ULTRA-THICK CROSS SECTION FIBER REINFORCED COMPOSITES
|
|
- Marcus Lawson
- 5 years ago
- Views:
Transcription
1 F98-D4 Page ULTRA-THICK CROSS SECTION FIBER REINFORCED COMPOSITES Project Code: F98-D4 Ivestigators: Yog K. Kim, Leader, Armad F. Lewis, Alex Fowler, UMass Dartmouth Graduate Studet: Joatha Reuss PROJECT OBJECTIVE The overall objective of this study is a fudametal examiatio of the basic fiber ad resi material parameters that are importat i the maufacture of thick cross-sectio composites. Advaced fabricatio methods will be used such as i-situ cosolidatio, o-autoclavig curig, ad advaced fiber placemet process for thermoset composites. The thickess rage of iterest is from.9 cm (.75 iches) to 7.6 cm (3. iches) i thickess, which would be adaptable to marie vessels ad other load bearig egieerig structures. INTRODUCTION Limitatios exist i the field of fiber reiforced Orgaic Polymer Egieerig Composite (OPEC) materials with referece to the process fabricatio of composite shapes havig a thick cross sectio (ca. greater tha -3 cm). Problems arise from the exotherm that occurs durig the cure of the composite. This ca lead to ueve curig, ueve mass shrikage ad improper cosolidatio of the fiber/resi composite media. This results i composite material structures of uacceptable egieerig quality. Furthermore, if the reactio exotherm is too high, thermal degradatio of the matrix material ca occur resultig i potetially hazardous processig coditios. Complex textile structures such as two ad three dimesioal wove fabrics ad braided structures offer the potetial to produce et shape preforms cotiuously, as well as provide a product form with improved damage tolerace ad impact resistace at low cost. [Mohamed ad El-Shiekh 994] Combied with resi trasfer moldig (RTM), these techologies also offer a cost reductio potetial for Orgaic Polymer Egieerig Composites (OPEC) for marie applicatios (e.g. ship hulls, submersible structural members) ad costructio egieerig structures. The proposed study will ehace competitiveess of high-tech textile preform maufacturig sectors by providig quality ad cost-effective fiber reiforcemet materials to these relevat idustries. Furthermore, the uses ad markets for polyaramid, glass ad carbo fiber would be greatly impacted if techiques could be developed for fabricatig OPEC materials of thick cross sectio. This year, the focus of ivestigatio is the heat trasfer i bulk fiber/resi masses durig the curig process.
2 F98-D4 Page THERMO-CHEMICAL MODEL OF CURING PROCESS Goverig Equatios The followig assumptios are made at this time: All material properties (specific heat, desity, thermal coductivity) are costat durig the curig process. This assumptio will be relaxed to accommodate more realistic measured properties i experimetal study phase. Temperature ad degree of cure are fuctios oly of time ad directio ormal to the part surface. The composite is a ifiite medium ad ca therefore be assumed to be oe dimesioal. Deformatio of the material is egligible. There is o covectio heat trasfer. Iitially, the heat trasfer durig the process was described by the followig heat coductio equatio [Beja 99] without takig the degree of cure ito accout. ρc T t T = k + q& () x The variables T, ρ, C, k, ad q are temperature distributio i thickess directio X, the desity, specific heat, thermal coductivity, ad rate of volumetric heat geeratio, respectively. I order to give the model superior flexibility the ext step was to replace the curret variables with dimesioless quatities. The ew dimesioless equatio reads as follows: C K θ θ = τ X + Q () Where the dimesioless variables are defied as, C C = ρ ρrefc p pref (3) θ = T T Tmold T (4) τ α = t L (5)
3 F98-D4 Page 3 K = k (6) kref & Q = ql T k ( Tmold ) ref (7) where, T, L, ad α are the ambiet temperature, the slab thickess, ad thermal diffusivity of the material, respectively. The dimesioless C ad K terms ca obviously be set to by assumig the referece values are equivalet to the curret values. The other dimesioless quatities are also show here ad their values ca be calculated by usig reported values for glass/polyester (see Table ). Other specific reiforcemet/matrix resi cases for give experimetal setup, these parameters are to be calculated with measured values. Table. Material properties ad Processig parameters for glass/polyester umerical results. [Kim 995] Property/Parameter Variable (Uits) Value R / 98 Heat of Reactio (Resi) H ( kj kg) Resi Mass Fractio γ.6 Desity ( 3 ρ kg / m ) 89. * 3 Thermal Coductivity k( kw / m C) 63. * 4 Cp kj / kg C.6 Specific Heat ( ) Frequecy Factor Z( s ) E( J / mol) 674. * 5 Kietic Expoets, m.476,.54 Ambiet Temperature T ( C) 5 Mold Temperature Tmold ( C) 6 The ext step i the thermo-chemical model is to vary the rate of heat geeratio as a fuctio of cure. The heat geeratio (equatio 8) ca ow be writte as a fuctio of the heat of reactio HR, the resi mass fractio γ, ad the cure rate c as a fuctio of time. The cure rate is calculated i equatio 9, where Z is a frequecy factor, E is the activatio eergy, ad R is the uiversal gas costat. [Kim 995] &q = ργh c R t (8)
4 F98-D4 Page 4 c t E = Zexp RT ( c) m c (9) Oce agai it was ecessary to form o-dimesioal quatities which could the be icorporated ito the previous o-dimesioal equatio. The ew dimesioless quatities are described fully i equatios - 5. Their values ca also be calculated from Table for glass/polyester composites. [Kim 995] Q = H c τ () H = γh Cp T R ( mold T) () c τ E = Z exp θ + R ( c) m c () Z = Z L α (3) E = E T R ( Tmold ) (4) R = T ( Tmold T) (5) Numerical Results Procedure A fiite differece method usig Trapezoidal Rule was used to itegrate the goverig equatio (). The TR method is secod order accurate i time ad space as well as beig ucoditioally stable. The TR method calculatesθ at time step + ad spatial step i for all i ad by usig both kow ad ukowθ as show i equatio 6. This is doe by placig the ukowθ's ad their coefficiets ito matrix form, settig it equal to the kow θ's, ad solvig it. The matrix formed is a tri-diagoal matrix ad a tri-diagoal matrix solver ca be used, which greatly icreases the speed of the program.
5 F98-D4 Page 5 + k τ k τ θi = θi + ( θ i+ θi + θi ) + ( θ i+ θ i + θ i ) + Q τ (6) X X The oliear equatio for calculatig the degree of cure (Eq. ) was solved usig a Forward Euler approach. It is oly first order accurate i time ad space, but it is stable for small time steps. The Forward Euler method simply uses the previous time step values at ay give spatial step ad therefore does t require a matrix solver. I order to fid the degree of cure at spatial step i ad time step + you simply implemet + E ci = ci + τ Z exp θ R i ( ci ) ( c i ) m (7) Numerical Results Although the umerical model is a prelimiary oe with limited coditios imposed by the assumptios, we have produced several graphs i order to test its accuracy ad validity. As stated i previous sectios, the dimesioless parameters were calculated usig data preseted i Table. First, a graph of the degree of cure ad θ ceter vs. τ is show i Figure. This model predicts that the cure occurs completely i a very short period of time, which is clearly oly true for ideal situatios. More accurate cure models will be icorporated at a later time. A thermal spike the occurs as a result of the sudde curig. The secod graph show i Figure is of θ max vs. τ. θ max is the maximum temperature withi the composite thickess at ay give time iterval. The thermal spike is idetical to that show i the θ ceter graph, however, it starts at because it is our set boudary coditio. It heats up quickly as would be expected from this type of idealized curig process. It the cools approximately 73 degrees i 4+ miutes, which also appears to be reasoable cosiderig the idealized model we are usig for a ich bulk thickess. The third ad fial graph show i Figure 3 is of θ max vs. Z. Z is the oly dimesioless variable that is directly affected by the thickess ad is therefore used as a dimesioless thickess. The results as Z gets larger become less ad less accurate. This is because the portio of the code cotaiig the Z is highly sesitive to the time step. It is therefore ecessary to reduce the size of the time step as the Z gets larger, which results i much larger data processig times. I view of this accuracy problem further refiemets i the model ad/or the umerical algorithm are eeded.
6 F98-D4 Page 6 Figure : Thetaceter & Cure vs. Tau Thetaceter & Cure Tau thetaceter cure Figure : Thetamax vs. Tau Thetamax Tau
7 F98-D4 Page 7 Figure 3 Thetamax vs. Z.5 Thetamax.5.5.E+.E+5.E+5 3.E+5 4.E+5 5.E+5 6.E+5 7.E+5 Z CONCLUSION There are several improvemets that eed to be made to the curret umerical model. First, a fluid flow model eeds to be icorporated ito the curret thermo-chemical model i order to model the resi flow resultig from the dowward pressure applied by the hydraulic press plates. Although variatios i thermal coductivity of the material durig the cure have bee show to be uimportat i certai composites [Twardowski 993], we ited to explore whether or ot they are ever sigificat i thick composites. A experimetal apparatus cosistig of a heated copper plates withi a Lexa holder has bee fabricated to measure the thermal coductivity of liquid thermoset matrix resis at differet curig stages. Oce the revised model is completed, various techiques of lowerig the maximum temperature of curig ca be umerically tested ad the checked experimetally. The plaed research activities for this year ad beyod are: Modificatio of model to iclude possible variatios i thermal properties, resi flow durig cosolidatio, ad aisotropic coductivity due to fiber arragemet. Experimetal verificatio of the improved model. Use of the model ad experimetal results to develop testig ad aalytical techiques for uderstadig the behavior of the thick wall composites as a fuctio of curig method ad reiforcemet structure. To develop a processig sciece ad techology basis for the maufacture of these materials ad to develop a uderstadig of the iteractio betwee processig ad performace.
8 F98-D4 Page 8 REFERENCES Beja, Adria. Heat Trasfer, Joh Wiley ad Sos, New York, 99. Kim, Cheol., The Cotiuous Curig Process for Thermoset Polymer Composites. Part : Modelig ad Demostratio, Joural of Composite Materials, vol. 9(9): pp3-53 (995) Mohamed, M, Aly EI-Shkieh et al, NTC Project Code S9-: Textile Structures for Composite Natioal Textile Ceter Aual Report, September 994 Twardowski, T.E, Curig i Thick Composite Lamiates: Experimet ad Simulatio, Joural of Composite Materials, 7(3): 6-5 (993)
Boundary layer problem on conveyor belt. Gabriella Bognár University of Miskolc 3515 Miskolc-Egyetemváros, Hungary
Boudary layer problem o coveyor belt Gabriella Bogár Uiversity of Miskolc 355 Miskolc-Egyetemváros, Hugary e-mail: matvbg@ui-miskolc.hu Abstract: A techologically importat source of the boudary layer pheomeo
More informationA STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD
IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: 39-63 pissn: 3-7308 A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD Satish
More informationThermal characterization of an insulating material through a tri-layer transient method
Advaced Sprig School «Thermal Measuremets & Iverse techiques», Domaie de Fraço, Biarritz, March -6 05 T6 Thermal characterizatio of a isulatig material through a tri-layer trasiet method V. Félix, Y. Jaot,
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationPILOT STUDY ON THE HORIZONTAL SHEAR BEHAVIOUR OF FRP RUBBER ISOLATORS
Asia-Pacific Coferece o FRP i Structures (APFIS 2007) S.T. Smith (ed) 2007 Iteratioal Istitute for FRP i Costructio PILOT STUDY ON THE HORIZONTAL SHEAR BEHAVIOUR OF FRP RUBBER ISOLATORS T.B. Peg *, J.Z.
More informationA numerical Technique Finite Volume Method for Solving Diffusion 2D Problem
The Iteratioal Joural Of Egieerig d Sciece (IJES) Volume 4 Issue 10 Pages PP -35-41 2015 ISSN (e): 2319 1813 ISSN (p): 2319 1805 umerical Techique Fiite Volume Method for Solvig Diffusio 2D Problem 1 Mohammed
More informationμ are complex parameters. Other
A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria.
More informationTeaching Mathematics Concepts via Computer Algebra Systems
Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) E-ISSN: 4767 P-ISSN: - 4759 Volume 4 Issue 7 September. 6 PP-- Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,
More informationMonte Carlo Optimization to Solve a Two-Dimensional Inverse Heat Conduction Problem
Australia Joural of Basic Applied Scieces, 5(): 097-05, 0 ISSN 99-878 Mote Carlo Optimizatio to Solve a Two-Dimesioal Iverse Heat Coductio Problem M Ebrahimi Departmet of Mathematics, Karaj Brach, Islamic
More informationThe axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.
5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationReliability and Queueing
Copyright 999 Uiversity of Califoria Reliability ad Queueig by David G. Messerschmitt Supplemetary sectio for Uderstadig Networked Applicatios: A First Course, Morga Kaufma, 999. Copyright otice: Permissio
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationDETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.
ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics
More informationNumerical Simulation of Thermomechanical Problems in Applied Mechanics: Application to Solidification Problem
Leoardo Joural of Scieces ISSN 1583-0233 Issue 9, July-December 2006 p. 25-32 Numerical Simulatio of Thermomechaical Problems i Applied Mechaics: Applicatio to Solidificatio Problem Vicet Obiajulu OGWUAGWU
More informationThe target reliability and design working life
Safety ad Security Egieerig IV 161 The target reliability ad desig workig life M. Holický Kloker Istitute, CTU i Prague, Czech Republic Abstract Desig workig life ad target reliability levels recommeded
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationPrecalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions
Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group
More informationFree Radical Polymerization
Free Radical Polymerizatio Referece: Aspe Polymers: Uit Operatios ad Reactio Models, Aspe Techology, Ic., 2013. 1. Itroductio The free-radical bulk/solutio polymerizatio model is applicable to bulk ad
More informationActivity 3: Length Measurements with the Four-Sided Meter Stick
Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationSECTION 2 Electrostatics
SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial
More informationEXPERIMENT OF SIMPLE VIBRATION
EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic
More informationMeasurement uncertainty of the sound absorption
Measuremet ucertaity of the soud absorptio coefficiet Aa Izewska Buildig Research Istitute, Filtrowa Str., 00-6 Warsaw, Polad a.izewska@itb.pl 6887 The stadard ISO/IEC 705:005 o the competece of testig
More informationProvläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE
TECHNICAL REPORT CISPR 16-4-3 2004 AMENDMENT 1 2006-10 INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE Amedmet 1 Specificatio for radio disturbace ad immuity measurig apparatus ad methods Part 4-3:
More informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationNumerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter In The Porous Medium
IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): 78-879 Vol. 5, Issue 4 (April. 5), V PP 8-7 www.iosrje.org Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter
More informationAnalysis of composites with multiple rigid-line reinforcements by the BEM
Aalysis of composites with multiple rigid-lie reiforcemets by the BEM Piotr Fedeliski* Departmet of Stregth of Materials ad Computatioal Mechaics, Silesia Uiversity of Techology ul. Koarskiego 18A, 44-100
More informationOrthogonal Gaussian Filters for Signal Processing
Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios
More informationThe following data were obtained in a homogenous batch reactor for the esterification of butanol (B) with acetic acid (A):
Departmet of Eerg Politecico di Milao ia Lambruschii 4-256 MILNO Exercises of udametals of hemical Processes Prof. Giapiero Groppi EXERISE Reactor for the esterificatio of butaol The followig data were
More informationA New Recursion for Space-Filling Geometric Fractals
A New Recursio for Space-Fillig Geometric Fractals Joh Shier Abstract. A recursive two-dimesioal geometric fractal costructio based upo area ad perimeter is described. For circles the radius of the ext
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationTemperature Distribution in a Cylindrical Core and Coil Assembly with Heat Generation
Temperature Distributio i a Cylidrical Core ad Coil Assembly with Heat Geeratio P. S. Yeh, Ph.D. Abstract A residetial distributio trasformer cosists of a cylidrical steel tak ad a core-ad-coil assembly
More informationTHE KALMAN FILTER RAUL ROJAS
THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationMicroscopic traffic flow modeling
Chapter 34 Microscopic traffic flow modelig 34.1 Overview Macroscopic modelig looks at traffic flow from a global perspective, whereas microscopic modelig, as the term suggests, gives attetio to the details
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationEVALUATION OF GLASS FIBER/EPOXY INTERFACIAL STRENGTH BY THE CRUCIFORM SPECIMEN METHOD
EVALUATION OF GLASS FIBER/EPOX INTERFACIAL STRENGTH B THE CRUCIFORM SPECIMEN METHOD Ju KOANAGI, Hajime KATO, Akihiro KASHIMA, uichi IGARASHI, Keichi WATANABE 3, Ichiro UENO 4 ad Shiji OGIHARA 4 Istitute
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationDEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES
DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES Icreasig ad Decreasig Auities ad Time Reversal by Jim Farmer Jim.Farmer@mq.edu.au Research Paper No. 2000/02 November 2000 Divisio of Ecoomic ad Fiacial
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationWhat is Physical Chemistry. Physical Chemistry for Chemical Engineers CHEM251. Basic Characteristics of a Gas
7/6/0 hysical Chemistry for Chemical Egieers CHEM5 What is hysical Chemistry hysical Chemistry is the study of the uderlyig physical priciples that gover the properties ad behaviour of chemical systems
More informationStatistical Pattern Recognition
Statistical Patter Recogitio Classificatio: No-Parametric Modelig Hamid R. Rabiee Jafar Muhammadi Sprig 2014 http://ce.sharif.edu/courses/92-93/2/ce725-2/ Ageda Parametric Modelig No-Parametric Modelig
More informationChE 471 Lecture 10 Fall 2005 SAFE OPERATION OF TUBULAR (PFR) ADIABATIC REACTORS
SAFE OPERATION OF TUBULAR (PFR) ADIABATIC REACTORS I a exothermic reactio the temperature will cotiue to rise as oe moves alog a plug flow reactor util all of the limitig reactat is exhausted. Schematically
More informationFinite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw
Fiite Differece Derivatios for Spreadsheet Modelig Joh C. Walto Modified: November 15, 2007 jcw Figure 1. Suset with 11 swas o Little Platte Lake, Michiga. Page 1 Modificatio Date: November 15, 2007 Review
More informationANALYSIS OF DAMPING EFFECT ON BEAM VIBRATION
Molecular ad Quatum Acoustics vol. 7, (6) 79 ANALYSIS OF DAMPING EFFECT ON BEAM VIBRATION Jerzy FILIPIAK 1, Lech SOLARZ, Korad ZUBKO 1 Istitute of Electroic ad Cotrol Systems, Techical Uiversity of Czestochowa,
More informationLecture 2: Finite Difference Methods in Heat Transfer
Lecture 2: Fiite Differece Methods i Heat Trasfer V.Vuorie Aalto Uiversity School of Egieerig Heat ad Mass Trasfer Course, Autum 2016 November 1 st 2017, Otaiemi ville.vuorie@aalto.fi Overview Part 1 (
More information9.4.3 Fundamental Parameters. Concentration Factor. Not recommended. See Extraction factor. Decontamination Factor
9.4.3 Fudametal Parameters Cocetratio Factor Not recommeded. See Extractio factor. Decotamiatio Factor The ratio of the proportio of cotamiat to product before treatmet to the proportio after treatmet.
More informationANALYSIS OF EXPERIMENTAL ERRORS
ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder
More informationDYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS
DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS Ivaa Štimac 1, Ivica Kožar 1 M.Sc,Assistat, Ph.D. Professor 1, Faculty of Civil Egieerig, Uiverity of Rieka, Croatia INTRODUCTION The vehicle-iduced
More informationVector Quantization: a Limiting Case of EM
. Itroductio & defiitios Assume that you are give a data set X = { x j }, j { 2,,, }, of d -dimesioal vectors. The vector quatizatio (VQ) problem requires that we fid a set of prototype vectors Z = { z
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationNUMERICAL INVESTIGATION OF FEEDBACK CONTROL IN PLASMA PROCESSING REACTORS INTRODUCTION
NUMERICAL INVESTIGATION OF FEEDBACK CONTROL IN PLASMA PROCESSING REACTORS Shahid Rauf ad Mark J. Kusher Departmet of Electrical ad Computer Egieerig Uiversity of Illiois 1406 W. Gree Street, Urbaa, Illiois
More informationModeling of Plasmas and Neutrals Including Plasma-Wall Interaction for Long Term Tokamak Operation
odelig of Plasmas ad Neutrals Icludig Plasma-Wall Iteractio for Log Term Tokamak Operatio Akiyoshi Hatayama 1, Kousuke Okamoto 1, Ryoko Tatsumi 1, Kazuhiro. Abe 1, ad Kazuaki Haada 2 1. Backgroud/otivatio
More informationMath 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
More informationFormation of A Supergain Array and Its Application in Radar
Formatio of A Supergai Array ad ts Applicatio i Radar Tra Cao Quye, Do Trug Kie ad Bach Gia Duog. Research Ceter for Electroic ad Telecommuicatios, College of Techology (Coltech, Vietam atioal Uiversity,
More information577. Estimation of surface roughness using high frequency vibrations
577. Estimatio of surface roughess usig high frequecy vibratios V. Augutis, M. Sauoris, Kauas Uiversity of Techology Electroics ad Measuremets Systems Departmet Studetu str. 5-443, LT-5368 Kauas, Lithuaia
More informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More informationBACKMIXING IN SCREW EXTRUDERS
BACKMIXING IN SCREW EXTRUDERS Chris Rauwedaal, Rauwedaal Extrusio Egieerig, Ic. Paul Grama, The Madiso Group Abstract Mixig is a critical fuctio i most extrusio operatios. Oe of the most difficult mixig
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationMechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationInformation-based Feature Selection
Iformatio-based Feature Selectio Farza Faria, Abbas Kazeroui, Afshi Babveyh Email: {faria,abbask,afshib}@staford.edu 1 Itroductio Feature selectio is a topic of great iterest i applicatios dealig with
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationA NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS
Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet
More informationTrue Nature of Potential Energy of a Hydrogen Atom
True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365-w, 33+p Abstract I cosiderig the potetial
More informationOn an Application of Bayesian Estimation
O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma
More information5. A formulae page and two tables are provided at the end of Part A of the examination PART A
Istructios: 1. You have bee provided with: (a) this questio paper (Part A ad Part B) (b) a multiple choice aswer sheet (for Part A) (c) Log Aswer Sheet(s) (for Part B) (d) a booklet of tables. (a) I PART
More informationControl chart for number of customers in the system of M [X] / M / 1 Queueing system
Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationMATHEMATICAL MODELLING OF ARCH FORMATION IN GRANULAR MATERIALS
6 th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE MATHEMATICAL MODELLING OF ARCH FORMATION IN GRANULAR MATERIALS Istva eppler SZIE Faculty of Mechaics, H-2103 Gödöllő Páter. 1., Hugary Abstract: The mathematical
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationInfinite Series and Improper Integrals
8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to
More informationKolmogorov-Smirnov type Tests for Local Gaussianity in High-Frequency Data
Proceedigs 59th ISI World Statistics Cogress, 5-30 August 013, Hog Kog (Sessio STS046) p.09 Kolmogorov-Smirov type Tests for Local Gaussiaity i High-Frequecy Data George Tauche, Duke Uiversity Viktor Todorov,
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationLecture 4 Conformal Mapping and Green s Theorem. 1. Let s try to solve the following problem by separation of variables
Lecture 4 Coformal Mappig ad Gree s Theorem Today s topics. Solvig electrostatic problems cotiued. Why separatio of variables does t always work 3. Coformal mappig 4. Gree s theorem The failure of separatio
More informationSOLUTIONS: ECE 606 Homework Week 7 Mark Lundstrom Purdue University (revised 3/27/13) e E i E T
SOUIONS: ECE 606 Homework Week 7 Mark udstrom Purdue Uiversity (revised 3/27/13) 1) Cosider a - type semicoductor for which the oly states i the badgap are door levels (i.e. ( E = E D ). Begi with the
More informationVALIDATION OF A CFD MODEL OF A THREE- DIMENSIONAL TUBE-IN-TUBE HEAT EXCHANGER
Third Iteratioal Coferece o CFD i the Mierals ad Process Idustries CSIRO, Melboure, Australia 1-1 December 3 VALIDATION OF A CFD MODEL OF A THREE- DIMENSIONAL TUBE-IN-TUBE HEAT EXCHANGER Hilde VAN DER
More information9.3 Power Series: Taylor & Maclaurin Series
9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationCALCULATION IN THE FIELD OF SEGMENTAL ROTOR MACHINES TAKING INTO ACCOUNT WINDING HARMONICS AND ROTOR AIRGAP IRREGULARITIES
CLCULTION IN THE FIELD OF SEGENTL OTO CHINES TKING INTO CCOUNT WINDING HONICS ND OTO IGP IEGULITIES Y STCT The stator mmf over a segmet of the segmetal rotor reluctace machie is treated as a ifiite array
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a
More informationDECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan
Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed
More informationResearch Article A New Second-Order Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More informationMicroscopic Theory of Transport (Fall 2003) Lecture 6 (9/19/03) Static and Short Time Properties of Time Correlation Functions
.03 Microscopic Theory of Trasport (Fall 003) Lecture 6 (9/9/03) Static ad Short Time Properties of Time Correlatio Fuctios Refereces -- Boo ad Yip, Chap There are a umber of properties of time correlatio
More informationChemical Kinetics CHAPTER 14. Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop. CHAPTER 14 Chemical Kinetics
Chemical Kietics CHAPTER 14 Chemistry: The Molecular Nature of Matter, 6 th editio By Jesperso, Brady, & Hyslop CHAPTER 14 Chemical Kietics Learig Objectives: Factors Affectig Reactio Rate: o Cocetratio
More informationTHE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE ABSTRACT
Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN 56-586 THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE Atif Nazir, Tahir Mahmood ad
More informationPolynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
More information