Search results for: three dimensional finite element modelling.
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 3305

Search results for: three dimensional finite element modelling.

2435 The Influence of Internal and External Damping on Turbocharger Stability

Authors: Zdeňka Rendlová

Abstract:

This paper presents the mathematical description of the high-speed rotating system taking into account the influence of internal and external damping. The mathematical model is obtained by using the finite element method. The analyzed system is an automotive turbocharger understood as a rotor-bearing system. The circular cross-section shaft is equipped with one compressor wheel, one turbine wheel and is supported by two floating ring bearings. Based on the model, the dynamical analysis of a turbocharger is performed and stability conditions are evaluated.

Keywords: External damping, internal damping, journal bearing, stability, turbocharger.

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2434 Study of the Flow Structure in a Penstock in Unsteady Regime

Authors: F. Nkontchou Ngongang, M. Tchawe Tchawe, B. Djeumako, B. Kenmeugne

Abstract:

In this work, the flow structure in the Songloulou dam, is visualized in a time interval to observe the different fluid layers in our structure. Firstly, the three-dimensional modelling of the penstock is carried out in the software Gambit, followed by calculations in Fluent that proceeds introduction of boundary conditions. After calculation, we identified four periods corresponding to four regimes. In the first, spanning from 0.00 to 1.50s, we have the non-developed hydraulically rough turbulent regime, characterized by abrupt variations with modifications of the velocity fields. The second extends from 1.50 to 3.50s, where we have the transition regime characterized by slight variations and modifications of the velocity fields but with a great difference of the values of the current lines. From 3.50 to 5.00s, we encounter the third, which is the fully developed turbulent hydraulically rough regime, characterized by fields that vary no more, but have minute differences in the streamlines. The last period is from 5.00s and more, where we have a flow that is almost stationary, hence there are no changes in the fields.

Keywords: Unsteady flow, penstock, friction coefficient, hydroelectric dam.

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2433 Approximate Tension Buckling Capacity of Thin Edge-Cracked Web Plate Subjected to Pure Bending

Authors: Sebastian B. Mendes

Abstract:

The presence of a vertical edge-crack within a web plate subjected to pure bending induces local compressive stresses about the crack which may cause tension buckling. Approximate theoretical expressions were derived for the critical far-field tensile stress and bending moment capacity of an edge-cracked web plate associated with tension buckling. These expressions were validated with finite element analyses and used to investigate the possibility of tension buckling in web-cracked trial girders. It was found that tension buckling is an unlikely occurrence unless the web is relatively thin or the crack is very long.

Keywords: Fatigue crack, tension buckling, Rayleigh-Ritz, structural stability.

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2432 A Numerical Modeling of Piping Phenomenon in Earth Dams

Authors: N. Zaki Alamdari, M. Banihashemi, A. Mirghasemi

Abstract:

To estimate the risks of dam failure phenomenon, it is necessary to understand this phenomenon and the involved governing factors. Overtopping and piping are the two main reasons of earthdam failures. In the piping context, the piping is determined as a phenomenon which is occurred between two phases, the water liquid and the solid soil. In this investigation, the onset of piping and its development, as well as the movement of water in soil, are numerically approached. In this regard, a one-dimensional numerical model based on the mass-conserving finite-volume method is developed and applied in order to simulate the piping phenomenon in a continuous circular tunnel of given initial length and radius, located between upstream and downstream. The simulation result includes the time-variations of radius along the tunnel until the radius value reaches its critical and the piping phenomenon converts to overtopping.

Keywords: Earth dam, dam break, piping, internal erosion

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2431 The Robust Clustering with Reduction Dimension

Authors: Dyah E. Herwindiati

Abstract:

A clustering is process to identify a homogeneous groups of object called as cluster. Clustering is one interesting topic on data mining. A group or class behaves similarly characteristics. This paper discusses a robust clustering process for data images with two reduction dimension approaches; i.e. the two dimensional principal component analysis (2DPCA) and principal component analysis (PCA). A standard approach to overcome this problem is dimension reduction, which transforms a high-dimensional data into a lower-dimensional space with limited loss of information. One of the most common forms of dimensionality reduction is the principal components analysis (PCA). The 2DPCA is often called a variant of principal component (PCA), the image matrices were directly treated as 2D matrices; they do not need to be transformed into a vector so that the covariance matrix of image can be constructed directly using the original image matrices. The decomposed classical covariance matrix is very sensitive to outlying observations. The objective of paper is to compare the performance of robust minimizing vector variance (MVV) in the two dimensional projection PCA (2DPCA) and the PCA for clustering on an arbitrary data image when outliers are hiden in the data set. The simulation aspects of robustness and the illustration of clustering images are discussed in the end of paper

Keywords: Breakdown point, Consistency, 2DPCA, PCA, Outlier, Vector Variance

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2430 Feasibility Study of a BLDC Motor with Integrated Drive Circuit

Authors: Jun-Hyuk Choi, Joon Sung Park, Jung-Moo Seo, In-Soung Jung

Abstract:

A brushless DC motor with integrated drive circuit for air management system is presented. Using magnetic equivalent circuit model a basic design of the motor is determined, and specific configurations are inspected thanks to finite element analysis. In order to reduce an unbalanced magnetic force in an axial direction, induced forces between a stator core and a permanent magnet are calculated with respect to the relative positions of them. For the high efficiency, and high power density, BLDC motor and drive are developed. Also vibration mode and eccentricity of a rotor are considered at the rated and maximum rotational speed Through the experimental results, a validity of the simulated one is confirmed.

Keywords: blower, BLDC, inverter

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2429 Detection ofTensile Forces in Cable-Stayed Structures Using the Advanced Hybrid Micro-Genetic Algorithm

Authors: Sang-Youl Lee

Abstract:

This study deals with an advanced numerical techniques to detect tensile forces in cable-stayed structures. The proposed method allows us not only to avoid the trap of minimum at initial searching stage but also to find their final solutions in better numerical efficiency. The validity of the technique is numerically verified using a set of dynamic data obtained from a simulation of the cable model modeled using the finite element method. The results indicate that the proposed method is computationally efficient in characterizing the tensile force variation for cable-stayed structures.

Keywords: Tensile force detection, cable-stayed structures, hybrid system identification (h-SI), dynamic response.

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2428 Color Image Segmentation and Multi-Level Thresholding by Maximization of Conditional Entropy

Authors: R.Sukesh Kumar, Abhisek Verma, Jasprit Singh

Abstract:

In this work a novel approach for color image segmentation using higher order entropy as a textural feature for determination of thresholds over a two dimensional image histogram is discussed. A similar approach is applied to achieve multi-level thresholding in both grayscale and color images. The paper discusses two methods of color image segmentation using RGB space as the standard processing space. The threshold for segmentation is decided by the maximization of conditional entropy in the two dimensional histogram of the color image separated into three grayscale images of R, G and B. The features are first developed independently for the three ( R, G, B ) spaces, and combined to get different color component segmentation. By considering local maxima instead of the maximum of conditional entropy yields multiple thresholds for the same image which forms the basis for multilevel thresholding.

Keywords: conditional entropy, multi-level thresholding, segmentation, two dimensional image histogram

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2427 Integral Domains and Their Algebras: Topological Aspects

Authors: Shai Sarussi

Abstract:

Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. Thus, the algebraic structure of W can be viewed from the point of view of topology. It is shown that every nonempty open subset of W has a maximal element in it, which is also a maximal element of W. Moreover, a supremum of an irreducible subset of W always exists. As a notable connection with valuation theory, one considers the case in which S is a valuation domain and A is an algebraic field extension of F; if S is indecomposed in A, then W is an irreducible topological space, and W contains a greatest element.

Keywords: Algebras over integral domains, Alexandroff topology, valuation domains, integral domains.

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2426 EML-Estimation of Multivariate t Copulas with Heuristic Optimization

Authors: Jin Zhang, Wing Lon Ng

Abstract:

In recent years, copulas have become very popular in financial research and actuarial science as they are more flexible in modelling the co-movements and relationships of risk factors as compared to the conventional linear correlation coefficient by Pearson. However, a precise estimation of the copula parameters is vital in order to correctly capture the (possibly nonlinear) dependence structure and joint tail events. In this study, we employ two optimization heuristics, namely Differential Evolution and Threshold Accepting to tackle the parameter estimation of multivariate t distribution models in the EML approach. Since the evolutionary optimizer does not rely on gradient search, the EML approach can be applied to estimation of more complicated copula models such as high-dimensional copulas. Our experimental study shows that the proposed method provides more robust and more accurate estimates as compared to the IFM approach.

Keywords: Copula Models, Student t Copula, Parameter Inference, Differential Evolution, Threshold Accepting.

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2425 Number of Parametrization of Discrete-Time Systems without Unit-Delay Element: Single-Input Single-Output Case

Authors: Kazuyoshi Mori

Abstract:

In this paper, we consider the parametrization of the discrete-time systems without the unit-delay element within the framework of the factorization approach. In the parametrization, we investigate the number of required parameters. We consider single-input single-output systems in this paper. By the investigation, we find, on the discrete-time systems without the unit-delay element, three cases that are (1) there exist plants which require only one parameter and (2) two parameters, and (3) the number of parameters is at most three.

Keywords: Linear systems, parametrization, Coprime Factorization, number of parameters.

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2424 Investigation on Nanoparticle Velocity in Two Phase Approach

Authors: E. Mat Tokit, Yusoff M. Z, Mohammed H.

Abstract:

Numerical investigation on the generality of nanoparticle velocity equation had been done on the previous published work. The three dimensional governing equations (continuity, momentum and energy) were solved using finite volume method (FVM). Parametric study of thermal performance between pure water-cooled and nanofluid-cooled are evaluated for volume fraction in the range of 1% to 4%, and nanofluid type of gamma-Al2O3 at Reynolds number range of 67.41 to 286.77. The nanofluid is modeled using single and two phase approach. Three different existing Brownian motion velocities are applied in comparing the generality of the equation for a wide parametric condition. Deviation in between the Brownian motion velocity is identified to be due to the different means of mean free path and constant value used in diffusion equation.

Keywords: Brownian nanoparticle velocity, heat transfer enhancement, nanofluid, two phase model.

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2423 An Adaptive Dynamic Fracture for 3D Fatigue Crack Growth Using X-FEM

Authors: S. Lecheb, A. Nour, A. Chellil, A. Basta, D. Belmiloud, H. Kebi

Abstract:

In recent years, a new numerical method has been developed, the extended finite element method (X-FEM). The objective of this work is to exploit the (X-FEM) for the treatment of the fracture mechanics problems on 3D geometries, where we showed the ability of this method to simulate the fatigue crack growth into two cases: edge and central crack. In the results we compared the six first natural frequencies of mode shapes uncracking with the cracking initiation in the structure, and showed the stress intensity factor (SIF) evolution function as crack size propagation into structure, the analytical validation of (SIF) is presented. For to evidence the aspects of this method, all result is compared between FEA and X-FEM.

Keywords: 3D fatigue crack growth, FEA, natural frequencies, stress intensity factor (SIF), X-FEM.

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2422 Mathematical Properties of the Viscous Rotating Stratified Fluid Counting with Salinity and Heat Transfer in a Layer

Authors: A. Giniatoulline

Abstract:

A model of the mathematical fluid dynamics which describes the motion of a three-dimensional viscous rotating fluid in a homogeneous gravitational field with the consideration of the salinity and heat transfer is considered in a vertical finite layer. The model is a generalization of the linearized Navier-Stokes system with the addition of the Coriolis parameter and the equations for changeable density, salinity, and heat transfer. An explicit solution is constructed and the proof of the existence and uniqueness theorems is given. The localization and the structure of the spectrum of inner waves is also investigated. The results may be used, in particular, for constructing stable numerical algorithms for solutions of the considered models of fluid dynamics of the Atmosphere and the Ocean.

Keywords: Fourier transform, generalized solutions, Navier-Stokes equations, stratified fluid.

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2421 Non-Circular Carbon Fiber Reinforced Polymers Chainring Failure Analysis

Authors: A. Elmikaty, Z. Thanawarothon, L. Mezeix

Abstract:

This paper presents a finite element model to simulate the teeth failure of non-circular composite chainring. Model consists of the chainring and a part of the chain. To reduce the size of the model, only the first 11 rollers are simulated. In order to validate the model, it is firstly applied to a circular aluminum chainring and evolution of the stress in the teeth is compared with the literature. Then, effect of the non-circular shape is studied through three different loading positions. Strength of non-circular composite chainring and failure scenario is investigated. Moreover, two composite lay-ups are proposed to observe the influence of the stacking. Results show that composite material can be used but the lay-up has a large influence on the strength. Finally, loading position does not have influence on the first composite failure that always occurs in the first tooth.

Keywords: CFRP, composite failure, FEA, non-circular chainring.

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2420 Unsteady Temperature Distribution in a Finite Functionally Graded Cylinder

Authors: A. Amiri Delouei

Abstract:

In the current study, two-dimensional unsteady heat conduction in a functionally graded cylinder is studied analytically. The temperature distribution is in radial and longitudinal directions. Heat conduction coefficients are considered a power function of radius both in radial and longitudinal directions. The proposed solution can exactly satisfy the boundary conditions. Analytical unsteady temperature distribution for different parameters of functionally graded cylinder is investigated. The achieved exact solution is useful for thermal stress analysis of functionally graded cylinders. Regarding the analytical approach, this solution can be used to understand the concepts of heat conduction in functionally graded materials.

Keywords: Functionally graded materials, unsteady heat conduction, cylinder, Temperature distribution.

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2419 A Method for Modeling Flexible Manipulators: Transfer Matrix Method with Finite Segments

Authors: Haijie Li, Xuping Zhang

Abstract:

This paper presents a computationally efficient method for the modeling of robot manipulators with flexible links and joints. This approach combines the Discrete Time Transfer Matrix Method with the Finite Segment Method, in which the flexible links are discretized by a number of rigid segments connected by torsion springs; and the flexibility of joints are modeled by torsion springs. The proposed method avoids the global dynamics and has the advantage of modeling non-uniform manipulators. Experiments and simulations of a single-link flexible manipulator are conducted for verifying the proposed methodologies. The simulations of a three-link robot arm with links and joints flexibility are also performed.

Keywords: Flexible manipulator, transfer matrix method, linearization, finite segment method.

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2418 Design Optimization of a Double Stator Cup- Rotor Machine

Authors: E. Diryak, P. Lefley, L. Petkovska, G. Cvetkovski

Abstract:

This paper presents the optimum design for a double stator, cup rotor machine; a novel type of BLDC PM Machine. The optimization approach is divided into two stages: the first stage is calculating the machine configuration using Matlab, and the second stage is the optimization of the machine using Finite Element Modeling (FEM). Under the design specifications, the machine model will be selected from three pole numbers, namely, 8, 10 and 12 with an appropriate slot number. A double stator brushless DC permanent magnet machine is designed to achieve low cogging torque; high electromagnetic torque and low ripple torque.

Keywords: Permanent magnet machine, low- cogging torque, low- ripple torque, high- electromagnetic torque, design optimization.

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2417 Electromagnetic Flow Meter Efficiency

Authors: Andrey D. Andreev, Ilona I. Iatcheva, Dimitar N. Karastoyanov, Rumena D. Stancheva

Abstract:

A study of electromagnetic flow meter is presented in the paper. Comparison has been made between the analytical and the numerical results by the use of FEM numerical analysis (Quick Field 5.6) for determining polarization voltage through the circle cross section of the polarization transducer. Exciting and geometrical parameters increasing its effectiveness has been examined. The aim is to obtain maximal output signal. The investigations include different variants of the magnetic flux density distribution around the tube: homogeneous field of magnitude Bm, linear distribution with maximal value Bm and trapezium distribution conserving the same exciting magnetic energy as the homogeneous field.

Keywords: Effectiveness, electromagnetic flow meter, finite element method, polarization voltage.

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2416 An Approach to Concerns and Aspects Mining for Web Applications

Authors: Carlo Bellettini, Alessandro Marchetto, Andrea Trentini

Abstract:

Web applications have become very complex and crucial, especially when combined with areas such as CRM (Customer Relationship Management) and BPR (Business Process Reengineering), the scientific community has focused attention to Web applications design, development, analysis, and testing, by studying and proposing methodologies and tools. This paper proposes an approach to automatic multi-dimensional concern mining for Web Applications, based on concepts analysis, impact analysis, and token-based concern identification. This approach lets the user to analyse and traverse Web software relevant to a particular concern (concept, goal, purpose, etc.) via multi-dimensional separation of concerns, to document, understand and test Web applications. This technique was developed in the context of WAAT (Web Applications Analysis and Testing) project. A semi-automatic tool to support this technique is currently under development.

Keywords: Aspect Mining, Concepts Analysis, Concerns Mining, Multi-Dimensional Separation of Concerns, Impact Analysis.

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2415 Entropy Generation for Natural Convection in a Darcy – Brinkman Porous Cavity

Authors: Ali Mchirgui, Nejib Hidouri, Mourad Magherbi, Ammar Ben Brahim

Abstract:

The paper provides a numerical investigation of the entropy generation analysis due to natural convection in an inclined square porous cavity. The coupled equations of mass, momentum, energy and species conservation are solved using the Control Volume Finite-Element Method. Effect of medium permeability and inclination angle on entropy generation is analysed. It was found that according to the Darcy number and the porous thermal Raleigh number values, the entropy generation could be mainly due to heat transfer or to fluid friction irreversibility and that entropy generation reaches extremum values for specific inclination angles.

Keywords: Porous media, entropy generation, convection, numerical method.

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2414 Slip Limit Prediction of High-Strength Bolt Joints Based on Local Approach

Authors: Chang He, Hiroshi Tamura, Hiroshi Katsuchi, Jiaqi Wang

Abstract:

In this study, the aim is to infer the slip limit (static friction limit) of contact interfaces in bolt friction joints by analyzing other bolt friction joints with the same contact surface but in a different shape. By using the Weibull distribution to deal with microelements on the contact surface statistically, the slip limit of a certain type of bolt joint was predicted from other types of bolt joint with the same contact surface. As a result, this research succeeded in predicting the slip limit of bolt joins with different numbers of contact surfaces and with different numbers of bolt rows.

Keywords: Bolt joints, slip coefficient, finite element method, Weibull distribution.

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2413 Parallel Explicit Group Domain Decomposition Methods for the Telegraph Equation

Authors: Kew Lee Ming, Norhashidah Hj. Mohd. Ali

Abstract:

In a previous work, we presented the numerical solution of the two dimensional second order telegraph partial differential equation discretized by the centred and rotated five-point finite difference discretizations, namely the explicit group (EG) and explicit decoupled group (EDG) iterative methods, respectively. In this paper, we utilize a domain decomposition algorithm on these group schemes to divide the tasks involved in solving the same equation. The objective of this study is to describe the development of the parallel group iterative schemes under OpenMP programming environment as a way to reduce the computational costs of the solution processes using multicore technologies. A detailed performance analysis of the parallel implementations of points and group iterative schemes will be reported and discussed.

Keywords: Telegraph equation, explicit group iterative scheme, domain decomposition algorithm, parallelization.

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2412 Clustered Signatures for Modeling and Recognizing 3D Rigid Objects

Authors: H. B. Darbandi, M. R. Ito, J. Little

Abstract:

This paper describes a probabilistic method for three-dimensional object recognition using a shared pool of surface signatures. This technique uses flatness, orientation, and convexity signatures that encode the surface of a free-form object into three discriminative vectors, and then creates a shared pool of data by clustering the signatures using a distance function. This method applies the Bayes-s rule for recognition process, and it is extensible to a large collection of three-dimensional objects.

Keywords: Object recognition, modeling, classification, computer vision.

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2411 Dimensional Accuracy of CNTs/PMMA Parts and Holes Produced by Laser Cutting

Authors: A. Karimzad Ghavidel, M. Zadshakouyan

Abstract:

Laser cutting is a very common production method for cutting 2D polymeric parts. Developing of polymer composites with nano-fibers makes important their other properties like laser workability. The aim of this research is investigation of the influence different laser cutting conditions on the dimensional accuracy of parts and holes from poly methyl methacrylate (PMMA)/carbon nanotubes (CNTs) material. Experiments were carried out by considering of CNTs (in four level 0,0.5, 1 and 1.5% wt.%), laser power (60, 80, and 100 watt) and cutting speed 20, 30, and 40 mm/s as input variable factors. The results reveal that CNTs adding improves the laser workability of PMMA and the increasing of power has a significant effect on the part and hole size. The findings also show cutting speed is effective parameter on the size accuracy. Eventually, the statistical analysis of results was done, and calculated mathematical equations by the regression are presented for determining relation between input and output factor.

Keywords: Dimensional accuracy-PMMA-CNTs-laser cutting.

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2410 The Cracks Propagation Monitoring of a Cantilever Beam Using Modal Analysis

Authors: Morteza Raki, Abolghasem Zabihollah, Omid Askari

Abstract:

Cantilever beam is a simplified sample of a lot of mechanical components used in a wide range of applications, including many industries such as gas turbine blade. Due to the nature of the operating conditions, beams are subject to variety of damages especially crack propagates. Crack propagation may lead to catastrophic failure during operation. Therefore, online detection of crack presence and its propagation is very important and may reduce possible significant cost of the whole system failure. This paper aims to investigate the effect of cracks presence and crack propagation on one end fixed beam`s vibration. A finite element model will be developed for the blade in which the modal response of the structure with and without crack will be studied. 

Keywords: Blade, crack propagation, health monitoring, modal analysis.

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2409 A TISM Model for Structuring the Productivity Elements of Flexible Manufacturing System

Authors: Sandhya Dixit, Tilak Raj

Abstract:

Flexible Manufacturing System (FMS) is seen as an option for industries which want to boost productivity as well as respond quickly to an increasingly changing marketplace. FMS produces in mid variety, mid volume range and can meet the changing market demands very quickly. But still the impact of adoption of FMS on the productivity of any industry is not very clear. In this paper an attempt has been made to model the various factors affecting the productivity of FMS installation using Total Interpretive Structural Modelling (TISM) Technique.

Keywords: Flexible manufacturing system, productivity, total interpretive structural modelling.

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2408 Earth Potential Rise (EPR) Computation for a Fault on Transmission Mains Pole

Authors: M. Nassereddine, J. Rizk, A. Hellany, M. Nagrial

Abstract:

The prologue of new High Voltage (HV) transmission mains into the community necessitates earthing design to ensure safety compliance of the system. Conductive structures such as steel or concrete poles are widely used in HV transmission mains. The earth potential rise (EPR) generated by a fault on these structures could result to an unsafe condition. This paper discusses information on the input impedance of the over head earth wire (OHEW) system for finite and infinite transmission mains. The definition of finite and infinite system is discussed, maximum EPR due to pole fault. The simplified equations for EPR assessments are introduced and discussed for the finite and infinite conditions. A case study is also shown.

Keywords: Coupling Factor, Earth Grid, EPR, Fault Current Distribution, High Voltage, Line Impedance, OHEW, Split Factor, Transmission Mains.

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2407 A Simplified Solid Mechanical and Acoustic Model for Human Middle Ear

Authors: Adarsh Venkataraman Ganesan, Sundaram Swaminathan, Rama Jayaraj

Abstract:

Human middle-ear is the key component of the auditory system. Its function is to transfer the sound waves through the ear canal to provide sufficient stimulus to the fluids of the inner ear. Degradation of the ossicles that transmit these sound waves from the eardrum to the inner ear leads to hearing loss. This problem can be overcome by replacing one or more of these ossicles by middleear prosthesis. Designing such prosthesis requires a comprehensive knowledge of the biomechanics of the middle-ear. There are many finite element modeling approaches developed to understand the biomechanics of the middle ear. The available models in the literature, involve high computation time. In this paper, we propose a simplified model which provides a reasonably accurate result with much less computational time. Simulation results indicate a maximum sound pressure gain of 10 dB at 5500 Hz.

Keywords: Ear, Ossicles, COMSOL, Stapes.

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2406 Dissipation of Higher Mode using Numerical Integration Algorithm in Dynamic Analysis

Authors: Jin Sup Kim, Woo Young Jung, Minho Kwon

Abstract:

In general dynamic analyses, lower mode response is of interest, however the higher modes of spatially discretized equations generally do not represent the real behavior and not affects to global response much. Some implicit algorithms, therefore, are introduced to filter out the high-frequency modes using intended numerical error. The objective of this study is to introduce the P-method and PC α-method to compare that with dissipation method and Newmark method through the stability analysis and numerical example. PC α-method gives more accuracy than other methods because it based on the α-method inherits the superior properties of the implicit α-method. In finite element analysis, the PC α-method is more useful than other methods because it is the explicit scheme and it achieves the second order accuracy and numerical damping simultaneously.

Keywords: Dynamic, α-Method, P-Method, PC α-Method, Newmark method.

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