Dynamic Equations Research Articles

Cohesive phase-field model for dynamic fractures in coal seams

A novel rate-dependent cohesive phase-field model that can simulate fluid-driven dynamic quasibrittle fracture in soft coal rocks is presented. The coal matrix and fractures are described by a unified fluid continuity equation, and the Poiseuille flow in the fractures is modeled by deformation-dependent permeability. Dynamic evolution equations for porosity, permeability and the Biot coefficient during phase-field evolution are developed. The phase-field evolution equation is derived from the total energy generalization based on the Francfort–Marigo variational principle. To simulate the rate-dependent dynamic fracture problem accurately, the effects of the phase-field evolution dissipation energy and inertia energy are considered. The above multifield coupled system is solved via a staggered approach within the finite element framework via the Newton–Raphson iterative algorithm. The independence and convergence of the proposed model were verified, and its accuracy was validated via an analytical solution for crack width and a benchmark test for dynamic crack propagation. The propagation law of single/multicluster hydraulic fractures in coal seams with interlayers was investigated via the proposed model. The simulations revealed that the number of clusters and the mechanical properties of the interlayers significantly influence the fracture morphology. Finally, hydraulic fracturing of fracture-developed coal seams was simulated, confirming the potential of the proposed model.

Finite element model to investigate the dynamic instability of ring stiffened conical shells subjected to flowing fluid

In this study, the vibration stability (i.e., static divergence) and critical velocity of fluid-conveying, ring-stiffened, truncated conical shells are investigated under various boundary conditions. The shell is characterized using Sanders’ theory, while the fluid is modeled using a velocity potential approach with the impermeability condition at the fluid-shell interface. Using linear superposition, the natural frequencies corresponding to each flow velocity are determined by satisfying the dynamic characteristic equation and boundary conditions. Critical velocities are identified where the natural frequencies vanish, indicating static divergence. Parametric studies are conducted to investigate the effect of ring stiffeners on the critical velocities with respect to the semi-cone angle, number of rings, and boundary conditions. The proposed model is validated through comparison with published data. It is found that the rings significantly affect the stability of the cone under different boundary conditions. Instability in stiffened shells occurs at higher critical fluid velocities than in unstiffened shells across all boundary conditions. An increase in the vertex angle leads to a decrease in critical flow discharge.

Inventory Optimization Model with Quadratic Demand Pattern and Non-instantaneous Deterioration Rate with Mixed Cash and Advance Payment Scheme

Objectives: To develop a mathematical model for cost minimization by examining a scenario involving a seasonal product retail market, evaluating the outcome, and providing solutions to unanswered queries of existing models are the highlights of this present work. Methods: 1. Using dynamic differential equations with initial and boundary conditions, we mathematically model a retail market scenario for perishable commodities and determine the retailer's total cost. 2. We consider a quadratic function of time to precisely describe the maintenance cost and demand function. Also, a combination of cash and advance payment schemes is implemented in our model. 3. Our model was numerically formulated using a dataset deduced from earlier research. We have included a sensitivity analysis and a numerical example with a graphical depiction of the total cost function that highlights the most and least influential factors. 4. Based on the Hessian matrix condition and the calculus derivative approach, we developed an algorithm to calculate the optimal cost. Findings: This model provides retailers with insights for effective stock management while optimizing the cost of retailing perishable commodities with non-linear demand patterns. Sensitivity analysis showed that the two factors that have the biggest impact on retailers' overall costs are non-linear demand and holding cost parameters. The smaller optimal inventory cycle time and the large value of time at which a shortage happens raise the overall cost. Additionally, because of higher demand, the time-dependent deterioration rate has the least impact on overall cost. Additionally, a three-dimensional graph of cost minimization is offered to illustrate the convexity of total cost. Novelty: Implementation of Quadratic time-dependent demand and non-linear holding costs under a mixed cash advance payment structure, along with time-dependent deterioration and partial backlog shortages. Keywords: Quadratic Demand pattern, Non­instantaneous decay rate, Cash and Advance Payment scheme, Partial Backlog Shortages

Research on multi-beam survey line system based on dynamic programming algorithm

This paper establishes a calculation model for the coverage width and overlap rate of a multi-beam survey line system [1], and optimizes the multi-beam bathymetric survey plan for marine areas. Through the dynamic programming algorithm, a reasonable survey line plan is designed based on the actual marine area depth. The research content includes: Model Establishment: Construct a calculation model for coverage width and overlap rate, determining the coverage width and overlap rate at each position. Three-Dimensional Space Model: Using the survey vessel as the origin and combining marine area information, solve for the coverage width at different angles. Shortest Survey Line Plan: Using the dynamic programming algorithm, determine the shortest survey line plan, with the dynamic transfer equation determined by the overlap rate index. The total length of the survey line in this plan is calculated to be 114,824 meters. Specific Survey Line Plan Design: Using sea depth data and cubic spline interpolation to complete the discrete point plane, construct approximate width and slope. Under the condition of relaxing the overlap rate to 5%-30%, the shortest survey line length is obtained as 398,059 meters, with a missing survey area ratio of 0%, and the total length of parts with an overlap rate exceeding 20% is 25,759 meters, accounting for 6.4%. This study provides theoretical basis and calculation methods for optimizing multi-beam bathymetric survey plans.

Structural damage identification by using physics-guided residual neural networks

In recent years, vibration-based structural damage identification has made significant progress by exploiting data-driven deep learning techniques, which can efficiently extract damage-sensitive features from a large amount of data. However, in some practical engineering applications, large volumes of measurement data are not readily available. This paper proposes a novel physics-guided residual neural network (PhyResNet) framework to improve the robustness and accuracy of structural damage identification under data-scarce conditions. In contrast to the state-of-the-art purely data-driven ResNet, the proposed method embedded available physics knowledge (e.g., governing equations of dynamics) of structures into the feature learning process via a novel physics-based loss function. The input-output relationship of the network is constrained to retain its physical meaning implicitly while the demand for large amounts of labeled training data is reduced. Notably, even with only 5 % of the dataset used for training, PhyResNet achieves a 13.1 % improvement in R-Value. The performance of the proposed approach is evaluated through both numerical and experimental verifications. Results demonstrate that damage localization and quantification are achieved with high accuracies and good robustness.

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.

Computational framework for the generation of one-dimensional vascular models accounting for uncertainty in networks extracted from medical images.

One-dimensional (1D) cardiovascular models offer a non-invasive method to answer medical questions, including predictions of wave-reflection, shear stress, functional flow reserve, vascular resistance and compliance. This model type can predict patient-specific outcomes by solving 1D fluid dynamics equations in geometric networks extracted from medical images. However, the inherent uncertainty in in vivo imaging introduces variability in network size and vessel dimensions, affecting haemodynamic predictions. Understanding the influence of variation in image-derived properties is essential to assess the fidelity of model predictions. Numerous programs exist to render three-dimensional surfaces and construct vessel centrelines. Still, there is no exact way to generate vascular trees from the centrelines while accounting for uncertainty in data. This study introduces an innovative framework employing statistical change point analysis to generate labelled trees that encode vessel dimensions and their associated uncertainty from medical images. To test this framework, we explore the impact of uncertainty in 1D haemodynamic predictions in a systemic and pulmonary arterial network. Simulations explore haemodynamic variations resulting from changes in vessel dimensions and segmentation; the latter is achieved by analysing multiple segmentations of the same images. Results demonstrate the importance of accurately defining vessel radii and lengths when generating high-fidelity patient-specific haemodynamics models. KEY POINTS: This study introduces novel algorithms for generating labelled directed trees from medical images, focusing on accurate junction node placement and radius extraction using change points to provide haemodynamic predictions with uncertainty within expected measurement error. Geometric features, such as vessel dimension (length and radius) and network size, significantly impact pressure and flow predictions in both pulmonary and aortic arterial networks. Standardizing networks to a consistent number of vessels is crucial for meaningful comparisons and decreases haemodynamic uncertainty. Change points are valuable to understanding structural transitions in vascular data, providing an automated and efficient way to detect shifts in vessel characteristics and ensure reliable extraction of representative vessel radii.

A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory.

Dynamic response and stability analysis of vehicle systems with double time-delay feedback control

In this paper, the vehicle suspension dynamic response of time delay feedback control is analyzed considering the suspension of the seat. An active suspension vibration reduction control method based on double time-delay feedback control is proposed to further improve the ride comfort. Firstly, the double time-delay dynamic response equation of the system with three degrees of freedom is established. Then, the root mean square values of seat acceleration and body acceleration are used as the objective functions. Meanwhile, the RMS values of suspension dynamic deflection and tire dynamic displacement are used as constraints, and the optimal feedback parameters are obtained using particle swarm optimisation. The Routh-Hurwitz criterion and frequency domain scanning methods are used to verify the stability of the double time-delay system. Finally, simulation in the time domain and frequency domain is accomplished. It is verified that this feedback control method has strong robustness and anti-interference ability.

Protocol and preregistration for the CODEC project: measuring, modelling and mechanistically understanding the nature of cognitive variability in early childhood

BackgroundChildren’s cognitive performance fluctuates across multiple timescales. However, fluctuations have often been neglected in favour of research into average cognitive performance, limiting the unique insights into cognitive abilities and development that cognitive variability may afford. Preliminary evidence suggests that greater variability is associated with increased symptoms of neurodevelopmental disorders, and differences in behavioural and neural functioning. The relative dearth of empirical work on variability, historically limited due to a lack of suitable data and quantitative methodology, has left crucial questions unanswered, which the CODEC (COgnitive Dynamics in Early Childhood) study aims to address.MethodThe CODEC cohort is an accelerated 3-year longitudinal study which encompasses 600 7-to-10-year-old children. Each year includes a ‘burst’ week (3 times per day, 5 days per week) of cognitive measurements on five cognitive domains (reasoning, working memory, processing speed, vocabulary, exploration), conducted both in classrooms and at home through experience sampling assessments. We also measure academic outcomes and external factors hypothesised to predict cognitive variability, including sleep, mood, motivation and background noise. A subset of 200 children (CODEC-MRI) are invited for two deep phenotyping sessions (in year 1 and year 3 of the study), including structural and functional magnetic resonance imaging, eye-tracking, parental measurements and questionnaire-based demographic and psychosocial measures. We will quantify developmental differences and changes in variability using Dynamic Structural Equation Modelling, allowing us to simultaneously capture variability and the multilevel structure of trials nested in sessions, days, children and classrooms.DiscussionCODEC’s unique design allows us to measure variability across a range of different cognitive domains, ages, and temporal resolutions. The deep-phenotyping arm allows us to test hypotheses concerning variability, including the role of mind wandering, strategy exploration, mood, sleep, and brain structure. Due to CODEC’s longitudinal nature, we are able to quantify which measures of variability at baseline predict long-term outcomes. In summary, the CODEC study is a unique longitudinal study combining experience sampling, an accelerated longitudinal ‘burst’ design, deep phenotyping, and cutting-edge statistical methodologies to better understand the nature, causes, and consequences of cognitive variability in children.Trial registrationClinicalTrials.gov - NCT06330090

Time Scale Calculus: a new approach to multi-dose pharmacokinetic modeling.

In this paper, we use Time Scale Calculus (TSC) to formulate and solve pharmacokinetic models exploring multiple dose dynamics. TSC is a mathematical framework that allows the modeling of dynamical systems comprising continuous and discrete processes. This characteristic makes TSC particularly suited for multi-dose pharmacokinetic problems, which inherently feature a blend of continuous processes (such as absorption, metabolization, and elimination) and discrete events (drug intake). We use this toolkit to derive analytical expressions for blood concentration trajectories under various multi-dose regimens across several flagship pharmacokinetic models. We demonstrate that this mathematical framework furnishes an alternative and simplified way to model and retrieve analytical solutions for multi-dose dynamics. For instance, it enables the study of blood concentration responses to arbitrary dose regimens and facilitates the characterization of the long-term behavior of the solutions, such as their steady state.

Dynamic Modeling and Experimental Verification of an RPR Type Compliant Parallel Mechanism with Low Orders

Efficiency of calculating a dynamic response is an important point of the compliant mechanism for posture adjustment. Dynamic modeling with low orders of a 2R1T compliant parallel mechanism is studied in the paper. The mechanism with two out-of-plane rotational and one lifting degrees of freedom (DoFs) plays an important role in posture adjustment. Based on elastic beam theory, the stiffness matrix and mass matrix of the beam element are established where the moment of inertia is considered. To improve solving efficiency, a dynamic model with low orders of the mechanism is established based on a modified modal synthesis method. Firstly, each branch of the RPR type mechanism is divided into a substructure. Subsequently, a set of hypothetical modes of each substructure is obtained based on the C-B method. Finally, dynamic equation of the whole mechanism is established by the substructure assembly. A dynamic experiment is conducted to verify the dynamic characteristics of the compliant mechanism.

Inferring dynamical models from time-series biological data using an interpretable machine learning method based on weighted expression trees

The growing time-series data make it possible to glimpse the hidden dynamics in various fields. However, developing a computational toolbox with high interpretability to unveil the interaction dynamics from data remains a crucial challenge. Here, we propose a new computational approach called automated dynamical model inference based on expression trees (ADMIET), in which the machine learning algorithm, the numerical integration of ordinary differential equations and the interpretability from prior knowledge are embedded into the symbolic learning scheme to establish a general framework for revealing the hidden dynamics in time-series data. ADMIET takes full advantage of both machine learning algorithm and expression tree. Firstly, we translate the prior knowledge into constraints on the structure of expression tree, reducing the search space and enhancing the interpretability. Secondly, we utilize the proposed adaptive penalty function to ensure the convergence of gradient descent algorithm and the selection of the symbols. Compared to gene expression programming, ADMIET exhibits its remarkable capability in function fitting with higher accuracy and broader applicability. Moreover, ADMIET can better fit parameters in nonlinear forms compared to regression methods. Furthermore, we apply ADMIET to two typical biological systems and one real data with different prior knowledge to infer the dynamical equations. The results indicate that ADMIET can not only discover the interaction relationships but also provide accurate estimates of the parameters in the equations. These results demonstrate ADMIET’s superiority in revealing interpretable dynamics from time-series biological data.

Influence of surface roughness on vibration response characteristics of flexible structures on offshore floating platforms

The expanding interest in floating offshore renewable energy technologies, including floating wind turbines and solar platforms, is attributed to their multifarious benefits. One pivotal factor influencing the power generation efficiency of these technologies is the vibration response of the flexible structures connected to the platforms, such as mooring systems, dynamic power cables, and flexible pipelines. In this research, a numerical investigation is conducted on the response characteristics of vortex-induced vibration (VIV) of cylinders with varying surface roughness in oscillatory flow. The unsteady Reynolds-averaged Navier-Stokes (URANS) equations and dynamic equation are used for fluid-structure interaction. The results reveal that for rough cylinders, multiple frequencies are involved in the vibration. For most simulated cases, the vibration frequency in the cross-flow (CF) direction is an integer multiple of the frequency of the oscillatory flow and associated with the quantity of shedding vortex pairs during a single period of the oscillatory flow. For rough cylinders, the CF vibration frequency differs by one from the number of shedding vortex pairs within one period of the oscillatory flow. With the increase of surface roughness, the maximum vibration amplitude shows an increasing trend (12%), then a decreasing trend (21%). Furthermore, the vortex shedding flow pattern presents C + S mode or S mode. This study offers valuable insights for analyzing the vibration response characteristics of authentic flexible structures.

Random flutter analysis of a novel binary airfoil with fractional order viscoelastic constitutive relationship

The fractional order model emerges naturally when the integer order model fails to meet practical requirements and finds extensive applications in engineering practice. However, there is still limited research on the random flutter of fractional order viscoelastic binary airfoils. This study aims to explore the random flutter of such airfoils under the influence of Gaussian white noise excitation by utilizing the pth moment Lyapunov exponent. Firstly, a two-degree-of-freedom fractional order stochastic dynamic equation is established for the viscoelastic binary airfoil through the introduction of fractional order damping. The singular perturbation method is then employed to derive the pth moment Lyapunov exponent of the system. An approximate analytical solution of the pth moment Lyapunov exponent is obtained through Fourier series expansion, which exhibits good agreement with numerical results obtained from Monte Carlo simulations. Subsequently, a comprehensive analysis is conducted to examine the impacts of fractional order, airflow velocity, noise intensity, and key structural parameters on the stochastic flutter of the fractional order viscoelastic binary airfoil. The findings indicate that compared with the typical binary airfoil model, the flutter speed of the fractional-order viscoelastic binary airfoil is significantly increased, the stability of the system is highly sensitive to the natural frequency of the system, and the fractional-order factor has a positive effect on the stochastic stability of the binary airfoil before the critical value. The above research results can guide the design optimization of the fractional-order viscoelastic airfoil.

Dynamics of a cross beam with a straight crack in a mining linear vibrating screen

Cross beam fracture is one of the common failures of vibrating screens, and crack is the early manifestation of fracture, which is hard to detect. In order to meet the screening requirement of the vibrating screen and improve the service life of the cracked beam, the cracked Euler-Bernoulli beam model is established to investigate the dynamics of the cross beam with a straight crack under different weights of eccentric block, processing capacities, and Rayleigh damping coefficients based on the finite element method in this paper. The local flexibility coefficients are derived from the principles of fracture mechanics and strain release energy and solved by the adaptive five-point Gaussian Legende algorithm. The stiffness matrix of the cracked beam element is calculated through the inverse method of total flexibility. The four order Runge-Kutta algorithm and MATLAB tools are used to solve the dynamic equation of the cracked cross beam. The relationship between the vibration amplitude of the cracked cross beam and the weight of the eccentric block is studied by fitting formulas using the least squares method. The influence of different weights of eccentric block, processing capacities, and Rayleigh damping coefficients on the vibration amplitude and service life of the cracked beam are discussed. The results show that the greater the weight of the eccentric block, the shorter service life of the beam. When the damping is greater, the service life of the cracked beam is longer.

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

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

A novel study of analytical solutions of some important nonlinear fractional differential equations in fluid dynamics

The space-time fractional Burger-like equation and the space-time coupled Boussinesq equation with conformable derivative, both of which are significant in fluid dynamics, are investigated in this work using the improved [Formula: see text] method. The process works effectively and produces soliton solutions. The method was successfully and consistently implemented with Maple, a symbolic computing tool. The solutions also contain a few of graphics. Numerous novel exact solutions to these equations, distinct from those found earlier with the proposed approach, have been provided. The study’s findings add to the body of knowledge by offering insightful justifications for various types of nonlinear systems. The results demonstrated the value of the proposed method as a mathematical tool and the ease, dependability, and speed increases that result from carrying out these tasks using a symbolic computing program. Notably, it applies to many nonlinear evolution problems in mathematical physics.

Distributed Model Predictive Control Based on Bus Voltage Derivative and SoC Dynamic Model for Shipboard DC Microgrids

State-of-charge (SoC) consistency and bus voltage regulation are two major control objectives of shipboard DC microgrids. To achieve these objectives, this paper presents a novel distributed model predictive control (DMPC) strategy with multiple cost functions. Firstly, based on the bus voltage derivative and SoC dynamic model, the voltage and SoC control equations in discrete form are established. Subsequently, considering the safe operation of battery energy storage systems (BESSs), a DMPC taking the energy storage current as the control set is designed. Finally, the average voltage compensation is taken to achieve precise control of the average voltage. The proposed method can avoid the complex process of adjusting weight coefficients, thereby simplifying controller design. Furthermore, the robustness and practicality of the proposed DMPC method are verified through MATLAB/Simulink 2021a simulations and hardware-in-the-loop (HiL) experiments.

Non-linear dynamics and bandgap control in magneto-rheological elastomers metamaterials with inertial amplification

Sandwich plate structures are extensively utilized in engineering due to their favorable stiffness-to-weight ratio. Nonetheless, these lightweight thin-walled structures frequently encounter challenges related to inadequate low-frequency vibration performance, which significantly restricts their applications, particularly in the realm of precision instruments which is sensitive to vibration. This study introduces an innovative design of an active nonlinear metamaterial, to achieve tunable broadband low-frequency bandgaps for sandwich plates. The nonlinear oscillator incorporates an inertia amplification mechanism (IAM), Euler-buckled beams, mass elements, and magneto-rheological elastomers (MREs), which are modulated via external magnetic fields to adjust the material's stiffness dynamically. Employing Hamilton's principles and the plate wave expansion method (PWE), the dispersion relations for the metamaterial plate are derived, elucidating the dispersion surfaces and the band structures within its sandwich-like plates.The dynamical equations of the metamaterial plate are formulated and validated through numerical simulations using the Galerkin method, confirming the theoretical predictions. The results demonstrate effective control over low-frequency and broadband bandgaps under low mass ratio conditions through strategic manipulation of the inertia amplification factor and magnetic flux. The study extensively explores the nonlinear dynamic responses of the metamaterial, highlighting the significant impact of excitation amplitudes on the amplitude-dependent bandgaps.