Dynamic Equations Of System Research Articles

Dynamic simulation of cargo transport along cislunar suspension tether by single node coupling model

A cislunar suspension tether (CST) is a kind of ultra-long tether anchored to the lunar surface and extends to the vicinity of the Earth, looks like hanging to the Moon, could be used as cislunar pathway for interstellar freightage easier. Its feasibility on material mechanics and dynamics has been verified by the authors. This paper focuses on the process of the cargo transport along the CST. A single coupling model is established, which couples the cargo with one single node of the discrete CST model. Merely the interactive force between the coupled node and cargo is considered, with a proper definition description of the cargo's velocity along the CST. The description defines the direction of the cargo's velocity to the next one node on the CST all the time. The dynamic equations of the whole system are established based on afore stipulations. Several examples will be simulated and presented, their results will prove the effectiveness of the single coupling model, while, reflect some interesting phenomena of the CST's work. This paper also explains the advantage of the model by contrasting to some other models with multi nodes actions to the cargo.

Representing turbulent statistics with partitions of state space. Part 2. The compressible Euler equations

This is the second part of a two-part paper. We apply the methodology of the first paper (Souza, J. Fluid Mech., vol. 997, 2024, A1) to construct a data-driven finite-volume discretization of the Liouville/Fokker–Planck equation of a high-dimensional dynamical system, i.e. the compressible Euler equations with gravity and rotation evolved on a thin spherical shell. We show that the method recovers a subset of the statistical properties of the underlying system, steady-state distributions of observables and autocorrelations of particular observables, as well as revealing the global Koopman modes of the system. We employ two different strategies for the partitioning of a high-dimensional state space, and explore their consequences.

Dissipativity of dynamical systems on time scales and the relationship between dissipative differential and dynamical systems

This work is devoted to study the dissipativity property of dynamical systems on time scales and relationship between the dissipativity of systems of dynamic equations on time scales and the corresponding systems of ordinary differential equations. It is established that the dissipativity property is preserved when transitioning from equations on time scales $\mathbb{T}_\lambda$ to the corresponding ordinary differential equations and vice versa, provided that the graininess function $\mu_\lambda$ converges to zero as $\lambda\to0$.

Existence of solutions for systems of conformable fractional dynamic equations on time scales

Existence of solutions for systems of conformable fractional dynamic equations on time scales

Dynamic optimization algorithms for enhancing blockchain network resilience against distributed attacks

This research introduces a dynamic optimization algorithm designed to enhance blockchain network resilience against distributed attacks such as Distributed Denial of Service (DDoS), Sybil, and eclipse attacks. The primary objective is to develop a real-time, adaptive control strategy that minimizes network performance degradation while dynamically responding to evolving threats. The research design integrates multi-objective optimization, game theory, and reinforcement learning to formulate a defense strategy that adapts to adversarial conditions. The methodology is based on a modified state-space model, where the blockchain's performance is represented by a system of dynamic equations influenced by both control actions (defensive measures) and attack vectors. The optimization problem is formulated to minimize a cost function that balances network resilience and resource usage. A numerical example is presented to validate the model, demonstrating the algorithm’s effectiveness in maintaining network performance under attack by adjusting defense mechanisms in real-time. The main results indicate that the proposed method significantly reduces the impact of distributed attacks while ensuring efficient resource allocation. In conclusion, this research offers a novel framework for enhancing blockchain security, with implications for real-world applications in decentralized systems, financial services, and critical infrastructure. Future work will address the scalability of the algorithm and explore more advanced reinforcement learning techniques to handle more complex and unpredictable attack patterns.

On the Energy Cost for Nitrogen Fixation in DC Glow Discharges in Atmospheric‐Pressure Air

ABSTRACTModeling of the production of nitrogen oxide molecules in DC discharges in laminar longitudinal flows of atmospheric‐pressure air is performed. Two‐dimensional nonequilibrium discharge model includes the system of gas dynamic equations and balance equations for various plasma components. Simulations are performed for the discharge currents 50–600 mA and gas flow velocities 1–10 m/s. In considered conditions, the gas temperature in the discharge core varies in the range of 3000–6000 K. Spatial profiles of the densities and fluxes of produced species are obtained, both inside the discharge and in the afterglow region. The energy cost for the production of nitrogen oxides is calculated, depending on the discharge current, gap length, and gas flow velocity. The results of the simulation are compared with available experimental data.

KAN-ODEs: Kolmogorov–Arnold network ordinary differential equations for learning dynamical systems and hidden physics

Kolmogorov–Arnold networks (KANs) as an alternative to multi-layer perceptrons (MLPs) are a recent development demonstrating strong potential for data-driven modeling. This work applies KANs as the backbone of a neural ordinary differential equation (ODE) framework, generalizing their use to the time-dependent and temporal grid-sensitive cases often seen in dynamical systems and scientific machine learning applications. The proposed KAN-ODEs retain the flexible dynamical system modeling framework of Neural ODEs while leveraging the many benefits of KANs compared to MLPs, including higher accuracy and faster neural scaling, stronger interpretability and generalizability, and lower parameter counts. First, we quantitatively demonstrated these improvements in a comprehensive study of the classical Lotka–Volterra predator–prey model. We then showcased the KAN-ODE framework’s ability to learn symbolic source terms and complete solution profiles in higher-complexity and data-lean scenarios including wave propagation and shock formation, the complex Schrödinger equation, and the Allen–Cahn phase separation equation. The successful training of KAN-ODEs, and their improved performance compared to traditional Neural ODEs, implies significant potential in leveraging this novel network architecture in myriad scientific machine learning applications for discovering hidden physics and predicting dynamic evolution.

Impact of compliant electrodes on the dynamics of electromagnetoactive membranes

The dynamics of electromagnetoactive polymer (EMAP) membranes have attracted much attention recently because of their wide range of modern robotic applications. Such applications majorly centered on how the dynamics of this novel class of membranes are affected by the mechanical behavior of the compliant electrode. This article presents the dynamic modeling and analysis of EMAP membranes, examining how the inertia of the electrode, coupled with its inherent viscoelastic properties, impacts its dynamic performance. Both the compression and suspension stages of the membrane are covered here in broad terms. An Euler–Lagrange equation of motion is implemented to deduce the governing dynamic model equation of the membrane system. The findings of the model solutions provide preliminary insights to characterize the dynamic response, instability analysis, periodic behavior, and resonance properties across varying parameters such as inertia, electric field, magnetic field, and prestress. Moreover, the study also evaluates the periodicity and stability of the nonlinear oscillations using Poincaré maps and phase portraits, facilitating an assessment of quasi-periodic to periodic transitions.

Dynamic characteristics analysis and vibration control of coupled bending-torsional system for axial flow hydraulic generating set

Aiming at the vibration problems of shaft system for axial flow hydraulic generating sets caused by complex excitations, the coupled bending-torsional dynamic equations of rotor-runner system are established considering multi-effect including hydraulic, mechanical, electrical and other external vibration sources. Whether taking the torsional degree of freedom into account, the stability and disparate destabilization laws of system periodic solutions are analyzed using the shooting method combined with Floquet theory. On this basis, the influence of coupling bending-torsional effect on system dynamic behaviors is discussed through bifurcation diagrams, Poincaré maps, trajectories, time domains and frequency spectrums. Furthermore, in order to alleviate the unsteady vibration of rotor as well as runner, the magnetorheological fluid damper is introduced into the shaft system, and numerical simulations are performed on models of coupled bending-torsional systems with or without the magnetorheological fluid damper. The analyses indicate that the interaction between bending and torsional vibration of the shaft system will expand the chaotic sustained range for rotor. Meanwhile, the amplitude of the rotor-runner system is magnified owing to the torsional vibration, which leads to local rubbing faults of the system under specific operating conditions, while the intervention of the magnetorheological fluid damper can significantly suppress the nonlinear motion of rotor and runner, reduce the amplitude of system vibration, so as to avoid the occurrence of friction defects effectively. The above research results can provide conducive references for fault diagnosis, optimized design, and vibration control of axial flow hydraulic generating sets.

Physics-Informed Neural Networks for Prediction of a Flow-Induced Vibration Cylinder

Abstract Flow-induced vibration (FIV) is a common phenomenon in ocean engineering for subsea structures with circular-shaped cross sections. A large amount of computational resources or experimental efforts are required to predict or measure the complicated motions and flow field surrounding a circular cylinder undergoing vortex-induced vibration (VIV). Physics-informed neural networks (PINNs) are powerful deep learning techniques for solving governing partial differential equations (PDEs) of dynamic systems as an alternative to complex numerical methods. In the present study, a framework is built employing PINNs for solving the Navier–Stokes equations to predict flows past an FIV cylinder using sparsely distributed spatiotemporal data inside the domain. The training process involves minimizing the supervised loss of flow data at these sparse points and the residuals of the governing PDEs. For the training of the PINN model, a moving frame around an FIV cylinder is used to collect the training flow data from two-dimensional direct numerical simulation results at a low Reynolds number. The structural displacements of the cylinder are also implemented in the residuals of the equations of the developed PINN. The performance of the PINN is evaluated by comparing the predicted contours of the surrounding flow velocities with the training data. The hydrodynamic forces prediction is achieved using the PINN-obtained flow field predictions combined with the force partitioning method.

On metallic-type asteroid rotation moving in magnetic field (introducing magnetic second-grade YORP effect)

A novel physical effect has been found to be illuminated of being physically self-consistent to take into account in further calculating the dynamics of magnetically-induced asteroid rotation, surface of which has mostly a conducting structure (whereas such mostly metallic-formed asteroid is assumed to be moving in an external magnetic field of planet but in the meantime to be orbiting preferably outside its sphere of effective attraction). The last condition of asteroid surface's having conducting structure means the existence of the applied external torques stemming from the physical interactions between external magnetic field and iron asteroid itself (including long-term effect due to torques stemming from arising eddy-currents on metallic surface of asteroid). Such eddy-currents should heat the conducting surface of asteroid with further non-uniformly re-directing the heat flow from surface of asteroid into outer space via fluxes of thermal photons which carry momentum (in orthogonal direction to the surface which is in most cases far from the ideal surface of sphere). This leads by taking into account the overall outcome of heat flow (from the non-ideal surface of asteroid) to a kind of long-term magnetic second-grade YORP effect. System of Euler equations for aforementioned dynamics of asteroid magnetic rotation has been investigated in regard to existence of semi-analytical solution. Various perturbations (such as collisions, YORP effect) may destabilize the rotation of asteroid deviating it from the current spin state, whereas the electric(eddy)current-induced dissipation of energy reduces kinetic one of asteroid spin. So, dynamics of the asteroid rotation should result in a spinning about maximal-inertia axis with the proper spin state corresponding to minimal energy with a fixed angular momentum.

Using reservoir computing to solve FPK equations for stochastic dynamical systems under Gaussian or Non-Gaussian excitation

This paper presents a new approach that uses the Reservoir Computing Algorithm to solve Fokker-Planck-Kolmogorov (FPK) equation excited by both Gaussian white noise and non-Gaussian noise. Unlike typical numerical methods, this methodology does not necessitate spatial reconstruction or numerical supplementation. The novelty of this paper lies in the modifications made to the conventional Reservoir Computing algorithm. We altered the approach for calculating values of the input weight matrix and incorporated autoregressive techniques in the reservoir layer. In addition, we applied data normalization to the training data before training the algorithm to avoid a zero solution. The efficacy of this approach was verified through multiple arithmetic examples, showcasing its practicality and efficiency in solving FPK equations. Moreover, the Reservoir Computing-FPK algorithm is capable of solving high-dimensional and fractional-order FPK equations with a smaller training set than earlier algorithms. Finally, we analyzed how values of the input weight matrix and regularization parameter affected the performance of the algorithm. The findings suggest that the careful selection of hyperparameters can greatly improve the performance of the Reservoir Computing algorithm.

Rigid-flexible coupling dynamic modelling and dynamic accuracy analysis of planar cam four-bar mechanism with multiple clearance joints

Combination mechanism of conjugate cam and four-bar is a complex type of multi-body system, in which the coupling problems such as cam profile law, flexible deformation of connecting rods, and multiple joint clearance collision make it difficult to model and optimise its dynamics. Just as the beating-up mechanism of high-speed rapier looms for multi-layer weaving is a representative conjugate cam four-bar mechanism. In response to the current development of textile machinery in the direction of high-speed and precision, and in order to improve the work efficiency and operational accuracy of the conjugate cam four-bar beating-up mechanism, the following work is done for the purpose of taking into account the multi-joint clearance and the flexible rod based on the mechanism. Firstly, based on Kane's equation, a multi-body dynamic model with multiple clearances is established combined with clearance collision theory. Secondly, a dynamic model of a rigid-flexible coupling system with multiple revolute clearances is established using finite element method combined with modal synthesis technology. Subsequently, the variables separation method and mode superposition method are used to solve the dynamic equations of the coupled system. Finally, numerical examples are used to analyse the effects of clearance quantity, clearance size, mode truncation order, cam speed, and component material properties on the dynamic response and accuracy of the system. The results show that, in the rotational speed range of 600–800 rpm, the dynamic performance of the system is little affected by the speed; Through comparative analysis of multiple materials, it is found that selecting carbon fibre composite materials has the smallest impact on the motion accuracy of the system. In this paper, the clearance collision model, finite element model and rigid-flexible coupling dynamic model are integrated, which are applied to the cam-linkage combined multi-body mechanism, and the dynamic problems of the actual machine are analysed and solved.

Monitoring and Reconstruction of Actuator and Sensor Attacks for Lipschitz Nonlinear Dynamic Systems Using Two Types of Augmented Descriptor Observers

Fault data injection attacks may lead to a decrease in system performance and even a malfunction in system operation for an automatic feedback control system, which has motive to develop an effective method for rapidly detecting such attacks so that appropriate measures can be taken correspondingly. In this study, a secure descriptor estimation technique is proposed for continuous-time Lipschitz nonlinear cyber physical systems affected by actuator attacks, sensor attacks, and unknown process uncertainties. Specifically, by forming a new state vector composed of original system states and sensor faults, an equivalent descriptor dynamic system is built. A proportional and derivate sliding-mode observer is presented so that the system states, sensor attack, and actuator attack can be reconstructed successfully. The observer gains are obtained by using linear matrix inequality to secure robustly stable estimation error dynamics. Moreover, a robust descriptor fast adaptive observer estimator is presented as a complement. Finally, the efficacy levels of the proposed design approaches are validated using a vertical take-off and landing aircraft system. Comparison studies are also carried out to assess the tracking performances of the proposed algorithms.

Low-Computational-Complexity Zeroing Neural Network Model for Solving Systems of Dynamic Nonlinear Equations

Low-Computational-Complexity Zeroing Neural Network Model for Solving Systems of Dynamic Nonlinear Equations

Random attractors for three-dimensional stochastic globally modified Navier–Stokes equations driven by additive noise on unbounded domains

Abstract This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 𝒪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball’s idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time.

An adaptive energy-based sequential method for training PINNs to solve gradient flow equations

Recent developments in numerous scientific computing fields have been made possible by physics-informed neural networks (PINNs), a class of machine learning techniques that incorporate physical knowledge into neural networks. However, PINNs still encounter difficulties in simulating a kind of partial differential equations (PDEs), i.e. gradient flow equations of complex dynamic systems. Through analysis from the neural tangent kernel's (NTK) point of view, we find that directly training PINNs on the large temporal window may lead to failure. To tackle this problem, we utilize one of the gradient flow equations' significant physical properties, the energy dissipation law, to propose an adaptive energy-based sequential training method for PINNs. Specifically, the gradient of free energy is employed to determine the adaptive temporal steps of the training window, which also enhances the training efficiency. Numerical experiments are demonstrated among gradient flow equations including Cahn-Hilliard, heat conduction, and Allen-Cahn equations. Our method not only outperforms PINNs but also preserves the energy dissipation law, which shows the ability to solve complex long-time simulation problems of gradient flow systems.

Theoretical characterization of the dynamics of a new planetary gearbox

Based on the newly designed planetary gear gearbox with reciprocating linear motion of its output shaft, research on its dynamic characteristics was conducted. The “kinematic mechanism method” was used to calculate the system’s transmission ratio, revealing its characteristic of having a relatively large transmission ratio. Combined with the calculation of transmission efficiency, the characteristic of high transmission efficiency was verified. The “velocity vector method” was used to calculate the relationship between the angular velocities of various components of the gearbox. By conducting a stress analysis on the planetary gear transmission system, the forces and torques acting on each component, as well as the meshing forces of the gears were calculated. The “central parameter method” was adopted to simulate the meshing contact between the gears in the planetary gear transmission system as undamped springs. Considering the two translational vibrations along the radial direction and the rotational twisting along the circumferential direction of each component, a translational-torsional coupled dynamic theoretical model of the system was constructed. Dynamic differential equations of the planetary gear transmission system were derived and solved to obtain the support forces and torques of each component at the bearing support positions, enhancing the research on the dynamic characteristics of the planetary gear transmission system.

Nonlinear energy harvesting system with multiple stability

The nonlinear energy harvesting systems of the forced vibration with an electron-mechanical coupling are widely used to capture ambient vibration energy and convert mechanical energy into electrical energy. However, the nonlinear response mechanism of the friction induced vibration (FIV) energy harvesting system with multiple stability and stick-slip motion is still unclear. In the current paper, a novel nonlinear energy harvesting model with multiple stability of single-, double- and triple-well potential is proposed based on V-shaped structure spring and the belt conveying system. The dynamic equations for the energy harvesting system with multiple stability and self-excited friction are established by using Euler-Lagrangian equations. Secondly, the static characteristics of the nonlinear restoring force, the friction force, and the potential energy surfaces are obtained to show the nonlinear stiffness, multiple equilibrium points, discontinuous behaviors and multiple well responses. Then, the equilibrium surface of bifurcation sets for the autonomous system is given to show the third-order quasi zero stiffness (QZS3), fifth-order quasi zero stiffness (QZS5), double well (DW) and triple well (TW). The co-dimension bifurcation sets of the self-excited vibration system are analyzed and the corresponding phase portraits for the coexistent of multiple limit cycles are obtained. Furthermore, the analytical formula of amplitude frequency response of the approximated system are obtained by the complex harmonic method. The response amplitudes of charge, current, voltage and power of the forced electron-mechanical coupled vibration system for QZS3, QZS5, DW and TW are analyzed by using the numerically solution. Finally, a prototype of FIV energy harvesting system is manufactured and the experimental system is setup. The experimental works of static restoring forces, damping forcse and the electrical outputs are well agreeable with the numerical results, which testified the proposed FIV energy harvesting model.

Discovering differential governing equations of hysteresis dynamic systems by data-driven sparse regression method

Discovering differential governing equations of hysteresis dynamic systems by data-driven sparse regression method