Deep-sea mining systems are a critical pathway for acquiring key strategic resources such as nickel and cobalt. The core conveying component, the mining rigid pipe, is susceptible to transverse vibrations under complex wave excitation, which threaten system safety, necessitating the development of efficient and reliable vibration control solutions. This paper proposes an improved Triple-spring nonlinear energy sink (TS-NES). An integrated dynamic model coupling the mining rigid pipe and the TS-NES is established using the vector form intrinsic finite element method and solved via the central difference method. The effectiveness and superiority of the TS-NES are verified through displacement, time–frequency, energy, and phase analyses. Subsequently, a systematic parameter sensitivity study is conducted. The results indicate that under both single-frequency and multi-frequency wave excitations, the TS-NES exhibits broadband, high-efficiency vibration suppression performance superior to that of the conventional tuned mass damper (TMD). It can substantially and uniformly dissipate vibration energy and maintain an approximately 90° phase lag with the primary structure. Parameter studies reveal that installing the TS-NES in the upper section of the pipe yields significant vibration reduction. The device is insensitive to stiffness variations, and appropriately increasing its mass, damping, and inclination angle can further enhance the vibration suppression effect.

Excellent compliance characteristics are demonstrated by rigid-flexible hybrid continuum robots that are made up of both elastic and flexible components. However, in conventional kinematic modeling techniques, their strong coupling nature results in high computational complexity and inadequate accuracy. In this work, kinematic models for cable-driven and pneumatically actuated continuum robots are developed using the Vector Form Intrinsic Finite Element (VFIFE) method. We developed a VFIFE-based forward kinematic model using finite spatial node simulation, which combines mass point discretization, the virtual reverse motion principle, and the central difference method. Accurate deformation prediction is made possible by this method, especially when examining large quasi-static deformations. The effectiveness of the suggested method is validated by simulation and experimental results, which show that the displacement error between the proposed model and the ADAMS/Simulink co-simulation model is less than 5% and that the VFIFE-based mechanical model exhibits over 90% agreement with the trajectory of the physical prototype experiments. This modeling framework offers theoretical support for real-time control and path planning in a variety of robotic systems made up of strut, cable, and beam elements while overcoming the drawbacks of conventional approaches in rigid-flexible hybrid systems.

Fourth-order hyperbolic equations model complex vibration and wave phenomena, and have broad applications in physics and engineering. This paper investigates a split mixed finite element method for such equations. By introducing intermediate variables, the fourth-order differential equation is reformulated as a system of lower-order equations. A semi-discrete split scheme is constructed, and the existence and uniqueness of its solution are established. Error estimates are derived using elliptic projection and the $L^2$-projection operator. A fully discrete split mixed finite element scheme is developed by applying the central difference method to discretize the time derivatives. The stability and convergence of the scheme are analyzed. Numerical experiments for a one-dimensional fourth-order hyperbolic equation validate the theoretical results.

This paper introduces a new explicit integration method for second-order ordinary differential equations (ODEs) commonly encountered in engineering applications. Traditionally, these problems are solved either by reformulating them as first-order systems to apply one-step methods such as Runge–Kutta schemes, or by using direct second-order approaches widely adopted in linear dynamics, including the generalized-α, central difference, and Newmark methods. The proposed method is derived from a Taylor series expansion truncated at the third derivative, resulting in a fully explicit algorithm that requires only one function evaluation per time step. Similar to Newmark’s formulation, it includes adjustable parameters that allow the user to balance accuracy and stability. For a specific parameter choice, the method exhibits convergence and stability properties comparable to those of the central difference scheme. An important advantage is that it remains explicit even when nonlinearities depend on first-derivative terms. The paper presents a theoretical analysis covering stability, local truncation error, spectral properties, numerical damping, and period elongation. The method is validated through four test cases from multibody dynamics, including linear and nonlinear problems. Results demonstrate that the Explicit Integration Grade 3 (EIG-3) method achieves accuracy comparable to existing explicit second-order integrators while significantly reducing computational cost, particularly in nonlinear applications.

Purpose The paper aims to introduce an innovative methodology to analyze the flutter velocity of a two-dimensional wing using numerical step-by-step integration methods. The primary objective is to apply the Theodorsen unsteady aerodynamic function to the two-degrees-of-freedom flutter equation and identify the flutter onset by assessing the decay rates at various reduced frequencies. Design/methodology/approach The proposed methodology transforms the flutter equation into a motion equation for a damped two-degrees-of-freedom system that integrates the Theodorsen function. Numerical step-by-step integration methods, including precise integration, Runge–Kutta, central difference and Newmark methods, are used to compute the time-dependent responses of displacement, velocity and acceleration. Findings The findings reveal that decay rates at different reduced frequencies can serve as a novel criterion for identifying flutter onset, marking a significant first in the field. The results obtained using step-by-step integration methods closely align with those using established eigenvalue calculation techniques, such as the V–g and p–k methods, confirming the accuracy and reliability of the numerical approach. Originality/value This study advances the understanding of flutter dynamics, consistent with established analysis methods while introducing a fresh perspective. The strong agreement among results from various numerical step-by-step integration methods underscores their robustness and reliability. This study offers an efficient and accurate tool for engineers and researchers to predict and analyze flutter in two-dimensional wing structures, essential for designing and ensuring the safety of aerospace vehicles.

To address the challenge of identifying damage in the hangers and bridge deck systems of long-span suspension bridges, this paper proposes a non-contact monitoring method based on video image recognition. This method extracts structural vibration displacement responses through video acquisition and image analysis, and combined with the strain mode change rate index, it achieves damage localization, type identification, and severity assessment. The principle of extracting displacement time-history data from video images is first elaborated, and MATLAB-based computational code is developed, including pixel tracking and time-history curve generation methods. The eigensystem realization algorithm is used to identify displacement mode shapes, which are then converted into strain mode shapes via the central difference method. The strain mode change rate and its deviation rate are proposed as damage indicators: under undamaged conditions, the curve is smooth; at damage locations, peaks appear; the distribution range of peaks can distinguish between hanger damage and bridge deck cracks; the deviation rate quantifies damage severity. The feasibility of the method is validated through finite element simulations and physical model experiments. The results show that hanger damage causes broad peaks, while bridge deck cracks present narrow peaks; the deviation rate increases monotonically with damage severity. Applied to an in-service suspension bridge, the method successfully identified hanger bending and weld cracking, with assessment results consistent with on-site inspections. This study demonstrates that the strain mode change rate analysis based on video images enables damage identification without prior knowledge of the structural health state, relying solely on the damaged state response. Offering advantages such as non-contact measurement, full-field monitoring, and no need for sensor deployment, it provides a new technical approach for the long-term monitoring of suspension bridge hanger systems.

Accurate crop phenology prediction is essential for modern agricultural management, irrigationscheduling, and climate change adaptation. This study develops a numerical-analysis-basedframework to predict maize (Zea mays L.) growth stages using daily meteorological data. Theproposed workflow integrates: (i) the non-linear Wang–Engel formulation to compute dailythermal units, (ii) cubic spline interpolation for data reconstruction under a missing-datavalidation scenario, (iii) Simpson’s 3/8 rule for numerical integration of cumulative thermalunits, (iv) the central difference method to analyze the accumulation-rate dynamics, and(v) Taylor series expansion for local approximation of the Wang–Engel function aroundthe optimum temperature. Daily meteorological data were obtained from the Open–MeteoHistorical API for Jakarta, Indonesia in 2025, comprising 348 observation days. Numericalintegration yields a cumulative thermal unit of 112.37 over the first 120 days. Derivativeanalysis identifies the maximum accumulation rate of 0.9784 per day at day 44. Using theadopted thermal thresholds, the model predicts the V3 stage at day 127 and the V6 stage atday 342.

This work is devoted to numerical analysis for transient heat transfer problems by the reduced integration and Richardson extrapolation (REQ method). This computationally efficient quadrature scheme is used to generate element matrices for functionally graded quadrilateral elements to analysis of unsteady state heat transfer. In the context of solving the finite element method (FEM) discrete formulations, the central difference method is considered for better accuracy, ensuring the reliability of the numerical solutions, since the central difference method posses stability and non-oscillatory nature, which are essential for achieving precise results. To assess the performance of the new numerical technique, the research focuses on validating the computational efficiency and accuracy that involves solving the benchmark reference problems and comparing the results with the outcomes obtained through conventional Gauss quadrature and other effective numerical methods from the existing literature. The validation process aims to demonstrate the superiority of the proposed REQ method in terms of computational speed and precision of the final results.

ABSTRACT The present paper deals with the analytical and numerical solution of the motion equation of a single degree of freedom (SDoF) oscillator with damping and negative stiffness (NS) due to seismic excitation at its base. The resulting solution shows that the structure's response displacement increases exponentially; thus, practically, there is no vibration. Moreover, a well‐known numerical procedure is selected to be modified in order to be suitable for the calculation of the response of the abovementioned NS‐SDoF oscillator. Specifically, an appropriate modification of the central difference method is presented, taking into account the damping and NS of the oscillator. The effectiveness of this proposed procedure is examined by comparing the extracted results to those obtained through the exact mathematical solution. The accuracy and precision of the proposed modified numerical method are investigated through numerical examples by comparing the results obtained with the analytical process. The results show that the new modified numerical method, as well as the analytical one, led to similar results; thus, the modified central difference method can be efficiently and safely used for the calculation of the response of NS‐SDoF systems.

The real concern, which represents the dangerous places in the tanks containing liquid materials, is those where cracks occurs in the walls of those tanks. To determine the zone areas where the maximum precipitation occurs, a study of those walls had to be conducted and the ideal dimensions of those walls had to be searched for in order to control the highest precipitation and thus reduced the possibility of cracks occurring as a result of the large precipitation of the walls and thus avoid being exposed to the risk of the liquids containing in those tanks penetrating into the walls and thus the possibility of the collapse of those and exposure to the loss of liquids and the economic losses and environmental risks that it causes. This paper deals with a new derivation to prediction the elastic behavior of tank wall supported with fixed ended on three edges and freely at the fourth and subjected to hydrostatic liquid triangular distributed load using central finite difference method. The new model was verified by compares with the classical model obtained by Timoshenko and Woinowsky-Krieger based on theory of plates and shells with a good agreement. The present parametric study deals with tank wall with length to depth ratios of 1, 2, 3 and 4 were considered. The results obtained from the analysis showed that the maximum deflection increased by 10.3%, 23.7% and 32.5% for L/h=2, 3, and 4 compares with L/h=1.

Guided wavefields in composite structures contain a wealth of information related to anomalies caused by interactions between waves and structural damage. Guided wave curvature has emerged as one of the most important damage indices due to its ability to extract spatial characteristics of the full wavefield. However, its practical application is limited by the classical modal curvature method, which requires central difference estimation. This approach is highly sensitive to measurement resolution and noise, resulting in poor stability and reduced reliability in damage identification. To address these limitations, a two-dimensional Fourier spectral method for guided wave curvature estimation is proposed. This study investigates the sensitivity of curvature derived from different guided wave modes, namely the symmetric and antisymmetric modes, in relation to damage detection. The two-dimensional Fourier transform is applied in the wavenumber domain to compute the modal curvature of the wavefields with improved numerical stability. The method is further assessed for its robustness to noise using experimental data obtained from composite structures. Compared with the classical central difference method, the proposed approach demonstrates significantly enhanced noise tolerance while preserving the accuracy of damage localization. The results validate that the twodimensional Fourier spectral curvature method provides a stable and reliable means for detecting invisible damage in composite materials.

This paper investigates the impact of lubrication and thermal effects on the performance of single-metal seals in roller cone bits, and it establishes the geometric, material, and operating parameter models for the single-metal seal. Based on the theory of statistics, the Greenwood–Williamson (G–W) model is employed to predict the contact stress of micro-protrusions on the sealing pair surface. This study establishes a Thermal Elastohydrodynamic Lubrication (TEHL) coupling model for single-metal seals, which utilizes the deformation matrix method to characterize the microscopic deformation of the sealing interface. The central difference method is applied to solve the oil film thickness and temperature distribution in the axial and film thickness directions of the sealing surface. The results indicate that the sealing zone is predominantly under rough peak contact pressure, operating in a mixed-lubrication state. Oil film thickness negatively correlates with static contact pressure, and seal pressure and pre-compression displacement significantly influence lubrication performance. Experiments validate the numerical simulation results, with a mean relative error of less than 15%, confirming the model’s effectiveness. This study offers a theoretical basis for optimizing single-metal seal design, enhancing the reliability and lifespan of roller cone bits in harsh conditions.

We investigate a backward problem for the ultraparabolic equation with nonlinear source function, which aims to determine the initial status of some physical field such as option pricing, multi-parameter Brownian motion, population dynamics, and so forth. This problem is well-known to be ill posed because the solution exhibits unstable dependence on the given data functions. To obtain the approximate solution, we apply the central difference method and establish a convergence rate of H\"{o}lder type under some a priori assumptions on the exact solution. Eventually, a numerical example is given to justify the theoretical analysis.

The paper is devoted to developing an innovative non-contact vibration measurement system for electrical equipment monitoring. A comprehensive comparative analysis of existing vibration monitoring methods is conducted, identifying their limitations and substantiating the necessity of transitioning to non-invasive diagnostic technologies for industrial unit predictive maintenance. To address the non-contact measurement challenge, high-speed imaging (≥1000 fps) of operating equipment is applied with subsequent frame-by-frame processing of reference points or areas displacement on the unit body. Technical limitations and equipment requirements are examined in detail including the necessity of satisfying Nyquist-Kotelnikov theorem for accurate high-frequency oscillation registration up to 500 Hz. For automatic detection and segmentation of target areas, identification using YOLOv4 neural network algorithms with CSPDarknet53 architecture trained via transfer learning on specialized industrial equipment dataset is proposed. The key point localization is performed using Shi-Tomasi corner detector with infrared fiducial markers support, ensuring sub-pixel positioning accuracy under variable illumination conditions. The acquired images undergo preprocessing for contrast enhancement using CLAHE method and noise suppression through cascaded Gaussian, median and bilateral filters. The system robustness to industrial operating conditions is achieved through synchronized pulsed IR illumination. Displacement tracking is performed by modified pyramidal Lucas-Kanade algorithm with inverse-compositional Gauss-Newton method providing metrological accuracy up to 0.01 pixel. Kinematic parameters calculation was performed using central differences method with subsequent Savitzky-Golay filtering. The vibration signal spectral analysis is implemented using fast Fourier transform and Morlet wavelet transform with preliminary adaptive Kalman filtering. The developed system demonstrates measurement error ≤5% in the 0-5 kHz range while achieving six-fold capital expenditure reduction as compared to laser vibrometers with two-year payback period.

Direct detection (DD) is a straightforward, cost-effective receiving scheme for medium- and short-range fiber-optic communication systems, yet directly accessing phase information presents inherent challenges. The temporal transport-of-intensity equation (T-TIE) enables phase recovery from intensity data, but the accuracy of this phase-retrieval method is constrained by finite difference approximation errors of intensity derivatives and electrical noise interference. In this paper, we propose a 4th-order central difference method for calculating intensity derivatives to enhance approximation accuracy and implement multiple intensity measurements to further mitigate electrical noise interference. The proposed method is validated in a 28 GBaud single-carrier 16-quadrature amplitude modulation (16QAM) direct detection system. The research results indicate that, under conditions of 10 nA dark current and 20 pA/Hz^1/2 thermal noise, our method achieves a receiver sensitivity gain of 14.85 dB compared with the 1st-order forward difference method and 8.47 dB compared with the 2nd-order central difference method at the 7% hard decision forward error correction (7% HD-FEC) threshold.

Purpose This study aims to elucidate the role of the main roof in highwall mining slope stability under localized coal pillar failure conditions. Prior research has not adequately addressed how large-scale pillar failures propagate instability through the mechanical linkage between the pillar support system and the slope sliding system. By analyzing fracture mechanisms and crack morphology of the main roof, this model seeks to clarify its control effect on highwall mining slope stability. Design/methodology/approach A CCCF (three edges clamped, one free) elastic foundation-supported thin rectangular plate model under trapezoidal load was established to analyze main roof fracturing in highwall mining. Using the derived governing equations and a second-order central difference method, the research examined key parameters influencing principal bending moments. Findings Pillar failure width and overburden depth increase all bending moments. Greater thickness and elastic modulus of the main roof amplify |M|z but reduce |M|c and |M|t, while higher foundation stiffness decreases |M|z and increases others. Three fracture sequences were identified, forming a horizontal “U-Y” pattern: U-shaped fractures develop around pillar failure zones, while Y-shaped fractures occur within failed pillars. Originality/value The mechanical model considers the CCCF boundary conditions, as well as the trapezoidal loads, and simplifies the edge of the coal pillar as Winkler elastic foundation to consider its mechanical effects. Validation through a case study demonstrated the model's accuracy in predicting fracture behavior aligned with actual mine pressure observations. The mechanical model and calculation method provide reliable theoretical support for assessing roof stability in similar engineering scenarios.

In this paper, a mass lumped, rate-type C 0 continuous finite element model of arbitrarily higher order is developed using well-known hypoelastic–plastic type constitutive relations for a shear deformable (Timoshenko) beam. Higher-order elements provide true curvature, improved performance in bending and buckling, and avoid hourglass modes and numerical locking without ad hoc treatments. Historically, however, first-order elements have been predominant in explicit finite element codes because of difficulties achieving robust and reliable mass-lumping. Although known to exhibit some deficiencies in certain regimes, hypoelastic type constitutive relations possess sufficient theoretical rigor and have been successfully and extensively used in production finite element codes for the nonlinear applications of interest herein. First, the governing equations of motion for a Timoshenko beam element capable of finite deformation are developed.We spatially discretize using a standard weak-form Galerkin finite element approximation and optimal C 0 Lagrange shape functions of arbitrarily higher order and integrate throughout time using a central-difference method and a lumped mass matrix. Then, the thermodynamical consistency of the hypoelastic-plastic rate type formulation is proved starting with a Gibbs potential and its viability for the considered elasto-plasticity applications is discussed.We present several numerical examples of nonlinear finite deformation demonstrating the accuracy and utility of the lumped-mass beam elements developed herein and evaluate the performance of increased order compared to standard linear elements.

This study evaluates the implicit, energy-conserving, and decaying scheme of Mamouri et al. for dynamic buckling analysis of geometrically exact 2D Reissner beams under non-impulsive, long-duration loading. While explicit schemes such as the central difference method excel in fast-dynamic events, they become computationally prohibitive in slow-to-moderate regimes due to stability constraints requiring millions of tiny time steps. Classical implicit methods like Newmark are efficient for smooth dynamics with small steps but fail to converge under large time steps in highly nonlinear regimes, particularly in the presence of high-frequency noise. In contrast, the proposed scheme combines unconditional stability with selective dissipation of high-frequency modes, enabling accurate and robust simulations using large time steps and drastically reducing the number of increments. Applied to three limit-point problems involving snap-through and snap-back, it consistently demonstrates superior efficiency: explicit schemes can incur excessive computational cost over long durations, and the Newmark scheme loses convergence under large steps in highly nonlinear contexts. In contrast, the proposed scheme delivers remarkable efficiency, accuracy, and stability; enabling reliable long-duration dynamic buckling analysis in highly nonlinear systems.

In this paper, we derive a semi-discrete scheme using the central difference method, which perfectly preserves the conservation of mass and energy for the Hirota equation. By applying the Crank–Nicolson method for temporal discretization, we develop the fully discrete scheme that conserves mass and energy. It is shown that the accuracy of the fully discrete scheme is of the second order in space and time. Because the Crank–Nicolson discretization leads to a nonlinear algebraic system, an efficient iterative solver is proposed that linearizes and solves the resulting five-diagonal matrix at each iteration while treating high-order contributions iteratively to reduce computational cost. Numerical experiments are presented to demonstrate the accuracy and verify the conservation properties.

This work presents a comparative study between two numerical approaches for modeling fracture in quasi-brittle materials: the truss-based Discrete Element Method (DEM) and the Phase Field Method (PFM), formulated within the Finite Element Method (FEM) framework. In the version of the Discrete Element Method used here (referred to as DEM), the spatial discretization is performed using a regular arrangement of pinned bars with masses concentrated at the nodes. The equivalent cross-sectional area of the diagonal and normal bars allows the representation of an equivalent elastic solid. To capture the nonlinear mechanical behaviour produced for the evolution of the material damage, a bilinear constitutive law is applied to each bar. This formulation enables the definition of a motion equation that must be integrated in time using an explicit scheme, such as the central difference finite difference method. An important feature of the present model is its ability to incorporate material properties as random fields. In contrast, the phase field model introduces a scalar damage field to describe fracture as a continuous transition in the medium, and in this work, it is implemented based on the Principle of Virtual Work. To compare the performance of both methods, three examples are presented. A Single Edge Notch Bending (SENB) performed in epoxy resin, a Notched Plate with Hole (NPWH) also built in cement mortar, and finally a parametric study where the influence of the material parameters of a specimen composed of a substrate, and an interface orthogonal to the crack propagation direction is analyzed. The comparison of the two approaches through these three examples allows the identification of the strengths and weaknesses of each method.