SPH simulation of fluid-structure interaction problems

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "SPH simulation of fluid-structure interaction problems"

Transcription

1 Diprtimento di ingegneri idrulic e mientle SPH simultion of fluid-structure interction prolems C. Antoci, M. Gllti, S. Siill

2 Reserch project Prolem: deformtion of plte due to the ction of fluid (lrge displcement of the structure, fluid free surfce) Numericl technique: SPH Lgrngin: it utomticlly follows moving interfces Reltively esy to include the simultion of different mterils

3 Scheme of the model Structure (SPH) Fluid (SPH) Coupling conditions on the interfce: Kinemtic condition: v f = v s v nˆ = f v s nˆ (perfect fluid) Dynmic condition: σ s T s nˆ s = σ s s f nˆ f f n ˆ σ nˆ = p (perfect fluid)

4 Continuum equtions Dρ v + ρ Dt x i i = 0 Dv σ ρ i = ρgi + Dt x ij j (i=1,3) (i,j=1,3) continuity eqution eqution of motion ρ t x i v i σ ij g i (isotherml conditions) density time position (i-component of the vector) velocity (i-component of the vector) stress (ij-component of the tensor) grvity ccelertion (i-component of the vector)

5 Stte eqution nd constitutive equtions pressure devitoric stress σ ij = p δ + ij S ij (i,j=1,3) p = c ( ρ ) 2 0 ρ0 fluid solid c = ε 0 ρ 0 c 0 = k ρ 0 (ε compressiility modulus) (k ulk modulus) S ij fluid solid ε ij = S ij = 0 (perfect fluid) ds dt ij 1 vi 2 x j 1 = 2µ εij δ ij ε 3 v + x j i Ω ij ij = + S 1 vi 2 x j ik Ω jk v x j i + Ω ik S kj (µ sher modulus)

6 2D SPH equtions (1) 2 Eqution of stte: = c ( ρ ) p 0 ρ0 Continuity eqution: Dρ = m ( v v ) Dt W ( nd : prticle lels)

7 2D SPH equtions (2) Eqution of motion: fluid solid Dv Dt Dv Dt i i p = p + W m δij + g 2 2 ρ x ρ j = σ ρ W x m rtificil viscosity (Monghn nd Gingold 1983) ij 2 σ ρ ij 2 δ ij R ij rtificil stress (Gry, Monghn nd Swift 2001) i f n j g i σ ij = p δ + ij S ij S ij clculted y n implicit scheme from the incrementl hypoelstic reltion

8 2D SPH equtions (3) Velocity grdient: v x i j = m ρ ( v v ) i i W x j spin rte of deformtion [or: velocity grdient normlized to ccount for non uniformity in prticle distriution]

9 - Two sets of prticles, fluid nd solid Fluid-structure interction - A simple pproch: to consider prticles in the equtions regrdless of the fct they re fluid or solid - Interpenetrtion of fluid nd solid prticles cn e prevented using XSPH ut the interction is not well reproduced (excessive dhesion) - Definition of the fluid-solid interfce (nd norml) - Dynmic condition (ction-rection principle) - Kinemtic condition

10 Definition of the fluid-solid interfce Università degli Studi di Pvi Since solid prticles mintin the sme regulr sptil distriution (no frgmenttion): ( ) = = ,, ˆ y x x x y y x x x x t t t ( ) x y t t n, ˆ = n d x x ˆ 2 int + = (for every solid prticle ner the interfce)

11 Dynmic condition p int = Ω f Ω m ρ f m ρ p W W ( x x, h) ( x x, h) int int 0.5 (constnt, in order to tke in ccount possile seprtion of the two medi) surfce term: F f s = pint Γ W F s f,* = - F f s, ( x x', h) dγ' int (liner interpoltion) (F f s /ρ dded term in the momentum eqution)

12 v iint Università degli Studi di Pvi * = Ω s Ω m viw ρ m W ρ s Kinemtic condition - fluid prticle * solid prticle (the nerest) - velocity of the interfce: ( x x, h) ( x x, h) int int * * - velocity distriution is ssigned to solid prticles in order to otin y SPH interpoltion (on the interfce) the interfce velocity (norml component): v n = v = t v v nint t * + d * nint ( v v ) * n d * = mx (d /d,2)

13 Other fetures of the code Correction of velocity: -XSPH (solid)(monghn 1989) -dissiptive correction (fluid) (Gllti nd Brschi 2000) (on the interfce prticles from oth medi hve to e included in the correction) Boundry conditions: - fluid - imginry prticles which reflect velocity - lyer of fixed prticles (2h) - solid (clmp) - lyer of fixed prticles which re clculted just like others ut with velocity equl to zero Time integrtion scheme: Euler explicit (stggered) Kernel: cuic spline (Monghn 1992)

14 Exmple: Elstic gte Dimensions A H L S 0.1 m 0.14 m m m Sluice-gte (ruer) ρ s E (Young modulus) 1100 kg/m P

15 Simultion ε = N/m 2 (compressile) ρ f = 1000 kg/m 3 K = N/m 2 µ = N/m 2 ν = 0.4 (Poisson coefficient) E= N/m 2 h/d=1.5 n P =6012 dt= s t=0 s

16 Comprison etween simultion nd experiment t=0 s

17 t=0.04 s

18 t=0.08 s

19 t=0.12 s

20 t=0.16 s

21 t=0.2 s

22 t=0.24 s

23 t=0.28 s

24 t=0.32 s

25 t=0.36 s

26 t=0.4 s

27 Free end of the plte: displcements (1) horizontl displcement 0,06 0,05 0,04 h.d. (m) 0,03 experiment simultion 0,02 0, ,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 time (s)

28 Free end of the plte: displcements (2) verticl displcement 0,025 0,02 v.d. (m) 0,015 0,01 experiment simultion 0, ,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 time (s)

29 Free surfce (1) wter level ehind the gte 0,16 0,14 0,12 0,1 y (m) 0,08 experiment simultion 0,06 0,04 0, ,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 t (s)

30 Free surfce (2) wter level 5 cm fr from the gte 0,16 0,14 0,12 0,1 y (m) 0,08 experiment simultion 0,06 0,04 0, ,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 time (s)

31 Future work To complete the normliztion of the elstic equtions with explicit tretment of oundry conditions To improve the model in order to mnge lrger deformtions To write the code in cylindricl coordintes in order to simulte xilly symmetric prolems ( flex-flow vlves, sloshing in cylindricl tnks ) To use two different sptil discretiztions for the fluid nd the solid (memrnes, hemodynmics) This reserch ws finnced y Dresser Itli

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

HYDRODYNAMIC FORCES. Drag Force and Drag Coefficient. Eq. (4) Eq. (2)

HYDRODYNAMIC FORCES. Drag Force and Drag Coefficient. Eq. (4) Eq. (2) HYROYNAMI FORES rg Force nd rg oefficient A prticle suspended in fluid is sujected to hydrodynmic forces. For low ynolds numer, the Stokes drg force on sphericl prticle is given y F = πµud, (1) where d

More information

Math 22B Solutions Homework 1 Spring 2008

Math 22B Solutions Homework 1 Spring 2008 Mth 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A sphericl rindrop evportes t rte proportionl to its surfce re. Write differentil eqution for the volume of the rindrop s function of time. Solution

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s

1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s Crete ssignment, 99552, Homework 5, Sep 15 t 10:11 m 1 This print-out should he 30 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. The due time

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Homework 10. Problems: 19.29, 19.63, 20.9, 20.68

Homework 10. Problems: 19.29, 19.63, 20.9, 20.68 Homework 0 Prolems: 9.29, 9.63, 20.9, 20.68 Prolem 9.29 An utomoile tire is inlted with ir originlly t 0 º nd norml tmospheric pressure. During the process, the ir is compressed to 28% o its originl volume

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

Matrix Inverse and Condition

Matrix Inverse and Condition Mtrix Inverse nd Condition Berlin Chen Deprtment of Computer Science & Informtion Engineering Ntionl Tiwn Norml University Reference: 1. Applied Numericl Methods with MATLAB for Engineers, Chpter 11 &

More information

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl

More information

Perfect competition model (PCM)

Perfect competition model (PCM) 18/9/21 Consumers: Benefits, WT, nd Demnd roducers: Costs nd Supply Aggregting individul curves erfect competition model (CM) Key ehviourl ssumption Economic gents, whether they e consumers or producers,

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING Sung Joon Kim*, Dong-Chul Che Kore Aerospce Reserch Institute, 45 Eoeun-Dong, Youseong-Gu, Dejeon, 35-333, Kore Phone : 82-42-86-231 FAX

More information

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

More information

SOLUTIONS TO CONCEPTS CHAPTER 5

SOLUTIONS TO CONCEPTS CHAPTER 5 1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

Theories of light and Interference S BALASUBRAMANYA SGL IN PHYSICS SARVODAYA PU COLLEGE, TUMKUR Important paints

Theories of light and Interference S BALASUBRAMANYA SGL IN PHYSICS SARVODAYA PU COLLEGE, TUMKUR Important paints Theories of light nd nterference S BALASUBRAMANYA SGL N PHYSCS SARVODAYA PU COLLEGE, TUMKUR mportnt pints Theories of Light Newton s Corpusculr theory (1675) Christin Huygen s Wve theory (1678) Mxwell

More information

Theory of Forces. Forces and Motion

Theory of Forces. Forces and Motion his eek extbook -- Red Chpter 4, 5 Competent roblem Solver - Chpter 4 re-lb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics - Everything

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply? Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

Null Similar Curves with Variable Transformations in Minkowski 3-space

Null Similar Curves with Variable Transformations in Minkowski 3-space Null Similr Curves with Vrile Trnsformtions in Minkowski -spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. -mil: mehmet.onder@yr.edu.tr

More information

Lecture 24: Laplace s Equation

Lecture 24: Laplace s Equation Introductory lecture notes on Prtil Differentil Equtions - c Anthony Peirce. Not to e copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 24: Lplce s Eqution

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

The Calculus of Variations: An Introduction. By Kolo Sunday Goshi

The Calculus of Variations: An Introduction. By Kolo Sunday Goshi The Clculus of Vritions: An Introduction By Kolo Sundy Goshi Some Greek Mythology Queen Dido of Tyre Fled Tyre fter the deth of her husbnd Arrived t wht is present dy Liby Irbs (King of Liby) offer Tell

More information

2D Finite Element Formulation for 3D Temperature Analysis of Layered Hybrid Structures

2D Finite Element Formulation for 3D Temperature Analysis of Layered Hybrid Structures D Finite Element Formultion for 3D emperture Anlysis of Lyered Hyrid Structures Rimund Rolfes, Jn eßmer Institut für Struturmechni, DLR e.v., Brunschweig, Germny Summry: hree-dimensionl (3D) finite element

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

2m + V ( ˆX) (1) 2. Consider a particle in one dimensions whose Hamiltonian is given by

2m + V ( ˆX) (1) 2. Consider a particle in one dimensions whose Hamiltonian is given by Teoretisk Fysik KTH Advnced QM SI2380), Exercise 8 12 1. 3 Consider prticle in one dimensions whose Hmiltonin is given by Ĥ = ˆP 2 2m + V ˆX) 1) with [ ˆP, ˆX] = i. By clculting [ ˆX, [ ˆX, Ĥ]] prove tht

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless

More information

Phys 207. Announcements. Hwk3 is posted on course website Quizzes & answers will be posted on course website Formula sheets.

Phys 207. Announcements. Hwk3 is posted on course website Quizzes & answers will be posted on course website Formula sheets. Phys 07 Announcements Hwk3 is posted on course website Quizzes & nswers will be posted on course website ormul sheets Newton s 3 lws Tody s Agend How nd why do objects move? Dynmics 1 Dynmics Isc Newton

More information

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1

More information

1. 0 m/s m/s m/s m/s

1. 0 m/s m/s m/s m/s Version PREVIEW Kine Grphs PRACTICE burke (1111) 1 This print-out should he 30 questions. Multiple-choice questions m continue on the next column or pge find ll choices before nswering. Distnce Time Grph

More information

Lecture Notes for Math 251: ODE and PDE. Lecture 35: 10.8 Laplace s Equation

Lecture Notes for Math 251: ODE and PDE. Lecture 35: 10.8 Laplace s Equation Lecture Notes for Mth 51: ODE nd PDE. Lecture 35: 1.8 Lplce s Eqution Shwn D. Ryn Spring 1 Lst Time: We studied nother fundmentl eqution in the study of prtil differentil equtions, which is the wve eqution.

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

Jackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.23 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson.3 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: A hollow cube hs conducting wlls defined by six plnes x =, y =, z =, nd x =, y =, z =. The wlls z =

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

More information

FEASIBILITY OF USING PRESSED SUGAR CANE STALK FOR THE PRODUCTION OF CHARCOAL. D Ffoulkes, R Elliott and T R Preston

FEASIBILITY OF USING PRESSED SUGAR CANE STALK FOR THE PRODUCTION OF CHARCOAL. D Ffoulkes, R Elliott and T R Preston Trop Anim Prod 1980 5:2 125 FEASIBILITY OF USING PRESSED SUGAR CANE STALK FOR THE PRODUCTION OF CHARCOAL D Ffoulkes, R Elliott nd T R Preston Fcultd de Medicin Veterinri y Zootecni, University of Yuctn,

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

ROMAI J., 2, 1(2006), Consider three system in fig.1.

ROMAI J., 2, 1(2006), Consider three system in fig.1. ROMAI J.,, 1(6), 19 5 AN INVESTIGATION OF MATHEMATICAL MODELS OF A PIPELINE - PRESSURE SENSOR MECHANICAL SYSTEM Petr A. Velmisov, Yuliy V. Pokldov Ulynovsk Stte Technicl University, Russi velmisov@ulstu.ru

More information

Chapter G - Problems

Chapter G - Problems Chpter G - Problems Blinn College - Physics 2426 - Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ10-5 T nd is directed t n

More information

Einstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on.

Einstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on. Cmbridge University Press 978-0-521-68359-3 - Study nd Mster Physicl Sciences Grde 11 Lerner s Book Krin Kelder More informtion MODULE 1 Einstein Mechnics motion force Glileo Newton decelerte moment of

More information

Version 001 Summer Review #03 tubman (IBII20142015) 1

Version 001 Summer Review #03 tubman (IBII20142015) 1 Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03

More information

Heat & Wave Equation in a Rectangle. Section 12.8

Heat & Wave Equation in a Rectangle. Section 12.8 Het & Wve Eqution in Rectngle Section 1.8 1 Het Eqution in Rectngle In this section we re concerned with ppliction of the method of seprtion of vriles pplied to the het eqution in two sptil dimensions.

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

2 Curvature of the Spine: Hydrostatic Pressure as an Indicator of Scoliosis

2 Curvature of the Spine: Hydrostatic Pressure as an Indicator of Scoliosis Report on problem studied t the UK Mthemtics-in-Medicine Study Group Nottinghm 2001 < http://www.mths-in-medicine.org/uk/2001/spine-curvture/ > MMSG2001 1 2 Curvture of the Spine: Hydrosttic Pressure s

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Diffraction and Interference of Light

Diffraction and Interference of Light rev 12/2016 Diffrction nd Interference of Light Equipment Qty Items Prt Number 1 Light Sensor CI-6504 1 Rotry Motion Sensor CI-6538 1 Single Slit Set OS-8523 1 Multiple Slit Set OS-8523 1 Liner Trnsltor

More information

Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems

Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

The area of the larger square is: IF it s a right triangle, THEN + =

The area of the larger square is: IF it s a right triangle, THEN + = 8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht

More information

DAM BREAK FLOW SIMULATION ON GRID

DAM BREAK FLOW SIMULATION ON GRID Mterils Physics nd Mechnics 9 (2010) 96-104 Received: Mrch 17, 2010 DAM BREAK FLOW SIMULATION ON GRID Arns Kčeniusks 1*, Rusln Pcevič 1 nd Toms Ktkevičius 2 1 Lbortory of Prllel Computing, Vilnius Gedimins

More information

Homework Assignment 1 Solutions

Homework Assignment 1 Solutions Dept. of Mth. Sci., WPI MA 1034 Anlysis 4 Bogdn Doytchinov, Term D01 Homework Assignment 1 Solutions 1. Find n eqution of sphere tht hs center t the point (5, 3, 6) nd touches the yz-plne. Solution. The

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Effect of microscopic damage events on static and ballistic impact strength of triaxial braid composites

Effect of microscopic damage events on static and ballistic impact strength of triaxial braid composites Astrct Sumitted to CompTest 2008 4th Interntionl Conference on Composites Testing nd Model Identifiction 20-22 Octoer, 2008 Dton, OH Effect of microscopic dmge events on sttic nd llistic impct strength

More information

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

Volumes of solids of revolution

Volumes of solids of revolution Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

More information

Rigid bodies moving in a stratified fluid

Rigid bodies moving in a stratified fluid Rigid bodies moving in strtified fluid Jules B. Kjtr School of Mthemticl Sciences Monsh University, Clyton Melbourne, Austrli jules.kjtr@monsh.edu Joe J. Monghn School of Mthemticl Sciences Monsh University,

More information

Continuous Random Variables: Derived Distributions

Continuous Random Variables: Derived Distributions Continuous Rndom Vriles: Derived Distriutions Berlin Chen Deprtment o Computer Science & Inormtion Engineering Ntionl Tiwn Norml Universit Reerence: - D. P. Bertseks, J. N. Tsitsiklis, Introduction to

More information

2.016 Hydrodynamics Prof. A.H. Techet

2.016 Hydrodynamics Prof. A.H. Techet .01 Hydrodynics Reding #.01 Hydrodynics Prof. A.H. Techet Added Mss For the cse of unstedy otion of bodies underwter or unstedy flow round objects, we ust consider the dditionl effect (force) resulting

More information

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s R 2 s(t).

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s R 2 s(t). 4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The

More information

, and the number of electrons is -19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow.

, and the number of electrons is -19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow. Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge

More information

Design Example 1 Special Moment Frame

Design Example 1 Special Moment Frame Design Exmple 1 pecil Moment Frme OVERVIEW tructurl steel specil moment frmes (MF) re typiclly comprised of wide-flnge bems, columns, nd bem-column connections. Connections re proportioned nd detiled to

More information

ProAdvice 3: AILERON SIZING

ProAdvice 3: AILERON SIZING ProAdvice 3: AILERON SIZING Introduction The purpose of the ilerons is to provide control out the irplne s roll xis. There re three common types of ilerons used in modern irplnes; Plin Flp Ailerons, Frise

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information

Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Final of Math 121, Fall 2014 Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

More information

Lecture 25: More Rectangular Domains: Neumann Problems, mixed BC, and semi-infinite strip problems

Lecture 25: More Rectangular Domains: Neumann Problems, mixed BC, and semi-infinite strip problems Introductory lecture notes on Prtil ifferentil Equtions - y Anthony Peirce UBC 1 Lecture 5: More Rectngulr omins: Neumnn Prolems, mixed BC, nd semi-infinite strip prolems Compiled 6 Novemer 13 In this

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Simulation of operation modes of isochronous cyclotron by a new interative method

Simulation of operation modes of isochronous cyclotron by a new interative method NUKLEONIKA 27;52(1):29 34 ORIGINAL PAPER Simultion of opertion modes of isochronous cyclotron y new intertive method Ryszrd Trszkiewicz, Mrek Tlch, Jcek Sulikowski, Henryk Doruch, Tdeusz Norys, Artur Srok,

More information

Equations between labeled directed graphs

Equations between labeled directed graphs Equtions etween leled directed grphs Existence of solutions Grret-Fontelles A., Misnikov A., Ventur E. My 2013 Motivtionl prolem H 1 nd H 2 two sugroups of the free group generted y X A, F (X, A). H 1

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Mechanics Cycle 1 Chapter 5. Chapter 5

Mechanics Cycle 1 Chapter 5. Chapter 5 Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

An Off-Center Coaxial Cable

An Off-Center Coaxial Cable 1 Problem An Off-Center Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2010

AAPT UNITED STATES PHYSICS TEAM AIP 2010 2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD

More information

Solving Linear Equations - Formulas

Solving Linear Equations - Formulas 1. Solving Liner Equtions - Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Surface Area and Volume

Surface Area and Volume Surfce Are nd Volume Student Book - Series J- Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book - Series J Contents Topics Topic - Surfce re of right prism Topic 2 - Surfce re of right

More information

Lie Group and Lie Algebra Variational Integrators for Flexible Beam and Plate in R 3

Lie Group and Lie Algebra Variational Integrators for Flexible Beam and Plate in R 3 Lie Group nd Lie Algebr Vritionl Integrtors for Flexible Bem nd Plte in R 3 THÈSE N O 5556 PRÉSENTÉE le 6 novembre À LA FACULTÉ DE L'ENVIRONNEMENT NATUREL, ARCHITECTURAL ET CONSTRUIT LABORATOIRE DE CONSTRUCTION

More information

In the following there are presented four different kinds of simulation games for a given Büchi automaton A = :

In the following there are presented four different kinds of simulation games for a given Büchi automaton A = : Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

CUBIC-FOOT VOLUME OF A LOG

CUBIC-FOOT VOLUME OF A LOG CUBIC-FOOT VOLUME OF A LOG Wys to clculte cuic foot volume ) xylometer: tu of wter sumerge tree or log in wter nd find volume of wter displced. ) grphic: exmple: log length = 4 feet, ech section feet in

More information