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To improve the application of the parametric curve tool path in surface machining, a method for generating arc-length parameterized toolpaths based on short-line toolpaths is proposed. The method first uses commercial CAM software to plan the short-line tool path. Then, the machining tool path with smooth continuous features is extracted. The smooth and continuous features are determined according to the chord length of the short-line tool path, and the extraction method is based on Chebyshev’s inequality of large numbers. After obtaining the smooth short-line trajectory, the iterative arc-length parameterized curve toolpath generation method is used to fit the discrete short-line toolpath. In the iterative process, the B-spline toolpath with parameterized chord length is obtained first. It is discretized according to the arc length difference information of the chord length B-spline. Finally, by continuously checking and inserting B-spline nodes, and fitting according to discrete points, the arc-length parameterized B-spline tool path is obtained. The feasibility of the method is verified with a part containing a sculpted surface, and two smooth arc-length parametric toolpaths are generated. In the end, the part is machined with the arc-length parameterized curve tool path.

Liquid‐liquid extraction is an important separation technique used in several chemical engineering processes. When experimental data are unavailable, reliable prediction of liquid‐liquid equilibrium (LLE) data is essential for the optimal design of these processes. This paper describes a new methodology based on the application of the arc‐length continuation numerical technique in order to compute LLE data using non‐random two liquid or universal quasi‐chemical activity coefficient models. This technique is straightforward to implement in Mathematica©. In addition, the arc‐length method avoids the main disadvantage of root finding methods, such as Newton's method, for which the convergence often depends on the proper choice of an initial guess. Another advantage of the proposed approach, based on arc‐length continuation, is that all LLE data are generated in one single calculation. Several case studies, involving the three main types of ternary systems (ie, Type I, II and 0), illustrate the proposed method. For type I, we study the following three ternary systems: (1) water‐ethanol‐benzene, (2) 5‐hydroxymethylfurfural‐1‐butanol‐water, and (3) methanol‐benzene‐water. We have selected, for type II systems, the following mixtures: (1) isobutyl alcohol‐cyclohexane‐water, (2) methanol‐cyclohexane‐water, and (3) 4‐methyl‐2‐pentanone‐2‐butanol‐water. For type 0 systems, we have chosen a mixture composed of DMSO‐THF‐water. Finally, for each case, our predicted results are benchmarked against the corresponding experimental data. This paper proposes a new method for the prediction of Liquid‐Liquid Equilibrium data using arc‐length continuation. The method, implemented in MATHEMATICA©, is tested considering several case studies, involving the three main types of ternary systems (ie, Type I, II, and 0). For each case, the predicted results are benchmarked against the corresponding experimental data.

Purpose. The aim is to build a mathematical model of the electric arc of arc furnace (EAF). The model should clearly show the relationship between the main parameters of the arc. These parameters determine the properties of the arc and the possibility of optimization of melting mode. Methodology. We have built a fairly simple model of the arc, which satisfies the above requirements. The model is designed for the analysis of electromagnetic processes arc of varying length. We have compared the results obtained when testing the model with the results obtained on actual furnaces. Results. During melting in real chipboard under the influence of changes in temperature changes its properties arc plasma. The proposed model takes into account these changes. Adjusting the length of the arc is the main way to regulate the mode of smelting chipboard. The arc length is controlled by the movement of the drive electrode. The model reflects the dynamic changes in the parameters of the arc when changing her length. We got the dynamic current-voltage characteristics (CVC) of the arc for the different stages of melting. We got the arc voltage waveform and identified criteria by which possible identified stage of smelting. Originality. In contrast to the previously known models, this model clearly shows the relationship between the main parameters of the arc EAF: arc voltage Ud, amperage arc id and length arc d. Comparison of the simulation results and experimental data obtained from real particleboard showed the adequacy of the constructed model. It was found that character of change of magnitude Md, helps determine the stage of melting. Practical value. It turned out that the model can be used to simulate smelting in EAF any capacity. Thus, when designing the system of control mechanism for moving the electrode, the model takes into account changes in the parameters of the arc and it can significantly reduce electrode material consumption and energy consumption during smelting.

Chemical equilibrium calculations—as they are required to evaluate the evolution limits of any reactive system—are fundamentally performed using either a reaction based stoichiometric (S) approach or a non‐stoichiometric (NS) method. One objective of the present paper is to check and discuss whether both methods lead to comparable solutions in the case of complex chemical equilibria encountered in high temperature and pressure combustion of hydrazine and propane and in steam gasification of glucose and cellulose. Within both calculation models various numerical methods can be used. We apply several techniques for each approach and discuss their easiness of implementation and usage, in particular when chemical equilibria are computed for reacting systems under a multitude of temperature and pressure conditions and feed compositions. In addition, we present a novel approach to perform sensitivity analyses based on a combination of S or NS methods and arc‐length continuation technique. For this purpose, three industrially relevant case studies are exposed: (1) synthesis of ammonia using the Haber process, (2) steam gasification of a typical biomass surrogate, glucose, and (3) steam gasification of cellulose. For all the above reacting systems, our results are benchmarked against their counterparts obtained either from the ubiquitous process simulator: ASPEN‐Plus® or from data available in the open literature. The article starts with a discussion of the stoichiometric and non‐stoichiometric methodologies, which are used in order to compute the chemical equilibria for complex reacting systems at fixed conditions. For each one of these two methodologies, author compares the equilibrium compositions obtained using several numerical techniques. Then, various parametric studies are presented using a novel numerical approach based on Lagrangian multipliers and arc‐length continuation. This approach is applied to three important industrial processes including the Haber synthesis of ammonia and the gasification of two biomass surrogates.

The advantages of intraoral model scanning have yielded recent developments. However, few studies have explored the orthodontic clinical use of this technique particularly among young patients. This study aimed to evaluate the reliability, reproducibility and validity of the orthodontic measurements: tooth width, arch length and arch length discrepancy in each digital model obtained by model scanner and intraoral scanner, relative to a plaster model. Arch length measured using two methods: curved arch length (CAL) measured automatically by digital program and sum of sectional liner arch length (SLAL) measured sum of anterior and posterior liner arch lengths. Arch length discrepancy calculated each arch length measurement methods: curved arch length discrepancy (CALD) and sum of sectional liner arch length discrepancy (SLALD). Forty young patients were eligible for the study. A plaster model (P), model-scanned digital model (MSD) and intraoral scanned digital model (ISD) were acquired from each patient. The reliability of the measurements was evaluated using Pearson’s correlation coefficient, while the reproducibility was evaluated using the intraclass correlation coefficient. The validity was assessed by a paired t-test. All measurements measured in P, MSD and ISD exhibited good reliability and reproducibility. Most orthodontic measurements despite of CAL in MSD exhibited high validity. Only the SLAL and SLALD in ISD group differed significantly, despite the good validity of the tooth width, CAL and CALD. The measurements based on the digital program appeared high reliability, reproducibility and accurate than conventional measurement. However, SLAL and SLALD in ISD group appeared shorter because of distortion during intraoral scanning. However, this could be compensated by using digital programed curved arch. Although the validity of SLAL and SLALD in the ISD group differed statistically, the difference is not considered clinically significant. Although MSD and ISD are acceptable for a clinical space analysis, clinicians should be aware of digital model-induced errors.

Gas tungsten arc (GTA) is a heat source commonly used in wire arc additive manufacturing (WAAM) and arc welding. During the WAAM and arc welding process, the GTA arc length has a significant influence on the quality of deposition layer and deposition conditions. Therefore, the monitoring of arc length is conducive to the automatic control of the manufacturing process. This paper studies the characteristics of arc acoustic signals with varying arc length in GTA-WAAM, and it provides a method to identify the arc length through the detection and processing of the arc acoustic signals. The signal decomposition method based on wavelet analysis is used to extract the trend signal of the variable arc acoustic signal, and the quantitative relationship between arc length and arc acoustic signal is deduced. The results show that the arc acoustic signal fluctuates regularly with the continuous change of arc length. The basic mathematical relationship between arc length and arc acoustic signal indicates the possibility of identifying the arc length. Meanwhile, the verification experimental results show that the proposed calculation method and signal processing flow are effective for arc length identification. This result provides the possibility for the application of arc acoustic signal in arc length identification and GTA-WAAM process control.

As the operating speed of electric multiple units (EMUs) in high-speed railways increases, pantograph–catenary (PC) detachment arcing occurs frequently. The resulting vehicular-grounding electromagnetic transients are related to the dynamic characteristics of the arc length. During large detachment, the processes of arc extinction and arc reignition may occur, resulting in more severe train body (TB) over-voltages and adverse effects on some vehicular electronic devices. As an extension of the previous works, this paper aims to establish a suitable PC arc model to examine the TB transient voltages. To begin with, the arc length dynamic characteristics are reasonably analyzed to deduce the relationship between the detachment distance and the arc length via the chain arc model. Then, the dynamic characteristics of the arc length are introduced, and an arc modeling scheme is proposed to elaborate the vehicle-grid electric power model for EMUs encountering various arcing scenarios. Based on this, the transient over-voltages are analyzed, accounting for both the arc extinction and arc reignition, as well as the mutual influences of multiple detachments in a short time. The influential factors, including arc length characteristics, phase angle, excitation inductance, and grounding parameters, are also involved in the performed analyses.

The arc behaviour of short, low current switching arcs is not well understood and lacks a reliable model. In this work, the behaviour of an arc in the air is studied during contact separation at low DC currents (0.5 A to 20 A) and for small gap lengths (0 mm to 6 mm). The experiments are performed on a low voltage relay with two different electrode configurations. The arc voltage is measured during the opening of the contacts at constant current. The arc length is determined optically by tracing the mean path of the arc over time from a series of high-speed images. From the synchronised data of voltage vs. distance, first a sudden jump of the voltage at the start of contact opening is observed. Secondly, a sudden change in the voltage gradient occurs as the arc is elongated. Short arcs with a length up to approximately 1.25 mm show an intense radiation in the overall gap region and high voltage gradients. An unexpected behaviour never reported before was observed for longer arcs at low current: Two characteristic regions occur, a region in front of the cathode, with a length of approximately 1.25 mm, having an intense radiation and a high voltage gradient as well as a region of much lower radiation intensity and a comparatively lower voltage gradient in the remaining gap area despite a small anode spot region. The characteristic border of approximately 1.25 mm is almost independent of the current. A generalised arc voltage model is proposed based on the assumption that a constant sheath voltage and two discrete field regions exist, which are modelled as two independent linear functions of voltage vs. length. The data for various currents is combined to yield a general non-linear function for predicting the arc voltage vs. arc length and current.

Multi-harmonic excitations are commonly observed in rotor systems and affect their nonlinear characteristics significantly. However, most of the published nonlinear studies on rotating structures only consider single-harmonic excitation. Compared with single-harmonic issues, multi-harmonic excitations increase the difficulty of calculation and solution exponentially. The purpose of this paper is to establish the nonlinear coupled mechanical model and analyze nonlinear forced vibrations of rotating shells subjected to multi-harmonic excitations in thermal environment. The nonlinear governing equations, considering the Coriolis forces, centrifugal force, initial hoop tension and thermal effect, are obtained by the improved Donnell nonlinear shell theory and Hamilton principle, and then, the multi-mode Galerkin technique is introduced to transform the partial differential equations into multi-degree-of-freedom nonlinear ordinary differential equations (ODEs). Afterward, numerical simulations are conducted by the pseudo-arc-length continuation algorithm. The verification of the solutions with available results in the literature and the convergency of the results are presented. At last, the effects of main factors on nonlinear dynamic response of rotating shells are evaluated. It can be observed that since the multi-DOF coupled system, which is excited by multi-harmonics, exhibits complex nonlinear dynamic responses of rotating shells, the nonlinear multiple internal resonances occur.

For evaluating the probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based on the Gauss-Laplace law. The latter will be considered here as an element of the newly-introduced family of (p,q) -spherical distributions. Based on a suitably-defined non-Euclidean arc-length measure on (p,q) -circles, we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly to with elliptically-contoured distributions and more general homogeneous star-shaped ones. This is demonstrated by the generalization of the Box-Muller simulation method. In passing, we prove an extension of the sector and circle number functions.