Nonlinear Dynamic Behavior Research Articles

Investigation of nonlinear dynamic behaviors of vertical rotor system supported by aerostatic bearings

Abstract A common issue associated with gas bearing-rotor systems is the tendency to generate self-excited vibrations, leading to instability. To address this problem, the fluid-structure coupling model of an aerostatic bearing-rotor system is established in this paper. Then a hybrid method combining the finite difference method and direct integration method, is employed to solve the bearing lubrication equation and rotor motion equation simultaneously. Furthermore, based on the orbit of rotor center,frequency spectrum diagram, Poincaré map, waterfall diagram and bifurcation diagram, the effects of rotational speed, rotor mass, orifice diameter and nominal clearance on the nonlinear dynamic behaviors of the bearing-rotor system are investigated. The results indicate that the system exhibits rich nonlinear behaviors with increasing rotational speed and rotor mass, including the occurrence of typical half-speed whirl. However, the nonlinear vibrations of the system can be restricted by selecting appropriate bearing structural parameters, providing theoretical guidance for the design of aerostatic bearing-rotor systems.

Analysis of the Nonlinear Complex Response of Cracked Blades at Variable Rotational Speeds

The operation of an aero-engine involves various non-stationary processes of acceleration and deceleration, with rotational speed varying in response to changing working conditions to meet different power requirements. To investigate the nonlinear dynamic behaviour of cracked blades under variable rotational speed conditions, this study constructed a rotating blade model with edge-penetrating cracks and proposes a component modal synthesis method that accounts for time-varying rotational speed. The nonlinear response behaviours of cracked blades were examined under three distinct operating conditions: spinless, steady speed, and non-constant speed. The findings indicated a competitive relationship between the effects of rotational speed fluctuations and unbalanced excitation on crack nonlinearity. Variations in rotational speed dominated when rotational speed perturbation was minimal; conversely, aerodynamic forces dominated when the effects of rotational speed were pronounced. An increase in rotational speed perturbation enhanced the super-harmonic nonlinearity induced by cracks, elevated the nonlinear damage index (NDI), and accentuated the crack breathing effect. As the perturbation coefficient increased, the super-harmonic nonlinearity of the crack intensified, resulting in a more complex vibration form and phase diagram.

Research on the complex dynamical behavior of H-bridge inverter with RLC load

Complex dynamical behaviors such as bifurcation and chaos exist in H-bridge inverter with RLC load, and these nonlinear behaviors will greatly increase the harmonic content of the output current and reduce the stability and reliability of the system. In this paper, a PI controller is added to widen the stable operation domain of the system. The stroboscopic mapping theory is used to model the system, the nonlinear dynamic behavior of the inverter is investigated by the bifurcation diagram, the folding diagram and the phase trajectory diagram are used for comparative verification, and the TDFC method is introduced to inhibit the chaotic behavior of the inverter, which further improves the stable range of the system operation. The fast-change stability theorem is used to analyze the stability of the system theoretically and verify the correctness of the numerical simulation. Therefore, the conclusions of the study provide a reliable theoretical basis for the design of the inverter system, which has important theoretical significance and practical value.

Nonlinear behavior and energy harvesting performance of a new tunable quasi-zero stiffness system.

Energy harvesting from vibrations is a popular research topic. However, it is difficult to change the values of its dynamic stiffness characteristics during the operation of device. This paper analyzes the nonlinear dynamical behavior and energy harvesting performance of an energy harvesting device with a gear unit and an inertial amplifier. The Melnikov method is used to discriminate the chaos behavior of the system and is illustrated by numerical simulations. Chaotic dynamics analysis provides a simplified analytical idea that offers more insight into the performance of this energy harvesting device. In addition, the basins of attraction of the coexistence solution of the system are plotted, and the trajectory and energy harvesting efficiency of the system are analyzed for changes in the initial value. The results show that the device has the convenience to change the form and efficiency of system energy harvesting. The theory of Melnikov functions recognizes the presence of chaos and can provide solutions for the parameterization of the system. The choice of initial value greatly affects the energy harvesting efficiency.

Evaluating dynamic models for rigid-foldable origami: unveiling intricate bistable dynamics of stacked-Miura-origami structures as a case study.

Recent advances in origami science and engineering have particularly focused on the challenges of dynamics. While research has primarily focused on statics and kinematics, the need for effective and processable dynamic models has become apparent. This paper evaluates various dynamic modelling techniques for rigid-foldable origami, particularly focusing on their ability to capture nonlinear dynamic behaviours. Two primary methods, the lumped mass-spring-damper approach and the energy-based method, are examined using a bistable stacked Miura-origami (SMO) structure as a case study. Through systematic dynamic experiments, we analyse the effectiveness of these models in predicting bistable dynamic responses, including intra- and interwell oscillations, in different loading conditions. Our findings reveal that the energy-based approach, which considers the structure's inertia and utilizes dynamic experimental data for parameter identification, outperforms other models in terms of validity and accuracy. This model effectively predicts the dynamic response types, the rich and complex nonlinear characteristics and the critical frequency where interwell oscillations occur. Despite its relatively increased complexity in model derivation, it maintains computational efficiency and shows promise for broader applications in origami dynamics. By comparing model predictions with experimental results, this study enhances our understanding of origami dynamics and contributes valuable insights for future research and applications. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Nonlinear dynamic behaviors of a shaft-bearing-pedestal system with outer ring slip and damage

Nonlinear dynamic behaviors of a shaft-bearing-pedestal system with outer ring slip and damage

Exploring nonlinear dynamics in brain functionality through phase portraits and fuzzy recurrence plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

The Time Series Classification of Discrete-Time Chaotic Systems Using Deep Learning Approaches

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of discrete-time chaotic systems by integrating deep learning methods and classification algorithms. The most important innovation of this study is the use of a unique dataset created using the time series of discrete-time chaotic systems. In this context, a large and unique dataset representing various dynamic behaviors was created for nine discrete-time chaotic systems using different initial conditions, control parameters, and iteration numbers. The dataset was based on existing chaotic system solutions in the literature, but the classification of the images representing the different dynamic structures of these systems was much more complex than ordinary image datasets due to their nonlinear and unpredictable nature. Although there are studies in the literature on the classification of continuous-time chaotic systems, no studies have been found on the classification of discrete-time chaotic systems. The obtained time series images were classified with deep learning models such as DenseNet121, VGG16, VGG19, InceptionV3, MobileNetV2, and Xception. In addition, these models were integrated with classification algorithms such as XGBOOST, k-NN, SVM, and RF, providing a methodological innovation. As the best result, a 95.76% accuracy rate was obtained with the DenseNet121 model and XGBOOST algorithm. This study takes the use of deep learning methods with the graphical representations of chaotic time series to an advanced level and provides a powerful tool for the classification of these systems. In this respect, classifying the dynamic structures of chaotic systems offers an important innovation in adapting deep learning models to complex datasets. The findings are thought to provide new perspectives for future research and further advance deep learning and chaotic system studies.

Nonlinear dynamic analyses of sandwich porous FG cylindrical shells with dual-layered FG porous cores

This research examines the nonlinear vibrations and dynamic postbuckling (DPB) analyses of sandwich porous functionally graded (SPFG) cylindrical shells with dual-layered FG porous (FGP) cores exposed to external excitation, using a semi-analytical method. The study investigates two types of FG layers: one with evenly distributed porosities (FG-EPD) and another with unevenly distributed porosities (FG-UEPD). It also studies the FGP cores in four configurations: one featuring symmetric porosity distribution with stiffening in the surface areas (SPD-Stiff), another with softening in the surface areas (SPD-Soft), a third with nonsymmetric porosity distribution (NSPD), and a fourth with uniform porosity distribution (UPD). Therefore, the SPFG cylindrical shells with dual-layered FGP cores with eight different configurations are investigated. Utilizing the classical shell theory (CST) alongside the geometrical nonlinearity in von Kármán–Donnell framework, and Galerkin’s method, this study addresses the nonlinear dynamic problem. An approximate solution for the deflection shape using three terms is selected, and the relationship between frequency and amplitude in nonlinear vibration is clearly defined. The nonlinear dynamic behaviors including vibration and DPB responses are examined using the P-T method, which is named for its use of the piecewise constant argument in conjunction with the Taylor series expansion. The critical dynamic buckling load of SPFG cylindrical shells with dual-layered FGP cores is investigated via the Budiansky–Roth criterion.

Exploring Nonlinear Dynamics In Brain Functionality Through Phase Portraits And Fuzzy Recurrence Plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Soliton molecules, bifurcation solitons and interaction solutions of a generalized (2 + 1)-dimensional korteweg-de vries system for the shallow-water waves

The central purpose of this paper is exploring the soliton molecules, bifurcation solitons and interaction solutions of the Korteweg–de Vries system based on the Hirota bilinear method. The studied system acts as an extension of the classic KdV system for the shallow-water waves, and is very useful to contribute in nonlinear wave phenomena. Firstly, the soliton molecules are obtained by adding resonance parameters in N-soliton. Then the interaction solutions between soliton/breather and soliton molecules are studied, as well as the interaction between two soliton molecules by using N-soliton. Moreover, a class of novel bifurcation solitons are derived, including Y-type bifurcation solitons, X-type bifurcation solitons and multiple-bifurcation solitons. In the end, the dynamic properties of soliton molecules, bifurcation solitons as well as the interaction solutions are presented graphically. The developed solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the generalized (2+1)-dimensional Korteweg–de Vries system better.

Spatiotemporal complexity analysis of a discrete space-time cancer growth model with self-diffusion and cross-diffusion

We investigate spatiotemporal pattern formation in cancer growth using discrete time and space variables. We first introduce the coupled map lattices (CMLs) model and provide a dynamical analysis of its fixed points along with stability results. We then offer parameter criteria for flip, Neimark–Sacker, and Turing bifurcations. In the presence of spatial diffusion, we find that stable homogeneous solutions can experience Turing instability under certain conditions. Numerical simulations reveal a variety of spatiotemporal patterns, including patches, spirals, and numerous other regular and irregular patterns. Compared to previous literature, our discrete model captures more complex and richer nonlinear dynamical behaviors, providing new insights into the formation of complex patterns in spatially extended discrete tumor models. These findings demonstrate the model’s ability to capture complex dynamics and offer valuable insights for understanding and treating cancer growth, highlighting its potential applications in biomedical research.

A data-driven technique for discovering the dynamical system with rigid impact characteristic

We propose a data-driven technique for discovering the equation of motion of the dynamical system with rigid impact. The method first discovers a dynamical system close to impact by the Fourier series with a high rate of convergence, known as the double-even extended series. Then, we use the system to construct an impact mapping, which maps the data close to impact to an estimated impact instant. By minimizing the error of impact mapping, we find the location of impact surface and energy lost during impact that generally satisfies the data close to impact. Finally, we discover the equation of motion without impact by the double-even extended series The analyzed data can be collected at equal time intervals with measurement error, and there is no need to deliberately collect data at the impact instant. The technique is able to capture the impact characteristic when there is a lack of knowledge about the critical changes of the dynamics at the impact instant and the non-linear dynamical behaviors without impact. We test the identification ability of the new technique using impact dynamical systems connected with cubic damping term and strong non-linear damping, respectively. The identified systems accurately capture impact dynamics such as the long-time prediction with impact, multistable dynamical phenomenon, and chattering dynamics.

Effect of relative phase of loose tie rods on nonlinear dynamic behavior of rod-fastening rotor-bearing-seal system

Loosening of multiple tie rods in a circumferential rod-fastening rotor caused by thermal deformation and centrifugal loads is very common in engineering. The relative position of loose tie rods in the circumferential direction will also have an important impact on rotor dynamics. The relevant research has not been done yet. In this paper, a contact model of the rough machined surface is firstly established by combining the probability functions describing rough surfaces and fractal contact theory of asperities, then the contact stiffness model between disks is established based on the contact model. Finally, the dynamic model of the circumferential rod-fastening rotor-bearing-seal system is established and solved. The effect of relative phase of loose tie rods on the nonlinear dynamic characteristics of rod-fastening rotor-bearing-sealing system is studied. It can be concluded that relative phase of the loose tie rod has an important influence on the frequency, bifurcation, and periodic motion of the rotor system. At lower speed, the anisotropy of contact stiffness caused by loose tie rods has little effect on the rotor trajectory, while the generalized bending moment has a greater influence on dynamics of the system. At higher speed, the anisotropy of contact stiffness has a great influence on the axis rail of the rod-fastening rotor.

Synchronization evaluation of memristive photosensitive neurons in multi-neuronal systems

At the forefront of brain-computer interface (BCI) technology exploration, the synchronization phenomenon in neural networks demonstrates significant potential for application. This paper focuses on the memory characteristics and nonlinear dynamic behavior of memristors, an emerging electronic component. By innovatively integrating memristors, phototubes, and FitzHugh-Nagumo (FHN) circuits, we construct a memristive photosensitive neuron (MPN) model, aiming to explore its unique role in neural network synchronization. Utilizing the memristors' inherent memory capabilities and nonlinear properties, the MPN model mimics the dynamic behavior of biological neurons. Experiments show that synchronization is highly sensitive to the memristor's initial values, revealing intricate synchronization regulation mechanisms. Furthermore, we find that chaotic electrical currents, as environmental disturbances, have dual effects—either promoting or inhibiting MPN network synchronization—depending on specific parametric conditions. To simulate a realistic biological neural network and achieve efficient coupling among multiple MPN units, this paper adopts a Hopfield network structure. The results indicate that this structure significantly enhances the synchronization stability of the system, reduces sensitivity to initial conditions, and mitigates the adverse effects of chaotic currents. This research not only enhances our understanding of neural network synchronization but also provides novel theoretical support and technical pathways for the development of BCI technology, indicating broad application prospects in neuroscience, rehabilitative medicine, and other fields.

Dynamics Modeling and Analysis of Small-Diameter Pipeline Inspection Gauge during Passing Through Elbow

Abstract With small diameters, numerous elbows and complex internal environment, gathering and transportation pipelines are widely applied in the oil and gas pipeline network. To guarantee their safety, a pipeline inspection gauge (PIG) driven by the pressure differential is used. When the PIG collides with the elbows in the pipeline, it exhibits complex nonlinear dynamic behaviors and its reliability and stability may also be greatly affected. To study the vibration response characteristics passing through the elbow, the dynamic modeling of the small-diameter PIG is conducted first in this paper based on the ADAMS software. Then, the fluid-structure coupling is achieved by combining the dynamic model and the numerical calculation program of the fluid equation in Simulink. The simulation results indicate that the axial and swing acceleration amplitudes of the two sections increase with the increasing speed of the double-section detector, while the opposite trend is observed for the vertical acceleration amplitudes. By comparison, it is found that the acceleration response in the second section is generally greater than that in the first section. The results also show that the axial offset of the PIG at the elbow is inversely proportional to the speed and radius of the elbow, which is the main cause of the lift-off values of sensors.

Study on the seismic wave input method for earth-rock dam on overburden foundation with dynamic nonlinearity based on nonlinear artificial boundary

Accurately obtaining the seismic response of earth-rock dams on overburden foundations is crucial for seismic safety assessment and design, and a reasonable seismic input method is necessary. The seismic wave input method based on artificial boundaries is widely used in elastic foundations. However, the overburden with obvious dynamic nonlinearity is an insurmountable obstacle for the traditional method. In this research, relying on a specific earth-rock dam project, the essence of the seismic wave input method for nonlinear foundations is analyzed in detail from equivalent loads and viscous boundaries, respectively. The mechanism of inapplicability of the traditional seismic wave input method is illustrated, and an effective modified method is validated and presented. The research indicated that equivalent loads for the nonlinear dynamic response of the overburden foundation can be accurately acquired with the shear box model. The dynamic parameters of soil can be captured dynamically by the nonlinear viscous coupling boundary. The states of the lateral boundary of the overburden foundation are consistent with the free field precisely with the modified seismic wave input method by combining the shear box model and the nonlinear viscous coupling boundary. A large error will be caused by neglecting the dynamic nonlinear behavior of the overburden foundation, where the maximum deviation of peak acceleration may be over 55 % or more. A numerical model with a relatively small lateral range of overburden foundation will be obtained when the modified seismic wave input method is engaged. Moreover, the computational accuracy is enhanced remarkably, with a maximum deviation of no more than 5 %. This study provides effective theoretical support for seismic analysis and safety assessment of earth-rock dams on deep overburden foundations.

Impact of environmental stochastic fluctuations on the evolutionary stability of imitation dynamics.

To show the impact of environmental noise on imitation dynamics, the stochastic stability and stochastic evolutionary stability of a discrete-time imitation dynamics with random payoffs are studied in this paper. Based on the stochastic local stability of fixation states and constant interior equilibria in a two-phenotype model, we extend the concept of stochastic evolutionary stability to the stochastic imitation dynamics, which is defined as a strategy such that, if all the members of the population adopt it, then the probability for any mutant strategy to invade the population successfully under the influence of natural selection is arbitrarily low. Our main results show clearly that the stochastic evolutionary stability of the system depends only on the properties of the mean matrix of the random payoff matrix and is independent of the randomness of the random payoff matrix. Moreover, as two examples, we show also that under the framework of stochastic imitation dynamics, the noise intensity affects the evolution of cooperative behavior in a stochastic prisoner's dilemma game and the system's nonlinear dynamic behavior.

Nonlinear dynamic study of FG-GFRC-NPR structures

The functionally graded glass fibre-reinforced polymer composite(FG-GFRC) is one of the contemporary sophisticated materials that may be employed for a variety of technical purposes. The shock absorption capacity of FG-GFRP laminates can be improved by incorporating auxetic property (i.e., negative Poisson's ratio NPR) in the structures, which can be attained by a particular stacking sequence of glass fibres (GF) and a different percentage of GF distribution along the thickness direction within the FG-GFRP structure. The nonlinear dynamic behaviour of such structures must be investigated. The clamped-clamped FG-GFRP-NPR structure is stimulated under a uniform transverse load distribution in the current work. Based on the higher-order shear deformation theory (SDT), the displacement field and nonlinear stress–strain relationship are derived. The nonlinear differential equation with cubic non-linear components were derived using Hamilton’s principle and transformed using Galerkin’s technique and obtained governing equation of motion. MATLAB has been used to conduct all numerical simulations. A time series, a phase picture and a Poincare diagram (PCM) were generated to characterize the vibration behavior of such auxetic structures.

Real-Time Massive Parallel Generation of Physical Random Bits Using Weak-Resonant-Cavity Fabry-Perot Laser Diodes

We experimentally demonstrate a scheme for generating massively parallel and real-time physical random bits (PRBs) by using weak-resonant-cavity Fabry-Perot laser diodes (WRC-FPLDs) with optical feedback. By using external optical feedback to modify the nonlinear dynamic behavior of the longitudinal modes in WRC-FPLDs, the chaotic behavior of each channel can be induced under suitable feedback strength. By filtering these longitudinal modes, a real-time PRBs at 10 Gbits/s can be generated by using field programmable gate array (FPGA) board for the real-time post-processing of a single-channel chaotic signal. Considering the presence of up to 70 longitudinal modes within a broad spectral range exceeding 40 nm, each of these modes can be used to extract chaotic time sequences for random number generation. Therefore, our PRB generation scheme has the potential to achieve a data throughput of over 700 Gbits/s.