Reliable Estimation of Minimum Embedding Dimension Through Statistical Analysis of Nearest Neighbors

False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay-coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Here, nearest-neighbor statistics are studied for uncorrelated random time series and contrasted with the corresponding deterministic and stochastic statistics. New composite FNN metrics are constructed and their performance is evaluated for deterministic, correlates stochastic, and white random time series. In addition, noise-contaminated deterministic data analysis shows that these composite FNN metrics are robust to noise. All FNN results are also contrasted with surrogate data analysis to show their robustness. The new metrics clearly identify random time series as not having a finite embedding dimension and provide information about the deterministic part of correlated stochastic processes. These metrics can also be used to differentiate between chaotic and random time series

Smooth Local Subspace Projection for Nonlinear Noise Reduction

Many nonlinear or chaotic time series exhibit an innate broad spectrum, which makes noise reduction difficult. Local projective noise reduction is one of the most effective tools. It is based on proper orthogonal decomposition (POD) and works for both map-like and continuously sampled time series. However, POD only looks at geometrical or topological properties of data and does not take into account the temporal characteristics of time series. Here, we present a new smooth projective noise reduction method. It uses smooth orthogonal decomposition (SOD) of bundles of reconstructed short-time trajectory strands to identify smooth local subspaces. Restricting trajectories to these subspaces imposes temporal smoothness on the filtered time series. It is shown that SOD-based noise reduction significantly outperforms the POD-based method for continuously sampled noisy time series

Dynamical systems approach to fatigue damage identification

In this paper, we present a dynamical systems approach to material damage identification. This methodology does not depend on knowledge of the particular damage physics of material fatigue. Instead, it provides experimental means to determine what are practically observable and observed facts of damage accumulation, thus making it possible to develop or experimentally verify appropriate damage evolution laws. A concept of phase space warping is used to develop new damage tracking feature vectors from measured time series through phase space reconstruction. These feature vectors provide critical information about the dimensionality of a damage process that was missing from an old scalar metric described in previous work. Damage identification is achieved by applying either proper orthogonal decomposition or smooth orthogonal decomposition to these vectors. The method is experimentally validated using an elasto-magnetic system, in which a harmonically forced cantilever beam in a non-linear magnetic field accumulates fatigue damage. Both damage identification methods yield a single active damage mode that shows power-law-type monotonic trends, which is also consistent with a result obtained using the old scalar metric. These results are shown to be in good agreement with the Paris fatigue crack growth model during the time period of macroscopic crack growth. Using this simple model, accurate time-to-failure predictions are shown well ahead of actual beam fracture. In addition, this identification scheme provides a much-needed insight into the dimensionality of incipient or early fatigue damage accumulation process, which is shown to be describable by only one scalar damage variable for this specific experiment

Robust and Dynamically Consistent Model Order Reduction for Nonlinear Dynamic Systems

There is a great importance for faithful reduced order models (ROMs) that are valid over a range of system parameters and initial conditions. In this paper, we demonstrate through two nonlinear dynamic models (pinned–pinned beam and thin plate) that are both randomly and periodically forced that smooth orthogonal decomposition (SOD)-based ROMs are valid over a wide operating range of system parameters and initial conditions when compared to proper orthogonal decomposition (POD)-based ROMs. Two new concepts of subspace robustness—the ROM is valid over a range of initial conditions, forcing functions, and system parameters—and dynamical consistency—the ROM embeds the nonlinear manifold—are used to show that SOD, as opposed to POD, can capture the low order dynamics of a particular system even if the system parameters or initial conditions are perturbed from the design case

Persistent Model Order Reduction for Complex Dynamical Systems Using Smooth Orthogonal Decomposition

Full-scale complex dynamic models are not effective for parametric studies due to the inherent constraints on available computational power and storage resources. A persistent reduced order model (ROM) that is robust, stable, and provides high-fidelity simulations for a relatively wide range of parameters and operating conditions can provide a solution to this problem. The fidelity of a new framework for persistent model order reduction of large and complex dynamical systems is investigated. The framework is validated using several numerical examples including a large linear system and two complex nonlinear systems with material and geometrical nonlinearities. While the framework is used for identifying the robust subspaces obtained from both proper and smooth orthogonal decompositions (POD and SOD, respectively), the results show that SOD outperforms POD in terms of stability, accuracy, and robustness

New Invariant Measures to Track Slow Parameter Drifts in Fast Dynamical Systems

Estimates of quantitative characteristics of nonlinear dynamics, e.g., correlation dimension or Lyapunov exponents, require long time series and are sensitive to noise. Other measures (e.g., phase space warping or sensitivity vector fields) are relatively difficult to implement and computationally intensive. In this paper, we propose a new class of features based on Birkhoff Ergodic Theorem, which are fast and easy to calculate. They are robust to noise and do not require large data or computational resources. Application of these metrics in conjunction with the smooth orthogonal decomposition to identify/track slowly changing parameters in nonlinear dynamical systems is demonstrated using both synthetic and experimental data

Dynamic Model for Fatigue Evolution in a Cracked Beam Subjected to Irregular Loading

The coupling of vibration and fatigue crack growth in a simply supported uniform Euler–Bernoulli beam containing a single-edge crack is analyzed. The fatigue crack length is treated as a generalized coordinate in a model for the mechanical system. This coupled model accounts for the interaction between the beam oscillations and the crack propagation dynamics. Nonlinear characteristics of the beam motion are introduced as loading parameters to the fatigue model to match experimentally observed failure dynamics. The method of averaging is utilized both as an analytical and numerical tool to: (1) show that, for cyclic loading, our fatigue model reduces to the Paris\u27 law and (2) compare the predicted fatigue damage accumulation with the experimental data for chaotic and random loadings. A utility of the fatigue model is demonstrated in estimating fatigue life under irregular loadings

Toward a unified interpretation of the “proper”/“smooth” orthogonal decompositions and “state variable”/“dynamic mode” decompositions with application to fluid dynamics

A common interpretation is presented for four powerful modal decomposition techniques: “proper orthogonal decomposition,” “smooth orthogonal decomposition,” “state-variable decomposition,” and “dynamic mode decomposition.” It is shown that, in certain cases, each technique can be interpreted as an optimization problem and similarities between methods are highlighted. By interpreting each technique as an optimization problem, significant insight is gained toward the physical properties of the identified modes. This insight is strengthened by being consistent with cross-multiple decomposition techniques. To illustrate this, an inter-method comparison of synthetic hypersonic boundary layer stability data is presented

Observed mode shape effects on the vortex-induced vibration of bending dominated flexible cylinders simply supported at both ends

The effect of varying the structural mode excitation on bending-dominated flexible cylinders undergoing vortex-induced vibrations was investigated. The response of the bending-dominated cylinders was compared with the response of a tension-dominated cylinder using multivariate analysis techniques. Experiments were conducted in a recirculating flow channel with a uniform free stream with Reynolds numbers between 650 and 5500. Three bending-dominated cylinders were tested with varying stiffness in the cross-flow and in-line directions of the cylinder in order to produce varying structural mode shapes associated with a fixed 2:1 (in-line:cross-flow) natural frequency ratio. A fourth cylinder with natural frequency characteristics determined through applied axial tension was also tested for comparison. The spanwise in-line and cross-flow responses of the flexible cylinders were measured through motion tracking with high-speed cameras. Global smooth-orthogonal decomposition was applied to the spatio-temporal response for empirical mode identification. The experimental observations show that for excitation of low mode numbers, the cylinder is unlikely to oscillate with an even mode shape in the in-line direction due to symmetric drag loading, even when the system is tuned to have an even mode at the expected frequency of vortex shedding. In addition, no mode shape changes were observed in the in-line direction unless a mode change occurs in the cross-flow direction, implying that the in-line response is a forced response dependent on the cross-flow response. The results confirm observations from previous field and laboratory experiments, while demonstrating how structural mode shape can affect vortex-induced vibrations

Nonlinear System Identification and Modeling of a New Fatigue Testing Rig Based on Inertial Forces

A novel fatigue testing rig based on inertial forces is introduced. The test rig has capacity to mimic various loading conditions including high frequency loads. The rig design allows reconfigurations to accommodate a range of specimen sizes, and changes in structural elements and instrumentation. It is designed to be used as a platform to study the interaction between fatigue crack propagation and structural dynamics. As the first step to understand this interaction, a numerical model of testing rig is constructed using nonlinear system identification approaches. Some initial testing results also are reported