Equivalent Rectangle Research Articles

Study on the vibration characteristics of worm helical gear drive

The application of worm helical gear in smart car, smart home and other fields can significantly reduce the weight of products. Accordingly, the requirements for vibration and noise are also increased. In this paper, the lumped mass method is used to establish the dynamic model of point contact bending-twisting shaft meshing coupling worm helical gear drive. In this model, the contact ellipse is transformed into an equivalent rectangle, and the nonlinear factors such as time-varying meshing stiffness, tooth meshing error and meshing damping are considered. The vibration characteristics of worm helical gear drive in three directions and at meshing point are obtained for the first time. Machining accuracy, tooth surface modification and assembly errors are transformed into the error in the dynamic model respectively, and their effects on the vibration characteristics are analyzed. The closed vibration and noise performance test bench of worm helical gear drive is built, and the influence of three kind of errors on the vibration characteristics of worm helical gear drive are tested to verify the correctness of the dynamic model and its numerical analysis results. Both the theoretical analysis and the experimental results show that machining accuracy, tooth surface modification and the assembly error have a great influence on the vibration characteristics.

Study on the meshing stiffness of plastic helical gear meshing with metal worm via point-contact

Time-varying meshing stiffness is the main internal excitation source for gear dynamic responses. A calculation method for meshing stiffness of point-contact transmission pairs is proposed to analysis the dynamic responses of the plastic helical gear metal worm drive. Based on the curvilinear motion trajectories and the contact ellipse changes of helical gear worm drive, a method of equivalent area rectangle is introduced to convert the contact ellipse into contact line, and the relationship between the contact line and the worm angle is established. The meshing stiffness of plastic helical gear metal worm drive is calculated and compared with the finite element method (FEM) to verify its accuracy. The results show that the meshing stiffness of plastic helical gear metal worm drive is affected by the meshing impact, and the time-varying meshing stiffness value of double-tooth meshing will be close to or even lower than that of single-tooth meshing. In the plastic helical gear metal worm drive, there is no stable stiffness value whether it is single-tooth meshing or double-tooth meshing. The single-tooth meshing stiffness has an upward trend under the influence of the pressure angle. The calculation results of theoretical model are basically consistent with the FEM results, but the theoretical model cannot fully reflect the stiffness change during meshing impact.

A Width Parameter Estimation Through Equivalent Rectangle Methodology for Hydraulic Modeling Applications

EPA SWMM hydraulic modeling requires the estimation of different parameters allowing the determination of the basin's response upon a given precipitation event. Some physical parameters, such as area or perimeter, are measurable and can be accurately determined; however, other parameter estimation presents greater uncertainty, such as the width parameter. For regular and uniform drainage areas, width parameter estimation is relatively simple; however, when a complete irregular basin analysis is required, width determination presents greater uncertainty, and its representativeness becomes complicated to define. Width determination is idealized with the representation of a rectangle, where, for an equal area, a higher width will result in a faster response of the basin, while a lower width will result in a slower response of the basin. This paper attempts to estimate a representative value of width for a realistic, irregularly shaped basin by defining the equivalent rectangle, which takes into account the area, perimeter, and compactness index of the basin. The compactness index can be used to classify the basin by its shape. The shape of the basin is an indicator of how the precipitation histograms are temporally distributed and how the water flows through the basin, i.e., it defines the response speed of the basin, as the width parameter does in modeling. The width parameter has a high sensitivity in the EPA SWMM modeling results; therefore, an inaccurate estimation of the parameter leads to unrepresentative results. For this reason, this study seeks to find an optimal methodology to reduce modeling uncertainty and achieve more accurate simulations of an irregular watershed's hydrological response.

A Comparative Study of the Mechanical Properties of Basalt Fiber and Basalt Grille Reinforced Concrete Composites and Theoretical Prediction

ABSTRACT In this study, the chopped basalt fiber (CBF) with different content and basalt fiber grilles (BFG) with different layers were composed with concrete matrix, respectively. The compressive, splitting tensile and flexural properties of concrete composites were studied by universal material testing machine. The cracks evolution law of specimens was analyzed to reveal the reinforcement mechanism of CBF and BFG. Results showed that under the same fiber volume ratio, BFG were better than CBF to reinforce mechanical properties of concrete. The three kinds of mechanical properties of BFG reinforced concrete were improved highest by 41.46%, 28.26% and 46.13%, respectively. BFG can play a crucial role in stress transmission, bearing load and absorbing energy during the loading process of the specimens. Based on the equivalent stress rectangle theory, an analytic model to predict the ultimate flexural capacity of BFG reinforced concrete was proposed. The experimental results from the flexural test were compared with the predicted values, and a good accuracy of the analytic model was obtained.

Preparation and characterization of basalt fabrics coated with zirconia obtained by sol-gel method for textile reinforced concrete

The poor alkali resistance of basalt fabric greatly affects the mechanical properties of textile reinforced concrete. In this study, dense zirconia coating obtained by sol-gel method was coated on the basalt fabrics surface as additional sizing to improve the alkali resistance of basalt fabrics. The morphology, wettability and phase composition of the fabric surfaces was characterized by scanning electron microscopy (SEM), static water contact angle (SWCA), X-ray powder diffraction (XRD), fourier transform infrared spectra (FI-IR), X-ray photoelectron spectroscopy (XPS) and Raman. The tensile properties of various basalt fabrics immersed in alkaline solution for different time and the mechanical property of concrete composites were examined by universal material testing machine. Results showed that a protective membrane was formed on the surface of basalt fabrics to prevent the diffusion of hydroxide ions. The final strength retention ratio of BF-c-ZrO2 (Basalt fabric-coated-Zirconia) fabrics could maintain at 89.92% compared with that of pristine basalt fabrics, which reached below 40%. The compressive strength and flexural strength of BF-c-ZrO2 fabrics reinforced concrete increased by 10.15% and 19.0% compared with that of pristine basalt fabrics reinforced concrete. A theoretical model was proposed to predict the ultimate flexural bearing capacity of BF-c-ZrO2 fabric reinforced concrete based on the equivalent stress rectangle theory. The corresponding predicted value compared with experimental result indicated a good coincidence and a reasonable accuracy, which was less than 5% of their discrepancy.

Mass center equivalent rectangle model for universal correlation of hydrothermal features of helical baffle heat exchangers

The prediction using a universal correlation of the shell side Nusselt number with the Reynolds number for helical baffle heat exchangers (HBHXs) has been in vain, because of the impropriate formula of the cross-section area of helical channel or the flow velocity. The shell side friction factor is of similar situation. A new model of mass center equivalent rectangle (MCER) is proposed to calculate the cross-sectional area of the helical flow channel of the HBHX with the hypothesis that the parameters on the mass center cylinder can represent the mean values of the spiral channel. Taking helix pitch as hypotenuse and two sides of the right triangle are along the mass center equivalent helix angle and its normal direction, and the latter and the radius of the shell are postulated to be the two edges of the windward rectangle. The flow cross-section area of the helical channel is the rectangle area subtracting the total ellipse area of the tubes in the rectangle area. The verification of the model is quite satisfactory for quadrant HBHXs by both the heat exchanger design software HTRI and the existing experimental results in a master thesis with the deviations of ho·Pro−1/3 ~ Reo are respectively within +17% to −10% and ±18%. The verification of the model is also satisfied with experimental results of a trisection circumferential overlap HBHX. The coefficients in the correlation formulas especially for the friction factor need be further consummated relating to the helical baffle type and baffle connection style of the HBHXs.

Method for Extracting the Effective Tensor Surface Impedance Function From Nonuniform, Anisotropic, Conductive Patterns

A patterning technique known as the point shifting method has enabled the generation of smoothly varying and highly anisotropic impedance surfaces with a wide range of patch sizes and shapes. Previously, the surface impedances of different shapes of unit cells were assumed by the impedance of similar size rectangle cells. In this paper, we study an approach to calculate the surface impedances for anisotropic polygon unit cells more accurately, based on the area moment of inertia equations. We define an alternative unit cell called the equivalent rectangle, which has the same tensor impedance properties of a general polygon unit cell in the surface impedance pattern. The size of the equivalent rectangle cell is calculated by using the moment of inertia equations between the polygon and the rectangle. The extracted surface impedance from the equivalent rectangle is compared to the surface impedance of polygon in the unit cell simulation, validating our method. We also verify the method by comparing the results between PEC patterns and impedance boundary sheets to which the extracted impedances are applied. Simulations of the patterns are verified by measurements as well.

Deflection of C section beams with circular web openings

The deflection of steel beams with circular openings is dependent on the loss of bending stiffness and shear area of the web, which adds to the deflection of the solid web beam. The additional deflection is a function of the ratio of opening diameter, ho, to section depth, h, and also of the spacing of the openings. This paper presents derived formulae for the additional deflection of beams with circular web openings that are expressed as a function of the pure bending deflection of the unperforated beam. The formulae are calibrated against finite element models and against short span beam tests with opening diameters, ho, of 0.6 h and 0.72 h, in which the additional defection is mainly due to the effects of shear. Further simplified design formulae for additional deflection are also presented which are shown to be accurate for C-section beams with a span: depth ratio exceeding 15. It is also shown that circular openings may be represented for analysis purposes as an equivalent rectangle of width x depth equal to 0.75 ho x ho.

COMPARISON BETWEEN THE DOUBLE BUFFER METHOD AND THE EQUIVALENT RECTANGLE METHOD FOR THE QUANTIFICATION OF DISCREPANCIES BETWEEN LINEAR FEATURES

Abstract Currently, in Brazil, for the assessment of the Positional Accuracy of non-point features (lines and polygons), there is no standard norm of execution. This work aims to compare the results of two methodologies that allow determining the average value of the discrepancies between linear features. The first, Equivalent Rectangle Method, aims to determine the discrepancy by considering an equivalent rectangle for the polygon obtained from the two homologous lines. The second, Double Buffer Method applies a buffer on both lines and obtains the average discrepancy value based on the relation of the areas of the generated polygons. These methods were compared in two steps. Initially, an experiment was performed with features of known measurements, where the displacement of the homologous lines was controlled in azimuth and distance. In this step, it was verified that the shape of the feature and the direction of the displacement interfere in the results of both methods when compared to the traditional procedure of measurement of discrepancies by homologous points. In the second stage, we evaluated the vector data of the OpenStreetMap (class of roads), with reference to a more accurate vector dataset produced for the Mapping of the State of Bahia. As a result, for the 1:25,000, 1:50,000, 1:100,000 and 1:250,000 scales, it was obtained, respectively, the PEC-PCD for the Equivalent Rectangle Method "C", "B", "A" and "A" and the PEC-PCD for the Double Buffer Method "R", "C", "B" and "A", where "R" means that it has not achieved the minimum PEC-PCD classification.

El concepto de Rectángulo Equivalente para la caracterización de la forma de una cuenca

En este trabajo se rescata la importancia de la enseñanza de la geometría como modeladora del espacio en el que estamos inmersos, para lo cual se presenta la resolución de un problema relacionado con cierto tipo de disciplinas como la cartografía, la geografía, la hidrología a través de la caracterización de la forma de una cuenca hidrográfica. Para ello se hace uso del concepto de rectángulo equivalente. Es importante aclarar que en el desarrollo de este trabajo, se presentan dos variables asociadas a la morfometría: el área y el perímetro las cuales también permiten describir la forma de una cuenca y los efectos asociados a la misma. A su vez se el trabajo también incluye el desarrollo de las funciones cuadráticas para su resolución. Las conclusiones demuestran que no son los “colegios emblemáticos” los que obtienen las mejores calificaciones, tampoco los que se ubican en las grandes ciudades, ni los más caros, sino aquellos que sirven a las capas medias de la población y que se encuentran en las ciudades intermedias del país.

The description of parcel geometry and its application in terms of land consolidation planning

Land consolidation is the current issue occurring around the world. In the course of consolidation works the state of fragmentation of farms and its impact on their productivity is analyzed. Various methods of fragmentation assessing and variants of implementation of consolidation procedures are being developed. There are many factors of a different nature affecting the need for consolidation. One of the basic elements needed in these works is a quantitative description of flawed shape of plots based on the parameters of their spatial shape. GIS systems allow quick collection of spatial data of the plots or any other surface structures, including parcels. In this paper we focus only on the geometric shape of a parcel as a condition to carry out the consolidation. The paper proposes a methodology for obtaining parameters of the geometry of parcel description, which supports the process of decision-making in the field of land consolidation works. In this paper we proposed an algorithm to replace parcel with equivalent rectangle (ER). The presented solution enables quick processing of the large amounts of data. The results of algorithm use are shown on the example of the fragment of village in the Southern Poland.

Reduction of computational times using the equivalent rectangle concept in the physics-based surface - subsurface models

La interacción entre la superficie y subsuperficie de una cuenca se puede estudiar utilizando modelos físicos, pero debido a su compleja geometría los procesos de calibración y validación requieren altos tiempos computacionales. Para ello, se propone una alternativa que permita reducir los tiempos de calibración del modelo, mediante la simplificación de la geometría de la cuenca y empleando el concepto del rectángulo equivalente. El rectángulo equivalente es usado para definir los parámetros del suelo de la cuenca del río Lerma, a través de un análisis de sensibilidad, que posteriormente son empleados dentro de la calibración. El modelo simplificado identifica con una gran precisión los rangos y valores iniciales de los parámetros utilizados para los procesos de calibración y además reduce significativamente los tiempos computacionales. Por lo tanto, el concepto de rectángulo equivalente ofrece una alternativa para acelerar la calibración de modelos acoplados superficie-subsuperficie.

Experimental and numerical study of fluid flow and heat transfer characteristics in microchannel heat sink with complex structure

Thermal management has become crucial to ensure the performance and reliability of high power chips and micro-cooling systems. The forced convective heat transfer of microchannel heat sink is a very promising method. In this paper, experiment is used to perform temperature and pressure drops and numerical simulation is used to understand and interpret the complex thermal behavior by presenting the flow field in the current complex corrugation microchannel heat sink. The comprehensive performance is evaluated by total thermal resistance and thermal enhancement factor. Compared with the equivalent rectangle microchannel heat sink, the average temperature and maximum temperature is reduced obviously and temperature distribution is more uniform albeit with higher pressure penalty for flow rates larger than 100ml/min. It is observed that the vortex becomes bigger and moves to the middle of channel with increasing of flow rate. The enhance heat transfer mechanisms can be contributed to the heat transfer area enlarged, thermal boundary interrupted and redeveloped, chaotic advection, hot and cooling fluid better mixed by vortex formed in the reentrant cavity. The pumping power is reduced 18.99% when total thermal resistance equals to 0.446K/W, compared with rectangle microchannel heat sink. The thermal enhancement factor can reach 1.24 for Reynolds number of 611. Therefore, complex corrugation microchannel heat sink is more economical for chip cooling system.

Implementing into GIS a Tool to Automate the Calculation of Physiographic Parameters of River Basins

The physiographic characterization of a basin is a fundamental element as it defines the hydrological behavior of that basin. The present work deals with the development and implementation of a tool that allows calculating in an automated manner the physiographic parameters of a basin, as well as those of the surface runoff and main river, besides other graphic elements: hypsometric curve, equivalent rectangle and profile of the main river. Such a tool was developed under Visual Basic 6 programming language and the spatial geographic component ArcObjects by ESRI; they enabled the development of a library as a final product (.dll), which can be loaded and implemented in ArcMap software. In the methodology a Conceptual Model was established, from which it was possible to identify the requirements and methods to obtain the parameters, as well as the conception and implementation of the Logical Model that includes the specific functions and also the input structures, processes and data output. Finally, the tool was tested with actual data from El Caracol river basin, located in central-southern Mexico, which showed the easiness and usefulness of it, besides the effectiveness of the results, not leaving aside the time and resources saved by the user when characterizing a basin, compared with other conventional processes.

Quasi-Axial GRIN Lens Implemented in Wedge-Shaped Fiber Coupling With InP-Based PLC

A 3-D equivalent rectangle approximation staircase concatenation method to analyze the optical mode evolution in a wedge-shaped fiber (WSF) is proposed in this paper. A scheme with an axial gradient-index (AGRIN) lens between the WSF and InP-based planar lightwave circuit (PLC) chip, yielding higher transmission efficiency than those with two cascaded lenses and one radial GRIN lenses, by 2.282 and 7.816 dB, respectively, is devised and experimentally demonstrated. The WSF-AGRIN-PLC-single-mode fiber coupling linkage supplies a preferable solution for solving the problem of index mismatching in fiber-to-InP-based PLC assembly to achieve a high-efficiency connection.

Experimental and numerical studies for the forming groove and separating groove design in slit rolling process

This paper presents a design method of the forming groove-to-separating groove rolling sequence used in slit rolling process. Based on the equivalent rectangle approximation, we have designed the forming groove which produces a peanut-shaped cross section via rolling from workpiece with an inlet (incoming) square cross section. The separating groove is designed by lowering its width and height in the ratio of 1:3 for the given reduction ratio. With the forming groove and separating groove designed by the proposed design method, we have performed a pilot rolling test at high temperature (around 1000°C) using low carbon (0.1%C) steel and have also carried out finite element analysis to examine the expansibility of the proposed design method. The Results reveal that if the ratio of the roll diameter over the side length of a square is in the range of 4.0 ∼ 8.0, the design method proposed in this study can be systematically applied to design the forming groove-to-separating groove without relying on in-house databases and/or on-site experience.

Equivalent rectangle approximation-staircase concatenation method for wedge shaped fiber

Wedge shaped fiber (WSF) is the key component to acquire high coupling efficiency between input/output ports of planar lightwave circuit chips and fiber arrays. It is very important to analyze the mode spot and the mode field evolution of the lightwave propagating in WSF using numerical simulation tools. A three_dimensional equivalent rectangle approximation-staircase concatenation method (ERA-SCM) is proposed to fractionize the wedged tip of WSF along the propagating direction and introduce rectangular waveguide approximation so that effective index of the segmentations is obtained and SCM analysis model is established. Lightwave propagation and mode field evolution is analyzed by ERA-SCM and compared with finite difference-beam propagation method (FD-BPM), showing that the former is more precise to solve mode field evolution for asymmetric fiber and waveguide structures. The measurement of the output optical spot verifies the ERA-SCM simulation error is within 1.9%, while that for FD-BPM method is 4.5%. Thus, ERA-SCM is an effective method for analyzing asymmetric fiber and waveguide structures.

Analytical model of pass-by-pass strain in bar rolling and its verification

This paper presents an analytical model for predicting the pass-by-pass strain, defined as maximum averaged effective (equivalent) plastic strain at a given pass, in a bar (or rod) rolling process. The model has been based on the equivalent rectangle approximation method which makes the curved configuration of the deformed section into a simplified rectilinear configuration and the hypothesis of parallelepiped deformation of material. Experimental and numerical studies were carried out to investigate the validity of the analytical model. For experimental verification, plate rolling and bar rolling experiments were designed that consisted of a four-pass oval-round (or round-oval) rolling sequence such that during rolling the material experiences the same amount of the strain at each pass. The microstructures and mechanical behaviors of specimens acquired from the two types of experiment were compared. A commercial finite element analysis program, DEFORM, was also used to examine the validity of the analytical model numerically. A similarity between the mechanical behaviors of the plate-rolled and bar-rolled specimens was found at each pass. It was also seen that the strains calculated by the analytical model were in agreement with the averaged effective plastic strain computed by finite element simulation.

Exact Riemann Solution and Weighted Average Flux Method for Power-law Channel Section

The aim of this work is to extend the Weighted Averaged Flux (WAF) method by E.F. Toro from rectangular to generic regular cross sections, in order to represent the geometric effects of the cross-section on wave propagation along channels. Assuming a power-law variation of the channel width, the 1D conservative shallow water equations, their characteristic form and the shock propagation equations are presented. The exact Riemann solver is derived and is applied to the dam-break problem in valleys with different shape in order to test its efficiency and to check the accuracy order of solutions obtained by approximating the real cross-section with an equivalent rectangle. The WAF method is extended to a power-law channel section and is used to solve the 1D proposed equations taking into account all the source terms that are incorporated into the local Riemann problem. A code based on this method has been developed and results of numerical applications to a Venturi channel and to the attenuation of waves are presented, in order to verify, for well known situations, how accurately the source terms are represented. The code is applied to reproduce one of Brock's experiments (1969; 1970) on roll waves generation in a rectangular channel and results are compared with those obtained with a Godunov-type code developed at CEMAGREF, which is based on a Roe scheme with a later average evaluation of source terms.

Analytic model for the prediction of mean effective strain in rod rolling process

This paper presents an analytic model for predicting the mean effective strain, defined as a maximum average effective (equivalent) plastic strain at a given pass, in bar (or rod) rolling process. The proposed model has been based on the elementary theory of plasticity and the equivalent rectangle approximation method which transforms a non-rectangular cross-sectional shape into a rectilinear one. Then, the mean effective strains are calculated using the hypothesis of parallelepiped uniform deformation of material during rolling. A numerical analysis using three-dimensional finite element method was also carried out to examine the validity of the analytic model proposed. The analysis shows that considering the assumption of uniform deformation of workpiece introduced in the formulation of the model, the mean effective strains calculated by the analytic model are in agreement with those computed by finite element analysis. The results of this study might be important for the study of the pass-by-pass microstructural evolution and temperature increment due to plastic deformation associated with rod rolling where the mean effective strain becomes a critical parameter.