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Item type:Publication, Access status: with files , From internal resonance to kinking and stacking in the response of nonlinear oscillatorsIn 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.Publicationtype: Journal Article39 46 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Building functional networks for complex response analysis in systems of coupled nonlinear oscillatorsSome 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.Publicationtype: Journal Article20 57 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Machine learning-based state maps for complex dynamical systems: applications to friction-excited brake system vibrationsIn this work, a purely data-driven approach to mapping out the state of a dynamical system over a set of chosen parameters is presented and demonstrated along a case study using real-world experimental data from a friction brake system. Complex engineering systems often exhibit a rich bifurcation behavior with respect to one or several parameters, which is difficult to grasp using experimental approaches or numerical simulations. At the same time, the growing need for energy-efficient machines that can operate under varying or extreme environmental conditions also calls for a better understanding of these systems to avoid critical transitions. The proposed method combines machine learning techniques with synthetic data augmentation to create a complete state map for a dynamical system. First, a machine learning model is trained on experimental data, picking up hidden mechanisms and complex parametric relations of the underlying dynamical system. The model is then exploited to assess the state of the system for a set of synthetically generated data to obtain a state map over the complete space spanned by the chosen parameters. In addition, an extension of the concept to a probability state map is introduced. The results indicate that the proposed method can uncover hidden variables which drive dynamical transitions between different states of a system that were previously inaccessible.Publicationtype: Journal ArticleScopus© Citations 2 36 77 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Surface similarity parameter : a new machine learning loss metric for oscillatory spatio-temporal dataSupervised machine learning approaches require the formulation of a loss functional to be minimized in the training phase. Sequential data are ubiquitous across many fields of research, and are often treated with Euclidean distance-based loss functions that were designed for tabular data. For smooth oscillatory data, those conventional approaches lack the ability to penalize amplitude, frequency and phase prediction errors at the same time, and tend to be biased towards amplitude errors. We introduce the surface similarity parameter (SSP) as a novel loss function that is especially useful for training machine learning models on smooth oscillatory sequences. Our extensive experiments on chaotic spatio-temporal dynamical systems indicate that the SSP is beneficial for shaping gradients, thereby accelerating the training process, reducing the final prediction error, increasing weight initialization robustness, and implementing a stronger regularization effect compared to using classical loss functions. The results indicate the potential of the novel loss metric particularly for highly complex and chaotic data, such as data stemming from the nonlinear two-dimensional Kuramoto–Sivashinsky equation and the linear propagation of dispersive surface gravity waves in fluids.Publicationtype: Journal ArticleScopus© Citations 5 52 57 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Towards neural network-based numerical friction models(2022-08-18); ; Friction contacts can be found in almost all mechanical systems and are often of great technical importance. However, they are usually difficult to describe, and their behavior and influence on the whole system are hard to predict accurately. Modern product design and system operation strongly benefit from numerical simulation approaches today, but reliable friction models still represent a major challenge in this context. To tackle this problem, we employ neural network regression to capture the characteristics of frictional contacts and make them accessible for numerical methods in a minimal intrusive fashion. In particular, we test our approach using a Finite Element model of a 2D cantilever beam subject to stick-slip vibrations induced by a moving conveyor belt at its free end. As a reference solution, we perform a transient analysis based on a simple analytical friction model, where the kinetic friction force only depends on the normal load and the relative sliding velocity. We take the same friction model, add some artificial noise to mimic uncertainties coming with experimental measurements, and pick a limited set of data points to train a regression neural network. The machine learning friction model is then deployed in the Finite Element code to predict the kinetic friction force acting on the beam tip during the slip phases. The deflection curves obtained by the transient numerical analysis using the new neural network friction model agree well with the reference solution based on the underlying analytical model. The results indicate that data-driven approaches may also be capable of capturing more complex frictional contacts, including effects of temperature, humidity, and load history. The trained neural network friction models can then be employed in numerical simulations in a minimally intrusive manner. This approach opens up new possibilities to predict individual mechanical system behavior as accurately as possible.Publicationtype: Conference Presentation122 192 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systemsThe pervasiveness of multi-stability in nonlinear dynamical systems calls for novel concepts of stability and a consistent quantification of long-term behavior. The basin stability is a global stability metric that builds on estimating the basin of attraction volumes by Monte Carlo sampling. The computation involves extensive numerical time integrations, attractor characterization, and clustering of trajectories. We introduce bSTAB, an open-source software project that aims at enabling researchers to efficiently compute the basin stability of their dynamical systems with minimal efforts and in a highly automated manner. The source code, available at https://github.com/TUHH-DYN/bSTAB/, is available for the programming language Matlab featuring parallelization for distributed computing, automated sensitivity and bifurcation analysis as well as plotting functionalities. We illustrate the versatility and robustness of bSTAB for four canonical dynamical systems from several fields of nonlinear dynamics featuring periodic and chaotic dynamics, complicated multi-stability, non-smooth dynamics, and fractal basins of attraction. The bSTAB projects aims at fostering interdisciplinary scientific collaborations in the field of nonlinear dynamics and is driven by the interaction and contribution of the community to the software package.Publicationtype: Journal ArticleScopus© Citations 13 110 257 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Hyperchaos co-existing with periodic orbits in a frictional oscillator(2020-04-28); ; ; This work reports observations of complex dynamical behavior and chimera-alike dynamics in a self-excited frictional oscillator with a weak stiffness nonlinearity. Multi-stability in the form of several co-existing stable limit cycle solutions are identified for a wide range of system parameters. For a particular system configuration, an unstable periodic orbit is found that gives rise to irregular long-term behavior. Trajectories starting from this orbit turn out to be hyper-chaotic with multiple positive Lyapunov exponents. Trajectories starting from different regions in phase space converge to stable limit cycles. Hence, this numerical study reveals co-existing regular and irregular dynamics at a fixed system configuration. This sensitive dependence of the qualitative system dynamics on initial conditions adds new aspects to a better understanding of the rich dynamic behavior of structures subjected to friction-induced vibrations.Publicationtype: Journal Article156 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry(Elsevier, 2016-08-09); ; ; ; Snaking bifurcations in a chain of mechanical oscillators are studied. The individual oscillators are weakly nonlinear and subject to self-excitation and subcritical Hopf-bifurcations with some parameter ranges yielding bistability. When the oscillators are coupled to their neighbours, snaking bifurcations result, corresponding to localised vibration states. The snaking patterns do seem to be more complex than in previously studied continuous systems, comprising a plethora of isolated branches and also a large number of similar but not identical states, originating from the weak coupling of the phases of the individual oscillators.Publicationtype: Journal Article73