Dynamic Equations Of System Research Articles

Universal embedding of autonomous dynamical systems into a Lotka-Volterra-like format

We show that the ordinary differential equations (ODEs) of any deterministic autonomous dynamical system with continuous and bounded rate-field components can be embedded into a quadratic Lotka-Volterra-like form by turning to an augmented set of state variables. The key step consists in expressing the rate equations by employing the Universal Approximation procedure (borrowed from the machine learning context) with logistic sigmoid ‘activation function’. Then, by applying already established methods, the resulting ODEs are first converted into a multivariate polynomial form (also known as generalized Lotka-Volterra), and finally into the quadratic structure. Although the final system of ODEs has a dimension virtually infinite, the feasibility of such a universal embedding opens to speculations and calls for an interpretation at the physical level.

An Energy Transfer-Based Bifurcation Detection Method for Nonlinear Rotating Systems: Enables Accurate Capture of Period-Doubling Bifurcation and Instability

Rotor systems are widely used in industrial power generation and propulsion. Once the nonlinear contact stiffness and oil film force are taken into account, the dynamics and stability of rotor systems become quite complex, often accompanied by super-harmonic and chaotic motions. Furthermore, conventional methods face limitations in real-time detection of the bifurcations and complex nonlinear motions. This research investigates the bifurcations and stability induced by nonlinear factors in a rotor-bearing system from an energy perspective. Dynamic equations of a rotor-bearing system considering cubic term stiffness are established, the steady responses are obtained by the fourth-order Runge–Kutta method. The relationship between the bifurcations and energy transfers is analyzed numerically, the proposed stability criterion is validated by comparing the Lyapunov exponents. The bistable phenomenon of period-3[Formula: see text] ([Formula: see text]) motion is discussed in terms of numerical results and experiments which corresponds to the asymmetric jumps of the generalized energy. It is found that bifurcations and unstable motions of the nonlinear system can be captured accurately by detecting the energy transfers, the proposed generalized energy curve exhibits more detailed information than the conventional speed-up curve. These findings provide a new perspective on studying bifurcations and stability of rotating systems, which can be further applied in the condition monitoring, stability prediction as well as the design of nonlinear energy sink, representing significant progress in converting theory into engineering applications for rotor systems.

Dynamic pricing strategies for multi‐type knowledge payment products: A system dynamics approach

AbstractWith the emergence of knowledge economy, knowledge payment has become an essential means of knowledge circulation. This study employs a system dynamics approach and integrates signal theory and customer value perception to investigate the dynamic pricing strategies for various knowledge payment products from the perspective of knowledge producers. First, this study analyzes the categories of knowledge payment products and proposes a theoretical research framework for designing dynamic pricing strategies for different products categories. Second, this study identifies the key factors that impact the profitability of knowledge payment platforms and construct a system dynamics model of product pricing strategies for knowledge payment platforms; next, three typical knowledge payment products on Zhihu Live are chosen as research objects, and system dynamics equations are constructed to simulate the profitability changes of different types of products in different periods. Finally, the results of simulation and sensitivity analysis are combined to select dynamic pricing strategies for different products types. This study offers an innovative exploration of multi‐stage pricing strategies for different types of knowledge payment products, which fills the gap in existing research and provides more scientific suggestions for knowledge producers and consumers in terms of knowledge transactions.

Natural Frequency Maximization of Beam with an Internal Sliding Joint of Translational Restraint

The natural frequency variation of a uniform Euler–Bernoulli beam with an internal sliding joint of the translational restraint is studied intensively such that the optimal design of the connecting point is pursued for maximization of a natural frequency of the system. First, the compatibility conditions at the linkage are presented in detail, and the dynamic characteristic equation of the beam system is accordingly derived. Next, the influences of both the translational restraint stiffness and the position of the sliding joint upon the natural frequency and mode shape are described comprehensively. Then, the optimal position and the corresponding minimum translational stiffness of the internal joint are explored exclusively for maximizing a natural frequency of the beam structure. In this context, the essential requirements for the optimization of the internal sliding joint are presented, respectively, in two different ways to reveal their mechanical implications. It turns out that the optimal joint location can be determined readily from the corresponding mode shape of the equivalent continuous beam. Afterward, numerical results are provided with two classical beams of a single internal sliding joint to demonstrate the variations of their dynamic characteristics as well as to offer significant physical insight into the vibratory behaviors of the beam system with maximization of a natural frequency of interest.

Seismic reduction principle and response analysis of variable damping viscous damper system

PurposeA new type of variable damping viscous damper is developed to meet the settings of different damping parameter values at different working stages. Its main principle and design structure are introduced, and the two-stage and multi-stage controllable damping methods are proposed.Design/methodology/approachThe theoretical calculation formulas of the damping force of power-law fluid variable damping viscous damper at elongated holes are derived, aiming to provide a theoretical basis for the development and application of variable damping viscous dampers. For the newly developed variable damping viscous damper, the dynamic equations for the seismic reduction system with variable damping viscous dampers under a multi-degree-of-freedom system are established. A feasible calculation and analysis method is proposed to derive the solution process of time history analysis. At the same time, a program is also developed using Matlab. The dynamic full-scale test of a two-stage variable damping viscous damper was conducted, demonstrating that the hysteresis curve is complete and the working condition is stable.FindingsThrough the calculation and analysis of examples, the results show that the seismic reduction effect of high and flexible buildings using the seismic reduction system with variable damping viscous dampers is significant. The program developed is used to analyze the seismic response of a broadcasting tower using a variable damping TMD system under large earthquakes. The results indicate that the installation of variable damping viscous dampers can effectively control the maximum inter-story displacement response of TMD water tanks and can effectively consume seismic energy.Originality/valueThis method can provide a guarantee for the safe and effective operation of TMD in wind and vibration control.

A new method for finding the proper initial conditions in passive locomotion of bipedal robotic systems

In this paper, the walking motion of a biped robot consisting of upper and lower body members (arms, legs, upper trunk and neck) on an inclined surface is analyzed kinematically and dynamically. Some of the challenges faced by this research include the dual-phase dynamics that govern the periodic gait of this robotic system and the consideration of all the body parts that are involved in the walking motion; which make it extremely difficult to derive the kinetic equations of such robotic systems. To deal with these challenges, the Gibbs-Appell formulation and the Newton's impact laws are employed to derive the most general form of the system's dynamic equations in the swing and transient phases of motion. Subsequently, by using the obtained motion equations and implementing a systematic procedure, we achieve an eigenvalue problem. By solving this problem, the suitable initial conditions that are necessary for the passive gait of this biped robot on a sloping surface are determined. By considering the obtained initial conditions as well as the dynamic response of the system to the Earth's gravity, we describe a general method for designing a biped robot's walking trajectory on an inclined surface. The most important feature of the designed trajectory is its close agreement with the trajectory followed by a biped robot walking passively down a sloping ramp. This close match between the said trajectories enables us to guide the robot in tracking a desired path by applying very small control torques to the robot joints at the start of each step. This claim is corroborated by the simulations carried out for a biped walking robot composed of 6 rigid links, in which the control torques that are applied to the actuated robot joints are obtained by employing a feedback linearization method and via a designed LQR controller.

Preface

The 2023 7th International Conference on Mechanics, Mathematics and Applied Physics (ICMMAP 2023) took place in Dalian, China during 8th to 10th September 2023. The conferences in this series started in 2017 as an academic conference and gradually expanded to include wider topic fields as well. Now the International Conference on Mechanics, Mathematics and Applied Physics has been developed into an annual international conference based in China with topics encompassing the whole area of mechanics, mathematics and applied physics.The aim of the Conference was to provide a platform for the exploration of both fundamental topics and new applications of research fields related to mechanics, mathematics and applied physics, bring different scientific communities together, and thus facilitate the contacts between science, technology and industry. All the attendees were welcomed to give comments and raise questions to the speeches and presentations during the Conference.More than 100 participants attended the Conference which consisted of ten keynote speeches, as well as plenty of oral and poster contributions. Among them, Prof. Zhenxin Liu (Dalian University of Technology, China), who is interested in the researches on Stochastic Dynamical Systems and Stochastic Differential Equations, delivered a keynote speech on The Averaging Principle for SDEs with Monotone Coefficients. And Prof. Chongjun Li (Dalian University of Technology, China) made a report on Revised SBFEM on Arbitrary Polygons and Faceted Polyhedrons with the Second Order Completeness by Elimination of Bubble Functions. He focuses his research on Numerical Approximation, Computational Geometry, Multivariate Spline Theory, and Spline Finite Elements. And the other speakers performed their speeches one by one, sharing research experience and professional insights on specific topics. It is really a great opportunity to promote academic communication and collaboration worldwide.This volume gathers papers from regular contributions of ICMMAP 2023; each contribution submitted for publication has been peer reviewed and the Editors are very grateful to the referees for their careful support in improving the original manuscripts. The manuscripts accepted for publication cover different topics of mechanics, mathematics and applied physics: Computational Mechanics, Multiphase Fluid Mechanics, Partial Differential Equation, Simulation and Monte-Carlo Methods, Nuclear Electronics, etc.It is a great honor to present this volume of Journal of Physics: Conference Series and we deeply thank the authors for their enthusiastic and high-grade contribution.Finally, we would like to thank the Conference Chair, the Publication Chair, the members of the Technical Program Committee as well as Organizing Committee, the conference secretariat and the support from Dalian Jiaotong University that allowed the success of ICMMAP 2023.The Committee of ICMMAP 2023List of Committee Member are avaible in this pdf.

Application of the Local Discontinuous Galerkin Method to the Solution of the Quasi-Gas Dynamic System of Equations

Application of the Local Discontinuous Galerkin Method to the Solution of the Quasi-Gas Dynamic System of Equations

Index Matrix-Based Modeling and Simulation of Buck Converter

The approach described in this paper handles the parameters and characteristics (analog and discrete ones) of a Buck DC-DC converter (in its power and control parts) in a common manner. The usage of probably complicated differential equations for discrete dynamical systems is avoided by means of index matrix equations, which can be easily understood. Compared to classical matrix models, the proposed index matrix models are more descriptive and also smaller in size. Such functionality is widely applied by the authors, and a new operation is defined and used as well. The relevance of the proposed techniques in power electronics, because of switching topologies and a limited numbers of components, is argued. Respective examples of functioning modes of the considered converter, in which power circuits and controllers are modeled jointly, are given. Estimations of analog values are based on partly linear dependencies, which are shown to be adequate in first-order analyses. Specific expressions for a Buck DC-DC converter are presented. A model-solving technique and an exhaustive search on a parameter space are considered in detail and automated via well-formalized algorithms. Nested parameter intervals and verifications with normal probability distributions are used in an optimization procedure. The full agreement of implementation via MATLAB source code with results obtained via Simulink is demonstrated. The short simulation times of this software (compared to Simulink and a .NET desktop application developed by the authors) justify the search for optimal variants in a wide multi-dimensional space. A max–min procedure with 10,000 simulations in each verification step is presented.

Influence of two kinds of clearance joints on the dynamics of planar mechanical system based on a modified contact force model

This study takes the slider-crank mechanism with revolute joint and translational joint as the research object and studies the contact force model of the clearance joint and the influence of the hybrid clearance joints on the nonlinear dynamic behavior of the mechanism. A modified contact force model is established based on the simplified elastic oscillator model, which can be used as a normal force in clearance joint. In the new contact force model, the component n of the indentation depth can be arbitrarily selected and it can support the calculation of contact force for both fully elastic recovery, non-elastic recovery and fully inelastic recovery. Based on the LuGre friction model, the tangential friction model of the clearance joint is given. Thus, the normal force and tangential force during the dynamic contact of the clearance joint are formed. Combining Lagrange’s equations of the first kind with the modified normal force and tangential friction force, the dynamic equations of the multi-body system with clearance joints are established. The Baumgarte stabilization method is used to improve the numerical stability. The correctness of the dynamic prediction model in the mechanism with clearance joint is verified by experiment. The dynamic analysis of the slider-crank mechanism with mixed clearance joints shows that the revolute clearance joint has a greater influence on the mechanism than the translational clearance, and the revolute clearance joint plays a leading role in the dynamic response.

Vibration reduction performance for the novel grounded inerter-based dynamic vibration absorber controlling a primary structure under random excitation

This paper investigates the control performance of a novel grounded inerter-based dynamic vibration absorber (GI-DVA) for random vibration reduction. First, the dynamics equations of the coupled system are written and the transfer function is obtained based on the Laplace transform. Then, assuming that the primary structure is under random white noise excitation, the displacement variance of the primary structure is calculated based on the exact definition of the H2 norm, by solving the Lyapunov equation. By imposing that the partial derivatives of the response variance of the primary structure with respect to the system parameters are simultaneously equal to zero, the optimum design values of the H2 optimized GI-DVA were derived numerically for given different mass ratio. It can be found that the optimal design parameters as the inerter-to-mass ratio, the stiffness ratio and the damping ratio increase, while the frequency ratio decreases with the increase in the mass ratio. Under the optimal conditions, the response analysis showed that the primary structure response controlled by the H2 optimized GI-DVA can decrease with the increase in the mass ratio, but is less sensitive to large mass ratios, and can be robust to the mistuning on the optimum design parameters. Then, the control performance evaluation is first performed in the frequency domain, which reveal that the dynamic response reduction capacity of the H2 optimized GI-DVA are significantly 62% and 48% superior to the H2 optimized classic DVA (CDVA) and high-performance passive nontraditional inerter-based DVA (NIDVA-C4), respectively. Furthermore, for the root mean square response evaluation ( H2 performance), the H2 optimized GI-DVA can provide significantly H2 performance 58% and 50% than the H2 optimized CDVA and NIDVA-C4, respectively. To obtain more realistic results, the time domain simulation is performed, which showed that the GI-DVA can provide significantly performance for random vibration reduction.

Brayton-Moser passivity-based controller for an on-board integrated electric vehicle battery charger

This paper explores the Brayton Moser passivity-based controller design for an on-board integrated battery charger system. The proposed controller is based on energy concept theory which ensures asymptotic stability against load variability. To design proposed controller, first, average dynamic equations of the proposed system have been developed through energy-based formulation namely Brayton Moser. In order to obtain robust controller and correspondingly robust charger against dynamics and external disturbances, a PI controller is integrated with the proposed controller. The proposed system is employed for electric vehicles battery charger which is charged by using CC/CV technique and their effectiveness has been evaluated through Simulink toolbox of MATLAB 2014. Furthermore, the benchmark PI controller is preferred for comparison and discussion. Additionally, the effectiveness of the system is verified via OP-5142 real time simulator.

Evaluation of energy saving and emission reduction effect of distributed energy system

The conventional evaluation methods of energy conservation and emission reduction lack the discussion of quantification index of energy conservation and emission reduction potential, and the evaluation effect is poor. Therefore, an evaluation method for energy conservation and emission reduction of distributed energy systems is proposed, which considers system dynamics and demand response. According to the requirements of system dynamics analysis, the relational variables in the system dynamics model are determined. The input-output relationship of energy saving, and emission reduction effect evaluation is described by system dynamics equation, and the input-output relationship of energy saving, and emission reduction effect evaluation is determined. Based on the maximum profit objective function and the minimum cost objective function of the user, the continuum equation of the medium transmission process in the distributed energy system is obtained. The evaluation indexes of energy saving, and emission reduction effect are selected from the three aspects of power consumption side, power generation side and the system itself to realize the evaluation of energy saving and emission reduction effect. The experimental results show that the evaluation curve obtained by this method is basically consistent with the actual curve, and the evaluation accuracy is high.

The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. II. Harmonic waves

AbstractThe paper presents a comparative analysis of three systems of dynamic equations for fluid‐saturated solids: the exact equations and two simplified versions known as the u‐p approximations obtained by neglecting certain acceleration terms in the exact equations. The constitutive relations for the solid skeleton are written in the general anisotropic incrementally linear form without considering any specific constitutive model or a particular type of anisotropy. The dynamic equations are compared in relation to the existence of solutions in the form of plane harmonic waves. Emphasis is placed on finding conditions for the non‐existence or existence of growing waves whose amplitude increases in time or space as the wave propagates. The conditions are formulated in terms of the acoustic tensor of the skeleton and the compressibility of the pore fluid. In particular, it is shown that for a hyperelastic skeleton, the exact equations and one of the u‐p approximations do not have growing wave solutions, whereas the other u‐p approximation may have such solutions even if the skeleton is hyperelastic.

Data-driven dynamics reconstruction using RBF network * *This work was supported by NSFC (No. 32171094) and National Key R&D Program of China (2019YFA0709503).

Constructing the governing dynamical equations of complex systems from observational data is of great interest for both theory and applications. However, it is a difficult inverse problem to explicitly construct the dynamical equations for many real complex systems based on observational data. Here, we propose to implicitly represent the dynamical equations of a complex system using a radial basis function (RBF) network trained on the observed data of the system. We show that the RBF network trained on trajectory data of the classical Lorenz and Chen system can faithfully reproduce the orbits, fixed points, and local bifurcations of the original dynamical equations. We also apply this method to electrocardiogram (ECG) data and show that the fixed points of the RBF network trained using ECG can discriminate healthy people from patients with heart disease, indicating that the method can be applied to real complex systems.

Governing equation discovery based on causal graph for nonlinear dynamic systems

The governing equations of nonlinear dynamic systems is of great significance for understanding the internal physical characteristics. In order to learn the governing equations of nonlinear systems from noisy observed data, we propose a novel method named governing equation discovery based on causal graph that combines spatio-temporal graph convolution network with governing equation modeling. The essence of our method is to first devise the causal graph encoding based on transfer entropy to obtain the adjacency matrix with causal significance between variables. Then, the spatio-temporal graph convolutional network is used to obtain approximate solutions for the system variables. On this basis, automatic differentiation is applied to obtain basic derivatives and form a dictionary of candidate algebraic terms. Finally, sparse regression is used to obtain the coefficient matrix and determine the explicit formulation of the governing equations. We also design a novel cross-combinatorial optimization strategy to learn the heterogeneous parameters that include neural network parameters and control equation coefficients. We conduct extensive experiments on seven datasets from different physical fields. The experimental results demonstrate the proposed method can automatically discover the underlying governing equation of the systems, and has great robustness.

Noise-driven signal study of power systems based on stochastic partial differential equations

The exploration of stochastic partial differential equations in noisy perturbations of dynamical systems remains a major challenge at this stage. The study analyzes the effective dynamical system combining degenerate additive noise-driven stochastic partial differential equations, firstly in the first class of stochastic partial differential equations, the terms in the non-nuclear space formed by nonlinear interactions are overcome by effectively replacing the elements in the non-nuclear space through the ItÔ formulation, and thus the final effective dynamical system is obtained. The effective dynamical system is then obtained in the second type of stochastic partial differential equation using the O-U process similar to the terms in the non-nuclear space. At noise disturbance amplitudes of 5%, 10%, 15% and 20% AC voltage maxima in that order, the effective dynamical systems for the first type of stochastic partial differential equation and the second type of stochastic partial differential equation are more stable compared to the other types of partial differential equation dynamical systems, with the maximum range of error rate improvement for the sampling points 0–239 voltage rms and voltage initial phase value being 3.62% and 26.85% and 2.13% and 19.86% for sampling points 240–360, respectively. The effective dynamic system and stochastic partial differential equation obtained by the research have very high approximation effect, and can be applied to mechanical devices such as thermal power machines.

RETRACTED: Analysis of governing equations of multibody dynamic systems through kinematic properties of joints and effects of kinematic chains on closed-form mechanism

In this investigation, the fundamental equations dictating the behavior of dynamic systems were ascertained by means of an innovative approach that amalgamated the kinematic properties of joints and the intricate kinematic chains of multibody systems into a series of governing equations. Subsequently, the equations that govern the behavior of multibody systems were converted into ordinary differential equations (ODEs) utilizing the calculus of matrix-valued functions. This computational technique is adept at swiftly acquiring recursive differential equations of motion for multibody systems. Thus, the expenditure of computational costs associated with the simulation was efficaciously minimized. The mechanisms of Andrew's squeezing and carpet scrapping were employed in conjunction with kinematic constraints to authenticate the validity of the proposed methodology. The findings suggest that, notwithstanding the fact that the root mean square error (RMSE) in the generalized coordinates between the two ODE and differential algebraic equation (DAE) techniques is small in the Andrew's squeezing mechanism (at approximately 0.02) and in the carpet scraping mechanism (at approximately 0.7), there are still discrepancies between the two methods. The degree of convergence in the aforementioned methodologies, despite the slight RMSE difference between them, shows a clear disparity. By way of illustration, when employing the conventional DAE technique, the duration required to resolve the Andrew's squeezing and carpet scraping mechanisms is 5 and 11.34 s, respectively. However, by utilizing the ODE methodology in the aforementioned mechanisms, the time required to determine a solution is truncated to 1.19 and 2.1 s, sequentially.

ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE

In this survey, we consider the dynamics of a contagious disease spread by employing a nonlinear dynamical control system of differential equations. It considers treatment and vaccination as key control parameters to discern their influence on disease control. The study, approximate the attainable sets of a given control system and presents visual results, while also discussing potential biological applications of their findings.

Theoretical and Experimental Research on Active Suspension System with Time‐Delay Control

AbstractTime‐delay significantly impacts the control system's performance and even leads to the unexpected instability of the control system. This paper aims at the adverse effect of time‐delay on the control effect and stability of the active suspension system. This paper proposes to suppress vehicle body vibration by applying time‐delay displacement feedback. In this paper, the time‐delay control characteristics of the system are studied by combining theory with experiment. This paper first establishes the suspension system's dynamic Equation with time‐delay. Then, the frequency domain sweep method is used to analyze the stability of the control system. Finally, the time‐delay simulation model is established, and the single‐frequency excitation is used as the external excitation to compare with passive and backstepping control. The simulation results verify the superiority of the time‐delay control under external excitation. In this paper, through the 1/4 model bench test of the active suspension system, the simulation results and the test results under the same working conditions are compared, and it is found that the two have good consistency. The results show that the research is credible and can provide a reference for the actual design and application of active vehicle suspension controllers.