Nonlinear Dynamic Equations Research Articles

Partial‐differential‐algebraic equations of nonlinear dynamics by physics‐informed neural‐network: (I) Operator splitting and framework assessment

Partial‐differential‐algebraic equations of nonlinear dynamics by physics‐informed neural‐network: (I) Operator splitting and framework assessment

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features.

Studying Dynamical and Hydraulic Characteristics of the Hydraulically Suspended Passive Shutdown Subassembly (HS-PSS) and Validating with a Prototypic Test Sample

The pool-type demonstrative China Fast Reactor 600 (CFR-600) adopted a series of improved safety designs, among which the hydraulic suspended passive shutdown subassembly (HS-PSS) is employed for inherent safety enhancement, especially suitable against unprotected loss of flow accident (ULOF). In this article, functional requirements for HS-PSS design in hydro-dynamic aspects are proposed, with the corresponding performance indicators discussed. To address these functional requirements, qualitative analysis on the equilibrium solution properties of the nonlinear dynamical model equation of the hydraulic moving body (HMB) in the HS-PSS are conducted, which leads to the determination of an applicable design parameter domain of the HMB for its practical design from the broad range of structural and parametric design options available for HS-PSS. Furtherly, hydraulically characterizing and modeling the constituent paths and consequent fluid network, hydraulic characteristics of the HS-PSS, as well as the coupled hydro-dynamic motion behaviors of the HMB for suspension and dropping states, were simulated and test-validated. Considering the HS-PSS hydro-dynamic behaviors, key indicators such as critical flowrate, drop time, hydraulic self-tightening performance, as well as the hydraulic characteristics curve are for fulfilling the functional requirements. Meanwhile, through sensitivity study of some structural parameters‘ impact on hydraulic characteristics, some most sensible structural parameters for adjusting and optimizing detailed design are observed. The work is quite significant in supporting the conceptual design of the HS-PSS as well as its engineering improvement.

A Near-Infrared Retinomorphic Device with High Dimensionality Reservoir Expression.

Physical reservoir-based reservoir computing (RC) systems for intelligent perception have recently gained attention because they require fewer computing resources. However, the system remains limited in infrared (IR) machine vision, including materials and physical reservoir expression power. Inspired by biological visual perception systems, the study proposes a near-infrared (NIR) retinomorphic device that simultaneously perceives and encodes narrow IR spectral information (at ≈980nm). The proposed device, featuring core-shell upconversion nanoparticle/poly (3-hexylthiophene) (P3HT) nanocomposite channels, enables the absorption and conversion of NIR into high-energy photons to excite more photo carriers in P3HT. The photon-electron-coupled dynamics under the synergy of photovoltaic and photogating effects influence the nonlinearity and high dimensionality of the RC system under narrow-band NIR irradiation. The device also exhibits multilevel data storage capability (≥8 levels), excellent stability (≥2000 s), and durability (≥100 cycles). The system accurately identifies NIR static and dynamic handwritten digit images, achieving recognition accuracies of 91.13% and 90.07%, respectively. Thus, the device tackles intricate computations like solving second-order nonlinear dynamic equations with minimal errors (normalized mean squared errorof 1.06 × 10⁻3 during prediction).

Optimization method of vibration isolation performance of the 6DOF vibration isolation platform based on different configurations

The configuration optimization methods are developed to design the six degree of freedom (6DOF) passive vibration isolation platform with superior isolation performance. According to different position arrangements, various configuration vibration isolation platforms, including the 33 type, 63 type, 66–1 type, and 66–2 type, are designed and systematically analyzed. Through analysis of the geometric characteristics of the configuration platforms, the nonlinear dynamical equations for each configuration vibration isolation platform are established using Hamilton's principle. Utilizing the transmission rate as an evaluation index, the isolation performance of each platform in six directions (three translational and three rotational) is systematically studied. The results indicate that: (a) Through the comparative analysis of different configuration vibration isolation platforms, it can be found that the 66–1 configuration not only has the largest loading capacity but also owns the best isolation performance in the x, y, and γ directions. The 33 configuration has better isolation performance in the z, α, and β directions, while having greater loading capacity; (b) Through configuration optimization, a 6DOF vibration isolation platform can be obtained with high static stiffness in the vertical direction and low dynamic stiffness in other external excitation directions. This kind of HSLDS characteristics will significantly enhance the isolation performance of the vibration isolation platform; (c) Workspace analysis under different configurations reveals that the 66–1 configuration offers the largest workspace in all six directions, followed by the 63 configuration, while the workspaces of the 33 and 66–2 configurations are similar.

Investigation of the nonlinear dynamics of thin sandwich shells composed of functionally graded materials with double curvature in thermal environments

Double curvature structures play a crucial role as load-bearing components across various engineering fields. Functionally graded materials, blending metals and ceramics, boast superior material characteristics, fueling their expanding applications. This study pioneers the non-linear dynamic analysis of sandwich shells that feature functional gradient compositions in thermal settings. We assume that both the upper and lower shells of these double curvature structures are crafted from functionally graded materials, with a ceramic core. This variation in material exhibits a ceramic layer on the inner surface and a metallic layer on the outer surface, showing properties that vary along the shell thickness in a power-law gradient. Non-linear dynamic equations are derived using third-order shear theory, encompassing geometric non-linearity and shear deformation. Employing the Galerkin method, we discretize the equations of motion into a non-linear dynamic system with five degrees of freedom, and subsequently give an analytical expression for the nonlinear naturalfrequency by means of multiscale analysis. Our discussion examines the impacts of structural parameters, porosity volume fraction, volume fraction index, and temperature differences on the non-linear/linear frequency ratio of doubly curved sandwich shells. Calculations reveal a sharp rise followed by a rapid decline in the non-linear/linear frequency ratio with increasing structural aspect ratio b/a. Conversely, it decreases with increasing thickness-to-length and radius-to-length ratios, with temperature differences initially reducing and later increasing it. These findings offer practical insights for designing functional gradient double curvature sandwich shells.

Dynamic buckling active control of FGM spherical shell with piezoelectric sensor and actuator layers

The mechanical snap-through phenomenon represents one of the buckling modes inherent in shallow shells, exemplifying the shell's reaction to external loads, which can potentially have detrimental effects on structural performance. This article delves into the study of this phenomenon within the context of spherical shells. Leveraging functionally graded material (FGM) in the main layer of this structure proves instrumental in enhancing the shell's performance against rapid mechanical loads. Furthermore, the strategic placement of piezoelectric layers, both symmetrically and asymmetrically, coupled with the implementation of a proportional and derivative control system, facilitates the control of the shell's response. The formulation of displacement field and electric potential is grounded in the first-order shear deformation theory (FSDT) and second-order distribution, respectively. The strong form of nonlinear dynamic and electrostatic equations is derived using Hamilton's principle. Solving these equations is achieved through the implementation of the generalized differential quadrature (GDQ) method in the spatial domain and Newmark in the time domain. Additionally, the Newton-Raphson method is employed to tackle the inherent nonlinearity of the problem. The Budiansky's criterion is employed for the assessment of the load necessary for the buckling of the shell. Following the validation of results against numerical, analytical, and laboratory outcomes, a set of parametric results is presented. These results underscore the remarkable impact of the control system in mitigating shell deformation under time-dependent mechanical loads, as well as in postponing the occurrence of the snap-through phenomenon and damping the vibrations.

Mass superinflation in the Reissner-Nordström black hole

Ever since Penrose and Simpson's prediction contradicted Novikov's idea that an infalling person would emerge into an asymptotically flat universe, there have been a continued interest in predicting the nature of singularity at the inner horizon of a Reissner-Nordström black hole. This prediction was first confirmed by Poisson and Israel using cross-stream of massless particles, leading to the phenomenon coined as mass inflation. On the other hand, Ori obtained a weaker singularity using a null shell of radiation. Thus it is important to capture the nature of singularity at the inner horizon. We therefore consider in this work a massive scalar field coupled to the Reissner-Nordström spacetime. The ensuing field equations lead to a coupled set of nonlinear dynamical equations. For definiteness, we analytically solve these equations employing the Adomian decomposition method. This facilitates in obtaining a closed form solution that exhibits an unbounded double-exponential growth in the mass function, giving rise to a novel phenomenon coined herein as mass superinflation. The scalar field is also found to undergo a very strong blueshift at the inner horizon.

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.

Stability analysis for passive robots walking on inclined surfaces with local angles

For common-legged robots with a kind of stiff joints, the human-like and consecutive gait shown by passive walking robots is a bit difficult to achieve because of varying degrees of control, which is usually accompanied by computational cost. Of course, passive dynamic walking undoubtedly has its inevitable disadvantages because of the lack of control, which is stability. At the same time, the significant nonlinearity of the passive gait increases the difficulty of realizing the walking stability of the passive robot. Therefore, the passive robot still needs certain control to achieve stable walking even on the natural plane or smooth slope. In this paper, a passive robot model walking on an inclined plane with a local angle is studied. First of all, the approximate solution of the nonlinear dynamic equation is given by the perturbation method, and the conditions for the robot to achieve stable walking without external forces are obtained. Further, the input-output feedback linearization control based on hybrid zero dynamics is employed to carry out virtual constraints on the passive robot during the swing stage of walking, facilitating the transition from the unstable state to the stable periodic state. From the result of the experiments, the walking stability of the passive robot is improved over a larger range of walking status values compared with the condition without external force control, and the use of this method reduces the complicated work of solving the fixed point. These findings possibly have reference value in passive walking stabilization control. Meanwhile, the simulation results obtained by designing and studying the minimum controlled walking model show that human walking is based on the uncontrolled mechanical process. The exploration of the mechanism of passive gait can provide some enlightenment to the research of human texture and the application of exoskeleton.

Nonlinear Vibrations of the Piezoelectric-Laminated Composite Cantilever Rectangular Plate Subjected to the Transverse and Parametric Excitation

The intelligent structure will be the key to the industrial manufacturing field in the future, piezoelectric materials are the most commonly used active materials in many mechatronic and vibration control systems. In this paper, we illustrated the nonlinear vibration behaviors of the piezoelectric-laminated composite cantilever rectangular plate subjected to transverse and parametric excitation. Two piezoelectric layers are attached to the upper and lower surfaces of graphite/epoxy composite laminated plate. The nonlinear dynamic equations are derived via the classical laminated plate theory, von Karman large deformation theory, the constitutive equation of the piezoelectric materials, and the Hamilton principle. Considering the Galerkin method, the nonlinear ordinary differential equations for the two-order vibration mode of the piezoelectric laminated composite cantilever rectangular plate are achieved. The multiple scale method is applied to obtain the average equations to study the nonlinear vibration characteristic of the stable motion. The primary resonance and the 1:1 internal resonance of the piezoelectric laminated composite cantilever rectangular plate are investigated. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase curves, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristic of the piezoelectric laminated composite cantilever rectangular plate. The detailed parametric analyses show that bubble response curves are separated from the amplitude-frequency response curves with the continuous increase of the external excitation due to the internal resonance. In addition, the results reveal the energy transform between the various order vibration modes of the system. This work is expected to provide theoretical guidance for the nonlinear vibration of the smart piezoelectric laminated composite structure.

Transient Synchronous Stability Modeling and Comparative Analysis of Grid-Following and Grid-Forming New Energy Power Sources

New energy power sources can be categorized into grid-following and grid-forming types based on their synchronization characteristics with the grid. Due to the different basic control operation principles of these two types of new energy power sources, they can lead to the instability of the grid-connected system through different paths. A mathematical model is established to describe the dynamic characteristics of the grid-following and the grid-forming new energy power resources, and the control block diagram of the nonlinear dynamic characteristic equation is compared and analyzed. The similarity between grid-following and grid-forming new energy power supply is revealed, and we preliminarily discuss the influencing factors of the stability of new energy sources. Using a single-machine infinite system model, the influence of key control parameters on transient stability is separately investigated for grid-following and grid-forming types. The influence of the converter synchronous loop control parameters on the system is verified, and the factors affecting the stability are discussed in combination with current limiting and frequency limiting. The research will provide reference for the design of a new energy power grid connection system.

Nonlinear static and dynamic response of a metastructure exhibiting quasi-zero-stiffness characteristics for vibration control: an experimental validation

This work introduces a novel metastructure designed for quasi-zero-stiffness (QZS) properties based on the High Static and Low Dynamic Stiffness mechanism. The metastructure consists of four-unit cells arranged in parallel, each incorporating inclined beams and semicircular arches. Under vertical compression, the inclined beams exhibit buckling and snap-through behavior, contributing negative stiffness, while the semicircular arches provide positive stiffness through bending-dominated behavior. The design procedure to achieve QZS is established and validated through finite element analysis and experimental investigations. The static analysis confirms a QZS region for specific displacement. Dynamic behavior is analyzed using a nonlinear dynamic equation solved using the Harmonic Balance Method, validated experimentally with transmissibility curves showing sudden jump down with effective vibration isolation. Parametric studies with varied payload masses and excitation amplitudes further verify the ability to of metastructure to attenuate vibrations effectively in low-frequency ranges.

Oscillatory behavior of second‐order nonlinear delay dynamic equations with multiple sublinear neutral terms utilizing canonical transformation

This paper aims to present innovative criteria for characterizing the oscillatory behavior of second‐order nonlinear delay dynamics expressed in noncanonical form with sublinear neutral terms. We achieve this by converting the noncanonical equation into an equivalent canonical form. The practical examples presented throughout this paper illustrate the significance of our findings and contribute to enhancing, generalizing, and augmenting the existing literature on this subject.

An Approach to Solve Gas Dynamic Equation by Fuzzy Hilfer Fractional Differential Equation

In this work, Adomian decomposition method (ADM) based on Hilfer derivative is proposed to solve nonlinear fuzzy fractional gas dynamic equation. We reform the provided fractional gas dynamic equation into fuzzy Hilfer fractional differential equation. Subsequently with description of Hilfer fractional derivative and suitable initial conditions under fuzzy sense, we obtain approximate solution to the given problem. This method provides an analytical solution in the form of infinite power series. The behavior of the solution is illustrated using graphical representation.

Bifurcation study, phase portraits and optical solitons of dual-mode resonant nonlinear Schrodinger dynamical equation with Kerr law non-linearity

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

On the qualitative study of solutions of a class of nonlinear abstract dynamic equations on time scales and applications

On the qualitative study of solutions of a class of nonlinear abstract dynamic equations on time scales and applications

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

Investigation on parametric excitation vibration and piezoelectric active control of axially moving viscoelastic beams

The nonlinear dynamics equation governing the behavior of axially moving viscoelastic beams is established in this paper, based on Hamilton’s principles. The Galerkin method is employed to discretize the equation for numerical analysis. Nonlinear parametric vibration control for axially moving viscoelastic beams is explored using piezoelectric laminated actuators based on LQR optimal linear control and SMC sliding mode control strategies. The stability of the moving beam is examined in respect of moving speed and active control. The influence of actuator position and control schemes on vibration control performance for the moving beam is also examined. In respect to linear and nonlinear equations, two control strategies of LQR are given to illustrate the control performances. The sliding mode control (SMC) is also given to reduce the vibration of the moving viscoelastic beam and exhibit the control performance and its robustness.

Modulation instability analysis, and characterize time-dependent variable coefficient solutions in electromagnetic transmission and biological field

This study integrates the telegraph model with the traveling wave coefficient. This model characterizes planar random motion of particles in fluid flow, pressure waves from pulsatile blood flow in arteries, electrical impulses in nerve and muscle cell axons, and electromagnetic waves in superconducting media. Using the efficient and well-established Enhanced Modified Simple Equation (EMSE) method, we investigate solitary wave solution with the variable coefficient of the telegraph model. To investigate the variable coefficient solitary wave solution, we apply a transformation variable η = h(t)x + g(t), which is dependent on the time variable function. The diverse functions of h(t) and g(t) yield many novel and exclusive solutions. Numerical solutions are plotted in 3D, density, and 2D. instanton soliton, kinky periodic wave, lump wave, kink wave, anti-kink wave, double periodic wave, diverse types periodic wave, lump with bell wave periodic wave, periodic lump wave, etc. Solitary waves explain complex wave phenomena through nonlinear effects like self-interaction and dispersion. Due to the classical nonlinear telegraph model's wave dynamics, they are used in brain function and communication system design. We conclude by testing the governing model's modulation instability. Dispersive and nonlinear processes modulating the stable state cause instability in high-order nonlinear equations. Examining the wave movement role and modulation instability analysis to assess solution stability shows that all solutions are accurate and stable. The computational difficulties and results demonstrate the approaches' clarity, effectiveness, and simplicity, suggesting they can be applied to evolutionary dynamic and static nonlinear equations in computational physics, other real-world situations, and numerous academic disciplines.