DYNSTALL: Subroutine Package with a Dynamic stall model


 Roland Griffin
 2 years ago
 Views:
Transcription
1 FFAPV110 DYNSTALL: Subroutie Package with a Dyamic stall model Aders Björck Steady data Experimet Simulatio FLYGTEKNISKA FÖRSÖKSANSTALTEN THE AERONAUTIAL RESEARH INSTITUTE OF SWEDEN
2 FLYGTEKNISKA FÖRSÖKSANSTALTEN THE AERONAUTIAL RESEARH INSTITUTE OF SWEDEN Box 11021, SE Bromma, Swede Phoe , Fax
3 Summary A subroutie package, called DYNSTALL, or the calculatio o 2D usteady airoil aerodyamics is described. The subrouties are writte i FORTRAN. DYNSTALL is a basically a implemetatio o the Beddoes Leishma dyamic stall model. This model is a semiempirical model or dyamic stall. It icludes, however, also models or attached low usteady aerodyamics. It is complete i the sese that it treats attached low as well as separated low. Semiempirical meas that the model relies o empirically determied costats. Semi because the costats are costats i equatios with some physical iterpretatio. It requires the iput o 2D airoil aerodyamic data via tables as uctio o agle o attack. The method is iteded or use i a aeroelastic code with the aerodyamics solved by blade/elemet method. DYNSTALL was writte to work or ay 2D agles o attack relative to the airoil, e.g. low rom the rear o a airoil. 3
4 4
5 otet 1 INTRODUTION THE BEDDOESLEISHMAN MODEL ATTAHED FLOW SEPARATED FLOW THE MODEL IN STEPS SEMIEMPIRIAL ONSTANTS THE DYNSTALL UNSTEADY AIRFOIL AERODYNAMIS MODEL A HANGE TO WORK IN THE WIND REFERENE SYSTEM ATTAHED FLOW The agle o attack Attached low equatios (LPOTMETH=3) Shed wake eects with accout or a varyig lit curve slope (LPOTMETH=4) SEPARATED FLOW A shit i the agle o attack Lit as uctio o the separatio poit positio The Øye separatio poit model (lmeth=2) A versio o the Kircho low (lmeth=4) Vortex lit (lvormeth=2) A alterative versio o vortex lit (lvormeth=1) UNSTEADY DRAG UNSTEADY PITHING MOMENT WAYS TO MAKE THE MODEL WORK FOR ALL ANGLES OF ATTAK ROBUSTNESS OF THE DYNSTALLMETHODS Methods that work or all agles o attack INPUT TO THE FORTRAN SUBROUTINE PAKAGE SUBROUTINES ONVENTIONS FOR VARIABLE NAMES ABOUT HISTORY OF VARIABLE NAMES OMMUNIATION WITH THE SUBROUTINES OMMON AREAS IN DYNL_.IN Storage o static airoil data Rage o agle o attack i iput airoil tables Lit curve slope ad zero lit agle o attack hoices betwee submodels oeiciets ad time costats Variables used or state variables ad other variables eeded or iteral commuicatio
6 4.5.5 Variables i commo areas that must be set at each time step outside o the DYNSTALLpackage GENERATION OF FNSTIN, NINVIN AND NSEPIN TABLES SUBMETHODS AND VALUES FOR SEMIEMPIRI ONSTANTS AN OPTIMIZATION STUDY TO FIND OPTIMUM VALUES LDDYN EFFETS OF VARYING VELOITY ON THE SEPARATION POINT REOMMENDED VALUES OF SUBMODELS AND SEMIEMPIRIAL PARAMETERS Eect o aerodyamic dampig, atigue ad extreme loads Dampig o lapwise vibratios i stall Dampig o edgewise vibratios i stall Extreme loads
7 Symbols ad otatios c hord cv Vortex eed (equatio (2.2.3) ad ( )) N Lit curve slope.[1/rad] Same as L L Lit curve slope.[1/rad] Same as N L D M T N Lit coeiciet Drag coeiciet Momet coeiciet Tagetial orce coeiciet. Normal orce coeiciet. Note that capital subscripts or lit, drag, pitchig momet, tagetial ad ormal orce coeiciets are used eve though it reers to twodimesioal quatities. apital letters or the subscripts is ot to mix with other idices, e.g. or time step umber, ' separatio poit locatio IB IR q Idex or blade umber IB Idex or radial elemet IR Pitch rate [rad/s] u p Pluge velocity (equatio )) Vrel w s Velocity relative to twodimesioal airoil elemet. Normal compoet o velocity relative to the airoil (equatio ( )) Dimesioless time, Dimesioless with halchord ad V rel t E Time Agle o attack Eective agle o attack (equatio (2.1.6) ad equatio ( )) i Shed wake iduced agle o attack (equatio (2.1.5)) 7
8 g Geometric agle o attack (equatio ( )) 75 ¾ chord agle o attack (equatio ( )) 2 Γ irculatio [ m / s ] Tp T Time costat or leadig edge pressure lag (equatio (2.3.1)) Time costat or separatio poit positio lag (equatio (2.2.2)) Tv Time costat or vortex lit decay (equatio (2.2.4)) 2D 3D Twodimesioal Three dimesioal Subscripts qs I c pot v stat dy quasi steady values time step Free stream coditios or coditios at iiity Impulsive loads irculatory loads Potetial loads (sum o impulsive ad circulatory loads) Vortex part o lit, drag et.c. Related to separatio poit locatio Static (or quasistatic) coditios Usteady coditios 8
9 1 Itroductio This report describes the usteady proile aerodyamics model used i the blade elemet/mometum (BEM) code AERFORE [1]. Usteady aerodyamics is o great importace or wid turbies. Both or low i the attached low agle o attack regio ad or low i the separated low regio. Most attetio is ote give to the usteady separated low aects, commoly amed dyamic stall. The stallig behavior o a airoil will be quite dieret rom its steady stallig behavior i the agle attack is chaged ast eough. The maximum lit coeiciet ca e.g. sigiicatly exceed its static value. For wid turbies, dyamic stall also largely aects the aerodyamic dampig i stall. Geerally the dampig would be largely uderpredicted i steady stall data were used i aeroelastic calculatios. Also, or yawed turbies, dyamic stallig plays a great role with icreased ad dieret phase L, max o the aerodyamic orces as a uctio o rotor azimuth agle relative to usig steady data. May papers ad reports have bee writte dealig with dyamic stall or wid turbies. A recet paper is [2] i which urther reereces ca oud. Dyamic stall has also bee subject o two E research projects [3] ad [4]. Usteady aerodyamics is, however, also importat or attached low. Usteady aerodyamics eects are importat e.g. or stability calculatios at turbie overspeedig, or bladetower iteractios ad or ast pitchig actio. The model i the DYNSTALLpackage is a implemetatio o the BeddoesLeishma dyamic stall model described i e.g. [5], [6] ad [7]. This model is a 2D semiempirical model or dyamic stall. It icludes, however, also models or attached low usteady aerodyamics. It is thus complete i the sese that it treats attached low as well as separated low 9
10 1.1 Structure o the report The BeddoesLeishma model i its origial versio i preseted i chapter 2. hages to this model that are itroduced i the FFA DYNSTALL versio are described i chapter 3. I chapter 4 details i the Fortra implemetatio are give. Fially some recommedatios o the use are give i chapter 5. 10
11 2 The BeddoesLeishma model The BeddoesLeishma model is a semiempirical model. Semiempirical meas that the model relies o empirically determied costats. Semi because the costats are costats i equatios with some physical iterpretatio. The model is based o tables o steady aerodyamic data ( L ( ), D () ad M ( ) ). The usteady behavior is described by dieret processes as e.g. the shed wake eect o the ilow ad a dyamic delay i the separatio process. The dieret dyamic processes are modeled as dieretial equatios that eed userset values or costats to get best agreemet with experimets or theoretical models. The BeddoesLeishma model ca be described as a idicial respose model or attached low exteded with models or separated low eects ad vortex lit. The orces are computed as ormal orce, tagetial orce ad pitchig momet. Idicial respose meas that the model works with a series o small disturbaces. The attached low part reproduces the liear theory o [9] or orce respose or a pitchig airoil, a plugig airoil ad the wid gust case. See e.g. [8], sectio 2.6 or a geeral descriptio o the attached low solutio. The usteady 2D airoil aerodyamic respose scales with time made dimesioless with speed relative to the airoil ad the chord o the airoil. The time i the equatios 1 is thereore replaced with dimesioless time: 1 Oe exceptio is the impulsive loads which scale with the chord ad the speed o soud 11
12 Vrel t s = (2.1) c 2 The model uses steady L ( ) ad D ( ) data as iput or a speciic airoil or which the dyamic orces are to be calculated. I the BeddoesLeishma model, the orces i the body ixed system, N ad T, are used, where: = cos( ) + si( ) (2.2) N L D T = cos( ) si( ) (2.3) D L The BeddoesLeishma model also treats the usteady eects o the pitchig momet. Dyamic stall has a dramatic eect o the pitchig momet ad o the dampig or pitchoscillatios. This is very importat or helicopter rotors but is less importat or wid turbie blades. The treatmet o the pitchig momet is thereore let out i the descriptio below. The descriptio below o the BeddoesLeishma model ollows as ar as how to solve equatios, the methods described i [5]. Alterative solutios usig a statespace model are described i [7]. 2.1 Attached low For attached low, two dieret aspects are modeled. Whe the boud vorticity o the airoil is time varyig, vorticity is shed i the wake. This shed vorticity iduces a low over the airoil, so that the airlow sesed by the airoil is ot the ree stream velocity. I or example the lit (ad the boud circulatio o the airoil) has bee icreasig or some time, the the shed wake will cause a dowwash over the airoil, resultig i less lit tha would be aticipated i steady low. 12
13 The other eect is the impulsive load eect. This eect is also called the apparet mass eect. This eect causes a lit orce or a ast pitch or ast dowward pluge motio. The shed wake eects ilueces the circulatory lit. The impulsive loads are added to get the total attached low lit (lit is here the same as ormal orce sice the agle o attack is assumed small or the attached low part) The low coditios are simulated by the superpositio o idicial resposes. irculatory part For the circulatory part this is writte as: [ ( M ) φ ( s, ] = ) (2.1.1) N, c N c M The idicial respose or the shed wake eects is approximated by φ c = A exp( b βs) A exp( b β ) (2.1.2) s where β = 2 1 M Suggestios or costats A 1, A 2, b 1 ad b 2 are give i [5] as A 1 =0.3 A 2 =0.7 b 1 =0.14 ad b 2 =0.53 The shed wake eect is computed by usig a lagged eective agle o attack: E = i where i ca be see as the shed wake iduced agle o attack. The umerical method to compute step: X i give i [5] is or the :th time = X 1 exp( b1 β s) + A1 exp( b1 β s / 2) (2.1.3) 13
14 Y = Y 1 exp( b2β s) + A2 exp( b2β s / 2) (2.1.4) = X + Y (2.1.5) i, The circulatory lit (or really the ormal orce) is take as ( ) (2.1.6) N, c, = N E 0 where E = i Impulsive load The impulsive lit is calculated rom pisto theory: 4 N, I = φ I ( s, M ) (2.1.7) M The traser uctio is approximated by I t K T = e I φ ( t) (2.1.8) where c T I = where c is the chord ad a is the speed o soud. a K is a uctio o the mach umber as give i [5]. Equatio (2.1.7) ad (2.1.8) are solved to give the impulsive lit at time step as N 4 K c, I, = D (2.1.9) V t rel where the deiciecy uctio is give by 14
15 D t t K T I 1 2KT I = D 1 e + e (2.1.10) t Total attached low lit The total attached low lit (potetial low lit) is the obtaied by addig the circulatory ad impulsive lit. N, pot N, c + N, I = (2.1.11) 2.2 Separated low The simulated separated low eects are: 1) a dyamic delay i the movemet o the boudary layer separatio positio ad 2) the dyamic leadig edge stall with a vortex travellig dowstream alog the chord. The irst part is based o the physical cocept that N is a uctio o the separatio poit positio ad that the movemet o the separatio poit has a dyamic delay. The secod part results i vortex lit that is added to the ormal orce. The vortex lit has a dramatic eect o the pitchig momet. Much attetio has thereore bee give to estimate the coditios at which the vortex lit is shed over the airoil ad the speed at which it travels. The coditio at which the dyamic stall vortex is shed, is i the model cotrolled by the coditio that the ormal orce coeiciet exceeds a certai value. The coditio at whe leadig edge stall occurs, is really cotrolled by a critical leadig edge pressure coeiciet. To lik this critical leadig edge pressure coeiciet to, the model works with a lagged. (See N e.g. [5] or details ad [10] or commets o the dierece betwee pluge ad pitchig motio). N 15
16 Delay i the movemet o the separatio poit. To model the eect o the delay i the movemet o the separatio poit positio requires a N ( ) relatioship. Oe such relatioship is the Kircho low model as described i [5]. The relatioship betwee ad N i the Kircho low model is: 2 ( 1+ ( )) ( ) ( = N ) N (2.2.1) Where 0 is the agle o attack or zero lit ad N is the attached low lit curve slope. is the chordwise positio o separatio. is made dimesioless with the chord legth ad = 1 represets ully attached low ad = 0 represets ully separated low. The static ( ) ca i priciple be obtaied as a table by solvig equatio (2.2.1) with the steady N ( ) curve. However, i the BeddoesLeishma model as described i [5], a parametric ( )  curve that is a best it to the steady ( ) curve is suggested. N I order to obtai the dyamic value o the positio o separatio poit, the model is that the dyamic lags behid its steady value accordig to a irst order lag equatio: d ds T stat = (2.2.2) s is here the dimesioless time rom equatio (2.1). T is a semiempirical time costat (time costat i the sspace). By usig the dimesioless time, s, the model or the lag i the separatio poit works or ay requecy o oscillatio or ay time history with T as a costat. The dyamic lit, N,, is obtaied rom the Kircho low equatio with the dyamic value o the separatio poit positio. 16
17 Vortex lit The model urther icludes a model or vortex lit, with the amout o vortex lit determied by a timelag uctio with a time costat, T. It is assumed that the vortex lit cotributio ca be viewed as a v excess circulatio, which is ot shed ito the wake util some critical coditio is reached. The vortex lit,, is determied by the ollowig equatios: N v The eed o vortex lit is proportioal to the dierece i usteady circulatory lit ad the oliear lit give rom the Kircho low equatio with the dyamic value o the separatio poit positio. c v N, c N, = (2.2.3) At the same time, the total accumulated vortex lit is allowed to decay expoetially with time, but may also be updated with ew vortex lit eed. d N, v = (2.2.4) T dcv N, v ds ds v The last equatio is solved at the :th time step as: s s Tv 2T v N, v, N, v, 1e + ( cv, cv, 1 ) e = (2.2.5) oditios or whe the additio o ew vortex lit eed should stop ad coditios or how T v should vary with ad the positio o the travellig vortex, are give i [5]. 2.3 The model i steps The model works as a ope loop model so that the whole model is programmed i steps i which the output rom earlier submodels are iput to the ext submodel. 17
18 Step 1 alculate attached low eects (impulsive loads ad shed wake eects) => eective agle o attack, ad impulsive orce E N, imp Step 2 ompute a shit i agle o attack due to the lag i leadig edge pressure respose. d ' N, pot = (2.3.1) T ' N, pot N, pot ds p Where T p is a empirical time costat. ' N, pot is the used to get substitute value o the eective agle o attack: = ' N, pot N 0 (2.3.2) Step 3 Use stat = stat ( ) ad compute the usteady separatio poit positio, dy, rom the lag equatio or the separatio poit. Use dy i the Kircho ( ) relatioship to obtai a dyamic value o the ormal orce coeiciet,,. N Step 4 ompute vortex lit rom equatios (2.2.3) ad (2.2.4) Step 5 Add compoets to get total ormal orce: N = N, I + N, + N, v 18
19 2.4 Semiempirical costats The mai semiempirical costats are the three time costats or the leadig edge pressure lag, T p, the lag i separatio poit movemet, T, ad or the vortex lit decay, T v. Values or these costats are suggested i [5]. I the BeddoesLeishma model, semiempirical costats are also eeded to cotrol the coditios or the dyamic stall vortex sheddig ad to cotrol its travel speed. For a explaatio o these costats, N, 1 ad T vl, see [5]. I [5], several coditios are also suggested or which some o the time costats should be halved or doubled depedig o e.g. the value o the separatio poit positio, or the positio o the travelig vortex. 19
20 20
21 3 The DYNSTALL usteady airoil aerodyamics model The model i the DYNSTALLpackage is a implemetatio o the BeddoesLeishma model, but with a ew chages made relative to the descriptio above ad as give i [5], [6] ad [7]. The chages to the model are made to make it more robust or use i wid turbie aeroelastic simulatios with turbulet wid. The model ow works or agles o attack i the whole rage [180,180 ]. Some chages were also itroduced durig the Stallvib project [3] where the usteady aerodyamics or leadlag blade oscillatios where studied. I the ollowig text, whe there is metio o the curret versio o DYNSTALL it is reerred to the subrouties i the iles: dycl_c., v_esti_c. ad vo_sub_c. 3.1 A chage to work i the wid reerece system The BeddoesLeishma model as described i [5] works i the body ixed rame. Forces are described i the directio ormal to the airoil,, ad i the chordwise directio,. The attached low N part origiates rom liearised theory at small agles o attack where si( ) ta( ), so that o distictio ca really be made betwee N ad L i that theory. For the oliear part (separated low) the BeddoesLeishma model also works with N ad T. There might be advatages by settig up the separated low equatios i the body ixed system but diiculties arise whe the dyamic behavior o T at high agles o attack should be modeled with the model rom [5]. T The curret versio o DYNSTALL thereore uses the wid reerece system. Most equatios i i the BeddoesLeishma N 21
22 model are treated as beig equatios i L. Oe exceptio is that the vortex lit still is treated as beig ormal to the airoil chord. Oe advatage o workig i the wid reerece system is that the delay i the separatio poit positio automatically aects the tagetial orce. This happes sice T has a rather large compoet o L at stall agles o attack. Most o the eect o the dyamic delay o the separatio poit o T is thereore automatically a result o the delay i the lit. The eect o the dyamic delay o the separatio poit o D is also modeled i DYNSTALL. Still the mai eect o comes rom the lit part at stall agles o attack. T The vortex lit is, however, assumed to act oly i the ormal orce directio. A icrease o T will icrease the dyamic stall loop width o the N ( ) curve as well as the T ( ) curve, whereas a icrease o T V will icrease oly the width o the N ( )  curve ad leave the T ( )  curve uaected. The act that T V oly aects N makes it possible to tue the time costats, T ad T V, to obtai a good it to both the ( ) as well as the ( ) curve. 3.2 Attached low T The attached low equatios are basically the same as i [5] ad give i chapter 2 above. N I DYNSTALL, two methods are however available or how to calculate the shed wake eect. I the alterative method explaied i sectio the shed wake eect is made a uctio o the actual circulatio history ad ot the circulatio history that would occur or attached low as i the origial model The agle o attack The agle o attack that is iput to the equatios icludes the pitch rate eects. 22
23 A geometrical agle o attack, g, is deied as the agle betwee the chordlie ad the relative velocity to the airoil. The relative velocity to the airoil icludes here the traslatio motio o the airoil. No dierece is hece made betwee the airoil movig i still air or the airoil beig at a stadstill at a icidece i a airlow. u y V x q u x V y V rel Figure 1 I the picture above, the airoil is at a geometrical icidece ad also has a heavig velocity, u y, ad a lead lag velocity, u x. g is the calculated rom the agle o the relative velocity to the chord lie as: Vy u y ta( g ) = ( ) V u x x I order to automatically iclude the pitch rate eect i the lit, the agle o attack at the ¾ chord positio ca be used. The DYNSTALL package thereore also works with the agle o attack at the ¾ chord positio. This agle o attack is called 75. With the velocity relative to the airoil, V rel, ad the pitch rate q, the c q 75 = g + ( ) 2 V rel 23
24 75 is the used or i equatios or the circulatory lit i sectio Attached low equatios (LPOTMETH=3) The attached low equatios are available i two versios. The choice o versio is cotrolled by a iput parameter LPOTMETH. (LPOTMETH=1 used to be the attached low equatios i sectio 2.1. but is o loger implemeted sice it is equivalet to LPOTMETH=3 or costat V rel ) irculatory part The equatios are reormulated to work with the ormal velocity at the ¾ poit chord positio. This is the same as reormulatig the equatios i circulatio rather tha i lit coeiciet. With this ormulatio, the circulatory eect will be correct or leadlag motios ad or a varyig wid speed. Equatio (2.1.1) the becomes c Γ = L φc w ( ) 2 w is the ormal compoet o the relative velocity to the airoil at the ¾ poit chord positio. I this case the ormal directio is take as ormal to the zerolit lie ad the small agle approximatio is used. ( 75 0 V rel ) w = ( ) The equatios to solve or the eective agle o attack are the at the :th time step X = X 1 exp( b1 β s) + A1 w exp( b1 β s / 2) ( ) Y = Y 1 exp( b2 β s) + A2 w exp( b2 β s / 2) ( ) 24
25 ( X Y V + ( ) E = 75 ) / rel 0 This method is obtaied by choosig LPOTMETH=3. 2 β = 1 M is i the curret versio set to uity sice Mach umber eects or wid turbies or the shed wake eect are assumed small. I order to avoid queer results whe the liearised theory is used at high agles o attack, w i equatios ( ) ad ( ) are actored with a actor ade. Where ade = cos ( 75 ) ( ) 2 Impulsive load The impulsive load should be calculated rom the airoil pluge velocity at the ¾ poit chord positio. This pluge velocity could come rom airoil plugig but could also be due to a pitch rate iduced velocity at the ¾ poit chord positio. With, u p as the velocity o the ¾ poit chord positio directed i the egative ydirectio, the equatios or impulsive loads at the :th time step are writte: L, I, 4 K c V U u p t, = D rel, mea ( ) with the deiciecy uctio give by D t t K T u p, u p, 1 I 2KT I = D 1 e + e t ( ) ad U mea is the mea o V rel durig the curret ad the previous time step. 25
26 K is i the curret versio o DYNSTALL set to which is the value take rom [5] or M=0.15) I the curret versio the impulsive ormal orce rom equatio ( ) is take as the impulsive lit orce eve though this is strictly correct oly or = 0. I order to avoid queer results whe the liearised theory is used at high agles o attack, u i equatios ( ) ad ( ) are actored with the actor ade rom equatio ( ). p Pitchig momet The attached low eects o the pitchig momet are icluded by addig the impulsive load compoet to the pitchig momet L, I M, I = ( ) Shed wake eects with accout or a varyig lit curve slope (LPOTMETH=4) The above equatios (used with LPOTMETH=3) are valid or attached low. The shed wake eect is the calculated as i the low were attached with a lit curve slope, which is the lit curve slope aroud the zero lit agle o attack. L At agles o attack where the airlow starts to separate, the steady lit curve slope is o loger the same as or attached agles o attack. The lit may eve decrease or a icrease i agle o attack above stall. It is the questioable i the shed wake eect (circulatory lit) could be calculated with a costat. L O way to overcome this would be to calculate the shed wake eects rom the circulatio history ( or lit history or costat V rel ) istead. 26
27 Oe problem is, however, that the lit is ot kow util the dyamic eects o the separatio poit positios is determied. This is solved by irst calculatig a estimate o L,. This estimate, L,, est, is calculated as i step 2 ad 3 o the BeddoesLeishma model but as explaied i the sectios o separated low below. The shed wake eect is the made a uctio o the dierece i,, istead o the dierece i 75 : L est ( ) L,, est, = L,, est, L,, est, 1 Further, L, est, must be based o the agle o attack without the shed wake eects i the model should give correct results or attached low with the equatios correspodig to ( )( ). is thereore set to 75 whe the estimate o the dyamic lit, E L, est,, is calculated. The method used to accout or the act that a reduced lit curve slope due to separated low give reduced shed wake eects is: exchage i equatios (2.1.1)(2.1.4) with L,, est L. To, at the same time accout or a varyig circulatio due to a varyig velocity, w is used istead o with w as i equatio ( ) w to be put i equatios ( )( ) is the calculated, by dieretiatio, as w = L,, est, mea L V rel + L,, est L U mea ( ) L, est, mea, with, beig the mea o L, est at the curret ad the previous time step. As or LPOTMETH=3 equatio ( ) w is multiplied with the actor ade rom 27
28 The so calculated eective agle o attack will be exactly the same as calculated with LPOTMETH=3 i the low is attached. Attached low meas here that =1 ad ( ) = ( ). L, E L, stat E For separated low it will approximately accout or the act that the shed wake history is a uctio o the lit history rather tha o the agle o attack history. The lit that is used to estimate the chage o lit excludes the vortex lit sice the vortex lit is cosidered ot to aect the shed wake history. The extra computig or LPOTMETH=4 relative to LPOTMETH=3, is that step 2 ad 3 has to be ru a extra time every time step to obtai L,, est. 3.3 Separated low A shit i the agle o attack The separatio poit positio is statically a uctio o the agle o attack. What really ilueces the separatio positio is the pressure gradiets i the boudary layer. To use the shited agle o attack,, as calculated rom equatio (2.3.2) is a way to accout or the usteady pressure gradiet. I DYNSTALL, the shit i the agle o attack to compesate or the lag i the leadig edge pressure is used just as i the origial model. What is ew to the FFA versio, is to accout or chages i the usteady pressure gradiet due to a varyig ree stream velocity. This was itroduced durig the STALLVIB project where leadlag airoil oscillatios were studied [3], [11]. By studyig the usteady Beroulli equatio it ca be see that the pressure gradiet alog the airoil will be dieret i the relative velocity is costat, icreasig or decreasig (see [3] or [11]). I Vrel 28
29 is icreasig, the the eective usteady pressure gradiet is oud to be more avorable which i priciple should result i less separatio. The questio is how this eect o the pressure gradiet ad the correspodig eect o the separatio poit should be icorporated i the dyamic stall model. Oe way to do this is to see the resultig pressure gradiets as gradiets occurrig at a dieret agle o attack. (A lower agle o attack or a acceleratig ree stream.) The poit o separatio,, could the be determied usig a corrected agle o attack as step 2b i the dyamic stall model. The assumptio is that, at some agle o attack, = +, the pressure gradiet or the case with varyig V rel, is similar to the pressure gradiet at a agle o attack, or the case with costat V rel. The equatios to derive a estimate o the shit are give i [11] ad [3]. The result is that the shit should be proportioal to the dimesioless velocity chage rate c Vrel γ = ( ) 2 2 V t rel The amout o shit is urther cotrolled by a empirical parameter,. u A estimate o = 0. 5 is give i [11]. u Step 2, as i the descriptio o the BeddoesLeisha model i sectio 2.3, is i the FFA model: Equatio (2.3.1) is solved at the :th time step as dp = dp 1 e s Tp + s 2 T p ( ) e N, pot, N, pot, 1 ( ) ad 29
30 , ade dp ( ) ' N pot, = N, pot, The actor ade rom equatio ( ) is icluded sice the cocept ayway breaks dow at high agles o attack. The shited agle o attack is take rom equatio (2.3.2) I the FFAmodel the agle o attack is also shited due to a varyig velocity. is thereore take as ' N, pot = 0 γ N u ( ) Lit as uctio o the separatio poit positio The steady l ( ) curve is i the dyamic stall model exchaged with a static ( ) relatioship as explaied i sectio 2.2. The dyamic value (usteady value) o the lit coeiciet is the obtaied by puttig i a dyamic value o i the ( ) relatioship. Two dieret types o ( ) ca be used i DYNSTALL. The choice betwee models is cotrolled by a variable lmeth. Oe model is the relatioship itroduced by Øye [12]. The other is a versio o the Kircho low model. I either case, the dyamic value o is calculated by solvig equatio (2.2.2) The Øye separatio poit model (lmeth=2) The ormula likig L, ad origiatig rom Øye is used. L L, iv + ( 1 ) L, sep = ( ) 30
31 This relatioship requires steady curves o L iv ( ) ad L sep ( ) i order to obtai either a steady L, stat ( ) curve rom a steady ( ) or a dyamic L rom a dyamic.,, Equatio ( ) ca be used to id a static ( ) relatioship. stat L, stat ( ) L, sep ( ) ( ) = ( ) ( ) ( ) L, iv L, sep ( ), ( ) ad ( ) are iput to the program as L, iv L, sep stat tabulated values as uctio o. should perhaps ot be see strictly as the separatio poit positio, but rather as a iterpolatio actor betwee ( ) ad ( ). L, iv L, sep L, iv could be take as L ( 0 ) ad L, sep as the curve or ully separated low. The choice or the latter curve is more arbitrary. Øye [12] suggests a curve a curve startig with a slope o hal value o the useparated curve ad gradually ittig to the steady L ( ) curve at approximately 30 agle o attack. ' To solve the dieretial equatio (2.2.2), a value o is obtaied rom the stat ( ) table with the agle o attack as iput. (2.2.2) is the solved at the :th time step as d = d 1 e s T + s ' ' 2 T ( ) e 1 ( ), d ( dy = ', is obtaied rom equatio ( ) as L ) = ( ) + (1 ) ( ) ( ) L, ( g dy L, iv E dy L, sep E 31
32 A versio o the Kircho low (lmeth=4) The Kircho low equatio writte i equatio (2.2.1) is used, but with some modiicatios. First, is used rather tha. L N With =1 the umodiied Kircho low equatio with L will give rather high values o L at large agles o attack. The DYNSTALL model should work over the whole rage o agles o attack, ad modiicatios are itroduced to limit queer results at high agles o attack. The Kircho low equatio is thereore modiied to limit L or =1. Aother ix is itroduced to overcome the act that i equatio (2.2.1) (with N replaced by L, stat ) is used to solve or the static ( ), the this will oly work as log as stat 1 L, stat < L ( 0 ). 4 With ( ) L, iv = L 0 ad L, sep = L, iv 4, the Kircho equatio ca be writte to use the expressios L, iv ( ) ad ( ) as i the Øye model. L, sep 1 = L, sep + L, iv ( 2 ) ( ) 4 L + To limit I c L or =1, equatio ( ) is modiied or > c. >, the the liear ( ) is exchaged or a L, iv = L 0 sie curve that beds dow almost to zero at 90 : 1 = d ( ) L, iv si( d1 ( 0 )) + d1 2 d 1 is a costat, somewhat arbitrarily, chose to be 1.8. The costat d 2 assures cotiuity o the two L, iv expressios at = c with 32
INVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationGENERATING A FRACTAL SQUARE
GENERATING A FRACTAL SQUARE I 194 the Swedish mathematicia Helge vo Koch(187194 itroduced oe o the earliest ow ractals, amely, the Koch Sowlae. It is a closed cotiuous curve with discotiuities i its derivative
More informationWindWise Education. 2 nd. T ransforming the Energy of Wind into Powerful Minds. editi. A Curriculum for Grades 6 12
WidWise Educatio T rasformig the Eergy of Wid ito Powerful Mids A Curriculum for Grades 6 12 Notice Except for educatioal use by a idividual teacher i a classroom settig this work may ot be reproduced
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationSection 7: Free electron model
Physics 97 Sectio 7: ree electro model A free electro model is the simplest way to represet the electroic structure of metals. Although the free electro model is a great oversimplificatio of the reality,
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationForecasting techniques
2 Forecastig techiques this chapter covers... I this chapter we will examie some useful forecastig techiques that ca be applied whe budgetig. We start by lookig at the way that samplig ca be used to collect
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationMeasuring Magneto Energy Output and Inductance Revision 1
Measurig Mageto Eergy Output ad Iductace evisio Itroductio A mageto is fudametally a iductor that is mechaically charged with a iitial curret value. That iitial curret is produced by movemet of the rotor
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationSolving Inequalities
Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK12
More informationCantilever Beam Experiment
Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Catilever Beam Experimet Backgroud A disk drive maufacturer is redesigig several disk drive armature mechaisms. This is the result of evaluatio
More informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module The Sciece of Surface ad Groud Water Versio CE IIT, Kharagpur Lesso 8 Flow Dyamics i Ope Chaels ad Rivers Versio CE IIT, Kharagpur Istructioal Objectives O completio of this lesso, the studet shall
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationHOSPITAL NURSE STAFFING SURVEY
2012 Ceter for Nursig Workforce St udies HOSPITAL NURSE STAFFING SURVEY Vacacy ad Turover Itroductio The Hospital Nurse Staffig Survey (HNSS) assesses the size ad effects of the ursig shortage i hospitals,
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationChapter XIV: Fundamentals of Probability and Statistics *
Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationIran. J. Chem. Chem. Eng. Vol. 26, No.1, 2007. Sensitivity Analysis of Water Flooding Optimization by Dynamic Optimization
Ira. J. Chem. Chem. Eg. Vol. 6, No., 007 Sesitivity Aalysis of Water Floodig Optimizatio by Dyamic Optimizatio Gharesheiklou, Ali Asghar* + ; MousaviDehghai, Sayed Ali Research Istitute of Petroleum Idustry
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
More informationThroughput of Ideally Routed Wireless Ad Hoc Networks
Throughput of Ideally Routed Wireless Ad Hoc Networks Gábor Németh, Zoltá Richárd Turáyi, 2 ad Adrás Valkó 2 Commuicatio Networks Laboratory 2 Traffic Lab, Ericsso Research, Hugary I. INTRODUCTION At the
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationTime Value of Money, NPV and IRR equation solving with the TI86
Time Value of Moey NPV ad IRR Equatio Solvig with the TI86 (may work with TI85) (similar process works with TI83, TI83 Plus ad may work with TI82) Time Value of Moey, NPV ad IRR equatio solvig with
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationStatement of cash flows
6 Statemet of cash flows this chapter covers... I this chapter we study the statemet of cash flows, which liks profit from the statemet of profit or loss ad other comprehesive icome with chages i assets
More informationHCL Dynamic Spiking Protocol
ELI LILLY AND COMPANY TIPPECANOE LABORATORIES LAFAYETTE, IN Revisio 2.0 TABLE OF CONTENTS REVISION HISTORY... 2. REVISION.0... 2.2 REVISION 2.0... 2 2 OVERVIEW... 3 3 DEFINITIONS... 5 4 EQUIPMENT... 7
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationPENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.
PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.
More informationTopics in Probability Theory and Stochastic Processes Steven R. Dunbar. The Weak Law of Large Numbers
Steve R. Dubar Departmet o Mathematics 203 Avery Hall Uiversity o NebrasaLicol Licol, NE 685880130 http://www.math.ul.edu Voice: 4024723731 Fax: 4024728466 Topics i Probability Theory ad Stochastic
More informationAccountability of teachers and schools: A valueadded approach
Iteratioal Educatio Joural, 6, 7(), 174188. ISSN 1441475 6 Shao esearch Press. http://iej.cjb.et 174 Accoutability o teachers ad schools: A valueadded approach I Gusti Ngurah Darmawa School o Educatio,
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationSum of Exterior Angles of Polygons TEACHER NOTES
Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).
More informationBond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond
What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixedicome security that typically pays periodic coupo paymets, ad a pricipal
More informationA Resource for Freestanding Mathematics Qualifications Working with %
Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationCONTROL CHART BASED ON A MULTIPLICATIVEBINOMIAL DISTRIBUTION
www.arpapress.com/volumes/vol8issue2/ijrras_8_2_04.pdf CONTROL CHART BASED ON A MULTIPLICATIVEBINOMIAL DISTRIBUTION Elsayed A. E. Habib Departmet of Statistics ad Mathematics, Faculty of Commerce, Beha
More informationUM USER SATISFACTION SURVEY 2011. Final Report. September 2, 2011. Prepared by. ers eresearch & Solutions (Macau)
UM USER SATISFACTION SURVEY 2011 Fial Report September 2, 2011 Prepared by ers eresearch & Solutios (Macau) 1 UM User Satisfactio Survey 2011 A Collaboratio Work by Project Cosultat Dr. Agus Cheog ers
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More information