Explain shortly the meaning of the following eight words in relation to shells structures.
|
|
- Aldous David Underwood
- 5 years ago
- Views:
Transcription
1 Delft University of Technology Fculty of Civil Engineering nd Geosciences Structurl Mechnics Section Write your nme nd study number t the top right-hnd of your work. Exm CIE4143 Shell Anlysis Tuesdy 15 August 017, 13:30 16:30 hours Problem 1 Explin shortly the mening of the following eight words in reltion to shells structures. Membrne forces Edge bem Snders-Koiter equtions Knockdown fctor Gussin curvture Elpr Love Abutment force Problem Figure 1 shows prt of shell nd edge bem with coordinte system. Drw in this figure the shell loding, membrne forces, moments nd sher forces in the positive directions. (You cn hnd in this pge with your nswers.) Figure 1. Prt of shell b Does k xx hve positive vlue or negtive vlue? (fig. 1) Does k yy hve positive vlue or negtive vlue? (fig. 1) Does k G hve positive vlue or negtive vlue? (fig. 1) 1
2 Problem 3 Which vribles re in the Snders-Koiter equtions? Choose A, B, C or D. A shpe xuv (, ), y( uv, ), zuv (, ) B slopes z z ( uv, ), ( uv, ) x y C curvtures kxx ( uv, ), kyy ( uv, ), kxy ( uv, ) D Lme prmeters x( uv, ), y( uv, ) Consider Tylor expnsion of k xx in u = v = 0. k = b + bu+ bv+ bu + buv+ bv + bu + buv+ buv + b v xx b Which constnts will end up in the solution of p z? Choose A, B, C or D. A b 1 to b 10 B b to b 10 C b 4 to b 10 D b 7 to b 10 c We wnt to mke the expnsion of kxx s short s possible. We lso wnt to compute the exct vlue of p z in point u = v = 0. Which constnts should be included in the expnsion? Choose A, B, C or D. A b 1 B b 1 to b 3 C b 1 to b 6 D b 7 to b 10 d Consider two shell prts tht re connected in the line u = 0. The curvtures of the prts hve Tylor expnsions with the sme vlues of b 1 nd b. The other constnts in the Tylor expnsions re different. Wht is the level of continuity of the connection? Choose A, B, C or D. A B C D C 0 continuity C 1 continuity C continuity C 3 continuity e Wht level of continuity in the shpe is required to prevent ny edge disturbnce? A B C D C 0 continuity C 1 continuity C continuity C 5 continuity
3 f Wht is the level of continuity between shell finite elements? A B C D C 0 continuity C 1 continuity C continuity C 5 continuity Consider Tylor expnsion of x in u = v = x = b1+ bu + bv 3 + bu 4 + buv 5 + bv 6 + bu 7 + buv 8 + buv 9 + b10v. g Which constnts will end up in the solution of p z? Choose A, B, C or D. A b 1 to b 10 B b to b 10 C b 4 to b 10 D b 7 to b 10 h We wnt to mke the expnsion of x s short s possible. We lso wnt to determine the exct vlue of p z in point u = v = 0. Which constnts should be included? Choose A, B, C or D. A b 1 B b 1 to b 3 C b 1 to b 6 D b 7 to b 10 i Consider improving the Snders-Koiter equtions with geometricl nonlinerity (lrge deformtions). In theory, how cn this be done? Choose A, B, C or D. A Replce xyz,, by x + ux, y + uy, z + u z. B Replce kxx, kyy, k xy by κ κ κ 1 xx xx, κyy yy, κ xy ρ xy. C Replce kxx, kyy, k xy by κxx +κ xx, κyy +κ yy, κ xy +ρxy. D Replce kxx, kyy, k xy by κxx, κyy, ρxy. 3
4 Problem 4 A shell structure hs the shpe of hlve torus (fig. ). The dimeter is m. The dimeter of the torus centre line is 8 m. The shell wll is 4 mm thick. One edge is fixed. A point lod of 75 kn is pplied to the free edge in the positive z direction. The principl membrne forces re shown in figure 3. Sher deformtion hs not been included in the computtion (Kirchhoff insted of Mindlin-Reissner). dimeter of the torus centre line 8000 mm smll dimeter of the torus 000 mm thickness t = 4 mm Young s modulus E = 10 5 N/mm Poisson s rtio ν = 0. mss density ρ = 7850 kg/m 3 yield strength 500 N/mm grvittionl ccelertion g = 9.8 m/s element size 100 mm x 100 mm lod F = 75 kn theoreticl buckling force n cr = 0.6 E t / knockdown fctor C = 1/6 Is this shell thin? Explin your nswer. b Clculte the influence length. c Are the elements sufficiently smll? Explin your nswer. d Will this shell buckle? Explin your nswer. e Wht other wys exist to check the buckling strength? Figure. Hlf torus with fixed support nd point lod 4
5 Figure 3. Principl membrne forces in the torus y x z Figure 4. Shell shpe, = 3, -1 < u < 1, -1 < v < 1, 5
6 Problem 5 Consider the prmeteristion (fig. 4) v x = u(1 ) 1+ v + 1+ v y = v xy z = Wht is the nme of this shpe? b Wht is the physicl mening of prmeter? Wht is the unit of? Wht re the units of u nd v? c Is this prmeteristion orthogonl? Explin your nswer. (You cn use Mple or nother mthemticl progrm.) Mple hs been used to clculte Lme s prmeters nd the curvtures of this shpe. These equtions re not provided becuse they re lrge. Insted the Tylor expnsions t u = v = 0 re provided. = = x y (1 u uv u +...) kxx = 0 kyy = ( uv +...) kxy = (1 u v +...) 1 1 k1= (1 u uv v +...) k = ( 1 + u uv + v +...) 1 1 km = ( uv +...) kg = ( 1+ u + v...) d Is the prmeteristion in the principl curvture directions? Explin your nswer. e Check the provided Gussin curvture. f Is the deformtion of this shpe extensionl? Choose A, B, C or D. A Yes becuse the Gussin curvture chnges; it depends on u nd v. B Yes becuse the distnce between the prmeter lines vries. C No becuse flt sheet cn be twisted in this shpe without stretching. D There is no deformtion; it is just shpe. g Check the eqution of Guß for this shpe t u = v = 0. 6
7 Problem 6 In engineering we often cn choose between clculting something nd using tble. Let us pply this to finite element nlysis of shell structures. A computer cn compute n element stiffness mtrix or it cn look it up in tble. The question is; which costs less? Consider liner elstic shell element with 3 nodes nd 6 dofs per node. The following dt pplies. Computtion time costs 0.30 euro/hour (mintennce, electricity nd write off). Hrd disk spce costs 0.0 euro/gb/yer. In one yer, on one computer, times shell element is computed or looked up. Finding one number in list tkes log n evlutions of <, where n is the number of steps in the domin of the input vrible. The domin of ech input vrible of the tble is divided in 10 steps. A computer cn perform evlutions of < per second. Computing 3 node shell element stiffness mtrix tkes 4000 multiplictions or dditions. A computer cn perform multiplictions nd dditions per second. Which input vribles should the tble hve? For exmple shell thickness, How mny input vribles hve you selected? b How mny output vribles (numbers) should the tble hve? Explin your nswer. c Give smrt wys to reduce the number of input nd output vribles of the tble. d Suppose tht the thicknesses of shell elements vry between 0.01 mm nd 1 m. For the tble we need to divide this domin in steps. (Proposed re 10 steps. This is n estimtion.) Wht determines the ctul step size? How cn computer proceed to determine the correct step size? e How mny numbers need be stored in the tble? How much GB is this? f How much time does it tke to find the right element stiffness mtrix in the tble? g How much time does it tke to compute n element stiffness mtrix? h Which costs less; computing or looking up in tble? Explin your nswer. i Wht hs most influence on nswer h? 7
8 Answers to problem 1 Membrne forces.. in plne norml forces nd sher forces nxx, nyy, nxy, n yx Edge bem. bem t the edge of thin shell Snders-Koiter equtions 1 equtions tht describe the behviour of thin shells Knockdown fctor. reduction of the theoreticl liner buckling lod to include the effect of imperfections Gussin curvture.. kk 1 Elpr ellipticl prboloid Love. Augustus Love ws the first to propose shell theory. Abutment force.. horizontl lod on the support of n rch Answers to problem b kxx is positive. kyy is negtive. k G is negtive. Answers to problem 3 C b A c C d D e D f A g A h C i B Answer to problem 4 / t = 1000 / 4 = 50 > 30 So it is thin. b.4 t = = 15 mm c The elements re too lrge t the edges. About 6 should be in the influence length. d Et ncr = 0.6 = 0.6 = 190 N/mm = 190 kn/m 1000 n n = cr ult = 30 kn/m < 1557 kn/m 6 It will buckle. 8
9 e - Perform liner buck buckling nlysis with the softwre (nd pply the knock down fctor). - Add imperfections nd perform sttic nonliner nlyses. Answers to problem 5 Hypr (The figure ppers strnge becuse of the edges creted by this prticulr prmeteristion nd the very lrge domin of u nd v. Nevertheless, the equtions show tht this is hypr.) b Rdius of curvture in the origin. hs the unit length, for exmple m. u nd v hve no unit. c Substitution in x x + y y + z z u v u v u v gives zero. So it is orthogonl. d No becuse k xy 0. e 1 1 kg = k1k = (1 u uv v +...) ( 1 + u uv + v +...) = 1 ( u uv+ v + u u + uv uv + uv uv+ uv uv + v uv + uv v +...) = 1 ( u + v u v u v +...) = 1 ( 1 + u + v...) correct f D g 1 1 y 1 k = ( ) + ( x G ) xy u x u v y v 1 ( 1 + u + v +...) = = 1 ( (1 u uv u...)) 8 1 ( ) + ( ) (1 u uv u...) u u v (1 u uv u...) v = ( ( u uv u...)) + ( 0) 4 (1 + 1u uv 1u +...) u v (1 1u uv 1u...) 8 1 = v u... = ( 1+ u + v +...) (1 u uv u...) 8 9
10 Answers to problem 6 l1, l, l3, kxx, kyy, kxy, t, E, ν These re 9 input vribles. (If you considered flt element, thus 6 input vribles, then this would be correct too. ) b Output vribles: A stiffness mtrix hs (3 x 6) = 34 elements. c Key words: - Stiffness is liner in E. - Dimensionless prmeters - Eliminte element rigid body modes - Symmetry of the stiffness mtrix d Keywords: - Accurcy of the output - Interpoltion For exmple, computer cn compute for smll steps nd disregrd the step results for which the difference with n interpoltion is smll. e (10+1) 9 x 34 = x 8 = byte = 6000 GB f log 10 x 9 / = s g 4000 / = s h Computing: x / 3600 x 0.30 = 1 euro/yer Looking up: 6000 x x / 3600 x 0.30 = = 100 euro/yer Computing costs less. i Hrd disk spce 10
Math 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More information+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0
Notes for Cosmology course, fll 2005 Cosmic Dynmics Prelude [ ds 2 = c 2 dt 2 +(t) 2 dx 2 ] + x 2 dω 2 = c 2 dt 2 +(t) [ 2 dr 2 + S 1 κx 2 /R0 2 κ (r) 2 dω 2] nd x = S κ (r) = r, R 0 sin(r/r 0 ), R 0 sinh(r/r
More informationPlates on elastic foundation
Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More information99/105 Comparison of OrcaFlex with standard theoretical results
99/105 Comprison of OrcFlex ith stndrd theoreticl results 1. Introduction A number of stndrd theoreticl results from literture cn be modelled in OrcFlex. Such cses re, by virtue of being theoreticlly solvble,
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationShear and torsion interaction of hollow core slabs
Competitive nd Sustinble Growth Contrct Nº G6RD-CT--6 Sher nd torsion interction of hollow core slbs HOLCOTORS Technicl Report, Rev. Anlyses of hollow core floors December The content of the present publiction
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationJackson 2.7 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson.7 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: Consider potentil problem in the hlf-spce defined by, with Dirichlet boundry conditions on the plne
More informationPhysics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:
Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More information1 Bending of a beam with a rectangular section
1 Bending of bem with rectngulr section x3 Episseur b M x 2 x x 1 2h M Figure 1 : Geometry of the bem nd pplied lod The bem in figure 1 hs rectngur section (thickness 2h, width b. The pplied lod is pure
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationLecture XVII. Vector functions, vector and scalar fields Definition 1 A vector-valued function is a map associating vectors to real numbers, that is
Lecture XVII Abstrct We introduce the concepts of vector functions, sclr nd vector fields nd stress their relevnce in pplied sciences. We study curves in three-dimensionl Eucliden spce nd introduce the
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationPhysics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016
Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationChapter 5 Bending Moments and Shear Force Diagrams for Beams
Chpter 5 ending Moments nd Sher Force Digrms for ems n ddition to illy loded brs/rods (e.g. truss) nd torsionl shfts, the structurl members my eperience some lods perpendiculr to the is of the bem nd will
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationCBE 291b - Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationHQPD - ALGEBRA I TEST Record your answers on the answer sheet.
HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property
More informationKirchhoff and Mindlin Plates
Kirchhoff nd Mindlin Pltes A plte significntly longer in two directions compred with the third, nd it crries lod perpendiculr to tht plne. The theory for pltes cn be regrded s n extension of bem theory,
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More information5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict
More informationMATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2
MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points
More informationData Provided: A formula sheet and table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester
Dt Provided: A formul sheet nd tble of physicl constnts re ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS & ASTRONOMY Spring Semester 2016-2017 MEDICAL PHYSICS: Physics of Living Systems
More informationA - INTRODUCTION AND OVERVIEW
MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS A - INTRODUCTION AND OVERVIEW INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Course Content: A INTRODUCTION AND
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More informationMathematics for Physicists and Astronomers
PHY472 Dt Provided: Formul sheet nd physicl constnts Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY DEPARTMENT
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationThe Form of Hanging Slinky
Bulletin of Aichi Univ. of Eduction, 66Nturl Sciences, pp. - 6, Mrch, 07 The Form of Hnging Slinky Kenzi ODANI Deprtment of Mthemtics Eduction, Aichi University of Eduction, Kriy 448-854, Jpn Introduction
More informationProf. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015
Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be
More informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationThe 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 informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationMATH SS124 Sec 39 Concepts summary with examples
This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationdt. However, we might also be curious about dy
Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For
More informationJURONG JUNIOR COLLEGE
JURONG JUNIOR COLLEGE 2010 JC1 H1 8866 hysics utoril : Dynmics Lerning Outcomes Sub-topic utoril Questions Newton's lws of motion 1 1 st Lw, b, e f 2 nd Lw, including drwing FBDs nd solving problems by
More informationModule 1. Energy Methods in Structural Analysis
Module 1 Energy Methods in Structurl Anlysis Lesson 4 Theorem of Lest Work Instructionl Objectives After reding this lesson, the reder will be ble to: 1. Stte nd prove theorem of Lest Work.. Anlyse stticlly
More informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationThe steps of the hypothesis test
ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationSAINT IGNATIUS COLLEGE
SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationLecture 21: Order statistics
Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for
More informationIndefinite Integral. Chapter Integration - reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationSample Problems for the Final of Math 121, Fall, 2005
Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationThe momentum of a body of constant mass m moving with velocity u is, by definition, equal to the product of mass and velocity, that is
Newtons Lws 1 Newton s Lws There re three lws which ber Newton s nme nd they re the fundmentls lws upon which the study of dynmics is bsed. The lws re set of sttements tht we believe to be true in most
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationStatically indeterminate examples - axial loaded members, rod in torsion, members in bending
Elsticity nd Plsticity Stticlly indeterminte exmples - xil loded memers, rod in torsion, memers in ending Deprtment of Structurl Mechnics Fculty of Civil Engineering, VSB - Technicl University Ostrv 1
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationUNIT 3 Indices and Standard Form Activities
UNIT 3 Indices nd Stndrd Form Activities Activities 3.1 Towers 3.2 Bode's Lw 3.3 Mesuring nd Stndrd Form 3.4 Stndrd Inde Form Notes nd Solutions (1 pge) ACTIVITY 3.1 Towers How mny cubes re needed to build
More informationPurpose of the experiment
Newton s Lws II PES 6 Advnced Physics Lb I Purpose of the experiment Exmine two cses using Newton s Lws. Sttic ( = 0) Dynmic ( 0) fyi fyi Did you know tht the longest recorded flight of chicken is thirteen
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationME 309 Fluid Mechanics Fall 2006 Solutions to Exam3. (ME309_Fa2006_soln3 Solutions to Exam 3)
Fll 6 Solutions to Exm3 (ME39_F6_soln3 Solutions to Exm 3) Fll 6. ( pts totl) Unidirectionl Flow in Tringulr Duct (A Multiple-Choice Problem) We revisit n old friend, the duct with n equilterl-tringle
More informationScientific notation is a way of expressing really big numbers or really small numbers.
Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific
More information