Application of an Analytical Method to Locate a Mixing Plane in a Supersonic Compressor

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Application of an Analytical Method to Locate a Mixing Plane in a Supersonic Compressor"

Transcription

1 Journal of Energy and ower Engineering 9 (015) doi: / / D DAVID UBLISHING Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor Emmanuel Benichou 1 and Isabelle Trébinjac 1. Turbomeca, Groupe SAFRAN, Bordes 64511, France. Laboraoire de Mécanique des Fluides e d Acousique, UMR NRS 5509, Ecole enrale de Lyon, Ecully edex 69134, France Received: Sepember 1, 014 / Acceped: November 03, 014 / ublished: January 31, 015. Absrac: In order o achieve greaer pressure raios, compressor designers have he opporuniy o use ransonic configuraions. In he supersonic par of he incoming flow, shock waves appear in he fron par of he blades and propagae in he upsream direcion. In case of muliple blade rows, seady simulaions have o impose an azimuhal averaging (mixing plane) which prevens hese shock waves o exend upsream. In he presen paper, several mixing plane locaions are numerically esed and compared in a supersonic configuraion. An analyical mehod is used o describe he shock paern. I enables o ake a criical look a he FD (compuaional fluid dynamics) seady resuls. Based on his mehod, he shock losses are also evaluaed. The good agreemen beween analyical and numerical values shows ha his mehod can be useful o wisely forecas he mixing plane locaion and o evaluae he shif in performances due o he presence of he mixing plane. Key words: Supersonic compressor, shock wave, pressure loss, RANS, mixing plane. Nomenclaure Symbol V V n V, V r a 1 Saic pressure Sagnaion pressure in he impeller frame ircumferenially averaged sagnaion pressure ircumferenial pich Densiy Velociy in he impeller frame Normal velociy componen Tangenial velociy componens Radius Speed of sound erfec gas consan Roaion speed osiion of he bow shock on he profile x 0 symmery axis x, y oordinaes of a poin in he profile frame M Mach number orresponding auhor: Emmanuel Benichou, h.d. suden, research fields: aerodynamic insabiliies in cenrifugal compressors, including roor-saor ineracions and flow conrol issues using boundary layer aspiraion. µ Mach angle Angle of he flow Angle of he bow shock e B Axial disance beween poins x 0 and B d Deachmen disance of he shock wave ichwise disance on which he flow is considered isenropic θ ircumferenial direcion K Toal pressure loss coefficien Subscrip B, Relaive o poins B and 1, alculaed in Secion 1 or Secion Value of he quaniy a infinie upsream Exponen Value of he quaniy a M = 1 1. Inroducion The need for compac, efficien high pressure raio compressors ofen resuls in high roaion speeds. In some cases, he enry flow may herefore be supersonic over he enire- or upper-span. The resuling physics of he flow field in he enry zone can be complex because

2 9 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor of he ineracion beween compression and expansion waves [1-3]. Besides, shock waves propagaing upsream he blades are dissipaive and mus necessarily be aken ino accoun in he predicion of he sage performances [4]. urren FD (compuaional fluid dynamics) offers hree main caegories for simulaions: RANS (Reynolds-averaged Navier-Sokes), LES (large eddy simulaion) and DNS (direc numerical simulaion). RANS simulaions approximae he mean effec of urbulence, while DNS enables a full resoluion of he Navier-Sokes equaions, from he smalles urbulence scale (Kolmogorov scale) up o he inegral scale. LES corresponds o a filered DNS: only he larges urbulence scales are resolved, he smalles ones being modeled. The more urbulence scales are resolved, he finer he mesh mus be, and hus he more expensive he simulaion becomes. In a curren engine design process, only RANS simulaions can be carried ou. This sor of simulaion is based on he Reynolds decomposiion and urbulence models are added o close he se of equaions. U-RANS (unseady RANS) simulaions are generally no affordable in a concepion approach, because of heir high U cos. Tha is why he only ool ypically available for designers oday consiss in seady RANS simulaions. In he case of muli-row urbomachinery, hese simulaions rely mos of he ime on he use of mixing planes, which average he daa in he circumferenial direcion, and hus do no le he non-uniformiies in he flow field ransmi in he upsream or downsream direcion. This aricle focuses on he enry zone of a supersonic compressor. The filraion of he shock paern upsream he blades by he mixing plane raises he issue of he mixing plane locaion. The presen paper compares numerically differen locaions o poin ou he influence of he mixing plane on he flow field, noably in erms of sagnaion pressure change. In a firs par, he numerical resuls are qualiaively analyzed and he role of he mixing plane is highlighed. An analyical model is hen used o reproduce he shape of he shock waves a he leading edge of he blades. Finally, a mehod is given ha enables o judge he reliabiliy of seady simulaions on supersonic configuraions.. Tes ase and Numerical rocedure The es case is a cenrifugal unshrouded impeller designed and buil by Turbomeca. In he presen sudy, only he fron par of i is concerned. There is herefore no need giving he compressor geomery and performance in his paper. Only one operaing poin is examined, and i corresponds o he sonic blockage region. The flow, supersonic over 60% of he span, is examined a several secion heighs beween h 1 and h in Fig. 1. A simulaion performed wihou any mixing plane is used as reference (Fig. a). In his one, he inle block is roaing wih he impeller, and here is no paricular inerface. In order o evaluae he change in performance induced by he mixing plane approach, hree differen mixing plane locaions are numerically esed (Fig. b). They are labeled a, b and c in Fig. 1. In hose hree simulaions, he inle block is fixed, like a saor row and he periodiciy enables o use a smaller domain since he flow upsream of he mixing plane is circumferenially uniform. ompuaions were performed wih he elsa sofware developed a ONERA [5]. The code is based on a cell-cenered finie volume mehod and solves he h 1 h Air inle Fig. 1 Meridional skech of he compressor inle par.

3 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 93 Inle Oule Splier blade Main blade eriodic boundaries (a) for convecive fluxes and a nd-order cenered scheme for viscous fluxes. An LU implici phase (lower upper decomposiion) is associaed o he backward-euler scheme for ime inegraion. The near-wall region is described wih y + < 13. The inle condiion imposes he velociy angles and he sandard sagnaion pressure and emperaure. The urbulen values are deermined from a free-sream urbulence rae of 5% (resuling from previous measuremens). The oule condiion imposes a uniform value of saic pressure. The walls are described wih non-slip and adiabaic condiions. The seady sae enables o simulae only one blade passage, he azimuhal boundaries being periodic. Blade-o-blade surfaces are hen exraced from all simulaions a he same secion heighs beween h 1 and h. A he mixing plane inerface, a circumferenial average using Riemann invarians is compued a boh upsream and downsream faces. The resuling values (m 1 -m 5 ) are hen applied o he adjacen face wih a non-reflecive boundary condiion: Mixing plane = 0 wih 1 m 1 ( av )d S n S 1 m ( avn )d S S 1 m 3 ( a )d S S 1 m V ds 4 1 S 1 m V ds 5 S S ds he oal surface of he inerface. (b) Fig. (a) 3D view of he domain wihou mixing plane; and (b) 3D view of he domain wih mixing plane. compressible RANS equaions on muli-block srucured meshes. The k-l model of Smih [6] (chosen according previous work [7]) is used for urbulence modeling. The se of equaions are resolved in he relaive frame of each row, using he Roe space scheme 3. Numerical Resuls Fig. 3 shows he relaive Mach number in he reference case (wihou any mixing plane), in a blade-o-blade surface. The whie lines indicae he hree mixing plane locaions and he purple line shows he iso-conour M = 1. Wih he mixing plane locaed in (c), he subsonic zone beween he leading edge and he shock wave is clearly cu.

4 94 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor Secion 1 (a) (b) (c) Secion Wihou mixing plane ase (a) Fig. 3 Relaive Mach number wihou mixing plane. Fig. 4 shows ha in he four configuraions, he deachmen disance of he shock remains consan, which ends o prove ha he mixing plane gives a correc value of he averaged Mach number. Indeed, as explained in he following, he deachmen disance only depends on he blade geomery and inle Mach number. The whie lines represen Mach iso-conours from 0.7 o 1.4. However, he shape of he subsonic zone is seriously affeced. The more he mixing plane is locaed downsream, he less he shock waves can exend upsream. Thereby, according o he posiion of he mixing plane, he oal pressure loss is under-esimaed. The change in sagnaion pressure can be quanified wih he value of he loss coefficien K calculaed as: K 1 wih, he momenum-averaged relaive sagnaion pressure inegraed on he whole surface a Secion (locaed a he blade leading edge) and on he whole surface a Secion 1 (locaed upsream) as: S 1 V ds S n n V ds ase (b) ase (c) Fig. 4 Relaive Mach number in he four es cases. As is expeced, he more he mixing plane is locaed downsream, he lower he losses are, and consequenly, he more he massflow is over-esimaed. Table 1 gives he difference beween overall inle massflow in cases

5 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 95 Table 1 erformance shif o he reference configuraion. Massflow K/Kref Mixing plane locaed in (a) +0.04% -1.% Mixing plane locaed in (b) +0.35% -8.3% Mixing plane locaed in (c) +0.80% -1.% (a), (b), (c) compared o he reference configuraion wihou mixing plane. The effec of he mixing plane can also be shown by he evoluion of he enropy along a consan span heigh in he supersonic region (black arrow in he meridional view in Fig. 5). By prevening he shock wave o exend upsream, he mixing plane inroduces a shif in he global level of enropy. These daa show ha he resul of he seady sae simulaion significanly depends on he mixing plane locaion boh in erms of performance (massflow and loss) and flow opology. The objecive of he following par is o propose a mehod which can be used o: (1) forecas he locaion of he mixing plane minimizing he shif in performance; () forecas he change in performance for a given mixing plane locaion. 4. Analyical Descripion Le us consider a supersonic incoming flow compressor wih subsonic axial velociy componen. Depending on he inle Mach number and on he back pressure level, wo differen regimes can exis: he unsared regime, characerized by a deached, quasi-normal shock across he passage; he sared regime, characerized by an aached oblique shock. In case of a blun leading edge, he shock canno be sricly aached and a small subsonic area exiss upsream he blade. The deachmen disance of he shock is obviously smaller in he case of sared regime han ha of unsared regime. The model presened hereafer is only valid for a sared regime. This is he firs reason why he sudy akes place near he sonic blockage: he shock wave has o remain aached o he leading edge of he blades. Fig. 5 Evoluion of enropy along he roaion axis z, a a consan span heigh. Moreover, he presence of splier blades here is likely o influence he shock sysem a he leading edge of he main blades hrough poenial effecs. Thus, he back pressure has o be imposed very low. The wo inpus of his model are he upsream Mach number and he geomery of he blade leading edge. The deachmen disance is calculaed wih Moeckel s mehod [8], which assumes ha he deached shock has a hyperbolic shape (Fig. 6). The equaion of he hyperbola is specified by is asympoe (of angle ) and he posiion of poin locaed on he sonic line [B], which is supposed o be sraigh. y M > 1 O Wihou mixing plane µ x 0 (a) M < 1 e B Fig. 6 Skech of a deached shock. d A (b) B (c) B LE x

6 96 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor In order o apply Moeckel s model, he upsream flow is supposed o be wo-dimensional, uniform and he profile is approximaed by a symmeric shape. The hermodynamic shock relaions for ideal gas are used. The maximum enropy rise, corresponding o a normal shock, is locaed on he sreamline passing by he leading edge. This line and he sonic line are assumed o be sraigh. The available equaions are: The equaion of he hyperbola: y ( x x ) g c c The equaion of he angen a poin : 0 (1) x g g () y which, combined wih Eq. (1), gives: x 0 y g g g (3) The equaion of he disance e B : g eb y ( y y ) g B g y g g g (4) In order o calculae he ordinae of poin, y, he coninuiy equaion is wrien beween he segmen [O ] upsream he shock wave and he sonic line [B]: y yb Vy ab a (5) cos which leads o: y y B V 1 a 1 a a cos Since a shock wave is isenhalpic, we can wrie: V a M 1 1 (1 M ) 1 (1 ) (6) (7) a a (8) I is imporan o noice ha a choice is possible a ha Eq. (8) in he way he sagnaion pressure raio is calculaed. In he presen case, a normal shock is considered a poin : a 1 1 M a 1 1 M (9) I would also be hinkable o consider an oblique shock o evaluae he hermodynamic sae along he sonic line. This is a he same ime a source of uncerainy in his mehod and a degree of freedom for he user (playing on his raio enables o fi FD or measuremens bu i is raher difficul o give a formula which works for all ypes of blades). The seps of he deachmen disance calculaion are hus: From he value of he inle Mach number M, is calculaed wih: 1 arcsin (10) M and he values of he deviaion and shock angle are deduced from he shock relaions. oin B which belongs o he profile is idenified from is angen B which has o be equal o. Eqs. (3), (4) and (6) are hen used o calculae he deachmen disance d: B B A d e x x (11) This procedure, iniially hough for symmeric isolaed profiles, leads o a geomeric represenaion of he subsonic zone (in grey in Fig. 6), beween he hyperbola and he sonic line. The par of he shock wave confining his subsonic region is responsible for mos of he oal pressure losses. Therefore, i is crucial o le i fully exend upsream he blades. The consequences of a mixing plane cuing he subsonic pocke are examined in he following.

7 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor omparison beween Analyical and Numerical Resuls The analyical mehod previously described for sared regimes enables o draw he shape of he deached bow shock and he subsonic zone. Fig. 7 superimposes he analyical shape (in black) on he FD resuls wihou any mixing plane (he yellow line is he iso-conour M = 1). Qualiaively, he subsonic area is well approximaed. Assuming ha he major par of he losses are due o he par of he shock confining he subsonic par, i is seen ha he analyical calculaion gives a prey good evaluaion of he minimal disance o be pu beween he mixing plane and he leading edge (d min in Fig. 7). However, wo weaknesses should be poined ou. Firsly, he model is very sensiive o he upsream Mach number, in paricular in he low supersonic Mach numbers region (beween M = 1.0 and M = 1.1). Fig. 8 shows he ypical evoluion of he deachmen disance as a funcion of he upsream Mach number. Furhermore, when he upsream Mach number decreases, he shape of he subsonic pocke ends o become more ellipic. Fig. 9 shows ha wih M = 1.1, he deachmen disance and he orienaion of he shock wave are sill well esimaed bu Moeckel s mehod canno reproduce he numerical resul near he leading edge. Secondly, he choice in he calculaion of he loss across he shock leads o he hermodynamic sae of he sonic line, because in he end, i direcly drives he value of he deachmen disance. Since Eq. (9) is non linear, he higher he upsream Mach number, he more imporan he way he loss is calculaed becomes. The main benefis of Moeckel s model are ha only he geomery of he profile and he upsream Mach number are needed. As a consequence, his is a ool which can easily be employed in a pre-design phase. I should be underlined ha he sonic blockage is he mos favorable case because when he mass flow decreases, he compressor will pass from sared d min Fig. 7 Applicaion of Moeckel s mehod wih M 1.3. Fig. 8 Evoluion of he deachmen disance for a given profile. Fig. 9 Applicaion of Moeckel s mehod wih M 1.1.

8 98 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor regime o unsared regime. Thus, he shock paern moves upsream, so ha he impac of he mixing plane can only become sronger. 6. Shock Loss redicion Alhough curren blade design relies on numerical opimizaion, including ransonic bladings [9], many analyical shock loss models exis, as described in Refs. [10, 11], for example. Bloch, openhaver and O Brien also give an ineresing approach based on Moeckel s mehod [1], adaped o unsared regime. Since he oher objecive of his paper is o analyically forecas he change in pressure loss for a given mixing plane locaion compared wih he reference case (wihou any mixing plane), he shock loss associaed o a sared regime has o be evaluaed. Fig. 10 shows he relaive oal pressure calculaed in he reference case (wihou mixing plane): he low relaive sagnaion pressure zone a he leading edge plane corresponds o he projecion of he sronges par of he shock on he leading edge plane in he mean flow direcion. In order o be consisen wih he numerical fields, only he par of he shock wave which is downsream he mixing plane locaion is kep and his porion of he upper branch of he hyperbola is projeced in he plane of he leading edge (Fig. 11). The losses are hen calculaed in he plane of he leading edge as if he fluid going hrough he bow shock was seeing a normal shock and he res of he incoming flow remained unperurbed. The losses are evaluaed wih he coefficien K defined as: K 1 (1) 1 where, is he lengh of he projeced hyperbola, is he pich and / 1 refers o he oal pressure raio across a normal shock (see Eq. (9)). The value of K is of course overesimaed, since a par of he incoming fluid acually goes across an oblique shock and he bow shock exending upsream becomes rapidly evanescen. Bu i enables o evaluae he shock loss in he mos unfavorable siuaion. Noe Secion 1 Secion max min Fig. 10 Relaive sagnaion pressure in a blade-o-blade surface. Mixing plane Incoming flow Incoming flow LE Isenropic par : no pressure loss Normal shock loss Bow shock cu by he mixing plane Fig. 11 Loss calculaion for a given Mach number wih a given mixing plane locaion.

9 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 99 ha in case of no mixing plane, he model is equivalen o considering a normal shock over he whole pich. This choice keeps he presen ool simple, bu i would be possible o make a more sophisicaed model: Firs, by improving he shock wave approximaion. A good example is given by Oavy [13], by coupling Levine s mehod [14] o predic he unique incidence seen by he blade profile wih Moeckel s deachmen disance calculaion. By discreizing he shock wave, so ha he flow angle and he pressure loss change from a normal shock on he blade axis o an oblique shock and inegraing he resul on a pich o he neighboring blade. For a given profile, i is possible o plo he loss as a funcion of he upsream Mach number in he differen configuraions (Fig. 1). The case wihou mixing plane corresponds o he shock loss across a normal shock. Depending on is locaion, he mixing plane has no more impac beyond a cerain Mach number. For example, wih a mixing plane locaed in Secion (b) (green curve), i can be seen ha beyond M = 1.46, he mixing plane has no more influence. I means ha if he upsream Mach number is higher han his value, he seady resuls can be considered reliable. For a given mixing plane locaion, he discrepancy compared o he reference case firsly increases wih he Mach number. Indeed, he lengh coninuously increases bu he pressure loss is increasing faser due o he shock wave. Then he upper branch of he hyperbola is more and more sraighened up ogeher wih a decrease in he deachmen disance unil is projecion covers he whole pich. A ha sep, here is no more difference wih or wihou a mixing plane and a seady RANS simulaion can be considered as reliable (Fig. 13). This analyical ool has been applied o he hree mixing plane posiions and o he reference case wih no mixing plane. The inpu daa are he circumferenially-averaged Mach numbers in Secion (b) coming from he FD wih no mixing plane and he geomery of he fron par of he blades. (a) (b) Fig. 1 (a) Evoluion of K for a given profile; and (b) zoom from Fig. 1a. M = 1.0 M = 1.50 M = 1.80 Fig. 13 Effec of he mixing plane on he bow shock for differen upsream Mach numbers.

10 100 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor Fig. 14 Analyical and numerical loss as funcion of he upsream Mach number for differen mixing plane locaions. Fig. 14 compares he analyical and numerical values obained for he losses. Differen secion heighs beween h 1 and h have been esed, so ha boh he profile and he Mach number were changing. The values of K are ploed as a funcion of he upsream Mach number and are calculaed as: where, K 1 π 0 π 0 1 V rd n n V rd In Fig. 14, he analyical evoluion corresponding o case (a) is he same as he reference one for Mach numbers greaer han This means ha he major par of he hyperbola is conained downsream he mixing plane for he corresponding secion heighs. According o his crierion, any mixing plane should be locaed upsream Secion (a) (Fig. 3) for he presen impeller. Neverheless, despie he good qualiaive agreemen beween analyical and numerical resuls in Fig. 7, we can observe quaniaive discrepancies in he shock loss. Firs of all, he simulaion wihou mixing plane describes a differen loss evoluion around M = 1.0 han hose wih a mixing plane. This is a well-known problem wih mixing planes in general: he loss is radially redisribued. The influence of he locaion of he mixing plane is clearly visible in he numerical curves. The slopes are no so far from he analyical ones. Bu here is a sor of offse beween he numerical and he analyical resuls. I is probable ha for he low Mach numbers, he shock loss is low compared o he viscous loss. To compare properly he wo curves families, i would be necessary o ake from he Navier-Sokes simulaions only he shock loss, as done in Ref. [1] by subracing he fricion loss from measuremens. The discrepancies are also due o he srong hypoheses made in he analyical mehod, which consiss in a wo-dimensional approach and due o he choice made for he loss calculaion. Real blade profiles are also ofen cambered and no symmeric in order o produce lif, which is no aken ino accoun in he presen model. And finally, he numerical loss a he highes Mach numbers (close o he secion heigh h ) is suddenly increasing, near he shroud boundary layer. I is likely ha fricion loss dominaes shock loss in his area. Thanks o his simple model, he order of magniude of he under predicion of he losses due o he inroducion of a mixing plane is easily evaluaed. I has been esed ha his approach gives accepable resuls from inle Mach number larger han 1.1. Once again, he major drawbacks of his mehod are ha i gives wrong predicions for lower Mach numbers and ha he shock formulas which are used in i are very sensiive. This is maybe one of he reasons why Bloch e al. [1] had o ake ino accoun an effecive leading edge radius in heir model dedicaed o predic he shock loss hrough he lower branch of Moeckel s hyperbola, in supersonic compressor cascades operaing in unsared regime. Indeed, hey increased he leading edge hickness unil he analyical resuls

11 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 101 mached he experimenal ones. This reminds us of he difficuly of implemening a shock loss model ha fis all ypes of profiles, under various operaing condiions. 7. onclusion Seady sae numerical simulaions performed wih a mixing plane approach show ha he resuls, in erms of mass flow and losses, significanly depend on he mixing plane posiion. The operaing poin chosen here corresponds o he sonic blockage of he compressor bu for near-surge poins, i would be even more imporan. The fac ha his sudy akes place near he blockage enables o propose an analyical mehod in order o forecas his change in performance. The validiy of his analyical mehod is checked by comparing is resuls wih he numerical ones in he enry zone of a ransonic compressor. Analyical and numerical resuls show good agreemen. This ool may be useful for ransonic compressor design: firs, o have an idea of he minimal disance ha should be pu beween he mixing plane and he leading edge of he blades, and hen o know how represenaively he seady simulaions can be expeced. Acknowledgemens We would like o hank Turbomeca which suppored his sudy, ogeher wih ONERA which collaboraed on he numerical simulaion. This work was graned access o he H resources of INES under he allocaion 013-a6356. References [1] Lichfuss, J. J., and Sarken, H Supersonic ascade Flow. rogress in Aerospace Sciences 15: [] Kanrowiz, A The Supersonic Axial-Flow ompressor. NAA echnical repor. [3] hauvin, J Supersonic Turbo-Je ropulsion Sysems and omponens. AGARD repor No. 10. [4] Trébinjac, I., Oavy, X., Rochuon, N., and Bulo, N On he Validiy of Seady alculaions wih Shock-Blade Row Ineracion in ompressors. In roceedings of he 9h Inernaional Symposium on Experimenal and ompuaional Aerohermodynamics of Inernal Flows, ISAIF9-06. [5] ambier, L., and Gazaix, M. 00. elsa: An Efficien Objec-Oriened Soluion o FD omplexiy. resened a 00 he 40h AIAA Aerospace Science Meeing and Exhibi, Reno, USA. [6] Smih, B. R redicion of Hypersonic Shock Wave Turbulen Boundary Layer Ineracions wih k-l Two-Equaions Turbulence Model. resened a 1995 he 33rd AIAA Aerospace Sciences Meeing and Exhibiion, Reno, USA. [7] Rochuon, N Analysis of he Three-Dimensional Unseady Flow in a High ressure Raio enrifugal ompressor. h.d. hesis, École cenrale de Lyon. [8] Moeckel, W. E Approximae Mehod for redicing Form and Locaion of Deached Shock Waves. NAA echnical repor. [9] Burguburu, S., Toussain,., Bonhomme,., and Leroy, G Numerical Opimizaion of Turbomachinery Bladings. Journal of Turbomachinery 16 (1): [10] König, W. M., Hennecke, D. K., and Foner, L Improved Blade rofile Loss and Deviaion Angle Models for Advanced Transonic ompressor Bladings: ar II A Model for Supersonic Flow. Journal of Turbomachinery 118 (1): [11] Schobeiri, M. T Advanced ompressor Loss orrelaions, ar I: Theroreical Aspecs. Inernaional Journal of Roaing Marchinery 3 (3): [1] Bloch, G. S., openhaver, W. W., and O Brien, W. F A Shock Loss Model for Supersonic ompressor ascades. Journal of Turbomachinery 11 (1): [13] Oavy, X Laser Anemomery Measuremens in an Axial Transonic ompressor. Analysis of he Unseady eriodic Srucures. h.d. hesis, École cenrale de Lyon. [14] Levine, The Two-Dimensional Inflow ondiions for a Supersonic ompressor wih urved Blades. WAD echnical repor

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

Computation of the Effect of Space Harmonics on Starting Process of Induction Motors Using TSFEM

Computation of the Effect of Space Harmonics on Starting Process of Induction Motors Using TSFEM Journal of elecrical sysems Special Issue N 01 : November 2009 pp: 48-52 Compuaion of he Effec of Space Harmonics on Saring Process of Inducion Moors Using TSFEM Youcef Ouazir USTHB Laboraoire des sysèmes

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

15. Vector Valued Functions

15. Vector Valued Functions 1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,

More information

Linear Response Theory: The connection between QFT and experiments

Linear Response Theory: The connection between QFT and experiments Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

Structural Dynamics and Earthquake Engineering

Structural Dynamics and Earthquake Engineering Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp://www.c.up.ro/users/aurelsraan/

More information

Theory of! Partial Differential Equations!

Theory of! Partial Differential Equations! hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and

More information

Numerical investigation of Ranque-Hilsch energy separation effect A.S. Noskov 1,a, V.N. Alekhin 1,b, A.V. Khait 1,a

Numerical investigation of Ranque-Hilsch energy separation effect A.S. Noskov 1,a, V.N. Alekhin 1,b, A.V. Khait 1,a Applied Mechanics and Maerials Online: 2013-01-11 ISSN: 1662-7482, Vol. 281, pp 355-358 doi:10.4028/www.scienific.ne/amm.281.355 2013 Trans Tech Publicaions, Swizerland Numerical invesigaion of Ranque-Hilsch

More information

V AK (t) I T (t) I TRM. V AK( full area) (t) t t 1 Axial turn-on. Switching losses for Phase Control and Bi- Directionally Controlled Thyristors

V AK (t) I T (t) I TRM. V AK( full area) (t) t t 1 Axial turn-on. Switching losses for Phase Control and Bi- Directionally Controlled Thyristors Applicaion Noe Swiching losses for Phase Conrol and Bi- Direcionally Conrolled Thyrisors V AK () I T () Causing W on I TRM V AK( full area) () 1 Axial urn-on Plasma spread 2 Swiching losses for Phase Conrol

More information

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec

More information

Lab 10: RC, RL, and RLC Circuits

Lab 10: RC, RL, and RLC Circuits Lab 10: RC, RL, and RLC Circuis In his experimen, we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors. We will sudy he way volages and currens change in

More information

IB Physics Kinematics Worksheet

IB Physics Kinematics Worksheet IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?

More information

Numerical Dispersion

Numerical Dispersion eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal

More information

Air Traffic Forecast Empirical Research Based on the MCMC Method

Air Traffic Forecast Empirical Research Based on the MCMC Method Compuer and Informaion Science; Vol. 5, No. 5; 0 ISSN 93-8989 E-ISSN 93-8997 Published by Canadian Cener of Science and Educaion Air Traffic Forecas Empirical Research Based on he MCMC Mehod Jian-bo Wang,

More information

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion

More information

STATE-SPACE MODELLING. A mass balance across the tank gives:

STATE-SPACE MODELLING. A mass balance across the tank gives: B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing

More information

NEWTON S SECOND LAW OF MOTION

NEWTON S SECOND LAW OF MOTION Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during

More information

Sterilization D Values

Sterilization D Values Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once,

More information

Time series Decomposition method

Time series Decomposition method Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,

More information

Displacement ( x) x x x

Displacement ( x) x x x Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh

More information

Second Law. first draft 9/23/04, second Sept Oct 2005 minor changes 2006, used spell check, expanded example

Second Law. first draft 9/23/04, second Sept Oct 2005 minor changes 2006, used spell check, expanded example Second Law firs draf 9/3/4, second Sep Oc 5 minor changes 6, used spell check, expanded example Kelvin-Planck: I is impossible o consruc a device ha will operae in a cycle and produce no effec oher han

More information

Multi-scale 2D acoustic full waveform inversion with high frequency impulsive source

Multi-scale 2D acoustic full waveform inversion with high frequency impulsive source Muli-scale D acousic full waveform inversion wih high frequency impulsive source Vladimir N Zubov*, Universiy of Calgary, Calgary AB vzubov@ucalgaryca and Michael P Lamoureux, Universiy of Calgary, Calgary

More information

Advanced Organic Chemistry

Advanced Organic Chemistry Lalic, G. Chem 53A Chemisry 53A Advanced Organic Chemisry Lecure noes 1 Kineics: A racical Approach Simple Kineics Scenarios Fiing Experimenal Daa Using Kineics o Deermine he Mechanism Doughery, D. A.,

More information

STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN

STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN 001-004 OBARA, Takashi * Absrac The

More information

Estimation of Kinetic Friction Coefficient for Sliding Rigid Block Nonstructural Components

Estimation of Kinetic Friction Coefficient for Sliding Rigid Block Nonstructural Components 7 Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens Cagdas Kafali Ph.D. Candidae, School of Civil and Environmenal Engineering, Cornell Universiy Research Supervisor:

More information

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species

More information

International Industrial Informatics and Computer Engineering Conference (IIICEC 2015)

International Industrial Informatics and Computer Engineering Conference (IIICEC 2015) Inernaional Indusrial Informaics and Compuer Engineering Conference (IIICEC 015) Effec of impeller blades on waer resisance coefficien and efficiency of mied-flow pump Du Yuanyinga, Shang Changchunb, Zhang

More information

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx. . Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.

More information

Lecture #8 Redfield theory of NMR relaxation

Lecture #8 Redfield theory of NMR relaxation Lecure #8 Redfield heory of NMR relaxaion Topics The ineracion frame of reference Perurbaion heory The Maser Equaion Handous and Reading assignmens van de Ven, Chapers 6.2. Kowalewski, Chaper 4. Abragam

More information

Vanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law

Vanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law Vanishing Viscosiy Mehod. There are anoher insrucive and perhaps more naural disconinuous soluions of he conservaion law (1 u +(q(u x 0, he so called vanishing viscosiy mehod. This mehod consiss in viewing

More information

10.6 Parametric Equations

10.6 Parametric Equations 0_006.qd /8/05 9:05 AM Page 77 Secion 0.6 77 Parameric Equaions 0.6 Parameric Equaions Wha ou should learn Evaluae ses of parameric equaions for given values of he parameer. Skech curves ha are represened

More information

EXPLICIT TIME INTEGRATORS FOR NONLINEAR DYNAMICS DERIVED FROM THE MIDPOINT RULE

EXPLICIT TIME INTEGRATORS FOR NONLINEAR DYNAMICS DERIVED FROM THE MIDPOINT RULE Version April 30, 2004.Submied o CTU Repors. EXPLICIT TIME INTEGRATORS FOR NONLINEAR DYNAMICS DERIVED FROM THE MIDPOINT RULE Per Krysl Universiy of California, San Diego La Jolla, California 92093-0085,

More information

Traveling Waves. Chapter Introduction

Traveling Waves. Chapter Introduction Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from

More information

NCSS Statistical Software. , contains a periodic (cyclic) component. A natural model of the periodic component would be

NCSS Statistical Software. , contains a periodic (cyclic) component. A natural model of the periodic component would be NCSS Saisical Sofware Chaper 468 Specral Analysis Inroducion This program calculaes and displays he periodogram and specrum of a ime series. This is someimes nown as harmonic analysis or he frequency approach

More information

Linear Surface Gravity Waves 3., Dispersion, Group Velocity, and Energy Propagation

Linear Surface Gravity Waves 3., Dispersion, Group Velocity, and Energy Propagation Chaper 4 Linear Surface Graviy Waves 3., Dispersion, Group Velociy, and Energy Propagaion 4. Descripion In many aspecs of wave evoluion, he concep of group velociy plays a cenral role. Mos people now i

More information

Two Coupled Oscillators / Normal Modes

Two Coupled Oscillators / Normal Modes Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own

More information

12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j =

12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j = 1: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME Moving Averages Recall ha a whie noise process is a series { } = having variance σ. The whie noise process has specral densiy f (λ) = of

More information

2.4 Cuk converter example

2.4 Cuk converter example 2.4 Cuk converer example C 1 Cuk converer, wih ideal swich i 1 i v 1 2 1 2 C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode C 1 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28 Analysis sraegy This converer

More information

FloEFD simulation of micro-turbine engine

FloEFD simulation of micro-turbine engine FloEFD simulaion of micro-urbine engine T.V. Trebunskikh, A.V. Ivanov, G.E. Dumnov Menor Graphics, Moscow, Russia Absrac Keywords: micro-urbine engine, CFD, combusion, compressor, urbine Turboje engines

More information

A Dynamic Model of Economic Fluctuations

A Dynamic Model of Economic Fluctuations CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model

More information

Dam Flooding Simulation Using Advanced CFD Methods

Dam Flooding Simulation Using Advanced CFD Methods WCCM V Fifh World Congress on Compuaional Mechanics July 7-12, 2002, Vienna, Ausria Dam Flooding Simulaion Using Advanced CFD Mehods Mohamed Gouda*, Dr. Konrad Karner VRVis Zenrum für Virual Realiy und

More information

( ) Eight geological shape curvature classes are defined in terms of the Gaussian and mean normal curvatures as follows: perfect saddle, plane

( ) Eight geological shape curvature classes are defined in terms of the Gaussian and mean normal curvatures as follows: perfect saddle, plane Geological Shape Curvaure Principal normal curvaures, and, are defined a any poin on a coninuous surface such ha:, + and + () The Gaussian curvaure, G, and he mean normal curvaure, M, are defined in erms

More information

where the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).

where the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP). Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness

More information

Zhihan Xu, Matt Proctor, Ilia Voloh

Zhihan Xu, Matt Proctor, Ilia Voloh Zhihan Xu, Ma rocor, lia Voloh - GE Digial Energy Mike Lara - SNC-Lavalin resened by: Terrence Smih GE Digial Energy CT fundamenals Circui model, exciaion curve, simulaion model CT sauraion AC sauraion,

More information

A car following model for traffic flow simulation

A car following model for traffic flow simulation Inernaional Journal of Applied Mahemaical Sciences ISSN 0973-076 Volume 9, Number (206), pp. -9 Research India Publicaions hp://www.ripublicaion.com A car following model for raffic flow simulaion Doudou

More information

Calculation of the Two High Voltage Transmission Line Conductors Minimum Distance

Calculation of the Two High Voltage Transmission Line Conductors Minimum Distance World Journal of Engineering and Technology, 15, 3, 89-96 Published Online Ocober 15 in SciRes. hp://www.scirp.org/journal/wje hp://dx.doi.org/1.436/wje.15.33c14 Calculaion of he Two High Volage Transmission

More information

Starting from a familiar curve

Starting from a familiar curve In[]:= NoebookDirecory Ou[]= C:\Dropbox\Work\myweb\Courses\Mah_pages\Mah_5\ You can evaluae he enire noebook by using he keyboard shorcu Al+v o, or he menu iem Evaluaion Evaluae Noebook. Saring from a

More information

Summer Term Albert-Ludwigs-Universität Freiburg Empirische Forschung und Okonometrie. Time Series Analysis

Summer Term Albert-Ludwigs-Universität Freiburg Empirische Forschung und Okonometrie. Time Series Analysis Summer Term 2009 Alber-Ludwigs-Universiä Freiburg Empirische Forschung und Okonomerie Time Series Analysis Classical Time Series Models Time Series Analysis Dr. Sevap Kesel 2 Componens Hourly earnings:

More information

CENTRALIZED VERSUS DECENTRALIZED PRODUCTION PLANNING IN SUPPLY CHAINS

CENTRALIZED VERSUS DECENTRALIZED PRODUCTION PLANNING IN SUPPLY CHAINS CENRALIZED VERSUS DECENRALIZED PRODUCION PLANNING IN SUPPLY CHAINS Georges SAHARIDIS* a, Yves DALLERY* a, Fikri KARAESMEN* b * a Ecole Cenrale Paris Deparmen of Indusial Engineering (LGI), +3343388, saharidis,dallery@lgi.ecp.fr

More information

Topic 1: Linear motion and forces

Topic 1: Linear motion and forces TOPIC 1 Topic 1: Linear moion and forces 1.1 Moion under consan acceleraion Science undersanding 1. Linear moion wih consan elociy is described in erms of relaionships beween measureable scalar and ecor

More information

6.003 Homework #8 Solutions

6.003 Homework #8 Solutions 6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d

More information

AP Chemistry--Chapter 12: Chemical Kinetics

AP Chemistry--Chapter 12: Chemical Kinetics AP Chemisry--Chaper 12: Chemical Kineics I. Reacion Raes A. The area of chemisry ha deals wih reacion raes, or how fas a reacion occurs, is called chemical kineics. B. The rae of reacion depends on he

More information

UNIVERSITY OF TRENTO MEASUREMENTS OF TRANSIENT PHENOMENA WITH DIGITAL OSCILLOSCOPES. Antonio Moschitta, Fabrizio Stefani, Dario Petri.

UNIVERSITY OF TRENTO MEASUREMENTS OF TRANSIENT PHENOMENA WITH DIGITAL OSCILLOSCOPES. Antonio Moschitta, Fabrizio Stefani, Dario Petri. UNIVERSIY OF RENO DEPARMEN OF INFORMAION AND COMMUNICAION ECHNOLOGY 385 Povo reno Ialy Via Sommarive 4 hp://www.di.unin.i MEASUREMENS OF RANSIEN PHENOMENA WIH DIGIAL OSCILLOSCOPES Anonio Moschia Fabrizio

More information

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H.

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H. ACE 56 Fall 005 Lecure 5: he Simple Linear Regression Model: Sampling Properies of he Leas Squares Esimaors by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Inference in he Simple

More information

Position, Velocity, and Acceleration

Position, Velocity, and Acceleration rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose

More information

LAPLACE TRANSFORM AND TRANSFER FUNCTION

LAPLACE TRANSFORM AND TRANSFER FUNCTION CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions

More information

0 time. 2 Which graph represents the motion of a car that is travelling along a straight road with a uniformly increasing speed?

0 time. 2 Which graph represents the motion of a car that is travelling along a straight road with a uniformly increasing speed? 1 1 The graph relaes o he moion of a falling body. y Which is a correc descripion of he graph? y is disance and air resisance is negligible y is disance and air resisance is no negligible y is speed and

More information

Matlab and Python programming: how to get started

Matlab and Python programming: how to get started Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,

More information

Computation of hull-pressure fluctuations due to non-cavitating propellers

Computation of hull-pressure fluctuations due to non-cavitating propellers Firs Inernaional Symposium on Marine Propulsors smp 9, Trondheim, Norway, June 9 Compuaion of hull-pressure flucuaions due o non-caviaing propellers Frans Hendrik Lafeber, Erik van Wijngaarden, Johan Bosschers

More information

4.5 Constant Acceleration

4.5 Constant Acceleration 4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),

More information

EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits

EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive

More information

The Paradox of Twins Described in a Three-dimensional Space-time Frame

The Paradox of Twins Described in a Three-dimensional Space-time Frame The Paradox of Twins Described in a Three-dimensional Space-ime Frame Tower Chen E_mail: chen@uguam.uog.edu Division of Mahemaical Sciences Universiy of Guam, USA Zeon Chen E_mail: zeon_chen@yahoo.com

More information

Flow-Induced Vibration Analysis of Supported Pipes with a Crack

Flow-Induced Vibration Analysis of Supported Pipes with a Crack Flow-Induced Vibraion Analsis of Suppored Pipes wih a Crack Jin-Huk Lee, Samer Masoud Al-Said Deparmen of Mechanical Engineering American Universi of Sharjah, UAE Ouline Inroducion and Moivaion Aeroacousicall

More information

= ( ) ) or a system of differential equations with continuous parametrization (T = R

= ( ) ) or a system of differential equations with continuous parametrization (T = R XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

1. Kinematics I: Position and Velocity

1. Kinematics I: Position and Velocity 1. Kinemaics I: Posiion and Velociy Inroducion The purpose of his eperimen is o undersand and describe moion. We describe he moion of an objec by specifying is posiion, velociy, and acceleraion. In his

More information

Article from. Predictive Analytics and Futurism. July 2016 Issue 13

Article from. Predictive Analytics and Futurism. July 2016 Issue 13 Aricle from Predicive Analyics and Fuurism July 6 Issue An Inroducion o Incremenal Learning By Qiang Wu and Dave Snell Machine learning provides useful ools for predicive analyics The ypical machine learning

More information

Acceleration. Part I. Uniformly Accelerated Motion: Kinematics & Geometry

Acceleration. Part I. Uniformly Accelerated Motion: Kinematics & Geometry Acceleraion Team: Par I. Uniformly Acceleraed Moion: Kinemaics & Geomery Acceleraion is he rae of change of velociy wih respec o ime: a dv/d. In his experimen, you will sudy a very imporan class of moion

More information

LAB 5 - PROJECTILE MOTION

LAB 5 - PROJECTILE MOTION Lab 5 Projecile Moion 71 Name Dae Parners OVEVIEW LAB 5 - POJECTILE MOTION We learn in our sudy of kinemaics ha wo-dimensional moion is a sraighforward exension of one-dimensional moion. Projecile moion

More information

Lecture 4 Notes (Little s Theorem)

Lecture 4 Notes (Little s Theorem) Lecure 4 Noes (Lile s Theorem) This lecure concerns one of he mos imporan (and simples) heorems in Queuing Theory, Lile s Theorem. More informaion can be found in he course book, Bersekas & Gallagher,

More information

I. OBJECTIVE OF THE EXPERIMENT.

I. OBJECTIVE OF THE EXPERIMENT. I. OBJECTIVE OF THE EXPERIMENT. Swissmero raels a high speeds hrough a unnel a low pressure. I will hereore undergo ricion ha can be due o: ) Viscosiy o gas (c. "Viscosiy o gas" eperimen) ) The air in

More information

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling?

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling? 256 MATHEMATICS A.2.1 Inroducion In class XI, we have learn abou mahemaical modelling as an aemp o sudy some par (or form) of some real-life problems in mahemaical erms, i.e., he conversion of a physical

More information

EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?

EE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal? EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of

More information

9231 FURTHER MATHEMATICS

9231 FURTHER MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for he May/June series 9 FURTHER MATHEMATICS 9/ Paper, maximum raw mark This mark scheme is published as an aid o eachers and candidaes,

More information

Numerical Evaluation of an Add-On Vehicle Protection System

Numerical Evaluation of an Add-On Vehicle Protection System Numerical Evaluaion of an Add-On Vehicle Proecion Sysem Geneviève Toussain, Amal Bouamoul, Rober Durocher, Jacob Bélanger*, Benoî S-Jean Defence Research and Developmen Canada Valcarier 2459 Bravoure Road,

More information

13.3 Term structure models

13.3 Term structure models 13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)

More information

Linear Time-invariant systems, Convolution, and Cross-correlation

Linear Time-invariant systems, Convolution, and Cross-correlation Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An

More information

Hall effect. Formulae :- 1) Hall coefficient RH = cm / Coulumb. 2) Magnetic induction BY 2

Hall effect. Formulae :- 1) Hall coefficient RH = cm / Coulumb. 2) Magnetic induction BY 2 Page of 6 all effec Aim :- ) To deermine he all coefficien (R ) ) To measure he unknown magneic field (B ) and o compare i wih ha measured by he Gaussmeer (B ). Apparaus :- ) Gauss meer wih probe ) Elecromagne

More information

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, 65-70 ISSN: 2241-0384 (prin), 2241-0376 (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models

More information

Lecture Notes 2. The Hilbert Space Approach to Time Series

Lecture Notes 2. The Hilbert Space Approach to Time Series Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship

More information

Section 4.4 Logarithmic Properties

Section 4.4 Logarithmic Properties Secion. Logarihmic Properies 5 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies

More information

Math 4600: Homework 11 Solutions

Math 4600: Homework 11 Solutions Mah 46: Homework Soluions Gregory Handy [.] One of he well-known phenomenological (capuring he phenomena, bu no necessarily he mechanisms) models of cancer is represened by Gomperz equaion dn d = bn ln(n/k)

More information

Comparing Means: t-tests for One Sample & Two Related Samples

Comparing Means: t-tests for One Sample & Two Related Samples Comparing Means: -Tess for One Sample & Two Relaed Samples Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion

More information

Involute Gear Tooth Bending Stress Analysis

Involute Gear Tooth Bending Stress Analysis Involue Gear Tooh Bending Sress Analysis Lecure 21 Engineering 473 Machine Design Gear Ineracion Line of Ceners Line Tangen o s Line Normal o Line of Ceners 1 s Close Up of Meshed Teeh Line of Conac W

More information

PHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections

PHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections PHYSICS 220 Lecure 02 Moion, Forces, and Newon s Laws Texbook Secions 2.2-2.4 Lecure 2 Purdue Universiy, Physics 220 1 Overview Las Lecure Unis Scienific Noaion Significan Figures Moion Displacemen: Δx

More information

Estimation of Poses with Particle Filters

Estimation of Poses with Particle Filters Esimaion of Poses wih Paricle Filers Dr.-Ing. Bernd Ludwig Chair for Arificial Inelligence Deparmen of Compuer Science Friedrich-Alexander-Universiä Erlangen-Nürnberg 12/05/2008 Dr.-Ing. Bernd Ludwig (FAU

More information

3.1 More on model selection

3.1 More on model selection 3. More on Model selecion 3. Comparing models AIC, BIC, Adjused R squared. 3. Over Fiing problem. 3.3 Sample spliing. 3. More on model selecion crieria Ofen afer model fiing you are lef wih a handful of

More information

This document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC

This document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,

More information

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004 ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform

More information

Topic Astable Circuits. Recall that an astable circuit has two unstable states;

Topic Astable Circuits. Recall that an astable circuit has two unstable states; Topic 2.2. Asable Circuis. Learning Objecives: A he end o his opic you will be able o; Recall ha an asable circui has wo unsable saes; Explain he operaion o a circui based on a Schmi inverer, and esimae

More information

Learning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure

Learning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure Lab 4: Synchronous Sae Machine Design Summary: Design and implemen synchronous sae machine circuis and es hem wih simulaions in Cadence Viruoso. Learning Objecives: Pracice designing and simulaing digial

More information

MATHEMATICAL MODELING OF THE TRACTOR-GRADER AGRICULTURAL SYSTEM CINEMATIC DURING LAND IMPROVING WORKS

MATHEMATICAL MODELING OF THE TRACTOR-GRADER AGRICULTURAL SYSTEM CINEMATIC DURING LAND IMPROVING WORKS Bullein of he Transilvania Universiy of Braşov Series II: Foresry Wood Indusry Agriculural Food Engineering Vol. 5 (54) No. 1-2012 MATHEMATICA MODEING OF THE TRACTOR-GRADER AGRICUTURA SYSTEM CINEMATIC

More information

CHAPTER 6: FIRST-ORDER CIRCUITS

CHAPTER 6: FIRST-ORDER CIRCUITS EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions

More information

MATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence

MATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,

More information

Georey E. Hinton. University oftoronto. Technical Report CRG-TR February 22, Abstract

Georey E. Hinton. University oftoronto.   Technical Report CRG-TR February 22, Abstract Parameer Esimaion for Linear Dynamical Sysems Zoubin Ghahramani Georey E. Hinon Deparmen of Compuer Science Universiy oftorono 6 King's College Road Torono, Canada M5S A4 Email: zoubin@cs.orono.edu Technical

More information

Mechanical Fatigue and Load-Induced Aging of Loudspeaker Suspension. Wolfgang Klippel,

Mechanical Fatigue and Load-Induced Aging of Loudspeaker Suspension. Wolfgang Klippel, Mechanical Faigue and Load-Induced Aging of Loudspeaker Suspension Wolfgang Klippel, Insiue of Acousics and Speech Communicaion Dresden Universiy of Technology presened a he ALMA Symposium 2012, Las Vegas

More information

CHAPTER 12 DIRECT CURRENT CIRCUITS

CHAPTER 12 DIRECT CURRENT CIRCUITS CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As

More information

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow 1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering

More information

Chapter 7 Response of First-order RL and RC Circuits

Chapter 7 Response of First-order RL and RC Circuits Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial

More information

Exponential Smoothing

Exponential Smoothing Exponenial moohing Inroducion A simple mehod for forecasing. Does no require long series. Enables o decompose he series ino a rend and seasonal effecs. Paricularly useful mehod when here is a need o forecas

More information