Dynamic Equations Research Articles

Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation

Quasi-zero stiffness (QZS) metastructures with simple structures exhibit a high low-frequency vibration isolation performance, but the bearing capacity and working regions are limited due to the linear positive stiffness elements. In this study, a novel hyperbolic QZS metastructure coupled with nonlinear stiffness is designed by combining a pair of cosine-shaped beams and arc-shaped beams for low-frequency vibration isolation. Finite element models (FEM) and experimental analysis are utilized to examine the static characteristics of the metastructure, revealing a wide QZS region and significant load-bearing capacity. The metastructure can achieve effective vibration isolation properties through reasonably designing the unit cells. The influence of geometrical parameters on the QZS region is investigated to optimize and realize desirable QZS features. The dynamic equation and response of the vibration isolation system are determined using the harmonic balance method. Moreover, the corresponding displacement transmissibility is confirmed by numerical simulation. The results of the dynamic sweep frequency experiment further verify the validity of the theoretical analysis model. The hyperbolic QZS metastructure system possesses good vibration isolation performance due to the introduction and compensation of a nonlinear positive stiffness arc beam element. The proposed hyperbolic QZS metastructure provides a significant route for the stiffness compensation of metastructures.

A meshless computational framework for a modified dynamic system of vehicle coupled with plate structure

In previous simulations of train–bridge interaction systems (TBIS), the supporting system for the train are commonly treated as beam structures, leading to less accurate results, particularly for small-span cases. To address this limitation, a modified vertical TBIS is proposed. In the presented TBIS, the supporting system is considered as a Reissner–Mindlin plate, and the displacement field is described by first-order shear deformation theory (FSDT). To establish the model, radial point interpolation method (RPIM), a meshless method, is employed. Finally, a coupled dynamic equation is established to calculate various responses of the system. Several numerical examples are presented to illustrate the disparities between the system based on plate model and traditional beam model. The results indicate that the beam model yields higher estimates of the mid-span vertical displacement of the bridge, while the peak of the mid-span vertical acceleration is smaller compared to the plate model; additionally, it is observed that the carbody is primarily influenced by rail irregularities. Consequently, the proposed plate model offers distinct advantages over the beam model in providing comprehensive structural response information, thereby offering novel insights into bridge design and analysis. Additionally, this marks the inaugural application of the meshless method in the field of TBIS, which further extends the application scope of meshless methods.

On Grüss, Ostrowski and trapezoid-type inequalities via nabla integral on time scales

AbstractOstrowski inequality gives the absolute deviation of the function from its integral mean. Delta and nabla calculi are first two approaches to study time scales calculus. This article presents the Ostrowski inequality for univariate first order nabla differentiable function by using Montgomery identity established for nabla integrals. Some extensions of dynamic Ostrowski-type inequality are investigated with the help of integration by parts for nabla integrals, properties of the modulus and polynomials on time scales. Furthermore, dynamic Grüss and trapezoid-type inequalities are established in their generalized form for twice nabla differentiable functions by utilizing the Montgomery identity. In addition, the obtained inequalities are discussed for continuous and discrete time scales.

Location-scale modeling as an integrative approach to symptom dynamics during psychotherapy: An illustration with depressive symptoms.

Depressive symptom dynamics, including change trajectories and symptom variability, have been related to therapy outcomes. However, such dynamics have often been examined separately and related to outcomes of interest using two-step analyses, which are characterized by several limitations. Here, we show how to overcome these limitations using location-scale models in a dynamic structural equation modeling framework. We introduce location-scale modeling in an accessible manner to pave the way for its use in research integrating within-person dynamics and intervention-related change in psychopathology, and we illustrate this modeling approach in a large-scale internet-based intervention for depression (N = 1,656). Using eight data points sampled across about 8 weeks, we predicted improvement across the intervention (50% symptom reduction) as a function of early change and symptom variability. Early symptom change was associated with a more likely improvement across therapy. Variability of symptoms beyond change trajectories during the intervention was associated with less likely improvement. Location-scale models, and dynamic structural equation modeling more generally, are well suited to modeling how patterns of symptom change during psychotherapy are related to important (e.g., therapy) outcomes. Our illustrative application of location-scale modeling showed that symptom variability was associated with less overall improvement in depressive symptoms. However, this finding requires replication with more intensive sampling of symptoms before final conclusions can be drawn on when and how to distinguish maladaptive from adaptive variability during psychotherapy. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Data-driven reconstruction of chaotic dynamical equations: The Hénon–Heiles type system

In this study, the classical two-dimensional potential VN=12mω2r2+1NrNsin(Nθ), N∈Z+, is considered. At N=1,2, the system is superintegrable and integrable, respectively, whereas for N>2 it exhibits a richer chaotic dynamics. For instance, at N=3 it coincides with the Hénon–Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy E and the parameter N. Concrete results for the lowest cases N=3,4 are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.

Prescribed Asymptotic Behavior of Nonlinear Dynamic Equations Under Impulsive Perturbations

Prescribed Asymptotic Behavior of Nonlinear Dynamic Equations Under Impulsive Perturbations

Exact Maxwell evolution equation of resonator dynamics: temporal coupled-mode theory revisited

Despite its widespread significance, the temporal coupled-mode theory (CMT) lacks a foundational validation based on electromagnetic principles and stands as a phenomenological theory relying on fitted coupling coefficients. We employ an ab initio Maxwellian approach using quasinormal-mode theory to derive an “exact” Maxwell evolution (EME) equation for resonator dynamics. While the resulting differential equation bears resemblance to the classical one, it introduces novel terms embodying distinct physics, suggesting that the CMT predictions could be faulted by dedicated experiments, for instance carried out with short and off-resonance pulses, or with resonators of sizes comparable to or greater than the wavelength. Nonetheless, our examination indicates that, despite its inherent lack of strictness, the CMT enables precise predictions for numerous experiments due to the flexibility provided by the fitted coupling coefficients. The new EME equation is anticipated to be applicable to all electromagnetic resonator geometries, and the theoretical approach we have taken can be extended to other wave physics.

Opto-magnonic reservoir computing coupling nonlinear interfered spin wave and visible light switching

Physical reservoir computing is a promising approach to realize high-performance artificial intelligence systems utilizing physical devices. Recently, it has been experimentally found that nonlinear interfered spin wave multi-detection shows excellent performance for processing nonlinear time-series data due to its outstanding features: nonlinearity, short-term memory, and the ability to map in high dimensional space. However, said performance is considerably inferior to reservoir computing utilizing an optical circuit with a large volume. Herein, we develop reservoir computing with nonlinear interfered spin wave coupled with light switching, namely opto-magnonic reservoir computing. The spin wave was modulated through a crystal field transition that occurred in two different Fe3+ sites of Y3Fe5O12 by visible light switching, and it was found that the spin wave modulated by visible light switching dramatically reduced normalized mean square errors to 4.96 × 10−3, 0.163, and 3.66 × 10−5 for NARMA2, NARMA10, and second-order nonlinear dynamical equation tasks. Said excellent performance results from the strong nonlinearity caused by chaos and large memory capacity induced by reservoir states diversified by visible light switching.

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

GESO based RISE Controller for Active Flutter Suppression for Aeroelastic System

Abstract The paper presents a Robust Integral of Signum of Error (RISE) controller for active suppression of flutter of a two dimensional aerofoil and it is integrated with a General Extended State Observer (GESO). Towards this, aerofoil model in state space is first changed into a canonical form. Then, the controller is designed using RISE control technique. As equations of system dynamics are not into integral chain form, a GESO is formulated to observe system states and disturbances. The controller is implemented using estimated states and it is robustified using estimated disturbance. Stability of the proposed GESO-RISE controller is established using the Lyapunov theory. Simulations are performed to check the proposed controller's performance against variations in airspeed, uncertainties in model parameters, external disturbances, time delay and unmodeled dynamics. Comparison of the proposed GESO-RISE controller is carried out with existing controllers using two performance criteria i.e. Control Efforts and Integral of Absolute Error (IAE). It is discovered from simulations that the proposed GESO-RISE controller significantly reduces IAE and control efforts. Additionally, Monte Carlo method is utilized to check robustness of the proposed GESO-RISE controller. The proposed GESO-RISE controller significantly enhances flutter boundary of aerofoil and is completely implementable.

A Kalman filter approach for state estimation in weakly monitored active distribution networks

Distribution networks are becoming active to improve security of operation under high net load volatility. Yet, the deployment of active devices creates new challenges to the operation of the weakly monitored distribution networks. One such challenge is to improve state estimation accuracy. This paper proposes to improve the estimation process of active distribution networks by embedding the active devices dynamics into the state dynamic equations, instead of translating them into the static measurement models. This is carried out with the Kalman filter (KF) by making use of the dynamic models and the static measurement models we have proposed earlier for active distribution networks. The paper provides the main steps to implement the estimation framework and illustrates its advantages under slow-responsive distribution network interactions. Results obtained show that the proposed KF approach has better performance than the static counterparts for networks subjected to multiple control decisions — the case where the relevance information on dynamics surpasses the relevance of setpoint information on static models alone.

A Daily Diary Study of the Reciprocal Relation between Parental Psychological Aggression and Adolescent Anxiety in China.

The daily reciprocal relations between parental psychological aggression and adolescent anxiety and the heterogeneity, i.e., whether these relations vary across different adolescents, remain unclear. This study examined this issue with a 15-day daily diary study among 326 Chinese adolescents (Mage = 14.53 years, SD = 0.60, 47.2% girls). Dynamic structural equation models revealed that parental psychological aggression co-fluctuated with adolescent anxiety within a day. For lagged associations, only father-driven effects were supported but not mother-driven effects, whereas child-driven effects were supported for both parents. These within-person associations were heterogeneous across adolescents. Moreover, adolescents with more parental psychological aggression reported higher anxiety. This study revealed the reciprocal relations between parental psychological aggression and adolescent anxiety at the micro timescale and also highlighted that the within-person associations were heterogeneous across different adolescents.

A Framework for Rapidly Predicting the Dynamics of Flexible Solar Arrays in the China Space Station with a Verification Based on On-Orbit Measurement Data

The Chinese space station is a complex structure with large flexible appendages. Obtaining the on-orbit response characteristics of such a structure under different working conditions is a traditional and classic challenge in the field of dynamics. To address the on-orbit dynamics of the China Space Station, the basic equations for dynamic reduction, assembly and data recovery of linear and nonlinear substructures are derived based on the reduction and recovery theory, and a fast coupling analysis framework for flexible systems with nonlinear attachments is formed. This coupling analysis framework is adopted to quickly acquire the dynamic response of the China Space Station during in-orbit operation, thereby guiding the design. Taking SZ-15 radial docking to the Chinese Space Station as the object, the substructure of six nonlinear flexible arrays is reduced, the full flexible dynamic equation of the space station is assembled, and the response of each part of the flexible wing during the docking process is analyzed and recovered. By designing a reasonable and reliable flexible wing test scheme in-orbit, the acceleration at the root and top of the flexible wing during the docking of SZ-15 is obtained. The measured data in-orbit show that the acceleration analysis results of the typical parts of the flexible wing have a good agreement, which verifies the correctness of the fast coupling analysis framework of the flexible system. Hence, the dynamic coupling characteristics analysis of the main structure of the space station and the flexible wing based on this method can better guide the rationality of the design of the dynamic characteristics of the Chinese Space Station.

Risk Analysis and Simulation of Large Bridge Construction Based on System Dynamics

To reduce risk incidents in large bridge construction, it is necessary to study the interaction mechanisms and dynamic changes among various risk factors during the construction of large bridges. First, the evolution mechanism of construction risks for large bridges is analyzed, and a risk factor structure system is established. Then, based on system dynamics theory, a causal loop diagram and flow–stock diagram are constructed, and system dynamics equations for each variable in the flow diagram are established to build a system dynamics model. Finally, taking a large bridge construction project as an example, the variable equations are assigned actual values, and the construction risk level is simulated and analyzed. The results show that, ➀ continuous investment in safety funds can effectively reduce the overall risk level of the system, ➁ changes in the management risk subsystem have a significant impact on the overall risk level of large bridge construction, and ➂ increasing the safety investment ratio in both the personnel risk subsystem and the management risk subsystem can effectively reduce the total risk level of the system.

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

Constructing the soliton wave structure to the nonlinear fractional Kairat-X dynamical equation under computational approach

In this paper, the nonlinear fractional Kairat-X equation is investigated on the basis of computational simulation. The nonlinear fractional Kairat-X equation is an integrable equation and is used to explain the differential geometry of curves and equivalence aspects. Several kinds of solitary wave structures of the nonlinear fractional Kairat-X equation are established successfully via the implantation of the extended simple equation method. Here, we explore the interesting, novel and general solutions in trigonometric, exponential, and rational types, which represent periodic wave solitons, mixed solitons in the shape of bright–dark solutions, kink wave solitons, peakon bright and dark solitons, anti-kink wave solutions, bright solitons, dark solitons, and solitary wave structure. The physical structures of secured results, aided by numerical simulation, have numerous applications in applied sciences such as optical fiber, geophysics, laser optics, mathematical physics, nonlinear optics, nonlinear dynamics, communication system, and engineering. This study explores the physical behavior of models through the visualization of solutions in contour, 2D and 3D plots by revealing that these solutions yield profitable results in the field of mathematical physics. The study demonstrates that the proposed technique is more reliable, efficient, and powerful in analyzing nonlinear evolution equations in various domains of science.

On stability of equations with an infinite distributed delay

For scalar equations of population dynamics with an infinite distributed delay x′(t)=r(t)[∫−∞tf(x(s))dsR(t,s)−x(t)], x(t)=φ(t), t⩽t0, where f is the delayed production function, we consider asymptotic stability of the zero and a positive equilibrium K. It is assumed that the initial distribution is an arbitrary continuous function. Introducing conditions on the memory decay, we characterize functions f such that any solution with nonnegative nontrivial initial conditions tends to a positive equilibrium. The differences between finite and infinite delays are outlined, in particular, we present an example when the weak Allee effect (meaning that f′(0)=1 together with f(x)>x , x∈(0,K) ) which has no effect in the finite delay case (all solutions are persistent) can lead to extinction in the case of an infinite delay.

Vibration Isolation and Launch Performance Enhancement of the Spacecraft In-Orbit Launch Design Using the Nonlinear Dynamic Feature

This paper proposes a new spacecraft in-orbit launch design using a nonlinear configuration to utilize nonlinear dynamics for the enhancement of vibration isolation and launch performance. The in-orbit launch device has four springs, where the stroke directions of two springs are perpendicular to the launch direction so as to produce nonlinearity with negative stiffness for enhancing the launch velocity. The other two springs are designed to counterbalance the above negative stiffness when the launch outlet is shut down, leading to quasi-zero dynamic stiffness for vibration isolation enhancement. The dynamic equations of the in-orbit launch device for both the on- and off-launch are presented. Then the performance enhancement of both the vibration isolation and launch performance is thoroughly investigated via comparative study and parametric study. The resonance peak is reduced by 4.16 dB, the effective vibration isolation bandwidth is increased by 57%, and the launch speed is increased 1.64 times. This validates the performance improvement of the new launch device design and presents a useful guideline for application.

Analyzing the Effect of Tethered Cable on the Stability of Tethered UAVs Based on Lyapunov Exponents

In the working process of the tethered unmanned aerial vehicle (UAV), there is interference from the tethered cable, which can easily lead to the instability of the UAV. To solve the above problems, a method based on the Lyapunov exponent is proposed to analyze the stability of tethered cables for tethered UAVs. The dynamics equation of the UAV platform is established using the Euler–Poincare equation. The tension formula of the tethered cable is derived from the catenary theory and the principle of micro-segment equilibrium. Based on the Lyapunov exponential method, the stability changes of the tethered UAV in the takeoff, hovering, and landing stages are simulated and analyzed in a MATLAB environment. Prototype tests are carried out to prove the correctness of the simulation model and calculation conclusions. The results show that with an increase in the density of the tethered cable, the stability of the tethered UAV tends to decrease. At the same time, stability is affected by the density of the tethered cable more often during takeoff than during landing.

A Novel Investigation on Propagation of Envelop Optical Soliton Structure Through a Dispersive Medium in the Nonlinear Whitham–Broer–Kaup Dynamical Equation

A Novel Investigation on Propagation of Envelop Optical Soliton Structure Through a Dispersive Medium in the Nonlinear Whitham–Broer–Kaup Dynamical Equation