A Tool for Converting FEM Models into Representations Suitable for Control Synthesis

 Hubert Thornton
 8 months ago
 Views:
Transcription
1 Proceedings o the 17th Word Congress The Internationa Federation o Automatic Contro Seou, Korea, Juy 611, 8 A Too or Converting FEM Modes into Representations Suitabe or Contro Synthesis Syed Fawad Raza Ai Bokhari, Sauat S. Chughtai, Herbert Werner Institute o Contro Systems, Hamburg University o Technoogy, Germany Abstract: Finite Eement Methods (FEM) pay an important roe in modeing o compex systems. Modes generated rom FEM are o very high dimension and are diicut to hande or contro system design. Inthis paper, an agorithm is presentedthat converts asystem generated rom FEM into the state space mode o an interconnected system, thus vasty reducing the compexity o synthesizing a distributed contro strategy. A simpe cantiever beam probem is used as an exampe to iustrate the proposed method. A state space representation o this system is obtained through FEM anaysis. This mode is converted rom umped state space orm to interconnected orm. Adecentraized controer is then designed usinga homotopy approach. Keywords: Sotware too, Interconnected Systems, Finite Eement Methods (FEM). 1. INTRODUCTION Engineering systems in which subsystems interact with each other are o considerabe practica interest. With advances in design techniques and computationa power compex system modes o very high dimensions can be deveoped, incuding umped approximations o partia dierentia equations (PDEs) exampes incudethedeection obeams, pates andmembranes, andthetemperature distribution o thermay conductive materias. Soutions to these types o probems are possibe or very simpe systems; or more reaistic systems the Finite Eement Method (FEM)(Becker, 4) is a very eective too or soving such probems numericay (Huebner et a., 1). Currenty FEM is used in structura anaysis, heat transer, uid mechanics etc. In FEM a compete system is divided into sma eements which are spatiay distributed and interact with each other, based on their spatia ocation. In structura anaysis behavior a system is described by the dispacement o eements satisying materia aws (constitutive equation) (Becker, 4). A eements are assembed together and the requirements o continuity and equiibrium are satisied between neighboring eements. A unique soution can be ound or a given boundary conditions. A state space representation obtained rom FEM generay has a arge number o states aong with arge number o inputs andoutputs. Forsuchsystems moderncontro design techniques may ai due to the size o the probem. One way o deaing with such systems is to reduce the size o system using standard mode order reduction techniques. However, this reduction may eiminate some aspects o the dynamic behavior which may get exited in the cosed oop, thus degrading the achieved perormance. In this paper we propose a dierent approach, which is to decompose the mode used or contro synthesis into subsystems. Each subsystem has ony a sma number o states and its own sensor and actuator. This decomposition can aciitate the use o modern controer synthesis techniques to arge systems, using the ramework o interconnected systems. A genera interconnected systems is composed o N subsystems interconnected with each other. The state space reaization o i th subsystem is given by ẋ i (t) = A i x i (t) + N j=1,j i B vij x j (t) + B i w i (t) z i (t) = C i x i (t) + D i w i (t) (1) where x i R ni, w i R ki, z i R oi are the state vector, disturbance inputs and outputs, respectivey. The matrices A i, B vij, B i, C i and D i are constant matrices o appropriate dimensions. Here, the matrix B vij represents the interconnection o the i th subsystem with the j th subsystem. Modes in the orm o (1) have been extensivey studied, and eicient synthesis toos have been proposed to design controers or such systems. Some earier work on such systems is presented in (Sijak, 1978), whie (Sijak and Zecevic, 5) presents some new approaches based on inear matrix inequaities. In (Chen et a., 5) the authors have presented a homotopy based approach to design robust controers or (1). In (D Andrea and Duerud, 3) the authors have shown that the Kaman YaqubovichPopov emma can be generaized or acass o interconnected systems with identica subsystems having interconnections ony with their near neighbors, which can be considered as a specia case o system (1). In this paper we present a sotware too that can decompose an FEM generated mode into the orm o (1), /8/$. 8 IFAC /8765KR
2 17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 611, 8 thus making it suitabe or contro system design. The paper is organized as oows. A genera matrix decomposition unction is presented in Section. This unction is appied in Section 3 to a FEM generated system and a compete agorithm to decompose the given system is discussed. Insection4auniorm cantiever exampe is used to demonstrate the appication o the too. In section 5 the homotopy approach o (Chen et a., 5) is used to designdecentraized controer orthedecomposedsystem. Concuding remarks and uture directions are given in Section 5.. THE BASIC FUNCTION In this section the basic unction decomp() is introduced. The unction decomposes any matrix G R p q into the bock structure G 11 G 1 G 1n G 1 G G n G n1 G n G nn. () In the oowing G i wi represent the i th diagona bock G(i,i), where G i R ( hi ji) such that, n p = hi (3) q = i=1 n ji (4) i=1 The unction requires two urther inputs rom the user. These are T h = [ h1 h hn ] T (5) T j = [ j1 j jn ] T (6) These vectors speciythedimension oeach bock, andor the genera case ( hi ji ). Let us deine G vi as G vi = [ G (i,1) G (i,i 1), G (i,i+1) G (i,n) ] In order to extract G i and G vi rom G we need upper and ower row and coumn indices in terms o the eements o vectors h and j. Next we wi ind these indices or G i and G vir, where r = 1,...,i 1,i + 1,...,n..1 Indices o G i Let e h be the ower row index and e j be the ower coumn index o G i respectivey, then these can be ound rom vectors h and j as Then we have i 1 e h = hk + 1 (7) k=1 i 1 e j = jk + 1 (8) k=1 G i = G(e h+ i,e j+ g i ), where. Indices o G vir i =,1,...,( hi 1) g i =,1,...,( ji 1) (9) Let G vir = G( vir,g vir ), then vir = e h + i. For the case when r < i, i 1 g vir = e j jk + {,1,..., jr 1} k=r and or the case when r > i, r 1 g vir = e j + jk + {,1,..., jr 1} k=i The above approach is coded into a MATLAB unction decomp(g, h, j,i,m) where the binary input m can be used to get G i i m = or G vi i m. 3. APPLICATION TO FEM GENERATED SYSTEMS A system mode generated via FEM is given as M q + C q + Kq = F d + F (1) where q represents the generaized coordinates and M = Mass matrix K = Stiness matrix C = Damping matrix F d = Appied noda orces (externa orces) F = Boundary conditions These matrices are obtained by combining the equations o motion o each eement o the FEM anaysis. Foowing this the continuity constraints between neighboring eements are appied. The above equation can be written in state space orm as [ ] [ ] I x = M 1 K M 1 x + C M 1 F(t) }{{}}{{} Ã y = Cx + Du (11) where x R n, u R k, y R o. The state vector in (11) is arranged as x T = [ x T 1 x T x T n ẋ T 1 ẋ T ẋ T n ]T This can be transormed into x T = [ x T 1 ẋ T 1 x T ẋ T x T n ẋ T n ]T (1) Deining a unitary transormation matrix T r such that x = T r x we obtain A = T r ÃTr 1, B = T r B, C = 1 CT r and D = D. Next we show how to decompose the transormed system into the rom (1) with N subsystems. Let or the i th subsystem ẋ i,x i, R xi, z i R zi and w i R wi. Deine the vectors 667
3 17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 611, 8 F x = ( x1,..., xn ) (13) F z = ( z1,..., zn ) (14) F w = ( w1,..., wn ) (15) The unction decomp() canthen be appied toa, B, C and D to decompose them to get the state space representation (1). This is expressed in the oowing agorithm. 3.1 Agorithm For an FEM generated state space in the rom (11) Step 1 Find the transormation matrix T r and appy this simiarity transormation to get A, B, C, D. Step 3 Deine the vectors F x, F z and F w. Step Obtain the other matrices by A i = decomp(a,f x,f x,i,); B vi = decomp(a,f x,f x,i,1); B i = decomp(b,f x,f w,i,); C i = decomp(c,f z,f x,i,); D i = decomp(d,f z,f w,i,); 4. EXAMPLE: CANTILEVER BEAM This section briey presents the FEM treatment o a uniorm cantiever (Juang and Phan, 4). Fig. 1. Beam cantiever FEM o singe cantiever beam eement Firstconsider abeam with camped boundary condition at one end and ree boundary condition at the other as shown in Fig 1. In order to iustrate the generation o an FEM mode, a singe eement with two nodes o a campedree uniorm beam is taken. Let an impuse act downwards at the node as shown in the Fig 1. This orce resuts in a dispacement d(x,t) and an anguar rotation, θ = d(x,t) x. The equation o motion can be written as ρ d(x,t) t + EM I 4 d(x,t) x 4 = (16) Here ρ, E, M I are density, Young moduus o easticity and second moment o inertia, respectivey. The behavior o dispacement and rotation can be approximated interms o interpoation unctions as d 1 (t) θ d(x,t) = [ N 1 (x) N (x) N 3 (x) N 4 (x) ] 1 (t) d (t) θ (t) where the subscripts 1 and denote the irst and second node o the eement. The N i (x) are the interpoation unctions o the corresponding variabe, which can be approximated as N k (x) = c + c 1 x + c x + c 3 x 3 k = 1,,3,4 Vaues o these coeicients can be determined by appying geometric boundary conditions on d and θ at node 1 and. Let us deine N T (x) = [ N 1 (x) N (x) N 3 (x) N 4 (x) ] Substituting in (16) we obtain ρn T (x) q(t) 4 N T (x)q(t) t + EM I x 4 = (x,t) where q T = [d 1,θ 1,d,θ ] T. Soving this equation resuts in which is equivaent to ρn(x)n T (x)dx }{{} M q(t) t + N(x) N T (x) EM I x x q(t) }{{} = K N(x)(x,t)dx + F (17) } {{ } F d M q + Kq = F d + F F in terms o shear orce and moment m is given as 1 m F = 1 = m EM 3 d(x,t) x= I x 3 EM d(x,t) x= I x EM 3 d(x,t) x= I x 3 EM d(x,t) = I x = For the above interpoation unction the oowing oca mass and stiness matrices can be cacuated: 668
4 17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 611, 8 M = ρ K = EM I Generaization to cantiever with a arge number o eements The above expressions or K and M represent oca eements in FEM which are assembed in a goba mass and stiness matrix to generate umped matrices. The goba equation or the whoe assemby can be obtained by combining the matrix contribution o a individua eements, suchthatcoeicients beongingtocommon nodes are added together. Let us deine k (i) = [ k (i) 11 k(i) 1 k (i) 1 k(i) ] which is the stiness matrix o i th eement, then k (1) 11 k (1) 1 k (1) 1 k(1)... K = + k() 11 k (i 1) + k (i) 11 k (i) 1 k (i) 1 k (i) + k(i+1) 11 Simiary the goba mass matrix M can aso be ound. These goba mass and stiness matrices are used in (11) to generate a state space representation. In this paper a cantiever beam consisting o 1 nodes, as shown in Fig, is considered. Each node has our states (d, θ, d, θ), and the overa system has ive inputs and outputs at every second node aso shown in Fig. The inputs are the orces, and the outputs are the dispacements d o these eements. It is desired to convert the mode into an interconnected system with ive subsystems, each consisting o two nodes (shown by dashed boxes in Fig ). Thus we have xi = 8 and zi, wi = 1 i = 1,...,5. The numerica vaues o the physica parameters used or the simuation are isted in Tabe 1. By appying the too presented here to this umped system, an interconnected system is created. Both umped and interconnected systems are simuated. The resuting dispacement and ange o the ast node are shown in Fig 3 and 4. It is observed that the error between umped and interconnected system remain beow the numerica precision, which shows that the decomposition retains a... w 1 w z 1 z Fig.. Beam cantiever Tabe 1. Physica parameters o the cantiever exampe Fied w 3 z 3 w 4 z 4 Vaues Area o crossection m Length 1 m Length o eement.1 m Moduus o Easticity N m the dynamics o the system. Note that each subsystem has ony 8 states, as compared to a tota o 4 states. Dispacement [m] Times [sec] Fig. 3. Dispacement (d) o the 1 th node Anguar dispacement [rad] Times [sec] Fig. 4. Anguar dispacement (θ) o the 1 th node 5. CONTROLLER SYNTHESIS The active vibration contro o acantiever can be considered as an input disturbance rejection probem as shown w 5 z 5 669
5 17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 611, 8 in Figure 5, where, W 1 and W are the weighting iters. Let the state space o the weighted generaized pant be given as, ẋ = Ax + Bw + Bu z = C 1 x + D 1 u (18) y = C x + D 1 w where, d,z 1,z are the disturbance,inputs and outputs acting on every node. In the cantiever exampe we have considered state eedback case and d incude both orce and torque disturbances acting on each node. Using the d K(s) u G(s) Fig. 5. Generaized pant used or oopshaping. approach presented in section the umped system (18) is decomposed, where dynamics o the i th subsystem can be written as ẋ i (t) = A ii x i (t) + B 1i w i (t) + B i u i (t) + z i (t) = C 1i x i (t) + D 1 u i (t) y i (t) = C i x i (t) + D 1i w i (t) y W1 W N j=1,j i z1 z A ij x j (t) i = 1,,...N (19) x i R ni, w i R ri, u i R mi, z i R (i+mi) and y i R i are the states, disturbance input, controed input, controed output and measured output o the i th subsystem. The matrices A ii, B 1i, B i, C 1i, C i, D 1i and D 1i are o appropriate dimensions ound by decomp(). A stricty proper i th output eedback controer is deined by ẋ ci = A ci x ci + B ci y i u i = C ci x ci () i = 1,,...,N From (19) and () the cosed oop system rom w to z is given by where ẋ c = A c x c + B c w [ ] A B C A c = c B c C A c z = C c x c (1) B c = [ B1 B c D 1 ] C c = [C 1 D 1 C c ] () Decentraized contro o the system (18) requires that A c = diag{a c1,...a cn }, B c = diag{b c1,...b cn }, C c = diag{c c1,...,c cn }. In order to obtain a decentraized contro aw one needs to impose structura constraints on the bounded rea emma. In (Chen et a., 5) it has been proposed that or a given constant γ >, the system (18) is stabiizabe with the disturbance attenuation eve γ by a decentraized controer (), i there exist positive deinite bockdiagonamatrices X,Y andbockdiagona matrices F,L,Q such that T <, where T = and J 11 J1 T B 1 XC 1 J 1 J Y B 1 + LD 1 C1 T B1 T B1 T Y + D1L T T γi C 1 X + D 1 F C 1 γi [ ] X I > I Y (3) Positive deinite bock diagona matrices X and Y and diagona matrices F, L, Q are obtained by soving the above biinear matrix inequaity (BMI). Bock diagona coeicient matrices are then given by A c = V 1 QU T, B c = V 1 L, C c = FU T (4) where J 11 = AX + XA T + B F + F T B T, J 1 = A T + Y AX + LC X + Y B F + Q, J = Y A + A T Y + LC + C T L T. The size o submatrices in the bock diagona are compatibe with the dimensions o the state, input, and output vectors o the subsystems. The approach presented in (Chen et a., 5) invoves ixing a group o variabes to convert it into an LMI. First the variabes L, and Y and then the variabes X, F are ixed. Soving these two LMIs aternatey is then a way o inding a soution. In the proposed approach the path rom centraized controer to decentraized controer is divided into M steps and the structura constraints are graduay appied, by deining a rea number λ such that λ = k M, where k is graduay increased rom to M. This deines a matrix unction H as H(X,Y,F,L,Q,λ) = T(X,Y,F,L,(1 λ)q F + λq) < This is same as (3) except that J 1 is repaced by A T + Y AX + LC X + Y B F + (1 λ)q F + λq. Then { T(X,Y,F,L,QF ), λ = H(X,Y,F,L,Q,λ) = T(X,Y,F,L,Q), λ = 1 By using the approach o (Chen et a., 5) a state eedback decentraized controer or the active vibration contro o exibe beam is designed or W 1 = 1 and W =.1. The requency domain cosedoop response between theorce input, ineardispacement andtheorce disturbance or the ast node are as shown in Figure 6 and 7. These igures show that the open oop system wi exhibit an osciatory response whereas in the cosed oop system the disturbance is rejected, whie the contro gain remains in reasonabe imits. One point which must be kept in mind is that or simpicity these requency domain resuts are generated rom the state space representation without imposing the appropriate boundary conditions. 67
6 17th IFAC Word Congress (IFAC'8) Seou, Korea, Juy 611, 8 Singuar Vaues (db) Singuar Vaues 1 Frequency 1 1 (rad/sec) Fig. 6. Frequencyresponse o the openoop system (upper curve) and the cosedoop system (ower curve) Singuar Vaues (db) N. Chen, M. Ikeda, and W. Gui. Design o robust H contro or interconnected systems: A homotopy method. Int. J. o Contro, Automation and Systems, 3(), 5. R. D Andrea and G. E. Duerud. Distributed contro design or spatiay interconnected systems. IEEE Transactions on Automatic Contro, 48(9): , 3. K. Huebner, D. Dewhirst, D. Smith, and T. Byrom. The Finite Eement Method or Engineers. A Wiey Interscience Pubication, 1. ISBN J. Juang and M. Phan. Identiication and Contro o Mechanica Systems. Cambridge Universitypress, Cambridge, 4. ISBN D. Sijak. Large Scae Dynamic Systems  Stabiity and Structure. The Netherands: North Hoand, Amsterdam, D. Sijak and A. Zecevic. Contro o argescae systems: Beyond decnetraized eedback. Annua Reviews in Contro, 9(): , 5. G. Zhai, M. Ikeda, and Y. Fujisaki. Decentraized H controer design: A matrix inequaity approach using a homotopy method. Automatica, 37(4), Frequency 1 4 (rad/sec) Fig. 7. Contro sensitivity 6. CONCLUSION The main idea presented in this paper is that instead o reducing the mode order o a high order arge system one can decompose the overa system into sma subsystems, where each o these subsystems interacts with its neighboring eements. This approach is suitabe or FEM generated modes since these modes are oten o very high order and are generated rom a combination o sma eements. A simpe yet eicient too is presented here to carry out such a decomposition. This aows the appication o anaysis and synthesis toos or arge interconnected systems, and enabes the designer to synthesize controers or such systems in a systematic way. As an iustrative exampe, the too presented in this paper is appied to a cantiever mode. Resuts shows that the time responses o the origina system and the one decomposed into interconnected orm give identica resuts, up to numerica precision. Furthermore a previousy presented approach o designingdecentraized controer is appied to the decomposed system to demonstrate that by using the decomposition one can systematicay design a distributed contro strategy or FEMgenerated systems without reducing the mode order. REFERENCES A. Becker. An introductory Guide to inite eement anaysis. Proessiona Engineering Pubishing, 4. ISBN
PERFORMANCE COMPARISON OF OPEN AND CLOSED LOOP OPERATION OF UPFC
OL. 3, NO. 5, OCTOE 008 ISSN 896608 AN Journa o Engineering and Appied Sciences 006008 Asian esearch ubishing Networ (AN. A rights reserved. www.arpnournas.com EFOMANCE COMAISON OF OEN AND CLOSED LOO
More informationCombining reaction kinetics to the multiphase Gibbs energy calculation
7 th European Symposium on Computer Aided Process Engineering ESCAPE7 V. Pesu and P.S. Agachi (Editors) 2007 Esevier B.V. A rights reserved. Combining reaction inetics to the mutiphase Gibbs energy cacuation
More informationFirstOrder Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems FirstOrder Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationLecture 6: Moderately Large Deflection Theory of Beams
Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey
More informationCOMPENSATION FOR DISTORTION OF DYNAMIC SENSOR DATA USING RECEPTANCE COUPLING
COMPENSATION FO DISTOTION OF DYNAMIC SENSO DATA SIN ECEPTANCE COPIN Tony. Schmitz niversity o Forida, Department o Mechanica and Aerospace Engineering, ainesvie, F 36. INTODCTION Sensors can be used to
More informationA Simple and Efficient Algorithm of 3D SingleSource Localization with Uniform Cross Array Bing Xue 1 2 a) * Guangyou Fang 1 2 b and Yicai Ji 1 2 c)
A Simpe Efficient Agorithm of 3D SingeSource Locaization with Uniform Cross Array Bing Xue a * Guangyou Fang b Yicai Ji c Key Laboratory of Eectromagnetic Radiation Sensing Technoogy, Institute of Eectronics,
More informationImproving the Reliability of a SeriesParallel System Using Modified Weibull Distribution
Internationa Mathematica Forum, Vo. 12, 217, no. 6, 257269 HIKARI Ltd, www.mhikari.com https://doi.org/1.12988/imf.217.611155 Improving the Reiabiity of a SeriesParae System Using Modified Weibu Distribution
More informationPhysics 235 Chapter 8. Chapter 8 CentralForce Motion
Physics 35 Chapter 8 Chapter 8 CentraForce Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationLeaderFollower Consensus Modeling Representative Democracy
Proceedings o the Internationa Conerence o Contro, Dynamic Systems, and Robotics Ottawa, Ontario, Canada, May 78, 2015 Paper o. 156 LeaderFoower Consensus Modeing Representative Democracy Subhradeep
More informationStrain Energy in Linear Elastic Solids
Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua
More informationTeecommunication Network Contro and Management (EE E694) Prof. A. A. Lazar Notes for the ecture of 7/Feb/95 by Huayan Wang (this document was ast LaT E Xed on May 9,995) Queueing Primer for Muticass Optima
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationCE601Structura Anaysis I UNITIV SOPEDEFECTION METHOD 1. What are the assumptions made in sopedefection method? (i) Between each pair of the supports the beam section is constant. (ii) The joint in
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
More informationApproximated MLC shape matrix decomposition with interleaf collision constraint
Approximated MLC shape matrix decomposition with intereaf coision constraint Thomas Kainowski Antje Kiese Abstract Shape matrix decomposition is a subprobem in radiation therapy panning. A given fuence
More informationXI PHYSICS. M. Affan Khan LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress.com
XI PHYSICS M. Affan Khan LECTURER PHYSICS, AKHSS, K affan_414@ive.com https://promotephysics.wordpress.com [TORQUE, ANGULAR MOMENTUM & EQUILIBRIUM] CHAPTER NO. 5 Okay here we are going to discuss Rotationa
More informationA SIMPLIFIED DESIGN OF MULTIDIMENSIONAL TRANSFER FUNCTION MODELS
A SIPLIFIED DESIGN OF ULTIDIENSIONAL TRANSFER FUNCTION ODELS Stefan Petrausch, Rudof Rabenstein utimedia Communications and Signa Procesg, University of ErangenNuremberg, Cauerstr. 7, 958 Erangen, GERANY
More informationNumerical Simulation for Optimizing Temperature Gradients during Single Crystal Casting Process
ISIJ Internationa Vo 54 (2014) No 2 pp 254 258 Numerica Simuation for Optimizing Temperature Gradients during Singe Crysta Casting Process Aeksandr Aeksandrovich INOZEMTSEV 1) Aeksandra Sergeevna DUBROVSKAYA
More informationCALIBRATION OF RIVER BED ROUGHNESS
CALIBRATION OF RIVER BED ROUGHNESS A. M. Wasantha La 1, M. ASCE Singuar vaue decomposition (SVD) is used to caibrate the Manning s roughness coefficients in a 1D unsteady fow mode of the Upper Niagara
More informationPHYS 110B  HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B  HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationIdentification of macro and micro parameters in solidification model
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vo. 55, No. 1, 27 Identification of macro and micro parameters in soidification mode B. MOCHNACKI 1 and E. MAJCHRZAK 2,1 1 Czestochowa University
More informationThe influence of temperature of photovoltaic modules on performance of solar power plant
IOSR Journa of Engineering (IOSRJEN) ISSN (e): 22503021, ISSN (p): 22788719 Vo. 05, Issue 04 (Apri. 2015), V1 PP 0915 www.iosrjen.org The infuence of temperature of photovotaic modues on performance
More informationMultigrid Method for Elliptic Control Problems
J OHANNES KEPLER UNIVERSITÄT LINZ Netzwerk f ür Forschung, L ehre und Praxis Mutigrid Method for Eiptic Contro Probems MASTERARBEIT zur Erangung des akademischen Grades MASTER OF SCIENCE in der Studienrichtung
More informationDuality, Polite Waterfilling, and Optimization for MIMO BMAC Interference Networks and itree Networks
Duaity, Poite Waterfiing, and Optimization for MIMO BMAC Interference Networks and itree Networks 1 An Liu, Youjian Eugene) Liu, Haige Xiang, Wu Luo arxiv:1004.2484v3 [cs.it] 4 Feb 2014 Abstract This
More informationSchool of Electrical Engineering, University of Bath, Claverton Down, Bath BA2 7AY
The ogic of Booean matrices C. R. Edwards Schoo of Eectrica Engineering, Universit of Bath, Caverton Down, Bath BA2 7AY A Booean matrix agebra is described which enabes man ogica functions to be manipuated
More information6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17. Solution 7
6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17 Soution 7 Probem 1: Generating Random Variabes Each part of this probem requires impementation in MATLAB. For the
More informationAsynchronous Control for Coupled Markov Decision Systems
INFORMATION THEORY WORKSHOP (ITW) 22 Asynchronous Contro for Couped Marov Decision Systems Michae J. Neey University of Southern Caifornia Abstract This paper considers optima contro for a coection of
More informationMA 201: Partial Differential Equations Lecture  10
MA 201: Partia Differentia Equations Lecture  10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over
More informationBALANCING REGULAR MATRIX PENCILS
BALANCING REGULAR MATRIX PENCILS DAMIEN LEMONNIER AND PAUL VAN DOOREN Abstract. In this paper we present a new diagona baancing technique for reguar matrix pencis λb A, which aims at reducing the sensitivity
More informationACTIVE electrical networks, which require energizing sources for their operation, are most widely
1 Stabiization heory For Active Mutiport Networks Mayuresh Bakshi Member, IEEE,, Virendra R Sue and Maryam Shoejai Baghini, Senior Member, IEEE, arxiv:1606.03194v1 [cs.sy] 10 Jun 2016 Abstract his paper
More informationTIME DEPENDENT TEMPERATURE DISTRIBUTION MODEL IN LAYERED HUMAN DERMAL PART
VOL. 8, No. II, DECEMBER, 0, pp 6676 TIME DEPENDENT TEMPERATURE DISTRIBUTION MODEL IN LAYERED HUMAN DERMAL PART Saraswati Acharya*, D. B. Gurung, V. P. Saxena Department of Natura Sciences (Mathematics),
More informationEfficiently Generating Random Bits from Finite State Markov Chains
1 Efficienty Generating Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More informationUnit 48: Structural Behaviour and Detailing for Construction. Deflection of Beams
Unit 48: Structura Behaviour and Detaiing for Construction 4.1 Introduction Defection of Beams This topic investigates the deformation of beams as the direct effect of that bending tendency, which affects
More informationModel Predictive Control of Interconnected Linear and Nonlinear Processes
Ind. Eng. Chem. Res. 2002, 41, 801816 801 Mode Predictive Contro of Interconnected Linear and Noninear Processes GuangYan Zhu and Michae A. Henson* Department of Chemica Engineering, Louisiana State
More informationarxiv: v2 [hepph] 18 Apr 2008
Entanged maxima mixings in U PMNS = U U ν, and a connection to compex mass textures Svenja Niehage a, Water Winter b arxiv:0804.1546v [hepph] 18 Apr 008 Institut für Theoretische Physik und Astrophysik,
More informationApplied Nuclear Physics (Fall 2006) Lecture 7 (10/2/06) Overview of Cross Section Calculation
22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Overview of Cross Section Cacuation References P. Roman, Advanced Quantum Theory (AddisonWesey, Reading, 1965), Chap 3. A. Foderaro, The Eements
More informationReliability Improvement with Optimal Placement of Distributed Generation in Distribution System
Reiabiity Improvement with Optima Pacement of Distributed Generation in Distribution System N. Rugthaicharoencheep, T. Langtharthong Abstract This paper presents the optima pacement and sizing of distributed
More informationDynamic Matching Markets and Voting Paths
Dynamic Matching Markets and Voting Paths David J. Abraham Teikepai Kavitha Abstract We consider a matching market, in which the aim is to maintain a popuar matching between a set o appicants and a set
More informationBending Analysis of Continuous Castellated Beams
Bending Anaysis of Continuous Casteated Beams * Sahar Eaiwi 1), Boksun Kim ) and Longyuan Li 3) 1), ), 3) Schoo of Engineering, Pymouth University, Drake Circus, Pymouth, UK PL4 8AA 1) sahar.eaiwi@pymouth.ac.uk
More informationNEW DEVELOPMENT OF OPTIMAL COMPUTING BUDGET ALLOCATION FOR DISCRETE EVENT SIMULATION
NEW DEVELOPMENT OF OPTIMAL COMPUTING BUDGET ALLOCATION FOR DISCRETE EVENT SIMULATION HsiaoChang Chen Dept. of Systems Engineering University of Pennsyvania Phiadephia, PA 904635, U.S.A. ChunHung Chen
More informationSemidefinite relaxation and BranchandBound Algorithm for LPECs
Semidefinite reaxation and BranchandBound Agorithm for LPECs Marcia H. C. Fampa Universidade Federa do Rio de Janeiro Instituto de Matemática e COPPE. Caixa Posta 68530 Rio de Janeiro RJ 21941590 Brasi
More informationCodes between MBR and MSR Points with Exact Repair Property
Codes between MBR and MSR Points with Exact Repair Property 1 Toni Ernva arxiv:1312.5106v1 [cs.it] 18 Dec 2013 Abstract In this paper distributed storage systems with exact repair are studied. A construction
More informationThe EM Algorithm applied to determining new limit points of Mahler measures
Contro and Cybernetics vo. 39 (2010) No. 4 The EM Agorithm appied to determining new imit points of Maher measures by Souad E Otmani, Georges Rhin and JeanMarc SacÉpée Université Pau VeraineMetz, LMAM,
More informationSymbolic models for nonlinear control systems using approximate bisimulation
Symboic modes for noninear contro systems using approximate bisimuation Giordano Poa, Antoine Girard and Pauo Tabuada Abstract Contro systems are usuay modeed by differentia equations describing how physica
More informationTwoStage Least Squares as Minimum Distance
TwoStage Least Squares as Minimum Distance Frank Windmeijer Discussion Paper 17 / 683 7 June 2017 Department of Economics University of Bristo Priory Road Compex Bristo BS8 1TU United Kingdom TwoStage
More informationDelay Analysis of Maximum Weight Scheduling in Wireless Ad Hoc Networks
1 Deay Anaysis o Maximum Weight Scheduing in Wireess Ad Hoc Networks ong Bao e, Krishna Jagannathan, and Eytan Modiano Abstract This paper studies deay properties o the weknown maximum weight scheduing
More informationLegendre Polynomials  Lecture 8
Legendre Poynomias  Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More informationTensegrity Structures Prestressability Investigation
Tensegrity Structures Prestressabiity Investigation Corne Sutan* and Robert Sketon** *Harvard University Boston MA 0115 USA **University of Caifornia La Joa CA 9090411 USA (Received 10th October 001 revised
More informationConsistent linguistic fuzzy preference relation with multigranular uncertain linguistic information for solving decision making problems
Consistent inguistic fuzzy preference reation with mutigranuar uncertain inguistic information for soving decision making probems Siti mnah Binti Mohd Ridzuan, and Daud Mohamad Citation: IP Conference
More informationBCCS TECHNICAL REPORT SERIES
BCCS TCHNICAL RPORT SRIS The CrouzeixRaviart F on Nonmatching Grids with an Approximate Mortar Condition Taa Rahman, Petter Bjørstad and Xuejun Xu RPORT No. 17 March 006 UNIFOB the University of Bergen
More informationSimple models for rope substructure mechanics: application to electromechanical lifts
Journa of Physics: Conference Series PAPER OPEN ACCESS Simpe modes for rope substructure mechanics: appication to eectromechanica ifts To cite this artice: I Herrera and S Kaczmarczyk 016 J. Phys.: Conf.
More informationConditions for SaddlePoint Equilibria in OutputFeedback MPC with MHE
Conditions for SaddePoint Equiibria in OutputFeedback MPC with MHE David A. Copp and João P. Hespanha Abstract A new method for soving outputfeedback mode predictive contro (MPC) and moving horizon
More informationarxiv: v2 [astroph.sr] 13 Aug 2013
Astronomy & Astrophysics manuscript no. ms c ESO 2013 August 14, 2013 PORTA: A threedimensiona mutieve radiative transfer code for modeing the intensity and poarization of spectra ines with massivey parae
More informationLaboratory Exercise 1: Pendulum Acceleration Measurement and Prediction Laboratory Handout AME 20213: Fundamentals of Measurements and Data Analysis
Laboratory Exercise 1: Penduum Acceeration Measurement and Prediction Laboratory Handout AME 20213: Fundamentas of Measurements and Data Anaysis Prepared by: Danie Van Ness Date exercises to be performed:
More informationMethods for Ordinary Differential Equations. Jacob White
Introduction to Simuation  Lecture 12 for Ordinary Differentia Equations Jacob White Thanks to Deepak Ramaswamy, Jaime Peraire, Micha Rewienski, and Karen Veroy Outine Initia Vaue probem exampes Signa
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationVersion 2.2 NE03  Faraday's Law of Induction
Definition Version. Laboratory Manua Department of Physics he University of Hong Kong Aims o demonstrate various properties of Faraday s Law such as: 1. Verify the aw.. Demonstrate the ighty damped osciation
More informationExperimental Investigation and Numerical Analysis of New MultiRibbed Slab Structure
Experimenta Investigation and Numerica Anaysis of New MutiRibbed Sab Structure Jie TIAN Xi an University of Technoogy, China Wei HUANG Xi an University of Architecture & Technoogy, China Junong LU Xi
More informationAvailable online at ScienceDirect. Procedia Computer Science 96 (2016 )
Avaiabe onine at www.sciencedirect.com ScienceDirect Procedia Computer Science 96 (206 92 99 20th Internationa Conference on Knowedge Based and Inteigent Information and Engineering Systems Connected categorica
More informationAn approximate method for solving the inverse scattering problem with fixedenergy data
J. Inv. IPosed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixedenergy data A. G. Ramm and W. Scheid Received May 12, 1999
More information2.1. Cantilever The Hooke's law
.1. Cantiever.1.1 The Hooke's aw The cantiever is the most common sensor of the force interaction in atomic force microscopy. The atomic force microscope acquires any information about a surface because
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The yaxis is directed down, xaxis is directed
More informationThe Hidden Beauty of Structural Dynamics
BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS DEPARTMENT OF STRUCTURAL MECHANICS Róbert K. NÉMETH Attia KOCSIS The Hidden Beauty of Structura Dynamics Budapest, 213 ISBN 978963313889 CONTENTS Contents
More informationPosition Control of Rolling Skateboard
osition Contro of Roing kateboard Baazs Varszegi enes Takacs Gabor tepan epartment of Appied Mechanics, Budapest University of Technoogy and Economics, Budapest, Hungary (emai: varszegi@mm.bme.hu) MTABME
More informationEfficient Generation of Random Bits from Finite State Markov Chains
Efficient Generation of Random Bits from Finite State Markov Chains Hongchao Zhou and Jehoshua Bruck, Feow, IEEE Abstract The probem of random number generation from an uncorreated random source (of unknown
More informationREVISITING AND EXTENDING INTERFACE PENALTIES FOR MULTIDOMAIN SUMMATIONBYPARTS OPERATORS
REVISITING AND EXTENDING INTERFACE PENALTIES FOR MULTIDOMAIN SUMMATIONBYPARTS OPERATORS Mark H. Carpenter, Jan Nordström David Gottieb Abstract A genera interface procedure is presented for mutidomain
More informationIterative Decoding Performance Bounds for LDPC Codes on Noisy Channels
Iterative Decoding Performance Bounds for LDPC Codes on Noisy Channes arxiv:cs/060700v1 [cs.it] 6 Ju 006 ChunHao Hsu and Achieas Anastasopouos Eectrica Engineering and Computer Science Department University
More informationREACTION BARRIER TRANSPARENCY FOR COLD FUSION WITH DEUTERIUM AND HYDROGEN
REACTION BARRIER TRANSPARENCY FOR COLD FUSION WITH DEUTERIUM AND HYDROGEN Yeong E. Kim, JinHee Yoon Department of Physics, Purdue University West Lafayette, IN 4797 Aexander L. Zubarev Racah Institute
More informationVALIDATED CONTINUATION FOR EQUILIBRIA OF PDES
VALIDATED CONTINUATION FOR EQUILIBRIA OF PDES SARAH DAY, JEANPHILIPPE LESSARD, AND KONSTANTIN MISCHAIKOW Abstract. One of the most efficient methods for determining the equiibria of a continuous parameterized
More informationEXACT CLOSED FORM FORMULA FOR SELF INDUC TANCE OF CONDUCTOR OF RECTANGULAR CROSS SECTION
Progress In Eectromagnetics Research M, Vo. 26, 225 236, 22 EXACT COSED FORM FORMUA FOR SEF INDUC TANCE OF CONDUCTOR OF RECTANGUAR CROSS SECTION Z. Piatek, * and B. Baron 2 Czestochowa University of Technoogy,
More informationIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 2, FEBRUARY
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 857 Optima Energy and Data Routing in Networks With Energy Cooperation Berk Gurakan, Student Member, IEEE, OmurOze,Member, IEEE,
More informationMultiscale Multilevel Approach to Solution of Nanotechnology Problems
Mathematica Modeing and Computationa Physics 2017 Mutiscae Mutieve Approach to Soution of Nanotechnoogy Probems Sergey Poyakov 1,2, and Viktoriia Podryga 1,3, 1 Kedysh Institute of Appied Mathematics of
More informationArticle An Efficient Method to Construct ParityCheck Matrices for Recursively Encoding Spatially Coupled LDPC Codes
entropy Artice An Efficient Method to Construct ParityCheck Matrices for Recursivey Encoding Spatiay Couped LDPC Codes Zhongwei Si, Sijie Wang and Junyang Ma Key Laboratory of Universa Wireess Communications,
More informationDesign of Rectangular Openings in Unbonded PostTensioned Precast Concrete Walls
Design o Rectanguar Openings in Unbonded PostTensioned Precast Concrete Was Apri 21 Michae Aen and Yahya C. Kurama Report #NDSE11 Structura Engineering Research Report Department o Civi Engineering
More informationSupplementary Appendix (not for publication) for: The Value of Network Information
Suppementary Appendix not for pubication for: The Vaue of Network Information Itay P. Fainmesser and Andrea Gaeotti September 6, 03 This appendix incudes the proof of Proposition from the paper "The Vaue
More informationMaryam Tabatabaei, Dy Le, Satya N. Atluri
Neary Exact and Highy Efficient EasticPastic Homogenization and/or Direct Numerica Simuation of LowMass Metaic Systems with Architected Ceuar Microstructures Maryam Tabatabaei, Dy Le, Satya N. Aturi
More informationKonradZuseZentrum für Informationstechnik Berlin Heilbronner Str. 10, D Berlin  Wilmersdorf
KonradZuseZentrum für Informationstechnik Berin Heibronner Str. 10, D10711 Berin  Wimersdorf Raf Kornhuber Monotone Mutigrid Methods for Eiptic Variationa Inequaities II Preprint SC 93{19 (Marz 1994)
More informationNonperturbative Shell Correction to the Bethe Bloch Formula for the Energy Losses of Fast Charged Particles
ISSN 0023640, JETP Letters, 20, Vo. 94, No., pp. 5. Peiades Pubishing, Inc., 20. Origina Russian Text V.I. Matveev, D.N. Makarov, 20, pubished in Pis ma v Zhurna Eksperimenta noi i Teoreticheskoi Fiziki,
More informationSupporting Information for Suppressing Klein tunneling in graphene using a onedimensional array of localized scatterers
Supporting Information for Suppressing Kein tunneing in graphene using a onedimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes,
More informationSpring Gravity Compensation Using the Noncircular Pulley and Cable For the LessSpring Design
The 14th IFToMM Word Congress, Taipei, Taiwan, October 2530, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.PS3.010 Spring Gravity Compensation Using the Noncircuar Puey and Cabe For the LessSpring Design M.C.
More informationMigration of Ground Penetrating Radar data in heterogeneous and dispersive media
New Strategies for European Remote Sensing, Oui (ed.) 25 Mipress, Rotterdam, ISBN 9 5966 3 X Migration of Ground Penetrating Radar data in heterogeneous and dispersive media Armando R. Sena, Pau L. Stoffa
More informationhttps://doi.org/ /epjconf/
HOW TO APPLY THE OPTIMAL ESTIMATION METHOD TO YOUR LIDAR MEASUREMENTS FOR IMPROVED RETRIEVALS OF TEMPERATURE AND COMPOSITION R. J. Sica 1,2,*, A. Haefee 2,1, A. Jaai 1, S. Gamage 1 and G. Farhani 1 1 Department
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the zdirection
More informationRandom maps and attractors in random Boolean networks
LU TP 0443 Rom maps attractors in rom Booean networks Björn Samuesson Car Troein Compex Systems Division, Department of Theoretica Physics Lund University, Sövegatan 4A, S3 6 Lund, Sweden Dated: 0050507)
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationThe Weighted Sum Rate Maximization in MIMO Interference Networks: Minimax Lagrangian Duality and Algorithm
1 The Weighted Sum Rate Maximization in MIMO Interference Networks: Minimax Lagrangian Duaity and Agorithm Lijun Chen Seungi You Abstract We take a new approach to the weighted sumrate maximization in
More informationarxiv: v1 [math.ca] 6 Mar 2017
Indefinite Integras of Spherica Besse Functions MITCTP/487 arxiv:703.0648v [math.ca] 6 Mar 07 Joyon K. Boomfied,, Stephen H. P. Face,, and Zander Moss, Center for Theoretica Physics, Laboratory for Nucear
More informationSMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA. Hiroyuki Fujioka and Hiroyuki Kano. Magnus Egerstedt. Clyde F. Martin
Internationa Journa of Innovative Computing, Information and Contro ICIC Internationa c 5 ISSN 49498 Voume x, Number x, x 5 pp. SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA Hiroyuki Fujioka and
More informationModel Calculation of n + 6 Li Reactions Below 20 MeV
Commun. Theor. Phys. (Beijing, China) 36 (2001) pp. 437 442 c Internationa Academic Pubishers Vo. 36, No. 4, October 15, 2001 Mode Cacuation of n + 6 Li Reactions Beow 20 MeV ZHANG JingShang and HAN YinLu
More informationQuantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18
Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms
More informationModelbased Clustering by Probabilistic Selforganizing Maps
EEE TRANATN N NERA NETRK, V. XX, N., odebased ustering by robabiistic eforganizing aps hihian heng, Hsinhia u, ember, EEE, and Hsinin ang, ember, EEE Abstract n this paper, we consider the earning
More informationMCCDMA CDMA Systems. Introduction. Ivan Cosovic. Stefan Kaiser. IEEE Communication Theory Workshop 2005 Park City, USA, June 15, 2005
On the Adaptivity in Down and Upink MC Systems Ivan Cosovic German Aerospace Center (DLR) Institute of Comm. and Navigation Oberpfaffenhofen, Germany Stefan Kaiser DoCoMo EuroLabs Wireess Soution Laboratory
More informationarxiv: v1 [physics.fludyn] 2 Nov 2007
A theoretica anaysis of the resoution due to diffusion and sizedispersion of partices in deterministic atera dispacement devices arxiv:7.347v [physics.fudyn] 2 Nov 27 Martin Heer and Henrik Bruus MIC
More informationA New Algorithm for the Weighted Sum Rate Maximization in MIMO Interference Networks
A New Agorithm for the Weighted Sum Rate Maximization in MIMO Interference Networks Xing Li, Seungi You 2, Lijun Chen, An Liu 3, Youjian Eugene Liu Abstract We propose a new agorithm to sove the nonconvex
More informationDifferential Complexes in Continuum Mechanics
Archive for Rationa Mechanics and Anaysis manuscript No. (wi be inserted by the editor) Arzhang Angoshtari Arash Yavari Differentia Compexes in Continuum Mechanics Abstract We study some differentia compexes
More informationTitle: Crack Diagnostics via Fourier Transform: Real and Imaginary Components vs. Power Spectral Density
Tite: rack Diagnostics via ourier Transform: ea and maginary omponents vs. Power pectra Density Authors: Leonid M. Geman Victor Giurgiutiu van V. Petrunin 3 rd nternationa Workshop on tructura Heath Monitoring
More informationAcousticStructure Simulation of Exhaust Systems by Coupled FEM and Fast BEM
AcousticStructure Simuation of Exhaust Systems by Couped FEM and Fast BEM Lothar Gau, Michae Junge Institute of Appied and Experimenta Mechanics, University of Stuttgart Pfaffenwadring 9, 755 Stuttgart,
More informationData Mining Technology for Failure Prognostic of Avionics
IEEE Transactions on Aerospace and Eectronic Systems. Voume 38, #, pp.388403, 00. Data Mining Technoogy for Faiure Prognostic of Avionics V.A. Skormin, Binghamton University, Binghamton, NY, 1390, USA
More information