Finite Element Modeling of Seismic Response of Field Fabricated Liquefied Natural Gas (LNG) Spherical Storage Vessels
|
|
- Daniela Sanders
- 5 years ago
- Views:
Transcription
1 Egieerig, 013, 5, doi:10.436/eg Published Olie Jue 013 ( Fiite Elemet Modelig of Seismic Respose of Field Fabricated Liquefied Natural Gas (LNG) Spherical Storage Vessels Oludele Adeyefa, Oluleke Oluwole Departmet of Mechaical Egieerig, Uiversity of Ibada, Ibada, Nigeria Received April 5, 013; revised May 6, 013; accepted May 16, 013 Copyright 013 Oludele Adeyefa, Oluleke Oluwole. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. ABSTRACT All real physical structures behave dyamically whe subjected to loads or displacemets. This research paper, therefore, presets seismic respose of field fabricated liquefied atural gas spherical storage vessels usig fiite elemet aalysis. The seismic aalysis procedure used represets a practical approach i quatifyig the respose of spherical storage vessel with its cotet whe it is subjected to seismic loadig. I the fiite elemet method approach, six degrees of freedom per ode is used for legs/colum of the spherical storage taks. Lumped mass procedure is employed to determie system mass matrix of the structure. Computer programme code is developed for the resultig matrix equatio form fiite elemet aalysis of the structure usig FORTRAN 90 programmig laguage. The modelig of the seismic load utilies the groud acceleratio curve of a site. From the results of the modal aalysis, the system is ucoupled thereby gives way to the applicatio of Newmark s method. Newmark s method as oe of the widely used time-step approach for the seismic respose is applied. The developed programme codig is validated with aalytical results (P > 0.5). It shows that the approach i this research work ca be successfully used i determie the stability of large spherical storage vessels agaist seismic loadigs whe base acceleratio spectral of the site are kow. This approach gives better results tha the static-force approach which gives coservative results. While the approach used i this research treats seismic loads as time evet, static-force approach assumed that the full groud force due to seismic motio is applied istataeously. Keywords: FEM; LNG; Seismic; FORTRAN; Newmark; UBC; ASME 1. Itroductio I more ad more egieerig situatios today, it is ecessary to obtai approximate umerical solutios to problems rather tha exact closed-form solutios. It is discovered that may of the egieerig problems faced by egieers i today s world preset o simple aalytical solutios. Though, desig egieer may have bee able to write dow the goverig equatios ad boudary coditios for the problem, yet he sees immediately that o simple aalytical solutio ca be foud. The difficulty i these types of egieerig problems lies i the fact that either the geometry or some other feature of the problem is irregular or arbitrary. Aalytical solutios to problems of this type seldom exist; yet these are the kids of problems that desig egieers are called upo to solve. The resourcefuless of the aalyst usually comes to the rescue ad provides several alteratives to overcome this dilemma. Oe possibility is to simplify the difficulties ad reduce the problem to oe that ca be hadled. Sometimes this procedure works; but, more ofte tha ot, it leads to serious iaccuracies or wrog aswers. Now that high-speed computers are widely available, a more viable alterative is to retai the complexities of the problem ad fid a approximate umerical solutio. Fiite elemet method is oe the widely used approximate umerical method ad its model of a problem gives a piecewise approximatio to the goverig equatios. The basic premise of the fiite elemet method is that a solutio regio ca be aalytically modeled or approximated by replacig it with a assemblage of discrete elemets. Sice these elemets ca be put together i a variety of ways, they ca be used to represet ex- Copyright 013 SciRes.
2 544 O. ADEYEFA, O. OLUWOLE ceedigly complex shapes. Though, the fiite elemet method ca be used to solve a very large umber of complex problems, there are still some practical egieerig problems that are difficult to address because we lack a adequate theory of failure, or because we lack appropriate material data. This is ot a fiite elemet problem, but is of serious cocer to ay egieer who wats to supplat testig with aalysis. The use of aalysis usually permits faster desig turaroud, the exploratio of widely varyig eviromets, ad the use of optimiatio tools. The seismic aalysis procedures outlie i Uiversal Buildig Code (UBC) is static-force procedures, which assume that the etire seismic force due to groud motio is applied istataeously. This assumptio is coservative but greatly simplifies the calculatio procedures. I reality earth quakes are time-depedet evets ad the full force is ot realied istataeously. The UBC allows, ad i some cases requires, that a dyamic aalysis be performed i lieu of the static force method. Although much more sophisticated, ofte the seismic loadigs are reduced sigificatly.. Fiite Elemet Methodology The most commo ad effective approach for seismic aalysis of liear structural systems is the mode superpositio method. After a set of orthogoal vectors have bee evaluated, this method reduces the large set of global equilibrium equatios to a relatively small umber of ucoupled secod order differetial equatios. The umerical solutio of those equatios ivolves greatly reduced computatioal time..1. Model Assumptios Below are the assumptios made i the model i this research work: 1) The storage tak is full of water, hydrotest, for the worst case sceario. This gives maximum compressive load/force per leg support. ) I determiig the system mass matrix, lumped sum mass matrix is employed. 3) The oly exterally applied load/force is the weight of the cotet (tak full of water) ad shell. Exterally applied force o each of the leg is the sum of the weight of cotet (tak full of water) ad shell divided by the umber of legs. While there is base acceleratio applied to the support due to seismic effect Figure 1. 4) Seismic effect o oe of the storage tak support legs is represetative of the effect of seismic load o the etire storage tak structure. If oe of the legs fails due to seismic load, the etire structure fails Figure. Figure 1. Typical vibratio model with compressive force ad earth/support motio. Figure. Showig sectio of spherical tak with leg support... Dyamic Equilibrium Takig ito accout the equivalet static force, the geeral equatios of equilibrium for a structure is as follows Ku F F 0 (1) where {F} is the force vector represetig the exteral applied forces as a fuctio of time. {F} dy, is the assembled force vector for the complete structure due to iertia ad dampig forces. [K] is the system static stiffess matrix ad {u} is odal displacemet. For may structural systems, the approximatio of liear structural behaviour is made to covert the physiccal equilibrium statemet, Equatio (1), to the followig set of secod-order, liear, differetial equatios, geeral equatio of motio M u C u K u F () dy where M is the mass matrix (lumped or cosistet), C is a Copyright 013 SciRes.
3 O. ADEYEFA, O. OLUWOLE 545 viscous dampig matrix (which is ormally selected to approximate eergy dissipatio i the real structure) ad K is the static stiffess matrix for the system of structural elemets. The time-depedet vectors, u, u ad u are ode acceleratios, velocities ad displacemets respectively..3. System Stiffess Matrix To determie the stiffess matrix for the beam-colum support of spherical storage vessels. It is assumed that a typical member leg ca be thought of a 3-D space frame havig four actios: two bedig actios, a twistig actio, ad a axial actio. Thus, the displacemet of each ode is described by three traslatioal ad three rotatioal compoets of displacemet, givig six degrees of freedom at each urestraied ode. Correspodig to these degrees of freedom are six odal loads. The otatios use for the displacemet ad force vectors at each ode are, respectively, u u v w (3a) T x y f f f f m m m T (3b) e x y x y O the local level, for each member, the forces are related to the displacemets by the partitioed matrices Fe ku (4) k a a b b 0 b b c1 0 c c1 0 c 0 d d1 0 0 c c 0 c4 0 b3 0 b b4 a b b c1 0 c 0 d1 0 0 c3 0 b (5) where EA 1EI 1EI a1 b1 c EI 4EI 6EI y b 3 c c 3 b c EI y 4 b4, EI L e GI x d L e 4EI y E is the elastic modulus, G is the modulus of rigidity, L e is the legth of beam-colum elemet, I x, I y, ad I are the secod momet of iertia about x, y, ad respectively..4. System Mass Matrix It is useful to realie that usig lumped mass method to geerate the system mass matrix, the methodology does ot ivolve derivatives of the shape fuctios. We ca therefore be more lax about the choice of shape fuctio for the determiatio of system mass matrix tha for the stiffess matrix. I may applicatios, it is preferable to L e use a lumped mass (ad dampig) approximatio where the oly oero terms are o the diagoal. The simplest mass model is to cosider oly the ger traslatioal iertias, which are obtaied simply by dividig the total mass by the umber of odes ad placig this value of mass at the ode. Thus, the diagoal terms for the 3-D frame ca be give as 1 m AL e (6) where ρ is material desity of beam elemet, A is the beam-colum cross sectioal area ad L e is the legth of beam-colum ad [M] is the mass matrix. These eglect the rotatioal iertias of the flexural actios. Geerally, may researchers i the field of fiite elemet methods had proved that the cotributios of the rotatioal iertia of the flexural actios to the system mass matrix are egligible ad matrix Equatio (6) above is quite accurate especially whe the elemets are small. There is, however, a very importat circumstace whe a ero diagoal mass is uacceptable ad reasoable o- Copyright 013 SciRes.
4 546 O. ADEYEFA, O. OLUWOLE ero values are eeded. I the preset research, a ero diagoal mass is uacceptable because this leads to computer programme error whe applyig Newmark s method for the seismic respose aalysis. Therefore, Equatio (6) is the modified to give equatio below with o-ero diagoal. 1 m AL e (7) L where e. 40 where is typically take as uity. This scheme has the merit of correctly givig the traslatioal iertias. Therefore, estimate the effective legth L A/π as basically the radius of a disk of the same area as the triagle. Agai, α is typically take as uity..5. Free Udamped Vibratio Mode The structure is ot excited exterally i free vibratio mode i.e. o force or support motio acts o it. So, uder the coditio of free motio, dyamic aalysis ca be carried out ad the importat properties like atural frequecies ad mode shapes correspodig to the atural frequecy ca be obtaied. Sice free vibratio mode is cosidered, the structure is ot uder the ifluece of ay exteral force. Hece, the force ad dampig force vectors i stiffess equatios are take as ero. By takig the above coditio ito cosideratio, the stiffess equatio ca be represeted as, M u K u (8) 0 Thus u u (9) substitutig Equatio (9) ito Equatio (8) gives K M u 0 (10) This is liear eigevalue problem whose solutios, whe arraged i order of magitude of the eigevalues, are approximatio to the correspodig atural frequecies ad ormal modes of the vibratio of the system. To covert the geeralied eigevalue problem to the stadard form, the followig steps eed to be take: The geeral equatio is writte as or or Rearragig Equatio (11) gives K M u 0 (11) K u Mu 1 M K u u (11a) (11b) K M u u 1 1 Equatio (11c) may be rewritte as K M I u 1 1 (11c) 0 (1) where [I] is a idetity matrix The oly problem with Equatio (1) is that although [K] ad [M] are symmetric, the product [K] 1 [M] is geerally ot symmetric. To preserve symmetry, the Cholesky s method may be used [1]. The Cholesky method decomposes a square, symmetric matrix to the product of a upper triagular matrix ad the traspose of the upper triagular matrix. Applyig the Cholesky decompositio to the matrix [K] gives K CC T (13) Substitutig Equatio (13) ito (1) ad covertig it ito the stadard form gives (Griffiths ad Smith, 1991) 1 C M C I u 1 T 1 0 (14) Equatio (14) is i the form of a stadard eigevalue problem. It ca be expressed more closely to Equatio (11) if it is rewritte as where S Iu 0 (15) 1 1 T S C M C ad 1 Sice the matrices i eigevalue problem ofte become very large, the matrices may be coverted to a simpler form usig Householder s method before solvig for the eigevalues [1]. Householder s method coverts a symmetric matrix ito a tridiagoal matrix. A tridiagoal matrix has o-ero elemets oly o the diagoal plus or mius oe colum []. Solutios are obtaied usig FORTRAN 90 programmig laguage. The output of the program cosists of the atural frequecies of the system with the correspodig mode shapes. Sice the matrices i eigevalue problem ofte become very large, the matrices may be coverted to a simpler form usig Householder s method before solvig for the eigevalues [1]. Householder s method coverts a symmetric matrix ito a tridiagoal matrix. A tridiagoal matrix has o-ero elemets oly o the diagoal plus or mius oe colum []. Solutios are obtaied usig FORTRAN 90 programmig laguage. The output of the program cosists of the atural frequecies of the system with the correspodig mode shapes..6. Trasformatio of Modal Equatio The dyamic force equilibrium Equatio () ca be re- Copyright 013 SciRes.
5 O. ADEYEFA, O. OLUWOLE 547 writte i the followig form as a set of N secod order differetial equatios: M u C u K u F f g t j (16) J j1 All possible types of time-depedet loadig, icludig wid, wave ad seismic, ca be represeted by a sum of J space vectors f j, which are ot a fuctio of time, ad J time fuctios g(t) j. The umber of dyamic degrees of freedom is equal to the umber of lumped masses i the system. The fudametal mathematical method that is used to solve Equatio (16) is the separatio of variables. This approach assumes the solutio ca be expressed i the followig form: u t Y t (17a) where is a N by N matrix cotaiig N spatial vectors that are ot a fuctio of time, ad Y(t) is vector cotaiig N fuctios of time. From Equatio (17a), it follows that: u t Y t (17b) u t Y t (17c) Before solutio, it is require that the space fuctios satisfy the followig mass ad stiffess orthogoality coditios: T T M I ad K (18) where I is a diagoal uit matrix ad Ω is a diagoal matrix i which the diagoal terms are. The term has the uits of radia per secod. After substitutio of Equatios (17) ito Equatio (16) T ad the pre-multiplicatio by, the followig matrix of N equatios is produced: J j I Y d Y Y F P g t (19) j1 For three-dimesioal seismic motio, this equatio ca be writte as: T where Pj f j ad are defied as the modal participatio factors for load fuctio j. The term P j is associated with the th mode. There is oe set of N modal participatio factors for each spatial load coditio f j. For all real structures, the N by N matrix is ot diagoal; however, to ucouple the modal equatios, it is ecessary to assume classical dampig where there is o couplig betwee modes. Therefore, the diagoal terms of the modal dampig are defied by: d (0) where is defied as the ratio of the dampig i mode to the critical dampig of the mode [3]. A typical j j ucoupled modal equatio for liear structural systems is of the followig form: y t y t y t F P g t (1) j j j1 For three-dimesioal seismic motio, this equatio ca be writte as: yt y t yt () Put Put Put x gx y gy g where the three-directioal modal participatio factors, or i this case seismic excitatio factors, are defied by T Pj M j i which j is equal to x, y, or ad is equal to the mode umber. By examiig Equatio () above, the dampig term costat ca be writte as c (3) It has bee show that dampig ratio of a structure is proportioal to the structural stiffess. Therefore, this is the adopted i the preset work ad Equatio (3) the becomes where c is the mode dampig ratio. J (4).7. Step by Step Numerical Itegratio Procedure The step by step aalysis is based o the powerful iterative techique developed by [4] kow as Newmark s parameter method. The acceleratio at the ed of a time step is estimated, ad the velocity ad displacemet are the calculated by Pa (5) u, u, ad u represet the displacemet, velocity, ad acceleratio of ay poit. The suffixes i ad i + 1 represetig the eds of the time itervals i ad i + 1. The above expressios are substituted i the geeral equatios of motio ad a ew estimate for the acceleratio at the ed of the time step i + 1 is determied. The process is repeated util successive values of the acceleratio agree withi a specified tolerace. The parameter β i the Newmark s equatio govers the ifluece of the acceleratio at the ed of the time iterval i + 1 o i + 1 the displacemet at that istat. Furthermore, the value selected for β determies the variatio of the acceleratio durig the iterval Δt from i to i + 1, the method becomes a liear acceleratio assumptio [4]. A β value of 1/4 represets a costat acceleratio throughout the iterval ad β = 1/8 may be iterpreted as a step fuctio havig acceleratio u i over the first half of the time iterval ad u through the last Copyright 013 SciRes.
6 548 O. ADEYEFA, O. OLUWOLE half [4]..8. Participatig Mass Ratio This requiremet is based o a uit base acceleratio i a particular directio ad calculatig the base shear due to that load. The steady state solutio for this case ivolves o dampig or elastic forces; therefore, the modal respose equatios for a uit base acceleratio i the x-directio ca be writte as: y P (6) The ode poit iertia forces i the x-directio for that mode are by defiitio: f Mu t M y P M (7) x x x The resistig base shear i the x-directio for mode is the sum of all ode poit x forces. Or: V P I M P (8) T x x x x The total base shear i the x-directio, icludig N modes, will be: V x N Px 1 (9) For a uit base acceleratio i ay directio, the exact base shear must be equal to the sum of all mass compoets i that directio. Therefore, the participatig mass ratio is defied as the participatig mass divided by the total mass i that directio. Or: Z X Y mass mass mass N Px 1 M x N Py 1 M y N P 1 M (30a) (30b) (30c) A examiatio of values givig by the factors i Equatios (30) gives desig egieer a idicatio of the value of the base shear associated with each mode. The agle with respect to the x-axis of the base shear associated with the first mode is give by: P 1 1x 1 ta P 1y 3. Seismic Respose Aalysis (31) Case 1: To validate the computer program developed, modal aalysis is performed o a -degree of freedom structural system subjected to free udamped vibratio with the followig characteristics ad the results are compared with the aalytical method. M 0 0 1, K= 6 1 Case : Spherical pressure vessel leg support with the characteristics below is subjected to the seismic loadig. The leg colum is carbo steel, cold hollow circular pipe of outside diameter 33.9mm. The seismic loadig i this model is base acceleratio of Table 1. Compressive vertical Load/force per leg (N) = ; Momet of iertia about x-axis, I x (m 4 ) = ; Momet of iertia about y-axis, I y (m 4 ) = ; Momet of iertia about -axis, I (m 4 ) = ; gth of the support (m) =.0; Modulus of Elasticity, E (N/m ) = ; Poisso Ratio, v = 0.3; Modulus of Rigidity, G (N/m ) = E/((1 + v)); Material Desity (Kg/m 3 ) = 7850; Structural Dampig proportioal costat = Results ad Discussios The results of the Case 1 i Table show that the FOR- TRAN codig developed i this research give better results of the atural frequecy of free udamped vibratio cosidered. I case two cosidered, Figure 3 gives the variatio of atural frequecies agaist for differet modes. Mode 7 gives highest atural frequecy of value of over 4500 H while the least of all is mode 1 which gives atural frequecy with value of ero H. Figures 3-6 give relative respose of the secod mode of vibratio for acceleratio, velocity ad displacemet. I the computer simulatio for the Case, the same value of dampig ratio is assumed for all modes. The structural system uder cosideratio has the same value for the dampig costat i all modes. Also, the results show above for the case two simulatio cosidered seismic effect i oe directio aloe the vertical directio. I reality, the effects of base acceleratio i differet directios at the site o the structural system have to be cosidered ad superimposed. By doig this, the desig Figure 3. Showig frequecy (H) agaist mode. Copyright 013 SciRes.
7 O. ADEYEFA, O. OLUWOLE 549 Figure 4. Showig relative acceleratio respose of secod mode. Figure 6. Showig relative displacemet respose of secod mode. egieer will be able to have better uderstatig of the behavioral or respose patter of the structure to the seismic loadig. Figure 5. Showig relative velocity respose of secod mode. Table 1. Showig typical earthquake groud acceleratio with time. Time (Sec) Mode Acceleratio Spectrum (g) Time (Sec) Acceleratio Spectrum (g) Table. Comparig FEA ad aalytical solutios. Natural Frequecy (H) (FEA) Natural Frequecy (H) (Aalytical) Percetage Error (%) Coclusios ad Recommedatios The use of fiite elemet method i the seismic respose aalysis of field fabricated spherical pressure vessels caot be over emphasied. This is also i lie with the recommedatio from UBC which allows desig egieer to use other method such as fiite elemet method i carryig out the seismic respose of a structure. This is because; it is believed that static-force used by UBC code is coservative, ad to get accurate respose of ay structure subjectig to seismic load the use of fiite elemet aalysis is desirable. While it is oted that i ASME code sectio VIII Div 1, recommedatio is made to cosider other loads apart from iteral ad exteral pressures. Such other loads are wid, local ad seismic yet it silet o procedures for determiig the effects of such loads o pressure vessels. The decisio, therefore, lies i the hads of desig egieer to use ay suitable method available at his disposal i aalyig effects of such loads o pressure vessels. Oe of such methods available to desig egieers is fiite elemet method. If this method is properly applied, it will give soud ad safe desig. This research work has shed light o the methodology of applyig fiite elemet method to seismic respose aalysis of field fabricated spherical pressure vessels. The seismic aalysis procedure used represets a practical approach i quatifyig the respose of spherical storage vessel with its cotet whe it is subjected to seismic loadig. The result shows that the approach i this research work ca be successfully used i determie the stability of large spherical storage vessels agaist seismic loadigs whe base acceleratio spectral of the site is kow. This approach gives better results tha the static-force approach which gives coservative results. Copyright 013 SciRes.
8 550 O. ADEYEFA, O. OLUWOLE REFERENCES [1] D. V. Griffiths ad I. M. Smith, Numerical Methods for Egieers, Blackwell Scietific Publicatios, Lodo, [] W. H. Press, Numerical Recipes i C: The Art of Scietific Computig, Cambridge Uiversity Press, Cambridge, 199. [3] L. W. Edward, Three-Dimesioal Static ad Dyamic Aalysis of Structures, A Physical Approach with Emphasis o Earthquake Egieerig, Computers ad Structures Ic., Berkeley, 00. [4] N. M. Newmark, A Method of Computatio for Structural Dyamics, ASCE Joural of the Egieerig Mechaics Divisio, Vol. 85, 1959, pp Copyright 013 SciRes.
2C09 Design for seismic and climate changes
2C09 Desig for seismic ad climate chages Lecture 02: Dyamic respose of sigle-degree-of-freedom systems I Daiel Grecea, Politehica Uiversity of Timisoara 10/03/2014 Europea Erasmus Mudus Master Course Sustaiable
More informationCALCULATION OF STIFFNESS AND MASS ORTHOGONAL VECTORS
14. CALCULATION OF STIFFNESS AND MASS ORTHOGONAL VECTORS LDR Vectors are Always More Accurate tha Usig the Exact Eigevectors i a Mode Superpositio Aalysis 14.1 INTRODUCTION The major reaso to calculate
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationDYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS
DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS Ivaa Štimac 1, Ivica Kožar 1 M.Sc,Assistat, Ph.D. Professor 1, Faculty of Civil Egieerig, Uiverity of Rieka, Croatia INTRODUCTION The vehicle-iduced
More informationDamped Vibration of a Non-prismatic Beam with a Rotational Spring
Vibratios i Physical Systems Vol.6 (0) Damped Vibratio of a No-prismatic Beam with a Rotatioal Sprig Wojciech SOCHACK stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa,
More informationPROPOSING INPUT-DEPENDENT MODE CONTRIBUTION FACTORS FOR SIMPLIFIED SEISMIC RESPONSE ANALYSIS OF BUILDING SYSTEMS
he 4 th World Coferece o Earthquake Egieerig October -7, 008, Beiig, Chia PROPOSING INPU-DEPENDEN ODE CONRIBUION FACORS FOR SIPLIFIED SEISIC RESPONSE ANALYSIS OF BUILDING SYSES ahmood Hosseii ad Laya Abbasi
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationProblem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:
2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium
More informationStopping oscillations of a simple harmonic oscillator using an impulse force
It. J. Adv. Appl. Math. ad Mech. 5() (207) 6 (ISSN: 2347-2529) IJAAMM Joural homepage: www.ijaamm.com Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics Stoppig oscillatios of a simple harmoic
More informationLoad Dependent Ritz Vector Algorithm and Error Analysis
Load Depedet Ritz Vector Algorithm ad Error Aalysis Writte by Ed Wilso i 006. he Complete eigealue subspace I the aalysis of structures subected to three base acceleratios there is a requiremet that oe
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More information15. DYNAMIC ANALYSIS USING RESPONSE SPECTRUM SEISMIC LOADING
From Static ad Dyamic of Structures by Ed Wilso 1 Modified November 5, 13 15. DYNAMIC ANALYSIS USING RESPONSE SPECTRUM SEISMIC LOADING Before the Existece of Iexpesive Persoal Computers, the Respose Spectrum
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationSenvion SE Franz-Lenz-Str. 1, Osnabrück, Germany
Iteratioal Wid Egieerig Coferece IWEC 014 WAVE INDUCED FATIGUE LOADS ON MONOPILES - NEW APPROACHES FOR LUMPING OF SCATTER TABLES AND SITE SPECIFIC INTERPOLATION OF FATIGUE LOADS M. SEIDEL Sevio SE Fraz-Lez-Str.
More informationPOWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS
Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationDETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.
ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationMechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationStreamfunction-Vorticity Formulation
Streamfuctio-Vorticity Formulatio A. Salih Departmet of Aerospace Egieerig Idia Istitute of Space Sciece ad Techology, Thiruvaathapuram March 2013 The streamfuctio-vorticity formulatio was amog the first
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More information15. DYNAMIC ANALYSIS USING RESPONSE SPECTRUM SEISMIC LOADING
From Static ad Dyamic of Structures by Ed Wilso 1 Modified July 13, 14 15. DYNAMIC ANALYSIS USING RESPONSE SPECTRUM SEISMIC LOADING Before the Existece of Iexpesive Persoal Computers, the Respose Spectrum
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More informationChapter 7 z-transform
Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time
More informationANALYSIS OF EXPERIMENTAL ERRORS
ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationReliability and Queueing
Copyright 999 Uiversity of Califoria Reliability ad Queueig by David G. Messerschmitt Supplemetary sectio for Uderstadig Networked Applicatios: A First Course, Morga Kaufma, 999. Copyright otice: Permissio
More informationEXPERIMENT OF SIMPLE VIBRATION
EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414-423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationResearch Article Some E-J Generalized Hausdorff Matrices Not of Type M
Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E.
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationUsing Spreadsheets as a Computational Tool in Teaching Mechanical. Engineering
Proceedigs of the th WSEAS Iteratioal Coferece o COMPUTERS, Vouliagmei, Athes, Greece, July 3-5, 6 (pp35-3) Usig Spreadsheets as a Computatioal Tool i Teachig Mechaical Egieerig AHMADI-BROOGHANI, ZAHRA
More informationANALYSIS OF DAMPING EFFECT ON BEAM VIBRATION
Molecular ad Quatum Acoustics vol. 7, (6) 79 ANALYSIS OF DAMPING EFFECT ON BEAM VIBRATION Jerzy FILIPIAK 1, Lech SOLARZ, Korad ZUBKO 1 Istitute of Electroic ad Cotrol Systems, Techical Uiversity of Czestochowa,
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationModified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
More informationMETHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS
Please cite this article as: Staisław Kula, Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 009,
More informationProblem Cosider the curve give parametrically as x = si t ad y = + cos t for» t» ß: (a) Describe the path this traverses: Where does it start (whe t =
Mathematics Summer Wilso Fial Exam August 8, ANSWERS Problem 1 (a) Fid the solutio to y +x y = e x x that satisfies y() = 5 : This is already i the form we used for a first order liear differetial equatio,
More informationResearch Article A New Second-Order Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More informationFREE VIBRATIONS OF SIMPLY SUPPORTED BEAMS USING FOURIER SERIES
Abdullah : FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS FREE VIBRATIONS OF SIMPY SUPPORTED BEAMS USING FOURIER SERIES SAWA MUBARAK ABDUAH Assistat ecturer Uiversity of Mosul Abstract Fourier series will be
More informationAnalysis of composites with multiple rigid-line reinforcements by the BEM
Aalysis of composites with multiple rigid-lie reiforcemets by the BEM Piotr Fedeliski* Departmet of Stregth of Materials ad Computatioal Mechaics, Silesia Uiversity of Techology ul. Koarskiego 18A, 44-100
More informationFIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
More informationResearch Article Health Monitoring for a Structure Using Its Nonstationary Vibration
Hidawi Publishig Corporatio Advaces i Acoustics ad Vibratio Volume 2, Article ID 69652, 5 pages doi:.55/2/69652 Research Article Health Moitorig for a Structure Usig Its Nostatioary Vibratio Yoshimutsu
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationPILOT STUDY ON THE HORIZONTAL SHEAR BEHAVIOUR OF FRP RUBBER ISOLATORS
Asia-Pacific Coferece o FRP i Structures (APFIS 2007) S.T. Smith (ed) 2007 Iteratioal Istitute for FRP i Costructio PILOT STUDY ON THE HORIZONTAL SHEAR BEHAVIOUR OF FRP RUBBER ISOLATORS T.B. Peg *, J.Z.
More informationDynamic Response of Second Order Mechanical Systems with Viscous Dissipation forces
Hadout #b (pp. 4-55) Dyamic Respose o Secod Order Mechaical Systems with Viscous Dissipatio orces M X + DX + K X = F t () Periodic Forced Respose to F (t) = F o si( t) ad F (t) = M u si(t) Frequecy Respose
More information1 Adiabatic and diabatic representations
1 Adiabatic ad diabatic represetatios 1.1 Bor-Oppeheimer approximatio The time-idepedet Schrödiger equatio for both electroic ad uclear degrees of freedom is Ĥ Ψ(r, R) = E Ψ(r, R), (1) where the full molecular
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationmx bx kx F t. dt IR I LI V t, Q LQ RQ V t,
Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationA numerical Technique Finite Volume Method for Solving Diffusion 2D Problem
The Iteratioal Joural Of Egieerig d Sciece (IJES) Volume 4 Issue 10 Pages PP -35-41 2015 ISSN (e): 2319 1813 ISSN (p): 2319 1805 umerical Techique Fiite Volume Method for Solvig Diffusio 2D Problem 1 Mohammed
More informationThe Mathematical Model and the Simulation Modelling Algoritm of the Multitiered Mechanical System
The Mathematical Model ad the Simulatio Modellig Algoritm of the Multitiered Mechaical System Demi Aatoliy, Kovalev Iva Dept. of Optical Digital Systems ad Techologies, The St. Petersburg Natioal Research
More informationFREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING
Mechaical Vibratios FREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING A commo dampig mechaism occurrig i machies is caused by slidig frictio or dry frictio ad is called Coulomb dampig. Coulomb dampig
More informationLemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.
15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic
More informationTMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationAvoidance of numerical singularities in free vibration analysis of Euler-Bernoulli beams using Green functions
WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut Avoidace of umerical sigularities i free vibratio aalysis of Euler-Beroulli beams usig
More informationVibratory Motion. Prof. Zheng-yi Feng NCHU SWC. National CHung Hsing University, Department of Soil and Water Conservation
Vibratory Motio Prof. Zheg-yi Feg NCHU SWC 1 Types of vibratory motio Periodic motio Noperiodic motio See Fig. A1, p.58 Harmoic motio Periodic motio Trasiet motio impact Trasiet motio earthquake A powerful
More informationCourse Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B.
Amme 3500 : System Dyamics ad Cotrol Root Locus Course Outlie Week Date Cotet Assigmet Notes Mar Itroductio 8 Mar Frequecy Domai Modellig 3 5 Mar Trasiet Performace ad the s-plae 4 Mar Block Diagrams Assig
More informationChapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io
Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationRotationally invariant integrals of arbitrary dimensions
September 1, 14 Rotatioally ivariat itegrals of arbitrary dimesios James D. Wells Physics Departmet, Uiversity of Michiga, A Arbor Abstract: I this ote itegrals over spherical volumes with rotatioally
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:
More informationCS322: Network Analysis. Problem Set 2 - Fall 2009
Due October 9 009 i class CS3: Network Aalysis Problem Set - Fall 009 If you have ay questios regardig the problems set, sed a email to the course assistats: simlac@staford.edu ad peleato@staford.edu.
More informationAnalytical solutions for multi-wave transfer matrices in layered structures
Joural of Physics: Coferece Series PAPER OPEN ACCESS Aalytical solutios for multi-wave trasfer matrices i layered structures To cite this article: Yu N Belyayev 018 J Phys: Cof Ser 109 01008 View the article
More informationComplex Number Theory without Imaginary Number (i)
Ope Access Library Joural Complex Number Theory without Imagiary Number (i Deepak Bhalchadra Gode Directorate of Cesus Operatios, Mumbai, Idia Email: deepakm_4@rediffmail.com Received 6 July 04; revised
More informationNANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
More informationThe target reliability and design working life
Safety ad Security Egieerig IV 161 The target reliability ad desig workig life M. Holický Kloker Istitute, CTU i Prague, Czech Republic Abstract Desig workig life ad target reliability levels recommeded
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationA PROCEDURE TO MODIFY THE FREQUENCY AND ENVELOPE CHARACTERISTICS OF EMPIRICAL GREEN'S FUNCTION. Lin LU 1 SUMMARY
A POCEDUE TO MODIFY THE FEQUENCY AND ENVELOPE CHAACTEISTICS OF EMPIICAL GEEN'S FUNCTION Li LU SUMMAY Semi-empirical method, which divides the fault plae of large earthquake ito mets ad uses small groud
More informationCHAPTER 8 SYSTEMS OF PARTICLES
CHAPTER 8 SYSTES OF PARTICLES CHAPTER 8 COLLISIONS 45 8. CENTER OF ASS The ceter of mass of a system of particles or a rigid body is the poit at which all of the mass are cosidered to be cocetrated there
More informationMathematical Statistics - MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationPOD-Based Analysis of Dynamic Wind Load Effects on a Large Span Roof
POD-Based Aalysis of Dyamic Wid Load Effects o a Large Spa Roof Xi-yag Ji, Yi Tag ad Hai Ji 3 Professor, Wid Egieerig Research Ceter, Chia Academy of Buildig Research, Beiig 3, Chia, ixiyag@cabrtech.com
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationSECTION 2 Electrostatics
SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial
More informationAdded mass estimation of open flat membranes vibrating in still air
Added mass estimatio of ope flat membraes vibratig i still air *Yua-qi Li 1),Yi Zhou ), Akihito YOHIDA 3) ad Yukio TAMURA 4) 1), ) Departmet of Buildig Egieerig, Togji Uiversity, haghai, 9, Chia 1) liyq@togji.edu.c
More informationTHE EIGENVALUE PROBLEM
November 5, 2013 APPENDIX D HE EIGENVALUE PROLEM Eigevalues ad Eigevectors are Properties of the Equatios that Simulate the ehavior of a Real Structure D.1 INRODUCION he classical mathematical eigevalue
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More information4. Hypothesis testing (Hotelling s T 2 -statistic)
4. Hypothesis testig (Hotellig s T -statistic) Cosider the test of hypothesis H 0 : = 0 H A = 6= 0 4. The Uio-Itersectio Priciple W accept the hypothesis H 0 as valid if ad oly if H 0 (a) : a T = a T 0
More informationResearch Article Health Monitoring for a Structure Using Its Nonstationary Vibration
Advaces i Acoustics ad Vibratio Volume 2, Article ID 69652, 5 pages doi:.55/2/69652 Research Article Health Moitorig for a Structure Usig Its Nostatioary Vibratio Yoshimutsu Hirata, Mikio Tohyama, Mitsuo
More information