Slowness vector vs ray direction in polar anisotropic media

Summary The inverse problem of finding the slowness vector from a known ray direction in general anisotropic elastic media is a challenging task, needed in many wave/ray-based methods, in particular, solving two-point ray bending problems. The conventional resolving equation set for general (triclinic) anisotropy consists of two fifth-degree polynomials and a sixth-degree polynomial, resulting in a single physical solution for quasi-compressional (qP) waves and up to 18 physical solutions for quasi-shear waves (qS). For polar anisotropy (transverse isotropy with a tilted symmetry axis), the resolving equations are formulated for the slowness vectors of the coupled qP and qSV waves (quasi-shear waves polarized in the axial symmetry plane), and independently for the decoupled pure shear waves polarized in the normal (to the axis) isotropic plane (SH). The novelty of our approach is the introduction of the geometric constraint that holds for any wave mode in polar anisotropic media: The three vectors—the slowness, ray velocity and medium symmetry axis—are coplanar. Thus, the slowness vector (to be found) can be presented as a linear combination of two unit-length vectors: the polar axis and the ray velocity directions, with two unknown scalar coefficients. The axial energy propagation is considered as a limit case. The problem is formulated as a set of two polynomial equations describing: a) the collinearity of the slowness-related Hamiltonian gradient and the ray velocity direction (third-order polynomial equation), and b) the vanishing Hamiltonian (fourth-order polynomial equation). Such a system has up to twelve real and complex-conjugate solutions, which appear in pairs of the opposite slowness directions. The common additional constraint, that the angle between the slowness and ray directions does not exceed ${90^{\rm{o}}}$, cuts off one half of the solutions. We rearrange the two bivariate polynomial equations and the above-mentioned constraint as a single univariate polynomial equation of degree six for qP and qSV waves, where the unknown parameter is the phase angle between the slowness vector and the medium symmetry axis. The slowness magnitude is then computed from the quadratic Christoffel equation, with a clear separation of compressional and shear roots. The final set of slowness solutions consists of a unique real solution for qP wave and one or three real solutions for qSV (due to possible triplications). The indication for a qSV triplication is a negative discriminant of the sixth-order polynomial equation, and this discriminant is computed and analyzed directly in the ray-angle domain. The roots of the governing univariate sixth-order polynomial are computed as eigenvalues of its companion matrix. The slowness of the SH wave is obtained from a separate equation with a unique analytic solution. We first present the resolving equation using the stiffness components, and then show its equivalent forms with the well-known parameterizations: Thomsen, Alkhalifah and ‘weak-anisotropy’. For the Thomsen and Alkhalifah forms, we also consider the (essentially simplified) acoustic approximation for qP waves governed by the quartic polynomials. The proposed method is coordinate-free and can be applied directly in the global Cartesian frame. Numerical examples demonstrate the advantages of the method.

An Attempt to Modify the Crop Coefficient Value of Wheat and Chickpea for Humid Subtropical Climate of Varanasi District, Uttar Pradesh

Crop water requirement is one of the essential components that should always be taken into account wherever water management strategies are adopted for effective utilization of water. In India agriculture is considered as the backbone of economy but nothing technical or advance technology has being adopted for its advancement and still major portion of agriculture are dependent on the verge of monsoon. Therefore crop coefficient (Kc) value for wheat and Chickpea as a function of relative humidity, wind speed and crop height has been formalized in this study for efficient use of available water. The empirical formula was applied for estimating Kc(mid) and Kc(end) values for humid subtropical climate of Varanasi. The corrected crop coefficient value of wheat for Kc(mid) was found to be 1.12 and for end season Kc(end) corrected was found to be 0.42. Mean maximum height of wheat crop in mid-season was obtained as 0.80 m and for the end season mean maximum height was found to be 0.99 m. The corrected crop coefficient value of Chickpea for midseason was found to be 0.98 whereas it was found out to be 0.38 at Kc(end). Mean maximum height of chickpea crop in mid-season was obtained as 0.39 m and for the end season of the crop mean maximum height was found to be 0.52 m. Crop coefficient curve for wheat crop was prepared in which X-axis represents time in days and Y- axis represents crop coefficient value for wheat. A third-order polynomial equation (y = -3E - 06x3 + 0.0005x2 - 0.0039x + 0.3702) has been obtained for wheat crop with correlation coefficient of 0.97. Similarly, graphical plot was constructed for chickpea crop and a third-order polynomial equation (y = -2E - 06x3 + 0.0002x2 - 0.0004x + 0.3732) has been obtained with correlation coefficient of 0.98. Furthermore, third-order polynomials were fitted well to predict the crop coefficient values as function of growing degree-days (GDD).

Digestibility, fermentation and microbiological characteristics of Calotropis procera silage with different quantities of grape pomace

ABSTRACT Preserving forage plants adapted to a semi-arid climate as silage may minimize the animal feed deficit during drought. The objective of this study was to evaluate the effects of different quantities of grape pomace added to Calotropis procera silage on its fermentation, in vitro digestibility, total digestible nutrients and microbiology. A completely randomized experimental design was used with four treatments (0, 10, 20 and 40% fresh matter) and four replicates. The silos were opened after 90 days of ensilage, and the soluble carbohydrate, ethanol, organic acid and ammoniacal nitrogen concentrations; pH; fermentation loss; dry matter (DM) recovery; DM density, and microbial populations were determined. The pH (3.96-3.87) was adequate for ensiling in all silage samples. The soluble carbohydrate concentration decreased (p<0.05), and the ethanol concentration increased with increasing quantities of grape pomace. The lactic acid concentration decreased (p<0.05) from 5.3 to 1.94% DM, and the acetic, propionic and butyric acid concentrations increased with increasing quantities of grape pomace. The lactic acid bacteria decreased linearly (p <0.05), varying from 6.43 to 5.82 log CFU/g silage. The mold and yeast population variations fit best using a third-order polynomial equation (p <0.05). Enterobacteria and Clostridium spp were not observed. Adding grape pomace to the silage increased the effluent and gas loss; the latter varied from 5.35 to 14.4%. The total digestible nutrient (TDN) variation fit best using a second-order polynomial equation, and the maximum value was estimated at 82.95% DM with 3.5% grape pomace using the regression equation. The percent digestibility decreased linearly (p<0.05) with increasing quantities of grape pomace. We show that Calotropis procera has potential as silage even without adding grape pomace.

Influence of the Trajectory Planning on the Accuracy of the Orthoglide 5-Axis

Usually, the accuracy of parallel manipulators depends on the architecture of the robot, the design parameters, the trajectory planning and the location of the path in the workspace. This paper reports the influence of static and dynamic parameters in computing the error in the pose associated with the trajectory planning made and analyzed with the Orthoglide 5-axis. An error model is proposed based on the joint parameters (velocity and acceleration) and experimental data coming from the Orthoglide 5-axis. Newton and Gröbner based elimination methods are used to project the joint error in the workspace to check the accuracy/error in the Cartesian space. For the analysis, five similar trajectories with different locations inside the workspace are defined using fifth order polynomial equation for the trajectory planning. It is shown that the accuracy of the robot depends on the location of the path as well as the starting and the ending posture of the manipulator due to the acceleration parameters.

Preparation of a monolithic sorbent using a response surface methodology

A 17-run Box-Behnken Design (BBD) was introduced to optimize the synthesis conditions of a monolithic sorbent. The effects of the amount of monomer (mL), crosslinker (mL) and porogen (mL) were investigated. The experimental data was fitted to a second - order polynomial equation by multiple regression analysis, which was examined using statistical methods. The adjusted coefficient of determination (R2Adj) in this model was 0.9867. The probability value (p<0.0001) revealed the high significance of the regression model. A mean amount of 5057.4 mg polymer was produced under the following optimized synthesis conditions: 0.51 mL monomer, 2.94 mL crosslinker and 2.84 mL porogen. The actual experimental result was in good agreement with the predicted model value.

Formulation and optimization of biological removal of flue gas pretreatment wastewater and sulfur recycling process by Box–Behnken design

The aim of this study was to investigate optimum conditions for biological removal of flue gas pretreatment wastewater and achieve maximum elemental sulfur yield. A three-factor, three-level Box–Behnken design was used to derive a second-order polynomial equation and construct contour plots to predict responses. The independent variables selected were hydraulic retention time (X1), inlet sulfate concentration (X2), and air flow (X3). Fifteen batches were done in a biological united system and evaluated for elemental sulfur yield (Y1). The transformed values of the independent variables and Y1 were subjected to a full-model second-order polynomial equation. The equation was modified based on Fisher's F- and probability P-values. The computer optimization process and contour plots predicted the values of independent variables X1, X2 and X3 (16 h, 1,348 mg L−1 and 165 L h−1 respectively), for maximized response of Y1. The experimental results at predicted conditions demonstrate that the modified model equation has good applicability to the practical system.

Effects of Crystallization Degree on Rate during Isothermal Crystallization from PET Melt

In this study, one new function is defined as change of relative crystallization degree in unit time and named relative crystallization rate (1/min.). The curve of heat flow rate to time is transmitted to that of to . The produced curve was fitted using one high-order polynomial equation with a variable of and the coefficient vector （Ai ,in this paper，the values of i were from 0 to 9）was produced. It was found that, even during the accelerated stage of crystallization from PET melt，both aspects to promote and delay the relative crystallization rate existed， furthermore， both aspects of promotion and delay declined with the crystallization process and appeared “internal exhaustion”.