Search Results (7,717)

The surface tension of a liquid droplet can be determined by fitting its actual profiles using the Young–Laplace equation, effectively reducing the measurement of surface tension to an accurate determination of the droplet’s profiles. Spectral confocal sensors are high-precision, interference-resistant, non-contact measurement systems for droplet surface profiling, employing a light source together with a dispersive objective lens and a spectrometer to acquire depth-dependent spectral information. The accuracy and stability of surface tension measurements can be effectively enhanced by spectral confocal sensors measuring the droplet surface profile. Although existing spectral confocal sensors have significantly improved measurement range and accuracy, their angular measurement performance remains limited, and deviations may arise at droplet edges with large inclinations or pronounced surface profile variations. This study presents the optical design of a large-angle spectral confocal sensor. By theoretically analyzing the conditions for generating linear axial dispersion in the dispersive objective lens, a front-end dispersive objective lens was designed by combining positive and negative lenses. Based on a Czerny–Turner (C-T) configuration, the back-end spectrometer was designed under the astigmatism-free condition, taking into account both central and edge wavelength effects. Zemax was employed for simulation optimization and tolerance analysis of each optical module. The results show that the designed system achieves an axial dispersion of 1.5 mm over the 430–700 nm wavelength range, with a maximum allowable object angle of ±40° and a theoretical resolution of 3 μm. The proposed spectral confocal sensor maintains high measurement accuracy over a wide angular range, facilitating precise measurement of droplet surface tension at large inclination angles. Full article

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium ($E_1$). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (${\lambda }_u>{\lambda }_s$), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article

As sustainable tourism gains global momentum, extended reality (XR) technologies have emerged as important tools for enhancing visitor experiences at overburdened World Heritage Sites while mitigating physical deterioration through non-consumptive engagement. However, existing research on immersive technologies in heritage tourism has largely relied on single-cultural samples and has paid limited attention to theoretically grounded boundary conditions in post-adoption behaviour. To address these gaps, this study extends the Expectation–Confirmation Model (ECM) by incorporating cultural distance (CD) and prior visitation experience (PVE) as moderating variables, and empirically tests the proposed framework using a mixed domestic–international sample exposed to an on-site XR application at the Badaling Great Wall World Heritage Site. Data were collected immediately after the XR experience and analysed using structural equation modelling. The results validate the core relationships of ECM while identifying significant moderating effects. Cultural distance attenuates the positive effects of confirmation on perceived usefulness as well as the effect of perceived usefulness on continuance intention, while prior visitation experience weakens the influences of enjoyment and visual appeal on satisfaction. These findings establish important boundary conditions for ECM in immersive heritage contexts. From a practical perspective, the study demonstrates that high-quality, culturally responsive XR can complement physical visitation and support sustainable conservation strategies at large-scale linear heritage sites. Full article

During the last decade, F-contraction has been a widely investigated problem in the fixed point theory. There are various outcomes regarding the extensions and generalizations of F-contraction in different perspectives, along with the findings concerning the application of those ideas, mostly in the area of differential and difference equations, fractional calculus, etc. The present article concludes some existence and uniqueness outcomes on fixed points for (φ, F)–contractions in the context of a metric space endowed with a local class of transitive binary relations. Some illustrative examples are furnished to justify that our contraction conditions are more general than many others in this area. The findings presented herein are used to obtain a unique solution to certain fractional boundary value problems. Full article

We present two complementary approaches to the Gorini–Kossakowski–Sudarshan–Lindblad equation for an open qubit. First, based on linearity, yields solutions illustrated by mixed-state trajectories in the Bloch ball, including non-random asymptotic fixed points and exceptional points. Second, exploiting the SU$\left(2\right)$ symmetry, leads to a nonlinear dynamical system that separates angular dynamics from radial dissipation. This symmetry-based perspective presents a promising route toward generalization to open qudits. Full article

The accuracy of dynamics parameters in the transmission system is essential for high-performance motion trajectory planning and stable operation of heavy-duty servo presses. To mitigate the performance degradation and potential overload risks caused by deviations between theoretical and actual parameters, this paper proposes a dynamics model accuracy enhancement method that integrates multi-objective global sensitivity analysis and ant colony optimization-based calibration. First, a nonlinear dynamics model of the eight-bar mechanism was constructed based on Lagrange’s equations, which systematically incorporates generalized external force models consistent with actual production, including gravity, friction, balance force, and stamping process load. Subsequently, six key sensitive parameters were identified from 28 system parameters using Sobol global sensitivity analysis, with response functions defined for torque prediction accuracy, transient overload risk, thermal load, and work done. Based on the sensitivity results, a parameter calibration model was formulated to minimize torque prediction error and transient overload risk, and solved by the ant colony algorithm. Experimental validation showed that, after calibration, the root mean square error between predicted and measured torque decreased significantly from 1366.9 N·m to 277.7 N·m (a reduction of 79.7%), the peak error dropped by 72.7%, and the servo motor’s effective torque prediction error was reduced from 7.6% to 1.4%. In an automotive door panel stamping application on a 25,000 kN heavy-duty servo press, the production rate increased from 11.4 to 11.6 strokes per minute, demonstrating enhanced performance without operational safety. This study provides a theoretical foundation and an effective engineering solution for high-precision modeling and performance optimization of heavy-duty servo presses. Full article

In this article, we investigate a nonlinear $\psi$-Hilfer fractional order Volterra integro-delay differential equation ($\psi$-Hilfer FRVIDDE) and a nonlinear $\psi$-Hilfer fractional Volterra delay integral equation ($\psi$-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the $\psi$-Hilfer FRVIDDE and $\psi$-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article

The study concerns the Duffing oscillator (Duffing equation) with softening stiffness. Using numerical simulations, the upper and lower limits of the unstable region for damping equal to 0 were identified, and analytic formulas describing them were developed. The analysis shows that the developed formulas are effective for all combinations of stiffness coefficient values that were tested. The curve of the upper limit of the unstable region is a jump-down curve, and in the theory of nonlinear systems, this curve for damping equal to zero is identified as a backbone curve (the curve of natural frequency of a system). However, the classical backbone curve calculated via a formula that is commonly known and used differs visibly from that actually obtained via numerical simulations of the upper boundary of the unstable region at large amplitudes. It could therefore be concluded that the backbone curve is not equal to the upper boundary of the unstable solution region. Moreover, the paper shows that the use of the scale relative to a critical oscillation amplitude leads to the conclusion that for damping equal to 0, systems with different parameters have the same instability regions in dimensionless space. Full article

This work presents the design of passivity based non-fragile retarded sampled data control (NFRSDC) for the wind energy system using permanent magnet synchronous generator. At first, the proposed system is characterized in terms of non-linear dynamical equations, which is later expressed in terms of linear sub-systems via fuzzy membership functions using the Takagi–Sugeno fuzzy approach. After that, a more applicative NFRSDC is proposed along with the delay involved during signal transmission as well as randomly occurring controller gain perturbations (ROCGPs). Here, the ROCGPs are modeled accordingly using stochastic variable which obeys the certain Bernoulli distribution sequences. Folowing that, an appropriate Lyapunov–Krasovskii functionals are constructed to obtain the sufficient conditions in the form of linear matrix inequalities. These obtained conditions are then used to ensure the global asymptotic stability of the given system with the exogenous disturbances. Finally, numerical simulations are performed using MATLAB/Simulink and the obtained results have clearly demonstrated the efficacy of the proposed controller. Full article

This paper investigates the optimal control problem of orbital rendezvous for spacecraft in near-circular orbits with a low-thrust propulsion system. Two optimality criteria are considered: time-optimal and motor-time-optimal control. A linearized mathematical model of relative motion between the active and passive spacecraft is employed, which is formulated in dimensionless variables that characterize secular, periodic, and lateral motion components of the relative motion. By applying Pontryagin’s Maximum Principle, the equations governing the optimal relative motion of the spacecraft are derived. To address the discontinuities associated with the bang–bang switching function inherent in the motor-time-optimal problem, and the lack of a suitable initial guess, a homotopy method is adopted, in which the solution to the rendezvous time-optimal problem is used as an initial guess and is gradually deformed into the motor-time-optimal control. Considering the errors introduced by the linearization of the relative motion model, the obtained control law is validated via numerical simulations based on the original nonlinear dynamics of the system. Simulation results demonstrate that the proposed trajectory optimization methodology achieves high success rates and rapid convergence, providing valuable theoretical support and practical guidance for mission scenarios with similar trajectory design requirements. Full article

The nonlinear Bagley–Torvik equation is of fundamental importance, as it captures a realistic and intricate interplay among memory effects, nonlinearity, and functional dependence—making it a powerful model for a wide range of natural and engineered systems. Its analysis contributes significantly to both the theoretical development of fractional differential equations and their practical applications across science and technology. In this paper, we employ the inverse operator method, the multivariate Mittag-Leffler function, and several classical fixed-point theorems to establish sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions to the nonlinear Bagley–Torvik equation with functional initial conditions. Finally, we present several examples by explicitly computing values of the multivariate Mittag-Leffler functions to illustrate the main results. Full article

The diffusion equation models a wide variety of physical and chemical processes and has significant interest in many scientific disciplines. Analytical and numerical methods found in the literature for solving the diffusion equation consider a constant diffusion coefficient. However, in reality, it is not constant. In this paper, we present a numerical approach to solve the diffusion equation when the diffusion coefficient is not constant. Unlike existing methods that require solving non-linear systems with iterative schemes, our approach transforms the problem into a linear system, drastically reducing computational cost while preserving temporal accuracy. Full article

The present paper investigates the dynamic behavior of an unbalanced rotor mounted in a balancing machine. Differential equations of motion are derived without linearization using Lagrange equations of the second kind to determine the nonlinear nature of the system. This study proposes a method for using differential equations in balancing to determine important parameters, such as the coordinates of the center of mass and the products of inertia of the rotor. An analysis of the interactions between the periodicities of the individual terms in the differential equations is carried out in order to eliminate terms with difficult-to-determine moments of inertia. Full article

The Parameter Expansion Method (PEM) is employed to study nonlinear Jerk equations, which are often difficult to solve because of their strong nonlinearity. This method provides higher accuracy and broader applicability, enabling analytical insights and closed-form approximations. This study explores the use of Prof. He’s PEM to derive approximate analytical solutions of the nonlinear third-order Jerk equation, this model is commonly encountered in the analysis of complex dynamical systems across physics and engineering. Owing to the strong nonlinearity inherent in Jerk equations, exact solutions are often unattainable. The PEM provides a simple, effective framework by expanding the solution with respect to an embedding parameter, allowing accurate approximations without the need of small parameters or linearization. The method’s reliability and precision are validated through comparisons with numerical simulations, demonstrating its practicality and robustness in tackling nonlinear problems. The results indicate that PEM provides highly accurate approximations of nonlinear Jerk equation, showcasing greater simplicity and efficiency relative to other analytical methods, along with excellent concordance with numerical simulations. Additionally, the nonlinear Jerk equation demonstrates exact approximate solutions via PEM, closely mirroring numerical results and surpassing several contemporary analytical techniques in efficiency and usability. Furthermore, the study indicates that PEM is a straightforward and effective approach in solving nonlinear Jerk equation. It generates accurate estimates that nearly align with numerical simulations and surpass numerous other analytical methods. Full article

This study investigated twin-propeller-induced scour on sandy seabeds with varying grain sizes (d₅₀ = 0.11, 0.5, and 0.95 mm) through a series of laboratory experiments. The effects of propeller rotation speed (rpm), offset height (y₀), propeller diameter (D_p), and sediment grain size (d₅₀) on scour development were examined. Results indicated that sediment grain size significantly influences scour patterns. A key objective was to develop predictive expressions for primary scour characteristics at equilibrium: maximum scour depth (S_max), scour hole length (L_max), and maximum scour width (B_max). Using a nonlinear regression approach, the proposed expressions demonstrated strong predictive performance. Findings show that equilibrium scour depth increases with higher Froude numbers (F₀) but decreases with larger sediment size (d₅₀) and higher propeller offset (y₀). Additionally, empirical equations were formulated to predict the temporal evolution of scour depth, achieving high correlations with experimental data (R² > 0.97). These results enhance understanding of scour induced by unconfined twin-propeller jets in harbors or navigation channels and provide valuable data for the design and protection of harbor basins. Full article