Surface Equation Research Articles

Precursor sticking coefficient determination from indented deposits fabricated by electron beam induced deposition.

A fast simulation approach for focused electron beam induced deposition (FEBID) numerically solves the diffusion-reaction equation (continuum model) of the precursor surface on the growing nanostructure in conjunction with a Monte Carlo simulation for electron transport in the growing deposit. An important requirement in this regard is to have access to a methodology that can be used to systematically determine the values for the set of precursor parameters needed for this model. In this work we introduce such a method to derive the precursor sticking coefficient as one member of the precursor parameter set. The method is based on the analysis of the different growth regimes in FEBID, in particular the diffusion-enhanced growth regime in the center region of an intentionally defocused electron beam. We employ the method to determine the precursor sticking coefficient for bis(benzene)chromium, Cr(C6H6)2, and trimethyl(methylcyclopentadienyl)platinum(IV), Me3CpPtMe, and find a value of about 10-2 for both precursors, which is substantially smaller than the sticking coefficients previously assumed for Me3CpPtMe (1.0). Furthermore, depositions performed at different substrate temperatures indicate a temperature dependence of the sticking coefficient.

Influence of thermal flux on Marangoni convection in weakly viscoelastic thin films on uniformly heated substrate

This study examines the hydrodynamic stability of weakly viscoelastic thin film of Walters liquid B″ type on a uniformly heated rigid inclined plane. The current model incorporates a heat flux (HF) boundary condition, which is a realistic representation. This approach accounts for heat flux and heat loss at the wall–liquid and wall–air interfaces, allowing for an examination of the effects of a nonvanishing temperature gradient at the top of the rigid wall. This consideration is essential, as maintaining a constant temperature throughout the wall in laboratory settings poses significant challenges. A Benney-type evolution equation is derived using the long-wave expansion of the flow variables, describing the film thickness as a function of x, t. A linear stability analysis in normal mode is conducted to determine the onset of instability to the critical Reynolds number. The linear study indicates the dual role of the wall film Biot number (Bw). For small values of (Bw), a destabilizing effect is observed; however, upon reaching a critical value, a stabilizing effect is produced. The linear study confirms that the presence of Bw consistently results in a destabilizing effect on the viscoelastic parameter. The nonlinear free surface equation is numerically simulated using the Fourier Spectral method. The temporal evolution of hmax and hmin in the presence of (Bw) proves the viscoelastic parameter's destabilizing effect.

The influence of thermal aging on the interlaminar mechanical properties and failure mechanisms of CFRP under complex stress states

The fiber/matrix interlayer structure of carbon fiber reinforced polymer (CFRP) is temperature-sensitive, so the aging is likely to cause degradation of interlayer performance. Aging experiments were conducted under high-temperature and thermal-cycling environments. The aged CFRP were performed to investigate the variation patterns of interlayer failure strength under different temperatures and stress states. The study found that thermal-cycling aging resulted in less severe degrees of post-curing and thermal oxidative aging. The failure of aged CFRP was a combined effect of fibers and matrix, and high temperature exacerbated the damage. A surface equation of the quadratic stress failure criterion was established.

Experimental Study on the Mechanical Properties and Damage Mechanism of Saturated Coal-Measure Sandstone in an Open Pit Mine Under the Freeze–Thaw Effect

The damage and degradation of coal-measure sandstone in cold-region open-pit mines due to freeze–thaw effects has become one of the significant factors inducing instability in the rock mass of open-pit mine slopes. This study conducts experiments on the physical and mechanical properties of saturated coal-measure sandstone under varying freeze–thaw cycle counts and freezing temperatures, revealing the intrinsic mechanisms of damage and degradation in saturated coal-measure sandstone due to freeze–thaw effects. The experimental results indicate that, with an increase in the number of freeze–thaw cycles and a decrease in the freezing temperature, the elastic modulus and peak compressive strength of the specimens exhibit an exponential decrease. In contrast, the peak strain shows an exponential increase. However, compared to the freezing temperature, the increase in the freeze–thaw cycle frequency leads to a more significant change in the mechanical parameters of the specimens, indicating that the frequency of freeze–thaw cycles has a more pronounced effect on the deformation resistance of saturated coal-measure sandstone than the freezing temperature. The failure mode of coal-measure sandstone specimens under uniaxial compressive loading primarily exhibits shear failure; however, as the number of freeze–thaw cycles increases and the freezing temperature decreases, the specimens begin to exhibit tensile failure modes, which gradually develop into a combined tensile and shear failure mode. Based on the experimental data, two sets of surface equations were fitted to characterize the relationship between the mechanical properties (peak compressive strength, elastic modulus) of the specimens and the experimental parameters (number of freeze–thaw cycles, freezing temperature). The research findings can provide references and insights for engineering disasters caused by the degradation of coal-bearing sandstone in cold-region open-pit mines.

CONSTRUCTION OF A MINIMAL PARAMETRICALLY SPECIFIED SURFACE BY THE METHOD OF MINIMIZING THE DIRICHLET INTEGRAL

In this paper we consider the polynomial approximate solutions of the Dirichlet problem for minimal surface equation. It is shown that under certain conditions on the geometric structure of the domain the absolute values of the gradients of the solutions are bounded as the degree of these polynomials increases. The obtained properties imply the uniform convergence of approximate solutions to the exact solution of the minimal surface equation. In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions. The study of this issue is especially important for nonlinear equations since in this case there is a series of difficulties related with the impossibility of employing traditional methods and approaches used for linear equations. At present, a quite topical problem is to determine the conditions ensuring the uniform convergence of approximate solutions obtained by various methods for nonlinear equations and system of equations of variational kind. In this case, it is natural to employ variational methods of solving boundary value problems. And this is an issue on the justification of these methods arises, which is reduced to studying general properties of approximate solutions. In the course of this work, an algorithm for numerical modeling of minimal surfaces was developed based on the Weierstrass-Enneper representation. An algorithm was also implemented in Python that allows you to calculate approximate minimal surfaces in the class of polynomial vector functions defined on the unit disk and satisfying the boundary condition Dirichlet (Plateau problem).

Research on calibration method for line-structured light sensor based on spatial quadratic surface fitting

Abstract The calibration of the light plane serves as the fundamental prerequisite for accurate three-dimensional (3D) measurement using line-structured light sensor (LSLS). Aiming at the problem that the light plane projected by the line laser is not an ideal plane, this paper proposes an LSLS calibration method based on spatial quadratic surface fitting. In the LSLS measurement model, the standard conical quadratic surface equation is used to replace the plane equation in the traditional measurement model to solve the 3D coordinates of the light stripe. In the LSLS calibration process, the spatial standard conical quadratic surface fitting algorithm is also used to replace the traditional plane equation fitting method to achieve structural parameter calibration. The calibration experiment results based on general LSLS show that the calibration method described in this paper improves the fitting accuracy by 15.38% and the 3D measurement accuracy by 13.33% compared with the traditional calibration method based on light plane fitting. This not only provides a high-precision measurement solution for low-cost LSLS, but also enables its application in 3D measurements in the presence of lens refraction, where the improvement in accuracy may be even more significant.

Design and Improvement of Curved Envelope Meshing Pair Profile of Single Screw Compressor

Abstract In recent years, the wear problem of the wheel tooth in the single screw compressor (SSC had been solved due to the development and application of various envelope profiles. Among these profiles, the curved envelope profile shows performance advantages, which displays great application potential in the fields of high-temperature heat pump, steam compression, etc. However, SSC with curved envelop profile still faces the problem of incomplete design method, large leakage clearance, difficult processing and meshing pair interference under extreme conditions. Therefore, aiming at solving these problems, a coupling profile design method for the curved envelope meshing pair of the SSC was presented and studied. The profile equation of curved envelop meshing pair of a SSC with an ellipse profile as the characteristic line was established. The method of using standard tooth width was proposed for the quick geometric design of the SSC. The coupling profile design of the star wheel tooth angle and the groove bottom angle with variable diameter circular arc was invented. The equation of the transition arc surface of the bottom angle of the groove and the angle of the star wheel tooth was derived. It provided a theoretical basis for the design and improvement of meshing pair with curved envelop profile.

Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity

Abstract Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation and to quadratic polynomials for the Monge–Ampère equation for dimension n ≥ 3 n\geq 3 , with an extra logarithmic term for n = 2 n=2 . Via Kelvin transforms, we characterize remainders in the asymptotic expansions by a single function near the origin. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge–Ampère equation in even dimension, but only C n − 1 , α C^{n-1,\alpha} for the Monge–Ampère equation in odd dimension 𝑛, for any α ∈ ( 0 , 1 ) \alpha\in(0,1) .

Study on type of magnetic suspension rotor groove and wear of drop touchdown bearing

The active magnetic bearing (AMB) is widely used in the field of flywheel energy storage system (FESS) in wind power generation. This study mainly studies the magnetic suspension rotor groove (MSRG) of FESS and the wear of drop touchdown bearing (DTB). The mathematical model of air gap magnetic field waveform on stator surface and Plath equation of control in air gap region were established. Contrastive analysis of conventional, second harmonic and third harmonic ‘petal-shaped’ MSRG in AMB. The life test of DTB on magnetic suspension rotor was carried out for 24000r/min and lasted for 1500 h. The second harmonic ‘petal-shaped’ MSRG could not only generate reluctance torque and increased the torque density of the motor, but also increased the fundamental component and excitation torque component by reducing the peak value of the magnetic field. The temperature ranges of front and rear DTB were ‘30℃–46℃’ and ‘26℃–40℃’, respectively. After service, the maximum thickness of metamorphic layer in the DTB raceway was 6 μm. After service, the content of Fe in the metamorphic layer of DTB raceway decreased by 10.11% and the content of C increased by 2.41%. Raceway hardness of DTB after service was 59.67 HV higher than that before service.

Calculation and analysis of wet clutch sliding torque based on fluid-solid coupling dynamic behavior

The fluid dynamics equations of groove cavity and non-grooved gap are established and combined to obtain fluid load bearing model. Considering the chain effect among local deformation, the deformation equation of friction surface under fluid-solid coupling load is developed to characterize reconstructed friction gap. The internal pressure and external load are used to iteratively solve the distribution of film thickness and contact area. An improved sliding torque calculation model is established by the microscopic state of fluid motion and solid deformation instead of fitting friction coefficient by multiple conditions. The accuracy of torque model is proved by sliding test. The influence of applied pressure, relative speed and lubricant flow on torque is weakened in turn according to fluid-solid state analysis.

A novel way for generating special surfaces

Abstract In this paper, we propose a novel approach for generating some special surfaces. Our solution offers a possible way to obtain exact equations for certain surfaces, thus potentially providing more accurate models for applications arising from scientific or engineering fields.

Analytical and Experimental Study of the Start of the Chip Removal in Rotational Turning

The present challenges in the automotive industry require the development and practical implication of novel machining procedures, which will provide appropriate solutions. These procedures should still meet the requirements of productivity, surface quality and energy efficiency. The further development of novel machining procedures introduces new problems that did not occur (or occurred to a lesser extent) with traditionally applied procedures. Rotational turning has come to the attention of production engineers in the previous decade since it can be used to machine ground-like surfaces in an ecologically friendly and highly productive manner. However, the chip removal characteristic is slightly different from traditional turning due to the applied special kinematic relation and complex tool edge geometry. The run-in phase will take longer, which is the time period between the first contact of the tool and the formation of a constant chip cross-sectional area. The clarification of the chip formation is important in any machining procedure. To achieve this goal, the geometric parameters of the chip must be determined. Since the start of the chip removal is a crucial stage in rotational turning due to its length, the chip height, chip width and the cross-sectional area of the chip should be separately defined in the initial stage. Therefore, in this paper, the initial phase of chip removal in rotational turning is studied. The increasing cross-sectional area of the chip is determined analytically by the application of the previously elaborated equation of the cut surface. Calculating formulas are defined for the different stages of the start of the chip removal, which could be used in the forthcoming studies to analyze the chip formation. The effects of different determining parameters are analyzed theoretically by the deduced formulas of the run-in phase and practical experiments are also carried out. The analytical and experimental analyses showed that increasing feed also increases the dynamic load on the cutting edge, while the depth of cut lowers the growth of the characteristic parameters of the chip, which results in a lower dynamic load on the tool.

A response surface equation for the drag polar for an airplane flying in the vicinity of wavy sea surface

The drag polar equation is the key issue to calculate the performance-based parameters during various flight phases and determine the optimum conditions. This is an essential feature to be included in airplane flight control system to manage the flight. The aircraft under certain emergency conditions are instructed to land on sea water. This underlines the combinational role of ground effect and the surface wave shape on airplane aerodynamic responses. Up to now, many investigations have described the airfoil and wing aerodynamic behaviors near the ground but none was devoted to an airplane flying near a wavy surface. In this paper, intensive numerical simulations have been carried out on a general aviation airplane flying near the water wavy surface. The effects of ground proximity as well as the wave amplitude have been investigated on airplane 2D longitudinal forces and moment. Some limited wind tunnel tests have also been performed on the same model, in the vicinity of water surface to validate the numerical calculations. A statistical quadratic equation based on Response Surface Model, RSM, has been proposed for the airplane flying in the vicinity of the wavy water in which, the wave shape deterioration due to both the airplane flowfield and the viscous dissipation has been taken into account. The statistical measures approve the model quality and accuracy in comparison to the numerical simulation data. The results show that the oscillations in aerodynamic forces are strong functions of the wave amplitude. The time variations of both CL and CD follow the shape of the surface wave while it does not have any remarkable effect on their mean values.

Development of Mathematical Function Control-Based 3D Printed Tablets and Effect on Drug Release

PurposeThe application of 3D printing technology in drug delivery is often limited by the challenges of achieving precise control over drug release profiles. The goal of this study was to apply surface equations to construct 3D printed tablet models, adjust the functional parameters to obtain multiple tablet models and to correlate the model parameters with the in vitro drug release behavior.MethodsThis study reports the development of 3D-printed tablets using surface geometries controlled by mathematical functions to modulate drug release. Utilizing fused deposition modeling (FDM) coupled with hot-melt extrusion (HME) technology, personalized drug delivery systems were produced using thermoplastic polymers. Different tablet shapes (T1-T5) were produced by varying the depth of the parabolic surface (b = 4, 2, 0, -2, -4 mm) to assess the impact of surface curvature on drug dissolution.ResultsThe T5 formulation, with the greatest surface curvature, demonstrated the fastest drug release, achieving complete release within 4 h. In contrast, T1 and T2 tablets exhibited a slower release over approximately 6 h. The correlation between surface area and drug release rate was confirmed, supporting the predictions of the Noyes-Whitney equation. Differential Scanning Calorimetry (DSC) and Scanning Electron Microscope (SEM) analyses verified the uniform dispersion of acetaminophen and the consistency of the internal structures, respectively.ConclusionsThe precise control of tablet surface geometry effectively tailored drug release profiles, enhancing patient compliance and treatment efficacy. This novel approach offers significant advancements in personalized medicine by providing a highly reproducible and adaptable platform for optimizing drug delivery.Graphical

Multi-line laser 3D reconstruction method based on spatial quadric surface and geometric estimation

The traditional multi-line laser 3D reconstruction is based on binocular epipolar constraints and the equation of the laser optical plane. Firstly, matching points are identified based on the epipolar constraints. Then, the correct matching points are selected based on the optical plane equation of the multi-line laser. Finally, 3D reconstruction is performed using the selected matching points. In actual production processes involving multi-line lasers, noticeable distortion occurs due to the projection of Diffractive Optical Elements (DOE) at a large angle. This results in curved laser light planes and curved reflections on the measured objects. Under such circumstances, efficiently screening matching points using the traditional method based on the laser plane equation becomes challenging. Additionally, inherent noise in multi-line laser systems introduces errors in extracted laser center coordinates, making it impossible to directly obtain high-precision 3D data. To address these issues, this paper proposes a method that utilizes spatial quadric surfaces and geometric estimation for completing multi-line laser 3D reconstructions. By analyzing the distortion principle of DOE and the positional relationship after optical diffraction, the multi-line laser manifests as a quadric surface on the optical output plane. Consequently, by calibrating the equations of the quadric surface and applying binocular polar constraints, suitable matching points for the multi-line laser can be chosen. Once an accurate matching point is identified, a minimum geometric distance estimation can be established based on the distance constraint between the point and its corresponding epipolar line. This distance represents the separation between the left and right camera’s laser center points and their respective epipolar lines. Utilizing this estimate of the minimum geometric distance, a refined calculation can be performed to better satisfy epipolar constraints and obtain new matching points. Ultimately, utilizing these new matching points enables the completion of 3D reconstruction for multi-line lasers. In comparison with methods relying on spatial plane and epipolar constraints, our algorithm exhibits a superior degree of matchability and accuracy.According to multiple groups of test data, the average error before optimization is 0.3126 mm, and can be increased to 0.0336 mm after optimization.

A novel time-varying mesh stiffness model for orthogonal face gear drives considering EHL

PurposeThis study aims to research the time-varying mesh stiffness (TVMS) model for orthogonal face gear drives considering elastohydrodynamic lubrication (EHL) and provide a theoretical basis for understanding the dynamic characteristics of face gear drives.Design/methodology/approachConsidering EHL, a novel model is proposed to calculate the TVMS of orthogonal face gears using the deformation compatibility condition. First, the tooth surface equations of orthogonal face gears are derived according to the tooth surface generation principle. Then, the oil film thickness on the tooth surface of face gears is obtained by solving the governing equations of EHL. Furthermore, the proposed model is used to calculate the TVMS of face gears along the mesh cycle and is verified. Finally, the effects of module, tooth number of shaper cutter and pressure angle on mesh stiffness are analyzed.FindingsThe results indicate that when the contact ratio is greater than 1 and less than or equal to 2, the TVMS of face gears exhibits a phenomenon of double-single tooth alternating meshing where sudden changes occur. As the module increases, the overall mesh stiffness of face gears increases, and the magnitude of the sudden change at the moment of single-double tooth alternating meshing gradually increases. As the tooth number of shaper cutter and pressure angle increase, so does the TVMS of face gears. When the effect of oil film is considered, the calculated TVMS of face gears slightly increases overall and the increase in average oil film thickness leads to a rise in the TVMS. This study provides a theoretical basis for understanding the dynamic characteristics of face gear drives.Originality/valueThis study’s originality and value lie in its comprehensive approach, which includes conducting analysis based on loaded tooth contact, considering the influence of elastohydrodynamic lubrication, proposing a novel analytical–finite–element model, calculating TVMS of face gears, verifying the proposed model and analyzing the effects of typical structural parameters and oil film thickness.

Calculating traveltimes in 2D general tilted transversely isotropic media using fast sweeping method

Traveltime calculations play an important role in the field of exploration seismology, such as traveltime tomography and seismic imaging and so on. Seismic anisotropy poses a challenge for traveltime calculation, because anisotropic eikonal solvers are more complex than the isotropic counter part. To solve the eikonal equations in 2D tilted transversely isotropic (TTI) media, we have developed a fast algorithm combine with fast sweeping method to compute the first arrival traveltimes of quasi-P (qP)-, quasi-SV (qSV)-, and quasi-SH(qSH)-waves. For the qP- and qSV-waves, we analyzed the quartic coupled slowness surface equation derived from the Christoffel equation. Then, we constructed a local solver to relate traveltime and slowness. We found that in the local solver, one component of the slowness vector is known and the corresponding slowness equation is monotonic. This provides a strong basis for the fast iterative algorithm we proposed, where we use the Newton method to solve the qP- and qSV-wave slowness equation to determine the related traveltimes. For the qSH wave, the slowness equation is quadratic and simple to solve. Numerical experiments demonstrate that the proposed method can obtain accurate traveltimes for simple and complicated 2D TTI models.

A novel forward performance-driven design method for gear parameters

Gear is one of the most crucial components of the transmission system, and the performance of gear directly affects the efficiency and reliability of the transmission system. Conventional methods for designing gear parameters involve several time-consuming and complex steps, which may not guarantee optimal performance. Therefore, we propose a new method for designing gear parameters that aims to improve efficiency and accuracy. First, the tooth surface equations of spur and helical involute gears suitable for symmetric and asymmetric teeth are deduced based on the gear-forming machining principle. Second, the performance evaluation models for load capacity, dynamic performance, efficiency, and power density of the gears are established based on the precise gear surface. The design objectives are standardized and evaluated comprehensively using a linear weighting method. Finally, a forward performance-driven design method of gear parameters is established. The proposed method is applied to a helical gear pair design case, and the results show that 90.7% of the individuals in the Pareto optimal front are asymmetric gears, with 9.3% being symmetric gears. This suggests that asymmetric gears have more opportunities to be optimal than symmetric gears. The highest-ranked gear designed using the proposed method is superior to the gear designed using conventional methods.

Research on the Measurement and Estimation Method of Wheel Resistance on a Soil Runway

In view of the fact that there is no suitable measurement method for tire driving resistance during the operation of a transporter on soil pavement, this paper proposes a simple measurement and estimation method for driving resistance on a soil runway based on the Bekker settlement model. In this paper, using the driving resistance equation based on the Bekker model, a simplified equation of driving resistance related to mass is proposed. Using the principle of kinematics, a simple test device for driving resistance was developed, and the test data of the test device were used to determine the parameters of the driving resistance estimation equation of the road surface. The driving resistance of the wheel running state was tested by the simulated aircraft load loading vehicle through the mechanical sensor, and the consistency between the theoretical value and the actual value of the driving resistance value under 8T, 10T, and 12T loads was verified. This research shows that the test method and estimation method have low cost and good accuracy and are suitable for wide promotion. The results of this paper provide a new idea and method for estimating the driving resistance of a wheel when a transporter runs on a soil road and also provide a reference for designing the length of a soil runway.

An inverse problem for the minimal surface equation in the presence of a riemannian metric

In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold (Rn,g) where the metric is of the form g(x)=c(x)(g^⊕e) . Here g^ is a simple Riemannian metric on Rn−1 , e is the Euclidean metric on R and c a smooth positive function. We show that if the associated Dirichlet-to-Neumann maps corresponding to metrics g and c~g agree, then the Taylor series of the conformal factor c~ at xn=0 is equal to a positive constant. We also show a partial data result when n = 3.