Dynamic Equations Research Articles

Dynamic modeling of flexible-link manipulators with time-varying link lengths operating in fluid medium

This paper presents the derivation of dynamic equations of motion for flexible-link manipulators with time-varying link lengths operating in stable fluid environments. The modeling considers the interaction between the manipulator and the surrounding fluid during simultaneous rigid-body and elastic motions. The time-varying link lengths introduce additional translational movements, which contribute to the overall rigid-body motion, comprising rotational and reciprocating motions, as well as link oscillations due to elasticity. These motions are influenced not only by the motor excitations but also by the interaction between the manipulator links and the fluid medium. Notably, the fluid interactions introduce resistive forces during linear motions, leading to deviations from the behavior of manipulators with fixed link lengths. Consequently, the fluid-robot interaction affects both the elastic and rigid-body modes. Although the equations share similarities with those for time-varying structures studied previously, the presence of resistive forces results in a complex model with new non-conservative forces. The system dynamic equations are derived using a recursive Gibbs-Appell formulation, accounting for changes in the fluid's mechanical properties, link elasticity, and the effects of system mass variations. The results are verified using MATLAB simulations. The outcomes show that the link's deformation increased by 100% when the medium's density changed from vacuum to fluid with a density of 100 kg/ m3.

Spatiotemporal analysis of a modified Leslie–Gower model with cross-diffusion and harvesting

This paper considers a modified Leslie–Gower prey-predator reaction–diffusion model introducing harvesting of both species. Both the temporal and spatiotemporal dynamics of the model have been examined. We have found the stability regions and drawn bifurcation diagrams to determine the harvesting effect on the model, revealing that the harvesting has a stabilizing effect. Local bifurcations, such as transcritical and Hopf bifurcations, appear in the temporal system. For the spatiotemporal model, Turing instability conditions have been determined. The amplitude equation for the critical modes has been derived using multiple time scale analyses by taking the harvesting effort as the bifurcating parameter. Also, we have verified the theoretical results by plotting several kinds of stationary patterns, including stripes, spots, and a mix of stripes and spots. This study’s critical observation is that as harvesting effort rises, the patterns steadily turn into spots, i.e., harvesting influences pattern creation strongly. This fosters a dynamic equilibrium, allowing competitors to maintain distance, optimize resource use and survive.

Semi-analytical modeling and vibration characteristics analysis of the orthogonally stiffened cylindrical shell with variable cross-sections of stiffeners

The stiffened shell with variable cross-sections of stiffeners (SS-VC) has important applications in aerospace and other fields. Its excellent mechanical properties provide new possibilities for the design and performance improvement of stiffened structural parts. However, its dynamic modeling problems have been urgent to be solved. In this study, the dynamic model of the orthogonally stiffened cylindrical shell with variable cross-sections of stiffeners (OSCS-VC) is established by the semi-analytical method (SAM) and it can be described as follows. The displacement allowable functions of the structure are constructed by using the Gram-Schmidt orthogonalization method. Based on Sanders shell theory, the stress-strain relationships of the longitudinal and ring stiffeners with variable cross-sections are derived under the variable limit integration. The boundary spring stiffness is obtained by the inverse identification technique. The dynamic equation of OSCS-VC is established and solved by using the Lagrange equation. Then, a case study is carried out, the rationality of the semi-analytical dynamic model of OSCS-VC is verified by ANSYS engineering software, literature and the experiment system. Finally, based on the semi-analytical model of OSCS-VC, the influence of the characteristic parameters of cross-sectional functions (CSF) for the longitudinal and ring stiffeners on the natural frequencies is analyzed.

Radial basis function neural network based data-driven iterative learning consensus tracking for unknown multi-agent systems

This paper provides a novel data-driven-distributed-consensus control protocol for unknown nonlinear non-affine discrete-time multi-agent systems (MAS) with repetitive properties. The leader’s commands are directed to the followers in the topological graph. The dynamic linearization technology (DLT) is used to build the distributed iterative learning (IL) controller along the iteration axis. In the iterative process, the control gain is automatically adjusted by updating the weight matrix of the high-order radial basis function neural network (RBFNN, HORBFNN). In global control, the higher order parameter (HOP) Newton method is used to achieve global convergence and stability of the control process. All the above processes do not require the understanding of dynamical equations or physical models for each agent, and only use local communication information of multi-agent to achieve consistent tracking of MAS leaders and followers. Based on the strong connection, the convergence performance, stability and boundedness properties of the proposed control protocol in the fixed topology as well as in the iterative topology are validated by a rigorous theoretical analysis. Simulation experiments are conducted to verify the effectiveness of the control protocol.

Robust motion control for an underactuated wheeled bipedal robot utilizing sliding mode strategy

In recent years, wheeled bipedal robots (WBRs) have become one of the frontier fields of robotic research, attracting immense interest from scholars globally. They can not only move rapidly on flat ground through wheels but also possess satisfactory adaptability to uneven terrain due to the introduction of legs. With concise structure and strong agility, WBRs have broad prospects of application in logistics, routing inspection, home service, and so on. However, the inherent underactuation of such robots presents a significant challenge in terms of balance control, particularly in the face of uncertainties and external disturbances. To address this problem, this paper presents a sliding mode control strategy for WBRs, which guarantees robust balance performance even under the influence of various external disturbances. Specifically, the dynamic equations of WBRs are first rearranged in cascaded form, facilitating targeted control design. After that, the wheel torque and leg supporting force are carefully devised to ensure that the sliding surfaces converge to zero within a fixed time. Rigorous Lyapunov-based analysis has shown that the desired equilibrium point is asymptotically stable. Finally, extensive hardware experiments are undertaken, providing convincing evidence of the superior balance control performance achieved by the proposed method.

Improvement and validation of discrete-sectional method based code for fast reactor aerosol dynamics

The study of the dynamics of aerosols produced during a severe accident is vital to the safety analysis of a Sodium-cooled Fast Reactor (SFR). Sodium leakage into the containment following a severe accident may result in the production of sodium aerosols along with fission product aerosols. The in-containment radioactive source term depends upon the settling behaviour of fission product aerosols, which co-agglomerate with sodium fire aerosols. Hence, an in-depth understanding of the dynamics and deposition of aerosols within the containment is essential for the safety assessment of an SFR. The current work uses an open-source code based on the Discrete-Sectional (DS) method to solve the simplified form of General Dynamics Equation (GDE) for aerosols relevant to fast reactors. Aerosols of SrO2, CeO2 and sodium are considered in the present work. The DS code has been modified and further improved by including gravitational coagulation, turbulent coagulation, Brownian deposition, gravitational deposition and thermophoretic deposition so that the code can handle the different processes leading to the deposition of aerosols following a sodium fire. The code is also enhanced to account for the effect of relative humidity through the modified Cooper's equation. The improved code is then validated with different experiments conducted in the Aerosol Test Facility (ATF), India; Mini Sodium Fire Facility (MINA), India; and AHMED (Aerosol and Heat Transfer Measurement Device), Finland. Validation from different facilities confirms the applicability of the code to various scenarios. It is found that the modified DS code could predict the decay of suspended mass concentration of aerosols in different enclosures with reasonable accuracy.

A special memristive diode-bridge-based hyperchaotic hyperjerk autonomous circuit with three positive Lyapunov exponents

This work presents an autonomous hyperjerk type circuit where a generalized memristor consisting of a diode-bridge and an RC filter acts as nonlinear component. The dynamics equations of the proposed circuit are presented in the form of an infinitely differentiable (i.e. smooth) system of order six. A detailed analysis of the model, carried out using classic techniques for studying nonlinear systems, reveals surprising behaviors such as the coexistence of bifurcation modes, non-trivial transient behaviors, offset boosting, torus, chaos, as well as hyperchaos with three positive Lyapunov exponents. These results are obtained by varying both the initial states and the model parameters. This multitude of dynamic properties is verified in the laboratory by carrying out series of measurements on the prototype of the memristive circuit. To the best of our knowledge, the circuit proposed in this article represents the simplest memristor-based circuit known to date in the relevant literature which can generate hyperchaotic signals with three positive Lyapunov exponents.

Storm: Incorporating transient stochastic dynamics to infer the RNA velocity with metabolic labeling information

The time-resolved scRNA-seq (tscRNA-seq) provides the possibility to infer physically meaningful kinetic parameters, e.g., the transcription, splicing or RNA degradation rate constants with correct magnitudes, and RNA velocities by incorporating temporal information. Previous approaches utilizing the deterministic dynamics and steady-state assumption on gene expression states are insufficient to achieve favorable results for the data involving transient process. We present a dynamical approach, Storm (Stochastic models of RNA metabolic-labeling), to overcome these limitations by solving stochastic differential equations of gene expression dynamics. The derivation reveals that the new mRNA sequencing data obeys different types of cell-specific Poisson distributions when jointly considering both biological and cell-specific technical noise. Storm deals with measured counts data directly and extends the RNA velocity methodology based on metabolic labeling scRNA-seq data to transient stochastic systems. Furthermore, we relax the constant parameter assumption over genes/cells to obtain gene-cell-specific transcription/splicing rates and gene-specific degradation rates, thus revealing time-dependent and cell-state-specific transcriptional regulations. Storm will facilitate the study of the statistical properties of tscRNA-seq data, eventually advancing our understanding of the dynamic transcription regulation during development and disease.

Criteria for oscillation of noncanonical superlinear half-linear dynamic equations.

This article comes up with criteria to make sure that the solutions to superlinear, half-linear, and noncanonical dynamic equations oscillate in both advanced and delayed cases; these criteria are comparable to the Hille-type and Ohriska-type criteria for the canonical nonlinear dynamic equations; and also these results solve an open problem in many existing works in the literature on dynamic equations. To demonstrate the importance of the results, some examples have been introduced.

Affect and executive function dynamics in primary school classrooms: an intensive longitudinal study

ABSTRACT This study investigates the temporal dynamics and affective associations related to executive function (EF) performance in primary school classrooms using an intensive longitudinal design. Data were collected from 35 students aged 8.9 to 11.4 years. Participants reported their affective experiences and completed EF measures three times daily following a fixed sampling schedule. The data collection spanned two consecutive school weeks across three primary school classrooms. Utilising Dynamic Structural Equation Modeling (DSEM), we examined 505 measurements of EF tasks and self-reported affective states over two weeks. The findings reveal significant within-person variability in EF, accounting for 52% of the observed variance, with performance declining later in the day and week. At the within-person level, positive affect was associated with improved EF performance, while negative affect was associated with poorer EF. No significant between-person relationships were found. These results underscore the importance of considering within-person processes and affective experiences in educational settings and highlight the need for further research employing intensive longitudinal methods to better understand the nuanced dynamics of affect and EF in real-world classroom environments.

Modeling the delay propagation dynamics in high-speed railway networks with a traffic-driven SIS epidemic model

The propagation of delays is a prevalent issue in high-speed railway (HSR) networks. However, the underlying properties of delay propagation have not been fully explored, hindering the effective management of this problem. In this paper, considering the impact of traffic flow, we model the delay propagation as a traffic-driven susceptible–infected–susceptible (SIS) epidemic spreading process. We introduce a traffic frequency matrix to estimate the traffic rate and further derive the propagation dynamic equations using a continuous-time Markov chain method. We then derive the epidemic threshold of the model, which is proportional to the recovery rate and inversely proportional to the average path length. Finally, through simulation, we study the impacts of various factors including the infection rate, recovery rate, and the degree and number of initial delay stations. Our work provides some insights for the prediction and control of delay propagation in HSR networks.

A theoretical model for compressible bubble dynamics considering phase transition and migration

A novel theoretical model for bubble dynamics is established that simultaneously accounts for the liquid compressibility, phase transition, oscillation, migration, ambient flow field, etc. The bubble dynamics equations are presented in a unified and concise mathematical form, with clear physical meanings and extensibility. The bubble oscillation equation can be simplified to the Keller–Miksis equation by neglecting the effects of phase transition and bubble migration. The present theoretical model effectively captures the experimental results for bubbles generated in free fields, near free surfaces, adjacent to rigid walls, and in the vicinity of other bubbles. Based on the present theory, we explore the effect of the bubble content by changing the vapour proportion inside the cavitation bubble for an initial high-pressure bubble. It is found that the energy loss of the bubble shows a consistent increase with increasing Mach number and initial vapour proportion. However, the radiated pressure peak by the bubble at the collapse stage increases with decreasing Mach number and increasing vapour proportion. The energy analyses of the bubble reveal that the presence of vapour inside the bubble not only directly contributes to the energy loss of the bubble through phase transition but also intensifies the bubble collapse, which leads to greater radiation of energy into the surrounding flow field due to the fluid compressibility.

Dynamic modelling and predictive position/force control of a plant-inspired growing robot

This paper presents the development and control of a dynamic model for a plant-inspired growing robot, termed the 'vine-robot', using the Euler-Lagrangian method. The unique growth mechanism of the vine-robot enables it to navigate complex environments by extending its body. We derive the dynamic equations of motion and employ model predictive control to regulate the task space position, orientation, and interaction forces. Simulation experiments are conducted to evaluate the performance of the proposed model and control strategy. The results demonstrate that the model effectively achieves sub-millimeter precision in the position control in both static and time varying refrence trajectroies, and sub micronewton in force control.

Improvements on the Leighton‐type oscillation criteria for impulsive differential equations and extension to non‐canonical case

We investigate the oscillatory behavior of solutions of second‐order linear differential equations with impulses. These equations can be categorized into two types: canonical and non‐canonical. This study examines the canonical and non‐canonical impulsive differential equations in connection with the famous Leighton‐type oscillation theorem. The well‐known theorem states that every solution of where and are continuous with , is oscillatory if and This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition fails, that is, when the equation is of non‐canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non‐canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

Numerical Study of Complex Heat Transfer and Aerodynamics in a Tube Furnace with a Flat Fuel Combustion Mode

The object of the study is a tubular furnace for steam reforming of natural gas. The channel walls are formed by a refractory lining and a tubular screen with a known temperature. The chamber volume is filled with a selectively radiating, absorbing and weakly scattering medium of gaseous fuel combustion products. The research method is based on the numerical solution of a system of two-dimensional integro-differential equations of radiative gas dynamics, closed by a two-parameter model of turbulent convection. It is shown that when burners are located on the side walls of the radiation chamber, a significant restructuring of the heat flux distribution occurs. A decrease in the chamber width is accompanied by an increase in both radiant and convective heat fluxes to the tubular screen.

Molecular dynamics simulations of a multicellular model with cell-cell interactions and Hippo signaling pathway.

The transcriptional coactivator Yes-associated protein (YAP)/transcriptional co-activator with PDZ binding motif (TAZ) induces cell proliferation through nuclear localization at low cell density. Conversely, at extremely high cell density, the Hippo pathway, which regulates YAP/TAZ, is activated. This activation leads to the translocation of YAP/TAZ into the cytoplasm, resulting in cell cycle arrest. Various cancer cells have several times more YAP/TAZ than normal cells. However, it is not entirely clear whether this several-fold increase in YAP/TAZ alone is sufficient to overcome proliferation inhibition (contact inhibition) under high-density conditions, thereby allowing continuous proliferation. In this study, we construct a three-dimensional (3D) mathematical model of cell proliferation incorporating the Hippo-YAP/TAZ pathway. Herein, a significant innovation in our approach is the introduction of a novel modeling component that inputs cell density, which reflects cell dynamics, into the Hippo pathway and enables the simulation of cell proliferation as the output response. We assume such 3D model with cell-cell interactions by solving reaction and molecular dynamics (MD) equations by applying adhesion and repulsive forces that act between cells and frictional forces acting on each cell. We assume Lennard-Jones (12-6) potential with a softcore character so that each cell secures its exclusive domain. We set cell cycles composed of mitotic and cell growth phases in which cells divide and grow under the influence of cell kinetics. We perform mathematical simulations at various YAP/TAZ levels to investigate the extent of YAP/TAZ increase required for sustained proliferation at high density. The results show that a twofold increase in YAP/TAZ levels of cancer cells was sufficient to evade cell cycle arrest compared to normal cells, enabling cells to continue proliferating even under high-density conditions. Finally, this mathematical model, which incorporates cell-cell interactions and the Hippo-YAP/TAZ pathway, may be applicable for evaluating cancer malignancy based on YAP/TAZ levels, developing drugs to suppress the abnormal proliferation of cancer cells, and determining appropriate drug dosages. The source codes are freely available.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Optimal Inspection and Maintenance Policy: Integrating a Continuous-Time Markov Chain into a Homing Problem

The state of a machine is modeled as a controlled continuous-time Markov chain. Moreover, the machine is being serviced at random times. The aim is to maximize the time until the machine must be repaired, while taking the maintenance costs into account. The dynamic programming equation satisfied by the value function is derived, enabling optimal decision-making regarding inspection rates, and special problems are solved explicitly. This approach minimizes direct maintenance costs along with potential failure expenses, establishing a robust methodology for determining inspection frequencies in reliability-centered maintenance. The results contribute to the advancement of maintenance strategies and provide explicit solutions for particular cases, offering ideas for application in reliability engineering.

Trajectory Planning and Adaptive Fuzzy PID Control for Precision in Robotic Vertebral Plate Cutting: Addressing Force Dynamics and Deformation Challenges

This study explores trajectory planning and cutting force control for spinal surgical robots, focusing on managing cutting deformation and vibration during vertebral plate cutting—a critical challenge in robotic spinal surgery. Through theoretical analysis, simulation, and experimental validation, the research addresses the complexities of cutting force dynamics and the associated deformations. A dynamic equation for vertebral plate cutting is established, providing a theoretical foundation for trajectory planning. A new trajectory planning method using B-spline curves and an interpolation point constraint formula is introduced, enabling precise robot trajectory control. Additionally, an adaptive fuzzy PID control strategy with force feedback is proposed to dynamically adjust the feed rate, effectively suppressing cutting deformation and vibration in real time. Simulations and experiments validate the effectiveness of these methods in reducing force fluctuations and enhancing control stability. The findings offer valuable guidance for improving the performance and safety of spinal surgery robots, optimizing their trajectories, incorporating dynamic sensing, and enhancing safety controls. Future research should focus on refining control strategies for more complex surgical scenarios and validating their applicability under conditions closer to actual surgical environments. This work provides theoretical and practical insights into advancing robotic spinal surgery, contributing to better surgical outcomes and patient safety.

Scalable O(log2n) Dynamics Control for Soft Exoskeletons

Robotic exoskeletons are being actively applied to support the activities of daily living (ADL) for patients with hand motion impairments. In terms of actuation, soft materials and sensors have opened new alternatives to conventional rigid body structures. In this arena, biomimetic soft systems play an important role in modeling and controlling human hand kinematics without the restrictions of rigid mechanical joints while having an entirely deformable body with limitless points of actuation. In this paper, we address the computational limitations of modeling large-scale articulated systems for soft robotic exoskeletons by integrating a parallel algorithm to compute the exoskeleton’s dynamics equations of motion (EoM), achieving a computation with O(log2n) complexity for the highly articulated n degrees of freedom (DoF) running on p processing cores. The proposed parallel algorithm achieves an exponential speedup for n=p=64 DoF while achieving a 0.96 degree of parallelism for n=p=256, which demonstrates the required scalability for controlling highly articulated soft exoskeletons in real time. However, scalability will be bounded by the n=p fraction.