responses to external inputs, and harmful vibrations might be excited along with the desired motion. Understanding the relative importance of individual components or systems aspects to the overall system dynamics could be a vital step towards focused design and maintenance efforts. This work proposes a network-based approach to studying the dynamics of a mechanical system by representing the system as a network of coupled oscillators, where each node corresponds to a machine component and each link denotes a physical connection, such as a weld or bolt. Inspired by studies of dynamics in biological and social networks, we show how network measures can be used to predict the importance of a single oscillator, or component, for shaping the overall dynamics. We further demonstrate under which conditions these conclusions are possible, and where the metrics fail. This study hopes to contribute to the broader field of network-based methods in engineering and yield insights that help focus design and maintenance efforts in the future.

The influence of non-linear modeling of phase-resolved ocean wave fields on the extent of the accessible predictable area is investigated. Assuming that the ocean surface dynamics is known over a limited spatial domain, e.g. via radar backscatter reconstruction, the linear wave theory as well as the high-order spectral method with various orders of non-linearity are used to propagate the surface with different levels of physical fidelity. The prediction accuracy is quantified by comparing the predicted waves to a reference, i.e. a fully known wave field propagated with a high-fidelity wave model. By doing this, it is made possible to track the spatiotemporal evolution of the prediction accuracy and define the predictable area as the region over which the accuracy is higher than a threshold, here defined by a "surface similarity parameter"lower than 0.1. Different unidirectional wave field characteristics are studied, highlighting the effect of the wave steepness, water depth and wave energy spreading around the peak spectral frequency, all impacting significantly the prediction accuracy, thus the predictable area. It is shown that the extent of the predictable area is highly dependent on the order of the considered wave model, and that the third order generally leads to the largest reachable predictable area in all configurations.

In various applications, dry friction could induce vibrations. A well-known example is frictional braking systems in ground transportation vehicles involving a sliding contact between a rotating and a stationary part. In such scenarios, the emission of high-intensity noise, commonly known as squeal, can present human health risks based on the noise’s intensity, frequency, and occurrences. Despite the importance of squeal in the context of advancing urbanization, the parameters determining its occurrence remain uncertain due to the complexity of the involved phenomena. This study aims to identify a relevant operando indicator for predicting squeal occurrences. To this end, a pin-on-disc test rig was developed to replicate various contact conditions found in road profiles and investigate resulting squealing. Each test involves a multimodal instrumentation, complemented by surface observations. It is illustrated that the enhanced thermal indicator identified is relevant because it is sensitive to the thermomechanical and tribological phenomena involved in squealing.

The measurement of deep water gravity wave elevations using in-situ devices, such as wave gauges, typically yields spatially sparse data. This sparsity arises from the deployment of a limited number of gauges due to their installation effort and high operational costs. The reconstruction of the spatio-temporal extent of surface elevation poses an ill-posed data assimilation problem, challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN), aiming to reconstruct physically consistent wave fields between two designated measurement locations several meters apart. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schrödinger equation (NLSE) into the PINN's loss function. Using synthetic wave elevation time series from distinct locations within a wave tank, we initially achieve successful reconstruction quality by employing constant, predetermined NLSE coefficients. However, the reconstruction quality is further improved by introducing NLSE coefficients as additional identifiable variables during PINN training. The results not only showcase a technically relevant application of the PINN method but also represent a pioneering step towards improving the initialization of deterministic wave prediction methods.

Abstract This paper explores the applicability of machine learning techniques for the generation of tailored wave sequences. For this purpose, a fully convolutional neural network was implemented for relating the target wave sequence at the target location in time domain to the respective control signal of the wave board. The database was generated by means of extensive wave tank tests. The experimental campaign focused on the generation of very steep wave groups including wave breaking which cannot be covered by the simplified wave generation methods. The extensive experimental campaign was performed in a small wave tank with an fully automated approach including determination and control of the wave maker motion as well as data measurement. The training data set features wave groups of short duration based on JONSWAP spectra, where the parameters wave steepness, peak wave period and enhancement factor were systematically varied. At the end of the training process, the trained models are able to predict the wave maker control signal based on time series of the target wave defined for a specific target location in the wave tank. The accuracy of the trained models are evaluated by means of unseen validation data. In addition, the predictive accuracy of the trained models is compared with the classical linear transformation approach.

Some aspects of engineering dynamics, such as nonlinearities and transient motion of many interconnected parts, remain difficult to handle today. To comply with increasing demands on resilience and safety, the dynamics of large machines need to be better understood. Complex network methods, already present in many scientific disciplines, provide a tool set complementary to conventional methods of system analysis. This work aims at providing a new, function-based view on mechanical systems by generating functional networks. To this end, a network algorithm is applied to sets of cyclically coupled Duffing oscillators as a common example of a complex nonlinear mechanical system. In the functional network, each node represents an oscillator while the direction of the network edges represents a functional coupling. Results show that the network method is capable of identifying dynamical transitions and synchronization between components, as well as determining the number of different states present within a system. Additionally, the time evolution of the component interactions, especially in response to a disturbance, is studied via a sliding-window approach. The results of this analysis might hopefully open new ways for a more efficient system analysis through optional sensor placement, and for effective countermeasures against unwanted dynamics through improved analysis of transient dynamics.

Most ground transportation vehicles rely on braking systems that involve a sliding contact between a rotating and a stationarypart. In such scenarios, the emission of high-intensity noise, commonly known as squeal, can present human health risksbased on the noise’s intensity, frequency, and occurrences. Despite the importance of squeal in the context of advancingurbanization, the parameters determining its occurrence remain uncertain due to the complexity of the involved phenomena.This study aims to identify a relevant operando indicator for predicting squeal occurrences. To this end, a pin-on-disc test rigwas developed to replicate various contact conditions found in road profiles and investigate resulting squealing. Each testinvolves thermal and mechanical instrumentation, complemented by surface observations. Thermo-mechanical phenomenaand tribological behavior of the contact are investigated to highlight the relevance of the proposed indicator by establishing linksbetween contact conditions and squealing.

Data-driven reduced order modeling methods that aim at extracting physically meaningful governing equations directly from measurement data are facing a growing interest in recent years. The HAVOK-algorithm is a Koopman-based method that distills a forced, low-dimensional state-space model for a given dynamical system from a univariate measurement time series. This article studies the potential of HAVOK for application to mechanical oscillators by investigating which information of the underlying system can be extracted from the state-space model generated by HAVOK. Extensive parameter studies are performed to point out the strengths and pitfalls of the algorithm and ultimately yield recommendations for choosing tuning parameters. The application of the algorithm to real-world friction brake system measurements concludes this study.

In the age of big data availability, data-driven techniques have been proposed recently to compute the time evolution of spatio-temporal dynamics. Depending on the required a priori knowledge about the underlying processes, a spectrum of black-box end-to-end learning approaches, physics-informed neural networks, and data-informed discrepancy modeling approaches can be identified. In this work, we propose a purely data-driven approach that uses fully convolutional neural networks to learn spatio-temporal dynamics directly from parameterized datasets of linear spatio-temporal processes. The parameterization allows for data fusion of field quantities, domain shapes, and boundary conditions in the proposed U ^p -Net architecture. Multi-domain U ^p -Net models, therefore, can generalize to different scenes, initial conditions, domain shapes, and domain sizes without requiring re-training or physical priors. Numerical experiments conducted on a universal and two-dimensional wave equation and the transient heat equation for validation purposes show that the proposed U ^p -Net outperforms classical U-Net and conventional encoder–decoder architectures of the same complexity. Owing to the scene parameterization, the U ^p -Net models learn to predict refraction and reflections arising from domain inhomogeneities and boundaries. Generalization properties of the model outside the physical training parameter distributions and for unseen domain shapes are analyzed. The deep learning flow map models are employed for long-term predictions in a recursive time-stepping scheme, indicating the potential for data-driven forecasting tasks. This work is accompanied by an open-sourced code.

<p>It is known that the modulation instability (MI) is a focusing mechanism responsible for the formation of rogue waves (RWs). Such dynamics are initiated from the injection of sidebands, which translates into an amplitude modulation (AM) of the wave field. The nonlinear stage of unstable wave evolution can be described by exact breather solutions of the nonlinear Schrödinger equation (NLSE). In fact, the amplitude modulation of such coherent RW structures is connected to a particular phase-shift seeded in the carrier wave, i.e. a particular form of localized frequency modulation (FM). By seeding only the local FM information of a deterministic breather to a regular wave train, our experiments show that such an FM localization can indeed trigger pure breather-type RW dynamics. Results of an experimental study on identifying spontaneous RWs in a random wave field by isolating the respective FM and AM dynamics will also be discussed. </p>

Algorithms for reconstructing and predicting nonlinear ocean wave fields from remote measurements are presented. Three types of synthetic observations are used to quantify the influence of remote measurement modulation mechanisms on the algorithms’ performance. First, the observations correspond to randomly distributed surface elevations. Then, they are related to a marine radar model – the second type takes the wave shadowing modulation into account whereas the third one also includes the tilt modulation. The observations are numerically generated based on unidirectional waves of various steepness values. Linear and weakly nonlinear prediction algorithms based on analytical models are considered, as well as a highly nonlinear algorithm relying on the high-order spectral (HOS) method. Reconstructing surfaces from shadowed observations is found to have an impact limited to the non-visible regions, while tilt modulation affects the reconstruction more generally due to the indirect, more complex extraction of wave information. It is shown that the accuracy of the surface reconstruction mainly depends on the correct modelling of the wave shape nonlinearities. Modelling the nonlinear correction of the dispersion relation, in particular the frequency-dependent wave phase effects in the case of irregular waves, substantially improves the prediction. The suitability of the algorithms for severe wave conditions in finite depth and using non-perfect observations is assessed through wave tank experiments. It shows that only the third-order HOS solution predicts the right amplitude and phase of an emerging extreme wave, emphasizing the relevance of the corresponding physical modelling.