New method for determining singularities on enveloped surface and its application to study curvature interference theory of involute worm drive

In this paper, a more computationally convenient singularity condition of the enveloped surface is proposed using the theory of linear algebra. Its preconditions are only the tangential vector of the enveloping surface, the relative velocity vector, and the total differential of the meshing function. It avoids calculating the curvature parameters of the enveloping surface. It is proved that the singularity conditions of enveloped surface from different references are equivalent to each other and the relational equations among them are obtained. The curvature interference theory for the involute worm drive is established using the proposed singularity condition. The equation for the singularity trajectory is obtained. The calculation method for the singularity trajectory is proposed and its numerical result is obtained. The influence of the design parameters on the singularity trajectory is studied using the proposed curvature interference theory. The study results show that the risk of curvature interference is high when the transmission ratio is too small, especially in the case of the single-threaded worm and large modulus. The proposed singularity condition can also be applied to study the curvature interference mechanism in other types of the worm drive and to study the undercutting mechanism when machining the worm drive.

The symbol and alphabet of two-loop NMHV amplitudes from $$ \overline{Q} $$ equations

We study the symbol and the alphabet for two-loop NMHV amplitudes in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills from the $$ \overline{Q} $$ Q ¯ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use $$ \overline{Q} $$ Q ¯ equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

GENERATION OF NEW OPERATIONAL MATRICES FOR DERIVATIVE, INTEGRATION AND PRODUCT BY USING SHIFTED CHEBYSHEV POLYNOMIALS OF TYPE FOUR

While solving the fractional order differential equation the requirement of the higher-order derivative is obvious therefore, this paper gives a definite expression for constructing the operational matrices of derivative which is the direct method to find the derivative of higher-order according to the requirement of the total differential equation. The proposed work expands the Chebyshev polynomial of type four up to six degrees that could help get the accuracy for the numerical solution of a given differential equation. Previously Chebyshev polynomial of the third type has been used by cutting the domain from [-1, 1] to [0, 1]. This study also generates the integrational operational matrix for solving the integral equation as well as an integrodifferential equation by using the Chebyshev polynomial of type four and expand it up to six order and generate the matrix by cutting the domain from [-1, 1] to [0, 1]. This is the first attempt to generate an integrational operational matrix that has never been highlight nor generate by any researcher. Another contribution of this paper is the generation of categorical expressions for the product of two Chebyshev vectors that will help in solving the differential equation of several kinds.

MHD Casson non-Newtonian nanofluid over a nonlinear penetrable elongated sheet with thermal radiation and chemical reaction

Consider a steady flow in two-dimensional of a viscous, incompressible Casson nano liquid over a nonlinear penetrable elongated sheet with radiation and chemical reaction. The Casson liquid rheological model is used to explain the non-Newtonian liquid attributes. Similarity variables are utilized to evaluate the governing flow model into set of nonlinear total differential equations. The outcomes of the flow equations were gotten by using Runge-Kutta alongside the shooting techniques. In other to explain the physics of the problem, impact of flow parameters are presented in graphs while computations on engineering curiosity are presented in table. Ahike in the Casson liquid term is observed to degenerate the fluid velocity alongside the momentum layer thickness. The impact of the imposed magnetic is felt by decreasing the velocity owning to the Lorentz force.

Locating Error when Clamping Workpiece between Centers

The total machining error includes some distinct errors summed up by following certain rules. One of these errors is the workpiece locating error, which occurs when the workpiece reference base does not match the workpiece locating surface. The paper analyses different definitions of positioning error, as they were proposed by different researchers or groups of researchers and presents an application valid in the case of turning a cylindrical workpiece with locating between the centers. A definition of locating error that takes into account the total differential of the size of the vector that joins the orientation surface with an element of the fixture was preferred. A fixture version was considered to ensure the locating and clamping of the workpiece on the lathes, with workpiece locating between the centers. This fixture uses a mandrel with a sliding center and an elastic collet.

Preliminary investigation of the relationship between differential phase shift and path-integrated attenuation at the X band frequency in an Alpine environment

The RadAlp experiment aims at developing advanced methods for rainfall and snowfall estimation using weather radar remote sensing techniques in high mountain regions for improved water resource assessment and hydrological risk mitigation. A unique observation system has been deployed since 2016 in the Grenoble region of France. It is composed of an X-band radar operated by Météo-France on top of the Moucherotte mountain (1901 m above sea level; hereinafter MOUC radar). In the Grenoble valley (220 m above sea level; hereinafter a.s.l.), we operate a research X-band radar called XPORT and in situ sensors (weather station, rain gauge and disdrometer). In this paper we present a methodology for studying the relationship between the differential phase shift due to propagation in precipitation (Φdp) and path-integrated attenuation (PIA) at X band. This relationship is critical for quantitative precipitation estimation (QPE) based on polarimetry due to severe attenuation effects in rain at the considered frequency. Furthermore, this relationship is still poorly documented in the melting layer (ML) due to the complexity of the hydrometeors' distributions in terms of size, shape and density. The available observation system offers promising features to improve this understanding and to subsequently better process the radar observations in the ML. We use the mountain reference technique (MRT) for direct PIA estimations associated with the decrease in returns from mountain targets during precipitation events. The polarimetric PIA estimations are based on the regularization of the profiles of the total differential phase shift (Ψdp) from which the profiles of the specific differential phase shift on propagation (Kdp) are derived. This is followed by the application of relationships between the specific attenuation (k) and the specific differential phase shift. Such k–Kdp relationships are estimated for rain by using drop size distribution (DSD) measurements available at ground level. Two sets of precipitation events are considered in this preliminary study, namely (i) nine convective cases with high rain rates which allow us to study the ϕdp–PIA relationship in rain, and (ii) a stratiform case with moderate rain rates, for which the melting layer (ML) rose up from about 1000 up to 2500 m a.s.l., where we were able to perform a horizontal scanning of the ML with the MOUC radar and a detailed analysis of the ϕdp–PIA relationship in the various layers of the ML. A common methodology was developed for the two configurations with some specific parameterizations. The various sources of error affecting the two PIA estimators are discussed, namely the stability of the dry weather mountain reference targets, radome attenuation, noise of the total differential phase shift profiles, contamination due to the differential phase shift on backscatter and relevance of the k–Kdp relationship derived from DSD measurements, etc. In the end, the rain case study indicates that the relationship between MRT-derived PIAs and polarimetry-derived PIAs presents an overall coherence but quite a considerable dispersion (explained variance of 0.77). Interestingly, the nonlinear k–Kdp relationship derived from independent DSD measurements yields almost unbiased PIA estimates. For the stratiform case, clear signatures of the MRT-derived PIAs, the corresponding ϕdp value and their ratio are evidenced within the ML. In particular, the averaged PIA∕ϕdp ratio, a proxy for the slope of a linear k–Kdp relationship in the ML, peaks at the level of the copolar correlation coefficient (ρhv) peak, just below the reflectivity peak, with a value of about 0.42 dB per degree. Its value in rain below the ML is 0.33 dB per degree, which is in rather good agreement with the slope of the linear k–Kdp relationship derived from DSD measurements at ground level. The PIA∕ϕdp ratio remains quite high in the upper part of the ML, between 0.32 and 0.38 dB per degree, before tending towards 0 above the ML.

A line-integral-based method to partition climate and catchment effects on runoff

It is a common task to partition the synergistic impacts of drivers in the environmental sciences. However, there is no mathematically precise solution to this partition task. Here I present a line-integral-based method, which addresses the sensitivity to the drivers throughout the drivers' evolutionary paths so as to ensure a precise partition. The method reveals that the partition depends on both the change magnitude and pathway (timing of the change) but not on the magnitude alone unless used for a linear system. To illustrate this method, I applied the Budyko framework to partition the effects of climatic and catchment conditions on the temporal change in the runoff for 19 catchments from Australia and China. The proposed method reduces to the decomposition method when assuming a path in which climate change occurs first, followed by an abrupt change in catchment properties. The proposed method re-defines the widely used sensitivity at a point as the path-averaged sensitivity. The total-differential and the complementary methods simply concern the sensitivity at the initial and/or the terminal state, so they cannot give precise results. Although the path-averaged sensitivities varied greatly among the catchments, they can be readily predicted within the Budyko framework. As a mathematically accurate solution, the proposed method provides a generic tool for conducting quantitative attribution analyses.