COMPONENTS: COMBINED LOADING
|
|
|
- William Davidson
- 7 years ago
- Views:
Transcription
1 LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of Mterils Deprtment of Civil nd Environmentl Engineering University of Mrylnd, College Prk LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 1 Stresses in Thin-Wlled Vessels The thin-wlled pressure vessels provide n importnt ppliction of plne-stress nlysis. This their wlls offer little resistnce to bending, it my be ssumed tht the internl forces exerted on given portion of the wll re tngent to the surfce of the vessel, s shown in Fig. 32.
2 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 2 Stresses in Thin-Wlled Vessels Figure 32. Internl Forces re Tngent LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 3 Stresses in Thin-Wlled Vessels The resulting stresses on n element of the wll will thus be contined in plne tngent to the surfce of the vessel. Two types of thin-wlled vessels re investigted: Sphericl Pressure Vessels Cylindricl Pressure Vessels
3 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 4 Stresses in Thin-Wlled Vessels The stress in thin-wlled vessel vries from mximum vlue t the inside surfce to minimum vlue t the outside surfce of the vessel. It cn be shown tht if the rtio of the wll thickness to inner rdius of the vessel is less thn 0.1, the mximum norml stress is less thn 5% greter thn the verge stress. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 5 Definition A pressure vessel is defined s thinwlled when the rtio of the wll thickness to the rdius of the vessel is so smll tht the distribution of norml stress on plne perpendiculr to the surfce of the vessel is essentilly uniform throughout the thickness of the vessel.
4 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 6 Generl Types of Vessels The following types of vessels cn be nlyzed s thin-wlled elements: Boilers Gs Storge Tnks Pipelines Metl Tires Hoops LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 7 Generl Types of Vessels The following types of vessels cn be treted s thick-wlled elements: Gun Brrels Certin High-pressure Vessels in Chemicl Processing Industry Cylinders nd Piping for Hevy Hydrulic Pressure
5 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 8 Sphericl Pressure Vessels A typicl thin-wlled sphericl vessel used for gs storge is shown in Fig. 33. If the weights of the gs nd vessel re negligible (in most cses), symmetry of loding nd geometry requires tht stresses on sections tht pss through the center of the sphere be equl. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 9 Sphericl Pressure Vessels Figure 33
6 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 10 Sphericl Pressure Vessels Consider the element shown in Fig. 34. The stresses σ x, σ y, nd σ n re relted by the following eqution: σ σ = σ x = (40) y n LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 11 Sphericl Pressure Vessels r Figure 34 t σ σ n σ x y σ y () z x (b)
7 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 12 Sphericl Pressure Vessels Shering stresses on ny of these plnes re not present becuse there re no lods to induce them. The norml stress component in sphere is known s meridionl or xil stress nd is commonly denoted s σ m or σ LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 13 Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel Consider the thin-wlled sphericl pressure vessel with rdius r nd thickness t, shown in Fig. 34b. The free-body digrm of tht figure cn be used to compute the stresses σ = σ = σ = σ (41) x y n
8 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 14 Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel in terms of the pressure p, nd the inside rdius r nd thickness t of the sphericl vessel. The force R is the resultnt of the internl forces tht ct on the cross-sectionl re of the sphere tht exposed by pssing plne through the center of the sphere. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 15 Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel The force P is the resultnt of the fluid forces cting on the fluid remining within the hemisphere. Since the vessel is under sttic equilibrium, it must stisfy Newton's first lw of motion. In other words, the stress round the wll must hve net resultnt to blnce the internl pressure cross the cross-section
9 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 16 t Sphericl Pressure Vessels Derivtion of Axil or Meridionl Stress in Sphericl Vessel σ F = 0; R P = 0 R = P σ 2 ()( t 2πr ) = p( πr ) pr σ = 2t (42) LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 17 Stress on Sphericl Pressure Vessels σ = pr 2t p = pressure of gs or fluid r = inside rdius of sphere t = thickness of thin-wlled sphere (42)
10 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 18 Exmple 12 A steel pressure vessel of sphericl shpe hs the following specifictions: inside rdius r of 36 inches thickness t of 3/16" llowble yield stress σ y of 50 ksi modulus of elsticity E of 29,000 ksi Poisson s rtio ν of 0.25 ) Wht is the mximum pressure p crried by the tnk before yielding occurs? b) If p = 100 psi, wht is the new outer rdius of the tnk? LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 19 Exmple 12 (cont d) () Norml in-plne stresses re given by Eq. 42. Rewrite the eqution to solve for the mximum p pr 2tσ σ = ρ = 2t r p = = ksi = 521 psi 36
11 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 20 Exmple 12 (cont d) (b) First find the norml in-plne stress in the shell: pr 100( 36) σ = = = 9,600 psi 2 t Now pply Hooke s lw for plne stress: 1 1 ν 1 ν pr ε x = ( σ x νσ y ) = σ = E E E 2t ε x = ( 9,600) = The circumference, nd therefore the rdius, of the sphere will increse by 1 + e, so 3 New r outer = 36 ( ) = in. 16 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 21 Cylindricl Pressure Vessels A typicl thin-wlled sphericl vessel used for liquefied gs storge is shown in Fig. 35. Norml stresses, s shown in Fig. 36, re esy to evlute by using pproprite freebody digrm. Agin, The norml stress component on trnsverse plne is known s meridionl or xil stress nd is commonly denoted s σ m or σ.,
12 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 22 Cylindricl Pressure Vessels Figure 35 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 23 Cylindricl Pressure Vessels Figure 36 t r σ σ σ y () z x (b)
13 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 24 Cylindricl Pressure Vessels The norml stress component on longitudinl plne is known s hoop, tngentil, or circumferentil stress, nd commonly denoted s σ h, σ t, or σ c. Agin, there re no shering stresses on trnsverse or longitudinl plnes. Stress determintion in this cse will be the sme s in the cse of sphericl shpe. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 25 Cylindricl Pressure Vessels Derivtion of Norml Stress σ To determine the longitudinl stress σ, we mke cut cross the cylinder similr to nlyzing the sphericl pressure vessel. The free body, illustrted on the left (Fig. 36), is in sttic equilibrium. This implies tht the stress round the wll must hve resultnt to blnce the internl pressure cross the crosssection.
14 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 26 Cylindricl Pressure Vessels Derivtion of Norml Stress σ Applying sttics (Newton's first lw of motion, we hve Or Or σ F = 0, R P = 0 R = P x 2 ()( t 2πr ) = p( πr ) pr σ = 2t (43) LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 27 Cylindricl Pressure Vessels Derivtion of hoop or tngentil stress σ h To determine the hoop stress σ h, we mke cut long the longitudinl xis nd construct smll slice s illustrted Fig. 37. The free body is in sttic equilibrium. According to Sttics (Newton's first lw of motion), the hoop stress yields, 2( σ )( t)( dx) = p( 2r)( dx) h Therefore, σ = h pr t (44)
15 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 28 Cylindricl Pressure Vessels Derivtion of hoop or tngentil stress σ h 2 ( σ )( t)( dx) = p( 2r)( dx) h Therefore, σ = h pr t Figure 37 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 29 Stress on Cylindricl Pressure Vessels pr σ = 2t (45) σ = pr h t (46) p = pressure of gs or fluid r = inside rdius of sphere t = thickness of thin-wlled sphere
16 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 30 Exmple 1 A steel pipe with inside dimeter of 12 in. will be used to trnsmit stem under pressure of 1000 psi. If the hoop stress in the pipe must be limited to 10 ksi becuse of longitudinl weld in the pipe, determine the mximum stisfctory thickness for the pipe. LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 31 Exmple 1 (cont d) For the cylinder, the hoop stress is given given by Eq. 46 s Therefore, σ h = pr t pr 2 t = = σ 10,000 h = 0.6 in.
17 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 32 Remrks of Thin-Wlled Pressure Vessels 1. The bove formuls re good for thinwlled pressure vessels. Generlly, pressure vessel is considered to be "thinwlled" if its rdius r is lrger thn 5 times its wll thickness t (r > 5 t). LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 33 Remrks of Thin-Wlled Pressure Vessels 2. When pressure vessel is subjected to externl pressure, the bove formuls re still vlid. However, the stresses re now negtive since the wll is now in compression insted of tension.
18 LECTURE 24. COMPONENTS: COMBINED LOADING (8.4) Slide No. 34 Remrks of Thin-Wlled Pressure Vessels 3. The hoop stress is twice s much s the longitudinl stress for the cylindricl pressure vessel. This is why n overcooked hotdog usully crcks long the longitudinl direction first (i.e. its skin fils from hoop stress, generted by internl stem pressure).
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
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
Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
Uplift Capacity of K-Series Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu
Uplift Cpcity of K-Series Open Web Steel Joist Sets Perry S. Green, Ph.D, M.ASCE 1 nd Thoms Sputo, Ph.D., P.E., M.ASCE 2 1 Assistnt Professor, Deprtment of Civil nd Costl Engineering, University of Florid,
PROBLEM 4.1 SOLUTION. Knowing that the couple shown acts in a vertical plane, determine the stress at (a) point A, (b) point B.
PROBLEM.1 Knowing tht the couple shown cts in verticl plne, determine the stress t () point A, (b) point B. SOLUTON () (b) For rectngle: For cross sectionl re: 1 = bh 1 1 = 1 + + = ()(1.5) + ()(5.5) +
Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
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
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
6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
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
Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
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
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
Rotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
Vectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
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
Basic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
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
All pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
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
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
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 +
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
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
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
Assessment of the structural integrity of cracked cylindrical geometries applying the EVTUBAG program
Rev. Téc. ng. Univ. Zuli. Vol. 32, Nº 3, 190-199, 2009 Assessment of the structurl integrity of crcked cylindricl geometries pplying the EVTUBAG progrm Luis Héctor Hernández Gómez 1, Guilllermo Urriolgoiti
Lecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
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
LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.
Integration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
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
Section 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only
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
, 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
CET 1: Stress Analysis & Pressure Vessels. Lent Term 2005. Dr. Clemens Kaminski. Telephone: +44 1223 763135
CET : Stress Anlysis & Pressure Vessels Lent Term 5 Dr. Clemens Kminski Telephone: +44 765 URL: E-mil: clemens_kminski@cheng.cm.c.uk http://www.cheng.cm.c.uk/reserch/groups/lser/ Synopsis Introduction
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
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
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
Binary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
Increasing Q of Waveguide Pulse-Compression Cavities
Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide Pulse-Compression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque
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
. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
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
Project 6 Aircraft static stability and control
Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
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
Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
** 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
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
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
Brillouin Zones. Physics 3P41 Chris Wiebe
Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified
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.
2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
LECTURE #05. Learning Objectives. How does atomic packing factor change with different atom types? How do you calculate the density of a material?
LECTURE #05 Chpter : Pcking Densities nd Coordintion Lerning Objectives es How does tomic pcking fctor chnge with different tom types? How do you clculte the density of mteril? 2 Relevnt Reding for this
Project Recovery. . It Can Be Done
Project Recovery. It Cn Be Done IPM Conference Wshington, D.C. Nov 4-7, 200 Wlt Lipke Oklhom City Air Logistics Center Tinker AFB, OK Overview Mngement Reserve Project Sttus Indictors Performnce Correction
Helicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
MODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
AREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
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
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
10 AREA AND VOLUME 1. Before you start. Objectives
10 AREA AND VOLUME 1 The Tower of Pis is circulr bell tower. Construction begn in the 1170s, nd the tower strted lening lmost immeditely becuse of poor foundtion nd loose soil. It is 56.7 metres tll, with
FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University
SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility
4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.
4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.
MATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207.
MIT OpenCourseWre http://ocw.mit.edu Hus, Hermnn A., nd Jmes R. Melcher. Electromgnetic Fields nd Energy. Englewood Cliffs, NJ: Prentice-Hll, 1989. ISBN: 9780132490207. Plese use the following cittion
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
PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
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.
and thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
Calculating Principal Strains using a Rectangular Strain Gage Rosette
Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used
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
Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn
Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?
Week 7 - Perfect Competition and Monopoly
Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between
Lecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
Integration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
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
Econ 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
Welding procedure qualification tests of steels for hull construction and marine structures
(June 2005) (Rev.1 Nov 2006) (Rev.2 Mr 2012) Welding procedure qulifiction tests of steels for hull construction nd mrine structures 1. Scope 1.1 This document gives requirements for qulifiction tests
Exponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
Volumes as integrals of cross-sections (Sect. 6.1) Volumes as integrals of cross-sections (Sect. 6.1)
Volumes s integrls of cross-sections (ect. 6.1) Te volume of simple regions in spce Volumes integrting cross-sections: Te generl cse. Certin regions wit oles. Volumes s integrls of cross-sections (ect.
Physics 2102 Lecture 2. Physics 2102
Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other
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
PHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
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
Drawing Diagrams From Labelled Graphs
Drwing Digrms From Lbelled Grphs Jérôme Thièvre 1 INA, 4, venue de l Europe, 94366 BRY SUR MARNE FRANCE Anne Verroust-Blondet 2 INRIA Rocquencourt, B.P. 105, 78153 LE CHESNAY Cedex FRANCE Mrie-Luce Viud
Small Business Cloud Services
Smll Business Cloud Services Summry. We re thick in the midst of historic se-chnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly
Lesson 13 Inductance, Magnetic energy /force /torque
Lesson 3 nductnce, Mgnetic energy /force /torque 楊 尚 達 Shng-D Yng nstitute of Photonics Technologies Deprtment of Electricl Engineering Ntionl Tsing Hu Uniersity, Tiwn Outline nductnce Mgnetic energy Mgnetic
15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
Thin Walled Pressure Vessels
3 Thin Walled Pressure Vessels 3 1 Lecture 3: THIN WALLED PRESSURE VESSELS TABLE OF CONTENTS Page 3.1. Introduction 3 3 3.2. Pressure Vessels 3 3 3.3. Assumptions 3 4 3.4. Cylindrical Vessels 3 4 3.4.1.
CHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods