ORBITAL MANEUVERS USING LOWTHRUST


 Quentin Mills
 2 years ago
 Views:
Transcription
1 Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ORBIAL MANEUVERS USING LOWHRUS VIVIAN MARINS GOMES, ANONIO F. B. A. PRADO, HÉLIO KOII KUGA Ntionl Institute for Spce Reserch INPE  DMC Av. Dos Astronuts 1758 São José dos Cmpos SP BRAZIL Abstrct:  o perform the orbitl mneuvers, softwre tht clcultes n optiml mneuver is developed. his method will be used s reference for comprison nd nlises of the suboptiml methods to be used on bord. his method id bsed on n nlyticl development tht generte equtions tht cn be computed in shorter time, llowing rel time pplictions. KeyWords:  Orbitl mneuvers, low thrust, strodynmics, rtificil stellites, orbitl dynmics. 1 Introduction he problem of clculting orbitl mneuvers is very importnt topic in Orbitl Mechnics. hus, the problem of trnsferring spcecrft from n orbit to nother hs grown in importnce in recent yers. Applictions of this study cn be found in vrious spce ctivities, such s plcing stellite in geosttionry orbit, the mneuvers of spce sttion, orbit mintennce of stellite, mong others. In ctul pplictions, there my be need to mke n dditionl mneuver, both for trnsfer orbit or only for periodic corrections of lesser mgnitude. his issue of trnsfer is to chnge the position, velocity nd mss of the stellite from its current sttus to new stte predetermined. he trnsfer my be completely constrined or prtilly free (free time, free finl velocity, etc.). In the most generl cse, the choice of direction, sense nd mgnitude of the thrust to be pplied should be mde, respecting the limits of the vilble equipment. o crry out this trnsfer, it is intended to use optiml or suboptiml continuous mneuvers [1], [2]. So, to fulfill tht tsk, two methods for clculting mneuvers were developed. he first of them will get n optimiztion without worrying bout the processing time. he second method is suboptiml nd it will pproch the directions of ppliction of the thrust to llow fster clcultion of the control. he optimum method will be used to compre the consumption obtined by the suboptiml method, which involves simplifictions for ech specil sitution, in order to obtin high processing speed, fvoring the possibility of using it in reltime. In both methods, it will be ssumed tht the mgnitude of the thrust to be pplied is constnt nd smll nd the serch will be to find its direction. his direction cn be free (optiml method), [] or with some kind of constrints (suboptiml method). 2 Suboptiml Method he gol of this topic is to develop suboptiml method with highspeed computing for the clcultion of orbitl mneuvers bsed on continuous thrust nd smll mgnitude. he ide is to hve method tht genertes quick result nd, if possible, with result in terms of cost of fuel not much different from the optiml method described bove. his method should be used in cses of trnsfers with smll mgnitude, which usully re more frequent in the steps tht follow the insertion of the spcecrft in its nominl orbit. o solve this problem, it ws chosen in the literture bse method to mke expnsions nd djustments to the needs of this work. he method is described below. A ner optiml method for clculting orbitl trnsfer nd with minimum time (so, minimum consumption, s the mgnitude of the thrust is constnt, wht implies tht the time of ppliction of thrust nd consumption re directly proportionl) round the Erth for spcecrft with electric solr propellnt ws developed by [4]. It used technique of direct optimiztion to solve the problem of optiml control, with pproches towrd ppliction of thrust. he optiml trjectories clculted by the direct pproch present very close results to the optiml trjectories obtined from vritionl clcultion. he equtions of motion for the vehicle when the thrust is cting re shown below. he equtions re written in terms of nonsingulr equinotil elements to cover both circulr nd plnr orbits (i = 0, 180). he reltion between the equinotil elements (, h, k, p, q, F) nd the clssicl orbitl elements (, e, i, W, W, E) is given by: h = esen( ω Ω) (1) ISSN: ISBN:
2 Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION k = e cos( ω Ω) (2) i p = tn sen Ω 2 () i q = tn cos Ω 2 (4) F = Ω ω E (5) where: = semimjor xis, e = eccentricity, i = inclintion, W = longitude of the scending node, w = rgument of perigee, E = eccentric nomly nd F = eccentric longitude. For spcecrft moving in the grvittionl field nd subject to the propulsive force, the equtions of motion re s follows: ' x = Mαˆ (6) In eqution 6 the stte vector is x = [, h, k, p, q] nd the sign ( ') indictes the derivtive with respect to time. he vector ( x 1) is unit vector long the direction of thrust ppliction. he vlue is the mgnitude of the thrust ccelertion, given by: 2ηP0 = (7) mgisp where h is the efficiency of the propulsion system, P0 is the initil power given propulsion system, m is the mss of the vehicle, g is the grvity ccelertion t se level nd Isp is the specific impulse. he eqution of stte for F is not included becuse it hs been used the verge of orbitl elements nd thus only elements which vry slowly re considered. he processing time is significntly reduced when using orbitl verges. As ll orbitl elements used re vribles tht vry slowly, due to the fct tht the force of thrust hs little mgnitude, it cn be used mjor steps of integrtion, in the order of dys. he eqution of motion of the spcecrft cn be pproximted by clculting the increment of ech orbitl element in period nd dividing by such time. herefore, the vrition in time of the equinotil elements by complete orbit with the propeller cting cn be obtined from the eqution: π ' 1 dt x = ˆ α df M (8) df π where is the pproximtion of the stte nd is the orbitl period. he br on top of vribles mens tht they were evluted using the verge stte vector. he integrtion represents the chnge in orbitl elements in revolution with the orbitl elements kept constnt, unless the eccentric longitude F, which is vried between p e p. Setting the direction for the thrust ppliction by the components (id, jd, kd), the product shown inside the integrl symbol cn be obtined. As the ccelertion is held constnt, it mens tht this vlue cn be plced outside of the integrl symbol. hus, the nlyticl equtions used for the terms corresponding to ech of the elements re shown below, where id, jd nd kd represent the three components of the direction vector of the thrust ppliction. For the semimjor xis (): 2jdA1 2idA 2 & = (9) µ µ ( 1 k cosf hsen F) ( 1 k cosf hsen F) For the element h: ( B1 B2 ) jd 1 ( B ( B4 B5 ) sen F) id 1 h & = kkd( p B6 q B7 ) (10) 1 For the element k: ISSN: ISBN:
3 Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ( C1 C2 ) id 1 ( ( C cosf C4 ) C5 ) jd 1 k & = hkd( p*c6 q C7 ) (11) 1 For the element p: 2 hk cosf k kd(1 p q ) h 1 sen F 1 1 h k 1 1 h k p & = (12) µ 2 1 For the element q: 2 h hksen F kd(1 p q ) k 1 cosf 1 1 h k 1 1 h k q & = (1) µ 2 1 hese equtions re written in terms of equinotil orbitl elements. It is lso possible to write them ccording to the trditionl keplerin elements. hey cn be found in [5]. he clculus of integrtion shown in Eqution 8 genertes equtions tht re too long for rel pplictions, especilly tking into ccount the need to implement them in rel time for mneuver. hus, lthough these equtions, in their complete form shown here, generte new method for clculting orbitl mneuvers, their use will be focused on individul cses. his implies to define reference orbit, which my be the finl desired orbit, the initil orbit of the spcecrft or even n verge of those two orbits. As it will be considered only mneuvers with smll mplitudes, this restriction will not bring gret losses in terms of ccurcy. hen, with these pproches mde, numericl vlues cn be used, so tht, these functions re only functions of F nd numericl constnts. From there, the integrl used in eqution 8 cn be clculted nd it is obtined simple nlyticl equtions for the vrition of ech orbitl element considered s function of direction nd mgnitude of the thrust pplied. hus, the problem of obtining the lowest fuel consumption in mneuver cn be defined s to find the optiml direction for thrust ppliction tht minimizes: J = t f (14) Subject to the men equtions of motion nd the initil condition: x (0) = x 0 (15) nd lso subject to the ties in the finl stte: ψ x(t ),t = x(t ) x (16) [ ] 0 f f f f = Results Severl mneuvers were simulted to test the methods developed [5]. he first two mneuvers involve the sme initil orbit, but the directions of thrust ppliction nd time of the opertion re different, with the gol of reching finl orbit frther. he third mneuver involves greter rnge of vrition in semimjor xis nd it is good to demonstrte the pplicbility of the method in situtions like this..1 Mneuver 1 In this specific cse, the semimjor xis ws chnged nd eccentricity nd the rgument of perigee of the ISSN: ISBN:
4 Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION orbit were kept constnts. he chnges were smll in mgnitude (bout 47 meters in semimjor xis, the min objective of the mneuver) to be comptible with the method developed. he rgument of perigee ws kept constnt s 90 degrees, but it could be ny vlue. ble 1 shows the elements of the initil orbit nd the components of the orbit to be chieved fter mneuver 1. ble 2 shows the input dt required for the first mneuver simultion with the optiml method: orbitl elements of the initil orbit, the vehicle chrcteristics (initil mss, the mgnitude of thrust, initil position of the vehicle, true nomly), the condition imposed to the finl orbit nd estimtion of the solution to strt the process of itertion (beginning nd end of propulsion, ngles of pitch nd yw nd their rtes of initil chnge nd n estimtion of fuel consumption). ble 1: Elements of initil nd finl orbit for mneuver 1. Initil orbit Condition in the finl orbit Semimjor xis Semimjor xis 7259,650 km 7259,697 km 0,0629 0,0629 Inclintion 66,52º Long. of scending node 110º Optiml cse ble 2: Dt for the mneuver1 using the optiml method. otl mss (vehicle fuel) 2500 kg Avilble thrust = 1 N Spcecrft Initil position = 0 initil dt rue nomly = 0º Strt of the engime = 0º Stop of the engine = 5º Initil estimte of solution Suboptiml cse Initil pitch ngle = 0º Initil yw ngle = 0º Initil rte of vrition in pitch = 0 Initil rte of vrition in yw = 0 Fuel needed to mneuver = 2 kg urning the initil keplerin elements to non singulr elements, ccording equtions 1 to 5: = 7,25965x106 m h = 0, k = 0, p = 0, q = 0,208 herefore, now the numericl vlues of the orbit cn be used, which will be used s the reference orbit, crrying out the integrtion shown in eqution 8 nd, in this wy, it is possible to obtin set of equtions tht provide the vrition of ech of the elements used to describe the orbit by orbitl revolution with the propellers cting ll the time. herefore, they will become the equtions of motion of the spcecrft with the ssumptions dopted. hese equtions, lredy tking into ccount the fct tht it ws plnr mneuver, so, kd = 0, s function of the components of the vector tht defines the direction of thrust pplied, re: d= cel*( *id *jd) (17) dh= cel*(0,822*id0, *jd) (18) dk= cel*(0, *id0,851282*jd) (19) dp= =0 (20) dq= =0 (21) where ccel is the ccelertion imposed by the stellite propellnt. o show in detil the usefulness of these equtions, figures 1 to show the vrition of the elements by orbit s function of the direction of the thrust pplied. It is possible to get mny informtions bout the effect of the direction of the thrust pplied in orbitl elements. Figure 1, mde for the sitution where the direction of the thrust pplied is constnt, shows tht there is vlue of the component x for which the semimjor xis shows mximum vrition. his vlue is round 0.5. ht figure my be used for prior ssessment of the direction of thrust pplied depending on the objectives of the mission. d(km) id Fig. 1: Vrition of the semimjor xis s function of the direction of the thrust pplied. ISSN: ISBN:
5 Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION dh id ble : Finl keplerin elements obtined for mneuver 1. Finl elements Optiml Suboptiml method method Semimjor xis 7259, ,697 (km) 0, , Inclintion 66,52 66,52 Long. of scending node Consumption (kg) 0,1652 0,2808 Fig. 2: Vrition of the orbitl element h s function of the direction of the thrust pplied. Durtion of the mneuver (min) dk id Mneuver 2 For mneuver 2, the semimjor xis ws lso chnged nd the eccentricity nd the rgument of perigee of the orbit were kept constnt. he chnges were of mgnitude slightly higher thn in the previous cse, bout 120 meters in semimjor xis, the min objective of the mneuver, lso iming to be comptible with the method developed. he rgument of perigee ws kept constnt in vlue 90 degrees, but could be ny vlue. ble 4 shows the elements of the initil orbit nd the conditions imposed for mneuver 2. Fig. : Vrition of the orbitl element k s function of the direction of the thrust pplied. As p nd q re constnts, grphic is not shown. With these equtions, the softwre Mthemtic is used to solve the optimiztion problem nd to obtin the optiml solution. Severl ssumptions cn be mde bout the direction of the thrust pplied. he simplest of them is ssuming constnt direction. So, the problem becomes to find the vlue of id tht genertes the minimum fuel consumption, becuse kd = 0 (plnr mneuver) nd jd is obtined by the condition tht the vector tht defines the direction of the thrust pplied is unit. he solution found is id = 0.8. Considering liner or prbolic reltions cn reduce the consumption so much nd the time of mneuver obtined, but it is not studied in this prt of the work. ble shows the finl orbit chieved by the spcecrft fter the mneuver, for the optiml nd suboptiml method, s well s the fuel consumed nd time of mke the mneuver. ble 4: Elements of initil nd finl orbit for mneuver 2. Initil orbit Condition in the finl orbit Semimjor xis Semimjor xis 7259,650 km 7259,770 km 0, , Inclintion 66,52º Long. of scending node 110º Optiml cse ble 2 lso shows the prmeters used for mneuver 2, with the optimum method. ISSN: ISBN:
6 Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION Suboptiml cse As the initil orbit (which is lso used s the reference orbit) is the sme s the previous exmple, both the initil orbitl elements nd the pproximte equtions of motion re the sme. So, the softwre Mthemtic is gin used to solve the problem of optimiztion nd to get the optiml solution. Once more it will be ssumed constnt direction. In this wy, the problem becomes to find the vlue of id tht generte the minimum fuel consumption, becuse kd = 0 (plnr mneuver) nd jd is obtined by the condition tht the vector tht defines the direction of the thrust pplied is unit. he solution found is id = ble 6 shows the finl orbit chieved by the spcecrft fter the mneuver, for the optiml nd suboptiml methods, s well s the fuel consumed nd the time of the mneuver. References: [1] sien, H. S. keoff from stellite orbit. Journl of the Americn Rocket Society: 2(4), 226, Julgo, 195. [2] Lwden, D. F. Optiml progrmming of rocket thrust direction. Astronutic Act: 1(1), 4156, Jnfev, [] Prdo, A. F. B. A.; Neto, A. R. Um Estudo Bibliográfico Sobre o Problem de rnsferêncis de Órbits. Revist Brsileir de Ciêncis Mecânics, 15(1):6578, 199. [4] Kluever, C. A.; Oleson, S. R. A direct pproch for computing ner optiml lowthrust trnsfers. Advnces in the Astronuticl Sciences, v. 97, n. 2, p , [5] Gomes, V. M. Determinção de órbit e mnobrs utilizndo gps e motor com bixo empuxo. Phd hesis. São José dos Cmpos, INPE, bel 6  Finl keplerin elements obtined for mneuver 2. Finl elements Optiml method Suboptiml method Semimjor xis 7259, ,77 (km) 0, , Inclintion Long. of scending node Consumption (kg) Durtion of the mneuver (min) 66,52 66, ,2520 kg 0,900 kg 601,8 1020,2 4 Conclusion It ws studied nd developed method for the cse of suboptiml continuous mneuvers. his method is bsed on n nlyticl development, which genertes equtions tht cn be used for fst processing time, llowing its use in rel time. he gol is to find the direction of the thrust pplied to perform the orbitl mneuvers, with the ppliction of the liner direction of the pplied thrust. he time nd consumption re bout 20% higher when compred to the ones obtined from the optiml method, so the suboptiml method cn be used s first estimte. ISSN: ISBN:
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 information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationNet 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 information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More informationNewton 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 informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More informationTheory of Forces. Forces and Motion
his eek extbook  Red Chpter 4, 5 Competent roblem Solver  Chpter 4 relb 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 informationOn 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 informationPolynomial 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 informationPhysics 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. nmwide region t x
More informationExperiment 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 informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge 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 informationr 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 informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationON THE FRAMESTEWART ALGORITHM FOR THE TOWER OF HANOI
ON THE FRAMESTEWART ALGORITHM FOR THE TOWER OF HANOI MICHAEL RAND 1. Introduction The Tower of Hnoi puzzle ws creted over century go by the number theorist Edourd Lucs [, 4], nd it nd its vrints hve chllenged
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More informationName: Lab Partner: Section:
Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force
More informationGraphs 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 informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationSimulation 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 informationReasoning 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 informationHomework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.
Text questions, Chpter 5, problems 15: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed
More information6.2 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 soclled volumes of
More information** 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 informationProject 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
More informationVectors 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
More informationTITLE THE PRINCIPLES OF COINTAP METHOD OF NONDESTRUCTIVE TESTING
TITLE THE PRINCIPLES OF COINTAP METHOD OF NONDESTRUCTIVE TESTING Sung Joon Kim*, DongChul Che Kore Aerospce Reserch Institute, 45 EoeunDong, YouseongGu, Dejeon, 35333, Kore Phone : 824286231 FAX
More informationNOTES AND CORRESPONDENCE. Uncertainties of Derived Dewpoint Temperature and Relative Humidity
MAY 4 NOTES AND CORRESPONDENCE 81 NOTES AND CORRESPONDENCE Uncertinties of Derived Dewpoint Temperture nd Reltive Humidity X. LIN AND K. G. HUBBARD High Plins Regionl Climte Center, School of Nturl Resource
More informationIntegration. 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
More information5.2. 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 relvlued
More informationOperations 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 informationUse 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
More informationEcon 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
More informationReview 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 informationMath 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 informationEconomics 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,
More informationPHYS 110A  HW #7 Solutions by David Pace Any referenced equations are from Griffiths
PHYS 110A  HW #7 Solutions by Dvid Pce Any referenced equtions re from Griffiths [1.] Problem 4.11 from Griffiths A cylinder of length L nd rdius hs permnent polriztion prllel to its xis. Find the bound
More informationEQUATIONS 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 pointdirection nd twopoint
More informationEnterprise Risk Management Software Buyer s Guide
Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors
More informationJaERM SoftwareasaSolution Package
JERM SoftwresSolution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock
More informationCHAPTER 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
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure 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
More informationAMTH247 Lecture 16. Numerical Integration I
AMTH47 Lecture 16 Numericl Integrtion I 3 My 006 Reding: Heth 8.1 8., 8.3.1, 8.3., 8.3.3 Contents 1 Numericl Integrtion 1.1 MonteCrlo Integrtion....................... 1. Attinble Accurcy.........................
More informationHelicopter 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
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationIntegration 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
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire 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 informationEinstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on.
Cmbridge University Press 9780521683593  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 information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationSmall Businesses Decisions to Offer Health Insurance to Employees
Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employersponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults
More informationReview 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 informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationThe 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 informationCypress 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 informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
More informationClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment
ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationModeling POMDPs for Generating and Simulating Stock Investment Policies
Modeling POMDPs for Generting nd Simulting Stock Investment Policies Augusto Cesr Espíndol Bff UNIRIO  Dep. Informátic Aplicd Av. Psteur, 458  Térreo Rio de Jneiro  Brzil ugusto.bff@uniriotec.br Angelo
More information9 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
More informationA5682: 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 informationCHAPTER 5a. SIMULTANEOUS LINEAR EQUATIONS
CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskkf Spring 00 ENCE 0  Computtion Methods in Civil Engineering
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationAREA 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.
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More informationpersons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining:
COST BENEFIT ANALYSIS OF NARCOTIC ADDICTION TREATMENT PROGRAMS with Specil Reference to Age Irving Leveson,l New York City Plnning Commission Introduction Efforts to del with consequences of poverty,
More informationQuantity Oriented Resource Allocation Strategy on Multiple Resources Projects under Stochastic Conditions
Interntionl Conference on Industril Engineering nd Systems Mngement IESM 2009 My 1315 MONTRÉAL  CANADA Quntity Oriented Resource Alloction Strtegy on Multiple Resources Projects under Stochstic Conditions
More informationPlotting and Graphing
Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured
More informationVendor Rating for Service Desk Selection
Vendor Presented By DATE Using the scores of 0, 1, 2, or 3, plese rte the vendor's presenttion on how well they demonstrted the functionl requirements in the res below. Also consider how efficient nd functionl
More informationCost Functions for Assessment of Vehicle Dynamics
Cost Functions for Assessment of Vehicle Dynmics Dzmitry Svitski Automotive Engineering Deprtment Ilmenu University of Technology Ilmenu, Germny dzmitry.svitski@tuilmenu.de Pvel Nedom, Jroslv Mchn Skod
More informationMathematics. 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 informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationWeek 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 informationBinary 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
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationffiiii::#;#ltlti.*?*:j,'i#,rffi
5..1 EXPEDTNG A PROJECT. 187 700 6 o 'o' 600 E 500 17 18 19 20 Project durtion (dys) Figure 66 Project cost vs. project durtion for smple crsh problem. Using Excel@ to Crsh Project T" llt ffiiii::#;#ltlti.*?*:j,'i#,rffi
More informationChapter 4: Dynamic Programming
Chpter 4: Dynmic Progrmming Objectives of this chpter: Overview of collection of clssicl solution methods for MDPs known s dynmic progrmming (DP) Show how DP cn be used to compute vlue functions, nd hence,
More informationLesson 10. Parametric Curves
Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationPROBLEMS 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 informationSection 54 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 informationDouble 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 informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationDispersion in Coaxial Cables
Dispersion in Coxil Cbles Steve Ellingson June 1, 2008 Contents 1 Summry 2 2 Theory 2 3 Comprison to Welch s Result 4 4 Findings for RG58 t LWA Frequencies 5 Brdley Dept. of Electricl & Computer Engineering,
More informationDesign, Development and Testing of an Air Damper to Control the Resonant Response of a SDOF QuarterCar Suspension System
Modern Mechnicl Engineering, 0,, 849 doi:0.436/mme.0.0 Published Online November 0 (http://www.scirp.org/journl/mme) Design, Development nd Testing of n Air Dmper to Control the Resonnt Response of SDOF
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More informationHealth insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
More informationTreatment 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 ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationFactoring 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 information2.6.3 Characteristic of Compoundwound motor connection common use aids speed falls opposes speed increases ratio seriestoshunt field ampereturns
.6.3 Chrcteristic of Compoundwound motor A compoundwound motor hs both series nd shunt field winding, (i.e. one winding in series nd one in prllel with the rmture circuit), by vrying the number of turns
More informationAAPT 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 informationVolumes 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 xxis. There is strightforwrd technique which enbles this to be done, using
More informationApplications 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 informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
More informationUsing Definite Integrals
Chpter 6 Using Definite Integrls 6. Using Definite Integrls to Find Are nd Length Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: How
More informationSmall Business Cloud Services
Smll Business Cloud Services Summry. We re thick in the midst of historic sechnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly
More information